Band Structure, Density of States, and Optical Susceptibilities of a

Jul 30, 2009 - electronic structure and the linear optical properties for the novel lithium ... full-potential linear augmented plane wave method with...
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J. Phys. Chem. B 2009, 113, 11583–11588

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Band Structure, Density of States, and Optical Susceptibilities of a Novel Lithium Indium Orthoborate Li3InB2O6 Ali Hussain Reshak,* S. Auluck, and I. V. Kityk Institute of Physical Biology-South Bohemia UniVersity, NoVe Hrady 37333, Czech Republic, Physics Department, Indian Institute of Technology Kanpur, Kanpur 208016, India, and Electrical engineering Department, Technological UniVersity of Czestochowa, Al. Armii Krajowej 17/19, Czestochowa, Poland ReceiVed: May 1, 2009; ReVised Manuscript ReceiVed: July 6, 2009

By use of the structural parameters of the single crystal lithium indium orthoborate obtained by Penin et al. (Solid State Sci. 2001, 3, 461-468), from X-ray diffraction data, we present a first-principle study of the electronic structure and the linear optical properties for the novel lithium indium orthoborate Li3InB2O6. A full-potential linear augmented plane wave method within density functional theory with the Engel-Vosko exchange correlation was used. This compound has a wide direct energy band gap of about 3.8 eV with both the valence band maximum and conduction band minimum located at the center of the Brillouin zone. Our calculations of the partial density of states shows that the upper valence band originates predominantly from the O-p, B-p, and In-p states, and the lower conduction band is dominated by the O-s/p, In-p, and B-p states. Thus the O-p states in the upper valence band and lower conduction band has a significant effect on the energy band gap dispersion. The uniaxial anisotropy [δε )(ε0zz - ε0xx)/ε0tot] is about -0.041. 1. Introduction Borate compounds are currently very attractive to the scientific community owing to their wide range of applications.1-3 They exhibit a high transparency in the ultraviolet (UV) and near-infrared (IR) region, good chemical stability, and high optical quality. The cationic subsystem influences the transparency and mechanical properties. Borates are among the most interesting and therefore the most extensively studied materials with interesting properties ranging from phosphorescence4 ferroelectric, semiconducting behavior, and nonlinear optical (NLO) susceptibilities.2,5,6 They have a high second harmonic generation (SHG) coefficient, high optical damage threshold, high hardness, and can be used in laser frequency conversion.7,8 In addition, a boron atom may adopt triangular or tetrahedral oxygen coordination, the BO3 and BO4 groups may be further linked via common oxygen atoms to form isolated rings and cages or polymerize into infinite chains, sheets, and networks, leading to a rich structural chemistry.2,9 The oxoborates K2R2OB2O6 with R ) Al,6 Ga,6 or Cs2Ga2OB2O610 have structures that contain triangular BO3 and tetrahedral RO4 (R ) Al, Ga). With the view of finding new optical materials, similar to non rare-earth trivalent cations,11 it might seem interesting to investigate the properties of lithium borates such as Li6R2B4O12 with R ) Al12 and Ga,13 which contain [R2(BO3)4] chains with two different BO3 groups. In the system Li2OIn2O3-B2O3 only Li3In2B3O9 has been synthesized and its structure characterized by Penin et al.11 Theoretical exploration has shown that anionic groups and chemical bonding structures and coordination of boron atoms have an important influence on the nonlinear optical properties of these crystals.14,15 The theoretical investigation of the band structure may play a crucial role in understanding of the physical properties and give predictions of the nonlinear optical susceptibilities. Firstprinciple band structure calculations have been successfully used * To whom correspondence should be addressed. Phone: +420 777729583. Fax: +420-386 361231. E-mail: [email protected].

