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57ISSN 2072-5981 doi: 10.26907/mrsej
aänetic Resonance in Solids
Electronic Journal
Volume 21 Special Issue 4 Paper No 19406 1-11 pages
2019
doi: 10.26907/mrsej-19406
http: //mrsej. kpfu. ru http: //mrsej. ksu. ru
Established and published by Kazan University Endorsed by International Society of Magnetic Resonance (ISMAR) Registered by Russian Federation Committee on Press (#015140),
August 2, 1996 First Issue appeared on July 25, 1997
© Kazan Federal University (KFU)*
"Magnetic Resonance in Solids. Electronic Journal" (MRSey) is a
peer-reviewed, all electronic journal, publishing articles which meet the highest standards of scientific quality in the field of basic research of a magnetic resonance in solids and related phenomena.
Indexed and abstracted by Web of Science (ESCI, Clarivate Analytics, from 2015), Scopus (Elsevier, from 2012), RusIndexSC (eLibrary, from 2006), Google Scholar, DOAJ, ROAD, CyberLeninka (from 2006), SCImago Journal & Country Rank, etc.
Editor-in-Chief Boris Kochelaev (KFU, Kazan)
Honorary Editors
Jean Jeener (Universite Libre de Bruxelles, Brussels) Raymond Orbach (University of California, Riverside)
Executive Editor
Yurii Proshin (KFU, Kazan) mrsej@kpfu. ru
0 © I This work is licensed under a Creative
* m-we» Commons Attribution-Share Alike 4.0 International License.
This is an open access journal which means that all content is freely available without charge to the user or his/her institution. This is in accordance with the BOAI definition of open access.
Special Editor of Issue
Eduard Baibekov (KFU)
Editors
Vadim Atsarkin (Institute of Radio Engineering and Electronics, Moscow) Yurij Bunkov (CNRS, Grenoble) Mikhail Eremin (KFU, Kazan) David Fushman (University of Maryland, College Park) Hugo Keller (University of Zürich,
Zürich)
Yoshio Kitaoka (Osaka University,
Osaka)
Boris Malkin (KFU, Kazan) Alexander Shengelaya (Tbilisi State University, Tbilisi) Jörg Sichelschmidt (Max Planck Institute for Chemical Physics of Solids, Dresden) Haruhiko Suzuki (Kanazawa University, Kanazava) Murat Tagirov (KFU, Kazan) Dmitrii Tayurskii (KFU, Kazan) Valentine Zhikharev (KNRTU,
Kazan)
Technical Editors of Issue
Nurbulat Abishev (KFU) Maxim Avdeev (KFU) Eduard Baibekov (KFu) Alexander Kutuzov (KFU)
* In Kazan University the Electron Paramagnetic Resonance (EPR) was discovered by Zavoisky E.K. in 1944.
Short cite this: Magn. Reson. Solids 21, 19406 (2019)
doi: 10.26907/mrsej-19406
Phonon spectrum of Nd2Zr2O7 crystal: ab initio calculation
V.A. Chernyshev
Ural Federal University, Mira 19, Ekaterinburg 620002, Russia
E-mail: vladimir.chernyshev@urfu.ru
(Received May 25, 2019; accepted May 28, 2019; published June 6, 2019)
Crystal structure and phonon spectrum of rare-earth pyroclore oxide Nd2Zr2O7 were studied within the framework of density functional theory and MO LKAO approach. The calculations were performed by using hybrid functionals that take into account both local and nonlocal (at the Hartree-Fock formalism) exchanges. Calculations were performed with the functionals PBES0L0 and PBE0. The fundamental vibration frequencies of Nd2Zr207 were calculated. The calculations were performed in the CRYSTAL17 program designed to simulate periodic structures.
PACS: 61.50.Ah, 63.20.dk.
Keywords: phonon spectrum, DFT, hybrid functionals, pyrochlores.
