- Email: [email protected]
Zener-tunnelling of inversion layer electrons on small gap semiconductors
"~_
0038-1098/88 $3.00 + .00 Pergamon Journals Ltd.
PrintedS°lid StateinGreatC°mmunicati°nS'Britain. Vol.65,No. 8, pp. 805-808, 1988.
ZENER-TUNNELLING
OF INVERSION
LAYER ELECTRONS
ON SMALL GAP SEMICONDUCTORS
A. Ziegler, U. R6ssler Institut ffir Theoretische Physik, Universit£t Regensburg, D-8400 Regensburg (Received June 9, 1987 by M. Cardona)
A new numerical method for multiband effective-mass calculations of electron states in n-inversion layers is presented. This method allows to consider the resonant character of subband states in narrow-gap semiconductors due to Zener tunnelling and is not restricted to simplified versions of the k . p Hamiltonian. Results obtained for n-inversion layers on Hg0.sCd0.2Te from a 8 x 8 k . p Hamiltonian demonstrate that Zener-tunnelling results in shifts of the subband energies, which need to be considered in a self-consistent calculation.
1. Introduction
conditions for the envelope functions at the interface, but solved the coupled differential equation for InSb by selfconsistent numerical integration without considering these terms. For narrow-gap semiconductors like HgCdTe, when the band bending of the interface potential becomes larger than the confinement energy, tunnelling of the inversion layer electrons into the continuum of valence band states becomes possible [1,2] and leads to a shift and broadening of the discrete subband levels [4] as illustrated in Fig. 1. A treatment of Zener-tunnelling is not possible within the concepts of previous work [1-3]
More recent studies of electron states in inversionlayers on narrow-gap semiconductors demonstrated the necessity of quantitative subband calculations instead of more qualitative ones, which primarily aimed at the understanding of the underlying physical aspects of these materials [1-5]. Most of previous work on this problem is based on Kane's 6 x 6 Hamiltonian, which neglects the spin split-off band F7 and takes into account only the k . p coupling between conduction band F6 and valence band F6. This Hamiltonian does not provide a quantitative interpretation of magneto-opticai experiments in bulk semiconductors as InSb [61 or Hg, C d l _ , T e [71. For n-inversion layers on Hg.76Cd.21Te Ohkawa and Uemura [1] have solved the corresponding set of coupled differentim equations by eliminating the valence band envelope functions and treating the resulting Schr6dinger equations for the conduction band envelope functions in the WKB approximation. Takada et ai. [2] have diagonalized the Kane Hamiltonian by a k-dependent unitary transformation, which - applied to the interface potential U ( z ) - produces nondiagonal coupling terms due to the electric field in the interface. For the case of InSb they neglected the electric field coupling between conduction and valence bands and solved the resulting equation for the conduction envelope band by expanding the wavefunction in a complete set of functions which they 6btained from the solution of the parabolic subband problem. Marques and Sham [3] extended Kane's Hamiltonian by free electron-like terms in the diagonal, which are crucial in their discussion of boundary
cb
-~specfrot weight O(E)
0
zo
Fig. 1 Schematic representation of a subband state (energy, wavefunction, and spectral weight) without (dashed lines) and with tunnelling (solid lines). The spectral weight D(E) shows the shift and broadening due to tunnelling. 805
z
ZENER-TUNNELLINC OF INVERSION LAYER ELECTRONS
806
because of singularities, which appear in the differential equations for the conduction band envelope functions when the subband energy equals the interface potential. The Greens function approach of Ref. [4] is only approximate and does not solve the coupled differential equations of this problem. In order to overcome these shortcomings and to make possible a rigorous t r e a t m e n t of Zener-tunnelling we present a new m e t h o d of calculating subband states in n-inversion layers and demonstrate for Hg0.sCd0.2Te the importance of the resonant character of subband states.
