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Variation of the energy gap of the SbSI crystals at ferroelectric phase transition
ARTICLE IN PRESS
Physica B 371 (2006) 68–73 www.elsevier.com/locate/physb
Variation of the energy gap of the SbSI crystals at ferroelectric phase transition A. Audzijonisa,, R. Zˇaltauskasa, L. Zˇigasa, I.V. Vinokurovab, O.V. Farberovichb, A. Pauliukasa, A. Kvedaravicˇiusa a Department of Physics, Vilnius Pedagogical University, Studentu 39, 08106 Vilnius, Lithuania Voronezh State University, Solid state physics department, Universitetskaya Pl. 1, 394693 Voronezh, Russia
b
Received 31 August 2005; received in revised form 21 September 2005; accepted 21 September 2005
Abstract Variation of the forbidden gap of SbSI crystals in the phase transition region is analyzed on the pseudopotential method for antiferroelectric and ferroelectric phase. The band gap at several special points of the Brillouin zone and some characteristic parameters of the band are considered. During the phase transition, the most significant changes are observed with the valence band top at points Q, C, R, H, E and with the conduction band bottom at points H, T and E of the Brillouin zone. At the ferroelectric phase transition, the valence and conduction bands change due to displacement of Sb and S atoms with respect to I and with respect to each other as a result of order–disorder and displacement-type transition. The obtained band gap values agree quite well with the experiment. This is apparently due to application of neutral rather than ionic atomic functions and inclusion of sufficiently many plane waves in the basis set for calculation. r 2005 Elsevier B.V. All rights reserved. PACS: 71.20.b; 77.80.Bh Keywords: SbSI; Electronic band structure; Ferroelectric phase transition; Pseudopotential method
1. Introduction The great interest to the A5B6C7-type ferroelectrics is caused by the fact that they possess semiconductor properties at the same time. The most outstanding representative of this class of compounds is sulfoiodide of antimony SbSI. It is characterized by a number of unique properties, e.g. it has the largest piezomodulus among the compounds of its class, it is by an order more sensitive to the external pressure, etc. Such a combination of SbSI properties opens up wide possibilities to employ it in practice. SbSI is used for developing optical light modulators, electroacousto-optical transformers, piezoelements, sensitive low-pressure gauges, and so on. Besides the practical use, SbSI is an interesting object of fundamental research. Be it as it may, SbSI is the most studied Corresponding author. Tel.: +370 5 2 79 00 43; fax: +370 5 2 75 17 32.
E-mail address: [email protected] (L. Zˇigas). 0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.09.039
compound of its kind. At present there is much information on its physical properties [1,2]. All the experimental results require theoretical interpretation. Therefore, it is important to choose some theoretical method that would provide calculation of the properties of such crystals on the same basis and yielding the results that would be in sufficient agreement with the experiment. As known [2,3], a ferroelectric phase transition of the first order takes place in SbSI, though it is close to the phase transition of the second order. Moreover, it has been proved theoretically in Ref. [4] that the phase transition in SbSI is intermediate between order–disorder and displacement types. At such a phase transition, the equilibrium position of atom Sb changes in accordance with the rules of order–disorder and displacement types, and that of atom S shifts by the rules of the displacement type. Investigation of the total density of states of SbSI crystals [5] has shown that the absolute valence band top is formed in both phases of 3p-orbitals of S, while the absolute conduction band
ARTICLE IN PRESS A. Audzijonis et al. / Physica B 371 (2006) 68–73
bottom of 5p-orbitals of Sb. Therefore, the valence band as well as the conduction band undergo certain changes in the course of the phase transition, because the atoms Sb and S responsible for the formation of the energy gap sides are displaced. The changes occur in quite a wide temperature region, since the phase transition is rather smooth. The absorption edge of SbSI in the phase transition region was studied in Ref. [2] quite thoroughly, though its relation with the band gap variation was not disclosed. This requires a more detailed study of the energy gap variation caused by the phase transition at all special points of the Brillouin zone in order to determine more exactly the location of the absorption edge on the energy scale. 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 [6]. The empirical pseudopotential method for calculating the band structure of SbSI was used in Refs. [7–9]. In Ref. [7] purely ionic and partially covalent models were used. An indirect energy gap of 2.28 eV at point S was obtained. However, the accuracy was 0.2 eV. In Ref. [8], the pseudopotentials were adjusted using the data on direct gaps. The absorption band edge phase in the paraelectric ðCÞ was found at 1.82 eV for Ejjc S ðVÞ 56 ! S 78 and 1.91 eV ðCÞ for E ? c S ðVÞ 122 ! S 728 . The valence band top corre-
ðCÞ sponded to S ðVÞ 526 , and the conduction band edge to Z 1 with the energy of indirect gap of 1.41 eV. In the ferroelectric phase the energy gap sides corresponded to points Z and R. The ðCÞ indirect band gap of 1.82 eV between GðVÞ 6 and S 1 and the minimum direct gap of 2.08 eV were obtained in Ref. [9] by analyzing the light reflection spectra. In Ref. [10], 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. A purely ionic model was assumed. Thus, the pseudopotential method results generally suggest that there is an indirect gap in the SbSI crystal with the extreme localized at points G and S. However, the band structure and characteristics of interatomic interactions within the SbSI crystal are not yet entirely clear even in the antiferroelectric phase (ferroelectric phase ToT C ¼ 295 K, antiferroelectric phase T C oTo410 K, and paraelectric phase T4410 K [11,12]), and there is still considerable controversy in the literature. We have chosen the empirical pseudopotential method for our band structure investigations, because it provides a rather high accuracy (0.1 meV) of computations with the equipment in our reach and is rather readily applicable to such complicated objects as the SbSI crystal structure. The advantage of the method is that it combines the possibility to use pseudopotentials derived from ab initio calculations and high sensitivity to the positions of atoms within the lattice under study.
