A. H. Reshak*ab,
Oleg. V. Parasyukc,
H. Kamarudinb,
I. V. Kitykd,
Zeyad A. Alahmede,
Nasser S. AlZayede,
Sushil Auluckfg,
Anatolii O. Fedorchukh and
J. Chyskýi
aNew Technologies – Research Centre, University of West Bohemia, Univerzitni 8, 306 14 Pilsen, Czech Republic. E-mail: maalidph@yahoo.co.uk
bCenter of Excellence Geopolymer and Green Technology, School of Material Engineering, University Malaysia Perlis, 01007 Kangar, Perlis, Malaysia
cLviv National University of Veterinary Medicine and Biotechnologies, Pekarska Street 50, 79010 Lviv, Ukraine
dFaculty of Electrical Engineering, Czestochowa University Technology, Armii Krajowej 17, PL-42201, Czestochowa, Poland
eDepartment of Physics and Astronomy, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
fCouncil of Scientific and Industrial Research – National Physical Laboratory, Dr. K S Krishnan Marg, New Delhi 110012, India
gDepartment of Physics, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India
hUniversity of Veterinary and Biotechnology, Lviv, Pekarska 50, Ukraine
iDepartment of Instrumentation and Control Engineering, Faculty of Mechanical Engineering, CTU in Prague, Technicka 4, 166 07 Prague 6, Czech Republic
First published on 25th July 2016
We present measurements of the optical properties of crystalline PbIn6Te10. The samples are grown in the form of parallelepipeds of size ∼5 × 3 × 0.3 mm3. The absorption coefficient α(hν) shows an exponential behavior with energy in the energy range 0.82–0.99 eV followed by an abrupt increase in the absorption from 1.07–1.13 eV. According to the analysis of the data, the indirect gap is 0.88 eV while the direct gap is 1.05 eV. We have used our own measured X-ray diffraction data of the atomic positions in the unit cell as the input for the first principles calculations. Using the generalized gradient approximation (PBE − GGA) within the full potential linear augmented plane wave (FPLAPW + lo) method, the atomic positions are relaxed so as to minimize the forces acting on the atoms. We have used this relaxed geometry to calculate the electronic structure and related ground state properties using PBE − GGA and a recently modified Becke–Johnson potential (mBJ) and mBJ with spin–orbit coupling (mBJ + soc). The analysis of band energy dispersion shows that the valence band maximum (VBM) is located at Γ symmetry points, while the conduction band minimum (CBM) is situated at the M symmetry point of the first Brillouin zone, resulting in an indirect energy band gap of about 0.5 eV (PBE − GGA), 0.8 eV (mBJ) and 0.5 eV (mBJ + soc). It is clear that mBJ succeeds by a large amount in bringing the calculated energy gap in good agreement with the measured energy gap of 0.9 eV. However the inclusion of spin–orbit coupling modifies strongly the CBM and reduces the band by 0.3 eV. The anisotropy of space charge density distribution is analyzed with respect to the charge density distribution.
One of the principal factors which substantially restrains their applications in the mid-IR and in thermo-electrical devices is a large number of defects which interact with the phonon subsystem4 substantially enhancing photo-thermal heating. Of particular importance is the cationic substitution which may tailor the energy gaps and carrier mobility to desirable values.5 Moreover these substitutions may lead to enhanced polarizability of the anionic clusters defining the principal linear and nonlinear optical susceptibility dispersions.6 Following the equilibrium phase diagram for the PbTe–In2Te3 (ref. 6) it is melt congruently at 903 K and possesses a homogenous phase range within 70–83% mol of In2Te3. Its crystalline structure was determined by Deiseroth and Müller7 who found it to be trigonal (space group R32, a = 1.4971 nm, c = 1.8505 nm). The crystals were grown by the Bridgman–Stockbarger6 method. These crystals are transparent in the 1.5–30 μm spectral range8 with absorption coefficient which does not exceed 0.5 cm−1. The refractive index varies within 3–3.2. The magnitude of the experimental second order susceptibility tensor is about 51 pm V−1. This is substantially higher than for the single crystals of AgGaS2 (13 pm V−1) and AgGaSe2 (33 pm V−1). Generally the PbIn6Te10 single crystals could be very interesting materials for studies of the second-order nonlinear optical effects due to the presence of highly polarizable heavy cations, large phonon anharmonicity and charge density acentricity of the telluride anions.9 Also the lead cations, are expected to have charge density acentricity due to the presence of a lone pair.10,11 Another interesting point could be the existence of intrinsic cationic defects which may cause local enhancement of the local Lorentz field responsible for the optical susceptibilities. At the same time differences in mobility for different Brillouin Zone (BZ) directions may be substantial for the determination of charge transfer responsible for the corresponding hyperpolarizability. The further improvement of the nonlinear optical efficiency and photo-thermal parameters requires optimization of the charge density acentricity. These factors favor a detailed study of the band structure and carrier dispersion and related effective masses which define the photo carrier transport and mobility.