to obtain fundamental parameters of semiconductors and dielectrics. The structural parameters and dynamical properties of crystals determine a wide range of microscopic and macroscopic features, i.e., diffraction, sound velocity, elastic constants, Raman and infrared absorption, inelastic neutron scattering, specific heat, etc. To the best of our knowledge no previous work either experimental data or theoretical calculation on the fundamental optical functions or first principles electronic structure calculations of lithium indium borate Li3InB2O6 have appeared in the literature. A detailed depiction of the electronic and spectral features of the optical properties of lithium indium borate using the full potential method is very essential and would bring us important insights in understanding the origin of the electronic band structure and densities of states. The present study is aimed toward calculations using the full potential linear augmented plane wave (FP-LAPW) method, which has proven to be one of the most accurate methods16,17 for the computation of the electronic structure of solids within a framework of density functional theory (DFT). In section 2 are presented theoretical aspects of the calculation methods. The calculated band structure, charge density distribution, and density of states are given in section 3a. Section 3b is devoted to the optical properties. 2. Theoretical Calculations Li3InB2O6 belongs to a monoclinic space group P21/n, with unit cell parameters a ) 5.168(5) Å, b ) 8.899(9) Å, c ) 10.099(10) Å, β ) 91.112(17)°, Z ) 4.11 It exhibits a threedimensional framework of vertex sharing InO5 trigonal bipyraminds and BO3 triangles, which isolates Li ions in channels. The structure of this crystal was characterized11 by unusual oxygenated environment of In cations with one of the three Li cations, forming more or less regular trigonal bipyramids. In Figure 1a we show the crystal structure. We use a full-potential linear augmented plane wave method within DFT,18 as implemented in the package WIEN2k code.19 Keeping the lattice parameters fixed at the experimental values, we have optimized

10.1021/jp904043f CCC: $40.75  2009 American Chemical Society Published on Web 07/30/2009

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Figure 1. (a) The calculated crystal structure and (b) the calculated band structure of the lithium indium orthoborate.

TABLE I: Calculated Atomic Position in Comparison with the Experimental One12 atom In O(1) O(2) O(3) O(4) O(5) O(6) B(1) B(2) Li(1) Li(2) Li(3)

x exptl 0.14845 0.4043 0.7515 0.8264 0.5885 0.2046 0.2001 0.6629 0.3329 0.828 0.657 0.324

x calcd 0.15411 0.40279 0.75292 0.82573 0.59262 0.20524 0.18420 0.66434 0.32966 0.83181 0.66298 0.32595

y exptl

y calcd

0.32611 0.1539 0.3171 0.0550 0.1704 0.2992 0.0374 0.1736 0.1681 0.346 0.525 0485

0.33158 0.15185 0.31477 0.04803 0.16677 0.30540 0.04179 0.17071 0.17105 0.34048 0.52278 0.48623

z exptl 0.12343 0.0899 0.1502 0.1208 0.4109 0.3356 0.3612 0.1201 0.3689 0.381 0.169 0.399

z calcd 0.12216 0.08780 0.15261 0.11861 0.40611 0.33753 0.35857 0.11962 0.36982 0.38157 0.17758 0.40993

TABLE II: Calculated Bond Lengths in Comparison with Experimental Data12 bond type

bond length (Å) a

b

B(1)-O(1) -O(2) -O(3)

1.375 , 1.376 1.389a, 1.389b 1.397a, 1.352b

B(2)-O(4) -O(5) -O(6)

1.391a, 1.380b 1.3403a, 1.380b 1.391a, 1.352b

In-O(1) -O(2) -O(3) -O(4) -O(5) -O(6) a

a

b

2.051 , 2.057 2.082a, 2.076b 2.930a, 2.931b 2.162a, 2.163b 2.169a, 2.170b 2.039a, 2.041b

TABLE III: Calculated Bond Angles in Comparison with Experimental Data12

bond type

bond length (Å)

Li(1)-O(1) -O(2) -O(3) -O(4) -O(5) Li(2)-O(2) -O(3) -O(4) -O(5) Li(3)-O(1) -O(2) -O(3) -O(5) -O(6)

