Impurities in a Homogeneous Electron Gas - Physics at Oregon ...
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<strong>Impurities</strong> <strong>in</strong> a <strong>Homogeneous</strong> <strong>Electron</strong> <strong>Gas</strong><br />
by<br />
Jung-Hwan Song<br />
A THESIS<br />
submitted to<br />
<strong>Oregon</strong> St<strong>at</strong>e University<br />
<strong>in</strong> partial fulfillment of<br />
the requirements for the<br />
degree of<br />
Doctor of Philosophy<br />
Presented December 7, 2004<br />
Commencement June 2005
AN ABSTRACT OF THE THESIS OF<br />
Jung-Hwan Song for the degree of Doctor of Philosophy <strong>in</strong> <strong>Physics</strong> presented on<br />
December 7, 2004.<br />
Title: <strong>Impurities</strong> <strong>in</strong> a <strong>Homogeneous</strong> <strong>Electron</strong> <strong>Gas</strong><br />
Abstract approved:<br />
Henri J. F. Jansen<br />
Immersion energies for an impurity <strong>in</strong> a homogeneous electron gas with a uni-<br />
form positive background charge density have been calcul<strong>at</strong>ed numerically us<strong>in</strong>g<br />
density functional theory. The numerical aspects of this problem are very de-<br />
mand<strong>in</strong>g and have not been properly discussed <strong>in</strong> previous work. The numerical<br />
problems are rel<strong>at</strong>ed to approxim<strong>at</strong>ions of <strong>in</strong>f<strong>in</strong>ity and cont<strong>in</strong>uity, and they have<br />
been corrected us<strong>in</strong>g physics based on the Friedel sum rule and Friedel oscill<strong>at</strong>ions.<br />
The numerical precision is tested extensively. Immersion energies are obta<strong>in</strong>ed for<br />
non-sp<strong>in</strong>-polarized systems, and are compared with published d<strong>at</strong>a. Numerical<br />
results, such as phase shifts, density of st<strong>at</strong>es, dielectric constants, and compress-<br />
ibilities, are obta<strong>in</strong>ed and compared with analytical theories. Immersion energies<br />
for excited systems are obta<strong>in</strong>ed by vary<strong>in</strong>g the number of electrons <strong>in</strong> the bound<br />
st<strong>at</strong>es of an impurity. The model is extended to sp<strong>in</strong>-polarized systems and is<br />
tested <strong>in</strong> detail for a carbon impurity. The sp<strong>in</strong>-coupl<strong>in</strong>g with an external mag-<br />
netic field is considered ma<strong>in</strong>ly for a hydrogen impurity. These new results show<br />
very <strong>in</strong>terest<strong>in</strong>g behavior <strong>at</strong> low densities.
c○ Copyright by Jung-Hwan Song<br />
December 7, 2004<br />
All Rights Reserved
<strong>Impurities</strong> <strong>in</strong> a <strong>Homogeneous</strong> <strong>Electron</strong> <strong>Gas</strong><br />
by<br />
Jung-Hwan Song<br />
A THESIS<br />
submitted to<br />
<strong>Oregon</strong> St<strong>at</strong>e University<br />
<strong>in</strong> partial fulfillment of<br />
the requirements for the<br />
degree of<br />
Doctor of Philosophy<br />
Presented December 7, 2004<br />
Commencement June 2005
Doctor of Philosophy thesis of Jung-Hwan Song presented on December 7, 2004<br />
APPROVED:<br />
Major Professor, represent<strong>in</strong>g <strong>Physics</strong><br />
Chair of the Department of <strong>Physics</strong><br />
Dean of the Gradu<strong>at</strong>e School<br />
I understand th<strong>at</strong> my thesis will become part of the permanent collection of<br />
<strong>Oregon</strong> St<strong>at</strong>e University libraries. My sign<strong>at</strong>ure below authorizes release of my<br />
thesis to any reader upon request.<br />
Jung-Hwan Song, Author
ACKNOWLEDGMENT<br />
I would like to thank the people who helped me come to Corvallis and over-<br />
come culture differences.<br />
I cannot fully express my gr<strong>at</strong>itude to my Major Professor, Henri J. F. Jansen,<br />
who has been a most important contributor to this thesis, and gave me guidel<strong>in</strong>es<br />
and helps not only for physics problems but also for my personal problems. He has<br />
been very k<strong>in</strong>d, generous, and has answered all my questions, <strong>in</strong>clud<strong>in</strong>g the trivial<br />
ones. I really appreci<strong>at</strong>e his expertise and experience, and look forward to further<br />
collabor<strong>at</strong>ions.<br />
I would like to express my appreci<strong>at</strong>ion to Professors William W. Warren,<br />
Rub<strong>in</strong> H. Landau, Tom Giebultowicz, and Solomon C. S. Yim for serv<strong>in</strong>g on my<br />
committee.<br />
I am thankful to my friends and office-m<strong>at</strong>es, especially Jae-Hyuk Lee, Porn-<br />
r<strong>at</strong> W<strong>at</strong>tanakasiwich, and David M<strong>at</strong>usevitsch, for mak<strong>in</strong>g my stay a very pleasant<br />
one.<br />
And f<strong>in</strong>ally, I am deeply gr<strong>at</strong>eful to my parents and my wife for their uncon-<br />
ditional love and support. This thesis might have taken much more time to be<br />
f<strong>in</strong>ished without helps from my wife, JungIn.<br />
F<strong>in</strong>ancial support was partially provided by ONR.
TABLE OF CONTENTS<br />
Page<br />
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1<br />
2 DENSITY FUNCTIONAL THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4<br />
2.1 The Hohenberg-Kohn theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4<br />
2.1.1 Uniqueness of the external potential <strong>in</strong> terms of the density . 5<br />
2.1.2 Vari<strong>at</strong>ional pr<strong>in</strong>ciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7<br />
2.1.3 Universal functional FHK[n] . . . . . . . . . . . . . . . . . . . . . . . . . 8<br />
2.2 The Kohn-Sham equ<strong>at</strong>ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9<br />
2.2.1 Non-<strong>in</strong>teract<strong>in</strong>g particle system <strong>in</strong> an external potential vNI(r) 9<br />
2.2.2 Interact<strong>in</strong>g particle system <strong>in</strong> an external potential v(r) . . . . 10<br />
2.2.3 Ground st<strong>at</strong>e energy of an <strong>in</strong>teract<strong>in</strong>g system . . . . . . . . . . . . 12<br />
2.2.4 The chemical potential µ and Fermi surface . . . . . . . . . . . . . 12<br />
2.2.5 Excited st<strong>at</strong>es; Eigenvalues <strong>in</strong> the Kohn-Sham equ<strong>at</strong>ions . . . 14<br />
2.2.6 Sp<strong>in</strong>-polarized systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15<br />
2.2.7 Exchange-correl<strong>at</strong>ion; LDA . . . . . . . . . . . . . . . . . . . . . . . . . . 17<br />
2.2.7.1 A functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17<br />
2.2.7.2 Exchange-correl<strong>at</strong>ion energy . . . . . . . . . . . . . . . . . . . . . . 17<br />
2.2.7.3 Interpol<strong>at</strong>ion scheme; Sp<strong>in</strong>-<strong>in</strong>dependent . . . . . . . . . . . . . 21<br />
2.2.7.4 Interpol<strong>at</strong>ion scheme; Sp<strong>in</strong>-dependent . . . . . . . . . . . . . . . 23<br />
3 A MODEL AND ITS PROPERTIES: AN IMPURITY IN A HOMOGE-<br />
NEOUS ELECTRON GAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30<br />
3.1 An impurity <strong>in</strong> a homogeneous electron gas. . . . . . . . . . . . . . . . . . . . . . . . . 30<br />
3.2 Energy calcul<strong>at</strong>ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32<br />
3.2.1 Symmetry <strong>in</strong> potentials and phase shifts . . . . . . . . . . . . . . . . 33<br />
3.2.2 Density of <strong>in</strong>duced st<strong>at</strong>es <strong>in</strong> terms of phase shifts . . . . . . . . . 39<br />
3.2.3 Immersion energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39<br />
3.3 Friedel sum rule and Friedel oscill<strong>at</strong>ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43<br />
3.4 Dielectric functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
TABLE OF CONTENTS (Cont<strong>in</strong>ued)<br />
Page<br />
3.5 Virtual bound st<strong>at</strong>e resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47<br />
3.6 Sc<strong>at</strong>ter<strong>in</strong>g length and bound st<strong>at</strong>e energy . . . . . . . . . . . . . . . . . . . . . . . . . . 51<br />
4 IMPLEMENTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55<br />
4.1 Perspective scheme for self-consistent solutions . . . . . . . . . . . . . . . . . . . . . 55<br />
4.1.1 Non-sp<strong>in</strong> polarized system . . . . . . . . . . . . . . . . . . . . . . . . . . . 55<br />
4.1.2 Sp<strong>in</strong> polarized system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57<br />
4.2 Mix<strong>in</strong>g scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57<br />
4.3 Search for eigenst<strong>at</strong>es and eigenvalues of the <strong>in</strong>f<strong>in</strong>ite system . . . . . . . . 60<br />
4.3.1 Bound st<strong>at</strong>es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60<br />
4.3.2 Sc<strong>at</strong>tered st<strong>at</strong>es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63<br />
4.4 Numerical problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65<br />
4.4.1 Convergence of phase shifts . . . . . . . . . . . . . . . . . . . . . . . . . . 65<br />
4.4.2 Correction for potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67<br />
4.4.3 Calcul<strong>at</strong>ions for <strong>in</strong>duced electrons and energies . . . . . . . . . . . 70<br />
4.4.4 Mesh for r-space and k-space . . . . . . . . . . . . . . . . . . . . . . . . 71<br />
4.4.5 Cancel<strong>at</strong>ion of numerical errors <strong>in</strong> immersion energies: Number<br />
of r po<strong>in</strong>ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76<br />
4.4.6 Further numerical precision tests . . . . . . . . . . . . . . . . . . . . . . 79<br />
4.5 Self-consistent solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83<br />
4.6 Sp<strong>in</strong> paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85<br />
5 RESULTS AND DISCUSSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88<br />
5.1 Non-sp<strong>in</strong>-polarized system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89<br />
5.1.1 Behavior of a system dur<strong>in</strong>g the numerical calcul<strong>at</strong>ions . . . . 89<br />
5.1.2 Charge densities and potentials . . . . . . . . . . . . . . . . . . . . . . . 89<br />
5.1.3 Phase shift, Sc<strong>at</strong>ter<strong>in</strong>g length, and Density of <strong>in</strong>duced st<strong>at</strong>es 93<br />
5.1.4 Dielectric constant and compressibility . . . . . . . . . . . . . . . . . 101<br />
5.1.5 Immersion energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.1.6 Excited system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109<br />
5.2 Sp<strong>in</strong> polarized system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113<br />
5.2.1 Zero external magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . 113<br />
5.2.1.1 Carbon impurity <strong>at</strong> low background densities . . . . . . . . . 113<br />
5.2.1.2 Electrical Resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119<br />
5.2.1.3 Excited system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121<br />
5.2.2 Non-zero external magnetic field . . . . . . . . . . . . . . . . . . . . . . 148<br />
6 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166<br />
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169<br />
APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171<br />
APPENDIX A Atomic Rydberg units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172<br />
APPENDIX B Comparison with published d<strong>at</strong>a. . . . . . . . . . . . . . . . . . . . . . . . . . 174<br />
APPENDIX C Boundary conditions for non-spherical potentials <strong>at</strong> small r176<br />
APPENDIX D S<strong>in</strong>gular value decomposition(SVD) . . . . . . . . . . . . . . . . . . . . . . 179<br />
APPENDIX E Behavior of phase shifts for E → ∞ . . . . . . . . . . . . . . . . . . . . . . 182<br />
APPENDIX F Sc<strong>at</strong>ter<strong>in</strong>g length and bound st<strong>at</strong>e energy . . . . . . . . . . . . . . . . . 184
LIST OF FIGURES<br />
Figure Page<br />
2.1 The exchange ɛx , correl<strong>at</strong>ion ɛc, and Thomas-Fermi k<strong>in</strong>etic energies<br />
TT F = 2.2099/r 2 s per electron from the parametriz<strong>at</strong>ion of Hed<strong>in</strong>-<br />
Lundqvist. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20<br />
2.2 The exchange ɛx , correl<strong>at</strong>ion ɛc, and Thomas-Fermi k<strong>in</strong>etic energies<br />
TT F = 2.2099/r 2 s multiplied by the electron density n. . . . . . . . . . . . 21<br />
2.3 The exchange vx and correl<strong>at</strong>ion vc potentials. . . . . . . . . . . . . . . . . . 22<br />
2.4 The exchange-correl<strong>at</strong>ion energy per electron ɛxc from the parametriz<strong>at</strong>ion<br />
of von Barth and Hed<strong>in</strong>. m = n ζ where n is the total<br />
density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25<br />
2.5 The exchange-correl<strong>at</strong>ion energy nɛxc. . . . . . . . . . . . . . . . . . . . . . . 26<br />
2.6 The exchange-correl<strong>at</strong>ion ɛxc and Thomas-Fermi k<strong>in</strong>etic energies<br />
TT F = 2.2099/r 2 s multiplied by the electron density n. . . . . . . . . . . . 28<br />
2.7 The exchange-correl<strong>at</strong>ion and Thomas-Fermi k<strong>in</strong>etic energies for extremely<br />
low densities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28<br />
2.8 The exchange-correl<strong>at</strong>ion potential vxc for the sp<strong>in</strong>-up density. . . . . 29<br />
2.9 The exchange-correl<strong>at</strong>ion potential vxc for the sp<strong>in</strong>-down density. . . 29<br />
3.1 An impurity system consists of an impurity <strong>at</strong>om and a homogeneous<br />
electron gas. The electron density fluctu<strong>at</strong>es due to the impurity<br />
<strong>at</strong>om. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30<br />
3.2 The density of st<strong>at</strong>es ρ0(ɛ) <strong>in</strong> the conduction band and the correspond<strong>in</strong>g<br />
phase shift. ɛd = −0.3 and ∆(ɛ) = (−0.13) 2 πρ0(ɛ) are<br />
used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50<br />
3.3 There is a virtual bound st<strong>at</strong>e resonance <strong>at</strong> the solution of ɛ − ɛd −<br />
Λ(ɛ) = 0. The density of st<strong>at</strong>es ∆ρ(ɛ) <strong>in</strong>duced by the impurity.<br />
Calcul<strong>at</strong>ed from Fig. 3.2 <strong>in</strong> M<strong>at</strong>hem<strong>at</strong>ica. . . . . . . . . . . . . . . . . . . . . 50<br />
3.4 Shows how the density of st<strong>at</strong>es ∆ρ(ɛ) varies <strong>in</strong> the conduction band<br />
for the different strength of the potential. Calcul<strong>at</strong>ed from Fig. 3.2<br />
<strong>in</strong> M<strong>at</strong>hem<strong>at</strong>ica. The number of electrons <strong>in</strong> the conduction band<br />
is obta<strong>in</strong>ed numerically. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
LIST OF FIGURES (Cont<strong>in</strong>ued)<br />
Figure Page<br />
4.1 Flow chart for non-polarized systems. . . . . . . . . . . . . . . . . . . . . . . . 56<br />
4.2 Phase shift values of l = 0 for a zero potential. . . . . . . . . . . . . . . . . 65<br />
4.3 Phase shift values for an <strong>at</strong>tractive potential of Yukawa type and a<br />
repulsive potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66<br />
4.4 Typical example show<strong>in</strong>g defects <strong>in</strong> the output potential dur<strong>in</strong>g the<br />
numerical calcul<strong>at</strong>ions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68<br />
4.5 Typical examples to show how the effective potentials behave before<br />
and after the correction. These potentials are calcul<strong>at</strong>ed for a<br />
hydrogen impurity and n0 = 0.0025. . . . . . . . . . . . . . . . . . . . . . . . . . 70<br />
4.6 Calcul<strong>at</strong>ed for a hydrogen impurity and n0 = 0.0025. This example<br />
shows the oscill<strong>at</strong>ions of <strong>in</strong>tegral values of the <strong>in</strong>duced electronic<br />
density as a function of r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71<br />
4.7 Typical example for the effective potential. The logarithmic mesh<br />
is used for the rapid vari<strong>at</strong>ion of the potential <strong>in</strong> the vic<strong>in</strong>ity of the<br />
orig<strong>in</strong>. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72<br />
4.8 Example for phase shifts. The phase shift varies rapidly <strong>at</strong> the<br />
bottom of the conduction band if the energy of a bound st<strong>at</strong>e is<br />
close to the bottom of the conduction band. . . . . . . . . . . . . . . . . . . 73<br />
4.9 Integrand <strong>in</strong> (3.22). Calcul<strong>at</strong>ed for a Hydrogen impurity <strong>at</strong> 0.04<br />
background density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74<br />
4.10 The density of a free carbon <strong>at</strong>om and the density <strong>in</strong>duced by a<br />
carbon impurity <strong>in</strong> a homogeneous electron gas. The 1s and 2s core<br />
st<strong>at</strong>es rema<strong>in</strong> almost the same. These calcul<strong>at</strong>ions are performed<br />
for a sp<strong>in</strong> polarized system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76<br />
4.11 Comparison between immersion energies calcul<strong>at</strong>ed with a constant<br />
free <strong>at</strong>om energy and the free <strong>at</strong>om energy calcul<strong>at</strong>ed with the same<br />
r-mesh as the impurity system. The immersion energies calcul<strong>at</strong>ed<br />
with the same free <strong>at</strong>om energy show rel<strong>at</strong>ively large vari<strong>at</strong>ion with<br />
the different number of r-po<strong>in</strong>ts due to the numerical errors of core<br />
st<strong>at</strong>es. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
LIST OF FIGURES (Cont<strong>in</strong>ued)<br />
Figure Page<br />
4.12 Fitt<strong>in</strong>g by <strong>in</strong>verse power law to f<strong>in</strong>d the correct immersion energy.<br />
D<strong>at</strong>a from Table 4.1 and Table 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 79<br />
4.13 Extra densities <strong>in</strong>duced by an impurity. . . . . . . . . . . . . . . . . . . . . . . 81<br />
4.14 Immersion energy versus the <strong>in</strong>verse of the number of r po<strong>in</strong>ts <strong>in</strong><br />
the l<strong>in</strong>ear r-mesh. D<strong>at</strong>a from Table 4.4. . . . . . . . . . . . . . . . . . . . . . 82<br />
4.15 The density of st<strong>at</strong>es <strong>at</strong> absolute zero temper<strong>at</strong>ure. With an external<br />
magnetic field, the energy of the sp<strong>in</strong>s (magnetic-moment-up)<br />
parallel to the field is lowered by µB, while the energy of the sp<strong>in</strong>s<br />
(magnetic-moment-down) antiparallel to the field is raised by µB.<br />
The chemical potential of the sp<strong>in</strong>-up band is equal to th<strong>at</strong> of the<br />
sp<strong>in</strong>-down band. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85<br />
5.1 Vari<strong>at</strong>ion of the number of extra electrons for a system with a hydrogen<br />
impurity dur<strong>in</strong>g the iter<strong>at</strong>ive loops of the numerical calcul<strong>at</strong>ions.<br />
The background electronic density is 0.06. . . . . . . . . . . . . . . . . . . . 90<br />
5.2 The electronic density <strong>in</strong>duced by a hydrogen impurity. It is plotted<br />
as a function of 2kF r. Friedel oscill<strong>at</strong>ions are <strong>in</strong>dependent of the<br />
background density. The range of the background density is from<br />
0.0005 to 0.06. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91<br />
5.3 The electronic density <strong>in</strong>duced by a hydrogen impurity. . . . . . . . . . . 92<br />
5.4 The electronic density <strong>in</strong>duced by a hydrogen impurity <strong>in</strong> the bound<br />
st<strong>at</strong>es. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93<br />
5.5 The electronic density <strong>in</strong>duced by a hydrogen impurity <strong>in</strong> the conduction<br />
band. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94<br />
5.6 Phase shifts (l = 0) for a hydrogen impurity. Plotted for low background<br />
densities only. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95<br />
5.7 Inverse of sc<strong>at</strong>ter<strong>in</strong>g length versus background density for a hydrogen<br />
impurity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96<br />
5.8 Phase shifts for a hydrogen impurity for two different background<br />
densities (0.0025 and 0.06). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
LIST OF FIGURES (Cont<strong>in</strong>ued)<br />
Figure Page<br />
5.9 Phase shifts (l = 0) for a hydrogen impurity. . . . . . . . . . . . . . . . . . . 98<br />
5.10 The density of st<strong>at</strong>es for a hydrogen impurity for two different background<br />
densities (0.0025 and 0.06). . . . . . . . . . . . . . . . . . . . . . . . . . 98<br />
5.11 Phase shifts of l = 0 and l = 1 for a lithium impurity. . . . . . . . . . . . 100<br />
5.12 Dielectric constants which are numerically calcul<strong>at</strong>ed for a hydrogen<br />
impurity. The Thomas-Fermi dielectric constants are also plotted<br />
for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102<br />
5.13 Compressibility calcul<strong>at</strong>ed numerically for a hydrogen impurity. A<br />
dotted l<strong>in</strong>e which is taken from [6] is for the result predicted by<br />
theories such as Hubbard, VS, and SS. Both l<strong>in</strong>es are go<strong>in</strong>g through<br />
zero around rs = 5.0. The electronic density which corresponds to<br />
rs of the bottom x-axis is given <strong>in</strong> the top x-axis. . . . . . . . . . . . . . . 103<br />
5.14 Comparison of the compressibility calcul<strong>at</strong>ed numerically for a hydrogen<br />
impurity with sc<strong>at</strong>ter<strong>in</strong>g length. . . . . . . . . . . . . . . . . . . . . . 104<br />
5.15 Immersion energies for several different impurities. 1.0 <strong>in</strong> the legend<br />
<strong>in</strong>dic<strong>at</strong>es th<strong>at</strong> the number of electrons <strong>in</strong> the bound st<strong>at</strong>es is limited<br />
to 1.0. 465 po<strong>in</strong>ts for the logarithmic r-space and 0.81205 × 10 −5<br />
for the first value of r-mesh are used <strong>in</strong> these calcul<strong>at</strong>ions. . . . . . . . 106<br />
5.16 Vari<strong>at</strong>ion <strong>in</strong> the k<strong>in</strong>etic, Coulomb, and exchange-correl<strong>at</strong>ion energies<br />
vs the background density for a hydrogen impurity. 465 po<strong>in</strong>ts for<br />
the logarithmic r-space and 0.81205×10 −5 for the first vale of r-mesh<br />
are used <strong>in</strong> these calcul<strong>at</strong>ions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107<br />
5.17 Immersion energies for the different number of electrons <strong>in</strong> the<br />
bound st<strong>at</strong>es. 465 po<strong>in</strong>ts for the logarithmic r-space and 0.81205 ×<br />
10 −5 for the first value of r-mesh are used <strong>in</strong> these calcul<strong>at</strong>ions. . . . 109<br />
5.18 Vari<strong>at</strong>ion <strong>in</strong> the bound st<strong>at</strong>e energy for the different number of electrons<br />
<strong>in</strong> the bound st<strong>at</strong>e. The second plot is obta<strong>in</strong>ed us<strong>in</strong>g (5.7)<br />
and slopes between two adjacent immersion energies <strong>in</strong> Fig. 5.17.<br />
465 po<strong>in</strong>ts for the logarithmic r-space and 0.81205 × 10 −5 for the<br />
first value of r-mesh are used <strong>in</strong> these calcul<strong>at</strong>ions. . . . . . . . . . . . . . 110
LIST OF FIGURES (Cont<strong>in</strong>ued)<br />
Figure Page<br />
5.19 Comparison of immersion energies for a carbon impurity between a<br />
non-sp<strong>in</strong>-polarized system and a sp<strong>in</strong>-polarized system. The same<br />
reference energy for a free carbon <strong>at</strong>om is used for both calcul<strong>at</strong>ions. 113<br />
5.20 Plot of ∆N vs a background density where ∆N is the difference <strong>in</strong><br />
the number of electrons between sp<strong>in</strong>-up and sp<strong>in</strong>-down electrons <strong>in</strong><br />
a system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114<br />
5.21 The electronic density and the sp<strong>in</strong> density <strong>in</strong>duced by a carbon<br />
impurity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115<br />
5.22 The electronic density (sp<strong>in</strong>-up and sp<strong>in</strong>-down) <strong>in</strong>duced by a carbon<br />
impurity <strong>in</strong> the conduction band. . . . . . . . . . . . . . . . . . . . . . . . . . . . 116<br />
5.23 The density of st<strong>at</strong>es <strong>in</strong>duced by a carbon impurity. . . . . . . . . . . . . 117<br />
5.24 The values of Q versus Z for the background density of 0.01. The<br />
calcul<strong>at</strong>ions are performed only for Z = 1, 3, 6. Note th<strong>at</strong>, <strong>at</strong> this<br />
background density, there is no difference between sp<strong>in</strong>-polarized<br />
and non-sp<strong>in</strong>-polarized systems for these impurities. . . . . . . . . . . . . 120<br />
5.25 Immersion energies and ∆N for a carbon impurity vs the number<br />
of sp<strong>in</strong>-up electrons <strong>in</strong> a 2p bound st<strong>at</strong>e. ∆N is the difference <strong>in</strong> the<br />
number of electrons between sp<strong>in</strong>-up and sp<strong>in</strong>-down electrons <strong>in</strong> a<br />
system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122<br />
5.26 Bound st<strong>at</strong>e energies vs the number of sp<strong>in</strong>-up electrons <strong>in</strong> a 2p<br />
bound st<strong>at</strong>e. The second plot shows the vari<strong>at</strong>ion of the curv<strong>at</strong>ure,<br />
the energy m<strong>in</strong>imum, and the number of electrons <strong>in</strong> a 2p bound<br />
st<strong>at</strong>e correspond<strong>in</strong>g to the energy m<strong>in</strong>imum <strong>in</strong> the first plot. . . . . . . 123<br />
5.27 The density and the sp<strong>in</strong> density <strong>in</strong>duced by a carbon impurity <strong>at</strong><br />
0.0005 background density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124<br />
5.28 The density <strong>in</strong> the bound st<strong>at</strong>es of sp<strong>in</strong>-up and sp<strong>in</strong>-down electrons<br />
<strong>in</strong>duced by a carbon impurity <strong>at</strong> 0.0005 background density. . . . . . . 125<br />
5.29 The density and the sp<strong>in</strong> density <strong>in</strong> the conduction band <strong>in</strong>duced<br />
by a carbon impurity <strong>at</strong> 0.0005 background density. . . . . . . . . . . . . 126
LIST OF FIGURES (Cont<strong>in</strong>ued)<br />
Figure Page<br />
5.30 The density and the sp<strong>in</strong> density <strong>in</strong>duced by a carbon impurity <strong>at</strong><br />
0.00075 background density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127<br />
5.31 The density <strong>in</strong> the bound st<strong>at</strong>es of sp<strong>in</strong>-up and sp<strong>in</strong>-down electrons<br />
<strong>in</strong>duced by a carbon impurity <strong>at</strong> 0.00075 background density. . . . . . 128<br />
5.32 The density and the sp<strong>in</strong> density <strong>in</strong> the conduction band <strong>in</strong>duced<br />
by a carbon impurity <strong>at</strong> 0.00075 background density. . . . . . . . . . . . 129<br />
5.33 The density and the sp<strong>in</strong> density <strong>in</strong>duced by a carbon impurity <strong>at</strong><br />
0.001 background density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130<br />
5.34 The density <strong>in</strong> the bound st<strong>at</strong>es of sp<strong>in</strong>-up and sp<strong>in</strong>-down electrons<br />
<strong>in</strong>duced by a carbon impurity <strong>at</strong> 0.001 background density. . . . . . . . 131<br />
5.35 The density and the sp<strong>in</strong> density <strong>in</strong> the conduction band <strong>in</strong>duced<br />
by a carbon impurity <strong>at</strong> 0.001 background density. . . . . . . . . . . . . . 132<br />
5.36 The density and the sp<strong>in</strong> density <strong>in</strong>duced by a carbon impurity <strong>at</strong><br />
0.0011 background density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133<br />
5.37 The density <strong>in</strong> the bound st<strong>at</strong>es of sp<strong>in</strong>-up and sp<strong>in</strong>-down electrons<br />
<strong>in</strong>duced by a carbon impurity <strong>at</strong> 0.0011 background density. . . . . . . 134<br />
5.38 The density and the sp<strong>in</strong> density <strong>in</strong> the conduction band <strong>in</strong>duced<br />
by a carbon impurity <strong>at</strong> 0.0011 background density. . . . . . . . . . . . . 135<br />
5.39 The effective potential of the sp<strong>in</strong>-up band vs the number of sp<strong>in</strong>-up<br />
electrons <strong>in</strong> a 2p bound st<strong>at</strong>e for 0.0005 background density. . . . . . . 137<br />
5.40 The Coulomb potentials calcul<strong>at</strong>ed by <strong>in</strong>duced sp<strong>in</strong>-up and sp<strong>in</strong>down<br />
electrons vs the number of sp<strong>in</strong>-up electrons <strong>in</strong> a 2p bound<br />
st<strong>at</strong>e for 0.0005 background density. . . . . . . . . . . . . . . . . . . . . . . . . . 138<br />
5.41 The Coulomb potentials vs the number of sp<strong>in</strong>-up electrons <strong>in</strong> a 2p<br />
bound st<strong>at</strong>e for 0.0005 background density. The second plot shows<br />
the area which corresponds to the box <strong>in</strong> the first plot. . . . . . . . . . . 139<br />
5.42 The exchange-correl<strong>at</strong>ion potentials of <strong>in</strong>duced sp<strong>in</strong>-up and sp<strong>in</strong>down<br />
electrons vs the number of sp<strong>in</strong>-up electrons <strong>in</strong> a 2p bound<br />
st<strong>at</strong>e for 0.0005 background density. . . . . . . . . . . . . . . . . . . . . . . . . . 140
LIST OF FIGURES (Cont<strong>in</strong>ued)<br />
Figure Page<br />
5.43 Phase shifts of l = 0 and l = 1 for sp<strong>in</strong>-up and sp<strong>in</strong>-down electrons. 142<br />
5.44 The density of <strong>in</strong>duced st<strong>at</strong>es for sp<strong>in</strong>-up and sp<strong>in</strong>-down electrons<br />
<strong>at</strong> 0.0005 background density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143<br />
5.45 The density of <strong>in</strong>duced st<strong>at</strong>es for sp<strong>in</strong>-up and sp<strong>in</strong>-down electrons<br />
<strong>at</strong> 0.001 background density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144<br />
5.46 The exchange-correl<strong>at</strong>ion potential with different <strong>in</strong>duced densities<br />
<strong>in</strong> the sp<strong>in</strong>-up band. Background density is 0.0005. The <strong>in</strong>duced<br />
density <strong>in</strong> the sp<strong>in</strong>-down band is assumed to be 0.047. . . . . . . . . . . 145<br />
5.47 Partial wave decompositions of the Friedel sum rule. These plots<br />
show the change <strong>in</strong> the number of electrons <strong>in</strong>duced by a carbon<br />
impurity for each band. 0.0005 is used for a background density.<br />
Bound st<strong>at</strong>es are not <strong>in</strong>cluded. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146<br />
5.48 Bound st<strong>at</strong>e energies, the conduction band m<strong>in</strong>imums, the sp<strong>in</strong>moment,<br />
and the immersion energies vs the magnetic field for a<br />
hydrogen impurity with 0.0025 background density. . . . . . . . . . . . . . 148<br />
5.49 The exchange-correl<strong>at</strong>ion potential with different background densities<br />
of the sp<strong>in</strong>-up band. The total background density is 0.0025 and<br />
the <strong>in</strong>duced density is assumed to be 0.03 for each sp<strong>in</strong>-up and sp<strong>in</strong>down<br />
band. This plot shows how the exchange-correl<strong>at</strong>ion potential<br />
for each band may vary with the <strong>in</strong>creas<strong>in</strong>g magnetic field. . . . . . . . 149<br />
5.50 The effective potentials for the different magnetic field 0.01 and 0.04.<br />
A hydrogen impurity with 0.0025 background density is used. . . . . . 151<br />
5.51 The effective potential for a hydrogen impurity. No external magnetic<br />
field is applied. The potential of the sp<strong>in</strong>-up band is identical<br />
to th<strong>at</strong> of the sp<strong>in</strong>-down band. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152<br />
5.52 Partial wave decompositions of the Friedel sum rule. These plots<br />
show the change <strong>in</strong> the number of electrons <strong>in</strong>duced by a hydrogen<br />
impurity for each band. 0.0025 is used for a background density. . . 153<br />
5.53 Partial wave decompositions of the Friedel sum rule. no external<br />
magnetic field is applied. This plot for the sp<strong>in</strong>-up band is identical<br />
with th<strong>at</strong> for the sp<strong>in</strong>-down band. . . . . . . . . . . . . . . . . . . . . . . . . . . 154
LIST OF FIGURES (Cont<strong>in</strong>ued)<br />
Figure Page<br />
5.54 The <strong>in</strong>duced electronic densities <strong>in</strong> the conduction band for different<br />
magnetic fields. A hydrogen impurity with 0.0025 background<br />
density is used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155<br />
5.55 The <strong>in</strong>duced electronic densities <strong>in</strong> the conduction band of sp<strong>in</strong>-up<br />
electrons for each angular momentum st<strong>at</strong>e. These plots correspond<br />
to the magnetic field 0.08 <strong>in</strong> Fig. 5.48 and Fig. 5.54. A hydrogen<br />
impurity with 0.0025 background density is used. . . . . . . . . . . . . . . 156<br />
5.56 The density of st<strong>at</strong>es of the sp<strong>in</strong>-up and sp<strong>in</strong>-down bands for different<br />
magnetic fields. The arrows <strong>in</strong> the first plot <strong>in</strong>dic<strong>at</strong>e the<br />
loc<strong>at</strong>ions of the sc<strong>at</strong>ter<strong>in</strong>g resonances. A hydrogen impurity with<br />
0.0025 background density is used. . . . . . . . . . . . . . . . . . . . . . . . . . . 157<br />
5.57 Comparison of kF of the sp<strong>in</strong>-down band with k values which correspond<br />
to the sc<strong>at</strong>ter<strong>in</strong>g resonances <strong>in</strong> Fig. 5.56. . . . . . . . . . . . . . . 158<br />
5.58 Bound st<strong>at</strong>e energies, the conduction band m<strong>in</strong>imums, the sp<strong>in</strong>moment,<br />
and the immersion energies vs the magnetic field for a<br />
hydrogen impurity with two different background densities, 0.01 and<br />
0.04. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160<br />
5.59 The exchange-correl<strong>at</strong>ion potential with different background densities<br />
of the sp<strong>in</strong>-up band. The total background density is 0.04 and<br />
the <strong>in</strong>duced density is assumed to be 0.03 for each sp<strong>in</strong>-up and sp<strong>in</strong>down<br />
band. This plot shows how the exchange-correl<strong>at</strong>ion potential<br />
for each band may vary with the <strong>in</strong>creas<strong>in</strong>g magnetic field. . . . . . . . 161<br />
5.60 The <strong>in</strong>tegr<strong>at</strong>ed values of the density of st<strong>at</strong>es of the sp<strong>in</strong>-up band<br />
for different magnetic fields to f<strong>in</strong>d the total <strong>in</strong>duced number of<br />
electrons <strong>in</strong> the sp<strong>in</strong>-up conduction band. A hydrogen impurity<br />
with 0.04 background density is used. The band m<strong>in</strong>imums are<br />
shifted <strong>in</strong> such a way th<strong>at</strong> they have the same m<strong>in</strong>imum. . . . . . . . . 162<br />
5.61 Partial wave decompositions of the Friedel sum rule. These plots<br />
show the change <strong>in</strong> the number of electrons <strong>in</strong>duced by a hydrogen<br />
impurity for each band. 0.04 is used for a background density. . . . . 163<br />
5.62 The density <strong>in</strong>duced by a hydrogen impurity for each conduction<br />
band. The background density is 0.04 and the magnetic field is 0.3. 164
LIST OF FIGURES (Cont<strong>in</strong>ued)<br />
Figure Page<br />
6.1 Behavior of phase shifts for k → ∞. . . . . . . . . . . . . . . . . . . . . . . . . . 182<br />
6.2 Sc<strong>at</strong>ter<strong>in</strong>g length versus 1/ |EB|, where EB is the bound st<strong>at</strong>e<br />
energy rel<strong>at</strong>ive to the conduction band m<strong>in</strong>imum. . . . . . . . . . . . . . . 184
LIST OF TABLES<br />
Table Page<br />
3.1 Energies of free <strong>at</strong>oms (<strong>in</strong> Rydbergs). . . . . . . . . . . . . . . . . . . . . . . . . 31<br />
3.2 Ioniz<strong>at</strong>ion energies of one electron from a neutral <strong>at</strong>om (<strong>in</strong> Rydbergs).<br />
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31<br />
4.1 Energies of a carbon impurity system ∆E and energies of a free<br />
carbon <strong>at</strong>om E<strong>at</strong>om (<strong>in</strong> Rydbergs). A carbon impurity <strong>in</strong> a sp<strong>in</strong><br />
polarized system and a background density of 0.002 are used for<br />
calcul<strong>at</strong>ions. The r-mesh <strong>in</strong> the table represents the number of rpo<strong>in</strong>ts<br />
used <strong>in</strong> the range from 0 to 8 a.u., which corresponds to the<br />
logarithmic r-mesh and <strong>in</strong>cludes most core st<strong>at</strong>es. (See Sec. 4.4.4.) . 77<br />
4.2 Example for a hydrogen impurity. (Non-sp<strong>in</strong> polarized system.) A<br />
background density of 0.0025 is used for these calcul<strong>at</strong>ions. . . . . . . . 78<br />
4.3 Immersion energies for a hydrogen impurity. 70 k po<strong>in</strong>ts, 550 r<br />
po<strong>in</strong>ts for the logarithmic r-mesh, and 0.1 for the <strong>in</strong>terval of the<br />
l<strong>in</strong>ear r-mesh are used for these calcul<strong>at</strong>ions. . . . . . . . . . . . . . . . . . . 80<br />
4.4 Immersion energies for a hydrogen impurity. 70 k po<strong>in</strong>ts, 80 for<br />
Rmax, and 550 r po<strong>in</strong>ts for the logarithmic r-mesh are used for<br />
calcul<strong>at</strong>ions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82<br />
4.5 Immersion energies for a hydrogen impurity. There is no difference<br />
with<strong>in</strong> 10 −5 Rydbergs for k po<strong>in</strong>ts more than 70. 80 for Rmax and<br />
550 r po<strong>in</strong>ts for the logarithmic r-mesh are used for these calcul<strong>at</strong>ions. 83
IMPURITIES IN A HOMOGENEOUS ELECTRON GAS<br />
1. INTRODUCTION<br />
The electronic structure of many-electron systems such as <strong>at</strong>oms, molecules,<br />
metals, and semiconductors is of gre<strong>at</strong> importance for analyz<strong>in</strong>g and develop<strong>in</strong>g<br />
m<strong>at</strong>erials. Hartree-Fock or second quantiz<strong>at</strong>ion calcul<strong>at</strong>ions for many-electron sys-<br />
tems, especially for solids, are, however, almost impossible or comput<strong>at</strong>ionally very<br />
demand<strong>in</strong>g. One of the most successful theoretical and practical approaches to such<br />
complex systems is density functional theory, <strong>in</strong> which systems are described by<br />
the electron density r<strong>at</strong>her than wave functions. Density functional theory has now<br />
been generalized for molecular dynamics, sp<strong>in</strong>-polarized systems, superconductors,<br />
excited systems, time-dependent systems, and so on.<br />
In 1964, Hohenberg and Kohn published density functional theory, <strong>in</strong> which<br />
the ground st<strong>at</strong>e properties of a many-electron system can be calcul<strong>at</strong>ed <strong>in</strong> terms<br />
of the electron density by m<strong>in</strong>imiz<strong>in</strong>g energy functionals. [1] The Hohenberg-Kohn<br />
theorem, however, has a critical shortcom<strong>in</strong>g <strong>in</strong> th<strong>at</strong> the energy functionals are<br />
impossible to determ<strong>in</strong>e or can only be approxim<strong>at</strong>ed for slowly vary<strong>in</strong>g charge dis-<br />
tributions, as <strong>in</strong> the Thomas-Fermi theory. The Kohn-Sham equ<strong>at</strong>ions form an ef-<br />
fective implement<strong>at</strong>ion of the Hohenberg-Kohn theorem. They were derived by <strong>in</strong>-<br />
troduc<strong>in</strong>g a non-<strong>in</strong>teract<strong>in</strong>g k<strong>in</strong>etic energy functional and the exchange-correl<strong>at</strong>ion<br />
energy functional and utiliz<strong>in</strong>g the energy vari<strong>at</strong>ional pr<strong>in</strong>ciple of the Hohenberg-<br />
Kohn theorem. [2] S<strong>in</strong>ce the exchange-correl<strong>at</strong>ion term <strong>in</strong>cludes every <strong>in</strong>teraction<br />
other than Coulomb <strong>in</strong>teraction and depends on the entire density distribution, it<br />
is, <strong>at</strong> present, not possible to calcul<strong>at</strong>e the exchange-correl<strong>at</strong>ion <strong>in</strong>teraction exactly.
