Quantum hydrodynamic models for the two-band Kane system

Quantum hydrodynamic models for the two-band Kane system

G. Al` ı (

¹

)(

²

) and G. Frosali(

³

)

(

¹

) CNR, Istituto per le Applicazioni del Calcolo “M. Picone”

Via P. Castellino 111, I-80131 Napoli, Italy

(

²

) Dipartimento di Matematica, Universit` a della Calabria and INFN-Gruppo c. Cosenza Ponte P. Bucci, I-87036 Arcavacata di Rende (CS), Italy

(

³

) Dipartimento di Matematica Applicata “G. Sansone”, Universit` a di Firenze Via S. Marta 3, I-50139 Firenze, Italy

(ricevuto il 20 Luglio 2005; revisionato il 23 Novembre 2005; approvato il 23 Novembre 2005;

pubblicato online l’1 Febbraio 2006)

Summary. — We consider Kane’s quantum model for two-band charged particles,

composed of two Schr¨ odinger-like equations coupled by a k · p term which describes interband tunneling. Using the WKB method we derive the hydrodynamic equa- tions both for a zero-temperature and for a nonzero-temperature two-band quantum fluid. In the latter case, we introduce a family of closure relations and write in full details the simplest two-band, isentropic, fluid-dynamical model. Finally, introduc- ing appropriate relaxation terms and performing drift-diffusive limits, we obtain the corresponding quantum drift-diffusion models for zero and nonzero-temperature quantum fluids.

PACS 72.10.Bg – General formulation of transport theory.

PACS 85.30.De – Semiconductor-device characterization, design, and modeling.

1. – Introduction

In recent years, much effort has been devoted to the study of hydrodynamic equations for new generations of semiconductor devices, for which quantum effects have to be taken into account. It is well known that fluid-dynamical models of semiconductors begin to fail when quantum phenomena become not negligible or even predominant. Nevertheless, the hydrodynamic approach presents notable advantages from the computational point of view.

In the classical framework, the literature on hydrodynamic models both from the theoretical and the numerical point of view is very wide. We refer the interested reader to one of the many review articles on the subject (see Anile et al. [1]).

Some very interesting results are present in literature, proposing quantum hydrody- namic equations, able to describe the behaviour of nanometric devices like resonant tunneling diodes. We recall here the “smooth” quantum hydrodynamic model proposed

c

Societ`a Italiana di Fisica

1279

by Gardner and Ringhofer [2], where the fluid-dynamical formulation is attained from moment expansion of a Wigner-Boltzmann equation and the approach by J¨ ungel [3], whose starting point is an appropriate representation of the solutions to the Schr¨ odinger equation. A new derivation of quantum hydrodynamic models starting from first princi- ples can be found in [4], where moments of the density matrix equation are considered and the resulting system is closed by an equilibrium density matrix.

The major part of published results is devoted to single-band problems. In the recent past new families of diodes, like Resonant Interband Tunneling Diodes (RITDs), have been produced, where the electrons in the valence band play an important role in the control of the current flow [5].

For such devices, we must consider the multi-band structure in the transport com- putation of the current. In this context, a simple model introduced by Kane [6] in the early 60’s describes the electron behaviour in a system equipped with two allowed energy bands separated by a forbidden region. The Kane model is a simple two-band model capable of including one conduction band and one valence band in the device material and it is formulated as the coupling of two Schr¨ odinger-like equations for the conduction and the valence band wave (envelope) functions [7]. The typical band diagram structure of a tunneling diode is characterized by a band alignment such that the valence band at the positive side of the semiconductor device lies above the conduction band at the negative one. The coupling term arises by the k · p perturbation method [8] using the fact that it is sufficient to obtain solutions of the single electron Schr¨ odinger equation in the neighbourhood of the bottom of the conduction and the top of the valence bands, where the major part of electrons and holes, respectively, are concentrated. Such model is of great importance for RITD, and is widely used in literature [9, 10].

The Kane model in the Schr¨ odinger-like form has been recently studied by Kefi [11].

An alternative approach to study this model is the Wigner formulation of the Kane system [7]. Moreover the Kane model in his Wigner formulation can be used to obtain hydrodynamic models in terms of macroscopic quantities [12, 13].

