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.

-4 -2 0 2 4101

102

103

104

L

(20-2) (200) (202)

1/2

c*

a*(200)

(201)

(202)

(20-1)

(20-2)

Q/2L

Real space

Recip. space

(001) surfacen̂

a

c

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

detector

Rik

rk

)r(

Q

r

dVe )( f inna,

rQr

dVe )( )( i rQrr FT

n i

nna, ef )( rQr FT

2

i

nna,

2nef E(R) I rQ

• sum over all n atoms at rn

• fa,n are atomic scattering factors

2

2

2e

2o*

det R

r E I rEE FT

Master Equation for X-ray Scattering

*** 2 2 cbaGQ LKH

ik rk

X-raySource

(plane wave)dete

ctor (

I)

2

sample

Q

• Q as a vector in reciprocal space

HKLd

2 2

GQ)2/2sin(

4 ||

Q

Express Q in terms of reciprocal lattice coordinates

a*

b*(1,0)

(1,1)(0,1)

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

(-1,0)

(-1,1)

Q/2

Real Space Reciprocal Space

• Q as a vector in real space

ir k - k Q

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:LKH

cba , ,

b

a

d(H=1,K=0)

d(H=0,K=1)

d(H=1,K=1)

HKLHKL 1/d || G

c b a

c b a*

c b a

a c b*

c b a

b a c*

1 *aa

0 ** ca ba

etc...

)( HKL,

*** cbaG LKH

Substitute for rn (real space coordinates) and Q (reciprocal space coordinates)

detector

R

ikrk

Q

n3

n2

(xyz) )nn(n j321cn rRr

cbar z y x j

• rj is the position of the j’th atom in the unit cell, expressed in terms of its fractional coordinates (xyz):

cb aR n n n )nn(n 321321c

• Rc is the origin of the (n1n2n3) unit cell w/r/to some arbitrary “center”:

jcn rQRQrQ

) n n n(2 321c LKH RQ

) z y x (2 j LKH rQ

• Dot products in sum become simple to evaluate

*** 2 cbaQ LKH

detector

R

ikrk

Q

n3

n2

)()()(FE 321 LSKSHSc

m

1j

M- ija,

jjeefF rQc

Structure factor of the unit cell

Slit Function

integer as N

)πsin(

)πNsin(e

1

11)/2(N

1)/2(N

n 2 i1

1

1

1

H

H

HS H

Simplify to:

n i

nna, ef )]([ )E( rQrR FT

Sum over n1

N1 total cells

Sum over the m atoms in the unit cell

Thermal disorder parameter

Sum over n2

N2 total cellsSum over n3

N3 total cells

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

3

2

2

2

1

1

1 eefeeeE rQLKH

Substitute for Q and rn master equation:

Scattering intensity at a Bragg point: (HKL) are integers

23

22

21

2

23

2

22

2

21

222 F

)π(sin

)πN(sin

)π(sin

)πN(sin

)π(sin

)πN(sinF |E| I NNN

L

L

K

K

H

Hcc

For (HKL) integer

ik rk

X-raySource

(plane wave)dete

ctor (

I)

2

sample

Q

a*

b*(1,0)

(1,1)(0,1)

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

(-1,0)

(-1,1)

Q/2

0 0.5 1 1.5 2 2.5-10

-5

0

5

10

integer as N )πsin(

)πNsin(e)( 3

31)/2(N

1)/2(N

n 2 i3

3

3

3

LL

LLS L

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 2

10-4

10-2

100

102

104

106

108

L

Intensity variation between Bragg peaks as a function of xtal dimension (H=integer, K=integer).

N3=1

23S

kikr

Q

0 0.5 1 1.5 2

10-4

10-2

100

102

104

106

108

L

N3=1N3=6

23S

ki

kr

Q

Intensity variation between Bragg peaks as a function of xtal dimension (H=integer, K=integer).

0 0.5 1 1.5 2

10-4

10-2

100

102

104

106

108

L

N3=1N3=6N3=30

23S

ki

kr

Q

Intensity variation between Bragg peaks as a function of xtal dimension (H=integer, K=integer).

0 0.5 1 1.5 2

10-4

10-2

100

102

104

106

108

L

N3=1N3=6N3=30

L2sin1

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!

Intensity variation between Bragg peaks as a function of xtal dimension (H=integer, K=integer).

5mm

200m

The crystal in this geometry appears infinite in-plane, and semi-infinite along the n3 direction

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 3

2

2

2

1

1

1 eeeFE LKHc

)e1(

1eFctr

2 i-

0n 2 i 3

LL

2

CTR

2

c22

21 )(F )(F NN I LHKL

c surface normal

ab0

-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 n3 from 0 (the surface) to -

)(sin4

1Fctr

2

2

L

0 0.5 1 1.5 210

-2

100

102

104

106

108

L

1/sin2(l)

(001) surface

c*

a*(200)

(201)

(202)

(20-1)

(20-2)

Q/2L

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(l)

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

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?

2

CTR

2

c22

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:

2

surf,cCTRbulk,c22

21 F )(FF NN I L

n

1j

M- ij

jjeefF rQc ) z y x (2 (xyz)j LKH rQ

L

0 0.5 1 1.5 210

-2

100

102

104

106

108

1/4sin2(l)

• 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

• 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 cell

Surface cell

ki kf

Q

H K

L

2

csurf,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

1

10

100

1000

-10 -5 0 5 10 15

( 1 0 L )

00.10.30.5

Pb occupation number

L(r.l.u.)

Fhk

l

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.)

Fhk

l

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.

-3 -2 -1 0 1 2 3

0.1

1.0

10

L

= 0 Å2

= 1 Å2

= 50 Å2

= 10 Å2

Roughness “kills” rod intensity

Robinson model

Scattering between different height features cause destructive interference

Distinguish roughness from structure because roughness is uniform decrease in intensity

ki

kf

Sample

Q

Six circle Kappa geometry diffractometer (Sector 13 APS)

Bulk cell

Surface cell

ki kf

Q

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

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

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

Surface scattering measurement:

• 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

Sample Environment

Detector Arm

Entrance Flight Path

General Purpose Diffractometer (APS sector 13)

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

L

KH

Measurement by rocking scans:

Q

Scan of rod through resolution function defined by the detector slits

Int

L

Measurement by rocking scans:

Q

Int

L

Measurement by rocking scans:

Scan of rod through resolution function defined by the detector slits

Q

Int

L

Measurement by rocking scans:

Scan of rod through resolution function defined by the detector slits

Q

Scan of rod through resolution function

Int

background

Integrated Intensity

L

Generally not too worried about L since rods are “slowly varying”, but can normalize data to constant value if the resolution function changes substantially

Measurement by rocking scans:

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

RodQ

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

2

surf,c22

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

• 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

Sample Environment

Detector Arm

Entrance Flight Path

Liquid cells

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

Data analysis based on least-squares structure factor routine (ROD)

Bulk unit cell

Surface unit cell

Semi-ordered overlayer

Ghose et al. 2007

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 -Al2O3 (0001) surface

Example: Ordering in Thermally Oxidized Silicon

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

Some new stuff

Baltes et. al. (1997) Phys. Rev. LettSaldin et. al. (2001-2002) J Phys Cond MattFenter and Zhang (2005) Phys. Rev. B, 081401.

Algorithms for rapid determination of interfacial electron density profiles

From Saldin et al.

From Fenter et al.

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

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., 076104Park C. and Fenter P.A. (2006) J. Appl. Cryst.

Some new stuff

From Park et al.

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)

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….

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

2

1

-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

ki kr

d

ki

kr

Q = kr-ki

(HKL))sin(d2

)2/2sin(4

d

2

HKL

HKL

• If Q integer HKL Bragg’s condition is satisfied

Bragg’s Law