Skip to main content
SpringerLink
Log in

Effect of Quantum Confinement on Electrons and Phonons in Semiconductors

  • Chapter
Fundamentals of Semiconductors

Abstract

In Chap. 5 we studied the Gunn effect as an example of negative differential resistance (NDR). This effect is observed in semiconductors, such as GaAs, whose conduction band structure satisfies a special condition, namely, the existence of higher conduction minima separated from the band edge by about 0.2–0.4 eV. As a way of achieving this condition in any semiconductor, Esaki and Tsu proposed in 1970 [9.1] the fabrication of an artificial periodic structure consisting of alternate layers of two dissimilar semiconductors with layer thicknesses of the order of nanometers. They called this synthetic structure a superlattice They suggested that the artificial periodicity would fold the Brillouin zone into smaller Brillouin zones or “mini-zones” and therefore create higher conduction band minima with the requisite energies for Gunn oscillations.

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 74.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

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

Chapter 9

  1. L. Esaki, R. Tsu: Superlattice and negative differential conductivity in semiconductors. IBM J. Res. Devel. 14, 61–65 (1970)

    Article  Google Scholar 

  2. K.K. Tiong, P.M. Amirtharaj, F.H. Pollak, D.E. Aspnes: Effects of As+ ion implantation on the Raman spectra of GaAs: “Spatial correlation” interpretation. Appl. Phys. Lett. 44, 122–124 (1984)

    Article  ADS  Google Scholar 

  3. N. Tomassini, A. d’Andrea, R. del Sole, H. Tuffigo-Ulmer, R.T. Cox: Center of mass quantization of excitons in CdTe/Cd1-xZnxTe quantum wells. Phys. Rev. B 51, 5005–5012 (1995)

    Article  ADS  Google Scholar 

  4. M. Asada, Y. Migamoto, Y. Suematsu: Gain and the threshold of three dimensional quantum-box lasers. IEEE J. QE-22, 1915–1921 (1986)

    Article  Google Scholar 

  5. C. Weisbuch, B. Vinter: Quantum Semiconductor Structures, Fundamentals and Applications (Academic, San Diego, CA 1991)

    Google Scholar 

  6. J. Singh: Physics of Semiconductors and their Heterostructures (McGraw-Hill, New York 1993) p.524

    Google Scholar 

  7. R. Dingle, W. Wiegmann, C.H. Henry: Quantum states of confined carriers in very thin AlxGa1-xAs-GaAs-AlxGa1-xAs heterostructures. Phys. Rev. Lett. 33, 827–830 (1974)

    Article  ADS  Google Scholar 

  8. M.H. Meynadier, C. Delalande, G. Bastard, M. Voos, F. Alexandre, J.L. Lievin: Size quantization and band-offset determination in GaAs-GaAlAs separate confinement heterostructure. Phys. Rev. B 31, 5539–5542 (1985)

    Article  ADS  Google Scholar 

  9. M.S. Skolnick, P.R. Tapster, S.J. Bass, A.D. Pitt, N. Apsley, S.P. Aldredy: Investigation of InGaAs-InP quantum wells by optical spectroscopy. Semicond. Sci. Technol. 1, 29–40 (1986)

    Article  ADS  Google Scholar 

  10. F. Capasso: Band-gap engineering: From physics and materials to new semiconductor devices. Science 235, 172–176 (1987)

    Article  ADS  Google Scholar 

  11. T.P. Pearsall (ed.): Strain-Layer Superlattices: Physics, Semiconductors and Semimetals, Vol.32 (Academic, New York, NY 1990)

    Google Scholar 

  12. L. Esaki: Semiconductor superlattices and quantum wells, in Proc. 17th Int’l Conf. on The Physics of Semiconductors, ed. by J. Chadi, W.A. Harrison (Springer, New York, Berlin 1984) pp.473–483

    Google Scholar 

  13. K. Ploog, G.H. Döhler: Compositional and doping superlattices in III–V semiconductors. Adv. Phys. 32, 285–359 (1983)

    Article  ADS  Google Scholar 

  14. J. Ihm: Effect of the layer thickness on the electronic character in GaAs-AlAs superlattices. Appl. Phys. Lett. 50, 1068–1070 (1987)

