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Ge superlattices
So/M-State Electronics Vol. 37, Nos 4-6, pp. 937-940. 1994 Copyright ~ 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1101/94 $6.00 + 0.00
Pergamon
ON THE TWO- AND THREE-DIMENSIONAL CHARACTER OF THE ABSORPTION SPECTRA OF Si/Ge SUPERLATTICES H. M. POLATOGLOU Physics Department, Aristotle University of Thessaloniki, Thessaloniki, GR-54006, Greece Abstract--The optical properties of some strained symmetrized Si~/Ge~ superlattices were studied. In particular the contribution of each direct transition to the absorption spectra and their two- or three-dimensional character is investigated. To achieve this we decomposed each contribution into the density of available superlattice states and the transition probability. It is found that the transition probability is as important is determining the absorption spectra as the density of available states. Furthermore, the signature of direct transitions from the Z point of the superlattice Brillouin zone is predicted to be presented by a structure in the absorption spectra at energies higher than about 0.3 eV above the absorption onset for the case of the Si5/GeSstrained symmetrized superlattice. The possibility arises that transitions from the Z point of the superlattice Brillouin zone can be detected experimentally.
1. INTRODUCTION Si/Ge superlattices conceived by Gnutzman and Clausecker[l] are by now the main ingredients of many special optoelectronic and electronic devices[2]. The growth of such structures is one of the great achievements of the available epitaxiai growth techniques[3]. In the beginning it was possible to grow finite repetitions of superlattices, usually 5[4,5]. During the last years the idea of symmetrically strained superlattices was utilized to produce high quality Si/Ge structures. A systematic study of the absorption spectra of strained symmetrized Sin/Gen for n = 4, 5 and 6 has been recently carried out by photoconductivity and photodiode measurements[6,7]. Clear absorption edges were observed at energies, where band-to-band transitions are expected. The energy of the absorption edge shifts to lower energies as the superlattice period increases and is always lower than that of the buffer alloy. Furthermore, the energy dependence of the absorption spectra close to the edge is found to obey a power law, with exponent equal to 2[6]. Theorectical calculations have confirmed many of the above experimental findings[8]. More specifically, the dependence of the absorption spectra due to the band-to-band electronic transitions on n, the buffer stoichiometry and the contribution from some specific transitions were studied. It was found that the dependence on the buffer stoichiometry is much larger that the dependence on the SL period and that theory predicts edges very close to the experimental ones. This supports the idea that the experimental spectra are due to direct band-to-band transitions. The energy dependence was found to be almost linear, in variance to the experimental results, although comparison is made for the experimental energy range, which is quite
restricted. The results of Pearsall favor a linear behavior[9]. Analysis of the spectra showed that their shape is the result of many band-to-band transitions, which make comparable contributions. The aim of the present work is to extend the theoretical study and correlate the structure of the absorption spectra to specific critical points of the Brillouin zone. Furthermore, we assess the character of the states which participate to the optical absorption spectra. We find that the transitions appear to be between states with predominant the three-dimensional character and that transitions from the Z point of the Brillouin zone have a great effect on the structure of the absorption spectra. 2. METHOD For the present study we apply the empirical tight-binding method within the complete three-center representation with integrals extending up to third neighbor distance. Very good description of the conduction and valence bands has been obtained for both Si and Ge. Instead of resorting to specific formulas in evaluating the Hamiltonian matrix elements we have implemented a more general scheme. The core of this method is the fact that many of the integrals are related through some symmetry operation. In this way possible errors in the sign of the different integrals are eliminated and the method can be easily applied to any crystal structure. The calculation of the optical properties needs more specific description. Firstly, it is very important to note that in the empirical tight-binding description, in the sense of Slater and Koster, the wavefunction as a function of the position is lacking. Only the amplitudes of the different unknown "atomic orbitals" of specific spherical symmetry around each
937
938
H.M. POLATOGLOU
crystalline site are calculated. Therefore, at first sight it seems impossible to obtain the momentum matrix elements, which are essential in finding the linear optical properties. This difficulty can be overcome by using the standard quantum mechanical result: im
p = -~--[r, H], and assuming that the basis functions are localized. The correctness of the formula has been tested by calculating the imaginary part of the dielectric function for different sets of tight-binding parametrizations. We find that the linear properties are very good as long as the conduction and the valence bands are described correctly• The present parametrization fulfills this condition. All linear optical properties like absorption, reflectance, etc., can be deduced from the imaginary part of the dielectric function, The absorption coefficient (~t) is related through the following simple relation to the imaginary part of the dielectric function (E:) and the index of refraction (r/):
hc
~l '
where the other symbols have their usual meaning. The refraction index is constant in the energy range we are about to consider, and increases very smoothly near the upper end of the energy interval. Thus the absorption spectrum simply reflects the imaginary part of the dielectric function times the energy. In calculating the absorption spectra near the absorption edge special care should be exercised. We have used the linear analytic tetrahedron method[10]. The details of the calculation have been presented previously[8]. A property closely related to the dielectric function is the Joint Density of States divided by the energy squared (JDOS/E2). It is defined in the present context as the density of available conduction and valence states at a given energy for wavevector conserving transitions. It is calculated by the same formula as the dielectric function but taking the momentum matrix elements equal to 1 / E 2 independent of the wavevector. From all the above we come to the conclusion that comparing the absorption spectra and the JDOS /E 2 can give us information about the behavior and the importance of the momentum matrix elements. It is well known from the work on the optical properties of bulk crystals that the momentum matrix elements are nearly equal to a constant divided by E 2 for all the transitions arising from a given pair of valence and conduction bands. Whether this is true or not for the Si/Ge SL is one of the questions which we intend to answer.
ent that in the energy range from 0 to 1.7 eV transitions which contribute significantly arise from the three upper valence and the two lower conduction bands[8]. These transitions give rise to the structure of the absorption spectra and here we will analyze this structure and relate it to specific critical points of the dispersion relations. To make the analysis concrete we will concentrate the study on the Sis/Ge5 strained symmetrized SL which according to the present theory and previous results[I I] is a direct gap semiconductor. In Fig. 1 the band structure of the above mentioned SL is displayed around the F and Z point of the SL Brillouin zone. The dispersion of the bands along the F - Z direction is a measure of the confinement of the states. Electrons with wavevectors on this direction are moving along the growth axis, i.e. traversing the two constituent materials. From Fig. 1 we can infer that the electronic states of the two lowest conduction bands are confined along the growth axis while the valence states are confined in a much lesser degree• From previous work it is known[12,13] that the upper valence bands are confined in the Ge layers, while the lower conduction bands in the Si layers. Thus the optical transitions are to a large extend spatially indirect. This means that the transition probability is largely determined by the amount that the electronic states confined in the Si layers penetrate in Ge layers and vice-versa. The question remains about the apparent dimensionality of the optical transitions. With this we want to denote the dimensional character of the critical structure present in the optical properties, like the absorption spectra. By looking at the conduction and valence bands it is obvious that the dispersion of the lowest conduction bands is very small while that of the upper valence bands is comparatively much larger. Therefore the critical points will be largely determined by the properties of the valence bands• One can notice that the three upper valence bands exhibit a maximum at point F and have a saddle point structure at point Z of the SL Brillouin zone. The dimensionality of these structures is determined by
-1
X~
3. RESULTSAND DISCUSSION From previous work of the present author on the absorption spectra of the Si/Ge SLs it became appar-
I"
Z
--X"
Fig. I. Band structure and band-to-band optical transitions for the Sis/Ge5strained symmetrizedsuperlattice. The Z-X' is a segment starting from Z and is parallel to the F-X segment of the Brillouin zone.
