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Nonparabolicity in the conduction band of II–VI semiconductors
~
0038-1098/93 $6.00+.00 Pergamon Press Ltd
Solid State Communications, Vol. 87, No. 2, PP. 81-84, 1993. Printed in Great Britain.
NONPARABOLICITY
IN THE CONDUCTION
B A N D O F IIoVI S E M I C O N D U C T O R S
H. Mayer and U. RSssler Institut fiir Theoretische Physik, Universit~it Regensburg, D-8400 Regensburg, Germany
(Received 21 April 1993, acceptedfor publication 6 May 1993 by P.H. Dedericbs)
The details of the conduction band structure (spin-splitting, nonparabolicity, and warping) for the II-VI semiconductors ZnTe, ZnSe, and CdTe are studied using an effective 2×2 k . p (or "envelon") Hamiltonian, which contains terms up to fourth order in the electron momentum operator. On the basis of known parameter sets for the II-VI compounds we make predictions for the spin-splitting of the conduction band and the anisotropy and spin-splitting of cyclotron resonance. The magnitude of these effects is compared with those obtained for GaAs and InP.
INTRODUCTION
The parameters which enter the multiband k . p model (energy gaps, Luttinger parameters, effective mass and g-factor) are available from a variety of independent experiments (exciton binding energies, polariton dispersion, cyclotron resonance of holes and electrons, impurity data etc.). 14 Though there is a larger uncertainty in the parameter sets for the II-VI compounds than in those for GaAs and InP, we believe that they are accurate enough to allow some reasonable predictions for the details of the conduction band structure.
The nonparabolicity and anisotropy of the conduction band of the III-V compounds GaAs and InP has been demonstrated in a series of cyclotron resonance experiments 1-3 and measurements of the inversionasymmetry induced spin-splittinga-8. The direct-gap IIVI semiconductors ZnTe, ZnSe, and CdTe are expected to show qualitatively the same effects. It is the purpose of the present paper to give a quantitative estimate of the nonparabolicity effects in these materials. A widely used method to describe the band structure of semiconductors in the vicinity of band extrema is k . p theory 7's. This method allows to separate symmetry properties from material-specific properties and may easily be extended to include also the effects of a magnetic field. The s-antibonding conduction band (F6¢) in direct gap zincblende semiconductors is only twofold (spin-) degenerate at the F-point and therefore shows naturally less structure at the Brillouin zone center than the fourfold degenerate p-antibonding valence band (Fs.). While the complexity of the latter is described by the Luttinger Hamiltonian 7, a description of the conduction band at the same level of k . p theory yields a simple spin-degenerate parabolic band. There are two approaches to extend this simple model for the conduction band. First, the conduction band can be embedded in a larger set of bands, with the k.p-coupling between these bands taken into account explicitly, thus leading to a multiband k . p model description 9--11. On the other hand, the 2 × 2 k . p model for the conduction band itself can be extended to include terms of higher order than quadratic in k. 12,13 Both extensions are capable to describe phenomena like spin splitting, nonparabolicity and band warping. Moreover, quasidegenerate perturbation theory (LSwdin partitioning) yields expressions for the 2 × 2 model parameters in terms of the multiband parameters, thus relating both models to each other n,13.
