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Giornata di lavoro sulle Nanoscienze
Firenze 16 dicembre 2005
Dipartimento di Matematica
Applicata Università di Firenze
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.
m* (isotropic) effective mass,
Multiband transport models for semiconductor devices
Eg
energy gap
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Dipartimento di Matematica
Applicata Università di Firenze
Giornata di lavoro sulle Nanoscienze
Firenze 16 dicembre 2005
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
Multiband transport models for semiconductor devices
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Dipartimento di Matematica
Applicata Università di Firenze
Giornata di lavoro sulle Nanoscienze
Firenze 16 dicembre 2005
• Going back in real space by Fourier transforming yields an equation for the
projection coefficiente  n ( Ri ) on the Wannier basis ( R atomic sites positions)
i
• Two approximations:
• Simplify the interband term around k  0 developing the periodic
part of the Bloch wavefunctions un (k , x) to the first order
• Effective mass approximation
MeF MODEL
Multiband transport models for semiconductor devices
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Giornata di lavoro sulle Nanoscienze
Firenze 16 dicembre 2005
Dipartimento di Matematica
Applicata Università di Firenze
Comments on the MeF MODEL
• The envelope functions  c 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
absence of the external field V.
Eg increases, and vanishes in the
• In the Kane model the coupling is present even for V=0 due to the basis choice
 
K
j
2
M
j
P
i
 hM ,
m0 ( E j  Eh )
Multiband transport models for semiconductor devices
j, h  c, v.
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Dipartimento di Matematica
Applicata Università di Firenze
Multiband transport models for semiconductor devices
Giornata di lavoro sulle Nanoscienze
Firenze 16 dicembre 2005
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