Hence approxim<strong>at</strong>ions are needed, for <strong>in</strong>stance, LDA(Local Density Approxima-<br />
tion) and GGA(Generalized Gradient Approxim<strong>at</strong>ion). 1 S<strong>in</strong>ce the Kohn-Sham<br />
equ<strong>at</strong>ions are a self-consistent set of one-particle equ<strong>at</strong>ions, they have been used<br />
widely for computer calcul<strong>at</strong>ions such as the band structure. [3] [4]<br />
Computer calcul<strong>at</strong>ions for a model of an impurity <strong>at</strong>om <strong>in</strong> a homogeneous<br />
electron gas were performed <strong>in</strong> 1981 by Puska, Niem<strong>in</strong>en, and Man<strong>in</strong>en. [5] This<br />
model has been developed orig<strong>in</strong>ally to study metal vacancies and has been ap-<br />
plied to chemisorbed <strong>at</strong>oms on metal surfaces and to molecular bond<strong>in</strong>g. [6] [7]<br />
The density of st<strong>at</strong>es <strong>in</strong>duced by impurities were calcul<strong>at</strong>ed, as well as the im-<br />
mersion energies. Recently, numerical calcul<strong>at</strong>ions have also been performed for<br />
non-spherical systems, such as an impurity mov<strong>in</strong>g through a uniform electronic<br />
gas, and di<strong>at</strong>omic molecules <strong>in</strong> an electronic gas, with<strong>in</strong> the Kohn-Sham scheme.<br />
[8] [9]<br />
The important properties of this model were already discussed by J. Friedel<br />
<strong>in</strong> 1950s <strong>in</strong> the context of metallic alloys. [10] [11] [12] For a given perturb<strong>at</strong>ion<br />
potential due to impurity <strong>at</strong>oms, a free-electron gas shows two important fe<strong>at</strong>ures,<br />
the Friedel sum rule and Friedel oscill<strong>at</strong>ions, which are rel<strong>at</strong>ed to the total <strong>in</strong>duced<br />
charge <strong>in</strong> the vic<strong>in</strong>ity of an impurity and to the long-range oscill<strong>at</strong>ions of the elec-<br />
tronic density <strong>at</strong> large distances. In Friedel’s papers, transition metal impurities <strong>in</strong><br />
metals such as Al and Cu were discussed. He considered the d shell of a transition<br />
metal impurity <strong>at</strong> the Fermi energy and used the concept of virtual bound st<strong>at</strong>es.<br />
1 The exchange-correl<strong>at</strong>ion <strong>in</strong>teraction <strong>in</strong> LDA depends only on the local electron density.<br />
In GGA it depends not only on the local density but also on the gradient of the local<br />
density.<br />
2
The density of st<strong>at</strong>es <strong>at</strong> the Fermi energy <strong>in</strong>duced by an impurity is used <strong>in</strong> the<br />
discussion of a variety of physical properties such as Pauli paramagnetism, orbital<br />
diamagnetism, electrical resistivity, and thermal properties.<br />
This model can be used to study the behavior of a s<strong>in</strong>gle impurity, for <strong>in</strong>-<br />
stance, to analyze the viol<strong>at</strong>ion of Hund’s second rule for a s<strong>in</strong>gle iron <strong>at</strong>om <strong>in</strong><br />
metals. This is important for properties like crystall<strong>in</strong>e magnetic anisotropy and<br />
quench<strong>in</strong>g of the orbital moment. It can also be considered as a first approxima-<br />
tion to analyze any system, consist<strong>in</strong>g of impurities and an electronic gas, when the<br />
impurities hardly <strong>in</strong>teract and l<strong>at</strong>tice vibr<strong>at</strong>ions can be neglected. This model can<br />
be used to <strong>in</strong>vestig<strong>at</strong>e the magnetic behavior of an electron gas due to an impurity.<br />
Analysis of the sp<strong>at</strong>ial distribution of an electron gas and the concentr<strong>at</strong>ion of<br />
impurities may also shed light on phenomena like diffusion and <strong>in</strong>direct exchange<br />
<strong>in</strong>teractions via conduction electrons.<br />
The calcul<strong>at</strong>ions <strong>in</strong> this thesis first follow the scheme used <strong>in</strong> the Puska, Niem-<br />
<strong>in</strong>en, and Man<strong>in</strong>en paper. LDA is used for the exchange-correl<strong>at</strong>ion <strong>in</strong>teraction <strong>in</strong><br />
this work. Their model is extended for excited systems and sp<strong>in</strong>-polarized systems<br />
with or without an external magnetic field. The sp<strong>in</strong>-moment for the model system<br />
is also calcul<strong>at</strong>ed, consider<strong>in</strong>g sp<strong>in</strong>-coupl<strong>in</strong>g only, with an external magnetic field.<br />
A variety of results such as immersion energies, phase shifts, potentials, dielec-<br />
tric functions, partial-wave decompositions of the Friedel sum rule, and electronic<br />
densities are obta<strong>in</strong>ed.<br />
Density functional theory and the Kohn-Sham equ<strong>at</strong>ion are briefly expla<strong>in</strong>ed<br />
<strong>in</strong> Chapter 2. The model used <strong>in</strong> this work is <strong>in</strong>troduced, and theoretical calcula-<br />
tions and general properties of an electronic gas are discussed <strong>in</strong> Chapter 3. The<br />
numerical methods developed <strong>in</strong> this work are expla<strong>in</strong>ed <strong>in</strong> Chapter 4. The results<br />
and discussion follow <strong>in</strong> Chapter 5. Chapter 6 is dedic<strong>at</strong>ed to conclusions.<br />
3
2. DENSITY FUNCTIONAL THEORY<br />
The Hohenberg-Kohn theorem provides the theoretical found<strong>at</strong>ion for func-<br />
tional calcul<strong>at</strong>ions of the electronic structure for <strong>in</strong>teract<strong>in</strong>g particle systems and<br />
suggests a universal functional FHK[n] which is <strong>in</strong>dependent of the n<strong>at</strong>ure of the<br />
external potential <strong>in</strong> a system. The Kohn-Sham equ<strong>at</strong>ions can be obta<strong>in</strong>ed us<strong>in</strong>g a<br />
vari<strong>at</strong>ion pr<strong>in</strong>ciple by compar<strong>in</strong>g with a Schröd<strong>in</strong>ger equ<strong>at</strong>ion and they <strong>in</strong>corpor<strong>at</strong>e<br />
FHK[n] via the exchange-correl<strong>at</strong>ion energy Exc[n]. Exc[n] can be approxim<strong>at</strong>ed as<br />
<strong>in</strong> LDA or GGA, s<strong>in</strong>ce FHK[n] is impossible to f<strong>in</strong>d. The Kohn-Sham equ<strong>at</strong>ions are<br />
a set of self-consistent one-particle equ<strong>at</strong>ions and have a variety of applic<strong>at</strong>ions,<br />
especially <strong>in</strong> a numerical way. In this chapter we will review the Hohenberg-Kohn<br />
theorem and the Kohn-Sham equ<strong>at</strong>ions briefly. The Kohn-Sham equ<strong>at</strong>ions will<br />
be extended for a sp<strong>in</strong>-polarized system. The exchange-correl<strong>at</strong>ion energies and<br />
potentials will be discussed and expressed explicitly with<strong>in</strong> the LDA scheme.<br />
2.1. The Hohenberg-Kohn theorem<br />
In 1964, Hohenberg and Kohn po<strong>in</strong>ted out th<strong>at</strong> it is possible to derive the<br />
ground st<strong>at</strong>e properties of a many-electron system <strong>in</strong> terms of the electron density.<br />
[1] The Hamiltonian for <strong>in</strong>teract<strong>in</strong>g particles under the <strong>in</strong>fluence of an external<br />
potential v(r) is of the form [13]<br />
where, <strong>in</strong> the second quantized not<strong>at</strong>ion,<br />
ˆH = ˆ T + ˆ V + ˆ W (2.1)<br />
4
ˆT = <br />
<br />
σ<br />
ˆV = <br />
<br />
σ<br />
ˆW = 1<br />
2<br />
<br />
<br />
σ,σ ′<br />
ψ † <br />
σ(r) − 2<br />
2m ∇2 <br />
(r) ψσ(r) dr (2.2)<br />
v σ (r)ψ † σ(r)ψσ(r) dr (2.3)<br />
ψ † σ(r)ψ †<br />
σ ′(r ′ )w(r, r ′ )ψσ ′(r′ )ψσ(r) drdr ′<br />
5<br />
(2.4)<br />
where σ is for sp<strong>in</strong> degeneracy and m is the mass of particles. Note th<strong>at</strong> ˆ W<br />
<strong>in</strong>dic<strong>at</strong>es a two-particle <strong>in</strong>teraction such as the Coulomb <strong>in</strong>teraction.<br />
We assume for simplicity th<strong>at</strong> the ground st<strong>at</strong>e is non-degener<strong>at</strong>e. (The<br />
theory can be, however, easily extended to the degener<strong>at</strong>e case [13].) S<strong>in</strong>ce a non-<br />
degener<strong>at</strong>e ground st<strong>at</strong>e φ is determ<strong>in</strong>ed by solv<strong>in</strong>g the Schröd<strong>in</strong>ger equ<strong>at</strong>ion with<br />
an external potential v(r)<br />
ˆH|φ〉 = ( ˆ T + ˆ V + ˆ W )|φ〉 = Egs|φ〉 , (2.5)<br />
a map C : V ↦→ Φ is def<strong>in</strong>ed from the set of external potentials V to the set of<br />
ground st<strong>at</strong>es Φ. Now, for any ground st<strong>at</strong>e, the electronic density <strong>in</strong> the ground<br />
st<strong>at</strong>e φ can be calcul<strong>at</strong>ed by<br />
n(r) = <br />
〈φ|ψ † σ(r)ψσ(r)|φ〉 , (2.6)<br />
σ<br />
which leads to a second map D : Φ ↦→ N from the set of ground st<strong>at</strong>es to the set<br />
of ground st<strong>at</strong>e densities. The map CD : V ↦→ N clearly tells us th<strong>at</strong> n(r) is a<br />
functional of v(r).<br />
2.1.1. Uniqueness of the external potential <strong>in</strong> terms of the density<br />
The Hohenberg - Kohn theorem st<strong>at</strong>es th<strong>at</strong> the map C and D are both<br />
<strong>in</strong>vertible uniquely, th<strong>at</strong> is, the external potential is uniquely determ<strong>in</strong>ed up to<br />
an additive constant by the ground st<strong>at</strong>e density. One can show this simply by
educio ad absurdum as <strong>in</strong> [1]. For the map C, assume th<strong>at</strong> two different external<br />
potentials give rise to the same ground st<strong>at</strong>e. Suppose<br />
( ˆ T + ˆ V1 + ˆ W )|φ〉 = Egs,1|φ〉 (2.7)<br />
and ( ˆ T + ˆ V2 + ˆ W )|φ〉 = Egs,2|φ〉 (2.8)<br />
where ˆ V1 = ˆ V2 + c by assumption. Note th<strong>at</strong> two potentials with only a constant<br />
difference are equivalent. By subtraction, one has, due to non-degeneracy,<br />
( ˆ V1 − ˆ V2)|φ〉 = (Egs,1 − Egs,2)|φ〉 (2.9)<br />
S<strong>in</strong>ce ˆ V1 and ˆ V2 are multiplic<strong>at</strong>ive oper<strong>at</strong>ors, one must have ˆ V1 − ˆ V2 = Egs,1 − Egs,2<br />
which is <strong>in</strong> contradiction with the assumption <strong>at</strong> the beg<strong>in</strong>n<strong>in</strong>g. More clearly, the<br />
st<strong>at</strong>ement can be proved as follows. Assume, for simplicity, th<strong>at</strong> the system has a<br />
s<strong>in</strong>gle particle with a sp<strong>in</strong> σ <strong>at</strong> position r. Then,<br />
〈rσ|( ˆ V1 − ˆ V2)|φ〉 = 〈rσ|(Egs,1 − Egs,2)|φ〉<br />
(v σ 1 (r) − v σ 2 (r))φσ(r) = (Egs,1 − Egs,2)φσ(r)<br />
(v σ 1 (r) − v σ 2 (r)) = (Egs,1 − Egs,2)<br />
assum<strong>in</strong>g th<strong>at</strong> |φ〉 does not vanish. F<strong>in</strong>ally,<br />
ˆV1 − ˆ V2 = <br />
<br />
dr (v σ 1 (r) − v σ 2 (r)))ψ † σ(r)ψσ(r)<br />
σ<br />
= (Egs,1 − Egs,2) <br />
<br />
= Egs,1 − Egs,2<br />
σ<br />
dr ψ † σ(r)ψσ(r)<br />
Two different potentials, hence, lead to the different ground st<strong>at</strong>es and the<br />
map C is <strong>in</strong>vertible.<br />
The proof for the map D follows by the same procedure. Assume th<strong>at</strong> two<br />
different ground st<strong>at</strong>es |φ1〉 and |φ2〉 yield the same ground st<strong>at</strong>e density, n1(r) =<br />
n2(r). Then one has<br />
6
Similarly<br />
Egs,1 = 〈φ1| ˆ H1|φ1〉 < 〈φ2| ˆ H1|φ2〉 = 〈φ2| ˆ H2 + ˆ V1 − ˆ V2|φ2〉<br />
<br />
= Egs,2 + dr n(r)(v1(r) − v2(r))<br />
Egs,2 = 〈φ2| ˆ H2|φ2〉 < 〈φ1| ˆ <br />
H2|φ1〉 = Egs,1 +<br />
Now, add<strong>in</strong>g of both <strong>in</strong>equalities leads to a contradiction aga<strong>in</strong><br />
Egs,1 + Egs,2 < Egs,1 + Egs,2<br />
dr n(r)(v2(r) − v1(r))<br />
7<br />
(2.10)<br />
and two different ground st<strong>at</strong>es |φ1〉 and |φ2〉, hence, yield different ground st<strong>at</strong>e<br />
densities. The map D must be <strong>in</strong>vertible. Due to this <strong>in</strong>vertibility, the ground st<strong>at</strong>e<br />
expect<strong>at</strong>ion value of any observable Ô (e.g. ˆ H, the Hamiltonian) is determ<strong>in</strong>ed<br />
by the ground st<strong>at</strong>e density and is thus a unique functional of the ground st<strong>at</strong>e<br />
density.<br />
〈φ[n]| Ô|φ[n]〉 = O[n] (2.11)<br />
The map CD is, therefore, <strong>in</strong>vertible and is thus the ground st<strong>at</strong>e density uniquely<br />
determ<strong>in</strong>es the external potential up to an additive constant.<br />
n,<br />
2.1.2. Vari<strong>at</strong>ional pr<strong>in</strong>ciple<br />
Now, the energy functional with the external potential ˆ V is, for any density<br />
E[n] = 〈φ[n]| ˆ T + ˆ V + ˆ W |φ[n]〉 . (2.12)<br />
S<strong>in</strong>ce there exist a non-degener<strong>at</strong>e ground st<strong>at</strong>e density ngs and ground st<strong>at</strong>e en-<br />
ergy Egs for this specific external potential via the map CD, we derive from the<br />
vari<strong>at</strong>ional pr<strong>in</strong>ciple
Egs = E[ngs] = 〈φ[ngs]| ˆ T + ˆ V + ˆ W |φ[ngs]〉<br />
E[n] = 〈φ[n]| ˆ T + ˆ V + ˆ W |φ[n]〉<br />
Egs < E[n] if n = ngs .<br />
The exact ground st<strong>at</strong>e density is, therefore, determ<strong>in</strong>ed by m<strong>in</strong>imiz<strong>in</strong>g the energy<br />
functional E[n]: Egs = m<strong>in</strong><br />
n∈N E[n].<br />
Therefore, one can conclude th<strong>at</strong> there exists a unique energy functional for<br />
a given external potential from the Hohenberg and Kohn theorem and the exact<br />
ground st<strong>at</strong>e density is determ<strong>in</strong>ed by the vari<strong>at</strong>ional equ<strong>at</strong>ion δE[n] = 0.<br />
2.1.3. Universal functional FHK[n]<br />
S<strong>in</strong>ce the map D is <strong>in</strong>dependent of the external potential of the particular<br />
system, one can write the energy functional E[n] as<br />
E[n] = 〈φ[n]| ˆ T + ˆ V + ˆ W |φ[n]〉<br />
<br />
(2.13)<br />
= FHK[n] + dr v0(r)n(r)<br />
where FHK[n] = 〈φ[n]| ˆ T + ˆ W |φ[n]〉. As one can see FHK[n] does not depend on the<br />
external potential (e.g. the <strong>in</strong>teraction between electrons and nuclei), and it is <strong>in</strong><br />
pr<strong>in</strong>ciple possible to calcul<strong>at</strong>e the universal functional FHK[n], which is the same for<br />
all electronic systems such as <strong>at</strong>oms, molecules and solids s<strong>in</strong>ce, for these systems,<br />
ˆW is the Coulomb repulsion between electrons. Now, the functional FHK[n] has<br />
a very important mean<strong>in</strong>g, s<strong>in</strong>ce once FHK[n] is known, one can apply it for any<br />
systems (any number of particles and any external potential).<br />
8
2.2. The Kohn-Sham equ<strong>at</strong>ions<br />
At first sight, density functional theory seems to lead to a new way for the<br />
physics of many-body systems. The universal functional FHK[n] is, however, <strong>in</strong><br />
most cases not possible to obta<strong>in</strong>. One can make some progress to f<strong>in</strong>d approxi-<br />
m<strong>at</strong>e expressions for FHK[n] for a gas of slowly vary<strong>in</strong>g density or almost constant<br />
density. [1] Kohn and Sham, <strong>in</strong> 1965, suggested an altern<strong>at</strong>ive way to solve the<br />
problem, us<strong>in</strong>g a set of self-consistent equ<strong>at</strong>ions for s<strong>in</strong>gle particles <strong>in</strong>clud<strong>in</strong>g the<br />
exchange-correl<strong>at</strong>ion potentials. [2] This s<strong>in</strong>gle particle picture, the Kohn-Sham<br />
equ<strong>at</strong>ions, of a many-body system is quite accessible especially <strong>in</strong> a numerical way<br />
and, thus, provides a new comput<strong>at</strong>ional way us<strong>in</strong>g s<strong>in</strong>gle particle aspects of solid<br />
st<strong>at</strong>e physics.<br />
The key po<strong>in</strong>t <strong>in</strong> deriv<strong>in</strong>g the Kohn-Sham equ<strong>at</strong>ions is th<strong>at</strong>, if n is the ground<br />
st<strong>at</strong>e density of an <strong>in</strong>teract<strong>in</strong>g particle system, one can assume there exists a unique<br />
external potential vNI of a non-<strong>in</strong>teract<strong>in</strong>g particle system correspond<strong>in</strong>g to the<br />
ground st<strong>at</strong>e density n. (Uniqueness of vNI is proven from the Hohenberg-Kohn<br />
theorem, but the existence of vNI must be assumed.) It is possible, now, to de-<br />
rive one-particle equ<strong>at</strong>ions of the <strong>in</strong>teract<strong>in</strong>g system by comparison with the non-<br />
<strong>in</strong>teract<strong>in</strong>g system.<br />
2.2.1. Non-<strong>in</strong>teract<strong>in</strong>g particle system <strong>in</strong> an external potential vNI(r)<br />
The Schröd<strong>in</strong>ger equ<strong>at</strong>ion for a non-<strong>in</strong>teract<strong>in</strong>g system <strong>in</strong> an external poten-<br />
tial vNI(r) is<br />
<br />
− 2<br />
2m ∇2 <br />
+ vNI(r) φi(r) = ɛiφi(r) . (2.14)<br />
The ground st<strong>at</strong>e is aga<strong>in</strong> assumed to be non-degener<strong>at</strong>e for simplicity.<br />
9
The ground st<strong>at</strong>e density nNI is<br />
nNI(r) =<br />
where φi(r) occupies the lowest eigenst<strong>at</strong>es.<br />
10<br />
N<br />
|φi(r)| 2 . (2.15)<br />
i<br />
If the ground st<strong>at</strong>e density nNI is given, the external potential vNI(r) is<br />
determ<strong>in</strong>ed up to a trivial constant from the Hohenberg-Kohn theorem. S<strong>in</strong>ce the<br />
wave functions φi(r) of the Schröd<strong>in</strong>ger equ<strong>at</strong>ion are, thus, unique functionals of<br />
the density nNI, φi(r) = φi([n]; r), the k<strong>in</strong>etic energy TNI[n] of the non-<strong>in</strong>teract<strong>in</strong>g<br />
system is a unique functional and hence so is the total energy. The k<strong>in</strong>etic energy<br />
and total energy functional with the ground st<strong>at</strong>e density nNI are<br />
TNI[n] = <br />
<br />
i,σ<br />
dr φ ∗ <br />
i,σ(r) − 2<br />
2m ∇2 <br />
(r) φi,σ(r)<br />
<br />
(2.16)<br />
and ENI[n] = TNI[n] + dr vNI(r)n(r) . (2.17)<br />
The ground st<strong>at</strong>e density, equivalently, can be obta<strong>in</strong>ed from the vari<strong>at</strong>ional<br />
pr<strong>in</strong>ciple δE = 0 [13] of the Hohenberg-Kohn theorem or the Euler equ<strong>at</strong>ions:<br />
<br />
<br />
δ<br />
ENI[n] − µNI<br />
δn(r)<br />
<br />
n(r) dr = δTNI[n]<br />
δn(r) + vNI(r) − µNI = 0 (2.18)<br />
where µNI is a Lagrange multiplier to keep the total number of particles constant,<br />
n(r) dr = N.<br />
2.2.2. Interact<strong>in</strong>g particle system <strong>in</strong> an external potential v(r)<br />
The energy functional for an <strong>in</strong>teract<strong>in</strong>g particle system is, from the<br />
Hohenberg-Kohn theorem,<br />
<br />
E[n] = FHK[n] + dr vI(r)n(r) (2.19)
If we def<strong>in</strong>e Exc[n] as<br />
Exc[n] = FHK[n] − 1<br />
<br />
2<br />
11<br />
dr dr ′ n(r)w(r, r ′ )n(r ′ ) − TNI[n] (2.20)<br />
where TNI[n] is the non-<strong>in</strong>teract<strong>in</strong>g k<strong>in</strong>etic energy, the energy functional follows<br />
the form<br />
E[n] = TNI[n] + 1<br />
<br />
2<br />
dr dr ′ n(r)w(r, r ′ )n(r ′ <br />
) + Exc[n] +<br />
dr vI(r)n(r) (2.21)<br />
The ground st<strong>at</strong>e density nI(r) of the <strong>in</strong>teract<strong>in</strong>g system can be obta<strong>in</strong>ed<br />
from the Euler equ<strong>at</strong>ion, [14]<br />
where<br />
δ<br />
δn(r) [EI[n]<br />
<br />
− µI dr n(r)] (2.22)<br />
= δTNI[n]<br />
δn(r) + vI(r)<br />
<br />
+ dr ′ w(r, r ′ )n(r ′ ) + vxc([n]; r) − µI = 0 (2.23)<br />
vxc([n]; r) = δExc[n]<br />
δn(r)<br />
As mentioned before, if there exists the unique external potential vNI(r) for<br />
a non-<strong>in</strong>teract<strong>in</strong>g system correspond<strong>in</strong>g to the ground st<strong>at</strong>e density nI(r) of the<br />
<strong>in</strong>teract<strong>in</strong>g system, one can f<strong>in</strong>d from (2.18) and (2.22) th<strong>at</strong><br />
<br />
vNI(r) = vI(r) +<br />
<br />
= vI(r) +<br />
dr ′ w(r, r ′ )n(r ′ ) + vxc([n]; r) − (µI − µNI)<br />
.<br />
dr ′ w(r, r ′ )n(r ′ ) + vxc([n]; r) (2.24)<br />
where the arbitrary constant <strong>in</strong> vNI(r) is chosen to cancel out the term (µI −<br />
µNI) and, of course, the potentials on both sides of the equ<strong>at</strong>ion s<strong>at</strong>isfy the same<br />
boundary conditions.<br />
One can f<strong>in</strong>ally write down, the Kohn-Sham equ<strong>at</strong>ions, a set of the self-<br />
consistent equ<strong>at</strong>ions of a non-<strong>in</strong>teract<strong>in</strong>g system with vNI(r) correspond<strong>in</strong>g to the<br />
<strong>in</strong>teract<strong>in</strong>g system.
[−<br />
12<br />
2<br />
2m ∇2 <br />
+ vI(r) + dr ′ w(r, r ′ )n(r ′ ) + vxc([n]; r)]φi(r) = ɛiφi(r) (2.25)<br />
N<br />
n(r) = |φi(r)| 2<br />
for the lowest eigenst<strong>at</strong>es i = 1, · · · , N (2.26)<br />
i<br />
These equ<strong>at</strong>ions are the self-consistent equ<strong>at</strong>ions <strong>in</strong> a sense th<strong>at</strong> one can solve<br />
(2.25) to obta<strong>in</strong> the ground st<strong>at</strong>e density and the effective s<strong>in</strong>gle particle potential<br />
depends on the density.<br />
2.2.3. Ground st<strong>at</strong>e energy of an <strong>in</strong>teract<strong>in</strong>g system<br />
The ground st<strong>at</strong>e energy for the Kohn-Sham equ<strong>at</strong>ions can be easily derived.<br />
From (2.25),<br />
<br />
ɛi =<br />
T [n] =<br />
N<br />
i<br />
dr φ ∗ i (r)(− 2<br />
ɛi −<br />
where vNI follows (2.24).<br />
Eg =<br />
<br />
2m ∇2 <br />
)φi(r) +<br />
dr |φi(r)| 2 vNI(r) (2.27)<br />
dr n(r)vNI(r) (2.28)<br />
By <strong>in</strong>sert<strong>in</strong>g (2.28) <strong>in</strong>to (2.21), the ground st<strong>at</strong>e energy Eg is<br />
N<br />
i<br />
ɛi − 1<br />
<br />
2<br />
dr dr ′ n(r)w(r, r ′ )n(r ′ <br />
) + Exc[n] −<br />
2.2.4. The chemical potential µ and Fermi surface<br />
The follow<strong>in</strong>g proof is taken from [13].<br />
dr vxc(r)n(r) . (2.29)<br />
With the def<strong>in</strong>ition of the energy functional (2.21), equ<strong>at</strong>ion (2.22) leads to<br />
δEI[n]<br />
δn(r)<br />
= µI<br />
(2.30)
If EN is the ground st<strong>at</strong>e energy and nN is the ground st<strong>at</strong>e density of a N-particle<br />
system,<br />
Then,<br />
∂EN<br />
∂N<br />
EN+ɛ − EN<br />
= lim<br />
ɛ→0 ɛ<br />
δEI[n]<br />
δn(r)<br />
<br />
<br />
<br />
nN<br />
<br />
1<br />
= lim<br />
ɛ→0 ɛ<br />
<br />
1<br />
= lim<br />
ɛ→0 ɛ<br />
= µI(N)<br />
13<br />
= µI[N] (2.31)<br />
dr δEI[n]<br />
δn(r)<br />
<br />
<br />
<br />
nN<br />
(nN+ɛ(r) − nN(r))<br />
dr µI(N)(nN+ɛ(r) − nN(r))<br />
The Lagrangian multiplier, therefore, equals the exact chemical potential of the<br />
<strong>in</strong>teract<strong>in</strong>g system.<br />
The eigenvalues of the Kohn-Sham equ<strong>at</strong>ion are less than or equal to the<br />
chemical potential µ. The Kohn-Sham Fermi surface <strong>in</strong> k space is, thus, def<strong>in</strong>ed<br />
by ɛFermi ≡ µ. The Kohn-Sham Fermi surface is, <strong>in</strong> general, not the same as<br />
the physical Fermi surface, which can be found from the Dyson equ<strong>at</strong>ion, s<strong>in</strong>ce<br />
veff(r) <strong>in</strong> the Kohn-Sham equ<strong>at</strong>ion is not the same as the self-energy Σ(r, r ′ ; ɛi)<br />
<strong>in</strong> the Dyson equ<strong>at</strong>ion. They are, however, identical <strong>in</strong> the case of a uniform<br />
system of <strong>in</strong>teract<strong>in</strong>g particles s<strong>in</strong>ce the Fermi surface must be spherical no m<strong>at</strong>ter<br />
wh<strong>at</strong> equ<strong>at</strong>ions are used. Even <strong>in</strong> the case of a non-uniform system, it turns<br />
out th<strong>at</strong> the difference between two Fermi surfaces is small s<strong>in</strong>ce the electrost<strong>at</strong>ic<br />
potential(external and Coulomb potentials) which is the primary term to cause the<br />
anisotropy of the Fermi surface is the same for both cases. (See Chapter 2, [14]<br />
and section 6.4, [13] for more details.)
