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The SbSI crystals are investigated in the paraelectric and ferroelectric phases. The calculations have been performed to find the symmetry and normal coordinates of vibrational modes. We have observed that potential energy with double well create the soft mode of B1u symmetry in the microwave range and semisoft modes in the IR range. The Au and Bg symmetry, top electronic levels of the highest valence band, are degenerate in the paraelectric phase. It is shown that the Jahn-Teller effect is caused by Au symmetry normal mode interacting with the degenerate Au symmetry electronic states in the valence band top. The pseudo-Jahn-Teller effect is induced due to the same mode interacting with Au symmetry electronic states in the valence band and Bg symmetry states in the conduction band bottom. Concerning these two effects, the normal mode force constant K decreases and vibrational constants undergo changes during the phase transition. The theoretical deformation along the crystallographic—x(a), y(b), and z(c)—axes were studied for Sb atoms. The major change of piezoelectric modulus takes place in the ferroelectric phase near the phase transition temperature. At lower temperatures piezoelectric modulus changes become slow. The value as well as anomalous temperature dependence of piezoelectric modulus and Δz(T) is influenced by the change of mean potential energy V¯pz of Sb atoms in soft mode.
Faculty of Science and Technology, Lithuanian University of Educational Sciences, Vilnius, Lithuania
Leonardas Žigas*
Faculty of Science and Technology, Lithuanian University of Educational Sciences, Vilnius, Lithuania
Raimundas Sereika
Faculty of Science and Technology, Lithuanian University of Educational Sciences, Vilnius, Lithuania
Raimundas Žaltauskas
Faculty of Science and Technology, Lithuanian University of Educational Sciences, Vilnius, Lithuania
*Address all correspondence to: leonardas.zigas@leu.lt
1. Potential energy of normal vibrational modes
1.1. Introduction
SbSI has a classical ferroelectric phase (FEF) transition at the temperature T_{C} = 295 K [1, 2]. This crystal has large interest to the fundamental physical properties as well as for the promising application possibilities. It is known that it has large temperature and electric field dependence of the optical characteristics [3], electronic structure [4, 5], and other features. Petzelt [6] observed in the IR reflectivity spectra a strong temperature dependence of the lowest-frequency dielectric constant. Sugawara et al. [7] has determined strong temperature dependence of reflectivity for E||c in the range of 0–70 cm^{−1}. According to the author [7, 8], the frequencies of the soft mode are 4.0 and 6.5 cm^{−1} at temperatures 298 and 318 K in the IR range of SbSI spectrum. Although some authors [9] demonstrate result
n (≈90%) of the frequency ≈ 9 cm^{−1} to the static dielectric constant
at the room temperature, others claim the soft mode with frequency 10 cm^{−1} to be insignificant [10]. The soft mode consisting of two components has been found in the SbSI [11], SbSI-SbSeI [12], SbSI-BiSI [13], and SbSI-SbSBr systems [14] in the microwave range. The first component is a soft microwave mode bringing the major contribution in , and the second is a semisoft mode in the IR range and contributes to , which is less than 10% [15].
In [16] we have described the symmetrical and normal coordinates which were used to calculate electronic potential. As known, in the adiabatic approximation the vibration energy E = T + V of a system of atoms can be expressed via normal coordinates Q as follows:
T=12∑kdQkdt2,V=12∑kλkQk2.E1
The force constants λ_{k} are solution of the characteristic equation
λaij-bij=0E2
where a_{ij} and b_{ij} are factors of the energy expression in orthogonal symmetry coordinates F_{i}:
T=12∑ijaijdFidFjdt,V=12∑ijbijFiFj.E3
Symmetry coordinates are defined as particular combinations of Cartesian components x_{i}, y_{i}, and z_{i} of displacements [17]. Similarly to the vibrations of Sb_{2}S_{3}, the ones of SbSI belong to the same irreducible representations Γ_{a}, since the space groups of both crystals are D2h16 in the paraelectric phase and C2ν9 in the ferroelectric phase. Symmetry operations of the space group D2h16 are S1÷S8 [18, 19]. Some of them, such as S_{1}, S_{3}, S_{6}, and S_{8}, are connected with the space group C2ν9. The effect of these symmetry operations is the same in the unit cell that consists of four molecules (SbSI)_{4} as in the Sb_{2}S_{3} unit cell [20] (Table 1).
Symmetry operations of the space groups D2h16 and C2ν9i n SbSI crystals.
The operator of projection is
ɛijα=fαg∑kΓαSkijSkE4
The Cartesian components K_{An} = x_{An}, y_{An}, z_{An} of atom displacements for all their linear combinations have certain symmetry properties with respect to operations of groups D2h16 and C2ν9:
KAmSΓα=∑manmKAn.E5
Here A = Sb, S, I; n = 1, 2, 3, 4; a_{nm} = ±1. These combinations of K_{An} are basis functions of Γ_{a}.
Group D2h16 has irreducible representations A_{ig(u)}, B_{ig(u)}. Here, the capital letters denote coordinates symmetric (A) or antisymmetric (B) with respect to all rotation C2z, C2y, andC2x, Table 2 [21]. Subscripts denote the coordinates symmetric (g), or antisymmetric (u) with respect to inversion I. The coordinates symmetric with respect to separate rotations around x, y, and z axes differ from each other by index i = 1, 2, 3. The obtained K_{An} combinations are also the basic functions of irreducible representations A_{i}, B_{i} of the space group C2ν9.
