**8. Electron-phonon interaction in one atomic chain**

**Figure 2** provides the dependence of one-electron energies in the valence band of a long SbSI chain on the number of atoms *N*, which is calculated by the unrestricted Hartree-Fock method (UHF) [33] in the basis set of *Hw* functions using the pseudopotential. As it is seen from **Figure 2**, the highest energy levels in the valence band degenerate when *N* turns to 40 or more. As it is shown in **Figure 3**, the highest *Au* and *Ag* one-electron energy levels are degenerated in the PEP close to the band gap. The lowest levels of the conduction band are set apart from the highest levels of the valence band by a band gap of *Eg* = 5.712 eV.

**Table 8** has been composed with reference to **Table 7**, where symmetry types of one-electron levels are presented. In **Table 8**, the energies of the degenerate *Au* and *Bg* symmetry electronic levels in the valence band are highlighted by boldface, where the energies of level 177 *Bg* and level 178 *Au* are −0.3344H, and the energies of level 179 *Au* and level 180 *Bg* are −0.3257H (assuming that 1 H = 27.21 eV).


**Table 8.** One-electron energies in the conduction band bottom and valence and top for an SbSI chain containing *N* = 60 atoms.

In the phase transition, i.e., when equilibrium positions of the Sb and S atoms alter along the z(*c*)-axis, the degeneracy is removed, and the band gap *E*g is narrowed. The decrease of the interatomic force factor *R* determines the decrease of *E*<sup>g</sup> (**Figure 8a**). However, the ionic charges *q* has no influence on the decrease of *E*g, as they remain almost constant throughout the phase transition (**Figure 8b**).

**Figure 8.** The variation of the interatomic bond strengths *R* and atomic charges *q* in the SbSI chain containing 60 atoms during the phase transition. Here *z*0 denotes the displacement of Sb atoms in the direction of *z*-axis.

The normal vibrational mode of symmetry *Au* (further for clarity denoted here as *u*) interacts

**Table 8** has been composed with reference to **Table 7**, where symmetry types of one-electron levels are presented. In **Table 8**, the energies of the degenerate *Au* and *Bg* symmetry electronic levels in the valence band are highlighted by boldface, where the energies of level 177 *Bg* and level 178 *Au* are −0.3344H, and the energies of level 179 *Au* and level 180 *Bg* are −0.3257H

**Table 8.** One-electron energies in the conduction band bottom and valence and top for an SbSI chain containing *N* = 60

In the phase transition, i.e., when equilibrium positions of the Sb and S atoms alter along the z(*c*)-axis, the degeneracy is removed, and the band gap *E*g is narrowed. The decrease of the interatomic force factor *R* determines the decrease of *E*<sup>g</sup> (**Figure 8a**). However, the ionic charges *q* has no influence on the decrease of *E*g, as they remain almost constant throughout the phase

**Figure 8.** The variation of the interatomic bond strengths *R* and atomic charges *q* in the SbSI chain containing 60 atoms

during the phase transition. Here *z*0 denotes the displacement of Sb atoms in the direction of *z*-axis.

(assuming that 1 H = 27.21 eV).

98 Piezoelectric Materials

atoms.

transition (**Figure 8b**).

**Valence band Conduction band**

Note: The degenerate electronic states in the valence band are denoted by boldface.

Symmetry Energy (H) Symmetry Energy (H) *Bg* −0.3363 181 *Au* −0.0926 *Bg* **−0.3344** 182 *Bg* −0.0925 *Au* **−0.3344** 183 *Au* −0.0847 *Au* **−0.3257** 184 *Bg* −0.0838 *Bg* **−0.3257** 185 *Au* −0.0821

with the electronic states of symmetry *Au* and *Bg* if *Au × Bg* = *u*. The electron-phonon interaction involving the highest degenerate *Au* and *Bg* symmetry levels of the valence band is called the