    Article  ADS  Google Scholar 

  15. S. Gopalan, N.E. Christensen, M. Cardona: Band edge states in short period (GaAs)m(AlAs)m superlattices. Phys. Rev. B 39, 5165–5174 (1989)

    Article  ADS  Google Scholar 

  16. L.I. Schiff: Quantum Mechanics (McGraw-Hill, New York 1955)

    MATH  Google Scholar 

  17. G. Bastard: Superlattice band structure in the envelope-function approximation. Phys. Rev. 24, 5693–5697 (1981)

    Article  ADS  Google Scholar 

  18. G. Bastard, J.A. Brum, R. Ferreira: Electronic states in semiconductor heterostructures. Solid State Physics 4, 229–415 (Academic, San Diego, CA 1991)

    Article  Google Scholar 

  19. E.L. Ivchenko, G. Pikus: Superlattices and Other Heterostructures: Symmetry and Optical Phenomena, Springer Ser. Solid-State Sci., Vol.110 (Springer, Berlin, Heidelberg 1995)

    Book  Google Scholar 

  20. G. Bastard, J.A. Brum: Electronic states in semiconductor heterostructures. IEEE J. QE-22, 1625–1644 (1986)

    Google Scholar 

  21. J.C. Hensel, G. Feher: Cyclotron resonance experiment in uniaxially stressed silicon: Valence band inverse mass parameters and deformation potentials. Phys. Rev. 129, 1041–1062 (1963)

    Article  ADS  MATH  Google Scholar 

  22. L. Liboff: Introductory Quantum Mechanics (Addison-Wesley, Reading, MA 1980) pp.267–279

    Google Scholar 

  23. L. Esaki: A perspective in superlattice development, in Symp. on Recent Topics in Semiconductor Physics (Prof. Y. Uemura’s Festschrift) (World Scientific, Singapore 1983) pp. 1–71

    Google Scholar 

  24. Y.C. Chang, J.N. Schulman: Interband optical transitions in GaAs-Ga1-xAlxAs and InAs-GaSb superlattices. Phys. Rev. B 31, 2069–2079 (1985)

    Article  ADS  Google Scholar 

  25. R. Dingle: Optical properties of semiconductor superlattices, in Proc. 13th Int’l Conf. on the Physics of Semiconductors, ed. by F.G. Fumi (Tipografia Marves, Rome 1976) pp.965–974

    Google Scholar 

  26. J.A. Kash, M. Zachau, M.A. Tischler: Anisotropic valence bands in quantum wells: Quantitative comparison of theory and experiment. Phys. Rev. Lett. 69, 2260–2263 (1992)

    Article  ADS  Google Scholar 

  27. G. Fasol, W. Hackenberg, H.P. Hughes, K. Ploog, E. Bauser, H. Kano: Continous-wave spectroscopy of femtosecond carrier scattering in GaAs. Phys. Rev. B 41, 1461–1478 (1990)

    Article  ADS  Google Scholar 

  28. A. Pinczuk, G. Abstreiter: Spectroscopy of free carries excitations in semiconductor quantum wells, in Light Scattering in Solids V, ed. by M. Cardona, G. Güntherodt, Topics Appl. Phys., Vol.66 (Springer, Berlin, Heidelberg 1989) pp.153–211

    Chapter  Google Scholar 

  29. P. Giannozzi, S.R. Gironcoli, P. Pavone, S. Baroni: Ab initio calculation of phonon dispersions in semiconductors. Phys. Rev. B 43, 7231–7242 (1991)

    Article  ADS  Google Scholar 

  30. S.M. Rytov: Acoustic properties of a thinly laminated medium. Sov. Phys. — Acoust. 2, 68–80 (1956)

    Google Scholar 

  31. B. Jusserand, M. Cardona: Raman spectroscopy of vibrations in superlattices, in Light Scattering in Solids V, ed. by M. Cardona, G. Güntherodt, Topics Appl. Phys., Vol.66 (Springer, Berlin, Heidelberg 1989)

