**15. Investigation by using the pseudopotential method**

The infinite crystal approximation, which states that the crystal properties obtained for the primary cell are extended then to the entire crystal employing periodical boundary conditions, has been employed. Hence, within the primary cell, the position of the *j*th atom of kind *α*(*α* = Sb, S, or I) is

$$\mathbf{R}\_{/}^{(a)} = \mathbf{R}\_{\rm coll} + \mathfrak{a}\_{/}^{(a)},\tag{33}$$

where **R**cell is the translation vector of the orthorhombic system. The vectors *τ <sup>j</sup>* **(***α***)** describe onprimitive translations that may be written as

$$\mathbf{r}\_{\rangle}^{a} = \lambda\_{\rangle}^{a}\mathbf{t}\_{x} + \mu\_{\rangle}^{a}\mathbf{t}\_{y} + \nu\_{\rangle}^{a}\mathbf{t}\_{z} \tag{34}$$

where *tx*, *ty*, and *t<sup>z</sup>* are the lattice parameters along the three coordinate axes. The pseudopo‐ tential of the crystal has been selected as a sum of atomic pseudopotentials:

**14. The energy gap of the SbSI crystals**

Investigation of the total density of states of SbSI crystals [41] has shown that the absolute valence band top is formed in both phases of 3p orbitals of S, while the absolute conduction

We have undertaken an attempt of a more detailed calculation of the electronic structure and some properties of SbSI from the first principles using the empirical pseudopotential method [42]. The method for calculating the band structure of SbSI was employed in Refs. [43–45]. In Ref. [43], purely ionic and partially covalent models were applied, and an indirect energy gap of 2.28 eV at point S was obtained. Nevertheless, the accuracy turned out to be 0.2 eV. In Ref. [44], the pseudopotentials were corrected applying the data on direct gaps. The absorption

*(C))*. The valence band top corresponded to *S*5-6

sides matched to points Z and R in the ferroelectric phase. Analyzing the light reflection

In Ref. [45], a purely ionic model was assumed. The form factors of the pseudopotential were adjusted by fitting the calculated band gap values to the ones obtained both experimentally

The infinite crystal approximation, which states that the crystal properties obtained for the primary cell are extended then to the entire crystal employing periodical boundary conditions, has been employed. Hence, within the primary cell, the position of the *j*th atom of kind *α*(*α* =

> ( ) ( ) cell , *j j*

α αα , *j jx jy jz*

 mn

*τ t tt* =++

where **R**cell is the translation vector of the orthorhombic system. The vectors *τ <sup>j</sup>*

 a

a

a

l

(*V* ) and *S*<sup>1</sup> (*V* ) *(V ) →S7-8*

, as well as the minimum direct

.

, whereas the conduc‐

(*V* )

(*C*) withthe energy of indirect gap of 1.41 eV. The energy gap

were used instead of I<sup>−</sup>

**R R**= + t (33)

(34)

*(C))* and

**(***α***)** describe

band edge in the paraelectric phase was determined at 1.82 eV for *E* || *c (S5-6*

**14.1. Introduction**

114 Piezoelectric Materials

1.91 eV for *E*﬩*c (S1-2*

Sb, S, or I) is

band bottom of 5p orbitals of Sb.

*(V ) →S7-8*

spectra, the indirect band gap of 1.82 eV between *Γ*<sup>6</sup>

and theoretically by other authors. Form factors for Cl<sup>−</sup>

**15. Investigation by using the pseudopotential method**

gap of 2.08 eV were obtained in reference [45].

onprimitive translations that may be written as

tion band edge complied with *Z*<sup>1</sup>

$$V(\mathbf{r}) = \mathbf{R}\_{\text{coll}} \sum\_{\alpha} \quad \sum\_{\text{cell}} \mathbf{R}\_{\text{cell}} \quad \sum\_{\text{cell}} \ \nu\_a \left( \mathbf{r} - \mathbf{R}\_{\text{cell}} - \mathbf{r}\_{\text{}}^{(a)} \right). \tag{35}$$

Here, *να*(**r**) is the atomic pseudopotential of atom *α*. It is assumed to be localized and energy independent. Weakness of the pseudopotential *V*ps(**r**) provides good convergence of the pseudo-wave functions expanded in terms of plane waves:

$$\Psi\_{\mathbf{k}}^{\mathrm{ps}}\left(\mathbf{r}\right) = \sum\_{\mathbf{C}} \mathbf{C}\_{\mathbf{k}}\left(\mathbf{G}\right) \mathbf{e}^{\mathbb{I}\left(\mathbf{k}\cdot\mathbf{G}\right)\mathbf{r}},\tag{36}$$

where **G** is the reciprocal lattice vector, *C***k***<sup>i</sup>* (**G**) are Fourier coefficients. The Schrödinger equation in the pseudopotential method has the form

$$\mathbb{E}\left(H+V^{\mathbb{P}^{\mathfrak{s}}}\right)\Psi^{\mathbb{P}^{\mathfrak{s}}}\_{\mathbb{k}l}=E\_{\mathbb{k}l}\Psi^{\mathbb{P}^{\mathfrak{s}}}\_{\mathbb{k}l}.\tag{37}$$