2. The m e t h o d of our calculation The subband problem for inversion layers can be formulated by using as kinetic energy operator a k • p Hamiltonian, which properly describes the bulk energy bands, and adding the interface potential in the diagonal of this matrix operator. We use in the following an 8 x 8 k . p matrix including the twofold F6 conduction band, the fourfold Fs valence band and the F7 spin splitoff band. By replacing k~ --* !c9 i z we obtain a set of coupled differential equations
(
Hs×s(kz --* = ~) + (U(z)
E). lsxs
)
tg(z) = 0 (1)
where tg(z) is an eight-component spinor. In order to find the eigenvalues E of eq. 1 numerically one has to integrate this set for a given E using the proper boundary conditions at z = 0. If the solution fits the proper b o u n d a r y conditions for large z, then E is an eigenvalue, if not, the integration procedure has to be repeated with a different value of E. This procedure requires the knowledge of the boundary conditions for all envelope functions of the spinor q ( z ) at z = 0, which are difficult to obtain even for simplified versions of the k ' p matrix [3]. Our m e t h o d to solve eq. 1 is based on the knowledge of the eigenvaiues Ei(kz) of Hsxs(kz) for real and imaginary kz. Moreover, we know the solution of eq. 1 for a constant potential U(z) = U to be of the form
f i ( z ) = o ~ e ik~z+/3e -ik"z,
fj(z)=OforjCi
Vol. 65, No. 8
which allows to define a local solution according to eq. 2 with kz = k~ and i obtained from E i ( k , ) + U(zn) - E = 0. The coefficients of the local solutions in neighbouring intervalls are determined by the flux conservation condition, i.e. continuity of f i ( z ) and its first derivative. We start our integration for a conduction band state at z = 0 with the boundary condition fc(0) = 0 for the conduction band envelope by assuming an infinite band offset. Together with eq. 2 this means that at z = 0 all envelope functions are zero. From Fig. 2 it can be seen that different branches of the dispersion relation are involved for different z. Starting in the conduction band at z = 0 with finite real kz = kl we approach the b o t t o m of this band with increasing z. By further increasing z we obtain imaginary kz = k2 from the dashed branch, connecting conduction and valence band. Finally, for the situation given in Fig. 2, we find again real k~ = k3 values by using the dispersion relation of the valence band. In this case, when z becomes larger than the depletion length, kz will be independent of z and we obtain the oscillating part of a resonant state solution. It should be mentioned, that for a given E the equation Ei(kz) + U(z) - E = 0 has more solutions of kz t h a n the kz = -I-kn considered in our m e t h o d if Ei(kz) depends on kz in higher than quadratic order. The additional kz values are imaginary and are connected with interface states [9], which become important at band offsets but are not relevant for a smooth potential and are, therefore, neglected here. The use of the dispersion relation of the bulk band structure corresponds to the concept of Takada et al. [2], who used a k-dependent unitary transformation to diagonalize Kane's Hamiltonian. However, their analytical transformation is restricted to a simplified k . p m a t r i x and real k~ and leads to coupled differential equations
Z - space:
k - spclce: cb
energy
vb
' - - ~
(2)
with i and kz being determined by the equation imo,g.
Ei(k~) + U - E = 0.
(3)
kz will be real (imaginary) if E - U is in a band (gap) of the bulk band structure. For a z-dependent interface potential we integrate eq. 1 by replacing U(z) by a piecewise constant potential
U(z) = U ( z , )
h
h
z,~ - ~ < z < zn + ~
(4)
cb
z~
z2
z3
z
kz
i
I I I'V
]
k2
I
I
k3
i \re"t
k1
kz
F i g . 2 S c h e m a t i c d e s c r i p t i o n of our m e t h o d of l o c a l s o l u t i o n s . The d i s p e r s i o n E(kz) is o b t a i n e d f r o m a d i a g o n a l i s a t i o n of the full 8 × 8k.p m a t r i x . For an e n e r g y A E n w i t h i n a b a n d we o b t a i n r e a l k z - v a l u e s (kl, k3 ) and for an e n e r g y in the gap c o m p l e x kz (k2).
Vol. 65, No. 8
ZENER-TUNNELLING OF INVERSION LAYER ELECTRONS
for the two (because of spin) conduction band envelope functions, which is solved in an approximate way. Instead, our procedure can be applied to any k.p Han~itonian and also to imaginary kz and is a direct numerical integration of eq. I, which converges with decreasing step width h with an error of order h 2 in the calculated wave function, i.e. it can be performed to any desired accuracy. As in Ref. [2] the diagonalization procedure, when applied to U(z), produces off-diagonal rnatrixelernents containing the electric field -lOzU(z) of the interface. These terms are known to be small for the coupling between conduction and valence band [2,8] and lead to a spin-splitting of the subband states at finite k H [1,2,3]. In order to restrict our calculation to the effects of Zener tunnelling we neglect all these off-diagonal couplings, also those leading to a spin-splitting. Our calculations will be restricted to the b o t t o m of the subbands (kll = 0) in a n-inversion channel. This means, that the solution, starting in the conduction band (Fig.2) with angular m o m e n t u m m -- 4-½ evolves into the light-hole band, which has the same m value.
807
( E 9 = 58 meV), Ns = 1.1016 m -2 and a n acceptor concentration NA = 5 - 1022 In -3 correspond to a situation of two occupied subbands and show resonant subband states at kll = 0 in comparison with corresponding discrete bound states without Zener-tunneUing. Due to the tunnelling the resonant states are broadened and shifted to higher energies. Both effects are stronger for the lower t h a n for the higher subband and, therefore, the tunnelling decreases the subband separations, a result which has been found already by Brenig and Kasai [4]. It is clear from this result that with increasing Ns or NA, which increase the electnc field in the inversion layer, as well as with decreasing gap energy Eg, which decreases the width of the barrier, the effects of Zener-tunnelling will become stronger. The magnitude of the shift in Fig. 3, which changes significantly the subband occupation and the self-consistent potential, demonstrates the necessity to include the effect of Zener-tunnelling in the self-consistent calculation, which is under way.
3. Zener-t unnelling It is clear from section 2., that for a large gap the subband energy we are looking for is the one for which the integration procedure yields an exponentially decaying solution for large z. For a small gap systern we can simulate this situation by ignoring the valence band. This defines a discrete bound state without Zener-tunnelling (Fig. 1). If we do not ignore the valence band, we obtain a solution of the subband problem at any energy E < Ev(z --* oc) with an oscillating part in the bulk region. It is known from previous studies for the triangular potential [3] that the amplitude of these solutions in the n-channel as compared to that in the bulk depends strongly on the energy E. We define as the spectral weight D(E) of these solutions the ratio between the probability density in the channel extending to z0 (Fig. 1) and in the bulk, whereby we apply plane wave normalization to the oscillating part of the solution. Fig. 1 shows schematically the shift and broadening which result as a consequence of Zener-tunnelling. In a concrete application to an n-inversion channel on p-HgCdTe we first perform a self-consistent calculation without Zener-tunnelling, i.e. by applying the exponential boundary condition mentioned at the beginning of this section. Using this self-consistent potential we consider Zener-tunnelling by integrating the coupled differential equations for varying energy E and computing the spectral weight D(E) along the lines described before. The results of Fig. 3 obtained for Hg0.sCd0.2Te
Acknowledgement - We would like to thank Prof F. Koch for helpful and clarifying discussions.
1.0 O.B 0.6 ¢-1
0.4 0.2 0
J
0.16
I
I
018
0.20
I
022
I
I
0.24 0.26 0.28 energy [eV]
Fig.3 R e s u l t s for Hg n ~Cd n 9Te for a ~ i k ~ acceptor con~nt~ati~ v'~NA= 5 . 1 0 ~ m -~ and N s = 1.10 m- . The solid line shows the spectral w e i g h t D(E) in the selfcons i s t e n t p o t e n t i a l of a c a l c u l a t i o n without tunnelling. The d a s h e d lines m a r k the b o t t o m of the d i s c r e t e subbands.
808
ZENER-TUNNELLINC OF II~ERSION LAYER ELECTRONS
Vol. 65, No. 8
References [1] F.J. Ohkawa, Y. Uemura, J. Phys. Soc. Jpn. 37, 1325 (1974) [2] Y. Takada, K. Arai, N. Uchimura, Y. Uemura, J. Phys. Soc. Japan 49, 1851 (1980). [3] G.E. Marquez, L.J. Sham, Surf. Sci. 113, 131 (1982). G.E. Marquez, Thesis, San Diego 1982. [4] W. Brenig, H. Kasai, Z. Phys. B 54, 191 (1984). [5] T. Ando, J. Phys. Soc. Jpn. 54, 2676 (1985).
[6] C.L. Littler, D.G. Seiler, R. Kaplan, R.J. Wagner, Phys Rev. B 27, 7473 (1983). [7] See e.g.R. Dornhaus, G. Nimtz in "Narrow-GapSemiconductors", Springer Tracts in Modern Physics, Vol. 98 (Springer, Berlin 1983) p. 119. [8] A. Ziegler, Diplomarbeit, Regensburg 1985. [9] M.F.H. Schuurmans, G.W. t'Hooft, Phys. Rev. B al, 8041 (1985).
