1. TWO-BAND MODELS FOR ELECTRON TRANSPORT IN SEMICONDUCTOR DEVICES
KINETIC EQUATIONS:
Direct and Inverse Problems
Università degli Studi di Pavia (sede di Mantova)
Mantova
May 15-17, 2005
TWO-BAND MODELS FOR ELECTRON TRANSPORT
IN SEMICONDUCTOR DEVICES
Giovanni Frosali
Dipartimento di Matematica Applicata “G.Sansone”

2. University of Florence research group on semicoductor modeling
Dipartimento di Matematica Applicata “G.Sansone”
Giovanni Frosali
Dipartimento di Matematica “U.Dini”
Luigi Barletti
Dipartimento di Elettronica e Telecomunicazioni
Stefano Biondini
Giovanni Borgioli
Omar Morandi
Università di Ancona
Lucio Demeio
Scuola Normale Superiore di Pisa (Munster)
Chiara Manzini
Others: G.Alì (Napoli), C.DeFalco (Milano), M.Modugno (LENS-INFM Firenze), A.Majorana(Catania), C.Jacoboni, P.Bordone et. al. (Modena)

3. SINGLE-BAND APPROXIMATION
In the standard semiconductor devices, like the Resonant Tunneling Diode, the single-band approximation, valid if most of the current is carried by the charged particles of a single band, is usually satisfactory. Together with the single-band approximation, the parabolic-band approximation is also usually made. This approximation is satisfactory as long as the carriers populate the region near the minimum of the band.
Also in the most bipolar electrons-holes models, there is no coupling mechanism between energy bands which are always decoupled in the effective-mass approximation for each band and the coupling is heuristically inserted by a "generation-recombination" term.
Most of the literature is devoted to single-band problems, both from the modeling and physical point of view and from the numerical point of view.

4. It is well known that the spectrum of the Hamiltonian of a quantum particle moving in a periodic potential is a continuous spectrum which can be decomposed into intervals called "energy bands". In the presence of external potentials, the projections of the wave function on the energy eigenspaces (Floquet subspaces) are coupled by the Schrödinger equation, which allows interband transitions to occur.
RITD Band Diagram
The single-band approximation is no longer valid when the architecture of the device is such that other bands are accessible to the carriers. In some nanometric semiconductor device like Interband Resonant Tunneling Diode, transport due to valence electrons becomes important.

5. It is necessary to use more sophisticated models, in which the charge carriers can be found in a super-position of quantum states belonging to different bands.
Different methods are currently employed for characterizing the band structures and the optical properties of heterostructures, such as
envelope functions methods based on the effective mass theory (Liu, Ting, McGill, Chao, Chuang, etc.)
tight-binding (Boykin, van der Wagt, Harris, Bowen, Frensley, etc.)
pseudopotential methods (Bachelet, Hamann, Schluter, etc.)
Various mathematical tools are employed to exploit the multiband quantum dynamics underlying the previous models:
the Schrödinger-like models (Sweeney,Xu, etc.)
the nonequilibrium Green’s function (Luke, Bowen, Jovanovic, Datta, etc.)
the Wigner function approach (Bertoni, Jacoboni, Borgioli, Frosali, Zweifel, Barletti, etc.)
the hydrodynamics multiband formalisms (Alì, Barletti, Borgioli, Frosali, Manzini, etc)

6. MULTI-BAND, NON-PARABOLIC ELECTRON TRANSPORT
Wigner-function approach
Formulation of general models for multi-band non-parabolic electron
transport
Use of Bloch-state decomposition (Demeio, Bordone, Jacoboni)
Envelope functions approach (Barletti)
Wigner formulation of the two-band Kane model (Borgioli, Frosali, Zweifel)
Numerical applications (Demeio, Morandi)
The Wigner function for thermal equilibrium of a two-band (Barletti)
Multiband envelope function models (MEF models) (Modugno, Morandi)
Two-band hydrodynamic models (Two-band QDD equations) (Alì, Biondini,
Frosali, Manzini)

7. QUANTUM MECHANICS LEVEL
In this talk we present different Schrödinger-like models. The first one is well-known in literature as the Kane model. The second, based on the Luttinger-Kohn approach, disregards the interband tunneling effect. The third, recently derived within the usual Bloch-Wannier formalism, is formulated in terms of a set of coupled equations for the electron envelope functions by an expansion in terms of the crystal wave vector k (MEF model).

