Neutron Reflectometry

Roger Pynn

Indiana University and the Spallation Neutron Source

Surface Reflection Is Very Different From Most Neutron Scattering

• Normally, neutrons are very WEAKLY scattered– One can ignore doubly scattered neutrons

• This approximation is called the Born Approximation

• Below an angle of incidence called the critical angle, neutrons are perfectly reflected from a smooth surface– This is NOT weak scattering and the Born Approximation is not applicable to

this case

• Specular reflection is used:– In neutron guides– In multilayer monochromators and polarizers– To probe surface and interface structure in layered systems

Why Use Neutron Reflectivity?

• Neutrons are reflected from most materials at grazing angles• If the surface is flat and smooth the reflection is specular

– Perfect reflection below a critical angle– Above the critical angle reflectivity is determined by the variation of scattering length

density perpendicular to the surface – i.e. we can determine the “average” density profile normal to the surface of a film on

the surface• Diffuse scattering gives information on surface & interface roughness

Images courtesy of M. Tolan & T. Salditt

Various forms of small angle neutron reflection

Viewgraph from M. R. Fitzsimmons

Randomly Chosen Examples of Neutron Reflectometry Studies

• Inter-diffusion of polymers• Phase separation in block copolymers• Amphiphilic molecules at air-water interfaces• Effect of shear on films of complex fluids• Grafting of polymers to surfaces (mushrooms and brushes)• Swelling of films exposed to vapor• Magnetic structure of multilayers• CMR/GMR films• Exchange bias and exchange springs• Nuclear polarization in spintronic materials

Refractive Index for Neutrons

materials.most from reflected externally are neutrons 1,ngenerally Since2/1

:get we)A 10(~ small very is and ),definition(by index refractive Since

4or 2222

givesenergy ofon Conservati

2 isenergy total themedium theInside

2

is in vaccuumneutron ofenergy total)(and kinetic The

SANSfor used weone same the- (SLD)Density Length Scatteringnuclear thecalled is

1 where2 :is medium theinside potential average theSo

nucleus. single afor )(2)( :bygiven is potentialneutron -nucleus The

2

2-6-0

220

2222220

2

22

20

2

2

2

<=

==

=−+=+=

+

=

==

=

∑

πρλ

ρ

πρρπ

ρ

ρρπ

δπ

-n

nk/k

kkmm

kVmk

mk

Vmk

mk

E

bvolumem

V

rbm

rV

ii

hhhh

h

h

h

rhr

nickelfor )(1.0)( :Note

quartzfor )(02.0)(

sin)/2(

A 1005.2 quartzFor

4 is reflection external for total of valuecritical The

4or 4sin'sin i.e.

4)sin(cos)'sin'(cos 4 SinceLaw sSnell'obey Neutrons i.e.

'/coscosn i.e 'cos'coscos:so surface the toparallellocity neutron ve thechangecannot surface The

/ :index Refractive

critical

critical

critical

1-3

00

20

2220

22

2220

22220

2

00

0

0

0

A

A

k

xk

kk

kkkk

kkkk

nkkk

nkk

o

o

critical

critical

zz

zz

z

z

λα

λα

αλπ

πρ

πρπραα

πρααααπρ

ααααα

≈

≈

⇒=

=

=

−=−=

−+=+−=

===

=

−

Only Neutrons With Very Low Velocities Perpendicular to a Surface Are Reflected

α

α’

k0

k

Reflection of Neutrons by a Smooth Surface: Fresnel’s Law

n = 1-λ2ρ/2π

TTRRII

TRI

kakaka

aaarrr

&

=+

=+⇒=

0zat & of

continuityψψ

)(2 :cetransmitan

)()( :ereflectanc

TzIz

Iz

I

T

TzIz

TzIz

I

R

kkk

aat

kkkk

aar

+==

+−

==

Reflectivity• We do not measure the reflectance, r, but the reflectivity, R given by:

R = # of neutrons reflected at Qz = r.r*# of incident neutrons

i.e., just as in diffraction, we lose phase information

Measured and Fresnel reflectivitiesfor water – difference is due to surfaceroughness

Fresnel Reflectivity

Penetration Depth

• In the absence of absorption, the penetration depth becomes infinite at large enough angles

• Because kz is imaginary below the critical edge (recall that ), the transmitted wave is evanescent

• The penetration depth

• Around the critical edge, one maytune the penetration depth to probedifferent depths in the sample

πρ420

2 −= zz kk

)Im(/1 k=Λ

Surface Roughness• Surface roughness causes diffuse

(non-specular) scattering and so reduces the magnitude of the specular reflectivity

k 1 k 2

k t 1

z

x

θ 1 θ 2 z = 0

• The way in which the specular reflection is damped depends on the length scale of the roughness in the surface as well as on the magnitude and distribution of roughness