Jahn-Teller effect. The pseudo-Jahn-Teller effect (PJTE) occurs when the normal *u* mode interacts with the *Au* electronic state at the top of the valence band and with the *Bg* electronic state at the bottom of the conduction band, as well as with the band gap of *E*g between them. Because of the PJTE, the dependence of the harmonic term on normal coordinates *Q* is split into two terms:

$$\mathcal{L}\_{f,\varepsilon}\left(\underline{Q}\right) = \frac{1}{2}\left(K\_0 - \frac{2F^2}{E\_g}\right)\underline{Q}^2\tag{16}$$

where *K*<sup>0</sup> represents nonvibronic part of the force constant (the bare force constant), *KV* stands for the vibronic coupling term. *K* = *K*0 – (2 *F*<sup>2</sup> /*E*g) = *K*0 – *Kv* denotes the force constant considering the PJTE. *F* which is found from the following equation represents the vibronic coupling constant:

$$F^2 = \frac{\Delta E E\_g}{2Q\_0^2}.\tag{17}$$

Here *Q*0 = *z*0 is the atomic displacement, and Δ*E* is the variation of the band gap during the phase transition. From **Figure 9** and Eq. (17), it follows that for *z*0 = 0.2 Å, Δ*E* = 2.23 eV, and *E*g = 5.7 eV, we would get *F* ≈ 13 eV/Å. From the experimental results of Ref. [34], we derived *F* ≈ 4 eV/Å. Therefore, the magnitude of the term 2*F*<sup>2</sup> = *E*g may attain as much as 2H/Å.

**Figure 9.** The variation of the one-electron energy spectrum of the SbSI chain containing 60 atoms during the phase transition. Here *z*0 denotes the displacement of Sb atoms in the direction of *z*-axis.

In the case of the normal mode *u*, the PJTE and JTE vibronic mixing leads to the force constant

$$K = K\_0 - \left(\frac{2F^2}{E\_g} + G\right) = K\_0 - K\_\nu^\*,$$


**Table 9.** The energies of CL of Sb, S, and I atoms calculated by UHF method at the edge of SbSI cluster.

where *G* is a quadratic vibronic constant from JTE, and *KV* \* is the vibronic coupling term of PJTE and JTE. The phase transition occurs when the low-frequency mode is unstable, i.e., when *K* < 0, see **Table 9**. The symmetry coordinates of this mode are Sb(+1), S(+1), and I(−2). In pursuance of finding *K*, the total energy *E*T of the *u* vibrational mode has to compute. Separate energy contributions constitute the total energy *E*T:

$$E\_{\rm T} = E\_{\rm K} + E\_{\rm ce} + E\_{\rm nc} + E\_{\rm m},\tag{18}$$

where the kinetic energy of electrons is

Electronic Structure and Piezoelectric Properties of SbSI Crystals http://dx.doi.org/10.5772/64223 101

$$E\_{\mathbf{x}} = \sum\_{i=1}^{n} \frac{\xrightarrow{\mathbf{p}}\_{p\_i}^{2}}{2m\_c},\tag{19}$$

the interelectron interaction energy is

In the case of the normal mode *u*, the PJTE and JTE vibronic mixing leads to the force constant

\*

**Left edge of cluster Right edge of cluster**

\* is the vibronic coupling term of

T K ee ne nn *EEEEE* =+++ , (18)

2

*g <sup>F</sup> KK G K K E* æ ö =- +=- ç ÷ è ø

**State AFEP (***T* **= 300 K) FEP (***T* **= 215 K)**

100 Piezoelectric Materials

0 0

*E***max (eV)** *E***min (eV)** *E***max (eV)** *E***min (eV)** *E***max (eV)** *E***min (eV)**