Dedicated to Boris Malkin, on the occasion of his 80th birthday
1. Introduction
The growing interest to the study of neodymium zirconate Nd2Zr2O7 has to do with varies of their properties and applications [1-3]. The crystal field on the rare-earth ion at pyrochlore structure have been studied at [4,5]. Neodymium zirconate Nd2Zr2O7 was experimentally investigated by methods of X-ray diffraction, Raman and IR spectroscopy [6-16]. The structural phase transformation from fluorite to pyrochlore phase was studied by Raman spectroscopy [10]. However, not all the phonon modes corresponding to the pyrochlore structure were detected during the experiments. Most of the measurements were carried out on polycrystals. Therefore, it is relevant to perform ab initio calculation of the phonon spectrum, which will determine the frequencies and types of the phonon modes for pyrochlore structures of Nd2Zr207. In this work, the phonon spectrum of the crystal Nd2Zr207 with the pyrochlore structure (Fd3m) is investigated in the framework of the MO LCAO approach with hybrid DFT functionals.
2. Calculations
Ab initio calculations were performed within the framework of the density functional theory (DFT) by using hybrid functionals which take into account both local and non-local (in the Hartree-Fock formalism) exchange. Calculations were performed with PBE0 [17] functional, as well as PBES0L0 functional, which is incremented in the program CRYSTAL17 [18,19]. The percentage of HF-exchange at PBES0L0 functional is 25% as well as in PBE0. By using the hybrid functionals that take into account both local and non-local (HF) exchanges, we can well describe the band structure, IR and Raman spectra, and elastic properties of compounds with an ion-covalent bond [20]. Comparison of PBE0 and other functionals with CCSD calculations has been performed recently (128 functionals of different levels were tested) [21]. It was shown that PBE0 is characterized by a rather small error for functionals of its level relative to the CCSD calculation when reproducing electron density and other parameters [21]. By using the PBE0 hybrid functional, we successfully described the structure and dynamics of the crystal lattice of rare-earth titanates with the pyrochlore structure R2Ti2O7 (R - rare-earth ion) in our previous work [22]. Oxygen is part of all structural units in the pyrochlore structure. It is located in two symmetrically nonequivalent positions (Table 1). Therefore, the reproduction of the structure and properties will depend essentially on the oxygen basis. The basis of TZVP type was used in work [23]. This basis is available on CRYSTAL website [24]. Zirconium
basis [25] is available on CRYSTAL website also. This basis was used by the authors of the CRYSTAL program to calculate the structure and IR spectrum of zirconium complexes with oxygen ligands [25]. Quasi-relativistic pseudopotential ECP49MWB was used to describe the inner shells of the neodymium (ECP - "effective core potential"; 49 is the number of internal electrons replaced by a pseudopotential; WB is "quasi-relativistic") [26,27]. Accordingly, the inner shells of the rare-earth ion were replaced by a pseudopotential on the 4f inclusive. TZVP type valence basis sets "ECP49MWB-II" were used to describe the outer shells, 5s25p6, involved in the formation of chemical bonds [26,28,29]. These pseudopotential and valence basis sets are available on the Stuttgart website [30]. Gaussian primitives with exponent values less than 0.1 were removed from the valence basis sets. The last diffuse orbital of the f type was also removed from the valence basis sets. The sequence of calculations was the following. Firstly, the optimization of the crystal structure was carried out. After that, the phonon spectrum (or the elastic constants) was calculated for the crystal structure corresponding to the minimum energy. The solving accuracy of self-consistent system of Kohn-Sham equations was set at 10-10 a.u. (TOLDEE = 10). The parameters "TOLINTEG", determining the accuracy of the calculation of the two-electron integrals were set equal to 8,8,8,8,16. The Monkhorst-Pack shrinking factor was taken to be 8. The phonon spectrum in the CRYSTAL program was calculated at the harmonic approximation. When calculating the Hessian matrix, the first derivatives were calculated analytically, while the second derivatives were calculated numerically. The Born charges were used by calculations of the Raman and infrared intensities at CRYSTAL code [31]. Electric dipole properties were calculated by using a periodic Coupled-perturbed Hartree-Fock (CPHF) or Kohn-Sham (CPKS) approach [32-34]. Details of the calculation algorithms are presented in [35]. The Placzek approximation was used to calculate the intensity of the Raman modes [33]. Non-resonant Raman intensities were calculated as well. According to [18,24,35], the mode intensity associated to an oriented single crystal is:
4 « C(4-)2. (1)
Here i, j = x, y, z, and aj is the element of the Raman tensor. The latter is given by
k = d3ETOT
aij dQkdeiSj. ( )
The element ak is calculated as the third derivative of the total electron energy. Here Qk is the normal mode coordinate, e is the external electric field. The value C defined by the laser frequency ul and temperature T is as follows:
1+ n(Uk), n 4 ^
C -(uL - Uk). (3)
30 Uk
Taking into account the Bose occupancy factor n(uk), we obtain
1 + n(Uk) =
1 - exp -
hujA
kBT)
1
(4)
For a powder polycrystalline sample, the integral intensity is averaged over the possible directions of an ideal bulk crystal [36]. The rotational invariants G^ are defined as:
j(0) _ 1 f„,k rk \2
G(1) = 1 " Gk =2
Gk = 3 yaxx + ayy + azzj , (5)
ak ak 2 + ak ak 2 + ak ak 2
xy - yx xz - zx zy - yz
(6)
G(2) _ 1 Gk _2
1
+ 3
ak + ak ^ + (ak + ak ^ + (ak + ak
Xy ' Kj-yx I 1 I "iz 1 "zi / 1 I ^zy 1 ^yz
+
(7)
k k 2 k k 2 axx ayy j + ( axx azz
+ ( ak _ ak I I L-t-yy ^zz
The Raman intensities of the two polarized components of the powder spectra, the parallel and perpendicular ones, are
I,
I
powder
k _ C (l0Gk0) + 4^) , powder _ C (5Gk1) + 3Gk2)) ,
_L,k
I
powder
tot,k
powder
IN,k + 1 ±,k
k
powder
(8) (9) (10)
The C value in the terms (8), (9) is defined in Eq. (3). The infrared intensity of the p-th mode [18] is defined as
Ip
kNa 3 c
dp lZp|2 '
(11)
where Na is Avogadro's number, c is the speed of light, dp is the degeneracy of the mode, and Zp is the mass-weighted effective-mode Born charge vector. The infrared intensity is calculated assuming an isotropic response.
3. Results and discussion
The coordinates of the ions in the unit cell of the crystal Nd2Zr2O7 with the pyrochlore structure (Fd3m) are given in Table 1.
The crystal structure of Nd2Zr2O7 was calculated with PBE0 and PBESOLO hybrid functionals. The results are shown in Table 2. The calculation results are in good agreement with the experimental data.
Rare-earth zirconate Nd2Zr2O7 with pyrochlore structure has phonon modes at the Г point: Г _ Aig + Eg + 2Fig + 4F2g + 3^2« + 3EU + 8Fiu + 4F2„. Here Aig, Eg and 4F2g are Raman active modes, 7F1u are infrared active modes, 4F2u, 3EU, 2F1g, 3A2u are silent modes. The results of the calculation of phonon modes at the Г point of Nd2Zr2O7 are given in Table 3. Frequencies and types of the phonon modes were determined from the ab initio calculation. From the analysis of displacement vectors obtained from this ab initio calculations, the degree of participation of each ion in a particular mode was estimated (Table 3, Fig. 1). The ions that are shifted significantly in the mode are listed in the column "Ion-participants" at Table 3. The "S" index is a strong shift ("Strong"), "W" is weak ("Weak"). The maximum displacement of ions reaches ~ 0.04 A. If the displacement of the ion is less than 0.01, the ion is not mentioned in the column "Ion-participants". If the value of its displacement is close to 0.01, it is indicated by the index "W". The results of the calculation of the intensity of Raman and infrared modes are shown in Tables 3-6 and Figures 2-3. We can distinguish between the modes in which only oxygen ions are involved, such as the infrared active mode F1u with a frequency of 234 cm-1. O1 ions, located at 48f position, characterized by the x coordinate, participate mainly in this mode. O1 is also predominantly involved in the most intense Raman mode F2g (309 cm-1). Only O1 ions are involved in the Raman Eg mode (328 cm-1). According to calculations, this is the second most intense mode in the Raman spectrum (Table 5, Fig. 2). Only O1 ions also participate in the Raman modes A1g. O1 ions participate mainly at the high-frequency F2g mode (711 cm-1). Thus, the behavior of these modes gives information about the value of the
2
x coordinate of the oxygen 01 under external impact on the crystal. The oxygen O2, located at 8b position, mainly participates in F2g modes with frequencies of 414 and 537 cm-1. The oxygen 02 participates mainly at infrared F1u mode (391 cm-1). All ions are involved in IR active modes (F1u), but their displacement values are different. Zirconium and both 01 and 02 oxygens are involved in the most intense F1u mode with a frequency of 333 cm-1, with 01 taking the most participation (Table 3). At the low-frequency F1u mode (98 cm-1), mostly neodymium is involved. Lanthanum and zirconium are substantially involved in silent modes. The calculation results are in satisfactory agreement with Raman experiments on the powder samples (Figure 4). According to the calculations, oxygen 01 is predominantly involved in the most intense Raman mode F2g (309 cm-1). This result agrees with the experimental data [11], where the Zr-06 bending was presumed at this mode.