As we have said, the above-mentioned multiband models are based on the single electron Schr¨ odinger equation, and the resulting equations are essentially linear. By applying the WKB method, it is possible to derive a zero-temperature hydrodynamic version of the Schr¨ odinger Kane model. Although application of this method leads to nonlinear equations, for regular solutions the resulting quantum hydrodynamic equations are basically equivalent to the original linear quantum equations. However, when it is desirable to model the dynamics of a family of electrons, the quantum description requires the introduction of a sequence of mixed states, with attached a probability measure. In this case, the WKB method leads to a sequence of hydrodynamic equations, which are not satisfactory in real application. Anyway, it is possible to derive a set of equations for a finite number of macroscopic averaged quantities, with hydrodynamic character. These hydrodynamic equations share a similar structure with the analogous equation for a single electron, the only difference being the appearance of terms that can be interpreted as thermal tensors, and of additional source terms. These new terms depend on all states, so the system is not closed unless appropriate closure conditions are provided. It is clear that the final hydrodynamic model with temperature is by no means equivalent to the original quantum model. We could say that the nonlinearity of the resulting hydrodynamic model is “genuine” and is the price to pay for keeping only a finite number of equations.

In this paper, we describe in details this approach, by applying it to the Schr¨ odinger

Kane model for mixed states. The issues of closure relations is not discussed in full extent

here, as it deserves an independent exposition. We limit ourselves to present the simplest

class of closure relations, based on some analogy with classical hydrodynamic models.

The paper is organized as follows. In sect. 2, we review the Kane model and write it in nondimensional form. Then we introduce the hydrodynamic quantities needed to study a two-band quantum system. In sect. 4, we obtain a system for the densities and the currents and we show how this system can be closed by a new equation for the phase difference. Such a system can be considered as the Madelung system for a two-band zero-temperature quantum fluid. Section 5 is devoted to the derivation of a nonzero- temperature model for the Kane system, by considering an electron ensemble represented by a mixed quantum-mechanical state. Then, in sect. 6, introducing a relaxation time term, we perform the drift-diffusive limit, obtaining the corresponding quantum drift- diffusion system. Finally, a short discussion on this model and on related problems, such as closure and numerical implementation, is proposed.

2. – Physical description of the Kane model

The Kane model consists of a couple of Schr¨ odinger-like equations for the conduction and the valence band envelope functions. It is used to describe the electron behaviour in a system equipped with two allowed energy bands separated by a forbidden region as a tunnel diode. The model is derived in the framework of the envelope function theory.

In this approach the envelope function is a smooth function which can be obtained by replacing the wave function by its average in each primitive cell, since it is not necessary to find the exact evolution of the full wave equation.

Let ψ

c

(x, t) be the conduction band electron envelope function and ψ

v

(x, t) be the valence band electron envelope function, where x ∈ R

³

is the space variable, and t ∈ R is the time. The Kane model reads as follows [6]:

(2.1)

 

 

 

 i ∂ψ

_c

∂t = − 

²

2 m ∆ψ

_c

+ V

_c

ψ

_c

− 

²

m P · ∇ψ

_v

, i ∂ψ

v

∂t = − 

²

2m ∆ ψ

v

+ V

v

ψ

v

+ 

²

m P · ∇ψ

v

.

Here, i is the imaginary unit,  is the Planck constant scaled by 2π, m is the bare mass of the carriers, V

_c

and V

_v

are the minimum of the conduction band energy and maximum of the valence band energy, respectively, and P is the coupling coefficient between the two bands. The coupling coefficient P is the matrix element of the gradient operator between the Bloch functions. It can be assumed to be real, and is given by

(2.2) P =



u−cell

u

^c₀

(r)∇u

^v₀

(r)dr ,

where u-cell is the unitary cell and u

^c₀

and u

^v₀

are the basis of the Bloch functions for the conduction and valence bands, respectively, evaluated at the wave vector k = 0.

In the Kane model the coupling parameter has to be considered constant. In realistic heterostructure semiconductor devices, the parameter P , approximatively expressed in terms of the effective electron mass and the energy gap, depends on the layer composition through the spatial coordinates.

In order to rewrite system (2.1) in a dimensionless form, we introduce the rescaled

Planck constant

 =  α ,

where the dimensional parameter α is given by α = mx

²_R

/t

R

, by using x

R

and t

R

as characteristic (scalar) length and time variables. We rescale system (2.1) to dimensionless units by introducing the scaled coordinates

t

^

= t

t

_R

, x

^

= x x

_R

,

and leaving the mass m unchanged. The band energy can be rescaled, taking new potential units V

₀

= mx

²_R

/t

²_R

. A dimensional argument shows that the original coupling coefficient (2.2) is a reciprocal of a characteristic lenght, thus we define a scaled coefficient by P

^

= P x

_R

.

