**7. Electron-phonon interaction and Jahn-Teller effect (JTE)**

#### **7.1. Introduction**

Examining the state of instability, the authors [30] have determined that in this respect the normal coordinate of the vibrational mode is *K* = *K*0 – *Kv* < 0, where *K*0 = 1/2 M 0 2 represents the bare force constant, while *Kv* stands for a vibrational coupling term, which is dependent on the vibrational coupling constant *F* and the energy gap Δ*E*. *C*2*<sup>h</sup>* 2 operations are *S*1 = *E*/(0,0,0), *S*2 = *C*2(*z*)/(0,0,1/ 2), *S*3 = *I*/(0,0,0), and *S*4 = *σz*/(0,0,1/2) for the PEP space group, whereas only *S*<sup>1</sup> and *S*2 are for the FEP space group *C*2*<sup>h</sup>* <sup>2</sup> . The movement of atoms within the chain is marked by energy states of independent normal vibrations, associated with vibrational symmetry coordinates that are composed of the projections of atomic displacements *Ki* = *Xi* , *Yi* , *Zi* on the axes of local coordinate systems [31, 32]:



**Table 6.** Characters of irreducible representations of symmetry space group *C*<sup>2</sup> 2 , and *C*2*<sup>h</sup>* 2 .

The coefficients *cji* are developed from the properties of irreducible representations *Γ*<sup>a</sup> obtained when coordinates *χ*(*α*) with symmetry operations *Si* are transformed (see **Table 6** for the characters of the representations). Symmetry coordinates that are composed of projections *Zi* of atomic displacements usually have symmetry *Au* and *Bg*, whereas the coordinates that are composed of projections *Xi* and *Yi* possess symmetry *Ag* and *Bu*. Consequently, by solving the system of vibrational equations, normal coordinates *Qi* (*α*) are obtained, in which *Xi* and *Yi* are interlinked and *Zi* participates separately. Thus, the normal vibrations in the SbSI chain along the *z*-axis can be characterized by *Au* and *Bg* symmetry coordinates, whereas the vibrations in the direction of *x*- and *y*-axes are related to *Bu* and *Ag* symmetry coordinates. The modes


*X*7(*Bu*) and *X*13(*Bu*) comply with acoustic vibrations, whereas the remaining ones correspond to the optical vibrations of the SbSI chain (**Table 7**).

Existence of the piezoelectric effect experimentally detected up to a temperature several degrees above the *TC* (**Figure 4**). The effect above *TC* (**Figure 7**) may be attributed to the

Examining the state of instability, the authors [30] have determined that in this respect the

bare force constant, while *Kv* stands for a vibrational coupling term, which is dependent on the

*S*2 = *C*2(*z*)/(0,0,1/ 2), *S*3 = *I*/(0,0,0), and *S*4 = *σz*/(0,0,1/2) for the PEP space group, whereas only *S*<sup>1</sup>

by energy states of independent normal vibrations, associated with vibrational symmetry

2 represents the

on the

2 operations are *S*1 = *E*/(0,0,0),

 = *Xi* , *Yi* , *Zi*

<sup>2</sup> . The movement of atoms within the chain is marked

<sup>=</sup> å (15)

*<sup>2</sup> )* **S1 S2 S3 S4**

2 , and *C*2*<sup>h</sup>* 2 .

possess symmetry *Ag* and *Bu*. Consequently, by solving the

(*α*)

are transformed (see **Table 6** for the

and *Yi*

are

are obtained, in which *Xi*

compositional inhomogeneity or the existence of the internal mechanical stresses.

**7. Electron-phonon interaction and Jahn-Teller effect (JTE)**

normal coordinate of the vibrational mode is *K* = *K*0 – *Kv* < 0, where *K*0 = 1/2 M 0

coordinates that are composed of the projections of atomic displacements *Ki*

( ) ( ) *j j <sup>i</sup> N cKi i*

 a

*A Ag* +1 +1 +1 +1 *B Bg* +1 −1 +1 −1 *A Au* +1 +1 −1 −1 *B Bu* +1 −1 −1 +1

The coefficients *cji* are developed from the properties of irreducible representations *Γ*<sup>a</sup> obtained

characters of the representations). Symmetry coordinates that are composed of projections *Zi* of atomic displacements usually have symmetry *Au* and *Bg*, whereas the coordinates that are

interlinked and *Zi* participates separately. Thus, the normal vibrations in the SbSI chain along the *z*-axis can be characterized by *Au* and *Bg* symmetry coordinates, whereas the vibrations in the direction of *x*- and *y*-axes are related to *Bu* and *Ag* symmetry coordinates. The modes

a

c

**Table 6.** Characters of irreducible representations of symmetry space group *C*<sup>2</sup>

and *Yi*

system of vibrational equations, normal coordinates *Qi*

with symmetry operations *Si*

vibrational coupling constant *F* and the energy gap Δ*E*. *C*2*<sup>h</sup>*

and *S*2 are for the FEP space group *C*2*<sup>h</sup>*

axes of local coordinate systems [31, 32]:

*2)* **<sup>Γ</sup>***<sup>α</sup> (C2h*

**7.1. Introduction**

96 Piezoelectric Materials

**<sup>Γ</sup>***<sup>α</sup> (C2*

when coordinates *χ*(*α*)

composed of projections *Xi*

**Table 7.** Vibrational symmetry coordinates in the simplified unit cell of the SbSI chain.
