Surface and Interface X-ray Scattering
Tom Trainor (fftpt@uaf.edu)
University of Alaska Fairbanks
SSRL Workshop on Synchrotron X-ray Scattering Techniques in
Materials and Environmental Sciences
Surface and Interface Scattering: Why bother?
• Interface electron density profiles (Å-scale resolution)
• Surface and interface roughness / correlation lengths
• Interface structure/surface crystallography (1-D & 3-D)
• Dependence on chemical/physical conditions
• Growth/dissolution mechanisms and kinetics
• Structure/binding modes of adsorbates
• Structure reactivity relationships
Advantages of x-rays scattering for surface and interface work
• Large penetration depth  experiments can be done in-situ
• Liquid water, controlled atmospheres, growth chamber, etc…
• Study buried interfaces
• Kinematic scattering  relatively straightforward analysis
Disadvantages
• Weak signals in general  need synchrotron x-rays
• Requires order
Outline
• A brief example
• Crystal Truncation Rod – what is it ??
• Influence of surface structure
• Measurements
• More examples
Example: Fe2O3 (0001) Surface Terminations and rxn with H2O
O3-Fe-Fe-R  Non-Stoichiometric, Lewis base
Fe-Fe-O3-R  Non-Stoichiometric, Lewis acid
Fe-O3-Fe-R  Stoichiometric, Lewis acid
Example: Hematite (0001) Surface Terminations and rxn with H2O
Surface scattering (CTR) data
and best fit model(s)
So what?
- Structural characterization of the
predominant chemical moieties
present at the solid-solution interface
 controls on interface reactivity.
What’s a crystal truncation rod? The short version…
Scattering intensity that arises
between bulk Bragg peaks due to the
presence of a sharp termination of the
crystal lattice (i.e. a surface). The
direction of the scattering intensity is
perpendicular to the surface and
sensitive to interface structure.
(20-2)
(200)
n̂
Real
space
(001) surface
c
a
(202)
104
103
Q/2p
I1/2
c*
102
101
-4
Recip.
space
-2
0
L
2
a*
(202)
L
(201)
(200)
(20-1)
4
(20-2)
What’s a crystal truncation rod? The long explanation…
Lets go back to some x-ray scattering basics and recall: the scattered
intensity is proportional to the square modulus of the Fourier transform
of the electron density
Q
r
kr
I det
R
ki
 (r )
E2o re2
2




E E 
FT

r
R2
*
FT  (r )    (r ) ei Qr dV
Master Equation for X-ray Scattering
I  E(R) 
2
f
2
a, n
e
i Qrn
n
• sum over all n atoms at rn
• fa,n are atomic scattering factors
FT  (r )   f a, n ei Qrn
n
f a,n    n (r ) e i Qr dV
Express Q in terms of reciprocal lattice coordinates
Reciprocal Space
Real Space
Q/2p
X-ray
Source
(plane wave)
Q
(-1,1)
kr
ki
(-1,0)
(0,1)
(1,1)
b*
(1,0)
a*
2
(-1,-1)
(1,-1)
(0,-1)
sample
• Q as a vector in real space
Q  k r - ki
|Q|
4p

sin( 2 / 2)
• Q as a vector in reciprocal space
Q  2p G  2p H a*  K b*  L c* 
Q  2p G 
2p
d HKL

Real Space Planes
Reciprocal Space Points
G(1,2)
(-1,1)
(0,1)
(1,1)
b*
(-1,0)
(1,0)
a*
(-1,-1)
(0,-1)
(1,-1)
d(H=0,K=1)
b
a
G  H a*  K b*  L c*
| GHKL |  1/d HKL ,  ( HKL)
d(H=1,K=1)
d(H=1,K=0)
(HKL) defines a plane with intercepts:
bc
abc
ca
b* 
abc
ab
c* 
abc
a* 
a b c
,
,
H K L
a  a*  1
a  b *  a  c*  0
etc...
Substitute for rn (real space coordinates) and Q (reciprocal space coordinates)
rn  R c (n 1n 2 n 3 )  rj (xyz)
• Rc is the origin of the (n1n2n3) unit
cell w/r/to some arbitrary “center”:
Q
kr
ki
n3
R c (n1n 2 n 3 )  n1 a  n 2 b  n 3 c
n2
• rj is the position of the j’th atom in the unit
cell, expressed in terms of its fractional
coordinates (xyz):
rj  x a  y b  z c
• Dot products in sum become
simple to evaluate
R