Hence, dropping the primes and without changing the name of the variables, we get the following scaled Kane system, which will be the object of our study:

(2.3)

 

 

 

 i ∂ψ

c

∂t = − 

²

2 ∆ψ

c

+ V

c

ψ

c

− 

²

P · ∇ψ

v

, i ∂ψ

_v

∂t = − 

²

2 ∆ψ

_v

+ V

_v

ψ

_v

+ 

²

P · ∇ψ

_c

. 3. – The hydrodynamic quantities

The first aim of this paper is to derive the hydrodynamic version of system (2.3) using the WKB method. This is the classical approach to put the Schr¨ odinger equation in hydrodynamic form [14]. It consists in characterizing the wave function by a quasi- classical limit expression a exp [iS/], where a is called the amplitude and S/ the phase of the wave. Using this approach the hydrodynamic limit is valid only for pure states, and in this sense, we say that the hydrodynamic limit is valid for a quantum system at zero temperature.

In the case under consideration of a two-band model, we look for solutions of the rescaled system (2.3) in the form

(3.1) ψ

c

( x, t) = 

n

c

( x, t) exp 

iSc(x,t)

 ,

ψ

v

(x, t) = 

n

v

(x, t) exp 

iSv(x,t)

 .

In the framework of the two-band Kane model, we introduce the particle densities n

ab

= ψ

_a

ψ

b

,

where ψ

_a

, with a = c, v, is the envelope function for the conduction and the valence band, respectively.

When a = b, the quantities n

ab

= n

a

= |ψ

a

|

²

are real and represent the quantum

probability densities for the position of conduction band and valence band electrons only

in an approximate sense, because ψ

_a

are envelope functions which mix the Bloch states.

Nevertheless, n = ψ

_c

ψ

_c

+ ψ

_v

ψ

_v

is exactly the total electron density in conduction and valence band, and, as expected, satisfies a continuity equation.

When a = b, the density ψ

_a

ψ

b

is a complex quantity, which does not have a precise physical meaning. Despite of this, as it will become clear in the next section, the complex quantities ψ

_a

ψ

b

appear explicitly in the evolution equation for the total density n.

Looking for solutions in the form (3.1), it is natural to write the coupling terms in a more manageable way, by introducing the complex quantity

(3.2) n

_cv

:= ψ

_c

ψ

_v

= √ n

_c

√ n

_v

e

^iσ

, where σ is the phase difference defined by

(3.3) σ := S

v

− S

c

 .

Now, it is apparent that, in order to study a zero-temperature quantum hydrodynamic model, we need to use only the three quantities n

_c

, n

_v

and σ to characterize the zero order moments (the “particle” densities).

The situation is more involved for the current densities. In analogy to the one-band case, we define quantum-mechanical electron current densities

J

_ab

=  Im

ψ

_a

∇ψ

_b

.

When a = b, using again the form (3.1), we recover the classical current densities (3.4) J

c

:= Im

ψ

_c

∇ψ

c

= n

c

∇S

c

, J

v

:= Im

ψ

_v

∇ψ

v

= n

v

∇S

v

,

whose physical meaning has to be interpreted similarly as for the particle densities. It is easy to get the identity

(3.5) ψ

_a

∇ψ

_b

= √ n

_a

√ n

_b

exp 

i S

_b

− S

_a

 

 ∇√n

b

√n

b

+ i∇S

_b

 .

Also, we introduce the complex velocities u

c

and u

v

, with

(3.6) u

_c

:= ∇ψ

c

ψ

_c

= ∇√n

c

√n

c

+ i∇S

_c

, u

_v

:= ∇ψ

v

ψ

_v

= ∇√n

v

√n

v

+ i∇S

_v

. We note that

(3.7) ∇n

cv

= n

cv

( u

c

+ u

v

) .

The real and imaginary parts of u

c

are called osmotic velocity and current velocity, respectively, and will be denoted by u

os,c

and the u

el,c

. Explicitly we have

(3.8) u

_os,c

:= ∇√n

c

√n

c

, u

_el,c

:= ∇S

_c

= J

_c

n

c

.

Analogous definitions hold for u

_os,v

and u

_el,v

, so that

(3.9) u

_c

= u

_os,c

+ iu

_el,c

, u

_v

= u

_os,v

+ iu

_el,v

.

Hence osmotic velocity and current velocity can be expressed solely in terms of n

c

, n

v

, J

_c

and J

_v

.

Coming back to the choice of the hydrodynamic quantities, we can maintain that for a zero-temperature quantum hydrodynamic system it is sufficient to take the usual quantities n

_c

, n

_v

, J

_c

and J

_v

, plus the phase difference σ. This will be verified in the next section.

