Quantum hydrodynamic models for the two-band Kane system

Transcription

1 IL NUOVO CIMENTO Online First Dicembre 2005 DOI /ncb/i Quantum hydrodynamic models for the two-band Kane system G Alì 1 2 andg Frosali 3 1 CNR Istituto per le Applicazioni del Calcolo M Picone Via P Castellino 111 I Napoli Italy 2 Dipartimento di Matematica Università della Calabria and INFN-Gruppo c Cosenza Ponte P Bucci I Arcavacata di Rende CS Italy 3 Dipartimento di Matematica Applicata G Sansone Università difirenze Via S Marta 3 I 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 Schrödinger-like equations coupled by a k p term which describes interband tunneling Using the WKB method we derive the hydrodynamic equations 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 introducing appropriate relaxation terms and performing drift-diffusive limits we obtain the corresponding quantum drift-diffusion models for zero and nonzero-temperature quantum fluids PACS 7210Bg General formulation of transport theory PACS 8530De 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 hydrodynamic equations able to describe the behaviour of nanometric devices like resonant tunneling diodes We recall here the smooth quantum hydrodynamic model proposed c Società Italiana di Fisica 1279

2 1280 G ALÌ and G FROSALI by Gardner and Ringhofer [2] where the fluid-dynamical formulation is attained from moment expansion of a Wigner-Boltzmann equation and the approach by Jüngel [3] whose starting point is an appropriate representation of the solutions to the Schrödinger equation A new derivation of quantum hydrodynamic models starting from first principles 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 computation 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 Schrödinger-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 Schrödinger 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 Schrödinger-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 formulatioan 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 Schrödinger 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 Schrödinger 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 Schrödinger 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

3 QUANTUM HYDRODYNAMIC MODELS FOR THE TWO-BAND KANE SYSTEM 1281 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 nonzerotemperature 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 driftdiffusion 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 Schrödinger-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 bethe 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]: 21 i ψ c i ψ v = 2 2m ψ c + V c ψ c 2 m P ψ v = 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 22 P = u cell u c 0r u v 0rdr where u-cell is the unitary cell and u c 0 and u v 0 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 21 in a dimensionless form we introduce the rescaled

4 1282 G ALÌ and G FROSALI Planck constant ɛ = 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 21 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 22 is a reciprocal of a characteristic lenght thus we define a scaled coefficient by P = Px 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: 23 iɛ ψ c iɛ ψ v 3 The hydrodynamic quantities = ɛ2 2 ψ c + V c ψ c ɛ 2 P ψ v = ɛ2 2 ψ v + V v ψ v + ɛ 2 P ψ c The first aim of this paper is to derive the hydrodynamic version of system 23 using the WKB method This is the classical approach to put the Schrödinger equation in hydrodynamic form [14] It consists iharacterizing the wave function by a quasiclassical limit expression a exp [is/ɛ] where a is called the amplitude and S/ɛ 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 23 in the form 31 ψ c x t = x texp ψ v x t = x texp [ iscxt ɛ [ isvxt ɛ ] ] 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

5 QUANTUM HYDRODYNAMIC MODELS FOR THE TWO-BAND KANE SYSTEM 1283 Nevertheless n = ψ c ψ c + ψ v ψ v is exactly the total electron density ionduction 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 31 it is natural to write the coupling terms in a more manageable way by introducing the complex quantity 32 v := ψ c ψ v = nv e iσ where σ is the phase difference defined by 33 σ := S v S c ɛ Now it is apparent that in order to study a zero-temperature quantum hydrodynamic model we need to use only the three quantities 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 electrourrent densities J ab = ɛ Im ψ a ψ b When a = b using again the form 31 we recover the classical current densities 34 J c := Im ɛψ c ψ c = nc S c J v := Im ɛψ v ψ v = nv S v whose physical meaning has to be interpreted similarly as for the particle densities It is easy to get the identity 35 ɛψ a ψ b = [ n a nb exp i S ] b S a ɛ n b + i S b ɛ nb Also we introduce the complex velocities u c and u v with 36 u c := ɛ ψ c ψ c = ɛ nc + i S c u v := ɛ ψ v nc ψ v = ɛ nv + i S v nv We note that 37 ɛ v = v 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 osc and the u elc Explicitly we have 38 u osc := ɛ nc u elc := S c = J c