D2h16
S1
S2
S3
S4
S5
S6
S7
S8
E
C2y
C2z
C2x
I
σ _{y}
σ _{z}
σ_{x}
σ
C2ν9
S1
S3
S6
S8
r 1
A_{1g}
r 1
A1
1
1
1
1
1
1
1
1
r 2
A_{1u}
r 2
A2
1
1
1
1
−1
−1
−1
−1
r 3
B_{2g}
r 3
B1
1
−1
1
−1
1
−1
1
−1
r 4
B_{2u}
r 4
B2
1
−1
1
−1
−1
1
−1
1
r 5
B_{1g}
r 2
A2
1
1
−1
−1
1
1
−1
−1
r 6
B_{1u}
r 1
A1
1
1
−1
−1
−1
−1
1
1
r 7
B_{3g}
r 4
B2
1
−1
−1
1
1
−1
−1
1
r 8
B_{3u}
r 3
B1
1
−1
−1
1
−1
1
1
−1
Table 2.
Group characters of irreducible representations of space groups D2h16 and C2ν9 [21].
Orthogonal symmetry coordinates of the entire unit cell can be formed of KAmSΓα by solving equations of orthonormality and are uniform with respect to A. Their expressions are as follows:
FSK,Γa=NS∑A∑mCmsKAmSΓαE6
Normalization factors N_{s} and coefficients C_{n} are presented in z direction and x, y directions in Table 3 [22]. The group theoretical analysis reveals that they are the unit cell of D2h16 point group with 12 atoms per cell and contains 33 optical modes and 3 acoustic modes. Among the optical modes, there are 18 Raman-active modes with the symmetries 6A_{g} + 6B_{1g} + 3B_{2g} + 3B_{3g}, 3 silent modes belong to the A_{u} irreducible representation, as well as 12 infrared active modes that have symmetries 2B_{1u} + 5B_{2u} + 5B_{3u}. In z direction, only 2B_{1u} modes are infrared active in the paraelectric phase (Table 3), whereas in the same direction, the modes 2A_{1} + 3A_{2} + 3B_{1} + 3B_{2} are Raman active in the ferroelectric phase. Note that 2A_{1} + 3B_{1} + 3B_{2} are also infrared active among these modes.
Taking into account the fact that SbSI crystal is made of two double chains with a weak interaction between them at T = 0 K (20–30 times weaker than within a chain; see Figure 1), the formation of dynamical matrix 3 becomes easier.
Γ_{α}
S
N_{S}
Sb_{1}
Sb_{2}
Sb_{3}
Sb_{4}
S_{1}
S_{2}
S_{3}
S_{4}
I_{1}
I_{2}
I_{3}
I_{4}
B_{1u}D2h16
1
112
+1
−1
−1
+1
+1
+1
+1
+1
+1
+1
+1
+1
A_{1}C2v9
2
18
+1
+1
+1
+1
−1
−1
−1
−1
0
0
0
0
3
124
+1
+1
+1
+1
+1
+1
+1
+1
−2
−2
−2
−2
B_{2g}D2h16
4
112
−1
+1
+1
−1
−1
+1
+1
−1
+1
−1
−1
+1
B_{1}C2ν9
5
18
−1
+1
+1
−1
+1
−1
−1
+1
0
0
0
0
6
124
−1
+1
+1
−1
−1
+1
+1
−1
−2
+2
+2
−2
B_{3g}D2h16
7
112
−1
+1
−1
+1
−1
+1
−1
+1
−1
+1
−1
+1
B_{2}C2v9
8
18
−1
+1
−1
+1
+1
−1
+1
−1
0
0
0
0
9
124
−1
+1
−1
+1
−1
+1
−1
+1
+2
−2
+2
−2
A_{u}D2h16
10
112
+1
+1
−1
−1
+1
+1
−1
−1
−1
−1
+1
+1
A_{2}C2v9
11
18
+1
+1
−1
−1
−1
−1
+1
+1
0
0
0
0
12
124
+1
+1
−1
−1
+1
+1
−1
−1
+2
+2
−2
−2
Table 3.
Symmetry coordinates of normal modes of SbSI crystals in z(c) direction.
In our research, force constants for a long chain of strongly bound atoms in the direction of z(c)-axis were defined by composing five simplified unit cells (30 atoms). The GAMESS program in the basis sets of atomic functions 3G, 3G + d, 21G, 21G + d were used to calculate force constants.
When atoms are vibrating, interaction between double chains strengthens. Therefore, following Furman et al. [23], we calculated normal models employing the Born-von Karman model and the elementary cell of four pseudomolecules SbSI (12 atoms). The binding constants and ionic charges, 0.19 (Sb), −0.01 (S), −0.18 (I), were also calculated. It appeared that covalent interactions prevailed in the SbSI crystal.
3. The lattice anharmonism on the vibrational spectrum along the z(c) direction
In [24] we have proposed the method for calculating the total potential energy. The potential energy (PE) is defined as follows:
Vr=4πΩ∑αss‐2fαsexp−ir+Rα+Qα⋅sexp−Mαs,E7
where Ω stands for the volume of the unit cell, Q represents the normal coordinate, R_{α} denotes the radius-vector of the atom position in the unit cell, s means the reciprocal lattice vector, α represents the number and kind of the atom in the unit cell, and exp[−M_{α}(s)] represents the Debye-Valler factor that is determined by the mean square amplitudes of atomic displacements (i.e., by the crystal temperature).