939
Absorption spectra of Si/Ge superlattices
0.6 "2.
r6
IO-Q
..... (Si),/(Ge)~
cu 0.4
onS~
w
4oo EII[aoo]
(/3 0
"~ ~
0 0 . 2 _~ "3
4-1
4-~
.1-
0
0.6
1.0
1.4
Energy (eV)
1% O.5
1.O
1.5
Energy (eV) Fig. 2. Sis/Ge5strain symmetrized SL. Decomposed absorption spectra into band-to-band transitions and the momentum matrix elements for transitions arising from the F (circles) and Z (rhombuses) point of the superlattice Brillouin zone. Labels i-j denote that the transitions are from the ith valence band to the jth conduction band. the dimensionality of the valence bands, which is clearly 3-D. Thus we are expecting the normal 3-D critical structures in the dielectric function, absorption spectra and JDOS/E 2. To verify this result we depict in Fig. 2 the analyzed absorption spectra in band-to-band contributions and the momentum matrix element of transitions from the upper three valence bands to the lowest two conduction bands specifically from the F and Z point. One can find a one-to-one correspondence to the critical structure of the components of the absorption spectra. In addition we observe that the structures from the saddle points are very close in pairs, thus giving rise to two pronounced structures in the total absorption spectra. We can conclude that the signature of transitions from the Z point appears in the absorption spectra of the Sis/Ges strain symmetrized SL and that the critical point structure appears to have 3-D character. The experimental situation is that the absorption spectrum is observed only up to 0.3 eV above the absorption edge and therefore it does not include the interesting region, where the saddle point structures appear. It would be interesting to have experimental results extended to higher energies. In this way it is possible to access experimentally states away from the Brillouin zone center. Absorption spectra obtained from GaAs/AIGaAs quantum wells[14] exhibit clear 2-D critical structures. Now we will study the effect of the momentum matrix elements on the optical properties. In Fig. 3 we present the JDOS/E 2 for the Sis/Ge5 strain symmetrized SL. The similarity with the absorption spectra is readily apparent. This means that the momentum matrix elements vary with energy like I/E2 and do not depend on the wavevector, but on the specific pair of valence and conduction bands. It
Fig. 3. Joint density of states divided by E: from some band-to-band transitions for the Sis/Ges strained symmetrized superlattice. Labels i-j denote that the transitions are from the ith valence band to the jth conduction band. is striking to note that bulk semiconductors exhibit similar behavior. In order to observe the change of the 3-D critical structures into the 2-D ones we have calculated the JDOS/E" for the case of Si~0/Gem strained symmetrized SL. The result is depicted in Fig. 4. One can notice that compared to the Sis/Ges case the onsets of the various band-to-band transitions occur at much lower energies, the structures related to the saddle points have almost disappeared (we estimate that the saddle points are about 0.1 eV above the edge) and that each contribution has a step-like shape. The latter is characteristic of the 2-D character of the optical transitions.
4. C O N C L U S I O N S
The critical point structure of the optical transitions is studied for the case of Sis/Ge s superlattice. We find that normal three dimensional minima and saddle structures are to be expected. The minima are related to transitions around the F point and the saddle points to transitions around the Z point of the SL Brillouin zone. In the energy range up to 1.7 eV a clear signature of the saddle points is present in the
(SiLo/(Ge)~o on Sio~Geo. ~
6
0 Cl "3
2
O
0.6
1
1.4
Energy ( e V l Fig. 4. Same as in Fig. 3 but for Si,0/Gel0 strained symmetrized superlattice.
H. M. POLATOGLOU
940
absorption spectra. This enables us to deduce inform a t i o n a b o u t the Z point, whenever experimental results would be available at that energy range. C o m p a r i s o n with the Joint Density o f States reveals that the effect o f the m o m e n t u m matrix elements is similar to the effect found in the bulk semiconductors. thus establishing the fact that the studied superlattice behaves as a normal anisotropic semiconductor. The calculated Joint Density o f States for the Sil0/Ge~0 superlattice shows that the 3-D character of the optical transitions has converted to a 2-D character. Acknowledgement--The author has been partially supported by the EEC Esprit Basic Research Action program N ° 7128.
REFERENCES
1. U. Gnutzman and K. Clausecker, Appl. Phys. 3, 9 (1974). 2. H. Presting, H. Kibbel, M. Jaros, R. M, Turton, U. Menczigar, G. Abstreiter and H. G. Grimmeiss, Semicond. Sci. Technol. 7, 1127 (1992).
3. L. Esaki, in Condensed Matter Physics and Future Prospects (Edited by L. Esaki), p. 55. Plenum Press. New York (1991). 4. T. P. Pearsall. J. Bevk, L. C. Fetdman, J. M. Bonar. J. P. Mannaerts and A. Ourmazd. Phys. Rev. Left. 58, 729 (1987). 5. T. P. Pearsall. CRC, Critical Rev. Solid St. mat. Sci. 15, 551 (1989): and references therein. 6. J. Olajos, J. Engvall, H. G. Grimmeiss, U. Menczigar, G. Abstreiter, H. Kibbel, E. Kasper and H. Presting, Phys. Rev. B 46, 12857 (1992). 7. U. Menczigar, G. Abstreiter, J. Olajos, H. Grimmeiss, H. Kibbel, H. Presting and E. Kasper, Phys. Rev. B 47, 4099 (1993). 8. H. M. Polatoglou, G. Theodorou and C. Tserbak, Phys. Rev. B. In press. 9. T. P. Pearsall, AppL Phys. Lett. 60, 1712 (1992). 10. O. Jepsen and O. K. Andersen, Solid St. Commun. 9, 1763 (1971). 11. U. Schmid, N. E. Christensen, M. Alouani and M. Cardona, Phys. Rer. B 43, 14597 (1991). 12. M. A. Gell, Phys. Rev. B 38, 7535 (1988). 13. H, M. Polatoglou, C. Tserbak and G. Theodorou, Thin Solid Films 222, 212 (1992). 14. S. Schmitt-Rink. D. S. Chemla and D. A. B. Miller, Adv. Phys. 38, 89 (1989).
Pergamon
ON THE TWO- AND THREE-DIMENSIONAL CHARACTER OF THE ABSORPTION SPECTRA OF Si/Ge SUPERLATTICES H. M. POLATOGLOU Physics Department, Aristotle University of Thessaloniki, Thessaloniki, GR-54006, Greece Abstract--The optical properties of some strained symmetrized Si~/Ge~ superlattices were studied. In particular the contribution of each direct transition to the absorption spectra and their two- or three-dimensional character is investigated. To achieve this we decomposed each contribution into the density of available superlattice states and the transition probability. It is found that the transition probability is as important is determining the absorption spectra as the density of available states. Furthermore, the signature of direct transitions from the Z point of the superlattice Brillouin zone is predicted to be presented by a structure in the absorption spectra at energies higher than about 0.3 eV above the absorption onset for the case of the Si5/GeSstrained symmetrized superlattice. The possibility arises that transitions from the Z point of the superlattice Brillouin zone can be detected experimentally.
1. INTRODUCTION Si/Ge superlattices conceived by Gnutzman and Clausecker[l] are by now the main ingredients of many special optoelectronic and electronic devices[2]. The growth of such structures is one of the great achievements of the available epitaxiai growth techniques[3]. In the beginning it was possible to grow finite repetitions of superlattices, usually 5[4,5]. During the last years the idea of symmetrically strained superlattices was utilized to produce high quality Si/Ge structures. A systematic study of the absorption spectra of strained symmetrized Sin/Gen for n = 4, 5 and 6 has been recently carried out by photoconductivity and photodiode measurements[6,7]. Clear absorption edges were observed at energies, where band-to-band transitions are expected. The energy of the absorption edge shifts to lower energies as the superlattice period increases and is always lower than that of the buffer alloy. Furthermore, the energy dependence of the absorption spectra close to the edge is found to obey a power law, with exponent equal to 2[6]. Theorectical calculations have confirmed many of the above experimental findings[8]. More specifically, the dependence of the absorption spectra due to the band-to-band electronic transitions on n, the buffer stoichiometry and the contribution from some specific transitions were studied. It was found that the dependence on the buffer stoichiometry is much larger that the dependence on the SL period and that theory predicts edges very close to the experimental ones. This supports the idea that the experimental spectra are due to direct band-to-band transitions. The energy dependence was found to be almost linear, in variance to the experimental results, although comparison is made for the experimental energy range, which is quite
restricted. The results of Pearsall favor a linear behavior[9]. Analysis of the spectra showed that their shape is the result of many band-to-band transitions, which make comparable contributions. The aim of the present work is to extend the theoretical study and correlate the structure of the absorption spectra to specific critical points of the Brillouin zone. Furthermore, we assess the character of the states which participate to the optical absorption spectra. We find that the transitions appear to be between states with predominant the three-dimensional character and that transitions from the Z point of the Brillouin zone have a great effect on the structure of the absorption spectra. 2. METHOD For the present study we apply the empirical tight-binding method within the complete three-center representation with integrals extending up to third neighbor distance. Very good description of the conduction and valence bands has been obtained for both Si and Ge. Instead of resorting to specific formulas in evaluating the Hamiltonian matrix elements we have implemented a more general scheme. The core of this method is the fact that many of the integrals are related through some symmetry operation. In this way possible errors in the sign of the different integrals are eliminated and the method can be easily applied to any crystal structure. The calculation of the optical properties needs more specific description. Firstly, it is very important to note that in the empirical tight-binding description, in the sense of Slater and Koster, the wavefunction as a function of the position is lacking. Only the amplitudes of the different unknown "atomic orbitals" of specific spherical symmetry around each
937
938
H.M. POLATOGLOU
crystalline site are calculated. Therefore, at first sight it seems impossible to obtain the momentum matrix elements, which are essential in finding the linear optical properties. This difficulty can be overcome by using the standard quantum mechanical result: im
p = -~--[r, H], and assuming that the basis functions are localized. The correctness of the formula has been tested by calculating the imaginary part of the dielectric function for different sets of tight-binding parametrizations. We find that the linear properties are very good as long as the conduction and the valence bands are described correctly• The present parametrization fulfills this condition. All linear optical properties like absorption, reflectance, etc., can be deduced from the imaginary part of the dielectric function, The absorption coefficient (~t) is related through the following simple relation to the imaginary part of the dielectric function (E:) and the index of refraction (r/):
hc
~l '
where the other symbols have their usual meaning. The refraction index is constant in the energy range we are about to consider, and increases very smoothly near the upper end of the energy interval. Thus the absorption spectrum simply reflects the imaginary part of the dielectric function times the energy. In calculating the absorption spectra near the absorption edge special care should be exercised. We have used the linear analytic tetrahedron method[10]. The details of the calculation have been presented previously[8]. A property closely related to the dielectric function is the Joint Density of States divided by the energy squared (JDOS/E2). It is defined in the present context as the density of available conduction and valence states at a given energy for wavevector conserving transitions. It is calculated by the same formula as the dielectric function but taking the momentum matrix elements equal to 1 / E 2 independent of the wavevector. From all the above we come to the conclusion that comparing the absorption spectra and the JDOS /E 2 can give us information about the behavior and the importance of the momentum matrix elements. It is well known from the work on the optical properties of bulk crystals that the momentum matrix elements are nearly equal to a constant divided by E 2 for all the transitions arising from a given pair of valence and conduction bands. Whether this is true or not for the Si/Ge SL is one of the questions which we intend to answer.
ent that in the energy range from 0 to 1.7 eV transitions which contribute significantly arise from the three upper valence and the two lower conduction bands[8]. These transitions give rise to the structure of the absorption spectra and here we will analyze this structure and relate it to specific critical points of the dispersion relations. To make the analysis concrete we will concentrate the study on the Sis/Ge5 strained symmetrized SL which according to the present theory and previous results[I I] is a direct gap semiconductor. In Fig. 1 the band structure of the above mentioned SL is displayed around the F and Z point of the SL Brillouin zone. The dispersion of the bands along the F - Z direction is a measure of the confinement of the states. Electrons with wavevectors on this direction are moving along the growth axis, i.e. traversing the two constituent materials. From Fig. 1 we can infer that the electronic states of the two lowest conduction bands are confined along the growth axis while the valence states are confined in a much lesser degree• From previous work it is known[12,13] that the upper valence bands are confined in the Ge layers, while the lower conduction bands in the Si layers. Thus the optical transitions are to a large extend spatially indirect. This means that the transition probability is largely determined by the amount that the electronic states confined in the Si layers penetrate in Ge layers and vice-versa. The question remains about the apparent dimensionality of the optical transitions. With this we want to denote the dimensional character of the critical structure present in the optical properties, like the absorption spectra. By looking at the conduction and valence bands it is obvious that the dispersion of the lowest conduction bands is very small while that of the upper valence bands is comparatively much larger. Therefore the critical points will be largely determined by the properties of the valence bands• One can notice that the three upper valence bands exhibit a maximum at point F and have a saddle point structure at point Z of the SL Brillouin zone. The dimensionality of these structures is determined by
-1
X~
3. RESULTSAND DISCUSSION From previous work of the present author on the absorption spectra of the Si/Ge SLs it became appar-
I"
Z
--X"
Fig. I. Band structure and band-to-band optical transitions for the Sis/Ge5strained symmetrizedsuperlattice. The Z-X' is a segment starting from Z and is parallel to the F-X segment of the Brillouin zone.
939
Absorption spectra of Si/Ge superlattices
0.6 "2.
r6
IO-Q
..... (Si),/(Ge)~
cu 0.4
onS~
w
4oo EII[aoo]
(/3 0
"~ ~
0 0 . 2 _~ "3
4-1
4-~
.1-
0
0.6
1.0
1.4
Energy (eV)
1% O.5
1.O
1.5
Energy (eV) Fig. 2. Sis/Ge5strain symmetrized SL. Decomposed absorption spectra into band-to-band transitions and the momentum matrix elements for transitions arising from the F (circles) and Z (rhombuses) point of the superlattice Brillouin zone. Labels i-j denote that the transitions are from the ith valence band to the jth conduction band. the dimensionality of the valence bands, which is clearly 3-D. Thus we are expecting the normal 3-D critical structures in the dielectric function, absorption spectra and JDOS/E 2. To verify this result we depict in Fig. 2 the analyzed absorption spectra in band-to-band contributions and the momentum matrix element of transitions from the upper three valence bands to the lowest two conduction bands specifically from the F and Z point. One can find a one-to-one correspondence to the critical structure of the components of the absorption spectra. In addition we observe that the structures from the saddle points are very close in pairs, thus giving rise to two pronounced structures in the total absorption spectra. We can conclude that the signature of transitions from the Z point appears in the absorption spectra of the Sis/Ges strain symmetrized SL and that the critical point structure appears to have 3-D character. The experimental situation is that the absorption spectrum is observed only up to 0.3 eV above the absorption edge and therefore it does not include the interesting region, where the saddle point structures appear. It would be interesting to have experimental results extended to higher energies. In this way it is possible to access experimentally states away from the Brillouin zone center. Absorption spectra obtained from GaAs/AIGaAs quantum wells[14] exhibit clear 2-D critical structures. Now we will study the effect of the momentum matrix elements on the optical properties. In Fig. 3 we present the JDOS/E 2 for the Sis/Ge5 strain symmetrized SL. The similarity with the absorption spectra is readily apparent. This means that the momentum matrix elements vary with energy like I/E2 and do not depend on the wavevector, but on the specific pair of valence and conduction bands. It
Fig. 3. Joint density of states divided by E: from some band-to-band transitions for the Sis/Ges strained symmetrized superlattice. Labels i-j denote that the transitions are from the ith valence band to the jth conduction band. is striking to note that bulk semiconductors exhibit similar behavior. In order to observe the change of the 3-D critical structures into the 2-D ones we have calculated the JDOS/E" for the case of Si~0/Gem strained symmetrized SL. The result is depicted in Fig. 4. One can notice that compared to the Sis/Ges case the onsets of the various band-to-band transitions occur at much lower energies, the structures related to the saddle points have almost disappeared (we estimate that the saddle points are about 0.1 eV above the edge) and that each contribution has a step-like shape. The latter is characteristic of the 2-D character of the optical transitions.
