X-ray and Neutron Scattering from Crystalline Surfaces and Interfaces

X-ray and Neutron Scattering from
Crystalline Surfaces and Interfaces
Paul F. Miceli,
Department of Physics and Astronomy
University of Missouri-Columbia
National School for X-ray and Neutron Scattering
Argonne and Oak Ridge National Laboratories
June 23, 2011
Many thanks to students and collaborators:
W. Elliott, C. Botez, S.W. Han, M. Gramlich, S. Hayden, Y. Chen,
C. Kim, C. Jeffrey, E. H. Conrad, C.J. Palmstrøm, P.W. Stephens;
and to NSF and DOE for funding.

How Crystal do we growth go from from this a vapor
Nucleation
Edge
diffusion
Step-Edge Barrier
Terrace diffusion

Morphology atomic scale mechanisms
Cu/Cu(001)
Zuo & Wendelken

To this ?
Y. Taur & T. H. Ning

Interplay between two regimes of Length Scales
Interatomicdistances
Structure, physics, chemistry Mechanisms
Mesoscale Nanoscale
Morphology Mechanisms

Example:
Rotation of graphene planes affect electronic properties
Graphene made from SiC
J. Phys.: Condens. Matter 20 (2008) 323202

Buried Interface Structure
to understand the growth and function of materials
http://www.tyndall.ie/research/electronic-theory-group/thin_film_simulation.html

Morphology atomic scale mechanisms
Nanoclusters
Interface roughness
Nucleation/growth/coarsening at interfaces
Crystalline morphology
Atomic interfacial structure
Vacancy clusters

Unique Advantages of X-ray Scattering:
Atomic-scale structure at a buried interface
Morphological structure at buried interfaces
Subsurface phenomena
Strains and defects near a surface
Accurate statistics of distributions
(eg. Island size distributions)
Neutrons: low intensity- limited to reflectivity
Soft Matter and Bio materials; H 2 O & D 2 O
Magnetic materials

Objective
An introduction to surface scattering techniques
Build a conceptual framework
Reciprocal Space is a large place: where do we look?

Scattering of X-rays and Neutrons:
2
k
2
E k
2
n
2
r E
Helmholtz Equation
X-rays Neutrons
0
n(r)=inhomogeneous refractive index
2 2
r
2
2m
2
k
m
EV 0
2
n 2
b
Refractive Index for neutron: r
1 V r
1
r
monoatomic Scattering length density: rb
r N
One language for both x-rays and neutrons
b
number density
scattering length:
re
f Q x rays
b
tabulated
neutrons

ki
k f
ki
f
f k
Wavevector Transfer:
Q k
k
4
Q sin
~
i
2
d
Probing Length Scale

Log Intensity
0
d
~
Regimes of Scattering
(Consider specular reflection)
2
Q
2
Q G
1 2
3 3
2
G
a
1. Grazing angle reflectivity: strong scattering d>>interatomic distances
Exact solution required. Neglect atomic positions: homogeneous medium
2. Bragg region: strong scattering; d~interatomic distances = a
Exact solution required. Atomic positions needed. Similar to e - band theory.
3. Everywhere else: weak scattering
Born approximation simplification. Atomic positions required.
d
~
ki
Q
Surface information 3 3D: surface info is
everywhere except
at Bragg points
Q
k f

Grazing Angles: Refraction and Total Reflection
Use average refractive index:
d>>a: consider homogenous medium
2
2
b
n 1
b
1
1
2
n 1
i
With absorption:
sin
Snell s Law:
2
2
sin 2
Critical Angle for
Total Reflection:
Wavevector transfer:
2 2
Q Q
Q
2
c
2
c
k z
(~ 10
5
Q
ki
)
kx f
k
k
Only k z component is affected by the surface.
k x is unchanged.
Q 16
2
c
b
i
k z
k z
k x
k x

Calculation of reflectivity
Q
Reflectivity, R, can be calculated, exacty,
by matching boundary conditions at
the interfaces.
Include an interfacial transition
Multi-slice method:
Numerical simulations

