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Ionic and covalent energy gaps of CsCl crystals
Volume 102A, number 3
PHYSICS LETTERS
7 May 1984
IONIC AND COVALENT ENERGY GAPS OF CsCI CRYSTALS V. KUMAR SRIVASTAVA Istituto Di Struttura Della Materia, Consiglio Nazionale Delle Ricerche, Via Enrico Fermi 38, 00044 Frascati, Rome, Italy Received 9 February 1984
Using the theory of plasma oscillations in solids, the ionic (C) and covalent (Eh) energy gaps of CsC1 crystals have been calculated. Our calculated values are in fair agreement with the values reported by earlier workers.
The calculation of ionic (C) and covalent (Eh) energy gaps has been an active field of theoretical and experimental physics for very many years. Several quantitative [ 1 - 6 ] and qualitative [7,8] explanations based on nearest-neighbor distance, coordination number, electronegativity etc. have been put forward to account for these gaps from time to time. The theories given by these workers require elaborate computation and have been developed only for limited semiconductors. Recently the author [9] has proposed a simple model based on the theory of plasmon oscillations in solids for the calculation of these gaps for rocksalt crystals. An excellent agreement between the calculated values and the values reported by earlier workers has been obtained. In this paper we present the calculation of ionic and covalent energy gaps of cesium and thallium compounds having CsC1 structures. It is now well established that the plasmon energy of a metal changes [10-12] when it undergoes a chemical combination and forms a compound. A plasmon is a collective excitation of the conduction electrons in a metal with an energy, ~Wp, which depends on the density of the conduction electrons. The density of the conduction electrons changes when a metal forms a compound. Recently the author has calculated the average energy gaps of alkali halides AIBVII and several AIIB Vl compounds using this idea [ 13,14]. The ionicity and covalency of the individual bonds can be determined by separating the average energy gap into ionic and covalent parts using the following equations: 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
E2 = C 2 +E 2 , =
c2/e
,
Ic = E h2/ E g2 ,
(1) (2)
C = 1 4 . 4 b [ Z A / r 0 + ( m / n ) (ZB/rO) ] exp(-Ksr0) eV,
(3) E h = 39.74/d 2"48 eV, 1
r 0 = ~ d and
b =0.089A7 C,
(4) (5)
where fi and fc are the fractional ionicity and covalency of the bonds, d is the nearest-neighbor distance (bond length), C and E h are the ionic (heteropolar) and covalent (homopolar) parts of average energy gap for An Bm compounds, b is the prescreening factor and exp(-Ksr0) is the Thomas-Fermi screening factor. The details of the above equations have been discussed elsewhere [2,6] and will not be pursued here. In eq. (3), C is given by the difference between the screened Coulomb potentials of atoms A and B having core charges Z A and Z B. The charge-transfer process is affected by the Thomas-Fermi screening factor exp(-Ksr0) which affects the chemical trend in a compound. This screening factor as well as the bond length d are related to the effective number of valence electrons in a compound. The plasmon energy also depends directly on the effective number of free electrons in the valence band or the valence electron density. This shows that there must be some correlation between the physical process which involves the ionic 127
Volume 102A, number 3
PHYSICS LETTERS
c o n t r i b u t i o n C to the average energy gap Eg, and the plasmon energy o f a c o m p o u n d (~Wp) C. The energy o f a plasma oscillation is given by (/~COp)2 =
4rmee2/m ,
(6)
f r o m which we have
n e = (m/47re 2)
(/~Wp) 2 ,
(7)
where n e is the effective n u m b e r o f free electrons taking part in the plasma oscillations, e and m are their charge and mass respectively. The above e q u a t i o n for the plasmon energy is valid for free electrons but to a first a p p r o x i m a t i o n it can be used for s e m i c o n d u c t o r s and insulators t o o [ 1 0 , 1 5 - 1 7 ] . The expression for n e can be written for single-bond c o m p o u n d s , as *~
V
(8)
where Z A is the n u m b e r o f valence electrons o f the cation, N C A is the c o o r d i n a t i o n n u m b e r o f a t o m A in the c o m p o u n d , A7C is the average c o o r d i n a t i o n n u m b e r , Z M is the n u m b e r o f molecules per unit and V is the unit cell volume. For CsC1 crystals, V = (8/3%/3)d 3 , NcA=NCB=A7 C=8
which, with eq. (7), we get the following relation between d and the plasmon energy : d ( A ) = t 9.2766 (hOOp) 2/3
(hOOp in e V ) .