6 1284 G ALÌ and G FROSALI Analogous definitions hold for u osv and u elv so that 39 u c = u osc + iu elc u v = u osv + iu elv Hence osmotic velocity and current velocity can be expressed solely in terms of 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 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 31 and using the first equation of the Kane system 23 we find ψ c = ψ c + ψ ψ c c = iɛ ψc ψ c ψ c ψ 2 c + ip ɛψc ψ v ɛψ c ψ v = Im ɛψ c ψ c 2P Im ɛψc ψ v In a similar way we get = Im ɛψ v ψ v +2P Im ɛψv ψ c Using the definition of the particle densities and of the complex velocities we find 41 ɛψ c ψ v = v u v ɛψ v ψ c = v u c and then recalling the definitions 34 of J c and J v the previous equations become 42 + J c = 2P Im v u v + J v = 2P Im v u c Explicitly in terms of osmotic and current velocities we can write { ncv u v = nv cos σ + i sin σu osv + iu elv v u c = nv cos σ i sin σu osc + iu elc Summing the equations in 42 and using the identity Im ɛψ c ψ v Im ɛψ v ψ c = ɛ Im v we obtain the balance law for the total density J c + J v = 2 ɛp Im v

7 QUANTUM HYDRODYNAMIC MODELS FOR THE TWO-BAND KANE SYSTEM 1285 It is convenient to write eq 43 in divergence form in order to derive the conservation of total density Using the fact that P = 0 43 gives + + J c + J v +2ɛP Im v =2ɛ Im v 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 interesting to note that this assumption ensures the self-adjointness of the k p Hamiltonian corresponding to the Schrödinger-type system 23 Next we derive equations for the phases S c S v Using 23 and 42 we have S c = iɛ ln ψc nc = ɛ2 2 ψc ψ c i Im ψ c ψ c V c + ɛ2 P ψ c ψ v i Im ψ c ψ v = ɛ2 2 Re ψc ψ c ψ c ψ c Vc + ɛ2 P Re ψ c ψ v It is possible to rewrite the previous equation as S c = 1 2 S c 2 + ɛ2 2 V c + ɛ P Re ɛψ c ψ v A similar equatioan be derived for S v Then using 41 the resulting equations are 44 nc S c S c 2 ɛ2 2 + V c = ɛ P Re v u v nv S v S v 2 ɛ2 2 + V v = ɛ P Re v u c Equations 42 and 44 are equivalent to the coupled Schrödinger equations 23 We would like to replace 44 with a system of coupled equations for the currents Using the definitions 34 we can evaluate 45 J c = j Re ψ c ψ c ψ ψ c c ψ x j x c ψ c V c + j ɛ 2 2 x j + ɛ 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 ψ ψ c c x j x j ψ c Re ψ c x j ψ c = ψ c 2 2 x j x j = 1 2 x j ɛ 2 S S c c x j

8 1286 G ALÌ and G FROSALI with j =1 2 3 which yield Re ψ x c ψ c ψ ψ c c = j j x j x j = [ 1 x j j 2 2 nc 2n ] c x j x j ɛ 2 S S c 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 45 becomes 46 J c Jc J c + ɛ 2 ɛ V c = ɛ 2 Re [ ψ c P ψ v 2 ψ c P ψ v ] A similar equatioan be written for J v 47 J v Jv J v + ɛ 2 ɛ V v = ɛ 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 na a = c v na The correction terms ɛ 2 2 n a na a = c v can be identified with the quantum Bohm potentials for each band in analogy with the single-band case Moreover using 41 and the identity ɛ 2 ψ a P ψ b =n ab P u b u a ab = c v the right-hand sides of eqs 46 and 47 can be expressed in terms of the hydrodynamic quantities