In fact, Eq. (7) at a certain point in the unit cell is the PE in terms of normal coordinates (or symmetrized plane waves). The atomic scattering form factor is
fαs=∑nlmnlmexp‐ir⋅snlm,E8
where nlm represents a set of quantum numbers for the atom α. It is noteworthy that the functions < nlm of all electronic states of an atom was used for calculating the form factors f_{α} (s) by Eq. (8). About 5000 vectors s were used for the sum in Eq. (7).
For SbSI, we studied the dependence of potential energy V(z) in the paraelectric phase at T = 420 K on the normal coordinates of all D2h16 symmetry normal modes that formed by displacements of the atoms along the z direction (along the polar axis c; Figure 2). This dependence was calculated by Eqs. (7) and (8). We determined that the curves V(z) can be described by the equation
Vz=V0+az+bz2+dz3+cz4,E9
where the coefficients a = d = 0 in the paraelectric phase. The displacements have been chosen in regard to the equilibrium positions of atoms. The curves V(z) of symmetry B_{1u}(3), B_{2g}(4, 5), B_{3g}(7), and A_{u}(10) are correspondent to coefficients b < 0 and c > 0. These curves V(z) have two side minima at distances of 0.3 a.u. from the equilibrium position. A potential barrier ΔV = b^{2}/4c separates the minima. Conversely, the curves V(z) of symmetry B_{lu}(2), B_{2g}(6), B_{3g}(8), B_{3g}(9), A_{u}(11), and A_{u} (12) have a single minima and the coefficients b > 0. The coefficient is b ≈ Mω^{2}, 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 ΔV. 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 depend on the temperature.
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 Germany in Karlsruhe University using the Bruker Fourier spectrometer.
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_{1u}(3), B_{2g}(4), B_{2g}(5), and B_{3g}(7) with a double-well V(z) are strongly anharmonic, whereas modes B_{1u}(2), B_{2g}(6), B_{3g}(8), and B_{3g}(9) with a single-well V(z) appear to be weakly anharmonic. The vibration frequencies of the former modes are lower compared to the latter ones. Therefore, the R(k) peaks are created by strongly anharmonic modes B_{1u}(3) → A_{1}, B_{2g}(4) → B_{1}, B_{2g}(5) → B_{1}, and B_{3g}(7) → B_{2} in the range of k = 10–100 cm^{−1} of the IR spectrum. In the range of k = 100–400 cm^{−1}, the R(k) peaks are created by weakly anharmonic modes. Following the optical selection rules, one R(k) peak in the paraelectric phase is created by the mode B_{1u}(2), while the peaks in the ferroelectric phase are created by modes B_{1u}(2) → A_{1}, B_{2g}(6) → B_{1}, B_{3g}(8) → B_{2}, and B_{3g}(9) → B_{2}. In the paraelectric phase, there are three silent modes of A_{u} symmetry: A_{u}(10), A_{u}(11), and A_{u}(12) (Table 3). Two of them, A_{u}(11) and A_{u}(12), comply with the out-of-phase motion of the infrared-active modes. It is expected that the anharmonic mode A_{u}(10) has a low frequency. This mode can be optically active and may cause a weak peak in the k <10 cm^{−1}IR range.
SbSI is a piezoelectric crystal that has a high-volume piezoelectric modulus d_{v} = 10 × 10^{−10} C/N and electromechanical coupling coefficient k_{3} = 0.87 [25]. The relation among polarization P, piezoelectric modulus e, d, deformation r = Δl/l, and elastic compliance coefficient is as follows:
ΔP=er,ΔP≈qΔz,E10
d=es,E11
where q is the atom’s ionic charge, s is the elastic compliance coefficient, Δz = z – z_{0}, z_{0} is the atomic equilibrium position, when crystal is not deformed (r = 0), and z is the atomic equilibrium 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 = Δa/a = Δb/b = Δc/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 d_{h} (the so-called “hydrostatic piezoelectric modulus”) were known. As d_{h} 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_{32} coefficient is of nearly the same value as d_{31} but opposite in sign. The high values of d_{h} = 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:
Also, the coordinates of all S and I atoms changes according to the equations: R_{α} = R_{0, α} + Q_{α}, 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_{α}= z_{α} by small steps from –Q_{α}(max) to +Q_{α}(max). The atomic form factor is
fαs=∑nlmnlmexp‐ir⋅snlm,E13
where nlm is a set of quantum numbers for the atom α.
For theoretical investigations, we used the most sensitive to temperature and deformations soft mode’s V¯pz dependence on normal coordinates [29]. It is handy to expand V¯pz using polynomial:
V¯pz=V0+a*z+b*z2+d*z3+c*z4,E14
where a*, b*, d*, and c* are polynomial expansion coefficients.
The V¯pz dependence on normal coordinates along z(c)-axis in paraelectric phase (PEF) at T = 300 K is shown in Figure 4 while deforming along x(a), y(b), and z(c) axes. The V¯pz in PEF has a shape of double-well (b* < 0, c* > 0). The barrier height when crystal is not deformed is ΔU_{C}≈ kT_{C}, where k is the Boltzmann constant. As seen from Figure 4, in paraelectric phase ∆z are equal to zero, because V¯pz of Sb atomic soft mode is symmetrical and double-well. In paraelectric phase, Sb atoms vibrate between two minima of double-well potential V¯pz, provided deformation r ≠ 0 and r = 0.
Due to the decrease in the temperature in ferroelectric phase, double-well V¯pz 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_{0} 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, Δz = z – z_{0}, and V¯pz scanning coefficients a*, b*, d*, and c* are shown in Tables 4 and 5. Temperature dependence of equilibrium position z_{0} of Sb atoms in ferroelectric phase (when r = 0) is calculated using Eq. (8):
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 = Δa/a = Δb/b = Δc/c = 0.03.