The main goal of this work is to establish some relations between the band structure features and the carrier mobility, anisotropy of the effective masses, etc. After comparison with the experimental optical data one can evaluate the role of defect states and their influence on perfect long-range ordered band states. Of principal importance is the role of the phonons and the type of the energy gap (direct and indirect). Such knowledge may give interesting inputs which can help to design better materials for devices.
Fig. 1 (a) As-grown single crystal of PbIn6Te10. (b) Experimental and theoretical diffraction patterns of the PbIn6Te10 crystals and their difference. |
Elements | Wyck. | x exp. | x opt. | y exp. | y opt. | z exp. | z opt. | Occ. | B |
---|---|---|---|---|---|---|---|---|---|
Pb | 9d | 0.7410(4) | 0.7408 | 0 | 0 | 0.259 | 0.2589 | 0.667 | 2.9(2) |
In1 | 18f | 0.0897(3) | 0.0895 | 0.3404(3) | 0.3401 | 0.0860(3) | 0.0861 | 1 | 0.96(15) |
In2 | 18f | 0.1905(2) | 0.1903 | 0.0808(2) | 0.0806 | 0.2356(3) | 0.2354 | 1 | 0.91(15) |
Te1 | 6c | 0 | 0 | 0 | 0 | 0.1630(4) | 0.1629 | 1 | 1.2(2) |
Te2 | 9d | 0.1657(3) | 0.1655 | 0 | 0 | 0 | 0 | 1 | 1.8(2) |
Te3 | 9e | 0.5927(3) | 0.5925 | 0 | 0 | 1/2 | 1/2 | 1 | 0.8(2) |
Te4 | 18f | 0.1838(3) | 0.1839 | 0.1387(3) | 0.1385 | 0.3750(2) | 0.3748 | 1 | 0.9(2) |
Te5 | 18f | 0.1030(2) | 0.1031 | 0.3015(3) | 0.3013 | 0.2319(2) | 0.2317 | 1 | 1.06(15) |
Inter-atomic distances are shown in Fig. 3. The lead atoms occupy a part of octahedral voids between the tellurium tetrahedron. As is clear from Table 1 the occupancy of the lead atoms is 2/3. Thus one third of the octahedral sites of the lead atoms in the pristine compound Pb3In12Te20 are un-occupied giving rise to PbIn6Te10. The space group is R32 (#155) which is rhombohedral in nature. In the rhombohedral symmetry the lead atoms are at the 3c sites (0, 0, 1/2). However one can easily switch from rhombohedral to hexagonal symmetry which has three times more atoms. This is what is shown in Table 1. In the hexagonal symmetry the lead atoms occupy the 9d site positions (see below) as is clear from Table 1. Inter-atomic distances within the borders of the tetrahedral and octahedral are less than the sum of the corresponding ionic radii. Due to such specific atomic architecture one can expect a possible space charge density acentricity which could be responsible for the enhanced observed second order nonlinear optical effects. It may be due to the isovalent substitution of the metallic atoms. More interesting it may be the heterovalent substitution of Pb2+ atoms by two atoms A+ (where A+ is one valence element). During such a substitution of Pb2+, the inserted atoms may occupy both lead atom positions as well as the corresponding voids. Generally it may be considered as a derivative of the phase PbII1−xAI2xIn6Te10.
The results presented in ref. 6 for PbIn6Te10 in the system PbTe–In2Te3 show that there exists a homogenous phase range within 13% mol. For such solid state alloys there occurs some heterovalence substitution of Pb2+ and In3+. So during the deviation of the content with respect to the perfect PbIn6Te10 single crystals for such materials the local site positions will be partially occupied both with respect to cationic site position as well as outside the site positions. The existence of the mentioned homogeneity for PbIn6Te10 may favor additional possibilities for the design of the materials with desired features. This could be done by controlling the number of defect states.
