**1. Potential energy of normal vibrational modes**

#### **1.1. Introduction**

SbSI has a classical ferroelectric phase (FEF) transition at the temperature *T*C = 295 K [1, 2]. This crystal has large interest to the fundamental physical properties as well as for the promising application possibilities. It is known that it has large temperature and electric field depend‐ ence of the optical characteristics [3], electronic structure [4, 5], and other features. Petzelt [6] observed in the *IR* reflectivity spectra a strong temperature dependence of the lowest-frequen‐ cy dielectric constant. Sugawara et al. [7] has determined strong temperature dependence of reflectivity for *E*||*c* in the range of 0–70 cm−1. According to the author [7, 8], the frequencies of the soft mode are 4.0 and 6.5 cm−1 at temperatures 298 and 318 K in the *IR* range of SbSI spectrum. Although some authors [9] demonstrate result n (≈90%) of the frequency ≈ 9 cm−1 to the static dielectric constant at the room temperature, others claim the soft mode with frequency 10 cm −1 to be insignificant [10]. The soft mode consisting of two components has been found in the SbSI [11], SbSI-SbSeI [12], SbSI-BiSI [13], and SbSI-SbSBr systems [14] in the microwave range. The first component is a soft microwave mode bringing the major contribution in , and the second is a semisoft mode in the *IR* range and contributes to , which is less than 10% [15].

### **2. Symmetrical and normal coordinates**

In [16] we have described the symmetrical and normal coordinates which were used to calculate electronic potential. As known, in the adiabatic approximation the vibration energy *E* = *T* + *V* of a system of atoms can be expressed via normal coordinates *Q* as follows:

$$T = \frac{1}{2} \sum\_{k} \left( \frac{dQ\_k}{dt} \right)^2,\\ V = \frac{1}{2} \sum\_{k} \lambda\_k Q\_k^2,\tag{1}$$

The force constants *λk* are solution of the characteristic equation

$$\left|\lambda a\_{ij} - b\_{ij}\right| = 0 \tag{2}$$

where *aij* and *bij* are factors of the energy expression in orthogonal symmetry coordinates *Fi* :

$$T = \frac{1}{2} \sum\_{\langle \rangle} a\_{\langle \rangle} \frac{dF\_i dF\_j}{dt}, \\ V = \frac{1}{2} \sum\_{\langle \rangle} b\_{\langle \rangle} F\_i F\_j. \tag{3}$$

Symmetry coordinates are defined as particular combinations of Cartesian components *xi* , *yi* , and *zi* of displacements [17]. Similarly to the vibrations of Sb2S3, the ones of SbSI belong to the same irreducible representations Γ*a*, since the space groups of both crystals are *D*2*<sup>h</sup>* 16 in the


paraelectric phase and *C*2*<sup>ν</sup>* <sup>9</sup> in the ferroelectric phase. Symmetry operations of the space group *D*2*<sup>h</sup>* 16 are *S*1÷*S*8 [18, 19]. Some of them, such as *S*1, *S*3, *S*6, and *S*8, are connected with the space group *C*2*<sup>ν</sup>* <sup>9</sup> . The effect of these symmetry operations is the same in the unit cell that consists of four molecules (SbSI)4 as in the Sb2S3 unit cell [20] (**Table 1**).

**Table 1.** Symmetry operations of the space groups *D*2*<sup>h</sup>* 16 and *C*2*<sup>ν</sup>* <sup>9</sup> i n SbSI crystals.

The operator of projection is

**1. Potential energy of normal vibrational modes**

**2. Symmetrical and normal coordinates**

SbSI has a classical ferroelectric phase (FEF) transition at the temperature *T*C = 295 K [1, 2]. This crystal has large interest to the fundamental physical properties as well as for the promising application possibilities. It is known that it has large temperature and electric field depend‐ ence of the optical characteristics [3], electronic structure [4, 5], and other features. Petzelt [6] observed in the *IR* reflectivity spectra a strong temperature dependence of the lowest-frequen‐ cy dielectric constant. Sugawara et al. [7] has determined strong temperature dependence of reflectivity for *E*||*c* in the range of 0–70 cm−1. According to the author [7, 8], the frequencies of the soft mode are 4.0 and 6.5 cm−1 at temperatures 298 and 318 K in the *IR* range of SbSI spectrum. Although some authors [9] demonstrate result n (≈90%) of the frequency ≈ 9 cm−1 to the static dielectric constant at the room temperature, others claim the soft mode with frequency 10 cm −1 to be insignificant [10]. The soft mode consisting of two components has been found in the SbSI [11], SbSI-SbSeI [12], SbSI-BiSI [13], and SbSI-SbSBr systems [14] in the microwave range. The first component is a soft microwave mode bringing the major contribution in , and the second is a semisoft mode in the *IR* range and contributes to , which is less than 10% [15].