2.133a, 2.136b 2.370a, 2.373b 2.025a, 2.027b 3.079a, 3.081b 2.053a, 2.051b 1.924a, 1.922b 2.138a, 2.138b 2.027a, 2.028b 3.071a, 3.073b 1.912a, 1.914b 1.929a, 1.928b 2.260a, 2.263b 1.869a, 1.874b 2.670a, 2.674b

This work. b Experimental data.12

the structure by minimization of the forces (1 mRy/au) acting on the atoms. From the relaxed geometry the electronic structure and the chemical bonding can be determined and various spectroscopies can be simulated and compared with experiment (see Tables I, II, and III). Once the forces are minimized in this construction one can then find the self-consistent density at these positions by turning off the relaxations and driving the system to self-consistency. The exchange and correlation potential used is the Engel-Vosko generalized gradient approximation (EV-GGA).20 We take the full relativistic effects for core states and use the scalar relativistic approximation for

bond type

bond angle (deg)

bond type

bond angle (deg)

O(1)-B(1)-O(2) O(1)-B(1)-O(3) O(2)-B(1)-O(3) O(4)-B(2)-O(5) O(4)-B(2)-O(6) O(5)-B(2)-O(6) O(4)-In-O(5) O(1)-In-O(4) O(1)-In-O(5) O(4)-In-O(6) O(2)-In-O(4) O(2)-In-O(5) O(5)-In-O(6) O(2)-In-O(6) O(1)-In-O(2) O(1)-In-O(6) O(1)-Li(1)-O(2) O(1)-Li(1)-O(3) O(1)-Li(1)-O(4)

118.3a, 118.7b 120.4a, 120.4b 120.7a, 120.9b 121.9a, 121.0b 119.6a, 120.8b 118.3a, 118.2b 175.5a, 174.5b 85.7a, 85.8b 88.4a, 90.4b 94.3a, 96.3b 91.7a, 90.4b 88.0a, 88.9b 89.5a, 89.0b 113.7a, 113.9b 129.9a, 129.3b 116.1a, 116.9b 173.0a, 173.9b 93.9a, 94.2b 88.1a, 87.6b

O(1)-Li(1)-O(5) O(2)-Li(1)-O(3) O(2)-Li(1)-O(4) O(2)-Li(1)-O(5) O(3)-Li(1)-O(4) O(3)-Li(1)-O(5) O(4)-Li(1)-O(5) O(2)-Li(2)-O(3) O(2)-Li(2)-O(4) O(2)-Li(2)-O(6) O(3)-Li(2)-O(4) O(3)-Li(2)-O(6) O(4)-Li(2)-O(6) O(1)-Li(3)-O(3) O(2)-Li(3)-O(3) O(1)-Li(3)-O(5) O(3)-Li(3)-O(5) O(3)-Li(3)-O(3) O(3)-Li(3)-O(5)

93.6a, 93.7b 91.4a, 91.7b 88.0a, 88.4b 84.1a, 84.1b 118.6a, 118.0b 120.9a, 123.9b 117.0a, 117.8b 102.3a, 102.4b 114.1a, 113.9b 107.9a, 107.1b 107.3a, 107.5b 101.1a, 100.2b 121.8a, 122.9b 109.0a, 109.4b 93.8a, 93.4b 120.1a, 120.9b 102.1a, 101.4b 99.5a, 99.7b 122.9a, 123.4b

a

This work. b Experimental data.12

the valence states. The muffin tins assumed to be 1.29 au (atomic units) for In, 1.75 au for Li, and 1.27 au for O and B. We used the parameters Kmax ) 9/RMT and lmax ) 10. The self-consistency was achieved by using of 300 k-points in the irreducible Brillouin zone (IBZ). The density of states and the optical properties are calculated using 500 k-points of the IBZ. The self-consistent interactions were performed until convergence only when the integrated charge distance per formula unit, ∫|Fn - Fn-1| dr, between the input charge charge density [Fn-1(r)] and the output [Fn(r)], is less than 0.0001. 3. Results and Discussion a. Band Structure, Density of States, and Electron Charge Density. The structure of lithium indium borate contains only two isolated BO3 triangles and no BO4 tetrahedron.11 The lithium indium borate belongs to the orthoborate class21 as in the M3RB2O6 family. The DFT calculations of the band structure and total and partial density of states are shown in Figures 1b and 2. This compound possesses a direct energy band gap of about 3.8 eV with valence band maximum (VBM) and the

