**16. Band structure at the first-order phase transition**

Direct (Brillouin zone point *U*) and indirect (jump *U*→*Z*) dependence of the forbidden band on the temperature that has been calculated by us is demonstrated in **Figure 25**. As shown in **Figures 25** and **26**, the width of the indirect forbidden band is 1.42 eV in antiferroelectric phase and 1.36 eV in ferroelectric phase. The following results match well to the results that have been obtained earlier [49]: 1.42 eV in antiferroelectric and 1.36 eV in ferroelectric phase. The indirect width of the forbidden band is 1.36 eV in that band structure, which was obtained by moving to the point of phase transition from ferroelectric phase point B1; and it is 1.41 eV when moving from antiferroelectric phase point A1. Hence, the indirect forbidden band moving from antiferroelectric to ferroelectric phase alters by 0.05 eV, according to our calculations. This number complies with the change 0.06 eV of the width of the indirect band, which has been experimentally set [48, 49]. **Figures 3** and **4** reveal that the direct forbidden band is narrowest in point U, 1.83 eV. The calculations presented in Ref. [49] demonstrate the value of 1.82 eV. The minimal direct forbidden band in ferroelectric phase occurs in point G, 1.94 eV, whereas this gap is 1.98 eV in point U. The narrowest forbidden band is in point U of Brillouin zone, approaching from both antiferroelectric and ferroelectric sides. This has been estimated in band structure in the area of phase transition (**Figures 26b** and **c**). In the first case, it turned out to be1.83 eV, point A2 (**Figure 25**), and in the second one it is 1.87 eV, point B2 [43] (Figure 25). For this reason, the observed jump of the direct forbidden band is 0.04 eV during the first-order phase transition. The change of the grating parameter along axis *c*(*y*), which affects the band structure of SbSI crystal, causes this jump. It is possible to find the width of the forbidden direct band in Brillouin zone point U in Ref. [49, 50], which is experimentally measured. Unfortunately, the SbSI exponential edge of absorption has not been taken into account. The exponential edge of absorption is explored in greater detail in antiferroelectric and ferroelectric phases and in the area of the temperatures in the phase transition in Ref. [34]. The edge of the absorption in antiferroelectric phase complies with the Urbach's rule:

$$K = K\_0 \exp\left[\frac{\sigma (E\_\mathrm{k} - E\_0)}{kT}\right],\tag{42}$$

**Figure 25.** We got the direct (2) and the indirect (1) forbidden band width's dependence on temperature. The brake of the curve in points *A*1 and *A*2 is got when moving to phase transition from antiferroelectric phase, and in points *B*1 and *B*2 when moving to phase transition from ferroelectric phase.

here

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

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].

and theoretical results of the SbSI crystal electronic structure sometimes differ from those

Direct (Brillouin zone point *U*) and indirect (jump *U*→*Z*) dependence of the forbidden band on the temperature that has been calculated by us is demonstrated in **Figure 25**. As shown in **Figures 25** and **26**, the width of the indirect forbidden band is 1.42 eV in antiferroelectric phase and 1.36 eV in ferroelectric phase. The following results match well to the results that have been obtained earlier [49]: 1.42 eV in antiferroelectric and 1.36 eV in ferroelectric phase. The indirect width of the forbidden band is 1.36 eV in that band structure, which was obtained by moving to the point of phase transition from ferroelectric phase point B1; and it is 1.41 eV when moving from antiferroelectric phase point A1. Hence, the indirect forbidden band moving from antiferroelectric to ferroelectric phase alters by 0.05 eV, according to our calculations. This number complies with the change 0.06 eV of the width of the indirect band, which has been experimentally set [48, 49]. **Figures 3** and **4** reveal that the direct forbidden band is narrowest in point U, 1.83 eV. The calculations presented in Ref. [49] demonstrate the value of 1.82 eV. The minimal direct forbidden band in ferroelectric phase occurs in point G, 1.94 eV, whereas this gap is 1.98 eV in point U. The narrowest forbidden band is in point U of Brillouin zone, approaching from both antiferroelectric and ferroelectric sides. This has been estimated in band structure in the area of phase transition (**Figures 26b** and **c**). In the first case, it turned out to be1.83 eV, point A2 (**Figure 25**), and in the second one it is 1.87 eV, point B2 [43] (Figure 25). For this reason, the observed jump of the direct forbidden band is 0.04 eV during

(*C*)

(*V* )

, and in ferroelectric phase at *R*3-4

, the valence band top in

. Our experimental

points *Q* and *C* (energy variation was 0.92 and 0.86 eV, respectively).