4. – Hydrodynamic formulation of the Kane model

Taking into account the wave form (3.1) and using the first equation of the Kane system (2.3), we find

∂n

c

∂t = ψ

_c

∂ψ

c

∂t + ψ

c

∂ψ

_c

∂t = i

2

ψ

_c

∆ ψ

c

− ψ

c

∆ ψ

_c

+ iP ·

ψ

_c

∇ψ

v

− ψ

c

∇ψ

_v

= −∇· Im

ψ

_c

∇ψ

_c

− 2P · Im

ψ

_c

∇ψ

_v

.

In a similar way, we get

∂n

v

∂t = −∇· Im

ψ

_v

∇ψ

v

+ 2 P · Im

ψ

_v

∇ψ

c

.

Using the definition of the particle densities and of the complex velocities, we find (4.1) ψ

_c

∇ψ

_v

= n

_cv

u

_v

, ψ

_v

∇ψ

_c

= n

_cv

u

_c

,

and then, recalling the definitions (3.4) of J

_c

and J

_v

, the previous equations become

(4.2)

 

 

 



∂n

c

∂t + ∇·J

c

= −2P · Im (n

cv

u

v

) ,

∂n

_v

∂t + ∇·J

_v

= 2P · Im (n

_cv

u

_c

) . Explicitly, in terms of osmotic and current velocities, we can write

 n

cv

u

v

= √ n

c

√ n

v

(cos σ + i sin σ)(u

os,v

+ iu

el,v

), n

_cv

u

_c

= √ n

_c

√ n

_v

(cos σ − i sin σ)(u

_os,c

+ iu

_el,c

).

Summing the equations in (4.2) and using the identity Im ( ψ

_c

∇ψ

v

) − Im (ψ

_v

∇ψ

c

) =

∇ Im n

_cv

, we obtain the balance law for the total density,

(4.3) ∂

∂t (n

_c

+ n

_v

) + ∇·(J

_c

+ J

_v

) = −2  P ·∇ Im n

_cv

.

It is convenient to write eq. (4.3) in divergence form, in order to derive the conservation of total density. Using the fact that ∇·P = 0, (4.3) gives

∂

∂t ( n

c

+ n

v

) + ∇·(J

c

+ J

v

+ 2 P Im n

cv

) = 2  Im n

cv

∇·P = 0 , which is just the quantum counterpart of the classical continuity equation.

We remark that ∇·P = 0 is obviously verified following the assumptions under which the Kane model is derived. On the other hand, the k · p approximation [8] which leads to the Kane model is performed under the assumption of null divergence. It is inter- esting to note that this assumption ensures the self-adjointness of the k · p Hamiltonian corresponding to the Schr¨ odinger-type system (2.3).

Next, we derive equations for the phases S

c

, S

v

. Using (2.3) and (4.2), we have

∂S

_c

∂t = −i ∂

∂t ln ψ

_c

√n

c



= 

²

2n

c

ψ

_c

∆ ψ

c

− i∇· Im (ψ

_c

∇ψ

c

)

− V

c

+ 

²

n

c

P ·

ψ

_c

∇ψ

v

− i Im (ψ

_c

∇ψ

v

)

= 

²

2 n

c

∇· Re (ψ

_c

∇ψ

_c

) − ∇ψ

_c

·∇ψ

_c

− V

_c

+ 

²

n

c

P · Re (ψ

_c

∇ψ

_v

).

It is possible to rewrite the previous equation as

∂S

_c

∂t = − 1

2 |∇S

_c

|

²

+ 

²

∆ √n

c

2 √n

c

− V

_c

+ 

n

c

P · Re (ψ

_c

∇ψ

_v

).

A similar equation can be derived for S

v

. Then, using (4.1), the resulting equations are

(4.4)

 

 



 

 

∂S

_c

∂t + 1

2 |∇S

_c

|

²

− 

²

∆√n

c

2 √n

c

+ V

_c

= 

n

c

P · Re (n

_cv

u

_v

),

∂S

v

∂t + 1

2 |∇S

v

|

²

− 

²

∆ √n

v

2√n

_v

+ V

v

= − 

n

_v

P · Re (n

cv

u

c

) .

Equations (4.2) and (4.4) are equivalent to the coupled Schr¨ odinger equations (2.3).

We would like to replace (4.4) with a system of coupled equations for the currents.

Using the definitions (3.4), we can evaluate

∂J

c

∂t = 

j



²

2

∂

∂x

_j

Re

ψ

_c

∇ ∂ψ

c

∂x

_j

− ∇ψ

_c

∂ψ

_c

∂x

_j



− ψ

_c

ψ

_c

∇V

_c

+ (4.5)

+ 

²

Re 

ψ

_c

∇(P ·∇ψ

v

) − ∇ψ

c

( P ·∇ψ

_v

)  .