“sparkling sea”model -- specular from many facets

each piece of surfacescatters indepedently-- Nevot Croce model

Note that roughnessintroduces a SLDprofile averaged overthe sample surface

212 σt

zIzkkFeRR −=

Fresnel’s Law for a Thin Film• r=(k0z-k1z)/(k1z+k0z) is Fresnel’s law

• Evaluate with ρ=4.10-6 A-2 gives thered curve with critical wavevectorgiven by k0z = (4πρ)1/2

• If we add a thin layer on top of thesubstrate we get interference fringes &the reflectance is given by:

and we measure the reflectivity R = r.r*

• If the film has a higher scattering length density than the substrate we get the green curve (if the film scattering is weaker than the substance, the green curve is below the red one)

• The fringe spacing at large k0z is ~ π/t (a 250 A film was used for the figure)

0.01 0.02 0.03 0.04 0.05

-5

-4

-3

-2

-1

Log(r.r*)

k0z

tki

tki

z

z

errerrr

1

1

21201

21201

1++

=

substrate

01 Film thickness = t2

V(z

)

z

One can also think about Neutron Reflection from a Surface as a 1-d Problem

V(z)= 2 π ρ(z) h 2/mn

k2=k02 - 4π ρ(z)

Where V(z) is the potential seen by the neutron & ρ(z) is the scattering length density

substrate

Film Vacuum

Multiple Layers – Parratt Iteration (1954)

• The same method of matching wavefunctions and derivatives at interfaces can be used to obtain an expression for the reflectivity of multiple layers

jjz

jjzjjz

zikjjj

zikjjjzik

j

jj

eXr

eXre

TR

X1,

1,,

211,

211,2

1 +

+

++

++−

+

+==

1,,

1,,1, where

+

++ +

−=

jzjz

jzjzjj kk

kkr

ave)incident w amplitudeunit & substrateinside fromback coming nothing (i.e.

1 and 0 withiterationStart

111 === ++ TXR NN

Image from M. Tolan

Dealing with Complex Density Profiles

• Any SLD depth profile can be “chopped” into slices

• The Parratt formalism allows the reflectivity to be calculated

• A thickness resolution of 1 Å is adequate – this corresponds to a value of Qz where the reflectivity has dropped below what neutrons can normally measure

• Computationally intensive!!

Image from M. Tolan

Typical Reflectivities

0.00 0.04 0.08 0.12 0.16 0.2010-6

1x10-51x10-4

10-310-210-1100

res: 2 x 10-3 Å-1

[30 Å Nb / 50 Å Fe]*12 on Si substrate

q [Å-1]

10-61x10-51x10-4

10-310-210-1100

500 Å Nb on Si substrate

Neu

tron

Ref

lect

ivity

10-61x10-51x10-4

10-310-210-1100

unpolarized

Si substrate

The Goal of Reflectivity Measurements Is to Infer a Density Profile Perpendicular to a Flat Interface

• In general the results are not unique, but independent knowledge of the system often makes them very reliable

• Frequently, layer models are used to fit the data• Advantages of neutrons include:

– Contrast variation (using H and D, for example)– Low absorption – probe buried interfaces, solid/liquid interfaces etc– Non-destructive– Sensitive to magnetism– Thickness length scale 10 – 5000 Å

• Issues include– Generally no unique solution for the SLD profile (use prior knowledge)– Large samples (~10 cm2) with good scattering contrast are needed

Analyzing Reflectivity Data• We want to find ρ(z) given a measurement of R(Qz)• This inverse problem is not generally solvable• Two methods are used:1. Modelling

– Parameterize ρ(z) and use the Parratt method to calculate R(Qz)– Refine the parameters of ρ(z)– BUT…there is a family of ρ(z) that produce different r(Qz) but exactly the

same R(Qz): many more ρ(z) that produce similar r(Qz). – This non-uniqueness can often be satisfactorily overcome by using additional

information about the sample (e.g. known order of layers)

2. Multiple measurements on the same sample– Use two different “backings” or “frontings” for the unknown layers– Allows r(Qz) to be calculated– R(Qz) can be inverted to give ρ(z) unless ρ(z) has bound states (unusual)

Perils of fitting

0

1x10-6

2x10-6

3x10-6

4x10-6

5x10-6

-50 0 50 100 150 200 250

Model 1Model 2

Scat

terin

g le

ngth

den

sity

[Å-2

]

Depth into sample [Å]

10-6

10-5

10-4

10-3

10-2

10-1

100

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Model 1Model 2

Ref

lect

ivity

Q [Å-1]

Lack of information about the phase of the reflected wave means that profoundly different scattering length density profiles can produce strikingly similar reflectivities.

