On the zero-temperature quantum hydrodynamic model for the two-band Kane system

Transcription

1 Recent Trends in Kinetic Theory and its Applications KYIV, Ukraine, May 11-15, 2004 On the zero-temperature quantum hydrodynamic model for the two-band Kane system Giuseppe Alì Istituto per le Applicazioni del Calcolo CNR Napoli Giovanni Frosali Dip. di Matematica Applicata G.Sansone Università di Firenze Firenze

2 Physical description of the KANE MODEL. The Kane model consists into a couple of Schrödinger-like equations for the conduction and the valence band envelope functions. Let ψ c x, t be the conduction band electron envelope function and ψ v x, t be the valence band electron envelope function i ψ c t = 2 2m 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 m is the bare mass of the carriers, V c V v is the minimum maximum of the conductioalence band energy P is the coupling coefficient between the two bands the matrix element of the gradient operator between the Bloch functions

3 SCALING In order to rewrite the Kane system in a dimensionless form, we introduce the rescaled Planck constant ɛ = α where α = mx2 R t R by using x R and t R as characteristic scalar lenght and time variables. We rescale t = t t R, x = x x R and we leave the mass m unchanged. rescaled, taking new potential units V 0 = mx2 R t 2. R The band energy can be The original coupling coefficient is a reciprocal of a characteristic lenght. P = P x R

4 THE SCALED KANE SYSTEM iɛ ψ c t iɛ ψ v t = 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. WIGNER APPROACH HYDRODYNAMIC APPROACH The first aim of this paper is to derive the hydrodynamic version of the Kane system using the WKB method. Using this approach the hydrodynamic limit is valid only for pure states, quantum system at zero temperature.

5 The HYDRODYNAMIC quantities Look for solutions in the form ψ c x, t = ψ v x, t = We introduce the particle densities x, t exp is c x,t x, t exp is v x,t ɛ n ij = ψ i ψ j Then n = ψ c ψ c + ψ v ψ v is exactly the electron density ionduction and valence bands. It is natural to write the coupling terms in a more manageable way, introducing the complex quantity v := ψ c ψ v = nv e iσ, ɛ with σ := S v S c. ɛ

6 We define quantum mechanical electrourrent densities J ij = ɛ Im ψ i ψ j. When i = j, we recover the classical current densities J c := Im ɛψ c ψ c = nc S c, J v := Im ɛψ v ψ v = nv S v. It is easy to get ɛψ i ψ j = n i nj exp i S j S i ɛ ɛ n j nj + i S j. We introduce the complex velocities u c := ɛ ψ c ψ c u v := ɛ ψ v ψ v = ɛ nc + i S c, = ɛ nv + i S v.

7 We name the real and imaginary part of u c current velocity respectively: osmotic velocity and so that u os,i := ɛ n i ni, u el,i := S i = J i n i. i = c, v, 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 in terms of,, J c and J v. CHOICE of the hydrodynamic quantities: 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.

8 HYDRODYNAMIC formulation of the Kane model Taking account of the wave form and using the first equation of the Kane system, we find t ψ c = ψ c t + ψ ψ c c t = Im ɛψ c ψ c 2P Im ɛψc ψ v. Analogously for t. Then, the previous equations become t + J c = 2P Im ɛψ c ψ v t + J v = 2P Im ɛψ v ψ c 10

9 and using the definitions of osmotic and current velocities: ɛψ c ψ v = v u v = nv cos σ + i sin σu os,v + iu el,v we have t + J c = 2 Im v P u v t + J v = 2 Im v P u c, 12 Summing the equations in 12, we obtain the balance law t + + J c + J v + 2ɛP Im v = 0, which is just the quantum counterpart of the classical continuity equation.

10 Next, we derive equations for phases S c, S v, obtaining a system equivalent to the coupled Schrödinger equations. To obtain a system of coupled equations for the currents, we get J c t = j ɛ 2 2 x j Re ψ c ψ c x j ψ c ψ c x j +ɛ 2 Re [ ψ c P ψ v ψ c P ψ v ]. similar equation for J v. ψ c ψ c V c The left-hand sides of such equations can be put in a more familiar form introducing the Bohm potentials for each band div n i n i 1 4 n i = n [ ] i ni 2, ni The correction terms ɛ2 n i i = c, v, can be called the quantum 2 ni Bohm potential for each band.

11 Moreover the right-hand sides of the previous equations can be expressed in terms of the hydrodynamic quantities, using the following relations for i, j = c, v ɛ ψ i P ψ j = n ij P u j ɛ 2 ψ i P ψ j = n ij P u j u i. Finally the resulting system takes the following form J c Jc t + div J c ɛ V c = ɛ Re v P u v 2 Re v P u v u c, J v t + div Jv J v ɛ V v = ɛ Re v P u c + 2 Re v P u c u v. 13

12 It is important to notice that, differently from the uncoupled model, 12 and 13 are not equivalent to the original equations, 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 fluids. 12 and 13 can be supplemented with the constraint ɛ σ = J v J c.

13 Now we are in position to rewrite the hydrodynamic system as follows t + divj c = 2 Im v P u v, t + divj v = 2 Im v P u c, J c t + div J v t + div Jc J c Jv J v ɛ V c = ɛ Re v P u v 2 Re v P u v u c, ɛ V v = ɛ Re v P u c + 2 Re v P u c u v, ɛ σ = J v J c, v, u v, and u v are given by the hydrodynamic quantities,, J c, J v, and σ.

