Neutron Reflectometry

Since the bulk of the present work deals with neutron reflectometry, the technique will be developed below in greater detail than any of the other analysis techniques. Neutron reflectometry is a technique in which neutrons at thermal energies are used to probe the near surface structure of material systems. In neutron reflectometry, neutrons are incident on a surface at a grazing angle of less than 3$^{\circ}$. At these small angles, the potential for scattering can be approximated by a continuous value, as will be shown later, and this value is called the scattering length density (SLD) usually denoted in equations as $\rho$. It is a product of the scattering length of each isotope and that isotope's number density.

For neutrons of thermal energies, the very small angles of incidence can be sufficient to cause the neutrons to be totally reflected from the surface. The position of this transition to total reflection yields information about the average SLD of the material. The calculation of this total reflection (critical) angle will be discussed below. The reflectivity away from the total reflection angle holds information about the change in scattering length density with depth. It can be analyzed to determine a film's total thickness, material composition, periodicity, and even roughness.

The geometry of a typical reflectometry experiment is shown in Fig. 1. The data are measured as

Figure 1: The geometry of a typical reflectometry experiment showing the incident and reflected angles, $\theta_i$ and $\theta_f$, and the incident and reflected wave vectors, $\ensuremath{\vec{k}}_i$ and $\ensuremath{\vec{k}}_f$, respectively.

intensity versus a quantity called the wave-vector transfer, . The wave-vector transfer is defined as the difference between the final and initial wave-vectors $\ensuremath{\vec{q}}= \vec{k}_f - \vec{k}_i$. Since neutron reflectometry assumes completely elastic scattering of neutrons the magnitudes of the incoming and outgoing wave vectors are the same and equal to the wave number, $\left\vert \vec{k}_f \right\vert = \left\vert \vec{k}_i \right\vert = 2\pi /\lambda$. The alignment of the two wave-vectors is also shown in Fig. 1. On this figure the components of in the $x$ and $z$ directions are

$$q_x = \frac{2\pi}{\lambda} \left( \cos \theta_f - \cos \theta_i \right)$$ (1)

$$q_z = \frac{2\pi}{\lambda} \left( \sin \theta_f - \sin \theta_i \right)$$ (2)

When $\theta_i = \theta_f$, $q_x = 0$, and the scattering is called specular. This is the condition that is used to determine the structure of the material in the $z-$direction, perpendicular to the surface. When dealing with specular reflectivities the data are usually presented as a function of $k_z=q_z / 2$.

Forward Scattering Geometry

The geometry that was assumed in the discussion above and shown in Fig. 1 is the forward scattering geometry. In this geometry, the direction perpendicular to the plane of reflection, the $y$-direction, is neglected. The slits which collimate the beam of neutrons are very wide in this direction (> 1 cm), so any measurements are assumed to be integrated over the out-of-plane direction. For this reason, all of the equations presented here will have the $y$ and $q_y$ dependence eliminated or integrated over.

Features of Reflectivity Data

Most reflectivity curves have three basic features. The first is the critical wave-vector transfer $k_c$. Neutrons are totally reflected for values of $k_z$ below $k_c$. For a uniform film, $k_c$ is given by

$$k_c = 2\sqrt{\pi \rho},$$ (3)

where the scattering length density, $\rho$, is given by

$$\rho=\sum_{i} N_{i} b_{i},$$ (4)

and $N_i$ and $b_i$ are the atomic number density and the isotopically averaged, bound coherent neutron scattering length of the $i^{th}$ element respectively. For a non-uniform film the calculation of critical wave vector is more complicated, but it is roughly a function of the average SLD of the film.

A second feature of reflectivity profiles is the decrease in intensity (or reflectivity) with $k_z$ which, for a smooth sample, becomes proportional to $k_{z}^{-4}$ as $k_{z}$ increases. A thin film can also show oscillations, the third feature, around this continuously decreasing reflectivity; the result of an interference effect between the air/film and film/substrate interfaces. The amplitude of these thickness oscillations is proportional to the difference in SLD between the thin film and the substrate, often referred to as the SLD contrast. An estimate of the film's thickness is given by $\pi/\Delta k_z$, where $\Delta k_z$ is the oscillation period.