T, K
Δz_{31}, a.u.
Δz_{32}, a.u.
Δz_{33}, a.u.
300
0
0
0
292
0.0759
−0.068310
0.097
289
0.0487
−0.04383
0.0694
274
0.0329
−0.02961
0.0384
250
0.0228
−0.02052
0.0284
225
0.0173
−0.01557
0.0219
195
0.0142
−0.01278
0.019
170
0.0141
−0.01269
0.0158
150
0.0124
−0.01116
0.0152
130
0.0107
−0.00963
0.0137
Table 5.
Temperature dependence of Sb atom’s equilibrium positions displacements’ Δz_{31}, Δz_{32}, and Δz_{33} along x(a), y(b), and z(c) axes when deformation is r = Δa/a = Δb/b = Δc/c = 0.03, respectively. Δz = 1 a.u. = 0.53 Å.
where T < T_{C} is the temperature of the SbSI crystal in ferroelectric phase.
Since e=ΔPr and r = const., then e ≈ ΔP ≈ Δz. Consequently, using the data from Table 5 we present Δz dependence on temperature, when crystal is being deformed equally along three axes: r = Δa/a = Δb/b = Δc/c (Figure 6).
Figures 4 and 6 show that close to T_{C} potential energy of Sb atoms V¯pz and Sb atom’s equilibrium position’s displacements Δz_{31}, Δz_{32}, and Δz_{33} are very sensitive to the temperature changes and deformation of SbSI crystal. In temperature range T < 250 K, the sensitivity of V¯pz and Δz_{31} ~ e_{31}, Δz_{32} ~ e_{32}, Δz_{33} ~ e_{33} to their temperature changes and deformation directly decreased.
In Figure 7, we compared theoretical results of Sb atoms Δz_{33} ~ e_{33} from Table 5 with experimental piezoelectric modulus e_{33} = d_{33}/s from [2, 11] dependence on temperature when crystal is deformed in direction of z(c)-axis.
As seen from Figure 7, the temperature dependences of Sb atom’s equilibrium position’s displacements Δz_{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 Δz_{33} is created by anharmonic Sb atom’s potential energy V¯pz.
Existence of the piezoelectric effect experimentally detected up to a temperature several degrees above the T_{C} (Figure 4). The effect above T_{C} (Figure 7) may be attributed to the compositional inhomogeneity or the existence of the internal mechanical stresses.
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} – K_{v} < 0, where K_{0} = 1/2 M 02 represents the bare force constant, while K_{v} stands for a vibrational coupling term, which is dependent on the vibrational coupling constant F and the energy gap ΔE. C2h2 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} = σ_{z}/(0,0,1/2) for the PEP space group, whereas only S_{1} and S_{2} are for the FEP space group C2h2. 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 K_{i} = X_{i}, Y_{i}, Z_{i} on the axes of local coordinate systems [31, 32]:
χjα=Nα∑icjiKi.E15
Γ_{α}C22
Γ_{α}C2h2
S_{1}
S_{2}
S_{3}
S_{4}
A
A_{
g
}
+1
+1
+1
+1
B
B_{
g
}
+1
−1
+1
−1
A
A_{
u
}
+1
+1
−1
−1
B
B_{u}
+1
−1
−1
+1
Table 6.
Characters of irreducible representations of symmetry space group C22, and C2h2.
The coefficients c_{ji} are developed from the properties of irreducible representations Γ_{a} obtained when coordinates χ^{(α)} with symmetry operations S_{i} are transformed (see Table 6 for the characters of the representations). Symmetry coordinates that are composed of projections Z_{i} of atomic displacements usually have symmetry A_{u} and B_{g}, whereas the coordinates that are composed of projections X_{i} and Y_{i} possess symmetry A_{g} and B_{u}. Consequently, by solving the system of vibrational equations, normal coordinates Qiα are obtained, in which X_{i} and Y_{i} are interlinked and Z_{i} participates separately. Thus, the normal vibrations in the SbSI chain along the z-axis can be characterized by A_{u} and B_{g} symmetry coordinates, whereas the vibrations in the direction of x- and y-axes are related to B_{u} and A_{g} symmetry coordinates. The modes X_{7}(B_{u}) and X_{13}(B_{u}) comply with acoustic vibrations, whereas the remaining ones correspond to the optical vibrations of the SbSI chain (Table 7).
Γ_{α}C22
Γ_{α}C2h2
N^{(α)}
Sb3
Sb4
S3
S4
I1
I2
K_{i}
A
A_{u}
16
+1
+1
+1
+1
+1
+1
z(1)
A
A_{u}
112
+1
+1
+1
+1
−2
−2
z(2)
A
A_{u}
12
+1
+1
−1
−1
0
0
z(3)
B
B_{g}
16
+1
−1
+1
−1
+1
−1
z(4)
B
B_{g}
112
+1
−1
+1
−1
−2
+2
z(5)
B
B_{g}
12
+1
−1
−1
+1
0
0
z(6)
B
B_{u}
16
+1
+1
+1
+1
+1
+1
x,y(7, 13)
B
B_{u}
112
+1
+1
+1
+1
−2
−2
x,y(8, 14)
B
B_{u}
12
+1
+1
−1
−1
0
0
x,y(9, 15)
A
A_{g}
16
+1
−1
+1
−1
+1
−1
x,y(10,16)
A
A_{g}
112
+1
−1
+1
−1
−2
+2
x,y(11, 17)
A
A_{g}
12
+1
−1
−1
+1
0
0
x,y(12, 18)
Table 7.
