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Charge transfer from chemical shifts at (110) surfaces of III–V compound semiconductors
Solid State Communications, Printed in Great Britain.
CHARGE TRANSFER
Vol. 58, No. 3, pp. 215-217,
1986.
0038-1098/86 $3.00 t .OO Pergamon Press Ltd.
FROM CHEMICAL SHIFTS AT (110) SEMICONDUCTORS
SURFACES OF III-V
COMPOUND
W. Month Laboratorium
f& Festkorperphysik,
Universitat
D&burg,
D-4100 Duisburg, Federal Republic of Germany
(Received 10 December 1985 by M. Ciudona) Surface charge-transfers are calculated from surface core-level shifts determined by Eastman et aL and Taniguchi et aL by using soft X-ray photoemission spectroscopy. A simple electrostatic model which was first applied by Shevchik et al. to the analysis of bulk data is adapted for surfaces of compound semiconductors. The charge transferred from cations to anions is found to be the same in the bulk and at the (1 10) surfaces of GaP, GaAs, GaSb, and InSb. IN MOLECULES AND SOLIDS the binding energies of core levels are known to depend on the chemical bonding of the atoms under study. In a simple approach, such chemical shifts, which probe changes of the electrostatic potential at the core, may be understood by a charge transfer between atoms exhibiting different electronegativities. Some years ago, additional shifts have been observed in the binding energies of the 3d and 4d core levels of surface atoms at cleaved (1 10) surfaces of some III-V compound semiconductors [l, 21. The levels of the surface metal atoms were found to have larger binding energies, while the levels of the group V surface-atoms are shifted by almost the same amount but to smaller binding energies, both relative to their bulk counter-parts. These results contain information on a possible deviation of the charge distribution at the surface from the one in the bulk. This might be expected since the (1 10) surfaces of the III-V compound semiconductors were found to be reconstructed. The type of surface rearrangement may be easily explained since in such surfaces the atoms only have three nearest neighbors compared to four in the bulk. The trivalent metal atoms, for example, tend to establish planar sp2 bonds with their three neighbors instead of the tetrahedrally oriented sp3 bonds in the bulk. The result is that the group III and group V atoms are shifted inward and outward, respectively, with reference to an ideally terminated bulk [3]. By using a simple electrostatic model, which was developed for explaining bulk chemical shifts [4] , it will be shown here that the core-level shifts observed with the metal atoms at (1 10) surfaces of GaP, GaAs, GaSb, and InSb are solely due to the difference in surface and bulk Madelung energy and no additional surface chargetransfer must be invoked. 215
In III-V compound semiconductors the bonds are partly ionic. The charge transfer between the two sublattices may be understood and estimated following one of Pauling’s concepts [S] . Since the group III atoms have a smaller electronegativity than the group V atoms, the metal atoms donate charge to the non-metal atoms. The electronegativity-model gives the charge transfered in a single bond as &I
= O.l6Ixrrr-xvI+O.O35Ixrrr-XVI’,
(1)
which is a refined version [6] of Pauling’s original scale. In a tetrahedrally coordinated III-V compound the charge transferred in the bulk may then be approximated by Aqb = 4Aqr.
