- Email: [email protected]
Energy bands, effective masses and g-factors of the lead salts and SnTe
Solid State Communications,
Vol. 8, PP. 569—575, 1970.
Pergamon Press.
Printed in Great Britain
ENERGY BANDS, EFFECTIVE MASSES AND g-FACTORS OF THE LEAD SALTS AND SnTe* R.L. Bernickt Department of Physics, University of Southern California, Los Angeles, Calif. and Leonard Kleinman Department of Physics, The University of Texas at Austin, Austin, Texas (Received 21 January 1970 by E. Burstein)
Using the pseudopotential scheme of Lin and Kleinman we have calculated the energy bands of PbTe, PbSe and PbS so as to obtain direct band gaps, effective masses and g-factors in very good agreement with experiment. Fitting the direct gap (the only piece of experimental data available) we have extrapolated the pseudopotential parameters to SnTe and obtained energy bands, effective masses and g-factors. Unlike other first principles and pseudopotential calculations we find the SnTe effective masses at the (~~ ~) point in the Brillouin Zone to be positive, i.e. the band extrema occur exactly at rather than just near the (~~ point.
j)
THERE is a vast amount of experimental data on the lead salts PbS, PbSe and PbTe. The energy gap widths, effective masses and g-factors have been determined from magneto-optical 2 studies,’ Shubnikov—de Haas experiments,photoRaman spin flip scattering experiments,3 emission,4 cyclotron resonance,5 6 and tunneling.7 Optical8 and electro-optical 9,10 studies provide information on the energy bands throughout the Brillouin Zone. On the other hand even though Shulxtikov—de ~ and optical data 8,12,13 are available for SnTe, there are no published values of effective masses or g-factors. In fact all that is known about SnTe is that the energy gap is about 0.022Ryd.,7~’3the p-type Fermi surface consists of four (111) prolate
surfaces ~ and the conduction and valence band edge symmetries are reversed from those in the lead salts. This follows from studies of the Pb~Sn1_~Tegap” and from the fact that the sign of the temperature dependence of the gap is opposite to that in the lead salts. First principles relativistic energy bands have been calculated for the lead salts ‘~‘~ and for SnTe. 17,16 These calculations have all been based on atomic rather than self-consistent crystal potentials and therefore cannot be relied upon for the exact ordering of nearby energy levels. Tung and Cohen 18 have made a pseudopotential calculation for PbTe and SnTe, choosing their parameters so as to get the correct energy gaps and fairly good agreement with the imaginary part of the frequency-dependent dielectric function. Lin and Kleinman’9 chose the parameters of their pseudopotential calculation of
* Research sponsored by the U.S. Air Force Office of Scientific Research Office of Aerospace Research under Grant No. AFOSR 68—1506
the lead salts to get correct energy gaps, gaps elsewhere in the Brillouin Zone in excellent agreement with peaks in the optical data8 and
Present address: Hughes Aircraft Co. Research Laboratories, Torrance, California 90505. t
569
570
LEAD SALTS AND SnTe
51
L~ L~ .103 021 L~ 000
L?-L.l3l
L
L1.089 012 LL 1+ .000
L1~.094 L3 .014 L~ 000 1+
-
.
Vol.8, No.7
.098
4J
L~ L31 .055 6+1 .072 L,~ .019 L~ .000
45+
L3~ -.058 L~
L~ 098 3+
-.076
—
L34. —.107
45+ L34 —.139 6+
L3+ —.144 L3~ —.,97 L~ .200 PbS -
PbSe
PbTe
SnTe
FIG. 1. Energy levels around gap at L. (Energy in Ry). fair agreement with experimental values of the effective masses. Because there are many more energy gaps than peaks in the optical data, this procedure is not unique. Indeed, some of the peaks may be due to contributions from several different portions of the Brillouin Zone. Therefore we here repeat the Lin and Kleinman calculation choosing our parameters to again obtain correct energy gaps while also obtaining excellent agreement with experimental effective masses but ignoring optical gaps. The crystal potential is made up from a superposition of atomic pseudopotentials each of which consists of a local part V
=
0, r
<
=
—(2Z/r)(1
—
V5~0~ =
—
a
s.o. j Im ~
E
jlrn>
ji so~I c/i
where cLi,~>is the outermost core state of the atom with angular momentum quantum numbers j, 1 and rn, E~°= H l/J~,> is obtained from atomic core spin-orbit splittings and E~is the energy of the outermost core state with I 0. E~and the atomic core spin-orbit splittings 2° the are takenarefrom Herman andbySkillman, approximated Slater analytic functions 21 and r 0 and 13 are determined 19 fromWethe Slaterfour functions by a parameters, simple prescription. have adjustable Z and a for the anion and cation plus the two a ~ In order ~
=
e~’~”0)),
r> r0
(1)
to reduce the number of parameters we took a = 1.2 for every atom. This gives a spinorbit splitting of the valence F level in PbTe in exact agreement with Herman’s ~ first ,~
and a non-local repulsive potential seen by s-electrons
This seemingly simple model for s~iinorbit coupling is actually quite accurate. ~ *
V5
=
—
(3)
T 0
V
plus a spin-orbit part*
aE3 çti5><çLi5
(2)
Vol.8, No.7
LEAD SALTS AND SnTe
571
Table 1. Core parwneters and lattice constants (in a.u.)
