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**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”

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**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)

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**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.

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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 RITD Band Diagram The single-band approximation is no longer valid when the architecture of the device is such that other bands are accessible to the carriers.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.

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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)

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**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)

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**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).

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**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.

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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)

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**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.

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**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).

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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:

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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.

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**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:

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**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

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**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 Vai a lla 19

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**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

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**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.

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**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.

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**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.

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**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

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**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

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**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

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**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

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**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

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**Recalling that and are given by the hydrodynamic quantities**

and , we have the HYDRODYNAMIC SYSTEM

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**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. Vai alla 31

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**ZER0-TEMPERATURE QUANTUM DRIFT-DIFFUSION MODEL**

for the Kane system.

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**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

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**HYDRODINAMIC SYSTEM for the KANE MODEL**

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**QUANTUM DRIFT-DIFFUSION for the KANE MODEL**

with Vai alla 32 Vai alla 33

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**HYDRODINAMIC SYSTEM for the MEF MODEL**

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**QUANTUM DRIFT-DIFFUSION for the MEF MODEL**

with

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**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

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**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

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

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

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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.

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**QUANTUM KINETICS LEVEL**

Vai alla 42

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Wigner picture Wigner function: MEF-Wigner Model Vai alla 42

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**Numerical results: Kane-Wigner**

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**Numerical results: MEF-Wigner**

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**Numerical results: MEF-WIGNER model**

Conduction band Valence band

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**Thanks for your attention !!!!!**

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Wigner picture Wigner function: Kane-Wigner Model

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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.

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