The conduction band bottom in both phases is located at point *Z*<sup>1</sup>

**16. Band structure at the first-order phase transition**

(*V* )

antiferroelectric phase is at the point *U*5-<sup>6</sup>

obtained by other authors.

118 Piezoelectric Materials

$$
\sigma = \sigma\_0 \frac{2kT}{a\_0} \text{th} \frac{a\_0}{2kT} \text{\textdegree} \tag{43}
$$

where *σ* stand for Urbach's parameter, which denotes the outspread of the absorption edge. The *σ*0 is a constant, which describes the intensity of interaction between electrons and phonons, *ħω*0 denotes effective phonons energy, *K*0 denotes "oscillator's strength" or the maximal absorption coefficient, *E*<sup>0</sup> represents characteristic "gap" of the forbidden band, and *EK* denotes light quantum energy for a particular absorption coefficient *K*. During the phase transition, the characteristic "gap" differs:

$$
\Delta E\_0 = E\_{\text{0F}} - E\_{0\text{AF}} \tag{44}
$$

where *E*0F and *E*0AF stand for the values of the energy "gap" in ferroelectric (F) and antiferro‐ electric (AF) phases, respectively. As seen from Eq. (43), temperature dependences *E*K (*T*) and σ/*kT* (*T*) should be measured when *K* = const. and *K*0 (*T*) in both phases in the area of the phase transition in order to determine *E*0F and *E*0AF. *K*0 (*T*) is not temperature dependent in antifer‐ roelectric phase and it is experimentally defined. Different temperatures are matched finding the point of crossing of the curve ln*K* (*E*). In ferroelectric phase, *K*0F is determined as follows:

$$\ln K\_{\rm{0F}} = \ln K\_0 - \mathcal{Y}\left(\frac{\sigma}{kT}\right)\_{\rm{AF}} P\_{\rm{S}\,\,\,\prime}^2 \tag{45}$$

**Figure 26.** The band structure of SbSI monocrystal: (a) in antiferroelectric phase (*T* = 308 K), (b) in points of phase tran‐ sition moving from antiferroelectric phase (**Figure 25** points *A*1 and *A*2; *TC* = 295 K), (c) in points of phase transition moving from ferroelectric phase (**Figure 25** points *B*1 and *B*2; *TC* = 295 K), and (d) in ferroelectric phase (**Figure 25** points *C*1 and *C*2; *T* = 278 K). Arrows show the width of the forbidden indirect band.

where ln*K*0 and (σ/*kT* )AF are the parameters in the antiferroelectric phase, and *γ* represents the coefficient of proportionality (polarization potential). It is possible to find the values of *γ*, *K*0, *P*S (*T*) and Eq. (42) in Ref. [34]. **Figure 27** demonstrates temperature dependencies of *E*K (*T*) and σ/*kT* (*T*), which have been measured experimentally employing the dynamic method with a continuously variable temperature [51]. Using experimental results provided in **Figure 27**, when temperature is 295 and 278 K, as well as employing the data indicated in Ref. [34], we estimate that according to Eqs. (43) and (46), Δ*E*0 = 0.12 ± 0.02 eV. The experimental Δ*E*0 value coincides with the theoretic variation of the width of the forbidden band in margins of error, which is 0.11 eV in Brillouin zone point U. It appears due to the variation of the grating parameters along the axis *c*(*y*), which affects the band structure of crystal (**Figures 26c** and **d**).

**Figure 27.** The experimental izo absorption curve energy *E*<sup>K</sup> (eV) dependence on temperature when *K* = 65 cm−1 (1) and the dependence σ/*kT* (T) (2) are measured in SbSI at phase transition.

### **17. Conclusions**

where *σ* stand for Urbach's parameter, which denotes the outspread of the absorption edge. The *σ*0 is a constant, which describes the intensity of interaction between electrons and phonons, *ħω*0 denotes effective phonons energy, *K*0 denotes "oscillator's strength" or the maximal absorption coefficient, *E*<sup>0</sup> represents characteristic "gap" of the forbidden band, and *EK* denotes light quantum energy for a particular absorption coefficient *K*. During the phase

where *E*0F and *E*0AF stand for the values of the energy "gap" in ferroelectric (F) and antiferro‐ electric (AF) phases, respectively. As seen from Eq. (43), temperature dependences *E*K (*T*) and σ/*kT* (*T*) should be measured when *K* = const. and *K*0 (*T*) in both phases in the area of the phase transition in order to determine *E*0F and *E*0AF. *K*0 (*T*) is not temperature dependent in antifer‐ roelectric phase and it is experimentally defined. Different temperatures are matched finding the point of crossing of the curve ln*K* (*E*). In ferroelectric phase, *K*0F is determined as follows:

g

0F 0 AF S

è ø

**Figure 26.** The band structure of SbSI monocrystal: (a) in antiferroelectric phase (*T* = 308 K), (b) in points of phase tran‐ sition moving from antiferroelectric phase (**Figure 25** points *A*1 and *A*2; *TC* = 295 K), (c) in points of phase transition moving from ferroelectric phase (**Figure 25** points *B*1 and *B*2; *TC* = 295 K), and (d) in ferroelectric phase (**Figure 25**

where ln*K*0 and (σ/*kT* )AF are the parameters in the antiferroelectric phase, and *γ* represents the coefficient of proportionality (polarization potential). It is possible to find the values of *γ*, *K*0, *P*S (*T*) and Eq. (42) in Ref. [34]. **Figure 27** demonstrates temperature dependencies of *E*K (*T*) and σ/*kT* (*T*), which have been measured experimentally employing the dynamic method with a

points *C*1 and *C*2; *T* = 278 K). Arrows show the width of the forbidden indirect band.

2

<sup>σ</sup> ln ln *KK P* , *kT* (45)

æ ö = - ç ÷

0 0F 0AF D = *EE E*– , (44)

transition, the characteristic "gap" differs:

120 Piezoelectric Materials

The potential energy of *Au*(10), *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) normal modes in the paraelectric phase are anharmonic with a double-well *V*(*z*), while *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) modes possess only one minimum. The semisoft 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 B3*g*(7) → *B***2** evoke experimental reflection *R*(*k*) peaks for *E*||*c* in the range of *k* = 10–100 cm−1 for both paraelectric and ferroelectric phases. The *R*(*k*) peak for *E*||*c* in the paraelectric phase is created by mode *B*1*<sup>u</sup>*(2), but in the ferroelectric phase the *R*(*k*) peaks are caused 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**. It has been determined that strong lattice anharmonisity, as well as interaction between chains, can split the mode *B*1*<sup>u</sup>*(3) into two components, from which one is soft in the microwave range and the other *B*1*<sup>u</sup>*(3) is semisoft in the *IR* range. The semisoft 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) increase the large (absorption) peak in the range *k* = 5–100 cm−1 and the dielectric contribution of 5000. The reflection spectra also show large peaks due to its strong temperature dependence. The strongest temperature dependence of reflection is observed in the ferroelectric phase. While in the paraelectric phase it becomes relatively weak.

In the long SbSI chains, the highest levels of one-electron energies in the valence band top are degenerate. Therefore, Jahn-Teller effect appears as an important factor. The ferroelectric phase transition in SbSI crystals is caused by electron-phonon and phonon-phonon interactions. The electron-phonon interaction reduce harmonic coefficient *K*, whereas phonon-phonon interaction reverses its sign, i.e., *K* < 0; *c* > 0. In the phase transition region, anomalous behavior is demonstrated by the coefficients of the polynomial expansion of the total energy *E*<sup>T</sup> dependence upon the *<sup>u</sup>* normal mode coordinate. The *<sup>u</sup>* mode temperature dependence of the coefficient *K* = *m<sup>τ</sup>* 2 is similar to the temperature dependence of the soft mode frequency, .