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Figure 2. (a-e) Calculated total and partial densities of states (states/eV unit cell), (f) total valence charge densities in the (110) plane (2D) and (3D).

conduction band minimum (CBM) at Γ point of the BZ. The electronic structure of the upper valence band originates predominantly from the O-p, B-p, and In-p states, while the lower conduction band is dominated by the O-s/p, In-p, and B-p states. The O-p states in the upper valence band and lower conduction band has significant effect on the energy band gap dispersion. From partial densities of states one can see that there exists a strong hybridization between O-p and In-p states at around 6.0 eV and above. In the energy region around -5.0 eV, B-s hybridizes with B-p, and In-p hybridized with O-s. At the CBM, Li-p strongly hybridizes with In-p states.

The nature of chemical bonding can be elucidated from the total and partial densities of states. We find that the densities of states, extending from -6.0 eV up to Fermi energy (EF) is larger from O-p states (1.0 electrons/eV), B-p states (0.3 electrons/eV), B-s states (0.29 electrons/eV), and In-p (0.07 electrons/eV). This is obtained by comparing the total densities of states with the angular momentum projected densities of states of O-p, B-p, B-s, and In-p states as shown in Figure 2. These results show that some electrons from O-p, B-p, B-s, and In-p states are transferred into valence bands (VBs) and contribute in weak covalence interactions between O-O, B-B, and In-In

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Figure 3. (a) Calculated ε2xx(ω) (dark solid curve) and ε2zz(ω) (light dashed curve) spectra. (b) Calculated ε1xx(ω) (dark solid curve) and ε1zz(ω) (light dashed curve). (c) Calculated Rxx(ω) (dark solid curve) and Rzz(ω) (light dashed curve) spectrum. (d) Calculated absorption coefficient Ixx(ω) (dark solid curve) and Izz(ω) (light dashed curve) spectrum. (e) Calculated Im σxx(ω) (light solid curve), Im σzz(ω)(dark dashed dotted curve), Re σxx(ω) (dark solid curve), and Re σzz(ω) (light long dashed curve) spectrum. (f) Calculated loss function of Lxx(ω) (dark solid curve) and Lzz(ω) (light dashed curve) spectrum.

atoms, and the substantial covalence interactions between B and O and In and O atoms. The bonding nature of the solids can be described accurately by using electronic density plots.22,23 The charge density in our calculation is derived from a reliable converged wave function and hence it can be used to study the bonding nature of the solid. To visualize the nature of the bonding character and to explain the charge transfer and bonding

properties we calculated the total valence charge density. In Figure 2f, we present the total charge densities in the (110) plane. The charge densities help us to analyze the nature of the bonds in this compound according to a classical chemical concept. This concept is very useful to classify compounds into different categories with different chemical and physical properties. Our calculated bond lengths (Table II) and the bond angles (Table III) show reasonable agreement with the

