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Band structure of α-Sn, InSb and CdTe including spin-orbit effects
Solid State Communications, Vol. 6, pp. 465 -467, 1968.
Pergamon Press. Printed in Great Britain
BAND STRUCTURE OF a-Sn, InSb AND CdTe INCLUDING SPIN-ORBIT EFFECTS S. Bloom RCA Laboratories, Princeton, New Jersey, U.S.A. and T.K. Bergstresser James Franck Institute, University of Chicago, Chicago Illinois, U.S.A. (Received 12 April 1968)
Full-zone band structures for a-Sn, InSb and CdTe are calculated by the pseudopotential method, with spin-orbit effects included. By empirically fitting the spin-orbit splitting at a selected point, the band structures at other points in the zone are found to agree well with experiment. A comparison with results of k. p calculations is made.
THE BAND structures of various elemental and binary semiconductors with spin-orbit coupling effects included are calculated by the empirical pseudopotential method. Because the spin-orbit splittings are an appreciable fraction of the energy gaps for the heavier elements, the calculations have been done first for the important and widely-studied grayare tin reported and its row neighbors, InSb andcases CdTe,of and here. The method is applicable to a wide variety of compounds.
For the row neighbors of Sn, we note that two contributions to ~A tend to cancel. The overlap of the relevant core state of the group V or VI atom with the valence pseudo-wavefunction is smaller than for the group III or II atom, but there is a greater concentration of valence electrons around the group V or VI atoms. 5 and With XA Sn asInSb theand standard, the form factors Afrom for CdTe may be estimated the p-state spin-orbit splittings4 of the constituent free atoms. Thus ~S is proportional to the sum of, and XA to the difference between, these freeatom splittings. A weighted average is more appropriate for some purposes,s but not for estimating ~A. Note that ~A contributes very little to the difference in ~ between Sn and CdTe. Only a slight readjustment of the estimated x -values is necessary to give the best overall agreement between experiment and the calculation in the valence band along t and at F. Details of the calculations, together with results for other semiconductors, will appear in a fulllength paper.
The calculations start with the spin-free crystal-field pseudopotential form factors found by Cohen and Bergstresser. 1 The spin- orbit contribution to the Hamiltonian is then added by our extension of Weisz’s2 method to cover the case of binary compounds. This requires that both symmetric (XS) and antisymmetric (XA) spin-orbit energy form factors be included. In addition, Weisz’s small-k approximation of the orthogonalization integrals is replaced here by analytic formulas valid as well for large k. rhe factor AS for Sn is chosen to yield agreement with optical experiments3 for the valence band splitting !~ = 0. 48 eV along the A symmetry line in the Brillouin zone,
The results of the present calculations are shown in the table and figures. The spinorbit splitting of the valence band along the A axis 465
466
BAND STRUCTURE OF a-Sn, LnSb AND CdTe I
o
I
I
I
Vol. 6, No. 7
III
Iii~
~
~.;.L
5 0
-i~.L
re
•2
Li
______
L
ili/
K _________________________
I
r
Ii
FIG.
“4
:
1
Band structure of a -Sn.
r
L
X
K
1’
FIG. 2
Band structure of InSb and CdTe. TABLE 1 Spin-orbit splittings, in eV. Experimental values are from reference 3 AS
AA (in Ryd.)
calc.
expt.
A~ caic.
A’0 expt. calc.
0. 73
0.48
0.48
0. 35
0.43
0.28
0. 19
0. 97
0.43
A (L~)
A15
k. p. present k. p. present calc. calc. caic. calc.
Sn
0. 00215
InSb
0. 0022
0. 0001
0. 82
0. 82
0. 50
0. 54
0. 33
0.42
0. 33
0. 16
0. 78
0. 38
CdTe 0. 0020
0. 0002
0. 91
0. 90
0. 57*
0. 57
-
0. 45
0.
0. 09
0.
0. 23
*
0
A expt.
-
M. Cardona and D. L. Greenaway, Phys. Rev. 131, 98 (1963).
t Reference 7.
Vol. 6, No. 7
BAND STRUCTURE OF a-Sn, InSb AND CdTe
is nearly constant from k = (2rr/a) (0.2, 0.2, 0.2) to the point L. The pertinent value of A1 is measured at the M1 critical point along A. For InSb, value of A the smallness of the experimental 1 is commented upon in reference 3. Thethe maximum calculated valueEs-axis of theissplitting of valence band along the recorded in Table 1 under ~ For Sn, this occurs at k = (2i-r/a) (0. 12, 0, 0). The calculated splitting becomes equal to the experimental value at k = (2Tr/a) (0.25, 0, 0). This point appears to be near a critical point in the bands 3 and 4 to band 6 joint density of states. In contrast, a k. p. calculation did not find a suitable critical point in this neighborhood. In InSb and CdTe the present calculations give a larger valence band spin-orbit splitting at X than the k. p. calculation. ~ .