2.2.5. Excited st<strong>at</strong>es; Eigenvalues <strong>in</strong> the Kohn-Sham equ<strong>at</strong>ions<br />
Koopmans’ theorem st<strong>at</strong>es th<strong>at</strong> the eigenvalue of the eigenst<strong>at</strong>e m <strong>in</strong> the<br />
Hartree-Fock equ<strong>at</strong>ion is the same as the neg<strong>at</strong>ive of the energy required to remove<br />
the electron <strong>in</strong> the st<strong>at</strong>e m. [15] (This theorem does not apply to small systems. See<br />
[15].) It is n<strong>at</strong>ural, thus, to th<strong>in</strong>k of higher eigenst<strong>at</strong>es <strong>in</strong> the Kohn-Sham equ<strong>at</strong>ion<br />
as those <strong>in</strong> the Hartree-Fock equ<strong>at</strong>ion and obta<strong>in</strong> excit<strong>at</strong>ion energies from the<br />
Kohn-Sham equ<strong>at</strong>ion.<br />
Density functional theory is, however, essentially a ground st<strong>at</strong>e and st<strong>at</strong>ic<br />
(frequency-<strong>in</strong>dependent) theory, which means th<strong>at</strong> eigenvalues or energy levels of<br />
the Kohn-Sham equ<strong>at</strong>ion do not represent excit<strong>at</strong>ion energies which are required<br />
to excite electrons. The primary reason is th<strong>at</strong> the effective exchange-correl<strong>at</strong>ion<br />
potentials must vary <strong>in</strong> the excited system. (See Section 4, [13].) One can also<br />
compare the real eigenvalues of the Kohn-Sham equ<strong>at</strong>ion with the excit<strong>at</strong>ion en-<br />
ergies of the Dyson equ<strong>at</strong>ion. The excit<strong>at</strong>ion energies of the Dyson equ<strong>at</strong>ion are,<br />
however, <strong>in</strong> general complex, which means the f<strong>in</strong>ite lifetime, and the real parts<br />
are <strong>in</strong> general different from the eigenvalues of the Kohn-Sham equ<strong>at</strong>ion as well.<br />
[14]<br />
In order to f<strong>in</strong>d the excit<strong>at</strong>ion energy <strong>in</strong> the Kohn-Sham scheme, one needs<br />
to solve two Kohn-Sham equ<strong>at</strong>ions and f<strong>in</strong>d two self-consistent solutions for the<br />
ground st<strong>at</strong>e and excited st<strong>at</strong>e. In this thesis, the excited st<strong>at</strong>es for a hydrogen<br />
and a carbon impurity will be exam<strong>in</strong>ed by vary<strong>in</strong>g the number of electrons <strong>in</strong> the<br />
bound st<strong>at</strong>es. It is, however, possible to f<strong>in</strong>d the approxim<strong>at</strong>e excit<strong>at</strong>ion energy<br />
us<strong>in</strong>g Janak’s theorem and the Sl<strong>at</strong>er transition st<strong>at</strong>e [13] and also, for an <strong>in</strong>f<strong>in</strong>ite<br />
system, derive a theorem of self-consistent equ<strong>at</strong>ions similar to Koopmans’ theorem<br />
(P. 142, [14]).<br />
14
2.2.6. Sp<strong>in</strong>-polarized systems<br />
If there’s an external magnetic field, one generally has to consider the elec-<br />
tronic orbital coupl<strong>in</strong>g as well as sp<strong>in</strong> coupl<strong>in</strong>g. In this thesis, only sp<strong>in</strong> coupl<strong>in</strong>g<br />
is discussed and the Kohn-Sham equ<strong>at</strong>ions are derived accord<strong>in</strong>gly.<br />
The external potential term, (2.1), is modified as follows. [13]<br />
ˆV = <br />
<br />
σ1σ2<br />
dr ψ † σ1 (r) [vσ1 (r)δσ1σ2 + µBB(r) · σσ1σ2(r)] ψσ2(r) (2.32)<br />
where µB = e/2mc is the Bohr magneton.<br />
The magnetic moment density oper<strong>at</strong>or is def<strong>in</strong>ed as<br />
<br />
ˆm(r) = −µB ψ † σ1 (r)σσ1σ2(r)ψσ2(r) (2.33)<br />
σ1σ2<br />
where σ denotes the vector of Pauli m<strong>at</strong>rices.<br />
With the density oper<strong>at</strong>or<br />
ˆn(r) = ˆn+(r) + ˆn−(r) = <br />
ˆnσ(r) = <br />
ψ † σ(r)ψσ(r) , (2.34)<br />
the external potential term can be rewritten as<br />
<br />
ˆV =<br />
σ<br />
σ<br />
dr [v(r)ˆn(r) − B(r) · ˆm(r)] . (2.35)<br />
Now the ground st<strong>at</strong>e density n(r) and magnetiz<strong>at</strong>ion density m(r) is<br />
n(r) = 〈φ|ˆn(r)|φ〉 (2.36)<br />
m(r) = 〈φ| ˆm(r)|φ〉 . (2.37)<br />
One can work either with (n(r), m(r)) or, equivalently, with the ground sp<strong>in</strong><br />
density m<strong>at</strong>rix nσ1σ2(r) = 〈φ|ψ † σ1 σσ1σ2(r)ψσ2(r)|φ〉. Note th<strong>at</strong> both cases have<br />
four <strong>in</strong>dependent real functions.<br />
15
For simplicity, it is assumed th<strong>at</strong> only z-components <strong>in</strong> the external magnetic<br />
field B(r) and the magnetiz<strong>at</strong>ion m(r) have non-vanish<strong>in</strong>g values. Note th<strong>at</strong>, for<br />
a z-component of m(r),<br />
m(r) = −µB(n+(r) − n−(r)) . (2.38)<br />
As <strong>in</strong> the case of the sp<strong>in</strong>-<strong>in</strong>dependent Hohenberg-Kohn theorem, two differ-<br />
ent non-degener<strong>at</strong>e ground st<strong>at</strong>es lead to two different sp<strong>in</strong> density m<strong>at</strong>rices nss ′(r)<br />
or, equivalently, to two different sets of (n(r), m(r)) or (n+(r), n−(r)).<br />
Now one can derive the Kohn-Sham equ<strong>at</strong>ions for a sp<strong>in</strong>-polarized system<br />
from the vari<strong>at</strong>ional pr<strong>in</strong>ciple of the Hohenberg-Kohn theorem as <strong>in</strong> the sp<strong>in</strong>-<br />
<strong>in</strong>dependent case, Sec. 2.2.<br />
[− 2<br />
2m ∇2 + vI(r) + σµBB(r) +<br />
nσ(r) =<br />
<br />
dr ′ w(r, r ′ )n(r ′ )<br />
+ v σ xc([n+, n−]; r)]φ σ i (r) = ɛ σ i φ σ i (r) (2.39)<br />
N<br />
|φ σ i (r)| 2 for the lowest eigenst<strong>at</strong>es i = 1, · · · , N (2.40)<br />
i<br />
where σ denotes + or − correspond<strong>in</strong>g to the sp<strong>in</strong> projection <strong>in</strong> the z direction.<br />
dent case,<br />
The exchange-correl<strong>at</strong>ion potential v σ xc([n+, n−]; r) is, as <strong>in</strong> the sp<strong>in</strong> <strong>in</strong>depen-<br />
v σ xc([n+, n−]; r) = δEσ xc[n+, n−]<br />
δnσ(r)<br />
16<br />
. (2.41)<br />
and the functional Eσ xc[n+, n−] is, as <strong>in</strong> the sp<strong>in</strong> <strong>in</strong>dependent case, given by<br />
E σ xc[n+, n−] = FHK[n+, n−] − 1<br />
<br />
2<br />
n(r)w(r, r ′ )n(r ′ ) − TNI[n+, n−] (2.42)<br />
The ground st<strong>at</strong>e energy with an analogy to (2.29) is given by<br />
N <br />
Eg = ɛ<br />
i σ<br />
σ i − 1<br />
<br />
dr n(r)w(r, r<br />
2<br />
′ )n(r ′ ) + Exc[n+, n−]<br />
− <br />
<br />
dr v σ xc([n+, n−]; r)nσ(r) . (2.43)<br />
σ
2.2.7. Exchange-correl<strong>at</strong>ion; LDA<br />
At this po<strong>in</strong>t the only problem <strong>in</strong> the Kohn-Sham scheme is to f<strong>in</strong>d a func-<br />
tional Exc[n] <strong>in</strong>stead of a functional FHK[n] <strong>in</strong> the Hohenberg-Kohn theorem.<br />
In this work, the local density approxim<strong>at</strong>ion(LDA) is used for the exchange-<br />
correl<strong>at</strong>ion functional Exc[n].<br />
2.2.7.1. A functional<br />
Suppose there is a functional E[n] for a system with a vary<strong>in</strong>g electron density.<br />
A functional E[n], <strong>in</strong> general, is nonlocal and, thus, E[n] depends on the whole<br />
distribution of the electron density. [16]<br />
If E[n] has the form E[n] = e[n] dr where e[n] is a density functional, for<br />
a very slowly vary<strong>in</strong>g density, e[n] can be expanded <strong>in</strong> Taylor series <strong>in</strong> terms of a<br />
density gradient.<br />
e[n] = e0(n(r)) + e1(n(r))∇n(r) + · · · (2.44)<br />
In LDA, us<strong>in</strong>g the first term only, the exchange-correl<strong>at</strong>ion functional becomes<br />
<br />
Exc[n] = n(r)ɛxc(n(r)) dr (2.45)<br />
where ɛxc(n(r)) is the exchange-correl<strong>at</strong>ion energy per electron for a uniform gas<br />
of <strong>in</strong>teract<strong>in</strong>g electrons with density n. [17] The problem is now reduced to f<strong>in</strong>d<strong>in</strong>g<br />
the exchange-correl<strong>at</strong>ion energy <strong>in</strong> a homogeneous electron gas.<br />
2.2.7.2. Exchange-correl<strong>at</strong>ion energy<br />
The energy of the Hartree-Fock model with plane-wave st<strong>at</strong>es can be easily<br />
obta<strong>in</strong>ed us<strong>in</strong>g the Sl<strong>at</strong>er determ<strong>in</strong>ant or us<strong>in</strong>g the field oper<strong>at</strong>ors <strong>in</strong> the second<br />
17
quantiz<strong>at</strong>ion not<strong>at</strong>ion (Ch. 5, [15]) or the exchange term of the self energy (Ch.<br />
5, [18]) s<strong>in</strong>ce the Hartree energy is zero. The exchange energy, subtract<strong>in</strong>g the<br />
k<strong>in</strong>etic energy, is given by<br />
Ex(k) = − e2 kF<br />
π<br />
<br />
2kF k ln<br />
<br />
kF<br />
<br />
+ k <br />
<br />
kF<br />
− k <br />
1 + k2 F − k2<br />
18<br />
(2.46)<br />
The average exchange energy contribution per particle to the ground st<strong>at</strong>e energy<br />
is easily obta<strong>in</strong>ed as well (See [15] and [18]). Allow<strong>in</strong>g two electrons (sp<strong>in</strong>-up and<br />
sp<strong>in</strong>-down) per st<strong>at</strong>e,<br />
ɛx = 2 1<br />
2N<br />
<br />
k<br />
nkEx(k) = − 3 e<br />
4<br />
2kF π<br />
(2.47)<br />
The exchange part (called the Kohn-Sham or Gáspár potential [17]) of the chemical<br />
potential is, us<strong>in</strong>g the Seitz’s theorem [18], given by<br />
µx = d<br />
dn (nɛx) = − e2<br />
π kF = 4<br />
3 ɛx<br />
(2.48)<br />
Collect<strong>in</strong>g the k<strong>in</strong>etic term and the exchange term gives the first two terms for the<br />
ground st<strong>at</strong>e energy per particle<br />
ɛgs = 2.2099<br />
r 2 s<br />
− 0.9163<br />
rs<br />
where rs is the Wigner-Seitz radius, def<strong>in</strong>ed by n 4π<br />
3 r3 s = 1.<br />
+ · · · Ryd. (2.49)<br />
The exclusion pr<strong>in</strong>ciple <strong>in</strong> the Hartree-Fock model <strong>in</strong>troduces the strong re-<br />
pulsion between the electrons of parallel sp<strong>in</strong> but the Hartree-Fock model fails to<br />
correl<strong>at</strong>e the motion of electrons of antiparallel sp<strong>in</strong> s<strong>in</strong>ce the electrons of antipar-<br />
allel sp<strong>in</strong> tends to separ<strong>at</strong>e due to the Coulomb repulsion. Now the difference<br />
between the exact energy and the Hartree-Fock energy is called the correl<strong>at</strong>ion<br />
energy and thus one can write<br />
ɛgs = 2.2099<br />
r 2 s<br />
− 0.9163<br />
rs<br />
+ ɛc Ryd. (2.50)
The high density limit of rs → 0 for ɛc was first given by Gell-Mann and Brueckner<br />
(1957) and the same result has been obta<strong>in</strong>ed by many other people. Qu<strong>in</strong>n and<br />
Ferrell (1958) used the Seitz’s theorem to f<strong>in</strong>d the correl<strong>at</strong>ion energy from the<br />
correl<strong>at</strong>ion chemical potential µc. They found µc by calcul<strong>at</strong><strong>in</strong>g the self-energy<br />
terms of an electron <strong>at</strong> the Fermi energy. The exchange-correl<strong>at</strong>ion terms <strong>in</strong> the<br />
self-energy can be c<strong>at</strong>egorized <strong>in</strong> the Feynman diagrams by the number of <strong>in</strong>ternal<br />
Coulomb l<strong>in</strong>es [18]. The exchange energy term has only one Coulomb l<strong>in</strong>e. The<br />
correl<strong>at</strong>ion terms have two or more Coulomb l<strong>in</strong>es. The correl<strong>at</strong>ion energy from<br />
one of self-energy diagrams was given by Onsager et al. (1966), which is 0.0436.<br />
The correl<strong>at</strong>ion energy <strong>in</strong> the RPA is<br />
ɛc = −0.142 + 0.0622 ln rs + · · · Ryd. (2.51)<br />
Collect<strong>in</strong>g these terms gives the result, with another term from Carr and<br />
Maradud<strong>in</strong> (1964),<br />
ɛc = −0.094 + 0.0622 ln rs + 0.018rs ln rs + · · · Ryd. (2.52)<br />
This result gives, however, positive values for low densities (rs >∼ 2.5), which<br />
is not correct s<strong>in</strong>ce the correl<strong>at</strong>ion energy must be neg<strong>at</strong>ive consider<strong>in</strong>g the pair<br />
distribution function. (The correl<strong>at</strong>ion effects lower the energy of an <strong>in</strong>teract<strong>in</strong>g<br />
electron system and <strong>in</strong>creases the magnitude of the b<strong>in</strong>d<strong>in</strong>g energy.) [18]<br />
Wigner (1934) found th<strong>at</strong> the system with the positive background charge<br />
ga<strong>in</strong>s energy by localiz<strong>at</strong>ion of electrons. The total energy for a simple system <strong>in</strong><br />
which there is an electron <strong>at</strong> the center of a sphere <strong>in</strong> the unit cell of the l<strong>at</strong>tice<br />
(Wigner-Seitz model) and the electric field is zero outside of each sphere due to<br />
the uniform positive charge is ∼ −1.8/rs. In the low density limit, subtract<strong>in</strong>g the<br />
exchange correl<strong>at</strong>ion energy −0.9163/rs,<br />
19
FIGURE 2.1. The exchange ɛx , correl<strong>at</strong>ion ɛc, and Thomas-Fermi k<strong>in</strong>etic energies<br />
TT F = 2.2099/r 2 s per electron from the parametriz<strong>at</strong>ion of Hed<strong>in</strong>-Lundqvist.<br />
ɛc = − 0.8757<br />
rs<br />
20<br />
Ryd. (2.53)<br />
Interpol<strong>at</strong>ion schemes between these two limits have been made by several<br />
people. For <strong>in</strong>stance, Nozières and P<strong>in</strong>es (1958) recommend, for the range of actual<br />
metallic densities, the <strong>in</strong>terpol<strong>at</strong>ion result ɛc −0.115 + 0.031 ln rs Ryd.<br />
The correl<strong>at</strong>ion energy also can be calcul<strong>at</strong>ed us<strong>in</strong>g dielectric functions for<br />
which there are several models such as Thomas-Fermi, RPA, Hubbard, S<strong>in</strong>gwi-<br />
Sjölander. The S<strong>in</strong>gwi-Sjölander correl<strong>at</strong>ion energy is considered to be the best<br />
due to its positive pair distribution function for most densities. [18]
FIGURE 2.2. The exchange ɛx , correl<strong>at</strong>ion ɛc, and Thomas-Fermi k<strong>in</strong>etic energies<br />
TT F = 2.2099/r 2 s multiplied by the electron density n.<br />
2.2.7.3. Interpol<strong>at</strong>ion scheme; Sp<strong>in</strong>-<strong>in</strong>dependent<br />
In this work, the <strong>in</strong>terpol<strong>at</strong>ion scheme by Hed<strong>in</strong> and Lundqvist (See [17] and<br />
[14]) is used. They used a model given by<br />
µxc(rs) = µx(rs) + µc(rs) = β(rs)µx(rs) . (2.54)<br />
where µxc(rs) is the exchange-correl<strong>at</strong>ion contribution to the effective potential<br />
veff and β(rs) is a correl<strong>at</strong>ion enhancement factor.<br />
With the expression<br />
one can obta<strong>in</strong>, us<strong>in</strong>g (2.48),<br />
<br />
β(rs) = 1 + Bx ln 1 + 1<br />
<br />
x<br />
21<br />
, (2.55)
FIGURE 2.3. The exchange vx and correl<strong>at</strong>ion vc potentials.<br />
<br />
µc(rs) = −C ln 1 + 1<br />
<br />
Ryd. (2.56)<br />
x<br />
<br />
ɛc(rs) = −C (1 + x 3 <br />
) ln 1 + 1<br />
<br />
+<br />
x<br />
x<br />
2 − x2 − 1<br />
<br />
Ryd. (2.57)<br />
3<br />
where x = rs/A and C = 2B((9π)/4) 1/3 /πA. The parameters<br />
22<br />
A = 21 and C = 0.045 (2.58)<br />
are chosen to reproduce the S<strong>in</strong>gwi et al. (1970) results for exc.<br />
S<strong>in</strong>ce β(rs) varies from 1.0 to 1.33 for 0 ≤ rs ≤ 6, the exchange effect of ɛxc<br />
dom<strong>in</strong><strong>at</strong>es the correl<strong>at</strong>ion contribution for the range of actual metallic densities.<br />
The behavior of the exchange and correl<strong>at</strong>ion energies is shown <strong>in</strong> Fig. 2.1 and<br />
Fig. 2.2. The correl<strong>at</strong>ion energies are always neg<strong>at</strong>ive as it should be. A system
of the <strong>in</strong>teract<strong>in</strong>g electrons reaches a lower exchange-correl<strong>at</strong>ion energy as the<br />
density <strong>in</strong>creases. A simple model, which <strong>in</strong>cludes the exchange-correl<strong>at</strong>ion and<br />
Thomas-Fermi k<strong>in</strong>etic energy only, as <strong>in</strong> Fig. 2.2, however, has a m<strong>in</strong>imum energy<br />
<strong>at</strong> n 0.015. One can argue th<strong>at</strong>, if Coulomb effects and the vari<strong>at</strong>ion of the<br />
wave functions are neglected, a system of <strong>in</strong>teract<strong>in</strong>g electrons favors the density<br />
n 0.015.<br />
2.2.7.4. Interpol<strong>at</strong>ion scheme; Sp<strong>in</strong>-dependent<br />
The LDA can be extended for the sp<strong>in</strong>-polarized case (called the local sp<strong>in</strong><br />
density approxim<strong>at</strong>ion, LSDA) and, by choos<strong>in</strong>g the z-axis along the local sp<strong>in</strong><br />
direction,<br />
<br />
Exc[n+, n−] =<br />
and the variables are now n+(r) and n−(r) or<br />
dr (n+(r) + n−(r))ɛxc(n+(r), n−(r)) (2.59)<br />
n(r) = n+(r) + n−(r)<br />
and ζ(r) = n+(r) − n−(r)<br />
n(r)<br />
where ζ describes the degree of local magnetiz<strong>at</strong>ion. For <strong>in</strong>stance, the exchange<br />
energy can be, us<strong>in</strong>g a simple superposition pr<strong>in</strong>ciple (Ch.5, [13]), written as<br />
Ex,pol[n, ζ] = 1<br />
2 Ex[(1 + ζ)n] + 1<br />
2 Ex[(1 − ζ)n] ,<br />
which can be expanded <strong>in</strong> a series us<strong>in</strong>g a gradient expansion. In lowest order, the<br />
exchange energy per particle is<br />
ɛ 0 x,pol(r) = 1<br />
2 [(1 + ζ(r))4/3 + (1 − ζ(r)) 4/3 ]ɛ 0 x<br />
where ɛ 0 x follows the form (2.47).<br />
23<br />
(2.60)
For the sp<strong>in</strong>-polarized case, there are several parametriz<strong>at</strong>ions, such as von<br />
Barth and Hed<strong>in</strong> (1972), Gunnarsson and Lundqvist (1976), and Monte Carlo<br />
results of Ceperley and Alder (1980). (See [13] or their papers for more details.)<br />
In this work, the parametriz<strong>at</strong>ion of von Barth and Hed<strong>in</strong> is used. They used a<br />
generalized random phase expression for ɛxc <strong>in</strong> terms of the polariz<strong>at</strong>ion propag<strong>at</strong>or<br />
and the two-bubble r<strong>in</strong>g approxim<strong>at</strong>ion for the irreducible propag<strong>at</strong>or. [19]<br />
The parametriz<strong>at</strong>ion is based on an <strong>in</strong>terpol<strong>at</strong>ion between the paramag-<br />
netic(unpolarized, ζ = 0) and the ferromagnetic(fully polarized, ζ = ±1) st<strong>at</strong>e:<br />
ɛxc(n(r), ζ(r)) = ɛx(n(r), ζ(r)) + ɛc(n(r), ζ(r)) (2.61)<br />
ɛi(n(r), ζ(r)) = ɛi(n, ζ = 0) + (ɛi(n, ζ = 1) − ɛi(n, ζ = 0))f(ζ(r)) (2.62)<br />
where i denotes the exchange (i = x) or correl<strong>at</strong>ion (i = c) contribution to the<br />
exchange-correl<strong>at</strong>ion energy per particle. The <strong>in</strong>terpol<strong>at</strong>ion function is:<br />
f(ζ(r)) = (1 + ζ(r))4/3 + (1 − ζ(r)) 4/3 − 2<br />
2 4/3 − 2<br />
24<br />
(2.63)<br />
from which one can obta<strong>in</strong> aga<strong>in</strong> the ζ-dependence of the exchange energy term<br />
(2.60). The exchange energy <strong>in</strong> the paramagnetic limit, ɛx(n, ζ = 0), is given by<br />
(2.47). In the ferromagnetic limit, ɛx(n, ζ = ±1) = 2 1/3 ɛx(n, ζ = 0).<br />
The correl<strong>at</strong>ion energy follows aga<strong>in</strong> the Hed<strong>in</strong>-Lundqvist form us<strong>in</strong>g the<br />
parameters<br />
(2.59).<br />
ζ = 0 : C P = 0.0504, A P = 30<br />
ζ = 1 : C F = 0.0254, A F = 75<br />
The exchange-correl<strong>at</strong>ion potential can be easily calcul<strong>at</strong>ed us<strong>in</strong>g (2.41) and<br />
v σ xc = ∂<br />
∂n σ ((n+ + n−)ɛxc(n+, n−)) (2.64)<br />
For <strong>in</strong>stance, v + xc = ɛxc(n+, n−)) + n ∂ɛxc<br />
∂n+<br />
(2.65)
FIGURE 2.4. The exchange-correl<strong>at</strong>ion energy per electron ɛxc from the parametriz<strong>at</strong>ion<br />
of von Barth and Hed<strong>in</strong>. m = n ζ where n is the total density.<br />
After some algebra,<br />
v + xc = ( 4<br />
3 ɛx(n, ζ = 0) + γ(ɛc(n, ζ = ±1) − ɛc(n, ζ = 0)))(1 + ζ) 1/3<br />
−C P <br />
ln 1 + AP<br />
<br />
− γ(ɛc(n, ζ = ±1) − ɛc(n, ζ = 0)) + τcf(ζ)<br />
rs<br />
where τc = −CF <br />
ln 1 + AF<br />
<br />
+ C rs<br />
P <br />
ln 1 + AP<br />
<br />
− rs<br />
4<br />
3 (ɛc(n, ζ = ±1) − ɛc(n, ζ = 0))<br />
and γ =<br />
4<br />
3(21/3−1) .<br />
S<strong>in</strong>ce the correl<strong>at</strong>ion energy for the sp<strong>in</strong> polarized case uses the Hed<strong>in</strong>-<br />
Lundqvist model which is already obta<strong>in</strong>ed for the non-sp<strong>in</strong> polarized case (al-<br />
though the coefficients are different) the behavior of ɛxc and vxc for the sp<strong>in</strong>-<br />
polarized case is not much different from those for the non-sp<strong>in</strong> polarized case.<br />
(See Fig. 2.4 along m = 0.) Note th<strong>at</strong> ɛxc is the exchange-correl<strong>at</strong>ion energy per<br />
electron.<br />
25
FIGURE 2.5. The exchange-correl<strong>at</strong>ion energy nɛxc.<br />
The behavior of the correl<strong>at</strong>ion effect <strong>in</strong> a polarized case (m = 0) is, however,<br />
different due to the degree of local magnetiz<strong>at</strong>ion. Notice th<strong>at</strong> the exchange-<br />
correl<strong>at</strong>ion potential for sp<strong>in</strong>-down has the mirror image of th<strong>at</strong> of sp<strong>in</strong>-up and<br />
s<strong>at</strong>isfies the symmetry rel<strong>at</strong>ion v + xc(ζ) = v − xc(−ζ). Along the l<strong>in</strong>e m = n (fully<br />
polarized limit) <strong>in</strong> Fig. 2.8 and Fig. 2.9, the exchange potential vanishes and the<br />
correl<strong>at</strong>ion potential opposes this effect. [13] As a result, the correl<strong>at</strong>ion effect<br />
reduces the polariz<strong>at</strong>ion dependence of vx, which is consistent with the behavior<br />
of the pair distribution function. [18]<br />
A system with a uniform density can reach a lower exchange-correl<strong>at</strong>ion<br />
energy as the magnitude of m <strong>in</strong>creases as shown <strong>in</strong> Fig. 2.5. The k<strong>in</strong>etic en-<br />
ergy n · TT F , however, makes the situ<strong>at</strong>ion opposite and dom<strong>in</strong><strong>at</strong>es the exchange-<br />
correl<strong>at</strong>ion energy especially for high densities. The system reaches a higher energy<br />
26
due to the term n · TT F as |m| <strong>in</strong>creases <strong>in</strong> most of the density range.(See Fig. 2.6)<br />
One can notice, however, <strong>in</strong> Fig. 2.7, th<strong>at</strong> for extremely low densities (n ≤ 0.001)<br />
the exchange-correl<strong>at</strong>ion energy dom<strong>in</strong><strong>at</strong>es the k<strong>in</strong>etic energy n · TT F as |m| <strong>in</strong>-<br />
creases and thus n · TT F does not keep the system away from a polarized st<strong>at</strong>e.<br />
The density n <strong>at</strong> which there’s no dom<strong>in</strong>ant term between n · TT F and n · exc and<br />
the system is <strong>in</strong> equilibrium is ∼ 0.00153.<br />
One can conclude th<strong>at</strong> the exchange energy favors a sp<strong>in</strong>-polarized solution<br />
and the k<strong>in</strong>etic energy opposes the exchange-correl<strong>at</strong>ion effects.<br />
27
FIGURE 2.6. The exchange-correl<strong>at</strong>ion ɛxc and Thomas-Fermi k<strong>in</strong>etic energies<br />
TT F = 2.2099/r 2 s multiplied by the electron density n.<br />
FIGURE 2.7. The exchange-correl<strong>at</strong>ion and Thomas-Fermi k<strong>in</strong>etic energies for<br />
extremely low densities.<br />
28
FIGURE 2.8. The exchange-correl<strong>at</strong>ion potential vxc for the sp<strong>in</strong>-up density.<br />
FIGURE 2.9. The exchange-correl<strong>at</strong>ion potential vxc for the sp<strong>in</strong>-down density.<br />
29
3. A MODEL AND ITS PROPERTIES: AN IMPURITY IN A<br />
HOMOGENEOUS ELECTRON GAS<br />
Immersion energy calcul<strong>at</strong>ions for an impurity <strong>in</strong> a homogeneous electron<br />
gas can be simplified us<strong>in</strong>g the symmetry of the potentials. Physical quantities,<br />
like density of st<strong>at</strong>es and phase shifts, are very important <strong>in</strong> the discussion of<br />
theoretical aspects, such as immersion energy calcul<strong>at</strong>ions and the behavior of an<br />
impurity, as well as <strong>in</strong> the discussion of numerical aspects, such as normaliz<strong>at</strong>ion<br />
conditions for sc<strong>at</strong>tered wave functions and criteria for the maximum number of<br />
angular momentum l for each momentum k. Various aspects of an impurity system<br />
can be exam<strong>in</strong>ed <strong>in</strong> terms of Friedel oscill<strong>at</strong>ions, dielectric functions, virtual bound<br />
st<strong>at</strong>e resonances, and sc<strong>at</strong>ter<strong>in</strong>g lengths.<br />
3.1. An impurity <strong>in</strong> a homogeneous electron gas<br />
The model system used <strong>in</strong> this work consists of an impurity <strong>at</strong>om immersed<br />
<strong>in</strong> a homogeneous electron gas with a uniform positive background charge density.<br />
(See Fig. 3.1.) This impurity system is neutral and also <strong>in</strong>f<strong>in</strong>ite. The immersion<br />
energy is now calcul<strong>at</strong>ed by subtract<strong>in</strong>g the energies of two isol<strong>at</strong>ed systems, an<br />
impurity <strong>at</strong>om and the background, from the impurity system.<br />
FIGURE 3.1. An impurity system consists of an impurity <strong>at</strong>om and a homogeneous<br />
electron gas. The electron density fluctu<strong>at</strong>es due to the impurity <strong>at</strong>om.<br />
30
Atoms V. B. H. Experiment 1 KS-KC 1 KS-X 1 HF 1<br />
Li −14.710 −14.956 −14.656 −14.349 −14.865<br />
Ne −256.747 −257.880 −256.349 −254.986 −257.094<br />
K −1195.943 −1199.97 −1196.215 −1193.387 −1198.329<br />
TABLE 3.1. Energies of free <strong>at</strong>oms (<strong>in</strong> Rydbergs).<br />
Atoms V. B. H. Experiment 1 KS-KC 1 KS-X 1 HF 1<br />
Li 0.417 0.396 0.396 0.331 0.393<br />
Ne 1.641 1.585 1.657 1.551 1.461<br />
K 0.347 0.319 0.330 0.271 0.295<br />
TABLE 3.2. Ioniz<strong>at</strong>ion energies of one electron from a neutral <strong>at</strong>om (<strong>in</strong> Rydbergs).<br />
Eimm = Eimp − Epure − E<strong>at</strong>om<br />
31<br />
(3.1)<br />
The energy of a free <strong>at</strong>om E<strong>at</strong>om is easily calcul<strong>at</strong>ed by solv<strong>in</strong>g the Kohn-Sham<br />
equ<strong>at</strong>ions for bound st<strong>at</strong>es. Some examples from this work (V. B. H.), <strong>in</strong> which<br />
sp<strong>in</strong>-polarized and spherically symmetric systems are used, are shown <strong>in</strong> Table 3.1<br />
and 3.2 and are compared with the results of Tong and Sham [20]. KS-KC denotes<br />
the exchange-correl<strong>at</strong>ion energy <strong>in</strong> which the correl<strong>at</strong>ion energy is simply <strong>in</strong>ter-<br />
pol<strong>at</strong>ed between Gell-Mann and Brueckner and Wigner schemes. KS-K and HF<br />
1 Ref. [20]
denote the exchange energy only (− 3 e<br />
4<br />
2kF ) and Hartree-Fock calcul<strong>at</strong>ions respec-<br />
π<br />
tively. (See [20] for more details.) Ioniz<strong>at</strong>ion energies can be found by evalu<strong>at</strong><strong>in</strong>g<br />
E[N] − E[N − 1], as mentioned <strong>in</strong> Sec. 2.2.5.<br />
Differences between numerical calcul<strong>at</strong>ions and experimental values for free<br />
<strong>at</strong>om energies are larger than those for ioniz<strong>at</strong>ion energies. LDA gives rel<strong>at</strong>ively<br />
large numerical errors for core st<strong>at</strong>es such as 1s and 2s st<strong>at</strong>es due to the rapid<br />
vari<strong>at</strong>ions <strong>in</strong> the density. Numerical errors are, however, reduced <strong>in</strong> ioniz<strong>at</strong>ion<br />
energy calcul<strong>at</strong>ions s<strong>in</strong>ce core st<strong>at</strong>es <strong>in</strong> E[N] and E[N − 1] do not vary much and<br />
thus errors are canceled out. The same reason<strong>in</strong>g applies to immersion energies,<br />
which is expla<strong>in</strong>ed <strong>in</strong> detail <strong>in</strong> Sec. 4.4.5.<br />
3.2. Energy calcul<strong>at</strong>ion<br />
S<strong>in</strong>ce this work focuses on spherically symmetric potentials, the Kohn-Sham<br />
equ<strong>at</strong>ions and density and energy calcul<strong>at</strong>ions can be gre<strong>at</strong>ly simplified and the<br />
comput<strong>at</strong>ional work can be reduced us<strong>in</strong>g this spherical symmetry. It is, however,<br />
important to extend this work to non-spherical systems, s<strong>in</strong>ce most systems such<br />
as molecules and solids have non-spherical potentials and even <strong>in</strong> the case of a<br />
s<strong>in</strong>gle impurity <strong>at</strong>om, the electronic structure and immersion energies depend on<br />
the non-spherical densities if the angular momentum shell of an impurity <strong>at</strong>om is<br />
not fully occupied and thus the spherical symmetry is broken. Phase shifts are<br />
very important quantities used to calcul<strong>at</strong>e density of st<strong>at</strong>es and energy and also<br />
provide boundary conditions for sc<strong>at</strong>tered st<strong>at</strong>es. In this section, the Kohn-Sham<br />
equ<strong>at</strong>ions for non-spherical potentials are briefly <strong>in</strong>troduced and phase shifts and<br />
energy calcul<strong>at</strong>ions will be reviewed.<br />
32
3.2.1. Symmetry <strong>in</strong> potentials and phase shifts<br />
S<strong>in</strong>ce angular momentum shells <strong>in</strong> general have cyl<strong>in</strong>drical symmetry (φ sym-<br />
metry), non-spherical potentials have the same symmetry. (Note th<strong>at</strong> vxc depends<br />
on only local density and is φ-symmetric. The readers may refer to [21], [9], and [8]<br />
for this section.) The Kohn-Sham equ<strong>at</strong>ions can be simplified accord<strong>in</strong>gly. There-<br />
fore, if the <strong>in</strong>put density used to calcul<strong>at</strong>e the potentials of the KS equ<strong>at</strong>ions is<br />
φ-symmetric, the output density from the KS equ<strong>at</strong>ions is also φ-symmetric s<strong>in</strong>ce<br />
[H, Lz] = 0 and the iter<strong>at</strong>ive procedure of the KS equ<strong>at</strong>ions does not break the<br />
symmetry. The Kohn-Sham equ<strong>at</strong>ions are, <strong>in</strong> Rydberg <strong>at</strong>omic units (APPENDIX<br />
B),<br />
−∇ 2 + Veff(r) ψi(r) = ɛiψi(r) . (3.2)<br />
The angular dependence of the Kohn-Sham st<strong>at</strong>es ψi(r) can be expanded <strong>in</strong> terms<br />
of spherical harmonics.<br />
ψi(r) = <br />
lm<br />
33<br />
uilm(r)<br />
Ylm(Ωr) . (3.3)<br />
r<br />
One can express the electronic density n(r) as a sum over bound st<strong>at</strong>es and sc<strong>at</strong>-<br />
tered st<strong>at</strong>es with a reference of zero-energy.<br />
The total density is, as <strong>in</strong> (2.15),<br />
n(r) = n bound (r) + n cond (r) =<br />
lim<br />
r→∞ Veff(r) = 0 . (3.4)<br />
N<br />
i<br />
|ψ bound<br />
i (r)| 2 +<br />
M<br />
j<br />
|ψ cond<br />
j (r)| 2<br />
(3.5)<br />
where i and j are order<strong>in</strong>g <strong>in</strong>dices for the lowest occupied st<strong>at</strong>es, and i <strong>in</strong> |ψbound i (r)|<br />
and j <strong>in</strong> |ψcond i (r)| denote the KS st<strong>at</strong>es <strong>in</strong> which ɛi < 0 and ɛj > 0, respectively.
Note th<strong>at</strong> m is a good quantum number s<strong>in</strong>ce [H, Lz] = 0 and the Kohn-Sham<br />
equ<strong>at</strong>ions can be written down for each ɛi and m value.<br />
Us<strong>in</strong>g (3.3), one obta<strong>in</strong>s coupled equ<strong>at</strong>ions, similar to radial equ<strong>at</strong>ions for a<br />
spherically symmetric potential,<br />
−∇ 2 + Veff(r) <br />
by projection onto Yl ′ m(Ωr),<br />
where<br />
d 2<br />
dr 2 + ɛi −<br />
V m<br />
<br />
ll ′ (r) =<br />
lm<br />
l(l + 1)<br />
r2 <br />
uilm(r)<br />
<br />
Ylm(Ωr) = ɛi<br />
r<br />
lm<br />
uilm(r) = <br />
l ′<br />
uilm(r)<br />
Ylm(Ωr) .<br />
r<br />
34<br />
V m<br />
ll ′ (r)uil ′ m(r) , (3.6)<br />
dΩ Y ∗<br />
lm(Ωr)Veff(r, θ)Yl ′ m(Ωr) . (3.7)<br />
Notice th<strong>at</strong> s<strong>in</strong>ce Veff(r, θ) does not depend on φ, V m<br />
ll ′ (r) and the solutions <strong>in</strong> 3.6<br />
are real.<br />
The l<strong>in</strong>early <strong>in</strong>dependent solutions u j<br />
lm (j is a simple order<strong>in</strong>g <strong>in</strong>dex2 vary<strong>in</strong>g<br />
from 0 to lmax) can be found from coupled equ<strong>at</strong>ions (3.6) and their behavior <strong>at</strong><br />
small r is<br />
lim<br />
r→0 uj<br />
lm (r) ∼ rj+|l−j|+1<br />
∞<br />
b l ir i−1 , (3.8)<br />
which provides the boundary conditions of <strong>in</strong>dependent solutions <strong>at</strong> small r. (See<br />
APPENDIX C.)<br />
One can now expand uilm <strong>in</strong> terms of l<strong>in</strong>early <strong>in</strong>dependent solutions.<br />
uilm(r) = <br />
j<br />
i=1<br />
α m j u j<br />
ilm (r) , (3.9)<br />
2 one has to use the cut-off maximum lmax <strong>in</strong> a computer program.
where the coefficients α m j can be determ<strong>in</strong>ed by the asymptotic behavior of the<br />
solutions and do not depend on l but on ɛi and m. One can determ<strong>in</strong>e α m j by<br />
boundary conditions, which uniquely determ<strong>in</strong>es the radial wave functions uilm(r)<br />
and Kohn-Sham eigenst<strong>at</strong>es ψi(r). For <strong>in</strong>stance, one has, for bound st<strong>at</strong>es, two sets<br />
of l<strong>in</strong>early <strong>in</strong>dependent of solutions which are u j<br />
ilm (r) from the outward <strong>in</strong>tegr<strong>at</strong>ion<br />
started from the orig<strong>in</strong> and v j<br />
ilm (r) from the <strong>in</strong>ward <strong>in</strong>tegr<strong>at</strong>ion started from the<br />
<strong>in</strong>f<strong>in</strong>ity. The asymptotic behavior of v j<br />
ilm (r) is<br />
lim<br />
r→∞ vj<br />
lm (r) ∼ δjl exp −√ −ɛr<br />
35<br />
. (3.10)<br />
The condition for the existence of bound st<strong>at</strong>es is th<strong>at</strong> the two sets of l<strong>in</strong>early<br />
<strong>in</strong>dependent solutions must m<strong>at</strong>ch <strong>at</strong> an <strong>in</strong>termedi<strong>at</strong>e r.<br />
<br />
j<br />
<br />
j<br />
α m j<br />
α m j u j<br />
ilm<br />
d<br />
dr uj<br />
ilm<br />
(r) = <br />
j<br />
(r) = <br />
j<br />
β m j v j<br />
ilm (r) (3.11)<br />
β m j<br />
d<br />
dr vj<br />
ilm (r) . (3.12)<br />
S<strong>in</strong>ce a nontrivial solution exist only if the determ<strong>in</strong>ant of (3.11) is zero, one needs<br />
to search for the zeroes of the determ<strong>in</strong>ant to f<strong>in</strong>d bound st<strong>at</strong>es. Notice the Eq.<br />
(3.11) becomes the standard Wronskian condition for the spherically symmetric<br />
potentials. Once the bound st<strong>at</strong>e energy has been determ<strong>in</strong>ed, one can use the<br />
s<strong>in</strong>gular value decomposition method (APPENDIX D) to f<strong>in</strong>d the eigenfunction<br />
ψi(r). For sc<strong>at</strong>tered st<strong>at</strong>es, one can use the asymptotic behavior of the sc<strong>at</strong>tered<br />
wave function for large r to determ<strong>in</strong>e the coefficients α m j . Notice th<strong>at</strong> the wave<br />
number vector k <strong>in</strong> uklm(r) also depends on the direction of the <strong>in</strong>cident plane<br />
wave, which thus leads to considerably more comput<strong>at</strong>ional work compared to<br />
spherical systems. (The readers refer to [9] and [8] for more details.)
In addition, it is known, <strong>in</strong> a case of a free <strong>at</strong>om calcul<strong>at</strong>ion, th<strong>at</strong> the energy<br />
difference between the non-spherical potential and the spherical potential is much<br />
smaller than th<strong>at</strong> between the non-sp<strong>in</strong> polarized and sp<strong>in</strong> polarized systems. [22]<br />
A spherical potential can, therefore, be considered the best for a first try<br />
and a sp<strong>in</strong>-polarized system with a spherical potential still has many <strong>in</strong>terest<strong>in</strong>g<br />
properties. If the potential is spherically symmetric, the angular momentum l<br />
is a constant of a motion and also a good quantum number s<strong>in</strong>ce [H, L 2 ] = 0.<br />
The eigenenergies do not depend on l and m and eigenst<strong>at</strong>es are degener<strong>at</strong>e. The<br />
coupled equ<strong>at</strong>ions (3.6) can be simplified to the radial equ<strong>at</strong>ions.<br />
2 d<br />
dr2 + ɛi,l<br />
l(l + 1)<br />
−<br />
r2 <br />
ui,l(r) = V (r)ui,l(r) . (3.13)<br />
For sc<strong>at</strong>tered st<strong>at</strong>es,<br />
d 2<br />
dr 2 + k2 −<br />
36<br />
l(l + 1)<br />
r2 <br />
uk,l(r) = V (r)uk,l(r) . (3.14)<br />
If the potential vanishes for the exterior region r > Rb, the solutions for the<br />
sc<strong>at</strong>tered st<strong>at</strong>es <strong>in</strong> (3.14) follow a form of free particle waves.<br />
Rk,l(r) = uk,l(r)<br />
r = Aljl(kr) + Blnl(kr) r > Rb (3.15)<br />
Us<strong>in</strong>g the asymptotic approxim<strong>at</strong>ions 3 ,<br />
uk,l(r)<br />
r<br />
s<strong>in</strong>(kr − l<br />
= Al<br />
π<br />
2 ) cos(kr − l<br />
− Bl<br />
kr<br />
π<br />
2 )<br />
kr<br />
for kr ≫ l . (3.16)<br />
If the potential vanishes <strong>at</strong> the orig<strong>in</strong> (r = 0), Bl must be zero as nl diverges <strong>at</strong><br />
the orig<strong>in</strong>. The <strong>in</strong>tensity of sc<strong>at</strong>ter<strong>in</strong>g by the potential, hence, can be compared<br />
with the r<strong>at</strong>io Bl/Al. The phase shift δl is now def<strong>in</strong>ed by<br />
Bl<br />
Al<br />
For large kr, (3.16) can be written as<br />
= − tan δl . (3.17)
uk,l(r)<br />
r<br />
π s<strong>in</strong>(kr − l + δl) 2 <br />
kr<br />
37<br />
for kr ≫ l , (3.18)<br />
from which one can deduce th<strong>at</strong> the quantity δl represents the phase difference<br />
of the radial wave function between the sc<strong>at</strong>ter<strong>in</strong>g (V = 0 and δl = 0) and the<br />
non-sc<strong>at</strong>ter<strong>in</strong>g (V = 0 and δl = 0) cases.<br />
The radial-wave solution of a free particle <strong>in</strong> the sc<strong>at</strong>ter<strong>in</strong>g case can be rewrit-<br />
ten, us<strong>in</strong>g (3.17), as<br />
Rk,l(r) = uk,l(r)<br />
r = eiδl [cos δk,ljl(kr) − s<strong>in</strong> δk,lnl(kr)] r > Rb , (3.19)<br />
where the normaliz<strong>at</strong>ion constant is chosen so th<strong>at</strong> Rk,l(r) is jl(kr) for V = 0.<br />
Note th<strong>at</strong> the wave function for r > Rb is [23]<br />
ψk(r) =<br />
1<br />
(2π) 3/2<br />
∞<br />
(2l + 1)Rk,l(r)Pl(cos θ) r > Rb (3.20)<br />
l=0<br />
and thus the electronic density is<br />
n(r) = 2<br />
(2π) 3<br />
kF<br />
dk (|ψk(r)|<br />
0<br />
2<br />
(3.21)<br />
= 8π<br />
(2π) 3<br />
<br />
kF<br />
(2l + 1) dk k 2 (Rk,l(r)) 2 . (3.22)<br />
l<br />
0<br />
The boundary condition for the <strong>in</strong>terior radial wave function (r < Rb) is the<br />
logarithmic deriv<strong>at</strong>ive <strong>at</strong> the boundary.<br />
<br />
1<br />
R<strong>in</strong>t dR<br />
k,l<br />
<strong>in</strong>t<br />
<br />
k,l<br />
1<br />
=<br />
dr<br />
Rext k,l<br />
r=Rb<br />
dR ext<br />
k,l<br />
dr<br />
where R<strong>in</strong>t k,l is a <strong>in</strong>terior radial wave function and Rext k,l<br />
tion which is given by (3.15).<br />
<br />
r=Rb<br />
One can easily obta<strong>in</strong>, us<strong>in</strong>g the phase shift def<strong>in</strong>ition (3.17),<br />
(3.23)<br />
a exterior radial wave func-<br />
tan δl = kRbj ′ l (kRb)uk,l(Rb) + jl(kRb)uk,l(Rb) − Rbjl(kRb)u ′ k,l (Rb)<br />
kRbn ′ l (kRb)uk,l(Rb) + nl(kRb)uk,l(Rb) − Rbnl(kRb)u ′ k,l (Rb)<br />
(3.24)
where uk,l(r) is a <strong>in</strong>terior solution (for r < Rb).<br />
Once the radial equ<strong>at</strong>ions (3.14) for r < Rb are solved, the phase shift can be<br />
obta<strong>in</strong>ed by (3.24) and normaliz<strong>at</strong>ion of the radial wave function can be performed<br />
by (3.19). For very small kr, one can use the asymptotic approxim<strong>at</strong>ions 3 aga<strong>in</strong><br />
and obta<strong>in</strong> the expression<br />
tan δl → alk 2l+1<br />
For s-waves, this equ<strong>at</strong>ion becomes<br />
38<br />
as k → 0 . (3.25)<br />
tan δ0 δ0 → a0k as k → 0 . (3.26)<br />
One sees th<strong>at</strong> the sc<strong>at</strong>ter<strong>in</strong>g length 4 a0 is a slope of δ0(k) for small k values, which<br />
can be used to determ<strong>in</strong>e whether the potential is <strong>at</strong>tractive or repulsive and also<br />
to determ<strong>in</strong>e the required <strong>in</strong>terval of a k-mesh 5 for small k values. If there’s a<br />
virtual bound st<strong>at</strong>e (sc<strong>at</strong>ter<strong>in</strong>g resonance) around the bottom of conduction band,<br />
the sc<strong>at</strong>ter<strong>in</strong>g length will be very large due to the large density of <strong>in</strong>duced st<strong>at</strong>es for<br />
small k values. In this case, one needs to use a correspond<strong>in</strong>g very small <strong>in</strong>terval<br />
<strong>in</strong> a k-mesh around k = 0.<br />
3 r → 0, krjl(kr) (kr) l+1 /(Π l i=0 2i + 1) and − krnl(kr) (Π l−1<br />
i=0 2i + 1)/(kr)l ,<br />
r → ∞, krjl(kr) s<strong>in</strong>(kr − lπ/2) and − krnl(kr) cos(kr − lπ/2).<br />
4 a1 is called the sc<strong>at</strong>ter<strong>in</strong>g volume.<br />
5 In the computer program, one has to use discrete values of k for the cont<strong>in</strong>uum st<strong>at</strong>es<br />
k = 0 · · · kF and the <strong>in</strong>terval between k values affects the numerical precision of <strong>in</strong>te-<br />
gr<strong>at</strong>ion <strong>in</strong> k-space. The larger the magnitude of a0 is , the smaller the <strong>in</strong>terval between<br />
small k values is.
