Interference phenomena of synchrotron radiation in TEY spectra for silicon-on-insulator structure

Andreeva, M.A.; Domashevskaya, E.P.; Odintsova, E.E.; Terekhov, V.A.; Turishchev, S.Y.

doi:10.1107/S0909049512022844

research papers

JOURNAL OF
SYNCHROTRON
RADIATION

ISSN: 1600-5775

Volume 19| Part 4| July 2012| Pages 609-618

doi:10.1107/S0909049512022844

Interference phenomena of synchrotron radiation in TEY spectra for silicon-on-insulator structure

M. A. Andreeva,^a ^* E. P. Domashevskaya,^b E. E. Odintsova,^a V. A. Terekhov ^b and S. Yu. Turishchev ^b

^aFaculty of Physics, M. V. Lomonosov Moscow State University, Leninskie Gory, 119991 Moscow, Russian Federation, and ^bVoronezh State University, Universitetskaya pl. 1, 394006 Voronezh, Russian Federation
^*Correspondence e-mail: mandreeva1@yandex.ru

(Received 1 November 2011; accepted 18 May 2012; online 12 June 2012)

The general matrix theory of the photoelectron/fluorescence excitation in anisotropic multilayer films at the total reflection condition of X-rays has been developed. In a particular case the theory has been applied to explain the oscillation structure of L_2,3 XANES spectra for a SiO₂/Si/SiO₂/c-Si sample in the pre-edge region which has been observed by a sample current technique at glancing angles of synchrotron radiation. Remarkably the phase of the oscillations is reversed by a ∼2° angle variation. The observed spectral features are found to be a consequence of waveguide mode creation in the middle layer of strained Si, which changes the radiation field amplitude in the top SiO₂ layer. The fit of the data required the correction of the optical constants for Si and SiO₂ near the Si L_2,3-edges.

Keywords: XANES; total electron yield; X-ray standing waves; X-ray reflectivity.

1. Introduction

Total electron yield (TEY) or quantum yield (Y) (the latter is proportional to the compensatory photocurrent in TEY measurements) are well established methods for measuring spectra of X-ray absorption near-edge structure (XANES). Usually TEY spectra give a spectral shape similar to that of absorption spectra. The advantage of TEY or Y spectrum measurements is that they do not require the special thin-sample preparations that are needed for the direct absorption measurements. The effective sampling depth of the TEY technique, especially in the case of soft X-rays, is rather small (∼2–5 nm) and is the reason for the excellent surface sensitivity of the method.

However, in recent years experimental evidence has been presented showing that in some cases an unexpected distortion of the TEY spectrum shape appears at X-ray glancing incidence angles and even at normal incidence (Ejima et al., 1999 ; Terekhov et al., 2010 , 2011 ; Domashevskaya et al., 2011 ). These results stimulated further investigations of TEY spectra at different glancing angles in order to find an explanation for the observed effects. For some samples extremely impressive modifications of the spectrum shape were obtained that show up as a periodic variation with the angle change (Domashevskaya et al., 2011).

It is well known that for small glancing angles TEY is influenced by reflection from the surface. In the 1960s a simple formula was published (Rumsh et al., 1961 ) that represented the number of created electrons dn in the differential layer dz,

$[{\rm{d}}n=({{{I_0}}/{E\,}})[1-R(\theta)]\exp[-(\mu/\sin\theta)z](\mu/\sin\theta)\,{\rm{d}}z,\eqno(1)]$

where I₀ = N₀hc/λ is the intensity of the incident radiation, E is the energy expenditure for the creation of one electron, λ is the incident photon wavelength, R is the reflectivity, θ is the glancing angle for the X-ray beam and μ is the linear absorption coefficient. In particular, from (1) it follows that if R(θ) is sufficiently small the TEY or Y from the surface is proportional to the absorption spectrum μ(λ). The similarity between the absorption and TEY spectra was demonstrated for the first time by Gudat & Kunz (1972 ).

The angular dependencies of TEY or fluorescence yield near the total external reflection angle have been well studied for soft and hard X-rays since the famous paper of Henke (1972 ), who presented a more accurate formula for dn,

$[\eqalign{{\rm{d}}n(\theta,z)&=S[1-R(\theta)]{{\sin\theta}\over{\sin\theta^{\,\prime}}}\exp(-{\mu}z/\sin\theta^{\,\prime})\rho{\tau}T(z)\,{\rm{d}}z,\cr T(z)&=\left[\exp(-\varepsilon_{\rm{e}}z/\sin\psi)\right]/\sin\psi,}\eqno(2)]$

where S = I₀A₀(ω/4π) is the instrument constant, A₀ is the illuminated area, θ′ is the glancing angle for the refracted wave, ρ is the volume density of the atoms, τ is the atomic cross section for the creation of the photoelectron, T(z) is the function of the electron yield at depth z, ∊_e is the linear electron-attenuation coefficient, and ψ is the glancing angle for the electron emission. In the simplest case, μ = ρτ; however, in (2) ρ and τ can refer to specific q-type photoelectrons τ_q and density of atoms which can emit such photoelectrons ρ_q. Later a similar theory was developed by Solomin & Kruglov (1984 ).

Note that in the case of the reflection from a semi-infinite mirror the following exact relation takes place,

$[[1-R(\theta)]{{\sin\theta }\over{\sin\theta^{\,\prime} }} ={\left|{1 + r(\theta)}\right|^2},\eqno(3)]$

where r(θ) is the Fresnel amplitude of reflectivity,

$[r=(\sin\theta-\eta)/(\sin\theta+\eta),\eqno(4)]$

$[\eta=\left(\sin^2\theta+\chi\right)^{1/2}.\eqno(5)]$

η is the normal component of the wavevector of the refracted wave in units of ω/c, sinθ′ = Re(η), χ = (∊ − 1) is the susceptibility of the reflecting medium, ∊ is its dielectric function. So TEY from the surface of a semi-infinite medium is initiated by the square module of the full electric field |1 + r(θ)|², which was later called the standing wave at the surface (Bedzyk et al., 1989 ).