    Google Scholar 

  32. J. Menéndez: Phonons in GaAs-AlxGa1-xAs superlattices. J. Luminesc. 44, 285–314 (1989)

    Article  Google Scholar 

  33. P. Molinas-Mata, M. Cardona: Planar force-constant models and internal strain parameter of Ge and Si. Phys. Rev. B 43, 9959–9961 (1991)

    Article  Google Scholar 

  34. R.A. Ghanbari, J.D. White, G. Fasol, C.J. Gibbings, C.G. Tuppen: Phonon frequencies for Si-Ge strained-layer superlattices calculated in a three-dimensional model. Phys. Rev. B 42, 7033–7041 (1990)

    Article  ADS  Google Scholar 

  35. W. Richter, D. Strauch: Lattice dynamics of GaAs/AlAs superlattices. Solid-State Commun. 64, 867–872 (1987)

    Article  ADS  Google Scholar 

  36. E. Molinari, A. Fasolino, K. Kunc: Superlattice effects on confined phonons. Phys. Rev. Lett. 56, 1751 (1986)

    Article  ADS  Google Scholar 

  37. R. Fuchs, K.L. Kliewer: Optical modes of vibration in an ionic crystal slab. Phys. Rev. A 140, 2076–2088 (1965)

    ADS  Google Scholar 

  38. M. Nakayama, M. Ishida, N. Sano: Raman scattering by interface-phonon polar-itons in GaAs-AlAs heterostructures. Phys. Rev. B 38, 6348–6351 (1988)

    Article  ADS  Google Scholar 

  39. R. Merlin, C. Colvard, M.V. Klein, H. Morkoc, A.Y. Cho, A.C. Gossard: Raman scattering in superlattices: Anisotropy of polar phonons. Appl. Phys. Lett. 36, 43–45 (1980)

    Article  ADS  Google Scholar 

  40. C. Trallero-Giner, F. Garciá-Moliner, V.R. Velasco, M. Cardona: Analysis of the phenomenological models for long-wavelength polar optical modes in semiconductor layered systems. Phys. Rev. B 45, 11944–11948 (1992)

    Article  ADS  Google Scholar 

  41. Z.V. Popovic, M. Cardona, E. Richter, D. Strauch, L. Tapfer, K. Ploog: Phonons in GaAs/AlAs superlattices grown along the (111) direction. Phys. Rev. B 41, 5904–5913 (1990)

    Article  ADS  Google Scholar 

  42. M.P. Chamberlain, M. Cardona, B.K. Ridley: Optical modes in GaAs/AlAs superlattices. Phys. Rev. B 48, 14356–14364 (1993)

    Article  ADS  Google Scholar 

  43. M. Zunke, R. Schorer, G. Abstreiter, W. Klein, G. Weimann, M.P. Chamberlain: Angular dispersion of confined optical phonons in GaAs/AlAs superlatices studied by micro-Raman spectroscopy. Solid State Commun. 93, 847–851 (1995)

    Article  ADS  Google Scholar 

  44. G. Scarmarcio, L. Tapfer, W. König, A. Fischer, K. Ploog, E. Molinari, S. Baroni, P. Giannozzi, S. de Gironcoli: Infrared reflectivity by transverse-optical phonons in (GaAs)m/(AlAs)n ultrathin-layer superlattices. Phys. Rev. B 43, 14754–14757 (1991)

    Article  ADS  Google Scholar 

  45. E. Jahne, A. Röseler, K. Ploog: Infrared reflectance and ellipsometric studies of GaAs/AlAs superlattices. Superlattices Microstruct. 9, 219–222 (1991)

    Article  ADS  Google Scholar 

  46. Yu. Pusep, A. Milekhin, A. Poropov: FTIR spectroscopy of (GaAs)n/(AlAs)m superlatices. Superlattices Microstruct. 13, 115–123 (1992)

    Article  ADS  Google Scholar 

  47. M. Krieger H. Sigg: Zone-edge gap in the folded acoustic phonon dispersion of an AlAs-GaAs semiconductor superlattice, in the Physics of Semiconductors, ed. by J. Lockwood (World Scientific, Singapore 1995) pp.959–962

