(in following context, please right click on the graphs to get a clearer view)

Chapter 2 Theory
 
 
 
 

2.1 Atomic Scattering Factor

The elastic scattering of x-rays from an atom is described in terms of the atomic scattering factor, defined as the ratio of the radiation amplitude scattered by the atom to that by a free electron, in the form

(2.1)
Where  is the atomic form factor calculated assuming the x-ray energy is large compared to the binding energy of all of the electrons in the atom and in the nonrelativistic approximation is given by:
(2.2)
yis the total wave function of the atom and i and f refer to the initial and final states. For elastic scattering they are the same. The vector s bisects the angle 180-2q . 2q is the angle between the incident and scattered wave vectors. The magnitude of s is 4p (sinq )/l , where l is the wavelength of the incident radiation and q the Bragg angle. In general,  is a function of q . In x-ray reflectivity measurements, however, s is usually close to zero and Eq. (2.2) can be simplified approximately:
(2.3)
where Z is the atomic number of the atom.

In Eq. (2.1),  and  are the real and imaginary part of the anomalous dispersion terms which account for the absorption of x-ray by electrons bound in an atom. The anomalous scattering effect is of interest for applications such as absorption and scattering experiments and phase determination of crystal structures, and is very important for RXR measurements. The  term represents in-phase scattering and the  term represents scattering shifted by 90° in phase. As for many types of dispersion phenomena  and  are linked by the Kramer-Kronig relation,

(2.4)
where (dg/dw )i is the oscillator density for oscillators of type i and the sum runs over all absorption edges. The oscillator density is related to the atomic absorption coefficient for the i-th oscillator m(w )i by
(2.5)
Close to an absorption edge  and  vary rapidly with wavelength. There is a sharp jump in  from below to above the edge while  dips to a large negative value through the edge.

The anomalous scattering factors can be calculated using the Cromer-Liberman method. A FORTRAN program based on this method is used to calculate  and  in this thesis. Fig. 2.1 shows the calculated curves of  and  for Ni and Fe around their K-edges. The anomalous scattering factors can also be obtained experimentally, for example, by measuring a set of Bragg reflections from a crystal of known structure that contains a suitable heavy atom.

Fig. 2.1 Calculated linear absorption and the anomalous scattering factor for Ni and Fe. The K-edge of Ni is at 8333 eV and Fe is at 7112 eV.
 
 

Even though the wave length l is comparable to atomic dimensions, specular reflectivity of x-ray can be described by the Fresnel theory of classical optics.20,25 For each material, a complex dielectric constant  can be defined:

(2.6)
Where Ni is the number density of atoms, lthe x-ray wavelength, re the classical electron radius, Na the Avogadro constant, rMi the mass density of the material and Ai the atomic mass. * is complex conjugate of , which is defined by Eq. (2.1) with the subscript referring to element i. Note that in this convention both  and ei have positive imaginary parts and the incident wave should be in the form of  . Substituting in (2.6) the numerical values for the constants, we will obtain:
(2.7)
In (2.7) the lis in unit of Å, rMi in unit of g/cm3 , and Ai in unit of g/mol .  is the effective electron density. From (2.7) we can see that for an x-ray beam with wave length of about 1 Å, (e i'-1) and ei" are of the order of 10-5 to 10-6 . Sometimes it is convenient to defined the refractive index in the medium by:
(2.8)
For a medium composed of N uniformly mixed elements, each with a mass fraction Pi , eis given by:
(2.9)
Where rM is the density of the medium. The last line of (2.9) defines the complex effective electron density r . Fig. 2.2 shows how the real part of the effective electron density and the linear absorption changes with the mass fraction of a two elements medium at different x-ray energies.

Fig. 2.2 Composition dependence of the electron density and linear absorption at different x-ray energies.
 
 
 
 

2.2 Dynamic Theory of X-ray Reflectivity

In this thesis, the concept of dynamic theory refers to the methods of solving the reflectivity problem for a layered structure in an exact way, such as the matrix formalism by Vidal and Vincent, and the recursion scheme by Parratt.20 Kinematical theory, on the other hand, is an approximation that neglect the multiple scattering and is valid only for the reflectivity at an incident angle larger than about twice the critical angle of total reflection. The Born approximation and the so called " single-scattering dynamical theory " are examples of kinematical theory. In solving the reflectivity problem for multilayer structures, the dynamical diffraction theory for perfect crystal developed by Darwin and Ewald are used by some authors. This method can be employed to calculate the reflected intensity about the Bragg peaks only and will not be discussed here.

Let's consider a layered structure with two rough interfaces at z1 and z2 (see Fig. 2.3). The real position of the interfaces can be described by a two dimensional surface function Zi(x,y) which satisfies:

(2.10)
Where the si is called the rms roughness of the interface. e1, e2 and e3 are the dielectric constants of the three homogeneous media divided by the two interfaces.

Fig. 2.3 Schematic representation of x-ray reflectivity from a two-interface structure.
 
 
 
 

The specular components of the x-ray waves in the three media e1, e2 and e3 can be represented by:

(2.11)
where  is the z-component of the wave vector in medium j, k0 the x-ray wave vector in vacuum and q0 the glancing angle of incidence.

For ideal interfaces (s= 0), the continuity of the electric field and its first derivative across the interface gives for s polarization (E perpendicular to the incident plane),

(2.12)
where d = z2-z1 and
(2.13)
A2 and B2 are given similarly by:
(2.14)
The reflectivity of the whole structure is given by B3/A3 . A1 in (2.14) is a constant and will not appear in the final result. The extension of above procedure to multilayers with n interfaces is straight forward,
(2.15)
where the M's are matrices defined in a similar way as in (2.12).

For a rough interface, when the roughness sis less than the thickness of the layer, a matrix form including the nonideal interface effects can be given by applying Green's theorem to the interface:38

(2.16)
where
(2.17)
The symbol á ñ x,y means in-plane average. Assuming the ergodicuty of the profile function h(x,y), (2.17) can be written as:
(2.18)
Where z is relative to the average position of the interface and w(z) is the height distribution function ( probability density ) at the interface. If, for example, the probability density of the roughness perturbation at the interface is assumed to be Gaussian:
(2.19)
Then the characteristic function  is real and given by,
(2.20)
To calculate the reflectivity from a single interface, set B2 in (2.16) to zero, then,
(2.21)
It is the well known Fresnel reflectivity equation for an ideal interface multiplied by a Gaussian form factor. In general, the reflectivity from a rough or graded interface can always be written in the form of a Fresnel equation times a form factor, which is determined by it's probability density w(z). We will come back to this point later.

For a two interface structure, from (2.16), one can get:

(2.22)
where
(2.23)
is the reflectivity from an individual interface. In (2.22) and (2.23) we have assumed that the probability densities are symmetric functions, i. e. , so the characteristic functions  are real functions. The last expression in (2.22) is actually the recursion formula used by Parrat20 if we replace the subscript 2 and 3 with N-1 and N, except that now the reflectivity for an individual interface is given by (2.23) instead of the simple Fresnel equation. The matrix formalism and the recursion scheme are equivalent in the case of a symmetric probability density.

From the recursion scheme we can derive a formula which is especially useful in calculating the reflectivity from a very thick multilayer with a constant repetition period. By "very thick " we mean the total thickness of the multilayer exceeds the sensitive depth of x-ray reflectivity measurements, which is mainly limited by the angular resolution of the experiment setup. For example, if the angular resolution is 0.01° , the sensitive depth estimated by the Bragg law is about 3000 Å. In this case, the rN and rN-2 defined in (2.22) are equal and the substrate information is not relevant, by using the recursion twice and set rN = rN-2 , we will have:

(2.24)
with
(2.25)
where d1 and d2 are the thickness of the two component layers in each repetition period, r12 and r21 are the reflectivities of individual interfaces, defined in a similar way as (2.23), assuming medium 2 is the top layer.