9 QUANTUM HYDRODYNAMIC MODELS FOR THE TWO-BAND KANE SYSTEM The resulting system takes the following form: J c J v Jc J c ɛ V c = = ɛ Re v P u v 2Rev P u v u c Jv J v ɛ 2 + V v = 2 = ɛ Re v P u c +2Rev P u c u v For the reader s convenience we express the right-hand sides of 48 in terms of osmotic and current velocities: Re v P u v = nv P u osv cos σ u elv sin σ Re v P u c = nv P u osc cos σ + u elc sin σ Re v P u v u c = nv [P u osc cos σ + u elc sin σu osv P u osc sin σ u elc cos σu elv ] Re v P u c u v = nv [P u osv cos σ u elv sin σu osc + + P u osv sin σ + u elv cos σu elc ] It is important to notice that at variance with the uncoupled model 42 and 48 are not equivalent to the original equation 23 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 44 to derive an evolution equation for σ =S v S c /ɛ 49 ɛ σ 1 2 J c J v 2 + ɛ2 2 ɛ2 nv 2 V c + V v = ɛ P Re v u c ɛ P Re v u v Equation 49 must be supplemented with the constraint 410 ɛ σ = J v J c Using 42 and 48 it is possible to prove that 49 and 410 are equivalent in the following sense: 49 implies that 410 is satisfied at all times if it holds at the initial time; 410 implies that 49 is satisfied up to a constant To see the truth of the above statement it is sufficient to take the gradient of 44 and afterwards account that 411 S c = J c S v = J v

10 1288 G ALÌ and G FROSALI Then noting that 49 was derived from the same equation 44 it is simple to see that 49 implies ɛ σ J v + J c =0 which yields the first part of the statement Similarly taking the time derivative of 410 and using again 411 we can recover the gradient of equation 49 which implies the second part of the statement The equivalence of 410 and 49 suggests that we can discard 49 and recover σ as a function of the other variables by solving the elliptic equation Jv 412 ɛ σ = J c which can be obtained immediately after derivation of the constraint 410 Another possibility is to regard v as an independent variable rather than σ Recalling the definition 32 and Kane s system 23 we find v = iɛ 2 ψ c ψ v ψ v ψ c + i ɛ V c V v ψ c ψ v iɛp ψ c ψ c + ψ v ψ v which using 35 and the definitions of osmotic and current velocities leads to 413 ɛ v = iɛ 2 vu v u c + iv V c V v iɛp u c + u v In addition to 413 the complex functio must satisfy the constraints v v = ɛ v =u c + u v v Alternatively we can use the identity 415 to derive a nonlinear elliptic equation for v ɛ ncv 416 div = divu c + u v v which must be solved together with the constraint 414

11 QUANTUM HYDRODYNAMIC MODELS FOR THE TWO-BAND KANE SYSTEM 1289 Now we are in position to rewrite the hydrodynamic system as follows: 417 +divj c = 2P Im v u v +divj v = 2P Im v u c J c J v ɛ σ = J v J c Jc J c ɛ V c = = ɛ Re v P u v 2Rev P u v u c Jv J v ɛ 2 + V v = 2 = ɛ Re v P u c +2Rev P u c u v where v u v andu v are expressed in the terms of the hydrodynamic quantities J c J v andσ by 32 and 39 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 k c andby the envelope function in the valence band ψ k v The occupation probabilities are such that λ k =1 The k-th state for the Kane system is described by the solutions of the system 51 iɛ ψk c iɛ ψk v = ɛ2 2 ψk c + V c ψ k c ɛ 2 P ψ k v = ɛ2 2 ψk v + V v ψ k v + ɛ 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 k c = n k c exp [ isc k /ɛ ] ψv k = n k v exp [ isv k /ɛ ]

12 1290 G ALÌ and G FROSALI the previous system is equivalent to 52 n k c +divj c k = 2P Im n k cvu k v n k v +divj v k = 2P Im n k cvu k c J k c J k v J k + div c Jc k n k n k ɛ 2 n c k c c 2 + n k n k c V c = c = ɛ Re n k cvp u k v 2Re n k cvp u k vu k c J k + div v Jv k n k n k ɛ 2 n v k v v 2 + n k v V v = n k v = ɛ Re n k cvp u k c +2Re n k cvp u k c u k v J ɛ σ k k = v n k v J k c n k c with Jc k = n k c Sc k Jv k = n k v Sv k σ k = Sk v Sc k ɛ n k cv = n k c n k v exp [ iσ k] u k c = ɛ n k c n k c + i J c k n k u k v = ɛ n k v + i J v k c n k v n k v The densities and the currents corresponding to the two mixed states for conduction and valence electrons can be defined as := λ k n k c := λ k n k v J c := λ k Jc k J v := λ k Jv k We also define σ := λ k σ k u c := ɛ nc + i J c v := nv exp[iσ] u v := ɛ nv + i J v nv