Q  2p H a*  K b*  L c*

Q  rn  Q  R c  Q  rj
Q  rj  2p ( x H  y K  z L)
Q  R c  2p (n1 H  n 2 K  n 3 L)
Substitute for Q and rn master equation:
E( R)  FT [  (r )] 
i Qrn
f
e
 a,n
kr
n3
Sum over n1
N1 total cells
E
e
R
ki
n
(N1 1)/2
Q
Sum over n2
N2 total cells
i 2p n1H
(N1 1)/2
Sum over n3
N3 total cells
n2
(N 3 1)/2
(N 2 1)/2
m
i 2p n 3 L
i 2p n 2 K
e
(N 2 1)/2
e
(N 3 1)/2
 fa, je
i Qrj - M j
e
j1
Thermal disorder
parameter
Sum over the m
atoms in the unit cell
Simplify to:
E  Fc S1 ( H ) S2 ( K ) S3 ( L)
Slit Function
Structure factor of the unit cell
m
Fc   fa, je
j1
i Qrj - M j
e
S1 
(N1 1)/2
e
i 2p n1H
(N1 1)/2
sin( N1πH )

sin( πH )
 N1 as H  integer
Scattering intensity at a Bragg point: (HKL) are integers
Q/2p
X-ray
Source
(plane wave)
Q
(-1,1)
kr
ki
(-1,0)
(0,1)
(1,1)
b*
(1,0)
a*
2
(-1,-1)
(0,-1)
sample
I  | E |  Fc
2
2
sin 2 ( N1πH ) sin 2 ( N 2 πK ) sin 2 ( N3 πL)
2
2
2
2

F
N
N
N
c
1
2
3
sin 2 (πH ) sin 2 ( πK ) sin 2 (πL)
For (HKL)  integer
(1,-1)
What about the scattering away from Bragg peak (slit functions)
S3 ( L ) 
(N 3 1)/2
i 2p n 3 L
e
(N 3 1)/2
sin( N 3πL)

 N 3 as L  integer
sin( πL)
100
10
N=10
S
5
S3
2
3
80
N=10
60
40
0
20
0
-5
0
0.5
1
1.5
2
2.5
L
-10
0
0.5
1
1.5
2
L
• Intensity is nominal for noninteger values.
• But its not zero if the xtal is
finite size!
2.5
10000
S
8000
2
3 6000
N=100
4000
2000
0
0.5
1
L
1.5
2
2.5
Intensity variation between Bragg peaks as a function of xtal dimension
(H=integer, K=integer).
S 32
8
10
6
10
Q
4
10
kr
N3=1
2
10
0
10
-2
10
-4
10
0
0.5
1
L
1.5
2
ki
Intensity variation between Bragg peaks as a function of xtal dimension
(H=integer, K=integer).
S 32
8
10
6
10
Q
4
10
kr
N3=1
N3=6
2
10
0
10
-2
10
-4
10
0
0.5
1
L
1.5
2
ki
Intensity variation between Bragg peaks as a function of xtal dimension
(H=integer, K=integer).
S
Q
2
3
kr
8
10
6
10
4
10
N3=30
2
10
N3=1
N3=6
0
10
-2
10
-4
10
0
0.5
1
L
1.5
2
ki
Intensity variation between Bragg peaks as a function of xtal dimension
(H=integer, K=integer).
S 32
For N=1 no oscillations,
scattering from a single
layer.
8
10
Oscillations for N>1 due
to interference between xrays scattering from the
top and bottom
1 sin 2 pL
6
10
4
10
N3=30
2
N3=1
N3=6
10
0
10
Intensity variation follows
the 1/sin2 profile
-2
10
-4
10
0
0.5
1
1.5
L
2
At mid-point (anti-Bragg)
the intensity is the same as
from a single layer!
The sharp boundaries of a finite size (i.e. small) crystal results in
intensity between Bragg peaks
However, for a large single crystal in the Bragg geometry a better
model for a surface is a semi-infinite stacking of slabs
5mm
200mm
n3
n2
n1
The crystal in this geometry
appears infinite in-plane, and
semi-infinite along the n3
direction
Return to the sums and take large N1 and N2 and sum n3 from 0 (the
surface) to -
E  Fc
(N1 1)/2
e
(N 2 1)/2
0
i 2p n1H
i 2p n 2 K
(N1 1)/2
e
(001) surface termination
i 2p n 3 L
e