Vibrational symmetry coordinates in the simplified unit cell of the SbSI chain.
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 H_{w} 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 A_{u} and A_{g} 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 E_{g} = 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 A_{u} and B_{g} symmetry electronic levels in the valence band are highlighted by boldface, where the energies of level 177 B_{g} and level 178 A_{u} are −0.3344H, and the energies of level 179 A_{u} and level 180 B_{g} are −0.3257H (assuming that 1 H = 27.21 eV).
Valence band
Conduction band
Symmetry
Energy (H)
Symmetry
Energy (H)
176 B_{g}
−0.3363
181 A_{u}
−0.0926
177 B_{g}
−0.3344
182 B_{g}
−0.0925
178 A_{u}
−0.3344
183 A_{u}
−0.0847
179 A_{u}
−0.3257
184 B_{g}
−0.0838
180 B_{g}
−0.3257
185 A_{u}
−0.0821
Table 8.
One-electron energies in the conduction band bottom and valence and top for an SbSI chain containing N = 60 atoms.
Note: The degenerate electronic states in the valence band are denoted by boldface.
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_{g} (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).
The normal vibrational mode of symmetry A_{u} (further for clarity denoted here as Ã_{u}) interacts with the electronic states of symmetry A_{u} and B_{g} if A_{u}× B_{g} = Ã_{u}. The electron-phonon interaction involving the highest degenerate A_{u} and B_{g} symmetry levels of the valence band is called the Jahn-Teller effect. The pseudo-Jahn-Teller effect (PJTE) occurs when the normal Ã_{u} mode interacts with the A_{u} electronic state at the top of the valence band and with the B_{g} 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:
εf.e(Q)=12(K0−2F2Eg)Q2E16
where K_{0} represents nonvibronic part of the force constant (the bare force constant), K_{V} stands for the vibronic coupling term. K = K_{0} – (2 F^{2}/E_{g}) = K_{0} – K_{v} denotes the force constant considering the PJTE. F which is found from the following equation represents the vibronic coupling constant:
F2=ΔEEg2Q02E17
Here Q_{0} = z_{0} is the atomic displacement, and ΔE 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, ΔE = 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 2F^{2} = E_{g} may attain as much as 2H/Å.
In the case of the normal mode Ã_{u}, the PJTE and JTE vibronic mixing leads to the force constant
K=K0-2F2Eg+G=K0-KV*,E101
State
AFEP (T = 300 K)
FEP (T = 215 K)
Left edge of cluster
Right edge of cluster
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.
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 Ã_{u} vibrational mode has to compute. Separate energy contributions constitute the total energy E_{T}:
ET=EK+Eee+Ene+Enn,E18
where the kinetic energy of electrons is
EK=∑i=1n→Pi22mcE19
the interelectron interaction energy is
Eee=12∑i,j≠ine2r⃗i-r⃗j,E20
the electron-nuclear interaction energy is
Ene=-∑i,A=1n,Ne2ZAR⃗A-r⃗i,E21
and the internuclear interaction energy is
Enn=12∑A,B≠ANe2ZAZBR⃗A-R⃗B.E22
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 Ã_{u} mode has been computed on the displacements z_{α} 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 Ã_{u} mode from their equilibrium positions z_{0α}. Vibrational displacements z_{α} of atoms (α = Sb, S, I) from their equilibrium positions z_{0α} are identified employing the equation
zα=zα±uα1mα⋅i20E23
where i = 0–19, u_{α}mα-1/2 is the vibrational amplitude of atoms (u_{α} is the normal coordinates of the Ã_{u} vibrational mode). Normal coordinates of the Ã_{u} vibrational mode are found by diagonalization of the dynamical matrix [23].
By expressing E_{T} = f (z) as a fourth-order polynomial
ET=ET0+Kz2+cz4.E24
For the Ã_{u} mode in PEP (300 K), K = 2.05 H/Å^{2} and c = 2.3 H/Å^{4}. Since the obtained K value is positive, it is supposed that the electron-phonon interaction will only decrease K. The interaction itself will not induce a phase transition because under the influence of the electron-phonon interaction the Ã_{u} mode becomes only slightly anharmonic. However, due to phonon-phonon interaction in the SbSI chain, the Ã_{u} 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 Ã_{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 A_{u} 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_{α} of atoms (α = Sb, S, I) from their equilibrium positions equal to z_{α} = 0, Eq. (24) is transformed into
ET0=EK0+Eee0+Ene0+Enn0.E25
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}.
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}.
In the FEP in the temperature range 280–295 K, a sharp increase of the unit cell volume V_{0} 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 Figure 11 demonstrating the growth of E_{T0} can also be considered as caused solely by the variation of Sb and S equilibrium positions.
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_{0} 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_{0} = constant. Consequently, the rapid change of K and c marked by the solid lines in Figures 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 = mτ2, where m_{τ} is the reduced mass of the Ã_{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 s2 [14]. Therefore, the temperature dependence of the soft mode Ã_{u} frequency s2 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 s2 can be obtained provided the anharmonicity caused by phonon-phonon interaction is considered.
10. The electronic structure of SbSI cluster at the phase transition region
10.1. Introduction
As it is known [25, 34, 35], a ferroelectric phase transition of the first kind occurs in SbSI, though it is close to the phase transition of the second kind. What is more, it has been theoretically proved in Ref. [36] that the phase transition in SbSI takes an intermediate position between order-disorder and displacement types. In general, it is known that ferroelectric phase is at T < T_{C} = 295 K, antiferroelectric phase exists at T_{C} < T < 410 K, and paraelectric phase at T > 410 K.