13 QUANTUM HYDRODYNAMIC MODELS FOR THE TWO-BAND KANE SYSTEM 1291 Multiplying 52 by λ k and summing over k we find 53 with +divj c = 2P Im R c +divj v = 2P Im R v J c Jc J c + θ c J v Jv J v + θ v J k ɛ σ = λ v k n k v J c k n k c ɛ V c = = ɛ P Re R c 2P Re Q cv ɛ V v = = ɛ P Re R v +2P Re Q vc R c = λ k n k cvu k v R v = λ k n k cvu k c Q cv = λ k n k cvu k v u k c Q vc = λ k n k cvu 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 osc +θ elc and θ v = θ osv +θ elv are the sum of osmotic temperature and electrourrent temperature given by We can write θ osc = θ elc = λ k n k c u k osc u osc u k osc u osc λ k n k c u k elc u elc u k elc u elc 54 R c = v αu v + β v R v = v αu c + β c with α := λ k n k cv v β v := λ k n k cv v u k v u v β c := λ k n k cv v u k c u c

14 1292 G ALÌ and G FROSALI These quantities are not independent due to 37 By taking the gradient of α we find ɛ ncv 55 ɛ α β v β c = α u v u c v and using this identity it is possible to show that R c + R v = ɛ αv Moreover recalling the definition of v u c and u v we find 56 ɛ v v u v u c = i ɛ σ J v + J c Using this identity in 55 we get 1 J v ɛ α βv β c = ɛ σ + J c iα which implies { 1 } Re ɛ α βv β c =0 α { 1 } Im ɛ α βv β c = ɛ σ + J v J c α 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 = v αu v u c + β v u c + u v β c + θ cv Q vc = v αu c u v + β c u v + u c β v + θ vc with θ cv := θ vc := λ k n k cv v u k v u v u k c u c λ k n k cv v u k c u c u k v u v

15 QUANTUM HYDRODYNAMIC MODELS FOR THE TWO-BAND KANE SYSTEM Ionclusion using the new quantities defined above system 53 becomes +divj c = 2P Im [v αu v + β v ] +divj v = 2P Im [v αu c + β c ] J c Jc J c ɛ 2 + θ c 2 = ɛp Re v αu v + β v + V c = 2P Re v αu v u c + β v u c + u v β c + θ cv J v Jv J v ɛ 2 + θ v 2 + V v = = ɛp Re v αu c + β c + +2P Re v αu c u v + β c u v + u c β v + θ vc ɛ σ J v + J { c 1 } = Im ɛ α βv β c α Unlike system 417 this system which can be considered as a nonzero-temperature quantum fluid model is not closed The quantities v u c andu v already present in 417 are expressed in terms of J c J v andσ while the new quantities α β c and β v are linked between them by Re { 1 } ɛ α βv β c =0 α and need appropriate closure relations Moreover we must assigonstitutive 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 σ and taking 58 β c =2 ᾱ u osc ᾱ σ u elc β v =2 α u osv + α σ u elv Then we have ɛ α β v β c =0 which implies ɛ σ J v + J c =0 In particular it is possible to choose 59 α =1 β c = β v =0

