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Band parameters of semiconductors with zincblende, wurtzite, and germanium structure
J. Phys. C%ern. Solids
Pergamon
Press 1963. Vol. 24, pp. 1543-1555.
BAND PARAMETERS ZINCBLENDE,
Printed in Great Britain.
OF SEMICONDUCTORS
WURTZITE,
WITH
AND GERMANIUM
STRUCTURE MANUEL RCA Laboratories,
CARDONA Princeton,
New Jersey
(Received 10 June 1963)
Abstract-The k * p method is used for calculating the band parameters of semiconductors with germanium, zincblende, and wurtzite structure at k = 0 and at the L and X points of the zincblende structure, At k = 0, the interaction between the highest valence band and the two lowest conduction bands is considered. The energy gaps between these states are obtained from reflectivity measurements. The wave functions of a polar material are obtained from the wave functions of the isoelectronic non-polar material by the apptication of an antisymmetric perturbing potential. The matrix elements of p between two given states of the non-polar material are assumed to be the same for all materials. The band parameters of the wurtzite materials are obtained by the application of a small hexagonal crystal-field to the corresponding zincblende-type compound. It is possible by this method to account for the existing experimental data and to predict the band parameters of many materials for which they have not been measured. It is suggested that the absolute valence band maximum in several II-VI compounds may not be at the center of the Briliouin zone.
action. The states have been labelfed according to 1. INTRODUCTION THE k p method(l9 2) has been very helpful for the standard notation, with the subindices c and v interpreting the effective masses and g-factors(s) of semiconductors with germanium and zincblende structure. By applying first and second order (Lzcd -2’ perturbation theory to the k p (k is the crystal momentum, p the linear momentum operator) term in the ~amiltoni~, one can show that the inverse effective mass parameters are functions of (LX?L3 squares of matrix elements of p divided by energy gaps. The magnitude of the various matrix elements of p between two given states seems to be approximately the same for all compounds of this family and differences in the corresponding band parameters are due only to differences in energy gaps.(*) The temperature dependence of the band parameters also indicates that the matrix elements of p are temperature independent.@? *) These matrix elements do not vary much from one pair of states to another, provided they are not zero due to symmetry.(7) For future reference we show in Fig. 1 the band FIG. 1. Band structure of germanium according to BA~SANIand YOSHIMINE@~. The .I,1 state, which according structure of germanium calculated by BASSANI to this calculation lies above the I’s’ state, is actually and ~OSH~MINE(*) neglecting the spin-orbit interknown to be below I’s’. l
l
1543
1544
MANUEL
added to distinguish between conduction and valence band states when two states of the same symmetry exist within the energy range of Fig. 1. We have indicated in brackets the symmetry of the corresponding states in zincblende-type materials. In semiconductors with germanium structure, the parameters of the I’z, minimum (lowest conduction band minimum in germanium and maybe gray tin) are determined almost exclusively by the k p interaction between this state and the I?ss* state (top of the valence band). The parameters at the I’s5, state are largely determined by the interaction with l?a,, which is responsible for the light hole mass and the mass at the lower valence band split by spin-orbit interaction, and the interaction with I?lsc, which determines the heavy hole mass.(r) The k - p interaction with the l?ls, conduction band produces a small correction on the l?s5, band parameters. BRAUNSTEINand KANE@)extended the approach described above to the III-V compounds. They calculated the light hole and split-off hole masses by taking into account only the k p interaction between the valence band (rlsv for the zincblende structure) and the lYleconduction band. They used the value of the magnitude square of the matrix element of p between these states calculated from the measured electron effective masses at l?rc assuming they are only determined by the interaction with !?lsc and neglecting the interaction of the r,, with the rise band. While this interaction vanishes in the germanium structure for parity reasons, it becomes appreciable in the zincblende structure due to the lack of inversion symmetry. If one neglects the I’rsc-171cinteraction, one obtains, from the fit of the experimental electron effective masses, a value of I(rlsvlpj r,, ) 121ower than the value found for germanium. Also the decrease in the rlav-l?lc interaction, due to the mixing of the rs5’ and the I15 states when the inversion symmetrv is lifted. gives for zincblende-tvne compounds a value’ofu I( l?~~v~pIr~,)12 lowei-than for the germanium-type materials. In this paper, we shall give a method for calculating band parameters which takes into account the k p interaction of the I?lsc band with rlsv and rre, provided the E(I’1se)-E(I’~5v) gap is known. This approach is based on the assumption that the energy bands of a zincblende-type compound can be obtained from the bands of the isoelectronic l
l
l
CARDONA
germanium-type material by the action of a purely antisymmetric perturbing potentiaL(71 1s) The E(lYlse)-E(rlav) gaps of most of these materials are known from optical reflectivity measurements.(79 11-14) This approach yields only a small correction to the light hole and split-off hole masses of the III-V compounds calculated by BRAUNSTEIN and KANE,(~) but is essential for calculating these masses in the II-VI compounds because of the larger 12a51-lY~5mixing. The treatment of the r~sc-l?rsv interaction also yields an estimate of the heavy hole masses in these materials. This approach can be extended to wurtzite-type compounds if one assumes that their band structure is obtained from the band structure of the corresponding zincblende-type material by the application of a small hexagonal perturbing crystal field.(rs) Since the unit cell of wurtzite has twice as many atoms as that of zincblende, the Brillouin zone of materials crystallized in that structure occupies only one-half of the volume of the Brillouin zone of the same material crystallized in the zincblende structure. Hence at k = 0 there are twice as many energy eigenstates for wurtzite as for zincblende. The interaction with the extra states contributes to the band parameters of the top valence band and it is possible to extract from the experimental effective masses the values of the matrix elements corresponding to this interaction. During the last few years a considerable amount of information has become available about the energy gaps of semiconductors with germanium, zincblende and wurtzite structure at points other than k = 0. (See Table 1 and references.) Hence it has become possible to extend the k p approach for calculating band parameters to points other than k = 0. In this paper we describe the application of this approach to states at the L and X points of the Brillouin zone. l
2. EFFECT OF AN ANTISYMMETRIC PERTURBING POTENTIAL ON THE BAND STRUCTURE OF A GERMANIUM-TYPE MATERIAL
The effect of an antisymmetric potential on the band structure of a germanium-type material has been discussed by CALLAWAY. The antisymmetric potential combines levels of opposite parity at a given point of the Brillouin zone. By this method, one can derive the band structure of
BAND
PARAMETERS
OF
1545
SEMICONDUCTORS
Table 1. Experimental and estimated values of the various energy gaps relevant for the determination of band parameters at low temperatures (liquid nitrogen or helium) unless otherwise indicated.
l?25'-l?z, or r15v-rle
LV-LlC
r25'-r15 or
Ao
r15v-r15c
or
L3’-LB A3-Al
A1
L3v-L1e
X4-&
Or
Or x5v-XlC
Lw-L3c
PO)
Si
3.6(a)
Ge
0*89(b)
3*43(c) 3.05(r# 0)
3.7(O)
0.044(a) 0.29(t)
2.0(U)
5.5(.% Y)
4.4(C)
2.20(C)
0*18(c)
5.9&
1.32(c) 3.7(a)
0.47(c)
4.4(“, I)
4*49(C) 3.5(C)
7 .O(Z. Y)
5.3(X, Y)
3.24(c)
0.14(C)
6.9(‘s y)
5 .O@, Y)
2*99(q)
0*24(q)
6.6(X, I)
5*12(q) 4*83(q) 4.5@)
I)
a-Sn
0.08(C)
2.9(c)
GaP
2.70(d)
O.l27(u)
InP
1*34(e)
3.76@) 4.1(P)
GaAs
1.55(j)
4.2(q)
0.35(j)
InAs
0.45(C) 1.8(d)
3.9(P)
0.41(C)
2.57(q)
0*28(q)
6.4(x, Y)
AlSb
4.3(a)
0.75(e)
2.88(c)
0.40(C)
6.5(a)
GaSb
0*81(b)
3.74(q)
2.08(q)
0.47(q)
5.7(9)
4.33(q)
InSb
0.24(g)
1*87(q)
0.58(q)
5.4(P)
4.20(q)
ZnO
3.44(h)
3.45(q) 5.3(‘1
ZnS
3.94(i)
5.7(P)
O*lO(“)
5.7(a)
9.5(P* Y)
7.O(P, Y)
ZnSe
2*80(5)
7.6(a)
0.43 (C)
4.85(c)
0.35(W)
9.1(W0 I)
6.4(w, Y)
ZnTe
2*38(k)
0.91 (k)
6.9&
5.4(k)
2.58(z)
0*068(Z)
3.71(k) 5.2(P)
0.57(k)
CdS
4.82(k) 6.1(P)
CdSe
1.84(m)
6.1(p)
0.43(m)
4.9(P)
0.28(P)
9.5(P, Y)
7.6(% y)
CdTe
5 *20(“) 5.2(a)
0.81(k)
0.57(k)
6.76&
H&e
1*59(k) 0.2(n)
HgTe
?
4.10(k)
2.6(u)
O.OO87(h)
3.48(k)
3.20(k) 2.()(k)
See Section 3. LAX B., ROTH L. M. and ZWERDLING S. J. Phys. Chem. Solids 8, 311 (1959). See Ref. 7. EHRENREICH H., J. appl. Phys. Suppl. 32, 2155
3 *44(k)
y)
9.5(P0 Y) ~1
7.l(P# Y)
2.91 (c)
0.31(C)
8.3(Z)
5.49(k) 5.7(Z)
2*21(k)
0.69(c)
7.5(Z)
4.95(k)
TAUC J. and ABRAHAMA., J. Phys. Chem. Solids 20 190 (1961). See Ref. 12. See Ref. 13. CARDONAM., unpublished work. (1961). ZWERDLING S., BUTTON K. J., LAX B. and ROTH See Ref. 9. L. M., Phys. Rev. Letters 4, 173 (1960). STURGE M., Phys. Rev. 127, 768 (1962). HOBDEN M. V., J. Phys. Chem. Solids 23,821 (1962). ZWERDLING S., KLEINER W. H. and THERIAULT See Ref. 29. J. P., J. appl. Phys. Suppl. 32,2118 (1961). See Ref. 28. See Ref. 20. AVEN M., MARPLE D. T. F. and SEGALL B., J. appl. PIPER W. W., JOHNSONP. D. and MARPLE D. T. F., Phys. Suppl. 32, 2261 (1961). J. Phys. Chem. Solids 8.457 (1959). EHRENREICHH., PHILIPP H. R., and PHILLIPS J. C. C&DO~A M. and HARE&E G.; J. appl. Phys. 34, Phys. Rev. Letters 8, 59 (1952). 813 (1963). Measured at room temperature. See Ref. 14. SCOULERW. J. and WRIGHT G. B., Bull. Amer. phys. See Ref. 18. Sot. Sec. II, 8, 246 (1963). See Ref. 19. (u) HODBY J. W., Proc. Phys. Sot. Lond. 82, 324 (1963). WRIGHT G. B., STRAUSSA. J. and HARMAN T. C., Phys. Rev. 125,1534 (1962).