k-p D E S C R I P T I O N OF T H E C O N D U C T I O N BAND: T H E E N V E L O N M O D E L Considering the basis of multiband k . p theory it is clear that the accuracy of a k - p model is determined in principal by the number of bands contained in the set of quasidegenerate basis states. It has been shown that at least a five-band 14 × 14 model is required for a satisfactory description of the nonparabolicity of the F6c conduction band of zinc-blende type semiconductors. TM In this model the set of quasidegenerate bands consists of the FT~@ F8~ p-bonding valence bands, the F6~ s-antibonding conduction band and the FT¢ @ Fsc p-antibonding conduction bands. A detailed dcscription of this model is given in Ref. 11. The material specific parameters used in this model are momentum matrix elements, band gaps at the F-point, Luttinger parameters, effective mass and g-factors and the off-diagonal spin-orbit coupling 15 A - . An alternative approach for the description of the conduction band has been introduced by Ogg 12. He formulated a 2 × 2 Hamiltonian including terms up to fourth order in k. As will be shown below, this model is equally sufficient to describe details of the conduction band structure at the F-point with the required accuracy. Its invariant expansion T M reads 81
H-VI S E M I C O N D U C T O R S
82
Vol. 87, No. 2 lOOq 161 ,
TABLE Table 1. Input parameter ]or the 14 x 14 model, taken from Ref. 14 E0 A0 E~ A~ A-/i m* g* 3,L .yL "y3 L
(eV) (eV) (eV) (eV) (eV) (m0)
q C C' P (meV)t) P~/i (meV/~) Q (meVA)
InP ZnTe 1.423 2.394 0.110 0.970 4.720 4.820 0.070 0.450 -0.130 0.0 0.0803 0.122 1.26 -0.40 4.95 3.90 1.65 0.83 2.35 1.30 0.97 0.14 0.0 0.001 -2.0 0.0 -0.02 0.0 8.850 9.964 2.866 5.221 7.216 7.247
ZnSe CdTe 2.820 1.606 0.403 0.949 7.330 5.360 0.090 0.250 0.0 0.0 0.160 0.090 1.06 -1.77 4.30 5.30 1.14 1.70 1.84 2.00 0.20 0.61 0.0 0.04 0.0 0.0 0.0 0.0 10.628 9.496 9.165 6.463 9.845 7.873
, .,.//
//
[111l , I ,
,
(2,~/a)~
/ k~" 0.0os 14
~12
GaAs 1.519 0.341 4.488 0.171 -0.05 0.0665 -0.44 6.85 2.10 2.90 1.20 0.01 -1.878 -0.02 10.493 4.780 8.165
,
~
-
-
ZlOl , -
_
x,I
, _ ~
/
____
~-'~k 2" 0.003 (2~la) z. ___
,.--k 2- 0.002 (2~/a) 2 2
L__
0
,,---k =- 0.001 (2xla) z ,~0 2'o
3Jo :0
5'0 s~o 7'0 (deg)
s~o 90
Fig. 1. Comparison of the conduction band anisotropy in ZnTe for k II ( 1 / v ~ s i n O, 1/v/2sin O, cos O) obtained from the 14 × 14 ( - - ) and envelon (- - -) model.
persion up to fourth order in k. From this point of view it is clear that the dispersions obtained from both models coincide as k ~ 0. 5
= E a': A=2
5
+ E a4, E A=I
(1)
t=z,y,z
where at are the Pauli spin matrices and K:(~'x)* are irreducible tensor components formulated as polynomials in the components of k. The explicit form of the latter is given in Tab. 1 of Ref. 13. The expansion coefficients a~,~ can be related to the 14 x 14 model parameters, if an analytical block diagonalization (LSwdin partitioning) of the 14 × 14 model is performed TM. For the parameters al,2 and a4,1 this procedure reproduces the formulae of Hermann and Weisbuch 16 for the effective mass and the effective g-factor in terms of the energy gaps and momentum matrix elements (plus additional terms induced by the off-diagonal spin-orbit coupling A - ) . Formally the 2 × 2 Hamiltonian describes a spin 1/2 particle with nonparabolic energy-momentum relation, which is intimately related to the original multi-band envelope function description of the conduction electron. The new "quasi-particle" described by the effective 2 x 2 conduction-band Hamiltonian will therefore further be referred to as "envelon". In order to demonstrate the accuracy of the envelon model with respect to the 14 × 14 model we examine the band structure E(k) for ZnTe in the case B = 0. A comparison of the nonparabolicities li2k2/2m * - E(k), calculated with the 14× 14 and the envelon model, respectively, is given in Fig. 1. We find that the error made by using the envelon model instead of the 14 x 14 model is less than 10 % of the conduction-band anisotropy for the range of k out to 7 % of the Brillouin zone. The dispersion obtained from the envelon model may also be understood as power expansion of the 14 x 14 model dis-
INVERSION-ASYMMETRY INDUCED SPIN-SPLITTING The lowest-order nonparabolic term in the envelon model is the term weighted by the coefficient a4,2. This term represents the inversion-asymmetry induced zerofield spin-splitting 6spin of the conduction band: 2 2 6spin= 2a4,2 [k 2 (kxky 2 2 + k z 2k x2) _ -'-=-r-zj ~k2k2k2]l/2 • (2) + kuk~
This splitting is experimentally accessible by investigation of the conduction electron spin-relaxation. 4-6 Using the parameter set given in Tab. , we find for GaAs la421 = 19.8 eV/~ a, which is in good agreement with the experimental data la421 = 20.9 eV• s (Ref. 4) and la421 = 24.5 eV/~ 3 (Ref. 5) . For InP we use A - = --130 meV, which is close to the LCAO value A - = - 1 6 0 meV of Cardona et al. 17, and obtain la421 = 9.5 eVA 3, which is within the experimental range la421 = 7.26... 9.5 eV/~ s. The spin-splitting parameter a4,2 is that which is most sensitively dependent on the off-diagonal spin-splitting A - . Because no values for A - are known for the II-VI compounds it is therefore not possible to make reasonable predictions for a4,2. Instead, we give the A - dependence of a4,2, which can be used for an experimental determination of A - from a measurement of a4,2. For ZnTe we find a4,2 = [34.6 + 59.3 x A-/(ieV)] eV~ 3, for ZnSe a4,2 = [14.3 + 53.2 × A-/(ieV)] eV/~3 and for CdTe a4,2 = [43.9 + 73.8 x A-/(ieV)] eV/~ 3. It should be noted that generally only the absolute value of a4,2 can be determined, thus leading to two possible values for A - . Additional information is needed to decide which of these values is preferable.