M-layer
http://www.ncnr.nist.gov/reflpak/

Born Approximation works if the
reflectivity is not too large.
R
2
Q Q
Q Q
4
2
Q
R
Qc
Born approx.
Exact
Single Interface
Q
ki
k
f
k
i

Differential Scattering Cross Section
( Born Approximation or Kinematic Approximation )
Reflectivity:
1 d
R d
A d
inc
A
d
P
d
2
S Q P
A Q
P is the polarization factor (x-ray case)
S(Q) is the structure factor
A(Q) is the scattering amplitude
f(Q) is the atomic form factor
is the scattering length density
b
iQr
iQ
r
Q d r r e be
3
Q
b
r
r
b r e f for x-rays or tabulated for neutrons

k
Grazing Incidence Diffraction
k
c
3D Diffraction
If or f
i
Q k
k
k
f
sin
i
2
are near grazing:
refraction
transmission
T ,
i f T
c
sin
2
2
Q
Evanescent wave

Perpendicular to Surface: internal Q and external Q are different
x
Q
Q
y
Qz
Q
Q
x
y
Parallel to Surface: internal Q and external Q are same
k
k
( f
x
( f
y
)
)
k
k
( i)
x
( i)
y
Q
H. Dosch, B.W. Batterman and D. C. Wack, PRL 56, 1144 (1986)

1
Im
Q z
H. Dosch, PRB 35, 2137 (1987)
Penetration Length
i c

H. Dosch, PRB 35, 2137 (1987)
Transmitted Beam Amplitude

Q
S
Fe 3 Al
H. Dosch, PRB 35, 2137 (1987)
Distorted Wave
Born Approximation
refraction
transmission
is the structure factor using
the internal wavevector transfer

A Six-Circle Diffractometer
H. You, J. Appl. Cryst. 32, 614-623 (1999)
Extra Angles give flexibility
Constraints are needed
Common working modes:
fixed
i
i
f
fixed
f

UHV Growth and Analysis Chamber
At Sector 6 at Advanced Photon Source
UHV 10 -10 Torr
Evaporation/deposition
Ion Sputtering
LEED
Auger
Low Temp: 55K
High Temp: 1500 C
Load Lock/sample transfer

Liquid Surface Diffractometer
M. Schlossman et. al., Rev. Sci. Inst. 68, 4372 (1997)

David Vaknin, Ames Lab

What is a crystal truncation rod?
First consider:
Large crystals; rough and irregular boundaries
Boundaries neglected
Real space
2
N 1
N 1
N 1
2 NQ a
iQ an iQ bn iQ cn sin 2
Q e e
2 Q a
sin
x x
y y
z z
S e
n
x
0
n
y
0
Q G
G
Large crystal
n
z
0
G is a reciprocal lattice vector
x
2
x
sin
sin
2
NQ b
2
Q b
2
y
2
y
sin
sin
Reciprocal space
Bragg points
2
NQ zc
2
Qc z
2
2

Now consider a crystal with one atomically flat boundary
z
Real space
Large crystal; flat boundary
Neglect boundary atz S
Rp
N 1
2 n iQ
cn
iQ
RR
Q b
e
e
n
z
0
z
1 N
iQzc
2
4sin
2
z
Very small attenuation
2
z
R R
p p
Reflected wave vanishes
N iQzcN
1
e
1
lim lim
c
1e
Q z
2
p
p
~
p
1
z
0
Crystal Truncation Rod Factor
1
2
Q G
2
c z
z

By neglecting the lateral boundaries:
e
R R
S
p p
and
Q
iQ
p
R
p
RpRp N
irr e
e
iQ
R
Rp
iQ
R
p
2
2
N
s
c
p
b
p
Q p Gp
2
2
s
c
G
N irr = the number of irradiated atoms at the surface
s = area per surface atom
c
G = an in-plane reciprocal lattice vector
p
1
p
p
p p
2
irr
2 Qzc
4sin
2 G
G
CTR
Intensity falls slowly
p
Q
Narrow reflection in-plane

Crystal Truncation Rods
Specular rod
(Q p =0)
Reciprocal Space
Crystal Surface
Q z
Q p

