NUMERICAL STUDY OF A QUANTUM-DIFFUSIVE SPIN MODEL FOR TWO-DIMENSIONAL ELECTRON GASES

Transcription

1 COMMUN. MATH. SCI. Vo. 13, No. 6, pp c 2015 Internationa Press NUMERICAL STUDY OF A QUANTUM-DIFFUSIVE SPIN MODEL FOR TWO-DIMENSIONAL ELECTRON GASES LUIGI BARLETTI, FLORIAN MÉHATS, CLAUDIA NEGULESCU, AND STEFAN POSSANNER Abstract. We investigate the time evoution of spin densities in a two-dimensiona eectron gas subjected to Rashba spin-orbit couping on the basis of the quantum drift-diffusive mode derived in [L. Baretti and F. Méhats, J. Math. Phys., 51, (20), 2010]. This mode assumes that the eectrons are in a quantum equiibrium state in the form of a Maxweian operator. The resuting quantum drift-diffusion equations for spin-up and spin-down densities are couped in a non-oca manner via two spin chemica potentias (Lagrange mutipiers) and via off-diagona eements of the equiibrium spin density and spin current matrices, respectivey. We present two space-time discretizations of the mode, one semi-impicit and one expicit, which aso comprise the Poisson equation in order to account for eectron-eectron interactions. In a first step, pure time discretization is appied in order to prove the we-posedness of the two schemes, both of which are based on a functiona formaism to treat the non-oca reations between spin densities. We then use the fuy space-time discrete schemes to simuate the time evoution of a Rashba eectron gas confined in a bounded domain and subjected to spin-dependent externa potentias. Finite difference approximations are first order in time and second order in space. The discrete functionas introduced are minimized with the hep of a conjugate gradientbased agorithm where the Newton method is appied in order to find the respective ine minima. The numerica convergence in the ong-time imit of a Gaussian initia condition towards the soution of the corresponding stationary Schrödinger Poisson probem is demonstrated for different vaues of the parameters ε (semicassica parameter), α (Rashba couping parameter), x (grid spacing), and t (time step). Moreover, the performances of the semi-impicit and the expicit scheme are compared. Key words. Rashba spin-orbit couping, spin reaxation, entropic quantum drift-diffusion, quantum Liouvie, Schrödinger Poisson drift-diffusion, spin diffusion. AMS subject cassifications. 65K10, 65M12, 65N25, 76Y05, 82C10, 82D37, 81R Introduction The purpose of this paper is the numerica study of the quantum diffusive mode for a spin-orbit system introduced in [2] with the aim of deveoping numerica toos for the investigation of spin-based eectronic devices. Diffusive modes offer a simpe, yet fairy accurate, description of charge transport, and, for this reason, they have a ong-standing tradition in semiconductor modeing. Cassica drift-diffusion equations for semiconductors [12] were first derived by van Roosbroeck [19] whie Poupaud [18] proved their rigorous derivation from the Botzmann equation. Quantum-corrected drift-diffusion equations were proposed in [1, 7] and were ater derived by using a quantum version of the maximum entropy principe by Degond, Méhats, and Ringhofer [5, 6]. Finay, fuy-quantum diffusive equations, sti based on the quantum maximum entropy principe, were proposed in [4, 5]. In view of recent progresses in controing the eectron spin, it is highy desirabe to extend the Received: January 9, 2013; accepted (in revised form): Juy 5, Communicated by Lorenzo Pareschi. Dipartimento di Matematica, Università di Firenze, Viae Morgagni, 67/a , Forence, Itaia (baretti@math.unifi.it). INRIA IPSO Team, IRMAR, Université de Rennes 1, Campus de Beauieu Cedex, Rennes, France (forian.mehats@univ-rennes1.fr). IMT, Université Pau Sabatier, 118 Route de Narbonne, 31400, Tououse, France (caudia.neguescu@math.univ-tououse.fr). IMT, Université Pau Sabatier, 118 Route de Narbonne, 31400, Tououse, France (Stefan.Possanner@math.univ-tououse.fr). 1347

2 1348 NUMERICAL STUDY OF A SPIN QDD MODEL drift-diffusion description to the spinoria case. The existing semicassica drift-diffusion modes for spin systems can be cassified into two categories: the two-component modes [9] and the spin-poarized, or matrix, modes [9, 17, 20]. Both modes have been used in practice; however, their mathematica derivation is sti at the very beginning. As far as we know, a fuy-quantum diffusive mode of a spin system was first reported in [2] where a two-component diffusive mode for a 2-dimensiona eectrons gas with spin-orbit interaction was derived. Such a mode, which wi be considered from the numerica point of view in the present work, is based on the quantum maximum entropy principe and concerns eectrons with a spin-orbit Hamitonian of Rashba type [3]: ( ) 2 H = 2 +V α ( x i y ) α ( x +i y ) 2 2 +V. (1.1) Here, (x,y) are the spatia coordinates of the 2-dimensiona region where the eectrons are assumed to be confined, α is the Rashba constant, and V is a potentia term which may consist of an externa part (representing e.g. a gate or an appied potentia) and a sef-consistent part, accounting for Couomb interactions in the mean-fied approximation. The Rashba effect [3, 21] is a spin-orbit interaction undergone by eectrons that are confined in an asymmetric 2-dimensiona we (here, perpendicuar to the z direction). Due to this interaction, the spin vector has a precession around a direction in the pane (x,y) perpendicuar to the eectron momentum p = (p x,p y ), the precession speed being α p. Since it does not invove buit-in magnetic fieds, and hence may be impemented by means of standard siicon technoogies, the Rashba effect is expected to be a key ingredient for the reaization of the so-caed S-FET (Spin Fied Effect Transistor) [21], a spintronic device in which the information is carried by the eectron spin rather than by the eectronic current (as in the usua eectronic devices). This may ead to eectronic devices of higher speed and ower power consumption. The purpose of this work is to contribute to the understanding of how the Rashba effect can be empoyed in order to contro the spin transport in these devices. Let us summarize briefy the derivation of the here investigated quantum diffusive mode. The starting point is the von Neumann equation (i.e. the Schrödinger equation for mixed states) for the Hamitonian (1.1) endowed with a coisiona term of BGK type i t ϱ(t) = [H,ϱ(t)]+ i τ (ϱ eq ϱ(t)), where ϱ(t) = (ϱ ij (t)) is the 2 2 density operator, representing the time-dependent mixed state of the system, and τ is the reaxation time. According to the theory deveoped in [6, 5], the oca equiibrium state ϱ eq is chosen as the maximizer of a free energyike functiona subject to the constraint of sharing with ϱ(t) the oca moments we are interested in, here the spin-up and spin-down (with respect to the z direction) eectron densities n 1, n 2 (or, equivaenty, the tota eectronic density n 1 +n 2 and the poarization n 1 n 2 ). Then, the maximizer, which has the form of a Maxweian operator, contains as many Lagrange mutipiers (chemica potentias) as the chosen moments. These mutipiers furnish the degrees of freedom necessary to satisfy the constraint equations. In our case, therefore, the oca equiibrium state contains two chemica potentias, A 1 and A 2, which depend on n 1 and n 2 through the constraint equations. The rigorous proof of reaizabiity of the quantum Maxweian associated to a given density and

