**12. Investigation of SbSI crystal valence band**

The paraelectric structure (*T* > *TC2* = 410 K) of SbSI is made of atom chains that belong to the paraelectric space group Pnam, which form square-pyramidal S3I2 groups with the Sb ion at the center of the pyramid base. All atoms lie on mirror planes normal to the *c*-axis (**Figure 15**).

**Figure 15.** Crystal structure of SbSI projected on the **a**–**b** plane (a), and one double chain of molecular cluster in projec‐ tion on the **a**–**c** plane (b).

On passage into the AFEP (*TC*2 > *T* > *TC*1 = 295 K) and FEP (*T* < *TC*1), the position parameters normal to the *c*-axis are substantially unchanged. In AFEP (*T* = 300 K) the displacements of equilibrium position occur for all Sb3 atoms along the *c*-axis *z*0 = 0, 02 Å and for all Sb4 atoms (*z*0 = −0, 02 Å) [37]. In FEP (*T* < *TC*1) all Sb and S atoms move along the *c*-axis toward these I sites, which leads to the removal of the mirror plane symmetry. In FEP (*T* = 215 K), the displacements of equilibrium position occur for all Sb atoms along the *c*-axis *z*0 = 0, 2 Å.

The molecular cluster model of one SbSI crystal chain was used to perform the theoretical *ab initio* calculation of energy levels (**Figure 15**). As seen from **Figure 16**, energy levels only slightly change with the increase of the cluster. **Figure 17** shows that the energy of some levels increases and of others decreases.

**Figure 16.** Dependence of energy levels of VB on the number *N* of SbSI molecules in a molecular cluster in AFEP (300 K).

**Figure 17.** Dependence of energy levels of VB on equilibrium positions *z*0 of Sb atoms in SbSI molecular cluster. The displacements of equilibrium position *z*0 of Sb atoms: for all Sb3 atoms in AFEP (300 K), *z*0 = 0, 02 Å, and for all Sb4 atoms *z*0 = −0, 02 Å; *z*0 = 0, 2 Å, for all Sb3 and Sb4 atoms in FEP (215 K).

We have calculated the bond orders between the atoms Sb3-S4, Sb4-S3, Sb3-I1, and Sb4-I2 along a cluster in AFEP and FEP (**Figures 18** and **19**). When the phase transition takes place, bond orders change. As a consequence, VB experiences the broadening of some energy levels and the narrowing of others.

(*z*0 = −0, 02 Å) [37]. In FEP (*T* < *TC*1) all Sb and S atoms move along the *c*-axis toward these I sites, which leads to the removal of the mirror plane symmetry. In FEP (*T* = 215 K), the displacements of equilibrium position occur for all Sb atoms along the *c*-axis *z*0 = 0, 2 Å.

The molecular cluster model of one SbSI crystal chain was used to perform the theoretical *ab initio* calculation of energy levels (**Figure 15**). As seen from **Figure 16**, energy levels only slightly change with the increase of the cluster. **Figure 17** shows that the energy of some levels increases

**Figure 16.** Dependence of energy levels of VB on the number *N* of SbSI molecules in a molecular cluster in AFEP

**Figure 17.** Dependence of energy levels of VB on equilibrium positions *z*0 of Sb atoms in SbSI molecular cluster. The displacements of equilibrium position *z*0 of Sb atoms: for all Sb3 atoms in AFEP (300 K), *z*0 = 0, 02 Å, and for all Sb4

atoms *z*0 = −0, 02 Å; *z*0 = 0, 2 Å, for all Sb3 and Sb4 atoms in FEP (215 K).

and of others decreases.

108 Piezoelectric Materials

(300 K).

**Figure 18.** Bond orders between the atoms Sb3–S4 (a) and Sb4–S3 (b) along the SbSI molecular cluster in AFEP and FEP.

**Figure 19.** Bond orders between the atoms Sb3–I1 (a) and Sb4–I2 (b) along the SbSI molecular cluster in AFEP and FEP.

While the temperature decreases further in the FEP, Sb and S atoms change their positions relative to I atoms, and VB becomes broader. In Grigas et al. [39], the authors present the Xray photoelectron spectra (XPS) of the VB and CL of the SbSI single crystals. The XPS are measured in the energy range 0–1400 eV and the temperature range 130–330 K. They compared experimentally obtained energies of CL with the results of theoretical *ab initio* calculations. However, the experimentally obtained energies of the VB were not compared with the results of theoretical *ab initio* calculations.