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2. Investigation of the electronic structure of SbSI-type crystals by the pseudopotential method Here, we describe some features of applying the pseudopotential method for the crystal lattices containing several sorts of atoms and particularly for SbSI-type crystals. First of all, the infinite crystal approximation was used. It states that the crystal properties obtained for the primary cell are extended then to the entire crystal employing periodical boundary conditions. Thus, the position of the jth atom of the kind a (a ¼ Sb, S, or I) within the primary cell is ðaÞ RðaÞ j ¼ Rcell þ sj ,
(1)
where Rcell is the translation vector of the orthorhombic system. The vectors sðaÞ j describe nonprimitive translations that may be written as ðaÞ ðaÞ ðaÞ sðaÞ j ¼ l j t x þ mj t y þ u j t z ,
(2)
where tx, ty and tz are the lattice parameters along the three co-ordinate axes. The co-ordinates of atoms and lattice parameters in antiferroelectric and ferroelectric phases for SbSI are given in works [11,12]. The pseudopotential of the crystal was chosen as a sum of atomic pseudopotentials: XXX V ðrÞ ¼ Rcell va ðr Rcell sðaÞ (3) j Þ. a
Rcell
j
Here, va ðrÞ is the atomic pseudopotential of atom a. It is assumed to be localized and energy independent. Weakness of the pseudopotential V ps ðrÞ provides good convergence of the pseudowave functions expanded in terms of plane waves: X Cps C ki ðGÞeiðkþGÞr , (4) ki ðrÞ ¼ G
where G is the reciprocal lattice vector, C ki ðGÞ are Fourier coefficients. The Schro¨dinger equation in the pseudopotential method has the form ps ðH þ V ps ÞCps ki ¼ E ki Cki .
(5)
The pseudo-wave functions of the valence electrons Cps ki ðrÞ approach the true ones outside the core region, though they show no oscillations within the core. The approximation is justified in greater part of the crystal, because the core usually occupies about 5% of the total volume. The Fourier component of this potential (or the matrix element in the plane-wave basis set) is expressed V ðGÞ ¼
X a
S a ðGÞ
Oa va ðGÞ, Oc
(6)
where Oa and Oc are the volume of the corresponding atom and of the primary cell as a whole. The atomic structural
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factor S a ðGÞ and atomic form factor va ðGÞ are defined as X S a ðGÞ ¼ eiGsj ðaÞ , (7) a
va ðGÞ ¼
1 Oa
Z
va ðrÞeiGr d3 r.
(8)
The integrals are taken over the entire volume Oa . The atomic structural factor may be readily evaluated provided the atomic co-ordinates within the cell are known. The pseudopotential forms factors for Sb, S and I was preliminary determined using the following equation [4,13]: X 4p va ðGÞ ¼ hnlmj exp ½iðGrÞjnlmi. (9) 2 Oa jGj nlm Here, nlm is the set of electron quantum numbers. Numerical evaluation of the form factors involved neutral functions for Sb, S and I [3,14,15]. It is important that the agreement between the theoretical and experimental data can be obtained only with neutral functions. Note that some other authors [9,10] that studied theoretically the band structure of the SbSI crystal used an ionic model of chemical bonding (Sb+3S2I1) instead. 3. The energy gap of SbSI crystals in antiferroelectric and ferroelectric phases The chemical bond in SbSI is of mixed kind, with contributions of both ionic and covalent components. As we have demonstrated in Ref. [3], it may be described by an approximate model formula Sb+0.3S0.2I0.1. The character of the chemical bonding does not change during the phase transition. Hence the form factors of the pseudopotential were taken for the case of neutral atoms Sb, S and I. For generating the reciprocal lattice vectors and calculating the form factors we used the computer programs described in Refs. [4,16]. It should be particularly noted that we used no adjustment of the initial pseudopotential form factors to fit between the theoretical and experimental data. Therefore, we may say that the calculations were performed from the first principles. The band structure for both phases was calculated in 27 points of the irreducible part of the eightfold Brillouin zone, which means the total of 216 points over the Brillouin zone. The points are depicted schematically in Fig. 1, and their co-ordinates are listed in Table 1. The basis set for the calculation included 600 plane waves. This number was chosen in order to get the stability of the obtained eigenvalues within 0.1 meV in both phases. At a lower number of the plane waves, convergence in ferroelectric phase was somewhat worse than in the antiferroelectric phase. The experimental energy gap values were found from the exponential light absorption tail at ln K ¼ 6, where K is the absorption coefficient [2]. As seen from Fig. 2, the most significant changes in the valence band at the phase transition take place at points Q and C (energy variation is 0.92 and 0.86 eV, respectively).
Fig. 1. The scheme and notation of 27 special points in the irreducible part of the Brillouin zone.
Table 1 Co-ordinates of the special points in the irreducible part of the Brillouin zone of SbSI (in relative units) No
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Point
G D Y L ZTYG H Z B T S GYSX C GXUZ O TYSR A ZTRU E X D S G URSX Q U F R
Co-ordinates kx
ky
kz
0 0 0 0 0 0 0 0 0 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
0 0.25 0.5 0 0.25 0.5 0 0.25 0.5 0 0.25 0.5 0 0.25 0.5 0 0.25 0.5 0 0.25 0.5 0 0.25 0.5 0 0.25 0.5
0 0 0 0.25 0.25 0.25 0.5 0.5 0.5 0 0 0 0.25 0.25 0.25 0.5 0.5 0.5 0 0 0 0.25 0.25 0.25 0.5 0.5 0.5
See also Fig. 1.
