Skip to main content
SpringerLink
Log in

Electron Spectrum in Crystals, Quantum Wells and Superlattices

  • Chapter
Superlattices and Other Heterostructures

Part of the book series: Springer Series in Solid-State Sciences ((SSSOL,volume 110))

  • 630 Accesses

Abstract

We shall briefly digress from group theory to get acquainted with the so-called k-p method. It is the simplest method for calculating carrier spectra near extreme points, i.e., the conduction-band minima and valence-band maxima; it essentially represents a variant of perturbation theory. The effective-mass approximation and the theory of deformation potential, which permit description of the effect of external, magnetic and electric, fields as well as of the interactions of carriers with lattice vibrations, may be considered a natural development of the k-p method. All these concepts enjoy widespread use in the theory of semiconductors and of materials where the carrier concentration is usually much lower than the number of lattice atoms, and therefore the electrons and holes cluster near the extrema.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. T. Ando, S. Wakahara, H. Akera: Phys. Rev. B 40, 11609 (1989)

    Google Scholar 

  2. S R. White, L.J. Sham: Phys. Rev. Lett. 47, 879 (1981)

    CAS  Google Scholar 

  3. Q.-G. Zhu, H. Kroemer: Phys. Rev. B 27, 3519 (1983)

    Google Scholar 

  4. K.B. Kahen, J.P. Leburton: Phys. Rev. B 33, 5465 (1986)

    Google Scholar 

  5. T. Ando, H. Akera: Phys. Rev. B 40, 11619 (1989)

    Google Scholar 

  6. R.A. Morrow, K.R. Brownstein: Phys. Rev. B 30, 678 (1984)

    Google Scholar 

  7. G.L. Bir, G.E. Pikus: Symmetry and Strain-Induced Effects in Semiconductors (Wiley, New York 1974)

    Google Scholar 

  8. W. Shockley: Phys. Rev. 78, 173 (1950)

    Google Scholar 

  9. R.J. Elliott: Phys. Rev. 96, 266 (1954)

    CAS  Google Scholar 

  10. R.J. Elliott: Phys. Rev. 96, 280 (1954)

    CAS  Google Scholar 

  11. G. Dresselhaus, A.F. Kip, C. Kittel: Phys. Rev. 98, 368 (1955)

    CAS  Google Scholar 

  12. J.M. Luttinger: Phys. Rev. 102, 1030 (1956)

    CAS  Google Scholar 

  13. E.O. Kane: J. Phys. Chem. Solids 1, 82 (1957)

    Google Scholar 

  14. E.O. Kane: J. Phys. Chem. Solids 1, 249 (1957)

    Google Scholar 

  15. R.A. Suris: Fiz. Tekh. Poluprovodn.20, 2008 (1986) [Sov. Phys. - Semicond. 20, 1258 (1986)]

    Google Scholar 

  16. R.A. Suris, A.B. Sokolskii: Fiz. Tekh. Poluprovodn. 21, 866 (1987) [Sov. Phys. - Semicond. 21, 529(1987)]

    Google Scholar 

  17. H.-R. Trebin, U. Rossler, R. Ranvaud: Phys. Rev. B 20, 686 (1979)

    Google Scholar 

  18. D.L. Smith, C. Mailhiot: Rev. Mod. Phys. 69, 173 (1990)

    Google Scholar 

  19. N.M. Ashkroft, N.D. Mermin: Solid State Physics (Holt, Rinehart and Winston, New York 1976)

    Google Scholar 

  20. I.A. Aleiner, E.L. Ivchenko: Fiz. Tekh. Poluprovodn. 27, 594 (1993) [Semiconductors 27, 330 (1993)]

    CAS  Google Scholar 

  21. Y. Fu, M. Willander, E.L. Ivchenko, A.A. Kiselev: Phys. Rev. B 47, 13498 (1993)

    Google Scholar 

  22. G. Bastard: Phys. Rev. B 24, 5693 (1981)

    Google Scholar 

  23. G. Bastard: Phys. Rev. B 25, 7584 (1982)

    Google Scholar 

  24. D.L. Smith, C. Mailhiot: Phys. Rev. B 33, 8345 (1986)

    Google Scholar 

  25. C. Mailhiot, D.L. Smith: Phys. Rev. B 33, 8360 (1986)

    Google Scholar 

  26. R. Eppenga, M.F.H. Schuurmans, S. Colak: Phys. Rev. B 36, 1554 (1987)

    Google Scholar 

  27. C.M. de Sterke, D.G. Hall: Phys. Rev. B 35, 1380 (1987)

    Google Scholar 

  28. C. Mailhiot, D.L. Smith: Phys. Rev. B 35, 1242 (1987)

    Google Scholar 

  29. Y.C. Chang, G.D. Sanders, D.Z.-Y. Ting: In Excitons in Confined Systems, ed. by R. Del Sole, A.D. Andrea, A. Lapiccirello. Springer Proc. Phys. 25, 159 (Springer, Berlin, Heidelberg 1988)