To write this equation in terms of hydrodynamic quantities, we use the identities ψ

_c

∇ ∂ψ

c

∂x

j

− ∇ψ

c

∂ψ

_c

∂x

j

= ∇

ψ

_c

∂ψ

c

∂x

j



− 2∇ √

n

c

∂√n

_c

∂x

j

− 2 n

c



²

∇S

c

∂S

c

∂x

j

, Re ∇

ψ

_c

∂ψ

c

∂x

_j



= 1 2 ∇ ∂n

c

∂x

_j

,

with j = 1, 2, 3, which yield



j

∂

∂x

_j

Re

ψ

_c

∇ ∂ψ

_c

∂x

_j

− ∇ψ

_c

∂ψ

_c

∂x

_j



=

= 

j

∂

∂x

_j

 1

2 ∇ ∂n

c

∂x

_j

− 2∇ √ n

_c

∂√n

c

∂x

_j

− 2n

c



²

∇S

_c

∂S

c

∂x

_j



.

Then, using

Re 

ψ

_c

∇(P ·∇ψ

_v

) − ∇ψ

_c

(P ·∇ψ

_v

) 

= Re 

∇(ψ

_c

(P ·∇ψ

_v

)) − 2∇ψ

_c

(P ·∇ψ

_v

)  ,

and the previous identities, eq. (4.5) becomes

∂J

_c

∂t + div

J

_c

⊗ J

_c

n

c

+ 

²

∇ √ n

_c

⊗ ∇ √ n

_c

− 

²

4 ∇ ⊗ ∇n

_c

 + (4.6)

+ n

c

∇V

c

= 

²

Re 

∇(ψ

_c

(P ·∇ψ

v

)) − 2∇ψ

_c

(P ·∇ψ

v

)  .

A similar equation can be written for J

v

.

∂J

v

∂t + div

J

v

⊗ J

v

n

_v

+ 

²

∇ √ n

_v

⊗ ∇ √ n

_v

− 

²

4 ∇ ⊗ ∇n

_v

 + (4.7)

+ n

v

∇V

v

= − 

²

Re 

∇(ψ

_v

( P ·∇ψ

c

)) − 2∇ψ

_v

( P ·∇ψ

c

)  .

The left-hand sides of the equations for the currents can be put in a more familiar form by using the identity

div

∇ √

n

a

⊗ ∇ √ n

a

− 1

4 ∇ ⊗ ∇n

a



= − n

a

2 ∇  ∆ √n

a

√n

_a



, a = c, v.

The correction terms



²

2

∆ √n

a

√n

a

, a = c, v ,

can be identified with the quantum Bohm potentials for each band, in analogy with the single-band case. Moreover, using (4.1) and the identity



²

∇ψ

_a

(P ·∇ψ

_b

) = n

_ab

P ·u

_b

u

_a

, a, b = c, v,

the right-hand sides of eqs. (4.6) and (4.7) can be expressed in terms of the hydrodynamic

quantities.

The resulting system takes the following form:

(4.8)

 

 

 

 

 

 

 

 

 



∂J

_c

∂t + div

J

_c

⊗ J

_c

n

_c



− n

_c

∇



²

∆ √n

c

2 √n

c



+ n

_c

∇V

_c

=

= −∇ Re (n

cv

P ·u

v

) − 2 Re (n

cv

P ·u

v

u

c

) ,

∂J

_v

∂t + div

J

_v

⊗ J

_v

n

v



− n

_v

∇



²

∆√n

v

2 √n

v



+ n

_v

∇V

_v

=

= −∇ Re (n

cv

P ·u

c

) + 2 Re (n

cv

P ·u

c

u

v

) .

For the reader’s convenience, we express the right-hand sides of (4.8) in terms of osmotic and current velocities:

 

 

 

 

 

 

 



 

 

 

 

 

 

 

Re ( n

cv

P ·u

v

) = √n

c

√n

v

P ·(u

os,v

cos σ − u

el,v

sin σ) , Re (n

_cv

P ·u

_c

) = √n

_c

√n

_v

P ·(u

_os,c

cos σ + u

_el,c

sin σ) , Re ( n

cv

P ·u

v

u

c

) = √n

c

√n

v

[ P ·(u

os,c

cos σ + u

el,c

sin σ)u

os,v

−

− P ·(u

os,c

sin σ − u

el,c

cos σ)u

el,v

] , Re (n

_cv

P ·u

_c

u

_v

) = √n

c

√n

v

[P ·(u

_os,v

cos σ − u

_el,v

sin σ)u

_os,c

+

+ P ·(u

os,v

sin σ + u

el,v

cos σ)u

el,c

] .