16 1294 G ALÌ and G FROSALI We still need to consider the temperature tensors θ c θ v θ cv and θ vc Heuristically following the analogy with the single-band fluid-dynamical model [3] the simplest closure relation is 510 θ c = 1 p c I θ v = 1 p v I θ cv = θ vc =0 where I is the identity tensor and the functions p c and p v can be interpreted as pressures In this way we obtain the simplest two-band isentropic fluid-dynamical model: 511 +divj c = 2P Im v u v +divj v = 2P Im v u c J c Jc J c + p c I ɛ V c = = ɛp Re v u v 2P Re v u v u c ɛ 2 J v Jv J v + p v I 2 + V v = = ɛp Re v u c +2P Re v u c u v ɛ σ J v + J c =0 We remark that if the classical pressures are linear functions of and respectively then we reduce to the so-called isothermal case 6 The drift-diffusive scaling In kinetic theory of gases it is customary to introduce a relaxation-time term in order to simulate the mechanisms which force the system towards the statistical mechanical equilibrium In semiconductor physics the collisional mechanisms are more intricate and difficult to simulate by simple phenomenological models Anyway also in this case it is classical to introduce momentum and energy relaxation times First we consider a modified version of the single state system 417 with additional relaxation terms for the currents by rewriting the current equations as J c Jc J c ɛ V c = 61 = ɛ Re v P u v 2Rev P u v u c J c τ J v Jv J v ɛ V v = = ɛ Re v P u c +2Rev P u c u v J v τ Here τ is a relaxation time which we assume equal for the two bands

17 QUANTUM HYDRODYNAMIC MODELS FOR THE TWO-BAND KANE SYSTEM 1295 In analogy with the classical diffusive limit for a one-band system [3] we introduce the following diffusive scaling: 62 t t τ J c τj c J v τj v Consequently from definition 33 the phase difference has to be rescaled in such a way that ɛ σ = J v J c τ J v τ J c = τɛ σ This simple consideration leads to the ansatz σ σ 0 + τσ where σ 0 is a constant phase to be determined Then we have v nv e iσ0 + i nv e iσ0 στ + Oτ 2 u c ɛ nc + i τj c u v ɛ nv + i τj v Using these relations for the coupling terms we find v u v nv e iσ0 u osv + i nv e iσ0 σu osv + u elv τ + Oτ 2 v u c nv e iσ0 u osc i nv e iσ0 σu osc u elc τ + Oτ 2 According to the previous scaling up to the first order in τ the balance equations for the densities become 63 { nc } τ +divj c = 2 nv sin σ 0 P u osv { τ 2 nv cos σ 0 σp u osv + P u elv { nv } τ +divj v = 2 nv sin σ 0 P u osc { τ 2 nv cos σ 0 σp u osc P u elc } + Oτ 2 } Formally as τ tends to zero the previous equations give sin σ 0 =0 that is σ 0 =0 Using this value as τ tends to zero the limit equations of 63 take the form 64 +divj c = 2 nv σp u osv + P u elv +divj v = 2 nv σp u osc P u elc

18 1296 G ALÌ and G FROSALI Finally performing the previous scaling on the current equations and expressing the osmotic and current velocities in terms of the other hydrodynamic quantities the hydrodynamic system will be formally reduced to 65 nc +divj c = 2ɛσ P 2 P J v nv +divj v = 2ɛσ P nv +2 P J c nc ɛ 2 J c = 2 V c + + ɛ 2 P 2ɛ 2 P ɛ 2 J v = 2 V v ɛ 2 P +2ɛ 2 P ɛ σ = J v J c This system represents the analog of the zero-temperature quantum drift-diffusion model for the Kane system We can use the same procedure to perform the diffusive limit of the non-zero temperature system 57 As before we consider a modified version of system 57 with additional relaxation terms for the currents Using the closure relation 58 we rewrite 57 as 66 +divj c = 2P Im [v αu v + β v ] +divj v = 2P Im [v αu c + β c ] J c Jc J c ɛ 2 + θ c 2 = ɛp Re v αu v + β v J v Jv J v + θ v 2 = ɛp Re v αu c + β c + + V c = 2P Re v αu v u c + β v u c + u v β c + θ cv J c τ ɛ 2 + V v = +2P Re v αu c u v + β c u v + u c β v + θ vc J v τ ɛ σ J v + J c =0 As done for the zero-temperature system we introduce the diffusive scaling 62 which again leads to the ansatz σ σ 0 + τσ where σ 0 is a constant phase to be