    Google Scholar 

  30. S.S. Nedorezov: Fiz. Tverd. Tela12, 2269 (1970) [Sov. Phys. - Solid State 12, 1814(1971)]

    CAS  Google Scholar 

  31. D.A. Broido, L.J. Sham: Phys. Rev. B 31, 888 (1985)

    Google Scholar 

  32. R. Wessel, M. Altarelli: Phys. Rev. B 40, 12457 (1989)

    Google Scholar 

  33. A.M. Cohen, G.E. Marques: Phys. Rev. B 41, 10608 (1990)

    Google Scholar 

  34. A. Matulis, K. Piragas: Fiz. Tekh. Poluprovodn. 9, 220 (1975) [Sov. Phys. - Semicond. 9, 1432 (1976)]

    Google Scholar 

  35. L.G. Gerchikov, A.V. Subashiev: Phys. Stat. Sol. (b) 160, 443 (1990)

    Google Scholar 

  36. J.C. Maan: In Two-Dimensional Systems. Heterostructures, and Superlattices, ed. by G. Bauer, F. Kuchar, H. Heinrich, Springer Ser. Solid State Sci., Vol.53 (Springer, Berlin, Heidelberg 1984) p. 183

    Google Scholar 

  37. G. Bastard, E.E. Mendez, L.L. Chang, L. Esaki: Phys. Rev. B 28, 3241 (1983)

    Google Scholar 

  38. G.H. Wannier: Rev. Mod. Phys. 34, 645 (1962)

    Google Scholar 

  39. W.V. Houston: Phys. Rev. 57, 184 (1940)

    CAS  Google Scholar 

  40. L.V. Keldysh: Zh. Exper. Teor. Fiz.33, 994 (1957) [Sov. Phys. - JETP 6, 763 (1958)]

    CAS  Google Scholar 

  41. A G. Aronov, G.E. Pikus: Zh. Exper. Teor. Fiz. 51, 281 (1966) [Sov. Phys. - JETP 24, 188(1967)]

    CAS  Google Scholar 

k p Method

  • Bouckaert L.P., R. Smoluchowski, E. Wigner: Theory of Brillouin zones and symmetry properties of wave functions in crystals. Phys. Rev. 50, 58 (1936)

    Google Scholar 

  • Cardona M, F. Pollak: Energy-band structure of germanium and silicon: The k p method. Phys. Rev. 142, 530 (1966)

    CAS  Google Scholar 

  • Gull F., M. Manas: The Dirac equation and integrable systems of k p type. J. Phys. A 29, 641 (1996)

    Google Scholar 

  • Lage F.C. von der, H.A. Bethe: A method for obtaining electronic eigenfunctions and eigenvalues in solids with an application to sodium. Phys. Rev. 71, 612 (1947)

    Google Scholar 

  • Lew Yan Voon L.C., M. Willatzen, M. Cardona: Terms linear in k in the band structure of wurzite-type semiconductors. Phys. Rev. B 53, 10703 (1996)

    Google Scholar 

  • Loehr J.P.: Parameter consistency in multienergetic k p models. Phys. Rev. B 52, 2374 (1995)

    Google Scholar 

  • Krasovskii E.E., W. Schattke: The extended-LAPW-based k p method for complex band structure calculations. Solid State Commun. 93, 775 (1995)

    CAS  Google Scholar 

  • Montasser S.S.: Analytical technique for extracting the eigenvalues of the k p matrix that represent the band structure of semiconductors. Phys. Rev. B 42, 7513 (1990)

    Google Scholar 

  • Rashba E.I.: Symmetry of energy bands in wurtzite-type crystals. I. Symmetry of bands neglecting the spin-orbit interaction. Fiz. Tverd. Tela 1, 407 (1959) [Sov. Phys. - Solid State 1, 368 (1959)]

    Google Scholar 

  • Rashba E.I., V.I. Sheka: Symmetry of energy bands in wurtzite-type crystals. II. Symmetry of bands taking into account the spin interactions. Fiz. Tverd. Tela, Sbornik II, p. 162 (1959) (in Russian)

    Google Scholar 

  • Wood D.M., A. Zunger: Successes and failures of the k p method: A direct assessment for GaAs/AlAs quantum structures. Phys. Rev. B 53, 7949 (1996)

    Google Scholar 

  • Zak J.: The k p representation in the dynamics of electron in solids.Solid State Physics 27, 1 (Academic, New York 1972)

    CAS  Google Scholar 

Effective Mass Theory

  • Brezini A., N. Zekri: Effective mass theory for abrupt heterojunctions. Solid State Commun. 86, 613 (1993)

    CAS  Google Scholar 

  • Foreman B.A.: Exact effective-mass theory for heterostructures. Phys. Rev. B 52, 12241 (1995)

    Google Scholar 

  • Geller M.R., W. Kohn: Quantum mechanics of electrons with graded composition. Phys. Rev. Lett. 70, 3103 (1993)

    CAS  Google Scholar 

  • Kittel C., A H. Mitchell: The theory of donor and acceptor states in silicon and germanium. Phys. Rev. 96, 1488 (1954)

    CAS  Google Scholar 

  • Luttinger J.M., W. Kohn: Motion of electrons and holes in perturbed periodic fields. Phys. Rev. 97, 869 (1955)

    CAS  Google Scholar 

  • Pidgeon C.R., R.N. Brown: Interband magneto-absorption and Faraday rotation in InSb. Phys. Rev. 146, 575 (1966)

    CAS  Google Scholar 

  • Wannier G.H.: The structure of electronic excitation levels in insulating crystals. Phys. Rev. 52, 191 (1937)

