30

CHAPTER 2

ELECTRONIC STRUCTURE AND GROUND STATE

PROPERTIES OF M2O (M: Li, Na, K, Rb)

2.1 INTRODUCTION

Oxides of alkali metals (M2O) (M: Li, Na, K, Rb) play an

important role in reducing the work function and thus enhancing the electrical

current of photo-cathodes (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 gas-detectors (Lee et al 1996). The materials with

fluorite (CaF2, SrF2, BaF2)-type structure have been extensively studied when

compared to antifluorite - type structure. The alkali-metal oxides (M2O) are

found to crystallize in the cubic antifluorite (anti-CaF2-type) structure (space

group no. 225) (Zintl et al 1934). This structure is antimorphous to the fluorite

(CaF2) structure. Materials with antifluorite-type 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

fluorite-type structure.

31

Experimentally, Hull et al (1988) measured the bulk modulus,

lattice parameter and the elastic constants of Li2O at ambient conditions and

temperature upto 1600 K. The transition temperature Tc ~ 1200 K (the melting

point is 1705 K) of Li2O was measured by Oishi et al (1979), who found that

the elastic constant C11 suddenly decreases at the transition temperature Tc.

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 Li2O studied by Liu et al (1996)

using the photoemission and electron-energy 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 X-ray

photoelectron spectroscopy measurements. Structural phase transition of Li2O

from antifluorite to anticotunnite (PbCl2-type structure) is identified by Laziki

et al (2006) and Kunc et al (2005) by using the x-diffraction study.

From theoretical point of view, Dovesi et al (1991) calculated the

lattice constants and elastic properties of Li2O, Na2O and K2O at zero pressure

via the ab - initio Hartree-Fock 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 Li2O. The Wannier function

based on LCAO formalism has been reported by Shukla et al (1998) on Li2O

and Na2O compounds. This study demonstrates the importance of the

correlation effects. The superionic behavior of Li2O 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 Li2O by using the LDA-pseudopotential

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 energy-loss near edge structure at the lithium K edge in Li2O using

the full potential linearized augmented plane wave (FP-LAPW) 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 M2O at ambient conditions as well as at high

pressure using the tight-binding linear muffin-tin orbital method (TB-LMTO)

(Andersen 1975, Andersen and Jepsen 1984).

2.3 CRYSTAL STRUCTURE

These compounds crystallize in the anti - CaF2 type structure.

Figure 2.1 shows the unit cell of fluorite (CaF2) - type structure (Galasso

1970). From Figure 2.1 it can be seen that the calcium ions occupy the corner

and face-centered 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 (CaF2) - 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 (CaF2) - type structure

Figure 2.2 Layer Sequence in Fluorite (CaF2) - type structure

34

Figure 2.3 Unit cell of antifluorite (anti-CaF2) - type structurepressure.

Figure 2.4 Layer Sequence in antifluorite (anti-CaF2) - type structure

35

Figure 2.3 shows the unit cell of antifluorite (anti - CaF2) - type

structure. In the antifluorite - type structure the fluorine ions occupy the

corner and face-centered 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 (anti-CaF2) - 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 (M2O) TB-LMTO method has been used (Andersen 1975,

Skriver 1984). von-Barth 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 non-

spherical shape of the atomic cell and the truncation of the higher partial

waves inside the sphere. The Wigner-Seitz 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 k-points 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: 2s1

2p0

3d0; 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, 2p-like states are well below the oxygen 2s-like states. In K2O and

Rb2O, there is hybridization of the semi core-like K-3p state and Rb-4p 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 core-like K-3p and Rb-4p

states as relaxed valence band states in K2O and Rb2O.

2.5 TOTAL ENERGY CALCULATIONS

In order to calculate the ground-state properties of alkali-metal

oxides (M2O) the total energies are calculated for all the four compounds as a

function of reduced volume ranging from 1.1 to 0.65V0, where V0 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

(B0). 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 (B0) and total energy (E0) are given in Tables 2.1 to 2.2 and are

compared with available experimental (Zintl et al 1934; Hull et al 1988)

ÍÝÍÜÛÍÌ

ÍËÊ ÕÖÔÄÅ

ÃÕÖÔÄÅ

à /?5/3

0

7/3

00

V

V

V

V

2

3BP (2.1)

37

Figure 2.5 Total energies as a function of reduced volume of

Li2O, Na2O, K2O and Rb2O

38

Table 2.1 Calculated lattice constant (a), bulk modulus (B0) and total

energy (E0) of Li2O and Na2O

Compounds a (Å) B0 (GPa) E0 (Ry)

Li2O

Present

Experiment

Other calculations

Na2O

Present

Experiment

Other calculations

4.533

4.619a

4.560b, 4.580

b, 4.580

d,

4.519d, 4.638

d, 4.584

d,

4.570e, 4.573

f, 4.570

g

5.465

5.560a

5.450b, 5.470

b, 5.497

c,

5.393c, 5.559

c, 5.498

c,

5.481e, 5.484

f

95.00

89.00h

102.76b, 97.91

b

94.60e, 105.00

f

59.00

-

62.18b

58.63b, 61.10

e

57.50f

-180.0488

-

-179.9254b

-180.8788b

-179.9286f

-179.9080g

-797.6686

-

-797.3796b

-799.7230b

-797.3862f

aZintl et al (1934)

bCancarevic et al (2006)

cMikajlo et al (2003)

dMikajlo et al (2002)

eShukla et al (1998)

fDovesi et al (1991)

gDovesi (1985)

hHull et al (1988)