It is important to notice that, at variance with the uncoupled model, (4.2) and (4.8) are not equivalent to the original equation (2.3), due to the presence of σ. There are many ways to “close” the system, in order to obtain an extension of the classical Madelung fluid equations to a two-band quantum fluid. One possibility is to use (4.4) to derive an evolution equation for σ = (S

_v

− S

_c

)/,

 ∂σ

∂t − 1 2

  J

c

n

_c

 

²

+ 1 2

  J

v

n

_v

 

²

+ 

²

∆ √n

c

2√n

_c

− 

²

∆ √n

v

2√n

_v

− (4.9)

−V

_c

+ V

_v

= − 

n

_v

P · Re (n

_cv

u

_c

) − 

n

_c

P · Re (n

_cv

u

_v

).

Equation (4.9) must be supplemented with the constraint

(4.10) ∇σ = J

_v

n

_v

− J

_c

n

_c

.

Using (4.2) and (4.8), it is possible to prove that (4.9) and (4.10) are equivalent, in the following sense: (4.9) implies that (4.10) is satisfied at all times if it holds at the initial time; (4.10) implies that (4.9) is satisfied (up to a constant).

To see the truth of the above statement, it is sufficient to take the gradient of (4.4) and, afterwards, account that

(4.11) ∇S

c

= J

c

n

_c

, ∇S

v

= J

v

n

_v

.

Then, noting that (4.9) was derived from the same equation (4.4), it is simple to see that (4.9) implies

∂

∂t

∇σ − J

v

n

_v

+ J

c

n

_c



= 0 ,

which yields the first part of the statement. Similarly, taking the time derivative of (4.10) and using again (4.11), we can recover the gradient of equation (4.9), which implies the second part of the statement.

The equivalence of (4.10) and (4.9) suggests that we can discard (4.9), and recover σ as a function of the other variables by solving the elliptic equation

(4.12) ∆σ = ∇·

J

v

n

_v

− J

c

n

_c

 ,

which can be obtained immediately after derivation of the constraint (4.10).

Another possibility is to regard n

cv

as an independent variable, rather than σ. Re- calling the definition (3.2) and Kane’s system (2.3), we find

∂n

cv

∂t = i

2 ∇·

ψ

_c

∇ψ

_v

− ψ

_v

∇ψ

_c

+ i

 (V

_c

− V

_v

)ψ

_c

ψ

_v

− iP ·

ψ

_c

∇ψ

_c

+ ψ

_v

∇ψ

_v

,

which, using (3.5) and the definitions of osmotic and current velocities, leads to

(4.13)  ∂n

cv

∂t = i

2 ∇·(n

_cv

(u

_v

− u

_c

)) + i n

_cv

(V

_c

− V

_v

) − iP ·(n

_c

u

_c

+ n

_v

u

_v

) .

In addition to (4.13), the complex function n

cv

must satisfy the constraints

n

_cv

n

_cv

= n

_c

n

_v

, (4.14)

∇n

_cv

= (u

_c

+ u

_v

)n

_cv

. (4.15)

Alternatively, we can use the identity (4.15) to derive a nonlinear elliptic equation for n

_cv

,

(4.16) div

∇n

cv

n

_cv



= div( u

c

+ u

v

) ,

which must be solved together with the constraint (4.14).

Now we are in position to rewrite the hydrodynamic system as follows:

(4.17)

 

 

 

 

 

 

 

 

 

 

 



 

 

 

 

 

 

 

 

 

 

 

∂n

c

∂t + divJ

_c

= −2P · Im (n

_cv

u

_v

),

∂n

_v

∂t + divJ

_v

= 2P · Im (n

_cv

u

_c

),

∂J

c

∂t + div

J

c

⊗ J

c

n

_c



− n

_c

∇



²

∆ √n

c

2 √n

c



+ n

_c

∇V

_c

=

= ∇ Re (n

cv

P ·u

v

) − 2 Re (n

cv

P ·u

v

u

c

) ,

∂J

_v

∂t + div

J

_v

⊗ J

_v

n

v



− n

_v

∇



²

∆√n

v

2 √n

v



+ n

_v

∇V

_v

=

= −∇ Re (n

_cv

P ·u

_c

) + 2 Re (n

_cv

P ·u

_c

u

_v

) ,

∇σ = J

_v

n

v

− J

_c

n

c

,

where n

_cv

, u

v

, and u

v

are expressed in the terms of the hydrodynamic quantities n

c

, n

v

, J

_c

, J

_v

, and σ by (3.2) and (3.9).

