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Energy gaps, polarisation and partial metallic valence
Solid State Communications, Vol. 51, No. 6, pp. 429-431, 1984. Printed in Great Britain.
0038-1098/8453.00 + .00 Pergamon Press Ltd.
ENERGY GAPS, POLARISATION AND PARTIAL METALLIC VALENCE J.A. Duffy Department of Chemistry, The University, Old Aberdeen, AB9 2UE, UK
(Received 28 February 1984 by R.A. Cowley) The metallic character of chemical bonding, for compounds that are inadequately described in a solely "ionic/covalent" framework, is discussed from the point of view of band gap electronegativity, ×*, and polarisation theory. With existing data for binary compounds, the plot of ×*(anion)-x*(cation) vs (1 --R/V), where R is the molar refractivity and V the molar volume (R/V being calculated from refractive index), shows fairly good correlation and appears to distinguish semiconductors from insulators. THE INTER-RELATIONSHIP between the three types of chemical bonding, ionic, covalent and metallic bonding, is not straightforward. For many simple binary compounds, M,,A,n, it is possible to ignore the metallic component and to consider just the ionic/covalent character which can then be dealt with in terms of Pauling electronegativity, x. Chemists appreciate the usefulness of being able to assess the degree of ionicity or covalency; it allows them to anticipate trends in properties or behaviour and to put unfamiliar compounds in the context of known patterns of behaviour. Unfortunately, at present there is no comparable facility for embracing metallic valence. There are many electronegativity scales, other than Pauling's, which are designed for dealing with ionic/ covalent character. Jgrgensen's "optical" electronegativity [ 1], on the other hand, is significantly different in that it is concerned with electron transfer energies in complex ions. Closely related to it is the recently introduced electronegativity, X*, which is designed to correlate the band gaps, Eg, of binary compounds. This "band gap" electronegativity is related to Eg by [2]: Eg = 3.72AX*,
(1)
where AX* = x*(anion)-×*(cation) and the constant, 3.72, was specially chosen (with Eft in eV) so that there is a direct connection between X* and J¢rgensen's optical electronegativity. If one imagines the departure from "true" metallic bonding to be a decoalescence of the partially filled energy band so that it divides into the valence band and the conduction band, then perhaps the band gap, Eg, is a measure of the decrease in the metallic component of bonding. It would follow, through equation (1), that AX* could be used as a means of exnressing the degree of metallic valence: increassing a,y* would correspond
to departure from metallic valence in an analogous way to which decreasing Ax corresponds to departure from ionicity. This idea is attractive since it allows band gap electronegativity to complement Pauling electronegativity, but, because the concept of a metallic bonding contribution in nonmetallic compounds is so elusive, it is difficult to decide how to test a possible relationship between AX* and metallic character. Recently, however, an alternative approach to metallic bonding has been highlighted by Edwards and Sienko [3], and