**3. The lattice anharmonism on the vibrational spectrum along the** *z***(***c***) direction**

In [24] we have proposed the method for calculating the total potential energy. The potential energy (PE) is defined as follows:

$$V\left(\mathbf{r}\right) = \frac{4\pi}{\Omega} \sum\_{as} \left|\mathbf{s}\right|^{-2} f\_a\left(\mathbf{s}\right) \exp\left[-i\left(\mathbf{r} + \mathbf{R}\_a + \mathbf{Q}\_a\right) \cdot \mathbf{s}\right] \exp\left[-M\_a\left(\mathbf{s}\right)\right],\tag{7}$$

where Ω stands for the volume of the unit cell, *Q* represents the normal coordinate, *R*<sup>α</sup> denotes the radius-vector of the atom position in the unit cell, *s* means the reciprocal lattice vector, *α* represents the number and kind of the atom in the unit cell, and exp[−*Mα*(*s*)] represents the Debye-Valler factor that is determined by the mean square amplitudes of atomic displacements (i.e., by the crystal temperature).

In fact, Eq. (7) at a certain point in the unit cell is the PE in terms of normal coordinates (or symmetrized plane waves). The atomic scattering form factor is

Electronic Structure and Piezoelectric Properties of SbSI Crystals http://dx.doi.org/10.5772/64223 89

$$\ln f\_a\left(\mathbf{s}\right) = \sum\_{nlm} nlm \left| \exp\left[\right.\left.i\left(\mathbf{r}\cdot\mathbf{s}\right)\right] nlm\right|,\tag{8}$$

where *nlm* represents a set of quantum numbers for the atom *α*. It is noteworthy that the functions *< nlm* of all electronic states of an atom was used for calculating the form factors *f<sup>α</sup>* (*s*) by Eq. (8). About 5000 vectors *s* were used for the sum in Eq. (7).

For SbSI, we studied the dependence of potential energy *V*(*z*) in the paraelectric phase at *T* = 420 K on the normal coordinates of all *D*2*<sup>h</sup>* 16 symmetry normal modes that formed by displacements of the atoms along the *z* direction (along the polar axis *c*; **Figure 2**). This dependence was calculated by Eqs. (7) and (8). We determined that the curves *V*(*z*) can be described by the equation

$$V\left(z\right) = V\_0 + az + bz^2 + dz^3 + cz^4,\tag{9}$$

**Figure 1.** Locations of atoms in the unit cell of SbSI in the *x*–*y* plane. Dotted lines restrict the simplified unit cell, as

**3. The lattice anharmonism on the vibrational spectrum along the** *z***(***c***)**

( ) ( ) ( ) ( ) <sup>4</sup> <sup>2</sup>

*V fi r s s rR s s Q M*

= é- + + × ù é- ù <sup>ë</sup> ûë û <sup>W</sup> å*<sup>s</sup>*

where Ω stands for the volume of the unit cell, *Q* represents the normal coordinate, *R*<sup>α</sup> denotes the radius-vector of the atom position in the unit cell, *s* means the reciprocal lattice vector, *α* represents the number and kind of the atom in the unit cell, and exp[−*Mα*(*s*)] represents the Debye-Valler factor that is determined by the mean square amplitudes of atomic displacements

In fact, Eq. (7) at a certain point in the unit cell is the PE in terms of normal coordinates (or

a

symmetrized plane waves). The atomic scattering form factor is

a

p-

In [24] we have proposed the method for calculating the total potential energy. The potential

a a

exp exp ,

a

(7)

formed by the central group of atoms Sb3, Sb4, S3, S4, I1, and I2.

**direction**

88 Piezoelectric Materials

energy (PE) is defined as follows:

(i.e., by the crystal temperature).

**Figure 2.** Dependence of the potential energy *V*(*z*) (*T* = 415 K) upon the normal coordinates in the *z*(*c*) direction for *D*2*<sup>h</sup>* 16 symmetry *B*1*<sup>u</sup>*(2), *B*1*<sup>u</sup>*(3), *B*2*<sup>g</sup>*(4), *B*2*<sup>g</sup>*(5), *B*2*<sup>g</sup>*(6), *B*3*<sup>g</sup>*(7), *B*3*<sup>g</sup>*(8), *B*3*<sup>g</sup>*(9), *Au*(10), *Au*(11), and *Au*(12) normal modes.