r~ui0~~
13catjon (3afl~0fl
a
PbS
PbSe
PbTe
SnTe
0.493 0.169
0.493 0.264
0.493 0.412
0.466 0.412
3.228 4.168
3.228 4.608
3.228 3.597
3.170 3.597
11.197
11.575
12.212
11.958
PbS
PbSe
PbTe
SnTe
2.816 3.272 0.129 0.430 1.2
2.816 3. 123 0.129 0.370 1.2
2.816 3.107 0.129 0.195 1.2
2.816 3.107 0.169 0.195 1.2
Table 2. Pseudopotential parameters.
~ anion acaijon
a a~0
Table 3. Amplitude of single-groups in double-group eigenfunctions. PbS cos@~ sinO + cosO sinO -
0.995 —0.099 0.859 0.511
PbSe
PbTe
SnTe
0.990 —0.138 0.819 0.573
0.978 —0.204 0.869 —0.493
0.987 —0.157 0.719 —0.695
principles calculation and fair agreement with both valence and conduction band F ~ splittings for all crystals. With the pseudopotential parameters given in Table 2 the energy bands of the lead salts were calculated expanding the wave functions in 181 plane waves* and diagonalizing the resulting secular determinent. In Fig. 1 we show the energy levels around the energy gap at the L point of the Brillouin Zone. In Table 3 cosO~ and sin6 ±arethe coefficients of the L 1~orL ~ and L3 ±single-group wave functions in the doublegroup functions L ~ or L~ as well as L ~ ±and L1~orL2_ in L~+.The energy gaps and the effective masses and g-factors calculated from the formulas of Mitchell and Wallis~t are * We used 181 plane waves at k = 0 and a comparable number at other points in the Brillouin Zone. .
The g-factor formulas in reference 19 are incorrect.
compared with experimental values in Table 4. The gaps and effective masses were used in determining the pseudopotential parameters, the g-factors, calculated afterwards, give added confirmation to the correctness of the results. Note in Fig. 1 that the states and L~ are reversed in PbTe from their order in PbS and PbSe. We have recalculated the transverse effective masses for PbTe assuming the ordering to be the same as PbS and PbSe (this causes sine in Table 3 to be positive) and found m~= 0.043 and m~’=0.052. Thus we are quite certain that our ordering of the bands at L is correct even though it contradicts all previous calculations 15,16,18 except Lin and Kleinman’s. Other experimental results attesting to the correctness 23 of the PbTe that results showing the include valence the band Knight shift, must be s-like about the lead atom, i.e. have L 5~symmetry and the photoemission studies~ which show (1) two valence levels at L
572
LEAD SALTS AND SnTe
Vol.8, No.7
Table 4. Effective masses, g-jac:ors and energy gaps. PbS
PbTe
0.100
0.0691
0.346
m1 (exp.)
0.105 ±0.015
0.068 ±0.015
0.31 ±0.05
(calc.) m1 (exp.) (calc.) ~J(exp.) gv (caic.) (exp.)
0.0709 0.075 ±0.01 14.3 13±3 16.5
0.0375 0.034 ±0.007 38.1 32±7 32.3
0.0279 0.022 ±0.003 48.9 51±8 15.2
~.
(caic.)
PbSe
(calc.)
—
—
SnTe 0.424 —
0.107 —
42.3 —
18.3
—
—
0.388
0.0891
0.0639
0.247
0.105 ±0.015 0.0774 0.080 ±0.01 16.2 12 ±3
0.070 ±0.015 0.038 0.040 ±0.008 36.3 27 ±7
0.24 ±0.05 0.0277 0.530 0.024 ±0.003 49.5 56.2 45 ±8, 57.5 ±.2~
(calc.) g~ (exp.)