Lukaszewicz et al. [17] determined the crystal structure of SbSI in the temperature region at 170–465 K. They found that there are three phases in the phase diagram of SbSI: ferroelectric phase (FEP) III (space group Pna2_{1}) below T_{C1} = 295 K, antiferroelectric phase (AFEP) II (P 2_{1}2_{1}2_{1}) in the temperature region (T_{C2} = 295–410 K), and high temperature paraelectric phase (PEP) I above 410 K. AFEP is characterized by the double polar chains [(SbSI)_{1}]_{2}, which are ordered antiferroelectrically: these cause a disorder in the crystal lattice. Experimental investigation of X-ray absorption, fluorescence, and electron emission [37, 38] adjusts the energy band structure of SbSxSe1−xI (x = 0.25; 0.5; 1), SbSBr. Its explanation necessitates for a more precise calculation of the contribution of separate atoms to the total density of states. The authors [39] had measured X-ray photoelectron emission spectra (XPS) of the SbSI crystals of valence bands (VB) and core levels (CL). XPS results showed the splitting of the CL. However, they have not explained theoretically the XPS spectra of the VB.
11. Ab initio method description
The UHF method in the MOLCAO approximation serves as the basis of the method of calculation. The molecular orbital (MO) (φ_{i}) can be expanded in the atomic orbital (AO) χ_{μ}(r) basis set
φir=∑μCiμχμr.E26
Here μ denotes the number of AO. Matrix C is obtained by solving the Hartree-Fock matrix equation
FC=ESC.E27
The roots of this secular equation lie in the molecular energies E_{i} of electrons, where F stands for the Fockian matrix and S for the matrix of overlap integrals:
Sμν=∫χμχνdrE28
The coefficients C_{iμ} allowed us to find the density matrix of the electron distribution:
Pμν=2∑i=1N/2CiμCiνE29
where N is the number of electrons, and the sum is over all occupied MO. We can find the bond order between the A and B atoms,
PAB=∑μ∈A∑ν∈BPμνE30
and the charge of atoms,
ρA=ZA-∑μ∈APμμE31
The calculations were performed by the GAMESS program [33]. Atomic coordinates have been used from the work of Lukaszewicz et al. [17].
12. Investigation of SbSI crystal valence band
The paraelectric structure (T > T_{C2} = 410 K) of SbSI is made of atom chains that belong to the paraelectric space group Pnam, which form square-pyramidal S_{3}I_{2} groups with the Sb ion at the center of the pyramid base. All atoms lie on mirror planes normal to the c-axis (Figure 15).
On passage into the AFEP (T_{C2} > T > T_{C1} = 295 K) and FEP (T < T_{C1}), the position parameters normal to the c-axis are substantially unchanged. In AFEP (T = 300 K) the displacements of equilibrium position occur for all Sb3 atoms along the c-axis z_{0} = 0, 02 Å and for all Sb4 atoms (z_{0} = −0, 02 Å) [37]. In FEP (T < T_{C1}) all Sb and S atoms move along the c-axis toward these I sites, which leads to the removal of the mirror plane symmetry. In FEP (T = 215 K), the displacements of equilibrium position occur for all Sb atoms along the c-axis z_{0} = 0, 2 Å.
The molecular cluster model of one SbSI crystal chain was used to perform the theoretical ab initio calculation of energy levels (Figure 15). As seen from Figure 16, energy levels only slightly change with the increase of the cluster. Figure 17 shows that the energy of some levels increases and of others decreases.
We have calculated the bond orders between the atoms Sb3-S4, Sb4-S3, Sb3-I1, and Sb4-I2 along a cluster in AFEP and FEP (Figures 18 and 19). When the phase transition takes place, bond orders change. As a consequence, VB experiences the broadening of some energy levels and the narrowing of others.
While the temperature decreases further in the FEP, Sb and S atoms change their positions relative to I atoms, and VB becomes broader. In Grigas et al. [39], the authors present the X-ray photoelectron spectra (XPS) of the VB and CL of the SbSI single crystals. The XPS are measured in the energy range 0–1400 eV and the temperature range 130–330 K. They compared experimentally obtained energies of CL with the results of theoretical ab initio calculations. However, the experimentally obtained energies of the VB were not compared with the results of theoretical ab initio calculations.
We have calculated the density of states of VB for Sb, S, and I atoms and the total molecular cluster in AFEP (T = 300 K) and FEP (T = 215 K). The density of states is
g=ΔNΔEE102
where ΔN is the number of states in the energy interval ΔE (eV). The dimension of g is (eV^{−1}). In Figure 20, the experimental results of XPS VB spectra of SbSI crystals are compared with the theoretically calculated total density of states of a molecular cluster that consists of 20 SbSI molecules in AFEP (T = 300 K) and FEP (T = 215 K) is the number of states in the energy interval ΔE (eV). The dimension of g is (eV^{−1}). In Figure 19, the experimental results of XPS VB spectra of SbSI crystals are compared with the theoretically calculated total density of states of a molecular cluster that consists of 20 SbSI molecules in AFEP (T = 300 K) and FEP (T = 215 K). The average total density of states is as follows:
<g>=1n∑i=1ngi,E103
where n is the number of normal modes (n = 15).
As seen from Figures 20 and 21, the experimental and theoretical results in AFEP and FEP are in good agreement in the energy range of 6–17 eV. In the range of 17–22 eV, there is a great difference between experimental and theoretical results.