19 QUANTUM HYDRODYNAMIC MODELS FOR THE TWO-BAND KANE SYSTEM 1297 determined Moreover we assume that α α 0 + τα α 0 = α 0 A justification of this ansatz can be found by inspecting the definition of α in terms of the mixed densities of the simple states n k cv Basically we are assuming that for all k we have σ k σ 0 + τσ k for the same σ 0 As a result the first term of the expansion of α around τ = 0 is real Then proceeding as for the zero-temperature system we find σ 0 =0and 67 +divj c = 2 nv α v σp u osv + α 0 P u elv +divj v = 2 nv α c σp u osc α 0 P u elc ɛ 2 J c = div θ c + 2 V c + + ɛp nv α v u osv 2 nv α cv P u osv u osc + P Re θ cv ɛ 2 J v = div θ v + 2 V v ɛp nv α c u osc +2 nv α cv P u osc u osv + P Re θ vc ɛ σ J v + J c =0 with α c := α 0 +2 α 0 α v := α 0 +2 α 0 α cv := α 0 +2 α 0 +2 α 0 7 Concluding remarks In this paper we have applied the WKB method in order to rewrite the two-band Schrödinger-like Kane s system in terms of conduction and valence band electron densities and currents J c and J v The strict analogies with the single-band case have allowed us to define the osmotic and current velocities as complex quantities which can be expressed solely by means of J c and J v In addition the coupling term v has been defined by introducing the phase difference σ Then we have obtained the hydrodynamic equations for a zero-temperature two-band quantum fluid in the form 417 using the usual quantities J c and J v plus the phase difference σ The extension to mixed quantum-mechanical state has given rise to the nonzerotemperature quantum hydrodynamic system 57 which is not closed The closure of this system can be achieved by different methods: in this paper we have limited ourselves to recall the isentropic and the isothermal assumptions As we have already said in the introduction quantum hydrodynamic models can be derived using different approaches but the classical WKB method has the advantage of handling directly mathematical quantities which can be easily extended to the two-band case still keeping a specific physical meaning

20 1298 G ALÌ and G FROSALI The results presented in this work are only formal and the mathematical analysis needs further investigations In particular existence uniqueness or non-uniqueness of the solutions to such models need to be studied In this paper we have also performed the diffusive limit and we have deduced the corresponding quantum drift-diffusion system The future research is oriented towards a physically-based closure of the hydrodynamic system and towards the numerical validation of the drift-diffusion model The authors are grateful to L Barletti and G Borgioli for many helpful discussions on the position of the problem This work was performed under the auspices of the National Group for Mathematical Physics of the Istituto Nazionale di Alta Matematica and was partly supported by the Italian Ministery of University MIUR National Project Mathematical Problems of Kinetic Theories Cofin2002 REFERENCES [1] Anile A M Mascali G and Romano V Recent development in hydrodynamical modeling of semiconductors inmathematical problems in semiconductor physics edited by Anile A M Lect Notes Math Vol1823 Springer 2003 pp 1-54 [2] Gardner C L and Ringhofer C Phys Rev E [3] Jüngel A Quasi-hydrodynamic Semiconductor Equations Birkhäuser Basel 2001 [4] Degond P and Ringhofer C J Stat Phys [5] Kluksdahl N C Kriman A M Ferry D K and Ringhofer C Phys Rev B [6] Kane E O J Phys Chem Solids [7] Borgioli B Frosali G and Zweifel P F Transport Theory Stat Phys [8] Wenckebach W T Essential of Semiconductor Physics J Wiley & Sons Chichester 1999 [9] Sweeney M and Xu J M Appl Phys Lett [10] Yang R Q Sweeny M Day D and Xu J M IEEE Trans Electron Devices [11] Kefi J Analyse mathématique et numérique de modèles quantiques pour les semiconducteurs PhD thesis Université Toulouse III - Paul Sabatier 2003 [12] Barletti L Boll Unione Mat Ital B 6-B [13] Biondini S Frosali G and Mugelli F Quantum hydrodynamic equations for a twoband Wigner-Kane model inproceedings WASCOM th Conference on Waves and Stability in Continuous Media edited by Monaco R et alworld Scientific 2004 pp [14] Landau L D and Lifshitz E M Quantum Mechanics: Non-relativistic Theory Pergamon Press Oxford 1977 [15] Gasser I Markowich P A and Unterreiter A Quantum Hydrodynamics in Modeling of Collisions edited by Raviart P A Series in Applied Mathematics Gauthier-Villars 1997 pp