    CAS  Google Scholar 

Deformation Potential Theory

  • Bir G.L., G.E. Pikus: Theory of deformation potential for semiconductors with the complicated band structure. Fiz. Tverd. Tela 2, 2287 (1960) [Sov. Phys. - Solid State 2, 2039 (1961)]

    CAS  Google Scholar 

  • Blacha A., H. Presting, M. Cardona: Deformation potentials of k=0 states in tetrahe-dral semiconductors. Phys. Status Solidi (b) 126, 11 (1984)

    CAS  Google Scholar 

  • Herring C.: Transport properties of a many-valley semiconductor. Bell. Syst. Techn. J. 34, 237 (1955)

    Google Scholar 

  • Herring C., E. Vogt: Transport and deformation-potential theory for many-valley semiconductors with anisotropic scattering. Phys. Rev. 101, 944 (1956)

    CAS  Google Scholar 

  • Shockley W., J. Bardeen: Energy bands and mobilities in monoatomic semiconductors. Phys. Rev. 77, 407 (1950)

    Google Scholar 

Method of Invariants

  • Koster G.F., H. Statz: Method of treating Zeeman splitting of paramagnetic ions in crystalline fields. Phys. Rev. 113, 445 (1959)

    CAS  Google Scholar 

  • Pikus G.E.: A new method of calculating of the energy spectrum of carriers in semiconductors. I. Neglecting spin-orbit interaction. Zh. Eksp. Teor. Fiz. 41, 1258 (1961) [Sov. Phys. - JETP 14, 898 (1962)]

    CAS  Google Scholar 

  • Pikus G.E.: A New method for calculating the energy spectrum of carriers in semiconductors. II. Account of spin-orbit coupling. Zh. Eksp. Teor. Fiz. 41, 1507 (1961) [Sov. Phys. - JETP 14, 1075 (1962)]

    CAS  Google Scholar 

  • Pikus G.E.: Polarization of exciton radiation of silicon in the presence of magnetic field or uniaxial strain. Fiz. Tverd. Tela 19, 1653 (1977) [Sov. Phys. - Solid State 19, 965 (1977)]

    CAS  Google Scholar 

  • Rashba E.I., V.I. Sheka: Combined resonance on acceptor centers. Fiz. Tverd. Tela, 6, 576 (1964) [Sov. Phys. - Solid State 6, 451 (1964)]

    CAS  Google Scholar 

  • Statz H., G.F. Koster: Zeeman splittings of paramagnetic atoms in crystalline fields. Phys. Rev. 115, 1568(1959)

    CAS  Google Scholar 

Electron and Hole Spectrum in Cubic Crystals

  • Braunstein R., E.O. Kane: The valence band structure of the III-V compounds. J. Phys. Chem. Solids 23, 1423 (1962)

    CAS  Google Scholar 

  • Cardona M., N.E. Christensen, G. Fasol: Relativistic band structure and spin-orbit splitting of zinc-blende-type semiconductors. Phys. Rev. B 38, 1806 (1988)

    Google Scholar 

  • Dresselhaus G.: Spin-orbit coupling effects in zinc blende structures. Phys. Rev. 100, 580 (1955)

    CAS  Google Scholar 

  • Groves S.H., R.N. Brown, C.R. Pidgeon: Interband magnetoreflection and band structure of HgTe. Phys. Rev. 161, 779 (1967)

    CAS  Google Scholar 

  • Kalt H.: Optical Properties of III-V Semiconductors. The Influence of Multi-Valley Band Structures, Springer Ser. Solid-State Sci., Vol.120 (Springer, Berlin, Heidelberg 1996)

    Google Scholar 

  • Montasser S.S.: Analytic approach to the inversion-asymmetry splitting of the valence band in zinc-blende-type semiconductor. Phys. Rev. B48, 12285 (1994)

    Google Scholar 

  • Pikus G.E., V.A. Marushchak, A.N. Titkov: Spin splitting of energy bands and spin relaxation of carriers in cubic III-V crystals (Review). Fiz. Tekh. Poluprovodn. 22, 185 (1988) [Sov. Phys. - Semicond. 22, 115 (1988)]

    CAS  Google Scholar 

  • Rashba E.I., V.I. Sheka: Combined resonance of band electrons in crystals with zinc- blende lattice. Fiz. Tverd. Tela 3, 1735 (1961) [Sov. Phys. - Solid State 3, 1257 (1961)]

    CAS  Google Scholar 

  • Wetcel C., R. Winkler, M. Drechsler, B.K. Meyer, U. Rössler, J. Scriba, J.P. Kotthaus, V. Härle, F. Scholz: Electron effective mass and nonparabolisity in Ga0 47In0 53 As/ InP quantum wells. Phys. Rev. B 53, 1038 (1996)

    Google Scholar 

  • Willatzen M., M. Cardona, N.E. Christensen: Linear muffin-tin-orbital and k p calculations of effective masses and band structure of semiconducting diamond. Phys. Rev. B 50, 18054(1994)

    CAS  Google Scholar 

  • Willatzen M., M. Cardona, N.E. Christensen: Spin-orbit coupling parameters and electron g factor of II-VI zinc-blende materials. Phys. Rev. B 51, 17993 (1995)

    Google Scholar 

  • Zawadski W., I.T. Yoon, C.L. Litter, X.N. Song, P. Pfeffer: Anisotropy of the conduction band of InSb: Orbital and spin properties. Phys. Rev. B 46, 9469 (1992)

    Google Scholar 

Electron Spectrum in Strained Cubic Crystals

  • Adams E.N.: Elastoresistance in p-type Ge and Si. Phys. Rev. 96, 803 (1954)

    CAS  Google Scholar 

  • Bahder T.B.: Eight-band k p model of strained zinc-blende crystals. Phys. Rev. B 41, 11992(1990)

    Google Scholar 

  • Bahder T.B.: Analytic dispersion relations near the T point in strained zinc-blende crystals. Phys. Rev. B 45, 1629 (1992)

    Google Scholar 

  • Bir G.L., G.E. Pikus: Effect of strain on energy spectrum and electrical properties of InSb-type semiconductors. Fiz. Tverd. Tela 3, 3050 (1961) [Sov. Phys. - Solid State 3, 2221 (1962)]

    CAS  Google Scholar 

  • Blacha A., H. Presting, M. Cardona: Deformation potentials of k=0 states of tetrahe-dral semiconductors. Phys. Status Solidi (b) 126, 11 (1984)

    CAS  Google Scholar 

  • Enders P., A. Bärwolf,M. Woerner, D. Suisky: k p theory of energy bands, wave functions, and optical selection rules in strained tetrahedral semiconductors. Phys. Rev. B 51, 16695 (1995)

    CAS  Google Scholar 

  • Montasser S.S.: Analytic approach to the inversion-asymmetry splitting of valence band in zinc-blende-type semiconductors. Phys. Rev. B 48, 12285 (1994)

    Google Scholar 

  • Pikus G.E., G.L. Bir: Effect of strain on energy spectrum and electrical properties of p-type germanium and silicon. Fiz. Tverd. Tela 1, 1642 (1959) [Sov. Phys. - Solid State 1, 1502(1960)]

    CAS  Google Scholar 

  • Pollak F.H., M. Cardona: Piezo-electroreflectance in Ge, GaAs and Si. Phys. Rev. 172, 816(1968)

    CAS  Google Scholar 

  • Silver M., W. Batty, E.P.O. Reilly: Strain-induced valence subband splitting in III-V semiconductors. Phys. Rev. B 46, 6781 (1992)

    Google Scholar 

Spectrum of Electrons and Holes in Quantum Wells and Superlattices, a) Nondegenerate Bands, Many-Band Models

  • Ando T.: Valley mixing in short-period superlattices and the interface matrix. Phys. Rev. B 47, 9621 (1993)

    Google Scholar 

  • Balian R., D. Bessis, G.A. Mezincescu: Form of kinetic energy in effective-mass Ham-iltonians for heterostructures. Phys. Rev. B 51, 17624 (1995)

    CAS  Google Scholar 

  • Burnett J.H., H.M. Cheong, W. Paul, E.S. Koteles, B. Elman: T-X mixing in GaAs/A1xGa1-xAs coupled double quantum wells under hydrostatic pressure. Phys. Rev. B 47, 1991 (1993)

    CAS  Google Scholar 

  • Burt M.G.: The justification for applying the effective-mass approximation to microstructures. J. Phys. Condens. Matter 4, 6651 (1992)

    Google Scholar 

  • Chu-liang Y., Y. Qing: Sublevels and excitons in GaAs/A1xGa1-xAs parabolic-quantum-well structures. Phys. Rev. B 37, 1364 (1988)

    Google Scholar 

  • Cnypers J.P., W. Van Haering: Coupling between T and X type envelope function at GaAs/Al(Ga)As interface. Phys. Rev. B 48, 11469 (1993)

    Google Scholar 

  • Dugaev V.K.: Electron energy spectrum and wave functions in quantum wells on the base of IV-VI narrow-gap semiconductors. Phys. Stat. Sol. (b) 184, 347 (1994)

    CAS  Google Scholar 

  • Einevoll G.T., L.T. Sham: Boundary conditions for envelope functions at interfaces between dissimilar materials. Phys. Rev. B 49, 10533 (1994)

    CAS  Google Scholar 

  • Elci A.: Effective band Hamiltonian in semiconductor quantum wells. Phys. Rev. B. 49, 7432 (1994)

    Google Scholar 

  • Ekenberg U.: Nonparabolicity effects in a quantum well: Sublevel shift, parallel mass, and Landau levels. Phys. Rev. B 40, 7714 (1989)

    Google Scholar 

  • Flatte M.E., P.M. Young, L.-H. Peng, H.E. Ehrenreich: Generalized superlattice theory and intersubband optical transitions. Phys. Rev. B 53, 1963 (1996)

    CAS  Google Scholar 

  • Forchhammer T., E. Veje, P. Tidemand-Petersson: Experimental determination of the conduction-band offset at GaAs/Ga1-xAlx As heterojunctions with the use of ballistic electrons. Phys. Rev. B 52, 14693 (1995)

    CAS  Google Scholar 

  • Foreman B.A.: Effective mass Hamiltonian and boundary conditions for the valence bands of semiconductor microstructures. Phys. Rev. B 48, 4964 (1993)

    CAS  Google Scholar 

  • Gershoni D., J. Oiknine-Schlesinger, E. Ehrenfreund, D. Ritter, R.A. Hamm, M.B. Panish: Minibands in the continuum of multi-quantum-well superlattices. Phys. Rev. Lett. 71,2975(1993)

