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Phys. Status Solidi B 247, No. 1, 176–181 (2010) / DOI 10.1002/pssb.200945288
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basic solid state physics
Optical spectra of bismuth sulfochloride crystals A. Audzijonis, R. Sereika*, R. Zˇaltauskas, and L. Zˇigas
Department of Physics, Vilnius Pedagogical University, Studentu 39, 08106 Vilnius, Lithuania Received 17 June 2009, revised 28 August 2009, accepted 4 September 2009 Published online 7 October 2009 PACS 71.15.Mb, 71.20.Nr, 77.22.d, 78.20.Ci * Corresponding
author: e-mail
[email protected], Phone: þ370 5 273 48 57, Fax: þ370 5 273 48 57
We present the results of the ab initio theoretical study of the optical properties for paraelectric BiSCl crystal using the full potential linearized augmented plane wave (FP-LAPW) method as implanted in the Wien 2k code. For theoretical calculations of optical constants and functions we used the generalized gradient approximation (PBE-GGA), an improvement of the local spin-density approximation (LSDA) and
recently Wu–Cohen (WC) proposed a new WC-GGA exchange-correlation energy functional. The dielectric function, refractive index, extinction coefficient, absorption coefficient, reflectivity, and energy loss function were calculated. The optical properties are analyzed and the origins of the peaks in the spectra are discussed in terms of the calculated density of states.
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1 Introduction Quasi-one-dimensional crystals attract attention due to a complicated chemical bonding and anisotropy of electron and phonon spectra. A5(Bi, Sb) B6(S, Se) C6(Cl, I, Br) crystals consist of chain along the [001]-axis. Crystal structure and physical properties are described in Refs. [1–3]. The crystals with Bi atoms (BiSCl, BiOCl, BiSI) have been the current focus due to special properties which have their potential applications in molecular based electronic devices, such as optical memory, display and data records [3], dielectrics for microelectronics, and nonlinear optical applications. BiSCl is isostructural to the well-known quasi-onedimensional ferroelectric SbSI, which exhibits a number of strongly coupled semiconductive and ferroelectric properties [2]. The band structure of BiSCl has been investigated using the self-consisted pseudopotential method [4, 5] in the energy region of one valence band (VB). The band structure was also obtained using the density functional theory and pseudopotential theory under generalized gradient approximation (GGA) concerning energy region of all VBs [6]. However, there is no ab initio calculation and experimental investigation of the optical properties of BiSCl crystals in the scientific literature. In the current paper we have investigated and calculated density of states of all VBs and optical properties of BiSCl crystals using the full potential linearized augmented plane
wave (FP-LAPW) with the GGA (Perdew–Burke–ErnzerhofGGA (PBE-GGA), Wu–Cohen-GGA (WC-GGA)) and local spin-density approximation (LSDA) by Wien 2k [7] package. 2 Computational methods The BiSCl crystal consists of chains of atoms along c(z)-axis and in paraelectric phase belongs to D16 2h space group. All atoms in the BiSCl crystal are on mirror planes normal to the c-axis. This crystal has four BiSCl molecules (12 atoms) in a unit cell. Each molecule of BiSCl extends in a chain-like fashion along the c-axis. The lattice constants and positions of the atoms in the unit cell were taken from the literature [4]. The positions of all 12 atoms in unit cell may be found by the symmetry operations: (xj, yj, zj); (xj, yj, zj þ 1=2); (xj þ 1=2, yj þ 1=2, zj þ 1=2); (xj þ 1=2, yj þ 1=2, zj ). The values of xj , yj , and zj are given in Table 1. The bond between the Bi and S atoms in the same chain is covalent, while the Cl ions are in an ionic bond with a covalently bound bridge (BiS). The interchain weak bond is Van der Waals-type. Therefore, a BiSCl-type crystal exhibits an anisotropy interatomic interaction and this creates the optical properties with strong optical anisotropy. The calculations reported in this work were carried out by means of the full-potential, linearized, augmented plane wave method using Wien 2k computer package [7]. This is ß 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Table 1 Lattice constants and relative atomic positions in the unit cell of the BiSCl crystal [4]. atoms