2.3 Kinematic Theory of X-ray Reflectivity

In the standard kinematic theory of x-ray reflectivity, Born Approximation, the reflected intensity is given by:

(2.26)
where  is the momentum transfer. This expression neglects both the multiple scattering of x-rays and the refraction effects. Therefore it is not valid for reflectivity at small angles and also not suitable for multilayers, as it will give a wrong Bragg peak position. (See Fig. 2.4)

Fig. 2.4 Calculated reflectivity curves from a multilayer with six bi-layer periods. The solid line is for the dynamical calculation (Matrix formalism), the dot line for the single-scattering theory and the dash-dot line for Born approximation.

The importance of the Born approximation is that it relates the measured intensity to the Fourier transform of the density profile derivative. From (2.26) the autocorrelation function (ACF) , or Patterson function can be derived by merely taking a Fourier transform:

(2.27)
(2.27) is useful in the qualitative interpretation of the experiment data,25,30 it also becomes the starting point of some Fourier-Synthesis methods trying to calculate the density profile directly from the reflectivity data.29,

There are several forms of pseudokinematical theories that make corrections over the standard Born approximation. These corrections usually involve two steps. First the prefactor in (2.26) is replaced with the Fresnel reflectivity of the substrate, thus avoid the divergence at small angles. The next step is to replace the vacuum path length Qzz in the phase factor of (2.26) with a refraction-corrected path length. The most straight forward and effective approach in doing this is given by the "single-scattering dynamical theory",45

(2.28)
for a multilayer with n sections. (2.28) can be obtained from (2.22) by simply dropping the r2 term, that is, the double scattering term. For a two-component periodical multilayer with n periods, (2.28) is reduced to:
(2.29)
For an arbitrary graded interface, the density profile can be approximated by n-steps (or boxes) function, applying (2.28) and let n tend to infinite, we will have:
(2.30)
where , and the [0, D] is defined as the region where the derivative of the density profile is not zero. Fig. 2.5 shows the reflectivity from a linear density profile calculated with (2.30), compared with those with other methods discussed in the next section. Note that for a linear density profile, the phase factor in (2.30) can be written explicitly as:
(2.31)
where a is the derivative of kz2 , a constant for a linear density profile.

Fig. 2.5 Reflectivity simulated from a linear density profile. (a) Form factor method (2.32), (b) Multi-step function (thin density box method), (c) exact solution, use Hankel functions as wave functions and (d) single scattering approximation (2.30). The interface model is shown in the inlet. The half width s is 12 angstrom.
 
 

2.4 Interface Model

In this section we will discuss different interface models and methods for calculating the reflectivity from these model interfaces.

Consider a single interface, the reflectivity coefficient r will be given by (2.23):
 
 

(2.32)
F12 is the form factor added to the Fresnel's reflectivity equation of an ideal interface. It can be calculated with (2.18) if the probability density w(z) is known. The probability density w(z), on the other hand, can be related to the derivative of the in-plane average of the effective electron density profile by:
(2.33)
Where r1 and r2 are the electron densities of the homogeneous media on the two sides of the interface. (2.33) implies a physical observation: the specular reflectivity measurements can not distinguish between the two kinds of interface broadening, namely, a rough interface with sharp local boundary and an overall smooth interface with graded density profile. In fact, the x-ray optics in this two cases is different. For a smooth interface, the radiation energy of the incident beam propagates in two directions only, that is, in the directions of reflection and refraction. The broadening of the interface decreases the reflected intensity while increase the refracted intensity. In the case of a rough interface, part of the incident energy flux will be scattered in to off-specular directions. This is called diffused scattering and can be observed by transverse scans. Fig. 2.6 shows an example of transverse diffuse scattering measurements.


Fig. 2.6 Transverse scan of diffuse and specular x-ray scattering for a Ni/W multi-layer across a superlattice Bragg peak. The narrow central potion represents the true specular folded with instrumental resolution.
 
 

To calculate the x-ray specular reflectivity from a rough interface, one has to assume a probability density function for the interface position in the direction of the surface normal. this often leads to a Gaussian distribution (2.19). For the reflectivity in the low Qz region, or in the case of a small interfacial roughness, the Gaussian distribution can reasonably account for almost any interface form since the higher cumulant terms are negligible. However, in the region of relatively higher Qz, and/or for an interface with large roughness or width, the Gaussian distribution is not necessarily the right choice. Eq. (2.18) and (2.32) can be used with other forms of probability density functions. Some examples are given in following paragraphs.
 
 

Constant distribution

(2.34)

(2.35)

A constant distribution corresponds to a linear density profile. the problem of x-ray propagation in a linear density medium can be solved exactly (cf. next section). Fig. 2.5 shows the calculated reflectivity curves from a linear medium with (2.32) ( rough boundary model), (2.30) ( graded interface model, single scattering approximation) and the exact solution (also graded interface model). The consistency between the three curves gives a numerical proof of the equivalence between a rough and a graded interface under specular reflectivity condition.

Lorentz distribution

(2.36)

(2.37)

The reflectivity from an interface with a Lorentz distribution calculated with (2.37) is shown in Fig. 2.7. For comparison, a step function (thin density boxes) calculation of the graded density profile is also shown in Fig. 2.7. The two calculations give consistent reflectivity curves.

Fig. 2.7 Calculated reflectivity from a Lorentz distribution (2.36) with s = 4.0 Angstrom. The squares are for the Form factor method and the triangles for thin density box method. The interface model is shown in the inlet.

Cosine distribution

(2.38)

(2.39)

Fig. 2.8 shows the reflectivity calculated with (2.37) and the graded density model (step function approximation).


Fig. 2.8 Calculated reflectivity from a Cosine distribution (2.38) with d = 8.0 Angstrom. The squares are for the Form factor method and the triangles for the thin density box method. The interface model is shown in the inlet.

The above examples demonstrated the flexibility of the form factor methods. They are also numerical proofs of the equivalence between the rough interface model and the graded density model in specular reflectivity calculation. In the analysis of x-ray reflectivity data, one can also assume other forms of distribution functions, and calculated the corresponding form factor. The effects of different interface models usually show up only at high Qz values of the reflectivity curves. (Fig. 2.9)

Fig. 2.9 Comparison of reflectivity curves calculated with different interface model. (A) shows the density profiles of the ideal sharp interface (a), Cosine distribution (b), Gaussian distribution (c) and the Lorentz distribution (d). (B) shows the reflectivity curves for a (), b (D ), c (¡ ) and d (¨ ).
 
 

The form factor method is based on a perturbation to the plane wave solution. The reflectivity calculated with this method is questionable when the thickness of the layer is less than or equal to the surface roughness of the interface.38 When there is a large in depth spatial variation region across the interface, one has to assume a stratified medium and solve the one dimensional Helmholtz equation for the electromagnetic fields.28

(2.40)
Where u(z) represents the electric field in the medium.

One way to solve (2.40) is to use a multi-step function (thin density boxes) to approach the real (in plane averaged) electron density. In each thin box, the plane wave solution is assumed and then (2.22) can be used to calculated the reflectivity. On the other hand, it is well known that (2.40) can be solved analytically for a linear density profile. In next section we will give a formalism for solving (2.40) by approaching the density profile with a series of straight line segments. This method has been used to analyze the reflectivity data from the Ni/Fe alloy thin film sample, which will be discussed in detail in Chapter 4.

2.5 The solution of the X-ray Reflectivity Problem for a Linear Stratified Media

Define: (see Fig. 2.10)

(2.41)

(2.42)

The equation satisfied by the E field in the media is:
(2.43)
Where  and  is the angle of incidence of the X-ray beam.

For z > z3:

(2.44)
For z2 < z < z3:
(2.45)
where:
(2.46)
and
(2.47)
is the 1/3 order Henkel function.

At z = z3, we apply the boundary conditions:

(2.48)
That is:
(2.49)
So:
(2.50)
For z1 < z < z2
(2.51)
The definition of  is similar to that of .

Applying the boundary conditions at z = z2, we have:

(2.52)
For z < z1
(2.53)

 

Boundary conditions at z = z1:

(2.54)

Combining (2.50), (2.52) and (2.54), we have:

(2.55)
The reflectivity coefficient r will be given by:
(2.56)
For a extended rough interface, by assuming that the line shape and the width of the extended interface are the same for all x, one can prove that the reflectivity is given by:
(2.57)
In following text we will derive (2.57) by applying the Green theorem onto an extended rough interface.

There are two ways of characterizing an interface. One way is to assume a zigzag but sharp boundary between two media (see Fig. 2.11A). The reflectivity from this kind of interface is given by (2.21) provided that . The other way is to assume a smooth but extended boundary and solve the Helmholtz equation (2.40). (see Fig. 2.11B) .It is well accepted that the Gaussian probability density function of the boundary position which leads to the Gaussian factor in (2.21) can be used to derive an effective refraction index profile. If this index function (or dielectric constant function) is inserted into (2.40), one get the same reflectivity by solving the differential equation. So it seems that the two way of characterizing the interfaces are equivalent and one can not tell the difference just from the specular reflectivity. However, these two kinds of interfaces are not completely equivalent because: (1) in case A , there is a limit  while in case B there is no limitation in the width of the interface. When there is a large in depth spatial variation region across the interface, it can not be described by a simple Gaussian factor. (2) In case A part of the energy flux goes into diffused scattering while in case B all the energy flux goes either into specular reflection or refracted beam.

We will try to demonstrate that the two ways (A and B) can be combined together to give a more complete description of an interface. (see Fig. 2.11C). We will prove that for a rough extended interface, the matrix is given by adding the same factor to the matrix elements of the smooth extended interface. The refraction index profile along z can be of any form. However, we assume that the interface has the same local, gradual surface profile, but that its boundary varies in the x-y plane as some function h(x,y).

Fig. 2.10 Linear segments description of a model interface.
 
 

Fig. 2.11 Characterization of interfaces.

For an ideal interface at z = z0, the electric field on the two sides can be written as:

(2.58)
Where .

For a smooth extended interface with z1-z2=d, we have:

(2.59)
where f(z), g(z) are two independent solutions of the Helmholtz equation (2.40) in the region of [z2, z1] (see Fig. 2.12). The relation between A', B' and etc. are:
(2.60)
Note that M(d) = M2(z2)M1(z1) depends on d only. Applying Green's theorem to the contour MNOP in Fig. 2.12,
(2.61)
Where  are defined in (2.58) and (2.59) respectively, with the subscripts ignored.

We then get:

(2.62)

Fig. 2.12 Schematic representation of the extended rough interface and the contour for Green's function.
 
 

Where  is a gradual function of z between [z2, z1] and  for z < z0 and  for z > z0 . And:

(2.63)
Now let
(2.64)
This will lead to:
(2.65)
And then,
(2.66)
Compare to (2.60), we have:
(2.67)
Now consider an extended rough interface. The specular component of the electric field is similar to (2.59):
(2.68)
Note that  is the Fourier component of  in the electric field solution within

region [z2, z1] and now z2, z1 are functions of x. Assume  are related to  by:

(2.69)
Applying Green's theorem to MNOP , we can get an equation in a form similar to (2.66):
(2.70)
Because we take the roughness as a perturbation, the function form in the inner parentheses should not deviate much from its zero order form, Eq.(2.67). Without a detail examination on the approximation error, we simply insert the right hand of Eq. (2.67) into (2.70) and end up with:
(2.71)
With the assumption that the interface has the same local, gradual refraction index profile, we have z1(x)-z1(x)=d independent of x, so the M matrix element can be factor out from the x average parentheses. Assuming a Gaussian probability density ,we will have:
(2.72)
So:
(2.73)
Where: 

In a similar way we can get:

(2.74)
So at  , the reflectivity coefficient r is given by:
(2.75)
(2.57) is thus proved.