Novel Lithium Indium Orthoborate Li3InB2O6 experimental data.11 This agreement is attributed to our use of the full potential calculations. b. First-Order (Linear) Optical Susceptibilities. The monoclinic space group P21/n of lithium indium orthoborate with four formula units per unit cell has two dominant components of the dielectric tensor. These frequency-dependent dielectric functions are ε2xx(ω) and ε2zz(ω) corresponding to the electric field direction parallel and perpendicular to the crystallographic c axis. The calculation of these frequency-dependent dielectric functions requires the precise values of energy eigenvalues and electron wave functions. These are natural outputs of a band structure calculation. Generally there are two contributions to frequency-dependent dielectric functions, namely, intraband and interband transitions. The contribution due to intraband transitions is crucial only for metals. The interband transitions of these frequency-dependent dielectric functions can be split into direct and indirect transitions. We neglect the indirect interband transitions involving scattering of phonons assuming that they give a small contribution to the frequency-dependent dielectric functions for the case of the dielectric borates.24 To calculate the direct interband contributions to the imaginary part of the frequency-dependent dielectric function, it is necessary to perform summation over the BZ structure for all possible transitions from the occupied to the unoccupied states. Taking the appropriate transition dipole matrix elements into account, we calculated the imaginary part of the frequency-dependent dielectric functions using the expressions in refs.25 and.26 Figure 3a depicts the variation of the imaginary part of the frequency dependent dielectric function. The spectral broadening is taken to be 0.01 eV.27 Following the ε2(ω) spectra one can conclude that the principal peaks are situated at energies around 8.5 eV for ε2xx(ω) and 8.0 and 10 eV for ε2zz(ω). We should emphasize that there is anisotropy between these two components. The peaks in the optical response are caused by the allowed electric-dipole transitions between the valence and conduction bands. To identify these structures we should consider the magnitude of the optical dipole matrix elements. The observed structures would correspond to those transitions which have larger optical matrix dipole transition elements. It would be worthwhile to attempt to identify the interband transitions that are responsible for the spectral features in ε2xx(ω) and ε2zz(ω) using our calculated band structure and density of states. The optical transitions occur between the occupied Ins/p, O-s/p, and B-s/p states and the unoccupied In-s/p/d, O-s/p, and B-s/p states. From the spectral dependences of imaginary parts of the dielectric function ε2xx(ω) and ε2zz(ω) the real parts ε1xx(ω) and ε1zz(ω) can be calculated using Kramers-Kronig relations.28 These are shown in Figure 3b. Again there is anisotropy between ε1xx(ω) and ε1zz(ω). The static dielectric constant ε1(0) is given by the low energy limit of ε1(ω). Note that we do not include phonon contributions to the dielectric screening. ε1(0) corresponds to the static optical dielectric constant (ε∞). The calculated value of ε1xx(0) is 2.4 and 2.3 for ε1zz(0). The uniaxial anisotropy [δε )(ε0zz - ε0xx)/ε0tot] is -0.041.29 For more details about the spectral features of the optical susceptibilities of the investigated crystal we have calculated the spectral dependences for reflectivity spectra R(ω), absorption coefficient I(ω), optical conductivity σ(ω), and loss function L(ω). Figure 3c shows the calculated reflectivity spectra. We notice that a reflectivity maximum around 9.0 eV arises from interband transitions. A reflectivity minimum at energies ranging from about 12.0 eV confirms the occurrence of a collective plasma resonance. The depth of the plasma minimum is

