Characterization and Modeling of Off-Specular Neutron

Scattering for Analysis of Two Dimensional Ordered Structures

Metting, C.J. and Briber, R.M.

Materials Science and Engineering,

University of Maryland, College Park, MD

Maranville, B.B., Kienzle, P., Majkrzak, C.F., and Dura,J.A.

NIST Center for Neutron Research, NIST, Gaithersburg, MD

(Dated: February 27, 2012)

Abstract

With high flux neutron sources like the Spallation Neutron Source coming on-line around the

world, new possibilities for neutron instrumentation are being realized as the limitations imposed by

low flux sources are lifted. This is especially true in the case of off-specular neutron reflectometry

where higher intensities will allow for a more in-depth study of the in-plane structure of thin

film samples. To interpret off-specular reflectometry data effectively, robust modeling tools are

required. Although much progress has been made in developing the mathematics required to model

such scattering data, no versatile, open-source software package for the modeling of off-specular

neutron reflectometry exists. Such software, in which the mechanics of calculations are exposed

and reviewable by the larger scientific community, is essential for reliable off-specular neutron

reflectometry data analysis. Here we present results from the current software. We also compare

theory functions created with different mathematical approximations to data taken from a system

of Au pillars to evaluate the modeling capabilities of each approximation under real scattering

conditions.

1

I. INTRODUCTION

Neutron reflectometry provides a scattering length density (SLD) depth profile of thin film

samples with sub-angstrom accuracy [1] [2] by measuring reflected neutron beam intensity

as a function of Qz; defined as:

Qz = 4π/λsin(θ) (I.1)

where θ is the glancing angle between the incident beam and the sample and λ is the

neutron beam wavelength. The SLD, ρ, is defined as:

ρ(z) =M∑j=1

N(z)jbj (I.2)

where N is the number density, b is the coherent nuclear isotopic scattering length, and

the sum is taken over all isotopes, j, for a given depth, z. Specular NR is not sensitive

to in-plane structures and interacts with a SLD which is averaged over any compositional

variations within that plane. The atomic dependence through bj allows the SLD profile

to be interpreted as a compositional profile when the constituent materials are known.

Furthermore, the isotopic dependence of bj allows for specific atoms in a system to be

tagged and traced using isotope substitution.

Due to a lack of neutron phase information, NR data cannot be directly inverted to de-

termine a SLD profile. To interpret NR data, theoretical reflectivity versus Qz is calculated

from model systems and compared to real data. Neutrons reflected from each interface in

multilayered thin films interfere constructively and destructively as a function of Q, resulting

in oscillating reflected amplitude with various periods versus Qz that are inversely propor-

tional to the thickness of the layers that caused them. Simple structures can be modeled as

single layers of constant SLD for each film in the system. More complicated profiles, such as

compositional gradients, can be modeled as a stack of arbitrarily thin layers to approximate

any SLD profile. The resulting interference fringes are highly dependent on SLD, thickness,

and interfacial roughness of the layers in the sample which lead to high structural sensitivity.

Specular NR is not sensitive to in-plane structures and interacts with a SLD which is

averaged over any compositional variations within that plane.The specular scattering can

be analytically solved for by either the Parratt[3] or matrix [4] method, but contains no

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detailed interpretation of scattering that results in a wave vector transfer with components

in the in-plane directions, which occurs when there is scattering contrast variation within

the plane of the film which is smaller than the projected neutron beam coherences length.

For nominally smooth films, diffuse scattering is treated as background scattering and is

removed from the specular curve[4]. When variations are corrolated, more robust treatments

of the diffuse scattering are required. Formalisms have been developed for special types of

roughness by Sinha et al.[5], which use a distorted wave Born approximation (DWBA) to

represent the diffuse scattering and allow for the extraction of information about in-plane cor-

related roughnesses. Work has also been carried out on the formulation of a DWBA, which

can model correlated nuclear and magnetic structures[6, 7] in two-dimensional reciprocal

space plots. The results of this approximation compare quite well with scattering data[8].

Despite these advancements, a well structured, user-friendly, and open source off-specular

neutron reflectometery modeling software has yet to be developed. The development, and

dissemination of, such software would help both foster and streamline new theoretical treat-

ments and facilitate discoveries in new areas of research where off-specular reflectometry

could uniquely contribute materials charactorization.