(2)
For the semiconductors to be discussed here, these values are listed in Table 4 and will be discussed at the end of this paper. Shevchik et al. [4] started from the valence-charge distribution of the cations in their neutral metallic form and considered the charge Aqb, which is transferred from the cations to the anions in the bulk of, for example, a III-V compound semiconductor, to be uniformly distributed within the cations which are approximated as spheres. The change of the potential seen by the cation core in the compound is then given by vb
c
_ ‘heo _ -.-
4nee
A &’
(3)
where d, is the nearest-neighbor distance in the metal which equals twice the atomic radius. The coefficient A is determined by the charge distribution and measures, for example, 2 or 3 for a sphere and a spherical shell, respectively. The corresponding energy shift AEE of a
CHARGE TRANSFER
216
FROM CHEMICAL SHIFTS
core level is then obtained by multiplying equation (3) by the electron charge e. . In an ionic compound, the electrostatic interaction of the cations and the anions has to be considered additionally. The corresponding Madelung energy is given by Eh
= --.-Awi he0
ffb d,’
(4)
Here, (Yb is the bulk Madelung constant of the specific lattice, and d, is the nearest-neighbor distance of the compound semiconductor under study. The total shift of a cation core-level in the bulk of the compound semiconductor with reference to the pure metal then results as
The same procedure may also be applied to a surface cation. The charge transfer is then labeled Aq,. For a (1 10) surface of a solid crystallizing in the zinc-blende lattice the surface Madelung constant ar, has been evaluated as 0.85 ob by Levine and Mark [7]. The next layer below the surface has almost the bulk value ob [8]. For the surface cations the energy shift of their core levels may thus be written as
Vol. 58, No. 3
Table 1. Binding energies of the Ga(3d5,z) and In(4dv2) core levels in the bulk of III-V compound semiconductors relative to the valence-band top E,” and the vacuum level E_; I = I&,, - E,” is the ionization energy, and d, is the nearest-neighbor distance III-V
d, nm
EE-E: eV
I eV
E VBC-E,” eV
GaP GaAs GaSb InSb
0.236 0.245 0.265 0.281
18.4’ 18.6a 18.7a 17.2b
6.01’ 5.56’ 4.91C 4.9 c
24.41 24.16 23.61 22.10
‘Ref. [l] bRef. 121 ‘Ref. [9] Table 2. Binding energies of the Ga(3ds,a) and In(4d5,2) core levels in the btdk of the metah rehtive to the Fermi level EF and the vacuum level Evclc; 6 is the work function and d, the nearest-neighbor distance III-V
d, nm
EF-e’ eV
@ eV
E,.,-rt eV
Ga In
0.271 0.333
18.3d 17.0e
4.2 ’ 4.12r
22.5 21.12
dRef. [lo
(6) The core-level binding the bulk then becomes
‘k:
[ii]
1
energy at the surface relative to
(7) Bulk and surface charge-transfer may now be evaluated from the observed energy shifts AE,b = (E,,, -rt) (Evac - Ei) and AP$’ = EE -PC by using equations (5) and (7). Here, the vacuum level E,,, is used as reference for both metals and compound semiconductors. The experimental data entering the analysis are listed in Tables 1 to 3. Following a recent paper by Falter et al. [131* we take A = 3.32. A more refined treatment should also consider a possible variation of this parameter at the surface. As Table 3 reveals, the charge transfers are found to be the same in the bulk and at the (1 10) surface of each of those four III-V compound semiconductors for which the corresponding core level shifts have been reported. This means that the difference in binding energy of the core levels in the bulk and at the (1 10) surface of GaP, GaAs, CaSb, and InSb - and thus most probably for all III-V compound semiconductors - is solely due to the difference of the Madelung energies in the bulk and at the surface.
Table 3. Binding-energy shifts of the (3ds,2) or (4ds,z) core levels, in the bulk and at (1 10) sueace of III- V compounds as well as the bulk and surface chargetransfers, Aqb and Aq,, calcukzted from these shifts by using equations (5) and (7). respectively III-V
Gal’ GaAs GaSb InSb
Cation
Anion
Eyt-E; eV
E,b-E’, eV
E,b-EC eV
Aqa
Aqs
1.50 1.66 1.11 0.98
0.28’ 0.28’ 0.30a 0.22b
-o.37a -0.36a --0.29b
0.291 0.207 0.127 0.165
0.240 0.205 0.140 0.166
‘Ref. [l] bRef. [2]
This finding is in excellent agreement with results of self-consistent pseudopotential calculations for the reconstructed and the unreconstructed (1 10) surface of GaAs by Chelikowsky and Cohen [14, 151. They also calculated the total valence charge density for both surface arrangements. With the reconstructed surface,
