30 CHAPTER 2 ELECTRONIC STRUCTURE AND GROUND STATE PROPERTIES OF M 2 O (M: Li, Na, K, Rb) 2.1 INTRODUCTION Oxides of alkali metals (M 2 O) (M: Li, Na, K, Rb) play an important role in reducing the work function and thus enhancing the electrical current of photocathodes (Esher 1981), and in promoting catalytic reactions and oxidation enhancement of various semiconductor surfaces (Campbell 1985). Hence, they appear to be promising candidates for technological applications in solid state batteries (Jamal et al 1999; Chao et al 2006), in fuel cells or in solid state gasdetectors (Lee et al 1996). The materials with fluorite (CaF 2, SrF 2, BaF 2 )type structure have been extensively studied when compared to antifluorite type structure. The alkalimetal oxides (M 2 O) are found to crystallize in the cubic antifluorite (anticaf 2 type) structure (space group no. 225) (Zintl et al 1934). This structure is antimorphous to the fluorite (CaF 2 ) structure. Materials with antifluoritetype are found to exhibit fast ionic conduction and they have attracted considerable attention due to their technological usefulness, and also by their other remarkable and interesting physical properties. Their high ionic conductivity arises as a consequence of Frenkel defect formation by metal atoms redistributing on their regular sites as well as on the interstitial sites without any significant distortion of the face centered cubic (FCC) oxygen sublattice. This is so mainly because of the less compact crystal structure of antifluorite type structure in comparison to that of fluoritetype structure.
31 Experimentally, Hull et al (1988) measured the bulk modulus, lattice parameter and the elastic constants of Li 2 O at ambient conditions and temperature upto 1600 K. The transition temperature T c ~ 1200 K (the melting point is 1705 K) of Li 2 O was measured by Oishi et al (1979), who found that the elastic constant C 11 suddenly decreases at the transition temperature T c. Mikajlo et al (2003, 2003a, 2002) performed the electronic structure of alkali metal oxides by using both the electron momentum spectroscopy measurement and the linear combination of atomic orbitals (LCAO) method. The surface and bulk electronic structure of Li 2 O studied by Liu et al (1996) using the photoemission and electronenergy loss spectroscopy measurement. The oxygen species formed in the presence of lithium, potassium and cesium have been studied by Jupille et al (1992), using the ultraviolet and Xray photoelectron spectroscopy measurements. Structural phase transition of Li 2 O from antifluorite to anticotunnite (PbCl 2 type structure) is identified by Laziki et al (2006) and Kunc et al (2005) by using the xdiffraction study. From theoretical point of view, Dovesi et al (1991) calculated the lattice constants and elastic properties of Li 2 O, Na 2 O and K 2 O at zero pressure via the ab initio HartreeFock LCAO method. This method (LCAO) is also applied by Cancarevic et al (2006) to study the stability of the alkali metal oxides under pressure. The electronic band structures of these materials at ambient conditions were discussed by Zhuravlev et al (2001) using the self consistent pseudopotential method (PP). Dovesi (1985) performed the LCAO formalism to study the electronic structure of Li 2 O. The Wannier function based on LCAO formalism has been reported by Shukla et al (1998) on Li 2 O and Na 2 O compounds. This study demonstrates the importance of the correlation effects. The superionic behavior of Li 2 O was investigated using the molecular dynamics (MD) simulation method (Goel et al 2004), from which the lattice constant and the elastic constants under ambient pressure were obtained. Rodeja et al (2001) and Wilson et al (2004) investigated the
32 structural and elastic properties of Li 2 O by using the LDApseudopotential plane wave method and the asphyrical ion model respectively. The structural, electronic and defects properties of lithium oxide have been studied by Islam et al (2006) using both the plane wave (PW) and the local combination of atomic orbitals (LCAO) methods. Mauchamp et al (2006) simulated the electron energyloss near edge structure at the lithium K edge in Li 2 O using the full potential linearized augmented plane wave (FPLAPW) method. 2.2 PRESENT STUDY To understand some of the physical and electronic properties of these compounds a detailed description of electronic structure and density of states (DOS) of these compounds is needed. In this chapter, we present the self consistent band structure for M 2 O at ambient conditions as well as at high pressure using the tightbinding linear muffintin orbital method (TBLMTO) (Andersen 1975, Andersen and Jepsen 1984). 2.3 CRYSTAL STRUCTURE These compounds crystallize in the anti CaF 2 type structure. Figure 2.1 shows the unit cell of fluorite (CaF 2 ) type structure (Galasso 1970). From Figure 2.1 it can be seen that the calcium ions occupy the corner and facecentered positions of the cubic unit cell. The fluorine ions are situated in calcium tetrahedra. Each calcium ion is surrounded by eight fluorine ions and each fluorine ion is coordinated with four calcium ions. Figure 2.2 shows the layer sequence in fluorite (CaF 2 ) type structure. Four calcium ion are at (0, 0, 0); (0, 0.5, 0.5); (0.5, 0, 0.5); (0.5, 0.5, 0) and eight fluorine ions at (0.25, 0.25, 0.25); (0.25, 0.75, 0.75); (0.75, 0.25, 0.75); (0.75, 0.75, 0.25); (0.75, 0.75, 0.75); (0.75, 0.25, 0.25); (0.25, 0.75, 0.25); (0.25, 0.25, 0.75).