has been applied to problems such as the natural existence of metals and nonmetals, solutions of alkali metals in liquid ammonia, etc. Briefly, the approach is based on the Clausius-Mossotti relationship: R/V = (e -- 1)/ (e + 2), where R is the molar refractivity (in the gaseous state for metallic elements) and V is the molar volume of the solid, and that when R/V = unity e becomes infinite. Thus, the closer R/V is to unity [or (1 - R / V ) is to zero], the nearer we are to the onset of metallisation. So far, simple binary compounds have not been considered from this viewpoint, but values of R / V should be readily obtainable from published refractive indices, n, by the Clausius-Mossotti-Maxwell relationship: R/V = (n 2 -- 1)/(n 2 + 2). Table 1 lists values of R/V obtained in this way for those binary compounds for which a value of AX* is available, either from Eg and equation (1) or from ×* values given in the Appendix. (Some of the X* values must be adjusted to take account of ligand field and other effects [4], and these adjusted values are used for expressing AX* in Table 1 - see footnote to Table 1). From the earlier discussion, it is anticipated that increasing AX* and increasing (1 --R~ V) both correspond to increasing departure from the metal/nonmetal transition and therefore to decreasing metallic character, and increasing ionic and/or covalent
429
430
ENERGY GAPS, POLARISATION AND PARTIAL METALLIC VALENCE
Table 1. Band gap, electronegativity and refractivity data o f binary compounds Compound LiF NaF SiO2 KF RbF CsF LiC1 A1203 NaCI KCI CsCI RbC1 LiBr KBr CsBr NaBr RbBr MgCI2 ZnCI2 CsI RbI KI NaI HgCI2 Ga203 CaS SrS BaS MgS ZnS NiO ZnO T1C1 Cr2Os CaSe SrSe BaSe CuC1 TiO2 CuI MgSe CaTe SrTe BaTe T1Br CuBr AgC1 HgI2 AgBr SiC
E e (eV) a 12.9 10.8 10.2 9.9 9.5 9.1
8.8 8.3 8.0 7.8 7.8 7.5 7.2 6.8 6.8 6.7 6.6
5.8 5.8 5.8 5.6 4.4
3.8 3.7 3.4 3.4 3.4
3.3 3.2 3.1
3.0 3.0 3.0 2.9 2.9
A× *b 3.45 2.9 2.75 2.65 2.55 2.45 2.35 2.25 2.15 2.1 2.1 2.0 1.95 1.85 1.85 1.8 1.75 1.75 1.75 1.55 1.55 1.55 1.5 1.3 1.2 1.15 1.15 1.15 1.05 1.0 1.0 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.85 0.85 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8
R~ V ~ 0.24 0.21 0.32 0.22 0.24 0.28 0.37 0.41 0.32 0.29 0.36 0.29 0.42 0.32 0.39 0.36 0.32 0.36 0.39 0.42 0.36 0.38 0.42 0.45 0.48 0.54 0.53 0.55 0.58 0.61 0.56 0.51 0.57 0.65 0.58 0.57 0.58 0.48 0.68 0.60 0.62 0.65 0.62 0.62 0.65 0.54 0.52 0.66 0.58 0.67
Vol. 51, No. 6
Table. Continued ZnSe PbO CdS CdO Cu20 GaP HgS Fe203 Sb2S3 A1Sb GaAs
2.8 2.6 2.6 2.3 2.2 2.2 2.1 2.0 1.7 1.5 1.4
0.75 0.7 0.7 0.6 0.6 0.6 0.55 0.55 0.45 0.4 0.4
0.71 0.66 0.64 0.63 0.68 0.71 0.73 0.72 0.82 0.75 0.77
a Values of Eg are from sources quoted in [21. b dx×* is obtained using equation (1) and the Eg values of the previous column. For compounds where E e is not known, dxX* is obtained using the values of x*(anion) and ×*(cation) quoted in the Appendix. Because of spin-pairing effects etc. in the case of NiO, Cr203 and F%O3, dxX* corresponds to the use of an adjusted ×* value for the cation [4]. e Values of R~ V are obtained using refractive indices taken from Handbook of Chemistry and Physics, 1981-1982, C.R.C. Press, U.S.A. and A.LP. Handbook, 3rd edn. (McGraw-Hill, New York).