where the coefficients *a* = *d* = 0 in the paraelectric phase. The displacements have been chosen in regard to the equilibrium positions of atoms. The curves *V*(*z*) of symmetry *B*1*<sup>u</sup>*(3), *B*2*<sup>g</sup>*(4, 5), *B*3*<sup>g</sup>*(7), and *Au*(10) are correspondent to coefficients *b <* 0 and *c >* 0. These curves *V*(*z*) have two side minima at distances of 0.3 a.u. from the equilibrium position. A potential barrier Δ*V* = *b*<sup>2</sup> */* 4*c* separates the minima. Conversely, the curves *V*(*z*) of symmetry *B*l*<sup>u</sup>*(2), *B*2*<sup>g</sup>*(6), *B*3*<sup>g</sup>*(8), *B*3*<sup>g</sup>*(9), *Au*(11), and *Au* (12) have a single minima and the coefficients *b >* 0. The coefficient is *b* ≈ *Mω*<sup>2</sup> , where *M* represents the reduced mass of the normal mode atoms, and *ω* denotes the vibration frequency of the normal mode. The frequency of a normal mode with a double-well *V*(*z*) is dependent on the height of the potential barrier Δ*V*. Temperature, electric field, pressure, and fluctuations of atoms in the *x*–*y* plane influence this potential barrier [15]. In all the three phases, *V*(*x*) and *V*(*y*) appear to be single-well with the coefficients *b >* 0 in Eq. (9). In the paraelectric phase, *a* = *d* = 0, whereas in the antiferro- and ferroelectric phases, *a* ≠ 0 and *d* ≠ 0. Single-well PE turns out to be weakly anharmonic and the frequencies of the *R*(*k*) peak should only slightly depend on the temperature.

The experimental results of *IR* reflectivity *R*(*k*) spectrum of SbSI crystals in the paraelectric (*T* = 415 K) and ferroelectric (*T* = 273 K) phases for *E*||*c* are presented in **Figure 3**. The reflectivity measurements of SbSI crystals have been repeated by Dr. Markus Goeppert in Germany in Karlsruhe University using the Bruker Fourier spectrometer.

**Figure 3.** The reflectivity spectra of SbSI for *E*||*c* in the paraelectric phase (*T* = 415 K) and in the ferroelectric phase (*T* = 273 K). Inset: the reflectivity spectra in range 50–95 cm−1.

In the reflectivity spectrum for *E*||*c* in the range of *k* = 10–100 cm−1, the number of peaks *R*(*k*) is equal for the paraelectric and ferroelectric phases. However, in the range of *k* = 100–200 cm −1, the number of peaks *R*(*k*) differs in both phases. Anharmonic modes with double-well *V*(*z*) are highly sensitive to dislocations, impurities, and fluctuations of chains in the *x*–*y* plane.

**Figure 2** demonstrates that modes *B*1*<sup>u</sup>*(3), *B*2*<sup>g</sup>*(4), *B*2*<sup>g</sup>*(5), and *B*3*<sup>g</sup>*(7) with a double-well *V*(*z*) are strongly anharmonic, whereas modes *B*1*<sup>u</sup>*(2), *B*2*<sup>g</sup>*(6), *B*3*<sup>g</sup>*(8), and *B*3*<sup>g</sup>*(9) with a single-well *V*(*z*) appear to be weakly anharmonic. The vibration frequencies of the former modes are lower compared to the latter ones. Therefore, the *R*(*k*) peaks are created by strongly anharmonic modes *B*1*<sup>u</sup>*(3) → *A***1**, *B*2*<sup>g</sup>*(4) → *B***1**, *B*2*<sup>g</sup>*(5) → *B***1**, and *B*3*<sup>g</sup>*(7) → *B***<sup>2</sup>** in the range of *k* = 10–100 cm−1 of the *IR* spectrum. In the range of *k* = 100–400 cm−1, the *R*(*k*) peaks are created by weakly anharmonic modes. Following the optical selection rules, one *R*(*k*) peak in the paraelectric phase is created by the mode *B*1*<sup>u</sup>*(2), while the peaks in the ferroelectric phase are created by modes *B*1*<sup>u</sup>*(2) → *A***1**, *B*2*<sup>g</sup>*(6) → *B***1**, *B*3*<sup>g</sup>*(8) → *B***2**, and *B*3*<sup>g</sup>*(9) → *B***2**. In the paraelectric phase, there are three silent modes of *Au* symmetry: *Au*(10), *Au*(11), and *Au*(12) (**Table 3**). Two of them, *Au*(11) and *Au*(12), comply with the out-of-phase motion of the infrared-active modes. It is expected that the anharmonic mode *Au*(10) has a low frequency. This mode can be optically active and may cause a weak peak in the *k <*10 cm−1 *IR* range.