J. Phys. Chem. B, Vol. 113, No. 34, 2009 11587 determined by the imaginary part of the dielectric function at the plasma resonance and is representative of the degree of overlap between the interband absorption regions. We should note that at high energies (at around 12.5 eV) this compound shows a rapidly increasing reflectivity. The calculated absorption coefficient I(ω) is shown in Figure 3d. The calculated optical conductivity σ(ω) is shown in Figure 3e. It also shows anisotropy between σxx(ω) and σzz(ω). The optical conductivity is related to the frequency-dependent dielectric function ε(ω) as ε(ω) ) 1 +(4πiσ(ω))/(ω). The peaks in the optical conductivity spectra are determined by the electric-dipole transitions between the occupied states to the unoccupied states. In Figure 3f, we present the electron energy loss L(ω) spectra. In the energy loss spectra we do not see any distinct maxima at low energy range. With the increase of the energy, ε2(ω) has a slow decrease. However, at higher energy ε2(ω) is smaller, and thus the amplitude of the energy loss becomes larger. The main peak in the energy loss function, which defines the screened plasma frequency ωp, is located at 12.0 eV. These main peaks correspond to the abrupt reduction of the reflectivity spectrum R(ω) and to the zero crossing of ε1(ω). As there are no experimental or theoretical results for the spectral features of the optical susceptibilities available for this compound, we hope that our work will stimulate more works. 4. Conclusions In summary, by use of an accurate full-potential density functional method, we have investigated the structural properties, electronic structure, and the spectral features of the linear optical properties of lithium indium orthoborate. The band structure and total densities of states along with the partial densities of states were calculated. The electronic structure calculation reveals that this crystal is a semiconductor with a direct energy gap (Γ - Γc) of about 3.8 eV. We have calculated the total valence charge density in the (110) plane in order to visualize the nature of the bonding and to explain the charge transfer. We have calculated the bond lengths and the bond angles. We find reasonable agreement with the experimental data. This agreement is attributed to our use of the full potential calculations. The frequency dependent dielectric function, reflectivity spectra, absorption coefficient, conductivity and the loss function were calculated. The linear optical properties are found to anisotropic. Acknowledgment. This work was supported from the institutional research concept of the Institute of Physical Biology, UFB (Grant No. MSM6007665808). References and Notes (1) Keszler, D. A. Curr. Opin. Solid State Mater. Sci 1996, 1, 204. (2) Becker, P. AdV. Mater. 1998, 10, 979. (3) Knitel, M. New Inorganic Scintillators and Storage Phosphors for Detection of Thermal Neutrons; Delft Univ. Press: Delft, 1998. (4) Chaminade, J.-P.; Gravereau, P.; Jubera, V.; Fouassier, C. J. Solid State Chem. 1999, 146, 189. (5) Hu, Z.-G.; Higashiyama, T.; Yoshimura, M.; Yap, Y. K.; Mori, Y.; Sasaki, T. Jpn. J. Appl. Phys. 1998, 37, L1093. (6) Smith, R. W.; Kennard, M. A.; Dudick, M. J. Mater. Res. Bull. 1997, 32, 649. (7) Chen, C. T.; Wu, B.; Jiang, A.; You, G. Sci. China B 1985, 18, 235. (8) Oseledchik, Yu. S.; Prosvirnin, A. L.; Pisarevskiy, A. I.; Starshenko, V. V.; Osadchuk, V. V.; Belokrys, S. P.; Svitanko, N. V.; Korol, A. S.; Krikunov, S. A.; Selevich, A. F. Opt. Mater 1995, 4, 669. (9) Grice, J. D.; Burns, P. C.; Hawthorne, F. C. Can. Mineral 1999, 37, 731. (10) Smith, R. W. Acta Cryst. C 1995, 51, 547.

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Reshak et al. (21) Heller, G. Top. Curr. Chem. 1986, 131, 39. (22) Hoffman, R. ReV. Mod. Phys 1988, 60, 801. (23) Gellatt, C. D., Jr.; Williams, A. R.; Moruzzi, V. L. Phys. ReV. B 1983, 27, 2005. (24) Smith, N. V. Phys. ReV. B 1971, 3, 1862. (25) (a) Hufner, S.; Claessen, R.; Reinert, F.; Straub, Th.; Strocov, V. N.; Steiner, P. J. Electron Spectrosc. Relat. Phenom. 1999, 100, 191. (b) Ahuja, R.; Auluck, S.; Johansson, B.; Khan, M. A. Phys. ReV. B 1994, 50, 2128. (26) Hussain Reshak, Ali; Kityk, I. V.; Auluck, S. J. Chem. Phys. 2008, 129, 074706. (27) Hussain Reshak, Ali; Chen, Xuean; Auluck, S.; Kityk, I. V. J. Chem. Phys. 2008, 129, 204111. (28) Tributsch, Z. Naturforsch. A 1977, 32A, 972. (29) Hussain Reshak, A. PhD Thesis, Indian Institute of TechnologyRookee; India, 2005.

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