Bi S Cl
˚ a ¼ 7:70 A
˚ b ¼ 9:87 A
˚ c ¼ 4:01 A
xj
yj
zj
0.140 0.770 0.500
0.138 0.040 0.790
1/4 1/4 1/4
an implementation of the density-functional theory (DFT) [8] with different possible approximations for the exchangecorrelation potential. Exchange and correlation were treated separately within the different approximations. We have used the LSDA [9], and two different GGA: PBE-GGA [10] and WC-GGA [11], in order to establish the influence of these approximations on the value of the energy gap, the density of states, and optical properties. Scalar relativistic equations are used to obtain self-consistency. In order to achieve energy eigen values convergence, the wave functionals in the interstitial region were expanded in plane waves with a cut-off parameter of RMT kmax ¼ 7.0, where RMT denotes the smallest atomic sphere radius and kmax largest k vector in the plane wave expansion. The selfconsistent calculations are considered to be converged when the total energy of the system is stable with 104 Ry. The MT muffin-tin radii were set to RMT Bi ¼ 2.94 for Bi, RS ¼ 1.90 for MT S, and RCl ¼ 1.81 for Cl. A mesh of 5000 k-points in the irreducible part of the Brillouin zone has been used. Broadening for optical spectra is set to 0.02 eV. The imaginary part of the dielectric tensor was calculated from the knowledge of the electronic band structure of a solid [12]: Z h2 e2 X finterg dkhck jpa jvk i vk pb ck Im eab ðvÞ ¼ 2 2 pm v c;v dðeck evk vÞ
Figure 1 (online color at: www.pss-b.com) The total density of states of BiSCl calculated with GGA (PBE), LSDA, and GGA (WC) approximations.
conduction band (CB) is dominated by Bi-p, S-p, and Cl-p. The density of states can be divided into three groups. The lowest VB group from 14 eV up to 8 eV has mainly Bi/S/Cl-s states with a small contribution from Bi-p and S-p states. The second VB group from 5 eV up to the Fermi energy (Ef ¼ 0 eV) are due to Bi-p, S-p and Cl-p with a small contribution from S-s states. The last group (CB) has contribution from Bi/S/Cl-p states.
3.2 Dielectric functions The optical properties of crystal can be described by mean of the dielectric function eðvÞ. The interband transitions can be split into direct and indirect transitions. The indirect transitions which involve scattering of phonon to give a small contribution to eðvÞ. Therefore, the imaginary parts of the dielectric function Im eðvÞ are sum of all transitions from the VBs to the CBs.
(1) where p is the momentum matrix element between states of bands a and b with crystal momentum k. In Eq. (1), ck and vk are the crystal wave functions corresponding to the conduction and the VBs with crystal wave vector k. The interband expansion in the corresponding real parts was obtained by Kramers–Kroning transformation: finterg Re eab ðvÞ
2 ¼ dab þ P p
Z1 0
v0 Im eab ðv0 Þ ðv0 Þ2 v2
dv0
(2)
3 Results and discussion 3.1 Density of states The total-DOS and partialDOS of BiSCl crystals, are shown in Figs. 1 and 2. The partial-DOS Bi-s/p, S-s/p, and Cl-s/p are shown in Fig. 2. The VB is dominated by Bi-s/p, S-s/p, and Cl-s/p while the www.pss-b.com
Figure 2 (online color at: www.pss-b.com) The partial density of states of BiSCl calculated with GGA (PBE) approximation. ß 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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A. Audzijonis et al.: Optical spectra of bismuth sulfochloride crystals
Figure 3 (online color at: www.pss-b.com) Theoretical imaginary part of dielectric function for BiSCl crystal (xx-direction) calculated with GGA (PBE), LSDA, and GGA (WC) approximations.
Figure 5 (online color at: www.pss-b.com) Theoretical real part of dielectric function for BiSCl crystal (xx-direction) calculated with GGA (PBE), LSDA, and GGA (WC) approximations.
The imaginary part of the dielectric function Im eðvÞ is presented in Fig. 3. The curve Im eðvÞ show that the threshold energy of the Im eðvÞ occurs at 1.9 eV. This energy gives the threshold for indirect optical transitions between the highest valence and the lowest CBs. Our calculations show that: (i) the top of the VB is located at some k-point along z-direction; (ii) the minimum of the CB is located at X point k-point along GX (the x-direction); (iii) crystals are semiconductors. Therefore BiSCl crystal has indirect forbidden gap. The value of the forbidden gap is 1.6 eV, next for direct transition 1.9 eV. Our results coincide with the experimental data given in Ref. [13] (1.89–1.93 eV) for direct transition. In the work [6] the electronic band structure and band gap of BiSCl crystal were calculated using density functional theory and pseudopotential theory under the
GGA. The indirect and direct gap values of BiSCl crystal are respectively 1.40 eV and 1.54 eV. In work [6] the band gap values are underestimated than the experimental values of [14]. It is an expected case because of the use of pseudopotential method. The Kohn-Sham density functional theory is a theory for the ground-state density and energy. Even if we had the exact density functional, it would not predict the exact band gap or optical properties [14]. Beyond the threshold, the curve Im eðvÞ increases rapidly. This is due to the fact that the number of critical points of BZ contributing to Im eðvÞ increases abruptly. According to the Kramers–Kroning dispersion relation the real part Re eðvÞ of the frequency dependent dielectric function eðvÞ is also obtained and displayed in Figs. 5–7.
Figure 4 (online color at: www.pss-b.com) Theoretical imaginary part of dielectric function for BiSCl crystal (zz-direction) calculated with GGA (PBE), LSDA, and GGA (WC) approximations.
Figure 6 (online color at: www.pss-b.com) Theoretical real part of dielectric function for BiSCl crystal (zz-direction) calculated with GGA (PBE), LSDA, and GGA (WC) approximations.
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Figure 7 (online color at: www.pss-b.com) Theoretical real part of dielectric function for BiSCl crystal (xx- and zz-directions) calculated with GGA (PBE) and LSDA approximations.
The peaks in the optical response are caused by the transitions between the valence and CBs. The first maximum of eðvÞ in the energy range 2–7 eV consists from four main peaks. These four main peaks are created by optical transition from p-states of VB of atoms Bi, S and Cl to the p-states of CB of Bi, S and Cl atoms. The second main maximum eðvÞ in energy range 7–14 eV consists from 6–7 main peaks. These main peaks are created by transitions from s-states of VB of atoms Bi, S and Cl to the p-states of CB of Bi, S and Cl atoms. In Fig. 7 we see the comparison between spectra of Re eðvÞ for BiSCl crystal (xx- and zz-directions) calculated with GGA (PBE) and LSDA approximations in energy range
0–5 eV. The static dielectric constant Re eð0Þ is given by the low energy limit of Re eðvÞ. Re eð0Þ corresponds to the static optical dielectric constant e0 . The GGA (PBE) calculated optical dielectric constants Re exx ð0Þ ¼ Re ezz ð0Þ ¼ 10.5, but LSDA calculated Re exx ð0Þ ¼ Re ezz ð0Þ ¼ 11.2 (see Fig. 7). The knowledge of Im eðvÞ and Re eðvÞ allows the calculations of refractive indices nðvÞ, refraction coefficient kðvÞ, reflectivity R(v), absorption coefficient K(v), and energy loss function Im½e1 ðvÞ [12]. 3.3 Refractive index, extinction coefficient, and absorption coefficient The calculated n(v), k(v), and K(v) with GGA (PBE) approximation are shown in Fig. 8. As
Figure 8 (online color at: www.pss-b.com) Theoretical refractive index n(E), extinction coefficient k(E), and absorption coefficient K(E) for BiSCl crystal.