A more general theory of TEY at the diffraction conditions from crystals has been developed (Afanas'ev & Kohn, 1978 ; Koval'chuk & Kohn, 1986 ). It was shown that the TEY angular dependency in the vicinity of the Bragg angle reflects the detailed structure of the radiation field inside the crystal (X-ray standing waves). It was demonstrated how it can be used for depth-selective analysis of imperfections of the crystal structure. Later (Bedzyk et al., 1989; de Boer, 1991 ; Dev et al., 2000 ), the standing-wave influence created at the total reflection conditions on the fluorescence or photoelectron yield was analysed as a function of the incidence angle. However, the problem of the proper determination of the function of the electron yield T(z) (see, for example, Liljequist et al., 1978 ; Kovalchuk et al., 1986 ) imposes some limitations on the wide application of the TEY method. The fluorescence yield measurements have shown more advantages in the X-ray standing-wave method mostly due to its element sensitivity and also due to a much simpler form of the yield function T(z) [see reviews by Vartanyants & Koval'chuk (2001 ) and von Bohlen (2009 ), and also recent papers (Lee et al., 2006 ; Bera et al., 2007 ; Gupta et al., 2007 ; Novikova et al., 2009 ; Andreeva et al., 2009 )].

The interpretation of the spectral dependencies of TEY from multilayers at glancing angles, where the reflectivity could not be ignored, has attracted minor attention up to the present. Reflectivity spectra near absorption edges in the soft X-ray region have been investigated in just a few works (Barchewitz et al., 1978 ; Bremer et al., 1980 ; Kaihola & Bremer, 1981 ; Jones & Woodruff, 1982 ; Andrė et al., 1984 ; Lyakhovskaya et al., 1988 ; van Brug et al., 1989 ; Waki & Hirai, 1989 ; Filatova et al., 1995 , 1999 , 2006 ). However, experiments at synchrotrons have presented ample opportunities for such measurements. Later, the magnetic effects at absorption edges, revealing themselves in the asymmetry of the reflectivity spectrum for different X-ray polarizations, attracted more attention (see, for example, Kao et al., 1994 ; Knabben et al., 1998 ; Sacchi, 1999 ; Goering et al., 2001 ; Kim & Kortright, 2001 ; Oppeneer et al., 2003 ; Andreeva et al., 2006 ; Bergmann et al., 2006 ). To our knowledge just one theoretical paper devoted to the TEY spectra at the total reflection conditions from multilayers has been published (Ejima, 2003 ). Later, this theory was applied to the interpretation of the experimental data (Watanabe et al., 2006 ). Following the theory developed earlier for the case of thin films (Pepper, 1970 ), the waves in direct and opposite directions in each sublayer, as well as the influence of the interference of these waves on TEY, were taken into account. The formalism for the standing-wave description by Ejima (2003) was similar to that used by de Boer (1991) and Dev et al. (2000) devoted to the fluorescence emission. Note that the possibility to study the magnetic effects near the absorption edges requires the more general development of the standing-wave formalism for the case of anisotropic (magnetic) interactions of X-rays with multilayers at total reflection conditions.

A similar problem was considered earlier for Mössbauer radiation. Effects of the spectrum shape distortions for the conversion electron yield at grazing angles have been reported in several papers (Andreeva et al., 1991 , 1994 , 1996 , 1999 ). In these works the general theory of reflectivity from anisotropic multilayers was adjusted for consideration of the conversion electron Mössbauer spectra. This approximation could be easily used in the soft X-ray region as well. Use of the propagation matrices operating with the full field vectors simplifies the consideration of the standing-wave influence on the photoelectron/fluorescence excitation in an isotropic case and in addition allows the theory to be generalized to the case of anisotropic multilayers. The main difference in the matrix theory from the earlier theories (Ejima, 2003; de Boer, 1991; Dev et al., 2000) (applicable only to the case of isotropic multilayers) is the rejection of consideration of waves in direct and opposite directions in each sublayer as well as their interference (a very cumbersome procedure especially in the anisotropic cases), and the performance of all computations with total field vectors.

Here we present the general 4 × 4-matrix theory of the standing-wave formation in anisotropic multilayers and their action on fluorescent quanta or photoelectron creation. We show as well how the propagation matrices are simplified in a particular case of isotropic interaction of X-rays with atoms. Then we apply our matrix algorithm for the interpretation of the quantum yield spectra measured from a SiO₂/Si/SiO₂ multilayer for the photon energy in the vicinity of the L_2,3 Si absorption edges at different glancing angles and find an explanation of the peculiarities of these spectra recently detected (Domashevskaya et al., 2011).

2. Theory

We calculate the total photoelectron yield Y(ω) using the general formula

$[Y(\omega,\theta)=\textstyle\int\limits_0^\infty {T(z)A(z,\omega,\theta)\,{\rm{d}}z},\eqno(6)]$

where A(z, ω, θ) is the number of electrons created at depth z and T(z) is the function of the electron yield from depth z. This is a common approach where the photoelectron/fluorescence yield is considered in three steps: (i) we calculate how many photoelectrons or fluorescence quanta are created at depth z by the electromagnetic radiation absorption; (ii) how these electrons or quanta will reach the detector or will be registered (e.g. in the induced-current technique); and (iii) perform the integration over all depths. Our efforts are mainly directed at the first step, and we rely upon famous papers by Henke (1972), Kasrai et al. (1996 ) and Ejima (2003) for the description of T(z).

For the determination of the number of photoelectrons or fluorescent quanta A(z, ω, θ) created in a differential layer dz we use the general energy conservation law (Born & Wolf, 1968 ; Fedorov, 1976 ),

$[{\rm{div}}\,{\bf{S}}+{{\partial W} \over {\partial t}}=-Q,\eqno(7)]$

where S = (c/4π)Re[ $[{\bf{E}}^*]$ × H] is the Pointing vector averaged over time (the × sign means the vector product), W = (c/8π)( $[{\bf{E}}^*\,\hat\varepsilon'\,{\bf{E}}]$ + $[|{\bf{H}}|^2]$ ) is the density of the electromagnetic energy, and

$[Q=\left({\bf{E}}^*\hat\sigma\,{\bf{E}}\right)\eqno(8)]$

is the energy loss in unit volume at unit time (the superscript * sign denotes a complex conjugate). In (7) for the separation of gyrotropic and absorption effects the dielectric tensor $[\hat\varepsilon]$ is separated into two parts,

$[\hat\varepsilon'= \left(\hat\varepsilon+\hat\varepsilon^{\times}\right)/2,\qquad\hat\sigma=({\omega/{4\pi i}})\,\left({{\hat\varepsilon-\hat\varepsilon^{\times}}/2}\right),\eqno(9)]$

where $[\hat\sigma]$ is the conductivity tensor, which determines the absorption processes (mainly photoabsorption for soft X-rays). The superscript × sign designates the Hermitian conjugated matrix. In non-gyrotropic media the tensor $[\hat\sigma]$ is simply the imaginary part of $[\hat\varepsilon]$ . In gyrotropic media the imaginary components of $[\hat\varepsilon]$ can be responsible for the polarization effects, so the extraction of the absorptive part from $[\hat\varepsilon]$ is more complicated as in (9).