(N 2 1)/2
c  surface normal

n1
0
Fctr   ei 2p n3L 

2
1
(1  e-i 2 p L )
1
-1
0
-2
0
b
a
-1
-2
-3
-4
-5
I N N
2
1
2
2
Fc ( HKL) FCTR ( L)
1
Fctr 
4 sin 2 (pL)
2
2
2
-2
-1
0
1
n2
2
3
n3
This is the origin of the crystal truncation rod:
• For integer H and K intensity is proportional to N1xN2xFctr(L)
• For non-integer H and K, S1 and S2 ~0, i.e. no sharp boundary in-plane
• Therefore, rods only occur in the direction perpendicular to the surface (n3 direction)
(001) surface
10
10
10
n̂
8
Real
space
6
4
1/sin2(pl)
1/4sin2(pl)
10
10
10
2
0
Q/2p
-2
0
0.5
1
1.5
2
L
Fctr lower at anti-Bragg than finite xtal.
Why? Finite xtal has scattering from
two sides, CTR is only from one side.
Recip.
space
c*
a*
(202)
L
(201)
(200)
(20-1)
(20-2)
The scattering between Bragg peaks along a CTR results from a sharp termination
of the crystal, and has a well defined functional form. But what does that tell us
about the interface structure?
I N N
2
1
2
2
2
Fc ( HKL) FCTR ( L)
2
Fc contains all the structure information (e.g. atomic coordinates). But so far we’ve
assumed all cells are structurally equivalent. What if we add a surface cell with a
different structure factor?
n3
1
surface cells
0
Ebulk  N1 N2 Fc, bulk( HKL) FCTR ( L)
-1
-2
-3
-4
E T  E b ulk  E surf
bulk cells
Esurf  N1 N2 Fc, surf ( HKL) ei2p L
Therefore final expression:
I  N N Fbulk,cFCTR ( L)  Fsurf,c
2
1
n
Fc   f je
2
2
i Qrj -M j
e
2
Q  rj (xyz)  2p ( x H  y K  z L)
j1
• In the mid-zone between Bragg peaks FCTR ~1
• Therefore the “bulk” scattering and “surface” are
of similar magnitude between Bragg peaks, ie
sensitive to one bulk cell (modified by Fctr) and
one surface cell
10
10
10
• The “surface” and “bulk” sum in-phase (i.e.
interfere if Fsurf, different from Fbulk)
10
• Near Bragg peak the surface is completely
swamped:
IBragg/ ICTR > 105
10
10
8
6
4
1/4sin2(pl)
2
0
-2
0
0.5
1
1.5
L
2
Influence of surface structure:
L
K
H
Q
2
kf
10
|FHKL |
ki
(0 0 L)
0
10
Surface cell
Bulk cell
0
0.5
1
1.5
L (r.l.u)
2
2.5
3
Known bulk structure and
modifiable surface cell
I
N12 N 22
Fbulk,c FCT R ( L)  Fsurf,c
2
Simulation of CTR profiles for a BCC bulk
and surface cell for (001) surface showing
sensitivity to occupancy and displacements
Influence of surface structure:
Bragg peak
Anti-Bragg
Observe several orders of magnitude intensity variation with changes in surface:
• atomic site occupancy
• relaxation (position)
• presence of adatoms
• roughness
Simulations of Pb/Fe2O3
1000
(10L)
A.
A. Calculations as a function of
surface coverage
Fhkl
100
Pb occupation number
0
0.1
0.3
0.5
10
1
-10
B. Calculations as a function of the
z-displacement (along the c-axis),
the Pb occupation number is fixed
at 0.3.
-5
0
5
10
15
L(r.l.u.)
1000
(10L)
B.
Fhkl
100
Pb displacement
0
-0.5Å
+0.5Å
10
1
-10
-5
0
L(r.l.u.)
5
10
15
Roughness “kills” rod intensity
Scattering between different
height features cause
destructive interference
Robinson b model
s = 0 Å2
10
s = 1 Å2
1.0
s = 10 Å2
s = 50 Å2
0.1
-3
-2
-1
0
1
2
3
L
Distinguish roughness from structure because roughness is
uniform decrease in intensity
Surface scattering measurement:
L
K
H
Q
ki
Surface cell
kf
Sample
kf
Q
ki
Bulk cell
• Goal is to measure the
intensity profile along one or
more rods.
• Sample orientation controls
reciprocal lattice orientation.
• Detector controls Q
Six circle Kappa geometry diffractometer
(Sector 13 APS)
Surface scattering measurement:
Scattered intensity is measured when
the rod intersects the Ewald Sphere
(from Schleputz, 2005)