We have calculated the density of states of VB for Sb, S, and I atoms and the total molecular cluster in AFEP (*T* = 300 K) and FEP (*T* = 215 K). The density of states is

$$\mathbf{g} = \frac{\Delta \mathbf{N}}{\Delta E}'$$

where Δ*N* is the number of states in the energy interval Δ*E* (eV). The dimension of *g* is (eV−1). In **Figure 20**, the experimental results of XPS VB spectra of SbSI crystals are compared with the theoretically calculated total density of states of a molecular cluster that consists of 20 SbSI molecules in AFEP (*T* = 300 K) and FEP (*T* = 215 K) is the number of states in the energy interval Δ*E* (eV). The dimension of *g* is (eV−1). In **Figure 19**, the experimental results of XPS VB spectra of SbSI crystals are compared with the theoretically calculated total density of states of a molecular cluster that consists of 20 SbSI molecules in AFEP (*T* = 300 K) and FEP (*T* = 215 K). The average total density of states is as follows:

$$ = \frac{1}{n} \sum\_{i} {}^{n} \mathcal{g}\_{i'} $$

**Figure 20.** Theoretical total density of states of SbSI molecular cluster and density of states of Sb, S, and I atoms in AFEP and FEP.

where *n* is the number of normal modes (*n* = 15).

As seen from **Figures 20** and **21**, the experimental and theoretical results in AFEP and FEP are in good agreement in the energy range of 6–17 eV. In the range of 17–22 eV, there is a great difference between experimental and theoretical results.

measured in the energy range 0–1400 eV and the temperature range 130–330 K. They compared experimentally obtained energies of CL with the results of theoretical *ab initio* calculations. However, the experimentally obtained energies of the VB were not compared with the results

We have calculated the density of states of VB for Sb, S, and I atoms and the total molecular

<sup>D</sup> <sup>=</sup> <sup>D</sup> , *<sup>N</sup> <sup>g</sup> <sup>E</sup>*

where Δ*N* is the number of states in the energy interval Δ*E* (eV). The dimension of *g* is (eV−1). In **Figure 20**, the experimental results of XPS VB spectra of SbSI crystals are compared with the theoretically calculated total density of states of a molecular cluster that consists of 20 SbSI molecules in AFEP (*T* = 300 K) and FEP (*T* = 215 K) is the number of states in the energy interval Δ*E* (eV). The dimension of *g* is (eV−1). In **Figure 19**, the experimental results of XPS VB spectra of SbSI crystals are compared with the theoretically calculated total density of states of a molecular cluster that consists of 20 SbSI molecules in AFEP (*T* = 300 K) and FEP (*T* = 215 K).

< >= å

**Figure 20.** Theoretical total density of states of SbSI molecular cluster and density of states of Sb, S, and I atoms in

<sup>1</sup> , *<sup>n</sup> <sup>i</sup> <sup>i</sup> g g n*

cluster in AFEP (*T* = 300 K) and FEP (*T* = 215 K). The density of states is

of theoretical *ab initio* calculations.

110 Piezoelectric Materials

The average total density of states is as follows:

where *n* is the number of normal modes (*n* = 15).

AFEP and FEP.

**Figure 21.** XPS of the VB of SbSI crystals in AFEP and FEP [13] (a). Theoretical total density of states (*g*) of SbSI molecu‐ lar cluster in AFEP and FEP (b). The averaged total density of states (*g*) at molecular cluster when all atoms participate of oscillations of all normal modes (c).

The peaks of **Figures 21(a)** and **(b)** have no distinct shape in the XPS (at *T* = 300 K). This difference is explicable because the theoretical calculation does not take into account vibra‐ tional displacements of atoms. Average densities of states shown in **Figure 21(c)** are more similar to the experimental XPS spectrum than those from **Figure 21(b)**. Comparison of **Figures 20** and **21** shows the contribution of atomic states to the total density of states.