The pseudo-wave functions of the valence electrons *Ψ***k***<sup>i</sup>* ps(**r**) approach the true ones outside the core region, though they show no oscillations within the core. The Fourier component of this potential (or the matrix element in the plane-wave basis set) is expressed

$$V\left(\mathbf{G}\right) = \sum\_{a} S\_{a}\left(\mathbf{G}\right) \xleftarrow{\alpha} \nu\_{a}\left(\mathbf{G}\right),\tag{38}$$

where *Ωα* and *Ωc* are the volume of the corresponding atom and of the primary cell as a whole. The atomic structural factor *Sα* (**G**) and atomic form factor *να*(**G**) are defined as

$$S\_a\left(\mathbf{G}\right) = \sum\_a e^{-i\mathbf{G}\mathbf{r}\_\parallel(a)}\,,\tag{39}$$

$$\nu\_a \left( \mathbf{G} \right) = \frac{1}{\Omega\_a} \int \nu\_a(\mathbf{r}) e^{-i\mathbf{G}\mathbf{r}} \mathbf{d}^3 \mathbf{r} \tag{40}$$

The integrals are taken over the whole volume *Ωα*. Provided the atomic coordinates within the cell are known, the atomic structural factor can be readily evaluated. The pseudopotential forms factors for Sb, S, and I, so the following equation [39, 43] was preliminary determined to be used:

$$\nu\_{\alpha} \left( \mathbf{G} \right) = \frac{4\pi}{\omega \left| \mathbf{G} \right|^{-2}} \sum\_{nlm} nlm \left| \exp \left[ -i \left( \mathbf{G} \mathbf{r} \right) \right] \right| nlm. \tag{41}$$

where *nlm* denotes the set of electron quantum numbers. Neutral functions for Sb, S, and I were involved by the numerical evaluation of the form factors [43, 45, 46]. It is significant that the agreement between the theoretical and experimental data shall be obtained solely employing neutral functions. It is noteworthy that some other authors [47, 48], who theoretically studied the band structure of the SbSI crystal, employed an ionic model of chemical bonding (Sb+3 S−2 I−1) instead.

**Figure 23.** The scheme and notation of 27 special points in the irreducible part of the Brillouin zone.



( ) <sup>2</sup>

a

p

a

**No Point Coordinates**

n

116 Piezoelectric Materials

chemical bonding (Sb+3 S−2 I−1) instead.

<sup>4</sup> exp ( ) .

where *nlm* denotes the set of electron quantum numbers. Neutral functions for Sb, S, and I were involved by the numerical evaluation of the form factors [43, 45, 46]. It is significant that the agreement between the theoretical and experimental data shall be obtained solely employing neutral functions. It is noteworthy that some other authors [47, 48], who theoretically studied the band structure of the SbSI crystal, employed an ionic model of

**<sup>G</sup>** (41)

*kx ky kz*

*nlm*

**Figure 23.** The scheme and notation of 27 special points in the irreducible part of the Brillouin zone.

1 Γ 0 0 0 Δ 0 0.25 0 Y 0 0.5 0 Λ 0 0 0.25 ZTY Γ 0 0.25 0.25 H 0 0.5 0.25 Z 0 0 0.5 B 0 0.25 0.5 T 0 0.5 0.5 Σ 0.25 0 0 ΓYSX 0.25 0.25 0 C 0.25 0.5 0 ΓXUZ 0.25 0 0.25 O 0.25 0.25 0.25 TYSR 0.25 0.5 0.25


*nlm i nlm*

**Table 12.** Coordinates of the special points in the irreducible part of the Brillouin zone of SbSI (in relative units), see also **Figure 23**.

**Figure 24.** A diagram of the forbidden gap of SbSI in antiferroelectric phase at 308 K and its change after transition to ferroelectric phase (at 278 K). *E*<sup>g</sup> antiferro in the indirect transition energy (1.42 eV) in antiferroelectric phase between points *U* and *Z* of the irreducible part of the Brillouin zone. *E*<sup>g</sup> antiferro is the indirect transition energy (1.36 eV) in fer‐ roelectric phase between points *R* and *Z* of the irreducible part of the Brillouin zone. The direct gap takes its minimum value at point *U* for both phases (a) and (b) correspond to different directions in the crystal.

The chemical bond in SbSI is of mixed kind, with contributions of both ionic and covalent components. As we have demonstrated in Ref. [43], it may be described by an approximate model formula Sb+0.3S−0.2I−0.1. The band structure for both phases was estimated in 27 points of the irreducible part of the eightfold Brillouin zone, which is the total of 216 points over the Brillouin zone. The points are schematically described in **Figure 23**, as well as their coordinates are provided in **Table 12**. A total of 600 plane waves were included in the basis set for the calculation. The experimental energy gap values were estimated from the exponential light absorption tail at ln *K* = 6, where *K* represented the absorption coefficient [2]. As it is seen from **Figure 24**, the most significant changes in the valence band at the phase transition occur at points *Q* and *C* (energy variation was 0.92 and 0.86 eV, respectively).

Moreover, the changes at points *R* (0.37 eV), *H* (0.55 eV), and *E* (0.42 eV) should be noted. As far as the conduction band is concerned, similar significant changes occur at points *H* (0.53 eV) and *E* (0.51 eV). At all the remaining points of the Brillouin zone, the band gap profile has changed insignificantly. All these changes have only a slight effect on the main characteristics of the band structure (except point *R*, obviously). As it is seen from **Figure 24**, the SbSI crystal has an indirect forbidden gap both in antiferroelectric phase and in ferroelectric phase [42]. The conduction band bottom in both phases is located at point *Z*<sup>1</sup> (*C*) , the valence band top in antiferroelectric phase is at the point *U*5-<sup>6</sup> (*V* ) , and in ferroelectric phase at *R*3-4 (*V* ) . Our experimental and theoretical results of the SbSI crystal electronic structure sometimes differ from those obtained by other authors.