14 The DRIFT-DIFFUSIVE scaling It is customary to introduce a relaxation time term in order to simulate all the mechanisms which force the system towards the statistical mechanical equilibrium. We rewrite the current equations in the previous system as J c Jc t + div J c ɛ V c = ɛ Re v P u v 2 Re v P u v u c J c τ, J v Jv t + div J v ɛ V v = ɛ Re v P u c + 2 Re v P u c u v J v τ, 16 where τ is a relaxation time, which we assume equal for the two bands.

15 In analogy with the classical diffusive limit for a one-band system, we introduce the scaling t t τ, J c τj c, J v τj v. 17 Consequently, the phase difference has to rescaled in a such way that Then, we have σ τσ. v nv + Oτ, u c ɛ nc u v ɛ nv + i τj c, + i τj v.

16 Moreover, the coupling terms has to be tackled with much care, as follows v P u v nv P u os,v + i nv σp uos,v + P u el,v τ + Oτ 2. Formally, as τ tends to zero, the hydrodynamic system with the current equations 16 reduces to t + divj c = 2 nv σp uos,v + P u el,v, t + divj v = 2 nv σp uos,c P u el,c, ɛ 2 J c = 2 V c + ɛ nv P u os,v 2 nc nv P u os,v u os,c, n c ɛ 2 J v = 2 V v ɛ nv P u os,c + 2 nc nv P u os,c u os,v, ɛ σ = J v J c. 19

17 Finally, after having expressed the osmotic and current velocities, in terms of the other hydrodynamic quantities, we have nc t + divj c = 2ɛσ P 2 P J v, nv t + divj v = 2ɛσ P nv + 2 P J c, nc ɛ 2 J c = 2 V c + ɛ 2 P 2ɛ 2 P, n c ɛ 2 J v = 2 V v ɛ 2 P + 2ɛ 2 P ɛ σ = J v J c. 20 This system represents the zero-temperature QUANTUM DRIFT- DIFFUSION MODEL for a Kane system.

18 NONZERO TEMPERATURE hydrodynamic model We consider an electron ensemble which is represented by a mixed quantum mechanical state, with a view to obtaining a nonzero temperature model for a Kane system. A mixed state is a sequence of single states with occupation probabilities λ k, k = 0, 1, 2,... k N for the k th single state. The k th state for the Kane system is described by the solutions of the system iɛ ψk c t = ɛ2 2 ψk c + V c ψ k c ɛ 2 P ψ k v, iɛ ψk v t = ɛ2 2 ψk v + V v ψ k v + ɛ 2 P ψ k c. 21 Using the Madelung-type transform, under the assumption of positivity of the densities n k c and n k v, ψc k = n k c exp isc k /ɛ, ψv k = n k v exp isv k /ɛ,

19 the previous system is equivalent to n k c t + divjk c = 2 Im n k cvp u k v, n k v t + divjk v = 2 Im n k cvp u k c, Jc k J k t + div c Jc k n k n k c c 2 Jv k J k t + div v Jv k n k v ɛ σ k = J k v n k v Jk c n k, c n k v ɛ2 n k c n k c + n k c V c = ɛ Re n k cvp u k v 2 Re n k cvp u k vu k c, ɛ2 n k v 2 n k v + n k v V v = ɛ Re n k cvp u k c + 2 Re n k cvp u k cu k v,

20 we define J k i = nk i Sk i, n k cv = σk = Sk v S k c ɛ n k c n k v exp iσ k,, u k i = ɛ n k i n k i + i Jk i n k i The densities and the currents corresponding to the two mixed states can be defined as We also define σ := k=0 n i := k=0 λ k n k i, J i := k=0 λ k J k i. λ k σ k, v := nv expiσ, u i := ɛ n i ni. + i J i.

21 Multiplying by λ k and summing over k, we find t + divj c = 2 Im R c, t + divj v = 2 Im R v, J c Jc t + div J c + θ c J v Jv t + div J v ɛ σ = k=0 λ k J k v n k v + θ v Jk c n k, c ɛ V c = ɛ Re R c 2P Re Q cv, ɛ V v = ɛ Re R v + 2P Re Q vc, 23

22 with R i = k=0 λ k n k ij P uk j, Q ij = k=0 λ k n k ij uk j uk j, i, j = c, v. Analogously with the one-band case, new terms containing the total temperature θ c and θ v, for each band, appear. 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 electrourrent temperature, We write θ os,c = θ el,c = k=0 k=0 λ k n k c u k os,c u os,c u k os,c u os,c, λ k n k c u k el,c u el,c u k el,c u el,c. R c = P v αu v + β v, R v = P v α u c + β c, 24

23 with α := k=0 λ k n k cv v, β i := k=0 These quantities are not independent: 1 iα λ k n k ij n ij u k i u i, i, j = c, v. ɛ α βv β c = ɛ σ + J v J c, Next, in order to obtain an expression of the coupling terms between the two bands by a sort of temperature tensors, we write with 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, θ cv := k=0 λ k n k cv v u k v u v u k c u c,

24 Ionclusion, using the new quantities defined above, system 23 can be written as t + divj c = 2P Im [v αu v + β v ], t + divj v = 2P Im [v α u c + β c ], J c Jc t + div J c ɛ 2 + θ c 2 + V c = ɛ P Re v αu v + β v 2P Re v αu v u c + β v u c + u v β c + θ cv, J v Jv t + div J v ɛ 2 + θ v 2 + V v = ɛ P 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 25

25 FINAL REMARKS We have derived a set of quantum hydrodynamic equations from the two-band Kane model. This system, which can be considered as a nonzero-temperature quantum fluid model, is not closed. In addition to other quantities, we have the tensors θ c, θ v, θ cv and θ vc only the first ones are similar to the temperature tensor of kinetic theory. Closure of the quantum hydrodynamic system Poor physical meaning of the envelope functions New basis for more physical envelope functions Heterogeneous materials Numerical treatment