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0.1 0.2 0.3 0.t~ 0.5 0.6 0.7 0.8 1- R/V
Fig. 1. Plot of electronegativity difference, A×*, vs (1 - - R / V ) for binary compounds in Table 1. For clarity, the data points are unlabelled, but can be identified from Table 1. character, in the bonding. The plot of AX* vs ( 1 - R / V ) is shown in Fig. 1, and the spread of data points in the case of the semiconducting compounds (compounds having E e less than, say, 4 eV) is sufficiently restricted to indicate that a meaningful relationship between dx×* and (1 - - R / V ) does indeed exist. It is noteworthy that if a
Vol. 51, No. 6
ENERGY GAPS, POLARISATION AND PARTIAL M ETALLIC VALENCE
straight line is drawn through the semiconductors, then it intersects the straight line through the insulators at around 1.1 to 1.3 for A×* (see Fig. 1), corresponding to a band gap of 4 to 5 eV which is normally taken to be the dividing line between semiconductors and insulators. APPENDIX In calculating R~ V from n, adjustment for infinite wavelength is an unnecessary refinement in view of the uncertainty in energy gap data which provide, through equation (1), most of the AX* values in Table 1. Where Eg is unknown, AX* is obtained from the X* values [2] of C1- (3.0), Br- (2.8), I- (2.5), S2- (2.15), Se2- (1.9), Te2- (1.8), Zn 2+ (1.1), Hg2+ (1.55), Mg~+ (1.1) and alkaline earth ions (1.0). X* values for Mg2+ and the alkaline earths have not previously been published, but from trends in the periodic table [2], and also the values of the alkali metal ions (0.95 to 1.0), a sensible choice is in
431
the range 1.0 to 1.1; bearing in mind that the visible/ ultra-violet border is 3.1 eV, the yellowness of MgSe and BaTe suggests a value of 1.1 for Mg2+ and 1.0 for the alkaline earths. Data for compounds having Eg less than approximately 1 eV are excluded owing to the unreliability of n for obtaining R/V. Acknowledgement - The author gratefully acknowledges valuable discussions with Dr J.H. Binks.
REFERENCES 1. 2. 3. 4.
C.K. Jgrgensen, Orbitals in Atoms and Molecules, Ch. 7, McGraw-Hill, New York (1962). J.A. Duffy, J. Phys. C13, 2979 (1980). P.P. Edwards & M.J. Sienko, Chemistry in Britain, 39 (1983); Acc. Chem. Res. 15, 87 (1982); J. A. Chem. Soc. 103, 2967 (1981). J.A. Duffy, J. Chem. Soc. Dalton Trans. 1475 (1983).
0038-1098/8453.00 + .00 Pergamon Press Ltd.
ENERGY GAPS, POLARISATION AND PARTIAL METALLIC VALENCE J.A. Duffy Department of Chemistry, The University, Old Aberdeen, AB9 2UE, UK
(Received 28 February 1984 by R.A. Cowley) The metallic character of chemical bonding, for compounds that are inadequately described in a solely "ionic/covalent" framework, is discussed from the point of view of band gap electronegativity, ×*, and polarisation theory. With existing data for binary compounds, the plot of ×*(anion)-x*(cation) vs (1 --R/V), where R is the molar refractivity and V the molar volume (R/V being calculated from refractive index), shows fairly good correlation and appears to distinguish semiconductors from insulators. THE INTER-RELATIONSHIP between the three types of chemical bonding, ionic, covalent and metallic bonding, is not straightforward. For many simple binary compounds, M,,A,n, it is possible to ignore the metallic component and to consider just the ionic/covalent character which can then be dealt with in terms of Pauling electronegativity, x. Chemists appreciate the usefulness of being able to assess the degree of ionicity or covalency; it allows them to anticipate trends in properties or behaviour and to put unfamiliar compounds in the context of known patterns of behaviour. Unfortunately, at present there is no comparable facility for embracing metallic valence. There are many electronegativity scales, other than Pauling's, which are designed for dealing with ionic/ covalent character. Jgrgensen's "optical" electronegativity [ 1], on the other hand, is significantly different in that it is concerned with electron transfer energies in complex ions. Closely related to it is the recently introduced electronegativity, X*, which is designed to correlate the band gaps, Eg, of binary compounds. This "band gap" electronegativity is related to Eg by [2]: Eg = 3.72AX*,
(1)