0038-1098/88 $3.00 + .00 Pergamon Journals Ltd.
PrintedS°lid StateinGreatC°mmunicati°nS'Britain. Vol.65,No. 8, pp. 805-808, 1988.
ZENER-TUNNELLING
OF INVERSION
LAYER ELECTRONS
ON SMALL GAP SEMICONDUCTORS
A. Ziegler, U. R6ssler Institut ffir Theoretische Physik, Universit£t Regensburg, D-8400 Regensburg (Received June 9, 1987 by M. Cardona)
A new numerical method for multiband effective-mass calculations of electron states in n-inversion layers is presented. This method allows to consider the resonant character of subband states in narrow-gap semiconductors due to Zener tunnelling and is not restricted to simplified versions of the k . p Hamiltonian. Results obtained for n-inversion layers on Hg0.sCd0.2Te from a 8 x 8 k . p Hamiltonian demonstrate that Zener-tunnelling results in shifts of the subband energies, which need to be considered in a self-consistent calculation.
1. Introduction
conditions for the envelope functions at the interface, but solved the coupled differential equation for InSb by selfconsistent numerical integration without considering these terms. For narrow-gap semiconductors like HgCdTe, when the band bending of the interface potential becomes larger than the confinement energy, tunnelling of the inversion layer electrons into the continuum of valence band states becomes possible [1,2] and leads to a shift and broadening of the discrete subband levels [4] as illustrated in Fig. 1. A treatment of Zener-tunnelling is not possible within the concepts of previous work [1-3]
More recent studies of electron states in inversionlayers on narrow-gap semiconductors demonstrated the necessity of quantitative subband calculations instead of more qualitative ones, which primarily aimed at the understanding of the underlying physical aspects of these materials [1-5]. Most of previous work on this problem is based on Kane's 6 x 6 Hamiltonian, which neglects the spin split-off band F7 and takes into account only the k . p coupling between conduction band F6 and valence band F6. This Hamiltonian does not provide a quantitative interpretation of magneto-opticai experiments in bulk semiconductors as InSb [61 or Hg, C d l _ , T e [71. For n-inversion layers on Hg.76Cd.21Te Ohkawa and Uemura [1] have solved the corresponding set of coupled differentim equations by eliminating the valence band envelope functions and treating the resulting Schr6dinger equations for the conduction band envelope functions in the WKB approximation. Takada et ai. [2] have diagonalized the Kane Hamiltonian by a k-dependent unitary transformation, which - applied to the interface potential U ( z ) - produces nondiagonal coupling terms due to the electric field in the interface. For the case of InSb they neglected the electric field coupling between conduction and valence bands and solved the resulting equation for the conduction envelope band by expanding the wavefunction in a complete set of functions which they 6btained from the solution of the parabolic subband problem. Marques and Sham [3] extended Kane's Hamiltonian by free electron-like terms in the diagonal, which are crucial in their discussion of boundary
cb
-~specfrot weight O(E)
0
zo
Fig. 1 Schematic representation of a subband state (energy, wavefunction, and spectral weight) without (dashed lines) and with tunnelling (solid lines). The spectral weight D(E) shows the shift and broadening due to tunnelling. 805
z
ZENER-TUNNELLINC OF INVERSION LAYER ELECTRONS
806
because of singularities, which appear in the differential equations for the conduction band envelope functions when the subband energy equals the interface potential. The Greens function approach of Ref. [4] is only approximate and does not solve the coupled differential equations of this problem. In order to overcome these shortcomings and to make possible a rigorous t r e a t m e n t of Zener-tunnelling we present a new m e t h o d of calculating subband states in n-inversion layers and demonstrate for Hg0.sCd0.2Te the importance of the resonant character of subband states.
2. The m e t h o d of our calculation The subband problem for inversion layers can be formulated by using as kinetic energy operator a k • p Hamiltonian, which properly describes the bulk energy bands, and adding the interface potential in the diagonal of this matrix operator. We use in the following an 8 x 8 k . p matrix including the twofold F6 conduction band, the fourfold Fs valence band and the F7 spin splitoff band. By replacing k~ --* !c9 i z we obtain a set of coupled differential equations
(
Hs×s(kz --* = ~) + (U(z)
E). lsxs
)
tg(z) = 0 (1)
where tg(z) is an eight-component spinor. In order to find the eigenvalues E of eq. 1 numerically one has to integrate this set for a given E using the proper boundary conditions at z = 0. If the solution fits the proper b o u n d a r y conditions for large z, then E is an eigenvalue, if not, the integration procedure has to be repeated with a different value of E. This procedure requires the knowledge of the boundary conditions for all envelope functions of the spinor q ( z ) at z = 0, which are difficult to obtain even for simplified versions of the k ' p matrix [3]. Our m e t h o d to solve eq. 1 is based on the knowledge of the eigenvaiues Ei(kz) of Hsxs(kz) for real and imaginary kz. Moreover, we know the solution of eq. 1 for a constant potential U(z) = U to be of the form
f i ( z ) = o ~ e ik~z+/3e -ik"z,
fj(z)=OforjCi
Vol. 65, No. 8
which allows to define a local solution according to eq. 2 with kz = k~ and i obtained from E i ( k , ) + U(zn) - E = 0. The coefficients of the local solutions in neighbouring intervalls are determined by the flux conservation condition, i.e. continuity of f i ( z ) and its first derivative. We start our integration for a conduction band state at z = 0 with the boundary condition fc(0) = 0 for the conduction band envelope by assuming an infinite band offset. Together with eq. 2 this means that at z = 0 all envelope functions are zero. From Fig. 2 it can be seen that different branches of the dispersion relation are involved for different z. Starting in the conduction band at z = 0 with finite real kz = kl we approach the b o t t o m of this band with increasing z. By further increasing z we obtain imaginary kz = k2 from the dashed branch, connecting conduction and valence band. Finally, for the situation given in Fig. 2, we find again real k~ = k3 values by using the dispersion relation of the valence band. In this case, when z becomes larger than the depletion length, kz will be independent of z and we obtain the oscillating part of a resonant state solution. It should be mentioned, that for a given E the equation Ei(kz) + U(z) - E = 0 has more solutions of kz t h a n the kz = -I-kn considered in our m e t h o d if Ei(kz) depends on kz in higher than quadratic order. The additional kz values are imaginary and are connected with interface states [9], which become important at band offsets but are not relevant for a smooth potential and are, therefore, neglected here. The use of the dispersion relation of the bulk band structure corresponds to the concept of Takada et al. [2], who used a k-dependent unitary transformation to diagonalize Kane's Hamiltonian. However, their analytical transformation is restricted to a simplified k . p m a t r i x and real k~ and leads to coupled differential equations
Z - space:
k - spclce: cb
energy
vb
' - - ~
(2)
with i and kz being determined by the equation imo,g.
Ei(k~) + U - E = 0.
(3)
kz will be real (imaginary) if E - U is in a band (gap) of the bulk band structure. For a z-dependent interface potential we integrate eq. 1 by replacing U(z) by a piecewise constant potential
U(z) = U ( z , )
h
h
z,~ - ~ < z < zn + ~
(4)
cb
z~
z2
z3
z
kz
i
I I I'V
]
k2
I
I
k3
i \re"t
k1
kz
F i g . 2 S c h e m a t i c d e s c r i p t i o n of our m e t h o d of l o c a l s o l u t i o n s . The d i s p e r s i o n E(kz) is o b t a i n e d f r o m a d i a g o n a l i s a t i o n of the full 8 × 8k.p m a t r i x . For an e n e r g y A E n w i t h i n a b a n d we o b t a i n r e a l k z - v a l u e s (kl, k3 ) and for an e n e r g y in the gap c o m p l e x kz (k2).
Vol. 65, No. 8
ZENER-TUNNELLING OF INVERSION LAYER ELECTRONS
for the two (because of spin) conduction band envelope functions, which is solved in an approximate way. Instead, our procedure can be applied to any k.p Han~itonian and also to imaginary kz and is a direct numerical integration of eq. I, which converges with decreasing step width h with an error of order h 2 in the calculated wave function, i.e. it can be performed to any desired accuracy. As in Ref. [2] the diagonalization procedure, when applied to U(z), produces off-diagonal rnatrixelernents containing the electric field -lOzU(z) of the interface. These terms are known to be small for the coupling between conduction and valence band [2,8] and lead to a spin-splitting of the subband states at finite k H [1,2,3]. In order to restrict our calculation to the effects of Zener tunnelling we neglect all these off-diagonal couplings, also those leading to a spin-splitting. Our calculations will be restricted to the b o t t o m of the subbands (kll = 0) in a n-inversion channel. This means, that the solution, starting in the conduction band (Fig.2) with angular m o m e n t u m m -- 4-½ evolves into the light-hole band, which has the same m value.
807
( E 9 = 58 meV), Ns = 1.1016 m -2 and a n acceptor concentration NA = 5 - 1022 In -3 correspond to a situation of two occupied subbands and show resonant subband states at kll = 0 in comparison with corresponding discrete bound states without Zener-tunneUing. Due to the tunnelling the resonant states are broadened and shifted to higher energies. Both effects are stronger for the lower t h a n for the higher subband and, therefore, the tunnelling decreases the subband separations, a result which has been found already by Brenig and Kasai [4]. It is clear from this result that with increasing Ns or NA, which increase the electnc field in the inversion layer, as well as with decreasing gap energy Eg, which decreases the width of the barrier, the effects of Zener-tunnelling will become stronger. The magnitude of the shift in Fig. 3, which changes significantly the subband occupation and the self-consistent potential, demonstrates the necessity to include the effect of Zener-tunnelling in the self-consistent calculation, which is under way.
3. Zener-t unnelling It is clear from section 2., that for a large gap the subband energy we are looking for is the one for which the integration procedure yields an exponentially decaying solution for large z. For a small gap systern we can simulate this situation by ignoring the valence band. This defines a discrete bound state without Zener-tunnelling (Fig. 1). If we do not ignore the valence band, we obtain a solution of the subband problem at any energy E < Ev(z --* oc) with an oscillating part in the bulk region. It is known from previous studies for the triangular potential [3] that the amplitude of these solutions in the n-channel as compared to that in the bulk depends strongly on the energy E. We define as the spectral weight D(E) of these solutions the ratio between the probability density in the channel extending to z0 (Fig. 1) and in the bulk, whereby we apply plane wave normalization to the oscillating part of the solution. Fig. 1 shows schematically the shift and broadening which result as a consequence of Zener-tunnelling. In a concrete application to an n-inversion channel on p-HgCdTe we first perform a self-consistent calculation without Zener-tunnelling, i.e. by applying the exponential boundary condition mentioned at the beginning of this section. Using this self-consistent potential we consider Zener-tunnelling by integrating the coupled differential equations for varying energy E and computing the spectral weight D(E) along the lines described before. The results of Fig. 3 obtained for Hg0.sCd0.2Te
Acknowledgement - We would like to thank Prof F. Koch for helpful and clarifying discussions.
1.0 O.B 0.6 ¢-1
0.4 0.2 0
J
0.16
I
I
018
0.20
I
022
I
I
0.24 0.26 0.28 energy [eV]
Fig.3 R e s u l t s for Hg n ~Cd n 9Te for a ~ i k ~ acceptor con~nt~ati~ v'~NA= 5 . 1 0 ~ m -~ and N s = 1.10 m- . The solid line shows the spectral w e i g h t D(E) in the selfcons i s t e n t p o t e n t i a l of a c a l c u l a t i o n without tunnelling. The d a s h e d lines m a r k the b o t t o m of the d i s c r e t e subbands.
808
ZENER-TUNNELLINC OF II~ERSION LAYER ELECTRONS
Vol. 65, No. 8
References [1] F.J. Ohkawa, Y. Uemura, J. Phys. Soc. Jpn. 37, 1325 (1974) [2] Y. Takada, K. Arai, N. Uchimura, Y. Uemura, J. Phys. Soc. Japan 49, 1851 (1980). [3] G.E. Marquez, L.J. Sham, Surf. Sci. 113, 131 (1982). G.E. Marquez, Thesis, San Diego 1982. [4] W. Brenig, H. Kasai, Z. Phys. B 54, 191 (1984). [5] T. Ando, J. Phys. Soc. Jpn. 54, 2676 (1985).
[6] C.L. Littler, D.G. Seiler, R. Kaplan, R.J. Wagner, Phys Rev. B 27, 7473 (1983). [7] See e.g.R. Dornhaus, G. Nimtz in "Narrow-GapSemiconductors", Springer Tracts in Modern Physics, Vol. 98 (Springer, Berlin 1983) p. 119. [8] A. Ziegler, Diplomarbeit, Regensburg 1985. [9] M.F.H. Schuurmans, G.W. t'Hooft, Phys. Rev. B al, 8041 (1985).