CONDENSON STATES DYNAMICS IN THE LAYERED CRYSTALS OF THE INDIUM SELENIDES UNDER ELASTIC DEFORMATIONS

The conditions of self-consistent localized state formation connected with the interaction between an electron and the longitudinal acoustic phonon in the In 4 Se 3 and  -InSe crystals have been investigated. In the consequence of studying of the deformation influence on the dispersion law with the low-energy non-parabolicity the evolution of appearance or disappearance of the condenson states have been established. The main characteristics such as binding energy and localization radii of the condenson states are


Introduction
As it is known [1][2][3], the interaction between an electron and the longitudinal acoustic phonon in the homopolar crystals leads to the formation of self-consistent localized state called as condensons. The dispersion law is cruel in the phenomena of the charge carrier localization. For the first time we have shown that the charge inhomogeneity in the form of the "multicondenson drops" (early it was possible for only the onedimensional materials) is realized in the layered In4Se3 crystal owing to its dispersion law with low-energy non-parabolicity and peak-like density of states of E -1/4 type [4,5]. Today, the developed by us condenson state concept for the In4Se3 crystal is used [8,9] for the explanation of its unique high thermoelectric figure of merit [10]. Considering the thermoelectric properties of the In4Se3 [8,9] we experimentally corroborate the existence of the condenson states in this crystal. With results obtained in works [8,9] it can be concluded that formation of the condenson state can be controlled by cationic (Sn) and anion (Te) substitution too.
It is of interest to investigate the charge carrier localization phenomenon in other InSe layered crystal where accordingly to our simulation [6,7] its band structure can be considered as initial to the band structure of In4Se3. As it follows from ab initio calculations of the energy spectra under pressure [6], the shape of the top of the valence band of deformed -InSe resembles the shape of the valence and conduction bands of the In4Se3 crystal in the x k -direction of the Brillouin zone, that it is the result of an increase of negative coefficients of the squared power terms and the possible decrease of the coefficients of the four power terms of the kvector components. Therefore, from this point of view, it is of interest to study the dynamics of the inhomogeneous states of the condenson type, their appearance or disappearance which connected with intensification or weakening of the low-energy non-parabolicity of the dispersion law owing to deformation induced by the external factors. In the second section of our work, the theory of the condenson state in In4Se3 crystal will be present. Investigation of the impact of deformation effects on the lowenergy non-parabolicity of the dispersion law and the condition of the condenson state formation are given in section 3. In section 4, investigation of the condenson states in the -InSe crystal is shown. We summarize our results in section 5.

Theory of condenson states in the 3D -In4Se3 layered crystal
A crystal In4Se3, in which it was founded the condenson related thermoelectric properties [8,9], has the complicated structure composed of one-dimensional chains of In and non-flat warped two-dimensional ion-covalent layers In -Se [11,12].
The dispersion law with the additional four power terms of the wave vector [4][5][6]13], which describes the dispersion dependences in the vicinity of the forbidden energy gap (Fig.1a), has the form: Here quadratic term coefficients A, B, and C have the negative values. Using equation (1), the especial condenson state in the framework of theory developed by Dejgen and Pekar [1] in this 3Dcrystal was theoretically predicted. The condenson state is a kind of a polaron state, where a conduction electron is localized in the region deformed by itself due to the deformational interaction between the electron and acoustic phonons. As it was shown in the continual and deformation potential approaches [4,5] the energy functional can be written as Where determines the kinetic energy of the charge carriers, the parameter A describes the electron-phonon interaction through the deformation potential [4,5]. At the checking of the probe function in the form: expression (2) leads to the appearance of the localized condenson states in the In4Se3 crystal. Localization radii c r of these states are determined from the relationship: Our investigations of the band structures for the In4Se3 crystals indicate on the transformation of both the conduction band and valence band under pressure ( Fig.1, b). In this case, in the next paragraph of our work, we will analyze the effect of the transformation of the dispersion law non-parabolicity under elastic deformation on the parameters of the condenson states in the In4Se3 crystal.

Impact of deformation effects on the low-energy non-parabolicity of the dispersion law and the condition of the condenson state formation in the In4Se3 crystal
It is noted that our first investigation of the deformation influence (namely, shear strain) on the dispersion law of the charge carriers in the vicinity of the energy gap for the In4Se3 crystal were carried out in [7,15]. In the framework of the group-theoretical analysis together the Pikus' method of invariants [16] for the model of interacting bands in In4Se3 it was shown [7,15] To determine the components of the dispersion law (6) and deformation potential tensor components ij b the calculations of the band structure of both the undeformed and deformed In4Se3 crystal was performed by us.
As it is known, the additional potential energy is obtained from the comparison of the energies corresponding to the initial localization of the conduction band minimum and the valence band maximum and their localization after crystal stresses: The dispersion law coefficients (6)    Using obtained coefficients, in the framework of the continuum and deformation potentials approaches on the basis of relationships (2)- (5) and (6) we determined the parameters of condenson states (see Table 3).