3.2.2. Density of <strong>in</strong>duced st<strong>at</strong>es <strong>in</strong> terms of phase shifts<br />
The density of <strong>in</strong>duced st<strong>at</strong>es can easily derived from the phase shifts as<br />
follows. [18] If one can assume all charges are conta<strong>in</strong>ed <strong>in</strong> a very large sphere of<br />
radius R0 (V (R0) → ∞), the radial wave functions for sc<strong>at</strong>tered st<strong>at</strong>es must vanish<br />
<strong>at</strong> r = R0. The asymptotic approxim<strong>at</strong>ions 3 can be used aga<strong>in</strong> due to the large R0<br />
and one can obta<strong>in</strong>, us<strong>in</strong>g (3.16), boundary conditions for free particle st<strong>at</strong>es and<br />
for sc<strong>at</strong>tered st<strong>at</strong>es. For free particle st<strong>at</strong>es,<br />
and for sc<strong>at</strong>tered st<strong>at</strong>es,<br />
kR0 − lπ<br />
2<br />
= nfπ<br />
kR0 − lπ<br />
2 + δl,m(k) = nsπ .<br />
One can assume th<strong>at</strong> k is cont<strong>in</strong>uous due to a very large radius R0 (kR0 ≫ l) and<br />
obta<strong>in</strong> the number of <strong>in</strong>duced st<strong>at</strong>es per k, △D(k),<br />
where dδl,m(k)<br />
dk<br />
△D(k)dk = d(ns − nf)<br />
dk =<br />
dk<br />
1<br />
π<br />
<br />
l,m,σ<br />
dδ σ l,m (k)<br />
dk<br />
39<br />
dk , (3.27)<br />
is summed over l, m, and σ to <strong>in</strong>clude the degener<strong>at</strong>e st<strong>at</strong>es.<br />
3.2.3. Immersion energies<br />
If E<strong>at</strong>om is given <strong>in</strong> the immersion equ<strong>at</strong>ion (3.1), the only term which one<br />
needs to f<strong>in</strong>d is Eimp − Epure. This is composed of the <strong>in</strong>duced k<strong>in</strong>etic energy, the<br />
electrost<strong>at</strong>ic energy, and the exchange-correl<strong>at</strong>ion energy.<br />
Eimp − Epure = △T + △C + △Exc<br />
(3.28)
1. The k<strong>in</strong>etic energy △T<br />
The k<strong>in</strong>etic energy which is similar to (2.28) can be written as<br />
△T =<br />
N<br />
i<br />
ɛi −<br />
<br />
dr n(r)veff(r)<br />
= △T bound + △T cond<br />
N<br />
<br />
= ɛi − dr n bound <br />
(r)veff(r) +<br />
i<br />
dr n cond <br />
(r)veff(r)<br />
where i <strong>in</strong>cludes the sp<strong>in</strong> s and bound and cond denote bound st<strong>at</strong>es and<br />
sc<strong>at</strong>tered st<strong>at</strong>es respectively.<br />
For bound st<strong>at</strong>es,<br />
△T bound =<br />
N<br />
i<br />
ɛi −<br />
where i is only for the occupied bound st<strong>at</strong>es.<br />
Similarly, for sc<strong>at</strong>tered st<strong>at</strong>es, s<strong>in</strong>ce k is cont<strong>in</strong>uous,<br />
△T cond =<br />
kF<br />
0<br />
<br />
k 2 <br />
△D(k) dk −<br />
,<br />
40<br />
dr n bound (r)veff(r) , (3.29)<br />
dr n cond (r)veff(r) , (3.30)<br />
where △D(k) is the number of st<strong>at</strong>es per k <strong>in</strong>duced by the potential veff.<br />
The first term can be rewritten, us<strong>in</strong>g (3.27) and E = k 2 , as<br />
kF<br />
0<br />
k 2 △D(k) dk = 1<br />
kF <br />
2<br />
k<br />
π 0<br />
= 1<br />
π<br />
= 1<br />
π<br />
<br />
l,m,σ<br />
<br />
l,m,σ<br />
l,m,σ<br />
EF<br />
0<br />
dδ σ l,m (k)<br />
dk<br />
E dδσ l,m (E)<br />
dE<br />
EF δ σ l,m(EF ) − 1<br />
π<br />
dk (3.31)<br />
dE (3.32)<br />
<br />
l,m,σ<br />
EF<br />
0<br />
δ σ l,m(E) dE , (3.33)<br />
where EF is the Fermi energy which is k2 F . Add<strong>in</strong>g the k<strong>in</strong>etic energies for<br />
the bound and sc<strong>at</strong>tered st<strong>at</strong>es together,
△T =<br />
N<br />
i<br />
+ 1<br />
π<br />
ɛi −<br />
<br />
l,m,σ<br />
<br />
dr n(r)veff(r)<br />
EF δ σ l,m(EF ) − 1<br />
π<br />
<br />
l,m,σ<br />
EF<br />
0<br />
41<br />
δ σ l,m(E) dE . (3.34)<br />
Note th<strong>at</strong> the first term <strong>in</strong> (3.33) is the energy, EF Z, required to add the<br />
extra Z electrons to the system, which is easily verified by the Friedel sum<br />
rule (See (3.45).) and the second term is the k<strong>in</strong>etic energy caused by the<br />
<strong>in</strong>teraction between an impurity and an electron gas <strong>in</strong> a metal.<br />
2. The electrost<strong>at</strong>ic energy ∆C<br />
S<strong>in</strong>ce the Hartree energy of the background system which has a uniform<br />
density n0 is zero, one can write the Coulomb energy as<br />
△C = 1<br />
2 e2<br />
<br />
dr dr ′ ∆n(r)∆n(r′ )<br />
|r ′ <br />
− e dr ∆n(r)Vext (r) (3.35)<br />
− r|<br />
= e 2<br />
<br />
1<br />
dr dr<br />
2<br />
′ ∆n(r′ )<br />
|r ′ <br />
Z<br />
− ∆n(r) , (3.36)<br />
− r| r<br />
where ∆n(r) = n(r) − n0(r) and the second term ( eZ ) is the external po-<br />
r<br />
tential due to the nucleus of an impurity <strong>at</strong>om. However, this can be further<br />
simplified us<strong>in</strong>g [24]<br />
<br />
dr ′ 1<br />
|r − r ′ | =<br />
∞<br />
r<br />
l=0<br />
l <<br />
r l+1<br />
><br />
Rb<br />
= 2π<br />
r0<br />
<br />
dr ′ r ′2<br />
Pl(cos θ)r ′2 dr ′ s<strong>in</strong> θdθdφ (3.37)<br />
∞<br />
l=0<br />
rl <<br />
r l+1<br />
1<br />
Pl(x)dx , (3.38)<br />
> −1<br />
where θ is the angle between r and r ′ and Pl(x) are the Legendre polynomials.<br />
Us<strong>in</strong>g the fact th<strong>at</strong> P0(x) = 1 and 1<br />
−1 dx Pl(x)P0(x) = 2 · δl,0,<br />
F<strong>in</strong>ally,<br />
<br />
dr ′<br />
1<br />
|r − r ′ |<br />
r<br />
1<br />
= 4π r<br />
r r0<br />
′2 dr ′ +<br />
Rb<br />
r<br />
r ′ dr ′<br />
<br />
. (3.39)
Rb<br />
∆C = 4π<br />
× e 2<br />
r0<br />
<br />
2π<br />
r∆n(r)<br />
r<br />
3. The exchange-correl<strong>at</strong>ion energy ∆Exc<br />
r0<br />
42<br />
∆n(r ′ )r ′2 dr ′ Rb<br />
+ 2πr ∆n(r<br />
r<br />
′ )r ′ dr ′ <br />
− Z dr . (3.40)<br />
The exchange-correl<strong>at</strong>ion energy for non-sp<strong>in</strong>-polarized systems is simply<br />
Exc[n]−Exc[n0], <strong>in</strong> which n0 is the background electron density and Exc[n] =<br />
dr n(r)ɛxc(n(r)).<br />
The exchange-correl<strong>at</strong>ion energy for sp<strong>in</strong>-polarized systems can be written<br />
as<br />
∆Exc = Exc[n + , n − ] − Exc[n + 0 , n − 0 ] for sp<strong>in</strong>-up (3.41)<br />
+ Exc[n − , n + ] − Exc[n − 0 , n + 0 ] for sp<strong>in</strong>-down, (3.42)<br />
where n + is the sp<strong>in</strong>-up density and n − is the sp<strong>in</strong>-down density.<br />
Now, the immersion energy is<br />
N<br />
<br />
Eimm = ɛi − dr n(r)veff(r)<br />
i<br />
+ 1 <br />
EF δ<br />
π<br />
l,m,σ<br />
σ l,m(EF ) − 1 <br />
π<br />
l,m,σ<br />
+ 4πe 2<br />
Rb<br />
r<br />
r∆n(r) 2π<br />
r0<br />
+ Exc[n + , n − ] − Exc[n + 0 , n − 0 ]<br />
EF<br />
δ σ l,m(E) dE<br />
0<br />
∆n(r<br />
r0<br />
′ )r ′2 dr ′ Rb<br />
+ 2πr<br />
r<br />
∆n(r ′ )r ′ dr ′ <br />
− Z dr<br />
+ Exc[n − , n + ] − Exc[n − 0 , n + 0 ] − E<strong>at</strong>om . (3.43)<br />
Note th<strong>at</strong> the effective potential energy is, by (2.24) and (3.39),<br />
Veff(r) = 4π e2 r<br />
1<br />
∆n(r<br />
r<br />
′ )r ′2 dr ′ Rb<br />
+ ∆n(r ′ )r ′ dr ′<br />
<br />
r0<br />
r<br />
− e2 Z<br />
r + vxc(n(r)) − vxc(n0) . (3.44)
3.3. Friedel sum rule and Friedel oscill<strong>at</strong>ions<br />
Two important phenomena <strong>in</strong> many-body problems are the screen<strong>in</strong>g of an<br />
impurity <strong>at</strong>om <strong>in</strong> an electron gas and the oscill<strong>at</strong>ory behavior of the electron density<br />
<strong>in</strong> the vic<strong>in</strong>ity of an impurity <strong>at</strong>om. [10] [11] [12]<br />
The number of <strong>in</strong>duced electrons due to an impurity <strong>at</strong>om can be easily<br />
obta<strong>in</strong>ed us<strong>in</strong>g (3.27).<br />
∆N = Z =<br />
kF<br />
0<br />
dk △D(k) =<br />
kF<br />
0<br />
dk 1<br />
π<br />
<br />
l,m,σ<br />
δ σ l,m (k)<br />
dk<br />
= 2<br />
π<br />
43<br />
<br />
(2l + 1)δl(kF ) , (3.45)<br />
where Z is the positive valence on an impurity, the factor of 2 is the sp<strong>in</strong> degeneracy<br />
σ, and (2l + 1) is the orbital degeneracy rel<strong>at</strong>ed to m. This rel<strong>at</strong>ion is called the<br />
Friedel sum rule and st<strong>at</strong>es th<strong>at</strong> the displaced electronic charge ∆N · e exactly<br />
cancels the charge Z · |e| of the impurity and th<strong>at</strong> the Coulomb potential of 1/r<br />
becomes a screened potential with the density of the screen<strong>in</strong>g electrons determ<strong>in</strong>ed<br />
<strong>in</strong> a self-consistent way. It should be noted th<strong>at</strong> if an impurity <strong>at</strong>om has one<br />
electron <strong>in</strong> bound st<strong>at</strong>es and thus has an excess valency Z − 1, one has to remove<br />
the contribution δ/π of the s<strong>in</strong>gle bound st<strong>at</strong>e on the right-hand side of (3.45).<br />
The value of the phase shift δ for a bound st<strong>at</strong>e should be π to be consistent. [15]<br />
This is actually another st<strong>at</strong>ement of Lev<strong>in</strong>son’s theorem.<br />
In order to see how the density varies <strong>at</strong> large distances (Friedel oscill<strong>at</strong>ions),<br />
one needs to take the asymptotic limit. (See Sec. 3.2.1, [15], and [18].) S<strong>in</strong>ce the<br />
radial wave function of free particles depends on only jl(kr)) as mentioned <strong>in</strong> Sec.<br />
3.2.1,<br />
l
n(r) − n0 = ∆n(r) = 2<br />
(2π) 3<br />
kF<br />
0<br />
= 8π<br />
(2π) 3<br />
1<br />
lim ∆n(r) =<br />
r→∞ π2r2 = 1<br />
π 2 r 2<br />
<br />
(2l + 1)<br />
l<br />
<br />
(2l + 1)<br />
l<br />
<br />
(2l + 1)<br />
l<br />
kF<br />
0<br />
kF<br />
0<br />
kF<br />
0<br />
44<br />
dk (|ψk(r)| 2 − |φk(r)| 2 ) (3.46)<br />
dk k 2 (uk,l(r)) 2<br />
(<br />
r2 − (jl(kr)) 2 ) (3.47)<br />
dk (s<strong>in</strong> 2 (kr + δl(k) − lπ/2) − (s<strong>in</strong> 2 (kr − lπ/2) (3.48)<br />
dk 1<br />
2 (cos(2kr − lπ) − (cos(2kr + 2δl(k) − lπ) , (3.49)<br />
where the normaliz<strong>at</strong>ion constant for radial solutions uk,l(r)<br />
r<br />
is chosen as <strong>in</strong> (3.19).<br />
Notice th<strong>at</strong> one can use (3.47) to f<strong>in</strong>d the density difference for sc<strong>at</strong>tered<br />
st<strong>at</strong>es numerically.<br />
Approxim<strong>at</strong>ely us<strong>in</strong>g δl(k) = δl(kF ) + (k − kF )(dδ/dk), [18]<br />
(−1)l<br />
lim ∆n(r) <br />
r→∞ 2π2r2 (−1)l<br />
4π 2 r 3<br />
− (−1)l<br />
4π 2 r 3<br />
<br />
(2l + 1)<br />
l<br />
kF<br />
× dk cos(2kr) − cos(2kr + 2δl(kF ) + 2(k − kF )<br />
0<br />
dδ<br />
dk )<br />
<br />
(3.50)<br />
<br />
(2l + 1) [s<strong>in</strong>(2kF r) − (s<strong>in</strong>(2kF r + 2δl(kF )))] (3.51)<br />
l<br />
<br />
l<br />
2(2l + 1) s<strong>in</strong>(δl(kF )) cos(2kF r + δl(kF )) + O( 1<br />
) (3.52)<br />
r4 The electronic density difference <strong>at</strong> large distances oscill<strong>at</strong>es with the wave number<br />
2kF and decreases as r −3 . The Friedel’s theorem is <strong>in</strong>dependent of the n<strong>at</strong>ure of<br />
the impurity <strong>at</strong>om and the impurity determ<strong>in</strong>es only the values of phase shifts. [18]<br />
The wave length λ ∼ π/kF is thus frequently used to verify the numerical results<br />
as <strong>in</strong> Ch. 4 and 5.<br />
3.4. Dielectric functions<br />
The screen<strong>in</strong>g of the Coulomb potential of an impurity and the behavior of<br />
the <strong>in</strong>duced electronic density can be derived us<strong>in</strong>g dielectric functions as well. [15]
[18] It is convenient to def<strong>in</strong>e the dielectric functions us<strong>in</strong>g potentials s<strong>in</strong>ce one has<br />
to use potentials <strong>in</strong> the Kohn-Sham equ<strong>at</strong>ions and the Fourier components of the<br />
potential are easily calcul<strong>at</strong>ed numerically. The dielectric function can be found<br />
<strong>in</strong> the usual def<strong>in</strong>ition,<br />
Eq + 4πPq = Dq = ɛ(w, q)Eq , (3.53)<br />
where Eq is the longitud<strong>in</strong>al electric field. If one writes the effective potential V eff<br />
as the external potential V ext plus the <strong>in</strong>duced potential V <strong>in</strong>d , V eff = V ext + V <strong>in</strong>d ,<br />
us<strong>in</strong>g eEq = −iqVq, one can obta<strong>in</strong> from (3.53)<br />
V eff<br />
q<br />
− V <strong>in</strong>d<br />
q<br />
ɛ(w, q) =<br />
= V ext<br />
q<br />
V ext<br />
q<br />
V eff<br />
q<br />
= ɛ(w, q)V eff<br />
q<br />
45<br />
(3.54)<br />
. (3.55)<br />
Note th<strong>at</strong> Pq = −E <strong>in</strong>d<br />
q /4π. S<strong>in</strong>ce the dielectric function is calcul<strong>at</strong>ed as the r<strong>at</strong>io<br />
of the external potential to the effective potential, one can estim<strong>at</strong>e the response<br />
of an electron gas due to an impurity <strong>at</strong>om, which is very important <strong>in</strong> a sense th<strong>at</strong><br />
one needs to determ<strong>in</strong>e the stability of the self-consistent procedure for a particular<br />
impurity <strong>at</strong>om. (See Sec. 4.2.)<br />
If the charge of the impurity is Z, the external potential becomes<br />
Z<br />
r<br />
1<br />
=<br />
(2π) 3<br />
<br />
The effective potential <strong>in</strong> the l<strong>in</strong>ear response region is<br />
V eff (r, w) = 1<br />
(2π) 3<br />
<br />
= 1<br />
(2π) 3<br />
= 1<br />
(2π) 3<br />
<br />
<br />
iq·r Z4π<br />
dq e<br />
q2 . (3.56)<br />
dq e iq·r V eff (w, q) (3.57)<br />
dq e iq·r V ext (w, q)<br />
ɛ(w, q)<br />
(3.58)<br />
dq e iq·r Z4π<br />
q2ɛ(w, q)<br />
. (3.59)
Density functional theory is essentially a st<strong>at</strong>ic theory and further more if the<br />
impurity is st<strong>at</strong>ionary, one can use w = 0. The Poisson equ<strong>at</strong>ion is<br />
∇ 2 V eff (r) = −4π[∆n(r) + Zδ(r)] , (3.60)<br />
where ∆n(r) is the charge <strong>in</strong>duced by Zδ(r). The screen<strong>in</strong>g electronic density is<br />
now<br />
∆n(r) = Z<br />
(2π) 3<br />
<br />
dq e iq·r<br />
<br />
1<br />
− 1<br />
ɛ(0, q)<br />
46<br />
. (3.61)<br />
One notices th<strong>at</strong> if there is a s<strong>in</strong>gularity <strong>in</strong> the dielectric constant, the screen<strong>in</strong>g<br />
electronic density will be oscill<strong>at</strong><strong>in</strong>g as the Friedel oscill<strong>at</strong>ions.<br />
The total <strong>in</strong>duced charge is<br />
<br />
∆N =<br />
dr ∆n(r) = Z<br />
(2π) 3<br />
<br />
dq dr e iq·r<br />
<br />
1<br />
ɛ(0, q)<br />
<br />
1<br />
= Z dq δ(q) − 1<br />
ɛ(0, q)<br />
<br />
1<br />
= Z − 1<br />
ɛ(0, 0)<br />
<br />
− 1<br />
(3.62)<br />
(3.63)<br />
(3.64)<br />
If one use the Thomas-Fermi dielectric constant for ɛ(0, q) ( [15], [18], and [25]),<br />
where k 2 s = 4e2 mkF<br />
2 π<br />
ɛT F (0, q) = q2 + k2 s<br />
q2 <br />
1<br />
∆N = Z<br />
ɛT F (0, 0)<br />
(3.65)<br />
<br />
− 1 = −Z , (3.66)<br />
= 4.6/rs and rs is the Wigner-Seitz radius. The Thomas-Fermi<br />
dielectric constant shows the correct Friedel sum rule result but does not <strong>in</strong>clude<br />
the Friedel oscill<strong>at</strong>ions. The effective potential screened by the electron gas is<br />
V eff (r, w) = 1<br />
(2π) 3<br />
<br />
= Z<br />
r e−ksr<br />
iq·r Z4π<br />
dq e<br />
q2 q 2<br />
q 2 + k 2 s<br />
(3.67)<br />
. (3.68)
Note th<strong>at</strong> this potential is also exactly the same as the Yukawa potential. The<br />
screen<strong>in</strong>g length is def<strong>in</strong>ed by (3.67) as ls = 1/ks ∼ r 1/2<br />
s .<br />
The L<strong>in</strong>dhard dielectric constant (<strong>at</strong> T = 0) can be derived by the self-<br />
consistent field [15] or RPA approxim<strong>at</strong>ions [18] or straightforward calcul<strong>at</strong>ions<br />
[25]. The dielectric constant <strong>in</strong> the L<strong>in</strong>dhard theory is<br />
ɛL<strong>in</strong>(q) = 1 + 4π<br />
q2 e2mkF 22π2 <br />
1 + 4k2 <br />
F − q2 <br />
ln <br />
q + 2kF <br />
<br />
4qkF q − 2kF<br />
47<br />
(3.69)<br />
Note th<strong>at</strong> the L<strong>in</strong>dhard dielectric constant is not analytic <strong>at</strong> q = 2kF , which results<br />
<strong>in</strong> th<strong>at</strong> the <strong>in</strong>duced density and the screened effective potential of an impurity<br />
charge oscill<strong>at</strong>e as 1<br />
r 3 cos 2kF r. The Friedel’s theorem has been verified aga<strong>in</strong> with<br />
dielectric constants. The limit of the L<strong>in</strong>dhard dielectric constant as q → 0 is just<br />
the Thomas-Fermi dielectric constant. As one can notice, the dielectric constant is<br />
<strong>in</strong> general gett<strong>in</strong>g close to 1 as q <strong>in</strong>creases, which means the effective potential <strong>at</strong><br />
high q’s will be almost same as the external potential and thus the <strong>in</strong>duced density<br />
<strong>at</strong> high q’s does not contribute much. However, if there is a virtual bound st<strong>at</strong>e <strong>at</strong><br />
high q values, one can see a small peak <strong>in</strong> the dielectric function <strong>at</strong> correspond<strong>in</strong>g<br />
q values s<strong>in</strong>ce there’s a contribution of the <strong>in</strong>duced density <strong>at</strong> the q values to the<br />
dielectric constant. (see Ch. 5.)<br />
3.5. Virtual bound st<strong>at</strong>e resonance<br />
If an impurity <strong>at</strong>om has bound st<strong>at</strong>es, the electrons <strong>in</strong> bound st<strong>at</strong>es will be<br />
localized <strong>in</strong> the vic<strong>in</strong>ity of the impurity ion. If the <strong>at</strong>tractive effective potential of<br />
the impurity is reduced or not sufficiently strong enough to have a bound st<strong>at</strong>e, the<br />
bound st<strong>at</strong>e will be merged <strong>in</strong>to the conduction band <strong>in</strong> a metal. In this case, the<br />
electrons with positive energy rel<strong>at</strong>ive to the bottom of the conduction band still<br />
tend to localize around the impurity ion and <strong>in</strong>duce a narrow peak <strong>in</strong> the density
of st<strong>at</strong>es of the conduction band, which is called a virtual bound st<strong>at</strong>e resonance.<br />
A virtual bound st<strong>at</strong>e can be seen very well <strong>in</strong> the non-<strong>in</strong>teract<strong>in</strong>g Anderson model<br />
by calcul<strong>at</strong>ions of density of st<strong>at</strong>es us<strong>in</strong>g Green’s functions. [26] The Hamiltonian<br />
takes the form, <strong>in</strong> the non-<strong>in</strong>teract<strong>in</strong>g Anderson model,<br />
H = <br />
σ<br />
ɛdc †<br />
d,σ cd,σ + <br />
k,σ<br />
ɛkc †<br />
k,σ<br />
48<br />
k,σck,σ + <br />
(Vkc †<br />
d,σck,σ + V ∗<br />
k c †<br />
k,σcd,σ) , (3.70)<br />
where Vk is an overlap m<strong>at</strong>rix element between the <strong>at</strong>omic d level of the impurity<br />
ion and the conduction electrons and ɛd is the energy of the d level of the impurity.<br />
By tak<strong>in</strong>g appropri<strong>at</strong>e m<strong>at</strong>rix elements of (ɛ ± is − H)G ± (ɛ) = I, one can f<strong>in</strong>d<br />
the coupled equ<strong>at</strong>ions of Green’s functions, which lead to the T m<strong>at</strong>rix element by<br />
compar<strong>in</strong>g with the expression G + = G + 0 + G + 0 T (ɛ + )G + 0 . Note th<strong>at</strong> ɛ ± = ɛ ± is.<br />
where<br />
〈k|T (ɛ)|k ′ 〉 = V ∗<br />
k G +<br />
d,d (ɛ)V ′ k , (3.71)<br />
G +<br />
d,d (ɛ) =<br />
1<br />
ɛ + is − ɛd − <br />
k<br />
|Vk| 2<br />
ɛ+is−ɛk<br />
. (3.72)<br />
One can f<strong>in</strong>d the phase shift from T m<strong>at</strong>rix element us<strong>in</strong>g the prescription<br />
lims→+0<br />
where<br />
1<br />
x−is<br />
1 = P + iπδ(x).<br />
x<br />
δ(ɛ) = arg det T (ɛ + ) (3.73)<br />
= π<br />
<br />
−ɛ + ɛd + Λ(ɛ)<br />
− tan−1<br />
(3.74)<br />
2 ∆(ɛ)<br />
Λ(ɛ) = P |Vk| 2<br />
, ∆(ɛ) = π<br />
ɛ − ɛk<br />
<br />
|Vk| 2 δ(ɛ − ɛk) . (3.75)<br />
k<br />
One can derive the density of st<strong>at</strong>es ρ(ɛ) from the expression of the Green’s function<br />
ρ(ɛ) = ∓ 1<br />
π ImTrG± (ɛ) [26] or from the Friedel sum rule (3.45). The density of st<strong>at</strong>e<br />
<strong>in</strong>duced by the impurity is given by<br />
k
∆ρ(ɛ) = 1 dδ(ɛ)<br />
π dɛ<br />
49<br />
, (3.76)<br />
where δ(ɛ) = 2 <br />
l (2l + 1)δl(ɛ) for spherically symmetric potentials. If one tries to<br />
f<strong>in</strong>d Λ(ɛ) and ∆(ɛ) with a simple density of st<strong>at</strong>es,<br />
ρ0(ɛ) = ρ0 for − D < ɛ < D , (3.77)<br />
by neglect<strong>in</strong>g the k dependence of Vk, one can obta<strong>in</strong><br />
∆(ɛ) = πρ0|V | 2 = C for − D < ɛ < D , (3.78)<br />
= 0 for ɛ ≤ −D or ɛ ≥ D (3.79)<br />
Λ(ɛ) = ρ0|V | 2 <br />
<br />
ln <br />
D + ɛ<br />
<br />
D<br />
− ɛ<br />
. (3.80)<br />
Note th<strong>at</strong> Λ(ɛ) can be easily calcul<strong>at</strong>ed by the Kramers-Kronig rel<strong>at</strong>ion.<br />
Λ(ɛ) = 1<br />
D<br />
dɛ<br />
π −D<br />
′ ∆(ɛ′ )<br />
ɛ − ɛ ′<br />
(3.81)<br />
It can be shown th<strong>at</strong> the density of <strong>in</strong>duced st<strong>at</strong>es approxim<strong>at</strong>ely follows a<br />
Lorentzian form and thus there’s a virtual bound st<strong>at</strong>e <strong>at</strong> ɛ ′ d .<br />
δ(ɛ) = π<br />
2 − tan−1 −ɛ + ɛ′ d<br />
C<br />
∆ρ(ɛ) = 1<br />
π<br />
dδ(ɛ)<br />
dɛ<br />
= 1<br />
π<br />
(3.82)<br />
C<br />
(ɛ − ɛ ′ d )2 + C 2 , (3.83)<br />
where ɛ ′ d − ɛd − Λ(ɛ ′ d ) = 0. Fig. 3.3 shows the density of st<strong>at</strong>es <strong>in</strong>duced by the<br />
impurity with the density of st<strong>at</strong>es (Fig. 3.2) <strong>in</strong> the conduction band. S<strong>in</strong>ce the<br />
potential (V = −0.13) is weak and ɛd = −0.3 is well with<strong>in</strong> the conduction band,<br />
there is a resonance around the energy of −0.3. This virtual bound st<strong>at</strong>e becomes<br />
a real bound st<strong>at</strong>e below the bottom of the conduction band for V < −0.6. (Fig.<br />
3.4) S<strong>in</strong>ce there is one electron <strong>in</strong> a bound st<strong>at</strong>e for V < −0.6, the number of<br />
electrons <strong>in</strong> the conduction band is zero as shown <strong>in</strong> Fig. 3.4. Note th<strong>at</strong> <strong>in</strong> this
FIGURE 3.2. The density of st<strong>at</strong>es ρ0(ɛ) <strong>in</strong> the conduction band and the correspond<strong>in</strong>g<br />
phase shift. ɛd = −0.3 and ∆(ɛ) = (−0.13) 2 πρ0(ɛ) are used.<br />
FIGURE 3.3. There is a virtual bound st<strong>at</strong>e resonance <strong>at</strong> the solution of<br />
ɛ − ɛd − Λ(ɛ) = 0. The density of st<strong>at</strong>es ∆ρ(ɛ) <strong>in</strong>duced by the impurity. Calcul<strong>at</strong>ed<br />
from Fig. 3.2 <strong>in</strong> M<strong>at</strong>hem<strong>at</strong>ica.<br />
case the system has always one extra electron due to the impurity, regardless of<br />
the number of virtual bound st<strong>at</strong>es. Actually, the transition between bound and<br />
unbound st<strong>at</strong>es is smooth and the properties such as the total energy are smooth<br />
functions of the strength of the potential as well. [27]<br />
50
FIGURE 3.4. Shows how the density of st<strong>at</strong>es ∆ρ(ɛ) varies <strong>in</strong> the conduction band<br />
for the different strength of the potential. Calcul<strong>at</strong>ed from Fig. 3.2 <strong>in</strong> M<strong>at</strong>hem<strong>at</strong>ica.<br />
The number of electrons <strong>in</strong> the conduction band is obta<strong>in</strong>ed numerically.<br />
3.6. Sc<strong>at</strong>ter<strong>in</strong>g length and bound st<strong>at</strong>e energy<br />
The purpose of this section is to give emphasis to the rel<strong>at</strong>ionship between<br />
the sc<strong>at</strong>ter<strong>in</strong>g length and the bound st<strong>at</strong>e energy, which easily can be derived for<br />
51
sc<strong>at</strong>ter<strong>in</strong>g by a spherically symmetric square-well potential <strong>at</strong> low energies (such<br />
as a deuteron system). [28] [29] Notice th<strong>at</strong> it is sufficient to consider only the<br />
S-wave for low energies. Suppose the potential has the form,<br />
V (r) = −V0<br />
for r < r0<br />
52<br />
= 0 for r > r0 . (3.84)<br />
One can immedi<strong>at</strong>ely write the correspond<strong>in</strong>g radial equ<strong>at</strong>ions and solutions for<br />
bound st<strong>at</strong>es of the energy EB,<br />
d 2 u(r)<br />
dr 2 + k2 u(r) = 0 and u(r) = A s<strong>in</strong>(kr) for r < r0 (3.85)<br />
d 2 u(r)<br />
dr 2 − k2 Bu(r) = 0 and u(r) = B exp(−kBr) for r > r0 , (3.86)<br />
with k 2 = V0 − |EB| and k 2 B = |EB|. The boundary condition <strong>at</strong> r = r0 gives rise<br />
to<br />
k cot(ka) = −kB . (3.87)<br />
S<strong>in</strong>ce the deuteron system has a small b<strong>in</strong>d<strong>in</strong>g energy |E|, us<strong>in</strong>g k0 k(k0 = V 1/2<br />
0 ),<br />
If k0r0 π/2, one obta<strong>in</strong>s<br />
k0 tan( π<br />
2 − k0r0) = −kB . (3.88)<br />
k0r0 = π kB<br />
+<br />
2 k0<br />
For low-energy sc<strong>at</strong>ter<strong>in</strong>g, similarly,<br />
. (3.89)<br />
u(r) = A s<strong>in</strong>(kr) for r < r0 (3.90)<br />
u(r) = B s<strong>in</strong>(k ′ r + δ) for r > r0 , (3.91)<br />
with k 2 = V0 + E and k ′2 = E. Us<strong>in</strong>g the boundary condition <strong>at</strong> r = r0,
k cot kr0 = k ′ cot(k ′ r0 + δ) (3.92)<br />
tan δ =<br />
k ′<br />
k tan kr0 − tan k ′ r0<br />
1 + k′<br />
k tan kr0 tan k ′ r0<br />
For k ′ → 0, one can use k → k0, which leads to<br />
δ tan δ k ′ <br />
tan k0r0<br />
r0 − 1<br />
k0r0<br />
Note th<strong>at</strong> the total cross section is given by<br />
lim<br />
k ′ 4π<br />
=<br />
→0 k ′2 s<strong>in</strong>2 δ0 4π( δ0<br />
k<br />
<br />
= 4π<br />
′ )2<br />
where the sc<strong>at</strong>ter<strong>in</strong>g length a is def<strong>in</strong>ed by<br />
r0<br />
tan k0r0<br />
k0r0<br />
53<br />
. (3.93)<br />
. (3.94)<br />
2 − 1<br />
(3.95)<br />
(3.96)<br />
= 4πa 2 , (3.97)<br />
δ0 → −ak ′<br />
and k0 cot k0r0 = 1<br />
as k ′ → 0 (3.98)<br />
r0 − a<br />
The quantity 4πa 2 only depends on the potential.<br />
One can also rewrite (3.93) to obta<strong>in</strong><br />
k ′ cot δ = k cot kr0 + k ′ tan k ′ r0<br />
1 − k<br />
k ′ cot kr0 tan k ′ r0<br />
. (3.99)<br />
(3.100)<br />
One f<strong>in</strong>ally can expand (3.100) <strong>in</strong> terms of k ′ keep<strong>in</strong>g terms up to k ′2 , with (3.99),<br />
k = k ′2 + k 2 0 k0 + k′2<br />
2k0 , and tan k′ r0 k ′ r0 + 1<br />
3 (k′ r0) 3 ,<br />
where reff = r0<br />
k ′ cot δ − 1 1<br />
+<br />
a<br />
<br />
1 − 1<br />
(k0r0) 2<br />
r0<br />
a<br />
1 r0 − ( 3 a )2<br />
expansion for k ′ cot δ for low-energy sc<strong>at</strong>ter<strong>in</strong>g.<br />
2 k′2reff + · · ·<br />
<br />
, (3.101)<br />
. This result is known as the effective range<br />
Now, <strong>in</strong> order to rel<strong>at</strong>e the bound st<strong>at</strong>e energy to the sc<strong>at</strong>ter<strong>in</strong>g length, one<br />
needs to write from (3.89), s<strong>in</strong>ce kB/k0 ≪ 1,
tan k0r0 = tan(π/2 + kB/k0) = −k0/kB . (3.102)<br />
One can now rewrite (3.100) with tan k ′ r0 k ′ r0 as<br />
k ′ cot δ − kB<br />
1 + r0kB<br />
r0<br />
54<br />
+ k<br />
1 + r0kB<br />
′2 . (3.103)<br />
By compar<strong>in</strong>g this equ<strong>at</strong>ion with the effective range expansion (3.101), one can<br />
obta<strong>in</strong><br />
a = r0 + 1<br />
kB<br />
. (3.104)<br />
This result provides a way to see whether the self-consistent solutions from the<br />
numerical calcul<strong>at</strong>ions are correct or not by compar<strong>in</strong>g the sc<strong>at</strong>ter<strong>in</strong>g length a and<br />
the b<strong>in</strong>d<strong>in</strong>g energy. The potential range r0 also can be estim<strong>at</strong>ed from the equ<strong>at</strong>ion<br />
above.
