Vibronic interaction in crystals with the Jahn-Teller centers in the elementary energy bands concept

The order-disorder type phase transition caused by the vibronic interaction (collective Jahn-Teller effect) in a monoclinic CuInP₂S₆ crystal is analyzed. For this purpose, a trigonal protostructure model of CuInP₂S₆ is created, through a slight change in the crystal lattice parameters of the CuInP₂S...

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Date:2015
Main Authors: Bercha, D.M., Bercha, S.A., Glukhov, K.E., Sznajder, M.
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Language:English
Published: Інститут фізики конденсованих систем НАН України 2015
Series:Condensed Matter Physics
Online Access:http://dspace.nbuv.gov.ua/handle/123456789/155266
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Cite this:Vibronic interaction in crystals with the Jahn-Teller centers in the elementary energy bands concept / D.M. Bercha, S.A. Bercha, K.E. Glukhov, M. Sznajder // Condensed Matter Physics. — 2015. — Т. 18, № 3. — С. 33705: 1–17. — Бібліогр.: 28 назв. — англ.

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spelling irk-123456789-1552662019-06-17T01:26:29Z Vibronic interaction in crystals with the Jahn-Teller centers in the elementary energy bands concept Bercha, D.M. Bercha, S.A. Glukhov, K.E. Sznajder, M. The order-disorder type phase transition caused by the vibronic interaction (collective Jahn-Teller effect) in a monoclinic CuInP₂S₆ crystal is analyzed. For this purpose, a trigonal protostructure model of CuInP₂S₆ is created, through a slight change in the crystal lattice parameters of the CuInP₂S₆ paraelectric phase. In parallel to the group-theoretical analysis, the DFT-based ab initio band structure calculations of the CuInP₂S₆ protostructure, para-, and ferriphases are performed. Using the elementary energy bands concept, a part of the band structure from the vicinity of the forbidden energy gap, which is created by the d-electron states of copper has been related with a certain Wyckoff position where the Jahn-Teller's centers are localized. A construction procedure of the vibronic potential energy matrix is generalized for the case of crystal using the elementary energy bands concept and the group theoretical method of invariants. The procedure is illustrated by the creation of the adiabatic potentials of the Γ₅-Γ₅ vibronic coupling for the protostructure and paraphase of the CuInP₂S₆ crystal. A structure of the obtained adiabatic potentials is analyzed, followed by conclusions on their transformation under a phase transition and the discussion on the possibility for the spontaneous polarization to arise in this crystal. Аналiзується фазовий перехiд типу порядок-безпорядок в моноклiнному кристалi CuInP₂S₆, спричинений вiбронними взаємодiями (ефект Яна-Телера). З цiєю метою створено модель тригональної протостроктури для CuInP₂S₆ шляхом невеликої змiни параметрiв гратки CuInP₂S₆ в параелектричнiй фазi. Одночасно з теоретикогруповим аналiзом, здiйснюється першопринципний розрахунок на основi методу функцiоналу густини зонної структури CuInP₂S₆ в протоструктурi, пара- i ферофазах. Використовуючи концепцiю елементарних енергетичних зон, встановлено зв’язок частини зонної структури в околi забороненої енергетичної зони, що створюється станами d-електронiв мiдi, з певним положенням Вiкоффа, де локалiзованi ян-телерiвськi центри. Процедура побудови матрицi вiбронної потенцiальної взаємодiї узагальнюється на випадок кристалу, використовуючи концепцiю елементарних енергетичних зон i теоретикогрупового методу iнварiантiв. Процедура iлюструється на прикладi створення адiабатичних потенцiалiв вiбронного зв’язку Γ₅-Γ₅ для протоструктури i парафази кристалу CuInP₂S₆. На основi аналiзу отриманої структури адiабатичних потенцiалiв зроблено висновки щодо їх перетворення при фазовому переходi i обговорено можливiсть виникнення в кристалi спонтанної поляризацiї. 2015 Article Vibronic interaction in crystals with the Jahn-Teller centers in the elementary energy bands concept / D.M. Bercha, S.A. Bercha, K.E. Glukhov, M. Sznajder // Condensed Matter Physics. — 2015. — Т. 18, № 3. — С. 33705: 1–17. — Бібліогр.: 28 назв. — англ. 1607-324X DOI:10.5488/CMP.18.33705 arXiv:1510.06671 PACS: 73.21.Cd, 33.20.Wr, 71.70.Ej, 71.15.Mb http://dspace.nbuv.gov.ua/handle/123456789/155266 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description The order-disorder type phase transition caused by the vibronic interaction (collective Jahn-Teller effect) in a monoclinic CuInP₂S₆ crystal is analyzed. For this purpose, a trigonal protostructure model of CuInP₂S₆ is created, through a slight change in the crystal lattice parameters of the CuInP₂S₆ paraelectric phase. In parallel to the group-theoretical analysis, the DFT-based ab initio band structure calculations of the CuInP₂S₆ protostructure, para-, and ferriphases are performed. Using the elementary energy bands concept, a part of the band structure from the vicinity of the forbidden energy gap, which is created by the d-electron states of copper has been related with a certain Wyckoff position where the Jahn-Teller's centers are localized. A construction procedure of the vibronic potential energy matrix is generalized for the case of crystal using the elementary energy bands concept and the group theoretical method of invariants. The procedure is illustrated by the creation of the adiabatic potentials of the Γ₅-Γ₅ vibronic coupling for the protostructure and paraphase of the CuInP₂S₆ crystal. A structure of the obtained adiabatic potentials is analyzed, followed by conclusions on their transformation under a phase transition and the discussion on the possibility for the spontaneous polarization to arise in this crystal.
format Article
author Bercha, D.M.
Bercha, S.A.
Glukhov, K.E.
Sznajder, M.
spellingShingle Bercha, D.M.
Bercha, S.A.
Glukhov, K.E.
Sznajder, M.
Vibronic interaction in crystals with the Jahn-Teller centers in the elementary energy bands concept
Condensed Matter Physics
author_facet Bercha, D.M.
Bercha, S.A.
Glukhov, K.E.
Sznajder, M.
author_sort Bercha, D.M.
title Vibronic interaction in crystals with the Jahn-Teller centers in the elementary energy bands concept
title_short Vibronic interaction in crystals with the Jahn-Teller centers in the elementary energy bands concept
title_full Vibronic interaction in crystals with the Jahn-Teller centers in the elementary energy bands concept
title_fullStr Vibronic interaction in crystals with the Jahn-Teller centers in the elementary energy bands concept
title_full_unstemmed Vibronic interaction in crystals with the Jahn-Teller centers in the elementary energy bands concept
title_sort vibronic interaction in crystals with the jahn-teller centers in the elementary energy bands concept
publisher Інститут фізики конденсованих систем НАН України
publishDate 2015
url http://dspace.nbuv.gov.ua/handle/123456789/155266
citation_txt Vibronic interaction in crystals with the Jahn-Teller centers in the elementary energy bands concept / D.M. Bercha, S.A. Bercha, K.E. Glukhov, M. Sznajder // Condensed Matter Physics. — 2015. — Т. 18, № 3. — С. 33705: 1–17. — Бібліогр.: 28 назв. — англ.
