Communications in Applied and Industrial Mathematics, DOI: 10.1685/2010CAIM474ISSN 2038-0909, 1, 1, (2010) 167–184

A WKB approach to the quantum multiband

electron dynamics in the kinetic formalism.

Omar Morandi1

1Institute of Theoretical and Computational Physics, TU Graz, Petersgasse 16, 8010

Graz, Austria

morandi@dipmat.univpm.it

Communicated by Roberto Natalini

Abstract

We derive a WKB-like asymptotic expansion of the multiband Wigner function. Themodel, derived in the envelope function theory, is designed to describe the dynamicsin semiconductor devices when the interband conduction-valence transition cannot beneglected. We derive a hierarchy of equations that to lowest order consist of two Hamilton-Jacobi equations corresponding to the classical dynamics of point particles with positiveand negative kinetic energy. Our methodology is based on the Van Vleck approach and aWKB-like asymptotic expansion procedure is used to reduce the numerical complexity ofthe Wigner multiband evolution system. An approximate closed-form solution is obtainedby an iterative procedure that exploits the different time scales on which the intrabandand interband dynamical variables evolve. The interband tunneling mechanism appearingto the first order of the expansion is expressed in a very simple mathematical form. Byexploiting the highly oscillating behaviour of the multiband Wigner functions we derive aasymptotic expression of the interband transition probability. The resulting formulationreveals particularly close to the classical description of the particles motion and thisformal analogy is useful to gain new physical insight and to profit of the numericalmethod developed for classical systems. The approximates evolution equations are usedto simulate the evolution of the Wigner quasi-distribution function in a IRTD diode.

Keywords: WKB, multiband kinetic dynamic, Wigner formalism.

AMS Subject Classification: 81S22, 81S30, 81Q20

1. Introduction

Recently much attention has been paid to quantum transport models.In particular, multiband transport is a topic of growing interest amongphysicists and applied mathematicians studying semiconductor devices ofnanometric size. Indeed, quantum effects cannot be neglected when the sizeof the electronic device becomes comparable to the electron wavelength, asin new generation devices. For this reason, quantum modeling becomesa crucial aspect in nanoelectronics research from the perspective of analogand digital applications. In the past several decades an increasing effort has

Received 2010 01 12, in final form 2010 06 08

Published 2010 06 21

Licensed under the Creative Commons Attribution Noncommercial No Derivatives

O. Morandi

been devoted in deriving new approach to the charge transport whose valid-ity spans form the classical transport to the full quantum dynamics. Despitethe progress obtained in this field, a general framework where quantum ef-fects can be easily included in a mesoscopic structure where also classicaltransport play an important role, is far from being achieved. Different ap-proaches based on the density matrix, non equilibrium Green’s functions,and the Wigner function have been proposed to achieve a quantum descrip-tion of electron transport [1]. Among them, the Wigner-function formalismis the one that bears the closest similarities to the classical Boltzmann equa-tion, which suggests the possibility of using this formalism in order to obtainquantum corrections to the classical phase-space motion. The phase-spaceformulation of quantum mechanics offers a framework in which quantumphenomena can be described in a classical language and the question of thequantum-classical correspondence can be directly investigated. In particu-lar, the visual representation of the quantum motion by quantum-correctedphases-plane trajectories, is a valuable aid to a conceptual understandingof the complex quantum dynamics.

We are interested in study semiconductor devices characterized by tun-neling effects between different bands, as the resonant interband tunnelingdiode (RITD) (see for instance [2]- [3]). Unlike the usual semiconductordevices where the bulk band structure is slightly modified by the presenceto some “external” electric field (due for example to the electrostatic inter-action of charges or to an applied source-drain voltage), and the electroniccurrent flows in a single band, the remarkable feature of a RITD is the pos-sibility to achieve a coupling between “conduction” and “valence” states,allowing an interband current. Hence, the description of electron transportin such quantum devices requires multiband models able to account fortunneling mechanisms between different bands induced by the heterostruc-ture design and the applied external bias. In electronics, the most popularmodel capable to describe the interaction of two bands is based on the KaneHamiltonian [4]. By means of a perturbative approach and an averaging pro-cedure, the effect of the periodic lattice potential is taken into account bysome phenomenological parameters, such as the energy-band gap and thecoupling coefficient (Kane momentum) between the two bands. It is wellknow that a strong formal analogy is present between the description ofthe quantum non-relativistic motion of a particle in a semiconductor ex-pressed in the kp envelope function framework and the relativistic Diracequation. This analogy has been for example widely exploited in graphenewhere the Fermi level is found near the gapless intersection of two approx-imatively linear branches of the graphene spectrum [5]. As demonstratedin transport measurements by Novoselov [6], and in spectroscopic mea-