5. – Nonzero-temperature hydrodynamic model

In this section we extend the considerations of the previous sections to an ensemble of electrons, in order to obtain a nonzero-temperature model for the Kane system. An ensemble of electrons is represented by a mixed quantum-mechanical state, which is a sequence of single states with occupation probabilities λ

k

for the k-th single state, k = 0, 1, 2, . . . , described by the envelope function in the conduction band, ψ

_c^k

, and by the envelope function in the valence band, ψ

_v^k

. The occupation probabilities are such that 

_∞

k=0

λ

_k

= 1.

The k-th state for the Kane system is described by the solutions of the system

(5.1)

 

 

 



 

 

  i ∂ψ

^k_c

∂t = − 

²

2 ∆ψ

_c^k

+ V

_c

ψ

_c^k

− 

²

P ·∇ψ

^k_v

, i ∂ψ

^k_v

∂t = − 

²

2 ∆ψ

_v^k

+ V

v

ψ

_v^k

+ 

²

P ·∇ψ

^k_c

.

Using the Madelung-type transforms, under the assumption of positivity of the densities n

^k_c

and n

^k_v

,

ψ

_c^k

=

 n

^k_c

exp  iS

_c^k

/ 

, ψ

^k_v

=

 n

^k_v

exp  iS

_v^k

/ 

,

the previous system is equivalent to

(5.2)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



∂n

^k_c

∂t + div J

_c^k

= − 2P · Im (n

^k_cv

u

^k_v

) ,

∂n

^k_v

∂t + divJ

_v^k

= 2P · Im (n

^k_cv

u

^k_c

),

∂J

_c^k

∂t + div

J

_c^k

⊗ J

_c^k

n

^k_c



− n

^k_c

∇

 

²

∆  n

^k_c

2 

n

^k_c



+ n

^k_c

∇V

c

=

= ∇ Re (n

^k_cv

P ·u

^k_v

) − 2 Re 

n

^k_cv

P ·u

^k_v

u

^k_c

 ,

∂J

_v^k

∂t + div

J

_v^k

⊗ J

_v^k

n

^k_v



− n

^k_v

∇

 

²

∆  n

^k_v

2 

n

^k_v



+ n

^k_v

∇V

v

=

= − ∇ Re (n

^k_cv

P ·u

^k_c

) + 2 Re

 n

^k_cv

P ·u

^k_c

u

^k_v

 ,

∆σ

^k

= ∇ J

_v^k

n

^k_v

− J

_c^k

n

^k_c

 ,

with

J

_c^k

= n

^k_c

∇S

^k_c

, J

_v^k

= n

^k_v

∇S

_v^k

, σ

^k

= S

_v^k

− S

_c^k

 , n

^k_cv

=

 n

^k_c

 n

^k_v

exp 

iσ

^k

 , u

^k_c

= ∇ 

n

^k_c

 n

^k_c

+ i J

_c^k

n

^k_c

, u

^k_v

= ∇  n

^k_v

 n

^k_v

+ i J

_v^k

n

^k_v

.

The densities and the currents corresponding to the two mixed states for conduction and valence electrons can be defined as

n

_c

:=



∞ k=0

λ

_k

n

^k_c

, n

_v

:=



∞ k=0

λ

_k

n

^k_v

,

J

_c

:=



∞ k=0

λ

_k

J

_c^k

, J

_v

:=



∞ k=0

λ

_k

J

_v^k

.

We also define

σ :=



∞ k=0

λ

k

σ

^k

, n

cv

:= √ n

c

√

n

v

exp[ iσ],

u

_c

:= ∇√n

c

√n

c

+ i J

_c

n

c

, u

_v

:= ∇√n

v

√n

v

+ i J

_v

n

v

.

Multiplying (5.2) by λ

_k

and summing over k, we find

(5.3)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



∂n

_c

∂t + divJ

_c

= −2P · Im R

_c

,

∂n

_v

∂t + divJ

_v

= 2P · Im R

_v

,

∂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

_c

=

= ∇(P · Re R

_c

) − 2P · 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

_v

=

= − ∇(P · Re R

_v

) + 2P · Re Q

_vc

,

∆σ = ∇



_∞



k=0

λ

_k

J

_v^k

n

^k_v

− J

_c^k

n

^k_c



,

with

R

c

=



∞ k=0

λ

k

n

^k_cv

u

^k_v

, R

v

=



∞ k=0

λ

k

n

^k_cv

u

^k_c

,

Q

cv

=



∞ k=0

λ

k

n

^k_cv

u

^k_v

⊗ u

^k_c

, Q

vc

=



∞ k=0

λ

k

n

^k_cv

u

^k_c

⊗ u

^k_v

.