3 L. BARLETTI, F. MÉHATS, C. NEGULESCU, AND S. POSSANNER 1349 current was obtained in [13, 14] for a scaar (i.e. non spinoria) Hamitonian. By assuming τ 1 and appying the Chapman Enskog method, the von Neumann equation eads in the imit to the quantum drift-diffusive system (2.1) for the unknown densities n 1 and n 2. Apart from the chemica potentias A 1 and A 2, which depend on n 1 and n 2 through the constraint, the system aso contains some extra moments, namey the off-diagona density n 21 and currents J21, x J y 21 which are computed via the equiibrium state and which depend on n 1 and n 2 as we. Note that, with respect to the origina Hamitonian (1.1), we sha work with a scaed version (see the Hamitonian (2.4) which aso contains the chemica potentias) in which ε is the scaed Panck constant and α is rescaed as εα. This is, therefore, a semicassica scaing with the additiona assumption of sma Rashba constant. Of course, the parameter ε is unimportant as ong as we are not interested in the semicassica behavior but becomes reevant when we ook for a semicassica expansion of the mode for sma ε. In summary, the diffusive Equation (2.1), equations (2.3) (2.7), which represent the equiibrium state and the constraints, and the Poisson equation (2.2) for the sefconsistent potentia, constitute the quantum diffusive mode we are going to anayze numericay in this work. Needess to say, the mode (2.1) (2.7) is rather impicit and invoved, and requires a very carefu numerica treatment. The aim of the present paper is thus to present two discrete versions of (2.1) (2.7) suitabe for time-resoved simuation of the spin densities n 1 and n 2 in a spatiay confined, two-dimensiona eectron gas. In both schemes, the finite-difference approximations of the occurring derivatives are first order in time and second order in space. At the core of the numerica study of the present mode is the minimization of a functiona that either maps from R 3P to R (in the first scheme) or from R 2P to R (in the second scheme) where P is the number of points on the space grid. We present an agorithm that uses a combination of the conjugate gradient method and the Newton method in order to find the minimum of the respective functiona at each time step. The paper is organized as foows. In Section 2, the continuous mode is introduced and is endowed with suitabe initia and boundary conditions. In Section 3, we perform two different time discretizations of the continuous mode and give a forma proof of the we-posedness of each of the two schemes. Then, in Section 4, two fuy discrete schemes (i.e. both in time and space) are introduced and anayzed as we. Finay, Section 5 is devoted to numerica experiments. Detais of the proofs and of the discretization matrices are deferred to the appendices. 2. The quantum spin drift-diffusion mode Let us start with the presentation of the quantum diffusive mode introduced in [2]. The mode describes the evoution of the spin-up and the spin-down densities n 1 and n 2, respectivey, of a two-dimensiona eectron gas by means of the foowing quantum drift-diffusion equations: t n 1 + (n 1 (A 1 V s ))+α(a 1 A 2 )Re(Dn 21 ) 2αRe(n 21 D(A 2 V s )) 2α ε (A 1 A 2 )Im(J x 21 ij y 21 ) = 0, t n 2 + (n 2 (A 2 V s ))+α(a 1 A 2 )Re(Dn 21 )+2αRe(n 21 D(A 1 V s )) (2.1) + 2α ε (A 1 A 2 )Im(J x 21 ij y 21 ) = 0. Here, = ( x, y ), D = x i y, A 1, and A 2 denote the two Lagrange mutipiers (A 1 V s and A 2 V s being the chemica potentias), V s stands for the sef-consistent potentia

4 1350 NUMERICAL STUDY OF A SPIN QDD MODEL arising from the eectron-eectron interaction, and n 21, J21, x and J y 21 are off-diagona eements of the spin-density matrix N and the spin-current tensor J written in (2.6) and (2.7), respectivey. The parameter α > 0 denotes the scaed Rashba constant and ε > 0 stands for the scaed Panck constant (for detais regarding the scaing we refer to [2]). The sef-consistent potentia V s is determined by the Poisson equation, γ 2 V s = n 1 +n 2, (2.2) where γ > 0 is proportiona to the occurring Debye ength. The system (2.1)-(2.2) is cosed through the fact that the eectrons are assumed to be in a quantum oca equiibrium state at a times. This constraint aows one to reate the Lagrange mutipiers A = (A 1,A 2 ) to the spin densities n 1 and n 2 as we as to the spin-mixing quantities n 21 and J 21, respectivey. More precisey, if H(A) denotes the system Hamitonian, the equiibrium state operator is given by ϱ eq = exp( H(A)), (2.3) where exp( ) here denotes the operator exponentia. In the present case, the Hamitonian is given by H(A) : D(H) (L 2 (,C)) 2 (L 2 (,C)) 2, D(H) (H 2 (,C)) 2, H(A) = ( ) ε2 2 +V ext,1 +A 1 ε 2 α( x i y ) ε 2 α( x +i y ) ε2 2 +V, (2.4) ext,2 +A 2 where R 2 denotes the bounded domain where the eectrons are assumed to be confined. In what foows, we sha denote x = (x,y). Hence, we have introduced two externa, time-independent potentias V ext,1 (x) and V ext,2 (x) for the spin-up and the spin-down eectrons, respectivey. Assuming that H(A) has a pure point spectrum, the eigenvaues and the eigenvectors of H(A), denoted by λ (A) and ψ (A) = (ψ 1 (A),ψ2 (A)), N, respectivey, are soutions of H(A)ψ (A) = λ (A)ψ (A) (2.5) and ink the Lagrange mutipiers to the spin-density matrix N as we as to the spincurrent matrix J, according to N = ( ) ( ) ψ 1 e λ 2 ψ 1 ψ2 n1 n 21 =, (2.6) ψ 2 ψ1 ψ 2 2 n 21 n 2 ( ) J = iε ψ 1 e λ ψ 1 ψ1 ψ1 ψ 2 ψ1 ψ1 ψ2 2 ψ 1 ψ2 ψ2 ψ1 ψ 2 ψ2 ψ2 ψ2 ( ) J1 J 21 =. (2.7) J 21 J 2 The formuas (2.6) and (2.7) are the standard textbook expressions for the spin-density and the spin-current, respectivey, corresponding to the density operator (2.3). The system (2.1)-(2.2) is now cosed through the non-oca reations N(A) and J(A) given by equations (2.4)-(2.7). As we do not have a proof of the invertibiity of these reations,

5 L. BARLETTI, F. MÉHATS, C. NEGULESCU, AND S. POSSANNER 1351 in other words whether it is possibe to compute A(n 1,n 2 ), Equation (2.1) can aso be viewed as evoution equations for the Lagrange mutipiers A 1 and A 2 rather than for the spin densities n 1 and n 2. Indeed, the two time-discretizations of the system (2.1)-(2.7), which wi be deveoped in Section 3, represent these two possibe viewpoints regarding the evoution Equation (2.1). Let us now come to the boundary conditions of our probem. The considered spatia domain R 2 is assumed to be bounded with reguar boundary. We sha impose Dirichet boundary conditions for the eigenvectors ψ, ψ (x) = 0 for x, hence the current across the domain boundary is zero. As we wi briefy show at the end of this section, the Hamitonian (2.4) is not hermitian in (L 2 (,C)) 2 when imposing Neumann boundary conditions. The study of this probem, as we as the impementation of transparent boundary conditions, can be matter for a future work. The sef-consistent potentia V s is suppemented with Dirichet conditions too: V s (x) = 0 for x. The Lagrange mutipiers A 1 and A 2 are aowed to vary freey at the boundary. Therefore, we take Neumann conditions, (A 1 (x) V s (x)) ν(x) = 0 for x, (A 2 (x) V s (x)) ν(x) = 0 for x. Here, ν(x) denotes the outward norma to the boundary at x. As far as initia conditions are concerned, one has two choices depending on the point of view of the evoution Equation (2.1). Since we do not know whether or not (2.6) is invertibe, the safe approach is to provide initia data for the chemica potentias. However, from the viewpoint of device modeing, it is more appeaing to start from initia spin densities. We sha take the atter approach and assume that n 1 (0,x) and n 2 (0,x) are smooth and bounded. In summary, we have the foowing quantum spin-drift-diffusion mode: t n 1 + (n 1 (A 1 V s ))+α(a 1 A 2 )Re(Dn 21 ) 2αRe(n 21 D(A 2 V s )) 2α ε (A 1 A 2 )Im(J21 x ij y 21 ) = 0, (2.8) t n 2 + (n 2 (A 2 V s ))+α(a 1 A 2 )Re(Dn 21 ) +2αRe(n 21 D(A 1 V s ))+ 2α ε (A 1 A 2 )Im(J21 x ij y 21 ) = 0, (2.9) γ 2 V s = n 1 +n 2, (2.10) H(A)ψ (A) = λ (A)ψ (A), (2.11) N = e λ (A) ( ψ 1 (A) 2 ψ 1(A)ψ2 (A) ), (2.12) ψ 2(A)ψ1 (A) ψ2 (A) 2

6 1352 NUMERICAL STUDY OF A SPIN QDD MODEL J 21 = iε ( e λ (A) ψ 1 2 (A) ψ2 (A) ψ 2 (A) ψ ). 1(A) (2.13) The Hamitonian H(A) is given by (2.4) and the probem is suppemented with the foowing initia and boundary conditions: n 1 (t = 0,x) = n 0 1(x), n 2 (t = 0,x) = n 0 2(x) for x, V s (x) = 0 for x, ψ (x) = 0 for x, (2.14) (A 1 (x) V s (x)) ν(x) = 0 for x, (A 2 (x) V s (x)) ν(x) = 0 for x. Let us finish this section by remarking that the Hamitonian (2.4) is not hermitian in (L 2 (,C)) 2 for Neumann boundary conditions. Indeed, et us consider ( ) ε 2 (χ,h(a)ψ) L 2 = (χ 1,χ 2 2 ) ψ1 +(V ext,1 +A 1 )ψ 1 +ε 2 α( x i y )ψ 2 dx, ε2 2 ψ2 +(V ext,2 +A 2 )ψ 2 ε 2 α( x +i y )ψ 1 where (, ) L 2 denotes the scaar product in (L 2 (,C)) 2. Specificay, et us ook at the Rashba couping terms, (χ 1 ( x i y )ψ 2 χ 2 ( x +i y )ψ 1 )dx = ψ 2 ( x i y )χ 1 dx+ ψ 1 ( x +i y )χ 2 )dx (2.15) + χ 1 ψ 2 (1, i) ν(x)dσ χ 2 ψ 1 (1,i) ν(x)dσ. Here, the boundary terms do not vanish when imposing Neumann conditions. However, if we considered the probem in the whoe space = R 2, the boundary terms woud vanish and the Hamitonian woud be hermitian. Considering the probem in the whoe R 2 means, from the numerica point of view, imposing transparent boundary conditions for ψ. 3. Semi-discretization in time In this section, we make a first step towards a fu space-time discretization of the system (2.8)-(2.14) by discretizing the time domain. The purpose of the semidiscretization is two-fod. Firsty, since the space discretization of the present twodimensiona spin mode is quite invoved, the functiona formaism which wi be appied in this work becomes more transparent in the semi-discrete case than in the fuy discrete case. Secondy, in contrast to the continuous case (2.8)-(2.14), existence and uniqueness of soutions of the semi-discrete system can be proven. Two different semidiscretizations wi be presented. The first one was studied in [8] for a scaar quantum diffusive mode (without the Rashba spin-orbit couping). We sha use some of the techniques eaborated in [8] and appy them to the present spin mode. The second semi-discrete scheme is an expicit one which reies heaviy on the abiity to invert the reation (2.12). Its benefits ie in the fact that, when passing to the fu discretization, its treatment is far ess invoved as compared to the first scheme.

7 L. BARLETTI, F. MÉHATS, C. NEGULESCU, AND S. POSSANNER 1353 In the subsequent anaysis, the identities (A 1 A 2 )Dn k 21 2n k 21D(A 2 ) = D(n k 21(A 1 A 2 )) n k 21D(A 1 +A 2 ), (A 1 A 2 )Dn k 21 +2n k 21D(A 1 ) = D(n k 21(A 1 A 2 ))+n k 21D(A 1 +A 2 ) (3.1) wi be hepfu A first semi-discrete system. Suppose T > 0 and et us discretize the tempora interva [0,T ] in the foowing homogeneous way: t k = k t, k {0,1,...,K, t := T K. Then, inspired by [8], we choose the foowing time-discretization of the continuous probem (2.8)-(2.13): n 1 (A k+1 ) n k 1 t + (n k 1 (A k+1 1 V k+1 s ))+αre[d(n k 21(A k+1 1 A k+1 2 ))] αre[n k 21D(A k+1 1 +A k+1 2 2V k+1 s )] 2α ε (Ak+1 1 A k+1 2 )Im(Jx 21,k ijy 21,k ) = 0, (3.2) n 2 (A k+1 ) n k 2 t + (n k 2 (A k+1 2 V k+1 s ))+αre[d(n k 21(A k+1 1 A k+1 2 ))] +αre[n k 21D(A k+1 1 +A k+1 2 2V k+1 s )] + 2α ε (Ak+1 1 A k+1 2 )Im(Jx 21,k ijy 21,k ) = 0, (3.3) γ 2 V k+1 s = n 1 (A k+1 )+n 2 (A k+1 ), (3.4) H(A k+1 )ψ k+1 = λ k+1 ψ k+1, (3.5) n 1 (A k+1 ) = e λk+1 ψ 1,k+1 2, n 2 (A k+1 ) = e λk+1 ψ 2,k+1 2. (3.6) In this scheme, one searches for the unknowns (A k+1,vs k+1 ) given (N k,j21). k The main difficuties concerning the soution of this system are the non-oca reations (3.5)-(3.6). We sha thus construct a mapping (A,V s ) (H 1 (,R)) 3 F(A,V s ) R whose unique minimum (A k+1,vs k+1 ) is the soution of system (3.2)-(3.6). Once A k+1 and the eigenvaues λ k+1, respectivey eigenvectors ψ k+1, are known, equations (2.12)-(2.13) can be used to compute (N k+1,j21 k+1 ) and the process can be repeated. Let us thus introduce the two functionas G : (L 2 (,R)) 2 R, F : (H 1 (,R)) 3 R defined by G(A) := e λ (A), A (L 2 (,R)) 2, (3.7)