where AX* = x*(anion)-×*(cation) and the constant, 3.72, was specially chosen (with Eft in eV) so that there is a direct connection between X* and J¢rgensen's optical electronegativity. If one imagines the departure from "true" metallic bonding to be a decoalescence of the partially filled energy band so that it divides into the valence band and the conduction band, then perhaps the band gap, Eg, is a measure of the decrease in the metallic component of bonding. It would follow, through equation (1), that AX* could be used as a means of exnressing the degree of metallic valence: increassing a,y* would correspond
to departure from metallic valence in an analogous way to which decreasing Ax corresponds to departure from ionicity. This idea is attractive since it allows band gap electronegativity to complement Pauling electronegativity, but, because the concept of a metallic bonding contribution in nonmetallic compounds is so elusive, it is difficult to decide how to test a possible relationship between AX* and metallic character. Recently, however, an alternative approach to metallic bonding has been highlighted by Edwards and Sienko [3], and has been applied to problems such as the natural existence of metals and nonmetals, solutions of alkali metals in liquid ammonia, etc. Briefly, the approach is based on the Clausius-Mossotti relationship: R/V = (e -- 1)/ (e + 2), where R is the molar refractivity (in the gaseous state for metallic elements) and V is the molar volume of the solid, and that when R/V = unity e becomes infinite. Thus, the closer R/V is to unity [or (1 - R / V ) is to zero], the nearer we are to the onset of metallisation. So far, simple binary compounds have not been considered from this viewpoint, but values of R / V should be readily obtainable from published refractive indices, n, by the Clausius-Mossotti-Maxwell relationship: R/V = (n 2 -- 1)/(n 2 + 2). Table 1 lists values of R/V obtained in this way for those binary compounds for which a value of AX* is available, either from Eg and equation (1) or from ×* values given in the Appendix. (Some of the X* values must be adjusted to take account of ligand field and other effects [4], and these adjusted values are used for expressing AX* in Table 1 - see footnote to Table 1). From the earlier discussion, it is anticipated that increasing AX* and increasing (1 --R~ V) both correspond to increasing departure from the metal/nonmetal transition and therefore to decreasing metallic character, and increasing ionic and/or covalent
429
430
ENERGY GAPS, POLARISATION AND PARTIAL METALLIC VALENCE
Table 1. Band gap, electronegativity and refractivity data o f binary compounds Compound LiF NaF SiO2 KF RbF CsF LiC1 A1203 NaCI KCI CsCI RbC1 LiBr KBr CsBr NaBr RbBr MgCI2 ZnCI2 CsI RbI KI NaI HgCI2 Ga203 CaS SrS BaS MgS ZnS NiO ZnO T1C1 Cr2Os CaSe SrSe BaSe CuC1 TiO2 CuI MgSe CaTe SrTe BaTe T1Br CuBr AgC1 HgI2 AgBr SiC
E e (eV) a 12.9 10.8 10.2 9.9 9.5 9.1
8.8 8.3 8.0 7.8 7.8 7.5 7.2 6.8 6.8 6.7 6.6
5.8 5.8 5.8 5.6 4.4
3.8 3.7 3.4 3.4 3.4
3.3 3.2 3.1
3.0 3.0 3.0 2.9 2.9
A× *b 3.45 2.9 2.75 2.65 2.55 2.45 2.35 2.25 2.15 2.1 2.1 2.0 1.95 1.85 1.85 1.8 1.75 1.75 1.75 1.55 1.55 1.55 1.5 1.3 1.2 1.15 1.15 1.15 1.05 1.0 1.0 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.85 0.85 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8
R~ V ~ 0.24 0.21 0.32 0.22 0.24 0.28 0.37 0.41 0.32 0.29 0.36 0.29 0.42 0.32 0.39 0.36 0.32 0.36 0.39 0.42 0.36 0.38 0.42 0.45 0.48 0.54 0.53 0.55 0.58 0.61 0.56 0.51 0.57 0.65 0.58 0.57 0.58 0.48 0.68 0.60 0.62 0.65 0.62 0.62 0.65 0.54 0.52 0.66 0.58 0.67
Vol. 51, No. 6
Table. Continued ZnSe PbO CdS CdO Cu20 GaP HgS Fe203 Sb2S3 A1Sb GaAs
2.8 2.6 2.6 2.3 2.2 2.2 2.1 2.0 1.7 1.5 1.4
0.75 0.7 0.7 0.6 0.6 0.6 0.55 0.55 0.45 0.4 0.4
0.71 0.66 0.64 0.63 0.68 0.71 0.73 0.72 0.82 0.75 0.77
a Values of Eg are from sources quoted in [21. b dx×* is obtained using equation (1) and the Eg values of the previous column. For compounds where E e is not known, dxX* is obtained using the values of x*(anion) and ×*(cation) quoted in the Appendix. Because of spin-pairing effects etc. in the case of NiO, Cr203 and F%O3, dxX* corresponds to the use of an adjusted ×* value for the cation [4]. e Values of R~ V are obtained using refractive indices taken from Handbook of Chemistry and Physics, 1981-1982, C.R.C. Press, U.S.A. and A.LP. Handbook, 3rd edn. (McGraw-Hill, New York).
