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Surface and Interface X-ray Scattering

Tom Trainor (fftpt@uaf.edu)University of Alaska Fairbanks

1st Annual 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• and ……

Why x-rays?

• Large penetration depth experiments can be done in-situ

• Liquid water, controlled atmospheres, growth chamber, etc…

• Kinematic scattering relatively straightforward analysis

• Downside?

• Weak signals in general need synchrotron x-rays

Outline

• A brief example

• Crystal Truncation Rod – what the heck is that ??

• Influence of surface structure

• Measurements

• More examples

Example: Hematite (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 is a sensitive function of surface/interface structure.

-4 -2 0 2 4101

102

103

104

L

(001) surface

(20-2) (200) (202)

Ι1/2

c*

a*(200)

(201)

(202)

(20-1)

(20-2)

Q/2π L

Real space

n̂

Recip. space

What’s a crystal truncation rod? The quantitative (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

detector

Rik

rk

)r(ρ

Q

r

dVe )( f inna,

rQr •∫= ρ

[ ] dVe )( )( i rQrr •∫∝ ρρFT

[ ] n i

nna, ef )( rQr •∑∝ρFT

2 i

nna,

2nef E(R) I rQ•∑∝=

Sum is over all atoms at rnfa,n are atomic scattering factors

( )[ ]22

2e

2o*

det R r E I rEE ρFT∝=

c

a

b

r1

r2

Atoms in a unit cell

cbar z y x j ++=

Position of the j’th atom in the cell is given by its fractional coordinates:

cb aR n n n )nn(n 321321c ++=

c

a b0-1-2

12

0-1-2 1 2 3

3

21

-2

-1

0

n1

n2

n3

Unit cells in a crystal

Position of the (n1n2n3) unit cell is given by:

(xyz) )nn(n j321cn rRr +=

The position of the n’th atom in the xtal is:

Take advantage of periodicity of a crystal to simplify rn

Reciprocal Space PointsReal Space Planes

a*

b*(1,0)

(1,1)(0,1)

(1,-1)(0,-1)(-1,-1)

(-1,0)

(-1,1)

G(1,2)

(HKL) defines a plane with intercepts:LKHcba , ,

b

a

d(H=1,K=0)

d(H=0,K=1)

d(H=1,K=1)HKLHKL 1/d || =G

c b ac b a*

××

=•

c b aa c b*

××

=•

c b ab a c*

××

=•

1=• *aa

0 == •• ** ca ba

etc...

)( HKL⊥,

*** cbaG LKH ++=

( )*** 2 2 cbaGQ LKH ++== ππ

ik rk

X-raySource

(plane wave)dete

ctor (I

)

θ2

sample

Q

• Q as a vector in reciprocal space

HKLd 2 2 ππ == GQ

)2/2sin(4 || θλπ

=Q

Express Q in terms of reciprocal lattice coordinates

jcn rQRQrQ •+•=•

) n n n(2 321c LKH ++=• πRQ

) z y x (2 j LKH ++=• πrQ

a*

b*(1,0)

(1,1)(0,1)

(1,-1)(0,-1)(-1,-1)

(-1,0)

(-1,1)

Q/2π

• Dot products in sum become simple to evaluate

n i

nna, ef )]([ )E( rQrR •∑=∝ ρFT

Substitute for rn and Q in the summation over all atoms:

Sum over n1 N1 total cells

Sum over the m atoms in the unit cell

Thermal disorder parameter

Sum over n2 N2 total cells

Sum over n3 N3 total cells

)()()(FE 321 LSKSHSc∝

∑∑∑∑=

•−

−−

−

−−

−

−−

∝m

1j

M- ija,

1)/2(N

1)/2(N

n 2 i1)/2(N

1)/2(N

n 2 i1)/2(N

1)/2(N

n 2 i jj3

3

32

2

21

1

1 eefeeeE rQLKH πππ

∑=

•=m

1j

M- ija,

jjeefF rQc

Structure factor of the unit cell

detector

Rik

rkQ

n3

n2

Slit Function

integer as N

)πsin(

)πNsin(e

1

11)/2(N

1)/2(N

n 2 i1

1

1

1

→→

== ∑−

−−

HH

HS Hπ

We can simplify to:

Scattering intensity at a Bragg point

23

22

21

22

32

22

2

21

222 F

)π(sin)πN(sin

)π(sin)πN(sin

)π(sin)πN(sinF |E| I NNN

LL

KK

HH

cc →∝∝

For (HKL) integer

ik rk

X-raySource

(plane wave)dete

ctor (I

)

θ2

sample

Q

a*

b*(1,0)

(1,1)(0,1)

(1,-1)(0,-1)(-1,-1)

(-1,0)

(-1,1)

Q/2π

0 0.5 1 1.5 2 2.5-10

-5

0

5

10

integer as N )πsin(

)πNsin(e)( 33

1)/2(N

1)/2(N

n 2 i3

3

3

3 →→== ∑−

−−

LL

LLS Lπ

What about the scattering away from Bragg peak (slit functions)

L

N=10

L0 0.5 1 1.5 2 2.5

0

20

40

60

80

100

N=10

0 0.5 1 1.5 2 2.5

2000

4000

6000

8000

10000

L

N=100

3S

23S

23S

Intensity is nominal for non-integer values. But its not zero if the xtal is finite size!

0 0.5 1 1.5 210-4

10-2

100

102

104

106

108

L

Intensity variation between Bragg peaks is more evident on log scale.

N=1N=6N=30

( )Lπ2sin1

23S For N=1 no oscillations,

scattering from a single layer.

Oscillations for N>1 due to interference between x-rays scattering from the top and bottom

Intensity variation follows the 1/sin2 profile

At mid-point (anti-Bragg) the intensity is the same as from a single layer!

5mm

200µm

The crystal in this geometry appears infinite in-plane, and semi-infinite along the n3direction

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

n1

n2

n3

∑∑∑∞−

−

−−

−

−−

∝0

n 2 i1)/2(N

1)/2(N

n 2 i1)/2(N

1)/2(N

n 2 i 32

2

21

1

1 eeeFE LKHc

πππ

)e1(1eFctr 2 i-

0n 2 i 3

LL

ππ

−== ∑

∞−

2CTR

2c

22

21 )(F )(F NN I LHKL∝

c = surface normal

a b0-1-2

12

0-1-2 1 2 3

0

-1-2

-5-4

-3

n1

n3

n2

(001) surface termination

Return to the sums and take large N1 and N2 and sum n3from 0 (the surface) to -∞

)(sin41Fctr 2

2

Lπ=

0 0.5 1 1.5 210-2

100

102

104

106

108

L

1/sin2(πl)

(001) surface

c*

a*(200)

(201)

(202)

(20-1)

(20-2)

Q/2π L

Real space

n̂

Recip. space

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)

1/4sin2(πl)

Fctr lower at anti-Bragg than finite xtal. Why? Finite xtal has scattering from two sides, CTR is only from one side.

surfbulkT EEE +=n3

0

1

-1

-2

-3

-4

surface cells

bulk cells

)( F)( FNN E CTRbulk c,21bulk LHKL=

e )( FNN E i2surf c,21surf

LHKL π=

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?

2CTR

2c

22

21 )(F )(F NN I LHKL∝

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?

Therefore final expression:

2surf,cCTRbulk,c

22

21 F )(FF NN I +∝ L

∑=

•=n

1j

M- ij

jjeefF rQc ) z y x (2 (xyz)j LKH ++=• πrQ

L0 0.5 1 1.5 2

10-2

100

102

104

106

108

1/4sin2(πl)

• In the mid-zone between Bragg peaks FCTR ~1

• Therefore the “bulk” scattering and “surface” are of similar magnitude between Bragg peaks, iesensitive to one bulk cell (modified by Fctr) and one surface cell

• The “surface” and “bulk” sum in-phase (i.e. interfere if Fsurf, different from Fbulk)

• Near Bragg peak the surface is completely swamped:

IBragg/ ICTR > 105

Bulk cellSurface cell

ki kfQ

H K

L

2csurf,CTRcbulk,

22

21 F )(FF NN I +∝ L

Known bulk structure and modifiable surface cell

0 0.5 1 1.5 2 2.5 3

100

102

(0 0 L)

L (r.l.u)

|FH

KL

|

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

Influence of surface structure:

Observe several orders of magnitude intensity variation with changes in surface:• atomic site occupancy• relaxation (position)• presence of adatoms• roughness

-3 -2 -1 0 1 2 3

0.1

1.0

10

L

σ = 0 Å2

σ = 1 Å2

σ = 50 Å2

σ = 10 Å2

Roughness “kills” rod intensity

Robinson β model

Interference between different height features cause destructive interference

Distinguish roughness from structure because roughness is uniform decrease in intensity

1

10

100

1000

-10 -5 0 5 10 15

( 1 0 L )

00.10.30.5

Pb occupation number

L(r.l.u.)

F hkl

A.

1

10

100

1000

-10 -5 0 5 10 15

( 1 0 L )

0-0.5Å+0.5Å

Pb displacement

L(r.l.u.)

F hkl

B.

Simulations of Pb/Fe2O3

A. Calculations as a function of surface coverage

B Calculations as a function of the z-displacement (along the c-axis), the Pb occupation number is fixed at 0.3.

ki

kf

Sample

Q

Six circle Kappa geometry diffractometer (Sector 13 APS)

Bulk cellSurface cell

ki kfQ

H K

L

Surface scattering measurement:

Goal is to measure the intensity profile along one or more rods.

Sample orientation controls reciprocal lattice orientation.

Detector controls Q

An incomplete list of practical details1. 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

2. 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….

3. Check your rod intensity and alignment- Miss-cut results in tilted rods: plan your scans

accordingly- Check for reconstruction/surface symmetry

c*

a*

L

n̂

miss-cut surface

c*

a*

Reconstruction

L

2surf,c

22

21 F NN I ∝

What’s the best way to figure out what rods to measure, what reflections to look for to align? Make a map!

a*

b*

• What’s the symmetry of reciprocal space?

• Where are the Bragg peaks on the rod?

• Whats Q max?

• How far to scan in L?• Min to max

• What rods to measure?• (00L) gives you z-information• (HKL) gives you x,y,z information

Multi-axis goniometer allows high degree of flexibility to access surface scattering features (from You, 1999)

Scattered intensity is measured when the rod intersects the Ewald Sphere(from Schleputz, 2005)

Q

Scan of rod through resolution function ∆η

Int

background

Integrated Intensity

∆L

Generally not too worried about ∆L since rods are “slowly varying” +∆η−∆η

Bragg peak Bragg peak

Anti-Bragg

• 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

ikrk

Single crystal mineral specimen

Q

LK

H

Measurement by rocking scans:

Sample cells/environmental chambers:

- Stable surface can be run in air

- UHV chamber/film growth scattering chamber

- Liquid / controlled atmosphere scattering cell

Sample Environment

Detector Arm

Entrance Flight Path

• 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 based area detectors• Analyzer crystal for high resolution diffraction and

inelastic scattering

Sample Environment

Detector Arm

Entrance Flight Path

General Purpose Diffractometer (APS sector 13)

Liquid cells

(a) Transmission and (b) thin film cells(Fenter 2004)

Example: Voltage dependant water structure at a Ag(111) electrode surface

Example: Structure of Mineral-Water Interfaces

(Eng et al., (2000) Science, 288 1029) (Guenard et al., (1997) Surf. Rev. Lett., 5 321)

Example: Hydrated vs. UHV prepared α-Al2O3 (0001) surface

Example: Ordering in Thermally Oxidized Silicon

A. Munkholm and S. Brennan (2004) Phys Rev. Lett. 93 036106

Some new stuff

Fenter and Zhang (2005) Phys. Rev. B, 081401.Saldin et. al. (2001-2002) J Phys Cond MattBaltes et. al. (1997) Phys. Rev. LettTweet 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 et. al. (2005) Phys Rev. Lett., 076104

Inversion algorithms for rapid determination of interfacial electron density profiles

Anomalous (E-dependant) surface scattering: phase constraints/chemical information

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

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.

ReviewsFenter 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, diffractometryYou 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….