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MeF MODEL (Morandi, Modugno, Phys.Rev.B, 2005)
Two coupled of Schrödinger-like equations for the
electron envelope function in the
conduction and valence bands
Interband coupling, proportional to the momentum
matrix element P. This is responsible for
tunneling between different bands caused by the
applied electric field proportional to the
x-derivative of V.
energy gap
(isotropic) effective mass,
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Derivation of the MeF MODEL

• Expansion of the wave function on the Bloch
  functions basis

• Introduction in the Schrödinger equation

• Separation of intra-band dynamics from
  inter-band coupling

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• Going back in real space by Fourier transforming
  yields an equation for the projection
  coefficiente on the Wannier basis (
  atomic sites positions)

• Two approximations

• Simplify the interband term around
  developing the periodic part of the Bloch
  wavefunctions to the first order
• Effective mass approximation

MeF MODEL
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Comments on the MeF MODEL

• The envelope functions have a direct
  physical meaning as they can be interpreted as
  the effective (cell averaged) wave functions of
  the electron in the conduction (valence) band.

• The presence of the effective masses (generally
  different in the two bands) implies a different
  mobility in the two bands.

• Similar to the Luttinger-Kohn model, but it
  includes a coupling beetween different bands due
  to the presence of an external (not constant)
  potential.

• The coupling reduces as the energy gap
  increases, and vanishes in the absence of the
  external field V.

• In the Kane model the coupling is present even
  for V0 due to the basis choice

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