8. The physical environment
Electromagnetic and spin effects are disregarded, just like the field generated by the charge carriers themselves. Dissipative phenomena like electron-phonon collisions are not taken into account.
The dynamics of charge carriers is considered as confined in the two highest energy bands of the semiconductor, i.e. the conduction and the (non-degenerate) valence band, around the point is the "crystal" wave vector. The point is assumed to be a minimum for the conduction band and a maximum for the valence band.
where
The Hamiltonian introduced in the Schrödinger equation is
where is the periodic potential of the crystal and an external potential.

9. KANE MODEL
The Kane model consists in a couple of Schrödinger-like equations for the conduction and the valence band envelope functions.
be the valence band envelope function
Let
be the conduction band electron envelope function and c
m is the bare mass of the carriers,
is the minimum (maximum) of the conduction (valence) band energy
P is the coupling coefficient between the two bands (the matrix element of the
gradient operator between the Bloch functions)

10. Interband Tunneling: PHYSICAL PICTURE
Interband transition in the 3-d dispersion diagram.
The transition is from the bottom of the conduction band to the top of the val-ence band, with the wave number becoming imaginary.
Then the electron continues propagating into the valence band.

11. Remarks on the Kane model
The envelope functions are obtained expanding the wave function on the basis of the periodic part of the Bloch functions evaluated at ,
where
The external potential affects the band energy terms , but it does not appear in the coupling coefficient P .
There is an interband coupling even in absence of an external potential.
The interband coefficient P increases when the energy gap between the two bands increases (the opposite of physical evidence).

12. LUTTINGER-KOHN model
This model is a model, i.e. the crystal momentum is used as a perturbation parameter of the Hamiltonian. The wave function is expanded on a different basis with respect to Kane model:
where n, n' are the band index and the bare electron mass.
As a result, if we limit ourselves to the two-band case, we have:

13. where are envelope functions in the conduction and valence bands, respectively and and are, respectively, the isotropic effective masses in the conduction and valence bands.
As it is manifest, disregarding the off-diagonal terms implies the achievement of two uncoupled equations for the envelope functions in the two bands. This means that the model, at this stage of approximation, is not able to describe an interband tunneling dynamics.

14. MEF MODEL (Morandi, Modugno, Phys.Rev.B, 2005)
The MEF model consists in a couple of Schrödinger-like equations as follows.
A different procedure of approximation leads to equations describing the intraband dynamics in the effective mass approximation as in the Luttinger-Kohn model, which also contain an interband coupling, proportional to the momentum matrix element P. This is responsible for tunneling between different bands induced by the applied electric field proportional to the x-derivative of V. In the two-band case they assume the form:

15. Which are the steps to attain MEF model formulation?
(and ) is the isotropic effective mass
and are the conduction and valence envelope functions
is the energy gap
P is the coupling coefficient between the two bands
Which are the steps to attain MEF model formulation?
Expansion of the wave function on the Bloch functions basis
Insert in the Schrödinger equation

16. Simplify the interband term in
Approximation
Simplify the interband term in
Introduce the effective mass approximation
Develope the periodic part of the Bloch functions to the first order
The equation for envelope functions in x-space is obtained by inverse Fourier transform
For more rigorous details:
See: Morandi, Modugno, Phys.Rev.B, 2005
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17. More rigorously MEF model can be obtained as follows:
projection of the wave function on the Wannier basis which depends on where are the atomic sites positions, i.e.
where the Wannier basis functions can be expressed in terms of Bloch functions as

18. The use of the Wannier basis has some advantages.
As a matter of fact the amplitudes that play the role of envelope functions on the new basis, can be obtained from the Bloch coefficients by a simple Fourier transform
Moreover they can be interpreted as the actual wave function of an electron in the n-band. In fact, ''macroscopic'' properties of the system, like charge density and current, can be expressed in term of averaging on a scale of the order of the lattice cell.
Performing the limit to the continuum to the whole space and by using standard properties of the Fourier transform, equations for the coefficients are achieved.

19. Comments on the MEF MODEL
The envelope functions can be interpreted as the effective wave functions of the electron in the conduction (valence) band
The coupling between the two bands appears only in presence of an external (not constant) potential
The presence of the effective masses (generally different in the two bands) implies a different mobility in the two bands.
The interband coupling term reduces as the energy gap increases, and vanishes in the absence of the external field V.

20. QUANTUM HYDRODYNAMICS LEVEL
From the point of view of practical applications, the approaches based on microscopic models are not completely satisfactory.
Hence, it is useful to formulate semi-classical models in terms of macroscopic variable. Using the WKB method, we obtain a system for densities and currents in the two bands.
In this context zero and nonzero temperatures quantum drift diffusion models are derived, for the Kane and MEF systems.