4. IMPLEMENTATION<br />
In general, the procedure to implement a theory and transl<strong>at</strong>e it <strong>in</strong>to com-<br />
puter codes is not easy due to the limit<strong>at</strong>ions of a computer. Two numerical<br />
problems which directly affect the numerical precision orig<strong>in</strong><strong>at</strong>e from approxima-<br />
tions to <strong>in</strong>f<strong>in</strong>ity and cont<strong>in</strong>uity, which frequently become the largest problems and<br />
can be expressed only approxim<strong>at</strong>ely <strong>in</strong> numerical calcul<strong>at</strong>ions. In most cases, one<br />
needs to use a large cut-off value for <strong>in</strong>f<strong>in</strong>ity and very small <strong>in</strong>terval for cont<strong>in</strong>uity<br />
<strong>in</strong> numerical calcul<strong>at</strong>ions to obta<strong>in</strong> a reasonable answer. It is important to f<strong>in</strong>d<br />
out how those parameters can be determ<strong>in</strong>ed. In this chapter, the iter<strong>at</strong>ive scheme<br />
to accomplish the self-consistent solutions and the numerical methods to f<strong>in</strong>d the<br />
eigenst<strong>at</strong>es will be <strong>in</strong>troduced. The technique to solve the numerical problems, <strong>in</strong><br />
which physical understand<strong>in</strong>g is <strong>in</strong>dispensable, will be discussed. It needs to be<br />
emphasized th<strong>at</strong> spherically symmetric potentials are used throughout this work.<br />
4.1. Perspective scheme for self-consistent solutions<br />
4.1.1. Non-sp<strong>in</strong> polarized system<br />
Fig. 4.1 shows one possible flow chart to f<strong>in</strong>d a self-consistent solution for<br />
the KS equ<strong>at</strong>ions. One can also start with the <strong>in</strong>itial electronic density, but <strong>in</strong><br />
this case the effective potential is more convenient to deal with s<strong>in</strong>ce the rel<strong>at</strong>ive<br />
numerical response of the effective potential <strong>in</strong> r-space is much larger than th<strong>at</strong> of<br />
the density. With an <strong>in</strong>put effective potential, one can solve the KS equ<strong>at</strong>ions to<br />
f<strong>in</strong>d the density, from which one can construct the output effective potential. A<br />
self-consistent solution means th<strong>at</strong> the <strong>in</strong>put effective potential must be the same<br />
as the output effective potential. In practice, one can stop the calcul<strong>at</strong>ions if the<br />
55
FIGURE 4.1. Flow chart for non-polarized systems.<br />
<strong>in</strong>put potential is very close to the output potential, when numerical precision<br />
required for the immersion energy can be achieved and thus further calcul<strong>at</strong>ions<br />
do not affect the f<strong>in</strong>al answers. In order to compare the <strong>in</strong>put potential with the<br />
output potential, one can use<br />
(∆V ) 2 = 1<br />
<br />
Ω<br />
56<br />
dr [V <strong>in</strong>put<br />
eff (r) − V output<br />
eff (r)] 2 , (4.1)<br />
where Ω is the total volume of the system. The potential difference ∆V can be<br />
used to f<strong>in</strong>d the physical quantities such as immersion energy, bound st<strong>at</strong>e energy,
and magnetiz<strong>at</strong>ion s<strong>in</strong>ce the self-consistent solution means th<strong>at</strong> ∆V = 0. One can<br />
<strong>at</strong>tempt to determ<strong>in</strong>e the physical quantities correspond<strong>in</strong>g to the zero value of<br />
∆V by us<strong>in</strong>g an appropri<strong>at</strong>e fitt<strong>in</strong>g method for the d<strong>at</strong>a po<strong>in</strong>ts.<br />
In order to make progress, one needs a numerical method to determ<strong>in</strong>e the<br />
<strong>in</strong>put potential from the output potential for the next iter<strong>at</strong>ion, which will be<br />
expla<strong>in</strong>ed <strong>in</strong> Sec. 4.2.<br />
4.1.2. Sp<strong>in</strong> polarized system<br />
One big difference between sp<strong>in</strong> polarized systems and non-sp<strong>in</strong> polarized sys-<br />
tems is the need to solve two different sets of KS equ<strong>at</strong>ions, for n+(r) and n−(r),<br />
which doubles the calcul<strong>at</strong>ion time. Only the exchange-correl<strong>at</strong>ion energies and po-<br />
tentials are, however, explicitly dependent upon the sp<strong>in</strong> density (n+(r) − n−(r))<br />
as <strong>in</strong> (2.41), (2.42), and Sec. 2.2.7.4. The Coulomb potentials and energies do not<br />
depend on the sp<strong>in</strong> density, but only on the total density (n+(r) + n−(r)).<br />
4.2. Mix<strong>in</strong>g scheme<br />
S<strong>in</strong>ce the KS equ<strong>at</strong>ions are solved self-consistently <strong>in</strong> this work and the time<br />
to f<strong>in</strong>d self-consistent solutions is directly proportional to the number of iter<strong>at</strong>ions<br />
of the self-consistent loop <strong>in</strong> Fig. 4.1, it is important to f<strong>in</strong>d an efficient method<br />
to construct the next <strong>in</strong>put potential from the output potential. The simplest<br />
mix<strong>in</strong>g scheme is V <strong>in</strong>itial,i+1<br />
eff<br />
57<br />
(r) = V f<strong>in</strong>al,i<br />
eff (r). This simple scheme, however, does<br />
not work s<strong>in</strong>ce the response <strong>in</strong> the output potential is usually enormous compared<br />
to the change <strong>in</strong> the <strong>in</strong>put potential, even for the high densities of an electron
gas. 1 The ma<strong>in</strong> purpose of a mix<strong>in</strong>g scheme is to make the convergence faster,<br />
th<strong>at</strong> is, to reduce the program-runn<strong>in</strong>g time to reach self-consistency while the<br />
solutions ma<strong>in</strong>ta<strong>in</strong> the stability such th<strong>at</strong> they don’t diverge. Another straight<br />
mix<strong>in</strong>g scheme is<br />
V <strong>in</strong>,i+1<br />
eff<br />
(r) = V <strong>in</strong>,i<br />
eff<br />
58<br />
out,i <strong>in</strong>,i<br />
(r) + a (V (r) − V (r)) (4.2)<br />
where i is the number of the iter<strong>at</strong>ion. The mix<strong>in</strong>g r<strong>at</strong>io a ensures the stability<br />
of the convergence dur<strong>in</strong>g the self-consistent calcul<strong>at</strong>ions. One can keep reduc<strong>in</strong>g<br />
a until stability is acquired. One way to speed up the convergence is Broyden’s<br />
method which is<br />
where F i (r) = V out,i<br />
eff<br />
eff<br />
eff<br />
V <strong>in</strong>,i+1<br />
eff (r) = V <strong>in</strong>,i<br />
eff (r) + a(r) F i (r)<br />
<strong>in</strong>,i <strong>in</strong>,i−1<br />
Veff (r) − Veff (r)<br />
a(r) =<br />
F i (r) − F i−1 (r)<br />
<strong>in</strong>,i<br />
(r) − V (r). One can obta<strong>in</strong> Broyden’s method by try-<br />
eff<br />
<strong>in</strong>g to f<strong>in</strong>d the po<strong>in</strong>t on the l<strong>in</strong>e of V <strong>in</strong><br />
out<br />
eff (r) = Veff (r) between the two po<strong>in</strong>ts,<br />
(V <strong>in</strong>,i−1<br />
eff<br />
(r), V out,i−1<br />
(r)) and (V <strong>in</strong>,i out,i<br />
(r), V (r)). S<strong>in</strong>ce the mix<strong>in</strong>g r<strong>at</strong>io <strong>in</strong> Broy-<br />
eff<br />
eff<br />
eff<br />
den’s method depends on the <strong>in</strong>put and output potentials of last two iter<strong>at</strong>ions,<br />
discont<strong>in</strong>uities <strong>in</strong> the potential of r-space may show up dur<strong>in</strong>g the calcul<strong>at</strong>ions and<br />
the <strong>in</strong>tegr<strong>at</strong>ion for wave functions based on the potential may go wrong as well.<br />
Another problem <strong>in</strong> this case is the long range tail of Friedel oscill<strong>at</strong>ion, which is<br />
usually strongly coupled to the electronic density <strong>in</strong> the vic<strong>in</strong>ity of an impurity<br />
due to the Coulomb potential. The idea to avoid this problem and speed up the<br />
1 Results for dielectric constants suggest the response of electron gases of high densities<br />
is small compared to those of low densities. See chapter 5.<br />
,
convergence <strong>at</strong> the same time is to use the straight mix<strong>in</strong>g scheme and the mix<strong>in</strong>g<br />
r<strong>at</strong>io of an exponential-like function.<br />
The mix<strong>in</strong>g scheme is now<br />
a(r) = a exp(−α r 2 ) . (4.3)<br />
V <strong>in</strong>,i+1<br />
eff (r) = V <strong>in</strong>,i<br />
eff (r) + a exp(−α r2 ) (V out,i<br />
eff<br />
59<br />
<strong>in</strong>,i<br />
(r) − V (r)) . (4.4)<br />
There are two parameters a and α <strong>in</strong> this mix<strong>in</strong>g scheme. a can be determ<strong>in</strong>ed<br />
by consider<strong>in</strong>g the response of a region <strong>in</strong> the vic<strong>in</strong>ity of an impurity and α is<br />
a parameter to suppress the strong response of the tail of Friedel oscill<strong>at</strong>ions.<br />
Now one can allow as large a change as possible <strong>in</strong> the vic<strong>in</strong>ity of an impurity<br />
by <strong>in</strong>creas<strong>in</strong>g the a value while one suppresses the response of the tail by us<strong>in</strong>g<br />
large value of α. In most cases, 0.02 for a and 0.002 for α have been found<br />
appropri<strong>at</strong>e. Another idea to make convergence even faster is to utilize the last<br />
two <strong>in</strong>put potentials and add the response of those to the mix<strong>in</strong>g scheme.<br />
V <strong>in</strong>,i+1<br />
eff<br />
(r) = V <strong>in</strong>,i<br />
eff<br />
= V <strong>in</strong>,i<br />
eff<br />
One can make two vectors V <strong>in</strong>,i<br />
eff<br />
out,i <strong>in</strong>,i <strong>in</strong>,i <strong>in</strong>,i−1<br />
(r) + a(r) (V (r) − V (r) + V (r) − V (r))<br />
eff<br />
eff<br />
out,i <strong>in</strong>,i−1<br />
(r) + a(r) (Veff (r) − Veff (r)) . (4.5)<br />
out,i<br />
<strong>in</strong>,i−1<br />
(r) → V (r) and V (r) → V <strong>in</strong>,i<br />
(r) for each r.<br />
eff<br />
If those vectors have the same directions, the mix<strong>in</strong>g scheme (4.5) will acceler<strong>at</strong>e<br />
the convergence (large mix<strong>in</strong>g r<strong>at</strong>io) compared to the previous mix<strong>in</strong>g scheme (4.4)<br />
and if they have opposite directions, (4.5) will deceler<strong>at</strong>e the convergence (small<br />
mix<strong>in</strong>g r<strong>at</strong>io) to avoid the possible oscill<strong>at</strong>ions of potentials between two iter<strong>at</strong>ions. 2<br />
2 The divergence can be noticed usually by fluctu<strong>at</strong>ions of potentials between iter<strong>at</strong>ions<br />
dur<strong>in</strong>g the numerical calcul<strong>at</strong>ions.<br />
eff<br />
eff<br />
eff<br />
eff<br />
eff
Once s<strong>at</strong>isfactory self-consistency is obta<strong>in</strong>ed <strong>in</strong> the vic<strong>in</strong>ity of an impurity,<br />
one can reduce a to avoid the possible large vari<strong>at</strong>ion of the density around the<br />
impurity and reduce α to allow now larger vari<strong>at</strong>ion of the tail of potentials to<br />
make convergence faster. One can implement this <strong>in</strong> the program by calcul<strong>at</strong><strong>in</strong>g<br />
the potential difference ∆V of (4.1) and have the program autom<strong>at</strong>ically vary<br />
the parameters a and α. If ∆V is small enough, especially <strong>in</strong> the vic<strong>in</strong>ity of an<br />
impurity, and the convergence is too slow, a and α can be reduced by a small<br />
amount dur<strong>in</strong>g calcul<strong>at</strong>ions. One idea is to calcul<strong>at</strong>e the rel<strong>at</strong>ive r<strong>at</strong>io of ∆V<br />
between two iter<strong>at</strong>ions.<br />
R<strong>at</strong>io =<br />
(∆Vi) 2 − (∆Vi−1) 2<br />
((∆Vi) 2 + (∆Vi−1) 2 )/2.0<br />
60<br />
. (4.6)<br />
If the r<strong>at</strong>io is too small, one can try to reduce a and α values. However one needs<br />
to f<strong>in</strong>d the appropri<strong>at</strong>e iter<strong>at</strong>ions <strong>at</strong> which the parameters can be reduced and the<br />
values of parameters by trial and error. It has been found <strong>in</strong> this work th<strong>at</strong> if the<br />
∆V 2 is smaller than 10 −8 ∼ 10 −9 and the r<strong>at</strong>io is smaller than 0.002 ∼ 0.005, a<br />
and α can be reduced by 10% ∼ 20%. A m<strong>in</strong>imum number of iter<strong>at</strong>ions between<br />
the iter<strong>at</strong>ions <strong>at</strong> which the parameters are reduced, however, often needs to be set<br />
to ensure the convergence of the calcul<strong>at</strong>ions.<br />
4.3. Search for eigenst<strong>at</strong>es and eigenvalues of the <strong>in</strong>f<strong>in</strong>ite system<br />
4.3.1. Bound st<strong>at</strong>es<br />
The method for the search for bound st<strong>at</strong>es is well established and calcu-<br />
l<strong>at</strong>ions are quite simple for spherically symmetric potentials. [30] S<strong>in</strong>ce only the<br />
pr<strong>in</strong>ciple quantum number n and the angular momentum number l for spherically<br />
symmetric potentials are necessary to dist<strong>in</strong>guish eigenst<strong>at</strong>es, each eigenst<strong>at</strong>e can
e c<strong>at</strong>egorized by n and l or the number of nodes nnodes and l. 3 One can then<br />
search for bound st<strong>at</strong>es from the m<strong>in</strong>imum energy to the maximum energy us<strong>in</strong>g<br />
nnodes and l. Note th<strong>at</strong> the possible lowest energy for the bound st<strong>at</strong>es can be set<br />
to − Z2<br />
n 2 Ryd., which is bound st<strong>at</strong>e energy for a pure Coulomb potential, s<strong>in</strong>ce once<br />
the pure Coulomb potential is screened, the energy eigenvalues are always higher<br />
than those for the pure Coulomb potential. An upper limit for bound st<strong>at</strong>es is<br />
zero with a reference of zero energy limr→∞ Veff(r) = 0.<br />
The method used <strong>in</strong> this work is known as the bisectional algorithm. [30]<br />
There is a better and faster method than the bisectional algorithm, which has<br />
never been tried <strong>in</strong> this work s<strong>in</strong>ce the time required to search for bound st<strong>at</strong>es is<br />
rel<strong>at</strong>ively quite small compared to th<strong>at</strong> for sc<strong>at</strong>tered st<strong>at</strong>es and is usually less than<br />
a few seconds for today’s computers. For the bisectional algorithm, one takes half<br />
the m<strong>in</strong>imum energy as the <strong>in</strong>itial energy for the energy eigenvalue one tries to<br />
achieve and <strong>in</strong>tegr<strong>at</strong>es the KS equ<strong>at</strong>ion with this energy. If the current energy is<br />
higher or smaller than the eigenenergy, one can use the current energy as an upper<br />
limit or a lower limit for the next iter<strong>at</strong>ion and <strong>in</strong>tegr<strong>at</strong>e the KS equ<strong>at</strong>ion with the<br />
new middle energy between the upper and lower limits. One can try to repe<strong>at</strong> the<br />
process until the bound st<strong>at</strong>e energy is obta<strong>in</strong>ed with<strong>in</strong> the required tolerance for<br />
the numerical error.<br />
In order to decide whether the current energy is higher or smaller than the<br />
eigenenergy, one uses two conditions, which both must be s<strong>at</strong>isfied for correct wave<br />
functions. The first one is th<strong>at</strong> the number of nodes <strong>in</strong> the wave functions must<br />
be the same as required <strong>in</strong> nnodes = n − l − 1. The second one is th<strong>at</strong> the first<br />
3 Sp<strong>in</strong>-orbit coupl<strong>in</strong>g is neglected <strong>in</strong> this work, and hence n = nnodes + l + 1.<br />
61
deriv<strong>at</strong>ive of one wave function <strong>in</strong>tegr<strong>at</strong>ed from the orig<strong>in</strong> outwards must m<strong>at</strong>ch <strong>at</strong><br />
a m<strong>at</strong>ch<strong>in</strong>g po<strong>in</strong>t with the first deriv<strong>at</strong>ive of the other wave function <strong>in</strong>tegr<strong>at</strong>ed from<br />
the maximum radius <strong>in</strong>wards. The numerical best po<strong>in</strong>t for a m<strong>at</strong>ch<strong>in</strong>g po<strong>in</strong>t is an<br />
<strong>in</strong>nermost classical turn<strong>in</strong>g po<strong>in</strong>t, s<strong>in</strong>ce one can expect th<strong>at</strong> the oscill<strong>at</strong>ory behavior<br />
of a solution becomes a decreas<strong>in</strong>g exponential-like behavior <strong>at</strong> this turn<strong>in</strong>g po<strong>in</strong>t.<br />
Note th<strong>at</strong> Friedel oscill<strong>at</strong>ions <strong>in</strong> the tail of potentials may give rise to a number of<br />
turn<strong>in</strong>g po<strong>in</strong>ts. First, one can narrow the range of the energy us<strong>in</strong>g the number<br />
of nodes. If the number of nodes <strong>in</strong> a wave function is larger or smaller than th<strong>at</strong><br />
required, the current energy is higher or lower than the eigenenergy. If the number<br />
of nodes is correct, one can check if first deriv<strong>at</strong>ives of wave functions m<strong>at</strong>ch <strong>at</strong><br />
the m<strong>at</strong>ch<strong>in</strong>g po<strong>in</strong>t. 4 If the magnitude of a first deriv<strong>at</strong>ive of an outward solution<br />
is larger or smaller than th<strong>at</strong> of an <strong>in</strong>ward solution, the current energy is higher<br />
or lower than the eigenenergy. If the energy is needed to be higher than the <strong>in</strong>itial<br />
maximum energy (zero as <strong>in</strong> (3.4)), one can stop the search s<strong>in</strong>ce there is no bound<br />
st<strong>at</strong>e.<br />
The KS equ<strong>at</strong>ions can be <strong>in</strong>tegr<strong>at</strong>ed only with appropri<strong>at</strong>e boundary condi-<br />
tions. For bound st<strong>at</strong>es,<br />
as r → 0<br />
un,l(r) → r l+1<br />
as r → ∞<br />
and u ′ n,l(r) → (l + 1)r l<br />
62<br />
, (4.7)<br />
un,l(r) → exp(− √ −Er) and u ′ n,l(r) → − √ −E exp(− √ −Er) . (4.8)<br />
4 S<strong>in</strong>ce the numerical solutions are not normalized, one has to ensure they both have<br />
the same numerical scale to compare first deriv<strong>at</strong>ives, for <strong>in</strong>stance, by us<strong>in</strong>g logarithmic<br />
deriv<strong>at</strong>ives.
The normaliz<strong>at</strong>ion condition for bound st<strong>at</strong>es is<br />
<br />
63<br />
dr |u(r)| 2 = 1 . (4.9)<br />
The density difference for spherically symmetric potentials due to the bound st<strong>at</strong>es<br />
of an impurity is given by<br />
n(r) − n0 = (2l + 1) u<br />
2<br />
4π<br />
2 n,l (r)<br />
r2 , (4.10)<br />
where the factor 2 is for sp<strong>in</strong> degeneracy 5<br />
as follows.<br />
4.3.2. Sc<strong>at</strong>tered st<strong>at</strong>es<br />
n,l<br />
The Fermi wave vector kF of a homogeneous electron gas can be calcul<strong>at</strong>ed<br />
2 Ω<br />
(2π) 3<br />
4π<br />
3 k3 F = N<br />
where the left hand side is the number of st<strong>at</strong>es available <strong>in</strong> a system of the volume<br />
Ω, the factor 2 <strong>in</strong> the left hand side is for sp<strong>in</strong> degeneracy, and N is the total number<br />
of electrons <strong>in</strong> a system. In the thermodynamic limit, the above equ<strong>at</strong>ion leads to<br />
kF = (3π 2 n0) 1/3 , (4.11)<br />
where n0 is the electronic density of the system. S<strong>in</strong>ce the system used <strong>in</strong> this<br />
work is extended to <strong>in</strong>f<strong>in</strong>ity, and thus there are no boundary conditions, there are<br />
5 The factor (2l + 1)/4π is obta<strong>in</strong>ed by us<strong>in</strong>g the addition theorem:<br />
l<br />
m=−l<br />
|Yl,m(θ, φ)| 2 =<br />
2l + 1<br />
4π<br />
.
no quantized k values and all k values up to kF are available. In addition, the<br />
chemical potential is assumed to be a constant and the same as the Fermi energy<br />
of a homogeneous electron gas <strong>at</strong> T = 0 s<strong>in</strong>ce the dimension of the system is much<br />
larger than th<strong>at</strong> of an impurity. Now one needs to <strong>in</strong>tegr<strong>at</strong>e the KS equ<strong>at</strong>ions<br />
<strong>in</strong> the k-mesh for the cont<strong>in</strong>uum st<strong>at</strong>es k = 0 · · · kF . The wave functions can be<br />
normalized by (3.24) and the density difference for sc<strong>at</strong>tered st<strong>at</strong>es is given by<br />
(3.47).<br />
The ma<strong>in</strong> numerical problem is th<strong>at</strong> the maximum number of angular mo-<br />
mentum values should be determ<strong>in</strong>ed for each k. Theoretically kR is a good value<br />
for the maximum l where R is the radius of the potential. [23] On the other hand,<br />
only 10 l values were used <strong>in</strong> the work of Puska, Niem<strong>in</strong>en, and Man<strong>in</strong>en.(1981)<br />
None of these are, however, good criteria for the maximum l value. The kR tends<br />
to neglect the low-energy sc<strong>at</strong>ter<strong>in</strong>g due to very small k values. Ten for the max-<br />
imum l value is more than enough for small k values and is too small for large k<br />
values. It is found th<strong>at</strong> us<strong>in</strong>g too small a number for the maximum l value gives<br />
an error <strong>in</strong> the energy and an <strong>in</strong>stability <strong>in</strong> terms of convergence. A better way<br />
is us<strong>in</strong>g the values of the phase shift to see how close the sc<strong>at</strong>tered st<strong>at</strong>e <strong>at</strong> the<br />
current l value is to the free particle st<strong>at</strong>e. In other words, if the phase shift is<br />
sufficiently small and thus l values larger than the current l value are not impor-<br />
tant, the calcul<strong>at</strong>ion for the current k value can be term<strong>in</strong><strong>at</strong>ed and one beg<strong>in</strong>s the<br />
calcul<strong>at</strong>ion for the next k value <strong>in</strong> the k-mesh. A good value for the smallest phase<br />
shift is 10 −8 , which turns out to give a good convergence.<br />
64
FIGURE 4.2. Phase shift values of l = 0 for a zero potential.<br />
4.4. Numerical problems<br />
4.4.1. Convergence of phase shifts<br />
The accuracy of the phase shifts needs to be tested to see if 10 −8 suggested<br />
for the smallest phase shift <strong>in</strong> the previous section is well with<strong>in</strong> the precision of<br />
the numerical error. One idea taken from [21] is to test the phase shift values for<br />
zero potential s<strong>in</strong>ce the phase shift is zero for the zero potential. One can see the<br />
peak <strong>in</strong> Fig. 4.2 around k = 0.6 and th<strong>at</strong> the phase shift devi<strong>at</strong>es more strongly<br />
from zero after this value. The test has been performed for the electronic density<br />
65
FIGURE 4.3. Phase shift values for an <strong>at</strong>tractive potential of Yukawa type and a<br />
repulsive potential.<br />
of n0 = 0.6. 6 One can conclude th<strong>at</strong> the numerical error <strong>in</strong> the phase shifts is<br />
less than 3 · 10 −11 for the range of actual metallic densities. It is found th<strong>at</strong> the<br />
numerical error of higher l values is smaller than th<strong>at</strong> of l = 0. This numerical<br />
behavior may be, however, dependent on the numerical libraries one uses. Fig.<br />
4.3 shows the rel<strong>at</strong>ionship between the maximum radius Rm used <strong>in</strong> the numerical<br />
calcul<strong>at</strong>ion and the phase shift values. As seen <strong>in</strong> Fig. 4.3, if Rm is too small and<br />
thus the potential does not go to zero fast enough, the phase shift calcul<strong>at</strong>ions<br />
fail. If the potential has such a long range, phase shift values <strong>at</strong> very low l show a<br />
6 Parameters for the test are R0 = 8.1250520·10 −6 , Rlog = 8.0, Rm = 31.02, Nlog = 465,<br />
and Nl<strong>in</strong> = 230, where Nlog is the number of po<strong>in</strong>ts used <strong>in</strong> the logarithmic r-space<br />
between R0 and Rlog and Nl<strong>in</strong> is the number of po<strong>in</strong>ts used <strong>in</strong> the l<strong>in</strong>ear r-space between<br />
Rlog and Rm.<br />
66
zig-zag behavior, which is similar to th<strong>at</strong> of hard sphere sc<strong>at</strong>ter<strong>in</strong>g for high energy<br />
approxim<strong>at</strong>ions. [23] They are, however, eventually approach<strong>in</strong>g zero as l <strong>in</strong>creases.<br />
Then one can stop the calcul<strong>at</strong>ion for each k if phase shift values are small enough,<br />
for <strong>in</strong>stance, less than 10 −8 as argued <strong>in</strong> the previous section.<br />
4.4.2. Correction for potentials<br />
The ma<strong>in</strong> problem <strong>in</strong> solv<strong>in</strong>g the KS equ<strong>at</strong>ions numerically is th<strong>at</strong> the effective<br />
potentials <strong>in</strong> the KS equ<strong>at</strong>ions have physically and numerically essential defects,<br />
especially <strong>at</strong> the beg<strong>in</strong>n<strong>in</strong>g of calcul<strong>at</strong>ions. These defects may not only slow down<br />
the convergence but also lead the calcul<strong>at</strong>ions to <strong>in</strong>correct self-consistent solutions.<br />
The first problem is th<strong>at</strong>, dur<strong>in</strong>g the numerical calcul<strong>at</strong>ions, s<strong>in</strong>ce the effective<br />
potential has not reached self-consistency yet, especially <strong>at</strong> the beg<strong>in</strong>n<strong>in</strong>g, there<br />
may be a surplus or deficiency of charge to screen the impurity and the system does<br />
not s<strong>at</strong>isfy the Friedel sum rule. The effective potentials <strong>in</strong> this case are oscill<strong>at</strong><strong>in</strong>g<br />
around some constant value due to the extra charge. This extra charge appears as<br />
a constant (<strong>in</strong>tersect<strong>in</strong>g po<strong>in</strong>t with r = 0) <strong>in</strong> the rV plot of Fig. 4.4. The second<br />
problem is th<strong>at</strong>, s<strong>in</strong>ce the system is extended to <strong>in</strong>f<strong>in</strong>ity, the upper limit of the<br />
<strong>in</strong>tegral <strong>in</strong> the Coulomb potential must be <strong>in</strong>f<strong>in</strong>ity as well. No m<strong>at</strong>ter how large<br />
a radius is used for the numerical calcul<strong>at</strong>ions, the <strong>in</strong>tegral from the maximum<br />
radius to <strong>in</strong>f<strong>in</strong>ity <strong>in</strong> the Coulomb potential is always miss<strong>in</strong>g, which appears as a<br />
slope <strong>in</strong> the rV plot <strong>in</strong> Fig. 4.4. If one uses naively this <strong>in</strong>correct output potential<br />
to mix with the <strong>in</strong>put potential for the next iter<strong>at</strong>ion, the phase shift calcul<strong>at</strong>ions<br />
cannot be performed s<strong>in</strong>ce the phase shift calcul<strong>at</strong>ions and thus the normaliz<strong>at</strong>ion<br />
for sc<strong>at</strong>tered wave functions are based on the fact th<strong>at</strong> the potentials are zero or<br />
67
FIGURE 4.4. Typical example show<strong>in</strong>g defects <strong>in</strong> the output potential dur<strong>in</strong>g the<br />
numerical calcul<strong>at</strong>ions.<br />
almost zero <strong>at</strong> the end of the maximum radius. 7 In addition, special phase shifts<br />
have to be used for Coulomb potentials. 8<br />
7 At the maximum radius, it is assumed th<strong>at</strong> an impurity is completely screened and<br />
the system s<strong>at</strong>isfies the Friedel sum rule.<br />
8 This may be much more difficult than zero Coulomb potentials. Besides, allow<strong>in</strong>g non-<br />
zero Coulomb potential <strong>at</strong> the maximum radius may be a wrong guide for a self-consistent<br />
solution of the system used <strong>in</strong> this work.<br />
68
One can simply make the potential zero <strong>at</strong> the maximum radius and run<br />
the program for many iter<strong>at</strong>ions until the extra charge (Constant <strong>in</strong> Fig. 4.4)<br />
disappears. This is, however, not only time consum<strong>in</strong>g, but also the potential is<br />
not likely to go to the zero <strong>at</strong> the maximum radius due to the Friedel oscill<strong>at</strong>ions.<br />
One can also try to remove the extra slope <strong>in</strong> the rV plot us<strong>in</strong>g an analytic or<br />
a numerical model for the miss<strong>in</strong>g <strong>in</strong>tegral, but this turns out to be not good<br />
enough. 9 The best way is simply to determ<strong>in</strong>e the extra constant and slope <strong>in</strong> the<br />
rV plot and subtract them from rV . One can not, however, simply subtract the<br />
constant from rV because the potential <strong>in</strong> the vic<strong>in</strong>ity of an impurity is affected by<br />
the <strong>in</strong>teraction between the impurity and an electron gas as well as by the Friedel<br />
oscill<strong>at</strong>ions and thus one has to allow for a vari<strong>at</strong>ion of the electronic charge around<br />
an impurity. S<strong>in</strong>ce <strong>at</strong> the same time the potential must be oscill<strong>at</strong><strong>in</strong>g around zero<br />
<strong>at</strong> the maximum radius, the formula to correct the potential can be written as<br />
r × V = r × V − r × (Slope + Constant/Rm) , (4.12)<br />
where Rm is the maximum radius r <strong>in</strong> the r-mesh for the numerical calcul<strong>at</strong>ions.<br />
One can try other functions <strong>in</strong>stead of a l<strong>in</strong>ear r <strong>in</strong> r × Constant/Rm. However<br />
other functions may suppress the vari<strong>at</strong>ion around an impurity too much or too<br />
little, either of which makes the convergence slower. The slope and constant can<br />
be easily calcul<strong>at</strong>ed by four po<strong>in</strong>ts of maximum and m<strong>in</strong>imum peaks near the end<br />
of the potential. Then one can set up two l<strong>in</strong>e equ<strong>at</strong>ions, from which one can f<strong>in</strong>d<br />
the average slope and constant <strong>at</strong> r = 0. Fig. 4.5 shows the typical examples of<br />
the effective potentials before and after the correction and the right figure <strong>in</strong> Fig.<br />
4.5 shows th<strong>at</strong> the potential is oscill<strong>at</strong><strong>in</strong>g around zero <strong>at</strong> large r values after the<br />
9 It has been found th<strong>at</strong> there is still a noticeable slope after correct<strong>in</strong>g the potential.<br />
69
FIGURE 4.5. Typical examples to show how the effective potentials behave before<br />
and after the correction. These potentials are calcul<strong>at</strong>ed for a hydrogen impurity<br />
and n0 = 0.0025.<br />
correction, which means an impurity is completely screened and Friedel sum rule<br />
is s<strong>at</strong>isfied as it should be.<br />
4.4.3. Calcul<strong>at</strong>ions for <strong>in</strong>duced electrons and energies<br />
One detail previous papers [5] have overlooked is th<strong>at</strong> after <strong>in</strong>clud<strong>in</strong>g all of<br />
the screen<strong>in</strong>g charge around impurity the electronic density is simply oscill<strong>at</strong><strong>in</strong>g<br />
due to the Friedel oscill<strong>at</strong>ions and thus the <strong>in</strong>tegr<strong>at</strong>ed values of the electronic<br />
density, needed to f<strong>in</strong>d the number of electrons <strong>in</strong>duced due to an impurity, are<br />
also oscill<strong>at</strong><strong>in</strong>g, which is shown <strong>in</strong> Fig. 4.6. In addition, the number of <strong>in</strong>duced<br />
electrons <strong>in</strong> the system should not depend on the maximum value of r s<strong>in</strong>ce the<br />
system goes to <strong>in</strong>f<strong>in</strong>ity and thus this number can not be found by simply <strong>in</strong>tegr<strong>at</strong><strong>in</strong>g<br />
from zero to the maximum of r. In order to f<strong>in</strong>d the charge <strong>in</strong>duced <strong>in</strong> the system,<br />
<strong>in</strong>tegr<strong>at</strong>ed values of the charge density as a function of r should be exam<strong>in</strong>ed as <strong>in</strong><br />
70
FIGURE 4.6. Calcul<strong>at</strong>ed for a hydrogen impurity and n0 = 0.0025. This example<br />
shows the oscill<strong>at</strong>ions of <strong>in</strong>tegral values of the <strong>in</strong>duced electronic density as a<br />
function of r.<br />
Fig. 4.6 and the average charge from the oscill<strong>at</strong>ion should be calcul<strong>at</strong>ed to f<strong>in</strong>d the<br />
total number of electrons <strong>in</strong>duced due to an impurity. It is important to note th<strong>at</strong><br />
the r dependence of energies such as the exchange-correl<strong>at</strong>ion and the Coulomb<br />
energy can be removed <strong>in</strong> the same way.<br />
4.4.4. Mesh for r-space and k-space<br />
In order to <strong>in</strong>tegr<strong>at</strong>e functions numerically, one needs to use small <strong>in</strong>terval to<br />
approxim<strong>at</strong>e the cont<strong>in</strong>uous space over which the functions are to be <strong>in</strong>tegr<strong>at</strong>ed.<br />
71
FIGURE 4.7. Typical example for the effective potential. The logarithmic mesh<br />
is used for the rapid vari<strong>at</strong>ion of the potential <strong>in</strong> the vic<strong>in</strong>ity of the orig<strong>in</strong>.<br />
The term, mesh, is def<strong>in</strong>ed <strong>in</strong> this work as how <strong>in</strong>tervals vary between two adjacent<br />
po<strong>in</strong>ts <strong>in</strong> a cont<strong>in</strong>uous space. If the <strong>in</strong>terval does not vary and is simply a constant,<br />
the <strong>in</strong>tegral space is called a l<strong>in</strong>ear mesh. S<strong>in</strong>ce the numerical precision directly<br />
depends on the mesh used for the <strong>in</strong>tegral <strong>in</strong> r or k-space, the behavior of functions<br />
which are to be <strong>in</strong>tegr<strong>at</strong>ed needs to be exam<strong>in</strong>ed to choose the appropri<strong>at</strong>e mesh for<br />
each space. The r-space can be c<strong>at</strong>egorized <strong>in</strong>to two regions, based on the density<br />
vari<strong>at</strong>ion. The first region covers the core st<strong>at</strong>es <strong>in</strong> which the electronic density<br />
usually varies rapidly. In the second region, the net electronic charge <strong>in</strong>duced<br />
by a electron gas is almost zero and the Friedel oscill<strong>at</strong>ions dom<strong>in</strong><strong>at</strong>e the density<br />
vari<strong>at</strong>ion. Therefore a logarithmic r-mesh is appropri<strong>at</strong>e for the first region which<br />
<strong>in</strong>cludes the core st<strong>at</strong>es <strong>in</strong> the vic<strong>in</strong>ity of an impurity and after th<strong>at</strong>, a simple l<strong>in</strong>ear<br />
72
FIGURE 4.8. Example for phase shifts. The phase shift varies rapidly <strong>at</strong> the<br />
bottom of the conduction band if the energy of a bound st<strong>at</strong>e is close to the<br />
bottom of the conduction band.<br />
r-mesh can be used for the second region as shown <strong>in</strong> Fig. 4.7. In this work, the<br />
logarithmic r-mesh always corresponds to the first region and the l<strong>in</strong>ear r-mesh<br />
corresponds to the second region.<br />
Phase shifts <strong>at</strong> l = 0 need to be exam<strong>in</strong>ed <strong>in</strong> order to see how the wave<br />
functions behave <strong>in</strong> k-space. Based on this <strong>in</strong>form<strong>at</strong>ion, one can choose the mesh <strong>in</strong><br />
k-space. The appropri<strong>at</strong>e mesh <strong>in</strong> k-space is, however, quite difficult to determ<strong>in</strong>e<br />
before calcul<strong>at</strong>ions s<strong>in</strong>ce <strong>at</strong> least the energies of bound st<strong>at</strong>es, or the sc<strong>at</strong>ter<strong>in</strong>g<br />
lengths are necessary to estim<strong>at</strong>e how large the slopes of phase shifts <strong>at</strong> l = 0 are<br />
<strong>in</strong> the vic<strong>in</strong>ity of the bottom of the conduction band. Note th<strong>at</strong> the sc<strong>at</strong>ter<strong>in</strong>g<br />
length is <strong>in</strong>versely proportional to the energy of bound st<strong>at</strong>es . (See Sec. 3.6.) One<br />
73
FIGURE 4.9. Integrand <strong>in</strong> (3.22). Calcul<strong>at</strong>ed for a Hydrogen impurity <strong>at</strong> 0.04<br />
background density.<br />
may try very low or high background densities of an electron gas first and try to<br />
determ<strong>in</strong>e the range of the background densities where the energies of bound st<strong>at</strong>es<br />
are close to the bottom of the conduction band. (See Fig. 4.8.) If the calcul<strong>at</strong>ions<br />
need to be performed for this range of the background density, the phase shifts <strong>in</strong><br />
k-space also need to be exam<strong>in</strong>ed <strong>in</strong> order to determ<strong>in</strong>e the range of k-space where<br />
the phase shifts vary fast. In th<strong>at</strong> case, a different k-mesh is necessary due to the<br />
different vari<strong>at</strong>ion of the phase shifts. One can try the logarithmic mesh for the<br />
fast vari<strong>at</strong>ion of phase shifts <strong>at</strong> the bottom of the conduction band as <strong>in</strong> r-space.<br />
However, it turns out th<strong>at</strong> a logarithmic mesh causes more numerical errors than<br />
the simple l<strong>in</strong>ear mesh. Fig. 4.9 shows how the wave functions <strong>in</strong> k-space vary <strong>at</strong><br />
different r values. A logarithmic mesh may work well <strong>at</strong> low r values but causes<br />
74
more numerical errors for the ord<strong>in</strong>ary oscill<strong>at</strong><strong>in</strong>g wave functions <strong>in</strong> k-space. A<br />
better way is simply to use two different l<strong>in</strong>ear k-meshes with different <strong>in</strong>tervals<br />
to deal with the rapid vari<strong>at</strong>ion <strong>at</strong> the bottom of the conduction band. The very<br />
small <strong>in</strong>terval should be used for the range of k <strong>in</strong> which the phase shifts vary fast.<br />
Because of the smooth behavior of the phase shifts, a large <strong>in</strong>terval can be used<br />
for large k values <strong>in</strong> order to reduce the time to <strong>in</strong>tegr<strong>at</strong>e <strong>in</strong> k-space.<br />
75
FIGURE 4.10. The density of a free carbon <strong>at</strong>om and the density <strong>in</strong>duced by a<br />
carbon impurity <strong>in</strong> a homogeneous electron gas. The 1s and 2s core st<strong>at</strong>es rema<strong>in</strong><br />
almost the same. These calcul<strong>at</strong>ions are performed for a sp<strong>in</strong> polarized system.<br />
4.4.5. Cancel<strong>at</strong>ion of numerical errors <strong>in</strong> immersion energies: Number of r<br />
po<strong>in</strong>ts<br />
In this section, the calcul<strong>at</strong>ions are based on the logarithmic r-mesh which is<br />
used for the range from 0 to 8 a.u. and on the l<strong>in</strong>ear r-mesh which is used after 8<br />
a.u. Due to the approxim<strong>at</strong>ions <strong>in</strong> LDA, the rapid vari<strong>at</strong>ion of the density of the<br />
core st<strong>at</strong>es <strong>in</strong> the vic<strong>in</strong>ity of an impurity may cause significant numerical errors.<br />
However, s<strong>in</strong>ce the core st<strong>at</strong>es hardly <strong>in</strong>teract with the environment, numerical<br />
errors of the core st<strong>at</strong>es of an impurity system can be reduced by those of a free<br />
<strong>at</strong>om <strong>in</strong> the immersion energy calcul<strong>at</strong>ions. Th<strong>at</strong> is,<br />
76<br />
Oimpuritysystem − Ofree<strong>at</strong>om 0 , (4.13)
where O denotes the numerical errors <strong>in</strong> the core st<strong>at</strong>es. Fig. 4.10 shows th<strong>at</strong> the<br />
core st<strong>at</strong>es of an impurity actually do not <strong>in</strong>teract with a homogeneous electron gas<br />
and hence do not contribute to the immersion energy significantly. Table 4.1 and<br />
Fig. 4.11 illustr<strong>at</strong>e how the r-mesh <strong>in</strong> the vic<strong>in</strong>ity of an impurity affects the energies<br />
and how numerical errors <strong>in</strong> the immersion energies can be reduced with<strong>in</strong> a few<br />
milli-Rydbergs by calcul<strong>at</strong><strong>in</strong>g the free <strong>at</strong>om energy with the same r-mesh as for<br />
the impurity system. These results may depend on the range of the logarithmic r-<br />
mesh which covers the core st<strong>at</strong>es. The hydrogen impurity for a non-sp<strong>in</strong> polarized<br />
system <strong>in</strong> Table 4.2 shows the same numerical behavior. Therefore it is important<br />
to use the same r-mesh <strong>in</strong> calcul<strong>at</strong><strong>in</strong>g the free <strong>at</strong>om energy and the energy of an<br />
impurity system to avoid large numerical errors due to the core st<strong>at</strong>es.<br />
In order to m<strong>in</strong>imize the rema<strong>in</strong>der of the numerical errors due to the <strong>in</strong>terval<br />
<strong>in</strong> the r-mesh and f<strong>in</strong>d the correct immersion energy, one can <strong>at</strong>tempt to use the<br />
l<strong>in</strong>ear rel<strong>at</strong>ionship between the immersion energies and the number of r-po<strong>in</strong>ts as<br />
r-mesh ∆E E<strong>at</strong>om Eimm = ∆E − E<strong>at</strong>om<br />
525 -73.43094 -73.29770 -0.13324<br />
550 -73.51473 -73.38030 -0.13443<br />
600 -73.64172 -73.50606 -0.13566<br />
650 -73.75003 -73.61365 -0.13638<br />
TABLE 4.1. Energies of a carbon impurity system ∆E and energies of a free<br />
carbon <strong>at</strong>om E<strong>at</strong>om (<strong>in</strong> Rydbergs). A carbon impurity <strong>in</strong> a sp<strong>in</strong> polarized system<br />
and a background density of 0.002 are used for calcul<strong>at</strong>ions. The r-mesh <strong>in</strong> the<br />
table represents the number of r- po<strong>in</strong>ts used <strong>in</strong> the range from 0 to 8 a.u., which<br />
corresponds to the logarithmic r-mesh and <strong>in</strong>cludes most core st<strong>at</strong>es. (See Sec.<br />
4.4.4.)<br />
77
-mesh ∆E E<strong>at</strong>om Eimm = ∆E − E<strong>at</strong>om<br />
450 -1.06535 -0.87278 -0.19257<br />
550 -1.07140 -0.87725 -0.19415<br />
650 -1.07576 -0.88037 -0.19539<br />
750 -1.07884 -0.88285 -0.19599<br />
850 -1.08129 -0.88458 -0.19671<br />
1050 -1.08471 -0.88707 -0.19764<br />
TABLE 4.2. Example for a hydrogen impurity. (Non-sp<strong>in</strong> polarized system.) A<br />
background density of 0.0025 is used for these calcul<strong>at</strong>ions.<br />
FIGURE 4.11. Comparison between immersion energies calcul<strong>at</strong>ed with a constant<br />
free <strong>at</strong>om energy and the free <strong>at</strong>om energy calcul<strong>at</strong>ed with the same r-mesh as the<br />
impurity system. The immersion energies calcul<strong>at</strong>ed with the same free <strong>at</strong>om<br />
energy show rel<strong>at</strong>ively large vari<strong>at</strong>ion with the different number of r-po<strong>in</strong>ts due to<br />
the numerical errors of core st<strong>at</strong>es.<br />
78
FIGURE 4.12. Fitt<strong>in</strong>g by <strong>in</strong>verse power law to f<strong>in</strong>d the correct immersion energy.<br />
D<strong>at</strong>a from Table 4.1 and Table 4.2 .<br />
<strong>in</strong> Fig. 4.12. Now s<strong>in</strong>ce the number of r-po<strong>in</strong>ts is <strong>in</strong> the denom<strong>in</strong><strong>at</strong>or, the number of<br />
r-po<strong>in</strong>ts can be easily extrapol<strong>at</strong>ed to <strong>in</strong>f<strong>in</strong>ity to make the <strong>in</strong>terval of r-mesh zero.<br />
Us<strong>in</strong>g this method, the immersion energy can be determ<strong>in</strong>ed up to 1 milli-rydberg.<br />
4.4.6. Further numerical precision tests<br />
The results of numerical calcul<strong>at</strong>ions may depend on other numerical param-<br />
eters such as the maximum radius Rmax and the number of po<strong>in</strong>ts <strong>in</strong> k and r-space.<br />
The effect of the logarithmic r-mesh <strong>in</strong> the vic<strong>in</strong>ity of an impurity is already tested<br />
<strong>in</strong> Sec. 4.4.5 and it is shown th<strong>at</strong> the immersion energy can be obta<strong>in</strong>ed with<strong>in</strong> 1<br />
milli-rydberg. Other important parameters are the maximum radius Rmax and the<br />
number of po<strong>in</strong>ts <strong>in</strong> k and r-space. The tests <strong>in</strong> this section have been performed<br />
for a Hydrogen impurity <strong>in</strong> a non-sp<strong>in</strong> polarized system of a homogeneous electron<br />
gas. The electronic density is 0.0025. The logarithmic r-mesh is used for the range<br />
from 0 to 8 a.u. and the l<strong>in</strong>ear r-mesh is used after 8 a.u. The maximum radius<br />
79
Rmax Immersion energy(Ry.)<br />
40 Not converged<br />
50 -0.19392<br />
60 -0.19449<br />
70 -0.19395<br />
80 -0.19402<br />
90 -0.19415<br />
120 -0.19411<br />
TABLE 4.3. Immersion energies for a hydrogen impurity. 70 k po<strong>in</strong>ts, 550 r po<strong>in</strong>ts<br />
for the logarithmic r-mesh, and 0.1 for the <strong>in</strong>terval of the l<strong>in</strong>ear r-mesh are used<br />
for these calcul<strong>at</strong>ions.<br />
is a very important parameter for convergence and stability of the calcul<strong>at</strong>ions.<br />
S<strong>in</strong>ce the phase shift calcul<strong>at</strong>ions are based on a zero potential <strong>at</strong> the maximum<br />
radius Rmax, the Coulomb potential due to the extra charge must vanish <strong>at</strong> least<br />
<strong>at</strong> Rmax. Therefore it is important to <strong>in</strong>clude enough Friedel oscill<strong>at</strong>ions and also<br />
not to loose any significant charge of the bound st<strong>at</strong>es. For <strong>in</strong>stance, the conver-<br />
gence required to obta<strong>in</strong> an immersion energy with<strong>in</strong> a error bar of 1 milli-Rydberg<br />
((∆V ) 2 10 −11 , See Sec. 4.1.1.) can not be accomplished for Rmax = 40 <strong>in</strong> Table<br />
4.3. S<strong>in</strong>ce the bound st<strong>at</strong>e has a very short range <strong>at</strong> this background density, the<br />
ma<strong>in</strong> reason is the long tail of Friedel oscill<strong>at</strong>ions which prevent the self-consistent<br />
calcul<strong>at</strong>ions from converg<strong>in</strong>g. (See Fig. 4.13.) One good barometer for the Friedel<br />
oscill<strong>at</strong>ions is to <strong>in</strong>clude <strong>at</strong> least 10 λ’s where λ = π/kF . As the background density<br />
<strong>in</strong>creases, the bound st<strong>at</strong>e plays a more important role and the extended bound<br />
st<strong>at</strong>e should be considered to determ<strong>in</strong>e the maximum radius Rmax.<br />
80
FIGURE 4.13. Extra densities <strong>in</strong>duced by an impurity.<br />
The immersion energies <strong>in</strong> Table 4.3 show small fluctu<strong>at</strong>ion, less than 1 milli-<br />
Rydberg, especially for Rmax larger than 70. Note th<strong>at</strong> the convergence process<br />
is smooth for Rmax larger than 70. The results for the number of r po<strong>in</strong>ts <strong>in</strong><br />
the l<strong>in</strong>ear r-mesh show a very regular p<strong>at</strong>tern <strong>in</strong> Table 4.4, similar to th<strong>at</strong> for the<br />
number of r po<strong>in</strong>ts <strong>in</strong> the logarithmic r-mesh. Fig. 4.14 shows th<strong>at</strong> the immersion<br />
energy decreases as the number of r po<strong>in</strong>ts for the l<strong>in</strong>ear r-mesh <strong>in</strong>creases. The<br />
difference is, however, not larger than 1 milli-Rydberg. One can use simply 0.1 for<br />
the <strong>in</strong>terval of the l<strong>in</strong>ear r-mesh to obta<strong>in</strong> the immersion energy with<strong>in</strong> an error<br />
bar of 1 milli-Rydberg. Another important parameter is the number of k po<strong>in</strong>ts<br />
<strong>in</strong> k-mesh. S<strong>in</strong>ce the k-mesh is a simple l<strong>in</strong>ear mesh, one can expect simply th<strong>at</strong><br />
a larger number of k po<strong>in</strong>ts give rise to the better results. However, for smooth<br />
81
Number of r po<strong>in</strong>ts Immersion energy(Ry.)<br />
320 -0.19353<br />
520 -0.19387<br />
720 -0.19402<br />
920 -0.19411<br />
1120 -0.19417<br />
TABLE 4.4. Immersion energies for a hydrogen impurity. 70 k po<strong>in</strong>ts, 80 for Rmax,<br />
and 550 r po<strong>in</strong>ts for the logarithmic r-mesh are used for calcul<strong>at</strong>ions.<br />
FIGURE 4.14. Immersion energy versus the <strong>in</strong>verse of the number of r po<strong>in</strong>ts <strong>in</strong><br />
the l<strong>in</strong>ear r-mesh. D<strong>at</strong>a from Table 4.4.<br />
82
Number of k po<strong>in</strong>ts Immersion energy(Ry.)<br />
60 -0.19403<br />
70 -0.19402<br />
80 -0.19402<br />
90 -0.19402<br />
100 -0.19402<br />
TABLE 4.5. Immersion energies for a hydrogen impurity. There is no difference<br />
with<strong>in</strong> 10 −5 Rydbergs for k po<strong>in</strong>ts more than 70. 80 for Rmax and 550 r po<strong>in</strong>ts for<br />
the logarithmic r-mesh are used for these calcul<strong>at</strong>ions.<br />
behavior of phase shifts, <strong>in</strong>creas<strong>in</strong>g the number of k po<strong>in</strong>ts does not improve the<br />
results significantly. The results for different number of k po<strong>in</strong>ts are given <strong>in</strong><br />
Table 4.5, which show there is no noticeable improvements with<strong>in</strong> 10 −5 Ry. <strong>in</strong><br />
the immersion energies for more than 70 k po<strong>in</strong>ts. It should be noted th<strong>at</strong> this<br />
result may be quite different for systems which have a large sc<strong>at</strong>ter<strong>in</strong>g length as<br />
mentioned <strong>in</strong> Sec. 4.4.4. Increas<strong>in</strong>g the number of k po<strong>in</strong>ts for those systems can<br />
adjust the significant charge amount correctly and yield better immersion energies.<br />
4.5. Self-consistent solutions<br />
The self-consistent solutions may depend on the methods to solve numerical<br />
problems as well as the boundary conditions. Incorrect numerical parameters or<br />
methods may lead to <strong>in</strong>correct self-consistent solutions. Th<strong>at</strong> is, self-consistency<br />
is not a sufficient condition for a correct solution. The only way to obta<strong>in</strong> the<br />
correct self-consistent solutions is to compare the solutions with the results of<br />
83
other theories or experiments. There are several ways to see if the iter<strong>at</strong>ive process<br />
converges correctly. The extra charge <strong>in</strong>duced by an impurity can be calcul<strong>at</strong>ed by<br />
the Friedel sum rule, <strong>in</strong>tegr<strong>at</strong><strong>in</strong>g the electronic density <strong>in</strong> r-space, or <strong>in</strong>tegr<strong>at</strong><strong>in</strong>g<br />
the density of st<strong>at</strong>es <strong>in</strong> energy space.<br />
<br />
Z = 4π<br />
dr r 2 ∆n(r) =<br />
ɛF<br />
−∞<br />
dɛ ∆ρ(ɛ) = Zb + 2<br />
π<br />
<br />
(2l + 1)δl(kF ) ,<br />
where Zb is the number of electrons <strong>in</strong> the bound st<strong>at</strong>es and the density of st<strong>at</strong>es<br />
∆ρ(ɛ) can be calcul<strong>at</strong>ed from the phase shift values as <strong>in</strong> (3.27):<br />
∆ρ(ɛ) = 2<br />
π<br />
<br />
l<br />
(2l + 1) dδl(ɛ)<br />
dɛ<br />
l<br />
84<br />
. (4.14)<br />
All three results for the extra charge must be very close to each other after the<br />
required convergence is achieved. The electronic density far away from an impurity<br />
should follow the form of the Friedel oscill<strong>at</strong>ions as mentioned <strong>in</strong> Ch. 3.<br />
∼ CF<br />
cos(2kF r + δ)<br />
r 3<br />
<strong>at</strong> large r ⇒ λ ∼ 2π/2kF<br />
Therefore the wave length of oscill<strong>at</strong>ions of the electronic density, λ, should be<br />
about 2π/2kF <strong>at</strong> large distances. There could be a virtual bound st<strong>at</strong>e if the<br />
number of bound st<strong>at</strong>e electrons is smaller than th<strong>at</strong> of the positive charge of the<br />
impurity. At low background densities, s<strong>in</strong>ce only low energies are allowed for the<br />
conduction band, only s-waves can contribute to the virtual bound st<strong>at</strong>e resonance<br />
<strong>in</strong> the density of st<strong>at</strong>es. The sc<strong>at</strong>ter<strong>in</strong>g length can be very important as <strong>in</strong> the case<br />
of a hydrogen impurity when the virtual bound st<strong>at</strong>e resonance is loc<strong>at</strong>ed near<br />
the conduction band m<strong>in</strong>imum and the non-zero angular momentum st<strong>at</strong>es can be<br />
neglected. In th<strong>at</strong> case, the sign of the sc<strong>at</strong>ter<strong>in</strong>g length can also be used to verify<br />
the behavior of the conduction band dur<strong>in</strong>g the iter<strong>at</strong>ive loops of the numerical<br />
calcul<strong>at</strong>ions.