Investigation of the condenson states in the -InSe crystal
As it follows from our earlier investigations [6], dispersion law with the four power terms of the k-vector components is not a unique one which characterizes only the In4Se3 crystal. Dispersion law with low-energy non-parabolicity and the abnormal anisotropy takes place in the deformed InSe crystal for  and modifications too [6,[18][19][20][21]. In [5] we performed the calculations of the band structure of both the undeformed and deformed -InSe crystal and obtained the analytical dependence of the type (1) which describes the dispersion relations in the vicinity of the energy gap along main Г-K, Г-М and Г-А directions in Brillouin zone of the hexagonal system. It was shown [6] the low-energy nonparabolicity in the -InSe crystal under hydrostatic pressure becomes more expressively similar to one in the In4Se3 crystal. It suggests the similarity of the dispersion laws for charge carriers in the In4Se3 and in the deformed γ -InSe and β -InSe crystals. Such dispersion law allows us to produce the specific condenson states due to the electron-phonon interaction in the -InSe layered crystal.
The dispersion relation in the form (1) for the -InSe layered crystal is confirmed too by the means of the investigations of the dispersion law in the vicinity of the Brillouin zone center in the framework of the Pikus' method of invariants [16]: ( ⃗ , ) = 0 + ( 2 + 2 ) + 2 + ( 2 + 2 ) 2 + 4 + ( + ) + The obtained coefficients of the dispersion law and the values for the parameters of deformational potentials in Tables 4, 5 are presented. In considered case of the interaction between an electron and the longitudinal acoustic phonon and using of the dispersion law (8) the energy functional can be written as Substituting the parameters A, В, C and D of the dispersion law from the Tabl. 4 in the equations (9), we obtain the condenson parameters for the -InSe crystal (Table 6). Here the integration in (9) is performed over the Brillouin zone for the hexagonal lattice. For this purpose, we adapt the following values of the elastic tensor components: C11 = C22=73.0 GPa, C12 = 27.0 GPa, C13 = 30.0 GPa, C33 = 36.0 GPa [23].
As it follows from our evaluations the condenson states ("electron + lattice deformation") are not formed at given parameters of the dispersion law (8). However, the localized states of "hole+lattice deformation" type can be realized in the -InSe crystal (Fig 2b).
The dependences for the localized electron energy for In4Se3 crystal (Fig. 2a) and the localized holes energy for InSe (Fig. 2b) versus the variational parameter  have the forms.

Таble 6
Localized states parameters for the holes in undeformed and deformed -InSe crystal

Conclusion
Basing on the first-principles calculations of the band structures under the pressure effect for the layered crystals of the indium selenides, characterized by unusual dispersion with the low-energy nonparabolicity, one concludes that the existing potential relief in both the conduction and valence bands for the In4Se3 and β-InSe crystals can be changed owing to the insignificant stresses when the spectrum transformation is observed. In favor of later there are testified the valuations of the coefficients of the quadratic and four power terms of the wave vector components in the dispersion laws (6) and (8) for the electrons and holes, and also the calculations of the effective masses [17,22] and the main characteristics (binding energy and localization radii) of the condenson states under elastic deformation.
Thus, our obtained results show that the charge inhomogeneity regions in the form of the condenson, which takes place in the In4Se3 and β-InSe three-dimensional semiconductors can be changed in the presence of the stresses. It explains why in the undeformed n-InSe hexagonal crystal, where according to our investigations the condenson states are not realized, the high thermoelectric efficiency similar to the In4Se3 crystal [24] has been not achieved. However, as it follows from [24], the increase of the dimensionless figure of merit for the heterostructure InSe/In4Se3 was discovered owing to the essential decrease of the thermal conductivity. In our opinion this fact can be connected with the charge carriers scattering on the self-consistent localized states. One waits that the lattice mismatch strains can appear in the result of the formation of this heterostructure. It leads to the significant changing of the electron structure and to the occurrence of the favorable conditions for the condensons accordingly. Our studies can be useful for the question solution of the further enhanced thermoelectricity in the indium selenides.