Multi-axis goniometer allows high degree
of flexibility to access surface scattering
features (from You, 1999)
• Sample motions control direction of rod
• Detector motions control Q
General Purpose Diffractometer (APS sector 13)
• Large Kappa-geometry six circle diffractometer
• Leveling table with 5-degrees of freedom
• High angular velocity (up to 8 deg/sec)
• Small sphere of confusion (< 50 microns)
• On the fly scanning
• Open sample cradle, capable of supporting large
sample environments weighting up to 10kg.
• Liquid/solid environment cells.
• Diamond Anvil Cell (DAC)
• Small UHV Chamber
• High temperature furnace
• Open geometry also allows for mounting solid
state fluorescence detectors and beam/sample
viewing optics on the Psi axis bench
• High load capacity detector arm supports a
variety of detectors
• Point detectors
• CCD and Si based area detectors
• Analyzer crystal for high resolution diffraction
and inelastic scattering
Detector Arm
Entrance
Flight Path
Sample
Environment
Measurement by rocking scans:
Bragg peak
L
Q
Anti-Bragg
K
H
ki
Bragg peak
kr
Single crystal
mineral specimen
• Given a fixed Q rock the sample so the rod cuts through Ewald sphere: provide
an accurate measure of the integrated intensity
• Integrated intensity is corrected for geometrical factors to produce experimental
structure factor (FE) for comparison with theory  e.g. lsq model fitting
• Symmetry equivalents are averaged to reduce the systematic errors
Measurement by rocking scans:
Q
DL
Dh
Scan of rod through resolution
function defined by the detector slits
Int
Dh
Measurement by rocking scans:
Q
Dh
DL
Dh
Scan of rod through resolution
function defined by the detector slits
Int
Dh
Measurement by rocking scans:
Q
DL
Dh
Scan of rod through resolution
function defined by the detector slits
Int
Dh
Measurement by rocking scans:
Q
Dh
DL
Dh
Scan of rod through resolution function
Int
Integrated
Intensity
background
Dh
Generally not too worried about DL since rods are “slowly varying”, but can normalize data to
constant value if the resolution function changes substantially
Pixel array detectors with high dynamic range and fast readout means data collection
speedup 10x or more
CTR
intersecting
Ewald Sphere
TDS from
nearby Bragg
peak
CTR
intersecting
Ewald Sphere
Powder ring
Pilatus 100k Specs: 20bit, ~500x200 pixels
(recently installed at APS sector 13)
http://pilatus.web.psi.ch/pilatus.htm
Direct Rod Scan
Detector
Sample
Q
Rod
An incomplete list of practical details
1.
2.
3.
What do you need:
High quality (mono-lithic) crystal (mosaic kills
intensity)
Sample sizes from 1mm to several cm
High quality surface (roughness kills intensity)
Goniometer and synchrotron
Know your bulk lattice parameters, coordinate
system for surface and Q’s of allowed Bragg peaks
Simulate before you measure
Sample orientation:
Find the optical surface (similar to reflectivity)
Find bulk reflections (usually you know the
approximate direction of the surface normal so
“dummy in” a reflection). Then hunt….
Check your rod intensity and alignment
Miss-cut results in tilted rods: plan your scans
accordingly
Check for reconstruction/surface symmetry
n̂
L
c*
a*
miss-cut surface
L
c*
a*
Reconstruction
I  N12 N 22 Fsurf,c
2
What’s the best way to figure out what rods to measure, what
reflections to look for to align? Make a map!
• What’s the symmetry of
reciprocal space?
• Where are the Bragg
peaks on the rod?
• Whats Q max?
• How far to scan in L?
b*
• Min to max
a*