FIGURE 4.15. The density of st<strong>at</strong>es <strong>at</strong> absolute zero temper<strong>at</strong>ure. With an<br />
external magnetic field, the energy of the sp<strong>in</strong>s (magnetic-moment-up) parallel to<br />
the field is lowered by µB, while the energy of the sp<strong>in</strong>s (magnetic-moment-down)<br />
antiparallel to the field is raised by µB. The chemical potential of the sp<strong>in</strong>-up<br />
band is equal to th<strong>at</strong> of the sp<strong>in</strong>-down band.<br />
4.6. Sp<strong>in</strong> paramagnetism<br />
A system with an external magnetic field is discussed <strong>in</strong> Sec. 2.2.6. In this<br />
section, details of the numerical implement<strong>at</strong>ion will be discussed. Note th<strong>at</strong> the<br />
sp<strong>in</strong>-up band is assumed to be parallel to an external magnetic field <strong>in</strong> this work. 10<br />
The energy shift by an external magnetic field <strong>in</strong> Fig. 4.15 is ±µB where − and<br />
+ should be taken for the sp<strong>in</strong>-up and sp<strong>in</strong>-down band respectively and µ is the<br />
Bohr magneton. Some electrons near the Fermi energy transfer from the sp<strong>in</strong>-down<br />
band to the sp<strong>in</strong>-up band such th<strong>at</strong> the chemical potential (Fermi energy) of the<br />
10 The term, sp<strong>in</strong>-up, <strong>in</strong> this work denotes sp<strong>in</strong>-magnetic-moment-up s<strong>in</strong>ce the magnetic<br />
moment is opposite <strong>in</strong> sign to the sp<strong>in</strong> S for an electron.<br />
85
sp<strong>in</strong>-up band equals th<strong>at</strong> of the sp<strong>in</strong>-down band. The range of k correspond<strong>in</strong>g to<br />
the range of the energy from the conduction band m<strong>in</strong>imum to the Fermi energy<br />
adjusts as well.<br />
sp<strong>in</strong>-up: 0 ↔ √ ɛF + u<br />
sp<strong>in</strong>-down: 0 ↔ √ ɛF − u ,<br />
where u = µB. The number of electrons is:<br />
N+ = V<br />
(2π) 3<br />
4<br />
3 π(ɛF + u) 3/2 for sp<strong>in</strong>-up (4.15)<br />
N− = V<br />
(2π) 3<br />
86<br />
4<br />
3 π(ɛF − u) 3/2 for sp<strong>in</strong>-down. (4.16)<br />
Now one can easily obta<strong>in</strong> the equ<strong>at</strong>ion for the Fermi energy ɛF .<br />
6π 2 n = (ɛF + u) 3/2 + (ɛF − u) 3/2 . (4.17)<br />
Note th<strong>at</strong> the Bohr magneton µ <strong>in</strong> <strong>at</strong>omic Rydberg unit is √ 2 and one <strong>at</strong>omic<br />
Rydberg unit for the magnetic <strong>in</strong>duction B corresponds to 3.324 × 10 5 T <strong>in</strong> SI.<br />
S<strong>in</strong>ce the Fermi wave vector kF of the sp<strong>in</strong>-up band is different from th<strong>at</strong><br />
of the sp<strong>in</strong>-down band with an external magnetic field, one can expect from the<br />
Friedel oscill<strong>at</strong>ions th<strong>at</strong> the electronic density for each band behaves differently<br />
even <strong>at</strong> large distances. Now, s<strong>in</strong>ce there are two different frequencies <strong>in</strong> the Friedel<br />
oscill<strong>at</strong>ions of the density, there should be p<strong>at</strong>terns of be<strong>at</strong>s 11 <strong>in</strong> the total electronic<br />
density <strong>in</strong>duced by an impurity. The Coulomb potential calcul<strong>at</strong>ed from the total<br />
11 The term, be<strong>at</strong>, is orig<strong>in</strong>ally used to describe fluctu<strong>at</strong>ions <strong>in</strong> the amplitude vari<strong>at</strong>ion<br />
caused by two sound waves of different frequency. In this work, be<strong>at</strong>s are used to<br />
emphasize the altern<strong>at</strong><strong>in</strong>g constructive and destructive <strong>in</strong>terference of two electronic<br />
density oscill<strong>at</strong>ions of different frequency.
electronic density, hence, has also p<strong>at</strong>terns of be<strong>at</strong>s as well. In th<strong>at</strong> case one can<br />
not use the method for the potential correction, which was <strong>in</strong>troduced <strong>in</strong> Sec.<br />
4.4.2 because the method takes advantage of the shape of the regular oscill<strong>at</strong>ions.<br />
However, one can avoid the be<strong>at</strong> p<strong>at</strong>terns by calcul<strong>at</strong><strong>in</strong>g the Coulomb potential<br />
separ<strong>at</strong>ely from the density of the sp<strong>in</strong>-up and sp<strong>in</strong>-down band and apply<strong>in</strong>g the<br />
method for each part of the Coulomb potential. After correct<strong>in</strong>g each Coulomb<br />
potential, one can f<strong>in</strong>d the total Coulomb potential by add<strong>in</strong>g both contributions<br />
to Coulomb potential and f<strong>in</strong>ally the effective potential for each band by add<strong>in</strong>g the<br />
exchange-correl<strong>at</strong>ion potential of each band to the total Coulomb potential. The<br />
Coulomb potential separ<strong>at</strong>ely calcul<strong>at</strong>ed from the density of each band, however,<br />
still may have irregular p<strong>at</strong>terns but not as strong as the total Coulomb potential.<br />
This is a second order effect s<strong>in</strong>ce the calcul<strong>at</strong>ions for the density are essentially<br />
based on an effective potential which has these be<strong>at</strong> p<strong>at</strong>terns. Therefore it is<br />
important to use a very large maximum radius to reduce the numerical error <strong>in</strong><br />
case th<strong>at</strong> the irregular p<strong>at</strong>tern causes a large numerical error. One often needs<br />
to exam<strong>in</strong>e the Coulomb potential and the effective potential dur<strong>in</strong>g the iter<strong>at</strong>ive<br />
loops of numerical calcul<strong>at</strong>ions and f<strong>in</strong>d the appropri<strong>at</strong>e maximum radius. Note<br />
th<strong>at</strong> the be<strong>at</strong> frequency kbe<strong>at</strong> is the difference of the two frequencies.<br />
kbe<strong>at</strong> = 2k +<br />
F<br />
− 2k−<br />
F<br />
87<br />
. (4.18)<br />
Thus one can estim<strong>at</strong>e how many be<strong>at</strong>s exist for a given range of r <strong>in</strong> the electronic<br />
density.
5. RESULTS AND DISCUSSION<br />
Two important quantities for the model used <strong>in</strong> this work are the positive<br />
charge of an impurity and the electronic background density. The Coulomb poten-<br />
tial is determ<strong>in</strong>ed by the positive charge of an impurity and the <strong>in</strong>duced electronic<br />
density while the exchange-correl<strong>at</strong>ion potential is determ<strong>in</strong>ed by the <strong>in</strong>duced elec-<br />
tronic density and the electronic background density. If the <strong>in</strong>duced electronic den-<br />
sity rema<strong>in</strong>s the same for the same impurity, 1 the exchange-correl<strong>at</strong>ion <strong>in</strong>teraction<br />
makes a difference between two different background densities, which, for a differ-<br />
ent background density, leads to a different self-consistent solution with different<br />
<strong>in</strong>duced charge densities, potentials, bound st<strong>at</strong>es, phase shifts, densities of st<strong>at</strong>es,<br />
and dielectric functions. The system still obeys the Friedel sum rule and exhibits<br />
Friedel oscill<strong>at</strong>ions. The results <strong>in</strong> this chapter are c<strong>at</strong>egorized by sp<strong>in</strong> polariza-<br />
tion. For a sp<strong>in</strong>-polarized system, the total sp<strong>in</strong>-moment will also be an important<br />
physical quantity, which may dist<strong>in</strong>guish between self-consistent solutions.<br />
1 This is not true. As the difference between background densities becomes larger, the<br />
response of an electron gas becomes more important due to the difference <strong>in</strong> Fermi<br />
energies and <strong>in</strong> Friedel oscill<strong>at</strong>ions. This effect is rel<strong>at</strong>ed to the amount of <strong>in</strong>duced<br />
charge <strong>in</strong> the vic<strong>in</strong>ity of an impurity as well as to the oscill<strong>at</strong>ions of the density <strong>at</strong> large<br />
distances.<br />
88
5.1. Non-sp<strong>in</strong>-polarized system<br />
The non-sp<strong>in</strong>-polarized system is tested ma<strong>in</strong>ly with a hydrogen impurity.<br />
The discussion <strong>in</strong> this section will proceed us<strong>in</strong>g a hydrogen impurity unless noted<br />
differently.<br />
5.1.1. Behavior of a system dur<strong>in</strong>g the numerical calcul<strong>at</strong>ions<br />
The number of electrons <strong>in</strong>duced by a hydrogen impurity should always be<br />
equal to one <strong>in</strong> order to screen the hydrogen impurity <strong>in</strong> a self-consistent manner,<br />
as mentioned <strong>in</strong> Ch. 3. Fig. 5.1 shows one example how the system evolves to<br />
s<strong>at</strong>isfy self-consistency. S<strong>in</strong>ce the <strong>in</strong>put effective potential for the first iter<strong>at</strong>ion is<br />
calcul<strong>at</strong>ed from the electronic density of a free hydrogen <strong>at</strong>om plus a background<br />
density of 0.06, the effective potential is sufficiently <strong>at</strong>tractive <strong>at</strong> the beg<strong>in</strong>n<strong>in</strong>g and<br />
thus there are two extra electrons <strong>in</strong> the bound st<strong>at</strong>es. The background density<br />
is, however, very high. S<strong>in</strong>ce the system with two electrons <strong>in</strong> the bound st<strong>at</strong>es<br />
of a hydrogen impurity can not be the self-consistent solutions for such a high<br />
background density, the system goes through a transition from hav<strong>in</strong>g a bound<br />
st<strong>at</strong>e to not hav<strong>in</strong>g a bound st<strong>at</strong>e and adds one extra electron <strong>in</strong> the conduction<br />
band. Notice <strong>in</strong> Fig. 5.1 th<strong>at</strong> the Friedel sum dur<strong>in</strong>g the transition responds to<br />
the change <strong>in</strong> the number of electrons <strong>in</strong> the bound st<strong>at</strong>es, while the conduction<br />
band cont<strong>in</strong>uously adds electrons to make one extra electron.<br />
5.1.2. Charge densities and potentials<br />
As already mentioned <strong>in</strong> Ch. 3, the Friedel oscill<strong>at</strong>ions are <strong>in</strong>dependent of<br />
the n<strong>at</strong>ure of an impurity. The <strong>in</strong>duced electronic densities <strong>at</strong> large distances<br />
89
FIGURE 5.1. Vari<strong>at</strong>ion of the number of extra electrons for a system with a<br />
hydrogen impurity dur<strong>in</strong>g the iter<strong>at</strong>ive loops of the numerical calcul<strong>at</strong>ions. The<br />
background electronic density is 0.06.<br />
90
FIGURE 5.2. The electronic density <strong>in</strong>duced by a hydrogen impurity. It is plotted<br />
as a function of 2kF r. Friedel oscill<strong>at</strong>ions are <strong>in</strong>dependent of the background<br />
density. The range of the background density is from 0.0005 to 0.06.<br />
show Friedel oscill<strong>at</strong>ions and they all have the same frequency 2kF , as illustr<strong>at</strong>ed<br />
<strong>in</strong> Fig. 5.2. The <strong>in</strong>duced electronic densities as a function of r are shown <strong>in</strong><br />
Fig. 5.3. Note th<strong>at</strong> although there is always one extra electron <strong>in</strong> the system<br />
of a hydrogen impurity, regardless of the background density, the amount of the<br />
electronic charge <strong>in</strong> the vic<strong>in</strong>ity of the impurity (r : 0 ∼ 4) is dependent upon<br />
the background density. There is rel<strong>at</strong>ively a large amount of electrons near the<br />
impurity for low background densities as a result of the effect of bound st<strong>at</strong>es<br />
and the Friedel oscill<strong>at</strong>ions <strong>in</strong> the conduction band. (Fig. 5.4 and Fig. 5.5) These<br />
phenomena are rel<strong>at</strong>ed to the behavior of the dielectric functions, different from<br />
91
FIGURE 5.3. The electronic density <strong>in</strong>duced by a hydrogen impurity.<br />
the Thomas-Fermi dielectric functions, which will be discussed <strong>in</strong> Sec. 5.1.4. The<br />
electronic density <strong>in</strong> the bound st<strong>at</strong>es becomes more extended and the maximum<br />
value of the density decreases as the background density <strong>in</strong>creases. (Fig. 5.4) The<br />
bound st<strong>at</strong>es disappear for the background density higher than about 0.03. The<br />
numerical calcul<strong>at</strong>ions for the background density between 0.02 and 0.03 are very<br />
difficult due to the gre<strong>at</strong>ly extended bound st<strong>at</strong>es. These results are not <strong>in</strong>cluded<br />
<strong>in</strong> Fig. 5.4 and Fig. 5.5 s<strong>in</strong>ce the required convergence could not be obta<strong>in</strong>ed. As<br />
discussed <strong>in</strong> Sec. 3.5, however, the transition from bound st<strong>at</strong>es to resonance st<strong>at</strong>es<br />
should not result <strong>in</strong> any non-analytic behavior of the properties of the system such<br />
as the immersion energy. The maximum value of the <strong>in</strong>duced electronic density <strong>in</strong><br />
92
FIGURE 5.4. The electronic density <strong>in</strong>duced by a hydrogen impurity <strong>in</strong> the bound<br />
st<strong>at</strong>es.<br />
the conduction band <strong>in</strong>creases <strong>in</strong> the vic<strong>in</strong>ity of the impurity as the background<br />
density <strong>in</strong>creases. (Fig. 5.5) As a result, the impurity is always screened sufficiently<br />
regardless of the existence of bound st<strong>at</strong>es.<br />
5.1.3. Phase shift, Sc<strong>at</strong>ter<strong>in</strong>g length, and Density of <strong>in</strong>duced st<strong>at</strong>es<br />
Phase shifts for l = 0 <strong>at</strong> zero energy have the value of π for low background<br />
densities, due to the presence of a bound st<strong>at</strong>e of a hydrogen impurity. The result<br />
is shown <strong>in</strong> Fig. 5.6. Note th<strong>at</strong> there are two electrons <strong>in</strong> the bound st<strong>at</strong>e due<br />
to the sp<strong>in</strong> degeneracy. The vari<strong>at</strong>ions of the phase shifts of p-waves <strong>in</strong> the case<br />
93
FIGURE 5.5. The electronic density <strong>in</strong>duced by a hydrogen impurity <strong>in</strong> the conduction<br />
band.<br />
of a hydrogen impurity are rel<strong>at</strong>ively very small compared to those of s-waves 2<br />
and thus the lead<strong>in</strong>g contribution to the change <strong>in</strong> the density of st<strong>at</strong>es arises<br />
from the phase shifts for l = 0. The slope of the phase shift (or the sc<strong>at</strong>ter<strong>in</strong>g<br />
length) for low background densities is neg<strong>at</strong>ive, from which one sees th<strong>at</strong> the<br />
2 This is only true if there are two electrons <strong>in</strong> the bound st<strong>at</strong>es. If the number of elec-<br />
trons is smaller than two, phase shifts of p-waves also become important and contribute<br />
significantly to the change <strong>in</strong> the number of electrons <strong>in</strong> the conduction band. See Sec.<br />
5.1.6.<br />
94
FIGURE 5.6. Phase shifts (l = 0) for a hydrogen impurity. Plotted for low<br />
background densities only.<br />
potential is <strong>at</strong>tractive enough to have a first bound st<strong>at</strong>e and, consider<strong>in</strong>g the<br />
density of st<strong>at</strong>es, the conduction band loses one electron to keep the number of<br />
total extra electrons one. The magnitude of the slope of the phase shifts <strong>in</strong> general<br />
becomes smaller <strong>in</strong> Fig. 5.6 as the background density decreases, which one can<br />
expect consider<strong>in</strong>g the rel<strong>at</strong>ionship between the strength of the potential and the<br />
sc<strong>at</strong>ter<strong>in</strong>g length. 3 However, as the background density decreases more, the phase<br />
3 One may expect th<strong>at</strong> the potentials become more <strong>at</strong>tractive as the background density<br />
decreases, which means th<strong>at</strong> the sc<strong>at</strong>ter<strong>in</strong>g length decreases <strong>in</strong> magnitude. [31]<br />
95
FIGURE 5.7. Inverse of sc<strong>at</strong>ter<strong>in</strong>g length versus background density for a hydrogen<br />
impurity.<br />
shifts near the conduction band maximum tend to have a larger neg<strong>at</strong>ive slope than<br />
those of the higher background densities. At extremely low background densities<br />
such as 0.0005, even the slope near the conduction band m<strong>in</strong>imum becomes larger.<br />
(See Fig. 5.7.) The reason for this is th<strong>at</strong> a smaller neg<strong>at</strong>ive slope <strong>in</strong> the phase<br />
shifts can not yield a correct neg<strong>at</strong>ive excess charge <strong>at</strong> such a low background<br />
density to s<strong>at</strong>isfy the Friedel sum rule as the conduction band becomes narrower. 4<br />
4 Moreover, as the background density decreases, the contribution of s-waves to the<br />
number of <strong>in</strong>duced electrons becomes larger, and thus the phase shifts of s-waves <strong>at</strong> the<br />
Fermi energy should respond accord<strong>in</strong>gly.<br />
96
FIGURE 5.8. Phase shifts for a hydrogen impurity for two different background<br />
densities (0.0025 and 0.06).<br />
As a result, the system behaves such th<strong>at</strong> the potentials are less <strong>at</strong>tractive <strong>at</strong> the<br />
extremely low background densities. For high background densities (≥ 0.03) for a<br />
hydrogen impurity, s<strong>in</strong>ce there are no electrons <strong>in</strong> the bound st<strong>at</strong>es, there should<br />
be one extra electron <strong>in</strong> the conduction band, for which the contribution of s-waves<br />
to a virtual bound st<strong>at</strong>e resonance is significantly larger than the higher angular<br />
momentum st<strong>at</strong>es. (Fig. 5.8) The sc<strong>at</strong>ter<strong>in</strong>g lengths correspond<strong>in</strong>g to the phase<br />
shifts <strong>in</strong> Fig. 5.9 are shown <strong>in</strong> Fig. 5.7. The transition from a bound st<strong>at</strong>e to a<br />
resonance st<strong>at</strong>e occurs <strong>at</strong> about 0.03 of the background density, close to when the<br />
sign of the sc<strong>at</strong>ter<strong>in</strong>g length changes. The <strong>in</strong>verse of the sc<strong>at</strong>ter<strong>in</strong>g length <strong>in</strong>creases<br />
as the background density <strong>in</strong>creases from 0.0015. The magnitude of the sc<strong>at</strong>ter<strong>in</strong>g<br />
length, however, rapidly <strong>in</strong>creases as the background density decreases from 0.0015<br />
and becomes large <strong>at</strong> extremely low background densities (≤ 0.001), which cre<strong>at</strong>es<br />
a m<strong>in</strong>imum po<strong>in</strong>t <strong>at</strong> about 0.0015 <strong>in</strong> Fig. 5.7. This behavior is already discussed<br />
with the phase shifts above.<br />
97
FIGURE 5.9. Phase shifts (l = 0) for a hydrogen impurity.<br />
FIGURE 5.10. The density of st<strong>at</strong>es for a hydrogen impurity for two different<br />
background densities (0.0025 and 0.06).<br />
98
Typical examples for the density of st<strong>at</strong>es are shown <strong>in</strong> Fig. 5.10. At the<br />
background density of 0.0025, there are two electrons <strong>in</strong> the bound st<strong>at</strong>e, and the<br />
lead<strong>in</strong>g contribution to the conduction density of st<strong>at</strong>es comes from s-st<strong>at</strong>es, which<br />
show a neg<strong>at</strong>ive sc<strong>at</strong>ter<strong>in</strong>g length and a sharp neg<strong>at</strong>ive peak <strong>at</strong> the conduction<br />
band m<strong>in</strong>imum <strong>in</strong> the density of st<strong>at</strong>es. The conduction band loses one electron,<br />
which removes the excess charge of the bound st<strong>at</strong>e electrons. At the background<br />
density of 0.06, there are no electrons <strong>in</strong> the bound st<strong>at</strong>es. The effective potential<br />
is, however, <strong>at</strong>tractive enough to <strong>in</strong>duce an extra electron <strong>in</strong> the conduction band,<br />
which appears as a narrow peak <strong>in</strong> the conduction density of st<strong>at</strong>es due to the<br />
rapid <strong>in</strong>crease of the phase shift of l = 0 <strong>in</strong> the region of the conduction band<br />
m<strong>in</strong>imum. Therefore one sees th<strong>at</strong> the Friedel sum rule is always s<strong>at</strong>isfied and can<br />
be achieved <strong>at</strong> self-consistency dur<strong>in</strong>g the numerical calcul<strong>at</strong>ions. The phase shifts<br />
for a lithium impurity are shown <strong>in</strong> Fig. 5.11. At the background density of 0.0005,<br />
the electron configur<strong>at</strong>ion of a lithium impurity is 1s 2 2s 2 and the phase shift of<br />
the zero energy must be 2π (third branch 5 of tan δ0) by Lev<strong>in</strong>son’s theorem. The<br />
phase shift <strong>at</strong> 0.0005 shows a large neg<strong>at</strong>ive slope, which gives rise to the neg<strong>at</strong>ive<br />
density of st<strong>at</strong>es <strong>in</strong> the conduction band to remove the excess charge due to the<br />
bound st<strong>at</strong>es. As the background density <strong>in</strong>creases, the effective potential becomes<br />
less <strong>at</strong>tractive and the system loses a bound st<strong>at</strong>e (second branch of tanδ0) and<br />
the sc<strong>at</strong>ter<strong>in</strong>g length becomes positive. (phase shifts <strong>at</strong> 0.0025 <strong>in</strong> Fig. 5.11) If the<br />
background density further <strong>in</strong>creases, the sc<strong>at</strong>ter<strong>in</strong>g length decreases (phase shifts<br />
<strong>at</strong> 0.03 <strong>in</strong> Fig. 5.11) and becomes neg<strong>at</strong>ive. At the higher background density the<br />
5 See (3.26) and [31].<br />
99
FIGURE 5.11. Phase shifts of l = 0 and l = 1 for a lithium impurity.<br />
100
same cycle of the phase shifts repe<strong>at</strong>s, after loos<strong>in</strong>g one more bound st<strong>at</strong>e. (first<br />
branch of tanδ0)<br />
5.1.4. Dielectric constant and compressibility<br />
101<br />
The dielectric constants can be easily calcul<strong>at</strong>ed us<strong>in</strong>g the def<strong>in</strong>ition <strong>in</strong> (3.55)<br />
and the Fourier transform.<br />
V ext<br />
q<br />
ɛ(w, q) =<br />
V eff<br />
q<br />
<br />
−Ze dr exp(−iqr)<br />
=<br />
2<br />
r<br />
dr exp(−iqr)V (r)<br />
=<br />
−e 2 Z<br />
q dr V (r)r s<strong>in</strong>(qr)<br />
(5.1)<br />
(5.2)<br />
. (5.3)<br />
The Thomas-Fermi dielectric constant, already <strong>in</strong>troduced <strong>in</strong> Sec. 3.4, is given by<br />
where q 2 T F = 4e2 mkF<br />
2 π<br />
ɛT F (0, q) = q2 + q 2 T F<br />
q 2 , (5.4)<br />
. The Thomas-Fermi dielectric constants <strong>in</strong> Fig. 5.12 show a<br />
simple behavior which gives a screen<strong>in</strong>g length 6 proportional to n −1/3 . One can ex-<br />
pect from the Thomas-Fermi dielectric constants th<strong>at</strong> a higher background density<br />
has a shorter screen<strong>in</strong>g length. The numerical results of the dielectric constants<br />
for a hydrogen impurity and the Thomas-Fermi dielectric constants both approach<br />
the value of one as q → ∞. The density functional calcul<strong>at</strong>ions, however, show<br />
quite different results for the dielectric constants, especially <strong>at</strong> small q values. Note<br />
th<strong>at</strong> the presence of electrons <strong>in</strong> the bound st<strong>at</strong>es and the long wave length of the<br />
6 The screen<strong>in</strong>g length for the Thomas-Fermi dielectric constants can be easily obta<strong>in</strong>ed<br />
from the Yukawa potential, l = 1/qT F . See (3.67).
102<br />
FIGURE 5.12. Dielectric constants which are numerically calcul<strong>at</strong>ed for a hydrogen<br />
impurity. The Thomas-Fermi dielectric constants are also plotted for comparison.
103<br />
FIGURE 5.13. Compressibility calcul<strong>at</strong>ed numerically for a hydrogen impurity.<br />
A dotted l<strong>in</strong>e which is taken from [6] is for the result predicted by theories such<br />
as Hubbard, VS, and SS. Both l<strong>in</strong>es are go<strong>in</strong>g through zero around rs = 5.0. The<br />
electronic density which corresponds to rs of the bottom x-axis is given <strong>in</strong> the top<br />
x-axis.<br />
Friedel oscill<strong>at</strong>ions result <strong>in</strong> a larger amount of screen<strong>in</strong>g charge near the impu-<br />
rity for low background densities than for high background densities, as already<br />
mentioned <strong>in</strong> Sec. 5.1.2. One sees th<strong>at</strong> dielectric constants may have very large<br />
neg<strong>at</strong>ive or positive values for a certa<strong>in</strong> q while the <strong>in</strong>tegral term <strong>in</strong> (5.3) <strong>in</strong>creases<br />
from neg<strong>at</strong>ive to positive values near zero. Actually, it turns out th<strong>at</strong> the <strong>in</strong>tegral<br />
term <strong>in</strong> (5.3) changes its sign from neg<strong>at</strong>ive to positive as the low background<br />
density (n 0.00191 or rs 5.0) decreases further. At extremely low background<br />
densities (≤ 0.0015), therefore, the dielectric constants for q → 0 are neg<strong>at</strong>ive, as
FIGURE 5.14. Comparison of the compressibility calcul<strong>at</strong>ed numerically for a<br />
hydrogen impurity with sc<strong>at</strong>ter<strong>in</strong>g length.<br />
illustr<strong>at</strong>ed by Fig. 5.12. This result can be verified by discuss<strong>in</strong>g compressibilities.<br />
The compressibility can be obta<strong>in</strong>ed <strong>in</strong> terms of the dielectric function by [13]<br />
lim<br />
q→0 ɛ(q, 0) = q2 + q2 T F<br />
q2 K<br />
Kf<br />
104<br />
, (5.5)<br />
where qT F = 6πne 2 /ɛF is the Thomas-Fermi wave number and the quantity Kf =<br />
3(3π 2 ) −2/3 n −5/3 [1/e 2 a0] is the isothermal compressibility of the non-<strong>in</strong>teract<strong>in</strong>g<br />
electron gas <strong>at</strong> zero temper<strong>at</strong>ure. The compressibility is already obta<strong>in</strong>ed by sev-<br />
eral theories. 7 The RPA gives the constant value 1.0 for Kf/K while many other<br />
7 Readers should refer to [18] for more details.
theories such as Hubbard, S<strong>in</strong>gwi-Sjölander, and Vashishta-S<strong>in</strong>gwi give results for<br />
Kf/K, which all go through 0 around rs = 5.0. [18] The density functional cal-<br />
cul<strong>at</strong>ions for the compressibility give rise to nearly the same result for Kf/K as<br />
those theories for low background densities but to much larger values for high<br />
background densities, which results <strong>in</strong> the curved l<strong>in</strong>e of a hyperbolic type <strong>in</strong> Fig.<br />
5.13. One sees from (5.5) and Fig. 5.13 th<strong>at</strong> the dielectric constant for the high<br />
background density as q → 0 has a smaller value than expected by theories 8 while<br />
the dielectric constant for the low background density has a larger value. 9 In other<br />
words, if the screen<strong>in</strong>g length can be def<strong>in</strong>ed for any potential 10 , it can be st<strong>at</strong>ed<br />
th<strong>at</strong> the screen<strong>in</strong>g length for the high background density has a larger value than<br />
expected by analytical theories while the screen<strong>in</strong>g length for a low background<br />
density has a smaller value.<br />
105<br />
As shown <strong>in</strong> Fig. 5.14, one sees th<strong>at</strong> the neg<strong>at</strong>ive m<strong>in</strong>imum (<strong>at</strong> rs 5.0)<br />
of a value of the sc<strong>at</strong>ter<strong>in</strong>g length corresponds to <strong>in</strong>f<strong>in</strong>ity of the compressibility<br />
and <strong>in</strong>f<strong>in</strong>ity (<strong>at</strong> rs 2.0) of a value of the sc<strong>at</strong>ter<strong>in</strong>g length corresponds to the<br />
compressibility close to Kf. Note th<strong>at</strong> <strong>in</strong>f<strong>in</strong>ity of a sc<strong>at</strong>ter<strong>in</strong>g length <strong>in</strong>dic<strong>at</strong>es the<br />
transition from a bound st<strong>at</strong>e to a resonance st<strong>at</strong>e. One may th<strong>in</strong>k th<strong>at</strong> the most<br />
<strong>at</strong>tractive potential corresponds to the maximum of the compressibility, which<br />
8 See the dotted l<strong>in</strong>e <strong>in</strong> Fig. 5.13. As rs → 0, the expected dielectric constant is the<br />
Thomas-Fermi dielectric constant.<br />
9 Note th<strong>at</strong>, <strong>in</strong> Fig. 5.13, the dielectric constants calcul<strong>at</strong>ed by theories are always larger<br />
than or equal to the Thomas-Fermi dielectric constants.<br />
10 One can try to f<strong>in</strong>d the screen<strong>in</strong>g length for a particular potential by compar<strong>in</strong>g with<br />
the Thomas-Fermi dielectric constant. l = 1/qT F = ((ɛ(q) − 1)q 2 ) −1/2 as q → 0
106<br />
FIGURE 5.15. Immersion energies for several different impurities. 1.0 <strong>in</strong> the<br />
legend <strong>in</strong>dic<strong>at</strong>es th<strong>at</strong> the number of electrons <strong>in</strong> the bound st<strong>at</strong>es is limited to 1.0.<br />
465 po<strong>in</strong>ts for the logarithmic r-space and 0.81205 × 10 −5 for the first value of<br />
r-mesh are used <strong>in</strong> these calcul<strong>at</strong>ions.<br />
may be rel<strong>at</strong>ed to the localiz<strong>at</strong>ion of electrons 11 . The rel<strong>at</strong>ionship between the<br />
sc<strong>at</strong>ter<strong>in</strong>g length and the compressibility, however, needs further <strong>in</strong>vestig<strong>at</strong>ion.<br />
5.1.5. Immersion energy<br />
Immersion energies for H, Li, and C are shown <strong>in</strong> Fig. 5.15. A m<strong>in</strong>imum <strong>in</strong><br />
the immersion energy exists for H and C, while calcul<strong>at</strong>ions for Li can not verify<br />
11 Wigner l<strong>at</strong>tice [18]
107<br />
FIGURE 5.16. Vari<strong>at</strong>ion <strong>in</strong> the k<strong>in</strong>etic, Coulomb, and exchange-correl<strong>at</strong>ion energies<br />
vs the background density for a hydrogen impurity. 465 po<strong>in</strong>ts for the<br />
logarithmic r-space and 0.81205 × 10 −5 for the first vale of r-mesh are used <strong>in</strong><br />
these calcul<strong>at</strong>ions.<br />
the existence of a m<strong>in</strong>imum immersion energy. 12 The background densities <strong>at</strong><br />
which the m<strong>in</strong>imum immersion energies exist are 0.0024 for H and 0.0033 for C.<br />
The immersion energy after the m<strong>in</strong>imum value <strong>in</strong>creases almost l<strong>in</strong>early as the<br />
background density <strong>in</strong>creases, which is due to the repulsion rel<strong>at</strong>ed to the <strong>in</strong>crease<br />
12 To verify the existence of the m<strong>in</strong>imum immersion energy for Li, it is necessary to<br />
perform the calcul<strong>at</strong>ions for very low background densities as shown <strong>in</strong> Fig. 5.15, which<br />
is, however, very difficult due to the long range wave length of the Friedel oscill<strong>at</strong>ions as<br />
mentioned <strong>in</strong> Ch. 4.