/Sh = 0.0258 (hOOp) 1 .6533 e V .
ZM : 1 .
Using these values, eq. (8) yields n e =
(I I )
The T h o n m s - F e r n a i m o n w n t u t n plasmon energy, can be written as
Ks,
in terms o f tile
Ks = ( 4Ki: /rraB ) l / 2 = ((4/aB)
[3m/(2rre) 2]
1/3}l/2(hCop)l/3
3V'-3/d3,
(9) from
,1 We have used the most general form of the expression, n e = nv/ub, where n v = (ZA/NCA + ZB/NCB), given by Lavine [61.
(131
The values o f eqs. (1 0) and (12) are substituted in eq. (3), the expression for the ionic gap for AIB VII CsC1 crystals, where 2xZ = Z A - Z B = 6, can be written as C = 51.06
(~Wp)U3exp[-7.88.13(l~Wp)(
1/3] eV,
(14) and for AIIIB VII CsCI crystals, where z3Z = 4, can be written as
(hOOp)(
1/3 ] e V .
(15) F r o m eqs. (11), (14) and (15), the general expression for E h and C can be written as ,~ /-,- ,1 6533 El,` = A1 ~nCOp)C" eV,
(16)
C=K2(PiCOp) 2/3 exp[-K3(lriCOp)C 1/3]
eV.
Table 1 Ionic and covalent energy gaps of CaC1 crystals. Compound
CsC1 CsBr Cs I TIC1 T1Br T1 I
128
(12)
where a B is the Bohr radius and Kt,, is the Fermi wave vector which is given by the relation
C = 34.04(~6Op) 2/3 exp [ - 7 . 8 8 1 3 and
( l 0)
F r o m eqs. (3) and (10), the covalent energy gap for the CsCI crystals can be written as
K/c = (3rr2ne)l/3 .
(ZA/NCA + ZB/NCB)JVcZ M ne =
7 May 1984
/~Wp (eV)
E h (eV) this work
ref. [6]
this work
ref. [6]
this work
ref. [6]
12.51 11.77 10.73 13.92 13.28 11.92
1.684 1.522 1.307 2.009 1.859 1.554
1.69 1.54 1.31 2.03 1.86 1.62
9.227 8.259 6.974 7.063 6.829 6.635
8.50 8.12 6.90 7.80 7.25 5.83
0.9677 0.9672 0.9660 0.9252 0.9309 0.9292
0.962 0.965 0.965 0.937 0.938 0.929
c (eV)
J'i
(17)
Volume 102A, number 3
PHYSICS LETTERS
where K1, K 2 and K 3 are constants depending upon the structure elements and AZ. In this paper we have taken six- and four-electron CsC1 crystals for which K 1 = 0.0258, K 2 = 51.06 and 34.04 and K 3 = 7.8813. Using eqs. (2), (16) and (17), the values of ionicity, ionic and covalent energy gaps of CsCI, CsBr, Cs I. T1CI, T1Br and TI 1 have been calculated and listed in table 1. Table 1 shows that the calculated values of ionicity, C and E h are in fair agreement with the values reported by Lavine [6]. Thus it is possible to predict the order of ionic and covalent energy gaps and hence the ionicity of semiconducting compounds from their plasmon energies. The author is grateful to Professor M. Piacentini for the helpful discussions. Thanks are also due to I.C.T.P., Trieste, for financial assistance.