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A. Audzijonis et al.: Optical spectra of bismuth sulfochloride crystals
Table 2 The positions of BiSCl main peaks of optical constants and functions in the photon energy range 0–7 eV along x- and z-axis. optical constants and functions calculated by FP-LAPW þ PBE-GGA
energy (in eV) of the main peaks E1
E2
E3
E4
Im ex Im ez kx kz nx nz Rx Rz Kx Kz
2.4 2.4 2.5 2.5 2.5 2.4 2.4 2.4 2.6 2.6
3.0 3.0 3.0 3.0 3.0 3.0 3.2 3.2 3.2 3.2
3.7 3.7 3.6 3.6 3.5 3.5 3.6 3.8 3.7 3.7
5.6 5.6 5.6 5.6 5.5 5.5 5.8 6.0 5.7 5.7
can be seen, all spectra can be divided in two energy regions. The first region is located in the energy range from 0 to 7 eV and the second from 7 to 14 eV. The first maximum of n(v), k(v), and K(v) in the first energy region, consist from four main peaks. These four main peaks are created by transitions from p-states of VB to the p-states of CB. The second maximum of n(v), k(v), and K(v) in the second energy region, consist of 6-7 small peaks. These peaks are created by transitions from s-states of VB to the p-states of CB. In this energy region, a considerable anisotropy is found between the parallel and perpendicular components of the frequencydependent optical properties. The positions of main peaks of the optical constants and functions in the first energy region (photon energy range 0–7 eV) along x- and z-axis, are illustrated in Table 2. As seen from Figs. 3–7, the main peaks of optical constants and functions spectra, calculated using LSDA
functional, are shifted to the low energy region, accordingly main peaks are calculated by PBE-GGA and WC-GGA, functional at about 0.2 eV. Theoretical investigations reveal that difference between the positions of main peaks of the optical constants and functions in the first energy region along x- and z-axis, calculated with FP-LAPW method using exchange-correlation energy functional PBE-GGA, WCGGA, and LSDA, are very small. 3.4 Reflectivity and energy loss function The reflectivity spectrum R(v) calculated with GGA (PBE) approximation along x- and z-axis is plotted in Fig. 9. From the reflectivity spectra, we note that at low energies 0–6 eV, R(v) along x- and z-axis increases up to about 50 and 65% respectively, forming a strong reflectivity maximum, created from transition of p-states of VB to p-states of CB. This maximum consists of four main peaks (Table 2). The second strong reflectivity maximum at high energies (7–14 eV) along x- and z-axis increase up to about 40% and are created from transitions of s-states of VB to p-states of CB. The calculated energy-loss function Im ðe1 ðvÞÞ is presented in Fig. 9. The function Im ðe1 ðvÞÞ describes the energy loss of the fast electron, traversing the material. The sharp maxima in the energy-loss function, is associated with the existence of plasma oscillations. The curves of xx- and zz-direction in Fig. 9, has a maximum near hvp ¼ 19 eV. At this energy of maxima, the Re eðvÞ goes through zero (Figs. 5 and 6). The plasma frequency hvp can be calculated by means of energy-loss function Im ðe1 ðvÞÞ and can be tested with the f-sum rule [15]: p 2 v ¼ 2 p
Z1
Im eðvÞv dv
(3)
0
Figure 9 (online color at: www.pss-b.com) Theoretical reflectivity and electron energy loss spectrum for BiSCl crystal.
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Numerical integration of Eq. (3) in the range from 0 to 30 eV, gives hvp ¼ 15.97 eV along x-axis and hvp ¼ 15.90 eV along z-axis. 4 Conclusions We have calculated the frequency dependent optical properties for the paraelectric BiSCl using the FP-LAPW method with exchange-correlation functionals PBE-GGA, WC-GGA, and LSDA. The calculations show that this crystal has indirect energy band gap. We present calculations of the frequency-dependent complex dielectric function eðvÞ. Also we calculate refractive index n(v), extinction coefficient k(v), absorption coefficient K(v), and energy loss function Im ðe1 ðvÞÞ along x- and z-axis. The optical properties are not scissors, corrected to match the calculated energy gap with the measured one. In the photon energy range 0–7 eV, the linear optical properties show a strong anisotropy. The optical properties are analyzed in terms of the calculated electronic band density structure. The energy of plasma resonance hvp has been observed by two methods (from spectra of Im ðe1 ðvÞÞ and f-sum). The positions of the main peaks of the optical constants and functions in the photon energy range 0–7 eV, along x- and z-axis of BiSCl, have very little differences (about 0.2 eV), using FP-LAPW method with LSDA, PBE-GGA, and WCGGA exchange-correlation energy functionals. We come to the conclusion that the theoretical calculations for the paraelectric BiSCl crystal using FP-LAPW method with functionals PBE-GGA, WC-GGA, and LSDA, gives very similar results.
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