Vacuum 101 (2014) 63e66 Contents lists available at ScienceDirect Vacuum journal homepage: www.elsevier.com/locate/vacuum Rapid communication Temperature dependence of DebyeeWaller factors of semiconductors Nguyen Van Hung a, *, Cu Sy Thang b, Nguyen Cong Toan a, Ho Khac Hieu c,1 a Department of Physics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam Institute of Geological Sciences (IGS), Vietnam Academy of Science and Technology (VAST), 18 Hoang Quoc Viet, Cau Giay, Hanoi, Viet Nam c National University of Civil Engineering, 55 Giai Phong, Hai Ba Trung, Hanoi, Viet Nam b a r t i c l e i n f o a b s t r a c t Article history: Received 16 June 2013 Received in revised form July 2013 Accepted 13 July 2013 Temperature dependence of DebyeeWaller factors such as mean square relative displacement (MSRD) and mean square displacement (MSD) in extended X-ray absorption fine structure (EXAFS) and related spectra of semiconductors have been studied based on statistical moment method This work illustrates our derived theory for calculation and analysis of temperature-dependent MSRD of semiconductors and our further developments of a previous work on MSD for zinc-blende type semiconductors by making detailed analysis and conclusions on their thermodynamic properties Numerical results for MSRD of Si, Ge having diamond structure, and MSD of GaAs, GaP, InP, InSb having zinc-blende structure, are found to be in good and reasonable agreement with experiment and with those of other theories It is found that the MSD of a semiconductor element changes when it is mixed by another semiconductor element to be compound and is about equal to the one of its another constituent element Ó 2013 Elsevier Ltd All rights reserved Keywords: Temperature-dependent DebyeeWaller factor StillingereWeber potential Statistical moment method Semiconductors Thermal vibrations and disorder in EXAFS and related spectroscopies give rise to DebyeeWaller factors (DWF) varying as eÀW(T), where W(T) z 2k2s2(T) containing MSRD s2(T) of the bond between absorber and backscattering atoms Such DWFs damp EXAFS and related spectra with respect to increasing temperature T and wave number k (or energy) [1e11] The EXAFS DWF is analogous to that for X-ray and neutron diffraction or the Mössbauer effect, where W(T) ¼ (1/2)k2 The difference is that the EXAFS DWF refers to correlated averages over relative displacements for the MSRD s2(T), while that for X-ray absorption or neutron diffraction refers to MSD of a given atom Due to their exponential damping, accurate DWFs are crucial to quantitative treatment of X-ray absorption spectra Consequently, the lack of the precise DWFs has been one of the biggest limitations to accurate structural determinations (e.g., the coordination numbers and the atomic distances) from EXAFS experiment Moreover, the DWF (MSD) plays an important role in calculation and analysis of Lindemann melting temperature of binary alloys [12] Therefore, investigation of temperature dependence of DWF is of great interest The purpose of this work is to study the temperature dependence of DWFs in EXAFS and related spectra of semiconductors such as MSRD and MSD based on statistical moment method (SMM) [13e16] that includes anharmonic effects Our contributions to this work are firstly, the derivation of a theory for calculation and analysis of the temperature-dependent MSRD s2(T) of semiconductors, and secondly, the next developments of our previous work [16] on MSD demonstrated by comparisons of calculated MSDs for zinc-blende type semiconductors with those of other theories [26] and by making detailed analysis of MSD characterizing their thermodynamic properties to show changes of MSD of a semiconductor element in compound depending on its another constituent element Based on the successes of anharmonic correlated Einstein model (ACEM) [6e11] thanks to using anharmonic effective potentials to include three-dimensional interaction in this work the empirical many-body StillingereWeber [17e19] potentials are used for describing interatomic interaction Numerical results for Si, Ge having diamond structure, and GaAs, GaP, InP, InSb having zinc-blende structure, are compared to experiment [20e22] and to those of other theories [23e26] with good and reasonable agreement, respectively To describe close relation between MSRD and MSD corresponding to DWF we express the MSRD as D s2 ẳ ẵR,ui u0 ị2 ẳ * Corresponding author E-mail addresses: hungnv@vnu.edu.vn, hung41043@gmail.com (N Van Hung) Duy Tan University, K7/25 Quang Trung, Danang, Vietnam 0042-207X/$ e see front matter Ó 2013 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.vacuum.2013.07.021 D E E D E R,ui ị2 ỵ R,u0 ị2 2hR,ui ÞðR,u0 Þi: (1) Here, u0 and ui are the atomic displacements of the zeroth and the ith sites from their equilibrium positions, R is the unit vector 64 N Van Hung et al / Vacuum 101 (2014) 63e66 Fig Temperature dependence of MSRD s2(T) calculated by the present theory a) for Si compared to the one of EXAFS calculation of M Benfatto et al [23] and b) for Ge compared to those of J J Rehr et al [24] at 300 K calculated by the methods LDA, GGA, hGGA, as well as to experimental results of A E Stern et al [20] at 300 K, of G Dalba et al [21], and of A Yoshiasa et al [22] at different temperatures, where the measurement of A Yoshiasa et al [22] is performed under the pressure of 0.1 MPa pointing from the zeroth site towards the ith site, and the brackets < > denote the thermal average The first two terms on the righthand side of Eq (1) are the uncorrelated MSDs, which are about equal to each other, and can be obtained from diffraction, while the third term describes the displacement correlation function Using the expressions of the second order moment in SMM [13e 16] for the case of dependence on temperature T we obtain the expression for the MSD as D E b Zu u2i ¼ hui i2 ỵ bA1 ỵ Z 1ị; Z ẳ zcoth z ; z ¼ ; b ¼ kB T; K 2b (2) # " 2 a2 g2 b Z Z ỵ 1ị ; K ẳ k ; ỵ A1 ẳ 1ỵ K 3g K ! X v fi hmu2 ; k ¼ vu2ix i s2 ðTÞz X j (3) g2 b K5 1ỵ Z 2bZ kK Z ỵ 1ị ỵ ỵ 2b ; k kK Fij ri ; rj ỵ X Wijk ri ; rj ; rk : eq where kB is Boltzmann constant, m is atomic mass, and u is atomic vibration frequency (6) j;k It is very important for this potential to define the two- and three-body terms Here the two-body term is given by h À4 i h À1 i r < εA B rij ; À exp sij À b À Á s Fij ri ; rj ¼ : r 0; sij ! b (4) (5) where at T / (cothz / 1) the zero-point energy contribution has resulted as s2 0ịhs20 ẳ Zu=k In the above expressions the atomic interaction potential fi plays an important role For calculating DWF of semiconductors we use the empirical many-body StillingereWeber potentials [17e19] described for the atom i as fi ¼ eq ! ! 39 = < X4 v4 fi v4 f i ; gẳ ỵ ; 12 : i vu4ix eq vu2ix vu2iy eq ! v3 f i a ¼ ; vuix vuiy vuiz Based on Eqs (1)e(4) we derive the expression for temperaturedependent MSRD and it is given by = s