Ambiguities may be resolved with additional information and physical intuition.Sample growersOther techniques, e.g., TEM, X-rayNeutron data of very high qualityWell-designed experiments (simulation is a key tool)

D. Sivia et al., J. Appl. Phys. 70, 732 (1991).

Direct Inversion of Reflectivity Data is Possible*

• Use different “fronting” or “backing” materials for two measurement of the same unknown film– E.g. D2O and H2O “backings” for an unknown film deposited on a quartz

substrate or Si & Al2O3 as substrates for the same unknown sample– Allows Re(R) to be obtained from two simultaneous equations for

– Re(R) can be inverted to yield a unique SLD profile

• Another possibility is to use a magnetic “backing” and polarized neutrons

Unknown film

Si or Al2O3 substrateSiO2

H2O or D2O

22

21 and RR

* Majkrzak et al Biophys Journal, 79,3330 (2000)

Planning a Reflectivity Measurement

• Simulation of reflectivity profiles using e.g. Parratt is essential– Can you see the effect you want to see?– What is the best substrate? Which materials should be deuterated?

• If your sample involves free liquid surface you will need to usea reflectometer with a vertical scattering plane

• Preparing good (i.e. low surface roughness) samples is key– Beware of large islands

• Layer thicknesses between 10 Å and 5000 Å– But don’t mix extremes of thickness

Reflection of Polarized Neutrons

• Neutrons are also scattered by variations of B (magnetization)• Only components of magnetization perpendicular to Q cause scattering• If the magnetization fluctuation that causes the scattering is parallel to

the magnetic moment (spin) of a polarized neutron, the neutron spin is not changed by the scattering (non-spin-flip scattering)

Potential V

Vnuc,mag+

Vnuc Vnuc,mag-

Depth z

Fermi pseudo potential:

V = 2π h2 N (bn+/-bmag)/mN

with bnuc: nuclear scattering length [fm]bmag: magnetic scattering length [fm]

(1 μB/Atom => 2.695 fm)N: number density [at/cm3]mN: neutron mass

Spin“up” neutrons see a high potential.Spin“down” neutrons see a low potential.

0.00 0.04 0.08 0.12 0.16 0.2010-6

1x10-51x10-4

10-310-210-1100

res: 2 x 10-3 �-1

[30 � Nb / 50 � Fe]*12 on Si substrate

q [�-1]

10-61x10-51x10-4

10-310-210-1100

500 � Fe on Si substrate

Neu

tron

Ref

lect

ivity

10-61x10-51x10-4

10-310-210-1100

polarized

Fe surface↑H

μN

R

0.00 0.04 0.08 0.12 0.16 0.2010-6

1x10-51x10-4

10-310-210-1100

res: 2 x 10-3 �-1

[30 � Nb / 50 � Fe]*12 on Si substrate

q [�-1]

10-61x10-51x10-4

10-310-210-1100

500 � Nb on Si substrate

Neu

tron

Ref

lect

ivity

10-61x10-51x10-4

10-310-210-1100

unpolarized

Si substrate

R

kz = kzo2 − 4π (b/V) kz

± = kzo2 − 4π[( b /V ) ± cB]qz = 2kzo

Courtesy of F. Klose

Typical Non-Spin-Flip Reflectivities

Summary so Far…

• Neutron reflectometry can be used to determine scattering length density profiles perpendicular to smooth, flat interfaces

• PNR (polarized neutron reflectometry) allows vector magnetic profiles to be determined

• Diffuse scattering gives information about surface roughness

What new things are we planning?

Viewgraph courtesy of Gian Felcher

Components of the Scattering Vector in Grazing Incidence Geometry

qx is very small

x

z

yαi

αf

ϕ

qx = (2π/λ)( cosαf - cosαi )qy = (2π/λ) sinϕ

Geometry at grazing incidence

αf Resolve with PSD

ϕ Resolve with Spin Echo

tight collimation

coarse collimation

+BO -BO

spinanalyser

spinpolarizer

1st spin-echoarm

2nd spin-echoarm

DIFFERENT PATH LENGTHS FOR THE DIFFERENT TRAJECTORIES !

P =

1The Classical Picture of Spin Precession

sampleposition

Viewgraphsequence byA. Vorobiev

+BO -BO

spinanalyser

spinpolarizer

DEPOLARIZEDBEAM !

ALL SPINSARE PARALLEL:

FULL POLARIZATION

0

1

0.5 Pi

0

1

0.5

ALL SPINSARE PARALLEL AGAIN:

POLARIZATIONIS BACK

0

1

0.5 Pf

Spin Echo Scattering Angle Measurement (SESAME)No Sample in Beam

Pi = Pf‘SPIN-ECHO’

sampleposition

+BO -BO

spinanalyser

spinpolarizer

FULL POLARIZATION

0

1

0.5 Pi

DEPOLARIZATION !0

1

0.5 Pf

Spin Echo Scattering Angle Measurement (SESAME)Scattering by the Sample

Pi > Pf

+BO -BO

spinanalyser

spinpolarizer

FULL POLARIZATION

0

1

0.5 Pi

DEPOLARIZATION !0

1

0.5 Pfthe same depolarization

Pi > Pfindependently ofinitial trajectory

Spin Echo Scattering Angle Measurement (SESAME)Scattering of a Divergent Beam