Analogously to the one-band case [15], new terms containing the total temperature θ

c

and θ

v

, for each band, appear in the current equations. The temperature tensors θ

c

= θ

_os,c

+θ

_el,c

and θ

_v

= θ

_os,v

+ θ

_el,v

are the sum of osmotic temperature and electron current temperature, given by

θ

_os,c

=



∞ k=0

λ

_k

n

^k_c

n

_c

(u

^k_os,c

− u

_os,c

) ⊗ (u

^k_os,c

− u

_os,c

), θ

_el,c

=



∞ k=0

λ

_k

n

^k_c

n

c

(u

^k_el,c

− u

_el,c

) ⊗ (u

^k_el,c

− u

_el,c

).

We can write

(5.4) R

_c

= n

_cv

(αu

_v

+ β

_v

), R

_v

= n

_cv

(α u

_c

+ β

_c

), with

α :=



∞ k=0

λ

k

n

^k_cv

n

_cv

, β

v

:=



∞ k=0

λ

k

n

^k_cv

n

_cv

(u

^k_v

− u

v

), β

c

:=



∞ k=0

λ

k

n

^k_cv

n

_cv

(u

^k_c

− u

c

).

These quantities are not independent, due to (3.7). By taking the gradient of α, we find

(5.5) ∇α − β

_v

− β

_c

= −α

∇n

_cv

n

cv

− u

_v

− u

_c

 ,

and using this identity, it is possible to show that R

_c

+ R

_v

= ∇(αn

_cv

).

Moreover, recalling the definition of n

cv

, u

c

and u

v

, we find

(5.6) ∇n

cv

n

_cv

− u

_v

− u

_c

= i

∇σ − J

v

n

_v

+ J

c

n

_c

 .

Using this identity in (5.5), we get

1 iα

∇α − β

v

− β

c

= −∇σ + J

v

n

v

− J

c

n

c

,

which implies

Re

 1 α

∇α − β

_v

− β

_c



= 0,

Im

 1 α

∇α − β

_v

− β

_c



= −∇σ + J

v

n

_v

− J

c

n

_c

.

Next, in order to obtain an expression of the coupling terms between the two bands by a sort of temperature tensors, we write

Q

cv

= n

cv

( αu

v

⊗ u

c

+ β

v

⊗ u

c

+ u

v

⊗ β

c

+ θ

cv

) , Q

vc

= n

cv

( αu

c

⊗ u

v

+ β

c

⊗ u

v

+ u

c

⊗ β

v

+ θ

vc

) ,

with

θ

cv

:=



∞ k=0

λ

k

n

^k_cv

n

cv

( u

^k_v

− u

v

) ⊗ (u

^k_c

− u

c

) , θ

vc

:=



∞ k=0

λ

k

n

^k_cv

n

_cv

(u

^k_c

− u

c

) ⊗ (u

^k_v

− u

v

).

In conclusion, using the new quantities defined above, system (5.3) becomes

(5.7)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



∂n

_c

∂t + divJ

_c

= − 2P · Im [n

_cv

(αu

_v

+ β

_v

)],

∂n

_v

∂t + divJ

_v

= 2P · Im [n

_cv

(α u

_c

+ β

_c

)],

∂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

_c

=

=  P ·∇ Re (n

cv

(αu

v

+ β

v

)) −

− 2P · Re

n

_cv

(αu

_v

⊗ u

_c

+ β

_v

⊗ u

_c

+ u

_v

⊗ β

_c

+ θ

_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

_v

=

= − P ·∇ Re (n

cv

(αu

c

+ β

c

)) + + 2P · Re

n

_cv

(αu

_c

⊗ u

_v

+ β

_c

⊗ u

_v

+ u

_c

⊗ β

_v

+ θ

_vc

) ,

∇σ − J

v

n

_v

+ J

c

n

_c

= − Im

 1 α

∇α − β

v

− β

c

 .

Unlike system (4.17), this system, which can be considered as a nonzero-temperature quantum fluid model, is not closed. The quantities n

cv

, u

c

, and u

v

, already present in (4.17), are expressed in terms of n

_c

, n

_v

, J

_c

, J

_v

, and σ, while the new quantities α, β

_c

, and β

_v

are linked between them by

Re

 1 α

∇α − β

_v

− β

_c



= 0.

and need appropriate closure relations. Moreover, we must assign constitutive relations for the tensors θ

_c

, θ

_v

, θ

_cv

and θ

_vc

, the first ones formally analogous to the temperature tensor of kinetic theory.

A simple class of closure conditions can be obtained by assigning a function α = α(n

_c

, n

_v

, σ) and taking

(5.8) 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

. Then, we have

∇α − β

_v

− β

_c

= 0, which implies

∇σ − J

_v

n

v

+ J

_c

n

c

= 0 . In particular, it is possible to choose

(5.9) α = 1, β

_c

= β

_v

= 0.