Surface Scattering - Spallation Neutron Source

X-ray and **Neutron** **Scattering** from

Crystalline **Surface**s 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

August 14, 2012

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,

M. Tringides; and to NSF and DOE for funding.

Buried Interface Structure

to understand the growth and function of materials

http://www.tyndall.ie/research/electronic-theory-group/thin_film_simulation.html

Crystal growth from a vapor

Nucleation

Edge

diffusion

Step-Edge Barrier

Terrace diffusion

Morphology → atomic scale mechanisms

Cu/Cu(001)

Zuo & Wendelken

Interplay between two regimes of Length Scales

• Interatomic distances

→Structure, physics, chemistry → Mechanisms

• “Mesoscale” – Nanoscale

→ Morphology → Mechanisms

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)

**Neutron**s: low intensity- limited to reflectivity

• Soft Matter and Bio materials; H 2O & D 2O

• Magnetic materials

Example:

Rotation of graphene planes affect electronic properties

Graphene made from SiC

J. Phys.: Condens. Matter 20 (2008) 323202

Morphology → atomic scale mechanisms

Nanoclusters

Interface roughness

Nucleation/growth/coarsening at interfaces

Crystalline morphology

Atomic interfacial structure

Vacancy clusters

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 **Neutron**s:

2

k

2

E k

2

n

2

r E

Helmholtz Equation

X-rays **Neutron**s

0

n(r)=inhomogeneous refractive index

2 2

r

2

m

EV 0

2m

2

n r

2 2

b

k

Refractive Index for neutron: 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

i

k

f

Q

k

f

f k

Wavevector Transfer:

Q

k

i

4

sin

~

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

n 1

1

b

With absorption:

sin

Snell’s Law:

2

n 1

i

2 sin 2

Critical Angle for

Total Reflection:

2

c

2

b

2

k

z

1

k

(~ 10

Q

ki

x

5

)

k

f

Wavevector transfer:

Only kz component is affected by the surface.

kx is unchanged.

2 2 2 Q Qc

2

Qc

16b

Q

k

i

k

z

k’ z

k

x

k

x

c

Total Reflection

k

c

Evanescent wave

2 2

sin 0sin 2

Critical Angle for Total External Reflection:

Q

c

4

c c

Beam does not transmit below

Frenel Reflectivity for a Single Interface

refl power

R

inc power

Exact Result for

A Single Interface

R

Q Q

Q Q

Critical angle for

Total External Reflection

2

Q

ki

k

f

k

i

H. Dosch, PRB 35, 2137 (1987)

Transmission Amplitude

1

ImQ

z

H. Dosch, PRB 35, 2137 (1987)

Penetration Length

i

c

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

L.G. Parratt, Phys Rev 95, 359 (1954)

M-layer

http://www.ncnr.nist.gov/reflpak/

Soft Matter: **Neutron** Reflectivity

• Thin film interfaces

Light elements

Isotopic substitutions

Polymers

lipid

D. J. McGillivray and M. Lösche

D. J. Vanderah and J. J. Kasianowicz

G. Valincius

Biophysical Journal Volume 95 November 2008 4845–4861

Magnetic Films: **Neutron** Reflectivity

• Magnetic thin films

Spin-polarized neutrons

FeCo/GaAs

S. Park, M. R. Fitzsimmons, X. Y. Dong,

B. D. Schultz, and C. J. Palmstrøm, Phys Rev B 70 104406 (2004)

Vortices in Thin-Film Superconductors

Studied by Spin-Polarized **Neutron** Reflectivity (SPNR)

H

S-W. Han, J.F. Ankner, H.Kaiser, P.F.Miceli,

E.Paraoanu, L.H.Greene, PRB 59, 14692 (1999)

Field parallel

to film surface

YBCO 600 nm Superconducting Film

Vortex Density

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

~

**Surface** information → “3”

ki

Q

Q

k

f

Differential **Scattering** Cross Section

Weak **Scattering**

“Born Approximation” or “Kinematic Approximation”

d

d

Reflectivity:

1 d

R d

A d

inc

P

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

d

k

2

d

Q

2

sin

p

f

Qz

k

F

f

Qy

Qx

A

Born Approximation:

simple sum over atomic positions

Sum over all atomic positions

3 iQr

iQr

Q d r r e b

e

Q b

b re

f for x-rays or tabulated for neutrons

r

r

Born Approximation works if the

reflectivity is not too large.