series Condensed Matter Physics
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fulltext Condensed Matter Physics, 2015, Vol. 18, No 3, 33705: 1–17 DOI: 10.5488/CMP.18.33705 http://www.icmp.lviv.ua/journal Vibronic interaction in crystals with the Jahn-Teller centers in the elementary energy bands concept D.M. Bercha1, S.A. Bercha1, K.E. Glukhov1, M. Sznajder2 1 Institute of Physics and Chemistry of Solid State, Uzhgorod National University, 54 Voloshin St., 88000 Uzhgorod, Ukraine 2 Faculty of Mathematics and Natural Sciences, University of Rzeszów, Pigonia 1, 35-959 Rzeszów, Poland Received April 6, 2015, in final form June 3, 2015 The order-disorder type phase transition caused by the vibronic interaction (collective Jahn-Teller effect) in a monoclinic CuInP2S6 crystal is analyzed. For this purpose, a trigonal protostructure model of CuInP2S6 is cre-ated, through a slight change in the crystal lattice parameters of the CuInP2S6 paraelectric phase. In parallelto the group-theoretical analysis, the DFT-based ab initio band structure calculations of the CuInP2S6 proto-structure, para-, and ferriphases are performed. Using the elementary energy bands concept, a part of the band structure from the vicinity of the forbidden energy gap, which is created by the d -electron states of copper, has been related with a certain Wyckoff position where the Jahn-Teller’s centers are localized. A construction proce- dure of the vibronic potential energy matrix is generalized for the case of crystal using the elementary energy bands concept and the group theoretical method of invariants. The procedure is illustrated by the creation of the adiabatic potentials of the Γ5–Γ5 vibronic coupling for the protostructure and paraphase of the CuInP2S6crystal. A structure of the obtained adiabatic potentials is analyzed, followed by conclusions on their transfor- mation under a phase transition and the discussion on the possibility for the spontaneous polarization to arise in this crystal. Key words: Jahn-Teller effect, adiabatic potentials, Wyckoff positions, group theory PACS: 73.21.Cd, 33.20.Wr, 71.70.Ej, 71.15.Mb Introduction The order-disorder type phase transitions occur in a series of compounds, including the CuInP2S6 one [1] and, in particular in the case of this crystal, they can be explained by the realization of the cooperative Jahn-Teller effect (see e.g., references [2, 3]). The theory of this effect described in literature [4] is based on the partially model approach. Namely, the effect of vibronic interaction in the Jahn-Teller’s centers in a unit cell of a crystal is considered, and further, the interaction between these centers is modeled. It should be emphasized, that a procedure describing the realization of the collective Jahn-Teller effect is discussed in book [4]. However, as opposed to the consideration here, it is not founded on the real band structure of crystal. Instead, the collective Jahn-Teller effect is studied in reference [4] in a model way, i.e., the effect is analyzed initially at the isolated structural unit that coincides with the Jahn-Teller center (unit cell of a crystal or another atoms formation which has a degenerated electronic state and is unstable with respect to Jahn-Teller effect), and next, minimization of the potential energy of interaction between structural units is performed. As opposed to molecules, where the degenerate or pseudodegenerate local electron states take part in the vibronic process, in crystals the band structure E(k) over the Brillouin zone (BZ) should be taken into account. This circumstance forces us into searching for a new approach to describe the Jahn-Teller effect in crystals which will be the subject of study in this paper. Herein, we present a theory of the Jahn-Teller effect illustrated for the CuInP2S6 crystal. The theory is based on the symmetry of the crystal band structure in the framework of the elementary energy bands (EEBs) concept, introduced in papers © D.M. Bercha, S.A. Bercha, K.E. Glukhov, M. Sznajder, 2015 33705-1 http://dx.doi.org/10.5488/CMP.18.33705 http://www.icmp.lviv.ua/journal D.M. Bercha et al. by Zak [5, 6]. The essence of this concept is that the information about the symmetry and topology of the band structure of a crystal is encoded in the site-symmetry group of a certain Wyckoff position, that was identified later on as the actual Wyckoff position [7, 8]. The physical meaning of the actual Wyckoff position has been demonstrated in our papers [7, 8], where it has been shown that the maximum of the spatial valence electron density distribution is focused in this position in the unit cell of a crystal. Moreover, representations of the irreducible band representation describing the symmetry of the EEBs that form the crystal valence band can be induced only from the irreducible representations (irrep) of the site-symmetry group of the actual Wyckoff position. The most evident example of a relation between the EEB’s symmetry, actual Wyckoff position, and the localization of maximum of the valence spatial density distribution in this position are germanium, silicon, A3B5 type crystals, and the superlattices based on them [8]. It should be expected that the band structure of crystals with the Jahn-Teller centers will be composed of the EEBs reflecting the local symmetry of certain Wyckoff positions which, in turn, coincide with cer- tain Jahn-Teller centers. Since the EEBs concept allows us to present the ‘spatial issue’ (i.e., the energy spectrum of crystal) as the issue concerning a point symmetry, it is possible to utilize in our approach the theory of the Jahn-Teller effect, elaborated for molecules [9]. The structure of this paper is as follows. In section 1, the information on the crystalline structure of CuInP2S6 and its phase transition related to the Jahn-Teller effect is presented. A low symmetry of the CuInP2S6 paraphase permits one to consider only the cooperative Jahn-Teller effect [4]. In order to demonstrate the existence of nearby-in-energy local electronic levels which are necessary to discuss this effect, a modelling of the high-symmetry CuInP2S6 ‘protostructure’ is performed in section 2, to- gether with the comparative group-theoretical analysis of energy states of all CuInP2S6 phases. Section 3 presents the results of the DFT-based ab initio band structure calculations of all phases of the CuInP2S6 crystal. Attention is paid to the presence of the EEB in the band structure of paraphase, which is related to the Wyckoff position d ( 2 3 , 1 3 , 1 4 ) where Cu atom is located. In section 4, the theory of Jahn-Teller effect is formulated, together with its generalization for the case of a crystal. A special role of the actual Wyck- off position as the information carrier about the electronic component of the vibronic instability is ex- plained. Additionally, the symmetry of normal vibrations which are active in the Jahn-Teller effect is dis- cussed for the CuInP2S6 protostructure, based upon the irreducible representations of the site-symmetry group of the actual Wyckoff position. In the vibronic instability analysis, the normal vibrations which are associated with the degenerate states near the energy gap have been checked with respect to their ac- tivities in the above mentioned group-theory sense, for several high-symmetry points in the BZ. This has been done to confirm the validity of using the site-symmetry group of the actual Wyckoff position, whose symmetry encodes the symmetry of the EEB associated with d -states of copper. Section 5 is devoted to the construction of the vibronic potential energy in a matrix form, as well as of the adiabatic potentials for protostructure and paraphase. The Pikus’method of invariants [10] is used there for the first time to solve such kind of a problem. The final section 6 presents the analysis on the completeness of our approach in the description of the Jahn-Teller effect, as a mechanism of the transition from the para- to ferriphase in the CuInP2S6 crystal. 