The changes at points R (0.37 eV), H (0.55 eV), and E (0.42 eV) may also be noted. For the conduction band, the similar significant changes take place at points H (0.53 eV) and E (0.51 eV). At all the remaining points of the Brillouin zone the profile of the band gap changes only slightly. All these changes have only small effect upon the main characteristics of the band structure (obviously, except point R). As it is seen from Fig. 2, the SbSI crystal has an indirect forbidden gap both in antiferroelectric phase and in ferroelectric phase. The conduction band bottom in both phases is located at point Z ðCÞ 1 , the valence band top in
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interpolation of the form factors, which was a common procedure for ions Sb+3, S2 and Cl (taken in this case instead of I). In contrast, we determined the form factors of neutral atoms Sb, S and I from the neutral atomic functions using the formulae of Refs. [4,16] and the corresponding ab initio calculations. 4. Change of band structure of SbSI during the first-order phase transition In this section we give the results of the research on the change of band structure during first-order phase transition in SbSI mono-crystal. It is explored, how the change of grating parameter along crystallographic axis and average co-ordinates of Sb and S atoms affects SbSI band structure. For this we calculated the SbSI crystals band structure in antiferroelectric, ferroelectric phases and in the region of phase transition. This structure is evaluated in a few special points of Brillouin zone. The average coordinates of Sb, S and I atoms in the case of order–disorder and displacement type transition were used from works [11,12]. Fig. 3 shows direct (Brillouin zone point U) and indirect (jump U ! Z) dependence of the forbidden band on the temperature calculated by us. As it is seen in Figs. 3 and 4, indirect forbidden band’s width is 1.42 eV in antiferroelectric phase and 1.36 eV in ferroelectric phase. The following results match well the results, obtained before Fig. 2. A diagram of the forbidden gap of SbSI in antiferroelectric phase at 308 K and its change after transition to ferroelectric phase (at 278 K). E antiferro is the indirect transition energy (1.42 eV) in antiferroelectric phase g between points U and Z of the irreducible part of the Brillouin zone. E ferro g is the indirect transition energy (1.36 eV) in ferroelectric phase between points R and Z of the irreducible part of the Brillouin zone. The direct gap takes its minimum value at point U for both phases (a) and (b) correspond to different directions in the crystal.
antiferroelectric phase is at the point U ðVÞ 526 , and in ferroelectric phase at RðVÞ 324 . Our experimental and theoretical results of the SbSI crystal electronic structure sometimes differ from those obtained by other authors. The reasons are as follows. We determined the experimental band gap not only from the energy position of the absorption band edge, because this edge is exponential and its slope in the phase transition region strongly depends on the temperature [2], but using a complex of several experimental methods [5,16–19] as well. Moreover, the absorption edge cannot be a priori assigned to transitions at any particular point of the Brillouin zone without having determined the entire spectrum of the band structure. On the other hand, our experimental data were obtained on very pure gas-phase-grown SbSI crystals with ideal light-reflecting surfaces. Differently from other authors, the theoretical band structure was obtained using the above-mentioned ioncovalent chemical bonding model [3]. Second, we used no
Fig. 3. We got the direct (2) and the indirect (1) forbidden band width’s dependence on temperature. The brake of the curve in points A1 and A2 is got when moving to phase transition from antiferroelectric phase, and in points B1 and B2 when moving to phase transition from ferroelectric phase.
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nential edge of absorption is not considered. The exponential edge of absorption is explored in details in the work [2] in antiferroelectric and ferroelectric phases and in the area of the temperatures in the phase transition. The edge of the absorption in antiferroelectric phase corresponds to the Urbach’s rule: sðE K E 0 Þ , (10) K ¼ K 0 exp kT s ¼ s0
2kT _o0 , th _o0 2kT
(11)
where s is Urbach’s parameter. It describes the outspread of the absorption edge. s0 —constant, describing the intensity of interaction between electrons and phonons, _o0 —effective phonons energy, K 0 —‘‘oscillators strength’’ or the maximal absorption coefficient, E 0 —characteristic ‘‘gap’’ of the forbidden band, EK—light quantum energy for a particular absorption coefficient K. During the phase transition the characteristic ‘‘gap’’ varies Fig. 4. The band structure of SbSI monocrystal: (a) in antiferroelectric phase (T ¼ 308 K). (b) in points of phase transition moving from antiferroelectric phase (Fig. 3 points A1 and A2; T C ¼ 295 K). (c) in points of phase transition moving from ferroelectric phase (Fig. 3. points B1 and B2; T C ¼ 295 K). (d) in ferroelectric phase (Fig. 3. points C1 and C2; T ¼ 278 K). Arrows shows the width of the forbidden indirect band.
in the work [10], (1.42 eV in antiferroelectric and 1.36 eV in ferroelectric phase). The indirect width of forbidden band is 1.36 eV in that band structure, which we got by moving to the point of phase transition from ferroelectric phase (point B1), and 1.41 eV when moving from antiferroelectric phase (point A1). So according to our calculations, the indirect forbidden band moving from antiferroelectric to ferroelectric phase changes by 0.05 eV.This number corresponds to the change 0.06 eV of the width of indirect band, which was set experimentally [10,20]. It is also seen from Figs. 3 and 4 that the direct forbidden band is the narrowest in point U-1.83 eV. Calculations made in work [10] gave the value which is 1.82 eV. The minimal direct forbidden band in ferroelectric phase is in point G-1.94 eV. And in point U this gap is 1.98 eV. The narrowest forbidden band is in Brillouin zone’s point U approaching from both antiferroelectric and ferroelectric sides. This was calculated in band structure in the area of phase transition (Figs. 4(b) and (c)). In the firs case it is 1.83 eV (Fig. 3, point A2) and in the second one it is 1.87 eV (Fig. 3. point B2). That is why during the first-order phase transition the observed jump of the direct forbidden band is 0.04 eV. The reason of this jump is the change of the grating parameter along axis c(y), which affects SbSI crystal’s band structure. The width of the forbidden direct band in Brillouin zone point U is possible to find in the work [10,21], which is measured experimentally. Unfortunately, the SbSI expo-
DE 0 ¼ E 0F E 0AF ,
(12)
where E0F and E0AF—the values of the energy ‘‘gap’’ in ferroelectric and in antiferroelectric phases. As it is seen from Eq. (10), we have to measure temperatures dependence EK (T ) and s/kT (T ) when K ¼ const., K0 (T ) in both phases and in the area of the phase transition in case to determine E0F and E0AF. K0 (T ) does not depend on the temperature in antiferroelectric phase and it is determined experimentally. Finding the point of crossing of the curve ln K(E ) matches different temperatures. In ferroelectric
Fig. 5. The experimental izoabsorption curve energy EK (eV) dependence on temperature when K ¼ 65 cm1 (1) and the dependence s/kT (T ) (2), is measured in SbSI at phase transition.
