**4. The piezoelectricity phenomena**

#### **4.1. Introduction**

where *M* represents the reduced mass of the normal mode atoms, and *ω* denotes the vibration frequency of the normal mode. The frequency of a normal mode with a double-well *V*(*z*) is dependent on the height of the potential barrier Δ*V*. Temperature, electric field, pressure, and fluctuations of atoms in the *x*–*y* plane influence this potential barrier [15]. In all the three phases, *V*(*x*) and *V*(*y*) appear to be single-well with the coefficients *b >* 0 in Eq. (9). In the paraelectric phase, *a* = *d* = 0, whereas in the antiferro- and ferroelectric phases, *a* ≠ 0 and *d* ≠ 0. Single-well PE turns out to be weakly anharmonic and the frequencies of the *R*(*k*) peak should only slightly

The experimental results of *IR* reflectivity *R*(*k*) spectrum of SbSI crystals in the paraelectric (*T* = 415 K) and ferroelectric (*T* = 273 K) phases for *E*||*c* are presented in **Figure 3**. The reflectivity measurements of SbSI crystals have been repeated by Dr. Markus Goeppert in

**Figure 3.** The reflectivity spectra of SbSI for *E*||*c* in the paraelectric phase (*T* = 415 K) and in the ferroelectric phase

In the reflectivity spectrum for *E*||*c* in the range of *k* = 10–100 cm−1, the number of peaks *R*(*k*) is equal for the paraelectric and ferroelectric phases. However, in the range of *k* = 100–200 cm −1, the number of peaks *R*(*k*) differs in both phases. Anharmonic modes with double-well *V*(*z*) are highly sensitive to dislocations, impurities, and fluctuations of chains in the *x*–*y* plane.

**Figure 2** demonstrates that 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) with a double-well *V*(*z*) are strongly anharmonic, whereas modes *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) with a single-well *V*(*z*) appear to be weakly anharmonic. The vibration frequencies of the former modes are lower

(*T* = 273 K). Inset: the reflectivity spectra in range 50–95 cm−1.

Germany in Karlsruhe University using the Bruker Fourier spectrometer.

depend on the temperature.

90 Piezoelectric Materials

SbSI is a piezoelectric crystal that has a high-volume piezoelectric modulus *dv* = 10 × 10−10 C/N and electromechanical coupling coefficient *k3* = 0.87 [25]. The relation among polarization *P*, piezoelectric modulus *e*, *d*, deformation *r =* Δ*l/l*, and elastic compliance coefficient is as follows:

$$
\Delta P = er, \Delta P \approx q \Delta \mathbf{z},\tag{10}
$$

$$d = \text{es.}\tag{11}$$

where *q* is the atom's ionic charge, *s* is the elastic compliance coefficient, Δ*z* = *z* – *z*0, *z*0 is the atomic equilibrium position, when crystal is not deformed (*r* = 0), and *z* is the atomic equili‐ brium position, when crystal is deformed (*r* ≠ 0). It is handy to calculate the piezoelectric modulus *d* along *z*(*c*)-axis for needle-like crystal deforming it along the crystallographic axes *x*(*a*), *y*(*b*), and *z*(*c*) when *r* = Δ*a*/*a* = Δ*b*/*b* = Δ*c*/*c…* Having *e* it is possible to calculate *d* according to Eq. (11). The dependence of elastic compliance coefficients on temperature in SbSI crystal is shown in [26]. From work [27], piezoelectric module *d*33 along *z*(*c*)-axis and *dh* (the so-called "hydrostatic piezoelectric modulus") were known. As *dh* has been found to be of the same magnitude as *d*33, it has been concluded that *d*31 is nearly equal to *d*32 and opposite in sign. Both quantities *d*32 and *d*31 are smaller than *d*33. In [28] it is found that *d*31 is of the same order of magnitude as *d*33. The *d*<sup>32</sup> coefficient is of nearly the same value as *d*<sup>31</sup> but opposite in sign. The high values of *dh* = *d*33 + *d*32 + *d*31 indicate that *d*31 and *d*32 are relatively small or of opposite in sign.

### **5. Potential energy of Sb atoms in anharmonic soft mode**

In [29] we described the method for calculating the potential energy at point *r* of the unit cell. The potential energy at the point *r* is given in Eq. (7). The average value of the potential energy of Sb atoms of unit cell in the normal mode of SbSI crystal may be written as follows:

$$\overline{V}\_{\rho} = \frac{V\_{\rho}\left(\mathbf{R}\_{\text{Sb1}}\right) + V\_{\rho}\left(\mathbf{R}\_{\text{Sb2}}\right) + V\_{\rho}\left(\mathbf{R}\_{\text{Sb3}}\right) + V\_{\rho}\left(\mathbf{R}\_{\text{Sb4}}\right)}{4},\tag{12}$$

where *R*Sb1 = *R*0,Sb1 + *Q*Sb1; *R*Sb2 = *R*0,Sb2 + *Q*Sb2; *R*Sb3 = *R*0,Sb3 + *Q*Sb3; and *R*Sb4 = *R*0,Sb4 + *Q*Sb4.

Also, the coordinates of all *S* and *I* atoms changes according to the equations: *Rα* = *R*0, *<sup>α</sup>* + *Qα*, *α* = *S*1; *S*2; *S*3; *S*4; *I*1; *I*2; *I*3; *I*4. For numerical evaluation of Eq. (12), we need to vary all *Q<sup>α</sup> = zα* by small steps from *–Qα*(max) to +*Qα*(max). The atomic form factor is

$$\begin{aligned} \, \_i f\_a \left( \mathbf{s} \right) &= \sum\_{nlm} nlm \left| \exp \left[ \right. \left. i \left( \mathbf{r} \cdot \mathbf{s} \right) \right] \right|nlm, \end{aligned} \tag{13}$$

where *nlm* is a set of quantum numbers for the atom *α*.

### **6. Antimony atoms' equilibrium positions**

For theoretical investigations, we used the most sensitive to temperature and deformations soft mode's *V*¯ *<sup>p</sup>*(*z*) dependence on normal coordinates [29]. It is handy to expand *V*¯ *<sup>p</sup>*(*z*) using polynomial:

$$
\overline{V}\_p \left( z \right) = V\_0 + a \ast z + b \ast z^2 + d \ast z^3 + c \ast z^4,\tag{14}
$$

where *a\**, *b\**, *d\**, and *c\** are polynomial expansion coefficients.

The *V*¯ *<sup>p</sup>*(*z*) dependence on normal coordinates along *z*(*c*)-axis in paraelectric phase (PEF) at *<sup>T</sup>* = 300 K is shown in **Figure 4** while deforming along *x*(*a*), *y*(*b*), and *z*(*c*) axes. The *V*¯ *<sup>p</sup>*(*z*) in PEF has a shape of double-well (*b\** < 0, *c\** > 0). The barrier height when crystal is not deformed is Δ*UC ≈ kTC*, where *k* is the Boltzmann constant. As seen from **Figure 4**, in paraelectric phase *∆z* are equal to zero, because *V*¯ *<sup>p</sup>*(*z*) of Sb atomic soft mode is symmetrical and double-well. In paraelectric phase, Sb atoms vibrate between two minima of double-well potential *V*¯ *<sup>p</sup>*(*z*), provided deformation *r* ≠ 0 and *r* = 0.

**5. Potential energy of Sb atoms in anharmonic soft mode**

*p*

*V*

In [29] we described the method for calculating the potential energy at point *r* of the unit cell. The potential energy at the point *r* is given in Eq. (7). The average value of the potential energy

> () ( ) () ( ) Sb1 Sb2 Sb3 Sb4 , <sup>4</sup> *pp p p*

Also, the coordinates of all *S* and *I* atoms changes according to the equations: *Rα* = *R*0, *<sup>α</sup>* + *Qα*, *α* = *S*1; *S*2; *S*3; *S*4; *I*1; *I*2; *I*3; *I*4. For numerical evaluation of Eq. (12), we need to vary all *Q<sup>α</sup> = zα* by

( )s exp ( ) ,

For theoretical investigations, we used the most sensitive to temperature and deformations

( ) 2 34 <sup>0</sup> \* \* \* \*, *V z V az bz dz cz <sup>p</sup>* =+ + + +

*<sup>T</sup>* = 300 K is shown in **Figure 4** while deforming along *x*(*a*), *y*(*b*), and *z*(*c*) axes. The *V*¯

paraelectric phase, Sb atoms vibrate between two minima of double-well potential *V*¯

PEF has a shape of double-well (*b\** < 0, *c\** > 0). The barrier height when crystal is not deformed is Δ*UC ≈ kTC*, where *k* is the Boltzmann constant. As seen from **Figure 4**, in paraelectric phase

*<sup>p</sup>*(*z*) dependence on normal coordinates [29]. It is handy to expand *V*¯

*<sup>p</sup>*(*z*) dependence on normal coordinates along *z*(*c*)-axis in paraelectric phase (PEF) at

*<sup>p</sup>*(*z*) of Sb atomic soft mode is symmetrical and double-well. In

*f nlm i nlm*

= é ×ù å ë û *r s*

(12)