18.4
33.9 —
14.3 15 ±ia
20.9
—
E gap (Ry) Caic. & exp.
0.021
0.012
0.014
001gb
m1 (exp.) (calc.) mt (exp.) g~ (calc.) (exp.)
—
—
—
All experimental data is from reference 2 except as noted a. reference 3 b. reference 25. approximately, 0.7 eV and 1.2 eV below the band edge, presumably our ~ and ~ 0.78eV and 1.33eV below the band edge (2) a second L conduction band up 1.3eV, presumably our L~ 1.28 eV above the valence band edge (3) a state at L between 8.1 and 8.6 eV above the principle conduction band, presumably our L41 8.53 eV above the principle conduction band (and wellabove the levels shown in Fig. 1). The excellent agreement obtained between the calculated levels and experiment is not too surprizing. Once the effective masses are fit, the energy levels at L are fairly uniquely determined (assuming the spin orbit term to be correct*). Although one might be able to fit the effective masses by making compensating errors in the spin ____________ *This seemingly simple model for spin orbit 19 coupling is actually quite accurate.
orbit term and the energy levels, the good agreement of the g-factors with experiment proves that this is not the case. We have no experimental evidence to compare our energy bands at other points of the Brillouin Zone. We are confident however that our non-local pseudopotential is better suited to extrapolating the energy bands throughout 18 the Zone than the local form used by Cohen. The fact that such good agreement with experiment was obtained for the lead salts using the same pseudopotential parameters for lead in all three salts, led us to believe that the Te parameters, obtained for PbTe, could be used in SaTe. Noting the difference between the Se and Te values of Z was very small, we decided to take Z = Zpb leaving only one parameter a 9~free toforfitSnTe, the one of experimental data available thepiece magnitude25 and sign14
Vol.8, No.7
LEAD SALTS AND SnTe
573
6-
K~
_____
.3
____
X~
2
L;: 4: L6+
w.
t_1+ ~6t_3~
0
L451W6 3+
_______
64
5
—.2
x:: x::
—.3 -.4
W~ -.5
-.6 L~1W. -.7 6+
K
I’
A
LWZX
r
~
FIG. 2. Energy bands of SnTe. (Energy in Ry). of the gap at L. It is gratifying that a50 turned out to be of the same magnitude as ave because one would expect the s-shifts of two atoms close together in theInsame the periodic table to to be similar. Fig. row 1 weofshow the energy levels at L and in Fig. 2 the energy bands throughout the Brillouin Zone for SnTe. The g-factors and effective masses are given in Table 4. Note that our effective masses are all positive which means that the band extrema occur exactly at L rather than just near L as found previously.’7 t8 Because in the lead salts all conduction bands shown in Fig. 1 are odd and all valence bands even, the energy denominators appearing in the effective mass formulas 22 are of constant sign, insuring all masses be positive. Because of the interchange of the valence and conduction band edge symmetries, the effective mass formulas for SnTe contain terms of both signs. This
assures that the SnTe masses~wi1lhave a larger magnitude than the PbTe masses* but leaves their sign uncertain, Noteis that the energy gap 25 used here 0.OO3Ry smaller of 0.019Ry than the gap measured by tunneling.7 This alone is sufficient to change the sign of the transverse conduction band mass. If the tunneling gap is correct as a function of temperature,7 it is quite possible that m~is positive at 150°K and negative at 4°K. On the other hand 1/me” is sufficiently large that small changes in the gaps cannot change its sign. The only way to change its sign is to reverse the ordering of the L~:and L: levels so they are like PbS and *
.
.
This would seem to indicate that the statement in reference 13 that, the electric susceptibility effective mass in p-type SnTe is half that in PbTe, is incorrect.
574
LEAD SALTS AND SnTe
PbSe rathe than PbTe. Not only must L2 lie below L3_ but it must be sufficiently far below that sinO, the coefficient of L3_ in L~, be small. While this is not impossible, we think it unlikely in view of our success with the lead salts. Thus we conclude that the valence band is conduction band is positive uncertainand andthat maythe actually change signmass with
Vol.8, No.7
It is interesting to note that there is a valence band edge in SnTe located along the (110) ~ axis 0.223eV below the L~ level (see Fig. 2) and a corresponding ~ edge in PbTe 0.130eV below the L~÷~ level. Such secondary band edges have been reported at primary approximately 26 and 0.17eV27 below the valence 0.3eV band edges.
temperature.
