**13. Core-level splitting in the antiferroelectric and ferroelectric phases**

The mechanisms of the XPS splitting in SbSI crystals were discussed by Grigas et al. [39]. After the breaking of the crystals under high vacuum conditions some bonds of atoms (see **Fig‐ ure 15b**) at the surface become open.

Employing the UHF method [40], the energy of CL and ionic charges of atoms Sb and S were calculated using the SbSI molecular cluster model. **Figure 22** demonstrates the calculated ionic charges of atoms Sb and S along a molecular cluster of 20 SbSI molecules.

**Figure 22.** Ionic charges *q* of atoms Sb (a) and S (b) along SbSI molecular cluster in AFEP (*T* = 300 K). Ionic charges *q* of atoms Sb in FEP: *T* = 273 K (c) and *T* = 215 K (d).

As it is seen from **Figures 22(a)** and **(b)**, the ionic charges at the edges of the SbSI cluster in AFEP considerably differ from those in the bulk of the sample. Besides, the differences of charges Δ*qi* between Sb3 and Sb4 and between S3 and S4 have increased at the cluster edges. The differences of Sb3 and Sb4 are equal at both edges in AFEP and unequal in FEP.

The difference of charge Δ*qi* at the cluster edges forms the binding energy difference (Δ*E*clust = *E*max − *E*min) between atoms Sb3 and Sb4, S3 and S4, as well as I1 and I2 (**Table 9**). Moreover, the differences (Δ*E*clust = *E*max − *E*min) at both cluster edges are equal in AFEP and unequal in FEP.


**Table 10.** The energies of CL of isolated neutral Sb, S, and I atoms and isolated Sb+1, Sb+2, Sb+3, S−1, S−2, and I−1 ions.

Pursuing to explain the splitting of the CL, the eigenvalues of isolated neutral Sb, S, and I atoms and isolated Sb+1, Sb+2, Sb+3, S−1, S−2, and I−1 ions have been calculated by the Hartree-Fock method using the *N21* orbital basis set (**Table 10**).

As shown in **Table 10**, the eigenvalues of isolated neutral atoms Sb and S and isolated ions Sb +1 and S−1 differ for all states. Besides, as demonstrated in **Table 10**, 3d state energies of an isolated Sb atom and Sb+1 ion vary by Δ*E* = 9.6 eV. The difference for 2p states of S and S−1 makes Δ*E* = 11 eV. Consequently, Δ*E* is proportional to ion charges Δ*qi* . The ion charges at the edge of the SbSI cluster vary between two Sb4 (and between two Sb3) by Δ*qi* = 0.23 a.u. and by Δ*qi* = 0.18 a.u. between two S4 (and between two S3; **Figure 22**). Hence, the energy difference Δ*E*\*clust of the 3d state between two Sb atoms and of the 2p state between two S atoms at the edges of an SbSI cluster can be calculated as follows:

$$
\Delta E\_{\text{class}}^{\*} = \frac{\Delta q\_{i}}{q\_{i}} \Delta E \tag{32}
$$

where Δ*E* is the difference between energies and *qi* is the difference between ionic charges of isolated atoms (**Table 10**).

The proportion between Δ*E*\*clust and Δ*qi* according to Eq. (33) allows us to explain the splitting, Δ*E*exp, of the CL in XPS. So in AFEP XPS along the *c*-axis, Δ*E*\*clust and Δ*qi* are equal at both edges of the cluster (**Figures 22a** and **b**). However, in FEP their values are different. At phase transition, Δ*E*\*clust changes differently at the left and right edges of the cluster (**Figures 22c** and **d**). Thus, at the left edge Δ*E*\*clust (AFEP) > Δ*E*\*clust (FEP) and at the right edge Δ*E*\*clust (AFEP) < Δ*E*\*clust (FEP). On the other hand, Δ*E*\*clust is in good agreement with the difference Δ*E*clust = *E*max − *E*min of the same CL energies of atoms at the edges of the cluster. For example, from **Table 4** we get Δ*E*clust = 1.93 eV between 3d of Sb3 and Sb4, 2.77 eV between 3d of I1 and I2, and 1.88 eV between 3d of S3 and S4 in AFEP (**Table 11**).

**Figure 22.** Ionic charges *q* of atoms Sb (a) and S (b) along SbSI molecular cluster in AFEP (*T* = 300 K). Ionic charges *q* of

As it is seen from **Figures 22(a)** and **(b)**, the ionic charges at the edges of the SbSI cluster in AFEP considerably differ from those in the bulk of the sample. Besides, the differences of charges Δ*qi* between Sb3 and Sb4 and between S3 and S4 have increased at the cluster edges.

(Δ*E*clust = *E*max − *E*min) between atoms Sb3 and Sb4, S3 and S4, as well as I1 and I2 (**Table 9**). Moreover, the differences (Δ*E*clust = *E*max − *E*min) at both cluster edges are equal in AFEP and

at the cluster edges forms the binding energy difference

The differences of Sb3 and Sb4 are equal at both edges in AFEP and unequal in FEP.

**State Energy (eV) State Energy (eV) State Energy (eV)** Sb 3d 562.41 S 2p 180.54 I 3d 657.86 Sb+1 3d 571.97 S−1 2p 169.58 I−1 3d 649.00

**Table 10.** The energies of CL of isolated neutral Sb, S, and I atoms and isolated Sb+1, Sb+2, Sb+3, S−1, S−2, and I−1 ions.

Pursuing to explain the splitting of the CL, the eigenvalues of isolated neutral Sb, S, and I atoms and isolated Sb+1, Sb+2, Sb+3, S−1, S−2, and I−1 ions have been calculated by the Hartree-Fock

atoms Sb in FEP: *T* = 273 K (c) and *T* = 215 K (d).

The difference of charge Δ*qi*

Sb+2 3d 58,175 S−2 2p 160.23

method using the *N21* orbital basis set (**Table 10**).

unequal in FEP.

112 Piezoelectric Materials

Sb+3 3d 593.13


**Table 11.** The calculated splitting of the deep CL Δ*E*\*clust and Δ*E*clust in SbSI cluster and experimentally observed Δ*E*exp of XPS levels in SbSI crystal.

As it is shown in **Table 11**, the experimentally observed Δ*E*exp of the XPS levels in an SbSI crystal is in good agreement with the calculated splitting of the deep CL Δ*E*\*clust and Δ*E*clust in an SbSI cluster. Thus, it can be concluded that both Δ*qi* and Δ*E*clust and the splitting of CL in an SbSI crystal are sensitive to the displacements of equilibrium position *z*<sup>0</sup> (depending on the temperature change).