The software development process focused on producing a well compartmentalized, ob-

ject oriented modeling software package which utilizes distributed programming to accom-

modate more complicated and computationally expensive algorithms. The software was also

developed in the python programming language, which has software libaries to support the

complicated mathematical algorithms required for off-specular theory function calculations.

Finally, because the formalism and modeling capabilities must be open and verifiable, the

source code for this software is publicly available for review.

Here, results from the formalism implemented in the off-specular neutron reflectometry

software are presented. This implementation utilizes a finite element process with formalisms

derived from a variety of sources [9] [7]. In section II, the mathematical formalism available

in the current software as well as the conditions under which each approximation is valid

is discussed. Section III discusses the fabrication and measurement of a well-charactorized,

well-ordered samples, which were used as standards for evaluating computed theory func-

tions. Section IV analyzes the theory functions produced by the modeling software and

evaluates their capabilities by modeling the data acquired from the standard sample. This

section will also highlight features in both the data and theory functions which are important

3

FIG. 1. Schematic representation of the off-specular scattering geometry. Specular scattering

geometry occurs when θi = θf so that qx = 0 and therefore Q = qz.

FIG. 2. Schematic representation of the two mathematical components used to calculate the

scattering from the sample. The form factor is a matrix of scattering length density values which

represents the single unit which is repeating across the sample. The structure factor contains

information about how the unit is repeating across a sample.

for accurately modeling experimental systems. Finally, section V will discuss requirements

which must be implemented for a more robust modeling software.

II. THEORETICAL BACKGROUND

The geometry of the off-specular reflectometry technique is shown in figure 1. Because

no closed-form formalism exists for the calculation of off-specular neutron scattering, the

theories being utilized in this software package are based on the Born approximation and

perturbations on it. For periodically repeating features, the system may be divided into two

components. These components are illustrated in figure 2 and in equation II.1

Intensity = |form factor|2 ∗ |structure factor|2 (II.1)

The formalism implemented in this software only solves for the coherent scattering and

can only model systems which contain features that are significantly smaller than the co-

4

herence length of the neutron beam where the coherent scattering dominates.

The first component in this equation, the form factor, is the Fourier transform of a single

repeated unit. This repeat unit is represented by a three-dimensional finite element matrix

of scattering length density (SLD) values with a discretization size that is commencerate

with the desired qx and qz range. This approch is valid for unit discretization sizes that

are smaller than the real space value of the maximum q in each direction respectively. The

second component, the structure factor, is the Fourier transform of the two-dimentional

in-plane repeat structure. Conceptually, this component could be eliminated if the finite

element matrix represented the entire coherently scattering surface area; however, the size

of a matrix which would describe the total sample volume would be prohibitively large. In

all of the approximations implemented in this software, the modeling is solved using form

factor and structure factor calculations.

A. Structure Factor

The structure factor is the Fourier transform of the repeating lattice structure in a model

system. The Fourier transform of a two dimensional lattice results in a delta function at

the Bragg conditions. Because the neutrons coherently scatter over a finite space, these

delta functions must be convolved with the coherence structure of the neutron. In this

work, the coherence length is treated as a Gaussian. The Fourier transform for the structure

factor is calculated by convolving the delta function Bragg peaks resulting from the in-plane

diffraction with the coherence length of the neutron beam as seen in equation II.2.

S = e−2π2q2σ2

FT (Vlattice) (II.2)

The Gaussian envelope applied to the neutron wave packet suppresses artifacts that result

from assuming a sharp cut-off. Figure 3 illustrates the difference between a square and a

Gaussian envelope.

B. Resolution Correction

The neutron beam resolution has a strong effect on the scattering. To account for these

resolution effects, a broadening function is applied to the theory function. There are two

5

FIG. 3. A comparison of calculations carried out in the Born approximation for a beam coherence

length with A) a square envelope and B) a Gaussian envelope.

main contributors to the neutron beam resolution. The first is wavelength distribution from

the monochromator, which exposes the sample to a finite distribution of energies. The

second is the angular divergence in the incoming beam, θ. The resolution is approximated

as the uncorrelated convolution of these effects.