1. Structure and symmetry of the CuInP2S6 crystal The CuInP2S6 crystal possesses a layered structure (see figure 1) with the atomic layers separated by the van der Waals gaps [11]. A single atomic layer is composed of the octahedral sulfur framework in which Cu, In, and P–P atom pairs fill the octahedral voids. A peculiarity of the CuInP2S6 crystal structure is the presence of three types of copper atoms sites which are partially occupied, i.e., (i) quasitrigonal Cu1, shifted from the centers of the octahedra, (ii) octahedral Cu2, located in the centers of the octahe- dra, and (iii) nearly tetrahedral Cu3, which penetrates into the interlayer space. The occupancy of these positions varies significantly with temperature [2]. Furthermore, there are two types of Cu1 positions: Cu1u which is shifted upward from the middle of the layer, and Cu1d, shifted downwards from it. At low temperatures (T < 153 K), copper atoms fully occupy positions Cu1u [1]. Upon heating, the occupancy of Cu1u position decreases while the occupancy of Cu1d begins to increase. A hopping motion between Cu1u and Cu1d positions leads to an increase in the atomic layer thickness, i.e., to an increase in the volume of 33705-2 Vibronic interaction in crystals with the Jahn-Teller centers Figure 1. (Color online) Projection of the CuInP2S6 crystal structure. Dashed line encompasses the primi- tive unit cell of the protocrystal. the elementary unit cell without change of the number of structural units [1]. In the region of the phase transition from the ferrielectric to paraelectric phase (Tc = 315 K), the po- sitions Cu1u and Cu1d are filled with equal probability and the polarity of both copper sublattices disap- pears. This phase transition is accompanied by the space symmetry group change, within the monoclinic system, from Cc (C 4 s ) (ferriphase) to C2/c (C 6 2h ) (paraphase). Above Tc = 315 K, the Cu1u and Cu1d sites be- come equivalent, while at higher temperatures the Cu1 occupancy decreases, and the Cu2 and Cu3 sites start to fill up (see figure 1). In our study below, we shall focus on the premises of the order-disorder type phase transition that occurs at Tc = 315 K. The lattice parameters of both ferrielectric and paraelectric phases of CuInP2S6 are as follows [2]: a = 6.09559 Å, b = 10.56450 Å, c = 13.6230 Å, β = 107.1011◦, while the primitive cell parameters are a1 = 13.6230 Å, a2 = a3 = 6.096846 Å, α= 120.0311◦, β= γ= 98.4508◦. Crystals of both phases belong to the monoclinic base-centered lattice. The basis vectors of the primitive lattice can be expressed by the lattice parameters in the following way a1 = c, a2 = (a−b)/2, a3 = (a+b)/2, and we associate with them a non-orthogonal x, y, z coordinate system shown in figure 2. The symmetry group C 6 2h (C12/c1) of the CuInP2S6 paraphase is chosen in such a way that the two-fold leading axis coincides with vector b. Moreover, the values of its lattice parameters allow us to construct Figure 2. Basis vectors of the base-centered monoclinic primitive lattice of the CuInP2S6 paraphase in the non-orthogonal x, y , z coordinate system. 33705-3 D.M. Bercha et al. Table 1. Atomic coordinates of the CuInP2S6 para- and ferriphases. Structure Coordinates Site Site-symmetry group paraphase Cu (0.5000,0.8355,0.2500) 4e 2 In (0.5000,0.5019,0.2500) 4e 2 P (0.5591,0.1682,0.1193) 8 f 1 S1 (0.7296,0.1620,0.1193) 8 f 1 S2 (0.7612,0.8304,0.1237) 8 f 1 S3 (0.2439,0.0117,0.1215) 8 f 1 ferriphase [11] Cu u (0.5957,0.8355,0.3869) 4a 1 Cu d (0.4310,0.8355,0.1490) 4a 1 In (0.5000,0.5019,0.2500) 4a 1 P1 (0.5686,0.1690,0.3491) 4a 1 P2 (0.4505,0.1674,0.1788) 4a 1 S1 (0.2808,0.1512,0.3950) 4a 1 S2 (0.2332,0.1645,0.8930) 4a 1 S3 (0.7845,0.0177,0.3950) 4a 1 S4 (0.7400,0.1727,0.1336) 4a 1 S5 (0.7555,0.1747,0.6404) 4a 1 S6 (0.2722,0.9943,0.6379) 4a 1 a hexagonal cell of the CuInP2S6 protostructure with c axis directed along the basis vector a of the mono- clinic lattice, using only a small displacements of atoms. To accomplish this task we have first calculated the atomic positions of the CuInP2S6 paraphase by symmetrizing those of the ferriphase, and preserving the same lattice parameters. The obtained atomic coordinates are presented in table 1. The model of the CuInP2S6 protostructure will be discussed in the next section. 2. Model of CuInP2S6 protostructure and comparative group-theoretical analysis A paraphase of the CuInP2Se6 crystal which is relative to CuInP2S6 belongs to the trigonal system with hexagonal Bravais lattice and is described by the D 2 3d space symmetry group [12]. Both this fact and the appropriate values of the CuInP2S6 primitive lattice parameters (a1, a2, a3,α,β,γ) urge us to create a model of the CuInP2S6 protostructure with the trigonal symmetry. This can be done in the following way. A three-fold leading axis can be directed along the primitive vector a1, whichwill become the basis vector c in the hexagonal unit cell, a1 = c. A slight deformation applied to the angles β and γ can transform them into the right angles. As a result, a hexagonal lattice is obtained with parameters c = 13.623 Å and the angle γ= 120◦, which roughly coincides with the angle α= 120.0311◦ of the monoclinic unit cell. Hence, the monoclinic a1 axis becomes the c axis of the hexagonal lattice, and two monoclinic basis vectors with length a2 = a3 spanning the angle 120◦ lie in the plane perpendicular to c axis. Next, the atomic coordinates of the CuInP2S6 paraphase are slightly changed from their original sites in such a way that the obtained trigonal CuInP2S6 protostructure is described by the D 2 3d (P3̄12/c) space group of the relative CuInP2Se6 crystal. The atomic coordinates of the CuInP2S6 protostructure are contained in table 2. As it is known, there exist 6 different kinds of the Brillouin zone for a monoclinic lattice [10], depend- ing on the relation between the basis vectors length of the direct and the reciprocal spaces, respectively. In our case, the BZ is presented in figure 3, together with the description of the high-symmetry points, ex- pressed by the combination of b1, b2, and b3 reciprocal lattice vectors [10]. Passing from themonoclinic to the hexagonal model cell. these three vectors are transformed in the following way, b1 → bhex3 , b2 → bhex2 , b3 → bhex1 , the angle ∠(bhex1 ,bhex3 ) = ∠(bhex2 ,bhex3 ) = 90◦, while ∠(bhex1 ,bhex2 ) = 60◦. The resulting hexago- nal BZ with high-symmetry points of the CuInP2S6 protostructure is presented in the right-hand part of figure 3. A correspondence between the high-symmetry points, their irreducible representations of the 33705-4 Vibronic interaction in crystals with the Jahn-Teller centers Table 2. The atomic coordinates of the trigonal model of CuInP2S6 protostructure. Atom Coordinates Site Site-symmetry group Cu ( 2 3 , 1 3 , 1 4 ) 2d 3 . 2 In (0,0, 1 4 ) 2a 3 . 