• What rods to measure?
• (00L) gives you zinformation
• (HKL) gives you
x,y,z information
• Simulate to test
sensitivity
Sample cells/environmental chambers:
- Stable surface can be run in air
- UHV chamber/film growth scattering chamber
- Liquid / Electrochemical cells
- Controlled atmosphere cells
Detector Arm
Entrance Flight Path
Sample Environment
Liquid cells
(a) Transmission and (b) thin film cells
(Fenter 2004)
Data analysis based on least-squares structure factor routine (ROD)
Semiordered
overlayer
Surface
unit cell
Bulk unit
cell
Ghose et al. 2007
Example: Voltage dependant water structure at a Ag(111) electrode surface
Example: Structure of Mineral-Water Interfaces
Example: Hydrated vs. UHV prepared a-Al2O3 (0001) surface
(Eng et al., (2000) Science, 288 1029)
(Guenard et al., (1997) Surf. Rev. Lett., 5 321)
Example: Ordering in Thermally Oxidized Silicon
A. Munkholm and S. Brennan (2004) Phys Rev. Lett. 93 036106
Some new stuff
Algorithms for rapid determination of interfacial electron density profiles
From Saldin et al.
From Fenter et al.
Baltes et. al. (1997) Phys. Rev. Lett
Saldin et. al. (2001-2002) J Phys Cond Matt
Fenter and Zhang (2005) Phys. Rev. B, 081401.
Some new stuff
Anomalous (E-dependant) surface scattering: phase constraints/chemical
information
From Park et al.
Tweet D. J., et. al. (1992) Physical Review Letters 69(15), 2236-9.
Walker F. J. and Specht E. D. (1994) In. Reson. Anomalous X-Ray Scattering, 365-87.
Park C. et. al. (2005) Phys Rev. Lett., 076104
Park C. and Fenter P.A. (2006) J. Appl. Cryst.
References (a very incomplete list)
Reference texts:
Warren B.E. (1969) X-ray Diffraction. New York: Addison-Wesley.
Als-Nielsen J. and McMorrow D. (2001) Elements of Modern X-ray Physics. New York: John Wiley.
Sands D.E. (1982) Vectors and Tensors in Crystallography. New York: Addison-Wesley.
A few surface scattering methods papers:
Robinson I. K. (1986) Phys. Rev. B 33(6), 3830-3836.
( original reference)
Andrews S.R. and Cowley R.A. (1985) J. Phys C. 18, 642-6439. ( original reference)
Vlieg E., et. al. (1989) Surf. Sci. 210(3), 301-321.
Vlieg E. (2000) J. Appl. Crystallogr. 33(2), 401-405. ( rod analysis code)
Trainor T. P., et. al.. (2002) J App Cryst 35(6), 696-701. ( rod analysis code)
Fenter P. and Park C. (2004) J. App Cryst 37(6), 977-987.
Fenter P. A. (2002) Reviews in Mineralogy & Geochemistry 49, 149-220. ( Excellent tech. review)
Reviews
Fenter P. and Sturchio N. C. (2005) Prog. Surface Science 77(5-8), 171-258.
Renaud G. (1998) Surf. Sci. Rep. 32, 1-90.
Robinson I.K. and Tweet D.J. (1992) Rep Prog Phys 55, 599-651.
Fuoss P.H. and Brennan S. (1990) Ann Rev Mater Sci 20 365-390.
Feidenhans’l R. (1989) Surf. Sci. Rep. 10, 105-188.
Coordinate transformations, reciprocal space, diffractometry
You H. (1999) J. App Cryst. 32 614-623.
Vlieg E. (1997) J. Appl. Crystallogr. 30(5), 532-543.
Toney M. (1993) Acta Cryst A49, 624-642.
…. And many more….
Take advantage of periodicity of a crystal to simplify rn
Atoms in a unit cell
Unit cells in a crystal
c
c
n1 -1-2
2
1
3
0
b
a
2
1
r1
0
-1
r2
-2
b
a
-2
-1
0
1
2
3
n2
Position of the j’th atom in the cell is given
by its fractional coordinates:
rj  x a  y b  z c
Position of the (n1n2n3) unit cell is
given by:
R c (n1n 2 n 3 )  n1 a  n 2 b  n 3 c
The position of the n’th atom in the xtal is:
rn  R c (n 1n 2 n 3 )  rj (xyz)
n3
kr
ki
ki

(HKL)
• If Q  integer HKL Bragg’s
condition is satisfied
Q = kr-ki
kr

d
2p
4p

sin( 2 / 2)
d HKL

  2d HKL sin(  )
Bragg’s Law