The resolution correction applied is:

(δq

q

)2

=

(δλ

λ

)2

+

(δθi

tan (θi)

)2

(II.3)

and the results are shown in figure 4.

C. Born Approximation (BA)

The kinematic approximation for calculating the theory function for a model assumes

that the reflected intensity of the plane wave impinging on a surface is so small that the

transmitted wave can be approximated as the original plane wave. This assumption is appli-

6

FIG. 4. A comparison of the Born Approximation results with A) no resolution correction and B)

resolution correction

cable at high q values where the reflectivity, and thus the probability of multiple reflections,

is small; however, in reflectometry scattering is inherently measured at low q values where

the kinimatic approximation is invalid. Theory functions calculated with the BA will not

represent distortions to the scattering resulting from multiple scattering effects, which con-

tribute significantly to the scattering at low q. The formalism for this approximation can

be written as:

qzS‖4π|r|2 =

∣∣∣∣∫∫∫ Vunit

0

ρ (r) eiqrunit

∣∣∣∣2︸ ︷︷ ︸Form Factor

∗

(e−2π2q2σ2 ∗ FT (Vlattice)

)2

︸ ︷︷ ︸Structure Factor

(II.4)

whereqzS‖4π

is a normalization factor, S‖ is the surface area covered by the probing beam, r

is the reflectivity, σ is the coherence length of the beam,Vlattice are the delta function lattice

points, and ρ is the SLD. The form factor of this calculation can be approximated as:

7

FF =

∣∣∣∣ 1

iq

(1− eiq∆runit

)∗ FT (ρ (r))

∣∣∣∣2 (II.5)

where FT is the Fourier transform of the SLD matrix. This is then multiplied with

the modulus squared of the structure factor. This variation is much less computationally

expensive. The calculation can be made even more efficient by substituting the FT with a

chirp-z transform [10]:

FF =

∣∣∣∣ 1

iq

(1− eiq∆runit

)∗ CZT (ρ (r))

∣∣∣∣2 (II.6)

which has a lower machine memory requirement because only the q-space region of interest

is calculated.

D. Substrate Modified Born Approximation (SMBA)

Although the theory functions calculated using the BA provides a scattering profile that

relates to the sample, the reflection geometry is susceptible to effects caused by perturbation

of the wave as it interacts with the sample, which cannot be captured using the kinomatic

approch. In the case of samples with a sparse array of features on a substrate where the dom-

inant contributor to dynamic scattering is the substrate, two corrections can be applied to

the BA. The algorithm which uses these corrections will be referred to as the substrate mod-

ified born approximation (SMBA). Because the dominant contributor to dynamic scattering

is the substrate/incident media interface, the scattering can be perturbed by the solution

to the wavefunction of the incident and scattered wave as it interacts with this interface. A

second correction is applied to scattering occurs below the angle at which the incident or

scattered beam enter or leave the substrate nearly orthogonal to the substrate sidewall. In

this case, the beam refracts resulting in the shift illustrated in figure 5. The scattering shifts

in the positive qx direction at qx > 0.0 and below θf < 0.0 and in the negative at qx < 0.0

and below θf < 0.0. Figure 5 illistrates this shift.

The form factor for the SMBA can then be written as:

FF =

∣∣∣∣∣−iq (1− eiq∆runit)

[Ψin

r∑n=0

{ρ (r) ∗ eiqr}Ψout

]∣∣∣∣∣2

(II.7)

8

FIG. 5. Data taken on AND/R which illustrate. A) the refractive shift caused by the substrate and

B) the ’horizons’ calculated through the perturbation of the scattering along with an illustration

of the scattering angles for i) negative qx and positive θi, ii)positive qx and positive θi, iii) negative

qx and negative θi, and iv) positive qx and negative θi.

The solution to the wavefunction is solved by first calculating the reflected and trans-

mitted beam resulting from the substrate/incident media interface. The reflected beam

is:

r =M1,1 + (i ∗ n0 ∗M0,1) + −i

nf∗ (−M1,0 − i ∗ n0 ∗M0,0)

−M1,1 + i ∗ n0 ∗M0,1−inf

(M1,0 − i ∗ n0 ∗M0,0)(II.8)

and the transmitted beam is:

t = 1.0− r (II.9)

Now the appropriate solutions can be calculated based on where the scattering occures

in q-space. For an incoming plane wave the solution can be calculated as:

9

FIG. 6. Schematic of the interactions accounted for in the SMBA.