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phase K0F is determined in the following way: s ln K 0F ¼ ln K 0 g P2 , kT AF s
(13)
where ln K0 and (s/kT )AF are parameters in antiferroelectric phase, g is coefficient of proportionality (polarization potential). The values of g, K0, PS (T ) and Eq. (13) is possible to find in the work [2]. Fig. 5 indicates temperature dependencies of EK (T ) and s/kT (T ), which were measured experimentally using dynamic method with a continuously variable temperature [22]. Using experimental Fig. 5 results, when temperature is 295 and 278 K and data from the work [2], we find from Eqs. (10)–(13), that DE 0 ¼ 0:12 0:02 eV. This experimental DE 0 value in margins of error coincides with the theoretic variation of the width of the forbidden band, which is 0.11 eV in Brillouin zone point U. It comes out because of the variation of the grating parameters along the axis cðyÞ, which affects the band structure of crystal (See Figs. 4(c) and (d)). 5. Conclusions Theoretical investigation of the SbSI crystals energy spectrum in antiferroelectric and ferroelectric phases showed that it has many energy bands which are located close to each other with a narrow forbidden band (less than 2 eV) in both phases. The valence and the conduction bands change due to displacement of Sb and S atoms with respect to I atom as a result of order–disorder and displacement type transition. The variations of the direct and indirect forbidden band order the variable interaction among 3p orbitals in S atom and 5p in Sb atom. The strongest variations during the phase transition are being observed at the top of the valence band in Brillouin zone points Q, C, R, H, E, and variations in conduction band are being observed at the bottom points H, T and E. The narrowest forbidden band is in Brillouin zone point U approaching from both antiferroelectric and ferroelectric sides. The values of the forbidden energy coincide well with experimental data. This happened because we verified the method of calculation. We calculated pseudo-potential form factors for neutral atoms Sb, S and I and we included sufficiently many plane waves in the basis set. SbSI crystal has an indirect forbidden gap both in antiferroelectric and in ferroelectric phases: the indirect transition (1.42 eV) in antiferroelectric phase between point U and Z, the indirect transition (1.36 eV) in ferroelectric phase between points R and Z on the Brillouin zone. According to our calculations, the indirect forbidden band decreases 0.05 eV moving from antiferroelectric to ferroelectric phase and the direct forbidden band increases 0.04 eV. The reason for the jump is the change of the
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grating parameter along the axis cðyÞ and the change of the average co-ordinates of Sb and S atoms. Using our experimental results and in respect to SbSI exponential edge of absorption, which corresponds to the Urbach’s rule, we found that the energy gap varies 0.12 eV. This experimental value of DE 0 coincide with the theoretical width of forbidden band which varies 0.11 eV. Acknowledgement This work was supported, in part, by a grant from the Soros Foundation awarded by the American Physical Society. References [1] V.M. Fridkin, Ferroelectrics-Semiconductors, Consultants bureau, New York, London, 1980. [2] K. Zˇicˇkus, A. Audzijonis, J. Batarunas, A. Sˇileika, Phys. Stat. Sol. B % 125 (1984) 645. [3] I.V. Vinokurova, O.V. Farberovich, A. Audzijonis, R. Zˇaltauskas, Lithuanian J. Phys. 35 (N4) (1995) 330. [4] S. Kvedaravicˇius, A. Audzijonis, N. Mykolaitiene’ , Ferroelectrics 150 (1993) 381. [5] I.V. Vinokurova, O.V. Farberovich, A. Audzijonis, R. Zˇaltauskas, Lithuanian J. Phys. 35 (N4) (1995) 306. [6] I.V. Vinokurova, O.V. Farberovich, A. Audzijonis, E. Cˇijauskas, R. Zˇaltauskas, Lithuanian J. Phys. 33 (N3) (1993) 140. [7] I.F. Alward, C.Y. Fong, M. El-Batanonny, F. Wooten, Solid State Commun. 25 (1978) 307. [8] M.L. Cohen, V. Heine, H. Ehenreich, et al. (Eds.), Academic Press, New York, 1970, p. 37. [9] C.Y. Fong, Y. Petroff, S. Kohn, et al., Solid State Commun. 14 (N8) (1974) 681. [10] K. Nakao, M. Balkanski, Phys. Rev. B 8 (1973) 5759. [11] K. Lukaszewicz, A. Pietraszko, J. Stepien-Damm, A. Kajokas, Polish J. Chem. 71 (1997) 1345. [12] K. Lukaszewicz, A. Pietraszko, J. Stepien-Damm, A. Kajokas, Polish J. Chem. 71 (1997) 1852. [13] J. Batarunas, A. Audzijonis, N. Mykolaitiene’ , K. Zˇicˇkus, Phys. Stat. % Sol. B 150 (1988) K31. [14] K. Zˇicˇkus, A. Audzijonis, J. Batarunas, V. Lazauskas, Solid State % Commun. 60 (N2) (1986) 143. [15] A. Audzijonis, K. Zˇicˇkus, J. Batarunas, V. Lazauskas, N. Mykolai% tiene’ , J. Grigas, The chemical bond of AVBVICVIIcrystals, In: Abstracts 6th International Meeting on Ferroelectricity, August 12, Kobe, Japan, 1985, p. 33. [16] N. Mykolaitiene’ , A. Audzijonis, J. Batarunas, K. Zˇicˇkus, Lithuanian % Phys. Coll. 28 (N6) (1988) 679. [17] Yu.V. Suchetskii, A.V. Soldatov, V.V. Likhacheva, K. Zˇicˇkus, A. Audzijonis, A.N. Gusatinskii, Lithuanian Phys. Coll. 26 (N1) (1986) 31. [18] Yu.V. Suchetskii, A.V. Soldatov, A.N. Gusatinskii, K. Zˇicˇkus, A. Audzijonis, Phys. Stat. Sol. B 132 (1985) K103. [19] A.V. Soldatov, A.N. Gusatinskii, Soviet Phys. Semicond. 17 (N10) (1983) 1221. [20] T.A. Pikka, V.M. Fridkin, Solid State Phys. 10 (N11) (1968) 3378. [21] G. Harbeke, J. Phys. Chem. Solids 24 (N7) (1963) 957. [22] A. Audzijonis, L. Audzijoniene’ , I. Levitas, A. Stasiukynas, E. Cˇijauskas, A way and a device to determine the temperature’s dependence of exponential edge of absorption in crystals. Invention 1565228 USSR, 1990.