References [1 ] J.C. Phillips, Phys. Rev. 20 (1968) 550; Rev. Mod. Phys. 42 (1970) 317,
7 May 1984
[2] J.A. Van Vechten, Phys. Rev. 182 (1969) 891; 187 (1969) 1007. [3] D.R. Penn, Phys. Rev. 128 (1962) 2093. [4] G. Srinivasan, Phys. Rev. 178 (1969) 1244. [5] J.P. Walter and M.L. Cohen, Phys. Rev. B2 (1970) 1821; 183 (1969) 763. [6] B.F. Lavine, Phys. Rev. B7 (1973) 2591, 2600; J. Chem. Phys. 59 (1973) 1463. [7] P.R. Sarode, Phys. Star. Sol. 88b (1978) K 35. [8] S.V. Adhyapak, S.M. Karnetkar and A.S. Nigavekar, Nuovo Cimento 35 (1976) 179. [9] V. Kumar Srivastava, submitted to Phys. Rev. [10] K.S. Srivastava, R.L. Shrivastava, O.K. Harsh and V. Kumar, Phys. B19 (1979) 4336; J. Phys. Chem. Solids 40 (1979) 489. [11] K.S. Srivastava et al., Phys. Rev. A25 (1982) 2838. [12] V. Kumar, A.R. Chetal and K.S. Srivastava, Phys. Stat. Sol. 70a (1982) K 107. [13] V. Kumar Srivastava, Phys. Stat. Sol. 114b (1982)667. [14] V. Kumar Srivastava and A.R. Chetal, Nuovo Cimento 2D (1983) 1014. [ 15] C. Kittel, Introduction to solid state physics, 4th Ed. 1971 (Second Wiley Eastern Reprint, New Delhi, 1974) p. 227. [16] H. Raether, Ergeb. Exakten Naturwiss. 38 (1965) 84. [ 17 ] H.R. Philippand H. Ehrenreich, Phys. Rev. 129 (1963) 1530.
129
PHYSICS LETTERS
7 May 1984
IONIC AND COVALENT ENERGY GAPS OF CsCI CRYSTALS V. KUMAR SRIVASTAVA Istituto Di Struttura Della Materia, Consiglio Nazionale Delle Ricerche, Via Enrico Fermi 38, 00044 Frascati, Rome, Italy Received 9 February 1984
Using the theory of plasma oscillations in solids, the ionic (C) and covalent (Eh) energy gaps of CsC1 crystals have been calculated. Our calculated values are in fair agreement with the values reported by earlier workers.
The calculation of ionic (C) and covalent (Eh) energy gaps has been an active field of theoretical and experimental physics for very many years. Several quantitative [ 1 - 6 ] and qualitative [7,8] explanations based on nearest-neighbor distance, coordination number, electronegativity etc. have been put forward to account for these gaps from time to time. The theories given by these workers require elaborate computation and have been developed only for limited semiconductors. Recently the author [9] has proposed a simple model based on the theory of plasmon oscillations in solids for the calculation of these gaps for rocksalt crystals. An excellent agreement between the calculated values and the values reported by earlier workers has been obtained. In this paper we present the calculation of ionic and covalent energy gaps of cesium and thallium compounds having CsC1 structures. It is now well established that the plasmon energy of a metal changes [10-12] when it undergoes a chemical combination and forms a compound. A plasmon is a collective excitation of the conduction electrons in a metal with an energy, ~Wp, which depends on the density of the conduction electrons. The density of the conduction electrons changes when a metal forms a compound. Recently the author has calculated the average energy gaps of alkali halides AIBVII and several AIIB Vl compounds using this idea [ 13,14]. The ionicity and covalency of the individual bonds can be determined by separating the average energy gap into ionic and covalent parts using the following equations: 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