Theory of Neutron Reflectometry

Wave Nature of the Neutron

Any theoretical development of neutron reflectometry must begin with the definition of the interacting particles as waves. In a collimated beam of neutrons, each neutron with an energy $E$, has an associated De Broglie wave with a wavelength, $\lambda$, given by

$$\lambda = \frac{h}{\sqrt{2mE}},$$ (5)

where $m$ is the mass of the neutron, and $h$ is Planck's constant. The time-dependent wave function of the neutrons with energy $E$ is

$$\phi \left( \ensuremath{\vec{r}}, t \right) = A \left( E \ri... ...k \hat{\imath} \cdot \ensuremath{\vec{r}}- \omega t \right)} ,$$ (6)

where $\ensuremath{\vec{r}}$ is the spatial position, $A \left( E \right)$ is the amplitude of the wave, $k$ is the wave number given by or , $\hat{\imath}$ is a unit vector in the direction of the wave propagation, and $\omega = E/\hbar$ is the angular frequency of the De Broglie wave. The product $k \hat{\imath}$ was called the incident wave vector, , in the previous section.

The 3-D steady state Schrödinger equation in three dimensions is given by

$$\ensuremath{\nabla_{\vec{r}}^{2}}\ensuremath{\psi \left( \ve... ...} \right)}\right] \ensuremath{\psi \left( \vec{r} \right)}= 0.$$ (7)

The potential used in neutron reflectometry is the equilibrium value of the Fermi pseudopotential given by[1]

$$\ensuremath{V \left( \vec{r} \right)}= \frac{2\pi\hbar^{2}}{m} \ensuremath{\rho \left( \vec{r} \right)},$$ (8)

where $\ensuremath{\rho \left( \vec{r} \right)}= N\left(\ensuremath{\vec{r}}\right)b\left(\ensuremath{\vec{r}}\right)$, the product of $N\left(\ensuremath{\vec{r}}\right)$ the number density of each nucleus, and $b\left(\ensuremath{\vec{r}}\right)$, the average scattering length at position $\ensuremath{\vec{r}}$ of the nucleus.

Since in specular reflection we are only concerned with the structure of the medium in the $z$ direction, perpendicular to the surface, and since off-specular scattering requires a special treatment, we will assume that is a function of $z$ only. This assumption is appropriate to the presentation of the basic development of the different equations, but it is not completely appropriate when calculations are made for real surfaces which usually have some kind of roughness. The only assumption that must be made about the SLD in the $z$ direction is that it has the limiting values

$$\ensuremath{\rho \left( {z} \right)}\rightarrow \left\{ \beg... ...ow \infty \\ 0, & z \rightarrow - \infty \end{array}\right. ,$$ (9)

where $N$ and $b$ are the values in the substrate. When this assumption is made the components of the wave vector in the $x$ and $y$ directions are not affected by the scattering process, and the solution of the Schrödinger equation Eq. 7 has the form

$$\ensuremath{\psi \left( \vec{r} \right)}= \exp \left[ i \lef... ...{y}y \right) \right] \ensuremath{{\cal X} \left( {z} \right)},$$ (10)

where $\ensuremath{{\cal X} \left( {z} \right)}$ satisfies the equation

$$\left(\frac{d^{2}}{dz^{2}} + k_{z}^{2} - 4\pi\ensuremath{\rh... ...z} \right)}\right)\ensuremath{{\cal X} \left( {z} \right)}= 0.$$ (11)

Since $\ensuremath{\rho \left( {z} \right)}\rightarrow 0$ as $z \rightarrow - \infty$, the above equation is just the equation for a wave in free space above the sample

$$\ensuremath{{\cal X} \left( {z} \right)}\rightarrow \exp \left( ik_{z}z \right) + r \exp \left(-ik_{z}z\right),$$ (12)

where $r$ is the reflectance of the wave from the surface. If the wave equation is determined above the surface of the sample it will have the form

$$\ensuremath{\psi \left( \vec{r} \right)}\rightarrow \exp \le... ...t( i \ensuremath{\vec{k}}_{f}\cdot\ensuremath{\vec{r}}\right).$$ (13)

The square modulus of the coefficient $r$, the complex amplitude of the reflected wave, is called the reflectivity, and it is this quantity that is measured. For specular reflectivity, since $k_x$ and $k_y$ are conserved, the momentum transfer vector has only one component: $q = q_z = 2k_z$.