CHARGE TRANSFER FROM CHEMICAL SHIFTS
•V o l . 58, No. 3
Table 4. Experimental charge transfers of I I I - V compound semiconductors compared with theoretical values from different ionizity scales III-V Experiment
Pauling 15]
Falter et al. [17]
Phillips [18]
GaP GaAs GaSb InSb
0.263 0.256 0.162 0.183
0.240 0.198 0.149 0.155
0.327 0.240 0.044 0.280
0.250 0.207 0.127 0.165
they found bulk features to dominate the density configuration while this is not true for the unreconstructed surface geometry. Davenport et al. [16] have also analyzed the corelevel shift observed with GaAs (1 1 0) surfaces. As here, they concluded that the relative shifts are accounted for mainly by the difference between the Madelung energies at the surface and in the bulk. In their evaluation, however, they ended up with a very large charge transfer of 0.28eo in bulk GaAs and they had to reduce the ionic charges at the surface atoms by 10% relative to those in the bulk. Those conclusions are not supported by the present analysis of the experimental data reported for four III-V compound semiconductors. The present paper has considered the most recent experimental data of photoemission from core levels in the compound semiconductors GaP, GaAs, GaSb and lnSb [1, 2] and in the corresponding metals Ga and In [10, 11]. Therefore, the charge transfers evaluated here shall be compared in Table 4 with those obtained from different ionizity scales. The values labeled "Pauling" were evaluated by using equations (1) and (2), i.e. with the assumption that in tetrahedrally coordinated compound semiconductors the charge transfer amounts to four times the charge transfer in a single bond. For the ionizity scales given by Falter etal. [17] and by Phillips [18] the charge transfers were taken from a most recent paper by Falter et aI. [13]. In that publication they have also adopted the approach by Shevchik et al. [4] to calculate the core-level shifts of the metal atoms in compounds crystallizing in the zinc-blende lattice. They had reached "good agreement" with a set of experimental data already published by Shevchik et aL [4] in 1974. In the present paper, experimental data taken with meanwhile improved techniques, which made possible to resolve the spin-orbit split components of the Ga(3d) and the As(3d) signals, have been analyzed. The theoretical results by Falter et al. [13] based on their
217
own ionizity scale now show an "improved agreement" with the newer experimental data. But nevertheless, Pauling's scale still is a very good and easy to use approximation. Note added in proof: Very recently, Baier et al. [19] observed the surface shifts of the In(4d) core levels as 0.3 and 0.26 eV, at (1 10) surfaces cleaved from lnP and lnAs, respectively. The evaluation of both the bulk and the surface corelevel shifts relative to metallic In gives Aq s = 0.255 and Aq b = 0.266 for InP and Aqs = 0.198 and Aqb = 0.202 for lnAs, which results excellently confirm the conclusion that the charge transfer is the same in the bulk and at the (I 1 0) surfaces of III-V compound semiconductors. REFERENCES 1.
D.E. Eastman, T.-C. Chiang, P. Heimann & F.J. Himpsel,Phy~ Rev. Lett. 45,656 (1980). 2. M. Taniguchi, S. Suga, M. Seki, S. Shin, K.L. Kobayashi & H. Kanzaki, J. Phys. C: Solid State Phys. 16, L45 (1983). 3. A. Kahn, Surf. Sci. Rep. 3, 193 (1983) and references cited therein. 4. N.J. Shevchik, J. Tejeda & M. Cardona, Phys. Rev. tl9, 2627 (1974) and references cited therein. 5. L. Pauling, The Nature of the Chemical Bond, Cornell Univ. Press, Ithaca, N.Y., (1940). 6. N.B. Hannay & C_P. Smith, J. Am. Cher~ Soc. 68, 171 (1946). 7. J.D. Levine & P. Mark, Phys. Rev. 144, 751 (1966). 8. R.E. Watson, J.W. Davenport, M.L. Perhnan & T.K. Sham, Phys. Rev. B24, 1791 (1981). 9. J. van Laar, A. Huijser & T.L. van Rooy, J. Vac. Sci. Technol. 14,894 (1977). 10. D. Schrneisser & K. Jacobi, Sur~. Sci. 108, 421 (1981). 11. J.C. Fuggle & N. Martensson, J. Electron. Speetrosc. 21,275 (1980). 12. H.B. Michaelson,J. Appl. Phys. 48, 4729 (1977). 13. C.F.alter, W. Ludwig & M. Selmke, Solid State Commun. 54, 321 (1985). 14. J.R. Chelikowsky & M.L. Cohen, Phys. Rev. BI3, 826 (1976). 15. J.R. Chelikowsky & M.L. Cohen, Phys. Rev. B20, 4150 (1979). 16. J.W. Davenport, R.E. Watson, M.L. Perlman & T.K. Sham, SolidState Commun. 40,999 (1981). 17. C. Falter, W. Ludwig, M. Selmke & W. Zieran, Phys. Lett. 105A, 139 (1984). 18. J.C. Phillips, Bonds and Bands in Semiconductors, Academic Press, New York (1973). 19. H.U. Baler, L. Koenders & W. M6nch, to be published.