2 P ( 1 3 , 2 3 ,0.1655) 4 f 3 . . S (0.3306,0.3401,0.1201) 12i 1 Figure 3. (Color online) Left: Brillouin zone for the para- and ferrielectric phases of the monoclinic CuInP2S6 crystal and its high symmetry points, Γ: 0, N : 1 2 b1, Q: 1 2 b2, P : 1 2 b3, F : 1 2 (b3 − b2), N1: 1 2 [b1 + (b2 −b3)], Q ′ : 1 2 (b1 −b2), P ′ : 1 2 (b1 −b3). Right: hexagonal BZ for the model protostructure. High- symmetry points: Γ: 0,M : 1 2 b1, A: 1 2 b3, L: 1 2 (b1+b3),K : 1 3 (b1+b2),H : 1 3 (b1+b2)+ 1 2 b3, P : 1 3 (b1+b2)+µb3. wave vector groups for the trigonal CuInP2S6 protostructure and the CuInP2S6 para- and ferriphases is displayed in table 3. The investigation on the additional degeneracy of energy states due to the time reversal symmetry for some high-symmetry points of the BZ, on the presence of extrema in E(k) dependencies in these points, as well as on the dispersion laws near these points have been performed for the symmetry groups C 6 2h (paraphase) and C 4 s (ferriphase) in paper [13]. In order to determine the changes in the energy spec- trum parameters when passing from the protostructure to the CuInP2S6 paraphase, we have performed a group-theoretical analysis of the E(k) dependencies in some high-symmetry points of the hexagonal BZ (D 2 3d space symmetry group). This comparative group-theoretical analysis will serve as an additional verification of the validity to simulate the phase transition by using the CuInP2S6 protostructure model. In our studies the Herring’s criterion [14], Rashba formula [15], as well as the Pikus’method of invariants [10] have been applied. Table 4 presents the investigation results concerning the presence of extrema in the E(k) dependencies in certain directions of the hexagonal and monoclinic BZs, as well as of the addi- tional degeneracy of energy states due to the time reversal symmetry, which is classified according to the Herring’s criterion. Comparing the results of paper [13] and Table IV it can be stated that when passing from the protostructure to the CuInP2S6 paraphase, neither changes in the presence of additional de- generacy of states at most high-symmetry points, nor in the presence of the E(k) extrema in particular directions of the reciprocal space can be observed. In order to solve the issue concerning the relationship between the energy spectrum and the insta- bility of the system caused by the vibronic interaction, attention should be paid to the following high- symmetry points of the hexagonal BZ Γ(0,0,0), A(0,0, 1 2 ), M( 1 2 ,0,0), L( 1 2 ,0, 1 2 ), H( 1 3 , 1 3 , 1 2 ), K ( 1 3 , 1 3 ,0), as well as to their counterparts from the BZ of a monoclinic crystal, as will be shown below. From table 4 it 33705-5 D.M. Bercha et al. Table 3. The correspondence between the high-symmetry points, as well as between the irreducible repre- sentations for the hexagonal BZ of protostructure and the monoclinic para- and ferriphases of CuInP2S6. Hexagonal BZ Irrep Monoclinic BZ Irrep Γ(0,0,0) Γ1 Γ2 Γ3 Γ4 Γ5 Γ6 Γ(0,0,0) Γ1 Γ2 Γ3 Γ4 Γ1 +Γ3 Γ2 +Γ4 A(0,0, 1 2 ) A1 A2 A3 N ( 1 2 ,0,0) N1 N2 N3 M( 1 2 ,0,0) M1 M2 M3 M4 P (0,0,± 1 2 ) P1 P2 L( 1 2 ,0, 1 2 ) L1 P ′( 1 2 ,0,± 1 2 ) P ′ 1 K ( 1 3 , 1 3 ,0) K1 K2 K3 V (µ1,µ2,µ2) V1 V2 V1 +V2 H( 1 3 , 1 3 , 1 2 ) H1 H2 H3 V (µ1,µ2,µ2) V1 V2 V1 +V2 P ( 1 3 , 1 3 ,µ) P1 P2 P3 V (µ1,µ2,µ2) V1 V2 V1 +V2 follows that the extrema in the E(k) dependencies can be observed in all three main directions of the BZ in the vicinity of certain high-symmetry points of hexagonal BZ and their counterparts, i.e., Γ (Γ), A (N ), L (P ′,Q ′), M (P,Q). The discussed D2 3d space symmetry group exhibits 6 irreducible representations in the BZ center [16], two of which (Γ5 (Eg ) and Γ6 (Eu) in notation by Kovalev [16]) are two-dimensional. Therefore, they are of particular interest for our study. These representations are reduced to two one- dimensional representations, describing split energy states, at the transition from C 6 2h (paraphase) to C 4 s (ferriphase). This is an important remark that validates the necessity to model the trigonal protostruc- ture of the CuInP2S6 crystal. A near-in-energy distance between the split energy states, described by the one-dimensional irreducible representations, can indicate slight changes in structural parameters when passing form the real paraphase to the trigonal CuInP2S6 protostructure. As will be shown below, the ab inito band structure calculation results confirm both the presence of a small splitting of the Γ5 and Γ6 degenerate states when passing form the trigonal to monoclinic CuInP2S6 crystal, and the occurrence of such splitting at other BZ points of the monoclinic lattice. The important information confirming the correct modelling of protostructure at which the parame- ters of the energy spectrum are changed slightly can be obtained by the analysis of the dispersion laws for charge carriers in the vicinity of certain high-symmetry points from the corresponding Brillouin zones. Below we present the investigation on the dispersion law near the point Γ for the doubly-degenerate Γ5 state of the CuInP2S6 protostructure. For this purpose, the Pikus’method of invariants [10] is used. In this 33705-6 Vibronic interaction in crystals with the Jahn-Teller centers Table 4. The effect of the time-reversal symmetry on the presence of band extrema in the main directions of the hexagonal and monoclinic BZs of CuInP2S6, as well as on the additional degeneracy of represen- tations expressed by the Herring’s criterion (cases a or b). The local coordinate systems have the same orientation of axes for all high symmetry points in both BZs. Hexagonal Monoclinic Point Irrep Case ∂En ∂ki = 0 Γ Γ1 Γ2 Γ3 Γ4 Γ5 Γ6 a1 i = x, y, z A A3 {A1 + A2} a1 b1 i = x, y − M M1 M2 M3 M4 a1 i = x, y, z L L1 a1 i = x, y, z K K1 K2 K3 a2 i = x, y, z H H1 H2 H3 a2 i = x, y, z P P1 P2 P3 a2 i = x, y, z Point Irrep Case ∂En ∂ki = 0 Γ Γ1 Γ2 Γ3 Γ4 Γ1 +Γ3 Γ2 +Γ4 a1 i = x, y, z N N1 N2 N3 a1 i = x, y P P1 P2 a1 i = x, y, z P ′ P ′ 1 a1 i = x, y, z V V1 V2 V1 +V2 a2 i = x, y V V1 V2 V1 +V2 a2 i = x, y V V1 V2 V1 +V2 a2 i = x, y method, the secular matrix D(k), which allows one to obtain the E(k) dependence, is presented as a sum of invariants. The invariants are products of the Ai s basis matrices and the f (k) functions depending on the wave vector components, whose symmetry is defined by means of the symmetric and antisymmetric decomposition of square character of the irreducible representation Γ5. The decomposition is defined as [10] n+ s = 1 2n ∑ g∈G χs (g ) { [χ(g )]2 +χ( g 2)} (2.1) and n− s = 1 2n ∑ g∈G χs (g ) { [χ(g )]2 −χ( g 2)}, (2.2) where n+ s and n− s denote numbers of irreducible representations (irreps) τ s in the considered symmetric (n+ s ) and antisymmetric (n − s ) squares of irreps, n is a number of elements in the space group, the summa- tion runs over elements of the wave vector group, and χs (g ) is a character of irreducible representations of the wave vector group in the center of the BZ. From equation (2.1) it follows that the even functions f+(k) are transformed according to the representations τs = Γ1, Γ5, while from equation (2.2), that the 33705-7 D.M. Bercha et al. uneven functions f−(k) are transformed according to τs = Γ3. The Ai s basis matrices which enter the sum of invariants can be obtained using a transformation rule of a matrix of the given order, under the effect of symmetry operations. In particular, for Γ1, the basis matrix will be identical to the second order unity matrix σ1, and for the representation Γ5, the basis matrices will be the σx and σz Pauli matrices. In order to determine the basis functions f+ = k2 x +k2 y (or f+ = k2 z ) for the representation Γ1, and f+ = kx kz (ky kz ) for Γ5, the projection operator method has been used [10]. It should be noted that the obtained D(k) matrix does not contain any components, which