The peaks of Figures 21(a) and (b) have no distinct shape in the XPS (at T = 300 K). This difference is explicable because the theoretical calculation does not take into account vibrational displacements of atoms. Average densities of states shown in Figure 21(c) are more similar to the experimental XPS spectrum than those from Figure 21(b). Comparison of Figures 20 and 21 shows the contribution of atomic states to the total density of states.
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 Figure 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.
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 Δq_{i} 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 Δq_{i} at the cluster edges forms the binding energy difference (ΔE_{clust} = E_{max} − E_{min}) between atoms Sb3 and Sb4, S3 and S4, as well as I1 and I2 (Table 9). Moreover, the differences (ΔE_{clust} = E_{max} − E_{min}) at both cluster edges are equal 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
Sb^{+2} 3d
58,175
S^{−2} 2p
160.23
Sb^{+3} 3d
593.13
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 ΔE = 9.6 eV. The difference for 2p states of S and S^{−1} makes ΔE = 11 eV. Consequently, ΔE is proportional to ion charges Δq_{i}. The ion charges at the edge of the SbSI cluster vary between two Sb4 (and between two Sb3) by Δq_{i} = 0.23 a.u. and by Δq_{i} = 0.18 a.u. between two S4 (and between two S3; Figure 22). Hence, the energy difference ΔE*_{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:
Eclust*=ΔqiqiE,E32
where ΔE is the difference between energies and q_{i} is the difference between ionic charges of isolated atoms (Table 10).
The proportion between ΔE*_{clust} and Δq_{i} according to Eq. (33) allows us to explain the splitting, ΔE_{exp}, of the CL in XPS. So in AFEP XPS along the c-axis, ΔE*_{clust} and Δq_{i} are equal at both edges of the cluster (Figures 22a and b). However, in FEP their values are different. At phase transition, ΔE*_{clust} changes differently at the left and right edges of the cluster (Figures 22c and d). Thus, at the left edge ΔE*_{clust} (AFEP) > ΔE*_{clust} (FEP) and at the right edge ΔE*_{clust} (AFEP) < ΔE*_{clust} (FEP). On the other hand, ΔE*_{clust} is in good agreement with the difference ΔE_{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 ΔE_{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).
State
ΔE (eV) of isolated neutral atoms and ions with q_{i} = 1 (a.u.)
The calculated splitting of the deep CL ΔE*_{clust} and ΔE_{clust} in SbSI cluster and experimentally observed ΔE_{exp} of XPS levels in SbSI crystal.
As it is shown in Table 11, the experimentally observed ΔE_{exp} of the XPS levels in an SbSI crystal is in good agreement with the calculated splitting of the deep CL ΔE*_{clust} and ΔE_{clust} in an SbSI cluster. Thus, it can be concluded that both Δq_{i} and ΔE_{clust} and the splitting of CL in an SbSI crystal are sensitive to the displacements of equilibrium position z_{0} (depending on the temperature change).
14. The energy gap of the SbSI crystals
14.1. Introduction
Investigation of the total density of states of SbSI crystals [41] has shown that the absolute valence band top is formed in both phases of 3p orbitals of S, while the absolute conduction band bottom of 5p orbitals of Sb.
We have undertaken an attempt of a more detailed calculation of the electronic structure and some properties of SbSI from the first principles using the empirical pseudopotential method [42]. The method for calculating the band structure of SbSI was employed in Refs. [43–45]. In Ref. [43], purely ionic and partially covalent models were applied, and an indirect energy gap of 2.28 eV at point S was obtained. Nevertheless, the accuracy turned out to be 0.2 eV. In Ref. [44], the pseudopotentials were corrected applying the data on direct gaps. The absorption band edge in the paraelectric phase was determined at 1.82 eV for E || cS5-6(V)→S7-8(C) and 1.91 eV for E﬩cS1-2(V)→S7-8(C). The valence band top corresponded to S5-6(V), whereas the conduction band edge complied with Z1(C) withthe energy of indirect gap of 1.41 eV. The energy gap sides matched to points Z and R in the ferroelectric phase. Analyzing the light reflection spectra, the indirect band gap of 1.82 eV between Γ6(V) and S1(V), as well as the minimum direct gap of 2.08 eV were obtained in reference [45].
In Ref. [45], a purely ionic model was assumed. The form factors of the pseudopotential were adjusted by fitting the calculated band gap values to the ones obtained both experimentally and theoretically by other authors. Form factors for Cl^{−} were used instead of I^{−}.