CHARGE TRANSFER
Vol. 58, No. 3, pp. 215-217,
1986.
0038-1098/86 $3.00 t .OO Pergamon Press Ltd.
FROM CHEMICAL SHIFTS AT (110) SEMICONDUCTORS
SURFACES OF III-V
COMPOUND
W. Month Laboratorium
f& Festkorperphysik,
Universitat
D&burg,
D-4100 Duisburg, Federal Republic of Germany
(Received 10 December 1985 by M. Ciudona) Surface charge-transfers are calculated from surface core-level shifts determined by Eastman et aL and Taniguchi et aL by using soft X-ray photoemission spectroscopy. A simple electrostatic model which was first applied by Shevchik et al. to the analysis of bulk data is adapted for surfaces of compound semiconductors. The charge transferred from cations to anions is found to be the same in the bulk and at the (1 10) surfaces of GaP, GaAs, GaSb, and InSb. IN MOLECULES AND SOLIDS the binding energies of core levels are known to depend on the chemical bonding of the atoms under study. In a simple approach, such chemical shifts, which probe changes of the electrostatic potential at the core, may be understood by a charge transfer between atoms exhibiting different electronegativities. Some years ago, additional shifts have been observed in the binding energies of the 3d and 4d core levels of surface atoms at cleaved (1 10) surfaces of some III-V compound semiconductors [l, 21. The levels of the surface metal atoms were found to have larger binding energies, while the levels of the group V surface-atoms are shifted by almost the same amount but to smaller binding energies, both relative to their bulk counter-parts. These results contain information on a possible deviation of the charge distribution at the surface from the one in the bulk. This might be expected since the (1 10) surfaces of the III-V compound semiconductors were found to be reconstructed. The type of surface rearrangement may be easily explained since in such surfaces the atoms only have three nearest neighbors compared to four in the bulk. The trivalent metal atoms, for example, tend to establish planar sp2 bonds with their three neighbors instead of the tetrahedrally oriented sp3 bonds in the bulk. The result is that the group III and group V atoms are shifted inward and outward, respectively, with reference to an ideally terminated bulk [3]. By using a simple electrostatic model, which was developed for explaining bulk chemical shifts [4] , it will be shown here that the core-level shifts observed with the metal atoms at (1 10) surfaces of GaP, GaAs, GaSb, and InSb are solely due to the difference in surface and bulk Madelung energy and no additional surface chargetransfer must be invoked. 215
In III-V compound semiconductors the bonds are partly ionic. The charge transfer between the two sublattices may be understood and estimated following one of Pauling’s concepts [S] . Since the group III atoms have a smaller electronegativity than the group V atoms, the metal atoms donate charge to the non-metal atoms. The electronegativity-model gives the charge transfered in a single bond as &I
= O.l6Ixrrr-xvI+O.O35Ixrrr-XVI’,
(1)
which is a refined version [6] of Pauling’s original scale. In a tetrahedrally coordinated III-V compound the charge transferred in the bulk may then be approximated by Aqb = 4Aqr.