REFERENCES 1. 2.
MITCHELL D.L., PALIK E.D. and ZEMEL J.N., Proc. mt. Conf. Physics of Semiconductors, p. 197, The Physical Society of Japan, Tokyo (1966). CUFF K.F., ELLET M.R., KUGLIN C.D. and WILLIAMS L.R., Proc. mt. Conf. Physics of Semiconductors, p. 677, Dunod, Paris (1964).
3.
PATEL C.K.N. and SLUSHER R.E., Phys. Rev. 177, 1200 (1969).
4.
SPICER W.E. and LAPEYRE G.J., Phys. Rev. 139, A565 (1965).
5. 6.
STILES P.J., BURSTEIN E. and LANGENBERG D.N., Proc. mt. Conf. Physics of Semiconductors, Exeter 1962, p. 577, The Institute of Physics and the Physical Society of London (1962). BERMON S., Phys. Rev. 158, 723 (1967).
7.
STILES P.J., J. Phys. Suppi. 29, C4—95 (1968).
8.
CARDONA M. and GREENAWAY D.L., Phys. Rev. 133, A1685 (1964).
9.
SERAPHIN B.O., J. Phys. Suppl. 29, C4—95 (1968).
10.
ASPNES D.E. and CARDONA M., Phys. Rev. 173, 714 (1968).
11. 12.
BURKE J.R. Jr., HOUSTON B., SAVAGE T.H., BABISKIN J. and SIEBERMAN P.G., Proc. ConJ. Physics oj Semiconductors, p. 384, The Physical Society of Japan, Tokyo (1966). RIEDL H.R., DIXON JR. and SCHOOLAR R.B., Phys. Rev. 162, 692 (1967).
13.
BURKE J.R. and RIEDL H.R., Phys. Rev. 184, 830 (1969).
14.
DIMMOCK J.O., MELNGAILIS I. and STRAUSS A.J., Phys. Rev. Lett. 16, 1193 (1966).
15.
CONKLIN J.B. Jr., JOHNSON L.E. and PRATT G.W. Jr., Phys. Rev. 137, A1282 (1965).
16. 17.
HERMAN F., KORTUM R.L., ORTENBURG I.B. and VAN DYKE J.P., J. Phys. Suppl. 29, C4—62 (1968). RABII S., Phys. Rev. 182, 821 (1969).
18. 19.
TUNG Y.W. and COHEN M.L., Phys. Rev. 180, 823 (1969). LIN P.J. and KLEINMAN L., Phys. Rev. 142, 478 (1965).
20. 21.
HERMAN F. and SKILLMAN S., Atomic Structure Calculations, Prentice-Hall, N.J. (1963) SLATER J.C., Phys. Rev. 36, 57 (1930).
22. 23.
MITCHELL D.L. and WALLIS R.F., Phys. Rev. 151, 581 (1966). WEINBERG I. and CALLAWAY J., Nuovo Cimento 24, 190 (1962).
24.
SPICER W.E. and LAPEYRE G.J., Phys. Rev. 139, A565 (1965).
Vol.8, No.7
LEAD SALTS AND SnTe
575
25.
MOLDOVONOVA M., SIMITROVA St. and DECHEVA St., Fiv. tverd Tela 6, 3717 (1964) [English translation: Soviet Phys. Solid State 7, 2032 (1966)].
26.
ANDREEV A.A., J. Phys. Suppl. 29, C4—95 (1968).
27.
ANDREEV A.A. and RADIANOV V.N., Fiz. Tekhn. Pol. 1, 183 (1967) [English translation: Soviet Phys. Semicond. 1, 145 (1967)]. Utilisant la théorie du pseudopotentiel de Lin et Kleinman, nous avons calculé les bandes d’énergie, les masses effectives et les facteurs g. Les résultats théoriques sont en trés bon accord avec les résultats experimentaux. Connaissant I‘intervalle direct (la seule donnée obtenue experimentalement),
nous avons adapté et
extrapolé les parametres du pseudopotentiel du SnTe et obtenu les bandes d’énergie, les masses effectives et les facteurs g. Contrairement aux résultats obtenus par calculs utilisant d’autres premiers principes et pseudopotentiels, nous trouvons que les masses effectives du SnTe situées au point (~ ~ ~) de la zàne de Brillouin sont positives, c’est dire que l’extrémum de la bande prend place exactement au point (~~ ~) au lieu de seulement prés de ce point.
a
Vol. 8, PP. 569—575, 1970.
Pergamon Press.
Printed in Great Britain
ENERGY BANDS, EFFECTIVE MASSES AND g-FACTORS OF THE LEAD SALTS AND SnTe* R.L. Bernickt Department of Physics, University of Southern California, Los Angeles, Calif. and Leonard Kleinman Department of Physics, The University of Texas at Austin, Austin, Texas (Received 21 January 1970 by E. Burstein)
Using the pseudopotential scheme of Lin and Kleinman we have calculated the energy bands of PbTe, PbSe and PbS so as to obtain direct band gaps, effective masses and g-factors in very good agreement with experiment. Fitting the direct gap (the only piece of experimental data available) we have extrapolated the pseudopotential parameters to SnTe and obtained energy bands, effective masses and g-factors. Unlike other first principles and pseudopotential calculations we find the SnTe effective masses at the (~~ ~) point in the Brillouin Zone to be positive, i.e. the band extrema occur exactly at rather than just near the (~~ point.
j)
THERE is a vast amount of experimental data on the lead salts PbS, PbSe and PbTe. The energy gap widths, effective masses and g-factors have been determined from magneto-optical 2 studies,’ Shubnikov—de Haas experiments,photoRaman spin flip scattering experiments,3 emission,4 cyclotron resonance,5 6 and tunneling.7 Optical8 and electro-optical 9,10 studies provide information on the energy bands throughout the Brillouin Zone. On the other hand even though Shulxtikov—de ~ and optical data 8,12,13 are available for SnTe, there are no published values of effective masses or g-factors. In fact all that is known about SnTe is that the energy gap is about 0.022Ryd.,7~’3the p-type Fermi surface consists of four (111) prolate
surfaces ~ and the conduction and valence band edge symmetries are reversed from those in the lead salts. This follows from studies of the Pb~Sn1_~Tegap” and from the fact that the sign of the temperature dependence of the gap is opposite to that in the lead salts. First principles relativistic energy bands have been calculated for the lead salts ‘~‘~ and for SnTe. 17,16 These calculations have all been based on atomic rather than self-consistent crystal potentials and therefore cannot be relied upon for the exact ordering of nearby energy levels. Tung and Cohen 18 have made a pseudopotential calculation for PbTe and SnTe, choosing their parameters so as to get the correct energy gaps and fairly good agreement with the imaginary part of the frequency-dependent dielectric function. Lin and Kleinman’9 chose the parameters of their pseudopotential calculation of
* Research sponsored by the U.S. Air Force Office of Scientific Research Office of Aerospace Research under Grant No. AFOSR 68—1506
the lead salts to get correct energy gaps, gaps elsewhere in the Brillouin Zone in excellent agreement with peaks in the optical data8 and
Present address: Hughes Aircraft Co. Research Laboratories, Torrance, California 90505. t
569
570
LEAD SALTS AND SnTe
51
L~ L~ .103 021 L~ 000
L?-L.l3l
L
L1.089 012 LL 1+ .000
L1~.094 L3 .014 L~ 000 1+
-
.
Vol.8, No.7
.098
4J
L~ L31 .055 6+1 .072 L,~ .019 L~ .000
45+
L3~ -.058 L~
L~ 098 3+
-.076
—
L34. —.107
45+ L34 —.139 6+
L3+ —.144 L3~ —.,97 L~ .200 PbS -
PbSe
PbTe
SnTe
FIG. 1. Energy levels around gap at L. (Energy in Ry). fair agreement with experimental values of the effective masses. Because there are many more energy gaps than peaks in the optical data, this procedure is not unique. Indeed, some of the peaks may be due to contributions from several different portions of the Brillouin Zone. Therefore we here repeat the Lin and Kleinman calculation choosing our parameters to again obtain correct energy gaps while also obtaining excellent agreement with experimental effective masses but ignoring optical gaps. The crystal potential is made up from a superposition of atomic pseudopotentials each of which consists of a local part V
=
0, r
<
=
—(2Z/r)(1
—
V5~0~ =
—
a
s.o. j Im ~
E
jlrn>
ji so~I c/i
where cLi,~>is the outermost core state of the atom with angular momentum quantum numbers j, 1 and rn, E~°= H l/J~,> is obtained from atomic core spin-orbit splittings and E~is the energy of the outermost core state with I 0. E~and the atomic core spin-orbit splittings 2° the are takenarefrom Herman andbySkillman, approximated Slater analytic functions 21 and r 0 and 13 are determined 19 fromWethe Slaterfour functions by a parameters, simple prescription. have adjustable Z and a for the anion and cation plus the two a ~ In order ~
=
e~’~”0)),
r> r0
(1)
to reduce the number of parameters we took a = 1.2 for every atom. This gives a spinorbit splitting of the valence F level in PbTe in exact agreement with Herman’s ~ first ,~
and a non-local repulsive potential seen by s-electrons
This seemingly simple model for s~iinorbit coupling is actually quite accurate. ~ *
V5
=
—
(3)
T 0
V
plus a spin-orbit part*
aE3 çti5><çLi5
(2)
Vol.8, No.7
LEAD SALTS AND SnTe
571
Table 1. Core parwneters and lattice constants (in a.u.)
r~ui0~~
13catjon (3afl~0fl
a
PbS
PbSe
PbTe
SnTe
0.493 0.169
0.493 0.264
0.493 0.412
0.466 0.412
3.228 4.168
3.228 4.608
3.228 3.597
3.170 3.597
11.197
11.575
12.212
11.958
PbS
PbSe
PbTe
SnTe
2.816 3.272 0.129 0.430 1.2
2.816 3. 123 0.129 0.370 1.2
2.816 3.107 0.129 0.195 1.2
2.816 3.107 0.169 0.195 1.2
Table 2. Pseudopotential parameters.
~ anion acaijon
a a~0
Table 3. Amplitude of single-groups in double-group eigenfunctions. PbS cos@~ sinO + cosO sinO -
0.995 —0.099 0.859 0.511
PbSe
PbTe
SnTe
0.990 —0.138 0.819 0.573
0.978 —0.204 0.869 —0.493
0.987 —0.157 0.719 —0.695
principles calculation and fair agreement with both valence and conduction band F ~ splittings for all crystals. With the pseudopotential parameters given in Table 2 the energy bands of the lead salts were calculated expanding the wave functions in 181 plane waves* and diagonalizing the resulting secular determinent. In Fig. 1 we show the energy levels around the energy gap at the L point of the Brillouin Zone. In Table 3 cosO~ and sin6 ±arethe coefficients of the L 1~orL ~ and L3 ±single-group wave functions in the doublegroup functions L ~ or L~ as well as L ~ ±and L1~orL2_ in L~+.The energy gaps and the effective masses and g-factors calculated from the formulas of Mitchell and Wallis~t are * We used 181 plane waves at k = 0 and a comparable number at other points in the Brillouin Zone. .
The g-factor formulas in reference 19 are incorrect.
compared with experimental values in Table 4. The gaps and effective masses were used in determining the pseudopotential parameters, the g-factors, calculated afterwards, give added confirmation to the correctness of the results. Note in Fig. 1 that the states and L~ are reversed in PbTe from their order in PbS and PbSe. We have recalculated the transverse effective masses for PbTe assuming the ordering to be the same as PbS and PbSe (this causes sine in Table 3 to be positive) and found m~= 0.043 and m~’=0.052. Thus we are quite certain that our ordering of the bands at L is correct even though it contradicts all previous calculations 15,16,18 except Lin and Kleinman’s. Other experimental results attesting to the correctness 23 of the PbTe that results showing the include valence the band Knight shift, must be s-like about the lead atom, i.e. have L 5~symmetry and the photoemission studies~ which show (1) two valence levels at L
572
LEAD SALTS AND SnTe
Vol.8, No.7
Table 4. Effective masses, g-jac:ors and energy gaps. PbS
PbTe
0.100
0.0691
0.346
m1 (exp.)
0.105 ±0.015
0.068 ±0.015
0.31 ±0.05
(calc.) m1 (exp.) (calc.) ~J(exp.) gv (caic.) (exp.)
0.0709 0.075 ±0.01 14.3 13±3 16.5
0.0375 0.034 ±0.007 38.1 32±7 32.3
0.0279 0.022 ±0.003 48.9 51±8 15.2
~.
(caic.)
PbSe
(calc.)
—
—
SnTe 0.424 —
0.107 —
42.3 —
18.3
—
—
0.388
0.0891
0.0639
0.247
0.105 ±0.015 0.0774 0.080 ±0.01 16.2 12 ±3
0.070 ±0.015 0.038 0.040 ±0.008 36.3 27 ±7
0.24 ±0.05 0.0277 0.530 0.024 ±0.003 49.5 56.2 45 ±8, 57.5 ±.2~
(calc.) g~ (exp.)
