Laue Diffraction of X-Rays in Gaas at the Zero Value of the Real Part of the Structure Factor for Quasiforbidden Reflections L

$Laue Diffraction of X-Rays in Gaas at the Zero Value of the Real Part of the Structure Factor for Quasiforbidden Reflections L$ Crystallography Reports, Vol. 45, No. 5, 2000, pp. 705–708. Translated from Kristallografiya, Vol. 45, No. 5, 2000, pp. 775–778. Original Russian Text Copyright © 2000 by Datsenko, Klad’ko. DIFFRACTION AND SCATTERING OF X-RAYS AND SYNCHROTRON RADIATION Laue Diffraction of X-rays in GaAs at the Zero Value of the Real Part of the Structure Factor for Quasiforbidden Reflections L. I. Datsenko and V. P. Klad’ko Institute of Physics of Semiconductors, National Academy of Sciences of Ukraine, pr. Nauki 45, Kiev 28, 252650 Ukraine e-mail: [email protected] Received October 28, 1998; in final form, February 18, 1999 Abstract—Specific features of the Laue diffraction of X-rays are considered for quasiforbidden reflections within the wavelength range between the K-absorption edges of Ga and As in a GaAs crystal. It is established that scattering for the 200 reflection is of the dynamical nature in the wavelength range where the real part of the structure factor F is zero and diffraction is determined by the imaginary part of the atomic form factor ω r200 λ f ''( ) alone. It is shown that the position of the minimum of the function Ri = f( ) is sensitive to the crystal nonstoichiometry predicted by the numerical calculations within the theory taking into account both the real and the imaginary parts of the structure factor. © 2000 MAIK “Nauka/Interperiodica”. INTRODUCTION edge of Ge [7]. An attempt to use this wavelength range to control the nonstoichiometry parameter of crystals in The study of X-ray scattering at the wavelengths the Bragg diffraction geometry was made in [8, 9], close to those of the K-absorption edge of Ga in GaAs where the authors assumed that X-ray scattering is of crystals with the use of quasiforbidden Laue reflections purely kinematical character, i.e., that the total inte- has experimentally confirmed the dynamical character grated reflectivity of a crystal consists of two compo- of the interaction between the X-ray radiation and the nents—those of coherent and diffuse scattering—and, crystal lattice (Pendellösung intensity oscillations) and according to [10], is independent of the structure allowed the determination of some parameters of struc- defects [10]. tural perfection of the samples such as the static Below, we describe the study of X-ray scattering for Debye–Waller factor L, coefficient of intensity reduc- quasiforbidden reflections in real GaAs crystals in the µ tion due to diffuse scattering ds, and the nonstoichiom- case where the real part of the structure factor vanishes. etry parameter [1–3]. Earlier [4, 5], it was shown that We also investigated the effect of structure defects and the intensity of the diffraction maximum in the vicinity crystal nonstoichiometry on the behavior of integrated of the K-absorption edge of the material should be intensities of the diffraction maxima in this wavelength described by the formulas of the dynamical theory of range. scattering, which takes into account the contributions of both real and imaginary parts of the atomic scattering function into the crystal polarizability χ. In the THEORETICAL CONSIDERATION studies of X-ray diffraction for quasiforbidden reflec- As is well known, the atomic scattering factor has tions, of special interest is the situation where the real the form part of the structure factor for the hth reflection with () ()ω ()ω due regard for the corrections for the atomic scattering ff=0h++f' if'' , (1) ω function for anomalous dispersion, f0(h) + f '( ), equals where f0(h) is the normal value of the atomic scattering zero. The modified theory of X-ray scattering for this factor for the h reflection, and f '(ω) and f ''(ω) the fre- case established the existence of the Pendellösung quency-dependent real and imaginary corrections to fringes of intensity oscillations on the rocking curve for f0(h) caused by anomalous dispersion. These correc- the Laue-diffraction [6]. It was also established that tions are especially important in the vicinity of the Pendellösung fringes are described by the imaginary K-absorption edges of the crystal components. It is seen ω part of the atomic scattering function alone. In the case from (1) that the condition f0(h) + f '( ) = 0 is fulfilled of Bragg diffraction, narrow rocking curves were for reflections with large absolute values of the diffrac- ω observed. The situation where f0(h) + f '( ) = 0 was tion vector (since f0(h) decreases with an increase of h) observed experimentally in the measurements of inte- in the vicinity of the ä-edge, where the corrections grated intensities in the vicinity of the K-absorption f '(ω) are especially pronounced. The case where the 1063-7745/00/4505-0705 $20.00 © 2000 MAIK “Nauka/Interperiodica” 706 DATSENKO, KLAD’KO χ χ χ × –7 2 2 rh, ih, i0 10 Ga length range. Here e /mc is the classical radius of an 0 frh electron, λ is the X-ray wavelength, and V is the unit- ω 1/4Fh cell volume of the crystal, and Fih ~ f ''( ). The disper- 1.2 Ga sion corrections to the atomic form factor f '(ω) and fih f As f ''(ω) were calculated by the method described in [11] ih with due regard for the oscillator strengths taken from As 5 1.0 frh [12]. It is seen that for the wavelength close to χ @ χ 0.109 nm, we have ih rh. The integrated reflectivity of an ideal crystal for this case can be calculated by the 4 formula of the modified theory [13] as 8.0 (γ γ )1/2( χ ϑ) y Ri = h/ 0 h /sin2 ∫Ri dy, 6.0 y ( µ ) δ Ri = exp – t' (1 – 2 psin ) (2) 1 × [sin2((t/Λ)ReM1/2) + sinh2((t/Λ)ImM1/2)], 4.0 ϑ β γ γ 2 where t is the crystal thickness; t' = tcos sin / 0 h; Λ λ γ γ 1/2 π|χ | γ γ = ( 0 h) / h is the extinction length; 0 and h are the directional cosines of the incident and reflected 2.0 ϑ β 3 X-ray beams, respectively; is the Bragg angle; and is the angle formed by the crystal surface and the reflecting planes. An important parameter entering (2) 0 χ χ 0.104 0.106 0.108 0.110 0.112 0.114 is the quantity M dependent on both rh and ih. All the λ remaining symbols correspond to those used in [13]. , nm The influence of defects on the reflectivity can be esti- Fig. 1. Energy dependence of the real (curves 1–3) and mated by the Molodkin formula [14], which takes into imaginary (curve 4) parts of the Fourier coefficient of GaAs account the contribution of diffuse scattering from polarizability for crystals with various nonstoichiometries χ structure imperfections. To confirm the validity of these (cGa – cAs): (1) –0.03, (2) 0, (3) 0.03, (5) the value of i0. 200 reflection. In the inset: the vector diagram of the contri- formulas for quasiforbidden reflections, we compared butions of the real and the imaginary parts of atomic Ga and the reflectivities calculated by (2) and by formulas for As form factors to the structure factor of the quasiforbidden an ideal crystal [15] with due regard for the contribu- χ 200 reflection. tions that come to h from both real and imaginary components. The satisfactory agreement of these values allowed the numerical evaluation of the effect of structure factor Fh is determined by the imaginary part structural imperfections on the reflectivity for quasifor- of f alone was considered for a Ge crystal [7]. The situ- bidden reflections at various values of the static Debye– ation where F 0 can readily be obtained for χ rh Waller factors L for the case where rh = 0. binary crystals, e.g., GaAs, by using the quasiforbidden 200 reflection and the wavelength range between the K-absorption edges of the crystal components (Frh ~ EXPERIMENTAL [cGa(fGa + f Ga' ) – cAs(fAs + f As' )] 0, where cGa and The integrated intensities of the symmetric 200 Laue reflections were measured on a single-crystal cAs are the atomic concentrations of Ga and As, respectively). For the reflections of the h + k + l = 4n + 2 type spectrometer by the method suggested in [16] with the with n = 0, 1, … the atoms in the sphalerite-type lattice use of a standard perfect silicon crystal. The parameters scatter the X-ray radiation in the counterphase. In this of the X-ray tube (20 kV, 35 mA) provided the elimina- tion of the parasitic effect of a harmonics multiple of wavelength range, the imaginary part of the atomic λ µ ω n /2. We used a 35- m-thick (100)-oriented GaAs scattering function, f ''( ), has quite a large value 3 –2 because of the jump in absorption at the K-absorption sample with the dislocation density ~10 cm . Special precautions were taken to avoid the elastic strains dur- edge of gallium (f ω ≥ f h f ω ). Figure 1 shows ''( ) 0( ) + '( ) ing sample positioning on a crystal holder. The energy the vector diagram of the structure factor of the λ resolution of the setup was characterized by the width 200 reflection for the wavelength = 0.1093 nm which of the spectral window ∆ω: shows that the resulting value, 1/4Fh, is imaginary. Fig- ure 1 also shows energy dependence of the Fourier ∆ω = ωcotϑ∆ϑ, (3) χ 2 2 λ2 π coefficients of polarizability, rh = –(e /mc )( / V)Frh where ϑ is the Bragg angle, ω is the frequency of the χ 2 2 λ2 π and ih = –(e /mc )( / V)Fih, for the quasiforbidden X-ray radiation, and ∆ϑ is the angular range of simul- 200 reflection in GaAs in the above indicated wave- taneously recorded diffracted waves.

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