According to the calculations, only 01 and 02 oxygens participate at F2g mode with frequency 414 cm-1. The result agrees with the experimental data [10], where it was assumed that the 0-Nd-0' vibrations are present in this mode (Table 7). The results of the simulation of the IR spectrum are shown in Fig. 1. The presence of the peak near 500 cm-1 agrees well with the measured IR transmittance spectrum of Nd2Zr207 [16]. It follows from the calculations that neodymium is involved in the modes with frequencies up to 250 cm-1. Zirconium participates in modes with frequencies up to 400 cm-1. 0xygen 02 participates in modes with frequencies up to 550 cm-1, while oxygen 01(48f) participates in all modes (Figure 1).
The results of the calculation of the elastic constants and the bulk modulus are given in Tables 8-10. The results are in good agreement with the experimental data [13]. The mechanical stability criterion [37] (Born stability criterion) is performed for Nd2Zr207.
Table 1. Ion coordinates in the R2Zr2O7 unit cell.
Ion x y z Wyckoff position
Zr 0 0 0 16 c
Nd 1/2 1/2 1/2 16 d
O1 x 1/8 1/8 48 f
O2 3/8 3/8 3/8 8 b
Table 2. The lattice constant, interionic distances (A), x coordinate of oxygen 01 (relative units) of the Nd2Zr207 crystal.
Calc. PBESOLO Calc. PBE0 Exp. [9] Exp. [1] Exp. [13] Exp. [5] Exp. [15]
Lattice constant 10.6526 10.7266 10.678 10.6134(1) 10.676 10.6611(1) 10.70
Nd-O1 x 6 2.5691 2.5914
Nd-O2 x 2 2.3064 2.3224
Zr-O1 2.0939 2.1056
Nd-Zr x 3.7663 0.3359 3.7924 0.3353 0.3357(2)
Table 3. Frequencies (cm-1) and types of phonon modes at the r point. The abbreviations in the "R" (Raman) and "IR" columns: "A" is an active mode, "I" is inactive one. In the last column, the "S" index designates a strong shift ("Strong"), "W" - weak ("Weak").
Calc.
Type IR R PBE-SOLO Calc. PBE0 Exp. [11] Exp. [10] Exp. [14] Exp. [9] Ions-participants
F2u I I 60 51 NdS
Eu I I 97 93 NdS, ZrW, O1W
Fiu A I 102 98 Nd, Zr, O1W, O2W
F2u I I 134 130 ZrS, O1W
Fiu A I 136 131 Nd, O1W, O2
Eu I I 138 133 NdW, ZrW, O1S
Fiu A I 202 199 215 Zr, O1
Fiu A I 238 234 O1S, O2W
A2u I I 249 242 Nd, O1
Fig I I 262 253 O1S
F2u I I 295 287 O1S
A2u I I 309 305 Zr
F2g I A 319 309 305 302 298 NdS
Eg I A 338 328 O1S
Fi u A I 345 333 365 ZrW, O1W, O2
A2u I I 394 392 O1
Fiu A I 402 391 400 O1W, O2S
Eu I I 410 402 ZrW, O1
F2g I A 422 414 399 400 392 O1, O2
Aig I A 518 511 506 501 503 O1
Fi u A I 522 512 525 O1, O2W
F2g I A 546 537 523 516 O1, O2S
Fig I I 587 573 O1
F2u I I 592 578 O1
F2g I A 782 771 O1
Table 4. Frequencies (cm x) and intensities of IR modes (km/mol). PBE0 calculation.