Crystal Truncation Rod Scattering for
Specular Reflection from the Ag(111) Surface
geometry CTR
1
Q
1
2 2
z z Q
G 0
geometry
Atomically smooth
Rough, 3Å
1
CTR
2
Qz z
1
2 QG G (
111)
Elliott et. al. PRB 54, 17938 (1996)

R
Reflectivity:
R
A
Q S d
1
A
inc
b
p
2 2
f inc c 2
d
k
2 2 Qzc
A s sin
2
d
Q
2
sin
p
f
Qz
QG
irr
p
2
k
d
2
sin
Q
For specular reflections:
R
(Two geometric factors of 1/Q)
spec
1
s
2
c
Q
2
z
4
Airr 2
kA Q
inc
z
Q 2
G
p
k
p
1
sin
2
f Qz
kF
f
Qy
Qx
Q c 4
2
QG 2
2 2
4 2
b
c c
2 Qzc
2 2 Qzc
2
2 4Qz
sin 2 Qz
z
sin
z

Step ordering
Q z
(typically semiconductors)
Miscut Surfaces
Tilted Rods
Q p
Q z
ns ns
G
n
G 0
s
L
Step bunching
(e.g. noble metals)

Specular
Diffuse
Q p
G
Qz
Q p
Q p
Angular Width of Specular Decreases with Q z
Q p
Q
Q
z
p
L~9000Å
Can determine terrace size, L
Q p
2
L

Specular Reflection from the Ag(111) Surface
Correct Crystal Truncation Rod Scattering for Terrace Size
Elliott et. al. PRB 54, 17938 (1996)

Special location along CTR: anti-Bragg
Bragg Position:
Q z
N 1
n 0
z
Q z
c
c
e
2
m
i 2n
m
2
z
1
2
1
0
N
i
nz
e
nz
For a smooth surface
N
Anti-Bragg Position:
(m odd)
2
1 atomic layer
Q z
Anti-Bragg Positions
out of phase
Q p

Synchrotron
X-ray Beam
In situ vapor deposition in UHV
evaporator

Specular Anti-Bragg Intensity
Nucleation and Growth
Step flow
multilayer
Layer-by-layer
Elliott et. al. PRB 54, 17938 (1996) Adv. X-ray Anal. 43, 169 (2000)
Specular and Diffuse
coverage
1.0 ML
0.5 ML
0 ML
1.5 ML
3 ML
2.5 ML
2.0 ML
Qp

A
Diffuse Scattering Q z
2
* iQrr
Q bb
e
r
r
r
r
mR
pmRp 2
iQp
Rp
R
p
iQzc
nn b e
e
Rp Rp
nn
2
Nirr
b
iQp
R
p iQz
h Rp
R
p
h
Rp
e e
iQ c
2
R
1e
z Rp
Neglect lateral boundaries
Caused by lateral structure
CTR factor
FT of average phase difference
due to lateral height-differences
h R
p
Rp
p
scan
Q p
Q p dependence
=S T (Q p )

S
S
Transverse Lineshape
Far regions of surface
Rp
Uncorrelated heights
hRR
hR p p
p
R
iQ 2
e
z
e
iQzh
Uncorrelated Roughness @ Large Distance Gives Bragg:
Diffuse
T
Bragg
T
Q QG
p
p
2
2
iQzh
p p e
s
c
Short-Range Correlations Give Diffuse Scattering:
2
p p
z p p p
iQzh
p
e
R
Rp
iQ
R
iQ hRRhR Q e e
p
2

Two Component Line Shape: Bragg + Diffuse
S
T
Bragg
Diffuse
Bragg
Diffuse
Q S
Q S
Q
p
T
Q p
p
T
Bragg due to laterally uncorrelated disorder at long distances
Diffuse due to short-range correlations
p
Q z
scan
Q p

Layer-by-layer growth
SpecularBragg Rod: intensity changes with roughness
Strong inter-island correlations seen in the diffuse
D
2
D
Qx
coverage
1.0 ML
0.5 ML
0 ML
1.5 ML
3 ML
2.5 ML
2.0 ML
Qx
Adv. X-ray Anal. 43, 169 (2000)