For the soft X-ray region the energy loss Q mainly determines the photon absorption and consequently the number of electrons or fluorescent quanta created at depth z in (6),

$[A\,\,\propto\,\,\tau\left({\bf{E}}^*\hat\sigma\,{\bf{E}}\right)\,\,\propto\,\,\tau\left({\bf{E}}^*\,{{\hat\chi-\hat\chi^{\times}}\over{\lambda}}\,{\bf{E}}\right),\eqno(10)]$

where $[\hat\chi]$ = $[\hat\varepsilon]$ − 1 is the tensor of susceptibility. If the medium contains different kinds of atoms then in (10) we should use the part of $[\hat\sigma]$ and $[\hat\chi]$ which is determined by the volume density ρ_q of the required kind of atoms, i.e. $[{\hat\sigma_q}]$ and $[\hat\chi_q]$ . In our case we deal with the limited energy interval of the incident photons (95–105 eV) and consider τ as a constant parameter determining the number of photoelectrons or fluorescent quanta for one absorbed photon (but keeping in mind that this number can be specific for different layers).

Vector E in (10) represents the total electric field of radiation. If we use (10) it is not necessary to consider separately the waves in the direct and opposite directions and their interference influence in each layer on photoelectron creation as done by Ejima (2003). The convolution of the conductivity tensor $[\hat\sigma]$ with the electric field vectors E and $[{\bf{E}}^*]$ takes into account the possible anisotropy of the absorption process, because not only is the absolute value of the Pointing vector at the atom position essential but also the ability of the atom to respond to a given polarization of the electric field. A similar problem of anisotropic media with multipole interactions was analysed by Vartanyants & Zegenhagen (1997 ).

To our knowledge the simple expression (10) is the most general expression for the photoelectron or fluorescent quantum creation which is valid for any anisotropic or gyrotropic media. The other approach based on the eigen states of the electromagnetic waves in anisotropic layered media should take into account the interaction of atoms with four eigen waves and their interference fluxes (4 + 6 terms in total) which makes the calculations very cumbersome. Note that the problem of the eigen wave calculations in anisotropic media is also rather complicated (see, for example, Odintsova & Andreeva, 2010 ).

If ∊ is a scalar function (or the polarization of the radiation electric field E is one of the eigen polarization of the reflectivity task) then $[\hat\chi]$ − $[\hat\chi^{\times}]$ = 2Im(χ) can be taken out of the convolution procedure and we easily obtain (ω = 2πc/λ)

$[\eqalignno{A(z,\omega,\theta)&\,\,\propto\,\, \tau \,\left\{{\rm{Im}}[\chi_q(z,\lambda)]/\lambda\right\}\,\left|E(z,\lambda,\theta)\right|^2 \cr&\,\,\propto\,\, \tau\mu_q(z,\lambda)\left|E(z,\lambda,\theta)\right|^2.&(11)}]$

The relation (11) establishes that the number of photoelectrons or fluorescent quanta created at depth z is proportional to the square module of the total radiation field amplitude at that depth, |E(z, ω, θ)|² (to the standing wave). In (11) we have taken into account that the imaginary part of the susceptibility χ(z) is directly connected to the linear absorption coefficient μ(z) [subscript q in (11) indicates that in some cases we are interested in the absorption by a definite kind of atom],

$[\mu=({{2\pi}/\lambda})\,{\rm{Im}}(\chi).\eqno(12)]$

For the case of a uniform mirror (semi-infinite medium) we can connect the number of created photoelectrons to the loss of the radiation intensity in a differential layer, as was done in the early papers (Henke, 1972; Solomin & Kruglov, 1984). For the reflection from a semi-infinite medium we have I(z) ∝ |E(z)|²Re(η) and E(z) = E₀exp(−iηkz) (k = 2π/λ = ω/c), where η is determined by (5). So

$[{{{\rm{d}}I}/{{\rm{d}}z}}\,\,\propto\,\,-2\,{\rm{Re}}(\eta)\,{\rm{Im}}(\eta)\,k\exp[-2\,{\rm{Im}}(\eta)\,kz]|E(0)|^2,\eqno(13)]$

but for η determined by (5) we have the exact relation

$[2\,{\rm{Re}}(\eta)\,{\rm{Im}}(\eta)={\rm{Im}}(\chi)\eqno(14)]$

and we come to (11). If we are far from the total reflection region,

$[{\rm{Im}}(\eta)\simeq{{\,{\rm{Im}}(\chi)}/({2\,\sin\theta})},\qquad{\rm{Re}}(\eta)\simeq\sin\theta,\eqno(15)]$

$[\eqalign{I(z)&=\left|E(0)\right|^2\sin\theta\exp(-\mu{z}/\sin\theta),\cr {{{\rm{d}}I}/{{\rm{d}}z}}&= -\mu\left|E(0)\right|^2\exp(-\mu{z}/\sin\theta),}\eqno(16)]$

and we again come to (11). Note that the additional factor 1/sinθ in expression (1) for dn appears when the full spot of radiation intensity on the surface becomes smaller with increasing angle. Expression (11) refers to the unit surface.

The most essential difference between a multilayer sample and a semi-infinite medium is the non-exponential z-dependence of the total radiation field amplitude E(z, ω, θ). For its calculation we use the propagation matrices which describe the evolution of the tangential components of the electrical and magnetic field of radiation with z. We consider the plane waves of radiation ∼exp[i(ω/c)κr − iωt] and, if the dielectric tensor in a multilayer changes just along the normal to the surface $[\hat\varepsilon]$ = $[\hat\varepsilon(z)]$ , we can write

$[\boldkappa={\bf{b}}+\eta{\bf{q}},\eqno(17)]$

where q is the unit vector along z and b is a tangential component of the wavevector (in units of ω/c). In the following we shall omit the dependence of the $[\hat\varepsilon]$ and field amplitude from ω (or λ) and θ for brevity. The homogeneity of the radiation field along the surface specifies the equality of the tangential components of the wavevectors for all reflected and transmitted waves in a multilayer, i.e. b = constant, |b| = cosθ for all sublayers in a multilayer. So we shall consider the radiation field amplitudes in the form

$[\left[\matrix{{\bf{E}}({\bf{r}},t)\cr {\bf{H}}({\bf{r}},t)}\right]=\left[\matrix{{\bf{E}}(z)\cr{\bf{H}}(z)}\right]\exp\{i\left[(\omega/c){\bf{br}}-\omega{t}\right]\}.\eqno(18)]$

Then the Maxwell equations for E(z), H(z) can be written in the following way,

$[\left({\bf{q}}^{\times}\,{{{\rm{d}}}\over{{\rm{d}}z}}+i\,{\omega\over{c}}\,{\bf{b}}^{\times}\right) \left[\matrix{{\bf{H}}(z)\cr{\bf{E}}(z)\cr}\right]\,\,=\,\,i\,{\omega\over{c}}\,\left[\matrix{-\hat\varepsilon(z)\,{\bf{E}}(z)\cr{\bf{H}}(z)}\right],\eqno(19)]$

where the superscript × symbol means the dual tensor (i.e. $[{\bf{q}}^{\times}{\bf{a}}]$ = $[{\bf{q}} \times {\bf{a}}]$ , × means the vector product). The system of equations (18) connects six components of the vectors E(z), H(z), but just four components are independent, because from (19) we can obtain two algebraic relations between their amplitudes,

$[({\bf{aH}})=({\bf{q}}\,\hat\varepsilon\,{\bf{E}}),\qquad\qquad({\bf{aE}})=-({\bf{qH}}),\eqno(20)]$

where a = $[{\bf{b}}^{\times}{\bf{q}}]$ = b × q. The tangential components of the vectors E(z), H(z),

$[\left(\matrix{\hat{I}\,{\bf{H}}\cr{\bf{q}}\times{\bf{E}}}\right)= \left(\matrix{H_x\cr H_y\cr -E_y\cr E_x}\right),\eqno(21)]$

should be continuous at the boundary crossing, and they are usually chosen for the task. In (21) we have used the projective operator on the surface plane $[\hat{I}]$ = $[-({\bf{q}}^{\times})^2]$ . The coordinate system used is presented in Fig. 1.

Figure 1
The coordinate system used.

The full vectors E, H can be expressed from the tangential components (Borzdov et al., 1976 ),

$[\left(\matrix{{\bf{H}}\cr{\bf{E}}}\right)= \left[\matrix{\hat{I} & -{\bf{q}}\circ{\bf{a}}\cr (1/\varepsilon _q)\,{\bf{q}}\circ{\bf{a}} & \hat{I}-(1/\varepsilon_q)\,{\bf{q}}\circ{\bf{q}}\,\hat\varepsilon\hat{I}}\right] \left(\matrix{{\bf{H}}_t \cr {\bf{E}}_t}\right),\eqno(22)]$

where the sign $[\circ]$ between vectors means the outer product (the diad), ∊_q = $[{\bf{q}}\,\hat\varepsilon\,{\bf{q}}]$ . Finally we have

$[{{\rm{d}}\over{{\rm{d}}z}}\left[\matrix{\hat{I}\,{\bf{H}}(z)\cr {\bf{q}}\times{\bf{E}}(z)}\right]= i\,{\omega\over{c}}\,\hat{\bf{M}}(z)\,\left[\matrix{\hat{I}\,{\bf{H}}(z)\cr {\bf{q}}\times{\bf{E}}(z)}\right], \eqno(23)]$

where $[\hat{\bf{M}}]$ is a differential propagation matrix, the explicit expression for which is given by, for example, Borzdov et al. (1976),

$[\hat{\bf{M}}=\left(\matrix{ {1\over{\varepsilon_q}}{\bf{q}}^{\times}{\hat\varepsilon}\,{\bf{q}}\circ{\bf{a}} & {1\over{\varepsilon_q}} \hat{I}\hat{\tilde\varepsilon}\hat{I}-{\bf{b}}\circ{\bf{b}}\cr&\cr \hat{I}-{1\over{\varepsilon_q}}{\bf{a}}\circ{\bf{a}} & -{1\over{\varepsilon_q}}{\bf{a}}\cdot{\bf{q}}\,\hat\varepsilon\,{\bf{q}}^{\times}}\right),\eqno(24)]$

or, in the coordinate system used (Azzam & Bashara, 1977 ),

$[\eqalignno{&\hat{\bf{M}}= \cr&\left(\! {\matrix{ { - {{{\varepsilon _{yz}}} \over {{\varepsilon _{zz}}}}\cos \theta } & 0 & {{1 \over {{\varepsilon _{zz}}}}({\varepsilon _{yy}}{\varepsilon _{zz}} - {\varepsilon _{zy}}{\varepsilon _{yz}})} & {{1 \over {{\varepsilon _{zz}}}}({\varepsilon _{yz}}{\varepsilon _{zx}} - {\varepsilon _{yx}}{\varepsilon _{zz}})} \cr &&&\cr {{{{\varepsilon _{xz}}} \over {{\varepsilon _{zz}}}}\cos \theta } & 0 & {{1 \over {{\varepsilon _{zz}}}}({\varepsilon _{zy}}{\varepsilon _{xz}} - {\varepsilon _{xy}}{\varepsilon _{zz}})} & {{1 \over {{\varepsilon _{zz}}}}({\varepsilon _{xx}}{\varepsilon _{zz}} - {\varepsilon _{zx}}{\varepsilon _{xz}}) - {{\cos }^2}\theta } \cr &&&\cr {1 - {{{{\cos }^2}\theta } \over {{\varepsilon _{zz}}}}} & 0 & { - {{{\varepsilon _{zy}}} \over {{\varepsilon _{zz}}}}\cos \theta } & {{{{\varepsilon _{zx}}} \over {{\varepsilon _{zz}}}}\cos \theta } \cr &&&\cr 0 & 1 & 0 & 0} } \!\!\right).\cr&&(24')}]$

Calculation of the total radiation field amplitude E(z) with the help of the propagation matrices is performed in two steps. Firstly the integration of (23) over the whole multilayer establishes the relation between four tangential field components on the top surface and at the substrate. This relation allows us to determine the reflected wave for a given incident wave and to obtain four tangential components of the total field (for incident plus reflected waves) on the top surface. In the second step we follow the evaluation of the tangential components of these vectors with depth by sequentially applying the propagation matrices specific for each sublayer. At each depth z we can restore the total field amplitude E(z) with help of (22) and use it for calculation of A(z, ω, θ) by (10). The whole algorithm is realized in the program package XFRAM (Odintsova & Andreeva, 2012 ) and examples of fluorescence calculations from an antiferromagnetic multilayer [Fe/Cr]_n at the L_2,3-edge of Fe have been published by Andreeva & Odintsova (2012 ).

Let us consider the calculation of E(z) for a simpler isotropic case applicable for our experimental data. When $[\hat\varepsilon]$ = $[\hat\varepsilon(z)]$ is a scalar function (or if this tensor is represented by a diagonal matrix in the used x,y,z basis), the σ- and π-polarized waves are independent. For σ-polarization of radiation (the electric field vector E for all waves is perpendicular to the reflection plane) we obtain the differential 2 × 2-propagation matrix which describes the evolution with z of the E_x = E and H_y components,

$[{{{\rm{d}}}\over{{\rm{d}}z}}\left[\matrix{ E_x(z)\cr H_y(z)\cr }\right]= i\,{\omega\over{c}}\left[\matrix{ 0 & 1 \cr \sin^2\theta+\chi_{xx}(z) & 0 }\right]\left[\matrix{ E_x(z)\cr H_y(z)\cr}\right]. \eqno(25)]$

In (25) we insert the designation of χ_xx(z) = ∊_xx(z) − 1. Note that, instead of the simple exponential attenuation of the amplitude of the radiation field, at the reflection conditions we need two variables in each layer and a 2 × 2 propagation matrix in order to take into account the existence of two waves in each sublayer. Here they are not the amplitudes of the waves in the direct and opposite directions but the total amplitude of the electric field (which is actually the coherent sum of the these waves) E = E_x and H_y. Often in the hard X-ray region the full electric field and its z-derivation is used, which is proportional to H_y at grazing angles.

For π-polarized radiation (the electric field vector E for waves in the direct and backward directions lies in the reflection plane) we obtain the differential 2 × 2-propagation matrix which describes the evolution of the H_x = H and −E_y components,

$[\hat{\bf{M}} = \left({\matrix{ 0 & {{\varepsilon _{yy}}} \cr {{{{{\sin }^2}\theta\,+\,{\chi _{zz}}} \over {{\varepsilon _{zz}}}}} & 0 \cr } } \right).\eqno(26)]$

In the thin nth sublayer we can put ∊_jj(z) = constant and the differential equation (25) will have the analytical solution

$[\left[{{\kern 1pt} \matrix{ {{E_x}(z + {d_n})} \cr {{\kern 1pt} {H_y}(z + {d_n})} \cr } {\kern 1pt} } \right] = {\exp[{i({\omega/c}){{\hat{\bf{M}}}_n}\,{d_n}}]}\,\left[{{\kern 1pt} \matrix{ {{E_x}(z)} \cr {{\kern 1pt} {H_y}(z)} \cr } {\kern 1pt} } \right], \eqno(27)]$

where the matrix exponential $[\exp[i(\omega/c)\hat{\bf{M}}d\,\!]]$ for σ-polarization is given by the expression

$[\exp[i(\omega/c)\hat{\bf{M}}d]= \left({\matrix{ {\cos Q} & {{i \over \eta }\sin Q} \cr {i\,\eta \,\sin Q} & {\cos Q} \cr } } \right), \eqno(28)]$

and Q = (ω/c)d_nη, η = (sin²θ + χ_xx,n)^1/2.

For π-polarized radiation we have

$[\exp[i(\omega/c)\hat{\bf{M}}d]= \left({\matrix{ {\cos Q} & {{{i\,{\varepsilon _{yy,n}}} \over \eta }\sin Q} \cr &\cr {{{i\,\eta \,\sin Q} \over {{\varepsilon _{yy,n}}}}} & {\cos Q}} } \right),\eqno(29)]$

where η = (∊_yy,n/∊_zz,n)^1/2(sin²θ + χ_zz,n)^1/2.

A multilayer characterized by a varying susceptibility χ = χ(z) is usually divided into thin sublayers; in each of them we can put ∊_jj,n(z) = 1 + χ_jj,n = constant. The total integral propagation matrix $[\hat{\bf{L}}(D)]$ is calculated as a product of the matrix exponentials,

$[\eqalign{\hat{\bf{L}}(D)&=\exp\left(i{\omega\over{c}}{d_N}\hat{\bf{M}}_N\!\right)\exp\left(i{\omega\over{c}}{d_{N-1}}\hat{\bf{M}}_{N-1}\!\right)\ldots\exp\left(i{\omega\over{c}}{d_1}\hat{\bf{M}}_1\right),\cr D&={d_1}+{d_2}+\ldots+{d_N}.}\eqno(30)]$

(Layers are numerated from a surface to a substrate having the number N + 1.) This matrix

$[\hat{\bf{L}}(D)=\left(\matrix{ {{L_{11}}} & {{L_{12}}} \cr {{L_{21}}} & {{L_{22}}}}\right)]$

allows us to find easily the reflected wave,

$[{E_{\rm{R}}} = {{{\eta _d}({\eta _0}{L_{12}} + L_{11}^{}) - ({\eta _0}{L_{22}} + L_{21}^{})} \over {{\eta _d}({\eta _0}{L_{12}} - L_{11}^{}) - ({\eta _0}{L_{22}} - L_{21}^{})}}\,{E_0}, \eqno(31)]$

where $[{\eta_0}]$ = $[\sin\theta]$ (in the external medium we put $[\varepsilon]$ = 1), $[{\eta_d}]$ = $[({{\sin^2}\theta+{\chi_d}})^{1/2}]$ , $[{\chi_d}]$ is the susceptibility of a substrate.

For the case of the σ-polarized radiation the total radiation field at the surface is

$[\eqalign{E(0)&= {E_x}(0) = {E_0} + {E_{\rm{R}}}, \cr {H_y}(0)& = \sin \theta \,\left({E_0} - {E_{\rm{R}}}\right),} \eqno(32)]$

and we can obtain the total radiation field amplitude E(z) at any depth z (being at each depth a coherent sum of the waves in the direct and opposite directions) by successive application of the propagation matrix (28) specific at each depth to the vector column

$[\left[{\matrix{ {E(z)} \cr {{H_y}(z)}} } \right]]$

[or (29) to

$[\left[{\matrix{ {H(z)} \cr { - {E_y}(z)} \cr } } \right]]$

in the case of the π-polarized radiation].

For calculation of the function of the electron yield T(z) we use the simplest exponential attenuation function along its way, $[\exp(-\varepsilon_{\rm{e}}l)]$ , $[\varepsilon_{\rm{e}}]$ = 1/λ^el, but suppose that the electron escape depth λ^el in each layer can be different and the path length l depends on the angle of emission. Using the approach of Kasrai et al. (1996) for the integrated electron yield over different angles of emission we describe T(z) from the nth layer in the following way,

$[T_n(z)= {C_1}{C_2}\ldots{C_{n-1}}\textstyle\int\limits_0^{2\pi} {\rm{d}}\varphi\textstyle\int\limits_0^{\pi/2} {\exp\left({-{Z/{\cos\psi}}}\right)\sin\psi\,{\rm{d}}\psi},\eqno(33)]$

where

$[Z = \left({{{d_1}} \over {\lambda _1^{\rm{el}}}} + {{{d_2}} \over {\lambda _2^{\rm{el}}}} +\ldots+{{{d_{n - 1}}} \over {\lambda _{n - 1}^{\rm{el}}}} + {{{\xi _n}} \over {\lambda _n^{\rm{el}}}}\right),]$

z = $[{d_1}+{d_2}+\ldots+{d_{n-1}}+{\xi_n}]$ , $[{\xi_n}]$ is the z coordinate for the considered nth layer calculated from its top; $[\lambda_n^{\rm{el}}]$ is the electron escape depth in the nth layer; ψ and φ are the polar and azimuth angles of the electron yield. The integral over ψ in (32) depends on the special function S(Z) called the `exponential integral',

$[\eqalign{T_n(z)&= 2\pi\left[{\exp\left({-Z}\right)-Z\,S(Z)}\right],\cr S(Z)&={\int\limits_1^\infty}\,{{{\exp(-Zt)}\over t}\,{\rm{d}}t}.}\eqno(34)]$

In our program we calculate S(Z) numerically.

According to Ejima (2003) we include in (33) the additional factors C_i describing the transmission probability through the interface between the (i − 1)th and ith layers originating from the difference in the electric potential in these layers. In the computing procedure we combine constant factors $[{C_1}{C_2}\ldots{C_{n-1}}]$ and the probability of the electron creation $[\tau_n]$ into one single factor for each nth layer.

Note that the exponential attenuation in (33) for the description of the electron propagation is not exactly correct, because the electrons change the direction of their propagation owing to the collisions with other electrons beside creating secondary electrons. The Monte Carlo method gives the most adequate picture of the electron propagation in the medium (see, for example, Liljequist et al., 1978; Kovalchuk et al., 1986; Bakaleinikov et al., 2001 ; Jablonski & Powell, 2002 ). However, in our case of very soft radiation the sampling depth of the electrons is so small that the exact function for the electron yield T(z) does not have much influence on the final result.

3. Experimental data

The experimental data have been obtained for a SiO₂/Si/SiO₂/c-Si sample at different glancing angles for σ-polarized soft X-rays which demonstrated the strong influence of the reflectivity on TEY spectra. The samples have a specific structure called `silicon-on-insulator' (Fig. 2). A ∼150 nm silicon oxide SiO₂ layer and then a strained silicon nanolayer, ∼100 nm thick, were deposited on a single-crystalline Si (100) substrate, about 300 µm thick. It is clear that the surface of the sample has been oxidized in the air and typically the thickness of the top native oxide layer is ∼2 nm. So in total we have the structure SiO₂/Si/SiO₂/c-Si. However, the reflectivity of the soft X-rays was actually determined by the three top layers SiO₂(2 nm)/Si(10–100 nm)/SiO₂, having SiO₂ as substrate. In this paper we consider the results for one sample with a strained Si layer of thickness ∼80 nm and characterized by the yellow colour of its surface.

Figure 2
Schematic view of the sample.

XANES spectra near the L_2,3 absorption edges of Si for this sample have been measured for different glancing angles at the SRC (Synchrotron Radiation Center, University of Wisconsin-Madison, Stoughton, USA) and BESSY II (Helmholtz Zentrum, Berlin, Germany) synchrotron radiation facilities. The sample current technique was used. Details of the experiment have been presented by Domashevskaya et al. (2011).

The experimental results are shown in Fig. 3 (left side). The normal-incidence spectrum is shown at the top of the figure. It is typical of the oxidized Si sample (Domashevskaya et al., 2011). The XANES spectrum of the standard sample (s-Si) consists of two basic absorption edges, the Si L_2,3-edges of elementary Si (100 eV) and the Si L_2,3-edges of natural oxide (105.5 eV). The fine doublet structure of the basic edges is caused by the spin-orbital splitting of the 2p_1/2,3/2 Si core level (0.6 eV) from which electrons transfer to the Si and SiO₂ conduction band under synchrotron beam excitation. The fine structure of the Si L_2,3 basic edges (∼100 eV and ∼103 eV) is characteristic of single-crystalline silicon independent of its conduction type and orientation [(100) or (111)] and disappears into amorphous silicon a-Si for which only one step (∼100.5 eV) remains. However, investigation of the XANES fine structure of pure Si for sample characterization is almost impossible owing to the existence of the SiO₂ top layer and the small sampling depth of the TEY method. It is not revealed in our normal-incidence spectrum. This is why we have attempted to obtain further information by changing the angle of incidence. The underlying layer of crystalline Si is revealed in XANES spectra measured at small glancing angles of incident radiation in a quite unexpected manner.

Figure 3
TEY spectra for the normal incidence of radiation (top panels) and at glancing angles of 5, 7, 9, 11, 13, 15, 17, 19, 21, 25° (below, vertically shifted). Experimental spectra are on the left-hand side and the results of our fit are on the right-hand side. Dashed lines (blue) on the right-hand side present the electron yield from the top oxidized layer only, ∼2.2 nm thick. The shaded area denotes the energy range for the fit of the glancing incidence spectra. Dotted vertical lines mark the energies for which the angular dependencies are calculated (see also Fig. 7

At small glancing angles (5, 7, 9, 11, 13, 15, 17, 19, 21, 25°) the measured TEY spectra dramatically differ from the normal-incidence spectrum. They exhibit oscillations at the per-edge energy range 90–100 eV. Surprisingly the phase of the oscillations switches sign if the glancing angle is changed by ∼2°. Besides, an abrupt jump (a rostrulum up or a rostrulum down) of the intensity is observed near 100 eV (Si L₃-edge). In the hard X-ray region, decreasing the glancing angle is commonly used in order to decrease the investigated depth of the sample. In our experiment the sampling depth is extremely small even at the normal incidence of radiation (∼1–2 nm). In the normal-incidence XANES spectrum mostly SiO₂ L_2,3 peaks dominate, whereas pure Si L_2,3 peaks have very low intensity. The specific features of the experimental TEY spectra at glancing angles do not occur in the normal-incidence spectra, so they are not the properties of the top layer of material itself but appear owing to the interaction of the incident radiation with the whole multilayer structure.

4. Interpretation of the data

The fit of the data was not simple because the reflectivity had not been measured either on the energy scale or on the angular scale. Information about the optical constants for Si and SiO₂ in the vicinity of the L_2,3-edges is quite unreliable in standard tables and in the literature.

Some data from different sources for the real and imaginary parts of the susceptibilities of Si and SiO₂ for the energy range under investigation are presented in Fig. 4. We see that the data are rather different. Therefore, during the fit procedure the optical constants for the Si, SiO₂-`substrate' and oxidized `cap' SiO₂-layer were varied. We tried to keep the overall shape of the susceptibility energy dependencies (and control the agreement of our calculations for θ = 90° with the normal-incidence spectrum), but used the additional variable factors (12 fit parameters) for the shift and multiplication for each curve. We partially used a manual fit of the parameters, but an automatic fit was also applied (like the AMOEBA code, sequent one-parameter descent and multi-dimensional NET calculations). We also changed the inclination of the low-energy `tail' of the susceptibility dependencies in the energy region 90–98 eV for pure Si in the middle layer and in the 90−103 eV region for SiO₂ in the top and third layers (additional six fit parameters) where the most interesting features appear on the experimental Y spectra. The resulting susceptibility dependencies for three layers are shown in Fig. 4 by the thick lines.

Figure 4
Real and imaginary parts of the Si and SiO₂ susceptibility in the vicinity of the L_2,3 absorption edges of Si for three layers in our structure. Dashed lines (blue) for Si and SiO₂ present data from the tables of Henke et al. (1993

); thin solid lines (magenta) for SiO₂ are the results from Filatova et al. (1999

); dash-dotted lines (green) are the absorption data from Kasrai et al. (1996

), normalized to the absolute values of Henke et al. (1993

) `at the tails'; dotted lines (dark blue) for Si are data from NIST (Chantler et al., 2009

); bold lines (black) are data obtained by the fit of the experimental data for three layers in our structure. The shaded area denotes the energy range for the fit of the glancing incidence spectra.

The susceptibility of the bottom SiO₂ layer does not influence the resulting TEY spectra heavily, so we have kept the dependencies presented by Filatova et al. (1999) but were forced to slightly shift the absolute value of the real part. For the middle Si layer we started from the data of Henke (1993 ), but a small energy shift is needed for a better correspondence of the experimental and theoretical normal-incidence Y spectrum. We have found as well that the absolute value of the imaginary part of χ in pure Si should be magnified [this is partly correlated with the data from NIST (Chantler et al., 2009 )]. The shape of the real part of χ in the Si layer in the pre-edge region (marked by the circle in Fig. 4) has been changed in order to obtain the appearance of the `rostrulum up' or `rostrulum down' features in the TEY dependencies at glancing angles.

The energy dependencies of susceptibilities in the top oxidized layer were essentially modified during the fit procedure. The essential change of the optical constants in the top layer was expected owing to the imperfections in the ultrathin layer structure (induced distortion of the elementary cell, defects, dislocations, impurities, etc.). Typically for hard X-rays the electronic density becomes lower (by ∼10–20%) in the top layer, but in our case we have found the opposite result: the absolute value of the real part of the susceptibility was found to be larger than the tabled values for SiO₂. The relation between the electronic density and the optical constants is more complicated in the soft X-ray region, e.g. oxidation leads to an increase in the absolute values of the real and imaginary parts of χ. The result obtained for the top layer can be explained by the reason that the top 2.2 nm-thick oxidized layer can contain some impurities, e.g. absorbed atoms C, N, etc. To be sure that these impurities can influence the optical constants in this way we have taken the tabled data from XOP (https://www.esrf.eu/UsersAndScience/Experiments/TBS/SciSoft/xop2.3/Main ) for Re(χ) and Im(χ) for different compounds of silicon (Fig. 5). It can be clearly seen that compounds like SiC or Si₃N₄ are characterized by an increase in the absolute values of Re(χ) and a decrease in Im(χ) in the pre-edge region compared with the optical parameters of SiO₂. By our fit we have obtained exactly such a tendency for the top SiO₂ layer.

Figure 5
Table dependencies from ESRF software XOP of the optical constants for different compounds of Si. In the inserts these dependencies are drawn in the energy range considered in our fit.

More accurate determination of the optical constants for the layers can be obtained by measurements of the angular dependencies of the reflectivity and TEY for different energies, as was carried out by, for example, Gupta et al. (2007) and Smekhova et al. (2010 ). Unfortunately we had no such experimental data; for soft X-rays such data can only be obtained at specialized synchrotron radiation beamlines.

The thickness d_1,2 of the top SiO₂ layer and strained Si layer were varied as well as the electron escape depth $[\lambda_n^{\rm el}]$ (additional four fit parameters). For $[\lambda_n^{\rm el}]$ we obtain the values 1.47 nm for SiO₂ (n = 1) and 0.81 nm for Si (n = 2), close to those used by Kasrai et al. (1996). The fit also gives the thickness of the top oxide layer d_SiO2 = 2.2 nm and the thickness of the strained Si layer d_Si = 85.8 nm. Finally, a reasonable agreement between the theoretical and experimental energy dependencies for all angles of incidence including 90° (Fig. 3, compare left- and right-hand parts) has been achieved. We have reproduced the oscillations in the pre-edge region of the Si L-edge, changing the phase with the angle variation, and have obtained rostrulum-up and rostrulum-down features at the Si absorption edge.

As long as our calculations are based on the exact solutions of the electromagnetic theory for a radiation field inside a multilayer, it is possible to explain the reasons for the intriguing energy dependencies of the TEY spectra at small glancing angles of radiation.

Initially we suppose that the peculiarities of the Y spectra at different angles are explained by the enhancement of the field in the middle Si layer because the main anomalous features were observed near the L_2,3-edges of pure Si but not at the top SiO₂. However, the very small escape depth for our low-energy electrons (λ_el $[\propto]$ 1 nm) contradicts this assumption. In Fig. 3, on the right-hand side, we also present the contribution from the top SiO₂ layer (dashed blue lines) in addition to the quantum yield Y and we see that the main features of the Y spectra in the pre-edge region are completely determined by this contribution. So the role of the underlying Si layer is different. The strong variation of the Si optical constants near the L_2,3-edges dramatically changes the total radiation field |E(z, ω, θ)|² in the top SiO₂ layer from which we actually measure the quantum yield Y. This explanation is illustrated in Fig. 6. The presented depth dependencies of |E(z)|² (standing waves) for the selected photon energies at two grazing angles of the incident radiation clearly show that the wisp or lapse on the TEY spectrum is caused by the specific waveguide modes in the middle Si layer. If on the top boundary of the Si layer we have the node of the standing wave, the measured TEY intensity decreases; if we have the anti-node then the TEY intensity from the top layer increases. This observed effect in the energy scale is similar to the calculated angular dependency of TEY from a thin layer in an earlier paper (Pepper, 1970).

Figure 6
Experimental (symbols) and theoretical (solid lines) energy dependencies of TEY spectra for the glancing angles 5° and 7°. In the inserts we show the depth dependencies of |E(z)|² (standing waves) for the photon energies, indicated by the vertical dashed lines. Function T(z) (green) presented in the inserts is almost unseen because it is concentrated in the top SiO₂ layer.

The abrupt change of the TEY intensity near 100 eV is explained by the substantial suppression of the waveguide mode in the Si layer by a huge increase in the absorption in this layer exactly at the Si absorption edge. The wave reflected from the bottom boundary of the Si layer becomes negligible, the enhancement of the intensity by the interference of the waves in direct and opposite directions disappears, and the next oscillation on the TEY spectrum drops off.

The suppression of the reflectivity from the bottom boundary of the Si layer and its influence on the TEY electron yield is also illustrated by the angular dependencies presented in Fig. 7. We see that the small change of the photon energy from 100 eV to 100.3 eV leads to the suppression of the oscillatory dependences of the reflectivity and TEY.

Figure 7
Angular dependencies of the reflectivity R (dashed blue lines), the square module of the field on the surface |E(0)|² (dotted black lines), (1 − R) dependencies (thin red lines) and calculated TEY angular dependencies (thick black lines) for different energies in the vicinity of the Si L_2,3 absorption edges indicated in Fig. 3

by dashed vertical lines.

In Fig. 7 we compare the (1 − R) and |E(0)|² angular dependencies. They are not identical because in the case of a multilayer the reflectivity is not described by the simple Fresnel formula and relation (3) does not take place. The TEY dependency is rather close to the |E(0)|² angular dependency (because the main contribution to our Y spectra is from the very surface), but not to (1 − R). This is an illustration of the non-applicability of formula (1) for the case of multilayers.

However, the energy dependencies of Y, (1 − R)[Im(χ₁)]/λ and |E(0)|²[Im(χ₁)]/λ are quite different (Fig. 8), so we should conclude that the contribution from the second pure Si layer noticeably influences the TEY energy dependences mainly due to the factor μ_Si(λ) $[\propto]$ Im[χ_Si(λ)]/λ.

Figure 8
The energy dependencies of the quantum yield Y, calculated by the exact theory (6)

, (11)

for the structure model, obtained by our fit (thick solid line), (1 − R)Im(χ₁)/λ (thin red solid line) and |E(0)|²Im(χ₁)/λ (dotted black line) for a 5° glancing angle. All curves are normalized.

5. Conclusions

We have shown that the remarkable features of the quantum yield spectra measured in the vicinity of the Si L_2,3 absorption edges at different glancing angles are not caused by some specific changes of the XANES spectra but originate from optical phenomena. TEY spectra are influenced by the reflectivity formed by the whole multilayer structure. The special structure of the investigated SiO₂/Si/SiO₂ multilayer has allowed us to enhance and reveal such influence in the most impressive way owing to the formation of the waveguide modes in the middle Si layer. In principle, such a change of the TEY spectra by the angle variation can be used as an additional source of information for determination of the absolute value of the optical constants because the precise matching of the optical parameters in the structure is required for the appearance of the waveguide effect. However, the task of determining the optical constants could be better solved if complimentary measurements of the reflectivity are performed.

Acknowledgements

This work is based in part upon research performed at the Synchrotron Radiation Center, University of Wisconsin-Madison, which is supported by the National Science Foundation under award No. DMR-0537588. Authors are grateful for support of this work to the Director, official BESSY administration, BESSY Beamtime Allocation Committee and Coordinators of Russian–German Beamline facility. This work is based upon research supported by the Federal Targeted Program `Scientific and Scientific Educational Personnel of the Innovative Russia'. The work was also supported by RFBR grants 09-02-01293-a, 10-02-00768-a, 12-02-00924-a.

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JOURNAL OF
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ISSN: 1600-5775

Volume 19| Part 4| July 2012| Pages 609-618

doi:10.1107/S0909049512022844

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research papers\(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

Interference phenomena of synchrotron radiation in TEY spectra for silicon-on-insulator structure

1. Introduction

2. Theory

3. Experimental data

4. Interpretation of the data

5. Conclusions

Acknowledgements

References

research papers