(13)

*<sup>p</sup>*(*z*) using

(14)

*<sup>p</sup>*(*z*) in

*<sup>p</sup>*(*z*),

of Sb atoms of unit cell in the normal mode of SbSI crystal may be written as follows:

*VVVV*

+++ <sup>=</sup> *RRRR*

where *R*Sb1 = *R*0,Sb1 + *Q*Sb1; *R*Sb2 = *R*0,Sb2 + *Q*Sb2; *R*Sb3 = *R*0,Sb3 + *Q*Sb3; and *R*Sb4 = *R*0,Sb4 + *Q*Sb4.

small steps from *–Qα*(max) to +*Qα*(max). The atomic form factor is

a

where *nlm* is a set of quantum numbers for the atom *α*.

**6. Antimony atoms' equilibrium positions**

where *a\**, *b\**, *d\**, and *c\** are polynomial expansion coefficients.

soft mode's *V*¯

92 Piezoelectric Materials

polynomial:

The *V*¯

*∆z* are equal to zero, because *V*¯

provided deformation *r* ≠ 0 and *r* = 0.

*nlm*

**Figure 4.** The SbSI mean potential energy dependence on soft mode normal coordinates at the Sb atoms sites in para‐ electric phase (*T* = 300 K), when deformation *r* = 0.03 along *x*(*a*), *y*(*b*), and *z*(*c*) axes and *r* = 0. *∆z* = 1 a.u. = 0.53 Å.

Due to the decrease in the temperature in ferroelectric phase, double-well *V*¯ *<sup>p</sup>*(*z*) gradually changes its form (see **Figure 2**) from double-well to single-well (*a\** ≠ 0, *b\** < 0, *d\** ≠ 0, *c\** > 0).

We calculated SbSI crystal's Sb atom potential energy's dependence on soft mode's normal coordinates in FEF at *T* = 289 K, when crystal is deformed along *x*(*a*), *y*(*b*), and *z*(*c*) axes, and when crystal is not deformed (*r* = 0; **Figure 5**). *z*<sup>0</sup> is the equilibrium position of Sb atom when *r* = 0 and *z* is the Sb atom's equilibrium position when *r* ≠ 0. Calculated temperature dependence of *z*, Δ*z = z – z*0, and *V*¯ *<sup>p</sup>*(*z*) scanning coefficients *a\**, *b\**, *d\**, and *c\** are shown in **Tables 4** and **5**. Temperature dependence of equilibrium position *z*<sup>0</sup> of Sb atoms in ferroelectric phase (when *r* = 0) is calculated using Eq. (8):

$$z\_0 = 0.24988 + 0.00221T - 3.13691E - \\$ \, T^2 + 1.42721E - 7 \, T^3 - 2.37137E - 10 \, T^4 \, \text{(Å)},$$

**Figure 5.** The SbSI mean potential energy dependence on soft mode normal coordinates at the Sb atoms sites in ferro‐ electric phase (*T* = 289 K), when deformation *r* = 0.03 along *x*(*a*), *y*(*b*), and *z*(*c*) axis and *r* = 0. ∆*z* = 1 a.u. = 0.53 Å.


**Table 4.** Scanning coefficients *a\**, *b\**, *c\**, and *d\** at different temperatures in SbSI crystal when crystal is not deformed (*r* = 0) and deformed along *x*(*a*), *y*(*b*), and *z*(*c*) axes, when *r =* Δ*a/a =* Δ*b/b =* Δ*c/c =* 0.03.


**Table 5.** Temperature dependence of Sb atom's equilibrium positions displacements' Δ*z*31, Δ*z*32, and Δ*z*33 along *x*(*a*), *y*(*b*), and *z*(*c*) axes when deformation is *r =* Δ*a*/*a* = Δ*b*/*b* = Δ*c*/*c* = 0.03, respectively. Δ*z* = 1 a.u. = 0.53 Å.

where *T* < *TC* is the temperature of the SbSI crystal in ferroelectric phase.

