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Volumn 59, Issue 23, 1999, Pages 14915-14925

Wave-packet dynamics in slowly perturbed crystals: gradient corrections and berry-phase effects

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EID: 0000952833     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.59.14915     Document Type: Article
Times cited : (1105)

References (65)
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    • See Solid State Physics, (Ref. 1), Chap. 12.
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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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    • 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.
    • 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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    • 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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    • This result can be verified for Hamiltonians of the form (Formula presented), the only types considered in this paper.
    • This result can be verified for Hamiltonians of the form (Formula presented), the only types considered in this paper.
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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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    • 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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    • 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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    • 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.
    • 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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    • 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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    • 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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    • 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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    • 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.
    • 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.
  • 64


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