(2)
For the semiconductors to be discussed here, these values are listed in Table 4 and will be discussed at the end of this paper. Shevchik et al. [4] started from the valence-charge distribution of the cations in their neutral metallic form and considered the charge Aqb, which is transferred from the cations to the anions in the bulk of, for example, a III-V compound semiconductor, to be uniformly distributed within the cations which are approximated as spheres. The change of the potential seen by the cation core in the compound is then given by vb
c
_ ‘heo _ -.-
4nee
A &’
(3)
where d, is the nearest-neighbor distance in the metal which equals twice the atomic radius. The coefficient A is determined by the charge distribution and measures, for example, 2 or 3 for a sphere and a spherical shell, respectively. The corresponding energy shift AEE of a
CHARGE TRANSFER
216
FROM CHEMICAL SHIFTS
core level is then obtained by multiplying equation (3) by the electron charge e. . In an ionic compound, the electrostatic interaction of the cations and the anions has to be considered additionally. The corresponding Madelung energy is given by Eh
= --.-Awi he0
ffb d,’
(4)
Here, (Yb is the bulk Madelung constant of the specific lattice, and d, is the nearest-neighbor distance of the compound semiconductor under study. The total shift of a cation core-level in the bulk of the compound semiconductor with reference to the pure metal then results as
The same procedure may also be applied to a surface cation. The charge transfer is then labeled Aq,. For a (1 10) surface of a solid crystallizing in the zinc-blende lattice the surface Madelung constant ar, has been evaluated as 0.85 ob by Levine and Mark [7]. The next layer below the surface has almost the bulk value ob [8]. For the surface cations the energy shift of their core levels may thus be written as
Vol. 58, No. 3
Table 1. Binding energies of the Ga(3d5,z) and In(4dv2) core levels in the bulk of III-V compound semiconductors relative to the valence-band top E,” and the vacuum level E_; I = I&,, - E,” is the ionization energy, and d, is the nearest-neighbor distance III-V
d, nm
EE-E: eV
I eV
E VBC-E,” eV
GaP GaAs GaSb InSb
0.236 0.245 0.265 0.281
18.4’ 18.6a 18.7a 17.2b
6.01’ 5.56’ 4.91C 4.9 c
24.41 24.16 23.61 22.10
‘Ref. [l] bRef. 121 ‘Ref. [9] Table 2. Binding energies of the Ga(3ds,a) and In(4d5,2) core levels in the btdk of the metah rehtive to the Fermi level EF and the vacuum level Evclc; 6 is the work function and d, the nearest-neighbor distance III-V
d, nm
EF-e’ eV
@ eV
E,.,-rt eV
Ga In
0.271 0.333
18.3d 17.0e
4.2 ’ 4.12r
22.5 21.12
dRef. [lo
(6) The core-level binding the bulk then becomes
‘k:
[ii]
1
energy at the surface relative to
(7) Bulk and surface charge-transfer may now be evaluated from the observed energy shifts AE,b = (E,,, -rt) (Evac - Ei) and AP$’ = EE -PC by using equations (5) and (7). Here, the vacuum level E,,, is used as reference for both metals and compound semiconductors. The experimental data entering the analysis are listed in Tables 1 to 3. Following a recent paper by Falter et al. [131* we take A = 3.32. A more refined treatment should also consider a possible variation of this parameter at the surface. As Table 3 reveals, the charge transfers are found to be the same in the bulk and at the (1 10) surface of each of those four III-V compound semiconductors for which the corresponding core level shifts have been reported. This means that the difference in binding energy of the core levels in the bulk and at the (1 10) surface of GaP, GaAs, CaSb, and InSb - and thus most probably for all III-V compound semiconductors - is solely due to the difference of the Madelung energies in the bulk and at the surface.
Table 3. Binding-energy shifts of the (3ds,2) or (4ds,z) core levels, in the bulk and at (1 10) sueace of III- V compounds as well as the bulk and surface chargetransfers, Aqb and Aq,, calcukzted from these shifts by using equations (5) and (7). respectively III-V
Gal’ GaAs GaSb InSb
Cation
Anion
Eyt-E; eV
E,b-E’, eV
E,b-EC eV
Aqa
Aqs
1.50 1.66 1.11 0.98
0.28’ 0.28’ 0.30a 0.22b
-o.37a -0.36a --0.29b
0.291 0.207 0.127 0.165
0.240 0.205 0.140 0.166
‘Ref. [l] bRef. [2]
This finding is in excellent agreement with results of self-consistent pseudopotential calculations for the reconstructed and the unreconstructed (1 10) surface of GaAs by Chelikowsky and Cohen [14, 151. They also calculated the total valence charge density for both surface arrangements. With the reconstructed surface,
