Knight Shift in Metals

(1)
Nuclear Magnetic Resonance (NMR)
Part 2: Knight Shift in Metals
shift of MNR resonance line due to polarisation of conduction electrons.
Consider polarisation of conduction electrons in a magnetic field B 0 .
B0 0
E
spin antiparallel B
spin parallel B
EF
}
2 B B0
E
F
k
1 meV
110eV
excess of electrons with s B
0
M n n
net magnetisation
0
e
B
1
n
f E DE B
B0
dE
2
f E
electron density parallel and antiparallel to B0
1
E
EF
1exp
"Fermi distribution function"
kT B
f E
k
B
T
E
D E B D E D E
B
0
B
E F

(2)
e
2
B
0
2
B
M f E D E dE
integration by parts
0
B f
E D E dE
0
DE
F
2
3N
Me B B0
DEF DEF
2E
T independent since E k T
F
0
F
pauli
Me
3N
B
2E
2
B
2
B
F
D E
F
Pauli spin susceptibility of conduction electrons
T-independent Knight shift
e at the
nuclear site!
this is not necessarily the average one.
15
Nuclei are very small a 10 m . They are surrounded by a cloud of polarized
conduction electrons.
however, the Knight shift is determined by magnetic field B r 0
Need to sum up the contributions to the local magnetic field at the nucleus site.
Consider homogenously magnetised sphere of magnetisation M
2
out 3
e r
r
B r
r a
5
r
in 2
B r
r a
3
a
have to consider lima
0
dipolar terms
"contact-term"

(3)
volume
in 3 2
4 3 8
B d r const a
3
a 3 3
so-called contact term:
8
3
r
2
for polarized electrons: er
0 el 0
e Sz
density of electrons at the nucleus site
Hhf N Be
2
3erNrNe
r
5
r
8
2
el 0 N re
r
r
3
S I
N e el
Paschen-Back regime
N e Sz Iz
Me
with e Sz e
N
pauli
B0
N
N S D E
e
2
pauli z B F
B0
N = electron density
Knight Shift K
0
0
Be
8 el
1
K B 3 N
pauli
Important: measure susceptibility at nuclear site.
Magnetic impurity phases are not important.
On the other hand, if I have a pure sample can determine
NMR
pauli
macroscopic
pauli
N
0 2
Examples: Knight Shift measurements in superconductors
B
B 0
SC
normal metal
T c
T

(4)
Superconducting state below T C
All electrons condense into a single state described by a macroscopic wave function.
i r
r r e
0
Fermi Dirac statistics does not apply!
electrons form pairs that are described by Bose statistics
Cooper-pairs, mediated by electron-phonon interaction
Concerning the spin there are two possibilities:
S 0
in most cases
S 1
spin-part
odd
even
orbital part of wavefunction
s- d- wave
L 0,2,...
even
with respect to inversion
p- f- wave
L 1,3,...
odd
in most superconductors, especially the classical ones:
SC
pauli 0 at T 0
SC
2
density of the SC condensate
T
K
T independent in the normal state D E
F
T c
T
K
0 for T 0
pairs

(5)
We can also learn about the symmetry of the orbital part of the wave function,
s-wave or d-wave:
–
s
wave
+
+
+
+
+
+
–
d
wave
Consider energy gap at the Fermi-surface:
energy required to break up Cooper pairs and create unpaired electrons
s-wave
k y
finite gap over entire
Fermi surface
k x
d-wave
–
nodes where the gap vanishes
electrons can be excited even at very small energy
+ +
–

(6)
SC
2
swave
d
wave
T c
T
K
d
wave
s
wave
T
This is the case for high T C superconductors
S 1
for example Sr 2 RuO 4 T C =1.5 K
where superconductivity may be mediated by ferromagnetic fluctuations
K
pwave L
1
T c
d wave L
swave L
2
0
T
– advantage: simple experiment that can be performed on polycrystalline samples
– impurity phases are not contributing local probe!

(7)
1
Spin lattice relaxation rate in metals
T1
due to spin-flip interaction between conduction electrons and nuclear spins.
i
e
k ,
nucleus
f
nucleus
k,
e
Concerning nucleus, consider
1
I , this corresponds to a transition
2
+ ½
- ½
E g
B
N
N
Transition rate is determined by Fermi’s golden rule (2nd order perturbation theory)
2
2
W
f H i E E
E
hf N k k
8
2
0
SI
3
with Hhf e e N
hyperfine coupling
constant a
very
small
Consider matrix elements fSIi
i ,
k
SI Sz Iz SyIy SxIy
1
SI
SI
2
1
with Sx
iSy
2
where S , I raising operator
S
, I
lowering operator
f ,
k
electron part
nuclear

(8)
S
S
S or 0
z
0
1
, k
IzSz
IS IS
,
k
2 2
8
fH 0
3
2 2
hf i
el e N
Finally we need to consider the density of available electronic states.
i must be occupied f must be empty
2
f
f E
kB
T
1
f E
E F
E
only this range of
energies contributes
1
W
dE dE D E f E
D E f E
since E E
E
EF
exp
2
kB
T
1
dE dE D E
with f E
E EF
E EF
1exp 1exp
kB
T kB
T
f E
kB
T
E
f
since dE E E
E
2
W
D E k T
F
B
F
8
2
considering the matrix element fHfh i e 0
e N
3 2
2
2
2
8
2
W
e0 e N DEFkB
T
3 2
density of available
states
2

(9)
1
T
1
W W 2W
2
4
8
2
e0 e N DEF kB
T
3 2
2
1
constant D EF e
T T
1
0 4
Korringa-relationship
8
with K
3
e
0
N
2
pauli
2
DE
B
F
1 e
1
T T 4k K
1
B
N
2
Knight shift
Experiment:
1 measurement on YBa Cu O
T
1
2 3 7
YBa 2 Cu 3 O 7

(10)
1
measured –relaxation rate on Cu and O site
T
1
The relaxation rates at all four oxygen sites and
both Cu sites in a 7.0T ( 17 O) or 7.4T ( 63 Cu) field
applied along the c-axis of the oriented powder,
versus temperature, plotted on a log-log scale.
The solid lines indicate a linear relationship
between relaxation rate and temperature. The
inset emphasizes the linear temperature
dependence of oxygen relaxation above T C .
P.C. Hammel et al., Phys. Rev. Lett. 63 (1989),
1992.
Cu
O
Cu
YBa 2
Cu 3
O 6
AF correlated Cu-spins
2
2
9
Cu 3d shell, 1 electron missing S
O
full p-shell
S
O
0
Cu
1
2
Korringa relation is fulfilled for O NMR signal, but violated for Cu NMR signal.
Short range AF magnetic correlations of Cu moments persist in the
superconducting state!
Many open questions
Once more it is very important to distinguish intrinsic signal from extrinsic (i.e.
from impurity phases).