Attenuation of the Bragg Rod and Surface Roughness
If height fluctuations are Gaussian: is rms roughness
e
iQ
z
h
2
e
Q
2
z
2
But crystal heights are discrete for a rough crystal:
e
iQ
h
2
z e
2
4
sin
2
c
2
Q
zc
2
Binomial distribution (limits to a Gaussian for large roughness)
Preserves translational symmetry in the roughness
h
Physica B 221, 65 (1996)

Sharper interface (real space) gives broader scattering
Gaussian roughness does not give translational symmetry
e
Q
2 2
z
(continuous)
No roughness
e
2
4
sin
2
c
2
Q
z
2
c
(discrete roughness)
Real space
z

Bragg
Diffuse
S
Q p
reciprocal space
Bragg
Q p
Q
Q p
1/Q p
Bragg is narrow:
it samples laterally uncorrelated roughness at long distances
b
1e
Transversely-integrated scattering shows no effect of roughness:
2
d Qp
S
2 2
2
iQ
z
c
2
e
b
Q
2
1e
4
sin
2
c
2
iQ
1/Q p
z
c
Q
zc
2
real space
(for 1 interface)

In practice, at every Q z the diffuse must be subtracted from
the total intensity to get the Bragg rod intensity:
Bragg
Diffuse
Q p
I total
Bragg rod intensity
I diffuse

X-ray Reflectivity from Si(111)7x7
small vertical displacements
1 st 3 atomic layers
oscillatory features
0
j
3
dj
e
j1
j
iQ
iQz z j
e
Bulk CTR + layers
z
Robinson & Vlieg, Surf.Sci 261, 123 (1992)
Qz

What would we expect from a thin film?
1 st let s recall Young s slit interference

Recall
N-Slit Interference and Diffraction Gratings
Principle maxima
d sin = m

Principle maxima always in
the same place for N slits:
But they narrow: width ~1/N
Double Slit
(no subsidiary maxima)
width ~ 1/N
Principle maxima
d sin = m
(N-2 subsidiary maxima)
1 subsidiary maximum
2 subsidiary maximum
N large:
Weak subsidiary maxima
Sharp principle maxima

5-slit interference of x-rays from 5 layers of atoms
Miceli et al., Appl. Phys. Lett. 62, 2060 (1992)

Reciprocal Space
Q z
Thin Films
Q x
Real Space
substrate

Q z =2L/c
Ag/Si(111)7x7
Specular Reflectivity: 0.3ML Ag/Si(111)7x7
Sensitive to Ag-Si distance
0.3 ML Ag
in 7x7 wetting layer
Si

specular
[0 H 0]
[H 0 0]
rods from 7x7 reconstruction
(thin surface layer)
L
Si In-Plane Rods
Si rods (from bulk crystal)
In plane
(6/7,0/7)
(8/7,0/7)

Specular Reflectivity: 0.9ML Ag/Si(111)7x7
=0.9ML
T=300K
Mainly 3-layer islands
Q z =2L/c
Ag/Si(111)7x7
Exposed Surface Frac
Height (ML)
3-layer islands
2ML above wetting layer
Wetting layer

=1.8ML
Q z =2L/c
Ag/Si(111)7x7
Specular Reflectivity: 1.8ML Ag/Si(111)7x7
Exposed Surface Frac
Height (ML)
Mainly 4-layer islands

Ag islands on a Ag 7x7 wetting layer: FCC = 2 ML
or Ag islands all the way to the substrate? FCC = 3 ML
Ag7x7 wetting layer
Si
?
Si

[-H 0 0]
[0 -H 0]
Ag(111) rods
[0 0 L]
[H -H 0]
[-H H 0]
[0 H 0]
[H 0 0]
Hex coordinates
(1 0 0) hex =1/3*(-4 2 2) cubic
(0 1 0) hex =1/3*(-2 -2 4) cubic
(0 0 1) hex =1/3*(1 1 1) cubic
Fcc 3 fold symmetry
One group: (1 0)rod=(-1 1)rod=(0 -1)rod
Another group: (0 1)rod=(-1 0)rod=(1 -1)rod

