1. Introduction

It is well known that for a great number of solids absorption near the optical absorption edge increases exponentially and the spectral dependences of the absorption coefficient at different temperatures are described by the Urbach rule[1]. Since the discovery of this universal rule, it was set for a number of ferroics: ferroelectrics (e.g.[2-6]), ferrielectrics (e.g.[7,8]), antiferroelectrics (e.g.[9]), and ferroelastics (e.g.[10,11]). However, despite its versatility, in many cases deviations from this rule are observed. Some of these materials are ferroelectrics with incommensurate phase. The aim of this paper is an attempt to explain the non-Urbach behaviour of the optical absorption edge in incommensurate phases of some ferroelectrics from the standpoint of disordering processes and the presence of the dynamic component of structural disordering caused by the structure modulation. As an example, we have chosen Sn₂Р₂Se₆ crystal which belongs to proper uniaxial ferroelectrics, in which at low temperatures two phase transitions exist, one of which is a phase transition from the paraelectric phase to the incommensurate phase at Т_i =221 K, the other is the phase transition from the incommensurate phase to the ferroelectric phase at Т_с=193 K[4, 5, 12].

2. Theory

In 1953, F. Urbach was the first to observe experimentally the exponential behaviour of absorption edge spectra in AgBr crystal, which formed a characteristic “bundle” at the temperature increase[1]:

(1)

where is the Urbach energy which is the value inverse to the absorption edge slope ; and are coordinates of the convergence point of the Urbach “bundle”. The temperature behaviour of the Urbach energy in the framework of the Einstein model is described by the equation[13]

(2)

where and are constants, is the Einstein temperature which corresponds to the average frequency of phonon excitations of non-interacting oscillators. It is known that the Urbach energy in solids is determined not only by temperature, but also by structural disordering[14]:

(3)

where K is a constant, and are mean-square deviations (displacements) of the atoms from their equilibrium positions, caused by the temperature disorder and the structural disorder of the solid-state system, respectively. As displacement of atoms from the equilibrium positions leads to a change in the electrical potential of the system, Eq. (3) can be written as

(4)

where is a constant, and are mean-square deviations from the electric potential of a perfectly ordered structure, caused by the temperature and the structural disorder, respectively. The contributions of the temperature disordering and structural disordering to the Urbach energy are considered independent, equivalent, and additive.

3. Experimental

Sn₂Р₂Se₆ single crystals were obtained by chemical transport[15]. The studies were performed for the samples of different thickness d=30÷500 µm in a broad temperature range 77–350 K. The incident light was linearly polarized, the electric field strength vector oscillating in planes parallel to X (coinciding with[100] crystallographic direction), or Y ([010] crystallographic direction). For transmission and reflection measurements a LOMO MDR-3 grating monochromator was used. The instrumental width was near 1 Å. A UTREX cryostat was applied providing temperature stabilization within 0.1 K. The experimental setup and technique are described in Ref.[10].

4. Results and Discussion

The studies of optical absorption edge of Sn₂P₂Se₆ ferroelectric have revealed three typical temperature ranges: (i) the range of the Urbach behaviour of the absorption edge at T<193 K, (ii) the range of parallel red shift of the optical absorption edge within the temperature interval 193≤T≤ 221 K and (iii) the range of the Urbach behaviour of the absorption edge at T>221 K (Fig.1). The parallel red shift of the optical absorption edge in Sn₂Р₂Se₆ and, consequently, the temperature invariance of the Urbach energy (Fig. 2), like in the case of other materials with modulated structures such as (Pb_ySn_1-y)₂P₂Se₆[16], CdP₂, α-ZnP₂[17], (N(CH₃)₄)₂ZnCl₄[18] (N(CH₃)₄)₂CuCl₄[19], are related to the presence of the structure modulation in the incommensurate phase.

Figure 1. Spectral dependences of logarithm of absorption coefficient for Sn2P2Se6 crystal at various temperatures T: 77 K (1), 110 K (2), 150 K (3), 180 (K), 190 K (5), 195 K (6), 210 K (7), 220 K (8), 230 K (9), 270 K (10), 310 K (11), 350 K (12)

Figure 2. Temperature dependences of the Urbach energy EU for Sn2P2Se6 crystal. The insert shows the temperature dependence of the contribution

To explain the invariance of the Urbach energy in the incommensurate phase (Fig. 2), we present the contribution of the structural disordering as a sum of two components – static structural disordering and dynamic structural disordering

(5)

thus the contribution of the temperature-independent static structural disordering is caused by the presence of various defects, impurities and inhomogeneities of the crystal structure. Thus, in Ref.[20] it is shown that the most probable defects in Sn₂P₂S_e6-type crystal lattice are the vacancies of Se atoms which result in the formation of deep levels in the band gap. The presence of the defects leads to the appearance of local non-uniform electrical fields which are revealed as the presence of a strong photorefractive effect and impurity-type photoconductivity[21, 22]. The contribution of the temperature-dependent dynamic structural disordering is caused by the presence of structure modulation in ferroics.

The values of these contributions were determined from Eq. (2), the parameters , and Einstein temperature being obtained from the temperature dependence of the Urbach energy . The calculations and further analysis showed that in the commensurate paraelectric (T>T_i) and ferroelectric (T<T_c) phases the temperature behaviour of is determined by the temperature behaviour of the contribution caused by lattice thermal vibrations, at a constant value of =const and in the absence of =0 (see Eq. (5)). In the incommensurate phase (T_c<T<T_i), with decreasing temperature the contribution of decreases; however, as the contribution increases and the contribution of remains constant, the combination of these factors leads to the temperature invariability of the Urbach energy =const (see Eq. (5), Fig. 2) and to the parallel red shift of the absorption edge (Fig. 1). Dynamic structural disordering in the incommensurate phase is caused by the appearance of a modulation wave formed by displacement of tin atoms in (010) symmetry plane along a direction close to[100] (See Ref.[23]). With the temperature variation, the changes of the amplitude and the period of the modulation wave are observed whereas the direction of the modulation wave vector remains unchanged[23].

Thus, the temperature range for which the condition ≠0 becomes true and the parallel red shift of the optical absorption edge as well the temperature invariance of the Urbach energy are observed, can be identified as the range of existence of the incommensurate phase in ferroics. The observed features of the absorption edge behaviour in the incommensurate phase of Sn₂P₂Se₆ crystal are characteristic properties of incommensurate superstructures, similarly to the invar effect[24] and anomalous hysteresis[25].

5. Conclusions

In Sn₂P₂Se₆ ferroelectric, the dynamic structural disordering of the crystal lattice, the parallel red shift of optical absorption edge and the temperature invariance of Urbach energy are observed in the incommensurate phase in the temperature range 193≤T≤221 K. The presence of the dynamic structural disordering in the incommensurate phase of Sn₂P₂Se₆ is caused by modulation wave, the parameters of which depend on temperature.