are transformed according to the irreducible rep- resentation Γ3. This is a consequence of parity of the corresponding basis function, whose application result does not coincide with that of an inversion element, for this representation. Our analysis allows us to present the secular matrix D(k) in the following form: D (k) = a ( k2 x +k2 y ) +bk2 z + ckx kz cky kz cky kz a ( k2 x +k2 y ) +bk2 z − ckx kz  . (2.3) By solving the resulting secular equation, the following dispersion law for charge carriers of the CuInP2S6 protostructure is obtained for the electron state described by the irreducible representation Γ5, E (k) = a ( k2 x +k2 y ) +bk2 z ± √ c2k2 z ( k2 x +k2 y ) . (2.4) Now, it can be checked how the obtained dispersion law is transformed when passing to the CuInP2S6 paraphase. By comparing the characters of irreducible representations of the C 6 2h symmetry group of the monoclinic crystal and the characters of Γ5 of D 2 3d (see tables 5 and 6 in appendix), we conclude that the Γ5 representation is reduced to two representations: Γ1 and Γ3 of the monoclinic crystal. For small changes in the energy spectrum parameters which are caused by the protostructure model used instead of the CuInP2S6 paraphase, the splitting between the energy states described by the irreducible representations Γ1 and Γ3 of the monoclinic crystal will be small. Hence, it can be concluded that these states interact with each other. For the joint representations which describe the split interacting states, the D(k)matrix, being the basis for the E(k) dependence, is two-dimensional [10]. The diagonal terms of this matrix are transformed according to the representation τs = |Γ1|2+|Γ3|2 = 2Γ1, while the off-diagonal terms, according to τs = Γ1 ×Γ3 +Γ3 ×Γ1 = 2Γ3. As a result, the D(~k)matrix takes the form: D (k) = ( a1k2 x +b1k2 y + c1k2 z + ∆ 2 αkx kz + f ky kz αkx kz + f ky kz a2k2 x +b2k2 y + c2k2 z − ∆ 2 ) , (2.5) where ∆ denotes the energetic distance between two interacting states. A comparison of equations (2.3) and (2.5) shows that bothmatrices are composed of functions of the samewave vector components. From the analysis presented above it follows that the comparison of the band structures of the paraphase and of the CuInP2S6 protostructure model is essential, in particular, in the Γ point. This issue will be discussed in the next section. 3. Ab initio band structure calculations of CuInP2S6 protostructure, para- and ferriphases Electronic energy structure of the CuInP2S6 protostructure, para-, and ferriphases has been calculated within the framework of the density functional theory [17, 18] in the local approximation (LDA) [19, 20], by means of the software packages ABINIT and SIESTA [21, 22]. In our calculations, plane waves and linear combination of atomic orbitals have been used correspondingly as a basis set for ABINIT and SIESTA programs. A periodic crystal structure has been taken into account through the boundary conditions at the boundaries of the unit cell. Ab initio norm conserving pseudopotentials [23, 24], for the following electron configurations of atoms have been utilized in calculations, Cu: [Ar] 3d 104s1 , In: [Kr] 5s25p1 , P: [Ne] 3s23p3 , and S: [Ne] 3s23p4 . The cutoff energy Ecut = 20 Ry of plane waves for the self-consistent calculation has been chosen to obtain a convergence in the total energy of the cell not worse than 0.001 Ry/atom. Such basis set consists 33705-8 Vibronic interaction in crystals with the Jahn-Teller centers Figure 4. (Color online) Ab initio band structure and density of states of the CuInP2S6 trigonal protostruc- ture. The calculated valence bandmaximum is in K –H (or P ) direction. Black line in the right-hand panel indicates the contribution of Cu atoms in the total DOS. of about 6000 plane waves. The total and partial electronic densities of states have been determined by a modified method of tetrahedra [25], for which the energy spectrum and wave functions are calculated on the 80 points k-mesh. Integration over the irreducible part of the Brillouin zone has been performed using the special k-points method [26, 27]. Finally, the optimization of the structural parameters has been done for all phases CuInP2S6. A contribution of particular atomic orbitals in the creation of various valence band ranges has been analyzed by means of the partial density of states function. Figures 4 and 5 show parts of the band spectrum of the protostructure and paraphase of CuInP2S6 crystal, respectively, together with the partial density of electronic states functions. As can be seen, the Figure 5. (Color online) Ab initio band structure and density of states of the CuInP2S6 paraelectric phase. 33705-9 D.M. Bercha et al. obtained valence band edges throughout the Brillouin zones are weakly dispersive for both phases of CuInP2S6. Both phases exhibit indirect bandgaps. In particular, the valence band top of the protostructure is located in the H–K direction, while the conduction band minimum is present in the Γ point. Moreover, the valence band top of the CuInP2S6 paraphase (figure 5) is composed of the elementary energy band which consists of four weakly split branches. Since the valence band top of the CuInP2S6 protostructure exhibits an identical topology (i.e., 4-branch EEB), it can be stated that this elementary en- ergy band is indeed suitable for analysis of protostructure— paraphase transformation of the CuInP2S6. In addition to the band structure calculations, we have performed the symmetry description of the ob- tained energy states near the energy gap in the Γ point of both phases. As a result, two highest energy states of the CuInP2S6 protostructure are described by two-dimensional irreducible representations Γ5 and Γ6 in the Γ point, in the direction of increasing energies. By comparing figures 4 and 5 it can be seen that these states undergo splitting when passing from the trigonal protostructure to the monoclinic para- phase. As a result, four nearby-in energy states described by one-dimensional irreducible representations from the monoclinic C 6 2h space group are obtained. Furthermore, the representations are reduced in the following way, Γ5 → Γ2 +Γ4, Γ6 → Γ1 +Γ3. The obtained representations exhibit different parity, i.e., Γ2 and Γ4 are uneven representations, while Γ1 and Γ3 are even ones. When the symmetry of the system is further lowered, i.e., the transition from paraelectric to ferrielectric phase occurs, the discussed four states becomemore distant in energy and the corresponding four-branch subband becomes more spread which indicates that the connection between the respective branches is weaker (see figure 6). Figure 6. (Color online) Ab initio band structure and density of states of the CuInP2S6 ferrielectric phase. Configuration of copper site: Cu u . 