    CAS  Google Scholar 

  • Hassenkam T., S. Pedersen, K. Balkanov, A. Kristensen, C.B. Sorensen, P.E. Liden- lof, F.G. Pikus, G.E. Pikus: Spin splitting and weak localization in (110) GaAs/ AlGaAs quantum wells. Phys. Rev. B, to be published

    Google Scholar 

  • Ivchenko E.L., A.A. Kiselev, Y. Fu, M. Willander: Fine structure of electron-transmission spectra across AlAs single barriers. Phys.Rev. B 50, 7747 (1994)

    Google Scholar 

  • Indjin D., V. Milanovic, Z. Ikonic: Bragg-confining structures with conventional and effective-mass superlattices. Phys. Rev. B 52, 16762 (1995)

    Google Scholar 

  • Knap W.,C. Skierbiszewski, A. Zduniak, E. Litvin-Staszevska, D. Berto, F. Kobbi, J.L. Robert, G.E. Pikus, F.G. Pikus, S.V. Iordanskii, V. Moser, K. Zekenes, Yu.B. Lyanda-Geller: Weak antilocalization and spin precession in quantum wells. Phys. Rev. B 53, 3918 (1996)

    Google Scholar 

  • Laikhtman B.: Boundary conditions for envelope functions in heterostructures. Phys. Rev. B 46, 4769(1992)

    Google Scholar 

  • Lommer G., F. Malcher, U. Rössler: Spin splitting in semiconductor heterostructures for B→0. Phys. Rev. Lett. 60, 728 (1988)

    CAS  Google Scholar 

  • Luo J., H. Munekata, F.F. Fang, P.Y. Stilles: Effects of inversion asymmetry on electron energy band structures in GaSb/InAs/GaSb quantum wells. Phys. Rev. B 41, 7685 (1990)

    CAS  Google Scholar 

  • Pfeiffer P., W. Zawadzki: Spin-splitting of conduction subbands in GaAs/Ga0 7 Al0 3 As heterostructures. Phys. Rev. B 41, 1561 (1995)

    Google Scholar 

  • Pikus F.G., G.E. Pikus: Conduction-band spin splitting and negative magnetoresistance in A3B5 heterostructures. Phys. Rev. B 51, 16928 (1995)

    CAS  Google Scholar 

  • Pötz W, D.K. Ferry: On the boundary conditions for envelope-function approaches for heterostructures. Superlatt. Microstruct. 3, 57 (1987)

    Google Scholar 

  • Rashba E.I., E.Ya. Sherman: Spin-orbital band splitting in symmetric quantum wells. Phys. Lett. A 129, 175 (1988)

    CAS  Google Scholar 

  • Santos P.V., M. Willatzen, M. Cardona, A. Cantarero: Tight-binding calculation of spin-splittings in semiconductor superlattices. Phys. Rev. B 51, 5121 (1995)

    CAS  Google Scholar 

  • Shi J., S. Pan: Envelope wave functions and subband energies in superlattices with complex bases: analytical solution and numerical examples. Phys. Rev. B 48, 8136 (1993)

    CAS  Google Scholar 

  • Ting D.Z.-Y., Y.-C. Chang: T-X mixing in GaAs/AlxGa1-XAs and A1xGa1-xAs/AlAs superlattices. Phys. Rev. B 36, 4359 (1987)

    CAS  Google Scholar 

  • Vecris G., J.J. Quinn: Novel diagrammatic method for analysis of finite periodic and aperiodic multilayer structures. Solid State Commun. 76, 1071 (1990)

    Google Scholar 

  • Winkler R., U. Rössler: General approach to the envelope function approximation based on a quadrature method. Phys. Rev. B 48, 8918 (1993)

    CAS  Google Scholar 

b) Degenerate Bands and Many-Band Models

  • Altarelli M., U. Ekenberg, A. Fasolino: Calculations of hole subbands in semiconductor quantum wells and superlattices. Phys. Rev. B 32, 5138 (1985)

    CAS  Google Scholar 

  • Cuypers J.P., W. van Haeringen: Connection rules for envelope functions at semicon- ductor-heterostructure interfaces. Phys. Rev. B47, 10310 (1993)

    CAS  Google Scholar 

  • Dyakonov M.I., A.V. Khaetskii: Size quantization of holes in a semiconductor with complicated valence band and of carriers in a gapless semiconductor. Zh. Eksp. Teor. Fiz. 82, 1584 (1982) [Sov. Phys. - JETP 55, 917 (1982)]

    CAS  Google Scholar 

  • Edwards G., J.C. Inkson: Hole states in GaAs/AlAs heterostructures and the limitations of the Luttinger model. Solid State Commun. 89, 595 (1994)

    CAS  Google Scholar 

  • Edwards G., E.C. Valadares, F.W. Sheard: Hole-subband-states of GaAs/A1xGaAs quantum wells within the 6×6 Luttinger model. Phys. Rev. B 50, 8493 (1994)

    CAS  Google Scholar 

  • Foreman B.A.: Effective-mass Hamiltonian and boundary conditions for the valence bands of semiconductor microstructures. Phys. Rev. B 48, 4964 (1993)

    CAS  Google Scholar 

  • Iconic Z., V. Milanovic, D. Tjapkin: Valence band structure of [100]-, [110]-, and [lll]-grown GaAs/(Al,Ga)As quantum wells and the accuracy of the axial approximation. Phys. Rev. B 46, 4285 (1992)

    Google Scholar 

  • Ivchenko E.L., A.Yu. Kaminski, U. Rössler: Heavy-light hole mixing at zinc-blende (001) interfaces under normal incidence. Phys. Rev. B 54, 5852 (1996)

    CAS  Google Scholar 

  • Johnson N.F., H. Ehrenreich, P.M. Hui, P.M. Young: Electronic and optical properties of III-V and II-VI semiconductor superlattices. Phys. Rev. B 41, 3655 (1990)

    CAS  Google Scholar 

  • Kriechbaum M.: Envelope function calculation for superlattices. In Two-Dimensional Systems and New Devices, ed. by G. Bauer, F. Kuchar, H. Heinrich, Springer Ser. Solid-State Sci., Vol.67 (Springer, Berlin, Heidelberg 1986) p. 120

    Google Scholar 

  • Meney A.T., B. Gonul, E.P. O’Reily: Evaluation of various approximations used in the envelope-functions method. Phys.Rev. B 50, 10893 (1994)

    CAS  Google Scholar 

  • Nachev I.: Bound states in inversion layers on p-Hg1-xCdxTe: Self-consistent results. Semicond. Sci. Technol. 3, 29 (1988)

    CAS  Google Scholar 

  • Nachev I.S.: Band-mixing and bound states in narrow-gap semiconductors. Physica Scripta 37, 825 (1988)

    CAS  Google Scholar 

  • Nojima S.: Anisotropy of optical transitions in [110] oriented quantum wells. Phys. Rev. B 47, 13535 (1993)

    CAS  Google Scholar 

  • Reboredo F.A., C.R. Proetto: Two-dimensional hole gas in acceptor δ-doped GaAs. Phys.Rev. B47, 4655 (1993)

    CAS  Google Scholar 

  • Richards D., J. Wagner, H. Schneider, G. Hendorfer, M. Maier, A. Fischer, K. Ploog: Two-dimensional hole gas and Fermi-edge singularity in Be-δ-doped GaAs. Phys. Rev. B 47, 9629(1993)

    CAS  Google Scholar 

  • Sobkowicz P.: Theory of n-inversion layers in narrow gap semiconductors: The role of the boundary conditions. Semicond. Sci. Technol. 5, 183 (1990)

    CAS  Google Scholar 

  • Szmulowicz F.: Derivation of a general expression for the momentum matrix element within the envelope-function approximation. Phys. Rev. B 51, 1613 (1994)

    Google Scholar 

c) Strained Quantum Wells and Superlattices

  • Anastassakis E.: Piezoelectric fields in strained heterostructures and superlattices. Phys.Rev. B 46, 4744(1992)

    Google Scholar 

  • Baliga A., D. Tzivedi, N.G. Anderson: Tensile-strain effects in quantum well and superlattice band structure. Phys. Rev. B 49, 10402 (1994)

    CAS  Google Scholar 

  • Bertho D., J.-M. Jancu, C. Jouanin: [001] strain-induced band mixing in zinc-blende semiconductors: Intervalence versus upper-conduction - valence band effect. Phys. Rev. 50, 16956(1994)

    CAS  Google Scholar 

  • Boring P., B. Gil, K.J. Moore: Optical properties and electronic structure of thin (Ga,In)As/AlAs multiple quantum wells and superlattices under internal and external strain fields. Phys. Rev. B 45, 8413 (1992)

    CAS  Google Scholar 

  • Caro De L., L. Tapfer: Strain and piesoelectric fields in orbitrary oriented semiconductor heterostructures: I. Multiple quantum wells. Phys. Rev. B 51, 4374 (1995)

    Google Scholar 

  • Chao C.Y.-P., S.L. Chuang: Spin-orbit-coupling effects on the valence-band structure of strained semiconductor quantum wells. Phys. Rev. B 46, 4110 (1992)

    Google Scholar 

  • Fantner E.J., G. Bauer: Strained layer IV-VI semiconductor superlattices. In Two- Dimensional Systems, Heterostructures, and Superlattices, ed. by G. Bauer, F. Kuchar, H. Heinrich, Springer Ser. Solid-State Sci., Vol.53 (Springer, Berlin, Heidelberg 1984) p.207

    Google Scholar 

  • Fishman G.: Hole subbands in strained quantum-well semiconductors in [hhk] directions. Phys. Rev. B 52, 11132 (1995)

    CAS  Google Scholar 

  • Foreman B.A.: Analytic model for the valence band structure of a straining quantum well. Phys. Rev. B 49, 1757 (1994)

    CAS  Google Scholar 

  • Gil B., P. Lefebvre, P. Bonnel, H. Mathieu, C. Deparis, J. Massies, G. Neu, Y. Chen: Uniaxial-stress investigation of asymmetrical GaAs-(Ga, Al)As double quantum wells. Phys. Rev. B 47, 1954(1993)

    CAS  Google Scholar 

  • Ilg M., K.H. Ploog, A. Trampert: Lateral piezoelectric field in strained semiconductor heterostructures. Phys. Rev. B 50, 17111 (1994)

    CAS  Google Scholar 

  • Jancu J.M., D. Bertho, C. Jouanin, B. Gil, N. Pelekanos, N. Magnea, H. Mariette: Upper-conduction-band effects in heavily strained low-dimensional zinc-blende semiconductor system. Phys. Rev. B 49, 10802 (1994)