Ψin =

t ∗ e−ik‖in∆qz if kin < 0.0

1 ∗ e−ik‖in∆qz + r ∗ e−ik

‖in∆qz if kin ≥ 0.0

(II.10)

and likewise, for the scattered wave the solution can be calculated as:

Ψout =

1 ∗ e−ik‖out∆qz + r ∗ e−ik

‖out∆qz if kout ≥ 0.0

t ∗ e−ik‖out∆qz if kout < 0.0

(II.11)

The process is illustrated in figure 6.

The qx that effect from the substrate refraction can now be applied to the finete element

matrix. The effective qx is:

qeffx =

qx − λρsub if qx < 0.0, θi < 0.0

qx + λρsub if qx > 0.0, θf < 0.0(II.12)

E. Distorted Wave Born Approximation (DWBA)

The distorted wave Born approximation is similar to the SMBA discussed previously. In

this approximation, the scattering is distorted by each change in scattering potential that the

beam encounters as it traverses the sample depth. The formalism applied in this calculation

is derived from that presented by Kentzinger [7]. In this treatment, the solution to the

wavefunction is obtained by assuming the in-plane interaction can be approximated as the

interaction between the wave and an in-plane averaged scattering profile. This solution is

then used to perturb the in-plane scattering results. Rather than using the lateral correlation

function discussed by Kentzinger, the structure factor described in equation II.2 is applied.

10

In this work only the off-specular scattering cross-section is calculated by taking the modulus

squared of the scattering amplitude, which is written as:

Ffi =∑

l

∫dzSfl(z)Fl(Q‖)Sil(z) (II.13)

where Fl(Q‖) is the lateral Fourier transform of the difference between the scattering

potential and the laterally averaged scattering potential and the S are the propagator func-

tions determined from the laterally averaged scattering potential. The specular scattering

must be calculated separately and can be empirically scaled to the data [7].

III. EXPERIMENTAL RESULTS

To validite the computational models against a known system, reference samples were

fabricated and their scattering profiles measured. Sample material and patterning were

optimized to provide high scattering intensity where the off-specular reflectometry technique

is most sensitive. For the non-magnetic sample, a 75mm diameter 5mm thick silicon wafer

was fully patterned with Au pillars on a Cr/Si substrate. The Si 5 mm thick substrate was

chosen to reduce the possibility of sample warping. Au was chosen for its large scattering

cross-section. The thin Cr layer was used as an adhesion layer and was left thin to reduce

its contribution to the scattering.

A. Sample Fabrication

The Au parallelepiped features were produced by sputter deposition of Cr followed by

Au onto a Si substrate using a Denton Discovery 22 sputtering system at the Center for

Nanoscale Science and Technology(CNST). The Au was patterned using standard lithogra-

phy techniques with a Suss Microtec MA8 Front Side Aligner also at the CNST. The Au

was etched using room temperature dilute aqua regia (HNO3 : 3HCl : 20H2O) until the Cr

was visible.

Systematic width variation of the features across the sample was found by optical mi-

croscopy to be almost 32% relative to the center feature size. This is due to the variation

in wet etch rate across the sample, which is generally observed with wet etch processes. Al-

though the full sample is, for most incident angles, not probed by the beam, this variation is

11

FIG. 7. A) a schematic of the fabricated sample, B) an optical microscopy image of the sample,

and C) the reduced data in qx/qz coordinates.

expected to have an effect on the neutron scattering. Sidewall roughness was also observed.

Although the mask contained a square pattern with sharp corners, the wet etching process

caused these corners to round, resulting in a more cylindrical feature.

Figure 7 shows the fabricated sample with the scattering.

B. Data Acquisition

Measurements were taken with the Advanced Neutron Diffractometer/Reflectometer

(AND/R) at the NIST Center for Neutron Research (NCNR) [11]. Due to the relaxed

collimation in the y direction, reflectometry measurements have no y and consequently no

qy resolution. This broadly divergent beam increases the amount of flux on the sample

without losing resolution along the x and z axis. The main beam was blocked using an

absorbing beam-stop. Data was acquired on a Position Sensitive Detector (PSD) [12].

Each data point contains multiple θf angles for a given θi. An example of data for a

given angle is shown in figure 8A. Because of the poor y resolution, no information can be

obtained from the y scattering and the data is integrated along the y-axis for each θi. The

x-axis is then converted to thetaf values. The collection of θi data is concatenated into one

data set to build a θ/2θ plot as seen in figure 8B. Because θ space plots can be difficult to

12

FIG. 8. A) a single data point from the PSD. B)A stack of multiple data points integrated over

the y-direction to form a θ/2θ plot. C) Transformation of the data into qx/qz space.

interpret, the data can be converted to different spacial coordinates [13]. In this work, qx/qz

plots are used; calculated using equation (III.1)

q = kf − ki =2π

λ

(cos(θf )− cos(θi))x

(sin(θf ) + sin(θi))z

(III.1)

Due to the q-space angular dependence, the rebinning process can be challenging and

requires sub-sampling the data matrix to reduce errors introduced by the spacial transform.

The result of this process are shown in figure 8.

IV. MODELING

The dataset obtained in section III can now be modeled using the approximations de-

scribed in section II. The the modeling results are shown in figure 9.

Figure 9 shows a comparison of the BA and SMBA results. The Bragg peaks are located

along the qx axis and correspond to the in-plane feature spacing. The beating pattern along

13

FIG. 9. Modeling a subset of data(B) collected from III with parameters(A), using the Born

approximation(C) and the substrate modified Born approximation(D)

the qz are the fringes corresponding to the sample’s depth profile [13]. This comparison

illustrates some improvement of the SMBA over the BA. In scattering data, the scattered

intensity is observed to be attenuated at θin = 0 and θout = 0, referred to in this paper

as the ’horizons’, which is primarally due to the substrate. This attenuation is observed in

the SMBA, but not in the BA. Also, below the ’horizons’, the neutron beam passes into

the substrate at a nearly orthogonal incident angle. This results in neutron beam refraction

which is also reproduced by the SMBA. The results for the SMBA still contain some of

the inaccuracies inherent to the BA. The approximation is still inaccurate at low q values,

especially toward the critical edge. The fringes are also slightly shifted in the qz direction as

is generally observed with the first order BA. Still, the SMBA shows improvement over the

general BA and, for the case of large structures below the coherence length of the probing

beam and at q-space values far away from 0, information about both the in-plane and depth

structure can be obtained.

Figure 10 shows the results from the DWBA calculation. The integrated slices show

remarkable qualitative agreement. These models do not take into account sample disorder

14

FIG. 10. Off-specular scattering calculation within the DWBA. The implementation of this cal-

culation does not include the specular scattering or disorder. The colored boxes on the theory

function plot match those around the integrated qx and qz plots. The averaged slices show good

agreement between theory and experimentation.

or instrumental background levels, which explains some of the mismatch between the the-

ory function and the data. Still, the results indicated that the scattering data from two

dimensionally ordered system may be modeled accurately with the DWBA.

V. CONCLUSION AND FUTURE WORK

A software infrastructure has been developed, which allows for the quick prototyping of

off-specular modeling formalisms and has allowed for the rapid modeling of a Au paral-

lelepiped system.

It is shown (with an open-source and verifiable set of calculations) that the BA can pro-

vide little information about the sample depth properties for a system of in-plane repeat

structures; however, better agreement may be achieved by treating the neutron beam coher-

ence length as a Gaussian and introducing the substrate refraction effects. By introducing

15

perturbation effects to the wave function from the substrate/incident media interface, a

more accurate theory function can be calculated at and below the horizons. By perturb-

ing the wavefuntion by interactions at each interface in the sample, much of the scattering

features are capture; however, combining the specular and off-specular scattering results

is challenging due to the nature of the approximation. Further work will implement the

ability to combine the specular and off-specular data within the framework of this software

infrastructure.

VI. ACKNOWLEDGMENTS

Research was performed in part at the NIST Center for Nanoscale Science and Technology

and at the NIST Center for Neutron Research. This project was funded by NSF Grant #:

DMR-0529547

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