In [16] we have described the symmetrical and normal coordinates which were used to calculate electronic potential. As known, in the adiabatic approximation the vibration energy

*k k*

è ø å å (1)

*a b* - = (2)

*dt* = = å å (3)

:

, *yi* ,

16 in the

l

*E* = *T* + *V* of a system of atoms can be expressed via normal coordinates *Q* as follows:

2 1 <sup>2</sup> 2 2 , <sup>1</sup> , *<sup>k</sup>*

æ ö = = ç ÷

The force constants *λk* are solution of the characteristic equation

*k k dQ T VQ dt*

0 *ij ij*

where *aij* and *bij* are factors of the energy expression in orthogonal symmetry coordinates *Fi*

1 1 , . 2 2 *i j*

Symmetry coordinates are defined as particular combinations of Cartesian components *xi*

same irreducible representations Γ*a*, since the space groups of both crystals are *D*2*<sup>h</sup>*

and *zi* of displacements [17]. Similarly to the vibrations of Sb2S3, the ones of SbSI belong to the

*dF dF T a V b FF*

*ij ij i j ij ij*

l

**1.1. Introduction**

84 Piezoelectric Materials

$$
\varepsilon\_{\boldsymbol{\eta}}^{\boldsymbol{a}} = \frac{\boldsymbol{f}\_{\boldsymbol{a}}}{\mathbf{g}} \sum\_{k} \Gamma\_{\boldsymbol{a}} \left( \boldsymbol{S}\_{k} \right)\_{\boldsymbol{\eta}} \boldsymbol{S}\_{k} \,. \tag{4}
$$

The Cartesian components *KAn* = *xAn*, *yAn*, *zAn* of atom displacements for all their linear combinations have certain symmetry properties with respect to operations of groups *D*2*<sup>h</sup>* 16 and *C*2*<sup>ν</sup>* 9 :

$$K\_{\mathcal{A}m}^{S}\left(\Gamma\_{\alpha}\right) = \sum\_{m} a\_{nm} K\_{\mathcal{A}n}.\tag{5}$$

Here *A* = Sb, *S*, *I*; *n* = 1, 2, 3, 4; *anm* = ±1. These combinations of *KAn* are basis functions of Γ*a*.

Group *D*2*<sup>h</sup>* <sup>16</sup> has irreducible representations *Aig*(*u*) , *Big*(*u*) . Here, the capital letters denote coordi‐ nates symmetric (*A*) or antisymmetric (*B*) with respect to all rotation *C*<sup>2</sup> *z* , *C*<sup>2</sup> *y* , and*C*<sup>2</sup> *x* , **Table 2** [21]. Subscripts denote the coordinates symmetric (*g*), or antisymmetric (*u*) with respect to inversion *I*. The coordinates symmetric with respect to separate rotations around *x*, *y*, and *z* axes differ from each other by index *i* = 1, 2, 3. The obtained *KAn* combinations are also the basic functions of irreducible representations *Ai* , *Bi* of the space group *C*2*<sup>ν</sup>* 9 .


**Table 2.** Group characters of irreducible representations of space groups *D*2*<sup>h</sup>* 16 and *C*2*<sup>ν</sup>* 9 [21].

Orthogonal symmetry coordinates of the entire unit cell can be formed of *KAm <sup>S</sup>* (*Γα*) by solving equations of orthonormality and are uniform with respect to *A*. Their expressions are as follows:

$$\left(F\_{\mathbb{S}}\left(\mathcal{K},\Gamma\_{\boldsymbol{\alpha}}\right) = \mathcal{N}\_{\mathbb{S}} \sum\_{A} \sum\_{m} \mathbb{C}\_{m} \mathcal{K}\_{Am}^{(\mathcal{S})} \left(\Gamma\_{\boldsymbol{\alpha}}\right) \tag{6}$$

Normalization factors *Ns* and coefficients *Cn* are presented in *z* direction and *x*, *y* directions in **Table 3** [22]. The group theoretical analysis reveals that they are the unit cell of *D*2*<sup>h</sup>* 16 point group with 12 atoms per cell and contains 33 optical modes and 3 acoustic modes. Among the optical modes, there are 18 Raman-active modes with the symmetries 6*Ag* + 6*B*1*<sup>g</sup>* + 3*B*2*<sup>g</sup>* + 3*B*3*<sup>g</sup>*, 3 silent modes belong to the *Au* irreducible representation, as well as 12 infrared active modes that have symmetries 2*B*1*<sup>u</sup>* + 5*B*2*<sup>u</sup>* + 5*B*3*<sup>u</sup>*. In *z* direction, only 2*B*1*<sup>u</sup>* modes are infrared active in the paraelectric phase (**Table 3**), whereas in the same direction, the modes 2*A*1 + 3*A*2 + 3*B*1 + 3*B*<sup>2</sup> are Raman active in the ferroelectric phase. Note that 2*A*1 + 3*B*1 + 3*B*<sup>2</sup> are also infrared active among these modes.