Type Frequency Intensity
Fiu 98 316
Fiu 131 1992
Fiu 199 6504
Fiu 234 32
Fiu 333 14933
Fiu 391 588
Fiu 512 3173
Table 5. Raman mode intensity for polycrystalline sample of Nd2Zr2O7 (relative units). The intensity of the Raman modes was calculated for A = 514 nm and T = 300 K. PBE0 calculation.
Type Frequency, cm 1 /tot /par Iperp
309 1000 571 429
Eg 328 205 117 88
414 27 15 12
Aig 511 54 54 0
F2g 537 69 40 29
F2g 771 25 14 11
Table 6. Raman mode intensity for a single crystal Nd2Zr2O7 (relative units). The intensity of the Raman modes was calculated for A = 514 nm and T = 300 K. PBE0 calculation.
Type Frequency, cm-1 Ixx Ixy Ixz Iyy Iyz Izz
F2g 309 0 1000 1000 0 1000 0
Eg 328 410 0 0 410 0 410
F2g 414 0 27 27 0 27 0
Aig 511 75 0 0 75 0 75
F2g 537 0 69 69 0 69 0
F2g 771 0 25 25 0 25 0
Table 7. The calculated and experimental Raman modes of the Nd2Zr2O7 crystal. Frequencies are given in cm-1. Types of modes are given according to the calculations. The calculated intensities (arb. un.) of the Raman modes for the polycrystalline sample are given in brackets.
Type Frequency, cm-1 Exp. [10] Exp. [11] Notes
F2g 309(1000) 302 305 Zr-O6 bending [11]
Eg 328(205)
F2g 414(27) 400 399 O-Nd-O' vib. [10]
A1g 511(54) 501 506
F2g 537(69) 516 523
F2g 771(25) 594
Table 8. Elastic constants and bulk modulus of Nd2Zr2O7 (GPa). PBESOLO calculation.
C11 C12 C44 B 318.8 122.3 102.3 187.8
Table 9. Elastic constants and bulk modulus of Nd2Zr2O7 (GPa) at hydrostatic pressure P = 2 GPa. PBESOL0 calculation.
C11 C12 C44 B
327.3 128.1 105.5 194.5
Table 10. Bulk modulus, Young's and shear modulus of Nd2Zr2O7, GPa. PBESOLO calculation.
Averaging scheme Bulk modulus Young's modulus Shear modulus Poisson's ratio
Voigt 187.8 256.2 100.7 0.27
Reuss 187.8 256.1 100.6 0.27
Hill 187.8 256.2 100.7 0.27
Exp. [13] 184 260 103 0.27
Figure 1. The displacements of ions at phonon modes.
Figure 2. Calculation results of the Raman spectrum of Nd2Zr207 crystal (PBES0L0 calculation). The intensity of the Raman modes was calculated for A = 488 nm and T = 298 K. Pseudo-Voigt functions with a damping factor of 8 cm-1 were used for modeling of the Raman spectrum based on the calculated frequencies and intensities for a polycrystal.
Figure 3. Calculation results of the IR spectrum of Nd2Zr207 (PBES0L0 calculation). All infrared modes are of F1u type.
J_11
200
400
600
800
1 r 200 400 600
800
F2g C
200 400 600 800 Frequency (cm1)
Figure 4. The calculated and experimental Raman modes of the Nd2Zr207 crystal. The calculated frequencies are indicated by vertical bars. Types of modes are given according to the calculations. PBE0 calculations are presented in (a) and (b). PBES0L0 calculation is shown in (c). The experimental data [10], [11], [12] are presented in (a), (b) and (c), respectively.
4. Summary
Ab initio calculation of the crystal structure and phonon spectrum Nd2Zr207 with the pyrochlore structure are carried out within the framework of the M0 LCA0 approach with a hybrid DFT functionals that takes into account the contribution of the HF exchange. The calculations reproduce the crystal structure in good agreement with the experimental data. Based on the analysis of displacement vectors obtained from these ab initio calculations, the degree of participation of each ion in a particular mode is estimated. It is shown that only oxygen ions are involved in Raman modes. The modes with absolute or predominant participation of oxygen at the 48/ position, characterized by the displacement textitx, are determined. The obtained results can be used for interpretation of the Raman and IR spectra of the crystal.
Acknowledgments
This study was supported by the Ministry of Education and Science of the Russian Federation (project No. 3.9534.2017/8.9).
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