33 Figure 2.1 Unit cell of Fluorite (CaF 2 ) type structure Figure 2.2 Layer Sequence in Fluorite (CaF 2 ) type structure
34 Figure 2.3 Unit cell of antifluorite (anticaf 2 ) type structure pressure. Figure 2.4 Layer Sequence in antifluorite (anticaf 2 ) type structure
35 Figure 2.3 shows the unit cell of antifluorite (anti CaF 2 ) type structure. In the antifluorite type structure the fluorine ions occupy the corner and facecentered positions and the calcium ions are situated in fluorine tetrahedra. Each fluorine ion is surrounded by eight calcium ions and each calcium ion is coordinated with four fluorine ions. Figure 2.4 shows the layer sequence in antifluorite (anticaf 2 ) type structure. Four fluorine ions are at (0, 0, 0); (0, 0.5, 0.5); (0.5, 0, 0.5); (0.5, 0.5, 0) and eight fluorine ions at (0.25, 0.25, 0.25); (0.25, 0.75, 0.75); (0.75, 0.25, 0.75); (0.75, 0.75, 0.25); (0.75, 0.75, 0.75); (0.75, 0.25, 0.25); (0.25, 0.75, 0.25); (0.25, 0.25, 0.75). 2.4 COMPUTATIONAL DETAILS To obtain the electronic structure and the ground state properties of alkali metal oxides (M 2 O) TBLMTO method has been used (Andersen 1975, Skriver 1984). vonbarth and Hedin parameterization scheme within the local density approximation (LDA) has been used to calculate the exchange correlation part of potential (von Barth and Hedin 1972). In this approximation, the crystal is divided into space filling spheres centered on each of the atomic site. To minimize the errors in the LMTO method combined correction terms are also included, which account for the nonspherical shape of the atomic cell and the truncation of the higher partial waves inside the sphere. The WignerSeitz sphere is chosen in such a way that the sphere boundary potential is minimum and the charge flow between the atoms is in accordance with the electro negativity criteria. The energy eigenvalues were calculated for 512 kpoints in the irreducible part of the Brillouin Zone. It is well known that the LMTO method gives accurate results only for the close packed structures. Hence required number of empty spheres are added. In this case one empty sphere is included at (0.5, 0.5, 0.5) without disturbing the crystal symmetry. The tetrahedron method of brillouin
36 zone integration has been used to calculate the density of states (Jepsen and Andersen 1971). The following basis orbitals were used in the calculation: Li: 2s 1 2p 0 3d 0 ; Na: 3s 1 3p 0 3d 0 ; K: 4s 1 3p 6 3d 0 ; Rb: 5s 1 4p 6 4d 0 ; O: 2s 2 2p 4 3d 0. In the case of Na, 2plike states are well below the oxygen 2slike states. In K 2 O and Rb 2 O, there is hybridization of the semi corelike K3p state and Rb4p states with the states on the other atoms. This effect is more pronounced in Rb than in K, so it becomes necessary to treat the semi corelike K3p and Rb4p states as relaxed valence band states in K 2 O and Rb 2 O. 2.5 TOTAL ENERGY CALCULATIONS In order to calculate the groundstate properties of alkalimetal oxides (M 2 O) the total energies are calculated for all the four compounds as a function of reduced volume ranging from 1.1 to 0.65V 0, where V 0 is the experimental equilibrium volume (Zintl et al 1934). The plots of calculated total energy versus reduced volume for these compounds are given in Figure 2.5. The calculated total energies were fitted to the Birch equation of state (Birch 1978) as a function of reduced volume to obtain equilibrium properties, such as the equilibrium lattice constant (a) and the bulk modulus (B 0 ). The pressure is obtained by taking the volume derivative of the total energy. The bulk modulus is calculated from the pressure volume relation equation (2.1). The theoretically calculated equilibrium lattice constant (a), bulk modulus (B 0 ) and total energy (E 0 ) are given in Tables 2.1 to 2.2 and are compared with available experimental (Zintl et al 1934; Hull et al 1988) 7/3 5/3 3B 0 V0 V0 P (2.1) 2 V V