<strong>in</strong> the k<strong>in</strong>etic energy. [5] The Coulomb and exchange-correl<strong>at</strong>ion energies actually<br />
become lower as the background density <strong>in</strong>creases. The r<strong>at</strong>e of the <strong>in</strong>crease <strong>in</strong> the<br />
k<strong>in</strong>etic energy is, however, large enough to compens<strong>at</strong>e this decrease, as illustr<strong>at</strong>ed<br />
<strong>in</strong> Fig. 5.16.<br />
108<br />
The zero background density limit <strong>in</strong> this <strong>in</strong>f<strong>in</strong>ite system may be different<br />
from a free <strong>at</strong>om without an electron gas. S<strong>in</strong>ce the system, <strong>at</strong> constant chem-<br />
ical potential, is extended to <strong>in</strong>f<strong>in</strong>ity, the impurity will be screened, no m<strong>at</strong>ter<br />
how small the background density, (but not zero) and, for this non-sp<strong>in</strong>-polarized<br />
<strong>in</strong>f<strong>in</strong>ite system, the impurity is not allowed to have partially occupied orbitals. 13<br />
Calcul<strong>at</strong>ions show th<strong>at</strong> a hydrogen impurity has no electrons <strong>in</strong> the bound st<strong>at</strong>es<br />
for high background densities( 0.03) and two electrons <strong>in</strong> the bound st<strong>at</strong>es for<br />
low background densities but no electrons aga<strong>in</strong> for extremely low background<br />
densities( 0.0001), which can be seen from the fact th<strong>at</strong> a free hydrogen <strong>at</strong>om<br />
can not have two electrons <strong>in</strong> the bound st<strong>at</strong>es <strong>in</strong> LDA. The <strong>in</strong>terest<strong>in</strong>g result<br />
is th<strong>at</strong> the immersion energy for a hydrogen impurity shows a l<strong>in</strong>ear behavior as<br />
illustr<strong>at</strong>ed <strong>in</strong> Fig. 5.15 when the hydrogen is forced to have only one electron <strong>in</strong><br />
the bound st<strong>at</strong>es regardless of the background density. Immersion energy <strong>in</strong> this<br />
case approaches zero as the background density goes to zero, which one can easily<br />
expect from the fact th<strong>at</strong> a free hydrogen <strong>at</strong>om has only one electron. A helium<br />
impurity which has two bound st<strong>at</strong>e electrons <strong>in</strong> a homogeneous electron gas shows<br />
the same l<strong>in</strong>ear behavior <strong>in</strong> Fig. 5.15.
109<br />
FIGURE 5.17. Immersion energies for the different number of electrons <strong>in</strong> the<br />
bound st<strong>at</strong>es. 465 po<strong>in</strong>ts for the logarithmic r-space and 0.81205 × 10 −5 for the<br />
first value of r-mesh are used <strong>in</strong> these calcul<strong>at</strong>ions.<br />
5.1.6. Excited system<br />
One can easily calcul<strong>at</strong>e the immersion energies of excited systems and obta<strong>in</strong><br />
the excit<strong>at</strong>ion energies by vary<strong>in</strong>g the number of electrons <strong>in</strong> the bound st<strong>at</strong>es as<br />
mentioned <strong>in</strong> Sec. 2.2.5. The bound st<strong>at</strong>e energy decreases as the number of<br />
electrons <strong>in</strong> an 1s st<strong>at</strong>e decreases while the immersion energy <strong>in</strong>creases. (See Fig.<br />
5.17.) This can be verified by Janak’s theorem:<br />
∂Eimm<br />
∂ni<br />
= ɛi(ni) , (5.6)<br />
13 For <strong>in</strong>stance, only two or zero electrons are allowed <strong>in</strong> an 1s st<strong>at</strong>e.
110<br />
FIGURE 5.18. Vari<strong>at</strong>ion <strong>in</strong> the bound st<strong>at</strong>e energy for the different number of<br />
electrons <strong>in</strong> the bound st<strong>at</strong>e. The second plot is obta<strong>in</strong>ed us<strong>in</strong>g (5.7) and slopes<br />
between two adjacent immersion energies <strong>in</strong> Fig. 5.17. 465 po<strong>in</strong>ts for the logarithmic<br />
r-space and 0.81205 × 10 −5 for the first value of r-mesh are used <strong>in</strong> these<br />
calcul<strong>at</strong>ions.
where ni denotes the occup<strong>at</strong>ion number of the level i. However, one needs to<br />
consider the change <strong>in</strong> the number of electrons <strong>at</strong> the Fermi energy as well as the<br />
bound st<strong>at</strong>es <strong>in</strong> order to compare the bound st<strong>at</strong>e energy with the vari<strong>at</strong>ion of<br />
the immersion energy us<strong>in</strong>g Janak’s theorem. Janak’s theorem (5.6) should be<br />
modified as<br />
∂Eimm<br />
∂ni<br />
111<br />
= −ɛF + ɛi(ni) , (5.7)<br />
where ɛF is the Fermi energy. Fig. 5.18 shows the result of (5.7) with the bound<br />
st<strong>at</strong>e energy plot. Bound st<strong>at</strong>e energies obta<strong>in</strong>ed from (5.7) are <strong>in</strong> error with<br />
respect to actually calcul<strong>at</strong>ed bound st<strong>at</strong>e energies by an energy difference smaller<br />
than 0.005 Rydbergs. 14 Bound st<strong>at</strong>e energies of a free hydrogen <strong>at</strong>om are <strong>in</strong>cluded<br />
<strong>in</strong> Fig. 5.18. One should, however, recall th<strong>at</strong> the zero background density limit<br />
may be different from a free <strong>at</strong>om as discussed <strong>in</strong> Sec. 5.1.5. One sees from the<br />
Friedel sum rule th<strong>at</strong> the screen<strong>in</strong>g <strong>in</strong> the model used <strong>in</strong> this work will always occur<br />
as long as there is a background density. The extra electrons <strong>in</strong> the conduction<br />
band will respond to the excess core charge, which makes the zero background<br />
density limit different from a free <strong>at</strong>om. A hydrogen impurity which has only<br />
one electron <strong>in</strong> a homogeneous electronic gas is, however, expected to reach a free<br />
hydrogen <strong>at</strong>om as the background density goes to zero. S<strong>in</strong>ce the extra electron<br />
<strong>in</strong> the conduction band <strong>in</strong> this case must be zero, as the background density goes<br />
to zero, the only response of the conduction band will be th<strong>at</strong> the wave length<br />
of the Friedel oscill<strong>at</strong>ion gets longer and it is found th<strong>at</strong> the amplitude of the<br />
Friedel oscill<strong>at</strong>ion near an impurity gets smaller. This st<strong>at</strong>ement is, however, not<br />
14 These differences are ma<strong>in</strong>ly due to <strong>in</strong>accur<strong>at</strong>e numerical deriv<strong>at</strong>ives.
verified numerically for a background density lower than 0.001 due to the numerical<br />
problems discussed <strong>in</strong> Ch. 4.<br />
112
113<br />
FIGURE 5.19. Comparison of immersion energies for a carbon impurity between a<br />
non-sp<strong>in</strong>-polarized system and a sp<strong>in</strong>-polarized system. The same reference energy<br />
for a free carbon <strong>at</strong>om is used for both calcul<strong>at</strong>ions.<br />
5.2. Sp<strong>in</strong> polarized system<br />
5.2.1. Zero external magnetic field<br />
5.2.1.1. Carbon impurity <strong>at</strong> low background densities<br />
A carbon impurity is ma<strong>in</strong>ly used for a sp<strong>in</strong>-polarized system <strong>in</strong> this work<br />
and <strong>in</strong>terest<strong>in</strong>g results are obta<strong>in</strong>ed <strong>at</strong> low background densities. Fig. 5.19 shows<br />
immersion energies for a sp<strong>in</strong>-polarized system and a non-sp<strong>in</strong>-polarized system of a<br />
carbon impurity. Overall, immersion energies of a sp<strong>in</strong>-polarized system are slightly<br />
lower than those of a non-sp<strong>in</strong>-polarized system. Note th<strong>at</strong> these calcul<strong>at</strong>ions are<br />
based on the same energy of a free <strong>at</strong>om, th<strong>at</strong> is, the same reference and thus only
FIGURE 5.20. Plot of ∆N vs a background density where ∆N is the difference<br />
<strong>in</strong> the number of electrons between sp<strong>in</strong>-up and sp<strong>in</strong>-down electrons <strong>in</strong> a system.<br />
rel<strong>at</strong>ive energies between two systems are mean<strong>in</strong>gful. One important difference<br />
between the two systems is the response of the conduction band to the number of<br />
extra electrons <strong>in</strong> the bound st<strong>at</strong>es <strong>at</strong> low background densities(≤ 0.005). For both<br />
systems, there are only four electrons <strong>in</strong> the bound st<strong>at</strong>es with the configur<strong>at</strong>ion<br />
1s 2 2s 2 , for a background density higher than 0.002. Note th<strong>at</strong> there should be<br />
two electrons <strong>in</strong> the conduction band due to the Friedel sum rule if there are<br />
four electrons <strong>in</strong> the bound st<strong>at</strong>es of a carbon impurity, which is true for both<br />
systems. However, a sp<strong>in</strong>-polarized system shows a non-zero magnetiz<strong>at</strong>ion for a<br />
background density lower than 0.005, which makes a difference <strong>in</strong> the immersion<br />
114<br />
energy compared to the non-sp<strong>in</strong>-polarized system. One sees th<strong>at</strong>, <strong>in</strong> Fig. 5.20,
FIGURE 5.21. The electronic density and the sp<strong>in</strong> density <strong>in</strong>duced by a carbon<br />
impurity.<br />
115
FIGURE 5.22. The electronic density (sp<strong>in</strong>-up and sp<strong>in</strong>-down) <strong>in</strong>duced by a carbon<br />
impurity <strong>in</strong> the conduction band.<br />
116
FIGURE 5.23. The density of st<strong>at</strong>es <strong>in</strong>duced by a carbon impurity.<br />
117
∆N (which is the difference <strong>in</strong> the number of electrons between sp<strong>in</strong>-up and sp<strong>in</strong>-<br />
down electrons <strong>in</strong> a system) becomes larger as the background density decreases<br />
and thus a sp<strong>in</strong>-polarized system favors a sp<strong>in</strong>-polarized solution for a background<br />
density lower than 0.005. Fig. 5.21 shows th<strong>at</strong> the sp<strong>in</strong> density becomes larger<br />
<strong>in</strong> the vic<strong>in</strong>ity of an impurity as the background density decreases while the total<br />
density <strong>in</strong>duced by an impurity rema<strong>in</strong>s almost the same. This sp<strong>in</strong> density also<br />
can be seen <strong>in</strong> Fig. 5.22. The maximum peak of the density <strong>in</strong>duced by an impurity<br />
<strong>in</strong> the conduction band of sp<strong>in</strong>-up electrons <strong>in</strong>creases <strong>in</strong> the vic<strong>in</strong>ity of an impurity<br />
between 0.002 and 0.005 background densities and decreases <strong>in</strong> the conduction<br />
band of sp<strong>in</strong>-down electrons. The density of st<strong>at</strong>es <strong>in</strong> the conduction bands of<br />
sp<strong>in</strong>-up and sp<strong>in</strong>-down electrons <strong>in</strong> Fig. 5.23 clearly show th<strong>at</strong> there is a resonance<br />
<strong>in</strong> the conduction band of sp<strong>in</strong>-up electrons between 0.002 and 0.005 background<br />
densities while there is none <strong>in</strong> the conduction band of sp<strong>in</strong>-down electrons. One<br />
other difference between a sp<strong>in</strong>-polarized system and a non-sp<strong>in</strong>-polarized system<br />
is th<strong>at</strong> a sp<strong>in</strong>-polarized system has a 2p bound st<strong>at</strong>e <strong>in</strong> the sp<strong>in</strong>-up band for a<br />
background density smaller than 0.002 and hence there are seven bound st<strong>at</strong>e<br />
electrons <strong>in</strong> a system while a non-sp<strong>in</strong>-polarized system can not have six electrons<br />
<strong>in</strong> a 2p bound st<strong>at</strong>e. 15 One sees now <strong>in</strong> Fig. 5.22 th<strong>at</strong>, due to a 2p bound st<strong>at</strong>e,<br />
the <strong>in</strong>duced density <strong>in</strong> the conduction band of sp<strong>in</strong>-up electrons sharply drops<br />
<strong>in</strong> the vic<strong>in</strong>ity of an impurity as the background density decreases from 0.002<br />
15 A non-sp<strong>in</strong>-polarized system is not tested for a background density lower than 0.0005.<br />
It is, however, found by consider<strong>in</strong>g the r<strong>at</strong>e of the vari<strong>at</strong>ion of the bound st<strong>at</strong>e energy<br />
as a background density decreases th<strong>at</strong> a non-sp<strong>in</strong>-polarized system does not have a 2p<br />
bound st<strong>at</strong>e <strong>at</strong> any background density.<br />
118
to 0.001. Note th<strong>at</strong> the <strong>in</strong>duced density <strong>in</strong> the conduction band of sp<strong>in</strong>-down<br />
electrons <strong>at</strong> 0.0004 background density has a rel<strong>at</strong>ively large hole (radius around<br />
10) compared to th<strong>at</strong> <strong>at</strong> 0.0005 background density, while there is a small difference<br />
of the <strong>in</strong>duced density <strong>in</strong> the conduction band of sp<strong>in</strong>-up electrons between 0.0004<br />
and 0.0005 background densities. One can expect th<strong>at</strong> ∆N <strong>in</strong> Fig. 5.20 is even<br />
larger than 3.0 <strong>at</strong> 0.0004 background density. 16 One possible explan<strong>at</strong>ion for this<br />
is th<strong>at</strong>, due to the strong effect of the exchange-correl<strong>at</strong>ion <strong>in</strong>teraction, a system<br />
may get a lower immersion energy by mak<strong>in</strong>g the magnetiz<strong>at</strong>ion larger <strong>at</strong> these<br />
extremely low background densities. (See Fig. 2.7, Fig. 2.8, and Fig. 2.9.)<br />
5.2.1.2. Electrical Resistivity<br />
The general idea is taken from [15] and [32].<br />
With<strong>in</strong> the scheme of density functional theory, the s<strong>in</strong>gle particle wave func-<br />
tions are used to calcul<strong>at</strong>e the ground st<strong>at</strong>e densities. The use of these wave func-<br />
tions for other purposes is not formally justified. However, the phase shifts (which<br />
are rel<strong>at</strong>ed to the wave functions) give reasonable agreement with experiment as<br />
discussed <strong>in</strong> [32].<br />
119<br />
The contribution of sc<strong>at</strong>ter<strong>in</strong>g centers to the electrical resistivity can be de-<br />
rived <strong>in</strong> terms of the phase shifts by<br />
∆ρ = 4πni<br />
e 2 kF Zh<br />
∞<br />
(l + 1) s<strong>in</strong> 2 [δl(ɛF ) − δl+1(ɛF )] , (5.8)<br />
l=0<br />
16 At 0.0004 background density, it is difficult to obta<strong>in</strong> a self-consistent solution with<br />
the required convergence due to the strong tendency toward a sp<strong>in</strong>-polarized system.
120<br />
FIGURE 5.24. The values of Q versus Z for the background density of 0.01. The<br />
calcul<strong>at</strong>ions are performed only for Z = 1, 3, 6. Note th<strong>at</strong>, <strong>at</strong> this background density,<br />
there is no difference between sp<strong>in</strong>-polarized and non-sp<strong>in</strong>-polarized systems<br />
for these impurities.<br />
where ni is the impurity concentr<strong>at</strong>ion and Zh is the number of valence electrons<br />
per host ion. 17 (See [32] and [15].) A useful expression is<br />
Q = 3<br />
∞<br />
(l + 1) s<strong>in</strong> 2 [δl(ɛF ) − δl+1(ɛF )] , (5.9)<br />
kF r 3 s<br />
l=0<br />
where rs is the Wigner-Seitz radius. The electronic stopp<strong>in</strong>g power for an ion<br />
slowly mov<strong>in</strong>g through the electron gas can be obta<strong>in</strong>ed from this equ<strong>at</strong>ion by<br />
multiply<strong>in</strong>g Q by vI, where vI is the velocity of an ion. Fig. 5.24 shows the Q<br />
values for H, Li, and C. The result agrees with Ref. [32]. The one important fe<strong>at</strong>ure<br />
17 CGS units are used <strong>in</strong> this section.
of such results is the oscill<strong>at</strong>ions of Q values as a function of Z. 18 These oscill<strong>at</strong>ions<br />
were observed experimentally <strong>in</strong> impurity residual resistivity measurements and <strong>in</strong><br />
the stopp<strong>in</strong>g power d<strong>at</strong>a for well-channeled slow ions. [33] The reader should refer<br />
to [32] for more details.<br />
5.2.1.3. Excited system<br />
121<br />
As discussed <strong>in</strong> Sec. 5.2.1.1, a sp<strong>in</strong>-polarized system with a carbon impurity<br />
has a very strong tendency toward a large sp<strong>in</strong>-moment solution(parallel to sp<strong>in</strong>-<br />
up) <strong>at</strong> low background densities(< 0.005). (See Fig. 5.20.) However, a strong<br />
tendency toward an anti-sp<strong>in</strong>-moment solution(parallel to sp<strong>in</strong>-down) <strong>in</strong> the con-<br />
duction band has been observed for excited systems as well. Fig. 5.25 shows the<br />
vari<strong>at</strong>ion of immersion energies and ∆N as a function of the number of sp<strong>in</strong>-up<br />
electrons <strong>in</strong> a 2p bound st<strong>at</strong>e. 19 The small vari<strong>at</strong>ion <strong>in</strong> the slope of the immersion<br />
energies <strong>in</strong> Fig. 5.25 can be verified with the bound st<strong>at</strong>e energies (Fig. 5.26) of a<br />
2p st<strong>at</strong>e us<strong>in</strong>g the Janak’s theorem as <strong>in</strong> Sec. 5.1.6. <strong>Electron</strong>s <strong>in</strong> the conduction<br />
band behave <strong>in</strong> such a way th<strong>at</strong> the difference <strong>in</strong> the total <strong>in</strong>duced density due to<br />
the change <strong>in</strong> a 2p bound st<strong>at</strong>e is as small as possible as illustr<strong>at</strong>ed <strong>in</strong> Fig. 5.27 ∼<br />
Fig. 5.38. However, one sees immedi<strong>at</strong>ely <strong>in</strong> Fig. 5.25 th<strong>at</strong> decreas<strong>in</strong>g the number<br />
of sp<strong>in</strong>-up electrons <strong>in</strong> a 2p bound st<strong>at</strong>e results <strong>in</strong> reduc<strong>in</strong>g the sp<strong>in</strong>-moment, th<strong>at</strong><br />
is, <strong>in</strong> a large anti-sp<strong>in</strong>-moment <strong>in</strong> the conduction band.<br />
18 In order to observe the oscill<strong>at</strong>ions of Q values, one needs to obta<strong>in</strong> the results for Z<br />
<strong>at</strong> least up to Z = 20. See [32].<br />
19 Note th<strong>at</strong> there is no 2p bound st<strong>at</strong>e for sp<strong>in</strong>-down electrons.
122<br />
FIGURE 5.25. Immersion energies and ∆N for a carbon impurity vs the number<br />
of sp<strong>in</strong>-up electrons <strong>in</strong> a 2p bound st<strong>at</strong>e. ∆N is the difference <strong>in</strong> the number of<br />
electrons between sp<strong>in</strong>-up and sp<strong>in</strong>-down electrons <strong>in</strong> a system.
123<br />
FIGURE 5.26. Bound st<strong>at</strong>e energies vs the number of sp<strong>in</strong>-up electrons <strong>in</strong> a 2p<br />
bound st<strong>at</strong>e. The second plot shows the vari<strong>at</strong>ion of the curv<strong>at</strong>ure, the energy<br />
m<strong>in</strong>imum, and the number of electrons <strong>in</strong> a 2p bound st<strong>at</strong>e correspond<strong>in</strong>g to the<br />
energy m<strong>in</strong>imum <strong>in</strong> the first plot.
FIGURE 5.27. The density and the sp<strong>in</strong> density <strong>in</strong>duced by a carbon impurity <strong>at</strong><br />
0.0005 background density.<br />
124
FIGURE 5.28. The density <strong>in</strong> the bound st<strong>at</strong>es of sp<strong>in</strong>-up and sp<strong>in</strong>-down electrons<br />
<strong>in</strong>duced by a carbon impurity <strong>at</strong> 0.0005 background density.<br />
125
FIGURE 5.29. The density and the sp<strong>in</strong> density <strong>in</strong> the conduction band <strong>in</strong>duced<br />
by a carbon impurity <strong>at</strong> 0.0005 background density.<br />
126
FIGURE 5.30. The density and the sp<strong>in</strong> density <strong>in</strong>duced by a carbon impurity <strong>at</strong><br />
0.00075 background density.<br />
127
FIGURE 5.31. The density <strong>in</strong> the bound st<strong>at</strong>es of sp<strong>in</strong>-up and sp<strong>in</strong>-down electrons<br />
<strong>in</strong>duced by a carbon impurity <strong>at</strong> 0.00075 background density.<br />
128
FIGURE 5.32. The density and the sp<strong>in</strong> density <strong>in</strong> the conduction band <strong>in</strong>duced<br />
by a carbon impurity <strong>at</strong> 0.00075 background density.<br />
129
FIGURE 5.33. The density and the sp<strong>in</strong> density <strong>in</strong>duced by a carbon impurity <strong>at</strong><br />
0.001 background density.<br />
130
FIGURE 5.34. The density <strong>in</strong> the bound st<strong>at</strong>es of sp<strong>in</strong>-up and sp<strong>in</strong>-down electrons<br />
<strong>in</strong>duced by a carbon impurity <strong>at</strong> 0.001 background density.<br />
131
FIGURE 5.35. The density and the sp<strong>in</strong> density <strong>in</strong> the conduction band <strong>in</strong>duced<br />
by a carbon impurity <strong>at</strong> 0.001 background density.<br />
132
FIGURE 5.36. The density and the sp<strong>in</strong> density <strong>in</strong>duced by a carbon impurity <strong>at</strong><br />
0.0011 background density.<br />
133
FIGURE 5.37. The density <strong>in</strong> the bound st<strong>at</strong>es of sp<strong>in</strong>-up and sp<strong>in</strong>-down electrons<br />
<strong>in</strong>duced by a carbon impurity <strong>at</strong> 0.0011 background density.<br />
134
FIGURE 5.38. The density and the sp<strong>in</strong> density <strong>in</strong> the conduction band <strong>in</strong>duced<br />
by a carbon impurity <strong>at</strong> 0.0011 background density.<br />
135
136<br />
Fig. 5.26 shows <strong>in</strong> this system th<strong>at</strong> the bound st<strong>at</strong>e energy as a function<br />
of the number of sp<strong>in</strong>-up electrons <strong>in</strong> a 2p bound st<strong>at</strong>e behaves unusually as a<br />
quadr<strong>at</strong>ic function. 20 Important parameters of quadr<strong>at</strong>ic functions are also shown<br />
<strong>in</strong> Fig. 5.26. The bound st<strong>at</strong>e energy decreases as the number of sp<strong>in</strong>-up electrons<br />
<strong>in</strong> a 2p bound st<strong>at</strong>e decreases from 3.0 electrons as expected. However, the bound<br />
st<strong>at</strong>e energy <strong>in</strong>creases as the number of sp<strong>in</strong>-up electrons <strong>in</strong> a 2p bound st<strong>at</strong>e<br />
decreases from the bound st<strong>at</strong>e energy m<strong>in</strong>imum due to the largely <strong>in</strong>duced sp<strong>in</strong>-<br />
down electrons near an impurity <strong>in</strong> the conduction band.<br />
This behavior is consistent with the potential.(Fig. 5.39) For <strong>in</strong>stance, the<br />
<strong>in</strong>duced sp<strong>in</strong> up and down electrons results <strong>in</strong> a behavior of the Coulomb potential<br />
for each band for 0.0005 background density as illustr<strong>at</strong>ed <strong>in</strong> Fig. 5.40. The total<br />
Coulomb potential near an impurity (r ≤ 1) <strong>in</strong> Fig. 5.41 decreases as the number of<br />
sp<strong>in</strong>-up electrons <strong>in</strong> a 2p bound st<strong>at</strong>e decreases and <strong>in</strong>creases as sp<strong>in</strong>-up electrons<br />
<strong>in</strong> a 2p bound st<strong>at</strong>e decreases from 2.5. One sees also th<strong>at</strong> the change <strong>in</strong> the total<br />
Coulomb potential (r ≤ 1) between 2.0 and 2.5 sp<strong>in</strong>-up electrons <strong>in</strong> a 2p bound<br />
st<strong>at</strong>e is not as large as th<strong>at</strong> between 2.5 and 3.0 sp<strong>in</strong>-up electrons <strong>in</strong> a 2p bound<br />
st<strong>at</strong>e. However, with the <strong>in</strong>crease of sp<strong>in</strong>-up exchange-correl<strong>at</strong>ion potential(Fig.<br />
5.42), one sees th<strong>at</strong> the total effective potential of sp<strong>in</strong>-up electrons <strong>at</strong> 2.0 electrons<br />
<strong>in</strong> a 2p bound st<strong>at</strong>e is now close to th<strong>at</strong> <strong>at</strong> 3.0 electrons. (See Fig. 5.39.) This<br />
potential behavior accounts for the vari<strong>at</strong>ion of the bound st<strong>at</strong>e energy <strong>in</strong> Fig.<br />
5.26 and the exchange-correl<strong>at</strong>ion <strong>in</strong>teraction <strong>in</strong> Fig. 5.42 causes the decrease of<br />
20 Note th<strong>at</strong> one can not use a simple first order of a Taylor series for this bound st<strong>at</strong>e<br />
energy as usual.
FIGURE 5.39. The effective potential of the sp<strong>in</strong>-up band vs the number of<br />
sp<strong>in</strong>-up electrons <strong>in</strong> a 2p bound st<strong>at</strong>e for 0.0005 background density.<br />
137
138<br />
FIGURE 5.40. The Coulomb potentials calcul<strong>at</strong>ed by <strong>in</strong>duced sp<strong>in</strong>-up and<br />
sp<strong>in</strong>-down electrons vs the number of sp<strong>in</strong>-up electrons <strong>in</strong> a 2p bound st<strong>at</strong>e for<br />
0.0005 background density.
139<br />
FIGURE 5.41. The Coulomb potentials vs the number of sp<strong>in</strong>-up electrons <strong>in</strong> a<br />
2p bound st<strong>at</strong>e for 0.0005 background density. The second plot shows the area<br />
which corresponds to the box <strong>in</strong> the first plot.
140<br />
FIGURE 5.42. The exchange-correl<strong>at</strong>ion potentials of <strong>in</strong>duced sp<strong>in</strong>-up and<br />
sp<strong>in</strong>-down electrons vs the number of sp<strong>in</strong>-up electrons <strong>in</strong> a 2p bound st<strong>at</strong>e for<br />
0.0005 background density.
the sp<strong>in</strong>-moment(the <strong>in</strong>crease of the anti-sp<strong>in</strong>-moment <strong>in</strong> the conduction band)<br />
s<strong>in</strong>ce sp<strong>in</strong>-up and down electrons experience the same Coulomb potential.<br />
141<br />
The phase shift <strong>in</strong> Fig. 5.43 shows a consistent behavior. Note <strong>in</strong> the overall<br />
sc<strong>at</strong>ter<strong>in</strong>g length and volume a decrease for sp<strong>in</strong>-up electrons and an <strong>in</strong>crease for<br />
sp<strong>in</strong>-down electrons. One knows th<strong>at</strong> the Friedel sum rule forces the phase shift<br />
<strong>at</strong> the Fermi energy to adjust to screen the excess charge of the core correctly.<br />
The way this system behaves due to the Friedel sum rule is th<strong>at</strong> the decrease <strong>in</strong><br />
the number of sp<strong>in</strong>-up electrons <strong>in</strong> a 2p bound st<strong>at</strong>e causes an <strong>in</strong>crease <strong>in</strong> the<br />
phase shift of the sp<strong>in</strong>-down electrons <strong>at</strong> the Fermi energy and the decrease <strong>in</strong><br />
the phase shift of the sp<strong>in</strong>-up electrons, which is consistent with the behavior of<br />
the sp<strong>in</strong>-density <strong>in</strong> the conduction band. With 3.0 electrons <strong>in</strong> a 2p bound st<strong>at</strong>e,<br />
the phase shift <strong>in</strong> Fig. 5.43 shows th<strong>at</strong> there is a sc<strong>at</strong>ter<strong>in</strong>g resonance <strong>in</strong> the high<br />
energy region (≥ ɛF ) of the phase shift of the sp<strong>in</strong>-down electrons. The resonance<br />
moves closer to the Fermi energy as the number of electrons <strong>in</strong> a 2p bound st<strong>at</strong>e<br />
decreases. Accord<strong>in</strong>gly, one sees large <strong>in</strong>creases <strong>at</strong> the Fermi energy <strong>in</strong> the density<br />
of <strong>in</strong>duced st<strong>at</strong>es of the sp<strong>in</strong>-down band, as illustr<strong>at</strong>ed <strong>in</strong> Fig. 5.44 and Fig. 5.45.<br />
These results can be expla<strong>in</strong>ed by self-consistency and the exchange-<br />
correl<strong>at</strong>ion <strong>in</strong>teraction. 21 If the <strong>in</strong>duced density <strong>in</strong> the bound st<strong>at</strong>es is reduced<br />
artificially, electrons <strong>in</strong> the conduction band simply can not be <strong>in</strong>duced more to<br />
ma<strong>in</strong>ta<strong>in</strong> the same total <strong>in</strong>duced density s<strong>in</strong>ce the same total <strong>in</strong>duced density yields<br />
the same Coulomb potentials and accord<strong>in</strong>gly the same <strong>in</strong>duced density <strong>in</strong> the con-<br />
21 These explan<strong>at</strong>ions need further <strong>in</strong>vestig<strong>at</strong>ion.
142<br />
FIGURE 5.43. Phase shifts of l = 0 and l = 1 for sp<strong>in</strong>-up and sp<strong>in</strong>-down electrons.
FIGURE 5.44. The density of <strong>in</strong>duced st<strong>at</strong>es for sp<strong>in</strong>-up and sp<strong>in</strong>-down electrons<br />
<strong>at</strong> 0.0005 background density.<br />
143
FIGURE 5.45. The density of <strong>in</strong>duced st<strong>at</strong>es for sp<strong>in</strong>-up and sp<strong>in</strong>-down electrons<br />
<strong>at</strong> 0.001 background density.<br />
144
145<br />
FIGURE 5.46. The exchange-correl<strong>at</strong>ion potential with different <strong>in</strong>duced densities<br />
<strong>in</strong> the sp<strong>in</strong>-up band. Background density is 0.0005. The <strong>in</strong>duced density <strong>in</strong> the<br />
sp<strong>in</strong>-down band is assumed to be 0.047.<br />
duction band. 22 The reduced density <strong>in</strong> the bound st<strong>at</strong>es, thus, tends to cause a<br />
more extended <strong>in</strong>duced density. Moreover, the potential due to the exchange-<br />
correl<strong>at</strong>ion <strong>in</strong>teraction 23 also can vary correspond<strong>in</strong>g to the change <strong>in</strong> the density.<br />
Fig. 5.46 shows one example of the exchange-correl<strong>at</strong>ion behavior <strong>in</strong> the case th<strong>at</strong><br />
the <strong>in</strong>duced density <strong>in</strong> the sp<strong>in</strong>-up band varies. The reduced density <strong>in</strong> the sp<strong>in</strong>-<br />
22 Once the density <strong>in</strong> the bound st<strong>at</strong>es is changed, the old bound st<strong>at</strong>es and sc<strong>at</strong>tered<br />
st<strong>at</strong>es which are orthogonal to each other do not s<strong>at</strong>isfy the self-consistency anymore.<br />
23 The exchange-correl<strong>at</strong>ion <strong>in</strong>teraction used <strong>in</strong> this work only depends on the local<br />
density.
146<br />
FIGURE 5.47. Partial wave decompositions of the Friedel sum rule. These plots<br />
show the change <strong>in</strong> the number of electrons <strong>in</strong>duced by a carbon impurity for each<br />
band. 0.0005 is used for a background density. Bound st<strong>at</strong>es are not <strong>in</strong>cluded.
up band gives rise to a more repulsive exchange-correl<strong>at</strong>ion potential and thus the<br />
sp<strong>in</strong>-down band has rel<strong>at</strong>ively more <strong>in</strong>duced electrons <strong>in</strong> the conduction band due<br />
to the Coulomb <strong>in</strong>teraction as the number of electrons <strong>in</strong> a 2p bound st<strong>at</strong>e of the<br />
sp<strong>in</strong>-up band decreases. Fig. 5.47 shows the response of the conduction bands <strong>in</strong><br />
terms of the phase shifts <strong>at</strong> the Fermi energy. One sees th<strong>at</strong>, as the the number of<br />
electrons <strong>in</strong> a 2p bound st<strong>at</strong>e of the sp<strong>in</strong>-up band decreases, the sp<strong>in</strong>-up conduction<br />
band loses more electrons <strong>in</strong> overall angular momentum st<strong>at</strong>es and the decrease of<br />
<strong>in</strong>duced electrons <strong>in</strong> the p st<strong>at</strong>e of the sp<strong>in</strong>-up band is compens<strong>at</strong>ed by ma<strong>in</strong>ly the<br />
same p st<strong>at</strong>e of the sp<strong>in</strong>-down band.<br />
147
148<br />
FIGURE 5.48. Bound st<strong>at</strong>e energies, the conduction band m<strong>in</strong>imums, the<br />
sp<strong>in</strong>-moment, and the immersion energies vs the magnetic field for a hydrogen<br />
impurity with 0.0025 background density.<br />
5.2.2. Non-zero external magnetic field<br />
A hydrogen impurity <strong>in</strong> a sp<strong>in</strong>-polarized system has been tested by apply<strong>in</strong>g<br />
an external magnetic field. Detailed <strong>in</strong>form<strong>at</strong>ion for the implement<strong>at</strong>ion can be<br />
found <strong>in</strong> Sec. 4.6. Results for three different background densities are obta<strong>in</strong>ed.<br />
Fig. 5.48 shows the <strong>in</strong>duced sp<strong>in</strong>-moment <strong>in</strong> a system, bound st<strong>at</strong>e energies, the<br />
conduction band m<strong>in</strong>imum, and the immersion energies for a background density<br />
of 0.0025. As the magnetic field <strong>in</strong>creases, the background density (conduction<br />
band width) of sp<strong>in</strong>-up (majority) electrons <strong>in</strong>creases and the background density<br />
of sp<strong>in</strong>-down (m<strong>in</strong>ority) electrons decreases. Therefore, ∆N <strong>in</strong> this case is the<br />
excess sp<strong>in</strong>-moment, the devi<strong>at</strong>ion from the moment due to the background by
149<br />
FIGURE 5.49. The exchange-correl<strong>at</strong>ion potential with different background densities<br />
of the sp<strong>in</strong>-up band. The total background density is 0.0025 and the <strong>in</strong>duced<br />
density is assumed to be 0.03 for each sp<strong>in</strong>-up and sp<strong>in</strong>-down band. This plot<br />
shows how the exchange-correl<strong>at</strong>ion potential for each band may vary with the<br />
<strong>in</strong>creas<strong>in</strong>g magnetic field.<br />
itself. There are two electrons <strong>in</strong> a 1s st<strong>at</strong>e without an external magnetic field as<br />
<strong>in</strong> a non-sp<strong>in</strong> polarized system. (See Sec. 5.1.) One sees th<strong>at</strong> bound st<strong>at</strong>e ener-<br />
gies behave as <strong>in</strong> non-sp<strong>in</strong> polarized systems <strong>in</strong> Fig. 5.48. Th<strong>at</strong> is, the bound st<strong>at</strong>e<br />
energy <strong>in</strong> the sp<strong>in</strong>-up band approaches the conduction band m<strong>in</strong>imum as the back-<br />
ground density of the sp<strong>in</strong>-up band <strong>in</strong>creases while the bound st<strong>at</strong>e energy <strong>in</strong> the<br />
sp<strong>in</strong>-down band becomes lower compared to the band m<strong>in</strong>imum. As the magnetic<br />
field <strong>in</strong> Fig. 5.48 approaches 0.09894, 24 where the sp<strong>in</strong>-down conduction band com-<br />
24 The unit for the magnetic field <strong>in</strong> this work is 3.3241346·10 5 T, which will be omitted<br />
<strong>in</strong> this paper for convenience.
pletely disappears, the calcul<strong>at</strong>ions are not performed correctly due to numerical<br />
problems (long wave length of Friedel oscill<strong>at</strong>ions) as mentioned <strong>in</strong> Ch. 4 and thus<br />
d<strong>at</strong>a are not available for a magnetic field between 0.08 and 0.10. One important<br />
phenomenon is the sp<strong>in</strong>-moment which decreases with the <strong>in</strong>creas<strong>in</strong>g magnetic field<br />
and then rapidly <strong>in</strong>creases after a magnetic field of 0.06, as illustr<strong>at</strong>ed <strong>in</strong> Fig. 5.48.<br />
Two important facts are, due to the difference between sp<strong>in</strong>-up and sp<strong>in</strong>-down<br />
electrons, the exchange-correl<strong>at</strong>ion <strong>in</strong>teraction and the Coulomb <strong>in</strong>teraction of the<br />
Friedel oscill<strong>at</strong>ions with two different Fermi wave vectors.<br />
150<br />
Fig. 5.49 shows th<strong>at</strong>, if the total background density rema<strong>in</strong>s the same, the<br />
exchange-correl<strong>at</strong>ion potential of the sp<strong>in</strong>-up band becomes more repulsive and<br />
the exchange-correl<strong>at</strong>ion potential of the sp<strong>in</strong>-down band rapidly becomes more<br />
<strong>at</strong>tractive due to the exchange-correl<strong>at</strong>ion <strong>in</strong>teraction as the background density<br />
of the sp<strong>in</strong>-down band approaches zero, which suggests th<strong>at</strong> there may be a rapid<br />
vari<strong>at</strong>ion <strong>in</strong> potentials of sp<strong>in</strong>-up and sp<strong>in</strong>-down electrons when the external mag-<br />
netic field <strong>in</strong>creases. The potential vari<strong>at</strong>ion <strong>in</strong> Fig. 5.49 is actually very large<br />
compared to the potential difference <strong>at</strong> the <strong>at</strong>omic 2p loc<strong>at</strong>ion (r ∼ 4) for a hy-<br />
drogen impurity between the high and low background densities (such as 0.04 and<br />
0.0025). As the external magnetic field <strong>in</strong>creases, this behavior yields not only a<br />
more repulsive potential <strong>in</strong> the vic<strong>in</strong>ity of an impurity but also a large peak <strong>in</strong> r<br />
times the potential near the 2p loc<strong>at</strong>ion for the background density of the sp<strong>in</strong>-<br />
up band, as illustr<strong>at</strong>ed <strong>in</strong> Fig. 5.50, which is the opposite behavior compared to<br />
the potentials of systems without a magnetic field <strong>in</strong> Fig. 5.51. 25 The number of<br />
25 The lower background density has a larger fluctu<strong>at</strong>ion <strong>at</strong> large distances <strong>in</strong> the r times<br />
potential for a system without the external magnetic field as <strong>in</strong> Fig. 5.51.
FIGURE 5.50. The effective potentials for the different magnetic field 0.01 and<br />
0.04. A hydrogen impurity with 0.0025 background density is used.<br />
151
152<br />
FIGURE 5.51. The effective potential for a hydrogen impurity. No external magnetic<br />
field is applied. The potential of the sp<strong>in</strong>-up band is identical to th<strong>at</strong> of the<br />
sp<strong>in</strong>-down band.<br />
<strong>in</strong>duced electrons, therefore, decreases <strong>in</strong> the sp<strong>in</strong>-up band due to the repulsive<br />
exchange-correl<strong>at</strong>ion <strong>in</strong>teraction and <strong>in</strong>creases <strong>in</strong> the sp<strong>in</strong>-down band due to the<br />
<strong>at</strong>tractive exchange-correl<strong>at</strong>ion <strong>in</strong>teraction and, as a result, the sp<strong>in</strong>-moment de-<br />
creases up to 0.05 magnetic field as shown <strong>in</strong> Fig. 5.48. The phase shifts <strong>at</strong> the<br />
Fermi energy show a consistent behavior <strong>in</strong> Fig. 5.52. Without any external mag-<br />
netic field, there is no difference between the sp<strong>in</strong>-up and sp<strong>in</strong>-down bands. This<br />
means th<strong>at</strong>, if the number of <strong>in</strong>duced electrons <strong>in</strong> the s-st<strong>at</strong>e of the conduction<br />
band is reduced by a higher background density, the higher angular momentum<br />
st<strong>at</strong>es must have a larger contribution to the number of <strong>in</strong>duced electrons, due to<br />
the Friedel sum rule as shown <strong>in</strong> Fig. 5.53. The phase shifts <strong>at</strong> the Fermi energy
153<br />
FIGURE 5.52. Partial wave decompositions of the Friedel sum rule. These plots<br />
show the change <strong>in</strong> the number of electrons <strong>in</strong>duced by a hydrogen impurity for<br />
each band. 0.0025 is used for a background density.