The pristine compound Pb3In12O20 is metallic in nature. In this compound the lead atoms at located at the 3c (0, 0, ½) Wyckoff positions in the rhombohedral unit cell. In the hexagonal unit cell, as given in Table 1, this corresponds to the 9d Wyckoff positions (0.741, 0.0, 0.259) and the other cyclic positions (0.0, 0.259, 0.741) and (0.259, 0.741, 0.0). The rhombohedral unit cell can be thought of as a hexagonal unit with three times the number of atoms. Hence there are 9 lead atoms in the pristine compound at the 9d positions. When we remove one lead atom (this amounts to three atoms in the hexagonal unit cell), the new compound is a defect compound Pb2In12O20 or PbIn6O10. The crystal structure of PbIn6Te10 single crystal is presented in Fig. 1. As mentioned above and as given in Table 1, the 9d site positions are not all occupied by the lead atoms. One third of the 9d site positions are empty. This will obviously change the symmetry group. We have removed one lead atom from the 3c position in the rhombohedral unit cell. This corresponds to three lead atoms being removed from the 9d site positions in the hexagonal unit cell. This is like putting empty spheres at these sites and preserving the hexagonal symmetry. We have performed the calculations in the hexagonal structures so as to be consistent with the experiment (see Table 1). Some codes require empty spheres to be explicitly mentioned but in WIEN2k it is implicit. The relaxed geometry is listed in Table 1 and compared with the experimental data. The potential for the construction of basis functions inside the sphere of the muffin-tin was spherically symmetric, whereas outside the sphere it was consistent with the crystalline symmetry.16 The muffin-tin radii (RMT) of the atoms were chosen in such a way that the interacting spheres did not overlap. The value of RMT is taken to be 2.5 a.u. for all the atoms. To achieve the total energy convergence, the basis functions in the interstitial region (IR) were expanded up to RMT × Kmax = 7.0 and inside the atomic spheres for the wave function. The maximum value of l was chosen to be lmax = 10, while the charge density is Fourier expanded up to Gmax = 12 (a.u.)−1. Self-consistency is obtained using 300 k-points in the irreducible Brillouin zone (IBZ). The self-consistent calculations are converged when the total energy of the system is stable within 0.0001 Ry. The electronic properties are calculated using 1500 k-points in the IBZ. The total and partial density of states (DOS) were calculated numerically by means of a modified tetrahedron method.17 The input required for calculating the DOS are the energy eigenvalues and eigenfunctions which are obtained from the band structure calculation. The total DOS and partial DOS are calculated for a large energy range covering principal inter-band transitions.
Fig. 4 Principal spectral dependence of the absorption coefficient near the energy gap edge for PbIn6Te10 at ambient temperature. |
The obtained results are explored with respect to their origin as direct and indirect absorption edge corresponding to inter-band transitions at different points of IBZ. For this reason the mentioned spectra dependences were plotted as α2 − hν dependences (Fig. 5(a)).19 The band energy gap is more close to indirect optical transitions. Analysis of the absorption data yields an indirect gap of 0.88 eV and a direct gap of 1.05 eV.
For estimation of the energy gap during indirect dipole allowed transition (Egi) (Fig. 5(b)) the straight lines (α)1/2 = f(hν) are extrapolated up to (α)1/2 = 0. The exponential part of the absorption usually is identified as Urbach tail.20 It is described by the equation (Fig. 5(c)):
(1) |
In Fig. 6 we present the temperature dependence of the conductivity. At ambient temperature the dark specific conductivity for the titled compound is equal to about 8 × 10−8 ohm−1 cm−1. Temperature dependence of the dark conductivity of the crystal (see Fig. 6) is described by exponential dependence:
σ = σ0exp(Ea/kT) | (2) |
We have calculated the effective mass of electrons () from the calculated band structure of PbIn6Te10 using mBJ and mBJ + soc. Usually we estimated the value of from the conduction band minimum curvature. The diagonal elements of the effective mass tensor, me, for the electrons in the conduction band are calculated following this expression;
(3) |
The effective mass of electron is assessed by fitting the electronic band structure to a parabolic function eqn (3). The calculated electron effective mass ratio () around M point of BZ is about 0.017 using mBJ and 0.020 using mBJ + soc, whereas the effective mass of the heavy holes () around Γ point the center of the BZ is about 0.029 using mBJ and 0.017 using mBJ + soc. Therefore, including the spin–orbit interaction strongly modifies of the CB, which arises primarily from Pb-6p orbitals. The spin–orbit interaction makes the CB almost isotropic and the effective mass is closer to the range typical of classical II–VI semiconductors.22
As PbIn6Te10 possesses enormous phonon acentricity described by the third rank polar tensors one can expect that this may be crucial for contributing to the charge density acentricity and the nonlinear optical susceptibility. In addition, the lead cations, are expected to have charge density acentricity due to the presence of a lone pair.10,11 The maximal carrier mobility is observed in the K–Γ direction of the BZ. Generally the mobility of the electrons (formed by conduction band dispersion) is higher. So this compound possesses a large difference in effective masses and we have a coexistence of different carriers which form non centro-symmetrical space charge density distribution. Due to the different mobility of holes and electrons there occurs some space charge separation, which is very important for susceptibilities due to the formation of charge transfer and related acentricity. Such factors may play a role in designing the optoelectronic devices using second harmonic generation. The principal role of the lead cations is to control the carrier mobility. This may be helpful in the manufacturing of useful optoelectronic materials. By varying the initial lead content near stoichiometry one can achieve large changes in the mobility and also in the local charge density acentricity, both of which have a strong influence on the nonlinear optical susceptibility (second and third order). So here we find a rare opportunity to tailor the mentioned features in the wide energy range.
