-
2
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85037894828
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-
See Solid State Physics, (Ref. 1), Chap. 12.
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See Solid State Physics, (Ref. 1), Chap. 12.
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-
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12
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77956944547
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D. Turnbull, Academic Press, New York F. Seitz
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E. I. Blount, in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic Press, New York, 1962), Vol. 13, p. 305.
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(1962)
Solid State Physics
, vol.13
, pp. 305
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Blount, E.I.1
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18
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36149023375
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W. Kohn, Phys. Rev. 115, 1460 (1959);
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(1959)
Phys. Rev.
, vol.115
, pp. 1460
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Kohn, W.1
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21
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36149017566
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Wannier and Fredkin also developed an effective Hamiltonian formalism; however, their method does not give corrections explicitly as a power series in field strengths; G. H. Wannier, Phys. Rev. 117, 432 (1960);
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(1960)
Phys. Rev.
, vol.117
, pp. 432
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Wannier, G.H.1
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32
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85037883259
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Geometry of the Time-Dependent Variational Principle in Quantum Mechanics, Lecture Notes in Physics No. 140 (Springer-Verlag, 1981).
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P. Kramer and M. Saraceno, Geometry of the Time-Dependent Variational Principle in Quantum Mechanics, Lecture Notes in Physics No. 140 (Springer-Verlag, 1981).
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Kramer, P.1
Saraceno, M.2
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33
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85037891665
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A. K. Pattanayak, Ph.D. dissertation, The University of Texas, Austin, 1994.
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A. K. Pattanayak, Ph.D. dissertation, The University of Texas, Austin, 1994.
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34
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0004765026
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See Ref. 23. However, the existence of a generalized type of Wannier functions with the required localization properties has been proved by M. Wilkinson, J. Phys.: Condens. Matter 10, 7407 (1998).
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(1998)
J. Phys.: Condens. Matter
, vol.10
, pp. 7407
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Wilkinson, M.1
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37
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85037922701
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The time scale associated with an energy gap is given by the uncertainty time (Formula presented) where (Formula presented) is the gap size. The length scale of an gap is given by (Formula presented) where (Formula presented) is the maximum size of the imaginary part of the wave vector in the gap.
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The time scale associated with an energy gap is given by the uncertainty time (Formula presented) where (Formula presented) is the gap size. The length scale of an gap is given by (Formula presented) where (Formula presented) is the maximum size of the imaginary part of the wave vector in the gap.
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38
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23944514543
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Our results are valid only in the time scale in which the effects of spreading may be ignored. For studies on the dynamics of higher order moments of a wave packet, see Ref. 25. A systematic study of wave-packet dynamics was initiated by E. J. Heller, J. Chem. Phys. 62, 1544 (1975).
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(1975)
J. Chem. Phys.
, vol.62
, pp. 1544
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Heller, E.J.1
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39
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85037901726
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This result can be verified for Hamiltonians of the form (Formula presented), the only types considered in this paper.
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This result can be verified for Hamiltonians of the form (Formula presented), the only types considered in this paper.
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40
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85037910694
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Lectures on Quantum Mechanics, Belfer Graduate School of Science (Yeshiva University, New York 1964);
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P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science (Yeshiva University, New York 1964);
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Dirac, P.A.M.1
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42
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85037919041
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Michael Tabor, Chaos and Integrability in Nonlinear Dynamical Systems: An Introduction (Wiley Interscience, New York, 1989).
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Michael Tabor, Chaos and Integrability in Nonlinear Dynamical Systems: An Introduction (Wiley Interscience, New York, 1989).
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45
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36149016365
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In addition to the works cited in Ref. 16 see L. M. Roth, Phys. Rev. 118, 1534 (1960);
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(1960)
Phys. Rev.
, vol.118
, pp. 1534
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Roth, L.M.1
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46
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84996190943
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E. H. Cohen and E. I. Blount, Philos. Mag. 5, 115 (1960), derive the orbital magnetic moment for an effective mass Hamiltonian.
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(1960)
Philos. Mag.
, vol.5
, pp. 115
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Cohen, E.H.1
Blount, E.I.2
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47
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3442880129
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D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982);
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(1982)
Phys. Rev. Lett.
, vol.49
, pp. 405
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Thouless, D.J.1
Kohmoto, M.2
Nightingale, M.P.3
den Nijs, M.4
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49
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4243106617
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J. Zak, Phys. Rev. Lett. 62, 2747 (1989), and Ref. 15. For centro-symmetric crystals, the Berry phase is quantized and equals 0 or (Formula presented); this corresponds to centers of Wannier functions being either at (Formula presented), where l is an integer.
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(1989)
Phys. Rev. Lett.
, vol.62
, pp. 2747
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Zak, J.1
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51
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0002800965
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Though Wannier-Stark ladder was predicted in the early 1960s, only in the late 1980s that it was confirmed by experiments. For details, see E. E. Mendes and G. Bastard, Phys. Today 46 (6), 34 (1993).
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(1993)
Phys. Today
, vol.46
, Issue.6
, pp. 34
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Mendes, E.E.1
Bastard, G.2
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53
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85037890101
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M. Lax (unpublished).
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Lax, M.1
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54
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0001433702
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L. Kleinman, Phys. Rev. B 24, 7412 (1981), has pointed out the difficulty in defining absolute potentials and deformation potentials for infinite crystals. We defer a more detailed consideration to a future publication.
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(1981)
Phys. Rev. B
, vol.24
, pp. 7412
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Kleinman, L.1
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55
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85037910226
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This definition corresponds to the Eulerian convention. In the Lagrangian convention one defines another displacement field (Formula presented) such that (Formula presented) In either convention, strains are of the same order of magnitude, which makes comparison with other works reasonable.
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This definition corresponds to the Eulerian convention. In the Lagrangian convention one defines another displacement field (Formula presented) such that (Formula presented) In either convention, strains are of the same order of magnitude, which makes comparison with other works reasonable.
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56
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36149012552
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J. Bardeen and W. Shockley, Phys. Rev. 80, 72 (1950), introduced the concept of deformation potential as the strain-induced shift in electron energy to simplify the computation of the matrix elements of the electron-phonon interaction at long wavelengths.
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(1950)
Phys. Rev.
, vol.80
, pp. 72
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Bardeen, J.1
Shockley, W.2
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57
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0000356144
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F. S. Khan and P. B. Allen, Phys. Rev. B 29, 3341 (1984), use Fermi energy as the reference point rather than the band bottom in their definition of the deformation potential, and include a term for the shift in the Fermi energy due the applied strain. However, this reference can be incorporated at the level of Hamiltonian itself by a suitable mapping, which result in band energies modified accordingly.
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(1984)
Phys. Rev. B
, vol.29
, pp. 3341
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Khan, F.S.1
Allen, P.B.2
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58
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35949022828
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Referred to as the deformation potential theorem, the result was first derived by Bardeen and Shockley for Bloch states near band edges where the term (Formula presented) is zero (Ref. 44). The exact result (4.10) was obtained by Khan and Allen (Ref. 45) and later confirmed by E. Kartheuser and S. Rodriguez, Phys. Rev. B 33, 772 (1986).
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(1986)
Phys. Rev. B
, vol.33
, pp. 772
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Kartheuser, E.1
Rodriguez, S.2
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61
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85037891553
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This can always be done as long as the Berry curvature (Formula presented) is zero everywhere. This condition is satisfied for centrosymmetric crystals in the absence of magnetic fields.
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This can always be done as long as the Berry curvature (Formula presented) is zero everywhere. This condition is satisfied for centrosymmetric crystals in the absence of magnetic fields.
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