E2 = C 2 +E 2 , =
c2/e
,
Ic = E h2/ E g2 ,
(1) (2)
C = 1 4 . 4 b [ Z A / r 0 + ( m / n ) (ZB/rO) ] exp(-Ksr0) eV,
(3) E h = 39.74/d 2"48 eV, 1
r 0 = ~ d and
b =0.089A7 C,
(4) (5)
where fi and fc are the fractional ionicity and covalency of the bonds, d is the nearest-neighbor distance (bond length), C and E h are the ionic (heteropolar) and covalent (homopolar) parts of average energy gap for An Bm compounds, b is the prescreening factor and exp(-Ksr0) is the Thomas-Fermi screening factor. The details of the above equations have been discussed elsewhere [2,6] and will not be pursued here. In eq. (3), C is given by the difference between the screened Coulomb potentials of atoms A and B having core charges Z A and Z B. The charge-transfer process is affected by the Thomas-Fermi screening factor exp(-Ksr0) which affects the chemical trend in a compound. This screening factor as well as the bond length d are related to the effective number of valence electrons in a compound. The plasmon energy also depends directly on the effective number of free electrons in the valence band or the valence electron density. This shows that there must be some correlation between the physical process which involves the ionic 127
Volume 102A, number 3
PHYSICS LETTERS
c o n t r i b u t i o n C to the average energy gap Eg, and the plasmon energy o f a c o m p o u n d (~Wp) C. The energy o f a plasma oscillation is given by (/~COp)2 =
4rmee2/m ,
(6)
f r o m which we have
n e = (m/47re 2)
(/~Wp) 2 ,
(7)
where n e is the effective n u m b e r o f free electrons taking part in the plasma oscillations, e and m are their charge and mass respectively. The above e q u a t i o n for the plasmon energy is valid for free electrons but to a first a p p r o x i m a t i o n it can be used for s e m i c o n d u c t o r s and insulators t o o [ 1 0 , 1 5 - 1 7 ] . The expression for n e can be written for single-bond c o m p o u n d s , as *~
V
(8)
where Z A is the n u m b e r o f valence electrons o f the cation, N C A is the c o o r d i n a t i o n n u m b e r o f a t o m A in the c o m p o u n d , A7C is the average c o o r d i n a t i o n n u m b e r , Z M is the n u m b e r o f molecules per unit and V is the unit cell volume. For CsC1 crystals, V = (8/3%/3)d 3 , NcA=NCB=A7 C=8
which, with eq. (7), we get the following relation between d and the plasmon energy : d ( A ) = t 9.2766 (hOOp) 2/3
(hOOp in e V ) .
/Sh = 0.0258 (hOOp) 1 .6533 e V .
ZM : 1 .
Using these values, eq. (8) yields n e =
(I I )
The T h o n m s - F e r n a i m o n w n t u t n plasmon energy, can be written as
Ks,
in terms o f tile
Ks = ( 4Ki: /rraB ) l / 2 = ((4/aB)
[3m/(2rre) 2]
1/3}l/2(hCop)l/3
3V'-3/d3,
(9) from
,1 We have used the most general form of the expression, n e = nv/ub, where n v = (ZA/NCA + ZB/NCB), given by Lavine [61.
(131
The values o f eqs. (1 0) and (12) are substituted in eq. (3), the expression for the ionic gap for AIB VII CsC1 crystals, where 2xZ = Z A - Z B = 6, can be written as C = 51.06
(~Wp)U3exp[-7.88.13(l~Wp)(
1/3] eV,
(14) and for AIIIB VII CsCI crystals, where z3Z = 4, can be written as
(hOOp)(
1/3 ] e V .
(15) F r o m eqs. (11), (14) and (15), the general expression for E h and C can be written as ,~ /-,- ,1 6533 El,` = A1 ~nCOp)C" eV,
(16)
C=K2(PiCOp) 2/3 exp[-K3(lriCOp)C 1/3]
eV.
Table 1 Ionic and covalent energy gaps of CaC1 crystals. Compound
CsC1 CsBr Cs I TIC1 T1Br T1 I
128
(12)
where a B is the Bohr radius and Kt,, is the Fermi wave vector which is given by the relation
C = 34.04(~6Op) 2/3 exp [ - 7 . 8 8 1 3 and
( l 0)
F r o m eqs. (3) and (10), the covalent energy gap for the CsCI crystals can be written as
K/c = (3rr2ne)l/3 .
(ZA/NCA + ZB/NCB)JVcZ M ne =
7 May 1984
/~Wp (eV)
E h (eV) this work
ref. [6]
this work
ref. [6]
this work
ref. [6]
12.51 11.77 10.73 13.92 13.28 11.92
1.684 1.522 1.307 2.009 1.859 1.554
1.69 1.54 1.31 2.03 1.86 1.62
9.227 8.259 6.974 7.063 6.829 6.635
8.50 8.12 6.90 7.80 7.25 5.83
0.9677 0.9672 0.9660 0.9252 0.9309 0.9292
0.962 0.965 0.965 0.937 0.938 0.929
c (eV)
J'i
(17)
Volume 102A, number 3
PHYSICS LETTERS
where K1, K 2 and K 3 are constants depending upon the structure elements and AZ. In this paper we have taken six- and four-electron CsC1 crystals for which K 1 = 0.0258, K 2 = 51.06 and 34.04 and K 3 = 7.8813. Using eqs. (2), (16) and (17), the values of ionicity, ionic and covalent energy gaps of CsCI, CsBr, Cs I. T1CI, T1Br and TI 1 have been calculated and listed in table 1. Table 1 shows that the calculated values of ionicity, C and E h are in fair agreement with the values reported by Lavine [6]. Thus it is possible to predict the order of ionic and covalent energy gaps and hence the ionicity of semiconducting compounds from their plasmon energies. The author is grateful to Professor M. Piacentini for the helpful discussions. Thanks are also due to I.C.T.P., Trieste, for financial assistance.
References [1 ] J.C. Phillips, Phys. Rev. 20 (1968) 550; Rev. Mod. Phys. 42 (1970) 317,
7 May 1984
[2] J.A. Van Vechten, Phys. Rev. 182 (1969) 891; 187 (1969) 1007. [3] D.R. Penn, Phys. Rev. 128 (1962) 2093. [4] G. Srinivasan, Phys. Rev. 178 (1969) 1244. [5] J.P. Walter and M.L. Cohen, Phys. Rev. B2 (1970) 1821; 183 (1969) 763. [6] B.F. Lavine, Phys. Rev. B7 (1973) 2591, 2600; J. Chem. Phys. 59 (1973) 1463. [7] P.R. Sarode, Phys. Star. Sol. 88b (1978) K 35. [8] S.V. Adhyapak, S.M. Karnetkar and A.S. Nigavekar, Nuovo Cimento 35 (1976) 179. [9] V. Kumar Srivastava, submitted to Phys. Rev. [10] K.S. Srivastava, R.L. Shrivastava, O.K. Harsh and V. Kumar, Phys. B19 (1979) 4336; J. Phys. Chem. Solids 40 (1979) 489. [11] K.S. Srivastava et al., Phys. Rev. A25 (1982) 2838. [12] V. Kumar, A.R. Chetal and K.S. Srivastava, Phys. Stat. Sol. 70a (1982) K 107. [13] V. Kumar Srivastava, Phys. Stat. Sol. 114b (1982)667. [14] V. Kumar Srivastava and A.R. Chetal, Nuovo Cimento 2D (1983) 1014. [ 15] C. Kittel, Introduction to solid state physics, 4th Ed. 1971 (Second Wiley Eastern Reprint, New Delhi, 1974) p. 227. [16] H. Raether, Ergeb. Exakten Naturwiss. 38 (1965) 84. [ 17 ] H.R. Philippand H. Ehrenreich, Phys. Rev. 129 (1963) 1530.
129