467
There is no unambiguous experimental evidence of the spin-orbit splitting of the p-like conduction bands, the electroreflectance 3 might give although some information between 4 of Sn and 5 eV. We compare in Table 1 our values3 6 withfind the much resultssmaller of full-zone k. p.than calculations. We splittings the k. p. calculations indicate. Perhaps this question could be resolved by the recent, sensitive denvative techniques. Acknowledgments This work was begun at Cambridge University and we wish to thank the Cavendish Laboratory for their hospitality. One of us (S. B.) also wishes to thank RCA Laboratories for sponsoring his year of study abroad, and the other thanks the National Science Foundation and the U. S. Office of Naval Research for support. -
References 1.
COHEN M. L. and BERGSTRESSER T.K.,
Phys. Rev. 141,
2.
WEISZ G.,
3.
CARDONA M., SHAKLEE K. L. and POLLAK F.H.,
4.
HERMAN F. and SKILLMAN S.,
5.
HERMAN F., KUGLIN C. D., (1963).
6.
CARDONA M., McELROY P.,
7.
319 (1966). POLLAK F. H.,
789 (1966).
Phys. Rev. 149, 504 (1966). Phys. Rev. 154, 696 (1967).
Atomic Energy Calculations, Prentice-Hall Inc.
CUFF K. F. and KORTUM H. L., POLLAK F.H. and
(1963).
Phys. Rev. Left. 11, 541
SHAKLEE K.L., Solid State Commun. 4, —
private communication, to be published.
Les structures de bande a zone complete ont èté calculées pour a -Sn, InSb et CdTe par la méthode des pseudopotentiels tenant compte des effects spin-orbite. En accordant empiriquement le splitting de spin-orbite a un point choisis, les structures de bande a d’autres points de la zone se trouvent en bon accord avec les resultats experimentaux. Une comparaison est faite avec les resultats obtenus par le calcul par méthode k. p.
7
Pergamon Press. Printed in Great Britain
BAND STRUCTURE OF a-Sn, InSb AND CdTe INCLUDING SPIN-ORBIT EFFECTS S. Bloom RCA Laboratories, Princeton, New Jersey, U.S.A. and T.K. Bergstresser James Franck Institute, University of Chicago, Chicago Illinois, U.S.A. (Received 12 April 1968)
Full-zone band structures for a-Sn, InSb and CdTe are calculated by the pseudopotential method, with spin-orbit effects included. By empirically fitting the spin-orbit splitting at a selected point, the band structures at other points in the zone are found to agree well with experiment. A comparison with results of k. p calculations is made.
THE BAND structures of various elemental and binary semiconductors with spin-orbit coupling effects included are calculated by the empirical pseudopotential method. Because the spin-orbit splittings are an appreciable fraction of the energy gaps for the heavier elements, the calculations have been done first for the important and widely-studied grayare tin reported and its row neighbors, InSb andcases CdTe,of and here. The method is applicable to a wide variety of compounds.
For the row neighbors of Sn, we note that two contributions to ~A tend to cancel. The overlap of the relevant core state of the group V or VI atom with the valence pseudo-wavefunction is smaller than for the group III or II atom, but there is a greater concentration of valence electrons around the group V or VI atoms. 5 and With XA Sn asInSb theand standard, the form factors Afrom for CdTe may be estimated the p-state spin-orbit splittings4 of the constituent free atoms. Thus ~S is proportional to the sum of, and XA to the difference between, these freeatom splittings. A weighted average is more appropriate for some purposes,s but not for estimating ~A. Note that ~A contributes very little to the difference in ~ between Sn and CdTe. Only a slight readjustment of the estimated x -values is necessary to give the best overall agreement between experiment and the calculation in the valence band along t and at F. Details of the calculations, together with results for other semiconductors, will appear in a fulllength paper.
