Wigner approach to a new two-band 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@unifi.it Giovanni Frosali Dipartimento di Matematica Applicata “G.Sansone” giovanni.frosali@unifi.it

Description of the model 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

Problem setting: unperturbed system Homogeneous periodic crystal lattice: Time-dependent evolution semigroup: No interband transition are possible if is a Bloch function

Multiband models: derivation Wannier envelope function Bloch envelope function Bloch function Wannier function

Multiband models: derivation Wannier envelope function Wannier function is the (cell. averaged) probability to find the electron in the site Ri and into n-th band Non Homogeneous lattices

Multiband models: derivation Wannier envelope function Wannier function High oscillating behaviour The direct use of Wannier basis is a difficult task!!

Multiband models: derivation In literature are proposed different approximations of We loose the simple interpretation of the envelope function L-K Kane “kp” methods

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)

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

MEF model: derivation

MEF model: derivation First approximation: By exploiting the periodicity of and for slow varying external potentials

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

MEF model: derivation

MEF model: derivation Our aim: simplify the above equation. “Interband term”:

We retain only the first MEF model: derivation Our aim: simplify the above equation. “Interband term”: We retain only the first order term Second approximation:

MEF model: derivation Approximate system We write it in coordinate space

Physical meaning of the envelope function: 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.

Effective mass dynamics: MEF model: first order Effective mass dynamics: Zero external electric field: exact electron dynamic intraband dynamic

MEF model: first order Coupling terms: intraband dynamic interband dynamic first order contribution of transition rate of Fermi Golden rule

Wigner picture: Wigner equation Liouville equation Classical limit Wigner function: Phase plane representation: pseudo probability function Wigner equation Liouville equation Classical limit Moments of Wigner function:

Wigner picture: General Schrödinger-like model Density matrix n-th band component matrix of operator General Schrödinger-like model Density matrix

Wigner picture: Multiband Wigner function Evolution equation Introduced by Borgioli, Frosali, Zweifel [1] Well-posedness of the two band Kane-Wigner System [1] G. Borgioli, G. Frosali and P. Zweifel, Wigner approach to the two-band Kane model for a tunneling diode, Transp. Teor.Stat. Phys. 32 3, 347-366 (2003).

Wigner picture: Two band MEF model Two band Wigner model Multiband Wigner function Evolution equation Two band MEF model Two band Wigner model

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.

Two band Wigner model Wigner picture:

Two band Wigner model Wigner picture: intraband dynamic: zero coupling if the external potential is null

Two band Wigner model Wigner picture: intraband dynamic: zero coupling if the external potential is null interband dynamic: coupling like G-R via

Mathematical setting 1 D problem: Hilbert space: Weighted spaces:

Mathematical setting If the external potential the two band Wigner system admits a unique solution

Mathematical setting If the external potential the two band Wigner system admits a unique solution Stone theorem unitary semigroup on Unbounded operator

Mathematical setting If the external potential the two band Wigner system admits a unique solution

Mathematical setting If the external potential the two band Wigner system admits a unique solution Symmetric bounded operators

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

Numerical implementation: splitting scheme Linear evolution semigroup Uniform mesh is a three element vector Discrete operator

Numerical implementation: splitting scheme f.f.t. Approximate solution of

Numerical implementation: splitting scheme f.f.t. Approximate solution of

Space coordinate Momentum coordinate x p Conduction band

x p Valence band Conduction band x p Space coordinate Space coordinate Momentum coordinate x p Valence band Conduction band x p Momentum coordinate Space coordinate

Stationary state: Thermal distribution Conduction band Valence band

Conclusion Next steps 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