I 3 s 1030.40 1027.72 1028.69 1025.72 1032.09 1029.87 Sb 3 s 917.51 916.66 918.17 915.10 919.59 918.17 I 3p 891.44 888.70 889.73 886.70 893.12 890.83 Sb 3p 787.89 786.98 785.73 785.42 789.97 788.52 I 3d 658.75 655.99 657.04 653.99 660.44 658.12 Sb 3d 567.67 565.74 565.41 565.19 569.13 568.24 S 2 s 244.78 243.29 242.96 241.21 246.45 245.35 I 4 s 198.47 195.77 196.78 193.76 200.16 197.93 S 2p 181.47 179.59 179.66 177.81 183.13 181.94 Sb 4 s 170.09 169.27 167.95 167.71 172.13 170.80 I 4p 147.59 144.70 145.92 142.72 149.26 146.55 Sb 4p 123.97 123.04 121.86 121.48 126.01 124.55 I 4d 67.13 64.11 65.45 62.14 68.82 66.23 Sb 4d 51.35 50.35 49.25 48.80 53.38 51.87

**Table 9.** The energies of CL of Sb, S, and I atoms calculated by UHF method at the edge of SbSI cluster.

PJTE and JTE. The phase transition occurs when the low-frequency mode is unstable, i.e., when *K* < 0, see **Table 9**. The symmetry coordinates of this mode are Sb(+1), S(+1), and I(−2). In pursuance of finding *K*, the total energy *E*T of the *u* vibrational mode has to compute. Separate

where *G* is a quadratic vibronic constant from JTE, and *KV*

energy contributions constitute the total energy *E*T:

where the kinetic energy of electrons is

<sup>2</sup> , *<sup>V</sup>*

$$E\_{\rm ce} = \frac{1}{2} \sum\_{i,j \neq i}^{n} \frac{e^2}{\left| \vec{r\_i} - \vec{r\_j} \right|},\tag{20}$$

the electron-nuclear interaction energy is

$$E\_{\rm nc} = -\sum\_{\ell,\ell=1}^{n,N} \frac{e^2 Z\_{\ell}}{\left|\vec{R}\_{\ell} - \vec{r\_{\ell}}\right|},\tag{21}$$

and the internuclear interaction energy is

$$E\_{\rm an} = \frac{1}{2} \sum\_{A,B \neq A}^{N} \frac{e^2 Z\_A Z\_B}{|\vec{R}\_A - \vec{R}\_B|}. \tag{22}$$

**Figure 10.** The dependence of the total energy *E*T of the *u* mode upon normal coordinates (relative displacements *Zα* of atoms (*α* = Sb, S, I) from equilibrium position), where *z* denotes the displacement of Sb atoms in the mode from their equilibrium position. The temperatures of the chain are 400 K in PEP and 284 and 248 K in FEP. In all cases *E*T0 = 0.

The aforesaid energies are dependent on nuclear and electronic coordinates . Their evaluation has been conducted with the unrestricted Hartree-Fock method using the computer program GAMESS described in [33]. In pursuance of finding *K*, the dependence of the total energy *E*T of the *u* mode has been computed on the displacements *zα* of atoms (*α* = Sb, S, I) from their equilibrium positions (**Figure 10**). **Figure 10** demonstrates that *z* measures the displacements of Sb atoms in the *u* mode from their equilibrium positions *z*0*<sup>α</sup>*. Vibrational displacements *zα* of atoms (*α* = Sb, S, I) from their equilibrium positions *z*0*<sup>α</sup>* are identified employing the equation

$$
\pi\_a = z\_a \pm \mu\_a \frac{1}{\sqrt{m\_a}} \cdot \frac{i}{20},
\tag{23}
$$

where *i* = 0–19, *u<sup>α</sup> m<sup>α</sup> -1/2* is the vibrational amplitude of atoms (*u<sup>α</sup>* is the normal coordinates of the *u* vibrational mode). Normal coordinates of the *u* vibrational mode are found by diagonalization of the dynamical matrix [23]. By expressing *E*T = *f* (*z*) as a fourth-order polynomial

$$E\_{\rm r} = E\_{\rm r0} + K\left(z\right)^2 + c\left(z\right)^4. \tag{24}$$

For the *u* mode in PEP (300 K), *K* = 2.05 H/Å2 and *c* = 2.3 H/Å4 . Since the obtained *K* value is positive, it is supposed that the electron-phonon interaction will only decrease *K*. The inter‐ action itself will not induce a phase transition because under the influence of the electron-

phonon interaction the *<sup>u</sup>* mode becomes only slightly anharmonic. However, due to phonon-

phonon interaction in the SbSI chain, the *<sup>u</sup>* mode is strongly anharmonic, as its *K* < 0 and *c* > 0 [14]. Simultaneous action of electron-phonon and phonon-phonon interactions increase even

more the anharmonicity of the *u* mode as compared to its anharmonicity determined by the phonon-phonon interaction alone. Therefore, the phase transition in the SbSI crystals is due to both electron-phonon and phonon-phonon interactions.