4. C O N C L U S I O N S
The critical point structure of the optical transitions is studied for the case of Sis/Ge s superlattice. We find that normal three dimensional minima and saddle structures are to be expected. The minima are related to transitions around the F point and the saddle points to transitions around the Z point of the SL Brillouin zone. In the energy range up to 1.7 eV a clear signature of the saddle points is present in the
(SiLo/(Ge)~o on Sio~Geo. ~
6
0 Cl "3
2
O
0.6
1
1.4
Energy ( e V l Fig. 4. Same as in Fig. 3 but for Si,0/Gel0 strained symmetrized superlattice.
H. M. POLATOGLOU
940
absorption spectra. This enables us to deduce inform a t i o n a b o u t the Z point, whenever experimental results would be available at that energy range. C o m p a r i s o n with the Joint Density o f States reveals that the effect o f the m o m e n t u m matrix elements is similar to the effect found in the bulk semiconductors. thus establishing the fact that the studied superlattice behaves as a normal anisotropic semiconductor. The calculated Joint Density o f States for the Sil0/Ge~0 superlattice shows that the 3-D character of the optical transitions has converted to a 2-D character. Acknowledgement--The author has been partially supported by the EEC Esprit Basic Research Action program N ° 7128.
REFERENCES
1. U. Gnutzman and K. Clausecker, Appl. Phys. 3, 9 (1974). 2. H. Presting, H. Kibbel, M. Jaros, R. M, Turton, U. Menczigar, G. Abstreiter and H. G. Grimmeiss, Semicond. Sci. Technol. 7, 1127 (1992).
3. L. Esaki, in Condensed Matter Physics and Future Prospects (Edited by L. Esaki), p. 55. Plenum Press. New York (1991). 4. T. P. Pearsall. J. Bevk, L. C. Fetdman, J. M. Bonar. J. P. Mannaerts and A. Ourmazd. Phys. Rev. Left. 58, 729 (1987). 5. T. P. Pearsall. CRC, Critical Rev. Solid St. mat. Sci. 15, 551 (1989): and references therein. 6. J. Olajos, J. Engvall, H. G. Grimmeiss, U. Menczigar, G. Abstreiter, H. Kibbel, E. Kasper and H. Presting, Phys. Rev. B 46, 12857 (1992). 7. U. Menczigar, G. Abstreiter, J. Olajos, H. Grimmeiss, H. Kibbel, H. Presting and E. Kasper, Phys. Rev. B 47, 4099 (1993). 8. H. M. Polatoglou, G. Theodorou and C. Tserbak, Phys. Rev. B. In press. 9. T. P. Pearsall, AppL Phys. Lett. 60, 1712 (1992). 10. O. Jepsen and O. K. Andersen, Solid St. Commun. 9, 1763 (1971). 11. U. Schmid, N. E. Christensen, M. Alouani and M. Cardona, Phys. Rer. B 43, 14597 (1991). 12. M. A. Gell, Phys. Rev. B 38, 7535 (1988). 13. H, M. Polatoglou, C. Tserbak and G. Theodorou, Thin Solid Films 222, 212 (1992). 14. S. Schmitt-Rink. D. S. Chemla and D. A. B. Miller, Adv. Phys. 38, 89 (1989).