4. Vibronic interaction as a mechanism of the configuration change of molecule and of the order-disorder phase transition in crystals with degenerate electron states As it is known, some symmetric molecules can undergo the Jahn-Teller effect [9], which consists in lowering the symmetry of molecule due to the electron-vibronic interaction. As a result of this interac- tion, a degenerate electronic term is split and a rearrangement of the vibronic spectrum occurs. For the molecule configuration to be stable, the molecule energy, which is a function of the distance between cores, should exhibit a minimum at the given configuration of cores. It means that the expression that 33705-10 Vibronic interaction in crystals with the Jahn-Teller centers describes the energy change due to a small shift of core positions cannot contain terms linear with re- spect to the normal coordinates. Neglecting the adiabatic approximation, the expression that describes the potential energy of an electron subsystem can contain terms dependent on the electron subsystem coordinates. This can lead to the instability of the molecule, change in its configuration, as well as to the degeneracy removal of the electronic state, which takes part in this vibronic interaction. A part of the Hamiltonian of electron subsystem related to the deviation from the adiabatic approxi- mation can be written with accuracy to quadratic terms of the normal coordinates, which are treated as parameters of the potential energy, in the following way: Ĥ1 = ∑ αi Vαi (r )Qαi + ∑ αβi k Wαβi k (r )Qαi Qβk + . . . , (4.1) where r denotes a set of coordinates of the electron subsystem. Expression (4.1) is at the same time the perturbation term of the Hamiltonian H0 that describes the electron subsystem of the symmetric config- uration of molecules. The first-order perturbation correction linear with respect to normal coordinates, which describes the electronic energy of a molecule, can be written, within the adiabatic approximation by means of the matrix element Vρσ =∑ αi Qαi ∫ ΨρVαi (r )Ψσdq, (4.2) where Ψρ and Ψσ denote wave functions of the electron subsystem describing a degenerate electron term. The Hamiltonian Ĥ1 is invariant with respect to the transformations of the symmetry group, of con- sidered system. Since it contains a linear term of decomposition in series with respect to Qαi , the coeffi- cients Vρσ(r ) andWαβi k (r ), dependent on the coordinates of electron subsystem, are transformed under the action of symmetry elements in the same way as the normal coordinates Qαi , or their products. The subscript α denotes the number of irreducible representation, while i is the number of the basis wave function of this representation. In analogy to the way how the dispersion law E(k) for charge carriers has been found, in our ap- proach a secular equation is used, where the Hamiltonian is presented in a matrix form with the ele- ments 〈Ψ∗ ρ |Ĥ1|Ψσ〉. This Hamiltonian is called the vibronic potential energy operator [4]. It should be noted that those vibrations which are related to the linear terms of the vibronic interaction operator Ĥ1 with respect to the normal coordinates are called active vibrations in the Jahn-Teller effect. By solving the secular equation, the so-called adiabatic potential is obtained, which can be used to predict the presence of some stable or unstable configurations of a molecule. Recently, a construction procedure of the vibronic potential energy and of the adiabatic potential for a high-symmetry molecule by means of the Pikus’method of invariants has been reported [28]. It has been demonstrated that the representations τs according to which the functions dependent on the normal coordinate components are transformed, together with the vibrations active in the Jahn-Teller effect, as well as the matrices Ai s contained in invariants [10], can be obtained only based upon a decomposition of the symmetric square of character of the irreducible representation, which describes a degenerate electron state. While investigating the Jahn-Teller effect in a crystal, a problem arises how to transform the criteria obtained for the degenerate electron states to the corresponding energy bands E(k) in the BZ. It is obvious that one should concentrate first on the energy states which form a vicinity of the forbidden energy band gap. In this case, a possibility occurs for exchanging energy between electrons from a degenerate state and the respective phonons, if this process is allowed by the selection rules. It has been demonstrated in the previous section that copper d -electron states create a connected so-called elementary energy band throughout the BZ at the valence band top of CuInP2S6. Therefore, as opposed to molecules, one should consider this EEB in the CuInP2S6 crystal, instead of a degenerate d -electron level. As it is mentioned in introduction, the symmetry of the EEB is described by the so-called irreducible band representation [5, 6]. Since neither periodicity of the CuInP2S6 crystal lattice nor the number of atoms in a unit cell is changed in its phase transition, it should be expected that a phonon with the symmetry described by the representation of the wave vector group in q = 0 will be active in the Jahn-Teller effect. Note that in order to identify a normal mode that is active in the vibronic interaction, we utilize in our study a 33705-11 D.M. Bercha et al. fact that the EEB of the CuInP2S6 valence band top originates from the d -electron states of copper atoms. These atoms, in turn, occupy the Wyckoff positon d( 1 3 , 2 3 , 1 4 ) in the protostructure of CuInP2S6 crystal. Representations of the corresponding irreducible band representation which describes the symmetry of the EEB can be induced from the irreducible representations of the site-symmetry group of one of the Wyckoff position d multiplicities. The coordinates of this position d( 1 3 , 2 3 , 1 4 ) coincide, in turn, with the localization of the Jahn-Teller center in the CuInP2S6 crystal. Hence, the actual Wyckoff position d can be regarded as a distinct center, where the information on the electronic band structure of the CuInP2S6 crystal is encoded. Therefore, in order to find the symmetry of a normal vibrational mode that is active in the Jahn-Teller effect, it is enough to consider a decomposition of the symmetric square of characters of the representation induced from the irreducible representations of the site-symmetry group of the actual Wyckoff position d . This site-symmetry group coincides with the factor group of space-symmetry group of CuInP2S6 at the point k = 0. Hence, it can be postulated that the above group-theoretical procedure to find a vibrational mode that is active in the Jahn-Teller effect coincides with the procedure elaborated for molecules. In order to confirm this statement, a direct calculation concerning the Jahn-Teller’s criterium can be performed, for all high-symmetry points of the BZ, describing the states from the vicinity of the forbidden energy gap of the discussed crystal. Generally, in the case of centrosymmetrical crystals, the wave vector groups can contain an inversion element I . If it is the case, then the action of this element on a wave vector is as follows, I k0 =−k0 � k0 +b, otherwise I k0 =−k0 , k0 +b, where b is a reciprocal lattice vector. However, the construction of the symmetric square of representation characters differs in both cases. In the first case, the symmetric square of representation characters of the k0 wave vector group can be decomposed into irreducible representations of the k0 wave vector group by means of the formula [15] ns = 1 2n ∑ g∈Gk0 {[ χk0 ( g )]2 +χk0 ( g 2)} χs ( g ) , (4.3) where ns is the number of irreps τ s in the considered direct product, n denotes the number of elements in the space group, and χs is the character of irreducible representations of the wave vector group in the center of the BZ. When the wave vector group does not contain the inversion element, the decomposition is as follows [15], ns = 1 2n ∑ g∈Gk0 { χs ( g ) χk0 ( g ) χk0 ( R−1g R )+χs ( Rg ) χk0 [ (Rg )2]}, (4.4) where R denotes an element that transforms the k0 into −k0 (inversion in our case, since it is present in the space group of a centrosymmetrical crystal). As we have demonstrated in section 