8 1354 NUMERICAL STUDY OF A SPIN QDD MODEL where λ (A) are the eigenvaues of the Hamitonian (2.4), and F(A,V s ) = G(A)+F 1 (A,V s )+F 2 (A,V s )+F 3 (A,V s )+F 4 (A), (3.8) where F 1 (A,V s ) := t n k 2 1 (A 1 V s ) 2 dx+ t n k 2 2 (A 2 V s ) 2 dx, (3.9) F 2 (A,V s ) := γ2 V s 2 dx+ n k 2 1(A 1 V s )dx+ n k 2(A 2 V s )dx, (3.10) { F 3 (A,V s ) := α tre n k 21(A 1 A 2 )D(A 1 +A 2 2V s )dx, (3.11) F 4 (A) := α t { ε Im (A 1 A 2 ) 2 (Jx 21,k ijy 21,k )dx. (3.12) The computation of the first and second Gateaux derivatives of the functionas (3.7)- (3.12) can be found in Appendix B and C, respectivey. One can immediatey see that a soution (A k+1,vs k+1 ) of the semi-discrete system (3.2)-(3.6) satisfies df(a k+1,v k+1 s )(δa,δv s ) = 0, (δa,δv s ) (H 1 (,R)) 3, and inversey. Thus, it remains to show that F has a unique extremum (minimum). This can be achieved in two steps as detaied in Appendix C. First we show that, under suitabe assumptions, the functiona F is stricty convex. Then it is sufficient to show that F is coercive to obtain the existence and uniqueness of the extremum (A k+1,vs k+1 ), a soution of the system (3.2)-(3.6) (see Appendix C) A second semi-discrete system. We suggest here an aternative way to discretize in time the quantum drift-diffusion mode (2.8)-(2.14). It is based on the point of view that one advances the spin densities in time rather than the chemica potentias. We sha impement a forward Euer scheme: n k+1 1 n k 1 + (n k t 1 (A k 1 Vs k ))+αre{d(n k 21(A k 1 A k 2)) αre(n k 21D(A k 1 +A k 2 2Vs k )) 2α ε (Ak 1 A k 2)Im(J x,k y,k ij21 ) =0, 21 (3.13) n k+1 2 n k 2 + (n k t 2 (A k 2 Vs k ))+αre{d[n k 21(A k 1 A k 2)] +αre[n k 21D(A k 1 +A k 2 2Vs k )]+ 2α ε (Ak 1 A k 2)Im(J x,k y,k ij21 ) =0, 21 (3.14) γ 2 V k s = n k 1 +n k 2, (3.15) H(A k )ψ k = λ k ψ k, (3.16)

9 L. BARLETTI, F. MÉHATS, C. NEGULESCU, AND S. POSSANNER 1355 N = J k 21 = iε 2 e λk e λk ( ) ψ 1,k 2 ψ 1,k ψ 2,k, (3.17) ψ 2,k ψ 1,k ψ 2,k 2 ( ) ψ 1,k ψ 2,k ψ 2,k ψ 1,k. (3.18) In this case, given the spin-densities (n k 1,n k 2), one first uses the Poisson equation (3.15) to get Vs k then inverts the reations (3.16)-(3.17) in order to get the chemica potentias (A k 1,A k 2). Finay, one advances in time using the drift-diffusion equations (3.13)-(3.14) in order to get the new spin densities (n k+1 1,n k+1 2 ), and then one repeats the steps. The inversion of the non-oca reation (3.16)-(3.17) can be achieved by minimizing the functiona G n : (L 2 (,R)) 2 R defined by G n (A) := G(A)+ n k 1A 1 dx+ n k 2A 2 dx. (3.19) Indeed, the first derivative of this functiona reads dg n (A)(δA) = e λ ( (A) ψ 1 (A) 2 δa 1 + ψ 2 (A) 2 ) δa 2 dx + n k 1δA 1 dx+ n k 2δA 2 dx, (3.20) which ceary impies that its zeros are soutions of (3.16)-(3.17). As shown in Appendix B, the functiona G n is stricty convex and coercive and hence admits a unique extremum. Remark 3.1. The two semi-discrete systems presented in this section conserve the tota mass (n 1 +n 2 ) because of the particuar choice of Dirichet boundary conditions for the eigenvectors ψ of the Hamitonian (2.4). This can be obtained by integrating the sum of the semi-discrete drift-diffusion equations for n 1 and n 2, equations (3.2)-(3.3) or equations (3.13)-(3.14), respectivey, over the domain. The remaining boundary term is of the form n 21 (A 1 A 2 )(1, i) ν(x)dσ, which does not vanish for Neumann boundary conditions. This is in accordance with the remark at the end of Section 2 where we showed that Neumann conditions for ψ ead to a non-hermitian Hamitonian (2.4) in (L 2 (,C)) Fuy discrete system This section is devoted to the fu discretization of the continuous spin QDD mode (2.8)-(2.14). The time discretization was done in the previous section; now we focus on the space discretization. Let x = [0,1] [0,1] with the discretization x ij = ((j 1) x, (i 1) y), j {1,2,...,M, i {1,2,...,N, x := 1 1, y := M 1 N 1. For functions f(x) on, we write f(x ij ) = f ij. A function f(x) that is subject to homogenous Dirichet boundary conditions on satisfies f 1j = f Nj = 0 j {1,2,...,M, f i1 = f im = 0 i {1,2,...,N.

10 1356 NUMERICAL STUDY OF A SPIN QDD MODEL We introduce the foowing index transformation: (i,j) p i {2,...,N 1, j {2,...,M 1, defined by p = (N 2)(j 2)+i 1, p = 1,...,P, P := (N 2)(M 2). in, the foowing vector notation wi be impe- For discrete functions (f ij ) N 1,M 1 i,j=2 mented: The corresponding Eucidean scaar product is denoted by ˆf := (f p ) P p=1 C P. (4.1) ( ˆf,ĝ) P = x y p N 1 f p g p = x y i=2 M 1 j=2 f ij g ij A first fuy discrete system (scheme 1). The discretization matrices D ± x, D ± y, D x, D y, Dx, Dy, and dir used in the foowing are defined in Appendix D. In view of the boundary conditions (2.14), we choose the foowing space discretization of the semi-discrete system (3.2)-(3.6): ˆn 1 (Âk+1 1,Âk+1 2 ) ˆn k 1 2α t ε (Âk+1 1 Âk+1 2 ) Im(Ĵ x 21,k iĵ y 21,k ) 1 2 (D+ x ) T [ˆn k 1 D + x (Âk (D+ y ) T [ˆn k 1 D + y (Âk+1 1 k+1 ˆV s )] 1 2 (D x ) T [ˆn k 1 Dx (Âk+1 k+1 1 ˆV s )] k+1 ˆV s )] 1 2 (D y ) T [ˆn k 1 Dy (Âk+1 k+1 1 ˆV s )] { { αre DT x [ˆn k 21 (Âk+1 1 Âk+1 2 )] +αre i D y T [ˆn k 21 (Âk+1 1 Âk+1 2 )] { αre ˆn k 21 [ D x (Âk+1 1 +Âk+1 k ˆV s )] { +αre iˆn k 21 [ D y (Âk+1 1 +Âk+1 k ˆV s )] = 0, (4.2) ˆn 2 (Âk+1 1,Âk+1 2 ) ˆn k 2 + 2α t ε (Âk+1 1 Âk+1 2 ) Im(Ĵ x 21,k iĵ y 21,k ) 1 2 (D+ x ) T [ˆn k 2 D + x (Âk (D+ y ) T [ˆn k 2 D + y (Âk+1 2 k+1 ˆV s )] 1 2 (D x ) T [ˆn k 2 Dx (Âk+1 k+1 2 ˆV s )] k+1 ˆV s )] 1 2 (D y ) T [ˆn k 2 Dy (Âk+1 k+1 2 ˆV s )] { { αre DT x [ˆn k 21 (Âk+1 1 Âk+1 2 )] +αre i D y T [ˆn k 21 (Âk+1 1 Âk+1 2 )] { +αre ˆn k 21 [ D x (Âk+1 1 +Âk+1 k ˆV s )]

11 L. BARLETTI, F. MÉHATS, C. NEGULESCU, AND S. POSSANNER 1357 { αre iˆn k 21 [ D y (Âk+1 1 +Âk+1 k ˆV s )] = 0, (4.3) γ 2 k+1 dir ˆV s = ˆn 1 (Âk+1 1,Âk+1 2 )+ ˆn 2 (Âk+1 1,Âk+1 2 ), (4.4) H(Âk+1 1,Âk+1 2 ) ( ) ˆψ1,k+1 = λ k+1 ˆψ 2,k+1 ( ) ˆψ1,k+1, (4.5) ˆψ 2,k+1 ˆn 1 (Âk+1 1,Âk+1 2 ) = ˆn 2 (Âk+1 1,Âk+1 2 ) = e λk+1 e λk+1 ˆψ1,k+1 1,k+1 ˆψ, (4.6) ˆψ2,k+1 2,k+1 ˆψ, (4.7) ˆn k 21 = e λk ˆψ2,k 1,k ˆψ, (4.8) Ĵ x,k 21 = iε 2 Ĵ y,k 21 = iε 2 e λk e λk [ ] 2,k 1,k 2,k 1,k D x ( ˆψ ) ˆψ ˆψ D x ( ˆψ ), (4.9) [ ] 2,k 1,k 2,k 1,k D y ( ˆψ ) ˆψ ˆψ D y ( ˆψ ). (4.10) Here, the operator symboizes the component by component mutipication of two vectors in C P, and the Hamitonian H(Âk+1 ) is given by H(Âk+1 1,Âk+1 2 ) = ε 2 2 dir +dg( ˆV ext,1 +Âk+1 1 ) ε 2 α(d x id y ) ε 2 α(d x +id y ) ε2 2 dir +dg( ˆV ext,2 +Âk+1 2 ), (4.11) where dg( ˆf) stands for a diagona P P matrix whose diagona eements are the components f p of ˆf. The scheme (4.2)-(4.10) is consistent with the continuous mode (2.8)- (2.14). It is of first order in time and of second order in space. Due to its rather impicit nature, the scheme (4.2)-(4.10) shows better stabiity properties than the forward Euer ˆV k+1 s scheme (c.f. numerica tests, Section 5). The soution (Âk+1 1,Âk+1 2, ) of the system (4.2)-(4.10) is the minimizer of the foowing discrete functiona F(Â1,Â2, ˆV s ) : R 3P R: F(Â1,Â2, ˆV s ) :=Ĝ(Â1,Â2)+ F 1 (Â1,Â2, ˆV s ) + F 2 (Â1,Â2, ˆV s )+ F 3 (Â1,Â2, ˆV s )+ F 4 (Â1,Â2), (4.12) where Ĝ(Â1,Â2) := 2P =1 e λ (Â1,Â2), (4.13)

12 1358 NUMERICAL STUDY OF A SPIN QDD MODEL F 1 (Â1,Â2, ˆV s ) := t [ (ˆn k 1 D x + (Â1 4 ˆV s ),D x + (Â1 ˆV s )) P +(ˆn k 1 Dx (Â1 ˆV s ),Dx (Â1 ˆV s )) P +(ˆn k 1 D y + (Â1 ˆV s ),D y + (Â1 ˆV s )) P +(ˆn k 1 D y (Â1 ˆV s ),D y (Â1 ˆV s )) P +(ˆn k 2 D + x (Â2 ˆV s ),D + x (Â2 ˆV s )) P +(ˆn k 2 D x (Â2 ˆV s ),D x (Â2 ˆV s )) P +(ˆn k 2 D + y (Â2 ˆV s ),D + y (Â2 ˆV s )) P +(ˆn k 2 D y (Â2 ˆV s ),D y (Â2 ˆV s )) P ], (4.14) F 2 (Â1,Â2, ˆV s ) :=(ˆn k 1,Â1 ˆV s ) P +(ˆn k 2,Â2 ˆV s ) P + [(D γ2 x b 2 ˆV s,dx b ˆV s ) P +(Dy b ˆV s,dy b ˆV ] s ) P + y N Vs,iM 2 + x M Vs,Nj 2 (4.15) x y F 3 (Â1,Â2, ˆV [( s ) := α tre i i=1 j=1 ˆn k 21 (Â1 Â2), D x (Â1 +Â2 2 ˆV ) s ) ) ( ˆn k 21 (Â1 Â2), D y (Â1 +Â2 2 ˆV s ) P P ] (4.16) F 4 (Â1,Â2) := α t [( ) ] ε Im (Â1 Â2) (Â1 Â2),Ĵ x 21,k iĵ y 21,k, (4.17) P and the further discretization matrices D b x and D b y are aso defined in Appendix D. Using the reation ( ˆV s, dir ˆVs ) P =(Dx b ˆV s,dx b ˆV s ) P +(Dy b ˆV s,dy b ˆV s ) P + y N Vs,iM 2 + x M V 2 x y s,nj, i=1 j=1 (4.18) it can be readiy verified that a soution (Âk+1 1,Âk+1 2, k+1 ˆV s ) of (4.2)-(4.10) satisfies d F(Âk+1 1,Âk+1 k+1 2, ˆV s )(δâ,δ ˆV s ) = 0 (δâ1,δâ2,δ ˆV s ) R 3P A second fuy discrete system (scheme 2). We chose the foowing space discretization of the forward Euer scheme (3.13)-(3.18): ˆn k+1 1 ˆn k 1 2α t ε (Âk 1 Âk 2) Im(Ĵ x 21,k iĵ y 21,k ) 1 2 (D+ x ) T [ˆn k 1 D + x (Âk 1 ˆV k s )] 1 2 (D x ) T [ˆn k 1 D x (Âk 1 ˆV k s )] 1 2 (D+ y ) T [ˆn k 1 D + y (Âk 1 ˆV k s )] 1 2 (D y ) T [ˆn k 1 D y (Âk 1 ˆV k s )] { αre{ DT x [ˆn k 21 (Âk 1 Âk 2)] +αre i D y T [ˆn k 21 (Âk 1 Âk 2)] { αre ˆn k 21 [ D x (Âk 1 +Âk 2 2 ˆV { s k )] +αre iˆn k 21 [ D y (Âk 1 +Âk 2 2 ˆV s k )]