    Google Scholar 

  48. C. Colvard, T.A. Gant, M.V. Klein, R. Merlin, R. Fischer, H. Morkoc, A.C. Gossard: Folded acoustic phonons and quantized optic phonons in (GaAl)As superlattices. Phys. Rev. B 31, 2080–2091 (1985)

    Article  ADS  Google Scholar 

  49. J. He, B. Djafari Rouhani, J. Sapriel: Theory of light scattering by longitudinal-acoustic phonons in superlattices. Phys. Rev. B 37, 4086–4098 (1988)

    Article  ADS  Google Scholar 

  50. J. He, J. Sapriel, H. Brugger: Semiconductor photoelastic constants measured by light scattering in superlattices. Phys. Rev. B 39, 5919–5923 (1989)

    Article  ADS  Google Scholar 

  51. J. Spitzer, Z.V. Popovic, T. Ruf, M. Cardona, R. Nötzel, K. Ploog: Folded acoustic phonons in GaAs/AlAs superlattices grown on non-(100)-oriented surface. Solid State Electron. 37, 753–756 (1994)

    Article  ADS  Google Scholar 

  52. T. Ruf, J. Spitzer, V.F. Sapega, V.I. Belitsky, M. Cardona, K. Ploog: Interface roughness and homogeneous linewidths in quantum wells and superlattices studied by resonant acoustic-phonon Raman scattering. Phys. Rev. B 50, 1792–1806 (1994)

    Article  ADS  Google Scholar 

  53. J.E. Zucker, A. Pinczuk, D.S. Chemla, A.C. Gossard, W. Wiegmann: Optical vibrational modes and electron-phonon interaction in GaAs quantum wells. Phys. Rev. Lett. 53, 1280–1283 (1984)

    Article  ADS  Google Scholar 

  54. A. Fainstein, P. Etchegoin, M.P. Chamberlain, M. Cardona, K. Tötemeyer, K. Eberl: Selection rules and dispersion of GaAs/AlAs multiple quatum well optical phonons studied by Raman scattering in right angle, forward and backscattering in-plane geometries. Phys. Rev. B 51, 14448–14458 (1995)

    Article  ADS  Google Scholar 

  55. A.J. Shields, M.P. Chamberlain, M. Cardona, K. Eberl: Raman scattering due to interface optical phonons in GaAs/AlAs multiple quantum wells. Phys. Rev. B 51, 17728–17739 (1995)

    Article  ADS  Google Scholar 

  56. A.K. Sood, J. Menéndez, M. Cardona, K. Ploog: Interface vibrational modes in GaAs-AlAs superlattices. Phys. Rev. Lett. 54, 2115–2118 (1985)

    Article  ADS  Google Scholar 

  57. K. Huang, B. Zhu: Dielectric continuum models and Fröhlich interaction in superlattices. Phys. Rev. B 38, 13377–13386 (1988)

    Article  ADS  Google Scholar 

  58. H. Rücker, E. Molinari, P. Lugli: Microscopic calculation of the electron-phonon interaction in quantum wells. Phys. Rev. B 45, 6747–6756 (1992)

    Article  ADS  Google Scholar 

  59. S. Rudin, T. Reinecke: Electron-LO-phonon scattering rates in semiconductor quantum wells. Phys. Rev. B 41, 7713–7717 (1991)

    Article  ADS  Google Scholar 

  60. K.T. Tsen, K.R. Wald, T. Ruf, P.Y. Yu, H. Morkoc: Electron-optical phonon interactions in ultrathin multiple quantum wells. Phys. Rev. Lett. 67, 2557–2560 (1991)

    Article  ADS  Google Scholar 

  61. L. Esaki: New phenomenon in narrow germanium p-n junctions. Phys. Rev. 109, 603–604 (1958)

    Article  ADS  Google Scholar 

  62. R. Tsu, L. Esaki: Tunneling in a finite superlattice. Appl. Phys. Lett. 22, 562–564 (1973)

    Article  ADS  Google Scholar 

  63. T.C.L.G. Sollner, W.D. Goodhue, P.E. Tannenwald, C.D. Parker, D.D. Peck: Resonant tunneling through quantum wells at frequencies up to 2.5 THz. Appl. Phys. Lett. 43, 588–590 (1983)