15. Investigation by using the pseudopotential method
The infinite crystal approximation, which states that the crystal properties obtained for the primary cell are extended then to the entire crystal employing periodical boundary conditions, has been employed. Hence, within the primary cell, the position of the jth atom of kind α(α = Sb, S, or I) is
Rj(α)=Rcell+τj(α),E33
where R_{cell} is the translation vector of the orthorhombic system. The vectors τj(α) describe onprimitive translations that may be written as
τjα=λjαtx+μjαty+νjαtz,E34
where t_{x}, t_{y}, and t_{z} are the lattice parameters along the three coordinate axes. The pseudopotential of the crystal has been selected as a sum of atomic pseudopotentials:
V(r)=Rcell∑α∑Rcell∑jναr-Rcell-τj(α).E35
Here, ν_{α}(r) is the atomic pseudopotential of atom α. It is assumed to be localized and energy independent. Weakness of the pseudopotential V^{ps}(r) provides good convergence of the pseudo-wave functions expanded in terms of plane waves:
Ψkipsr=∑GCkiGeik+Gr,E36
where G is the reciprocal lattice vector, C_{ki}(G) are Fourier coefficients. The Schrödinger equation in the pseudopotential method has the form
H+VpsΨkips=EkiΨkips.E37
The pseudo-wave functions of the valence electrons Ψkips(r) approach the true ones outside the core region, though they show no oscillations within the core. The Fourier component of this potential (or the matrix element in the plane-wave basis set) is expressed
VG=∑αSαGΩαΩcναG,E38
where Ω_{α} and Ω_{c} are the volume of the corresponding atom and of the primary cell as a whole. The atomic structural factor S_{α} (G) and atomic form factor ν_{α}(G) are defined as
SαG=∑αe-iGτj(α)E39
ναG=1Ωα∫να(r)e-iGrd3rE40
The integrals are taken over the whole volume Ω_{α}. Provided the atomic coordinates within the cell are known, the atomic structural factor can be readily evaluated. The pseudopotential forms factors for Sb, S, and I, so the following equation [39, 43] was preliminary determined to be used:
ναG=4πΩαG-2∑nlmnlmexp-i(Gr)nlmE41
where nlm denotes the set of electron quantum numbers. Neutral functions for Sb, S, and I were involved by the numerical evaluation of the form factors [43, 45, 46]. It is significant that the agreement between the theoretical and experimental data shall be obtained solely employing neutral functions. It is noteworthy that some other authors [47, 48], who theoretically studied the band structure of the SbSI crystal, employed an ionic model of chemical bonding (Sb^{+3} S^{−2} I^{−1}) instead.
No
Point
Coordinates
kx
ky
kz
1
Γ
0
0
0
2
Δ
0
0.25
0
3
Y
0
0.5
0
4
Λ
0
0
0.25
5
ZTY Γ
0
0.25
0.25
6
H
0
0.5
0.25
7
Z
0
0
0.5
8
B
0
0.25
0.5
9
T
0
0.5
0.5
10
Σ
0.25
0
0
11
ΓYSX
0.25
0.25
0
12
C
0.25
0.5
0
13
ΓXUZ
0.25
0
0.25
14
O
0.25
0.25
0.25
15
TYSR
0.25
0.5
0.25
16
A
0.25
0
0.5
17
ZTRU
0.25
0.25
0.5
18
E
0.5
0.5
0.5
19
X
0.5
0
0
20
D
0.5
0.25
0
21
S
0.5
0.5
0
22
G
0.5
0
0.25
23
URSX
0.5
0.25
0.25
24
Q
0.5
0.5
0.25
25
U
0.5
0
0.5
26
F
0.5
0.25
0.5
27
R
0.5
0.5
0.5
Table 12.
Coordinates of the special points in the irreducible part of the Brillouin zone of SbSI (in relative units), see also Figure 23.
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^{+0.3}S^{−0.2}I^{−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 points Q and C (energy variation was 0.92 and 0.86 eV, respectively).
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]. The conduction band bottom in both phases is located at point Z1(C), the valence band top in antiferroelectric phase is at the point U5-6(V), and in ferroelectric phase at R3-4(V). Our experimental and theoretical results of the SbSI crystal electronic structure sometimes differ from those obtained by other authors.
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=K0expσ(EK-E0)kT,E42
here
σ=σ02kTħ0thħ02kTE43
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_{0} represents characteristic “gap” of the forbidden band, and E_{K} denotes light quantum energy for a particular absorption coefficient K. During the phase transition, the characteristic “gap” differs:
ΔE0=E0F–E0AF,E44
where E_{0F} and E_{0AF} stand for the values of the energy “gap” in ferroelectric (F) and antiferroelectric (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 antiferroelectric phase and it is experimentally defined. Different temperatures are matched finding the point of crossing of the curve lnK (E). In ferroelectric phase, K_{0F} is determined as follows:
lnK0F=lnK0-γσkTPS2AFE45
where lnK_{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), ΔE_{0} = 0.12 ± 0.02 eV. The experimental ΔE_{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).
17. Conclusions
The potential energy of A_{u}(10), B_{1u}(3), B_{2g}(4), B_{2g}(5), and B_{3g}(7) normal modes in the paraelectric phase are anharmonic with a double-well V(z), while B_{1u}(2), B_{2g}(6), B_{3g}(8), and B_{3g}(9) modes possess only one minimum. The semisoft modes B_{1u}(3) → A_{1}, B_{2g}(4) → B_{1}, B_{2g}(5) → B_{1}, and B_{3g}(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_{1u}(2), but in the ferroelectric phase the R(k) peaks are caused by modes B_{1u}(2) → A_{1}, B_{2g}(6) → B_{1}, B_{3g}(8) → B_{2}, and B_{3g}(9) → B_{2}. It has been determined that strong lattice anharmonisity, as well as interaction between chains, can split the mode B_{1u}(3) into two components, from which one is soft in the microwave range and the other B_{1u}(3) is semisoft in the IR range. The semisoft modes B_{1u}(3), B_{2g}(4), B_{2g}(5), and B_{3g}(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_{T} dependence upon the Ã_{u} normal mode coordinate. The Ã_{u} mode temperature dependence of the coefficient K = mτ2 is similar to the temperature dependence of the soft mode frequency,
.
Acknowledgments
This works was supported by Science Foundation from Lithuanian University of Educational Sciences.
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Written By
Algirdas Audzijonis, Leonardas Žigas, Raimundas Sereika and
Raimundas Žaltauskas
Submitted: October 30th, 2015Reviewed: May 12th, 2016Published: August 24th, 2016