HYDRODYNAMIC MODELS FOR A TWO-BAND NONZERO-TEMPERATURE QUANTUM FLUID

HYDRODYNAMIC MODELS FOR A TWO-BAND NONZERO-TEMPERATURE QUANTUM FLUID

G. BORGIOLI ^∗ , G. FROSALI ^∗∗ , C. MANZINI ^∗∗

Dip. di Elettronica e Telecomunicazioni ^∗ , Dip. di Matematica Applicata ^∗∗

Universit` a di Firenze, Via S. Marta 3, I-50139 Firenze, Italy E-mail: giovanni.borgioli@unifi.it

In this paper a hydrodynamic set of equations is derived from a Schr¨ odinger-like model for the dynamics of electrons in a two-band semiconductor, via the Madelung ansatz. A diffusive scaling allows to attain a drift-diffusion formulation.

1. Introduction

Recent advances in semiconductor devices design have compelled the sci- entific community to provide theoretical models that take fully into ac- count the quantum dynamics of carriers. For example the Resonant Inter- band Tunneling Diode (RITD ¹⁵ ) is built on the quantum effect of tun- neling of electrons between conduction and valence bands. Multiband models ^11,12 derived from the Schr¨ odinger equation are the starting point of a recent series of articles ^3,4,6 that propose a two-band description in terms of Wigner functions. However, in the perspective of numerical sim- ulations, quantum hydrodynamic models ^5,7,10 are preferable, since they involve directly macroscopic quantities and they admit natural boundary conditions. The Madelung equations constitute the fluiddynamical equiv- alent of the Schr¨ odinger equation and they are formally identical to the Euler equations for a perfect fluid at zero temperature, apart for the Bohm potential ⁹ . Analogously, two-band zero-temperature quantum fluiddynam- ical models ^1,2 can be derived by applying the Madelung ansatz either to the two-band Schr¨ odinger-like model introduced by Kane ¹¹ , or to the MEF (Multiband Envelope Function) model ¹³ ; the latter one, at difference with the Kane model, seems to be reliable also in presence of heterostructures and impurities of the semiconductor material. Here, the derivation ² is ex- tended to the case when the electron ensemble is described by mixed states:

Madelung-like equations for each band are recovered, coupled by “interband

1

terms” and containing also temperature terms, as expected by comparison with the single-band mixed-state case ⁸ . A two-band drift-diffusion system is attained in the zero-relaxation time limit and the equations for each band differ from the QDD ¹⁴ only for the coupling interband terms.

2. Nonzero-temperature hydrodynamic model

In order to derive a nonzero-temperature model ⁸ , let us describe an electron ensemble by a mixed quantum state, i.e. by a sequence of pure states con- stituting an orthonormal basis for the electron ensemble state space, with occupation probabilities λ k ≥ 0, k ∈ N 0 , such that P

k λ k = 1. In a “macro- scopic” description ¹³ , a pure state of the system can be individuated by {ψ n } _n∈N

₀

with ψ n wave-function of the n-th band and n(x) = P

n |ψ n | ² (x) and J (x) = Im P

n ψ n (x)∇ψ n (x). Since we restrict to a conduction-valence band description, we individuate the k-th pure state by two wave-functions ψ _c ^k , ψ ^k _v , that are solutions of the (rescaled version of) MeF ¹³ system

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 i ∂ψ _c ^k

∂t = −  ²

2 ∆ψ ^k _c + (V c + V )ψ _c ^k −  ² K ψ ^k _v , i ∂ψ _v ^k

∂t =  ²

2 ∆ψ ^k _v + (V _v + V )ψ _v ^k −  ² K ψ ^k _c ,

(1)

where K = P · ∇V , P is the interband momentum matrix, V is the electro- static potential, V c , V v are the minimum and maximum of the conduction and the valence band energy, respectively, and  is the Planck constant.

Then, by using the Madelung ansatz ψ _b ^k = q

n ^k _b exp iS _b ^k /
with the band- index b = c, v, the hydrodynamic system corresponding to Eqs. (1) reads

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∂n ^k _c

∂t + divJ _c ^k = −2K Im n ^k _cv ,

∂n ^k _v

∂t − divJ _v ^k = 2K Im n ^k _cv ,

∂J _c ^k

∂t + div  J _c ^k ⊗ J _c ^k n ^k _c



− n ^k _c ∇  ² ∆pn ^k _c 2pn ^k _c

!

