Temperature dependence of anharmonic exafs oscillation of crystalline silicon

 No.19_Sep 2020|Số 19 – Tháng 12 năm 2020|p.95-102 
 TẠP CHÍ KHOA HỌC ĐẠI HỌC TÂN TRÀO 
 ISSN: 2354 - 1431 
 TEMPERATURE DEPENDENCE OF ANHARMONIC EXAFS 
 OSCILLATION OF CRYSTALLINE SILICON 
Tong Sy Tien1,*, Le Viet Hoang2 
1 Department of Basic Sciences, University of Fire, 243 Khuat Duy Tien, Thanh Xuan, Hanoi, Vietnam 
2 Department of Physics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam 
* E-mail: tongsytien@yahoo.com 
Article info Abstract: 
 In this work, the anharmonic extended X-ray absorption fine structure 
Recieved: (EXAFS) oscillation of crystalline silicon (c-Si) is presented in terms of the 
 Debye-Waller factor using the cumulant expansion approach up to the fourth-
16/9/2020 
 order. The first four EXAFS cumulant has been calculated based on the 
Accepted: 
 classical anharmonic correlated Einstein (ACE) model and suitable analysis 
10/12/2020 procedure, in which thermodynamic parameters are derived from the 
 anharmonic effective potential obtained using the first shell near-neighbor 
 contribution approach. The analysis of the temperature dependence of the 
 EXAFS oscillation is performed via evaluating the influence of the cumulants 
Keywords: 
 on the amplitude reduction and the phase shift of the anharmonic EXAFS 
EXAFS analysis; Debye- oscillation. The numerical results are found to be in good agreement with those 
Waller factor; Anharmonic obtained using the quantum ACE model and experiments at various 
correlated Einstein model; temperatures. The obtained results are useful in analyzing the experimental 
Crystalline silicon. EXAFS data of c-Si. 
 1. Introduction non-negligible errors in the structural parameters. 
 The use of the moments of the radial distribution 
 The extended X-ray absorption fine structure 
 function (or cumulants) to investigate local disorder 
(EXAFS) has been developed into a powerful 
 of EXAFS spectra was introduced by Rehr [3] who 
technique and is widely used to determine many 
 showed that the Debye-Waller (DW) factor of 
structural parameters and dynamic properties of 
 EXAFS spectra has a natural cumulant expansion in 
materials [1]. However, the position of atoms is not 
 powers of the photoelectron wavenumber. The 
stationary, and their interatomic distance always 
 connection between the DW factor and the EXAFS 
changes due to thermal vibrations that were 
 cumulants was described in detail in the cumulant 
detected by Beni & Platzman [2]. They cause 
 expansion approach (ratio method) by Bunker [4]. 
anharmonic effects on crystal vibrations and smear 
 The ratio method is particularly appealing because 
out the EXAFS oscillations. The anharmonicity of 
 it summarizes the relevant structural and dynamic 
the potential yields additional terms in the EXAFS 
oscillation, so if ignoring these terms, can lead to 
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 T.S.Tien et al/ No.19_Dec 2020|p.95-102 
information that is easily obtained from the 2. Basic formulae of EXAFS function and 
experimental EXAFS spectra. anharmonic effective potential of c-Si 
 Recently, a classical anharmonic correlated 2.1. Basic formulae of EXAFS function 
Einstein (ACE) model [5] was developed based on The EXAFS oscillation for a single coordination 
the anharmonic effective (AE) potential [6] and the shell, including thermal disorders has the form: 
classical statistical theory [7]. This model has the 
advantage that the expressions of the first four  k, T  A k , T sin ( k , T ) , (1) 
EXAFS cumulants are obtained in explicit and 
simple forms, so it is very convenient for where A k, T and (,)kT are the EXAFS 
anharmonic EXAFS data analysis in the range of 
 amplitude and phase, respectively, and k is the 
temperatures not too low. It has also been 
 photoelectron wavenumber, and T is the 
successfully applied to investigate the anharmonic 
 temperature. 