    Article  ADS  Google Scholar 

  64. A. Silbille, J.F. Palmier, H. Wang, F. Mollot: Observation of Esaki-Tsu negative differential velocity in GaAs/AlAs superlattices. Phys. Rev. Lett. 64, 52–55 (1990)

    Article  ADS  Google Scholar 

  65. J.F. Palmier: Miniband transport and resonant tunneling in superlattices, in Resonant Tunneling in Semiconductors, ed. by L.L. Chang (Plenum, New York 1991) pp.361–375

    Chapter  Google Scholar 

  66. E. Merzbacher: Quantum Mechanics (Wiley, New York 1961) pp.94–97

    MATH  Google Scholar 

  67. D.Z.-Y. Ting, S.K. Kirby, T.C. McGill: Interface roughness effects in resonant tunneling structures. Appl. Phys. Lett. 64, 2004–2006 (1994)

    Article  ADS  Google Scholar 

  68. T.P.E. Broekaert, W. Lee, C.G. Fonstad: Pseudomorphic In0.53Ga04.7As/AlAs/InAs resonant tunneling diodes with peak-to-valley current ratios of 30 at room temperature. Appl. Phys. Lett. 53, 1545–1547 (1988)

    Article  ADS  Google Scholar 

  69. K. von Klitzing, G. Dorda, M. Pepper: New method for high-accuracy determination of the fine-structure constant based on quantum Hall resistance. Phys. Rev. Lett. 45, 494–497 (1980)

    Article  ADS  Google Scholar 

  70. D.C. Tsui, H.L. Störnier, A.C. Gossard: Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 45, 1559–1562 (1982)

    Article  ADS  Google Scholar 

  71. L.D. Landau: Diamagnetism of metals (in German). Z. Physik 64, 629–637 (1930)

    Article  ADS  MATH  Google Scholar 

  72. L.D. Landau, E.N. Lifshitz: Quantum Mechanics, 3rd edn. (Pergamon, Oxford 1977)

    Google Scholar 

  73. J.M. Ziman: Principles of the Theory of Solids, 2nd edn. (Cambridge Univ. Press, Cambridge 1972) pp. 313–326

    Book  Google Scholar 

  74. M. Brodsky (ed.): Amorphous Semiconductors, 2nd edn., Topics Appl. Phys., Vol.36 (Springer, Berlin, Heidelberg 1985)

    Google Scholar 

  75. J. Tauc: Amorphous and Liquid Semiconductors (Plenum, New York 1974)

    Book  Google Scholar 

  76. R. Loudon: Raman effect in crystals. Adv. Phys. 13, 423–482 (1964); erratum ibid. 14, 621 (1965)

    Article  ADS  Google Scholar 

General Reading, Growth of Quantum Wells and Superlattices

  • Herman M.A., H. Sitter: Molecular Beam Epitaxy, 2nd edn., Springer Ser. Mater. Sci., Vol.7 (Springer, Berlin, Heidelberg 1996)

    Book  Google Scholar 

  • Panish M.B., H. Temkin: Gas Source Molecular Beam Epitaxy, Springer Ser. Mater. Sci., Vol.26 (Springer, Berlin, Heidelberg 1993)

    Book  Google Scholar 

  • Ploog K.: Formation of semiconductor interfaces during molecular beam epitaxy, in Semiconductor Interfaces: Formation and Properties, ed. by G. LeLay, J. Denien, N. Boccara (Springer, Berlin, Heidelberg 1987)

    Google Scholar 

  • Stradling R.A., P.C. Klipstein (eds.): Growth and Characterization of Semiconductors (Hilger, Bristol, UK 1990)

    Google Scholar 

Properties of Quantum Wells and Superlattices

  • Dekker A.J.: Solid State Physics (Prentice Hall, Englewood Cliffs, NJ 1957) p.244

    Google Scholar 

  • Ehrenreich H., D. Turnbull (eds.): Semiconductor Heterostructures and Nanostructures, Adv. Res. Appl., Vol.44 (Academic, San Diego, CA 1991)