o
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°° o
o
2 "~
o
-4-
i_~ IlJ r"
oo oco o oo
1
g
°
Oo\ ° o•°
I.x.l
o
0o
I
0
I
0.1 0.2 0.3 0.t~ 0.5 0.6 0.7 0.8 1- R/V
Fig. 1. Plot of electronegativity difference, A×*, vs (1 - - R / V ) for binary compounds in Table 1. For clarity, the data points are unlabelled, but can be identified from Table 1. character, in the bonding. The plot of AX* vs ( 1 - R / V ) is shown in Fig. 1, and the spread of data points in the case of the semiconducting compounds (compounds having E e less than, say, 4 eV) is sufficiently restricted to indicate that a meaningful relationship between dx×* and (1 - - R / V ) does indeed exist. It is noteworthy that if a
Vol. 51, No. 6
ENERGY GAPS, POLARISATION AND PARTIAL M ETALLIC VALENCE
straight line is drawn through the semiconductors, then it intersects the straight line through the insulators at around 1.1 to 1.3 for A×* (see Fig. 1), corresponding to a band gap of 4 to 5 eV which is normally taken to be the dividing line between semiconductors and insulators. APPENDIX In calculating R~ V from n, adjustment for infinite wavelength is an unnecessary refinement in view of the uncertainty in energy gap data which provide, through equation (1), most of the AX* values in Table 1. Where Eg is unknown, AX* is obtained from the X* values [2] of C1- (3.0), Br- (2.8), I- (2.5), S2- (2.15), Se2- (1.9), Te2- (1.8), Zn 2+ (1.1), Hg2+ (1.55), Mg~+ (1.1) and alkaline earth ions (1.0). X* values for Mg2+ and the alkaline earths have not previously been published, but from trends in the periodic table [2], and also the values of the alkali metal ions (0.95 to 1.0), a sensible choice is in
431
the range 1.0 to 1.1; bearing in mind that the visible/ ultra-violet border is 3.1 eV, the yellowness of MgSe and BaTe suggests a value of 1.1 for Mg2+ and 1.0 for the alkaline earths. Data for compounds having Eg less than approximately 1 eV are excluded owing to the unreliability of n for obtaining R/V. Acknowledgement - The author gratefully acknowledges valuable discussions with Dr J.H. Binks.
REFERENCES 1. 2. 3. 4.
C.K. Jgrgensen, Orbitals in Atoms and Molecules, Ch. 7, McGraw-Hill, New York (1962). J.A. Duffy, J. Phys. C13, 2979 (1980). P.P. Edwards & M.J. Sienko, Chemistry in Britain, 39 (1983); Acc. Chem. Res. 15, 87 (1982); J. A. Chem. Soc. 103, 2967 (1981). J.A. Duffy, J. Chem. Soc. Dalton Trans. 1475 (1983).