Abstract
The motion of a one-dimensional Bloch electron with mass m and charge —e in an external field F is governed by the Hamiltonian
The eigenvalue problem of Ĥ,known as the Wannier—Stark problem, has drawn the attention of generations of physicists.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
F. Bloch: Z. Physik 52, 555 (1928)
A. Sommerfeld, H. Bethe. In: Handbuch der Physik, ed by H. Geiger, K. Scheel, Band 24, Teil 2 ( Springer, Berlin 1933 ) pp 333–413
W. Franz: Z. Naturforschg. 13 a, 484 (1958)
L. V. Keldysh: ZETP 34, 1138 (1958) [Sov. Phys. JETP 34, 788 (1958)]
L. Fritsche: phys. stat. sol. 13, 487 (1966)
C. Zener, Proc. Roy. Soc. A 145, 523 (1934)
W.V. Houston: Phys. Rev. 57, 184 (1940)
P. Feuer: Phys. Rev. 88, 92 (1952)
E. N. Adams: J. Chem. Phys. 21, 2013 (1953)
E. N. Adams: Phys. Rev. 107, 698 (1957)
L. V. Keldysh: ZETP 33, 994 (1957) [Sov. Phys. JETP 6, 763 (1958)]
E. O. Kane: J. Phys. Chem. Solids. 12, 181 (1959)
G. H. Wannier: Phys. Rev. 117, 432 (1960)
E. O. Kane: J. Appl. Phys. 32, 83 (1961)
D. E. Aspnes, N. Bottka. In: Semiconductors and Semimetals, ed by R. K. Willardson, A. C. Beer, vol 9 ( Academic, New York 1972 ) pp 457–543
J. B. Krieger, G. J. Iafrate: Phys. Rev. B 33, 5494 (1986)
D. A.B. Miller, D. S. Chemla, T. C. Damen, A. C Gossard, W. Wiegmann, T. H. Wood, C. A. Burrus: Phys. Rev. B 32, 1043 (1985)
S. Katsura, T. Hatta, A. Morita, Sci. Rep. Tôhuku Imp. Univ. (Ser. I ) 34, 19 (1950)
K. Hacker, G. Obermair: Z. Physik 234, 1 (1970)
S. G. Davison, K. P. Tan: Z. Physik 251, 6 (1972)
G. C. Stay, G. Gusman: J. Phys. C 6, 650 (1973)
H. Fukuyama, R. A. Bari, H. C. Fogedby: Phys. Rev. B 8, 5579 (1973)
J. E. Avron: Phys. Rev. Lett. 37, 1568 (1976)
J. Zak: Phys. Rev. Lett. 20, 1477 (1968); G. H. Wannier: Phys. Rev. 181, 1364 (1969); J. Zak: ibid., 1366 (1969)
A. Rabinovitch, J. Zak: Phys. Rev. B 4, 2358 (1971); W. Shockley: Phys. Rev. Lett. 28, 349 (1972)
A. Rabinovitch, J. Zak: Phys. Lett. 40A, 189 (1972)
J. E. Avron, J. Zak, A. Grossmann, L. Gunther: J. Math. Phys. 18, 918 (1977)
F. Bentosela, R. Carmona, P. Duclos, B. Simon, B. Souillard, R. Weder: Comm. Math. Phys. 88, 387 (1983)
P. Ao: Phys. Rev. B 41, 3998 (1990)
D. Emin, C.F. Hart: Phys. Rev. B 36, 7353 (1987); C.F. Hart and D. Emin: Phys. Rev. B 37, 6100 (1988); C.F. Hart: Phys. Rev. B 38, 2158 (1988); L. Kleinman, Phys. Rev. B 41, 3857 (1990); D. Emin, C. F. Hart: ibid., 3859 (1990); D. A. Page, E. Brown: Phys. Rev. B 43, 2423 (1991); D. Emin, C.F. Hart: ibid., 2426 (1991); J. Zak: Phys. Rev. B 43, 4519 (1991); C.F. Hart, D. Emin: ibid., 4521 (1991); J. Leo, A. MacKinnon: Phys. Rev. B 43, 5166 (1991); D. Emin, C.F. Hart: ibid., 5169 (1991); C. F. Hart: Phys. Lett. A 132, 453 (1988); P.N. Argyres, S. Sfiat: Phys. Lett. A 146, 231 (1990)
J. J. Sakurai: Modern Quantum Mechanics, revised edn, ed by S. F. Tuan ( Addison-Wesley, Reading 1994 )
M. Reed, B. Simon: Methods of Mathematical Physics, vol IV: Analysis of Operators ( Academic, San Diego 1978 )
I. W. Herbst, J. S. Howland: Comm. Math. Phys. 80, 23 (1981)
J.-M. Combes, P. D. Hislop: Comm. Math. Phys. 140, 291 (1991)
F. Bentosela, V. Grecchi: Comm. Math. Phys. 142, 169 (1991)
P. D. Hislop, I. M. Sigal: Introduction to Spectral Theory With Application to Schrödinger Operators ( Springer, New York 1996 )
E. Merzbacher: Quantum Mechanics, 3rd edn ( Wiley, New York 1998 )
I. W. Herbst: Comm. Math. Phys. 75, 197 (1980)
V. Grecchi, M. Maioli, A. Sacchetti: J. Phys. A 26, L379 (1993); Comm. Math. Phys. 159, 605 (1994)
L. Esaki, R. Tsu, IBM J. Res. Dev. 14, 61 (1970)
E. E. Mendez, F. Agulló-Rueda, J. M. Hong: Phys. Rev. Lett. 60, 2426 (1988); F. Agulló-Rueda, E. E. Mendez, J. M. Hong: Phys. Rev. B 40, 1357 (1989)
P. Voisin, J. Bleuse, C. Bouche, S. Gaillard, C. Alibert, and A. Regreny: Phys. Rev. Lett. 61, 1639 (1988)
H. Schneider, H. T. Grahn, K. v. K.itzing, K. Ploog: Phys. Rev. Lett. 65, 2720 (1990)
A. Alexandrou, E. E. Mendez, J. M. Hong: Phys. Rev. B 44, 1934 (1991)
K. H. Schmidt, N. Linder, G. H. Döhler, H. T. Grahn, K. Ploog, H. Schneider: Phys. Rev. Lett. 72, 2769 (1994); N. Linder, K. H. Schmidt, W. Geisselbracht, G. H. Döhler, H. T. Grahn, K. Ploog, H. Schneider: Phys. Rev. B 52, 17352 (1995)
C. P. Holfeld, F. Löser, M. Sudzius, K. Leo, D.M. Whittaker, K. Köhler: Phys. Rev. Lett. 81, 874 (1998)
J. Feldmann, K. Leo, J. Shah, D. A. B. Miller, J. E. Cunningham, T. Meier, G. von Plessen, A. Schulze, P. Thomas, S. Schmitt-Rink: Phys. Rev. B 46, 7252 (1992)
K. Leo, P. Haring Bolivar, F. Brüggemann, R. Schwedler, K. Köhler: Solid State Commun. 84, 943 (1992)
C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Köhler: Phys. Rev. Lett. 70, 3319 (1993)
T. Dekorsy, R. Ott, H. Kurz, K. Köhler: Phys. Rev. B 51, 17275 (1995); P. Leisching, T. Dekorsy, H. J. Bakker, H. Kurz, K. Köhler: Phys. Rev. B 51, 18015 (1995)
K. Unterrainer, B. J. Keay, M. C. Wanke, S. J. Allen, D. Leonard, G. MedeirosRibeiro, U. Bhattacharya, M. J. W. Rodwell: Phys. Rev. Lett. 76, 2973 (1996)
V. G. Lyssenko, G. Valusis, F. Löser, T. Hasche, K. Leo, M. M. Dignam, K. Köhler: Phys. Rev. Lett. 79, 301 (1997)
M. M. Dignam, J. E. Sipe: Phys. Rev. B 41, 2865 (1990)
M. M. Dignam, J. E. Sipe: Phys. Rev. Lett. 64, 1797 (1990); Phys. Rev. B 43, 4097 (1991)
A. M. Fox, D. A. B. Miller, J. E. Cunningham, W. Y. Jan, C. Y. P. Chao, S. L. Chuang: Phys. Rev. B 46, 15365 (1992)
N. Linder, W. Geisselbrecht, G. Philipp, K. H. Schmidt, and G. H. Döhler: Journal de Physique IV, Colloque C5, vol 3, p 195 (1993); N. Linder: Phys. Rev. B 55, 13664 (1997)
D. M. Whittaker: Journal de Physique IV, Colloque C5, vol 3, p 199 (1993); D. M. Whittaker. In: 22nd International Conference on the Physics of Semiconductors, ed by D. J. Lockwood (World Scientific, Singapore 1995) vol 2, p 1420; D. M. Whittaker: Europhys. Lett. 31, 55 (1995)
T. Meier, G. von Plessen, P. Thomas, S.W. Koch: Phys. Rev. Lett. 73, 902 (1994); Phys. Rev. B 51, 14490 (1995); T. Meier, H. J. Kolbe, A. Thränhardt, G. Weiser, P. Thomas, S.W. Koch: Physica E 7, 267 (2000)
M. Dignam, J. E. Sipe, J. Shah: Phys. Rev. B 49, 10502 (1994)
Z. Barticevic, M. Pachecho, F. Claro: Phys. Rev. B 51, 14414 (1995)
R. Ferreira, B. Soucail, P. Voisin, G. Bastard: Phys. Rev. B 42, 11404 (1990)
J. Leo, A. MacKinnon: J. Phys.: Condens. Matt. 1, 1449 (1989)
A. M. Bouchard, M. Luban: Phys. Rev. B 47, 6815 (1993)
A. M. Bouchard, M. Luban: Phys. Rev. B 52, 5105 (1995)
R. Till, H. Kümmel, A. Philipp, G. Böhm, G. Weimann: Superlatt. Microstr. 24, 227 (1998)
K. Murayama, H. Nagasawa, S. Ozaki, M. Morifuji, C. Hamaguchi, A. Di Carlo, P. Vogl, G. Böhm, G. Weimann: Superlatt. M.crostr. 20, 493 (1996); M. Morifuji, K. Murayama, C. Hamaguchi, A. Di Carlo, P. Vogl, G. Böhm, M. Sexl: phys. stat. sol. (b) 204, 368 (1997)
A. Sibille, J.F. Palmier, F. Laruelle: Phys. Rev. Lett. 80, 4506 (1998)
M. Helm, W. Hilber, G. Strasser, R. De Meester, F. M. Peeters, A. Wacker: Phys. Rev. Lett. 82, 3120 (1999)
S. Glutsch, F. Bechstedt: Phys. Rev. B 60, 16584 (1999)
B. Rosam, D. Meinhold, F. Löser, V. G. Lyssenko, K. Leo, S. Glutsch, F. Bechstedt, K. Köhler: phys. stat. sol. 221, 463 (2000)
B. Rosam, D. Meinhold, F. Löser, V. G. Lyssenko, S. Glutsch, F. Bechstedt, F. Rossi, K. Köhler, K. Leo: Phys. Rev. Lett. 86, 1307 (2001); F. Löser, B. Rosam, D. Meinhold, V. G. Lyssenko, M. Sudzius, M. M. Dignam, S. Glutsch, F. Bechstedt, F. Rossi, K. Köhler, K. Leo: Physica E 11, 268 (2001)
S. Glutsch, F. Bechstedt, B. Rosam, K. Leo: Phys. Rev. B 63, 085307 (2001)
D. Meinhold, K. Leo, N. A. Fromer, D. S. Chemla, S. Glutsch, F. Bechstedt, K. Köhler: Phys. Rev. B 65, 161307 (2002)
M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, C. Salomon: Phys. Rev. Lett. 76, 4508 (1996)
S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. Niu, M. G. Raisen: Phys. Rev. B 76, 4512 (1996); C.F. Bharucha, K. W. Madison, P.R. Morrow, S.R. Wilkinson, B. Sundaram, M. G. Raizen: Phys. Rev. A 55, 857 (1997)
U. Peschel, T. Pertsch, F. Lederer, Optics Letters 23, 1701 (1998)
T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, F. Lederer: Phys. Rev. Lett. 83, 4752 (1999)
R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg: Phys. Rev. Lett. 83, 4756 (1999)
A. Di Carlo, P. Vogl, W. Pötz: Phys. Rev. B 50, 8358 (1994)
M. Holthaus, J. Opt. B 2, 589 (2000)
M. Glück, A. R. Kolovsky, H. J. Korsch, N. Moiseyev, Eur. Phys. J. D 4, 239 (1998); M. Glück, A. Kolovsky, H.J. Korsch: J. Phys. A 32, L49 (1999);
M. Glück, A. R. Kolovsky, H.J. Korsch, F. Zimmer: Phys. Rev. B 65, 115302 (2002)
G. Nenciu: Rev. Mod. Phys. 63, 91 (1991)
G. H. Döhler. In: Physics of Low-dimensional Structures, ed by P. Butcher, N. H. March, M. P. Tosi ( Plenum, New York London 1993 ) pp 375–406
R. de L. Kronig, W. G. Penney: Proc. Roy. Soc. A 130, 499 (1931)
S. Flügge, H. Marschall: Rechenmethoden der Quantentheorie, Teil 1 ( Springer, Berlin 1952 )
J. H. Davies: The Physics of Low-Dimensional Semiconductors: An Introduction (Cambridge University Press 1998 )
H. M. James: Phys. Rev. 76, 1602 (1949)
W. Kohn: Phys. Rev. 115, 809 (1959)
E. H. Lieb, D. C. Mattis: Mathematical Physics in One Dimension ( Academic, New York 1966 )
J. Yellin: Phys. Rev. B 52, 2208 (1995)
J. Callaway: Quantum Theory of the Solid State, 2nd edn ( Academic, San Diego 1991 )
W. Magnus, F. Oberhettinger, R. P. Sony, Formulas and Theorems for the Special Functions of Mathematical Physics ( Springer, Berlin 1966 )
M. Abramowitz, I. A. Stegun: Pocketbook of Mathematical Functions, abridged edn of Handbook of Mathematical Functions, Material selected by M. Danos, J. Rafelski ( Verlag Harry Deutsch, Thun Frankfurt/Main 1984 )
T. Pertsch, U. Peschel, F. Lederer (unpublished)
D. W. Hone, X.-G. Zhao: Phys. Rev. B 53, 4834 (1996)
J. Hader, P. Thomas, S.W. Koch: Prog. Quant. Electr. 22, 123 (1998)
S. Glutsch: J. Phys.: Condens. Matt. 11, 5533 (1999)
W. Franz, Z. Naturforschg. 14 a, 415 (1959)
G. Eilenberger: Z. Physik 164, 59 (1961)
M. Ritze, N. J. M. Horing, R. Enderlein: Phys. Rev. B 47, 10437 (1993)
S. Glutsch, F. Bechstedt: Phys. Rev. B 57, 11887 (1998)
A. Ahland, D. Schulz, E. Voges: Phys. Rev. B 60, 5109 (1999)
M. L. Zambrano, J. C. Arce: Phys. Rev. B 66, 155340 (2002)
A. B. Hummel, T. Bauer, H. G. Roskos, S. Glutsch, K. Köhler Phys. Rev. B 67, 045319 (2003)
L. Landau: Phys. Z. Sowjetunion 1, 88 (1932); 2, 46 (1932)
E. E. Nikitin, Annual Rev. Chem. 50, 1 (1999)
J. B. Krieger, G. J. Iafrate: Phys. Rev. B 35, 9644 (1987)
P. Ao, J. Rammer: Phys. Rev. B 44, 11494 (1991)
J. M. Ziman: Principles of the Theory of Solids ( Cambridge University Press, London 1979 )
L. D. Landau, E. M. Lifshitz: Course of Theoretical Physics, vol 3, 3rd edn ( Pergamon, Oxford 1999 )
M. V. Berry, R. Lim: J. Phys. A 26, 4737 (1993)
K. B. McAfee, E. J. Ryder, W. Shockley, M. Sparks: Phys. Rev. 83, 650 (1951)
W. Franz. In: Encyclopedia of Physics, ed by S. Flügge, vol 17 ( Springer, Berlin 1956 ) pp 155–263
Q. Niu, M. G. Raizen: Phys. Rev. Lett. 80, 3491 (1998)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Glutsch, S. (2004). Zener Breakdown in Superlattices. In: Excitons in Low-Dimensional Semiconductors. Springer Series in Solid-State Sciences, vol 141. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-07150-2_4
Download citation
DOI: https://doi.org/10.1007/978-3-662-07150-2_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-05781-6
Online ISBN: 978-3-662-07150-2
eBook Packages: Springer Book Archive