+ n ^k _c ∇V

=  ² ∇K Re n ^k _cv + K Re n ^k _cv (u ^k _v − u ^k _c )
,

∂J _v ^k

∂t − div  J _v ^k ⊗ J _v ^k n ^k _v



+ n ^k _v ∇  ² ∆pn ^k _v 2pn ^k _v

!

+ n ^k _v ∇V

=  ² ∇K Re n ^k _cv − K Re n ^k _cv (u ^k _v − u ^k _c )
,

∇σ ^k = J _v ^k n ^k _v − J _c ^k

n ^k _c ,

(2)

where J _b ^k = n ^k _b ∇S _b ^k , u ^k _b = ∇

q n ^k _b /

q

n ^k _b + i J _b ^k /n ^k _b , σ ^k = (S _v ^k − S _c ^k )/ and n ^k _cv = pn ^k _c pn ^k _v exp(iσ ^k ). In the mixed-state description ⁸ , densities and currents corresponding to the bands are n b := P

k λ k n ^k _b , J b := P

k λ k J _b ^k , u _b := ∇ √

n _b / √

n _b +iJ _b /n _b = u _os,b +iu _el,b , while the “interband” quantities are σ := P

k λ k σ ^k and n cv := √ n c

√ n v exp(iσ). Observe that we assume V b to be constant, as in the derivation ¹³ of Eqs. (1). Multiplying Eqs. (2) by λ k and summing over k, we find the equations for the hydrodynamic quantities n b , J b , and σ

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

∂n c

∂t + divJ c = −2K Im R cv ,

∂n v

∂t − divJ v = 2K Im R cv ,

∂J _c

∂t + div  J _c ⊗ J c

n c

+ n c θ c



− n c ∇   ² ∆ √ n c

2 √ n c



+ n c ∇V

=  ² ∇K Re R cv +  ² K Re Q cv ,

∂J _v

∂t − div  J _v ⊗ J v

n v

+ n v θ v



+ n v ∇   ² ∆ √ n v

2 √ n v



+ n v ∇V

=  ² ∇K Re R cv −  ² K Re Q cv ,

∇σ = X

k

λ _k  J _v ^k n ^k _v − J _c ^k

n ^k _c

 ,

(3)

with R _cv = P

k λ _k n ^k _cv , Q _cv = P

k λ _k n ^k _cv 

u ^k _v − u ^k _c 

. In analogy with the one-band case ⁸ , we introduce the temperatures θ b = θ os,b + θ el,b , b = c, v, with the osmotic parts θ _os,b defined by

θ os,b = X

k

λ k

n ^k _b n b

(u ^k _os,b − u os,b ) ⊗ (u ^k _os,b − u os,b )

and the current temperatures θ _el,b defined correspondingly. If we call α := X

k

λ k

n ^k _cv n cv

, β v := X

k

λ k

n ^k _cv n cv

(u ^k _v − u v ), β c := X

k

λ k

n ^k _cv n cv

(u ^k _c − u c ),

the coupling terms contain R _cv = αn _cv , Q _cv = n _cv α(u v − u c ) + β _v − β c  . In order to find a relation between α, β v and β c and the hydrodynamic quantities, we take the gradient of α and use the definition of n _cv , u _c , u _v and the identity

∇n _cv n cv

− u v − u c = i



∇σ − J _v n v

+ J _c n c



.

Accordingly

∇σ − J v

n _v + J c

n _c = i

α ∇α − β v − β c
. (4) The last equation of system (3) can be rephrased as

∇σ = J v

n _v − J c

n _c + X

k

λ k

 J _v ^k n ^k _v − J v

n _v



− X

k

λ k

 J _c ^k n ^k _c − J c

n _c

 (5) and, by comparison of (4) with (5), we get

X

k

λ _k  J _v ^k n ^k _v − J v

n _v



− X

k

λ _k  J _c ^k n ^k _c − J c

n _c



= i

α ∇α − β _v − β _c
, then Re 

∇α − β v − β c
/α = 0. Accordingly, Eqs. (3) can be written as

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

∂n c

∂t + divJ c = −2K Im (αn cv ),

∂n _v

∂t − divJ v = 2K Im (αn cv ),

∂J c

∂t + div  J c ⊗ J c

n c

+ n c θ c



− n c ∇   ² ∆ √ n _c 2 √

n c



+ n c ∇V

=  ² ∇K Re (αn cv ) +  ² K Re (n cv α(u v − u c ) + β v − β c ),

∂J v

∂t − div  J v ⊗ J v

n _v + n v θ v



+ n v ∇   ² ∆ √ n v

2 √ n _v



+ n v ∇V

=  ² ∇K Re (αn _cv ) −  ² K Re (n _cv α(u v − u _c ) + β _v − β _c ),

∇σ − J v

n _v + J c

n _c = − Im  1

α ∇α − β _v − β _c

 .