The calculations start with the spin-free crystal-field pseudopotential form factors found by Cohen and Bergstresser. 1 The spin- orbit contribution to the Hamiltonian is then added by our extension of Weisz’s2 method to cover the case of binary compounds. This requires that both symmetric (XS) and antisymmetric (XA) spin-orbit energy form factors be included. In addition, Weisz’s small-k approximation of the orthogonalization integrals is replaced here by analytic formulas valid as well for large k. rhe factor AS for Sn is chosen to yield agreement with optical experiments3 for the valence band splitting !~ = 0. 48 eV along the A symmetry line in the Brillouin zone,
The results of the present calculations are shown in the table and figures. The spinorbit splitting of the valence band along the A axis 465
466
BAND STRUCTURE OF a-Sn, LnSb AND CdTe I
o
I
I
I
Vol. 6, No. 7
III
Iii~
~
~.;.L
5 0
-i~.L
re
•2
Li
______
L
ili/
K _________________________
I
r
Ii
FIG.
“4
:
1
Band structure of a -Sn.
r
L
X
K
1’
FIG. 2
Band structure of InSb and CdTe. TABLE 1 Spin-orbit splittings, in eV. Experimental values are from reference 3 AS
AA (in Ryd.)
calc.
expt.
A~ caic.
A’0 expt. calc.
0. 73
0.48
0.48
0. 35
0.43
0.28
0. 19
0. 97
0.43
A (L~)
A15
k. p. present k. p. present calc. calc. caic. calc.
Sn
0. 00215
InSb
0. 0022
0. 0001
0. 82
0. 82
0. 50
0. 54
0. 33
0.42
0. 33
0. 16
0. 78
0. 38
CdTe 0. 0020
0. 0002
0. 91
0. 90
0. 57*
0. 57
-
0. 45
0.
0. 09
0.
0. 23
*
0
A expt.
-
M. Cardona and D. L. Greenaway, Phys. Rev. 131, 98 (1963).
t Reference 7.
Vol. 6, No. 7
BAND STRUCTURE OF a-Sn, InSb AND CdTe
is nearly constant from k = (2rr/a) (0.2, 0.2, 0.2) to the point L. The pertinent value of A1 is measured at the M1 critical point along A. For InSb, value of A the smallness of the experimental 1 is commented upon in reference 3. Thethe maximum calculated valueEs-axis of theissplitting of valence band along the recorded in Table 1 under ~ For Sn, this occurs at k = (2i-r/a) (0. 12, 0, 0). The calculated splitting becomes equal to the experimental value at k = (2Tr/a) (0.25, 0, 0). This point appears to be near a critical point in the bands 3 and 4 to band 6 joint density of states. In contrast, a k. p. calculation did not find a suitable critical point in this neighborhood. In InSb and CdTe the present calculations give a larger valence band spin-orbit splitting at X than the k. p. calculation. ~ .
467
There is no unambiguous experimental evidence of the spin-orbit splitting of the p-like conduction bands, the electroreflectance 3 might give although some information between 4 of Sn and 5 eV. We compare in Table 1 our values3 6 withfind the much resultssmaller of full-zone k. p.than calculations. We splittings the k. p. calculations indicate. Perhaps this question could be resolved by the recent, sensitive denvative techniques. Acknowledgments This work was begun at Cambridge University and we wish to thank the Cavendish Laboratory for their hospitality. One of us (S. B.) also wishes to thank RCA Laboratories for sponsoring his year of study abroad, and the other thanks the National Science Foundation and the U. S. Office of Naval Research for support. -
References 1.
COHEN M. L. and BERGSTRESSER T.K.,
Phys. Rev. 141,
2.
WEISZ G.,
3.
CARDONA M., SHAKLEE K. L. and POLLAK F.H.,
4.
HERMAN F. and SKILLMAN S.,
5.
HERMAN F., KUGLIN C. D., (1963).
6.
CARDONA M., McELROY P.,
7.
319 (1966). POLLAK F. H.,
789 (1966).
Phys. Rev. 149, 504 (1966). Phys. Rev. 154, 696 (1967).
Atomic Energy Calculations, Prentice-Hall Inc.
CUFF K. F. and KORTUM H. L., POLLAK F.H. and
(1963).
Phys. Rev. Left. 11, 541
SHAKLEE K.L., Solid State Commun. 4, —
private communication, to be published.
Les structures de bande a zone complete ont èté calculées pour a -Sn, InSb et CdTe par la méthode des pseudopotentiels tenant compte des effects spin-orbite. En accordant empiriquement le splitting de spin-orbite a un point choisis, les structures de bande a d’autres points de la zone se trouvent en bon accord avec les resultats experimentaux. Une comparaison est faite avec les resultats obtenus par le calcul par méthode k. p.
7