3, the valence band top of the CuInP2S6 crystal protostructure is composed of the EEB that is created by d -electron states of copper. Meanwhile, its symmetry is described by the following irreducible band representation, Γ5 +Γ6 − A1 + A2 + A3 −M1 +M2 +M3 +M4 −2L1 −K1 +K2 +K3 −H1 +H2 +H3 , (4.5) that is related to the actualWyckoff position d( 1 3 , 2 3 , 1 4 ) in which a copper atom is located. Note that the ab- solute maximum of the valence band of the CuInP2S6 protostructure is observed in the K P H direction. Hence, the analysis of energy states that are situated in these high-symmetry points of the BZ becomes crucial. The wave vector groups at points K ( 1 3 , 1 3 ,0) and H( 1 3 , 1 3 , 1 2 ) (see figure 3, right) do not contain the inversion element. Hence, the decomposition of their symmetric square of representation characters can be described by equation (4.4). However, from the viewpoint of the Jahn-Teller effect realization, only two-dimensional representations of the wave-vector groups which are contained in the irreducible band representation (4.5) are interesting for study. The wave vector groups of other points in the BZ (Γ(0,0,0), A(0,0, 1 2 ),M( 1 2 ,0,0), L( 1 2 ,0, 1 2 )) contain an inversion element. Hence, the decomposition of their symmetric square of characters of two-dimensional representations (in particular Γ5 and Γ6) is given by equation (4.3). Consequently, representations for the Γ point which can be found from equation (4.3) become a background to construct the adiabatic potential for the collective Jahn-Teller effect. 33705-12 Vibronic interaction in crystals with the Jahn-Teller centers 5. Vibronic potential energy and adiabatic potential of CuInP2S6 proto- structure and paraphase We shall find in the beginning of this section the normal vibrational modes of the CuInP2S6 crystal protostructure which take part in the vibronic instability. By analyzing the transformation of the atomic positions under the action of symmetry elements of the structure, the following expression that describes the character of the mechanical representation is obtained χM = 4Γ1 +6Γ3 +10Γ5 . (5.1) In order to construct a matrix of the vibronic potential energy that is connected with the doubly-dege- nerate electron states of the EEB (4.5), one should establish first the symmetry of the normal vibration (phonon) that is active in the Jahn-Teller effect. Correspondingly, by applying decomposition (4.3) to the representations Γ5, Γ6, and equation (4.4) to K3, H3 from the EEB (4.5), we obtain the expected result: ns , 0, only for τs = Γ1, and τs = Γ5. It means that these representations can describe the vibrations ac- tive in the Jahn-Teller effect. The same result can be found bymeans of the decomposition of the symmet- ric square of characters of the irreducible representations Γ5 and Γ6, from the extended site-symmetry group of the actual Wyckoff position d comprising its twomuliplicities d1( 1 3 , 2 3 , 1 4 ) and d2( 2 3 , 1 3 , 1 4 ). In these positions, there are the Jahn-Teller’s centers located, i.e., copper atoms. Hence, the normal vibrations that are active in the Jahn-Teller effect exhibit the symmetry described by the irreducible representations Γ1, and Γ5 in the center of the BZ. However, the vibration Γ1 should be excluded from the considerations, since it does not lead to a change in configurations of atoms in a unit cell. Therefore, the matrix of the vibronic interaction potential energy is built based on the normal coordinatesQ1 andQ2, being the func- tions which are transformed according to the representation Γ5 of the space group D 2 3d . The construction procedure of this matrix is analogous to the way how the dispersion law E(k)was obtained in section 2. In order to create the matrix of the vibronic interaction potential energy, it is necessary to construct some invariants in the form of matrices being the products of Q1 and Q2 functions, and their combinations. Theses matrices and functions are transformed according to the representation τs = Γ5. Similarly to the D(k) matrix, the above matrices should be chosen in the form of Pauli matrices, as well as a second- order unity matrix. It is obvious that the normal coordinates Q1 and Q2 are also transformed according to the representation Γ5. Using the projection operator technique and the matrix of representation Γ5 written in a real basis (see table 7 in appendix) one obtains that the functions Q2 1 −Q2 2 and 2Q1Q2 are transformed according to the representation Γ5, as well [28]. Moreover, these functions forming the ba- sis of the representation Γ5 are mutually transformed one into another under the action of the D3d space group symmetry elements. In order to obtain a result of the action of a D3d group element on the first of the above functions, one should utilize the first row of Γ5 matrix, while in the case of the second function, the second row. Finally, the resulting D(Q1,Q2) matrix for the vibronic coupling Γ5–Γ5, or Γ5–Γ6 can be written as a sum of invariants, D(Q1,Q2) = 1 2 ω2 ( Q2 1 +Q2 2 ) σ1 +V Q1σx +W 1 2 W Q1Q2 σx +V Q2σz +W ( Q2 1 −Q2 2 ) σz , (5.2) where ω denotes the frequency of a phonon in the harmonic approximation, V and W are matrix ele- ments of the linear and quadratic vibronic coupling, Γ1, and Γ6 denote representations which describe degenerate electron states of the CuInP2S6 protostructure. The solution of the secular equation |D(Q1,Q2)−ε| = 0, (5.3) leads to the adiabatic potential. In order to obtain a more convenient expression for further manipulations, one shall express equa- tion (5.2) in the Cartesian coordinates. It should be taken into account at the same time that the basis of representation Γ5 are the x2− y2 and 2x y functions. Next, changing to the polar coordinate system (ρ,ϕ) in which the ρ axis coincides with the c leading axis of the protostructure crystal, we are in position to 33705-13 D.M. Bercha et al. Figure 7. (Color online) Left: A dependence of the adiabatic potential ε1,2 of CuInP2S6 protostructure versus polar coordinates ρ and ϕ. Middle and right: cross sections of the adiabatic potential for various energy values marked by different colors. write the D(ρ,ϕ)matrix elements as follows: D11 = 1 2 ω2ρ4 +V ρ2 sin2ϕ+W ρ4 cos4ϕ, D22 = 1 2 ω2ρ4 −V ρ2 sin2ϕ−W ρ4 cos4ϕ, D12 = D21 =V ρ2 cos2ϕ+W ρ4 sin4ϕ. (5.4) The adiabatic potential ε calculated from the respective secular equation takes the form ε1,2 ( ρ,ϕ )= ω2ρ4 2 ± [ V 2ρ4 +2V W ρ6 sin6ϕ+W 2ρ8]1/2 . (5.5) A dependence of the adiabatic potential versus ρ and ϕ coordinates, together with its cross-sections for various energy values, is presented in figure 7. As can be seen from figure 7, the obtained adiabatic potential possesses 6 minima. Such a shape of the adiabatic potential allows a transition from the CuInP2S6 protostructure to the paraphase of the symmetry C 6 2h , since its symmetry with respect to the inversion operation is then preserved. It should be noted here that the symmetry of the CuInP2S6 protostructure crystal is the same as that of paraphase of a related CuInP2Se6 crystal. The shape of adiabatic potential (5.5) for the CuInP2Se6 para- phase allows, therefore, its phase transition to the ferrielectric phase of the symmetry C 2 3v , at the simul- taneous loss of symmetry with respect to the inversion operation. Moreover, the shape of (5.5) reflects the presence of the inversion element in the symmetry group D 2 3d of the CuInP2S6 protostructure. The ε(ρ,ϕ) function which exhibits 6 minima is transformed during the phase transition to the paraelectric phase