37 Figure 2.5 Total energies as a function of reduced volume of Li 2 O, Na 2 O, K 2 O and Rb 2 O
38 Table 2.1 Calculated lattice constant (a), bulk modulus (B 0 ) and total energy (E 0 ) of Li 2 O and Na 2 O Compounds a (Å) B 0 (GPa) E 0 (Ry) Li 2 O Present Experiment Other calculations 4.533 4.619 a 4.560 b, 4.580 b, 4.580 d, 4.519 d, 4.638 d, 4.584 d, 4.570 e, 4.573 f, 4.570 g 95.00 89.00 h 102.76 b, 97.91 b 94.60 e, 105.00 f 180.0488 179.9254 b 180.8788 b 179.9286 f 179.9080 g Na 2 O Present Experiment Other calculations 5.465 5.560 a 5.450 b, 5.470 b, 5.497 c, 5.393 c, 5.559 c, 5.498 c, 59.00 62.18 b 58.63 b, 61.10 e 797.6686 797.3796 b 799.7230 b 5.481 e, 5.484 f 57.50 f 797.3862 f a Zintl et al (1934) b Cancarevic et al (2006) c Mikajlo et al (2003) d Mikajlo et al (2002) e Shukla et al (1998) f Dovesi et al (1991) g Dovesi (1985) h Hull et al (1988)
39 Table 2.2 Calculated lattice constant (a), bulk modulus (B 0 ) and total energy (E 0 ) of K 2 O and Rb 2 O Compounds a (Å) B 0 (GPa) E 0 (Ry) K 2 O Present Experiment Other calculations Rb 2 O Present 6.362 6.449 a 6.430 b, 6.420 b, 6.466 c 6.168 c, 6.414 c, 6.360 c 6.466 d 6.819 33.46 40.74 b, 38.69 b 34.60 g 30.00 2553.6098 2546.3622 b 2549.9316 b 2546.3864 d 12063.0895 Experiment 6.742 a Other calculations 6.830 b, 6.800 b 35.84 b, 35.85 b 244.9996 b 244.8180 b a Zintl et al (1934) b Cancarevic et al (2006) c Mikajlo et al (2003a) d Dovesi et al (1991)
40 and other theoretical work (Mikajlo et al 2003, 2003a, 2002; Cancarevic et al 2006). The calculated groundstate properties of these compounds are in agreement with other theoretical and experimental results. The calculated lattice constant is 1.8%, 1.6% and 1.3% smaller than the experimental value for Li 2 O, Na 2 O and K 2 O, whereas for Rb 2 O calculated lattice constant is 1.1% greater than the experimental value. A comparison of the bulk modulus B 0 show a clear decrease with heavier atom (Li 2 O to Rb 2 O). This variation in the bulk modulus is similar to that of the pure metal atom bulk modulus B 0. The calculated total energy is compared with other theoretical results (Cancarevic et al 2006; Dovesi et al 1991; Dovesi (1985)). From the table, it can be seen that there is a small difference in total energies. This difference may be due to different exchange correlation schemes used in the calculations. 2.6 ELECTRONIC STRUCTURE AND DENSITY OF STATES The electronic band structures of alkalimetal oxides (M 2 O) have been calculated at ambient as well as at high pressure and are shown in Figures 2.6 to 2.9. The overall band profiles of all four compounds are found to have the same characteristic features. Bandgap occurring between the plike valence bands arising from the anion (O) and the slike conduction bands arising from the cation (M). In Figures 2.10 to 2.13, the DOS for (M 2 O) in anitifluorite structure at ambient pressure are presented. From DOS calculations, it can be found that the conduction bands arise due to the hybridization of anion and cation states. The bottom of the conduction band arises predominantly from the slike states of cations and is separated from the rest by an energy gap. Quite complicated hybridization effects between the anions and cations states form the uppermost conduction bands. In Li 2 O and Na 2 O the lowestlying band is dominated by the slike states of the anion, while the upper valence band is made up of predominantly plike states of the anion. In K 2 O, the lowestlying bands arise from the hybridization of the O2s
41 Figure 2.6 Band structure of Li 2 O at ambient and high pressure (V/V 0 =0.65) Figure 2.7 Band structure of Na 2 O at ambient and high pressure (V/V 0 =0.65)
42 Figure 2.8 Band structure of K 2 O at ambient and high pressure (V/V 0 =0.65) Figure 2.9 Band structure of Rb 2 O at ambient and high pressure (V/V 0 =0.65)
43 Figure 2.10 Density of states of Li 2 O at ambient and high pressure (V/V 0 =0.65)
44 Figure 2.11 Density of states of Na 2 O at ambient and high pressure (V/V 0 =0.65)
45 Figure 2.12 Density of states of K 2 O at ambient and high pressure (V/V 0 =0.65)
46 Figure 2.13 Density of states of Rb 2 O at ambient and high pressure (V/V 0 =0.65)
47 Table 2.3 Energy bandgap, valenceband width (VBW) and neighboring distance R (O O) Compounds Present work HF LDA GGA (ev) HY BRID Expt. (+0.2eV) R (OO) (Å) Li 2 O 6.722 3.266 X 5.809 VBW 1.402 2.690 a 2.120 a 2.040 a 2.270 a 1.3 a Na 2 O 2.421 3.931 X 5.283 VBW 0.485 1.250 b 1.010 b 1.020 b 1.100 b 0.6 b K 2 O 2.213 4.560 X 1.782 VBW 0.432 0.380 c 0.370 c 0.300 c 0.390 c 0.3 c Rb 2 O 2.553 4.767 X 1.496 VBW 0.405 a Mikajlo et al (2002) b Mikajlo et al (2003) c Mikajlo et al (2003a)
48 anion. In K 2 O, the lowestlying bands arise from the hybridization of the O2s and K3p states. In Rb 2 O, the lowest deeplying bands arises from O2s states, above this bands semi core like Rb4p state arises. Bandstructure results show that Li 2 O, K 2 O and Rb 2 O are indirect bandgap semiconductors, with their gap lying between the and X points, whereas, Na 2 O is a direct bandgap semiconductor with a gap occurring at the point. The calculated bandgap values, valenceband width and neighboring distance R (O O) are given in Table 2.3 and are compared with the other theoretical and experimental result (Mikajlo et al 2003, 2003a, 2002). From the table it can be seen that the bandgap value decreases in going from Li 2 O to Rb 2 O and the calculated valence bandwidth is in agreement with experiment results (Mikajlo et al 2003, 2003a, 2002). From the table, it can also be noted that the valenceband width decreases with increasing size of the metal ion (M: Li, Na, K, Rb). This is because as the size of the metal ion increases the interatomic (O O) distance increases in going from Li 2 O to Rb 2 O, leading to reduced overlap between the neighboring states. The variations of the bandgap values as a function of reduced volume are given in Table 2.4. Upon compression, the (M 2 O) bandgap increases. Since in the alkali metal oxides the bottom of the conduction band is predominantly of cation slike states on compression the conduction band moves away from the valence band. This behavior is also observed in the case of MgO (Kalpana et al 1995; Chang et al 1984), (Ca, Sr) F 2 (Kanchana et al 2003). 2.7 SUMMARY In summary, the electronic structure and groundsate properties of alkali metal oxides (M 2 O) calculated using the TBLMTO method. The theoretically calculated equilibrium lattice constant is 1.8%, 1.6% and 1.3% smaller than the experimental value for Li 2 O, Na 2 O and K 2 O, respectively,
49 Table 2.4 Variation of bandgap E g as a function of reduced volume in the alkalimetal oxides Bandgap (E g ) (ev) V/V 0 Li 2 O Na 2 O K 2 O Rb 2 O 1.00 5.809 2.421 1.782 1.496 0.95 5.918 2.653 2.040 1.727 0.90 6.040 2.925 2.340 2.054 0.85 6.163 3.224 2.666 2.394 0.80 6.285 3.578 3.034 2.802 0.75 6.408 3.986 3.442 3.088 0.70 6.544 3.959 3.891 3.102 0.65 6.680 3.932 3.918 3.115
50 whereas for Rb 2 O calculated equilibrium lattice constant is 1.1% greater than the experimental value. The bulk modulus is in agreement with the experimental value. The bulk modulus decreases from Li 2 O to Rb 2 O. The variation in the bulk modulus is similar to that of pure alkali metal atom bulk moduli. The calculated total energy is compared with other theoretical results. From the results of the electronic properties, Li 2 O, K 2 O and Rb 2 O are indirect bandgap semiconductors, whereas Na 2 O is a direct bandgap semiconductor. It can be seen that, at equilibrium volume, the bandgap decreases from Li 2 O to Rb 2 O. The valenceband width decreases with increasing size of the metal ion. Upon further compression, the Li 2 O, Na 2 O, K 2 O and Rb 2 O bandgap increases.