18.4
33.9 —
14.3 15 ±ia
20.9
—
E gap (Ry) Caic. & exp.
0.021
0.012
0.014
001gb
m1 (exp.) (calc.) mt (exp.) g~ (calc.) (exp.)
—
—
—
All experimental data is from reference 2 except as noted a. reference 3 b. reference 25. approximately, 0.7 eV and 1.2 eV below the band edge, presumably our ~ and ~ 0.78eV and 1.33eV below the band edge (2) a second L conduction band up 1.3eV, presumably our L~ 1.28 eV above the valence band edge (3) a state at L between 8.1 and 8.6 eV above the principle conduction band, presumably our L41 8.53 eV above the principle conduction band (and wellabove the levels shown in Fig. 1). The excellent agreement obtained between the calculated levels and experiment is not too surprizing. Once the effective masses are fit, the energy levels at L are fairly uniquely determined (assuming the spin orbit term to be correct*). Although one might be able to fit the effective masses by making compensating errors in the spin ____________ *This seemingly simple model for spin orbit 19 coupling is actually quite accurate.
orbit term and the energy levels, the good agreement of the g-factors with experiment proves that this is not the case. We have no experimental evidence to compare our energy bands at other points of the Brillouin Zone. We are confident however that our non-local pseudopotential is better suited to extrapolating the energy bands throughout 18 the Zone than the local form used by Cohen. The fact that such good agreement with experiment was obtained for the lead salts using the same pseudopotential parameters for lead in all three salts, led us to believe that the Te parameters, obtained for PbTe, could be used in SaTe. Noting the difference between the Se and Te values of Z was very small, we decided to take Z = Zpb leaving only one parameter a 9~free toforfitSnTe, the one of experimental data available thepiece magnitude25 and sign14
Vol.8, No.7
LEAD SALTS AND SnTe
573
6-
K~
_____
.3
____
X~
2
L;: 4: L6+
w.
t_1+ ~6t_3~
0
L451W6 3+
_______
64
5
—.2
x:: x::
—.3 -.4
W~ -.5
-.6 L~1W. -.7 6+
K
I’
A
LWZX
r
~
FIG. 2. Energy bands of SnTe. (Energy in Ry). of the gap at L. It is gratifying that a50 turned out to be of the same magnitude as ave because one would expect the s-shifts of two atoms close together in theInsame the periodic table to to be similar. Fig. row 1 weofshow the energy levels at L and in Fig. 2 the energy bands throughout the Brillouin Zone for SnTe. The g-factors and effective masses are given in Table 4. Note that our effective masses are all positive which means that the band extrema occur exactly at L rather than just near L as found previously.’7 t8 Because in the lead salts all conduction bands shown in Fig. 1 are odd and all valence bands even, the energy denominators appearing in the effective mass formulas 22 are of constant sign, insuring all masses be positive. Because of the interchange of the valence and conduction band edge symmetries, the effective mass formulas for SnTe contain terms of both signs. This
assures that the SnTe masses~wi1lhave a larger magnitude than the PbTe masses* but leaves their sign uncertain, Noteis that the energy gap 25 used here 0.OO3Ry smaller of 0.019Ry than the gap measured by tunneling.7 This alone is sufficient to change the sign of the transverse conduction band mass. If the tunneling gap is correct as a function of temperature,7 it is quite possible that m~is positive at 150°K and negative at 4°K. On the other hand 1/me” is sufficiently large that small changes in the gaps cannot change its sign. The only way to change its sign is to reverse the ordering of the L~:and L: levels so they are like PbS and *
.
.
This would seem to indicate that the statement in reference 13 that, the electric susceptibility effective mass in p-type SnTe is half that in PbTe, is incorrect.
574
LEAD SALTS AND SnTe
PbSe rathe than PbTe. Not only must L2 lie below L3_ but it must be sufficiently far below that sinO, the coefficient of L3_ in L~, be small. While this is not impossible, we think it unlikely in view of our success with the lead salts. Thus we conclude that the valence band is conduction band is positive uncertainand andthat maythe actually change signmass with
Vol.8, No.7
It is interesting to note that there is a valence band edge in SnTe located along the (110) ~ axis 0.223eV below the L~ level (see Fig. 2) and a corresponding ~ edge in PbTe 0.130eV below the L~÷~ level. Such secondary band edges have been reported at primary approximately 26 and 0.17eV27 below the valence 0.3eV band edges.
temperature.
REFERENCES 1. 2.
MITCHELL D.L., PALIK E.D. and ZEMEL J.N., Proc. mt. Conf. Physics of Semiconductors, p. 197, The Physical Society of Japan, Tokyo (1966). CUFF K.F., ELLET M.R., KUGLIN C.D. and WILLIAMS L.R., Proc. mt. Conf. Physics of Semiconductors, p. 677, Dunod, Paris (1964).
3.
PATEL C.K.N. and SLUSHER R.E., Phys. Rev. 177, 1200 (1969).
4.
SPICER W.E. and LAPEYRE G.J., Phys. Rev. 139, A565 (1965).
5. 6.
STILES P.J., BURSTEIN E. and LANGENBERG D.N., Proc. mt. Conf. Physics of Semiconductors, Exeter 1962, p. 577, The Institute of Physics and the Physical Society of London (1962). BERMON S., Phys. Rev. 158, 723 (1967).
7.
STILES P.J., J. Phys. Suppi. 29, C4—95 (1968).
8.
CARDONA M. and GREENAWAY D.L., Phys. Rev. 133, A1685 (1964).
9.
SERAPHIN B.O., J. Phys. Suppl. 29, C4—95 (1968).
10.
ASPNES D.E. and CARDONA M., Phys. Rev. 173, 714 (1968).
11. 12.
BURKE J.R. Jr., HOUSTON B., SAVAGE T.H., BABISKIN J. and SIEBERMAN P.G., Proc. ConJ. Physics oj Semiconductors, p. 384, The Physical Society of Japan, Tokyo (1966). RIEDL H.R., DIXON JR. and SCHOOLAR R.B., Phys. Rev. 162, 692 (1967).
13.
BURKE J.R. and RIEDL H.R., Phys. Rev. 184, 830 (1969).
14.
DIMMOCK J.O., MELNGAILIS I. and STRAUSS A.J., Phys. Rev. Lett. 16, 1193 (1966).
15.
CONKLIN J.B. Jr., JOHNSON L.E. and PRATT G.W. Jr., Phys. Rev. 137, A1282 (1965).
16. 17.
HERMAN F., KORTUM R.L., ORTENBURG I.B. and VAN DYKE J.P., J. Phys. Suppl. 29, C4—62 (1968). RABII S., Phys. Rev. 182, 821 (1969).
18. 19.
TUNG Y.W. and COHEN M.L., Phys. Rev. 180, 823 (1969). LIN P.J. and KLEINMAN L., Phys. Rev. 142, 478 (1965).
20. 21.
HERMAN F. and SKILLMAN S., Atomic Structure Calculations, Prentice-Hall, N.J. (1963) SLATER J.C., Phys. Rev. 36, 57 (1930).
22. 23.
MITCHELL D.L. and WALLIS R.F., Phys. Rev. 151, 581 (1966). WEINBERG I. and CALLAWAY J., Nuovo Cimento 24, 190 (1962).
24.
SPICER W.E. and LAPEYRE G.J., Phys. Rev. 139, A565 (1965).
Vol.8, No.7
LEAD SALTS AND SnTe
575
25.
MOLDOVONOVA M., SIMITROVA St. and DECHEVA St., Fiv. tverd Tela 6, 3717 (1964) [English translation: Soviet Phys. Solid State 7, 2032 (1966)].
26.
ANDREEV A.A., J. Phys. Suppl. 29, C4—95 (1968).
27.
ANDREEV A.A. and RADIANOV V.N., Fiz. Tekhn. Pol. 1, 183 (1967) [English translation: Soviet Phys. Semicond. 1, 145 (1967)]. Utilisant la théorie du pseudopotentiel de Lin et Kleinman, nous avons calculé les bandes d’énergie, les masses effectives et les facteurs g. Les résultats théoriques sont en trés bon accord avec les résultats experimentaux. Connaissant I‘intervalle direct (la seule donnée obtenue experimentalement),
nous avons adapté et
extrapolé les parametres du pseudopotentiel du SnTe et obtenu les bandes d’énergie, les masses effectives et les facteurs g. Contrairement aux résultats obtenus par calculs utilisant d’autres premiers principes et pseudopotentiels, nous trouvons que les masses effectives du SnTe situées au point (~ ~ ~) de la zàne de Brillouin sont positives, c’est dire que l’extrémum de la bande prend place exactement au point (~~ ~) au lieu de seulement prés de ce point.
a