of CuInP2S6 to a function having 2 minima, whereas at the phase transition paraphase–ferriphase of CuInP2Se6, this function attains 3 minima. The two-minima adiabatic potential can be obtained in an analogous way for the paraphase of CuInP2S6. It is responsible for the realization of the Jahn-Teller pseudoeffect, which is connected with the existence of two nearby-in-energy electron states. As has been demonstrated in section 3, when pass- ing from the model protostructure of the CuInP2S6 crystal to its paraphase, there occurs a splitting of the double degenerate electron states in points Γ, K , and H of the elementary energy band that forms the vicinity of the forbidden band gap (figure 4). Moreover, two doubly degenerate electron states which are described in the Γ point by representations Γ5 and Γ6 are reduced to representations of the monoclinic paraphase as follows: Γ5 → Γ2+Γ4, Γ6 → Γ1+Γ3. As a result, there occurs a splitting of two double degen- erate states into four energy states, described by one-dimensional representations with different parity. The adiabatic potential takes the form: ε (Q) = 1 2 Q2 ± √ ∆2 +V 2Q2 . (5.6) 33705-14 Vibronic interaction in crystals with the Jahn-Teller centers Due to the presence of pairs of states Γ1 and Γ2 as well as Γ3 and Γ4 with the opposite parity, the Jahn- Teller pseudoeffect can take place in the system. States with the opposite parity are connected by a phonon with odd parity. This leads to the appearance of the dipole moments in the CuInP2S6 crystal. As a consequence of the vibronic interaction between the quasi-degenerate electronic state and a polar phonon, the levels nearby-in-energy become apart from one another, a connection between them is lost, and the adiabatic potential becomes a single minimum one. In the case of double minimum adiabatic potential related to the CuInP2S6 paraphase, two possible atomic sites Cu u and Cu d of copper in the unit cell are filled with equal probability. When the transition to the single minimum adiabatic potential takes place, the ordering of dipoles occurs with the appearance of spontaneous polarization. 6. Conclusions It has been demonstrated in this paper that it is enough to use a local symmetry of the crystal’s ac- tual Wyckoff position where the Jahn-Teller center is located in order to obtain a correct theory of the order-disorder type phase transitions in crystals. The group-theoretical approach together with the Pikus’ method of invariants can be successfully utilized to construct the vibronic potential energy, and the adi- abatic potential of a crystal. At the same time, it should be taken into account that the information on the symmetry and topology of the whole energy band structure of a crystal is encoded in the symmetry of the actual Wyckoff position. The developed approach allows one to find the mechanisms of the effect on the parameters of the phase transition in the CuInP2S6 crystal, which will be the subject of the study elsewhere. Acknowledgements The authors would like to thank Prof. Kharkhalis L.Yu. for valuable discussions and her interest to the work. A. Tables of irreducible representations Table 5. Characters of irreducible representations of the D3d point group, as well as of the wave vector group in k = 0 for the D2 3d space group (notation of symmetry elements and representations in tables 5–7 is adapted from reference [16]. h13 denotes inversion element). h1 h3, h5 h8, h10, h12 h13 h15, h17 h20, h22, h24 Γ1 (Ag ) 1 1 1 1 1 1 Γ2 (Au) 1 1 1 –1 –1 –1 Γ3 (Bg ) 1 1 –1 1 1 –1 Γ4 (Bu) 1 1 –1 –1 –1 1 Γ5 (Eg ) 2 –1 0 2 –1 0 Γ6 (Eu) 2 –1 0 –2 1 0 Table 6. Characters of the irreducible representations of the C2h point group, as well as of the wave vector group in k = 0 for the C5 2h space group. (h25 denotes inversion element). h1 h4 h25 h28 Γ1 (Ag ) 1 1 1 1 Γ2 (Au) 1 1 –1 –1 Γ3 (Bg ) 1 –1 1 –1 Γ4 (Bu) 1 –1 –1 1 33705-15 D.M. Bercha et al. Table 7. Irreducible representations Γ5 (Eg ) and Γ6 (Eu ), written as real matrices (in cartesian coordi- nates). h1 h3 h5 h8 h10 h12 Γ5 ( 1 0 0 1 ) − 1 2 − p 3 2p 3 2 − 1 2   − 1 2 p 3 2 − p 3 2 − 1 2  ( 1 0 0 −1 ) − 1 2 p 3 2p 3 2 1 2  − 1 2 p 3 2p 3 2 1 2  h13 h15 h17 h20 h22 h24 Γ5 ( 1 0 0 1 ) − 1 2 − p 3 2p 3 2 − 1 2   − 1 2 p 3 2 − p 3 2 − 1 2  ( 1 0 0 −1 ) − 1 2 p 3 2p 3 2 1 2  − 1 2 p 3 2p 3 2 1 2  h1 h3 h5 h8 h10 h12 Γ6 ( 1 0 0 1 ) − 1 2 − p 3 2p 3 2 − 1 2   − 1 2 p 3 2 − p 3 2 − 1 2  ( 1 0 0 −1 ) − 1 2 p 3 2p 3 2 1 2  − 1 2 p 3 2p 3 2 1 2  h13 h15 h17 h20 h22 h24 Γ6 (−1 0 0 −1 )  1 2 p 3 2 − p 3 2 1 2   1 2 − p 3 2p 3 2 1 2  (−1 0 0 1 )  1 2 − p 3 2 − p 3 2 − 1 2   1 2 − p 3 2 − p 3 2 − 1 2  References 1. 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B, 1998, 58, No. 7, 3641; doi:10.1103/PhysRevB.58.3641. 25. Blöchl P.E., Jepsen O., Andersen O.K., Phys. Rev. B, 1994, 49, 16223; doi:10.1103/PhysRevB.49.16223. 26. Chadi D.J., Cohen M.L., Phys. Rev. B, 1973, 8, No. 12, 5747; doi:10.1103/PhysRevB.8.5747. 27. Monkhorst H.J., Pack J.D., Phys. Rev. B, 1976, 13, No. 12, 5188; doi:10.1103/PhysRevB.13.5188. 28. Bercha S.A., Rizak V.M., Uzhhorod University Scientific Herald. Series Physics, 2013, No. 33, 15 (in Ukrainian). Вiбронна взаємодiя в кристалах з ян-теллеровськими центрами в концепцiї мiнiмальних комплексiв зон Д.М. Берча1, С.А. Берча1, К.Є. Глухов1,М.Шнайдер2 1 Iнститут фiзики та хiмiї твердого тiла, Ужгородський нацiональний унiверситет, вул. Волошина 54, 88000 Ужгород, Україна 2 Факультет математичних та природничих наук, Унiверситет Жешува, вул. Пiгонiя 1, 35-959 Жешув, Польща Аналiзується фазовий перехiд типу порядок-безпорядок в моноклiнному кристалi CuInP2S6, спричине-ний вiбронними взаємодiями (ефект Яна-Телера). З цiєю метою створено модель тригональної прото- строктури для CuInP2S6 шляхом невеликої змiни параметрiв гратки CuInP2S6 в параелектричнiй фазi. Одночасно з теоретикогруповим аналiзом, здiйснюється першопринципний розрахунок на основi мето- ду функцiоналу густини зонної структури CuInP2S6 в протоструктурi, пара- i ферофазах. Використовуючи концепцiю елементарних енергетичних зон, встановлено зв’язок частини зонної структури в околi забо- роненої енергетичної зони, що створюється станами d -електронiв мiдi, з певним положенням Вiкоффа, де локалiзованi ян-телерiвськi центри. Процедура побудови матрицi вiбронної потенцiальної взаємодiї узагальнюється на випадок кристалу, використовуючи концепцiю елементарних енергетичних зон i те- оретикогрупового методу iнварiантiв. Процедура iлюструється на прикладi створення адiабатичних по- тенцiалiв вiбронного зв’язку Γ5–Γ5 для протоструктури i парафази кристалу CuInP2S6. На основi аналiзуотриманої структури адiабатичних потенцiалiв зроблено висновки щодо їх перетворення при фазовому переходi i обговорено можливiсть виникнення в кристалi спонтанної поляризацiї. Ключовi слова: ефект Яна-Телера, адiабатичнi потенцiали, положення Вiкоффа, теорiя груп 33705-17 http://dx.doi.org/10.1103/PhysRevB.58.3641 http://dx.doi.org/10.1103/PhysRevB.49.16223 http://dx.doi.org/10.1103/PhysRevB.8.5747 http://dx.doi.org/10.1103/PhysRevB.13.5188 Structure and symmetry of the CuInP2S6 crystal Model of CuInP2S6 protostructure and comparative group-theoretical analysis Ab initio band structure calculations of CuInP2S6 protostructure, para- and ferriphases Vibronic interaction as a mechanism of the configuration change of molecule and of the order-disorder phase transition in crystals with degenerate electron states Vibronic potential energy and adiabatic potential of CuInP2S6 protostructure and paraphase Conclusions Tables of irreducible representations