The calculated angular momentum resolved projected density of states (PDOS) using mBJ and mBJ + soc and are shown in Fig. 8(a, c, e, g) and (b, d, f, h), respectively. We noticed that the spin–orbit coupling has significant influence on the band gap. It is seen that VBM has contributions mainly from Te-5p with small admixture of Pb-5p, In-5s/5p and Te-4d states whereas CBM is formed prevailingly by Pb-5p, In-5s, Te-5p with small admixture of In-5p states. It has been found that there exists hybridization between Te-5p and In-5s, Pb-6p and In-5p as well as between Pb-6p and In-4d and Te-4d states. Therefore, we expected to observe dominated ionic bonding and partial valence bonds with some admixture of covalence bonds.
Fig. 8 Calculated partial density of states of PbIn6Te10; (a, c, e and g) using mBJ; (b, d, f and h) using mBJ + soc. The PDOS is in electrons per atom. |
The origin of chemical bonding can be elucidated from the calculated partial density of states (PDOS). Integrating the PDOS in the energy region between −6.0 eV and Fermi level (EF) we obtain the total number of electrons for the orbitals of each atom of PbIn6Te10. For instance Pb-6s state posses 0.1 electrons, Te-5p state 0.9 electrons, In-5s state 0.9 electrons, Pb-6p state 0.2 electrons, In-5p state 0.19 electrons, Te-4d state 0.04 electrons, Pb-5d state 0.02 electrons, In-4d state 0.01 electrons and Pb-4f state 0.005 electrons. In Fig. 8(c–h), we present the partial DOS of those atoms which have significant contributions. The PDOS is per atom while in total DOS all atoms are taken into account. The contributions of the atoms to the valence bands exhibit presence of some electrons originating from lead, indium and tellurium atoms which are transferred into valence bands and contribute to the interactions between the atoms. The covalent bond arises due to the significant degree of the hybridization and the electro-negativity differences between the atoms. Electro-negativity is powerful tool to describe the strength electron affinity of the chemical bonding. It has been found that with increasing the electro-negativity differences between the atoms, the ionic nature of the bonding increases.
Therefore, according to the electro-negativity values, the degree of hybridization and the ionic character of the chemical bonds we can expect that lead atoms form substantially covalent bonds with tellurium atoms and tellurium with indium atoms. Indium and lead atoms have respectively tetrahedral and octahedral coordination. The measured In–Te bond lengths are shown Fig. 3. The calculated In–Te and Pb–Te bond lengths are listed in Table 2 and compared with the experimental values. It is clear that our calculations show good agreement with the experimental data. However some deviations may be explained by the existence of defects of cationic origin.
Atoms | Exp. bond length (Å) | Calc. bond length (Å) | Atoms | Exp. bond length (Å) | Calc. bond length (Å) |
---|---|---|---|---|---|
In1–Te4 | 2.783 | 2.781 | In2–Te5 | 2.816 | 2.814 |
In1–Te2 | 2.775 | 2.772 | In2–Te1 | 2.825 | 2.823 |
In1–Te5 | 2.845 | 2.842 | In2–Te3 | 2.767 | 2.766 |
In1–Te5 | 2.808 | 2.807 | In2–Te4 | 2.754 | 2.753 |
Pb–Te1 | 3.298 | 3.296 | Pb–Te4 | 3.298 | 3.297 |
Pb–Te2 | 3.251 | 3.249 | Pb–Te5 | 3.404 | 3.403 |
Pb–Te3 | 3.251 | 3.249 | Pb–Te6 | 3.404 | 3.403 |
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