Specular reflectivity and rod give same thickness:
Island is FCC Ag all the way to the substrate
Specular Reflectivity
Specular Reflectivity
Probes Si substrate, Ag
wetting layer and Ag island
Ag (1 1) rod
Ag truncation rod
Only probes FCC Ag

Quantum-Size-Effects:
Pb Nanocyrstals on Si(111)7x7
Discoveries:
anomalously (10 4 ) fast kinetics
Non-classical coarsening
Unusual behavior:
fast growth => most stable
structures
Quantum Mechanics Influences Nanocrystal Growth
C. A. Jeffrey et al., PRL 96, 106105 (2006)
Si(111) substrate
Disordered
Pb wetting layer

metal
insulator
Pb
Si
F. K. Schulte, Surf. Sci. 55, 427 (1976)
P. J. Feibelman, PRB 27, 1991 (1983)
Electrons in a box
Electrons will Exhibit Discrete
Quantum Mechanical Energies
Electrons are confined to the metal
physchem.ox.ac.uk

Integrated Intensity
Si(111)
Si(111)
Pb(111)
Pb(111)
l (r.l.u.)
Reflectivity
Pb(222)

Islands consume the wetting layer
& move away from the interface
Specular Intensity Pb(111) (au) Intensity
Pb
Si(111)
Displacive Transition
Coverage, (ml)
C.A. Jeffrey et al., J. Superlat. & Micro. 41, 168 (2007)
0.4Å

Charge Density Oscillations
Due to Quantum Confinement
Pb/Si(111)
Czoschke et al., PRL 91, 226801 (2003)
Electron density in Sommerfeld model:

Coarsening
laist.com
greeneurope.org
Rain Drops On
Your Winshield

Surface Chamber
Advanced Photon Source
Mean island separation, :
L
2
L
Pb(111)
Experiment:
Deposit Pb (1.2 to 2 ml) at 208K
Island Density: Measure the island 1 density vstime
(flux
n
off)
2
L
Intensity
X-Rays Incident
Transverse Scan: In-plane Info.
CCD Image
Q x
Pb Source

Island Density
t
n t n
1
0
High Initial Density
Low Initial Density
Time
2
, m
m2 0,
1,
2
Long time: independent
of initial conditions
t n t
n 0
n
1
0
Relaxation time depends
only on the initial density

does not conform to the classical picture!
C. A. Jeffrey et al., PRL 96, 106105 (2006)
Island densities do not approach each
other at long times:
QSE Non-Oswald
Breakdown of Classical Coarsening
Time constant ~ 1/Flux
Strong flux dependence!
Unexpected!
Anomalously fast relaxation
~1000x faster than expected!
allows equilibrium!
n
1.2 ML of Pb at 208K
at various flux rates
t
n
0
t
1

Reciprocal Space is Superb for Obtaining
Good Statistics of Distributions
0.01 1 100
Equivalent ML Time = t*F
Flux
0.03 ml/min
0.14
0.28
0.56
1.5

Psuedomorphic Laminar Structures
Reciprocal Space
Q z
t 2
Q x
d 3
23
d 3
d 2
12
d 2
d 1

Averaging with Overlapping Rods
Sum 1 column of atoms at R p :
A
c
( R , Q
p
z
)
FT all columns:
S(
Q)
A ( R , Q
R
N
irr
p
c
p
j(
R )
z
) e
p
b
j
e
iQ
z
z
j
= + +
2
iQp
Rp
*
A
c ( Rp
, Qz
) Ac
( R
p
R R
p p
,
Q
iQp
R
p
*
e A
c ( Rp
Rp
, Qz
) Ac
( Rp
, Qz
)
Rp
R
p
= FT of an amplitude-amplitude correlation function
z
) e
iQ
(
R R
)
p
p
p

S
Familiar short-range order diffuse and
long-range uncorrelated disorder in the Bragg
Q
N irr
S
R
p
e
Bragg
iQ
R
p
Bragg:
Transversely
Integrated:
p
A
c ( Rp
, Qz
)
Rp
A
c ( Rp
Rp
, Q
Diffuse
Q S
Q
z
z
2
)
A
*
c
( R,
Q
p
z
)
R
p
A
c
( R,
Q
S
Bragg Qz
A ( R,
Q )
c p z
2
Rp
d Q
2
p S Q
2
Ac
( Rp
, Qz
)
R
p
p
z
)
2
R