II-VI SEMICONDUCTORS
Vol. 87, No. 2
100
S P I N - S P L I T T I N G A N D A N I S O T R O P Y OF CYCLOTRON RESONANCE
83 ,
,
,
,
,
90
The nonparabolicity and anisotropy of the conduction band leads to an anisotropy shift of the cyclotron resonance peak position and to a spin-splitting of the resonance line. These effects have been measured and quantitatively explained for the III-V semiconductors GaAs and InP. 1-3'11 To our knowledge no measurements of these effects have been done so far for the II-VI compounds ZnTe, ZnSe, and CdTe. It is possible, however, to estimate the size of the anisotropy shift and the spin-splitting for these materials using the enveton model and the parameters of T a b . . The off-diagonal spin-orbit coupling A - between the p-bonding valence bands and the p-antibonding conduction bands has been neglected in the calculations for the II-V semiconductors. In order to calculate the Landau-level ladders for a given magnetic field we expand the envelon spinorfunction into harmonic-oscillator eigenfunctions and diagonalize the resulting Hamilton matrix. We performed this calculation for magnetic fields in [001] and [111] directions. For the latter it is known that the spin-splitting and anisotropy shift are maximal for fixed magnitude of the magnetic field.T M The resulting spin-splittings for B II [001] and anisotropy shifts for the [111] resonance field with respect to the [001] resonance for GaAs, InP, ZnTe, ZnSe, and CdTe are shown in Figs. 2 and 3. The quantitative agreement with experimental values for GaAs and InP demonstrates the accuracy of the k . p models and the parameter sets. A simple rule for the magnitude of the spin-splittings
150
I
I
I
120 110
/
[~'100 ,
9O
/
,/ /
80
/
70
/
I
/
/
/ /
/
/
/~Y//
/~/
ZnSe
,.~
,'/
11~
~
200
6O ..l::
.~
~
300
4110
500
t//'a s
~, 50 O. O
o 4o t-03 30
"" CdTe 20 10
100
200
300
400
I 500
600
(transition energy) 2 (meV 2) Fig. 3. Calculated anisotropy shift B l u - B001 of cyclotron resonance for GaAs, InP, ZnTe, ZnSe, and CdTe vs. square of transition energy. Experimental values: O, GaAs (Ref. 3). can be deduced from Fig. 2. The spin-splitting is closely related to the strength of the spin-orbit coupling, most of which is represented by the valence band splitting A 0. Thus the tellurides ZnTe and CdTe show the largest splittings, InP the smallest. For all II-VI semiconductors considered in this paper the magnitude of the spin-splitting is comparable or even larger than that for GaAs and InP. For the anisotropy shift B m - Boot (Fig. 3) no chemical trend within the considered semiconductor compounds can be found. This may be due to a more complicated relation between the input parameters and the calculated anisotropy shift. Nevertheless, the magnitude of the anisotropy shift for the II-VI compounds is larger than that of GaAs, which has already been detected experimentally. CONCLUSIONS
"
,O,o 0
E
GaAs c ( J
/ /
s//
~ " 7O
/
///
so so~,)o I~o/
l
/ / CdTe/" / / / / /ZnTe / /
130
°~
I
1'
140
ZnTe//" InP
00
600
(transition energy) 2 (meV 2) Fig. 2. Calculated spin-splitting of cyclotron resonance for GaAs, InP, ZnTe, ZnSe, and CdTe vs. square of transition energy. Experimental values: El, GaAs (Refs. 2, 3), A, InP (Ref. 2).