13 L. BARLETTI, F. MÉHATS, C. NEGULESCU, AND S. POSSANNER 1359 =0, (4.19) ˆn k+1 2 ˆn k 2 + 2α t ε (Âk 1 Âk 2) Im(Ĵ x 21,k iĵ y 21,k ) 1 2 (D+ x ) T [ˆn k 2 D + x (Âk 2 ˆV k s )] 1 2 (D x ) T [ˆn k 2 D x (Âk 2 ˆV k s )] 1 2 (D+ y ) T [ˆn k 2 D + y (Âk 2 ˆV k s )] 1 2 (D y ) T [ˆn k 2 D y (Âk 2 ˆV k s )] { αre{ DT x [ˆn k 21 (Âk 1 Âk 2)] +αre i D y T [ˆn k 21 (Âk 1 Âk 2)] { +αre ˆn k 21 [ D x (Âk 1 +Âk 2 2 ˆV { s k )] αre iˆn k 21 [ D y (Âk 1 +Âk 2 2 ˆV s k )] =0, (4.20) ˆn k 1 = H(Âk 1,Âk 2) e λk γ 2 dir ˆV k s = ˆn k 1 + ˆn k 2, (4.21) ( ) ˆψ1,k = λ k ˆψ 2,k ˆψ1,k 1,k ˆψ, ˆn k 2 = ( ) ˆψ1,k, (4.22) ˆψ 2,k e λk ˆψ2,k 2,k ˆψ, (4.23) Ĵ x,k 21 = iε 2 Ĵ y,k 21 = iε 2 ˆn k 21 = e λk ˆψ2,k 1,k ˆψ, (4.24) [ ] 2,k 1,k 2,k 1,k D x ( ˆψ ) ˆψ ˆψ D x ( ˆψ ), (4.25) e λk e λk [ ] 2,k 1,k 2,k 1,k D y ( ˆψ ) ˆψ ˆψ D y ( ˆψ ). (4.26) Here, the Hamitonian H is the same discrete Hamitonian (4.11) as in the first fuy discrete scheme. Ceary, the scheme (4.19)-(4.26) is consistent with the continuous mode (2.8)-(2.14). It is of first order in time and of second order in space. A drawback of the expicit nature of the forward Euer scheme (3.13)-(3.18) is that its fu discretization is ony conditionay stabe; i.e. the space-time grid must be chosen in such a way that a CFL condition is fufied. The soution of this scheme requires the inversion of the non-oca reation (4.22)- (4.23) at each time step. For this, et us define the discrete version Ĝn : R 2P R of (3.19), Ĝ n (Â1,Â2) := Ĝ(Â1,Â2)+(ˆn k 1,Â1) P +(ˆn k 2,Â2) P. (4.27) It can be easiy verified that the first derivative of this functiona is given by ( ) dĝn(â1,â2)(δâ1,δâ2) = + ( e λk e λk ˆψ1,k ˆψ2,k 1,k ˆψ + ˆn k 1,δÂ1 2,k ˆψ + ˆn k 2,δÂ2 P ) P, (4.28)

14 1360 NUMERICAL STUDY OF A SPIN QDD MODEL whose zeros are hence the soutions of (4.22)-(4.23). Remark 4.1. The equations (3.13)-(3.14) contain a term of conservative form (n j A j ). Therefore, appropriate discretizations for conservation aws [11], such as Lax Friedrichs, shoud be used to ensure numerica stabiity. Nevertheess, we empoyed a forward Euer scheme with centra difference in space which is known to be unconditionay unstabe for (inear) hyperboic equations. In the numerics Section 5, the expicit scheme is found to be stabe for sma vaues of the semicassica parameter, i.e. ε 0.5. An expanation for the observed stabiity can be given by regarding the Lagrange mutipiers A ε j = Aε j (n 1,n 2 ) in the semi-cassica imit ε 0. As described in [2], the correct semi cassica expansion reads A ε j(n 1,n 2 ) = ogn j + ε2 n j +O(ε 2 α 2 ). (4.29) 6 nj Therefore, in the imit ε 0, the conservative term reads (n ε j A ε j) ε 0 (n j ogn j ). (4.30) Hence, for sma ε, equations (3.13)-(3.14) resembe a heat equation or a drift-diffusion equation, respectivey, where the diffusive term is written in the non-standard form (4.30). In this case, a forward Euer scheme with centra finite difference space derivatives is stabe with respect to a CFL condition of the form t d x 2 for some constant d Initiaization of scheme 1. As was briefy mentioned in Section 2, a natura way to initiaize the system (4.2)-(4.10) woud be to start from given initia chemica potentias Â0 1 and Â0 2, compute the corresponding spin and current densities, and subsequenty begin the iteration. However, from an experimenta point of view it is more appeaing to start from the initia spin densities ˆn 0 1 and ˆn 0 2. The probem in the atter approach is the ack of information about the initia spin-mixing quantities ˆn 0 21, Ĵ x,0 y,0 21, and Ĵ21 which are not directy reated to the spin densities. At t = t0 it is thus necessary to do a haf step of scheme 2, which means minimizing the functiona (4.27) in order to obtain the chemica potentias corresponding to the initia spin densities ˆn 0 1 and ˆn 0 2. One can then proceed according to scheme Numerica resuts This part deas with the numerica study of the two fuy discrete schemes introduced in the previous section. In the spin-ess case, it is we known that the steady-state soutions ( t n = 0) of quantum drift-diffusion modes are soutions of the corresponding stationary Schrödinger Poisson probem [15, 16]. We sha check whether this convergence in the ong-time imit is achieved by the numerica schemes deveoped in this work for the spin-dependent case. Tests are performed for different vaues of the semicassica parameters ε, the Rashba couping parameter α and the discretization parameters x and t. Moreover, the performance of the semi-impicit scheme 1 (c.f. Section 4.1) wi be compared to the performance of the expicit scheme 2 (c.f. Section 4.2). In what foows, the spin-dependent externa potentias are chosen to be V ext,1 = 0.0, V ext,2 = sin(2πx)sin(2πy). (5.1) Using these externa potentias, steady-state soutions were obtained by soving the stationary Schrödinger Poisson (SP) system (4.4)-(4.7). In the discrete Hamitonian

15 L. BARLETTI, F. MÉHATS, C. NEGULESCU, AND S. POSSANNER 1361 (4.11), we set Âk+1 1 = Âk+1 2 = ˆV s k where k now denotes the index of the SP-iteration. We use a space grid with N = M = 41 points in each direction and set γ = 1 (throughout this section). During SP-iteration, the tota mass density was renormaized to 1.0 after each soution of the eigenvaue probem. We set ˆV s 0 = 0.0 and iterated unti the change in the spin densities was ess than 10 6 from one iteration to the next. Obtained soutions are denoted with an index SP and are depicted in Figure 5.1 (for α = 1 and different vaues ε) and in Figure 5.2 (for ε = 0.2 and different vaues α). In view of the the externa potentias given in (5.1), it is reasonabe to obtain maxima of the tota mass density n SP tot = n SP 1 +n SP 2 at the minima of V ext,2, namey (x,y) = (1/4,3/4) and (x,y) = (3/4,1/4). Ceary, the spin poarization n SP po = nsp 1 n SP 2 has minima at those points. In Figure 5.1, we remark that the arger the semicassica parameter ε becomes the ess apparent is the infuence of the externa potentias on the steady state. This is expected since, in the quantum regime, eectrons have the abiity to tunne potentia barriers which therefore tend to have a esser impact on the eectron distribution. Additionay, from Figure 5.2, we earn that increased vaues of the Rashba couping parameter α ead to esser spin poarization in the steady state. This is physicay reasonabe because Rashba spin-orbit couping is a source of spin depoarization [21]. On the other hand, the tota mass density is hardy affected by a change of α. Let us present a few computationa detais concerning the numerics. The deveoped agorithms were impemented in the Fortran 90 anguage. Eigenvaue probems were soved using the routine zheev.f90 from the Lapack ibrary. The soution of scheme 1, equations (4.2)-(4.10), was achieved by minimizing the discrete functiona (4.12) at each time step t k, k > 0. At t 0, the system was initiaized as detaied in Subsection 4.3. Each minimization probem was soved by a conjugate gradient method in the parameter space R 3P (or R 2P for scheme 2, respectivey). We denote vectors in the parameter space by capita etters, i.e. X = (Â1,Â2, ˆV s ), X R 3P in scheme 1 and X = (Â1,Â2), X R 2P in scheme 2. In the parameter space, the gradient is denoted by X and the usua eucidean scaar product is denoted by. In order to find the ine minimum of X F Yn, where Y n denotes the search direction ( Y n = 1) during the n-th step of the conjugate gradient scheme, a Newton method was empoyed. The derivative of X F Yn in the direction Y n was computed numericay with a forward discretization and the sma step size ε NT = 10 3, ( X F(X) Yn ) X F(X +ε NT Y n ) Y n X F(X) Yn ε NT. (5.2) The same method was appied to the functiona Ĝn in scheme 2. The Newton method considered converged when X F(X) Yn < We estabished two convergence criteria for the conjugate gradient method. In scheme 1, the functiona (4.12) was considered optimized when the maxima change in the vector (Â1,Â2, ˆV s ) was ess than 10 5 from one conjugate gradient step to the next. On the other hand, in scheme 2 and during initiaization, the functiona (4.27) was considered optimized when X Ĝ n < (5.3) In a tests performed, the initia spin densities were two Gaussians centered at (x,y) = (0.5,0.5), n 0 1(x,y) = n 0 2(x,y) = 1 ) ( 0.12π exp (x 0.5)2 (y 0.5)2. (5.4)