1546
MANUEL
a zincblende-type material with its two atoms belonging to the same row in the periodic tabIe, from the band structure af the isoelectronic group IV material. The same approach can be extended to materials whose two atoms belong to different rows of the periodic table; one must take for the band structure of the corresponding group IV material the average of the band structures of the two group IV materials belonging to the two rows of the elements in the compound.(7) At the F point, the I’s5, level of the germanium structure interacts with the I’15 level via the antisymmetric potential giving the I’lsu and l?lse zincblende levels. Neglecting the interaction with other I’15 and Fa5, levels, one can write the following H~iltonian for the interaction between r15 and Fs5,:
11 qrlj)
Hz
(
I!v-
v/ I i/
(1)
0 j
where I’ is the matrix element of the perturbing potential between Fsy and I?ls. We have taken the origin of energies to be the unperturbed Fss* state. Diagonalizing equation (1) we obtain:
(we use the subindex p for wave functions and energies of zincblende-type compounds and no subindex for germanium-type materials). Equation (I) also yields the perturbed wave functions:
tt,(rlsc) = +(n5) - y(b) (3)
tfrp(r15v) = ~(r15) + 4ow with 1
a==z
Equations
qr15e)- qr15v)i-qr15)l/z .W’m)
-
J%(&sv)
(4) give the coefficients
1
(A\
a and b if we
CARDONA
know the gaps Ep(I’lsc)-Ep(I’~sv) and E(I’1s)E(I’ss,) of the polar and the corresponding nonpolar compound. The Fe, conduction band level interacts with the I’l, and 1‘1, levels via the antisymmetric potential. Both levels are more than 10 eV away from the Fst level and therefore the admixture of l?l wave function produced on the Fe, level by the antisymmetric potential is likely to be much smaller than the corresponding effect on the rz6+-r15levels. Since no experimental information is available about the position of the l?l levels, we shall assume that the polar potential does not appreciably mix Far and I’r levels. The possible effect of this mixing will be discussed in Section 3. At the L point the antisymmetric potential mixes the L1 and Ll, states. Since all the L1, states are ‘very far away in energy from the L1, state, we assume that we have no admixture of the L1, with 4, states. The L3 and Ls, states are strongly mixed by the antisymmetric potential giving Lsc and Lsv. The admixture coefficients can be obtained from equation (4) replacing ~~(rls~)-~~(r~s~,) by ~~(Ls~)-~~(Ls~) and E(Gs) by I-~(Ls‘). At the X point the antisymmetric potential splits the Xl germanium states into Xl and Xa and mixes the X4 and Xs states giving the Xs levels of the zincblende structure. Only the position of the Xjv, Xl, and Xs, levels is known experimentally at the X point. However, due to the large X4--Xs gap in germanium (- 13 eV) no large mixing of these states by the antisymmetr~c potential is expected to occur. 3. rzc CONDUCTION BAND PARAMETERS If we neglect the interaction of Flc with 17rse, the electron effective mass at Is, or Flc is given by:(a) m
m*(6)
= l+Ps/3
1 _5+_ i Eo Eo+Ao
(5)
where P2 = (2/39n)J(rz,ip(rzs,>lz, EO is the E(I’s,)--E(Fssf) energy gap, and As the spin-orbit splitting at rz5,. For diamond-type materials, there is no interaction of Fs, with Fls. In polar materials this interaction increases the effective mass at I’lc, since the l?lsc state is normally above I’r,. Using the wave functions for the polar material derived
BAND
PARAMETBRS
in Section 2, we can calculate the term contributed to equation (5) by the I’raCinteraction:
expression for g* becomes : g*(rQ
21(rl(rZ~)(pla~~(r15)-b~,(r25~)>
(6)
=pz
2v~b~
X
P2As Wp(r15e) - .qr15di
Ep(rIfrc)- .qrlsv) + .qr15b Efr2d Eo(Eo+ A)
+ qr15) - -wh5~)
- Edr15v)i[(Ep(~15c) - Edwi
f
P is obtained from the matrix element ofp between the l?ss, and the rs, states in the corresponding non-polar compound. The interaction of Fl, with l?lsv is also weakened, in polar materials, by the admixture of I’sa, and r15 wave functions at I’lr,,,. The resulting effective mass at 1‘1, is given by: ?n ----=l+ m*(rlC)
= 2 l-
1’
37$.qG5c) - =%(b)l
~~~~~~~~- qr15v)
1547
OF SEMICONDUCTORS
PZ 2[E,(r15,) - ~~(r~~~)]
Pp(rlSe) - Ep(rlsv) + E(r15) - E(r25’) 3
Ep(h5c)- Ep(rltiv)- E(r15)+ E(r25r) (g) [wrlBc) - ~mdi2
II
Table 1 lists all energy gaps required for the evaluation of equations (7) and (9) at low temperatures (liquid nitrogen or helium depending on the av~labili~ of data; this difference in temperature is not significant for the evaluation of band parameters). For energy gaps which have not been measured at low temperatures, their low temperature values have been estimated from the room temperature gaps and the known temperature coefficients of these gaps in other materials of the family,@*
13,14)
The gaps preceded by the N sign have not been (7) observed experimentally; they have been estimated > from the general trend followed by the corresponding gaps of the nearest compounds in the family. Ep(rlJc)- Ep(r15v) - qr15) -f-E(rw,) The Es gap of silicon has not been directly observed. We have estimated EO z 3.4eV in silicon Ep(rlBe)- Ep(r15dby extrapolating the direct gap observed in the In equation (7) we have neglected the spin-orbit germanium-silicon’alloys up to 10% silicon conAnother estimate of this gap has splitting of the I’lse state since it is always much centrations. smaller than the energy difference between this been obtained by calculating, from the values of Es in GaP and ZnS, Es for the hypothetical state and rle. If we neglect the l?sr,4?15 mixing we obtain the ordered GeSi compound. This compound should following expression for the effective g-factor at have approximately the average gaps of germanium rzt or r;3,+ and silicon and hence, from the values of EO for germanium and GeSi we estimate EO w 3.8eV P2Ao for silicon. In Table 1 we have listed the average g*k(rzt) = 2 i(8) of these two estimates (Es = 3.6 eV). > 3Eo(~o+Ao) In the calculations of m*(l’,,) and g*(I’lJ we In polar materials not only the spin-orbit splitting have used the values of As given in Table 1. at l?lsv contributes to g*(rle), but also the splitting When no experimental value of As is available, at l?lsc. This last contribution, however, is quite we use for As three-halves of the spin-orbit small since the energy gap Ep(I’~sc)-Ep(~~C) is splitting at Ll,, listed under Al in Table l.(7) If much larger than E,(l?&E,(l?~~,) and the contri- neither the value of As nor that of Al has been bution to g* is inversely proportional to the square obtained experimentally, we use the values of As estimated from atomic data.(7* 9) of these gaps. The contribution of Gsc tog*(rlJ can be estimated by assuming that the spinIn Table 2 we list the estimated values of orbit splitting at rlaC is the same as at I’lav. The m*(I’I,) and g*(l?& We have taken Pz = 23 eV
isp
I
1548
MANUEL
CARDONA
for group IV and III-V materials since this value gives the best agreement between experimental and calculated data. For the II-VI compounds good effective mass determinations are not very numerous. The best known electron masses seem to be those of CdTe, CdSe and CdS. Two of these materials (CdS and CdSe) have wurtzite structure, but, as we shall see in the next Section, the band parameters at ‘PI, are the same for zincblende and wurtzite-type materials. In order to obtain good agreement with the experimental values in these materials we have taken Ps = 21 eV. This lower value of P is probably due to some admixture of I’1 wave function of the non-polar material in the I?l, state of the zincblende.
We shall take Q2 = 15.5 eV, calculated from the value of M in germanium.(a) G is 1.6 in germanium.@) The E(I’~2~)-E(r~~~) gap is about 12 eV in germanium.(l? 8) The gaps of the materials of this family (with the exception of diamond) differ from the corresponding gaps in germanium by no more than 4 eV. In order to calculate the small correction due to the E(l?~~)-E(r~~,) interaction, we assume that the corresponding gap is the same for all materials. The value of G in any compound will thus be the value for germanium (1.6) decreased by the admixture of r25, and rls states in r15u:
4. VALENCE BAND PARAME TEFG AT THE l- POINT
Table 2 lists the values of A, B and 0 that we have calculated by using equations (ll)-(14). It also lists the values of the split-off hole effective masses “I,: and the average heavy and light hole masses m,$, and rnk obtained by averaging equation (8) over all possible directions of It:
(a) III-V Compounds In these materials the shape of the valence band around k = 0 is given by:(l)
G = -1.6
E&b) - Ep(kw) +E(h) - V'25') 2[EP(r15+
EP(rlBv)i
(14)
1 -m-,4(1+&) * ?h
;E = A’kZ-Aa where A = ~(F+ZG+ZM)+-
1
1
-T-
(11)
Ca = i[(F-G+M)z-(F+2G-M)a]
1
A’=
= -A’
Ch
B = +(F+ZG--M)
+I
F represents the interaction between PI_+, and I’r,, M the interaction between l(lsv and 171scand G the interaction between rlEv and I’ls. All other interactions, including those yielding linear terms in(z) k, have been neglected. F can be obtained from P by the method of Section 3 :
and likewise : M = _ Q2[E(r&--E(r2s)12 [Ep(r~5c)-EP(r~5w)13’
(13)
The agreement of the calculated W& and m,* with the few available experimental data is quite good, somewhat better than for the estimates of BRAUNSTEINand KANE,(a) as expected. The calculated w&, does not agree so well with the experimental values for the III-V compounds. These experimental values, however, are not very reliable since all available determinations are only indirect and not very sensitive to the value of T&,. The accuracy of our estimate of n& is not very good either: T& is obtained as the difference of two large terms almost equal in magnitude (see equation (15)). (b) II-VI Com~ounrJs The amount of experimental information available about the valence band of II-VI compounds is very small. Only the transverse masses
BAND
PARAMETERS
OF
of the top valence band at I’ (I’g double group representation) in CdS and CdSe (wurtzite structure) and the transverse mass of the second highest l? band (I’7 double group representation) in CdSe are known with a certain degree of reliability.@s* 19) It is possible to derive the band parameters of the wurtzite-type materials if we assume that their band structure can be obtained from the band structure of the zincblende-type materials by the application of a small hexagonal crystal_field.(l5) In most of the materials which can crystallize in the wurtzite structure (ZnS, &Se, CdS, CdSe), the spin-orbit splitting of the I’ valence band is larger than the crystal-field splitting (N 0.02 eV). In ZnO, however, the spin-orbit splitting is smaller than the crystal-field splitting.(2a) When going from the zincblende to the wurtzite structure, the size of the unit cell doubles and hence the size of the Brillouin zone is divided by two. As a result the number of k = 0 states is doubled: a l?a (doubly degenerate point group representation) and two I’s (non-degenerate) states are added to the valence states at l?. These new states do not interact with I’1 via k lp and hence do not affect the conduction band parameters of II+ Since the crystal field splitting of the other (l?15v and I’15c) states interacting with l?lc is very small,(ls) the band parameters at l?re must be nearly the same for the zincblende and the wurtzite structures. This conclusion explains the lack of anisotropy in the experimentally observed m*(l?rt) of CdS and CdSe.@ss 1% Linear terms in k, allowed by symmetry at I’lc in wur~~te-type materials, are too small to be of any significance.@) A I’s state is known to exist about 1 eV below the I’s (double group) top valence band state. The two highest valence band states at l? (rs and l?T double group representations) belong to the r5 and I’1 point group representation. Since I’s interacts with I’5 via the transverse component of p, it is expected that the presence of I’a will affect the transverse masses at Ia, I’7 and the lower l?T valence band. This effect will be absent in zincblende-type materials. Let us neglect for a moment the I’&5 interaction. For crystal-field splittings much smaller than the spin-orbit splitting, the lowest l?15v band, split off by spin-orbit interaction, has the same parameters in wurtzite as in zincblende-tie materials (see equation (15)). The effective masses 6
SEMICONDUCTORS
1549
of the two higher I’ujv bands, split now by the crystal field, are found as the solution of a 4 x 4 k p secular equation. The problem is analogous to the problem of a group IV semiconductor under the action of a uniaxial stress in the (111) direction.@) The highest of these bands (Pa symmetry in wurtzite) has the masses: l
m P-Z? mr(rg)
_,-
m
~mT(rgJ =
*
_---F 2
G 2
5M 6
M
(16) \ ,
-l-rG
where m, is the mass in the direction of the hexagonal axis and mi perpendicular to it. The masses at the lower (r7) valence band, split from l?lsv by the crystal field, are: m --------= mz(I’,)
---=
m
F M -_1----.____-r: 6 2 2F
-1-T-3-&f.
G
5 6(17)
m:(r7) It can be easily shown that the effect of the I’,&‘5 interaction is to add to F in the expressions of m* (equations (15)-(17)) a positive term W. The longitudinal masses are not affected by this interaction. W = 3 -9 can be calculated from the experimental values of rn: (I’?) and mT(I’g) for CdSe. Since both mT_(I’,) and m:(rg) can be fitted within the experimental accuracy with this value of W (see Table Z), we conclude that any possible contribution of the core d-electrons,@*) which lie about 10 eV below rg, is negligible. Let us now treat the case of a spin-orbit splitting much smaller than the crystal-field splitting (ZnO). We shall again first assume W = 0. For a finite W, we must add it to Fin the expressions of the transverse masses. The orbital wave functions which diagonalize the Hamiltonian are :
$0
=
&x+ Y+.q
where X,
Y and 2 are the orbital wave functions
m*O;rd
m*(rd
0.07(d) 0.026(f)
0.0050
0.13
0.072
0.084
0.026
0.11
0046
0.0152
a-Sn
GaP
InP
GaAs
InAS
AlSb
GaSb
InSb
-6.1
0.4
-12
0.32
0.60
I.76
-48(&b
-31.5
_ 10.2
-4.8
- 16.5
-5.5
-6.3
-4.7
-99
-172
-5.2
talc.
-12.1
_ 6.5(p)
exp.
g*FI,j
A
-2.22
1.95
talc.
g*;,,j
&?*(rz’) g*(rz’)
0~0155(~~ -44
0.047(P)
0.073(d)
0.040
Ge
0.041(a)
0.13
Si
CdC.
exp.
m*(rz’)
or
m*(rz’)
-29.3
-8.6
-4
-15.2
-4.5
-5.3
-2.6
-95
-8.0
-1.7
talc.
B
113
16
-5
12
-1
0
9
0.016
0.053
0.12
0.03 1
0.10
0.086
0.13
0~0051
0.047
96 1160
0.12
talc.
mlh
“*’
0*012(e)
0.06ta)
0.025
0.12(d)
0*045(a)
0.16(a)
exp.
* mlh
or
m*wy) m*m)
20
talc.
c2
0.11 0.27
0.68(d) 0.41(d) 0.gw 0*3w 0.25(d)
1.0 0.82 I.1 0.71 0.53
0.12
0.16
0.21
0.18
0.4w
0.075
1 *o
rr 0.2(C)
0.25
0.10
0.19
talc.
0.22
0.35(a)
0.35
or
* m,h
0.56
0.52(a)
exp.
“4
mbh
0.44
talc.
mhh
“;
0.083(d)
0.20(*)
exp.
ma*,
Or
m*(r7) m*(r7) m*w7) m*(h)
Table 2. Estimated values of the band parameters at I?z, and I’25, (germanium structure), l?lc and l?15~ (zincblende structure) and the equivalent points for the wurtzite structure (Fl for the conduction band, Fg, l3, and !J7 for the valence band). The eflective masses are given in units of the free electron mass.
0.18
o-13
o-11
CdS
CdSe
CdTe 0.018@)
O*ll@) -34
- 0.40
0.4
1.46
1.82
0*70(m)
l-78@)
-72
-27.9
-27.8
(p)
(k) (I) (m) (n) (0)
(j)
-14
5 5
0.018
0.18 /j 0.94 _.L
0.13
0*4S L(l)
0.11
0.24 11 1.21
2.5 II 0,561 2.5 II O-421
0.42
1
0.16 N S [J(k) 0,7J_(k’
0.67
0.85 11 0*55 I
0.56
0.25 j[ 21
-1.9
0.32 j] 1.11
10
0.67 11 0.63 _I._I_
0.25
- 1.9 11 0‘671
0.76 1.
0.67 11 0.63 _I_
The Institute of Physics London (1962).
and the Physical
Society,
ings of the International Conference on the Physics of Semiconductors July 1962, Exeter, England, p. 375.
AVEN M., MARPLE D. T. F. and SECALL B., J. uppl. Phys. Suppl. 32, 2261 (1961). See Ref. 18. See Ref. 19. HOPFIELDJ, J.,f. uppl. Phys. Suppl. 32,2277 (1961). MARPLE D. T. F,, Phys. Rew. 129,2466 (1963). WRIGHT G. B., STRAUSSA. J. and HARMAN T. C., Phys. Rev. 125, 1534 (1962). JOHNSONE. J., FILINSKI I. and FAN H. Y., Proceed-
-0.2
-4.0
-2.38
-1.2
4.5
-4.3
-3.9
-2.39
-1.93
-1.71
-1.81
seeRef. 1. ROTH L. M., LAX B. and ZWERDLINGS., Phys. Rev. 114, 90 (1959). See Ref. 7. EH~REICH H., J. appl. Phys. Suppl. 32, 2155 (1961), other references given in this paper. See Ref. 9. See Ref. 5. See Ref. 6. ZWERULING S., KLEINER W. H. and THERIAULT J. P.,J. uppl. Phys. Suppl. 32, 2118 (1961). HUTSONA. R., J. appl. Phys. Suppl. 32,2287 (1961).
0*016
0.84
0.130, m)
0.17
ZnTe
HgSe
1.89
0.20(k)
o-21
ZnSe *0.1(f)
0.39
ZnS(hex.)
1.82
0.39
1.99
ZnS(cubic)
0*32(Q
O-25
ZnO
Table 2-continued
1552
MANUEL
in the absence of crystal field splitting, which transform like the coordinates x, y and x. In the tight binding limit equation (18) represents angular momentum wave functions for I = 1 and magnetic quantum number equal to 1, - 1 and 0, in the direction of the hexagonal axis. When the spin-orbit interaction is applied, the wave functions which diagonalize the k p Hamiltonian, including spin-orbit coupling, are: l
where f and j, represent spin eigenfunctions referred to the hexagonal axis. Each pair of functions in equation (19) defines a doubly degenerate valence band. The two upper bands have the same effective masses : 1 -=
-=
m,
CARDONA
decrease in the I?r&‘15c interaction produced by the I’ssf-I’rs admixture. When this interaction becomes very small, the - 1 term in the expression for some inverse masses can become dominant and the corresponding bands are curved upwards like free electron bands. Under these conditions the highest valence band maximum does not occur at the I’ point, as shown in Fig. 2. There is considerable experimental data in the literature which
f *
,IC
-1-G-M F
M
G
(20)
-‘-T-T-z
and the lower $0 band has the masses: 1 -= ml,
-1-F
(21)
1
I -=
-1-G-M
m, We have listed in Table 2 the valence band parameters for the II-VI compounds, calculated with the rules discussed above. We have calculated zincblende parameters for 263, ZnSe, ZnTe, CdTe and HgSe and wurtzite parameters for ZnO, ZnS, CdS and CdSe. We have estimated Win these materials from the value of Win CdSe, by making the reasonable assumption that W is proportional to F. A salient feature of the calculated zincblende parameters is that m& is very large, and even negative for ZnSe. K& is only an average mass and actually the heavy hole mass in the (100) direction is also negative for ZnS, CdTe and HgSe. The small contribution to M of the cl-electrons, which we have neglected, is likely to make this mass even more negative and may be also negative in ZnTe. This effect is due to the
FIG. 2. Sketch of the band structure proposed in Section IV(b) for the II-VI compounds.
can be explained using this model for the valence band. Indirect transitions have been reported to occur below the direct exciton gap in ZnTe,(s4) CdTe@@ and ZnO.@s) These transitions would occur, in our model, between the highest valence band off k = 0 and the lowest conduction band at k = 0. In HgSe and HgTe the existence of a valence band off k = 0 overlapping the conduction band, has been postulated in order to explain the variation of the Hall constant with temperature.(s’J) The overlapping valence band may well be the highest maximum off k = 0 as shown in Fig. 2. While this estimate of the heavy masses is rather crude and other interactions which we have
BAND
PARAMETERS
OF
neglected may appreciably change this estimate, it indicates clearly the possibility of large negative hole masses at the top of the valence band. These comments must be considered speculative at present; more accurate calculations, based on a direct computation of the matrix elements of p for the calculated wave functions of the material, are needed. We expect them to yield negative heavy hole masses for some, if not all the II-VI compounds. We have not been able to apply this type of approach to the I-VII compounds with zincblende and wurtzite structure, While the various band gaps required are known experimentally, the values of F and M are probably strongly modified by the admixture of Q-electrons of the metal in the l?lsv top valence band. No experimental information on effective masses is available for these materials.
5. BAND PARAMET3iRS AT r15c
The same type of analysis applies to I’lsc of the zincblende structure. Equations (10), (11) and (1s) hold provided the proper values of F, M and G are used. In group IV materials F (PIED)is determined by the interaction with the very remote l’r bands. Since we have no experimental information about these bands, it is reasonable to take F(I’lsc) = 0. In polar compounds, the interaction witb the lowest conduction band (Tic) contributes the following term to F (l?r&:
(22)
has the same magni~de as ~~~P~5~)but opposite sign. In group IV materials, G(r~s,) = 0 since all interacting bands (I’&) are very far away in energy. In polar materials G(I’lse) is determined by the interaction with the l?is conduction band. compounds. The G(r15c) = 0.3 eV for the II-VI correction due to the d-electrons in the II-VI compounds A&(l?lsc) and G#lsc) can be estimated to be the same as for the PI,, bands. X(lYm)
1553
SE~~C~N~UCTU~S
6. BAND PAR-S
AT L
The La--La gap of the germanium structure (Lae-Lsv in zincblende) is known in many semiconductors of the family we are considering (see Table 1). The Ls+Llc (or Lsv-Llc) gap is known for CdTe, ZnTe, I-IgTe(l~~ and GaAs.@*) It is also known to be 2*0 eV from band calculations in germanium.~2s) In all these materials Lau-& is only slightly smaller f - O-3 eV) than the As-& gap obtained from reflectivity measurements in almost all materials of this family (see Table 1). We can estimate the Lav-Lle gap in those materials for which it is not known by assuming it is O-3 eV smaller than the As-Al gap. The method described in the previous Sections can now be used to determine the band parameters at LsFI El, and Lse_ The longitudinal masses at Lzc and I&, are given only by the interaction between these two states and possibly (for II-VI compounds) some contribution from d-states, The calculation of these masses is equivalent to the calculation of MI in Section 4. No experimental data are available for determining the square of the matrix elements for the Lsc-La8 interaction. One can reasonably assume that it is the same as for the l?f5v-F15c interaction (155 eV). Similarly to what happens at l?rsu, the curvature at Ls+,, which is downwards for group IV and III-V materials, becomes upwards in some II-VI compounds. The transverse mass at Ll, is given almost entirely by the interaction with Lac and La,,. The square of the matrix element for the transitions is also 23 eV@f (in Ref. 7 we found 25 eV for this square matrix element since we used the A374 1 gap instead of the Ls-L1 gap). rn:(L~,) is given by equation (7) with rlsc, I’rbvrI”ss, and Eop replaced by Lsc, Law,La, and E(L$-(i&). The longitudinal mass at L1, is determined by the interaction with other L1 or Lx1 states which are very far in energy. Silence this mass must be close to the free electron mass. The transverse mass at Lse is determined by the interaction with L1, La and other Ls states. It can only be qualitatively estimated from the interaction with LI, and Lsc since the interaction with Llo, which cannot be easily estimated, contributes significantly to m,(L3,). The longitudinal effective g-factor at LI can also be estimated by this rne~od.~s~~
1554
MANUEL
7. BAND PARAIvIETERS AT X The only energy gap at X known experimentally for the materials under consideration is the X,-X,, gap (see Fig. 1). The splitting at X1,(X1-Xa) is also known for several polar materials. The X*-Xl, gap determines the transverse effective mass at Xl,, No corrections due to the admixture of X4 and Xa wave functions in polar materials can be made since no information about the position of Xs is available. It has been shown in Section 2, however, that this correction is quite small, due to the large gaps involved. The square matrix element for the X4-Xre interaction can be estimated from the value of the transverse mass at the lowest conduction band minimum in silicon.@‘) This minimum occurs in the (111) direction near X but somewhat inside the Brillouin zone, At this point the matrix elements and the energy denominators are likely to be close to the corresponding values at X. The square matrix element for the XCX~, interaction obtained by this procedure, is 19 eV. The electron effective masses have been determined in two materials which are likely to have the lowest conduction band minimum at the X point@Q: AlSb and GaP.@st From the values of the average masses determined from the Faraday rotation and m:(x;l) z 1 with the reasonable assumption (like in silicon), one obtains@) m:(X,) = 0.22 for GaP and mT(Xl) = 0.25 for AlSb. The values of m(*Xl) estimated by the method described abo& are 0.22 for GaP and 0.19 for AlSb, This fact lends further support to the hypothesis that t&se two materials have (100) lowest conduction band minima: both the Flc and the LQ minima have much smaller transverse masses.
1.
3. 4. 5. 6.
CARDONA
7, CARDONA M. and
GRJENRWAY
D.
L.,
Pkys. Rev.
125, 1291 (1962). 8. ~ASSANI F. and YOSH~M~NEM.. Phvs. Rev. 130. 20 (1963). 9. BRAUNSTEIN R. and KANE E. O., J. Phvs. Chem. I
_
Solids 23, 1423 (1962). 10. HERM~ F., J. Electron. 1, 103 (1955). H., PHILLXP~ J. C, and PHILI~P XI. R,, 11. E~ZRFSREICH Pk-ys. Ren. Letters 8, 53 (1952). 12. CARDOr%% M., P&5. Rev. 129,1068 (1963). D. L. and CARDONAM., Proceedings af 13. GWENAWAY
the International
Conference on the Physics of
Scmiconductovs July 1962, Exeter, England, p. 666. The Institute of Physics and the Physical Society, London (1962). 14. CARDONAM. and G~AWAY D. I*., Phys. Rcz~.
131, 98 (1963). 15. 16. 17. Bnnmsyz~~
R., HERMAN F: and ‘MU&E
A., Phys.
Rev. 109, 395 jl958). 28. HOPFIELD J. J. and THONEASD. G., Phys. Rev. 122, 35 (1961). 19. DKMMOCKJ. 0. and WHEELER R. G., Phys. Rev. 125,
F;%Ili1962);
J.
appl.
Phys.
Suppl.
32, 2271
20. 22. 22. 23”
THOMAS D. G., J. Phys. Ckm. Solids 15,86 (1961)). WOPFIELDJ. J., J. appl. Phys. SuppI. 32,2277 (196l). PRILLXPS J. C_, P&ys. Rec., to be published. Pncus G. E. and RrR G. L., P&s. RPU. Letters 6, 103 (195ff. 24. ATEN A. C., VAN Dcmw C. 2. and VINK A. T.,
International Conference on the Physics of Semiconductors Tuly 1962. Exeter, England, P. 696. The Institu-te of Physics and the PhysicaT &c&y, London (1962). 23. DAVB D. W. and SHILLIDAY T. S., Bys. Rev. 118, 1020 (1960). 26.
27. HARMAN T. C. and S~RAXJ~~A. J., J. a#. Whys. Su@l. 32, 2265 (1962). D. L., P&s. Rezi. Letters 9, 97 (1962). 28. GREENAW~ZY 29. Bsus~ D., PHILLIPS J. C. and BASSANIF., Phys. Rew. Letters 9, 94 (1960). progr. RIP., 30. Rora L. M.. Lincoln Lab. wart. _ _-- Jan. 15, REFERENCES p. 45 (1959). DRESSEI..SAUS G., KIP A. F. and KITTEC C., Pplys. 31. PAUL W., J_ appf. Pkyr. Suppa. 32,2Og2 (1961). 32. Moss T. S.. WALTON A. K. and ELLIS B., fnternational Conference on the Physic ofSe~.~~d~tars~ July l962, Exeter, England, p. 295. The in&t&e of Physics and the Physical Society, London ROTA L. M., Limoln La&. punrt, progr. Rep. Novem(1962): ber 1 p* 45 (1957). 33, Moss T. S., J. appl. Phys. 30, 951 (1959). EHRENREKZ H., J. appl. Phys. Su#$& 32, 2155 34” G~ovss S. and PAUL W., F!z~ls. Rev. Letters 11, 194 (1961). (1963). CARDONA M., Phys. Rev. 121, 752 (1961). M. L. A. and S’XXAD35. X~AGGULW D. M. S., ROBERTSON SMITH S, D., PIDCEON C. R. and PROSSER V., UN% R. A., Phys. Rev. .Z&ters 6, 143 (1963). Proceedings ef the InternationaE Conference on the 36. RAEFZW. S. and DF%~BR R. N., BuK Amer. phys_ Physics of Semiconductors July 1962, Exeter, Sot. Ser. If, 8, 516 (l963), England, p. 301. The Institute of Physics and the 37. SAWAMOT~ fi.,J. p&s. Sot. Japan 18, 1224 (1963). Physical Society, London (1962).
BAND
PARAMETERS
OF
Note added in proof GROWS and PAIJL@~)have recently reinterpreted the existing data for c&n in terms of a negative Es gap the band parameters (Eo x -0.2 eV). Accordingly listed for this material in Table 2 should be recalculated for this value of Eo if this interpretation is correct. BAGGULEYet aZ.(35) have determined the valence band parameters of InSb by cyclotron resonance. They obtain
SEMICONDUCTORS
1555
* m Ih = 0.021 kO.005 and tni, = 0.4, in good agreement with our calculations. Cyclotron resonance measurements have also been reported for rr(ss) and ~(~7) type CdS. For n-type material rn: (PI) = 0.175 andm: (Pi) = 0,156. While these values agree well with the estimate in this paper, the anisotropy of m*(I’l) is larger than expected. For p-type material [m*(P,)m:(P,)]i/s = 1.87@7) was obtained, in reasonable agreement with our estimates.
Pergamon
Press 1963. Vol. 24, pp. 1543-1555.
BAND PARAMETERS ZINCBLENDE,
Printed in Great Britain.
OF SEMICONDUCTORS
WURTZITE,
WITH
AND GERMANIUM
STRUCTURE MANUEL RCA Laboratories,
CARDONA Princeton,
New Jersey
(Received 10 June 1963)
Abstract-The k * p method is used for calculating the band parameters of semiconductors with germanium, zincblende, and wurtzite structure at k = 0 and at the L and X points of the zincblende structure, At k = 0, the interaction between the highest valence band and the two lowest conduction bands is considered. The energy gaps between these states are obtained from reflectivity measurements. The wave functions of a polar material are obtained from the wave functions of the isoelectronic non-polar material by the apptication of an antisymmetric perturbing potential. The matrix elements of p between two given states of the non-polar material are assumed to be the same for all materials. The band parameters of the wurtzite materials are obtained by the application of a small hexagonal crystal-field to the corresponding zincblende-type compound. It is possible by this method to account for the existing experimental data and to predict the band parameters of many materials for which they have not been measured. It is suggested that the absolute valence band maximum in several II-VI compounds may not be at the center of the Briliouin zone.
action. The states have been labelfed according to 1. INTRODUCTION THE k p method(l9 2) has been very helpful for the standard notation, with the subindices c and v interpreting the effective masses and g-factors(s) of semiconductors with germanium and zincblende structure. By applying first and second order (Lzcd -2’ perturbation theory to the k p (k is the crystal momentum, p the linear momentum operator) term in the ~amiltoni~, one can show that the inverse effective mass parameters are functions of (LX?L3 squares of matrix elements of p divided by energy gaps. The magnitude of the various matrix elements of p between two given states seems to be approximately the same for all compounds of this family and differences in the corresponding band parameters are due only to differences in energy gaps.(*) The temperature dependence of the band parameters also indicates that the matrix elements of p are temperature independent.@? *) These matrix elements do not vary much from one pair of states to another, provided they are not zero due to symmetry.(7) For future reference we show in Fig. 1 the band FIG. 1. Band structure of germanium according to BA~SANIand YOSHIMINE@~. The .I,1 state, which according structure of germanium calculated by BASSANI to this calculation lies above the I’s’ state, is actually and ~OSH~MINE(*) neglecting the spin-orbit interknown to be below I’s’. l
l
1543
1544
MANUEL
added to distinguish between conduction and valence band states when two states of the same symmetry exist within the energy range of Fig. 1. We have indicated in brackets the symmetry of the corresponding states in zincblende-type materials. In semiconductors with germanium structure, the parameters of the I’z, minimum (lowest conduction band minimum in germanium and maybe gray tin) are determined almost exclusively by the k p interaction between this state and the I?ss* state (top of the valence band). The parameters at the I’s5, state are largely determined by the interaction with l?a,, which is responsible for the light hole mass and the mass at the lower valence band split by spin-orbit interaction, and the interaction with I?lsc, which determines the heavy hole mass.(r) The k - p interaction with the l?ls, conduction band produces a small correction on the l?s5, band parameters. BRAUNSTEINand KANE@)extended the approach described above to the III-V compounds. They calculated the light hole and split-off hole masses by taking into account only the k p interaction between the valence band (rlsv for the zincblende structure) and the lYleconduction band. They used the value of the magnitude square of the matrix element of p between these states calculated from the measured electron effective masses at l?rc assuming they are only determined by the interaction with !?lsc and neglecting the interaction of the r,, with the rise band. While this interaction vanishes in the germanium structure for parity reasons, it becomes appreciable in the zincblende structure due to the lack of inversion symmetry. If one neglects the I’rsc-171cinteraction, one obtains, from the fit of the experimental electron effective masses, a value of I(rlsvlpj r,, ) 121ower than the value found for germanium. Also the decrease in the rlav-l?lc interaction, due to the mixing of the rs5’ and the I15 states when the inversion symmetrv is lifted. gives for zincblende-tvne compounds a value’ofu I( l?~~v~pIr~,)12 lowei-than for the germanium-type materials. In this paper, we shall give a method for calculating band parameters which takes into account the k p interaction of the I?lsc band with rlsv and rre, provided the E(I’1se)-E(I’~5v) gap is known. This approach is based on the assumption that the energy bands of a zincblende-type compound can be obtained from the bands of the isoelectronic l
l
l
CARDONA
germanium-type material by the action of a purely antisymmetric perturbing potentiaL(71 1s) The E(lYlse)-E(rlav) gaps of most of these materials are known from optical reflectivity measurements.(79 11-14) This approach yields only a small correction to the light hole and split-off hole masses of the III-V compounds calculated by BRAUNSTEIN and KANE,(~) but is essential for calculating these masses in the II-VI compounds because of the larger 12a51-lY~5mixing. The treatment of the r~sc-l?rsv interaction also yields an estimate of the heavy hole masses in these materials. This approach can be extended to wurtzite-type compounds if one assumes that their band structure is obtained from the band structure of the corresponding zincblende-type material by the application of a small hexagonal perturbing crystal field.(rs) Since the unit cell of wurtzite has twice as many atoms as that of zincblende, the Brillouin zone of materials crystallized in that structure occupies only one-half of the volume of the Brillouin zone of the same material crystallized in the zincblende structure. Hence at k = 0 there are twice as many energy eigenstates for wurtzite as for zincblende. The interaction with the extra states contributes to the band parameters of the top valence band and it is possible to extract from the experimental effective masses the values of the matrix elements corresponding to this interaction. During the last few years a considerable amount of information has become available about the energy gaps of semiconductors with germanium, zincblende and wurtzite structure at points other than k = 0. (See Table 1 and references.) Hence it has become possible to extend the k p approach for calculating band parameters to points other than k = 0. In this paper we describe the application of this approach to states at the L and X points of the Brillouin zone. l
2. EFFECT OF AN ANTISYMMETRIC PERTURBING POTENTIAL ON THE BAND STRUCTURE OF A GERMANIUM-TYPE MATERIAL
The effect of an antisymmetric potential on the band structure of a germanium-type material has been discussed by CALLAWAY. The antisymmetric potential combines levels of opposite parity at a given point of the Brillouin zone. By this method, one can derive the band structure of
BAND
PARAMETERS
OF
1545
SEMICONDUCTORS
Table 1. Experimental and estimated values of the various energy gaps relevant for the determination of band parameters at low temperatures (liquid nitrogen or helium) unless otherwise indicated.
l?25'-l?z, or r15v-rle
LV-LlC
r25'-r15 or
Ao
r15v-r15c
or
L3’-LB A3-Al
A1
L3v-L1e
X4-&
Or
Or x5v-XlC
Lw-L3c
PO)
Si
3.6(a)
Ge
0*89(b)
3*43(c) 3.05(r# 0)
3.7(O)
0.044(a) 0.29(t)
2.0(U)
5.5(.% Y)
4.4(C)
2.20(C)
0*18(c)
5.9&
1.32(c) 3.7(a)
0.47(c)
4.4(“, I)
4*49(C) 3.5(C)
7 .O(Z. Y)
5.3(X, Y)
3.24(c)
0.14(C)
6.9(‘s y)
5 .O@, Y)
2*99(q)
0*24(q)
6.6(X, I)
5*12(q) 4*83(q) 4.5@)
I)
a-Sn
0.08(C)
2.9(c)
GaP
2.70(d)
O.l27(u)
InP
1*34(e)
3.76@) 4.1(P)
GaAs
1.55(j)
4.2(q)
0.35(j)
InAs
0.45(C) 1.8(d)
3.9(P)
0.41(C)
2.57(q)
0*28(q)
6.4(x, Y)
AlSb
4.3(a)
0.75(e)
2.88(c)
0.40(C)
6.5(a)
GaSb
0*81(b)
3.74(q)
2.08(q)
0.47(q)
5.7(9)
4.33(q)
InSb
0.24(g)
1*87(q)
0.58(q)
5.4(P)
4.20(q)
ZnO
3.44(h)
3.45(q) 5.3(‘1
ZnS
3.94(i)
5.7(P)
O*lO(“)
5.7(a)
9.5(P* Y)
7.O(P, Y)
ZnSe
2*80(5)
7.6(a)
0.43 (C)
4.85(c)
0.35(W)
9.1(W0 I)
6.4(w, Y)
ZnTe
2*38(k)
0.91 (k)
6.9&
5.4(k)
2.58(z)
0*068(Z)
3.71(k) 5.2(P)
0.57(k)
CdS
4.82(k) 6.1(P)
CdSe
1.84(m)
6.1(p)
0.43(m)
4.9(P)
0.28(P)
9.5(P, Y)
7.6(% y)
CdTe
5 *20(“) 5.2(a)
0.81(k)
0.57(k)
6.76&
H&e
1*59(k) 0.2(n)
HgTe
?
4.10(k)
2.6(u)
O.OO87(h)
3.48(k)
3.20(k) 2.()(k)
See Section 3. LAX B., ROTH L. M. and ZWERDLING S. J. Phys. Chem. Solids 8, 311 (1959). See Ref. 7. EHRENREICH H., J. appl. Phys. Suppl. 32, 2155
3 *44(k)
y)
9.5(P0 Y) ~1
7.l(P# Y)
2.91 (c)
0.31(C)
8.3(Z)
5.49(k) 5.7(Z)
2*21(k)
0.69(c)
7.5(Z)
4.95(k)
TAUC J. and ABRAHAMA., J. Phys. Chem. Solids 20 190 (1961). See Ref. 12. See Ref. 13. CARDONAM., unpublished work. (1961). ZWERDLING S., BUTTON K. J., LAX B. and ROTH See Ref. 9. L. M., Phys. Rev. Letters 4, 173 (1960). STURGE M., Phys. Rev. 127, 768 (1962). HOBDEN M. V., J. Phys. Chem. Solids 23,821 (1962). ZWERDLING S., KLEINER W. H. and THERIAULT See Ref. 29. J. P., J. appl. Phys. Suppl. 32,2118 (1961). See Ref. 28. See Ref. 20. AVEN M., MARPLE D. T. F. and SEGALL B., J. appl. PIPER W. W., JOHNSONP. D. and MARPLE D. T. F., Phys. Suppl. 32, 2261 (1961). J. Phys. Chem. Solids 8.457 (1959). EHRENREICHH., PHILIPP H. R., and PHILLIPS J. C. C&DO~A M. and HARE&E G.; J. appl. Phys. 34, Phys. Rev. Letters 8, 59 (1952). 813 (1963). Measured at room temperature. See Ref. 14. SCOULERW. J. and WRIGHT G. B., Bull. Amer. phys. See Ref. 18. Sot. Sec. II, 8, 246 (1963). See Ref. 19. (u) HODBY J. W., Proc. Phys. Sot. Lond. 82, 324 (1963). WRIGHT G. B., STRAUSSA. J. and HARMAN T. C., Phys. Rev. 125,1534 (1962).
1546
MANUEL
a zincblende-type material with its two atoms belonging to the same row in the periodic tabIe, from the band structure af the isoelectronic group IV material. The same approach can be extended to materials whose two atoms belong to different rows of the periodic table; one must take for the band structure of the corresponding group IV material the average of the band structures of the two group IV materials belonging to the two rows of the elements in the compound.(7) At the F point, the I’s5, level of the germanium structure interacts with the I’15 level via the antisymmetric potential giving the I’lsu and l?lse zincblende levels. Neglecting the interaction with other I’15 and Fa5, levels, one can write the following H~iltonian for the interaction between r15 and Fs5,:
11 qrlj)
Hz
(
I!v-
v/ I i/
(1)
0 j
where I’ is the matrix element of the perturbing potential between Fsy and I?ls. We have taken the origin of energies to be the unperturbed Fss* state. Diagonalizing equation (1) we obtain:
(we use the subindex p for wave functions and energies of zincblende-type compounds and no subindex for germanium-type materials). Equation (I) also yields the perturbed wave functions:
tt,(rlsc) = +(n5) - y(b) (3)
tfrp(r15v) = ~(r15) + 4ow with 1
a==z
Equations
qr15e)- qr15v)i-qr15)l/z .W’m)
-
J%(&sv)
(4) give the coefficients
1
(A\
a and b if we
CARDONA
know the gaps Ep(I’lsc)-Ep(I’~sv) and E(I’1s)E(I’ss,) of the polar and the corresponding nonpolar compound. The Fe, conduction band level interacts with the I’l, and 1‘1, levels via the antisymmetric potential. Both levels are more than 10 eV away from the Fst level and therefore the admixture of l?l wave function produced on the Fe, level by the antisymmetric potential is likely to be much smaller than the corresponding effect on the rz6+-r15levels. Since no experimental information is available about the position of the l?l levels, we shall assume that the polar potential does not appreciably mix Far and I’r levels. The possible effect of this mixing will be discussed in Section 3. At the L point the antisymmetric potential mixes the L1 and Ll, states. Since all the L1, states are ‘very far away in energy from the L1, state, we assume that we have no admixture of the L1, with 4, states. The L3 and Ls, states are strongly mixed by the antisymmetric potential giving Lsc and Lsv. The admixture coefficients can be obtained from equation (4) replacing ~~(rls~)-~~(r~s~,) by ~~(Ls~)-~~(Ls~) and E(Gs) by I-~(Ls‘). At the X point the antisymmetric potential splits the Xl germanium states into Xl and Xa and mixes the X4 and Xs states giving the Xs levels of the zincblende structure. Only the position of the Xjv, Xl, and Xs, levels is known experimentally at the X point. However, due to the large X4--Xs gap in germanium (- 13 eV) no large mixing of these states by the antisymmetr~c potential is expected to occur. 3. rzc CONDUCTION BAND PARAMETERS If we neglect the interaction of Flc with 17rse, the electron effective mass at Is, or Flc is given by:(a) m
m*(6)
= l+Ps/3
1 _5+_ i Eo Eo+Ao
(5)
where P2 = (2/39n)J(rz,ip(rzs,>lz, EO is the E(I’s,)--E(Fssf) energy gap, and As the spin-orbit splitting at rz5,. For diamond-type materials, there is no interaction of Fs, with Fls. In polar materials this interaction increases the effective mass at I’lc, since the l?lsc state is normally above I’r,. Using the wave functions for the polar material derived
BAND
PARAMETBRS
in Section 2, we can calculate the term contributed to equation (5) by the I’raCinteraction:
expression for g* becomes : g*(rQ
21(rl(rZ~)(pla~~(r15)-b~,(r25~)>
(6)
=pz
2v~b~
X
P2As Wp(r15e) - .qr15di
Ep(rIfrc)- .qrlsv) + .qr15b Efr2d Eo(Eo+ A)
+ qr15) - -wh5~)
- Edr15v)i[(Ep(~15c) - Edwi
f
P is obtained from the matrix element ofp between the l?ss, and the rs, states in the corresponding non-polar compound. The interaction of Fl, with l?lsv is also weakened, in polar materials, by the admixture of I’sa, and r15 wave functions at I’lr,,,. The resulting effective mass at 1‘1, is given by: ?n ----=l+ m*(rlC)
= 2 l-
1’
37$.qG5c) - =%(b)l
~~~~~~~~- qr15v)
1547
OF SEMICONDUCTORS
PZ 2[E,(r15,) - ~~(r~~~)]
Pp(rlSe) - Ep(rlsv) + E(r15) - E(r25’) 3
Ep(h5c)- Ep(rltiv)- E(r15)+ E(r25r) (g) [wrlBc) - ~mdi2
II
Table 1 lists all energy gaps required for the evaluation of equations (7) and (9) at low temperatures (liquid nitrogen or helium depending on the av~labili~ of data; this difference in temperature is not significant for the evaluation of band parameters). For energy gaps which have not been measured at low temperatures, their low temperature values have been estimated from the room temperature gaps and the known temperature coefficients of these gaps in other materials of the family,@*
13,14)
The gaps preceded by the N sign have not been (7) observed experimentally; they have been estimated > from the general trend followed by the corresponding gaps of the nearest compounds in the family. Ep(rlJc)- Ep(r15v) - qr15) -f-E(rw,) The Es gap of silicon has not been directly observed. We have estimated EO z 3.4eV in silicon Ep(rlBe)- Ep(r15dby extrapolating the direct gap observed in the In equation (7) we have neglected the spin-orbit germanium-silicon’alloys up to 10% silicon conAnother estimate of this gap has splitting of the I’lse state since it is always much centrations. smaller than the energy difference between this been obtained by calculating, from the values of Es in GaP and ZnS, Es for the hypothetical state and rle. If we neglect the l?sr,4?15 mixing we obtain the ordered GeSi compound. This compound should following expression for the effective g-factor at have approximately the average gaps of germanium rzt or r;3,+ and silicon and hence, from the values of EO for germanium and GeSi we estimate EO w 3.8eV P2Ao for silicon. In Table 1 we have listed the average g*k(rzt) = 2 i(8) of these two estimates (Es = 3.6 eV). > 3Eo(~o+Ao) In the calculations of m*(l’,,) and g*(I’lJ we In polar materials not only the spin-orbit splitting have used the values of As given in Table 1. at l?lsv contributes to g*(rle), but also the splitting When no experimental value of As is available, at l?lsc. This last contribution, however, is quite we use for As three-halves of the spin-orbit small since the energy gap Ep(I’~sc)-Ep(~~C) is splitting at Ll,, listed under Al in Table l.(7) If much larger than E,(l?&E,(l?~~,) and the contri- neither the value of As nor that of Al has been bution to g* is inversely proportional to the square obtained experimentally, we use the values of As estimated from atomic data.(7* 9) of these gaps. The contribution of Gsc tog*(rlJ can be estimated by assuming that the spinIn Table 2 we list the estimated values of orbit splitting at rlaC is the same as at I’lav. The m*(I’I,) and g*(l?& We have taken Pz = 23 eV
isp
I
1548
MANUEL
CARDONA
for group IV and III-V materials since this value gives the best agreement between experimental and calculated data. For the II-VI compounds good effective mass determinations are not very numerous. The best known electron masses seem to be those of CdTe, CdSe and CdS. Two of these materials (CdS and CdSe) have wurtzite structure, but, as we shall see in the next Section, the band parameters at ‘PI, are the same for zincblende and wurtzite-type materials. In order to obtain good agreement with the experimental values in these materials we have taken Ps = 21 eV. This lower value of P is probably due to some admixture of I’1 wave function of the non-polar material in the I?l, state of the zincblende.