Taking into account the fact that SbSI crystal is made of two double chains with a weak interaction between them at *T* = 0 K (20–30 times weaker than within a chain; see **Figure 1**), the formation of dynamical matrix 3 becomes easier.

Group *D*2*<sup>h</sup>*

86 Piezoelectric Materials

*D***2***<sup>h</sup>*

follows:

among these modes.

<sup>16</sup> has irreducible representations *Aig*(*u*)

functions of irreducible representations *Ai*

*C***2***<sup>ν</sup>*

nates symmetric (*A*) or antisymmetric (*B*) with respect to all rotation *C*<sup>2</sup>

**<sup>E</sup>** *<sup>C</sup>***<sup>2</sup>**

**Table 2.** Group characters of irreducible representations of space groups *D*2*<sup>h</sup>*

, *Big*(*u*)

of the space group *C*2*<sup>ν</sup>*

*<sup>z</sup> <sup>C</sup>***<sup>2</sup>**

[21]. Subscripts denote the coordinates symmetric (*g*), or antisymmetric (*u*) with respect to inversion *I*. The coordinates symmetric with respect to separate rotations around *x*, *y*, and *z* axes differ from each other by index *i* = 1, 2, 3. The obtained *KAn* combinations are also the basic

, *Bi*

*<sup>y</sup> <sup>C</sup>***<sup>2</sup>**

r 1 A1g r 1 A1 1 1 1 1 1 1 1 1 r 2 A1u r 2 A2 1 1 1 1 −1 −1 −1 −1 r 3 B2g r 3 B1 1 −1 1 −1 1 −1 1 −1 r 4 B2u r 4 B2 1 −1 1 −1 −1 1 −1 1 r 5 B1g r 2 A2 1 1 −1 −1 1 1 −1 −1 r 6 B1u r 1 A1 1 1 −1 −1 −1 −1 1 1 r 7 B3g r 4 B2 1 −1 −1 1 1 −1 −1 1 r 8 B3u r 3 B1 1 −1 −1 1 −1 1 1 −1

Orthogonal symmetry coordinates of the entire unit cell can be formed of *KAm*

equations of orthonormality and are uniform with respect to *A*. Their expressions are as

( ) ( ) ( )

**Table 3** [22]. The group theoretical analysis reveals that they are the unit cell of *D*2*<sup>h</sup>*

 , Γ = ∑∑ <sup>Γ</sup> *<sup>S</sup> S a S ms Am A m*

Normalization factors *Ns* and coefficients *Cn* are presented in *z* direction and *x*, *y* directions in

with 12 atoms per cell and contains 33 optical modes and 3 acoustic modes. Among the optical modes, there are 18 Raman-active modes with the symmetries 6*Ag* + 6*B*1*<sup>g</sup>* + 3*B*2*<sup>g</sup>* + 3*B*3*<sup>g</sup>*, 3 silent modes belong to the *Au* irreducible representation, as well as 12 infrared active modes that have symmetries 2*B*1*<sup>u</sup>* + 5*B*2*<sup>u</sup>* + 5*B*3*<sup>u</sup>*. In *z* direction, only 2*B*1*<sup>u</sup>* modes are infrared active in the paraelectric phase (**Table 3**), whereas in the same direction, the modes 2*A*1 + 3*A*2 + 3*B*1 + 3*B*<sup>2</sup> are Raman active in the ferroelectric phase. Note that 2*A*1 + 3*B*1 + 3*B*<sup>2</sup> are also infrared active

**16 S1 S2 S3 S4 S5 S6 S7 S8**

**9 S1 S3 S6 S8**

. Here, the capital letters denote coordi‐

, **Table 2**

*z* , *C*<sup>2</sup> *y* , and*C*<sup>2</sup> *x*

*x* **I σ <sup>y</sup> σ <sup>z</sup> σ<sup>x</sup> σ**

9 .

16 and *C*2*<sup>ν</sup>*

α

*FK N CK* (6)

9 [21].

*<sup>S</sup>* (*Γα*) by solving

16 point group


**Table 3.** Symmetry coordinates of normal modes of SbSI crystals in *z*(*c*) direction.

In our research, force constants for a long chain of strongly bound atoms in the direction of *z*(*c*)-axis were defined by composing five simplified unit cells (30 atoms). The GAMESS program in the basis sets of atomic functions 3*G*, 3*G* + *d*, 21*G*, 21*G* + *d* were used to calculate force constants.

When atoms are vibrating, interaction between double chains strengthens. Therefore, follow‐ ing Furman et al. [23], we calculated normal models employing the Born-von Karman model and the elementary cell of four pseudomolecules SbSI (12 atoms). The binding constants and ionic charges, 0.19 (Sb), −0.01 (*S*), −0.18 (*I*), were also calculated. It appeared that covalent interactions prevailed in the SbSI crystal.

**Figure 1.** Locations of atoms in the unit cell of SbSI in the *x*–*y* plane. Dotted lines restrict the simplified unit cell, as formed by the central group of atoms Sb3, Sb4, S3, S4, I1, and I2.