R

4

2

Q

R

Qc

Q Q

Q Q

Born approx.

2

Exact

Single Interface

f

ki

k Q

k

Specular Reflection

i

General Case: non-specular scattering

Grazing Incidence Effects

k

3D **Scattering**

Q

k

k

f

k

If i or f are near grazing:

• refraction of both beams

internal wavevector transfer:

• transmission of both beams:

i

Q

i T T ,

f

Q

Perpendicular to **Surface**: internal Q’ and external Q are different

Q

x

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

H. Dosch, B.W. Batterman and D. C. Wack, PRL 56, 1144 (1986)

Q

1

ImQ

z

H. Dosch, PRB 35, 2137 (1987)

Penetration Length

i

c

S

Q

Fe 3Al

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:

i

i

f

fixed

fixed

Detector angles

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

The Effect of a

Crystalline Boundary

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

G

Q G

Large crystal

n

z

0

G is a reciprocal lattice vector

x

2

x

a

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 z

• Neglect boundary at z

S

R

p

N 1

2 n iQ

cn

iQ

RR

Q b e

e

n

z

0

z

z

Very small attenuation

2

z

R R

1

N

iQzc

2

4sin

2

p p

Reflected wave vanishes

N iQzcN

1

e

1

lim lim

c

1

e

Q z

2

p

p

~

p

Crystal Truncation Rod Factor

1

1

2 G Q

2

c z

z

0

By neglecting the lateral boundaries:

R R

S

p p

and

e

iQ

p

R

p

RpRp

e

2 Q

iQp

R

p

Nirr

e

Rp

iQ

R

2

p

N

s

c

p

b

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

2

irr

2 Qzc

4sin

2 G

p Gp

CTR

Intensity falls slowly

p

Q Narrow reflection in-plane

Crystal Truncation Rods

Crystal **Surface**

Specular rod

(Q p=0)

Reciprocal Space

Real Space

Q z

Q p

Crystal Truncation Rod **Scattering** for

Specular Reflection from the Ag(111) **Surface**

geometry CTR

1 1

2 2

z z Q Q

G 0

geometry

Atomically smooth

Rough, 3Å

1

2

Qz z

1

2 Q G

G

CTR

( 111)

Elliott et. al. PRB 54, 17938 (1996)

Step ordering

Q z

(typically semiconductors)

Miscut **Surface**s

Tilted Rods

Q p

Q z

nˆ s

s

G

G

nˆ

0

s

L

Step bunching

nˆ

(e.g. noble metals)

Specular

Diffuse

G

Q p

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)

The Effect of a

Rough **Surface**

Sharper interface (real space) gives “broader” scattering

Atomically smooth

Rough, 3Å

Elliott et. al. PRB 54, 17938 (1996)

Special location along CTR: anti-Bragg

Bragg Position:

Q z

N 1

z

I e N

n

c

z

2

m

0

N 1

2

i2n

2

Anti-Bragg Position:

Q z

c

m

i

nz

I e

(m odd)

nz

0

For a smooth surface

2

1

1 atomic layer

Q z

Anti-Bragg Positions

I

0

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

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 b

b

e

r

r r

r

mR

pmRp 2

iQp

Rp

Rp

iQzc

nn b e

e

Rp Rp

n

n

2

Nirr

b

iQp

R

p iQz

h Rp

Rp

h

Rp

e e

iQ c

2

R

1

e

z Rp

• Neglect lateral boundaries

Caused by lateral structure

CTR factor

FT of average phase difference

due to lateral height-differences

h R

p

R

p

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

p

2

2

iQzh

p p e

s

Q QG p

iQ

R

iQ hRRhR

Q e e

p

c

Short-Range Correlations Give Diffuse **Scattering**:

2

p p

z p p p

iQzh

p

e

R

Rp

2

Two Component Line Shape: Bragg + Diffuse

S

T

Bragg

Diffuse

Bragg

Diffuse

Q S Q S Q

p

T

Q p

p

• Bragg due to laterally uncorrelated disorder at long distances

• Diffuse due to short-range correlations

T

p

Q z

scan

Q p

Layer-by-layer growth

• Specular Bragg Rod: intensity changes with roughness

• Strong inter-island correlations seen in the diffuse

D

2

D

Qx

1.0 ML

0.5 ML

0 ML

1.5 ML

3 ML

2.5 ML

2.0 ML

x Q

Adv. X-ray Anal. 43, 169 (2000)

Attenuation of the Bragg Rod and **Surface** Roughness

If height fluctuations are Gaussian: σ is rms roughness

e

iQ

z

h

2

e

Q

But crystal heights are discrete for a rough crystal:

e

iQ

h

2

z e

2

4

2

c

sin

2

2

z

Qzc

2

• Binomial distribution (limits to a Gaussian for large roughness)

• Preserves translational symmetry in the roughness

2

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

2

c

sin

2

Qzc

2

(discrete roughness)

Real space

z

Bragg

Diffuse

S

Q p

reciprocal space

Bragg

Transversely-integrated scattering shows no effect of roughness:

∆Q p

Q d

2

Q

∆Q p

p

S

1/∆Q p

b

1

e

2 2

2

iQ

z

c

2

e

b

Q 2

1

e

4

2

c

2

iQ

1/∆Q p

Bragg is narrow:

it samples laterally uncorrelated roughness at long distances

z

c

sin

Qzc

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

What do 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

Specular rods overlap

Q z

Thin Films

Non-specular rods:

No overlap (incommensurate)

Q x

Real Space

substrate

Q z=2πL/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

Yiyao Chen et al.

Specular Reflectivity: 0.9ML Ag/Si(111)7x7

Θ =0.9ML

T=300K

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

p2 p3 p4 p5

Mainly 3-layer islands

Q z=2πL/c

Ag/Si(111)7x7

Exposed **Surface** Frac

Height (ML)

3-layer islands

2ML above wetting layer

Wetting layer

Specular reflectivity cannot easily

distinguish between these two cases:

FCC Ag islands on

FCC Ag islands all the way

a Ag 7x7 wetting layer?

Ag7x7 wetting layer

to the substrate?

Reflectivity = 3 layers

FCC CTR = 2 layers

?

Reflectivity = 3 layers

FCC CTR = 3 layers

Si Si

Specular reflectivity and rod give same thickness:

Island is FCC Ag all the way to the substrate

Islands remove the wetting layer!

Specular Reflectivity

Specular Reflectivity

Probes Si substrate, Ag

wetting layer and Ag island

Ag (1 1) rod

Ag truncation rod

Only probes FCC Ag

Yiyao Chen et al.

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

Height Selection: “Magic” crystal heights

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

Coarsening

laist.com

greeneurope.org

Rain Drops On

Your Winshield

**Surface** Chamber

Advanced Photon **Source**

Mean island separation, :

2

L

L

Pb(111)

Experiment:

• Deposit Pb (1.2 to 2 ml) at 208K

Island Density: • Measure the island 1 density vs time

(flux

n

off)

2

L

Intensity

X-Rays Incident

Transverse Scan: In-plane Info.

CCD Image

Q x

Pb **Source**

Island Density

n t n

1

0

t

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

1

t

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

Summary

Materials research problems require

information on a broad range of length

scales, from atomic to mesoscale

**Scattering** from surfaces involves many

different types of measurements:

Reflectivity, Rods, Grazing Incidence

Diffraction, Diffuse **Scattering**

Unique ability of x-rays: surface and

subsurface structure simultaneously