    CAS  Google Scholar 

  • Kajikawa Y.: Anomaly in the in-plane polarization properties of (llO)-oriented quantum wells under [110] uniaxial stress. Phys. Rev. B 47, 3649 (1993)

    CAS  Google Scholar 

  • Li T., H.J. Lozykowski, J.L. Reno: Optical properties of CdTe/Cd1-xZnxTe strained-layer single quantum wells. Phys. Rev. B 46, 6961 (1992)

    CAS  Google Scholar 

  • Los J., A. Fasalino, A. Catellanni: Generalization of the k p approach for strained layered semiconductor structures grown of the high-index-planes. Phys. Rev. B 53, 4960 (1996)

    Google Scholar 

  • Moise T.S., L.J. Guido, R.C. Barker: Strain-induced heavy-hole-to-light-hole energy splitting in (lll)B pseudomorphic Iny Ga1-y As quantum wells. Phys. Rev. B 47, 6758 (1193)

    Google Scholar 

  • Shechter G., L.D. Shvartsman, J.E. Golub: Orientation as a key parameter in the val- ence-subband structure engineering of quantum wells. Phys. Rev. B 51, 10857 (1995)

    CAS  Google Scholar 

  • Sirenko Yu.M., J.-B. Jeon, K.W. Kim, M.A. Littlejohn: Envelope-function formalism for valence bands in wurzite quantum wells. Phys. Rev. B 53, 1997 (1996)

    CAS  Google Scholar 

  • Sugawaza M., N. Okazaki, T. Fujii, S. Yamazaki: Conduction-band and valence-band structures in strained In1-xGaxAs/InP quantum wells on (001) InP substrates. Phys. Rev. B 48, 8161 (1993)

    Google Scholar 

  • Talwar D.N., J.P. Loehr, B. Jogai: Comperative study of band-structure calculations for type-II InAs/InxGa1-xSb strained-layer superlattices. Phys. Rev. B 49, 10345 (1993)

    Google Scholar 

  • Valarades E.C.: Strong anisotropy of hole subbands in (311) GaAs/AlAs quantum wells. Phys. Rev. B 46, 3935 (1992)

    Google Scholar 

  • Voisin P.: Strained-layer superlattices. In Two-Dimensional Systems, Heterostructures, and Superlattices, ed. by G. Bauer, F. Kuchar, H. Heinrich, Springer Ser. Solid-State Sci., Vol.53 (Springer, Berlin, Heidelberg 1984) p. 192

    Google Scholar 

  • Weihofen R., G. Weiser, Ch. Starck, R.L. Simes: Energy gaps in strained In1-xGaxAs/In1_yGay AszP1-z quantum wells, grown on (00l)InP. Phys. Rev. B 51, 4296(1995)

    CAS  Google Scholar 

Quantum Wells and Superlattices in Magnetic and Electric Fields

  • Agullo-Rueda F., E.E. Mendez, J.A. Brum, J.M. Hong: Coherence and localization in superlattices under electric fields. Surf. Sci. 228, 80 (1990)

    CAS  Google Scholar 

  • Austin E.J., M. Jaros: Electronic structure of an isolated GaAs-GaAlAs quantum well in a strong electric field. Phys. Rev. B 31, 5569 (1985)

    CAS  Google Scholar 

  • Bastard G., R. Ferreira, S. Chelles, P. Voisin: Interaction between Wannier-Stark states in semiconductor superlattices. Phys. Rev. B 50, 4445 (1994)

    CAS  Google Scholar 

  • Bereford R.: Envelope functions for a three-band semiconductor in a uniform electric field. Phys. Rev. B 49, 13363 (1994)

    Google Scholar 

  • Di Carlo A., P. Vogl, W. Pötz: Theory of Zener tunneling and Wannier-Stark states in semiconductors. Phys. Rev. B 50, 8358 (1994)

    Google Scholar 

  • Fasolino A., Altarelli, M.: Subband structure and Landau levels in heterostructures. In Two-Dimensional Systems, Heterostructures, and Superlattices, ed. by G. Bauer, F. Kuchar, H. Heinrich, Springer Ser. Solid-State Sci., Vol.53 (Springer, Berlin, Heidelberg 1984) p. 176

    Google Scholar 

  • Ferreira R., G. Bastard: Wannier-Stark levels in the valence band of semiconductor multiple quantum wells. Phys. Rev. B 38, 8406 (1988)

    Google Scholar 

  • Fukuyama H., R.A. Bari, H.C. Fogedby: Tightly bound electrons in a uniform electric field. Phys. Rev. B 8, 5579 (1973)

    Google Scholar 

  • Goldoni G., A. Fasalino: Hole states in quantum wells in high in-plane magnetic fields: Implications for resonant magnetotunneling spectroscopy. Phys. Rev. B 48, 4948 (1993)

    CAS  Google Scholar 

  • Greene S.K., J. Singelton, P. Sobkowicz, T.D. Golding, M. Pepper, J.A.A.J. Perenboom, J. Dinan: Subband occupancies and zero-field spin splitting in InSb-CdTe het- erojunctions: Magnetotransport and self-consistent calculations. Semicond. Sci. Technol. 7, 1377 (1992)

    CAS  Google Scholar 

  • Hagon J.P., M. Jaros: Stark shifts in GaAs/Ga1-xA1xAs finite-length superlattices. Phys.Rev. B 41, 2900 (1990)

    Google Scholar 

  • Hembree C.E., B.A. Mason, A. Zhang, J.A. Slinkman: Subband spectrum of a parabolic quantum well in a perpendicular magnetic field. Phys. Rev. B 46, 7588 (1992)

    Google Scholar 

  • Ivchenko E.L., A.A. Kiselev: Electron g-factor of quantum wells and superlattices. Fiz. Techn. Poluprovodn. 26, 1471 (1992) [Sov. Phys. - Semicond. 26, 827 (1992)]

    Google Scholar 

  • Kajikawa Y.: Level anticrossing and related giant optical anisotropy caused by the Stark effect in a strained (110) quantum well. Phys. Rev. B 49, 8136 (1994)

    CAS  Google Scholar 

  • Krieger J.B., G.J. Iafrate: Time evolution of Bloch electrons in a homogeneous electric field. Phys. Rev. B 33, 5494 (1986)

    Google Scholar 

  • Matsuura M., T. Kamizato: Subbands and excitons in a quantum well in an electric field. Phys. Rev. B 33, 8385 (1986)

    Google Scholar 

  • Merlin R.: Subband-Landau-level coupling in tilted magnetic fields: Exact results for parabolic wells. Solid State Commun. 64, 99 (1987)

    Google Scholar 

  • Morifuji M., M. Yamaguchi, K. Taniguchi, C. Hamaguchi: Electric-field-induced T-X mixing between Stark ladders in short-period GaAs/AlAs supperlattices. Phys. Rev. B 50, 8722(1994)

    CAS  Google Scholar 

  • Müller W., H.T. Grahn, K. von Klitzing, K. Ploog: Resonant tunneling in crossed electric and magnetic fields in GaAs-AlAs superlattices. Phys. Rev. B 48, 11176 (1993)

    Google Scholar 

  • Nakayama M., M. Ando, I. Tanaka, H. Nishimura, H. Schneider, K. Fujiwara: Electric-field effect on above-barrier states in a GaAs/AlxGa1-xAs superlattice. Phys. Rev. B51,4236 (1995)

    CAS  Google Scholar 

  • Pacheco M., Z. Barticevic, F. Claro: Optical response of a superlattice in parallel magnetic and electric fields. Phys. Rev. B 46, 15200 (1993)

    Google Scholar 

  • Platero G., M. Altarelli: Valence-band levels and optical transitions in quantum wells in a parallel magnetic field. Phys. Rev. B 39, 3758 (1989)

    CAS  Google Scholar 

  • Ritze M., N.J.M. Horing, R. Enderlein: Density of states and Wannier-Stark levels of superlattices in an electric field. Phys. Rev. B 47, 10437 (1993)

    CAS  Google Scholar 

  • Schmidt K.H., N. Under, G.H. Döhler, H.T. Grahn, K. Ploog, H. Schneider: Coexistence of Wannier-Stark transitions and miniband Franz-Keldysh oscillations in strongly coupled GaAs/Al As superlattices. Phys. Rev. Lett. 72, 2769 (1994)

    CAS  Google Scholar 

  • Schwedler R., F. Brüggemann, A. Ziebell, B. Opitz, B., A. Kohl, H. Kurz: Valence band structure of fractional Wannier-Stark-effect shallow InGaAs/InGaAs superlattices. Superlatt. Microstruct. 15, 145 (1994)

    CAS  Google Scholar 

  • Smith T.P., F.F. Fang: g-Factor of electrons in an InAs quantum well. Phys. Rev B 35, 7729 (1977)

    Google Scholar 

  • Wollrab R., R. Sizmann, F. Koch, J. Ziegler, H. Maier: Spin splitting of electron subbands in the electrostatic interface potential on HgCdTe. Semicond. Sci. Technol 4 491 (1989)

    CAS  Google Scholar 

  • Wong S.L., R.W. Martin, M. Lakrimi, R.J. Nicholas, T.-Y. Seong, N.J. Mason, P.J. Walker: Optical and transport properties of piezoelectric [lll]-oriented strained Ga1-xInxSb/GaSb quantum wells. Phys. Rev. B 48, 17885 (1993)

    CAS  Google Scholar 

  • Xu W.: Electronic structure of AlxGa1-xAs-GaAs-AlyGa1-y As single quantum wells subjected to in-plane magnetic field. Phys. Rev. B 51, 9770 (1995)

    CAS  Google Scholar 

  • Zawadski W., S. Klahn, H. Merkt: Inversion electrons on narrow-band-gap semiconductors in crossed electric and magnetic fields. Phys. Rev. B 33, 6916 (1986)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. A.F. Ioffe Physico-Technical Institute, Politechnicheskaya 26, St. Petersburg, 194021, Russia

    Professor Eougenious L. Ivchenko & Professor Grigory E. Pikus

Authors
  1. Professor Eougenious L. Ivchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Professor Grigory E. Pikus
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Ivchenko, E.L., Pikus, G.E. (1997). Electron Spectrum in Crystals, Quantum Wells and Superlattices. In: Superlattices and Other Heterostructures. Springer Series in Solid-State Sciences, vol 110. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60650-2_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-60650-2_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-64493-1

  • Online ISBN: 978-3-642-60650-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation