Abstract

This paper predicts that the quantum Faraday effect manifests itself in a two-dimensional electron gas through a stepwise increase accompanied by a resonant behavior of the Faraday rotation angle when the quantum number of levels between which transition occurs varies. It is also shown that this effect can be demonstrated experimentally.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, and V. V. Yashcuk, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
    [CrossRef]
  2. A. V. Kavokin, M. R. Vladimirova, M. A. Kaliteevski, O. Lyngnes, J. D. Berger, H. M. Gibbs, and G. Khitrova, "Resonant Faraday rotation in a semiconductor cavity," Phys. Rev. B 56, 1087-1090 (1997).
    [CrossRef]
  3. D. A. Dahl and L. J. Sham, "Electrodynamics of quasi-two-dimensional electrons," Phys. Rev. B 16, 651-661 (1977).
    [CrossRef]
  4. Yu. Bychkov, C. Faugeras, and G. Martinez, "Multidielectric response of a two-dimensional electrons gas in tilted magnetic fields," Phys. Rev. B 70, 085306 (2004).
    [CrossRef]
  5. D. K. Ferry and S. M. Goodnick, Transport in Nanostructures (Cambridge U. Press, 1997).
    [CrossRef]
  6. M. A. Brummell, M. A. Hopkins, R. J. Nicholas, J. C. Portal, K. Y. Cheng, and A. Y. Cho, "Subband-Landau level coupling in a two-dimensional electron gas in tilted magnetic fields," J. Phys. C 19, L107-L112 (1986).
    [CrossRef]
  7. U. Zeitler, H. W. Schumacher, A. G. M. Jansen, and R. J. Haug, "Magnetoresistance anisotropy in Si-SiGe in tilted magnetic fields: experimental evidence for a stripe-phase formation," Phys. Rev. Lett. 86, 866-869 (2001).
    [CrossRef] [PubMed]
  8. D. J. Newson, K.-F. Berggren, M. Pepper, H. W. Myron, G. J. Davies, and E. G. Scott, "Magnetic depopulation of sub-bands in In0.53Ga0.47As-In0.52Al0.48As heterojunctions," J. Phys. C 19, L403-L410 (1986).
    [CrossRef]
  9. R. E. Doezema, M. Nealon, and S. Whitmore, "Hybrid quantum oscillations in a surface space-charge layer," Phys. Rev. Lett. 45, 1593-1596 (1980).
    [CrossRef]
  10. L. Smrcka, P. Vasek, J. Kolácek, T. Jungwirth, and M. Cukr, "Cyclotron effective mass of a two-dimensional electron layer at the GaAs-AlxGa1−xAs heterojunction subject to in-plane magnetic fields," Phys. Rev. B 51, 18011-18014 (1995).
    [CrossRef]
  11. G. Salis, B. Ruhstaller, K. Ensslin, K. Campman, K. Maranowski, and A. C. Gossard, "Subband densities in quantum wells under in-plane magnetic fields," Phys. Rev. B 58, 1436-1441 (1998).
    [CrossRef]
  12. P. K. Basu, Theory of Optical Processes in Semiconductors (Clarendon, 1997).
  13. D. Dragoman and M. Dragoman, Optical Characterization of Solids (Springer, 2002).
    [CrossRef]

2004 (1)

Yu. Bychkov, C. Faugeras, and G. Martinez, "Multidielectric response of a two-dimensional electrons gas in tilted magnetic fields," Phys. Rev. B 70, 085306 (2004).
[CrossRef]

2002 (1)

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, and V. V. Yashcuk, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

2001 (1)

U. Zeitler, H. W. Schumacher, A. G. M. Jansen, and R. J. Haug, "Magnetoresistance anisotropy in Si-SiGe in tilted magnetic fields: experimental evidence for a stripe-phase formation," Phys. Rev. Lett. 86, 866-869 (2001).
[CrossRef] [PubMed]

1998 (1)

G. Salis, B. Ruhstaller, K. Ensslin, K. Campman, K. Maranowski, and A. C. Gossard, "Subband densities in quantum wells under in-plane magnetic fields," Phys. Rev. B 58, 1436-1441 (1998).
[CrossRef]

1997 (1)

A. V. Kavokin, M. R. Vladimirova, M. A. Kaliteevski, O. Lyngnes, J. D. Berger, H. M. Gibbs, and G. Khitrova, "Resonant Faraday rotation in a semiconductor cavity," Phys. Rev. B 56, 1087-1090 (1997).
[CrossRef]

1995 (1)

L. Smrcka, P. Vasek, J. Kolácek, T. Jungwirth, and M. Cukr, "Cyclotron effective mass of a two-dimensional electron layer at the GaAs-AlxGa1−xAs heterojunction subject to in-plane magnetic fields," Phys. Rev. B 51, 18011-18014 (1995).
[CrossRef]

1986 (2)

D. J. Newson, K.-F. Berggren, M. Pepper, H. W. Myron, G. J. Davies, and E. G. Scott, "Magnetic depopulation of sub-bands in In0.53Ga0.47As-In0.52Al0.48As heterojunctions," J. Phys. C 19, L403-L410 (1986).
[CrossRef]

M. A. Brummell, M. A. Hopkins, R. J. Nicholas, J. C. Portal, K. Y. Cheng, and A. Y. Cho, "Subband-Landau level coupling in a two-dimensional electron gas in tilted magnetic fields," J. Phys. C 19, L107-L112 (1986).
[CrossRef]

1980 (1)

R. E. Doezema, M. Nealon, and S. Whitmore, "Hybrid quantum oscillations in a surface space-charge layer," Phys. Rev. Lett. 45, 1593-1596 (1980).
[CrossRef]

1977 (1)

D. A. Dahl and L. J. Sham, "Electrodynamics of quasi-two-dimensional electrons," Phys. Rev. B 16, 651-661 (1977).
[CrossRef]

Basu, P. K.

P. K. Basu, Theory of Optical Processes in Semiconductors (Clarendon, 1997).

Berger, J. D.

A. V. Kavokin, M. R. Vladimirova, M. A. Kaliteevski, O. Lyngnes, J. D. Berger, H. M. Gibbs, and G. Khitrova, "Resonant Faraday rotation in a semiconductor cavity," Phys. Rev. B 56, 1087-1090 (1997).
[CrossRef]

Berggren, K.-F.

D. J. Newson, K.-F. Berggren, M. Pepper, H. W. Myron, G. J. Davies, and E. G. Scott, "Magnetic depopulation of sub-bands in In0.53Ga0.47As-In0.52Al0.48As heterojunctions," J. Phys. C 19, L403-L410 (1986).
[CrossRef]

Brummell, M. A.

M. A. Brummell, M. A. Hopkins, R. J. Nicholas, J. C. Portal, K. Y. Cheng, and A. Y. Cho, "Subband-Landau level coupling in a two-dimensional electron gas in tilted magnetic fields," J. Phys. C 19, L107-L112 (1986).
[CrossRef]

Budker, D.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, and V. V. Yashcuk, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

Bychkov, Yu.

Yu. Bychkov, C. Faugeras, and G. Martinez, "Multidielectric response of a two-dimensional electrons gas in tilted magnetic fields," Phys. Rev. B 70, 085306 (2004).
[CrossRef]

Campman, K.

G. Salis, B. Ruhstaller, K. Ensslin, K. Campman, K. Maranowski, and A. C. Gossard, "Subband densities in quantum wells under in-plane magnetic fields," Phys. Rev. B 58, 1436-1441 (1998).
[CrossRef]

Cheng, K. Y.

M. A. Brummell, M. A. Hopkins, R. J. Nicholas, J. C. Portal, K. Y. Cheng, and A. Y. Cho, "Subband-Landau level coupling in a two-dimensional electron gas in tilted magnetic fields," J. Phys. C 19, L107-L112 (1986).
[CrossRef]

Cho, A. Y.

M. A. Brummell, M. A. Hopkins, R. J. Nicholas, J. C. Portal, K. Y. Cheng, and A. Y. Cho, "Subband-Landau level coupling in a two-dimensional electron gas in tilted magnetic fields," J. Phys. C 19, L107-L112 (1986).
[CrossRef]

Cukr, M.

L. Smrcka, P. Vasek, J. Kolácek, T. Jungwirth, and M. Cukr, "Cyclotron effective mass of a two-dimensional electron layer at the GaAs-AlxGa1−xAs heterojunction subject to in-plane magnetic fields," Phys. Rev. B 51, 18011-18014 (1995).
[CrossRef]

Dahl, D. A.

D. A. Dahl and L. J. Sham, "Electrodynamics of quasi-two-dimensional electrons," Phys. Rev. B 16, 651-661 (1977).
[CrossRef]

Davies, G. J.

D. J. Newson, K.-F. Berggren, M. Pepper, H. W. Myron, G. J. Davies, and E. G. Scott, "Magnetic depopulation of sub-bands in In0.53Ga0.47As-In0.52Al0.48As heterojunctions," J. Phys. C 19, L403-L410 (1986).
[CrossRef]

Doezema, R. E.

R. E. Doezema, M. Nealon, and S. Whitmore, "Hybrid quantum oscillations in a surface space-charge layer," Phys. Rev. Lett. 45, 1593-1596 (1980).
[CrossRef]

Dragoman, D.

D. Dragoman and M. Dragoman, Optical Characterization of Solids (Springer, 2002).
[CrossRef]

Dragoman, M.

D. Dragoman and M. Dragoman, Optical Characterization of Solids (Springer, 2002).
[CrossRef]

Ensslin, K.

G. Salis, B. Ruhstaller, K. Ensslin, K. Campman, K. Maranowski, and A. C. Gossard, "Subband densities in quantum wells under in-plane magnetic fields," Phys. Rev. B 58, 1436-1441 (1998).
[CrossRef]

Faugeras, C.

Yu. Bychkov, C. Faugeras, and G. Martinez, "Multidielectric response of a two-dimensional electrons gas in tilted magnetic fields," Phys. Rev. B 70, 085306 (2004).
[CrossRef]

Ferry, D. K.

D. K. Ferry and S. M. Goodnick, Transport in Nanostructures (Cambridge U. Press, 1997).
[CrossRef]

Gawlik, W.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, and V. V. Yashcuk, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

Gibbs, H. M.

A. V. Kavokin, M. R. Vladimirova, M. A. Kaliteevski, O. Lyngnes, J. D. Berger, H. M. Gibbs, and G. Khitrova, "Resonant Faraday rotation in a semiconductor cavity," Phys. Rev. B 56, 1087-1090 (1997).
[CrossRef]

Goodnick, S. M.

D. K. Ferry and S. M. Goodnick, Transport in Nanostructures (Cambridge U. Press, 1997).
[CrossRef]

Gossard, A. C.

G. Salis, B. Ruhstaller, K. Ensslin, K. Campman, K. Maranowski, and A. C. Gossard, "Subband densities in quantum wells under in-plane magnetic fields," Phys. Rev. B 58, 1436-1441 (1998).
[CrossRef]

Haug, R. J.

U. Zeitler, H. W. Schumacher, A. G. M. Jansen, and R. J. Haug, "Magnetoresistance anisotropy in Si-SiGe in tilted magnetic fields: experimental evidence for a stripe-phase formation," Phys. Rev. Lett. 86, 866-869 (2001).
[CrossRef] [PubMed]

Hopkins, M. A.

M. A. Brummell, M. A. Hopkins, R. J. Nicholas, J. C. Portal, K. Y. Cheng, and A. Y. Cho, "Subband-Landau level coupling in a two-dimensional electron gas in tilted magnetic fields," J. Phys. C 19, L107-L112 (1986).
[CrossRef]

Jansen, A. G. M.

U. Zeitler, H. W. Schumacher, A. G. M. Jansen, and R. J. Haug, "Magnetoresistance anisotropy in Si-SiGe in tilted magnetic fields: experimental evidence for a stripe-phase formation," Phys. Rev. Lett. 86, 866-869 (2001).
[CrossRef] [PubMed]

Jungwirth, T.

L. Smrcka, P. Vasek, J. Kolácek, T. Jungwirth, and M. Cukr, "Cyclotron effective mass of a two-dimensional electron layer at the GaAs-AlxGa1−xAs heterojunction subject to in-plane magnetic fields," Phys. Rev. B 51, 18011-18014 (1995).
[CrossRef]

Kaliteevski, M. A.

A. V. Kavokin, M. R. Vladimirova, M. A. Kaliteevski, O. Lyngnes, J. D. Berger, H. M. Gibbs, and G. Khitrova, "Resonant Faraday rotation in a semiconductor cavity," Phys. Rev. B 56, 1087-1090 (1997).
[CrossRef]

Kavokin, A. V.

A. V. Kavokin, M. R. Vladimirova, M. A. Kaliteevski, O. Lyngnes, J. D. Berger, H. M. Gibbs, and G. Khitrova, "Resonant Faraday rotation in a semiconductor cavity," Phys. Rev. B 56, 1087-1090 (1997).
[CrossRef]

Khitrova, G.

A. V. Kavokin, M. R. Vladimirova, M. A. Kaliteevski, O. Lyngnes, J. D. Berger, H. M. Gibbs, and G. Khitrova, "Resonant Faraday rotation in a semiconductor cavity," Phys. Rev. B 56, 1087-1090 (1997).
[CrossRef]

Kimball, D. F.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, and V. V. Yashcuk, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

Kolácek, J.

L. Smrcka, P. Vasek, J. Kolácek, T. Jungwirth, and M. Cukr, "Cyclotron effective mass of a two-dimensional electron layer at the GaAs-AlxGa1−xAs heterojunction subject to in-plane magnetic fields," Phys. Rev. B 51, 18011-18014 (1995).
[CrossRef]

Lyngnes, O.

A. V. Kavokin, M. R. Vladimirova, M. A. Kaliteevski, O. Lyngnes, J. D. Berger, H. M. Gibbs, and G. Khitrova, "Resonant Faraday rotation in a semiconductor cavity," Phys. Rev. B 56, 1087-1090 (1997).
[CrossRef]

Maranowski, K.

G. Salis, B. Ruhstaller, K. Ensslin, K. Campman, K. Maranowski, and A. C. Gossard, "Subband densities in quantum wells under in-plane magnetic fields," Phys. Rev. B 58, 1436-1441 (1998).
[CrossRef]

Martinez, G.

Yu. Bychkov, C. Faugeras, and G. Martinez, "Multidielectric response of a two-dimensional electrons gas in tilted magnetic fields," Phys. Rev. B 70, 085306 (2004).
[CrossRef]

Myron, H. W.

D. J. Newson, K.-F. Berggren, M. Pepper, H. W. Myron, G. J. Davies, and E. G. Scott, "Magnetic depopulation of sub-bands in In0.53Ga0.47As-In0.52Al0.48As heterojunctions," J. Phys. C 19, L403-L410 (1986).
[CrossRef]

Nealon, M.

R. E. Doezema, M. Nealon, and S. Whitmore, "Hybrid quantum oscillations in a surface space-charge layer," Phys. Rev. Lett. 45, 1593-1596 (1980).
[CrossRef]

Newson, D. J.

D. J. Newson, K.-F. Berggren, M. Pepper, H. W. Myron, G. J. Davies, and E. G. Scott, "Magnetic depopulation of sub-bands in In0.53Ga0.47As-In0.52Al0.48As heterojunctions," J. Phys. C 19, L403-L410 (1986).
[CrossRef]

Nicholas, R. J.

M. A. Brummell, M. A. Hopkins, R. J. Nicholas, J. C. Portal, K. Y. Cheng, and A. Y. Cho, "Subband-Landau level coupling in a two-dimensional electron gas in tilted magnetic fields," J. Phys. C 19, L107-L112 (1986).
[CrossRef]

Pepper, M.

D. J. Newson, K.-F. Berggren, M. Pepper, H. W. Myron, G. J. Davies, and E. G. Scott, "Magnetic depopulation of sub-bands in In0.53Ga0.47As-In0.52Al0.48As heterojunctions," J. Phys. C 19, L403-L410 (1986).
[CrossRef]

Portal, J. C.

M. A. Brummell, M. A. Hopkins, R. J. Nicholas, J. C. Portal, K. Y. Cheng, and A. Y. Cho, "Subband-Landau level coupling in a two-dimensional electron gas in tilted magnetic fields," J. Phys. C 19, L107-L112 (1986).
[CrossRef]

Rochester, S. M.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, and V. V. Yashcuk, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

Ruhstaller, B.

G. Salis, B. Ruhstaller, K. Ensslin, K. Campman, K. Maranowski, and A. C. Gossard, "Subband densities in quantum wells under in-plane magnetic fields," Phys. Rev. B 58, 1436-1441 (1998).
[CrossRef]

Salis, G.

G. Salis, B. Ruhstaller, K. Ensslin, K. Campman, K. Maranowski, and A. C. Gossard, "Subband densities in quantum wells under in-plane magnetic fields," Phys. Rev. B 58, 1436-1441 (1998).
[CrossRef]

Schumacher, H. W.

U. Zeitler, H. W. Schumacher, A. G. M. Jansen, and R. J. Haug, "Magnetoresistance anisotropy in Si-SiGe in tilted magnetic fields: experimental evidence for a stripe-phase formation," Phys. Rev. Lett. 86, 866-869 (2001).
[CrossRef] [PubMed]

Scott, E. G.

D. J. Newson, K.-F. Berggren, M. Pepper, H. W. Myron, G. J. Davies, and E. G. Scott, "Magnetic depopulation of sub-bands in In0.53Ga0.47As-In0.52Al0.48As heterojunctions," J. Phys. C 19, L403-L410 (1986).
[CrossRef]

Sham, L. J.

D. A. Dahl and L. J. Sham, "Electrodynamics of quasi-two-dimensional electrons," Phys. Rev. B 16, 651-661 (1977).
[CrossRef]

Smrcka, L.

L. Smrcka, P. Vasek, J. Kolácek, T. Jungwirth, and M. Cukr, "Cyclotron effective mass of a two-dimensional electron layer at the GaAs-AlxGa1−xAs heterojunction subject to in-plane magnetic fields," Phys. Rev. B 51, 18011-18014 (1995).
[CrossRef]

Vasek, P.

L. Smrcka, P. Vasek, J. Kolácek, T. Jungwirth, and M. Cukr, "Cyclotron effective mass of a two-dimensional electron layer at the GaAs-AlxGa1−xAs heterojunction subject to in-plane magnetic fields," Phys. Rev. B 51, 18011-18014 (1995).
[CrossRef]

Vladimirova, M. R.

A. V. Kavokin, M. R. Vladimirova, M. A. Kaliteevski, O. Lyngnes, J. D. Berger, H. M. Gibbs, and G. Khitrova, "Resonant Faraday rotation in a semiconductor cavity," Phys. Rev. B 56, 1087-1090 (1997).
[CrossRef]

Whitmore, S.

R. E. Doezema, M. Nealon, and S. Whitmore, "Hybrid quantum oscillations in a surface space-charge layer," Phys. Rev. Lett. 45, 1593-1596 (1980).
[CrossRef]

Yashcuk, V. V.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, and V. V. Yashcuk, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

Zeitler, U.

U. Zeitler, H. W. Schumacher, A. G. M. Jansen, and R. J. Haug, "Magnetoresistance anisotropy in Si-SiGe in tilted magnetic fields: experimental evidence for a stripe-phase formation," Phys. Rev. Lett. 86, 866-869 (2001).
[CrossRef] [PubMed]

J. Phys. C (2)

M. A. Brummell, M. A. Hopkins, R. J. Nicholas, J. C. Portal, K. Y. Cheng, and A. Y. Cho, "Subband-Landau level coupling in a two-dimensional electron gas in tilted magnetic fields," J. Phys. C 19, L107-L112 (1986).
[CrossRef]

D. J. Newson, K.-F. Berggren, M. Pepper, H. W. Myron, G. J. Davies, and E. G. Scott, "Magnetic depopulation of sub-bands in In0.53Ga0.47As-In0.52Al0.48As heterojunctions," J. Phys. C 19, L403-L410 (1986).
[CrossRef]

Phys. Rev. B (5)

A. V. Kavokin, M. R. Vladimirova, M. A. Kaliteevski, O. Lyngnes, J. D. Berger, H. M. Gibbs, and G. Khitrova, "Resonant Faraday rotation in a semiconductor cavity," Phys. Rev. B 56, 1087-1090 (1997).
[CrossRef]

D. A. Dahl and L. J. Sham, "Electrodynamics of quasi-two-dimensional electrons," Phys. Rev. B 16, 651-661 (1977).
[CrossRef]

Yu. Bychkov, C. Faugeras, and G. Martinez, "Multidielectric response of a two-dimensional electrons gas in tilted magnetic fields," Phys. Rev. B 70, 085306 (2004).
[CrossRef]

L. Smrcka, P. Vasek, J. Kolácek, T. Jungwirth, and M. Cukr, "Cyclotron effective mass of a two-dimensional electron layer at the GaAs-AlxGa1−xAs heterojunction subject to in-plane magnetic fields," Phys. Rev. B 51, 18011-18014 (1995).
[CrossRef]

G. Salis, B. Ruhstaller, K. Ensslin, K. Campman, K. Maranowski, and A. C. Gossard, "Subband densities in quantum wells under in-plane magnetic fields," Phys. Rev. B 58, 1436-1441 (1998).
[CrossRef]

Phys. Rev. Lett. (2)

R. E. Doezema, M. Nealon, and S. Whitmore, "Hybrid quantum oscillations in a surface space-charge layer," Phys. Rev. Lett. 45, 1593-1596 (1980).
[CrossRef]

U. Zeitler, H. W. Schumacher, A. G. M. Jansen, and R. J. Haug, "Magnetoresistance anisotropy in Si-SiGe in tilted magnetic fields: experimental evidence for a stripe-phase formation," Phys. Rev. Lett. 86, 866-869 (2001).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, and V. V. Yashcuk, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

Other (3)

P. K. Basu, Theory of Optical Processes in Semiconductors (Clarendon, 1997).

D. Dragoman and M. Dragoman, Optical Characterization of Solids (Springer, 2002).
[CrossRef]

D. K. Ferry and S. M. Goodnick, Transport in Nanostructures (Cambridge U. Press, 1997).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Geometry of the quasi-2DEG system in the presence of an applied magnetic field.

Fig. 2
Fig. 2

Energy dependence of the two terms that enter the expression of the Faraday rotation angle in Eq. (10): the resonant term (solid curve) and the constant term (dashed curve).

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

H = p x 2 2 M c + p y 2 2 m c + p z 2 2 m c + m c ω eff , c 2 2 ( z z ¯ ) 2 + g μ B s y B .
z ¯ = ( k x e B ) ( ω c , c 2 ω eff , c 2 ) .
E c , n , s y ( B ) = 2 k x 2 2 M c + 2 k y 2 2 m c + ω eff , c ( n + 1 2 ) + g μ B s y B ,
E v , n ( B ) = E g 2 k x 2 2 M v 2 k y 2 2 m v ω eff , v ( n + 1 2 ) ,
ω = E c , n , s y ( B ) E v , n ( B ) ,
ω = E g + g μ B s y B + ω eff ( n + 1 2 ) + 2 k x 2 2 M r + 2 k y 2 2 m r ,
α ( ω ) = n , s y α n ω d k x d k y δ [ E g + g μ B s y B + ω eff ( n + 1 2 ) + 2 k x 2 2 M r + 2 k y 2 2 m r ω ] ( f v f c ) ,
α ( ω ) = n , s y α n ω H [ ω E g g μ B s y B ω eff ( n + 1 2 ) ] ( f v f c ) ,
n ( ω ) = 1 + c π PV 0 α ( ω ) ( ω ) 2 ( ω ) 2 d ( ω ) ,
Δ n n ( E ) = c π α n E min d E E ( E 2 E 2 ) ,
Δ n n ( E ) = c π α n E 2 [ ln ( 1 E E min ) E E min ] ,
Δ θ n ( E ) = α n L 2 E [ ln ( 1 E E min 1 E E min + ) + E ( 1 E min 1 E min + ) ] .
Δ θ const . ( n ) = α n L g μ B B 2 [ E g + ω eff ( n + 1 2 ) ] 2 ( g μ B B 2 ) 2 α n L g μ B B 2 [ E g + ω eff ( n + 1 2 ) ] 2 .
Δ E c , n , s y = E c , n , s y ( B ) E c , n , s y ( 0 ) = ( n + 1 2 ) [ ( ω 0 2 + ω c , c 2 ) 1 2 ω 0 ] + g μ B s y B ,

Metrics