We have given a quantitative estimate of the spinsplitting and anisotropy of cyclotron resonance for the IIVI semiconductors ZnTe, ZnSe and CdTe. The estimate is based on a 14 x 14 k . p and envelon parametrization of the band structure. The material parameters used in the calculation are known F-point band gaps, the Luttinger parameters and effective electron masses and g-factors. We hope that our calculations will stimulate further experimental effort in exploring the details of the conduction band structure of II-VI semiconductors.
Acknowledgement- We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschafl (Schwerpunktprogramm II-VI-Halbleiter, Projekt Ro-
522/s)
84
II-Vl SEMICONDUCTORS
Vol. 87, No. 2
REFERENCES 1. V.G. Golubev, V.I. Ivanov-Omskii, I.G. Minervin, A.V. Osutin, D.G. Polyakov, Soy. Phys.-JETP 61, 1214 (1985). 2. M.A. Hopkins, R.J. Nicholas, P. Pfeffer, W. Zawadzki, D. Gauthier, J.C. Portal, M.A. di FortePoisson, Semicond. Sci. Technol. 2, 568 (1987). 3. H. Sigg, J.A.A.J. Perenboom, P. Pfeffer, W. Zawadzki, Solid State Commun. 61,685 (1987). 4. A.G. Aronov, G.E. Pikus, A.N. Titkov, Soy. Phys. JETP 57, 680 (1983). 5. V.A. Marushchak, M.N. Stepanova, A.N. Titkov, Soy. Phys.-Solid State 25, 2053 (1983) 6. A.T. Gorelenko, V.G. Gruzdov, V.A. Marushchak, A.N. Titkov, Soy. Phys.-Semicond. 20, 216 (1986). 7. J.M. Luttinger, W. Kohn, Phys. Rev. 97, 869 (1955); J.M. Luttinger, Phys. Rev. 102, 1030 (1956). 8. E.O. Kane in Semiconductors and Semimetals, Vol. 1, eds. R.K. Willardson and A.C. Beers, Academic Press, New York, 1966.
9. M. Cardona, J. Phys. Chem. Solids 24, 1543 (1963). 10. U. RSssler, Solid State Commun. 49, 943 (1984). 11. H. Mayer and U. RSssler, Phys. Rev. B44, 9048 (1991); H.Mayer and U. RSssler in High Magnetic Fields in Semiconductor Physics III, edited by G. Landwehr, Springer, Berlin, 1992, p. 589. 12. N.R. Ogg, Proc. Phys. Soc. London 89, 43 (1966). 13. M. Braun, U. RSssler, J. Phys. C: Solid State Phys. 18, 3365 (1985). 14. Landoldt-B6rnstein New Series: Semiconductors Vols. 17a, 22b , edited by O. Madelung, Springer, Berlin, 1982, 1986. 15. F. H. Pollak, C.W. Higginbotham, M. Cardona, I. Phys. Soc. Jpn. Suppl. 21, 20 (1966). 16. C. Hermann, C. Weisbuch, Phys. Rev. B15, 823 (1977). 17. M. Cardona, N. E. Christensen, G. Fasol, Phys. Rev. B38, 1806 (1988).
0038-1098/93 $6.00+.00 Pergamon Press Ltd
Solid State Communications, Vol. 87, No. 2, PP. 81-84, 1993. Printed in Great Britain.
NONPARABOLICITY
IN THE CONDUCTION
B A N D O F IIoVI S E M I C O N D U C T O R S
H. Mayer and U. RSssler Institut fiir Theoretische Physik, Universit~it Regensburg, D-8400 Regensburg, Germany
(Received 21 April 1993, acceptedfor publication 6 May 1993 by P.H. Dedericbs)
The details of the conduction band structure (spin-splitting, nonparabolicity, and warping) for the II-VI semiconductors ZnTe, ZnSe, and CdTe are studied using an effective 2×2 k . p (or "envelon") Hamiltonian, which contains terms up to fourth order in the electron momentum operator. On the basis of known parameter sets for the II-VI compounds we make predictions for the spin-splitting of the conduction band and the anisotropy and spin-splitting of cyclotron resonance. The magnitude of these effects is compared with those obtained for GaAs and InP.
INTRODUCTION
The parameters which enter the multiband k . p model (energy gaps, Luttinger parameters, effective mass and g-factor) are available from a variety of independent experiments (exciton binding energies, polariton dispersion, cyclotron resonance of holes and electrons, impurity data etc.). 14 Though there is a larger uncertainty in the parameter sets for the II-VI compounds than in those for GaAs and InP, we believe that they are accurate enough to allow some reasonable predictions for the details of the conduction band structure.
The nonparabolicity and anisotropy of the conduction band of the III-V compounds GaAs and InP has been demonstrated in a series of cyclotron resonance experiments 1-3 and measurements of the inversionasymmetry induced spin-splittinga-8. The direct-gap IIVI semiconductors ZnTe, ZnSe, and CdTe are expected to show qualitatively the same effects. It is the purpose of the present paper to give a quantitative estimate of the nonparabolicity effects in these materials. A widely used method to describe the band structure of semiconductors in the vicinity of band extrema is k . p theory 7's. This method allows to separate symmetry properties from material-specific properties and may easily be extended to include also the effects of a magnetic field. The s-antibonding conduction band (F6¢) in direct gap zincblende semiconductors is only twofold (spin-) degenerate at the F-point and therefore shows naturally less structure at the Brillouin zone center than the fourfold degenerate p-antibonding valence band (Fs.). While the complexity of the latter is described by the Luttinger Hamiltonian 7, a description of the conduction band at the same level of k . p theory yields a simple spin-degenerate parabolic band. There are two approaches to extend this simple model for the conduction band. First, the conduction band can be embedded in a larger set of bands, with the k.p-coupling between these bands taken into account explicitly, thus leading to a multiband k . p model description 9--11. On the other hand, the 2 × 2 k . p model for the conduction band itself can be extended to include terms of higher order than quadratic in k. 12,13 Both extensions are capable to describe phenomena like spin splitting, nonparabolicity and band warping. Moreover, quasidegenerate perturbation theory (LSwdin partitioning) yields expressions for the 2 × 2 model parameters in terms of the multiband parameters, thus relating both models to each other n,13.
k-p D E S C R I P T I O N OF T H E C O N D U C T I O N BAND: T H E E N V E L O N M O D E L Considering the basis of multiband k . p theory it is clear that the accuracy of a k - p model is determined in principal by the number of bands contained in the set of quasidegenerate basis states. It has been shown that at least a five-band 14 × 14 model is required for a satisfactory description of the nonparabolicity of the F6c conduction band of zinc-blende type semiconductors. TM In this model the set of quasidegenerate bands consists of the FT~@ F8~ p-bonding valence bands, the F6~ s-antibonding conduction band and the FT¢ @ Fsc p-antibonding conduction bands. A detailed dcscription of this model is given in Ref. 11. The material specific parameters used in this model are momentum matrix elements, band gaps at the F-point, Luttinger parameters, effective mass and g-factors and the off-diagonal spin-orbit coupling 15 A - . An alternative approach for the description of the conduction band has been introduced by Ogg 12. He formulated a 2 × 2 Hamiltonian including terms up to fourth order in k. As will be shown below, this model is equally sufficient to describe details of the conduction band structure at the F-point with the required accuracy. Its invariant expansion T M reads 81
H-VI S E M I C O N D U C T O R S
82
Vol. 87, No. 2 lOOq 161 ,
TABLE Table 1. Input parameter ]or the 14 x 14 model, taken from Ref. 14 E0 A0 E~ A~ A-/i m* g* 3,L .yL "y3 L
(eV) (eV) (eV) (eV) (eV) (m0)
q C C' P (meV)t) P~/i (meV/~) Q (meVA)
InP ZnTe 1.423 2.394 0.110 0.970 4.720 4.820 0.070 0.450 -0.130 0.0 0.0803 0.122 1.26 -0.40 4.95 3.90 1.65 0.83 2.35 1.30 0.97 0.14 0.0 0.001 -2.0 0.0 -0.02 0.0 8.850 9.964 2.866 5.221 7.216 7.247
ZnSe CdTe 2.820 1.606 0.403 0.949 7.330 5.360 0.090 0.250 0.0 0.0 0.160 0.090 1.06 -1.77 4.30 5.30 1.14 1.70 1.84 2.00 0.20 0.61 0.0 0.04 0.0 0.0 0.0 0.0 10.628 9.496 9.165 6.463 9.845 7.873
, .,.//
//
[111l , I ,
,
(2,~/a)~
/ k~" 0.0os 14
~12
GaAs 1.519 0.341 4.488 0.171 -0.05 0.0665 -0.44 6.85 2.10 2.90 1.20 0.01 -1.878 -0.02 10.493 4.780 8.165
,
~
-
-
ZlOl , -
_
x,I
, _ ~
/
____
~-'~k 2" 0.003 (2~la) z. ___
,.--k 2- 0.002 (2~/a) 2 2
L__
0
,,---k =- 0.001 (2xla) z ,~0 2'o
3Jo :0
5'0 s~o 7'0 (deg)
s~o 90
Fig. 1. Comparison of the conduction band anisotropy in ZnTe for k II ( 1 / v ~ s i n O, 1/v/2sin O, cos O) obtained from the 14 × 14 ( - - ) and envelon (- - -) model.