16 1362 NUMERICAL STUDY OF A SPIN QDD MODEL Fig Soutions of the stationary Schrödinger Poisson system (4.4)-(4.7) with externa potentias (5.1) and A 1 = A 2 = V s. The number of mesh points is N = M = 41. Parameters are α = 1.0 and ε ranging from 0.1 (ine 1) to 1.0 (ine 4). Additionay, n SP 1 +n SP 2 is the tota mass density, n SP 1 n SP 2 denotes the spin poarization, and Vs SP is the sef-consistent eectric potentia.

17 L. BARLETTI, F. MÉHATS, C. NEGULESCU, AND S. POSSANNER 1363 Fig Soutions of the stationary Schrödinger Poisson system (4.4)-(4.7) with externa potentias (5.1) and A 1 = A 2 = V s. The number of mesh points is N = M = 41. Parameters are ε = 0.2 and α ranging from 1.0 (ine 1) to 20.0 (ine 4). Additionay, n SP 1 +n SP 2 is the tota mass density, n SP 1 n SP 2 denotes the spin poarization, and Vs SP is the sef-consistent eectric potentia.

18 1364 NUMERICAL STUDY OF A SPIN QDD MODEL The initia data for n 1 and n 2 were discretized according to the conventions at the beginning of Section 4. The initia tota mass of the system was 1.0. We reca that, in a simuations, the externay appied potentias are given by (5.1). (a) Numerica convergence (in L 2 -norm) for α = 1.0 (scheme 1). (b) Numerica convergence (in L 2 -norm) for ε = 0.2 (scheme 1). Fig Numerica convergence (in L 2 -norm) over time of the initia state (5.4) to the steady states SP (soutions of the stationary Schrödinger Poisson probem depicted in Figures 5.1 and 5.2) computed with the semi-impicit scheme 1. Simuation resuts are shown in (a) for different vaues of ε (α = 1.0) and in (b) for different vaues of α (ε = 0.2). Numerica parameters were x = y = 0.05 and t = Long-time convergence towards steady state. We sha test the reaxation of the initia data (5.4) towards the steady-state soutions SP shown in Figure 5.1 and Figure 5.2, respectivey, for different vaues of the parameters ε and α. Simuations are performed with the semi-impicit scheme 1, equations (4.2)-(4.10). We choose a time step t = 10 2 and simuate unti the fina time t f = 0.2, i.e. 20 time steps. The number of grid points in each direction is N = M = 21, thus x = y = The expicit scheme 2, aong with different choices for the time and the space discretization, wi be tested in the next subsection. Figure 5.3 demonstrates the ong-time convergence of the Gaussian initia state towards the stationary Schrödinger Poisson states for various choices of the semicassica parameter ε and the Rashba couping parameter α. Convergence is with respect to the L 2 -norm. The three coumns of sub-figures show the L 2 -errors over time of the tota mass density n tot n SP tot, of the spin poarization n po n SP po, and of the sefconsistent potentia V s Vs SP, respectivey. As can be seen in subfigure (a), reaxation is faster and the reached accuracy (the pateau ) is more favorabe for arger vaues of ε (quantum regime). By contrast, from subfigure (b) one obtains that for ε = 0.2 the rate of convergence is hardy dependent on the Rashba couping parameter α. Moreover, the

19 L. BARLETTI, F. MÉHATS, C. NEGULESCU, AND S. POSSANNER 1365 reached accuracy (pateau) gets worse as α increases. In order to highight the obtained ong-time convergence, we show the evoution of the tota mass density n tot = n 1 +n 2, the spin poarization n po = n 1 n 2, and the sef-consistent potentia V s for parameters ε = 0.1 and α = 1.0 in figures 5.4 and 5.5, respectivey Parameter studies for the schemes 1 and 2. It is the aim of this subsection to compare the performances of the semi-impicit scheme 1, equations (4.2)-(4.10), and the expicit scheme 2, equations (4.19)-(4.26), by means of ong-time simuations unti the fina time t f = 0.2. Various vaues of the parameters ε and α wi be tested. Moreover, different space discretizations sha be appied in order to check for numerica convergence as x 0 1. As in the previous subsection, the externa potentias are given by (5.1) and we use the initia spin densities (5.4). scheme ε x t #CG (init.) avg.#cg t CP U semi-impicit semi-impicit semi-impicit semi-impicit expicit expicit expicit expicit semi-impicit semi-impicit semi-impicit semi-impicit expicit expicit expicit expicit semi-impicit semi-impicit semi-impicit semi-impicit Tabe 5.1. Comparison of the performance of the semi-impicit scheme 1, equations (4.2)-(4.10), with the expicit scheme 2, equations (4.19)-(4.26). The computationa time t CP U has been normaized to a run with the parameter set in ine four of this tabe. Fina simuation time was t f = 0.2. Moreover, α = 1.0 and x = y in a simuations. Additionay, #CG (init.) stands for the number of conjugate gradient steps during the initiaization and avg.#cg denotes the average number of conjugate gradient steps in one time step. Parameters shown here were used to obtain the resuts depicted in Figure 5.6. For α = 1.0 and various choices of the parameters ε and x, a comparison regarding the computationa cost of the schemes 1 and 2 is given in Tabe 5.1. The CPU-time t CP U 1 We remark that, due to the arge number of eigenvaue probems to be soved at each time step, mesh refinement beyond x = 0.05 was not feasibe. For exampe, for N = M = 41, the soution of one eigenvaue probem takes about 1500 seconds which is required approximatey times to arrive at t f = 0.2.

20 1366 NUMERICAL STUDY OF A SPIN QDD MODEL Fig Evoution of the tota mass density of the initia state (5.4) towards the steady state SP computed from the stationary Schrödinger Poisson system depicted in Figure 5.1. Parameters were ε = 0.1, α = 1.0, x = y = 0.05, and t = The simuation was performed with the semi-impicit scheme 1.

21 L. BARLETTI, F. MÉHATS, C. NEGULESCU, AND S. POSSANNER 1367 (a) Evoution of the spin poarization for ε = 0.1. (b) Evoution of the sef-consistent eectric potentia for ε = 0.1. Fig Evoution of the initia state (5.4) towards the steady state SP computed from the stationary Schrödinger Poisson system depicted in Figure 5.1. Parameters were α = 1.0, x = y = 0.05, and t = The simuation was performed with the semi-impicit scheme 1.