In order to begin a calculation of the reflectance, $r$, we first begin by setting the scattering length density equal to

$$\ensuremath{\rho \left( {z} \right)}= \ensuremath{\rho_{0} \left( {z} \right)}+ \ensuremath{\rho_{1} \left( {z} \right)}$$ (14)

where $\ensuremath{\rho_{0} \left( {z} \right)}$ is the scattering length density that will be used to calculate the form of the unperturbed wave and $\ensuremath{\rho_{1} \left( {z} \right)}$ is the scattering length density that will perturb the wave from its initial state and produce a scattering event.

If we define an operator ${\cal L}$ such that

$${\cal L} = \frac{d^{2}}{dz^{2}} + k_{z}^{2} - 4\pi\ensuremath{\rho_{0} \left( {z} \right)},$$ (15)

then when Eq. 14 is inserted into Eq. 11 the resulting wave equation is

$${\cal L}\ensuremath{{\cal X} \left( {z} \right)}= 4\pi\ensur... ...} \left( {z} \right)}\ensuremath{{\cal X} \left( {z} \right)}.$$ (16)

We let be the solution to the unperturbed wave equation,

$${\cal L}\ensuremath{{\cal X}_{0} \left( {z} \right)}= 0,$$ (17)

and we can define a Green's function which satisfies the equation

$${\cal L}\ensuremath{G \left( z \mid z^{\prime} \right)}= 4\pi\ensuremath{\delta \left( z - z^{\prime} \right)}.$$ (18)

This Green's function is a wave traveling from the point $z^{\prime}$, and Eq. 16 can be rewritten as an integral equation using this Green's function and the previous definitions

$$\ensuremath{{\cal X} \left( {z} \right)}= \ensuremath{{\cal ... ...t( {\ensuremath{z^{\prime}}} \right)}d\ensuremath{z^{\prime}}.$$ (19)

The solution to Eq. 17 has the asymptotic form

$$\ensuremath{{\cal X}_{0} \left( {z} \right)}\rightarrow \exp... ..._z z) + r_0 \exp(-ik_z z) \;\mbox{as} \;z \rightarrow -\infty,$$ (20)

and the solution to Eq. 18 has the asymptotic form

$$\ensuremath{G \left( z \mid z^{\prime} \right)}\rightarrow \... ...th{z^{\prime}}} \right)}\; \mbox{as} \;z \rightarrow -\infty .$$ (21)

When Eq. 20 and Eq. 21 are substituted into Eq. 19 is can be seen just by inspection that the equation has the form of Eq. 12 and that the reflection coefficient, $r$, can be expressed as

$$r = r_0 + \frac{2\pi}{i k_z} \int_{-\infty}^{\infty} \ensure... ...\left( {z} \right)}\ensuremath{{\cal X} \left( {z} \right)}dz.$$ (22)

To this point no assumptions have been made regarding the form of the wave function, so this is an exact equation for the scattered wave amplitude. After this point, assumptions must be made and this is where the different Born approximations are used.

Plane Wave Born Approximation

Also called the first Born approximation or just the Born approximation, this approximation assumes that the entire scattering event is just a perturbation of the incoming beam so we set[1]

$$\ensuremath{\rho_{0} \left( {z} \right)}= 0, \;\; \ensuremat... ..._{1} \left( {z} \right)}=\ensuremath{\rho \left( {z} \right)}.$$ (23)

With this density function, the solution to Eq. 17 is a plane wave

$$\ensuremath{{\cal X}_{0} \left( {z} \right)}=\exp (ik_z z),$$ (24)

and the Green's function is an outgoing wave from $z^{\prime}$

$$\ensuremath{G \left( z \mid z^{\prime} \right)}= \frac{2\pi}... ...\exp \left( i k_z \left\vert z-z^{\prime}\right\vert \right) .$$ (25)

When Eq. 24 and this Green's function are substituted into Eq. 22 the reflected amplitude becomes

$$r = \frac{2\pi}{ik_z} \int_{-\infty}^{\infty} \exp (ik_z z) ... ...\left( {z} \right)}\ensuremath{{\cal X} \left( {z} \right)}dz.$$ (26)

No assumption has been made about the form of the incoming wave yet in Eq. 26, so the above equation is still technically an exact result. For the plane wave Born approximation, we assume $\ensuremath{{\cal X} \left( {z} \right)}= \ensuremath{{\cal X}_{0} \left( {z} \right)}$ and the reflected amplitude is

$$r = \frac{2\pi}{iq_z} \int_{-\infty}^{\infty} \exp (iq_z z) \ensuremath{\rho \left( {z} \right)}dz ,$$ (27)

where $q_z= 2k_z$ since the reflection is assumed to be specular.

Distorted Wave Born Approximation

The DWBA assumes that the unperturbed wave is a wave reflected from a completely smooth surface so we set[1]

$$\ensuremath{\rho_{0} \left( {z} \right)}= \left\{ \begin{array}{ll} Nb, & z > 0 \\ 0, & z < 0 , \end{array}\right.$$ (28)

and $\ensuremath{\rho_{1} \left( {z} \right)}= \ensuremath{\rho \left( {z} \right)}$, where $Nb$ is the scattering length density of the substrate and is just the scattering length density of the film which may or may not be a function of $z$. We can redefine the ${\cal L}$ operator so that

$${\cal L} = \left\{ \begin{array}{ll} \vspace{2pt}\frac{d^2}{... ...2pt}\frac{d^2}{dz^2} + k_{z}^{2}, & z < 0 , \end{array}\right.$$ (29)

where $K_{z}^{2}= k_{z}^{2} - 4\pi Nb$.

We know that the solution of Eq. 17 has the form of a wave that is partially reflected and partially refracted at $z = 0$:

$$\ensuremath{{\cal X}_{0} \left( {z} \right)}= \left\{ \begin... ...{z} z) + r_{0} \exp (-i k_{z} z), & z < 0, \end{array} \right.$$ (30)

where $t_{0}$ and $r_{0}$ are the transmission and reflection coefficients of the smooth surface respectively. The Green's function will then vary depending on the values of $z$ and $\ensuremath{z^{\prime}}$, and, since the Green's function is just a wave originating at the source point $\ensuremath{z^{\prime}}$, the function and its first derivative must be continuous, and we get the result that

$$\ensuremath{G \left( z \mid z^{\prime} \right)}= \frac{2\pi}... ...ime}})], & \ensuremath{z^{\prime}}< 0 < z . \end{array}\right.$$ (31)

It should be noted here that the Green's function has the correct asymptotic property of Eq. 12, namely

$$\ensuremath{G \left( z \mid z^{\prime} \right)}= \frac{2\pi}... ...\mbox{if}\; z < \ensuremath{z^{\prime}}\; \mbox{and}\; z < 0 .$$ (32)

So the final expression for the amplitude of the reflected wave in the distorted wave Born approximation is

$$r = r_0 + \frac{2\pi}{ik_z} \int_{0}^{\infty} \{ \left[ \exp (-ik_z z) + r_0 \exp(ik_z z) \right]^{2} \ensuremath{\rho_{1} \left( {-z} \right)}$$ (33)

$$+ [t_0 \exp(iK_z z)]^{2} \ensuremath{\rho_{1} \left( {z} \right)}\} dz .$$

The use of this equation varies depending on the specific type of system under study. For a single thin film on a substrate, $\ensuremath{\rho_{0} \left( {z} \right)}$ is taken to be the scattering length density of the substrate, and $\ensuremath{\rho_{1} \left( {z} \right)}$ is taken to be the scattering length density of the film. The result of this is a simple equation:

$$r = r_0 + \frac{2\pi}{ik_z} \int_{-d}^{0} [\exp(ik_z z) + r_0 \exp(-ik_z z)]^{2}\ensuremath{\rho_{1} \left( {z} \right)}dz .$$ (34)

Parratt's Recursion Relation

The final method of calculating reflectivities presented here is an iteratively exact method called Parratt's recursion relation[2]. The basic principle for this relation is very simple. The model scattering length density profile is broken up into a series of layers as shown in Fig. 2. The $i^{th}$ layer has a thickness $\Delta z_{i}$ and the $z$ component of the wave number $k_{zi}=\sqrt{k_{z0}^{2} - 4\pi\rho_{i}}$, where $k_{z0}$ is the incident wave number in air in the $z$

Figure 2: Scattering length density profile used for calculating reflectivities from Parratt's recursion relation. The profile is divided into $N$ layers, and the wave is calculated in each layer.

direction. The qualification that the substrate is a semi-infinite medium in which the wave number is $k_{zs}$ is also made. Then the solution to the wave equation in the $i^{th}$ layer can be written as

$$\ensuremath{\psi_{i} \left( {z} \right)}= A_i \left\{ \exp [ik_{zi}(z-z_i)] + r_i \exp [-ik_{zi}(z-z_i)] \right\},$$ (35)

where $A_i$ is the amplitude of the wave in the $i^{th}$ layer and $r_i$ is the reflection coefficient of the wave reflected from the edge of the $i^{th}$ layer.

Each of these waves are originating from a depth $z_i$ in the medium. The first derivative of the wave in each layer is given by

$$\psi_{i}^{\prime}(x) = ik_{zi}A_i \left\{ \exp [ik_{zi}(z-z_i)] -r_i \exp [-ik_{zi}(z-z_i)] \right\},$$ (36)

and since the wave and its first derivative must be continuous across adjacent layers we get the two equations:

$$\ensuremath{\psi_{i} \left( {z} \right)} = \ensuremath{\psi_{i+1} \left( {z} \right)}$$ (37)

$$\psi_{i}^{\prime}(x) = \psi_{i+1}^{\prime}(x)$$ (38)

or

$${A_i [ 1 + r_i] = }$$

$$A_{i+1} [ \exp (ik_{zi+1}\Delta z_{i+1}) + r_{i+1} \exp (-ik_{zi+1} \Delta z_{i+1}) ],$$ (39)

$${ik_{zi}A_i [ 1-r_i] = }$$

$$ik_{zi+1}A_{i+1} [ \exp (ik_{zi+1} \Delta z_{i+1}) - r_{i+1} \exp (-ik_{zi+1}\Delta z_{i+1})].$$ (40)

If Eq. 40 is divided by Eq. 39 the result is

$$\frac{1-r_i}{1+r_i} = \frac{k_{zi+1}}{k_{zi}} \frac{1-r_{i+1... ...t) }{1+r_{i+1} \exp \left(2ik_{zi+1} \Delta z_{i+1}\right) } ,$$ (41)

and when this equation is solved for $r_i$, the reflected amplitude in each layer is

$$r_i = \frac{R_{i+1} + r_{i+1} \exp \left( 2ik_{zi+1} \Delta ... ..._{i+1} r_{i+1} \exp \left(2ik_{zi+1} \Delta z_{i+1}\right) } ,$$ (42)

where $R_{i+1}= \left( k_i - k_{i+1}\right) /\left( k_i + k_{i+1}\right)$ and $r_{N} = R_{S} = \left( k_N - k_S\right) /\left( k_N + k_S\right)$. The iteration is started from the film-substrate surface and each higher layer is calculated recursively until the final result of $r_0$ for the reflectance in air is obtained. The reflected amplitude, $I_r$, is the square modulus of the reflectance in air, $r_0$, and is the quantity measured in a reflectivity experiment.

Bibliography

1

V. F. Sears.
Neutron News, 3:26, 1992.

2

L. G. Parratt.
Phys. Rev., 95:359, 1954.

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