Since *<sup>e</sup>* <sup>=</sup> *<sup>Δ</sup><sup>P</sup> <sup>r</sup>* and *r* = const., then *e ≈* Δ*P ≈* Δ*z*. Consequently, using the data from **Table 5** we present Δ*z* dependence on temperature, when crystal is being deformed equally along three axes: *r =* Δ*a*/*a* = Δ*b*/*b* = Δ*c*/*c* (**Figure 6**)*.*

*T***, K** *a\* b\* c\* d\* r* 300 0 −0.058422 0.237293 0 0

292 −0.015845 −0.049218 0.221618 0.034324 0

289 −0.044289 −0.030932 0.207956 0.094969 0

0 −0.078223 0.260437 0 Δ*c*/*c* 0 −0.021201 0.188036 0 Δ*b*/*b* 0 −0.068322 0.248865 0 Δ*a*/*a*

−0.016505 −0.069437 0.245634 0.035488 Δ*a*/*a* −0.014301 0.013858 0.179530 0.030567 Δ*b*/*b* −0.030185 0.051400 0.226095 0.031457 Δ*c*/*c*

−0.046523 −0.049664 0.228457 0.099234 Δ*a*/*a* −0.039959 0.004717 0.164141 0.084533 Δ*b*/*b* −0.05896 −0.033813 0.210681 0.093388 Δ*c*/*c*

**Table 4.** Scanning coefficients *a\**, *b\**, *c\**, and *d\** at different temperatures in SbSI crystal when crystal is not deformed

**Table 5.** Temperature dependence of Sb atom's equilibrium positions displacements' Δ*z*31, Δ*z*32, and Δ*z*33 along *x*(*a*),

present Δ*z* dependence on temperature, when crystal is being deformed equally along three

*<sup>r</sup>* and *r* = const., then *e ≈* Δ*P ≈* Δ*z*. Consequently, using the data from **Table 5** we

*y*(*b*), and *z*(*c*) axes when deformation is *r =* Δ*a*/*a* = Δ*b*/*b* = Δ*c*/*c* = 0.03, respectively. Δ*z* = 1 a.u. = 0.53 Å.

where *T* < *TC* is the temperature of the SbSI crystal in ferroelectric phase.

Since *<sup>e</sup>* <sup>=</sup> *<sup>Δ</sup><sup>P</sup>*

94 Piezoelectric Materials

axes: *r =* Δ*a*/*a* = Δ*b*/*b* = Δ*c*/*c* (**Figure 6**)*.*

(*r* = 0) and deformed along *x*(*a*), *y*(*b*), and *z*(*c*) axes, when *r =* Δ*a/a =* Δ*b/b =* Δ*c/c =* 0.03.

*T***, K Δ***z***31, a.u. Δ***z***32, a.u. Δ***z***33, a.u.** 300 0 0 0 0.0759 −0.068310 0.097 0.0487 −0.04383 0.0694 0.0329 −0.02961 0.0384 0.0228 −0.02052 0.0284 0.0173 −0.01557 0.0219 0.0142 −0.01278 0.019 0.0141 −0.01269 0.0158 0.0124 −0.01116 0.0152 0.0107 −0.00963 0.0137

**Figure 6.** Temperature dependence of Sb atom's equilibrium position's displacements' Δ*z*31, Δ*z*32, and Δ*z*33 along *x*(*a*), *y*(*b*), and *z*(*c*) axes when deformation of SbSI crystal is *r* = Δ*a*/*a* = Δ*b*/*b* = Δ*c*/*c* = 0.03, respectively, Δ*z* = 1 a.u.

**Figures 4** and **6** show that close to *TC* potential energy of Sb atoms *V*¯ *<sup>p</sup>*(*z*) and Sb atom's equilibrium position's displacements Δ*z*31, Δ*z*32, and Δ*z*<sup>33</sup> are very sensitive to the temperature changes and deformation of SbSI crystal. In temperature range *T* < 250 K, the sensitivity of *V*¯ *<sup>p</sup>*(*z*) and Δ*z*31 ~ *e*31, Δ*z*32 ~ *e*32, Δ*z*33 ~ *e*<sup>33</sup> to their temperature changes and deformation directly decreased.

In **Figure 7**, we compared theoretical results of Sb atoms Δz33 ~ e33 from **Table 5** with experi‐ mental piezoelectric modulus *e*33 = *d*33/*s* from [2, 11] dependence on temperature when crystal is deformed in direction of *z*(*c*)-axis.

**Figure 7.** Temperature dependence of Sb atom's equilibrium position's displacement ∆*z*33 along *z*(*c*)-axis and experi‐ mental [2, 11] piezoelectric modulus *e*33 = *d*33/*s*. Δ*z* = 1 a.u.

As seen from **Figure 7**, the temperature dependences of Sb atom's equilibrium position's displacements Δ*z*33 ~ *e*33 coincide well with experimental piezoelectric modulus *e*33 = *d*33/*s* when crystal is deformed in direction of *z*(*c*)-axis because Δ*z*33 is created by anharmonic Sb atom's potential energy *V*¯ *<sup>p</sup>*(*z*).

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 compositional inhomogeneity or the existence of the internal mechanical stresses.