39

Table 2.2 Calculated lattice constant (a), bulk modulus (B0) and total

energy (E0) of K2O and Rb2O

Compounds a (Å) B0 (GPa) E0 (Ry)

K2O

Present

Experiment

Other calculations

Rb2O

Present

Experiment

Other calculations

6.362

6.449a

6.430b, 6.420

b, 6.466

c

6.168c, 6.414

c, 6.360

c

6.466d

6.819

6.742a

6.830b, 6.800

b

33.46

-

40.74b, 38.69

b

34.60g

30.00

-

35.84b, 35.85

b

-2553.6098

-

-2546.3622b

-2549.9316b

-2546.3864d

-12063.0895

-

-244.9996b

-244.8180b

aZintl et al (1934)

bCancarevic et al (2006)

cMikajlo et al (2003a)

dDovesi et al (1991)

40

and other theoretical work (Mikajlo et al 2003, 2003a, 2002; Cancarevic et al

2006). The calculated ground-state 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 Li2O, Na2O and K2O, whereas for Rb2O calculated lattice constant is 1.1%

greater than the experimental value. A comparison of the bulk modulus B0

show a clear decrease with heavier atom (Li2O to Rb2O). This variation in the

bulk modulus is similar to that of the pure metal atom bulk modulus B0. 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 alkali-metal oxides (M2O) 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

p-like valence bands arising from the anion (O) and the s-like conduction

bands arising from the cation (M). In Figures 2.10 to 2.13, the DOS for (M2O)

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 s-like 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 Li2O

and Na2O the lowest-lying band is dominated by the s-like states of the anion,

while the upper valence band is made up of predominantly p-like states of the

anion. In K2O, the lowest-lying bands arise from the hybridization of the O-2s

41

Figure 2.6 Band structure of Li2O at ambient and

high pressure (V/V0=0.65)

Figure 2.7 Band structure of Na2O at ambient and

high pressure (V/V0=0.65)

42

Figure 2.8 Band structure of K2O at ambient and

high pressure (V/V0=0.65)

Figure 2.9 Band structure of Rb2O at ambient and

high pressure (V/V0=0.65)

43

Figure 2.10 Density of states of Li2O at ambient and

high pressure (V/V0=0.65)

44

Figure 2.11 Density of states of Na2O at ambient and

high pressure (V/V0=0.65)

45

Figure 2.12 Density of states of K2O at ambient and

high pressure (V/V0=0.65)

46

Figure 2.13 Density of states of Rb2O at ambient and

high pressure (V/V0=0.65)

47

Table 2.3 Energy bandgap, valence-band width (VBW) and

neighboring distance R (O–O)

Presentwork

HF LDA GGAHY-

BRIDCompounds

(eV)

Expt.(+0.2eV)

R (O-O)(Å)

Li2O

dÎd

d–X

VBW

Na2O

dÎd

d–X

VBW

K2O

dÎd

d–X

VBW

Rb2O

dÎd

d–X

VBW

6.722

5.809

1.402

2.421

5.283

0.485

2.213

1.782

0.432

2.553

1.496

0.405

-

-

2.690a

-

-

1.250b

-

-

0.380c

-

-

-

-

-

2.120a

-

-

1.010b

-

-

0.370c

-

-

-

-

-

2.040a

-

-

1.020b

-

-

0.300c

-

-

-

-

-

2.270a

-

-

1.100b

-

-

0.390c

-

-

-

-

-

1.3a

-

-

0.6b

-

-

0.3c

-

-

-

3.266

3.931

4.560

4.767

aMikajlo et al (2002)

bMikajlo et al (2003)

cMikajlo et al (2003a)

48

anion. In K2O, the lowest-lying bands arise from the hybridization of the O-2s

and K-3p states. In Rb2O, the lowest deep-lying bands arises from O-2s states,

above this bands semi core like Rb-4p state arises. Band-structure results

show that Li2O, K2O and Rb2O are indirect bandgap semiconductors, with

their gap lying between the d and X points, whereas, Na2O is a direct bandgap

semiconductor with a gap occurring at the d point. The calculated bandgap

values, valence-band 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 Li2O to Rb2O 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 valence-band

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 Li2O to Rb2O, 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 (M2O) bandgap

increases. Since in the alkali metal oxides the bottom of the conduction band

is predominantly of cation s-like 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) F2 (Kanchana et al

2003).

2.7 SUMMARY

In summary, the electronic structure and ground-sate properties of

alkali metal oxides (M2O) calculated using the TB-LMTO method. The

theoretically calculated equilibrium lattice constant is 1.8%, 1.6% and 1.3%

smaller than the experimental value for Li2O, Na2O and K2O, respectively,

49

Table 2.4 Variation of bandgap Eg as a

function of reduced volume in the

alkali-metal oxides

Bandgap (Eg)(eV)

V/V0 Li2O Na2O K2O Rb2O

1.00

0.95

0.90

0.85

0.80

0.75

0.70

0.65

5.809

5.918

6.040

6.163

6.285

6.408

6.544

6.680

2.421

2.653

2.925

3.224

3.578

3.986

3.959

3.932

1.782

2.040

2.340

2.666

3.034

3.442

3.891

3.918

1.496

1.727

2.054

2.394

2.802

3.088

3.102

3.115

50

whereas for Rb2O 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 Li2O to Rb2O. 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, Li2O, K2O and Rb2O are indirect

bandgap semiconductors, whereas Na2O is a direct bandgap semiconductor. It

can be seen that, at equilibrium volume, the bandgap decreases from Li2O to

Rb2O. The valence-band width decreases with increasing size of the metal

ion. Upon further compression, the Li2O, Na2O, K2O and Rb2O bandgap

increases.