EXAFS oscillation for diamond crystals by Tien et 
al. [8]. However, the temperature dependence of The K-edge EXAFS oscillation for the distribution 
anharmonic EXAFS amplitude and phase has not of identical atoms is described within the framework 
been discussed in detail. Besides, it has been of single-scattering and plane-wave approximations. 
applied to successfully investigate crystalline Following the approach proposed by Tien [9], the 
germanium but has not yet been investigated logarithm of amplitude ratio 
 and the linear 
crystalline silicon (c-Si). Therefore, the analysis of M k, T1 , T 2 ln A k , T 2 A k , T 1 
the temperature dependence of anharmonic EXAFS 
 phase difference k,,,, T T  k T  k T 
oscillation for c-Si will be a necessary addition to 1 2 2 1 
evaluate the effectiveness of the classical ACE between temperatures T2 and T1 in the cumulant 
model in the EXAFS technique. expansion approach up to the fourth-order, which are 
 given as follows: 
 4
 2 2 2 2k 44 
 M k, T1 , T 2 2 k  T 2  T 1   T 2  T 1 , (2) 
 3 
 3
 1 1 1 ...  k3 and 
 k4 are local force constants giving asymmetry of 
 2.2. Thermodynamic parameters and 
 potential due to the inclusion of anharmonicity. 
anharmonic effective potential of c-Si 
 The Morse potential is assumed to describe the 
 To determine the thermodynamic parameters of 
 interatomic interaction model for the potential 
a system, it is necessary to specify its AE potential 
 energy of a diatomic molecule. Applying the Morse 
and force constants [10]. One considers a potential to calculate the interaction energy between 
monatomic system with an anharmonic effective each pair of atoms in cubic metals was proposed by 
potential (ignored the constant contribution) is Girifalco and Weizer [11]. In the present study, we 
extended up to the fourth-order: expand the Morse potential to the fourth-order: 
 1
 V(), x k x2 k x 3 k x 4 (4) 
 eff 2 0 3 4
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 T.S.Tien et al/ No.19_Dec 2020|p.95-102 
 7
 VxDe( ) 2 xx 2 e  DDxDx 2 2 3 3 Dx 4 4 . (5) 
 12
 where x is the same previously defined value, backscattering (B) atoms, including correlation 
 describes the width of the potential, and D is the effects and taking into account only the nearest-
dissociation energy. neighbor interactions, the AE potential [6] is given 
 by 
 In the relative vibrations of absorbing (A) and 
  ˆˆ
 Veff V(), x V xR AB R ij (6) 
 i A,, B j A B Mi
 vector, the sum i is the over absorbers ( iA ) and 
 where  MMMMABAB/ is the 
reduced mass of the absorber and backscatterer backscatterers ( iB ), and the sum j is over the 
 nearest neighbors. 
with masses MA and MB, respectively, Rˆ is a unit 
 Fig. 1. Structural model of c-Si. 
 Applying Eqs. (5) and (6) to the structure of c- Comparing Eq. (4) with Eq. (7), we deduce the 
Si with a mass of atoms is MAB M m, as seen local force constants , k3, and k4 as follows: 
in Fig. 1, the AE potential is written as 
 7 35 1519
 k D 2,,. k D 3 k D 4
 7 35 1519 03 3 36 4 2592
V() x D 2 x 2 D 3 x 3 4 x 4 . 
 eff 3 36 2592 (8) 
 (7) The thermal vibration of atoms is characterized 
 by the correlated Einstein frequency E and 
 The local force constants k0 , k3, and k4 are 
 temperature , which are calculated from the 
deduced from Eq. (7) as follows: E
 effective force constant k0 in the following forms: 
 k
 eff 10DDhE h 10
 EE ,  , (9) 
  m kBB k m
 where k is the Boltzmann constant, is the 
 B kk03, , and k4 are expressed in terms of the Morse 
reduced Planck constant. potential parameters via Eqs. (8) and (9). 