persion up to fourth order in k. From this point of view it is clear that the dispersions obtained from both models coincide as k ~ 0. 5
= E a': A=2
5
+ E a4, E A=I
(1)
t=z,y,z
where at are the Pauli spin matrices and K:(~'x)* are irreducible tensor components formulated as polynomials in the components of k. The explicit form of the latter is given in Tab. 1 of Ref. 13. The expansion coefficients a~,~ can be related to the 14 x 14 model parameters, if an analytical block diagonalization (LSwdin partitioning) of the 14 × 14 model is performed TM. For the parameters al,2 and a4,1 this procedure reproduces the formulae of Hermann and Weisbuch 16 for the effective mass and the effective g-factor in terms of the energy gaps and momentum matrix elements (plus additional terms induced by the off-diagonal spin-orbit coupling A - ) . Formally the 2 × 2 Hamiltonian describes a spin 1/2 particle with nonparabolic energy-momentum relation, which is intimately related to the original multi-band envelope function description of the conduction electron. The new "quasi-particle" described by the effective 2 x 2 conduction-band Hamiltonian will therefore further be referred to as "envelon". In order to demonstrate the accuracy of the envelon model with respect to the 14 × 14 model we examine the band structure E(k) for ZnTe in the case B = 0. A comparison of the nonparabolicities li2k2/2m * - E(k), calculated with the 14× 14 and the envelon model, respectively, is given in Fig. 1. We find that the error made by using the envelon model instead of the 14 x 14 model is less than 10 % of the conduction-band anisotropy for the range of k out to 7 % of the Brillouin zone. The dispersion obtained from the envelon model may also be understood as power expansion of the 14 x 14 model dis-
INVERSION-ASYMMETRY INDUCED SPIN-SPLITTING The lowest-order nonparabolic term in the envelon model is the term weighted by the coefficient a4,2. This term represents the inversion-asymmetry induced zerofield spin-splitting 6spin of the conduction band: 2 2 6spin= 2a4,2 [k 2 (kxky 2 2 + k z 2k x2) _ -'-=-r-zj ~k2k2k2]l/2 • (2) + kuk~
This splitting is experimentally accessible by investigation of the conduction electron spin-relaxation. 4-6 Using the parameter set given in Tab. , we find for GaAs la421 = 19.8 eV/~ a, which is in good agreement with the experimental data la421 = 20.9 eV• s (Ref. 4) and la421 = 24.5 eV/~ 3 (Ref. 5) . For InP we use A - = --130 meV, which is close to the LCAO value A - = - 1 6 0 meV of Cardona et al. 17, and obtain la421 = 9.5 eVA 3, which is within the experimental range la421 = 7.26... 9.5 eV/~ s. The spin-splitting parameter a4,2 is that which is most sensitively dependent on the off-diagonal spin-splitting A - . Because no values for A - are known for the II-VI compounds it is therefore not possible to make reasonable predictions for a4,2. Instead, we give the A - dependence of a4,2, which can be used for an experimental determination of A - from a measurement of a4,2. For ZnTe we find a4,2 = [34.6 + 59.3 x A-/(ieV)] eV~ 3, for ZnSe a4,2 = [14.3 + 53.2 × A-/(ieV)] eV/~3 and for CdTe a4,2 = [43.9 + 73.8 x A-/(ieV)] eV/~ 3. It should be noted that generally only the absolute value of a4,2 can be determined, thus leading to two possible values for A - . Additional information is needed to decide which of these values is preferable.