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surements by Zhou [7], the electronic properties of graphene are describedby the Dirac equation (where the speed of light is replaced by the Fermivelocity), even though the microscopic Hamiltonian of carbon atoms is non-relativistic. The standard multiband semiconductor framework open thusthe interesting possibility to test some mathematical procedure originallyderived in the contest of high energy physic in a framework, the solid statephysics, where a broad set of experimental data are available and the mostof the semiconductor parameters can be controlled.

In the present contribution we adapt the WKB procedure for relativisticparticles to the kinetic framework and we derive approximated equationsfor the quantum evolution of a multiband system constituted of chargedparticles. Historically, the early extension of the WKB method to the Diracequation is due to Pauli [8], who showed that the phase of a WKB spinoris given by a solution of the Hamilton-Jacobi equation for relativistic pointparticles. Although the Pauli approach has been successfully applied tonon-relativistic quantum mechanics, where it leads to the so-called Einstein-Brillouin-Keller quantization [9], in general it is possible to determine theamplitude of the semiclassical spinor only in some special cases. As pointedout in Ref. [10] in the full quantum-relativistic contest, the Dirac-like cou-pling among the different bands strongly affects the dynamics inside everysingle band. In particular at the semiclassical level it is possible to derive aneffective dynamics where the particles follow a quantum-corrected Hamil-tonian flux and the single-band classical Hamiltonian is replaced by theeigenvalue of the Dirac operator.

The main result of this work is to extend this approach in order to derivea WKB-like asymptotic expansion of the multiband Wigner function. Theapproximate evolution equations can be used to obtain the Wigner quasi-distribution function in a IRTD diode. The interband tunneling mechanismappearing to the first order of the expansion is expressed in a very simplemathematical form, in contrast with the highly complex formulation of theexact evolution equation. In particular, by exploiting the highly oscillatingbehaviour of the multiband Wigner functions that was already obserbedin Ref. [11], we derive a asymptotic expression of the interband transitionkernel which is formally close to the usual generation-recombination terms.This achievement allow to describe the interband transition phenomena atthe same level of the more common scattering effects present in a semicon-ductor material. We derive a hierarchy of equations that to lowest orderconsist of two Hamilton-Jacobi equations corresponding to the classical dy-namics of point particles with positive and negative kinetic energy, andthat represent the electron motion in conduction and in valence band re-spectively. The condition that arises in next-to-leading order can be reduced

169

O. Morandi

to two differential equations for 2× 2 matrices describing the transport ofthe band degrees of freedom along particle orbits. Out methodology is basedon the Van Vleck approach where a semiclassical representation of the timeevolution kernel of the Schrodinger wave function in term of a WBK ansatzis provided. Such a representation, usually derived from a Feynman pathintegral, to which the method of stationary phase is applied [12], makes useof a representation of the kernel in terms of an oscillatory integral.

1.1. WKB procedure

In this paper we adopt the multiband envelope function model (MEF)described in Ref. [13]. This model is derived within the k · p frameworkand is so far very general. This approach allows the description of electrontransport in devices where tunneling mechanisms between different bandsare induced by an external applied bias U . We consider a physical modelin which only the valence and the conduction band are taken into account.Under this hypothesis the MEF model is a 2 × 2 Schrodinger-like set ofequations

(1) i~∂Ψ

∂t= HΨ,

where

H =

Ec + U(x)− ~2

2m∗

∂2

∂x2− ~

m0

PKE(x)Eg

− ~

m0

PKE(x)Eg

Ev + U(x) +~2

2m∗

∂2

∂x2

(2)

is the Hamiltonian, Ψ = (Ψc,Ψv) with Ψc and Ψv the conduction and va-lence Wannier envelope functions. Ec (Ev) is the minimum (maximum) ofthe conduction (valence) energy band, PK is the Kane momentum and m0,m∗ are the bare and the effective mass of the electron. U is the “external”potential, which takes into account different effects, like the bias voltageapplied across the device, the contribution from the doping impurities andfrom the self-consistent field produced by the mobile electronic charge. Ac-cording to Ref. [14], the multiband Wigner function is defined as:

f(x, p, t) ≡(fcc fcvf cv fvv

)(3)

fij ≡1

2π

∫ ∞

−∞Ψi

(x+

~η

2

)Ψj

(x− ~η

2

)e−iηp dη ,(4)