(6)

The terms on the right hand side of the equations determine the coupling.

The system is not closed: the quantities α, β c and β v are not expressed in terms of the hydrodynamic quantites, but they are linked by

Re 

∇α − β _v − β _c
/α = 0. (7) In addition, we must assign constitutive relations for the tensors θ c and θ _v . A simple class of closure conditions can be obtained by assuming α = α(n c , n v , σ) and taking

β c = 2n c

∂ ¯ α

∂n _c u os,c − ∂ ¯ α

∂σ u el,c , β v = 2n v

∂α

∂n _v u os,v + ∂α

∂σ u el,v . (8)

Then ∇α − β v − β c = 0, thus Eq. (7) is fulfilled and moreover ∇σ −

J _v /n _v + J _c /n _c = 0. For the temperatures we assume n _b θ _b = p _b (n _b )I, where

I is the identity tensor and the functions p c , p v are pressures. In particular,

we can take p _b = θ ⁰ n _b , as for an ideal gas in isothermal conditions, and α = 1, β c = β v = 0 (which follows, e.g., from n ^k _cv ' n cv ). Accordingly, we obtain the simplest two-band isothermal QHD model

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∂n c

∂t + divJ c = −2K Im n cv ,

∂n v

∂t − divJ v = 2K Im n cv ,

∂J c

∂t + div J c ⊗ J c

n _c + θ ⁰ ∇n c − n c ∇   ² ∆ √ n _c 2 √

n _c − V



=  ² ∇K Re n cv +  ² K Re (n cv (u v − u c )),

∂J _v

∂t − div J _v ⊗ J _v n v

+ θ ⁰ ∇n v + n _v ∇   ² ∆ √ n v

2 √ n v

+ V



=  ² ∇K Re n cv −  ² K Re (n _cv (u _v − u c )),

∇σ − J v

n _v + J c

n _c = 0 .

(9)

Here, the only peculiarity of the mixed-state case is the presence of the temperature terms.

3. The drift-diffusion model

Now we perform the zero-relaxation time limit of the hydrodynamic models recovered. First, we start from Eqs. (9), we add relaxation terms for the currents and we introduce the diffusive scaling

t → t

τ , J c → τ J c , J v → τ J v , (10) that leads to the ansatz σ → σ 0 + τ σ , where σ 0 is a constant phase to be determined. In the limit τ → 0 the system reads

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∂n c

∂t + divJ c = −2K √ n c

√ n v σ,

∂n v

∂t − divJ _v = 2K √ n _c √

n _v σ, J _c = − θ ⁰ ∇n c + n _c



∇   ² ∆ √ n c

2 √ n c

− V +  ²

√ n v

√ n c

K



,

J v = θ ⁰ ∇n v − n v



∇   ² ∆ √ n v

2 √ n v

+ V −  ²

√ n c

√ n v

K



,

∇σ − J v

n _v + J c

n _c = 0,

(11)

where σ 0 = 0, due to the limit of the first equation. Alternatively, we start from the isothermal version of the Eqs. (6), closed with α = α(n c , n v ), β c := 2n c ∂α

∂n

c

u os,c , β v := 2n v ∂α

∂n

v

u os,v , we add relaxation terms for the currents and we consider the diffusive scaling in Eq. (10), with σ 0 = 0. In the limit τ → 0, we get Im α = 0 and the isothermal QDD system reads

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∂n c

∂t + divJ c = −2K √ n c

√ n v ασ,

∂n _v

∂t − divJ v = 2K √ n _c √

n _v ασ, J c = −θ ⁰ ∇n c + n c



∇   ² ∆ √ n c

2 √

n _c − V



+  ² α∇  √n v

√ n _c K



+ +  ² (β _v − β _c ) √

n _v √ n _c K, J _v = θ ⁰ ∇n v − n v



∇   ² ∆ √ n v

2 √ n v

+ V



−  ² α∇  √n c

√ n v

K



+

−  ² (β v − β c ) √ n c

√ n v K,

∇σ − J v

n v

+ J c

n c

= 0.

Acknowledgements The authors are grateful to Giuseppe Al´ı for helpful discussions. The paper was supported by GNFM and MIUR.

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