CHARGE TRANSFER FROM CHEMICAL SHIFTS
•V o l . 58, No. 3
Table 4. Experimental charge transfers of I I I - V compound semiconductors compared with theoretical values from different ionizity scales III-V Experiment
Pauling 15]
Falter et al. [17]
Phillips [18]
GaP GaAs GaSb InSb
0.263 0.256 0.162 0.183
0.240 0.198 0.149 0.155
0.327 0.240 0.044 0.280
0.250 0.207 0.127 0.165
they found bulk features to dominate the density configuration while this is not true for the unreconstructed surface geometry. Davenport et al. [16] have also analyzed the corelevel shift observed with GaAs (1 1 0) surfaces. As here, they concluded that the relative shifts are accounted for mainly by the difference between the Madelung energies at the surface and in the bulk. In their evaluation, however, they ended up with a very large charge transfer of 0.28eo in bulk GaAs and they had to reduce the ionic charges at the surface atoms by 10% relative to those in the bulk. Those conclusions are not supported by the present analysis of the experimental data reported for four III-V compound semiconductors. The present paper has considered the most recent experimental data of photoemission from core levels in the compound semiconductors GaP, GaAs, GaSb and lnSb [1, 2] and in the corresponding metals Ga and In [10, 11]. Therefore, the charge transfers evaluated here shall be compared in Table 4 with those obtained from different ionizity scales. The values labeled "Pauling" were evaluated by using equations (1) and (2), i.e. with the assumption that in tetrahedrally coordinated compound semiconductors the charge transfer amounts to four times the charge transfer in a single bond. For the ionizity scales given by Falter etal. [17] and by Phillips [18] the charge transfers were taken from a most recent paper by Falter et aI. [13]. In that publication they have also adopted the approach by Shevchik et al. [4] to calculate the core-level shifts of the metal atoms in compounds crystallizing in the zinc-blende lattice. They had reached "good agreement" with a set of experimental data already published by Shevchik et aL [4] in 1974. In the present paper, experimental data taken with meanwhile improved techniques, which made possible to resolve the spin-orbit split components of the Ga(3d) and the As(3d) signals, have been analyzed. The theoretical results by Falter et al. [13] based on their
217
own ionizity scale now show an "improved agreement" with the newer experimental data. But nevertheless, Pauling's scale still is a very good and easy to use approximation. Note added in proof: Very recently, Baier et al. [19] observed the surface shifts of the In(4d) core levels as 0.3 and 0.26 eV, at (1 10) surfaces cleaved from lnP and lnAs, respectively. The evaluation of both the bulk and the surface corelevel shifts relative to metallic In gives Aq s = 0.255 and Aq b = 0.266 for InP and Aqs = 0.198 and Aqb = 0.202 for lnAs, which results excellently confirm the conclusion that the charge transfer is the same in the bulk and at the (I 1 0) surfaces of III-V compound semiconductors. REFERENCES 1.
D.E. Eastman, T.-C. Chiang, P. Heimann & F.J. Himpsel,Phy~ Rev. Lett. 45,656 (1980). 2. M. Taniguchi, S. Suga, M. Seki, S. Shin, K.L. Kobayashi & H. Kanzaki, J. Phys. C: Solid State Phys. 16, L45 (1983). 3. A. Kahn, Surf. Sci. Rep. 3, 193 (1983) and references cited therein. 4. N.J. Shevchik, J. Tejeda & M. Cardona, Phys. Rev. tl9, 2627 (1974) and references cited therein. 5. L. Pauling, The Nature of the Chemical Bond, Cornell Univ. Press, Ithaca, N.Y., (1940). 6. N.B. Hannay & C_P. Smith, J. Am. Cher~ Soc. 68, 171 (1946). 7. J.D. Levine & P. Mark, Phys. Rev. 144, 751 (1966). 8. R.E. Watson, J.W. Davenport, M.L. Perhnan & T.K. Sham, Phys. Rev. B24, 1791 (1981). 9. J. van Laar, A. Huijser & T.L. van Rooy, J. Vac. Sci. Technol. 14,894 (1977). 10. D. Schrneisser & K. Jacobi, Sur~. Sci. 108, 421 (1981). 11. J.C. Fuggle & N. Martensson, J. Electron. Speetrosc. 21,275 (1980). 12. H.B. Michaelson,J. Appl. Phys. 48, 4729 (1977). 13. C.F.alter, W. Ludwig & M. Selmke, Solid State Commun. 54, 321 (1985). 14. J.R. Chelikowsky & M.L. Cohen, Phys. Rev. BI3, 826 (1976). 15. J.R. Chelikowsky & M.L. Cohen, Phys. Rev. B20, 4150 (1979). 16. J.W. Davenport, R.E. Watson, M.L. Perlman & T.K. Sham, SolidState Commun. 40,999 (1981). 17. C. Falter, W. Ludwig, M. Selmke & W. Zieran, Phys. Lett. 105A, 139 (1984). 18. J.C. Phillips, Bonds and Bands in Semiconductors, Academic Press, New York (1973). 19. H.U. Baler, L. Koenders & W. M6nch, to be published.