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where i, j = c, v. This definition is a straightforward extension of the single-band Wigner function to a multiband system. By differentiating Eq. (3)with respect to time, and by using Eq. (1), we obtain the evolution equationfor the multiband Wigner function (W-MEF system),

(5) i~∂f

∂t= [H, f ]# ,

where [H, f ]# = H#f−f#H denotes the Moyal commutator with respect tothe Moyal product #, and H(x, p) is the Hamiltonian symbol with respectto the Weyl quantization procedure. We recall that, given a differentialoperator A acting on the x variable, the Weyl quantization procedure es-tablishes a unique correspondence of A with a function A(x, p), which isthe symbol of the operator. The relationship between the operator A andits symbol A is given by

(6)(AΨ)(x) =

1

2π~

∫A(x+ y

2, p

)Ψ(y) e

i~(x−y)p dy dp .

The Moyal product can then be expanded with respect to ~ according to

A#B =∞∑

k=0

~k

(2i)k

∑

|α|+|β|=k

(−1)|α|

α!β!

(∂αx ∂

βpA)(

∂αp ∂

βxB).

To attack the quantum evolution problem by solving directly the systemof Eq. (5) is a difficult task: a preliminary theoretical study of the multi-band Wigner evolution system showed the occurrence of a very complexdynamics [15]. The numerical complexity of this system prevent its directapplication to a real device even in the simple cases where the one dimen-sion approximation of the motion can be considered sufficiently accurateto reproduce the statical and dynamical characteristics. Furthermore con-ceptual problems arise if we address to the practical question of couplinga multiband quantum system with a classical evolution system (that is ofparamount importance in a real device where typically quantum effectstake place in a very narrow region, the active region, and the remaining do-main can be well described by a classical distribution). It is very importantto provide an accurate modelization of the interplay between quantum ef-fects and classical transport (describing for example scattering mechanisms,heath flux, injection of charge and external driving voltage effects) to repro-duce the observed thermodynamical and transport properties of a device.Since in the Wigner framework the quantum kinetic transport operatorsare defined in term of Fourier transform, which is strongly non-local, con-straint are imposed to a possible numerical algorithm, making practically

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O. Morandi

impossible for example the use non structured mesh grids or, more simply,the definition of variable spatial and momentum discretization mesh size.For these reason, we are interested in reducing the numerical complexityof the Wigner multiband evolution system by using a WKB-like asymp-totic expansion procedure. Instead of deriving a ~ expansion directly onthe multiband Wigner function defined in Eq. (5), it is more convenientto expand the time evolution kernel of the Schrodinger wave function Ψand to use that the Wigner quasi-distribution function is defined througha bilinear transformation of Ψ.

We write the solution of Eq. (1) with initial condition ΨI as follow

(7) Ψ(x, t) =

∫K(x, x′, t)ΨI(x

′) dx′

where K denotes the kernel of the evolution operator of Ψ and it solves

(8) i~∂K∂t

= H(x)K

with initial condition

(9) K(x, x′, t0) = δ(x− x′) I2×2 ,

I2×2 denotes the 2 × 2 identity matrix. According to Eq. (7) the Wignerfunction can be expressed as

f(x, p, t) =

∫K(x+

~η

2, ϑ− ~τ

2

)fI(ϑ, p′

)K†

(x− ~η

2, ϑ+

~τ

2

)ei(pη−p′τ) dη dτ dϑ dp′ ,

(10)

where † denotes hermitian conjugate and fI is the initial condition for fij .Explicitly

fI(ϑ, p′

)=

1

2π

∫ΨI

(ϑ− ~τ

2

)Ψ†

I

(ϑ+

~τ

2

)e−iτp′ dτ

We are interested in deriving a asymptotic approximation of the evolutionkernel K. For future references we consider the following result: given atwo-band wave function in the WKB representation

Ψ(x, t) = a(x, t)ei~S(x,t) ,

we have

(HΨ)(x) = ei~S(x,t)

{H0

(x,

∂S

∂x

)a+

~

i

[∂H0

∂p

(x,

∂S

∂x

)∂a

∂x+

1

2

[∂

∂x

∂H0

∂p

(x,

∂S

∂x

)]a+H1

(x,

∂S

∂x

)a

]− ~

2

2

∂2H0

∂p2

(x,

∂S

∂x

)∂2a

∂x2

},

(11)

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where we have considered the following ~ expansion of the symbol H

(12) H(x, p) =∑

n

~nHn(x, p) .

Given two orthonormal x−dependent column vectors e+(x), e−(x) (andwhose specific values will be defined in the following) the 2 × 2 matrix Kcan de decomposed as follows

K = K+ +K−

K± ≡ e±[e±]†K .

We derive a semiclassical-like expansion of the evolution kernel in the spiritof the Van Vleck formula. In the present context it is convenient to representthe time evolution kernel in terms of the following oscillatory integral

K±(x, x′) =

∫K±

(x, ξ =

∂S±

∂x, t

)e

i~[S±(x,p′,t)+x′p′] dp′

K± (x, ξ, t) = e±(x, ξ)[a±]†(x, ξ, t) .

This method was developed for the study of scalar wave equations inthe context of microlocal analysis, and subsequently found application tothe development of several trace formulae [16]- [17]. In the case of theSchrodinger equation it leads to the same result as Gutzwillers originalderivation which employed a stationary phase approximation of a Feynmanpath integral [12]. This approach is similar to the usual WKB method andthe possibility of the band transition is reflected in these equations throughtheir matrix character.

In this formula K is represented as the quantum-mechanical transitionprobability between x and x′ averaged over all possible initial classical mo-menta p′. The explicit expressions of the vectors e±(x), the equations ofmotion of a± and of S±, can be obtained by solving Eq. (8). In the spirit ofthe WKB approximation we consider a ~-expansion of Eq. (8) and we sepa-rate the zeroth order term from the higher order terms. In particular we willtake advantage of the expansion of Eq. (11) and we fix the unknown e±(x)and of S± in order to satisfy the zeroth order equation. All the ~ correctionswill be take into account by the vectors a±. In this way we decompose thesolution into simpler elements each of which has a specific interpretation:S± describe the classical interband motion of the particles in term of point-like trajectory, the vectors e±(x) define some local in space and momentumprojectors in the conduction and valence band. This projection is similar towhat we found in the usual effective mass approximation where the mixing

173

O. Morandi

among different band branches, due to the presence of an external electricfield, are discarded and the spectrum of the particle is defined only by thematerial in the surrounding of its classical position. Finally the vectors a±

provide the quantum correction to the classical motion and are responsibleof the band transition. The zeroth order of Eq. (8) gives

−∫ {

ei~[S+(x,p′,t)+x′p′]K+∂S+

∂t+ e

i~[S−(x,p′,t)+x′p′]K−∂S−

∂t

}dp′ =

∫ {e

i~[S+(x,p′,t)+x′p′]H0

(x,

∂S+

∂x

)K+ + e

i~[S−(x,p′,t)+x′p′]H0

(x,

∂S−

∂x

)K−

}dp′ .

This expression is satisfied if the vectors e±(p, x)

e± ≡ v±

(x,

∂S±

∂x

)

are the eigenvector of the symbol H0(p, x) (see expansion of Eq. (12))

H0(x, p)v±(x, p) = λ±(x, p)v±(x, p)

and S± satisfy the Hamilton-Jacobi equations associated to the eigenvaluesof the Hamiltonian

∂S±

∂t+ λ±

(x,

∂S±

∂x

)= 0(13)

S±(t0) = −xp′ .(14)

After some algebra it is possible to show that Eq. (8) is equivalent to thefollowing system for the unknown a±

(15)(−1

2

∂

∂x

∂λ+

∂p− e+

†H1e+ − ∂λ+

∂p

∂

∂x+

∂

∂t

)[a+]†

= −ei~(S−−S+) ×

e+†(

∂

∂t−Q−

)e−[a−]†

(−1

2

∂

∂x

∂λ−

∂p− e−

†H1e− − ∂λ−

∂p

∂

∂x+

∂

∂t

)[a−]†

= −ei~(S+−S−) ×

e−†(

∂

∂t−Q+

)e+[a+]†

where

Q± =∂H0

∂p

(x,

∂S±

∂x

)∂

∂x+

1

2

[∂

∂x

∂H0

∂p

(x,

∂S±

∂x

)](16)

+H1

(x,

∂S±

∂x

)− ~

2

2

∂2H0

∂p2

(x,

∂S

∂x

)∂2

∂x2

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The initial condition of K given in Eq. (9) leads to the following initialcondition for a±

a±(x, ξ, t0) = e± (x, ξ) .

The system of Eq. (15) takes into account the band transition process. Ina rather general framework in Ref. [11]- [18] was showed that the bandtransition is a complex coherent phenomena where the interference effectsbetween the conduction and the valence components of the distributiongenerate a highly oscillating-in-time solution. Furthermore the usual ~ ex-pansion of the solution which is a characteristic features of the WKB ap-proach fail to give a good convergence. In fact high order terms play aimportant role in the band transition phenomena when the external poten-tial becomes strong enough to give a relevant contribution to the currentflux. In the practical application of a band-to-band model to semiconductordevices we are usually interested in the long time behaviour of the solution,and the main goal is to obtain a accurate estimation of the transition prob-ability. In this contest, in the same spirit of the Markovian approximationwhere the memory effects are integrated out by defining a two-time tempo-ral scale (the fast time scale including memory effect and the microscopicaldetail of the scattering processes, and the long time scale which describingthe evolution of the mean quantities), the fast processes which take placeon the femtosecond scale and describe all the interference effects of thewave function are considered of minor importance. Due to the presence ofhigh oscillating terms (whose frequency is proportional to the differences ofthe energy between the two band, evaluate along the pseudo-classical path)we introduce the following hierarchy of approximated solutions (fixed-pointlike expansion)

(17) a± = limn→∞

a±n

with(−1

2

∂

∂x

∂λ+

∂p− e±

†H1e± − ∂λ±

∂p

∂

∂x+

∂

∂t

)[a±n

]†=(18)

−ei~(S∓−S±)e±

†(

∂

∂t−Q∓

)e∓[a∓n−1

]†

This ansatz is equivalent to the asymptotic procedure in term of the “Dysonlike” expansion derived in Ref. [11]. Noting that |∂S+

∂t− ∂S−

∂t| > Eg, in the

case of in a wide gap semiconductor, the exponential term which appearsin Eq. (15) is a fast oscillating function and gives a relevant contributionto the motion only in the neighborhood of the minimum of the oscillationfrequency. Corresponding, the solution can be estimated by means of a

175

O. Morandi

stationary phase procedure. This iterative procedure is very similar to theDyson formalism, which is used to describe electron scattering phenomenain semiconductors. This analogy is useful in order to describe the tunnelingprocess in which a particle “disappears” from to a given band and it “ap-pears” in a different branch of the band diagram in a similar way as thegeneration of an electron-hole pair by the absorption of a photon.

To obtain a suitable approximated analytic solution of Eq. (15) in thegeneral case is a difficult task even in the stationary phases approxima-tion. A simple approximation can be obtained if we limit ourselves to thesimplified case of a uniform static electric field. In this case Eq. (13) gives

S± = −x(p′ + Et)±∫ t

t0

((p′ + Eτ)2

2+

Eg

2

)dτ

H1 = 0

and the system of Eq. (15) becomes

(19)

[(p′ + Et) ∂

∂x+

∂

∂t

] [a+]†

= −iPKEEgm0

e−i~ϕ(t)

[a−]†

[−(p′ + Et) ∂

∂x+

∂

∂t

] [a−]†

= −iPKEEgm0

ei~ϕ(t)

[a+]†

where

ϕ =

∫ t

t0

[(p′ + Eτ

)2+ Eg

]dτ ,

with initial conditions

a+(t0) =

(1

0

)a−(t0) =

(0

1

).

We denote with αi, βi, the components of a+ and a− respectively e. i.a+ = (α1, α2)

† ; a− = (β1, β2)†. It is easy to show that the quantity

|αn|2 + |βn|2 with n = 1, 2 is conserved. In fact(20)

∂(|αn|2 + |βn|2

)

∂t= 2

PKEEg

(ℜ{e−

i~ϕ(t)βnαn

}−ℜ

{e

i~ϕ(t)αnβn

})= 0

Since both the initial datum and the eigenvector of H0 does not depend onx, we can obtain a simple expression for the solution. Eq. (19) becomes

∂αn

∂t= −i

PKEEgm0

e−i~ϕ(t)βn

∂βn∂t

= −iPKEEgm0

ei~ϕ(t)αn

; n = 1, 2.

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If we take the substitution PK → −PK it is immediate to verify that β2 = α1

and α2 = β1. By approximating α1 up to the second order of the fixed pointexpansion of Eq. (17) we obtain

α1 = α1(t0)−(

PKEEgm0

)2 ∫ t

t0

∫ t′

t0

e−i~[ϕ(t′)−ϕ(t′′)]α1(t

′′) dt′′ dt′(21)

≃ 1−(

PKEEgm0

)2 ∫ t

t0

∫ t′

t0

e−i~[ϕ(t′)−ϕ(t′′)] dt′′ dt′

≃ 1− 2π2

(PK

3√E

Egm0

)2

Ai2(

Eg

3√E2

)θ

(t+

p′

E

).

In Appendix 4 we give the detail of the calculation. We note the strongsimilarity of this result with the transition integral obtained in Ref. [11]and Ref. [18] where a different approach is considered. In particular inRef. [18] a numerical study showed that these formulae provide a goodestimation of the exact interband transition probability for a realistic setof the semiconductor parameters.

1.2. Stationary phases approximation

The explicit form of the coefficients defining K can be used to obtainthe expression of the Wigner function. In particular Eq. (10) gives

f =∑

i,j=±

f ij

f ij(x, p, t) =

∫Ki

(x+

~η

2, P − r

2

)fI (ϑ,−P )Kj†

(x− ~η

2, P +

r

2

)

ei~[Si(x+ ~η

2,P− r

2,t)−Sj(x− ~η

2,P+ r

2,t)−ϑr+~pη] dη dϑ dP dr .

(22)

In this representation we have that K+ fI K+†describes both the intraband

conduction dynamics and the interband transition from the valence bando the conduction band, while K− fI K−†

describes the valence dynamicsincluding the transition from conduction to the valence band. Finally themixed terms K+ fI K−†

, K− fI K+†represent the non diagonal correlation

that in the original formulation was manly represented by the terms fcvand fvc. To obtain a manageable expression for the diagonal terms of thequasi-distribution function fij (representing the interesting quantities for aphysical point of view), we apply the stationary phases approximation to

177

O. Morandi

Eq. (22). In particular Eq. (22) for i = j = ± is of the form

f ii(x, p, t) =

∫G (x, η, r, ϑ, P ) e

i~[Si(x+ ~η

2,P− r

2,t)−Si(x− ~η

2,P+ r

2,t)−ϑr+~pη] dϑ dP dη dr ,

where G is assumed sufficiently smooth. To apply the stationary approxi-mation we expand the generating functions Si with respect to η and r andwe discard the (η, r) dependence of the function G. We obtain

f ii(x, p, t) ≃∫

G (x, ϑ, P ) Φ (x, ϑ, P ) dϑ dP

Φ =

∫e

i~

[(

∂Si

∂x

∣

∣

∣

(x,P )+p

)

~η+

(

∂Si

∂p′

∣

∣

∣

(x,P )+ϑ

)

r

]

dη dr

= 4π2~δ

(∂Si

∂x

∣∣∣∣(x,P )

+ p

)δ

(∂Si

∂p′

∣∣∣∣(x,P )

+ ϑ

)

Since the functions S± solve the Hamilton-Jacobi Equation (13), S± aregenerating functions for canonical transformations

(23)t0 t(

x′ = ∂Si

∂p′, p0 = P

)→(x, ∂S

j

∂x

) .

More specifically the spatial derivative ∂S±

∂x≡ p±cl(t) is the momentum of

a particle traveling along the trajectory connecting the initial point of co-

ordinate (x′, P ) for t = t0, with the final point(x, ∂S

j

∂x

). In the case of a

uniform electric field we thus obtain that the zeroth order diagonal Wignerfunctions f++, f−−, follow the trajectory

p(t) = P + Et

x(t) = ϑ±∫ t

t0

p(τ) dτ.

and (similar formula for f−−)

f++ ≃ 4π2~

(α1 α2

0 0

)fI

(x−

∫ t

t0

(P + Eτ) dτ, E(t− t0) + p

)(α1 0α2 0

).