154<br />
FIGURE 5.53. Partial wave decompositions of the Friedel sum rule. no external<br />
magnetic field is applied. This plot for the sp<strong>in</strong>-up band is identical with th<strong>at</strong> for<br />
the sp<strong>in</strong>-down band.<br />
with a non-zero magnetic field, however, <strong>in</strong> Fig. 5.52, show a different behavior<br />
for the higher angular momentum st<strong>at</strong>es. One sees <strong>in</strong> Fig. 5.52 th<strong>at</strong> the vari<strong>at</strong>ion<br />
<strong>in</strong> the number of <strong>in</strong>duced electrons due to a different magnetic field for the high<br />
angular momentum st<strong>at</strong>es of the sp<strong>in</strong>-down band (≥ p) is small up to 0.05 mag-<br />
netic field compared to th<strong>at</strong> for a system without a magnetic field(Fig. 5.53). One<br />
sees also th<strong>at</strong> the phase shifts <strong>at</strong> the p-st<strong>at</strong>e <strong>in</strong> the sp<strong>in</strong>-up band become lower as<br />
the magnetic field <strong>in</strong>creases and thus lose more electrons <strong>in</strong> the p-st<strong>at</strong>e while the<br />
s-st<strong>at</strong>e lose electrons as well. This behavior is also opposite to a system without a<br />
magnetic field(Fig. 5.53). The strong exchange-correl<strong>at</strong>ion <strong>in</strong>teraction due to the<br />
magnetic field for this background density is believed to cause these phenomena.
155<br />
FIGURE 5.54. The <strong>in</strong>duced electronic densities <strong>in</strong> the conduction band for different<br />
magnetic fields. A hydrogen impurity with 0.0025 background density is<br />
used.
156<br />
FIGURE 5.55. The <strong>in</strong>duced electronic densities <strong>in</strong> the conduction band of sp<strong>in</strong>-up<br />
electrons for each angular momentum st<strong>at</strong>e. These plots correspond to the magnetic<br />
field 0.08 <strong>in</strong> Fig. 5.48 and Fig. 5.54. A hydrogen impurity with 0.0025 background<br />
density is used.
157<br />
FIGURE 5.56. The density of st<strong>at</strong>es of the sp<strong>in</strong>-up and sp<strong>in</strong>-down bands for<br />
different magnetic fields. The arrows <strong>in</strong> the first plot <strong>in</strong>dic<strong>at</strong>e the loc<strong>at</strong>ions of the<br />
sc<strong>at</strong>ter<strong>in</strong>g resonances. A hydrogen impurity with 0.0025 background density is<br />
used.
FIGURE 5.57. Comparison of kF of the sp<strong>in</strong>-down band with k values which<br />
correspond to the sc<strong>at</strong>ter<strong>in</strong>g resonances <strong>in</strong> Fig. 5.56.<br />
158<br />
There is another important <strong>in</strong>teraction which causes the rapid <strong>in</strong>crease <strong>in</strong><br />
the sp<strong>in</strong>-moment (Fig. 5.48) and the fluctu<strong>at</strong>ions <strong>in</strong> the high angular momentum<br />
st<strong>at</strong>es of the sp<strong>in</strong>-up band as <strong>in</strong> Fig. 5.52. S<strong>in</strong>ce there is a difference <strong>in</strong> Fermi<br />
wave vector between the sp<strong>in</strong>-up and sp<strong>in</strong>-down bands, one may easily expect th<strong>at</strong><br />
there is a be<strong>at</strong> p<strong>at</strong>tern <strong>in</strong> the <strong>in</strong>duced density. The sp<strong>in</strong>-down band has a larger<br />
range of Coulomb holes and peaks <strong>in</strong> the density than the sp<strong>in</strong>-up band due to<br />
the smaller Fermi wave vector as shown <strong>in</strong> Fig. 5.54. The sp<strong>in</strong>-up band tries to<br />
compens<strong>at</strong>e a large range of these Coulomb fluctu<strong>at</strong>ions <strong>in</strong> the <strong>in</strong>duced density<br />
of the sp<strong>in</strong>-down band and results <strong>in</strong> the <strong>in</strong>creased values of the phase shifts of<br />
high angular momentum st<strong>at</strong>es <strong>at</strong> the Fermi energies. On the other hand, the<br />
sp<strong>in</strong>-down band also tries to keep the same Coulomb holes as usual by remov<strong>in</strong>g
sp<strong>in</strong>-down electrons. 26 One sees th<strong>at</strong> the <strong>in</strong>creases for l = 3, 4, and 5 of the<br />
sp<strong>in</strong>-up band <strong>in</strong> Fig. 5.52 and consistently the <strong>in</strong>duced densities <strong>in</strong> the sp<strong>in</strong>-up<br />
band for the high angular momentum st<strong>at</strong>es, which are <strong>at</strong> loc<strong>at</strong>ions of the large<br />
Coulomb holes of the sp<strong>in</strong>-down band, are unusually large as illustr<strong>at</strong>ed <strong>in</strong> Fig.<br />
5.54 and Fig. 5.55. Also note th<strong>at</strong>, for large magnetic fields, there are significant<br />
decreases <strong>in</strong> the l = 1, and 2 phase shifts of the sp<strong>in</strong>-down band <strong>in</strong> Fig. 5.52.<br />
This <strong>in</strong>teraction dom<strong>in</strong><strong>at</strong>es the exchange-correl<strong>at</strong>ion <strong>in</strong>teraction as the sp<strong>in</strong>-down<br />
background density approaches zero and yields a large sp<strong>in</strong>-moment as <strong>in</strong> Fig.<br />
5.48. One can see this <strong>in</strong>teraction <strong>in</strong> the density of st<strong>at</strong>es as well. (See Fig. 5.56.)<br />
There is a sc<strong>at</strong>ter<strong>in</strong>g resonance <strong>in</strong> the density of st<strong>at</strong>es for the sp<strong>in</strong>-up band which<br />
becomes large as the background density of the sp<strong>in</strong>-down band approaches zero.<br />
One sees also <strong>in</strong> Fig. 5.57 th<strong>at</strong> the momentum value correspond<strong>in</strong>g to the sc<strong>at</strong>ter<strong>in</strong>g<br />
resonance closely follows the Fermi wave vector of the sp<strong>in</strong>-down band, which is<br />
rel<strong>at</strong>ed to the Coulomb compens<strong>at</strong>ion of the sp<strong>in</strong>-up band. After the sp<strong>in</strong>-down<br />
conduction band disappears, the sp<strong>in</strong>-moment is simply -1 s<strong>in</strong>ce only the sp<strong>in</strong>-<br />
down band has a bound st<strong>at</strong>e. The immersion energy varies <strong>in</strong> Fig. 5.48 opposite<br />
to the sp<strong>in</strong>-moment s<strong>in</strong>ce the <strong>in</strong>duced sp<strong>in</strong>-up electrons have a lower energy when<br />
the energy is shifted by the magnetic field.<br />
26 This Coulomb <strong>in</strong>teraction of the Friedel oscill<strong>at</strong>ions makes obta<strong>in</strong><strong>in</strong>g a numerical self-<br />
consistent solution much more difficult.<br />
159
160<br />
FIGURE 5.58. Bound st<strong>at</strong>e energies, the conduction band m<strong>in</strong>imums, the<br />
sp<strong>in</strong>-moment, and the immersion energies vs the magnetic field for a hydrogen<br />
impurity with two different background densities, 0.01 and 0.04.
161<br />
FIGURE 5.59. The exchange-correl<strong>at</strong>ion potential with different background densities<br />
of the sp<strong>in</strong>-up band. The total background density is 0.04 and the <strong>in</strong>duced<br />
density is assumed to be 0.03 for each sp<strong>in</strong>-up and sp<strong>in</strong>-down band. This plot<br />
shows how the exchange-correl<strong>at</strong>ion potential for each band may vary with the<br />
<strong>in</strong>creas<strong>in</strong>g magnetic field.<br />
The calcul<strong>at</strong>ions for a background density of 0.01 yield similar results as<br />
shown <strong>in</strong> Fig. 5.58. 27 The background density 0.04, however, gives different results<br />
as shown <strong>in</strong> Fig. 5.58. Note th<strong>at</strong>, without an external magnetic field, there are no<br />
electrons <strong>in</strong> the bound st<strong>at</strong>es for a background density of 0.04 and a 1s bound st<strong>at</strong>e<br />
exists <strong>in</strong> the sp<strong>in</strong>-down band for the magnetic field larger than 0.25. The ma<strong>in</strong><br />
27 In the results for 0.01 background density, there is a discont<strong>in</strong>uity <strong>in</strong> the sp<strong>in</strong>-moment<br />
near 0.19 magnetic field as shown <strong>in</strong> Fig. 5.58, which is believed to be a numerical<br />
problem due to the large sp<strong>at</strong>ial extension of the bound st<strong>at</strong>e <strong>in</strong> the sp<strong>in</strong>-up band.
162<br />
FIGURE 5.60. The <strong>in</strong>tegr<strong>at</strong>ed values of the density of st<strong>at</strong>es of the sp<strong>in</strong>-up band<br />
for different magnetic fields to f<strong>in</strong>d the total <strong>in</strong>duced number of electrons <strong>in</strong> the<br />
sp<strong>in</strong>-up conduction band. A hydrogen impurity with 0.04 background density is<br />
used. The band m<strong>in</strong>imums are shifted <strong>in</strong> such a way th<strong>at</strong> they have the same<br />
m<strong>in</strong>imum.<br />
difference is th<strong>at</strong> the sp<strong>in</strong>-moment slightly <strong>in</strong>creases as the magnetic field <strong>in</strong>creases<br />
from zero but shows the same behavior as results for the background densities of<br />
0.0025 and 0.01 after 0.035 magnetic field. This may be expla<strong>in</strong>ed by the exchange-<br />
correl<strong>at</strong>ion <strong>in</strong>teraction for this high background density and the difference <strong>in</strong> the<br />
band width(kF ) between sp<strong>in</strong>-up and sp<strong>in</strong>-down bands. S<strong>in</strong>ce this is such a high<br />
background density, the exchange-correl<strong>at</strong>ion potentials vary rel<strong>at</strong>ively slowly as<br />
the magnetic field <strong>in</strong>creases from zero compared to Fig. 5.49 (See Fig. 5.59.) and<br />
the change <strong>in</strong> the band width due to the magnetic field becomes more important.<br />
In this case, the number of <strong>in</strong>duced electrons <strong>in</strong>creases as the Fermi wave vector
163<br />
FIGURE 5.61. Partial wave decompositions of the Friedel sum rule. These plots<br />
show the change <strong>in</strong> the number of electrons <strong>in</strong>duced by a hydrogen impurity for<br />
each band. 0.04 is used for a background density.
FIGURE 5.62. The density <strong>in</strong>duced by a hydrogen impurity for each conduction<br />
band. The background density is 0.04 and the magnetic field is 0.3.<br />
of the sp<strong>in</strong>-up electrons <strong>in</strong>creases, s<strong>in</strong>ce there are more st<strong>at</strong>es available which can<br />
contribute to the <strong>in</strong>duced electrons. One can <strong>in</strong>tegr<strong>at</strong>e the density of st<strong>at</strong>es to<br />
f<strong>in</strong>d the number of <strong>in</strong>duced electrons, which is shown <strong>in</strong> Fig. 5.60. The <strong>in</strong>tegr<strong>at</strong>ed<br />
number of <strong>in</strong>duced electrons as a function of energy <strong>in</strong> the sp<strong>in</strong>-up band is slightly<br />
different between a magnetic field of 0.1 and 0.35, but the higher Fermi wave vector<br />
results <strong>in</strong> a ga<strong>in</strong> of more electrons as illustr<strong>at</strong>ed <strong>in</strong> Fig. 5.60. The phase shifts <strong>at</strong><br />
the Fermi wave vector <strong>in</strong> Fig. 5.61 for values of the magnetic field between 0.1<br />
and 0.35 show a significant change for high angular momentum st<strong>at</strong>es (such as<br />
l = 1 ∼ 4) compared to the change <strong>in</strong> the s-st<strong>at</strong>e as the magnetic field <strong>in</strong>creases.<br />
The <strong>in</strong>duced density of the sp<strong>in</strong>-up band is much larger than th<strong>at</strong> of the sp<strong>in</strong>-down<br />
164<br />
band for a magnetic field gre<strong>at</strong>er than 0.35 as illustr<strong>at</strong>ed <strong>in</strong> Fig. 5.62. However, as
the magnetic field <strong>in</strong>creases, the sp<strong>in</strong>-moment <strong>in</strong> Fig. 5.58 decreases for a magnetic<br />
field larger than 0.035 28 where the exchange-correl<strong>at</strong>ion potential of the sp<strong>in</strong>-down<br />
band may decrease very rapidly and thus the difference <strong>in</strong> the exchange-correl<strong>at</strong>ion<br />
potentials becomes very large as shown <strong>in</strong> Fig. 5.59. The same reason<strong>in</strong>g which is<br />
applied to analyze the results for a background density of 0.0025 can be applied to<br />
the results for a magnetic field larger than 0.035 for this high background density<br />
as well.<br />
165<br />
28 0.035 magnetic field corresponds to 0.032796 background density of the sp<strong>in</strong>-up band.
6. CONCLUSIONS<br />
Immersion energies are calcul<strong>at</strong>ed for H, Li, C, and excited systems of H and C<br />
us<strong>in</strong>g the Kohn-Sham equ<strong>at</strong>ions <strong>in</strong> the LDA scheme. The self-consistent solutions<br />
are verified and compared with the Friedel sum rule, Friedel oscill<strong>at</strong>ions, phase<br />
shifts, the sc<strong>at</strong>ter<strong>in</strong>g length, bound st<strong>at</strong>e energies, potentials, Janak’s theorem,<br />
and compressibilities.<br />
166<br />
In order to obta<strong>in</strong> the correct self-consistent solutions, numerical problems<br />
must be solved first. The numerical and theoretical errors <strong>in</strong> LDA due to the rapid<br />
density vari<strong>at</strong>ion <strong>in</strong> the vic<strong>in</strong>ity of an impurity are canceled out by subtract<strong>in</strong>g<br />
the reference energy of a free <strong>at</strong>om. The Friedel sum rule and Friedel oscill<strong>at</strong>ions<br />
play a crucial role <strong>in</strong> solv<strong>in</strong>g numerical problems dur<strong>in</strong>g the numerical iter<strong>at</strong>ion<br />
process. The long tail of the Friedel oscill<strong>at</strong>ions is strongly coupled to the <strong>in</strong>duced<br />
density near an impurity by the Coulomb <strong>in</strong>teraction. A mix<strong>in</strong>g r<strong>at</strong>io as a function<br />
of r is used to acceler<strong>at</strong>e the convergence by suppress<strong>in</strong>g this coupled <strong>in</strong>teraction<br />
<strong>in</strong> the mix<strong>in</strong>g step <strong>in</strong> the self-consistent calcul<strong>at</strong>ion. In the numerical calcul<strong>at</strong>ion,<br />
<strong>in</strong>f<strong>in</strong>ity must be replaced with a cut-off large number and numerical results must<br />
be corrected us<strong>in</strong>g the behavior of the Friedel oscill<strong>at</strong>ions.<br />
The background density <strong>at</strong> the immersion energy m<strong>in</strong>imum is 0.0024 for a<br />
non-sp<strong>in</strong> polarized system of H and 0.0033 for a non-sp<strong>in</strong> polarized system of<br />
C. The behavior of immersion energies as a function of the background density<br />
depends on the number of bound st<strong>at</strong>es as well as on the impurity. For a neutral<br />
impurity, which is forced to have the same number of electrons as the impurity<br />
charge, the immersion energies show a simple l<strong>in</strong>ear behavior and approach zero as<br />
the background density becomes zero. The immersion energies for excited systems<br />
are verified with Janak’s theorem.
The impurity is completely screened, which is consistent with the Friedel<br />
sum rule, regardless of the number of bound st<strong>at</strong>es or the background density. In<br />
the low energy sc<strong>at</strong>ter<strong>in</strong>g cases, one can predict the behavior of the phase shifts<br />
from the Friedel sum rule and the number of bound st<strong>at</strong>es. The phase shifts may,<br />
however, show a complic<strong>at</strong>ed behavior due to the narrow width of the conduction<br />
band <strong>at</strong> an extremely low background density. This behavior can be expressed <strong>in</strong><br />
terms of sc<strong>at</strong>ter<strong>in</strong>g lengths and compressibilities, which is verified with analytical<br />
theories.<br />
167<br />
At low background densities, the calcul<strong>at</strong>ions show th<strong>at</strong> the immersion en-<br />
ergy for the sp<strong>in</strong>-polarized system of a carbon impurity cont<strong>in</strong>ues to decrease as the<br />
background density decreases and does not have the immersion energy m<strong>in</strong>imum<br />
near the background density where the non-sp<strong>in</strong>-polarized system of a carbon im-<br />
purity has a m<strong>in</strong>imum. For these sp<strong>in</strong>-polarized systems, the exchange-correl<strong>at</strong>ion<br />
<strong>in</strong>teraction <strong>at</strong> low background densities becomes very important, caus<strong>in</strong>g a very<br />
strong tendency toward a sp<strong>in</strong>-polarized solution. It <strong>in</strong>duces a 2p bound st<strong>at</strong>e <strong>in</strong><br />
the sp<strong>in</strong>-up band only, and can yield a non-zero magnetiz<strong>at</strong>ion even without an<br />
external magnetic field. These phenomena lead to an unusual angular momentum<br />
st<strong>at</strong>e behavior <strong>in</strong> terms of the number of <strong>in</strong>duced electrons for excited systems of<br />
a carbon impurity <strong>at</strong> low background densities.<br />
With an external magnetic field, the strong Coulomb <strong>in</strong>teraction appears due<br />
to two different wave lengths of Friedel oscill<strong>at</strong>ions, for sp<strong>in</strong>-up and sp<strong>in</strong>-down<br />
bands. With this Coulomb <strong>in</strong>teraction, several <strong>in</strong>teractions such as the exchange-<br />
correl<strong>at</strong>ion <strong>in</strong>teraction and the change <strong>in</strong> the band width due to an external mag-<br />
netic field give rise to a complic<strong>at</strong>ed behavior of the <strong>in</strong>duced sp<strong>in</strong>-moment. For<br />
<strong>in</strong>stance, there is a rapid <strong>in</strong>crease of the sp<strong>in</strong>-moment as the band width of the<br />
sp<strong>in</strong>-down band goes to zero. This magnetic behavior of a sp<strong>in</strong>-polarized system
due to the exchange-correl<strong>at</strong>ion <strong>in</strong>teraction <strong>at</strong> low background densities is driven by<br />
the response of an electron gas <strong>in</strong> the conduction band and, as a result, has a long<br />
range effect compared to usual l<strong>at</strong>tice constants. These results, therefore, suggest<br />
th<strong>at</strong> the separ<strong>at</strong>ion distance between impurities is important, and the magnetic<br />
behavior of impurities may largely depend on the separ<strong>at</strong>ion distance.<br />
The model used <strong>in</strong> this work is spherically symmetric, the sp<strong>in</strong>-orbit <strong>in</strong>ter-<br />
action is ignored, and the self-<strong>in</strong>teraction correction is not implemented. A non-<br />
spherical model must be used for partially occupied angular momentum shells and,<br />
hence, may yield angular dependent results especially with an external magnetic<br />
field. The sp<strong>in</strong>-orbit <strong>in</strong>teraction also may lead to very different results s<strong>in</strong>ce non-<br />
zero angular momentum st<strong>at</strong>es can show the complic<strong>at</strong>ed behavior as expla<strong>in</strong>ed<br />
<strong>in</strong> Ch. 5. In the future, a non-spherical model should be studied with the sp<strong>in</strong>-<br />
orbit <strong>in</strong>teraction and the self-<strong>in</strong>teraction correction. Further, a non-local density<br />
scheme for the exchange-correl<strong>at</strong>ion <strong>in</strong>teraction should be used to <strong>in</strong>vestig<strong>at</strong>e the<br />
important dependance of our results on the exchange-correl<strong>at</strong>ion potential.<br />
168
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APPENDIX<br />
171
APPENDIX A. Atomic Rydberg units<br />
energy:<br />
Atomic Rydberg units are def<strong>in</strong>ed by = 2me = e 2 /2 = 1.<br />
The unit for length is the Bohr radius:<br />
a0 = 2<br />
mee 2 = 1 0.529 · 10−8 cm .<br />
172<br />
The unit for time is the r<strong>at</strong>io of an angular momentum and the Rydberg<br />
t0 = <br />
ER<br />
= 23<br />
mee 4 = 1 4.84 · 10−17 s .<br />
The unit for energy is the Rydberg energy:<br />
ER = e2<br />
2a0<br />
The unit for charge is:<br />
= mee 4<br />
The unit for charge density is:<br />
= 1 Rydberg 13.6 eV .<br />
22 q0 = 1 = e<br />
√ 2 1.1329105 · 10 −19 C .<br />
ρ0 = 1 = 7.6452571 · 10 11 C/m 3 .<br />
Other units for important physical quantities are as follows.<br />
The unit for electric field:<br />
The unit for current:<br />
E0 = 1 3.6360903 · 10 11 V/m .<br />
I0 = 1 2.3418037 mA .
The unit for conductivity:<br />
The unit for resistance:<br />
The unit for magnetic <strong>in</strong>duction:<br />
The unit for magnetiz<strong>at</strong>ion:<br />
σ0 = 1 2.2999241 · 10 6 S/m .<br />
R0 = 1 8.2164712 kΩ .<br />
B0 = 1 3.3241346 · 10 5 T .<br />
M0 = 1 4.4253673 · 10 7 A/m .<br />
173
APPENDIX B. Comparison with published d<strong>at</strong>a<br />
This section is dedic<strong>at</strong>ed to focus on the ma<strong>in</strong> difference between this thesis<br />
and Puska, Niem<strong>in</strong>en, and Man<strong>in</strong>en’s work( [5]).<br />
B.1. Numerical aspects<br />
174<br />
In Puska, Niem<strong>in</strong>en, and Man<strong>in</strong>en’s work, only 10 partial waves were used.<br />
The Pr<strong>at</strong>t improvement scheme(Broyden’s method) is used for the mix<strong>in</strong>g r<strong>at</strong>io.<br />
They used a first order approxim<strong>at</strong>ion of Taylor series to remove the dependence<br />
of the cut-off radius <strong>in</strong> the calcul<strong>at</strong>ion of the exchange-correl<strong>at</strong>ion energy.<br />
In this thesis, a cut-off value of 10 −8 for the phase shifts is used to determ<strong>in</strong>e<br />
the number of partial waves for each k. Hence, the number of partial waves is<br />
variable. Exponential-like functions are used for the mix<strong>in</strong>g r<strong>at</strong>io s<strong>in</strong>ce discont<strong>in</strong>u-<br />
ities <strong>in</strong> the potential <strong>in</strong> r-space are found <strong>in</strong> Broyden’s method. The <strong>in</strong>form<strong>at</strong>ion<br />
of the last two <strong>in</strong>put potentials is utilized <strong>in</strong> the mix<strong>in</strong>g scheme to consider the<br />
response of iter<strong>at</strong>ion process. The cut-off radius dependance of the calcul<strong>at</strong>ions is<br />
removed by consider<strong>in</strong>g the oscill<strong>at</strong>ion of <strong>in</strong>tegr<strong>at</strong>ed values, which is believed to be<br />
accur<strong>at</strong>e due to the regular behavior of the Friedel oscill<strong>at</strong>ions <strong>at</strong> large distances.<br />
The output potentials also must be corrected after each iter<strong>at</strong>ion of the numerical<br />
calcul<strong>at</strong>ions. This is performed <strong>in</strong> such a way th<strong>at</strong> the system s<strong>at</strong>isfies the Friedel<br />
sum rule and correct Friedel oscill<strong>at</strong>ions. It is also po<strong>in</strong>ted out th<strong>at</strong> the system<strong>at</strong>ic<br />
numerical error of core st<strong>at</strong>es of an impurity <strong>in</strong> an electronic gas can be canceled<br />
out by th<strong>at</strong> of the reference of a free <strong>at</strong>om energy. In order to obta<strong>in</strong> this cance-<br />
l<strong>at</strong>ion, one must use the same r-mesh for both calcul<strong>at</strong>ions. Detailed <strong>in</strong>form<strong>at</strong>ion<br />
can be found <strong>in</strong> Ch. 4.
B.2. Results<br />
175<br />
The immersion energies obta<strong>in</strong>ed <strong>in</strong> this thesis are slightly lower than those<br />
<strong>in</strong> Puska, Niem<strong>in</strong>en, and Man<strong>in</strong>en’s work. For <strong>in</strong>stance, the background density is<br />
about 0.025 <strong>at</strong> zero immersion energy for H while it is 0.02 <strong>in</strong> Puska, Niem<strong>in</strong>en, and<br />
Man<strong>in</strong>en’s work. However, the result for the background density <strong>at</strong> the immersion<br />
energy m<strong>in</strong>imum is very close to th<strong>at</strong> of Puska, Niem<strong>in</strong>en, and Man<strong>in</strong>en’s work.<br />
They showed 0.0026 for H and 0.0035 for C while, <strong>in</strong> this work, 0.0024 for H and<br />
0.0033 for C are obta<strong>in</strong>ed. While they focused on the partial-wave decompositions<br />
of the density of <strong>in</strong>duced st<strong>at</strong>es, the results for the partial-wave decompositions of<br />
the Friedel sum rule 1 are obta<strong>in</strong>ed and used to analyze the behavior of the poten-<br />
tials especially for sp<strong>in</strong>-polarized systems. The follow<strong>in</strong>g new results are obta<strong>in</strong>ed.<br />
Dielectric functions and compressibilities for H are calcul<strong>at</strong>ed and compared with<br />
the theory <strong>in</strong> this work. Immersion energies for excited systems are also calcul<strong>at</strong>ed<br />
and verified with the Janak’s theorem, and the results for sp<strong>in</strong>-polarized systems<br />
and sp<strong>in</strong>-coupl<strong>in</strong>g with an external magnetic field are also newly obta<strong>in</strong>ed <strong>in</strong> this<br />
work, which have not been performed <strong>in</strong> the older work.<br />
1 This k<strong>in</strong>d of discussion also can be found <strong>in</strong> [32].
APPENDIX C. Boundary conditions for non-spherical potentials <strong>at</strong><br />
small r<br />
where<br />
The general idea is taken from [8]. The equ<strong>at</strong>ions to be solved are<br />
d 2<br />
dr 2 + ɛn −<br />
V m<br />
<br />
ll ′ (r) =<br />
l(l + 1)<br />
r2 <br />
unlm(r) = <br />
l ′<br />
176<br />
V m<br />
ll ′ (r)unl ′ m(r) , (C1)<br />
dΩ Y ∗<br />
lm(Ωr)Veff(r, θ)Yl ′ m(Ωr) . (C2)<br />
The coupl<strong>in</strong>g terms must be exam<strong>in</strong>ed <strong>in</strong> order to f<strong>in</strong>d the boundary conditions of<br />
a set of l<strong>in</strong>early <strong>in</strong>dependent solutions of the coupled equ<strong>at</strong>ions <strong>at</strong> small r. The<br />
potential can be expanded due to its cyl<strong>in</strong>drical symmetry as<br />
V (r) = <br />
Vi(r)Pi(cos θ) , (C3)<br />
i<br />
where Pi(cos θ) is a Legendre polynomial. The behavior of the components Vi(r)<br />
of the potential <strong>at</strong> small r can be shown, which is<br />
lim<br />
r→0 Vi(r) ∼<br />
Us<strong>in</strong>g the expansion for V (r),<br />
V m<br />
<br />
ll ′ (r) =<br />
Note th<strong>at</strong><br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
r i if i > 0 ,<br />
− Z<br />
r<br />
if i = 0 .<br />
(C4)<br />
dΩ [Y ∗<br />
lm(Ωr)V0(r)P0(cos θ)Yl ′ m(Ωr) + Y ∗<br />
lm(Ωr)V1(r)P1(cos θ)Yl ′ m(Ωr)<br />
<br />
+Y ∗<br />
lm(Ωr)V2(r)P2(cos θ)Yl ′ m(Ωr) + · · · ] . (C5)<br />
dΩ Y ∗<br />
lm(Ωr)Pv(cos θ)Yl ′ m(Ωr) = βδ(l−v)+2i,l ′ , (C6)<br />
where i = 0, 1, 2, · · · until l ′ ≤ (l + v).
177<br />
Now, us<strong>in</strong>g the form of Vi(r) <strong>at</strong> small r <strong>in</strong> (C4), each term <strong>in</strong> (C5) contributes<br />
to V m<br />
ll ′ (r) <strong>at</strong> small r as<br />
<br />
m 1<br />
lim Vll ′ (r) = α0 δl,l ′ + α1r<br />
r→0 r<br />
+ α2r 2<br />
<br />
Y ∗<br />
lm(Ωr)P1(cos θ)Yl ′ m(Ωr)<br />
Y ∗<br />
lm(Ωr)P2(cos θ)Yl ′ m(Ωr) + · · · (C7)<br />
1<br />
= β0<br />
r δl,l ′ + β1δl±1,l ′r + β2δl±2,l ′r2 + · · · , (C8)<br />
where only the lowest order terms of r are kept for each coupl<strong>in</strong>g term l ′ and βi<br />
simply represents constants produced <strong>in</strong> <strong>in</strong>tegrals and coefficients <strong>in</strong> Vi(r). For<br />
spherically symmetric potential, the form of ulm <strong>at</strong> small r is<br />
lim<br />
r→0 ulm(r) = Ar l+1<br />
Us<strong>in</strong>g this form, the coupl<strong>in</strong>g terms become <strong>at</strong> small r<br />
<br />
lim V<br />
r→0<br />
m<br />
ll ′ (r)ul′ l ′ 1<br />
m(r) = β0<br />
l ′<br />
r rl+1 + β1r l+3 + β1r l+1<br />
One can see th<strong>at</strong> the coupl<strong>in</strong>g terms follow the form of<br />
(C9)<br />
+ β2r l+5 + β2r l+1 + β3r l+7 + β3r l+1 + · · · . (C10)<br />
r l+1 if j < l ,<br />
r 2j+l+1 if j ≥ l ,<br />
(C11)<br />
where j = 0, 1, 2, · · · , lmax. 2 S<strong>in</strong>ce there exist solutions follow<strong>in</strong>g these form of<br />
coupl<strong>in</strong>g terms, one can f<strong>in</strong>ally write power series form for l<strong>in</strong>early <strong>in</strong>dependent<br />
solutions of coupled equ<strong>at</strong>ions.<br />
2 lmax is the maximum l value used <strong>in</strong> the numerical calcul<strong>at</strong>ion to trunc<strong>at</strong>e the partial<br />
wave expansion <strong>in</strong> (3.3).
u j<br />
lm (r) = rj+|l−j|+1<br />
178<br />
∞<br />
b l ir i−1 . (C12)<br />
The lowest order terms r l+1 s<strong>at</strong>isfy autom<strong>at</strong>ically the boundary conditions <strong>at</strong> small<br />
r. For the higher order terms than r l+1 , there are lower order terms to be coupled<br />
<strong>in</strong> (C1).<br />
i=1
APPENDIX D. S<strong>in</strong>gular value decomposition(SVD)<br />
179<br />
The s<strong>in</strong>gular value decomposition(or SVD) is very popular, used for many<br />
different areas such as image compression but essentially a technique to deal with<br />
m<strong>at</strong>rices which are s<strong>in</strong>gular or very close to s<strong>in</strong>gular and thus for which Gaussian<br />
elim<strong>in</strong><strong>at</strong>ion or other decomposition methods may fail.<br />
Any real m×n m<strong>at</strong>rix A can be decomposed uniquely with a m×m orthogonal<br />
m<strong>at</strong>rix U and an n × n orthogonal m<strong>at</strong>rix V such th<strong>at</strong><br />
A = U · D · V T , (D1)<br />
where D = diag(σ1, σ2, · · · , σn) and follows order of σ1 ≥ σ2 ≥ · · · ≥ σn. The<br />
scalars σ1, σ2, · · · , σn are called the s<strong>in</strong>gular values of A and columns of U and V<br />
are eigenvectors of AA T and A T A, respectively.<br />
Let’s consider a system of l<strong>in</strong>ear equ<strong>at</strong>ions.<br />
A · x = b . (D2)<br />
The nullspace is def<strong>in</strong>ed as a subspace of x such th<strong>at</strong> A · x = 0 if A is a s<strong>in</strong>gular.<br />
The range is def<strong>in</strong>ed as a subspace of b which can be obta<strong>in</strong>ed by mapp<strong>in</strong>g x with<br />
A. The dimension of the range is actually the rank of A. The dimension of the<br />
range is also can be obta<strong>in</strong>ed by subtract<strong>in</strong>g the dimension of the nullspace from<br />
m. If A is nons<strong>in</strong>gular, its range is all of the vector space b. If A is s<strong>in</strong>gular, the<br />
rank r is less than m and σr+1 = σr+2 = · · · = σn = 0. Suppose th<strong>at</strong> A is s<strong>in</strong>gular.<br />
If b = 0, any column of V which corresponds to zero σj becomes a solution of<br />
Ax = 0. A solution is <strong>in</strong> this case x = a1vr + a2vr+1 + · · · + an−rvn where vj is a<br />
j column of V and a 2 1 + a 2 2 + · · · + a 2 n−r = 1. If the vector b is not zero and lies <strong>in</strong><br />
the range of A, solutions can be found by<br />
x = V · [diag(1/σj)] · U T · b (D3)
with a condition of the smallest length |x| 2 which removes any redundant solutions<br />
s<strong>in</strong>ce any column of V with a zero σj(solutions correspond to the nullspace) can<br />
be added to x <strong>in</strong> any l<strong>in</strong>ear comb<strong>in</strong><strong>at</strong>ion. One important trick <strong>in</strong> (D3) is to equ<strong>at</strong>e<br />
1/σj to zero if σj = 0. The readers should refer to [34] for more details.<br />
• An example for A · x = 0 (calcul<strong>at</strong>ed <strong>in</strong> M<strong>at</strong>hem<strong>at</strong>ica):<br />
The m<strong>at</strong>rix A is given by<br />
⎛ ⎞<br />
1 3 6 10<br />
⎜ ⎟<br />
⎜ ⎟<br />
⎜ 2 4 5 3 ⎟<br />
A = ⎜ ⎟<br />
⎜ ⎟<br />
⎜ 2 6 12 20 ⎟<br />
⎝ ⎠<br />
4 8 10 6<br />
180<br />
. (D4)<br />
S<strong>in</strong>ce the last two rows of A are just twice the first two rows, the rank of A is 2.<br />
The m<strong>at</strong>rix A can be decomposed us<strong>in</strong>g SVD as mentioned before. U, D, and V<br />
are<br />
⎛<br />
⎞<br />
⎜<br />
−0.390888 0.217271 −0.729017 0.518202<br />
⎟<br />
⎜<br />
⎟<br />
⎜ −0.217271 −0.390888 0.518202 0.729017 ⎟<br />
U = ⎜<br />
⎟<br />
⎜ −0.781776 0.434542 0.364509 −0.259101<br />
⎟<br />
⎝<br />
⎠<br />
−0.434542 −0.781776 −0.259101 −0.364509<br />
⎛<br />
⎞<br />
⎜<br />
30.5886 0 0 0<br />
⎟<br />
⎜<br />
⎟<br />
⎜ 0 8.0212 0 0 ⎟<br />
D = ⎜<br />
⎟<br />
⎜ 0 0 0 0<br />
⎟<br />
⎝<br />
⎠<br />
0 0 0 0<br />
⎛<br />
⎞<br />
⎜<br />
−0.134925 −0.351883 0.695905 −0.611301<br />
⎟<br />
⎜<br />
⎟<br />
⎜ −0.333743 −0.568331 0.265892 0.703503 ⎟<br />
V = ⎜<br />
⎟<br />
⎜ −0.560942 −0.405685 −0.627282 −0.356765<br />
⎟<br />
⎝<br />
⎠<br />
−0.74549 0.623376 0.227011 0.064138<br />
, (D5)<br />
, and (D6)<br />
, (D7)
S<strong>in</strong>ce the rank of A is 2, σ3 = σ4 = 0 which correspond to the nullspace. The<br />
181<br />
calcul<strong>at</strong>ion can be confirmed as<br />
U · D · V T ⎛ ⎞<br />
⎜<br />
1 3 6 10<br />
⎟<br />
⎜ ⎟<br />
⎜ 2 4 5 3 ⎟<br />
= ⎜ ⎟ = A . (D8)<br />
⎜ ⎟<br />
⎜ 2 6 12 20 ⎟<br />
⎝ ⎠<br />
4 8 10 6<br />
Last two columns of V give solutions of A · x = 0, which can be confirmed as<br />
⎛ ⎞ ⎛<br />
⎜<br />
0.695905<br />
⎟ ⎜<br />
−1. × 10<br />
⎜ ⎟ ⎜<br />
⎜ 0.265892 ⎟ ⎜<br />
⎟ ⎜<br />
A · ⎜ ⎟ = ⎜<br />
⎜ −0.627282<br />
⎟ ⎜<br />
⎟ ⎜<br />
⎝ ⎠ ⎝<br />
0.227011<br />
−6<br />
1. × 10−6 −2. × 10−6 2. × 10−6 ⎞<br />
⎟<br />
⎠<br />
. (D9)<br />
⎛ ⎞ ⎛<br />
⎜<br />
−0.611301<br />
⎟ ⎜<br />
−2. × 10<br />
⎜ ⎟ ⎜<br />
⎜ 0.703503 ⎟ ⎜<br />
⎟ ⎜<br />
A · ⎜ ⎟ = ⎜<br />
⎜ ⎟ ⎜<br />
⎜ −0.356765 ⎟ ⎜<br />
⎝ ⎠ ⎝<br />
0.064138<br />
−6<br />
−1. × 10−6 −4. × 10−6 −2. × 10−6 ⎞<br />
⎟<br />
⎠<br />
. (D10)<br />
The right hand sides <strong>in</strong> the above equ<strong>at</strong>ions show very small values which are<br />
actually zero with<strong>in</strong> the numerical tolerance. Note th<strong>at</strong> for any column vj of V ,<br />
v T j · vj = 1.
APPENDIX E. Behavior of phase shifts for E → ∞<br />
FIGURE 6.1. Behavior of phase shifts for k → ∞.<br />
182<br />
The high energy behavior of phase shifts is tested numerically with a hydrogen<br />
impurity and a background density of 0.0025. A discussion of phase shifts <strong>in</strong><br />
Coulomb-like potentials can be found <strong>in</strong> several places. [31] [35] [36]<br />
For a potential of the type [36]<br />
V (r) = − ag(λr)<br />
r<br />
, (E1)<br />
approxim<strong>at</strong>e expressions for high energy phase shifts are, up to a first order,<br />
δl(k) δcl + a<br />
[ln(2k/λ)] , (E2)<br />
k
where δcl is the po<strong>in</strong>t-Coulomb phase shift, which is, as k is <strong>in</strong>creas<strong>in</strong>g, vanish<strong>in</strong>g<br />
much faster than the second term <strong>in</strong> (E2). The results <strong>in</strong> Fig. 6.1 are obta<strong>in</strong>ed<br />
with the assumption, λ = 1/Rm, which is correct for g(λr) = 1 − r/Rm. 3<br />
183<br />
One can see <strong>in</strong> Fig. 6.1 th<strong>at</strong> numerically obta<strong>in</strong>ed results well agree with the<br />
analytic form.<br />
3 The potential is zero for r ≥ Rm.
APPENDIX F. Sc<strong>at</strong>ter<strong>in</strong>g length and bound st<strong>at</strong>e energy<br />
FIGURE 6.2. Sc<strong>at</strong>ter<strong>in</strong>g length versus 1/ |EB|, where EB is the bound st<strong>at</strong>e<br />
energy rel<strong>at</strong>ive to the conduction band m<strong>in</strong>imum.<br />
184<br />
It is shown <strong>in</strong> 3.6 th<strong>at</strong> the sc<strong>at</strong>ter<strong>in</strong>g length a can be rel<strong>at</strong>ed to the bound<br />
st<strong>at</strong>e energy as <strong>in</strong><br />
a = r0 + 1<br />
kB<br />
. (F1)<br />
If the effective potential range r0 is sufficiently small, as <strong>in</strong> the screened cases, the<br />
sc<strong>at</strong>ter<strong>in</strong>g length can be estim<strong>at</strong>ed from the bound st<strong>at</strong>e energies us<strong>in</strong>g (F1).<br />
with (F1).<br />
Fig. 6.2 shows th<strong>at</strong> the numerical results calcul<strong>at</strong>ed <strong>in</strong> this work are consistent