    Google Scholar 

  • Ivchenko E.L., G. Pikus: Superlattices and Other Heterostructures: Symmetry and Other Optical Phenomena, Springer Ser. Solid-State Sci., Vol.110 (Springer, Berlin, Heidelberg 1995)

    Book  Google Scholar 

  • Janssen M., O. Viehweger, V. Fastenrath, J. Hajdu: Introduction to the Theory of the Quantum Hall Effect (VCH, Weinheim 1994)

    Google Scholar 

  • Jaros M.: Superlattices of Semiconductor Microstructures (Clarendon, Oxford 1985)

    Google Scholar 

  • Jusserand B., M. Cardona: Raman spectroscopy of vibrations in superlattices, in Light Scattering in Solids V, ed. by M. Cardona, G. Güntherodt, Topics Appl. Phys., Vol.66 (Springer, Berlin, Heidelberg 1989) pp. 49–152

    Chapter  Google Scholar 

  • Kroemer H.: Theory of heterojunction: A critical review, in Molecular Beam Epitaxy and Heterostructures, ed. by L.L. Chang, K. Ploog (Nijhoff, Dordrecht 1985) pp.331–379

    Chapter  Google Scholar 

  • Menéndez J.: Phonons in GaAs-AlxGa1-xAs superlattices. J. Luminesc. 44, 285–314 (1989)

    Article  Google Scholar 

  • Pearsall T.P. (ed.): Strain-Layer Superlattices: Physics, Semiconductors and Semimetals, Vol.32 (Academic, Orlando, FL 1990)

    Google Scholar 

  • Pinczuk A., G. Abstreiter: Spectroscopy of free carriers excitations in semiconductor quantum wells, in Light Scattering in Solids V, ed. by M. Cardona, G. Guntherod, Topics Appl. Phys., Vol.66 (Springer, Berlin, Heidelberg 1989) Chap.4, pp. 153–211

    Chapter  Google Scholar 

  • Sapriel J., B. Djafari Rouani: Vibrations in superlattices. Surf. Sci. Rept. 10, 189–275 (1989)

    Article  Google Scholar 

  • Singh J.: Physics of Semiconductors and their Heterostructures (McGraw-Hill, New York 1993)

    Google Scholar 

  • Wang S.: Fundamentals of Semiconductor Theory and Device Physics (Prentice Hall, Englewood Cliffs, NJ 1989)

    Google Scholar 

  • Weisbuch C., B. Vinter: Quantum Semiconductor Structures, Fundamentals and Applications (Academic, San Diego, CA 1991)

    Google Scholar 

Quantum Hall Effect

  • Chakraborty T., P. Pietiläinen: The Quantum Hall Effects — Fractional and Integral, 2nd edn., Springer Ser. Solid-State Sci., Vol.85 (Springer, Berlin, Heidelberg 1995)

    Book  Google Scholar 

  • Janssen M., O. Viehweger, U. Fastenrath, J. Hajdu: Introduction to the Theory of the Integer Quantum Hall Effect (VCH, Weinheim 1994)

    Google Scholar 

  • McDonald A.H. (ed.): Quantum Hall Effect: A Perspective (Kluwer, Boston 1989)

    Google Scholar 

  • Prange R.E., S.M. Girvin (eds.): The Quantum Hall Effect (Springer, New York, Berlin 1990)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, University of California, 94720-7300, Berkeley, CA, USA

    Professor Dr. Peter Y. Yu

  2. Max-Planck-Institut für Festkörperforschung, Heisenbergstrasse 1, D-70569, Stuttgart, Germany

    Professor Dr., Dres. h.c. Manuel Cardona

Authors
  1. Professor Dr. Peter Y. Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Professor Dr., Dres. h.c. Manuel Cardona
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Yu, P.Y., Cardona, M. (1996). Effect of Quantum Confinement on Electrons and Phonons in Semiconductors. In: Fundamentals of Semiconductors. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03313-5_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-03313-5_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61461-6

  • Online ISBN: 978-3-662-03313-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation