Presentazione di PowerPoint

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ICTT19 19th International Conference on Transport
Theory
Budapest, July 24-30, 2005
Wigner approach to a new two-band envelope
function model for quantum transport
Omar Morandi Dipartimento di Elettronica e
Telecomunicazioni omar.morandi_at_unifi.it
Giovanni Frosali Dipartimento di Matematica
Applicata G.Sansone giovanni.frosali_at_unifi.it
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Plain of the talk

• Description of the model

• Multiband (MEF) model

• Multiband-Wigner picture

• Mathematical problem

• Mathematical setting
• Well posedness of the Multiband-Wigner system

• Numerical applications

• Description of the numerical algorithm

• Application to IRTD

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Problem setting unperturbed system

• Homogeneous periodic crystal lattice

• Time-dependent evolution semigroup

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Multiband models derivation
Wannier envelope function
Wannier function
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Multiband models derivation
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MEF model derivation
Wannier function
Bloch function
Our approach

• To get our multiband model in Wannier basis
• We recover un approximate set of equation for
  in the Bloch basis
• (momentum space)
• We Fourier transform the equations obtained
  (coordinate space)

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MEF model characteristics
Hierarchy of kp multiband effective mass
models, where the asimptotic parmeter is the
quasi-momentum of the electron

• Direct physical meaning of the envelope function
• Easy approximation (cut off on the index band)
• Highlight the action of the electric field in
  the interband transition phenomena
• Easy implementation Wigner and
  quantum-hydrodynamic formalism

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MEF model derivation
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MEF model formal derivation
Evaluation of matrix elements
Kane momentum matrix
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MEF model derivation
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MEF model derivation

• Our aim simplify the above equation.

Interband term
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MEF model derivation
Approximate system
We write it in coordinate space
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MEF model first order
Physical meaning of the envelope function
The quantity represents the
mean probability density to find the electron
into n-th band, in a lattice cell.
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MEF model first order
Effective mass dynamics
Zero external electric field exact electron
dynamic

• intraband dynamic

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MEF model first order
Coupling terms

• intraband dynamic
• interband dynamic
• first order contribution of
• 
  transition rate of Fermi Golden rule

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Kane model
Problems in the practical use of the Kane model

• Strong coupling between envelope function
  related
• to different band index, even if the external
  field is null

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Wigner picture
Wigner function
Phase plane representation pseudo
probability function
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Wigner picture
n-th band component
matrix of operator
General Schrödinger-like model
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Wigner picture
Multiband Wigner function
Evolution equation
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Wigner picture
Multiband Wigner function
Evolution equation
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Two band Wigner model
Wigner picture
Moments of the multiband Wigner function
represents the mean
probability density to find the electron into
n-th band, in a lattice cell.
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Two band Wigner model
Wigner picture
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Two band Wigner model
Wigner picture

• intraband dynamic zero coupling if the external
  potential is null

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Two band Wigner model
Wigner picture

• intraband dynamic zero coupling if the external
  potential is null

• interband dynamic coupling like G-R via

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Mathematical setting
1 D problem
Hilbert space
Weighted spaces
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Mathematical setting
If the external potential
the two band Wigner system admits a unique
solution
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Mathematical setting
If the external potential
the two band Wigner system admits a unique
solution
Stone theorem
unitary semigroup on
Unbounded operator
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Mathematical setting
If the external potential
the two band Wigner system admits a unique
solution
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Mathematical setting
If the external potential
the two band Wigner system admits a unique
solution
Symmetric bounded operators
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Mathematical setting
If the external potential
the two band Wigner system admits a unique
solution
The operator generate
semigroup
The unique solution of (1) is
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Numerical implementation splitting scheme
Linear evolution semigroup
Uniform mesh
is a three element vector
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Numerical implementation splitting scheme
f.f.t.
Approximate solution of
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Numerical implementation splitting scheme
f.f.t.
Approximate solution of
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Conduction band
x
p
Momentum coordinate
Space coordinate
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Stationary state Thermal distribution
Conduction band
Valence band
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Conclusion

• Multiband-Wigner model

• Well posedness of the Multiband-Wigner system

• Application to IRTD

Next steps

• Extention of MEF model to more general
  semiconductor

• Well posedness of Multiband-Wigner model coupled
  with Poisson eq.

• Calculation of I-V IRDT characteristic for
  self-consistent model