2.1. GISAXS basic formulae

The basic setup of a GISAXS experiment referring to a medium of finite thickness supported on a semi-infinite substrate is sketched in Fig. 1. It is assumed that three media [air/vacuum, medium (m), substrate (s)] exist with relative indices of refraction

*n*
_{air} <

*n*
_{m} <

*n*
_{s}. The theoretical formulation of GISAXS derived recently (Lee, Park

*et al.*, 2005

; Lee

*et al.*, 2007

; Lee, Yoon

*et al.*, 2005

) is briefly reviewed here to establish the conceptual basis for the depth-profiling procedure.

For an incident plane wave of wavevector

**k**
_{i} and exit plane wave with wavevector

**k**
_{f}, the boundary conditions imposed by Maxwell’s equations yield two solutions (Lee

*et al.*, 2007

; Dosch, 1992

) for the electric field inside a uniform medium at

**r**, where the medium of thickness

*d* is bounded above by the vacuum and below by the substrate,

Following Lee

*et al.* (2007

),

,

is the wavevector component parallel to the interface,

(with

*j* = i, f) is the refracted component of the wavevector normal to the interface, and the amplitudes of the transmitted and reflected waves inside the medium are given by the Fresnel coefficients

*T*
_{j} and

*R*
_{j}. As usual, * indicates the complex conjugate. Conventional X-ray reflectivity information associated with substrate-supported homogeneous films is not accounted for in this formulation.

The above solutions are exact for smooth interfaces bounding a uniform medium with electron density

. If it is assumed that the medium includes local deviations in electron density that are sufficiently small to be treated as perturbations, the electron density is written as

. The deviations

give rise to a scattering potential (Lee

*et al.*, 2007

; Born & Wolf, 1999

)

, where

*v* is the total volume occupied by

and

*r*
_{o} is the classical electron radius (Als-Nielsen & McMorrow, 2001

). In the first-order Born approximation, the scattering amplitude for

**q** =

**k**
_{f} −

**k**
_{i} at a distance

*R* far from the scattering region is given by

where

*k* is the magnitude of the wavevector

and the integration is over the volume of the medium.

As noted by numerous authors (Lee, Park

*et al.*, 2005

; Lee

*et al.*, 2007

; Vartanyants

*et al.*, 2007

; Lee, Yoon

*et al.*, 2005

), when the incident angle α

_{i} is greater than the air–medium critical angle, α

_{c,m}, the total scattered wave amplitude can be written as the sum of four distinct scattering amplitudes resulting from the product of terms in equation (2)

. For the current case of a medium of finite thickness supported by a semi-infinite substrate (Lee, Park

*et al.*, 2005

; Lee

*et al.*, 2007

; Lee, Yoon

*et al.*, 2005

), this sum has been shown to be

where

is the in-plane momentum transfer vector. The individual

*z* components of the momentum transfer inside the medium,

,

,

and

, are identified with specific scattering processes by noting the condition yielding

*q*
_{i,z} = 0. In the case of

*q*
_{2,z} and

*q*
_{3,z}, this occurs when

, indicating scattering about the reflected beam (Stein

*et al.*, 2007

; Lee, Park

*et al.*, 2005

). Similarly,

*q*
_{1,z} and

*q*
_{4,z} are identified as representing scattering about the direct beam. Since

*q*
_{1,z} = −

*q*
_{4,z} and

*q*
_{2,z} = −

*q*
_{3,z}, the GISAXS amplitude may be identified as being the sum of two sets of scattering events about the direct and reflected beams.

Assuming that the cross terms resulting from the product of equation (2)

can be neglected (Lee, Park

*et al.*, 2005

; Lee, Yoon

*et al.*, 2005

) and that the incident angle is greater than the air–medium critical angle, the observed intensity is effectively the sum of two distinct sets of scattering events by the direct

and reflected

sources. In this work it is assumed that these scattering events can be distinguished in the measured GISAXS data (Lee, Park

*et al.*, 2005

) and only the results of scattering from the reflected beam will be considered. In addition, it will be assumed that consideration can be restricted to

*q*
_{3,z} terms only. As described previously (Stein

*et al.*, 2007

), scattering associated with

*q*
_{2,z} occurs before specular reflection from the substrate while scattering associated with

*q*
_{3,z} occurs after specular reflection. Ideally, GISAXS experiments relying on the reflected beam use an incident angle within the range α

_{c,m} < α

_{i} < α

_{c,s}, where α

_{c,s} is the critical angle of the substrate. Given that the materials of interest are assumed to be weak scatterers within the Born approximation, the large path lengths introduced by the shallow incident and exit angles of a typical GISAXS experiment imply that scattering associated with

*q*
_{2,z} will be negligible in comparison to scattering occurring

*after* specular reflection (

*q*
_{3,z}). In the latter case, absorption of the scattered radiation takes place only on the path out of the sample towards the detector (Lee, Yoon

*et al.*, 2005

). With these assumptions in place, the scattered wave amplitude of interest,

, is

where

represents the kinematic structure amplitude associated with the specific branch of GISAXS given by the momentum transfer

*q*
_{3,z}.

For incident radiation of wavelength λ the momentum transfer

inside the medium for the specific scattering process of interest has components (Dosch, 1992

)

Following the convention defined in Fig. 1, α

_{i} and α

_{f} are the incident and exit angles relative to the interface in the (

*xy*) plane,

and

are the refracted incident and exit angles seen inside the medium, and 2θ is the horizontal scattering angle (no refraction). Rewriting

*q*
_{3,z} in terms of the directly measured quantities, α

_{i} and α

_{f}, for a medium of uniform index of refraction,

*n*
_{m} = 1 − δ

_{m} +

*i*β

_{m} (Als-Nielsen & McMorrow, 2001

),

The scattering depth describing the interaction range of the X-ray probe inside the medium is given by the inverse of the imaginary component of

*q*
_{3,z} (Lee, Yoon,

*et al.*, 2005

; Dosch, 1992

),

With the real part of

*q*
_{3,z} written as

*Q*
_{z} Re(

*q*
_{3,z}), the scattering amplitude is written as

where

represents the in-plane structure amplitude at a position (depth)

*z* inside the medium (Dosch, 1992

).

Finally, the observed intensity of that part of the GISAXS profile under consideration is proportional to the absolute square of the scattered wave amplitude,

Consider now a representation of the integral for the structure amplitude over

*z* in discretized form, summing over layers of finite thickness, Δ

*z*, with

*N* layers extending from −

*d* <

*z* < 0. In each layer, located at

*z* =

*z*
_{j}, the ratio

*z*
_{j}/Λ is written as a constant and the possibility of an index of refraction varying with depth is neglected as a first-order approximation. The thickness Δ

*z* is chosen to be sufficiently large to give access to structural information within the layer and, as discussed later, to allow for incoherent addition of scattering amplitudes from different layers (Luo & Tao, 1996

). It should be noted that this construction is not intended to apply to a discrete layer system with sharp interfaces (Lee, Yoon

*et al.*, 2005

). The scattering amplitude becomes

The observed intensity can therefore be written as the sum over independent scattering contributions from the individual layers, weighted by the absorption terms, and the cross terms arising from the products of structure amplitudes of different layers.

A rigorous theoretical treatment dealing with the role played by the cross terms will not be attempted in this work. Instead, it will be assumed that for the applications of interest the intensity contribution from the interference term,

, may be neglected. That is, a minimum layer thickness, Δ

*z*, is chosen to allow for incoherent summation of the scattering amplitudes from the layers at

*z*
_{j} (Luo & Tao, 1996

). This situation may be realized when dealing with finite layer thickness in the absence of a highly coherent X-ray source with the assumption that the usual experimental difficulties associated with GISAXS measurements (Schlepütz

*et al.*, 2005

) have been successfully dealt with. The observed scattering about the reflected GISAXS beam is then simply expressed as the sum of scattering intensities from

*N* layers, weighted by the absorption factor,

.

The final expression for the GISAXS relative to the reflected beam can be compared to similar integral formulations that have been used in depth-resolved X-ray diffraction analysis (Broadhurst, Rogers, Lowe & Lane, 2005

; Luo & Tao, 1996

) and early reports on the application of GISAXS methods to the study of thin surface layers (Naudon

*et al.*, 1991

). For the simple case of a material with a constant linear absorption coefficient μ (μ = 2

*k*β

_{m}) referring to intensity attenuation rather than amplitude attenuation (Als-Nielsen & McMorrow, 2001

), the observed X-ray diffraction intensity (Broadhurst, Rogers, Lowe & Lane, 2005

) can be written as

In equation (12)

, α

_{i} is again the angle of incidence relative to the sample surface and 2θ

_{d} is the scattering angle in the plane of incidence relative to the direct beam. Diffraction data from the medium at depth

*z*,

*I*(

*z*, θ

_{d}), are recovered. Broadhurst, Rogers, Lowe & Lane (2005

) assumed that the medium was homogeneous in the direction parallel to the interface throughout the depth of the sample. For the purposes of this work, the procedure outlined by Broadhurst, Rogers, Lowe & Lane (2005

) is adapted to deal with GISAXS data obtained for varying incident angle, α

_{i}, typically using two-dimensional detection giving access to scattering information as a function of both 2θ

_{d} (alternatively α

_{f}) and

**q**
_{||}. In particular, information regarding variations of in-plane scattering as a function of depth below the sample surface is desired. The present goal is therefore reconstruction of scattering information

*I*(

**q**
_{||},

*z*) from specific depths

*z* below the sample surface for fixed

*Q*
_{z}. It must be noted that the condition of fixed

*Q*
_{z} generally implies variation of α

_{f} with varying α

_{i} [

*Q*
_{z} Re(

*q*
_{3},

*z*), where

*q*
_{3,z} is given in equation (6)

]. In accordance with the basic premise of the depth-profiling algorithm, the scattering depth Λ(α

_{i}, α

_{f}) will vary with α

_{i} when

*Q*
_{z} is held fixed. The original procedure proposed by Broadhurst, Rogers, Lowe & Lane (2005

) can, of course, still be applied to obtain information of the form

*I*(

*Q*
_{z},

*z*) for fixed

**q**
_{||}.

2.2. Depth-profiling algorithm

Equation (11)

, written now in integral form with

neglected, can be used as the starting point for applying the numerical methods proposed by Broadhurst, Rogers, Lowe & Lane (2005

) to extract intensities

from the measured scattering profiles obtained for a range of α

_{i} values. Including geometric factors associated with the beam cross section on the sample and explicit reference to the angle of incidence, equation (11)

is rewritten as

The intensity contribution,

, is a function only of the scattering power for the below-surface layer at

*z* for the momentum transfer, (

**q**
_{||},

*Q*
_{z}), independent of the α

_{i} geometrical factor. The scattering depth for the layer at a depth

*z*, Λ(α

_{i}, α

_{f}), is a function of incident and exit angle relative to the air–material interface and can be written as Λ(α

_{i},

*Q*
_{z}). In the following discussion it is assumed that the measured GISAXS data are processed to account for parasitic background and the |

*R*
_{i}
*T*
_{f}|

^{2}/sin α

_{i} prefactor in equation (13)

. For fixed (

**q**
_{||},

*Q*
_{z}) equation (13)

can then be identified as a standard Fredholm integral equation of the first kind (Broadhurst, Rogers, Lowe & Lane, 2005

) with a kernel given by

*K*(

*z*, α

_{i}) = exp(−2

*z*/Λ). The observed scattering intensity measured for varying incident angle α

_{i} can then be written as

The notation used here is intended to follow that of Broadhurst, Rogers, Lowe & Lane (2005

) in referring to

and

as

and

, respectively. Recovery of the scattering for a given value of (

**q**
_{||},

*Q*
_{z}) from the medium at a depth

*z* below the sample surface,

, is the goal of the depth-profiling procedure.

Inverting integrals of the form given in equation (14)

is known to be complicated by the fact that small perturbations in the measured quantity,

, can lead to large distortions in the recovered values for

(Svergun, 1992

; Broadhurst, Rogers, Lowe & Lane, 2005

). Broadhurst, Rogers, Lowe & Lane (2005

) have proposed a numerical method of determining

as a linear combination of

*n* Chebyshev polynomials,

*T*
_{j}(

*z*), using the method of collocation (Boyce & DiPrima, 1997

).

Following Broadhurst, Rogers, Lowe & Lane (2005

),

is written as

According to the method of collocation, values for

are calculated at depth values

*z* that coincide with the zeros of the

*n*th Chebyshev polynomial,

This approach is applied to ensure the best fit of equation (14)

for the observed data (after processing as noted above) at all α

_{i} values. Values for the scattering intensity for fixed (

**q**
_{||},

*Q*
_{z}) momentum transfer are thereby determined at specific depth locations,

. Defining the matrix

**T** of values of the Chebyshev polynomials calculated for a medium of thickness

*d* at the zeros of the

*n*th Chebyshev polynomial,

, the final solution of interest is written as

For

*m* incident angles α

_{i,k},

, equation (14)

can be rewritten as

The

*m* ×

*n* matrix of definite integrals,

**Z**, is in general not square. Following the solution prescription outlined by Broadhurst, Rogers, Lowe & Lane (2005

), the final goal of determining the vector of coefficients,

, is achieved using linear regularization. They identify a residual function which incorporates a stabilizer that is the product of a regularization parameter, γ, and a function of the solution,

. First-order regularization (Press

*et al.*, 1992

) is incorporated in the form of the (

*n* − 1) ×

*n* matrix

**B**, introduced below. The final result for the vector of coefficients

is the solution of the equations [see Broadhurst, Rogers, Lowe & Lane (2005

) for details of the derivation]

where

In equation (20)

the value of the constant,

*c*, is determined by a combination of the objective χ

^{2} criterion (Press

*et al.*, 1992

) and a subjective identification of a non-negative result with an acceptable degree of smoothness in the final solution (Svergun, 1992

). Broadhurst, Rogers, Lowe & Lane (2005

) report that a value of

*c* = 0.3 worked well for their numerical experiments.

Finally, the solution to equation (19)

yields the vector of coefficients,

, which is used in equation (17)

to calculate

for fixed (

**q**
_{||},

*Q*
_{z}). The matrices of equation (19)

,

**Z**,

**B** and

**T**, are independent of

**q**
_{||}. Therefore, for a given (fixed) value of

*Q*
_{z}, the calculation outlined in equation (19)

can be straightforwardly repeated for each vector of observed intensities

at different

**q**
_{||} to reconstruct the depth-specific scattering intensity in the horizontal plane,

.