Physica B 371 (2006) 68–73 www.elsevier.com/locate/physb
Variation of the energy gap of the SbSI crystals at ferroelectric phase transition A. Audzijonisa,, R. Zˇaltauskasa, L. Zˇigasa, I.V. Vinokurovab, O.V. Farberovichb, A. Pauliukasa, A. Kvedaravicˇiusa a Department of Physics, Vilnius Pedagogical University, Studentu 39, 08106 Vilnius, Lithuania Voronezh State University, Solid state physics department, Universitetskaya Pl. 1, 394693 Voronezh, Russia
b
Received 31 August 2005; received in revised form 21 September 2005; accepted 21 September 2005
Abstract Variation of the forbidden gap of SbSI crystals in the phase transition region is analyzed on the pseudopotential method for antiferroelectric and ferroelectric phase. The band gap at several special points of the Brillouin zone and some characteristic parameters of the band are considered. During the phase transition, the most significant changes are observed with the valence band top at points Q, C, R, H, E and with the conduction band bottom at points H, T and E of the Brillouin zone. At the ferroelectric phase transition, the valence and conduction bands change due to displacement of Sb and S atoms with respect to I and with respect to each other as a result of order–disorder and displacement-type transition. The obtained band gap values agree quite well with the experiment. This is apparently due to application of neutral rather than ionic atomic functions and inclusion of sufficiently many plane waves in the basis set for calculation. r 2005 Elsevier B.V. All rights reserved. PACS: 71.20.b; 77.80.Bh Keywords: SbSI; Electronic band structure; Ferroelectric phase transition; Pseudopotential method
1. Introduction The great interest to the A5B6C7-type ferroelectrics is caused by the fact that they possess semiconductor properties at the same time. The most outstanding representative of this class of compounds is sulfoiodide of antimony SbSI. It is characterized by a number of unique properties, e.g. it has the largest piezomodulus among the compounds of its class, it is by an order more sensitive to the external pressure, etc. Such a combination of SbSI properties opens up wide possibilities to employ it in practice. SbSI is used for developing optical light modulators, electroacousto-optical transformers, piezoelements, sensitive low-pressure gauges, and so on. Besides the practical use, SbSI is an interesting object of fundamental research. Be it as it may, SbSI is the most studied Corresponding author. Tel.: +370 5 2 79 00 43; fax: +370 5 2 75 17 32.
E-mail address: [email protected] (L. Zˇigas). 0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.09.039
compound of its kind. At present there is much information on its physical properties [1,2]. All the experimental results require theoretical interpretation. Therefore, it is important to choose some theoretical method that would provide calculation of the properties of such crystals on the same basis and yielding the results that would be in sufficient agreement with the experiment. As known [2,3], a ferroelectric phase transition of the first order takes place in SbSI, though it is close to the phase transition of the second order. Moreover, it has been proved theoretically in Ref. [4] that the phase transition in SbSI is intermediate between order–disorder and displacement types. At such a phase transition, the equilibrium position of atom Sb changes in accordance with the rules of order–disorder and displacement types, and that of atom S shifts by the rules of the displacement type. Investigation of the total density of states of SbSI crystals [5] has shown that the absolute valence band top is formed in both phases of 3p-orbitals of S, while the absolute conduction band
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bottom of 5p-orbitals of Sb. Therefore, the valence band as well as the conduction band undergo certain changes in the course of the phase transition, because the atoms Sb and S responsible for the formation of the energy gap sides are displaced. The changes occur in quite a wide temperature region, since the phase transition is rather smooth. The absorption edge of SbSI in the phase transition region was studied in Ref. [2] quite thoroughly, though its relation with the band gap variation was not disclosed. This requires a more detailed study of the energy gap variation caused by the phase transition at all special points of the Brillouin zone in order to determine more exactly the location of the absorption edge on the energy scale. 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 [6]. The empirical pseudopotential method for calculating the band structure of SbSI was used in Refs. [7–9]. In Ref. [7] purely ionic and partially covalent models were used. An indirect energy gap of 2.28 eV at point S was obtained. However, the accuracy was 0.2 eV. In Ref. [8], the pseudopotentials were adjusted using the data on direct gaps. The absorption band edge phase in the paraelectric ðCÞ was found at 1.82 eV for Ejjc S ðVÞ 56 ! S 78 and 1.91 eV ðCÞ for E ? c S ðVÞ 122 ! S 728 . The valence band top corre-
ðCÞ sponded to S ðVÞ 526 , and the conduction band edge to Z 1 with the energy of indirect gap of 1.41 eV. In the ferroelectric phase the energy gap sides corresponded to points Z and R. The ðCÞ indirect band gap of 1.82 eV between GðVÞ 6 and S 1 and the minimum direct gap of 2.08 eV were obtained in Ref. [9] by analyzing the light reflection spectra. In Ref. [10], 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. A purely ionic model was assumed. Thus, the pseudopotential method results generally suggest that there is an indirect gap in the SbSI crystal with the extreme localized at points G and S. However, the band structure and characteristics of interatomic interactions within the SbSI crystal are not yet entirely clear even in the antiferroelectric phase (ferroelectric phase ToT C ¼ 295 K, antiferroelectric phase T C oTo410 K, and paraelectric phase T4410 K [11,12]), and there is still considerable controversy in the literature. We have chosen the empirical pseudopotential method for our band structure investigations, because it provides a rather high accuracy (0.1 meV) of computations with the equipment in our reach and is rather readily applicable to such complicated objects as the SbSI crystal structure. The advantage of the method is that it combines the possibility to use pseudopotentials derived from ab initio calculations and high sensitivity to the positions of atoms within the lattice under study.
