Grazing Incidence X-Ray Reflectometry - Rigaku

The Rigaku Journal Vol. 14/ number 2/ 1997

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RECENT THEORETICAL MODELS IN GRAZING INCIDENCE X-RAY REFLECTOMETRY KRASSIMIR STOEV* AND KENJI SAKURAI National Institute for Research for Metals, 1-2-1 Sengen, Tsukuba, Ibaraki 305, Japan

1. Introduction The total reflection of X-rays from solid samples with flat and smooth surfaces was first reported by Compton in 1923 [1], which can be assumed to mark the birth of the experimental technique of X-ray specular reflectivity. Since the angle of incidence is very shallow and almost parallel to the surface, the measurement using X-ray total reflection is also called grazing incidence experiment. If the surface is not ideally smooth and somewhat rough, the X-rays can be diffusely scattered in any direction. The experimental technique is known as X-ray diffuse scattering (X-ray non-specular reflection). It started to develop immediately after the pioneering work in 1963 by Yoneda [2], who reported intensity modulation in the X-ray diffuse scattering, known as Yoneda wings or anomalous reflection. In 1971, Yoneda and Horiuchi [3] first reported X-ray fluorescence radiation from particles, deposited on a smooth substrate, and excited by X-rays impinging the surface at an angle below the critical angle of total reflection.

In this paper, recent advances in the theoretical models for grazing incidence X-ray reflectometry are reviewed. Special emphasis is placed on the discussion about the surface roughness and also on the DWBA theory in diffuse scattering. Total reflection X-ray fluorescence (TXRF) spectrometry is also discussed, because one of the most important recent trends is the combination of grazing incidence X-ray reflectometry and spectrometry. As several reviews of grazing incidence X-ray techniques have been already published [4-6], here, only some recent advances are introduced. Though grazing incidence Xray diffractometry is another interesting and important topic, it is out of the scope of this article.

Nowadays, X-ray reflectometry based on total reflection has become a powerful tool for analysis of surfaces and thin film interfaces, and still continues to grow up further. This is mainly due to the significant development of experimental techniques and instrumentation, especially the advent of synchrotron radiation and the progress achieved in detector technology. The advancement of the theoretical modeling and the techniques for analyzing experimental data are also important. This has been accelerated and aided by the rapid evolution of computers. Furthermore, theoretical modeling and experimentation are being stimulated by each other.

where the real term δ is associated with the dispersion, and the imaginary term β with the absorption. In the X-ray region, both δ and β are small quantities (about 10-4 to 10-8), and usually β is smaller than δ. These parameters are closely related to the atomic scattering factors f1 and f2:

* STA fellow at National Research Institute for Metals, Tsukuba, Japan. Permanent address: Bulgarian Academy of Sciences, Institute for Nuclear Research and Nuclear Energy, blvd. “Tzarigradsko shousse" 72, 1784 Sofia, Bulgaria.

where ρ is the density [g/cm3], E is the energy of Xray radiation in [keV], M is the molar weight [g/mol]

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2. Fundamentals and Physical Constants The complex index of refraction in the X-ray region can be written as n= 1 - δ- iβ

δ = 41516 . × 10 −4 ×

β = 41516 . × 10 −4 ×

(1)

ρ M ⋅E

N

2

ρ

∑C j =1

j

⋅ f1 j

(2)

⋅ f2 j

(3)

N

M ⋅E 2

∑C j =1

j

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of a compound of N different atoms, C, is the number of atoms of type j per molecule, and, f1j, and f2j are the atomic scattering factors for atoms of type j [7]. For all elements in the energy region away from the absorption edges, δ is a positive quantity, so the real part of the refractive index is smaller than unity, and therefore total reflection of X-rays can be observed in the region of incident angles below the critical angle of total reflection θ c = 2δ . For some elements, δ has negative values for energies around the absorption edges, and no total reflection process is present for these energies. The critical angle of total reflection, θc, is usually in order of few mrad to few tenths of mrad (i.e., in the region 0.1° to 1°). Databases of physical constants, necessary for the calculation of X-ray reflectivity curves, were published in [7-9]. There is a good agreement between the data for the atomic scattering factors published in [7] and [8], except for the energy region below 1 keV, where the difference is in some cases more than 50%. The Henke-GulliksonDavis database [7] covers the energy region 0.01-30.0 keV, and is available on the following WWW addresses: http://xray.uu.se/hypertext/henke.htm and http://grace.lbl.gov/optical-constants/index.html. The Chantler database [8] covers the energy region 0.1 keV-1.0 MeV. An example of the dependence of δ, β and θc on the atomic number Z for

Cu-Kα1 line (calculations are based on HenkeGullikson-Davis database [7]) is presented in Fig. 1. 3. X-Ray Specular Reflectivity In several reviews [5, 6], the theoretical concepts of the total reflection of X-rays from single surface and multilayers were presented. Basic description of the theory (known as Parratt's formalism) was published in 1954 [10]. In several recently published papers [11-13], the theory of neutron and X-ray reflectometry are treated similarly, employing Fresnel reflection and transmission coefficients as derived from electromagnetic wave-propagation theory. Schematic representation of a multilayer is given in Fig. 2. The reflectivity amplitude Rj from the bottom interface of the layer j is defined as a ratio of the amplitudes of the reflected Er,,j and the transmitted Et,,j electrical fields in layer j, i. e., k j ⋅dj  E r, j  , aj = e − i⋅ 2 R j = a 2j   E t, j 

(4)

where j is the index for the layer, t is the index for the transmitted incident radiation, r is the index for the reflected radiation, aj is the phase factor, dj is the thickness of layer j, and kj represents the z-component of the wave-vector for layer j, and is given by:

Fig. 1. Dependence of the real and imaginary correction terms (δ ανδ β) in the refractive index and of the critical angle of total reflection θc, on the atomic number Z of the scatterer. The energy is 8.047 keV (Cu-Kα1). Tabulated data is used for the density, and no data is presented for the gases.

Vol. 14 No. 2 1997

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Fig. 2. Multilayer representation for calculation of X-ray specular reflectivity. Each layer “j” is defined by its thickness dj, refraction index nj, and roughness of the bottom interface σj. Total of (NL-1) layers are considered plus the incident space (vacuum) and the substrate NL.

kj =

2π 2π sin 2θ 0 − 2δ j − 2iβ j n j sin θ j = λ λ

(5)

where n j=1-δj-iβj is the refractive index for layer j, and θ0 is the incident angle for the first interface. Here we have used the Snell's law: n 0 ⋅ cos θ 0 = n j ⋅ cos θ j . In the model used for the multilayer, the “0”-Iayer is assumed to be semi-infinite vacuum or air, and the last layer NL is also assumed to be semi-infinite, so that there will not be a recursive wave in this layer. Following recursive formula is used for calculation of the reflectivity amplitude: R j = a 4j

R j+1 + r j

k j − k j+1 ; rj =Qj R j+1 ⋅ r j + 1 k j + k j+1

(6)

where rj is the Fresnel reflection coefficient, and Qj is the roughness-modeling factor. One can start from the last layer NL (i.e., the substrate), where R N L is equal to zero, and can calculate the reflectivity amplitude Rj for all layers using the above recursion formula. The final reflectivity and transmission are defined as: R=

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IR = R0 I0

2

=

2 E t, N L E r, 0 ; TN L = E t, 0 E t, 0

2

(7)

The above formula is valid for an ideally parallel beam. If a divergent incident beam has certain spread expressed as I(θ), the reflectivity R(θ) will be given as a convolution of the reflectivity function and the intensity distribution: ∞

∫ R(θ − ϕ)I (ϕ)dϕ R(θ) = ∫ I (ϕ)dϕ −∞

∞

(8)

−∞

An example of the effect of the beam divergence on the reflectivity curve is presented in Fig. 3. Unfortunately, the Parratt's formalism requires recursive calculation of the intensity of the reflected beam, and is relatively difficult to be implemented for reconstruction of density profiles. Several analytical approximated formulas for calculating X-ray reflectivity have been also proposed [13-16]. An analytical formula for calculating X-ray and neutron reflection from thin surface films was proposed in [14]. An approximate analytical expression for X-ray reflectivity from one-dimensional scattering-length-density (SLD) profile ρ(z), based on the weighted superposition approximation, was derived in [15]:

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Fig. 3. Simulated reflectivity of Cu(50 C)/Al(200 C)/Co (100 C)/Si multilayer system at 8.047 keV for different angular divergence of the incident beam. The intensity distribution of the incident beam is considered constant (homogeneous). Calculations were performed with RefleX software [104].

∞

R = π ∫− d dz ⋅

[

z 1 dρ ⋅ exp 2i ∫−kd (z ) ⋅ dz k (z ) dz 2

]

(9)

where k (z ) = k 02 − 4π ⋅ ρ(z ), k 0 = ( 2π λ ) ⋅ sin θ

and the vacuum is in the z-region [-∞, d], the sample is in [-d, 0], and the substrate is in [0, ∞]. This model was compared to the Parratt's recursion formula, the Born approximation (BA), and the distorted-wave Born approximation (DWBA). It was concluded that the proposed formula is more accurate than the BA and DWBA, and is valid in the entire range of wave-vector transfer except the narrow region around the critical angle of total reflection. An analytical approximation for the calculation of grazing incidence X-ray reflectivity from a multilayers was derived in [16]. The approximation is correct in the low θ-range, and deviates by less than 1% from the rigorous Parratt's recurrence algorithm in the range 4πsin θ/λ > 0.28 C-1. 4.Modeling of Roughness in X-Ray Specular Reflectivity The treatment of the roughness is a very important part of the analysis of X-ray specular reflectivity. Usually this is done by the introduction of the roughness factor Qj in the Fresnel reflection coefficient rj. Following models are used:

Vol. 14 No. 2 1997

Qj=1 (i.e., an ideally smooth interface), Qj=e −2 σ j k j (Debye-Waller factor, [17]), 2

2

Qj= e −2 σ j k j k j +1 (Nevot-Croce factor, [17-19]), 2

where σj represents the root-mean-square (r.m.s.) of the vertical interface roughness. The Nevot-Croce roughness correction factor [17-19] was derived from the reciprocity principle and is based on the assumption for Gaussian distribution of the vertical roughness. Sinha et al. [53] used a different calculation model and have shown that the Debye-Waller factor can be derived within the frame of the Born Approximation (BA), while the Nevot-Croce factor can be derived within the frame of the Distorted Wave Born Approximation (DWBA). Thus they have introduced an unified way of description of the roughness for X-ray specular reflectivity and X-ray diffuse scattering. Later, de Boer has generalized this approach to include angle-dependent TXRF modeling [25]. Usually for low spatial frequencies of the roughness spectrum (