### **9. Physical parameters in the phase transition region**

The vibration frequencies of the normal mode *Au* are temperature dependent [10]. Therefore, the influence of the electron-phonon interaction and variation of unit cell parameters in the phase transition region on temperature dependence of frequency should be assessed properly. By putting vibrational displacements *z<sup>α</sup>* of atoms (*α* = Sb, S, I) from their equilibrium positions equal to *zα* = 0, Eq. (24) is transformed into

The aforesaid energies are dependent on nuclear and electronic coordinates . Their evaluation has been conducted with the unrestricted Hartree-Fock method using the computer program GAMESS described in [33]. In pursuance of finding *K*, the dependence of the total energy *E*T of the *u* mode has been computed on the displacements *zα* of atoms (*α* = Sb, S, I) from their equilibrium positions (**Figure 10**). **Figure 10** demonstrates that *z* measures the displacements of Sb atoms in the *u* mode from their equilibrium positions *z*0*<sup>α</sup>*. Vibrational displacements *zα* of atoms (*α* = Sb, S, I) from their equilibrium positions *z*0*<sup>α</sup>* are identified

<sup>1</sup> , <sup>20</sup>

*-1/2* is the vibrational amplitude of atoms (*u<sup>α</sup>* is the normal coordinates of

T T0 *E E Kz cz* =+ + . (24)

. Since the obtained *K* value is

(23)

a

() () 2 4

positive, it is supposed that the electron-phonon interaction will only decrease *K*. The inter‐ action itself will not induce a phase transition because under the influence of the electron-

phonon interaction the *<sup>u</sup>* mode becomes only slightly anharmonic. However, due to phonon-

phonon interaction in the SbSI chain, the *<sup>u</sup>* mode is strongly anharmonic, as its *K* < 0 and *c* > 0 [14]. Simultaneous action of electron-phonon and phonon-phonon interactions increase even

more the anharmonicity of the *u* mode as compared to its anharmonicity determined by the phonon-phonon interaction alone. Therefore, the phase transition in the SbSI crystals is due to

The vibration frequencies of the normal mode *Au* are temperature dependent [10]. Therefore, the influence of the electron-phonon interaction and variation of unit cell parameters in the phase transition region on temperature dependence of frequency should be assessed properly.

and *c* = 2.3 H/Å4

*<sup>i</sup> z zu <sup>m</sup>* aaa

the *u* vibrational mode). Normal coordinates of the *u* vibrational mode are found by

=± ×

employing the equation

102 Piezoelectric Materials

where *i* = 0–19, *u<sup>α</sup> m<sup>α</sup>*

diagonalization of the dynamical matrix [23].

For the *u* mode in PEP (300 K), *K* = 2.05 H/Å2

both electron-phonon and phonon-phonon interactions.

**9. Physical parameters in the phase transition region**

By expressing *E*T = *f* (*z*) as a fourth-order polynomial

$$E\_{\rm 70} = E\_{\rm 80} + E\_{\rm ec0} + E\_{\rm ac0} + E\_{\rm na0}.\tag{25}$$

It follows from Eqs. (18) and (22) that separate terms in Eq. (25) are functions of atomic coordinates and distances between atoms During the phase transition, due to variation of the volume of the simplified unit cell *V*0 = (*a*/2)*bc* [30]. Just this causes anomalies in the temperature dependences of *E*T0 and its separate components in the phase transition region **Figures 9** and **11**. **Figure 11** demonstrates temperature dependences of *E*T0 and of the unit cell volume *V*0.