22 1368 NUMERICAL STUDY OF A SPIN QDD MODEL has been normaized to a run with the semi-impicit scheme with parameters ε = 1.0, x = 0.1, and t = A few remarks concerning Tabe 5.1: 1. The expicit scheme is subjected to a stabiity condition with a critica time step t c for stabiity that depends on the grid size x as we as on the semicassica parameter ε. For a vaues of ε and x studied, it can be seen in Tabe 5.1 that the time step in the expicit scheme had to be chosen consideraby smaer than in the semi-impicit scheme in order to have stabiity. Moreover, we find that the expicit scheme is unstabe for arge vaues of ε, c.f. Remark 4.1. This can be seen from the fact that as ε passes from 0.1 to 1.0, for x = 0.07, the time step has to be increased by two orders of magnitude to have stabiity, a typica feature of unstabe schemes. Indeed, one observes instabiities when decreasing x further (and tightening the convergence criterion (5.3); see point 2. beow). 2. For ε 0.5, the time step for the expicit scheme had to be chosen equa to or smaer than the convergence criterion (5.3), thus ( t 10 5 ). This manifests itsef in the sma average of conjugate gradient steps per time step, avg.#cg, which means that the Functiona Ĝn might not have been propery minimized at each time step. Indeed, this expected ack of accuracy is seen in the figures 5.6(c) and 5.6(d). Improved accuracy can be achieved by tightening the convergence criterion which in turn eads to considerabe increase in computationa cost. As mentioned above, for ε 1 and x 0, the expicit scheme is unstabe. 3. Regarding CPU-time, the semi-impicit scheme is ceary favorabe compared to the expicit scheme for ong-time simuations for a parameter vaues considered. 4. The computationa cost depends strongy on x for both schemes, see footnote 1. For the parameter sets dispayed in Tabe 5.1, the numerica convergence in L 2 -norm over time of the initia state (5.4) towards the stationary Schrödinger Poisson states shown in Figure 5.1 is depicted in Figure 5.6. One observes that the expicit scheme, with a time step much smaer than the semi-impicit scheme, converges faster. The achieved accuracy in the steady state is amost identica for the two schemes for ε 0.2 whie it is much worse for the expicit scheme for ε = 1.0 as discussed in point 2 above. Regarding the semi-impicit scheme, it is obtained that mesh refinement eads to a better accuracy in the steady-state for a ε considered. Hence, one expects convergence of numerica soutions as x 0, the order of convergence being dependent on the semicassica parameter ε. Finay, in Figure 5.7, we show numerica convergence towards steady-sate in the case ε = 0.2 for the two schemes with different vaues of x and the Rashba couping parameter α. Other parameters were chosen as in Tabe 5.1 for ε = Concusion In this work we carried out a numerica investigation of the quantum diffusive spin mode introduced in [2] and summarized in equations (2.8)-(2.14). We formay proved (under suitabe assumptions) the existence and uniqueness of a soution of two time-discrete versions of this mode on the basis of a functiona argument. Furthermore, finite difference approximations of space derivatives resuted in two fuy discrete schemes which were appied to simuate the evoution of a Rashba eectron gas confined in a bounded domain and subjected to spin-dependent externa potentias. The first scheme is semi-impicit and advances in time the spin chemica potentias, whereas the second scheme is Euer expicit and advances in time the spin-up and spin-down densi-

23 L. BARLETTI, F. MÉHATS, C. NEGULESCU, AND S. POSSANNER 1369 (a) Numerica convergence (in L 2 -norm) for ε = 0.1. (b) Numerica convergence (in L 2 -norm) for ε = 0.2. (c) Numerica convergence (in L 2 -norm) for ε = 0.5. (d) Numerica convergence (in L 2 -norm) for ε = 1.0. Fig Numerica convergence (in L 2 -norm) over time of the initia state (5.4) to the steady states SP (soutions of the stationary Schrödinger Poisson probem depicted in Figure 5.1) for α = 1.0 and different vaues of ε. Resuts are shown for the semi-impicit scheme (S1) as we as for the expicit scheme (S2) for diffferent space discretizations. The numerica parameters concerning this study can be found in Tabe 5.1.

24 1370 NUMERICAL STUDY OF A SPIN QDD MODEL (a) Numerica convergence (in L 2 -norm) for α = 5.0. (b) Numerica convergence (in L 2 -norm) for α = (c) Numerica convergence (in L 2 -norm) for α = Fig Numerica convergence (in L 2 -norm) over time of the initia state (5.4) to the steady states SP (soutions of the stationary Schrödinger Poisson probem depicted in Figure 5.2) for ε = 0.2 and different vaues of α. Resuts are shown for the semi-impicit scheme (S1) as we as for the expicit scheme (S2) for diffferent space discretizations. The numerica parameters concerning this study were the same as those used to obtain Figure 5.4. ties, respectivey. The numerica convergence in the ong-time imit of a Gaussian initia condition towards the soution of the corresponding stationary Schrödinger Poisson probem was demonstrated for different vaues of the parameters ε (semicassica parameter), α (Rashba couping parameter), x (grid spacing), and t (time step). In contrast to the first scheme, the expicit scheme is subjected to a stabiity condition which makes it ess appeaing for ong-time simuations. Our resuts show that the quantum drift-diffusion mode considered here can be appied for the numerica study of spin-poarized effects due to Rashba spin-orbit couping and, thus, appears to benefit

25 L. BARLETTI, F. MÉHATS, C. NEGULESCU, AND S. POSSANNER 1371 the design of nove spintronics appications. Appendix A. Perturbed eigenvaue probem. This section is devoted to the computation of the derivatives dλ (A)(δA) and dψ (A)(δA) of the eigenvaues and eigenfunctions, respectivey, of the Hamitonian (2.4) when a sma perturbation δa of the chemica potentia A is appied. Let us define and start from δh = ( δa1 0 0 δa 2 ) (A.1) (H +δh)(ψ +dψ ) = (λ +dλ )(ψ +dψ ) (A.2) where H denotes the Hamitonian (2.4). Using Hψ = λ ψ one obtains, up to first order in the variations, Hdψ +δhψ = λ dψ +dλ ψ. (A.3) Taking now the scaar product with ψ k and using the orthonormaitiy of the eigenfunctions, (ψ k,ψ ) L 2 = (ψkψ 1 1 +ψ2 kψ 2)dx = δ k, (A.4) one obtains Since H is hermitian, we have and (A.5) can be written as (ψ k,hdψ ) L 2 +(ψ k,δhψ ) L 2 = λ (ψ k,dψ ) L 2 +dλ δ k. (A.5) For = k, we obtain dλ (A)(δA) = (ψ,δhψ ) L 2 = (ψ k,hdψ ) L 2 = (Hψ k,dψ ) L 2 = λ k (ψ k,dψ ) L 2, (A.6) (ψ k,δhψ ) L 2 = (λ λ k )(ψ k,dψ ) L 2 +dλ δ k. (A.7) ( ψ 1 (A) 2 δa 1 + ψ 2 (A) 2 δa 2 ) dx, (A.8) and for k, assuming that the spectrum of H is non-degenerate, i.e. λ λ k for k, one obtains (ψ k,dψ ) L 2 = (ψ k,δhψ ) L 2 λ λ k. (A.9) Since (A.9) is the projection of dψ onto the k-th basis vector of the eigenbasis of H we may write dψ (A)(δA) = k ψ k λ λ k (ψ k,δhψ ) L 2