 Consequently, the correlated Einstein frequency 
E and temperature E , and force constants 
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 T.S.Tien et al/ No.19_Dec 2020|p.95-102 
 3. Temperature dependence of EXAFS The analysis of the EXAFS spectra usually uses the 
oscillation within the classical ACE model first four EXAFS cumulants [10]. The expressions 
 of the first four cumulants of diamond crystals 
 The EXAFS cumulants are explicitly related to 
 within the classical ACE model are given as 
low-order moments of the distribution function. 
 follows [8]: 
 1 15k 5 2
 ; B T , (10) 
 28D 4
 (2)32kkBB 2
 22TT , (11) 
 7Dm E
 2
 2
 3 45kB 225 
 ; 23T , (12) 
 98D 2
 3 2
 3
 4 687kB 32229 
 ; 24T . (13) 
 686D 18
 Substituting the anharmonic EXAFS cumulants from Eqs. (10)-(13) into Eqs. (2) and (3) to calculated the 
logarithm of amplitude ratio M k,, T12 T and the linear phase difference  k,, T12 T , the results are obtained as 
 3
 6kkBB2 229 4 3 3
 M k,,, T1 T 2 2 k T 2 T 1 3 4 k T 2 T 1 (14) 
 7DD 343
 2
 15kBBB 12 k 11 30 k 3 2 2
  kTT,,1 2 kTT 2 1 2  kTT 2 1 2 3 kTT 2 1 . (15) 
 14D 7 D  r0  49 D 
 Thus, the EXAFS cumulants using the CACE (9) in Sec. 2 and Eqs. (10)-(15) in Sec. 3 to the 
model can be expressed in the explicit and simple numerical calculations. Firstly, we perform 
forms of the mean-square relative displacement numerical calculations for the force constants of the 
(MSRD)  2 or temperature T . Consequently, the AE potential, the thermodynamic parameters, and 
logarithm of amplitude ratio and the phase the temperature dependence of the first four 
difference of the EXAFS oscillation is also EXAFS cumulants in the range from 0 K to 1000 
expressed in simple forms of temperature via the K. Our results of the cumulants are compared with 
EXAFS cumulants. These obtained results can those obtained using the quantum ACE model [6] 
describe the influence of anharmonic effects on the in the range from 0 K to 1000 K by Hung et al. [12] 
classical limit at high temperatures, and show that and experiment at 80 K, 300 K, and 500 K by 
the cumulants are very useful for the quantitative Benfatto et al. [13]. Then, we analyze the logarithm 
treatment of the anharmonic EXAFS oscillation. of amplitude ratio and the phase difference of the 
 anharmonic EXAFS oscillation with reference 
 4. Numerical results and discussions -1
 value at E in the wavenumber range from 0 Å to 
 To discuss the effectiveness of the classical 20 Å-1 at 700 K, 800 K, and 900K. Lastly, we 
ACE model for the analysis of the EXAFS evaluate and comment on the results obtained using 
oscillation of c-Si in this work, we apply Eqs. (8)- the classical ACE model in this work. 
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 2 3 
  ()T , (c) the third cumulant  ()T , and (d) 
 Fig. 2. Temperature dependence of (a) the first, 4 
 the fourth cumulant  ()T of c-Si is calculated 
(b) second, (c) third, and (d) fourth EXAFS 
 by Eqs. (9)-(12) and shown in Fig. 2. Our obtained 
cumulants of c-Si obtained using the ACE model, 
 results using the classical ACE model agree well 
the quantum ACE model [12], and the experiments 
 with the results obtained using the quantum ACE 
[13]. 
 model [12] (for the first three cumulants) and the 
 experiments [13] (for the second cumulants) at high 
 The force constants k0 , k3, and k3, and the 
 temperatures, especially for high-order cumulants. 