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2. Investigation of the electronic structure of SbSI-type crystals by the pseudopotential method Here, we describe some features of applying the pseudopotential method for the crystal lattices containing several sorts of atoms and particularly for SbSI-type crystals. First of all, the infinite crystal approximation was used. It states that the crystal properties obtained for the primary cell are extended then to the entire crystal employing periodical boundary conditions. Thus, the position of the jth atom of the kind a (a ¼ Sb, S, or I) within the primary cell is ðaÞ RðaÞ j ¼ Rcell þ sj ,
(1)
where Rcell is the translation vector of the orthorhombic system. The vectors sðaÞ j describe nonprimitive translations that may be written as ðaÞ ðaÞ ðaÞ sðaÞ j ¼ l j t x þ mj t y þ u j t z ,
(2)
where tx, ty and tz are the lattice parameters along the three co-ordinate axes. The co-ordinates of atoms and lattice parameters in antiferroelectric and ferroelectric phases for SbSI are given in works [11,12]. The pseudopotential of the crystal was chosen as a sum of atomic pseudopotentials: XXX V ðrÞ ¼ Rcell va ðr Rcell sðaÞ (3) j Þ. a
Rcell
j
Here, va ðrÞ is the atomic pseudopotential of atom a. It is assumed to be localized and energy independent. Weakness of the pseudopotential V ps ðrÞ provides good convergence of the pseudowave functions expanded in terms of plane waves: X Cps C ki ðGÞeiðkþGÞr , (4) ki ðrÞ ¼ G
where G is the reciprocal lattice vector, C ki ðGÞ are Fourier coefficients. The Schro¨dinger equation in the pseudopotential method has the form ps ðH þ V ps ÞCps ki ¼ E ki Cki .
(5)
The pseudo-wave functions of the valence electrons Cps ki ðrÞ approach the true ones outside the core region, though they show no oscillations within the core. The approximation is justified in greater part of the crystal, because the core usually occupies about 5% of the total volume. The Fourier component of this potential (or the matrix element in the plane-wave basis set) is expressed V ðGÞ ¼
X a
S a ðGÞ
Oa va ðGÞ, Oc
(6)
where Oa and Oc are the volume of the corresponding atom and of the primary cell as a whole. The atomic structural
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factor S a ðGÞ and atomic form factor va ðGÞ are defined as X S a ðGÞ ¼ eiGsj ðaÞ , (7) a
va ðGÞ ¼
1 Oa
Z
va ðrÞeiGr d3 r.
(8)
The integrals are taken over the entire volume Oa . The atomic structural factor may be readily evaluated provided the atomic co-ordinates within the cell are known. The pseudopotential forms factors for Sb, S and I was preliminary determined using the following equation [4,13]: X 4p va ðGÞ ¼ hnlmj exp ½iðGrÞjnlmi. (9) 2 Oa jGj nlm Here, nlm is the set of electron quantum numbers. Numerical evaluation of the form factors involved neutral functions for Sb, S and I [3,14,15]. It is important that the agreement between the theoretical and experimental data can be obtained only with neutral functions. Note that some other authors [9,10] that studied theoretically the band structure of the SbSI crystal used an ionic model of chemical bonding (Sb+3S2I1) instead. 3. The energy gap of SbSI crystals in antiferroelectric and ferroelectric phases The chemical bond in SbSI is of mixed kind, with contributions of both ionic and covalent components. As we have demonstrated in Ref. [3], it may be described by an approximate model formula Sb+0.3S0.2I0.1. The character of the chemical bonding does not change during the phase transition. Hence the form factors of the pseudopotential were taken for the case of neutral atoms Sb, S and I. For generating the reciprocal lattice vectors and calculating the form factors we used the computer programs described in Refs. [4,16]. It should be particularly noted that we used no adjustment of the initial pseudopotential form factors to fit between the theoretical and experimental data. Therefore, we may say that the calculations were performed from the first principles. The band structure for both phases was calculated in 27 points of the irreducible part of the eightfold Brillouin zone, which means the total of 216 points over the Brillouin zone. The points are depicted schematically in Fig. 1, and their co-ordinates are listed in Table 1. The basis set for the calculation included 600 plane waves. This number was chosen in order to get the stability of the obtained eigenvalues within 0.1 meV in both phases. At a lower number of the plane waves, convergence in ferroelectric phase was somewhat worse than in the antiferroelectric phase. The experimental energy gap values were found from the exponential light absorption tail at ln K ¼ 6, where K is the absorption coefficient [2]. As seen from Fig. 2, the most significant changes in the valence band at the phase transition take place at points Q and C (energy variation is 0.92 and 0.86 eV, respectively).
Fig. 1. The scheme and notation of 27 special points in the irreducible part of the Brillouin zone.
Table 1 Co-ordinates of the special points in the irreducible part of the Brillouin zone of SbSI (in relative units) No
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Point
G D Y L ZTYG H Z B T S GYSX C GXUZ O TYSR A ZTRU E X D S G URSX Q U F R
Co-ordinates kx
ky
kz
0 0 0 0 0 0 0 0 0 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
0 0.25 0.5 0 0.25 0.5 0 0.25 0.5 0 0.25 0.5 0 0.25 0.5 0 0.25 0.5 0 0.25 0.5 0 0.25 0.5 0 0.25 0.5
0 0 0 0.25 0.25 0.25 0.5 0.5 0.5 0 0 0 0.25 0.25 0.25 0.5 0.5 0.5 0 0 0 0.25 0.25 0.25 0.5 0.5 0.5
See also Fig. 1.
The changes at points R (0.37 eV), H (0.55 eV), and E (0.42 eV) may also be noted. For the conduction band, the similar significant changes take place at points H (0.53 eV) and E (0.51 eV). At all the remaining points of the Brillouin zone the profile of the band gap changes only slightly. All these changes have only small effect upon the main characteristics of the band structure (obviously, except point R). As it is seen from Fig. 2, the SbSI crystal has an indirect forbidden gap both in antiferroelectric phase and in ferroelectric phase. The conduction band bottom in both phases is located at point Z ðCÞ 1 , the valence band top in
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interpolation of the form factors, which was a common procedure for ions Sb+3, S2 and Cl (taken in this case instead of I). In contrast, we determined the form factors of neutral atoms Sb, S and I from the neutral atomic functions using the formulae of Refs. [4,16] and the corresponding ab initio calculations. 4. Change of band structure of SbSI during the first-order phase transition In this section we give the results of the research on the change of band structure during first-order phase transition in SbSI mono-crystal. It is explored, how the change of grating parameter along crystallographic axis and average co-ordinates of Sb and S atoms affects SbSI band structure. For this we calculated the SbSI crystals band structure in antiferroelectric, ferroelectric phases and in the region of phase transition. This structure is evaluated in a few special points of Brillouin zone. The average coordinates of Sb, S and I atoms in the case of order–disorder and displacement type transition were used from works [11,12]. Fig. 3 shows direct (Brillouin zone point U) and indirect (jump U ! Z) dependence of the forbidden band on the temperature calculated by us. As it is seen in Figs. 3 and 4, indirect forbidden band’s width is 1.42 eV in antiferroelectric phase and 1.36 eV in ferroelectric phase. The following results match well the results, obtained before Fig. 2. A diagram of the forbidden gap of SbSI in antiferroelectric phase at 308 K and its change after transition to ferroelectric phase (at 278 K). E antiferro is the indirect transition energy (1.42 eV) in antiferroelectric phase g between points U and Z of the irreducible part of the Brillouin zone. E ferro g is the indirect transition energy (1.36 eV) in ferroelectric phase between points R and Z of the irreducible part of the Brillouin zone. The direct gap takes its minimum value at point U for both phases (a) and (b) correspond to different directions in the crystal.
antiferroelectric phase is at the point U ðVÞ 526 , and in ferroelectric phase at RðVÞ 324 . Our experimental and theoretical results of the SbSI crystal electronic structure sometimes differ from those obtained by other authors. The reasons are as follows. We determined the experimental band gap not only from the energy position of the absorption band edge, because this edge is exponential and its slope in the phase transition region strongly depends on the temperature [2], but using a complex of several experimental methods [5,16–19] as well. Moreover, the absorption edge cannot be a priori assigned to transitions at any particular point of the Brillouin zone without having determined the entire spectrum of the band structure. On the other hand, our experimental data were obtained on very pure gas-phase-grown SbSI crystals with ideal light-reflecting surfaces. Differently from other authors, the theoretical band structure was obtained using the above-mentioned ioncovalent chemical bonding model [3]. Second, we used no
Fig. 3. We got the direct (2) and the indirect (1) forbidden band width’s dependence on temperature. The brake of the curve in points A1 and A2 is got when moving to phase transition from antiferroelectric phase, and in points B1 and B2 when moving to phase transition from ferroelectric phase.
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nential edge of absorption is not considered. The exponential edge of absorption is explored in details in the work [2] in antiferroelectric and ferroelectric phases and in the area of the temperatures in the phase transition. The edge of the absorption in antiferroelectric phase corresponds to the Urbach’s rule: sðE K E 0 Þ , (10) K ¼ K 0 exp kT s ¼ s0
2kT _o0 , th _o0 2kT
(11)
where s is Urbach’s parameter. It describes the outspread of the absorption edge. s0 —constant, describing the intensity of interaction between electrons and phonons, _o0 —effective phonons energy, K 0 —‘‘oscillators strength’’ or the maximal absorption coefficient, E 0 —characteristic ‘‘gap’’ of the forbidden band, EK—light quantum energy for a particular absorption coefficient K. During the phase transition the characteristic ‘‘gap’’ varies Fig. 4. The band structure of SbSI monocrystal: (a) in antiferroelectric phase (T ¼ 308 K). (b) in points of phase transition moving from antiferroelectric phase (Fig. 3 points A1 and A2; T C ¼ 295 K). (c) in points of phase transition moving from ferroelectric phase (Fig. 3. points B1 and B2; T C ¼ 295 K). (d) in ferroelectric phase (Fig. 3. points C1 and C2; T ¼ 278 K). Arrows shows the width of the forbidden indirect band.
in the work [10], (1.42 eV in antiferroelectric and 1.36 eV in ferroelectric phase). The indirect width of forbidden band is 1.36 eV in that band structure, which we got by moving to the point of phase transition from ferroelectric phase (point B1), and 1.41 eV when moving from antiferroelectric phase (point A1). So according to our calculations, the indirect forbidden band moving from antiferroelectric to ferroelectric phase changes by 0.05 eV.This number corresponds to the change 0.06 eV of the width of indirect band, which was set experimentally [10,20]. It is also seen from Figs. 3 and 4 that the direct forbidden band is the narrowest in point U-1.83 eV. Calculations made in work [10] gave the value which is 1.82 eV. The minimal direct forbidden band in ferroelectric phase is in point G-1.94 eV. And in point U this gap is 1.98 eV. The narrowest forbidden band is in Brillouin zone’s point U approaching from both antiferroelectric and ferroelectric sides. This was calculated in band structure in the area of phase transition (Figs. 4(b) and (c)). In the firs case it is 1.83 eV (Fig. 3, point A2) and in the second one it is 1.87 eV (Fig. 3. point B2). That is why during the first-order phase transition the observed jump of the direct forbidden band is 0.04 eV. The reason of this jump is the change of the grating parameter along axis c(y), which affects SbSI crystal’s band structure. The width of the forbidden direct band in Brillouin zone point U is possible to find in the work [10,21], which is measured experimentally. Unfortunately, the SbSI expo-
DE 0 ¼ E 0F E 0AF ,
(12)
where E0F and E0AF—the values of the energy ‘‘gap’’ in ferroelectric and in antiferroelectric phases. As it is seen from Eq. (10), we have to measure temperatures dependence EK (T ) and s/kT (T ) when K ¼ const., K0 (T ) in both phases and in the area of the phase transition in case to determine E0F and E0AF. K0 (T ) does not depend on the temperature in antiferroelectric phase and it is determined experimentally. Finding the point of crossing of the curve ln K(E ) matches different temperatures. In ferroelectric
Fig. 5. The experimental izoabsorption curve energy EK (eV) dependence on temperature when K ¼ 65 cm1 (1) and the dependence s/kT (T ) (2), is measured in SbSI at phase transition.
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phase K0F is determined in the following way: s ln K 0F ¼ ln K 0 g P2 , kT AF s
(13)
where ln K0 and (s/kT )AF are parameters in antiferroelectric phase, g is coefficient of proportionality (polarization potential). The values of g, K0, PS (T ) and Eq. (13) is possible to find in the work [2]. Fig. 5 indicates temperature dependencies of EK (T ) and s/kT (T ), which were measured experimentally using dynamic method with a continuously variable temperature [22]. Using experimental Fig. 5 results, when temperature is 295 and 278 K and data from the work [2], we find from Eqs. (10)–(13), that DE 0 ¼ 0:12 0:02 eV. This experimental DE 0 value in margins of error coincides with the theoretic variation of the width of the forbidden band, which is 0.11 eV in Brillouin zone point U. It comes out because of the variation of the grating parameters along the axis cðyÞ, which affects the band structure of crystal (See Figs. 4(c) and (d)). 5. Conclusions Theoretical investigation of the SbSI crystals energy spectrum in antiferroelectric and ferroelectric phases showed that it has many energy bands which are located close to each other with a narrow forbidden band (less than 2 eV) in both phases. The valence and the conduction bands change due to displacement of Sb and S atoms with respect to I atom as a result of order–disorder and displacement type transition. The variations of the direct and indirect forbidden band order the variable interaction among 3p orbitals in S atom and 5p in Sb atom. The strongest variations during the phase transition are being observed at the top of the valence band in Brillouin zone points Q, C, R, H, E, and variations in conduction band are being observed at the bottom points H, T and E. The narrowest forbidden band is in Brillouin zone point U approaching from both antiferroelectric and ferroelectric sides. The values of the forbidden energy coincide well with experimental data. This happened because we verified the method of calculation. We calculated pseudo-potential form factors for neutral atoms Sb, S and I and we included sufficiently many plane waves in the basis set. SbSI crystal has an indirect forbidden gap both in antiferroelectric and in ferroelectric phases: the indirect transition (1.42 eV) in antiferroelectric phase between point U and Z, the indirect transition (1.36 eV) in ferroelectric phase between points R and Z on the Brillouin zone. According to our calculations, the indirect forbidden band decreases 0.05 eV moving from antiferroelectric to ferroelectric phase and the direct forbidden band increases 0.04 eV. The reason for the jump is the change of the
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grating parameter along the axis cðyÞ and the change of the average co-ordinates of Sb and S atoms. Using our experimental results and in respect to SbSI exponential edge of absorption, which corresponds to the Urbach’s rule, we found that the energy gap varies 0.12 eV. This experimental value of DE 0 coincide with the theoretical width of forbidden band which varies 0.11 eV. Acknowledgement This work was supported, in part, by a grant from the Soros Foundation awarded by the American Physical Society. References [1] V.M. Fridkin, Ferroelectrics-Semiconductors, Consultants bureau, New York, London, 1980. [2] K. Zˇicˇkus, A. Audzijonis, J. Batarunas, A. Sˇileika, Phys. Stat. Sol. B % 125 (1984) 645. [3] I.V. Vinokurova, O.V. Farberovich, A. Audzijonis, R. Zˇaltauskas, Lithuanian J. Phys. 35 (N4) (1995) 330. [4] S. Kvedaravicˇius, A. Audzijonis, N. Mykolaitiene’ , Ferroelectrics 150 (1993) 381. [5] I.V. Vinokurova, O.V. Farberovich, A. Audzijonis, R. Zˇaltauskas, Lithuanian J. Phys. 35 (N4) (1995) 306. [6] I.V. Vinokurova, O.V. Farberovich, A. Audzijonis, E. Cˇijauskas, R. Zˇaltauskas, Lithuanian J. Phys. 33 (N3) (1993) 140. [7] I.F. Alward, C.Y. Fong, M. El-Batanonny, F. Wooten, Solid State Commun. 25 (1978) 307. [8] M.L. Cohen, V. Heine, H. Ehenreich, et al. (Eds.), Academic Press, New York, 1970, p. 37. [9] C.Y. Fong, Y. Petroff, S. Kohn, et al., Solid State Commun. 14 (N8) (1974) 681. [10] K. Nakao, M. Balkanski, Phys. Rev. B 8 (1973) 5759. [11] K. Lukaszewicz, A. Pietraszko, J. Stepien-Damm, A. Kajokas, Polish J. Chem. 71 (1997) 1345. [12] K. Lukaszewicz, A. Pietraszko, J. Stepien-Damm, A. Kajokas, Polish J. Chem. 71 (1997) 1852. [13] J. Batarunas, A. Audzijonis, N. Mykolaitiene’ , K. Zˇicˇkus, Phys. Stat. % Sol. B 150 (1988) K31. [14] K. Zˇicˇkus, A. Audzijonis, J. Batarunas, V. Lazauskas, Solid State % Commun. 60 (N2) (1986) 143. [15] A. Audzijonis, K. Zˇicˇkus, J. Batarunas, V. Lazauskas, N. Mykolai% tiene’ , J. Grigas, The chemical bond of AVBVICVIIcrystals, In: Abstracts 6th International Meeting on Ferroelectricity, August 12, Kobe, Japan, 1985, p. 33. [16] N. Mykolaitiene’ , A. Audzijonis, J. Batarunas, K. Zˇicˇkus, Lithuanian % Phys. Coll. 28 (N6) (1988) 679. [17] Yu.V. Suchetskii, A.V. Soldatov, V.V. Likhacheva, K. Zˇicˇkus, A. Audzijonis, A.N. Gusatinskii, Lithuanian Phys. Coll. 26 (N1) (1986) 31. [18] Yu.V. Suchetskii, A.V. Soldatov, A.N. Gusatinskii, K. Zˇicˇkus, A. Audzijonis, Phys. Stat. Sol. B 132 (1985) K103. [19] A.V. Soldatov, A.N. Gusatinskii, Soviet Phys. Semicond. 17 (N10) (1983) 1221. [20] T.A. Pikka, V.M. Fridkin, Solid State Phys. 10 (N11) (1968) 3378. [21] G. Harbeke, J. Phys. Chem. Solids 24 (N7) (1963) 957. [22] A. Audzijonis, L. Audzijoniene’ , I. Levitas, A. Stasiukynas, E. Cˇijauskas, A way and a device to determine the temperature’s dependence of exponential edge of absorption in crystals. Invention 1565228 USSR, 1990.