Mosaic crystal gave transverse integration: must model with R Int
1000
100
10
R Integrated
Miceli,Palmstrom,Moyers, APL 61, 2060 (1992)
R specular
P. Miceli, in Semiconductor, Interfaces, Microstructures and Devices.
Ed. Z.C Feng (IOP Publishing, Bristol, 1993) pp. 87-114

Summary
Scattering Scattering from surfaces involves a range
of different types of measurements
Materials Materials research problems require
information on a broad range of length
scale from atomic to mesoscale
Unique Unique ability of x-rays: x rays: surface and
subsurface structure simultaneously

X-ray Diffraction Text Books:
Bibliography
B. E. Warren, X-ray Diffraction (Dover Publications, New York, 1990). Originally published by
Addison-Wesley (1969).
J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics (Wiley, New York, 2001).
U. Pietsch, V. Holy, T. Baumbach, High Resolution X-ray Diffraction: From Thin Films to
Lateral Nanostructures, (Springer, 2004).
R. W. James, The Optical Principles of the Diffraction of X-rays (Cornell University Press, Ithaca,
New York, 1965). Now available from Oxbow Press.
A. Guinier, X-ray Diffraction in Crystals, Imperfect Crystals, and Amorphous Bodies, (Freeman,
San Francisco, 1963).
W. H. Zachariasen, Theory of X-ray Diffraction in Crystals, (Wiley, New York, 1945) (and
subsequently by Dover in 1967).
J. M. Cowley, Diffraction Physics, (North-Holland, Amsterdam, New York, 1981).
Engineering-type books on x-ray diffraction:
B. D. Cullity, Elements of X-ray Diffraction, (Addison-Wesley, Reading MA, 1978).
H. P. Klug and L. E. Alexander, X-ray Diffraction Procedures (Wiley, New York, 1974).
Useful Reference Books:
International Tables of X-ray Crystallography Vol. 1, 2, 3, 4 and Vol. A (Published for The
International Union of Crystallography by Kluwer Academic Publishers, 1989).
P. Villars and L. D. Calvert, Pearson’s Handbook of Crystallographic Data for Intermetallic
Phases, (American Society for Metals, Metals Park, OH, USA, 1986).
Laue Atlas, Ed. Kernforschungsanlage Jülich (Wiley, 1975).
Elementary treatments of crystallography in solid state textbooks, such as
C. Kittel, Introduction to Solid State Physics, 7 th ed. (Wiley, 2005).
M. P. Marder, Condensed Matter Physics (Wiley, 2000) (nice treatment of quasicrystals)
N. W. Aschroft and N. D. Mermin, Solid State Physics (Saunders College, 1976).
1

Excellent (in depth) treatment of scattering from defects:
M. A. Krivoglaz, Theory of X-ray and Thermal Neutron Scattering by Real Crystals, (Plenum,
New York, 1969).
M. A. Krivoglaz, X-ray and Thermal Neutron Diffraction in Nonideal Crystals, (Springer, 1996)
M. A. Krivoglaz, Diffuse Scattering of X-rays and Neutrons by Fluctuations, (Springer, 1996)
Neutron Scattering:
G. L. Squires, Introduction to the Theory of Thermal Neutron Scattering, (Cambridbge
University Press, Cambridge, London, New York, 1978).
G. E. Bacon, Neutron Diffraction, (Clarendon Press-Oxford, 1975).
S. W. Lovesey, Theory of Neutron Scattering from Condensed Matter, (Clarendon Press-Oxford,
1984). Volumes 1 and 2.
Reviews Articles of Surface Scattering
R. Feidenhans’l, Surf. Sci. Rep. 10, 105 (1989).
I. K. Robinson and D. J. Tweet, Rep. Prog. Phys. 55, 599 (1992).
P. F. Miceli, in Seminconductor Interfaces, Microstructures and Devices: Properties and
Applications, ed. Z. C. Feng, (IOP Publishing, Bristol, 1993) p. 87-116.
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