21. Hydrodynamic version of the KANE MODEL
We can derive the hydrodynamic version of the Kane model using the WKB method (quantum system at zero temperature).
Look for solutions in the form
we introduce the particle densities
Then is the electron density in conduction and valence bands.
We write the coupling terms in a more manageable way, introducing the complex quantity
with

22. Quantum hydrodynamic quantities
Quantum electron current densities
when i=j , we recover the classical current densities
Osmotic and current velocities
Complex velocities given by osmotic and current velocities, which can be expressed in terms of
plus the phase difference

23. The quantum counterpart of the classical continuity equation
Taking account of the wave form, the Kane system gives rise to
Summing the previous equations, we obtain the balance law
where we have used the “interband density” and the complex velocities

24. The previous balance law is just the quantum counterpart of the classical continuity equation.
Next, we derive a system of coupled equations for phases , obtaining an equivalent system to the coupled Schrödinger equations. Then we obtain a system for the currents
(similar equation for ).
The left-hand sides can be put in a more familiar form with the quantum Bohm potentials

25. We express the right-hand sides of the previous equations in terms of the hydrodynamic quantities
It is important to notice that, differently from the uncoupled model, equations for densities and currents are not equivalent to the original equations, due to the presence of

26. Recalling that and are given by the hydrodynamic quantities
and , we have the HYDRODYNAMIC SYSTEM

27. The DRIFT-DIFFUSION scaling
We rewrite the current equations, introducing a relaxation time term in order to simulate all the mechanisms which force the system towards the statistical mechanical equilibrium characterized by the relaxation time
In analogy with the classical diffusive limit for a one-band system, we introduce the scaling
We express the osmotic and current velocities, in terms of the other hydrodynamic quantities.
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28. ZER0-TEMPERATURE QUANTUM DRIFT-DIFFUSION MODEL
for the Kane system.

29. NON ZERO TEMPERATURE hydrodynamic model
We consider an electron ensemble which is represented by a mixed quantum mechanical state, with a view to obtaining a nonzero temperature model for a Kane system.
We rewrite the Kane system for the k-th state
We use the Madelung-type transform
We define
We define the densities and the currents corresponding to the two mixed states

30. HYDRODINAMIC SYSTEM for the KANE MODEL

31. QUANTUM DRIFT-DIFFUSION for the KANE MODEL
with
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32. HYDRODINAMIC SYSTEM for the MEF MODEL

33. QUANTUM DRIFT-DIFFUSION for the MEF MODEL
with

34. Physical meaning of the envelope functions
A more direct physical meaning can be ascribed to the hydrodynamical variables derived from the second approach.
The envelope functions and are the projections of on the Wannier basis, and therefore the corresponding multi-band densities represent the (cell-averaged) probability amplitude of finding an electron on the conduction or valence bands, respectively.
The Wannier basis element arises from applying the Fourier transform to
the Bloch functions related to the same band index .
This simple picture does not apply to the Kane model.
The Kane envelope functions and the MEF envelope functions are linked by the relation

35. This fact confirms that even in absence of external potential , when no interband
transition can occur, the Kane model shows a coupling of all the envelope functions.
We consider a heterostructure which consists of two homogeneous regions separated by a potential barrier and which realizes a single quantum well in valence band.
NUMERICAL SIMULATION
See: Alì, F.,Morandi,
SCEE2005 Proceedings

36. Kane
MEF
The incident (from the left) conduction electron beam is mainly reflected by the barrier and the valence states are almost unexcited.

37. Kane
MEF
The incident (from the left) conduction electron beam is partially reflected by the barrier and partially captured by the well

38. Kane
MEF
When the electron energy approaches the resonant level, the electron can travel from the left to the right, using the bounded valence resonant state as a bridge state.

39. QUANTUM KINETICS LEVEL
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40. Wigner picture
Wigner function:
MEF-Wigner Model
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41. Numerical results: Kane-Wigner

42. Numerical results: MEF-Wigner

43. Numerical results: MEF-WIGNER model
Conduction band
Valence band

44. Thanks for your attention !!!!!

45. Wigner picture
Wigner function:
Kane-Wigner Model

46. REMARKS
We have derived a set of quantum hydrodynamic equations from the two- band Kane model, and from the MEF model. These systems, which can be considered as a nonzero-temperature quantum fluid models, are not closed.
In addition to other quantities, we have the tensors
and ,
similar to the temperature tensor of kinetic theory.
Closure of the quantum hydrodynamic system
Numerical treatment
Heterogeneous materials
Numerical validation for the quantum drift-diffusion equations (Kane and MEF models) are work in progress.