II-VI SEMICONDUCTORS
Vol. 87, No. 2
100
S P I N - S P L I T T I N G A N D A N I S O T R O P Y OF CYCLOTRON RESONANCE
83 ,
,
,
,
,
90
The nonparabolicity and anisotropy of the conduction band leads to an anisotropy shift of the cyclotron resonance peak position and to a spin-splitting of the resonance line. These effects have been measured and quantitatively explained for the III-V semiconductors GaAs and InP. 1-3'11 To our knowledge no measurements of these effects have been done so far for the II-VI compounds ZnTe, ZnSe, and CdTe. It is possible, however, to estimate the size of the anisotropy shift and the spin-splitting for these materials using the enveton model and the parameters of T a b . . The off-diagonal spin-orbit coupling A - between the p-bonding valence bands and the p-antibonding conduction bands has been neglected in the calculations for the II-V semiconductors. In order to calculate the Landau-level ladders for a given magnetic field we expand the envelon spinorfunction into harmonic-oscillator eigenfunctions and diagonalize the resulting Hamilton matrix. We performed this calculation for magnetic fields in [001] and [111] directions. For the latter it is known that the spin-splitting and anisotropy shift are maximal for fixed magnitude of the magnetic field.T M The resulting spin-splittings for B II [001] and anisotropy shifts for the [111] resonance field with respect to the [001] resonance for GaAs, InP, ZnTe, ZnSe, and CdTe are shown in Figs. 2 and 3. The quantitative agreement with experimental values for GaAs and InP demonstrates the accuracy of the k . p models and the parameter sets. A simple rule for the magnitude of the spin-splittings
150
I
I
I
120 110
/
[~'100 ,
9O
/
,/ /
80
/
70
/
I
/
/
/ /
/
/
/~Y//
/~/
ZnSe
,.~
,'/
11~
~
200
6O ..l::
.~
~
300
4110
500
t//'a s
~, 50 O. O
o 4o t-03 30
"" CdTe 20 10
100
200
300
400
I 500
600
(transition energy) 2 (meV 2) Fig. 3. Calculated anisotropy shift B l u - B001 of cyclotron resonance for GaAs, InP, ZnTe, ZnSe, and CdTe vs. square of transition energy. Experimental values: O, GaAs (Ref. 3). can be deduced from Fig. 2. The spin-splitting is closely related to the strength of the spin-orbit coupling, most of which is represented by the valence band splitting A 0. Thus the tellurides ZnTe and CdTe show the largest splittings, InP the smallest. For all II-VI semiconductors considered in this paper the magnitude of the spin-splitting is comparable or even larger than that for GaAs and InP. For the anisotropy shift B m - Boot (Fig. 3) no chemical trend within the considered semiconductor compounds can be found. This may be due to a more complicated relation between the input parameters and the calculated anisotropy shift. Nevertheless, the magnitude of the anisotropy shift for the II-VI compounds is larger than that of GaAs, which has already been detected experimentally. CONCLUSIONS
"
,O,o 0
E
GaAs c ( J
/ /
s//
~ " 7O
/
///
so so~,)o I~o/
l
/ / CdTe/" / / / / /ZnTe / /
130
°~
I
1'
140
ZnTe//" InP
00
600
(transition energy) 2 (meV 2) Fig. 2. Calculated spin-splitting of cyclotron resonance for GaAs, InP, ZnTe, ZnSe, and CdTe vs. square of transition energy. Experimental values: El, GaAs (Refs. 2, 3), A, InP (Ref. 2).
We have given a quantitative estimate of the spinsplitting and anisotropy of cyclotron resonance for the IIVI semiconductors ZnTe, ZnSe and CdTe. The estimate is based on a 14 x 14 k . p and envelon parametrization of the band structure. The material parameters used in the calculation are known F-point band gaps, the Luttinger parameters and effective electron masses and g-factors. We hope that our calculations will stimulate further experimental effort in exploring the details of the conduction band structure of II-VI semiconductors.
Acknowledgement- We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschafl (Schwerpunktprogramm II-VI-Halbleiter, Projekt Ro-
522/s)
84
II-Vl SEMICONDUCTORS
Vol. 87, No. 2
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