**Figure 11.** Temperature dependences of the unit cell volume *V*0 = 2 V (where V is the volume of the simplified unit cell) (1) and of the total energy *E*T0 (2) in the phase transition region. For curve 2, *V*0 is assumed to depend upon tempera‐ ture, and for curve 3 it is taken as constant: *V*0 = 354.25 Å<sup>3</sup> .

**Figure 12.** Temperature dependences of the total kinetic *E*TK0 and potential energy *E*TP0 in the phase transition region.

In PEP (or in the antiferroelectric phase, according to Ref. [14]), the anomalies of *E*T0 and *V* correlate in the temperature range of 295–400 K. It means that *E*T0 decreases due to the growth of *V*0, and vice versa. **Figures 12** and **13** show that the temperature dependence of *E*T0 is mainly determined by the components *E*TP0 = *E*ee0 + *E*ne0 + *E*nn0.

**Figure 13.** Temperature dependences of the total potential energy *E*TP0 components *E*ee0, *E*ne0, and *E*nn0 in the phase tran‐ sition region.

In the FEP in the temperature range 280–295 K, a sharp increase of the unit cell volume *V*<sup>0</sup> leads to a decrease of *E*T0, curves 1 and 2 in **Figure 11**. However, in FEP at the temperatures of 220– 280 K, *E*T0 grows if the temperature is decreased, while the unit cell volume *V*0 changes only slightly. Curve 3 in **Figure 11** shows the calculated temperature dependence of *E*T0 provided that *V*0 = constant. In this temperature range, when *V*0 = constant, the growth of *E*T0 is caused by the variable shift of the equilibrium positions of Sb and S atoms. Thus, curve 2 in **Fig‐ ure 11** demonstrating the growth of *E*T0 can also be considered as caused solely by the variation of Sb and S equilibrium positions.

**Figure 14.** Temperature dependences of the harmonic constant *K* = (*K*0 – *KV* \* ) Å and of the anharmonicity factor *c* (B) for the normal mode in the phase transition region. Dashed lines show the temperature dependences of *K* and *c* assuming that *V*0 = 354.25 Å<sup>3</sup> .

The three temperature intervals, i.e., 295–400 K in PEP, 280–295 K in FEP, and 220–280 K in FEP, clearly reveal the anomalies in the temperature dependences of the coefficients *K* and *c* of the polynomial (10) (**Figures 14a** and **b**). The temperature dependences of *K* and *c* are determined by the variation of the unit cell volume *V*0 in the PEP interval at 295–400 K, whereas the behavior of *K* and *c* is determined by *V*<sup>0</sup> and the variable shift of the equilibrium positions of Sb and S atoms in the FEP interval at 280–295 K.

Pursuing to separate the influence of the volume *V*0 and of the shift of Sb and S equilibrium positions on the anomalies of *K* and *c*, the calculated temperature dependences of *K* and *c* at a constant *V*0 are demonstrated in **Figures 14(a)** and **(b)** (dashed lines). In this temperature range, *K* and *c* rapidly change due to the variation of the equilibrium position of Sb and S atoms if *V*<sup>0</sup> = constant. Consequently, the rapid change of *K* and *c* marked by the solid lines in **Fig‐ ures 14(a)** and **(b)** is also caused merely by the shift of Sb and S atoms. The rapid variation of *K* in this temperature range occurs due to the decrease of *KV* \* that is determined by thevariation

of degeneracy (see **Figure 9**). Since *K* = *m<sup>τ</sup>* <sup>2</sup> , where *mτ* is the reduced mass of the *u* mode and *ω* is the frequency of this mode, then the dependence of *K* upon temperature (**Figure 14a**) is similar in its shape to the temperature dependence of the soft mode *<sup>s</sup>* <sup>2</sup> [14]. Therefore, the

temperature dependence of the soft mode *u* frequency *<sup>s</sup>* 2 is essentially affected by the variation of the unit cell volume *V* and of the polynomial coefficient *K* in the phase transition region: the total temperature dependence of *<sup>s</sup>* 2 can be obtained provided the anharmonicity caused by phonon-phonon interaction is considered.