We choose the following physical based initial condition for fij

fI =

(f0cc 00 f0

vv

),

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which represents a initial distribution of particles in conduction and in va-lence band where the interference effects are discarded. It is immediate toverify that with the previous initial conditions, the component of the multi-band Wigner function f++ (f−−) is a matrix with a single non-vanishingvalue f++

11 (f−−22 ) that we will denote with f+ (f−). In same way the mixed

term f+− (f+−) has a single out-of-diagonal component. Since we are inter-ested to obtain the equation of motion of the diagonal components we limitourselves to the function f+ and f−. We differentiate the integral equationwith respect to the time and we obtain the approximated effective evolutionsystem for the diagonal components of the multiband Wigner function

(24)

∂f+

∂t=

p

m∗

∂f+

∂x+ E ∂f

+

∂p+N(p, t)

(f− − f+

)

∂f−

∂t= − p

m∗

∂f−

∂x+ E ∂f

−

∂p+N(p, t)

(f+ − f−

)

with initial conditions f+(t0) = f0cc, f

−(t0) = f0vv. We defined

N(p, t) =∂|α1|2∂t

≃ 4π2

(PK

3√E

Egm0

)2

Ai2(

Eg

3√E2

)δ(p) ,

and we have used Eq. (20). This formulation reveals that the effect of theinterband transition is taken into account as an effective scattering processthat occurs when the particle is at rest.

2. Numerical Application

We use the system of Eq. (24) to simulate the steady state of a sim-ple IRDT diode. In particular we are interested to obtain the correction tothe stationary current due to the interband tunneling. The diode consistof two homogeneous regions separated by a region of constant electric fieldand realizing a barrier in x = 0 (corresponding to the source contact) forconduction electron and in x = L (corresponding to the drain contact) forholes in the valence band. At the boundary of the simulation domain weassume that the semiconductor is connected with metallic contacts, whichat the microscopical level are described by a flux of electrons injected intothe diode with a Fermi dispersion (see figure 1). We fix the potential ofthe source contact (corresponding to x = 0) equal to zero and the drainpotential equal to Vb + Vex where Vb is the built-in potential and Vex is theexternal potential. The resulting electric potential in the domain [0, L] isthus equal to (Vb+Vex)/L. In our simulation, we used the following param-eters: Eg = 0.02 eV, PK = 5 109m−1, L = 10 nm, the lattice temperature

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O. Morandi

−0.1

0

0.1

−200

20

0

0.1

0.2

0.3

0.4

Velocity 106 (m s−1)Length (nm)

−0.1

−0.05

0

0.05

0.1

−20

0

20

0

0.5

1

Velocity 106 (m s−1)Length (nm)

Fig. 1. Electron conduction distribution f+ (left hand side) and valence hole distribution1− f− (right hand side) for an applied potential Vext = 0.01 eV.

is T = 100 K, the doping concentration in the drain (source) contact isND = 1019 cm−3 (NA = 1019 cm−3) and Vb = 0.05 eV.

The results of the simulation can be interpreted as follows. f+ (f−)describes the motion of the electron ensemble in the conduction (valence)band. It shows that the conduction electron and valence beam is (mainly)reflected back by the potential barrier. Besides, the gradient of the potentialcouples conduction electrons with valence ones. Due to the tunneling a fluxof electrons can travel from the source to the drain contact. Electrons tunnelthrough the junction barrier because filled electron states in the conductionband on the side become aligned with empty valence band hole states. Asvoltage increases, the diode begins to operate as a normal diode, whereelectrons travel by diffusion across the junction.

In figure 1 we plot stationary values of the electron conduction distri-bution f+ and the valence hole distribution 1− f− for an external appliedpotential Vext = 0.01 eV. The figure show that the electrons cumulate near

0 0.005 0.01 0.015 0.02 0.025 0.030

2

4

6

8

10

12

Vex

(eV)

Cur

rent

(10

3 A c

m−

2 )

Fig. 2. Comparison between the Current-Voltage characteristic of the diode withoutinterband transition (dotted blue line) and the multiband correction (continuous greenline).

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DOI: 10.1685/2010CAIM474

the source contact and the holes near the drain contact. In figure 2 weshow the effects of the inclusion of the interband transition to the usualnon interactive two band motion. Since to each value of the the externalpotential Vex corresponds a flux of electron inside the device, by varyingVex we obtain the Current-Voltage characteristic of the diode. We plot theresults obtained if we discard the interband transitions (dotted blue line)(that is with N ≡ 0 in Eq. (24)) and if we include the multiband corrections(continuous green line). Since we are considering a highly doped diode theintraband diffusion process of electron inhibit the formation of a negativedifferential resistance in the diagram, nevertheless, even in this case thesimulation show that the tunneling processes cannot be discarded.

3. Conclusion

In this contribution we present a WKB-like asymptotic expansion of themultiband Wigner function designed to describe the dynamics in semicon-ductor devices in presence of the interband conduction-valence tunneling.The interband tunneling mechanism appearing to the first order of theexpansion is expressed in a very simple mathematical form. We derive ahierarchy of equations that to lowest order consist of two Hamilton-Jacobiequations corresponding to the classical dynamics of point particles withpositive and negative kinetic energy. We obtain the equation of motion forthe multiband Wigner functions in term of the (pseudo)classical trajecto-ries in the conduction and valence phase plane, related to the Hamilton-Jacobi functions S+ e S−. The resulting formulation reveals particularlyclose to the classical description of the particles motion and the approxi-mates evolution equations are used to simulate the evolution of the Wignerquasi-distribution function in a IRTD diode. Our finding pave the way toexplore the WKB approximation in the Wigner framework in order to ob-tain quantum-corrected equation of motion which are more tractable for anumerical point of view with respect to the full quantum formulation andthat can find application in the solid state field. The simulations show thatfor a highly doped n− p junction where of a strong electric field is present,our model predict a non-negligible correction to the total current of thedevice.

Acknowledgements.

This work has been supported by the Austrian Science Fund, Vienna,under the contract No. P 21326 - N 16.

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O. Morandi

4. Appendix

In this Appendix we give some detail of the approximation of Eq. (21).For sake of simplicity we assume as initial time t0 = −∞ and we consider

T (t) =

∫ t

−∞

∫ t′

−∞f(t′)g(t′′) dt′′ dt′(25)

f(t) = g = e−i~ϕ(t)(26)

ϕ =

∫ t

t∗

[(p′ + Eτ

)2+ Eg

]dτ .(27)

It is immediate to verify that T does not depend of the time t∗. For ourpurpose it is convenient to fix t∗ = −p′/E which is the stationary point

corresponding to ∂2ϕ∂t2

= 0. By using the following property of the Fouriertransform

∫ t′

−∞g(t′′) dt′′ =

∫ ∞

−∞

g(k)

ikeikt

′

dk +g(0)

2(28)

where g(k) denotes the Fourier transform, T becomes

T (t) =

∫ ∞

−∞

g(k)

ik

∫ t

−∞f(t′)eikt

′

dk dt′ +g(0)

2

∫ t

−∞f(t′) dt′ .(29)

In Ref. [18] some numerical test showed that in presence of a highly oscillat-ing function such as f(t), a very good estimate of the transition probabilityT which avoid time oscillation and preserves the limit t → ∞ can be ob-tained with the simple approximation

∫ t

−∞f(t′)eikt

′

dt′ ≃ θ(t− t∗)

∫ ∞

−∞f(t′)eikt

′

dt′ ,(30)

where θ is the step function. This formula express that the transition processtake place around the stationary point t∗, corresponding to the minimumoscillation frequency of ϕ(t′). We obtain

T (t) ≃ θ

(t+

p′

E

)[∫ ∞

−∞

∫ ∞

−∞

g(k)f(−k)

ikdk +

1

2

∫ ∞

−∞g(t′) dt′

∫ ∞

−∞f(t′) dt′

]

from the definition of f and g given in Eq. (26) we have that g, f are realfunctions and g(k) = f(−k). The first integral vanish and we obtain

T (t) =1

2θ

(t+

p′

E

) ∣∣∣∣∫ ∞

−∞f(t′) dt′

∣∣∣∣2

= θ

(t+

p′

E

)2π2 3

√E2

E2Ai2

(Eg

3√E2

)

where Ai denotes the Airy function Ai(x) ≡ 12π

∫exp

[i(xy + y3

3

)]dy.

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DOI: 10.1685/2010CAIM474

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