correlated Einstein frequency  and temperature 
 E For example, at 500 K, the results obtained using 
E are calculated via Eqs. (8)-(9). Our obtained the classical ACE model and quantum ACE model 
 1 3 2 3
 -2 -2 are  8.1 10 Å,  4.1 10 Å2, 
results are k0 10.39 eVÅ , k3 6.75 eVÅ , 
 3 5 3 4 6 4
 -2 13  6.7 10 Å , and  2,2 10 Å , 
 k4 6.35eVÅ , E 8.42 10 Hz, 
 1 3
 and  9.2 10 Å, 
and E 643.49 K, in which the use of the Morse 
 2 3 2 3 5 3
potential parameters D 1.83 eV and  4.7 10 Å , and  6.8 10 Å 
 1.56Å-1 in calculations were determined by [12], respectively, while the experimental value is 
 2 3
Swalin (1961) [14].  4.3 10 Å2 at 500 K [13]. Our 
 expressions of the first three EXAFS cumulants are 
 The temperature dependence of (a) the first 
 the same as the corresponding expressions 
 (1) 
cumulant  ()T , (b) the second cumulant calculated by the quantum ACE model [12] in the 
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 T.S.Tien et al/ No.19_Dec 2020|p.95-102 
high-temperature limit. Moreover, our results of the obtained using the classical ACE model, which 
EXAFS cumulants can agree with their satisfied all of their fundamental properties in 
experimental values at high temperatures although comparison with the quantum ACE model and 
we have only experimental values of the second experiment at temperatures above the correlated 
cumulant in comparisons. It is because the Einstein temperature. It is explained because 
experimental values of other cumulants can all be anharmonicity in EXAFS spectra appears from 
deduced from the second cumulant [15]. about room temperature. These results described the 
 influence of anharmonic effects on the classical 
 Thus, the results of the temperature dependence 
 limit via thermal vibration-contributions at high 
of the first four EXAFS cumulants of c-Si are 
 temperatures. 
 Fig. 3. The (a) logarithm of amplitude ratio and 0.573, and – 1.012 at Å-1, respectively. 
(b) phase difference with reference value at E of Moreover, in the temperature dependence, the value 
c-Si obtained using the classical ACE model at of decreases faster than the value of 
various temperatures. 
 . It is because high-order cumulants 
 The (a) logarithm of amplitude ratio 
 increase with temperature T, in which the third 
 M k, T ln A k , T A k ,E and (b) phase 
 cumulant reduces the value of and the 
difference  k,,, T  k T  k  of c-Si 
 E fourth cumulant increases the value of , 
at 700 K, 800 K, and 900 K are calculated by Eqs. 
 as seen Eqs. (14) and (15). 
(14)-(15) and shown in Fig. 3. It can be seen that 
the values of M k, T and  kT, decrease Thus, the results of the temperature dependence 
 of anharmonic EXAFS oscillation of c-Si are 
with increasing temperature T and decrease with 
 obtained using the classical ACE model, which 
fast-increasing wavenumber k. For example, at T = 
 shows that the even-order cumulants contribute to 
700 K, T = 800 K, and T = 900 K, the approximate 
 the amplitude reduction, and the odd-order 
results of M k, T are – 0.086, – 0.231, and – cumulants contribute primarily to the phase shift of 
0.372 at k 10 Å-1, and – 0.233, – 0.578, and – the anharmonic EXAFS oscillation. Accurate 
 calculation of the cumulants will allow us to 
0.833 at k 20 Å-1, respectively, while the 
 accurately analyze the change of the anharmonic 
corresponding results of  kT, are – 0.014, – EXAFS oscillation, and from which one will 
0.043, and – 0.080 at Å-1, and – 0.192, – determine the structural parameters from the 
 experimental EXAFS data. 
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 T.S.Tien et al/ No.19_Dec 2020|p.95-102 
 5 Conclusions [5] N.V. Hung, T.S. Tien, N.B. Duc, D.Q. 
 Vuong, High-order expanded XAFS Debye-Waller 
 In this work, the classical ACE model has been 
 factors of HCP crystals based on classical 
used successfully in the analysis of the anharmonic anharmonic correlated Einstein model, Mod. Phys. 
EXAFS oscillation of c-Si. The EXAFS analysis is Lett. B 28 (21) (2014) 1450174. 
performed based on evaluating the contribution of 
terms of the Debye-Waller factor in the cumulant [6] N.V. Hung and J.J. Rehr, Anharmonic 
expansion approach up to the fourth-order. The correlated Einstein-model Debye-Waller factors, 
 Phys. Rev. B 56 (1) (1997) 43-46. 
results of the first four EXAFS cumulants are not 
only expressed in explicit and simple forms of the [7] E.A. Stern, P. Livins, Z. Zhang, Thermal 
temperature T or MSRD but also satisfy all of their vibration and melting from a local perspective, 
fundamental properties in temperature dependence. Phys. Rev. B 43 (11) (1991) 8850-8860. 
The analytical results show the role and meaning of [8] T.S. Tien, N.V. Hung, N.T. Tuan, N.V. 
the EXAFS cumulants for the amplitude reduction Nam, N.Q. An, N.T.M. Thuy, V.T.K. Lien, N.V. 
and the phase shift of the anharmonic EXAFS Nghia, High-order EXAFS cumulants of diamond 
oscillation. The obtained results are very useful for crystals based on a classical anharmonic correlated 
analyzing the experimental data of the anharmonic Einstein model, J. Phys. Chem. Solids 134 (2019) 
EXAFS oscillation of c-Si. 307-312. 
 The good agreement of our numerical results with [9] T.S Tien, Advances in studies of the 
those obtained using the quantum ACD model and the temperature dependence of the EXAFS amplitude 
experiments at various temperatures show the and phase of FCC crystals, J. Phys. D: Appl. Phys. 
 53 (2020) 315303. 
effectiveness of the classical ACE model for 
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 Acknowledgments [11] L.A. Girifalco and V.G. Weizer, 
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 The authors are thankful to Professor Nguyen metals, Phys. Rev. 114 (3) (1959) 687-690. 
Ba Duc for helpful discussions and comments. 
 [12] V.V. Hung and H.K. Hieu, Study the 
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 SỰ PHỤ THUỘC VÀO NHIỆT ĐỘ CỦA DAO ĐỘNG EXAFS 
 PHÍ ĐIỀU HÒA CỦA TINH THỂ SILIC 
Tống Sỹ Tiến, Lê Việt Hoàng 
Thông tin bài viết Tóm tắt 
 Trong công việc này, dao động của phổ cấu trúc tinh tế mở rộng (EXAFS) phi 
Ngày nhận bài: điều hòa của tinh thể silic đã được biểu diễn qua các số hạng của hệ số Debye-
16/9/2020 Waller bằng phương pháp khai triển cumulant đến bậc bốn. Bốn EXAFS 
Ngày duyệt đăng: cumulant đầu tiên đã được tính toán dựa trên mô hình Einstein tương quan phi 
10/12/2020 điều hòa (ACE) cổ điển và qui trình phân tích phù hợp, trong đó các tham số 
 nhiệt động được rút ra từ hàm thế hiệu dụng phi điều hòa thu được bằng cách 
Từ khóa: tíếp cận các đóng góp lân cận của lớp nguyên tử đầu tiên. Việc phân tích sự 
Phân tích EXAFS; Hệ số phụ thuộc vào nhiệt độ của dao động EXAFS được thực hiện thông qua việc 
Debye-Waller; Mô hình đánh giá ảnh hưởng của các cumulant vào sự giảm biên độ và sự dịch pha. Các 
Einstein tương quan phi kết quả tính số được tìm thấy trùng hợp tốt với các kết quả thu được bằng mô 
điều hòa; Tinh thể silic. hình ACE lượng tử và thực nghiệm ở các nhiệt độ khác nhau. Các kết quả thu 
 được là rất hữu ích đối trong việc phân tích các dữ liệu EXAFS thực nghiệm 
 của c-Si. 
104