We shall take Q2 = 15.5 eV, calculated from the value of M in germanium.(a) G is 1.6 in germanium.@) The E(I’~2~)-E(r~~~) gap is about 12 eV in germanium.(l? 8) The gaps of the materials of this family (with the exception of diamond) differ from the corresponding gaps in germanium by no more than 4 eV. In order to calculate the small correction due to the E(l?~~)-E(r~~,) interaction, we assume that the corresponding gap is the same for all materials. The value of G in any compound will thus be the value for germanium (1.6) decreased by the admixture of r25, and rls states in r15u:
4. VALENCE BAND PARAME TEFG AT THE l- POINT
Table 2 lists the values of A, B and 0 that we have calculated by using equations (ll)-(14). It also lists the values of the split-off hole effective masses “I,: and the average heavy and light hole masses m,$, and rnk obtained by averaging equation (8) over all possible directions of It:
(a) III-V Compounds In these materials the shape of the valence band around k = 0 is given by:(l)
G = -1.6
E&b) - Ep(kw) +E(h) - V'25') 2[EP(r15+
EP(rlBv)i
(14)
1 -m-,4(1+&) * ?h
;E = A’kZ-Aa where A = ~(F+ZG+ZM)+-
1
1
-T-
(11)
Ca = i[(F-G+M)z-(F+2G-M)a]
1
A’=
= -A’
Ch
B = +(F+ZG--M)
+I
F represents the interaction between PI_+, and I’r,, M the interaction between l(lsv and 171scand G the interaction between rlEv and I’ls. All other interactions, including those yielding linear terms in(z) k, have been neglected. F can be obtained from P by the method of Section 3 :
and likewise : M = _ Q2[E(r&--E(r2s)12 [Ep(r~5c)-EP(r~5w)13’
(13)
The agreement of the calculated W& and m,* with the few available experimental data is quite good, somewhat better than for the estimates of BRAUNSTEINand KANE,(a) as expected. The calculated w&, does not agree so well with the experimental values for the III-V compounds. These experimental values, however, are not very reliable since all available determinations are only indirect and not very sensitive to the value of T&,. The accuracy of our estimate of n& is not very good either: T& is obtained as the difference of two large terms almost equal in magnitude (see equation (15)). (b) II-VI Com~ounrJs The amount of experimental information available about the valence band of II-VI compounds is very small. Only the transverse masses
BAND
PARAMETERS
OF
of the top valence band at I’ (I’g double group representation) in CdS and CdSe (wurtzite structure) and the transverse mass of the second highest l? band (I’7 double group representation) in CdSe are known with a certain degree of reliability.@s* 19) It is possible to derive the band parameters of the wurtzite-type materials if we assume that their band structure can be obtained from the band structure of the zincblende-type materials by the application of a small hexagonal crystal_field.(l5) In most of the materials which can crystallize in the wurtzite structure (ZnS, &Se, CdS, CdSe), the spin-orbit splitting of the I’ valence band is larger than the crystal-field splitting (N 0.02 eV). In ZnO, however, the spin-orbit splitting is smaller than the crystal-field splitting.(2a) When going from the zincblende to the wurtzite structure, the size of the unit cell doubles and hence the size of the Brillouin zone is divided by two. As a result the number of k = 0 states is doubled: a l?a (doubly degenerate point group representation) and two I’s (non-degenerate) states are added to the valence states at l?. These new states do not interact with I’1 via k lp and hence do not affect the conduction band parameters of II+ Since the crystal field splitting of the other (l?15v and I’15c) states interacting with l?lc is very small,(ls) the band parameters at l?re must be nearly the same for the zincblende and the wurtzite structures. This conclusion explains the lack of anisotropy in the experimentally observed m*(l?rt) of CdS and CdSe.@ss 1% Linear terms in k, allowed by symmetry at I’lc in wur~~te-type materials, are too small to be of any significance.@) A I’s state is known to exist about 1 eV below the I’s (double group) top valence band state. The two highest valence band states at l? (rs and l?T double group representations) belong to the r5 and I’1 point group representation. Since I’s interacts with I’5 via the transverse component of p, it is expected that the presence of I’a will affect the transverse masses at Ia, I’7 and the lower l?T valence band. This effect will be absent in zincblende-type materials. Let us neglect for a moment the I’&5 interaction. For crystal-field splittings much smaller than the spin-orbit splitting, the lowest l?15v band, split off by spin-orbit interaction, has the same parameters in wurtzite as in zincblende-tie materials (see equation (15)). The effective masses 6
SEMICONDUCTORS
1549
of the two higher I’ujv bands, split now by the crystal field, are found as the solution of a 4 x 4 k p secular equation. The problem is analogous to the problem of a group IV semiconductor under the action of a uniaxial stress in the (111) direction.@) The highest of these bands (Pa symmetry in wurtzite) has the masses: l
m P-Z? mr(rg)
_,-
m
~mT(rgJ =
*
_---F 2
G 2
5M 6
M
(16) \ ,
-l-rG
where m, is the mass in the direction of the hexagonal axis and mi perpendicular to it. The masses at the lower (r7) valence band, split from l?lsv by the crystal field, are: m --------= mz(I’,)
---=
m
F M -_1----.____-r: 6 2 2F
-1-T-3-&f.
G
5 6(17)
m:(r7) It can be easily shown that the effect of the I’,&‘5 interaction is to add to F in the expressions of m* (equations (15)-(17)) a positive term W. The longitudinal masses are not affected by this interaction. W = 3 -9 can be calculated from the experimental values of rn: (I’?) and mT(I’g) for CdSe. Since both mT_(I’,) and m:(rg) can be fitted within the experimental accuracy with this value of W (see Table Z), we conclude that any possible contribution of the core d-electrons,@*) which lie about 10 eV below rg, is negligible. Let us now treat the case of a spin-orbit splitting much smaller than the crystal-field splitting (ZnO). We shall again first assume W = 0. For a finite W, we must add it to Fin the expressions of the transverse masses. The orbital wave functions which diagonalize the Hamiltonian are :
$0
=
&x+ Y+.q
where X,
Y and 2 are the orbital wave functions
m*O;rd
m*(rd
0.07(d) 0.026(f)
0.0050
0.13
0.072
0.084
0.026
0.11
0046
0.0152
a-Sn
GaP
InP
GaAs
InAS
AlSb
GaSb
InSb
-6.1
0.4
-12
0.32
0.60
I.76
-48(&b
-31.5
_ 10.2
-4.8
- 16.5
-5.5
-6.3
-4.7
-99
-172
-5.2
talc.
-12.1
_ 6.5(p)
exp.
g*FI,j
A
-2.22
1.95
talc.
g*;,,j
&?*(rz’) g*(rz’)
0~0155(~~ -44
0.047(P)
0.073(d)
0.040
Ge
0.041(a)
0.13
Si
CdC.
exp.
m*(rz’)
or
m*(rz’)
-29.3
-8.6
-4
-15.2
-4.5
-5.3
-2.6
-95
-8.0
-1.7
talc.
B
113
16
-5
12
-1
0
9
0.016
0.053
0.12
0.03 1
0.10
0.086
0.13
0~0051
0.047
96 1160
0.12
talc.
mlh
“*’
0*012(e)
0.06ta)
0.025
0.12(d)
0*045(a)
0.16(a)
exp.
* mlh
or
m*wy) m*m)
20
talc.
c2
0.11 0.27
0.68(d) 0.41(d) 0.gw 0*3w 0.25(d)
1.0 0.82 I.1 0.71 0.53
0.12
0.16
0.21
0.18
0.4w
0.075
1 *o
rr 0.2(C)
0.25
0.10
0.19
talc.
0.22
0.35(a)
0.35
or
* m,h
0.56
0.52(a)
exp.
“4
mbh
0.44
talc.
mhh
“;
0.083(d)
0.20(*)
exp.
ma*,
Or
m*(r7) m*(r7) m*w7) m*(h)
Table 2. Estimated values of the band parameters at I?z, and I’25, (germanium structure), l?lc and l?15~ (zincblende structure) and the equivalent points for the wurtzite structure (Fl for the conduction band, Fg, l3, and !J7 for the valence band). The eflective masses are given in units of the free electron mass.
0.18
o-13
o-11
CdS
CdSe
CdTe 0.018@)
O*ll@) -34
- 0.40
0.4
1.46
1.82
0*70(m)
l-78@)
-72
-27.9
-27.8
(p)
(k) (I) (m) (n) (0)
(j)
-14
5 5
0.018
0.18 /j 0.94 _.L
0.13
0*4S L(l)
0.11
0.24 11 1.21
2.5 II 0,561 2.5 II O-421
0.42
1
0.16 N S [J(k) 0,7J_(k’
0.67
0.85 11 0*55 I
0.56
0.25 j[ 21
-1.9
0.32 j] 1.11
10
0.67 11 0.63 _I._I_
0.25
- 1.9 11 0‘671
0.76 1.
0.67 11 0.63 _I_
The Institute of Physics London (1962).
and the Physical
Society,
ings of the International Conference on the Physics of Semiconductors July 1962, Exeter, England, p. 375.
AVEN M., MARPLE D. T. F. and SECALL B., J. uppl. Phys. Suppl. 32, 2261 (1961). See Ref. 18. See Ref. 19. HOPFIELDJ, J.,f. uppl. Phys. Suppl. 32,2277 (1961). MARPLE D. T. F,, Phys. Rew. 129,2466 (1963). WRIGHT G. B., STRAUSSA. J. and HARMAN T. C., Phys. Rev. 125, 1534 (1962). JOHNSONE. J., FILINSKI I. and FAN H. Y., Proceed-
-0.2
-4.0
-2.38
-1.2
4.5
-4.3
-3.9
-2.39
-1.93
-1.71
-1.81
seeRef. 1. ROTH L. M., LAX B. and ZWERDLINGS., Phys. Rev. 114, 90 (1959). See Ref. 7. EH~REICH H., J. appl. Phys. Suppl. 32, 2155 (1961), other references given in this paper. See Ref. 9. See Ref. 5. See Ref. 6. ZWERULING S., KLEINER W. H. and THERIAULT J. P.,J. uppl. Phys. Suppl. 32, 2118 (1961). HUTSONA. R., J. appl. Phys. Suppl. 32,2287 (1961).
0*016
0.84
0.130, m)
0.17
ZnTe
HgSe
1.89
0.20(k)
o-21
ZnSe *0.1(f)
0.39
ZnS(hex.)
1.82
0.39
1.99
ZnS(cubic)
0*32(Q
O-25
ZnO
Table 2-continued
1552
MANUEL
in the absence of crystal field splitting, which transform like the coordinates x, y and x. In the tight binding limit equation (18) represents angular momentum wave functions for I = 1 and magnetic quantum number equal to 1, - 1 and 0, in the direction of the hexagonal axis. When the spin-orbit interaction is applied, the wave functions which diagonalize the k p Hamiltonian, including spin-orbit coupling, are: l
where f and j, represent spin eigenfunctions referred to the hexagonal axis. Each pair of functions in equation (19) defines a doubly degenerate valence band. The two upper bands have the same effective masses : 1 -=
-=
m,
CARDONA
decrease in the I?r&‘15c interaction produced by the I’ssf-I’rs admixture. When this interaction becomes very small, the - 1 term in the expression for some inverse masses can become dominant and the corresponding bands are curved upwards like free electron bands. Under these conditions the highest valence band maximum does not occur at the I’ point, as shown in Fig. 2. There is considerable experimental data in the literature which
f *
,IC
-1-G-M F
M
G
(20)
-‘-T-T-z
and the lower $0 band has the masses: 1 -= ml,
-1-F
(21)
1
I -=
-1-G-M
m, We have listed in Table 2 the valence band parameters for the II-VI compounds, calculated with the rules discussed above. We have calculated zincblende parameters for 263, ZnSe, ZnTe, CdTe and HgSe and wurtzite parameters for ZnO, ZnS, CdS and CdSe. We have estimated Win these materials from the value of Win CdSe, by making the reasonable assumption that W is proportional to F. A salient feature of the calculated zincblende parameters is that m& is very large, and even negative for ZnSe. K& is only an average mass and actually the heavy hole mass in the (100) direction is also negative for ZnS, CdTe and HgSe. The small contribution to M of the cl-electrons, which we have neglected, is likely to make this mass even more negative and may be also negative in ZnTe. This effect is due to the
FIG. 2. Sketch of the band structure proposed in Section IV(b) for the II-VI compounds.
can be explained using this model for the valence band. Indirect transitions have been reported to occur below the direct exciton gap in ZnTe,(s4) CdTe@@ and ZnO.@s) These transitions would occur, in our model, between the highest valence band off k = 0 and the lowest conduction band at k = 0. In HgSe and HgTe the existence of a valence band off k = 0 overlapping the conduction band, has been postulated in order to explain the variation of the Hall constant with temperature.(s’J) The overlapping valence band may well be the highest maximum off k = 0 as shown in Fig. 2. While this estimate of the heavy masses is rather crude and other interactions which we have
BAND
PARAMETERS
OF
neglected may appreciably change this estimate, it indicates clearly the possibility of large negative hole masses at the top of the valence band. These comments must be considered speculative at present; more accurate calculations, based on a direct computation of the matrix elements of p for the calculated wave functions of the material, are needed. We expect them to yield negative heavy hole masses for some, if not all the II-VI compounds. We have not been able to apply this type of approach to the I-VII compounds with zincblende and wurtzite structure, While the various band gaps required are known experimentally, the values of F and M are probably strongly modified by the admixture of Q-electrons of the metal in the l?lsv top valence band. No experimental information on effective masses is available for these materials.
5. BAND PARAMET3iRS AT r15c
The same type of analysis applies to I’lsc of the zincblende structure. Equations (10), (11) and (1s) hold provided the proper values of F, M and G are used. In group IV materials F (PIED)is determined by the interaction with the very remote l’r bands. Since we have no experimental information about these bands, it is reasonable to take F(I’lsc) = 0. In polar compounds, the interaction witb the lowest conduction band (Tic) contributes the following term to F (l?r&:
(22)
has the same magni~de as ~~~P~5~)but opposite sign. In group IV materials, G(r~s,) = 0 since all interacting bands (I’&) are very far away in energy. In polar materials G(I’lse) is determined by the interaction with the l?is conduction band. compounds. The G(r15c) = 0.3 eV for the II-VI correction due to the d-electrons in the II-VI compounds A&(l?lsc) and G#lsc) can be estimated to be the same as for the PI,, bands. X(lYm)
1553
SE~~C~N~UCTU~S
6. BAND PAR-S
AT L
The La--La gap of the germanium structure (Lae-Lsv in zincblende) is known in many semiconductors of the family we are considering (see Table 1). The Ls+Llc (or Lsv-Llc) gap is known for CdTe, ZnTe, I-IgTe(l~~ and GaAs.@*) It is also known to be 2*0 eV from band calculations in germanium.~2s) In all these materials Lau-& is only slightly smaller f - O-3 eV) than the As-& gap obtained from reflectivity measurements in almost all materials of this family (see Table 1). We can estimate the Lav-Lle gap in those materials for which it is not known by assuming it is O-3 eV smaller than the As-Al gap. The method described in the previous Sections can now be used to determine the band parameters at LsFI El, and Lse_ The longitudinal masses at Lzc and I&, are given only by the interaction between these two states and possibly (for II-VI compounds) some contribution from d-states, The calculation of these masses is equivalent to the calculation of MI in Section 4. No experimental data are available for determining the square of the matrix elements for the Lsc-La8 interaction. One can reasonably assume that it is the same as for the l?f5v-F15c interaction (155 eV). Similarly to what happens at l?rsu, the curvature at Ls+,, which is downwards for group IV and III-V materials, becomes upwards in some II-VI compounds. The transverse mass at Ll, is given almost entirely by the interaction with Lac and La,,. The square of the matrix element for the transitions is also 23 eV@f (in Ref. 7 we found 25 eV for this square matrix element since we used the A374 1 gap instead of the Ls-L1 gap). rn:(L~,) is given by equation (7) with rlsc, I’rbvrI”ss, and Eop replaced by Lsc, Law,La, and E(L$-(i&). The longitudinal mass at L1, is determined by the interaction with other L1 or Lx1 states which are very far in energy. Silence this mass must be close to the free electron mass. The transverse mass at Lse is determined by the interaction with L1, La and other Ls states. It can only be qualitatively estimated from the interaction with LI, and Lsc since the interaction with Llo, which cannot be easily estimated, contributes significantly to m,(L3,). The longitudinal effective g-factor at LI can also be estimated by this rne~od.~s~~
1554
MANUEL
7. BAND PARAIvIETERS AT X The only energy gap at X known experimentally for the materials under consideration is the X,-X,, gap (see Fig. 1). The splitting at X1,(X1-Xa) is also known for several polar materials. The X*-Xl, gap determines the transverse effective mass at Xl,, No corrections due to the admixture of X4 and Xa wave functions in polar materials can be made since no information about the position of Xs is available. It has been shown in Section 2, however, that this correction is quite small, due to the large gaps involved. The square matrix element for the X4-Xre interaction can be estimated from the value of the transverse mass at the lowest conduction band minimum in silicon.@‘) This minimum occurs in the (111) direction near X but somewhat inside the Brillouin zone, At this point the matrix elements and the energy denominators are likely to be close to the corresponding values at X. The square matrix element for the XCX~, interaction obtained by this procedure, is 19 eV. The electron effective masses have been determined in two materials which are likely to have the lowest conduction band minimum at the X point@Q: AlSb and GaP.@st From the values of the average masses determined from the Faraday rotation and m:(x;l) z 1 with the reasonable assumption (like in silicon), one obtains@) m:(X,) = 0.22 for GaP and mT(Xl) = 0.25 for AlSb. The values of m(*Xl) estimated by the method described abo& are 0.22 for GaP and 0.19 for AlSb, This fact lends further support to the hypothesis that t&se two materials have (100) lowest conduction band minima: both the Flc and the LQ minima have much smaller transverse masses.
1.
3. 4. 5. 6.
CARDONA
7, CARDONA M. and
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Solids 23, 1423 (1962). 10. HERM~ F., J. Electron. 1, 103 (1955). H., PHILLXP~ J. C, and PHILI~P XI. R,, 11. E~ZRFSREICH Pk-ys. Ren. Letters 8, 53 (1952). 12. CARDOr%% M., P&5. Rev. 129,1068 (1963). D. L. and CARDONAM., Proceedings af 13. GWENAWAY
the International
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Scmiconductovs July 1962, Exeter, England, p. 666. The Institute of Physics and the Physical Society, London (1962). 14. CARDONAM. and G~AWAY D. I*., Phys. Rcz~.
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International Conference on the Physics of Semiconductors Tuly 1962. Exeter, England, P. 696. The Institu-te of Physics and the PhysicaT &c&y, London (1962). 23. DAVB D. W. and SHILLIDAY T. S., Bys. Rev. 118, 1020 (1960). 26.
27. HARMAN T. C. and S~RAXJ~~A. J., J. a#. Whys. Su@l. 32, 2265 (1962). D. L., P&s. Rezi. Letters 9, 97 (1962). 28. GREENAW~ZY 29. Bsus~ D., PHILLIPS J. C. and BASSANIF., Phys. Rew. Letters 9, 94 (1960). progr. RIP., 30. Rora L. M.. Lincoln Lab. wart. _ _-- Jan. 15, REFERENCES p. 45 (1959). DRESSEI..SAUS G., KIP A. F. and KITTEC C., Pplys. 31. PAUL W., J_ appf. Pkyr. Suppa. 32,2Og2 (1961). 32. Moss T. S.. WALTON A. K. and ELLIS B., fnternational Conference on the Physic ofSe~.~~d~tars~ July l962, Exeter, England, p. 295. The in&t&e of Physics and the Physical Society, London ROTA L. M., Limoln La&. punrt, progr. Rep. Novem(1962): ber 1 p* 45 (1957). 33, Moss T. S., J. appl. Phys. 30, 951 (1959). EHRENREKZ H., J. appl. Phys. Su#$& 32, 2155 34” G~ovss S. and PAUL W., F!z~ls. Rev. Letters 11, 194 (1961). (1963). CARDONA M., Phys. Rev. 121, 752 (1961). M. L. A. and S’XXAD35. X~AGGULW D. M. S., ROBERTSON SMITH S, D., PIDCEON C. R. and PROSSER V., UN% R. A., Phys. Rev. .Z&ters 6, 143 (1963). Proceedings ef the InternationaE Conference on the 36. RAEFZW. S. and DF%~BR R. N., BuK Amer. phys_ Physics of Semiconductors July 1962, Exeter, Sot. Ser. If, 8, 516 (l963), England, p. 301. The Institute of Physics and the 37. SAWAMOT~ fi.,J. p&s. Sot. Japan 18, 1224 (1963). Physical Society, London (1962).
BAND
PARAMETERS
OF
Note added in proof GROWS and PAIJL@~)have recently reinterpreted the existing data for c&n in terms of a negative Es gap the band parameters (Eo x -0.2 eV). Accordingly listed for this material in Table 2 should be recalculated for this value of Eo if this interpretation is correct. BAGGULEYet aZ.(35) have determined the valence band parameters of InSb by cyclotron resonance. They obtain
SEMICONDUCTORS
1555
* m Ih = 0.021 kO.005 and tni, = 0.4, in good agreement with our calculations. Cyclotron resonance measurements have also been reported for rr(ss) and ~(~7) type CdS. For n-type material rn: (PI) = 0.175 andm: (Pi) = 0,156. While these values agree well with the estimate in this paper, the anisotropy of m*(I’l) is larger than expected. For p-type material [m*(P,)m:(P,)]i/s = 1.87@7) was obtained, in reasonable agreement with our estimates.