Spin relaxation of upstream electrons: beyond the drift diffusion model

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    a  r   X   i  v  :  c  o  n   d  -  m  a   t   /   0   5   1   1   0   2   7  v   1   [  c  o  n   d  -  m  a   t .  m  e  s  -   h  a   l   l   ]   1   N  o  v   2   0   0   5  Spin relaxation of “upstream” electrons: beyond the driftdiffusion model Sandipan Pramanik and Supriyo Bandyopadhyay  ∗ Department of Electrical and Computer Engineering,Virginia Commonwealth University, Richmond, VA 23284, USA Marc Cahay Department of Electrical and Computer Engineering and Computer Science,University of Cincinnati, Cincinnati, OH 45221, USA Abstract The classical drift diffusion (DD) model of spin transport treats spin relaxation via an empiricalparameter known as the “spin diffusion length”. According to this model, the ensemble averagedspin of electrons drifting and diffusing in a solid decays exponentially with distance due to spindephasing interactions. The characteristic length scale associated with this decay is the spindiffusion length. The DD model also predicts that this length is different for “upstream” electronstraveling in a decelerating electric field than for “downstream” electrons traveling in an acceleratingfield. However this picture ignores energy quantization in confined systems (e.g. quantum wires)and therefore fails to capture the non-trivial influence of subband structure on spin relaxation. Herewe highlight this influence by simulating upstream spin transport in a multi-subband quantum wire,in the presence of D’yakonov-Perel’ spin relaxation, using a semi-classical model that accounts forthe subband structure rigorously. We find that upstream spin transport has a complex dynamicsthat defies the simplistic definition of a “spin diffusion length”. In fact, spin does not decayexponentially or even monotonically with distance, and the drift diffusion picture fails to explainthe qualitative behavior, let alone predict quantitative features accurately. Unrelated to spintransport, we also find that upstream electrons undergo a “population inversion” as a consequenceof the energy dependence of the density of states in a quasi one-dimensional structure. PACS numbers: 72.25.Dc, 85.75.Hh, 73.21.Hb, 85.35.Ds ∗ Corresponding author. E-mail: sbandy@vcu.edu  I. INTRODUCTION Spin transport in semiconductor structures is a subject of much interest from the perspec-tive of both fundamental physics and device applications. A number of different formalismshave been used to study this problem, primary among which are a classical drift diffusionapproach [1, 2, 3], a kinetic theory approach [4], and a microscopic semiclassical approach[3, 5, 6, 7, 8, 9, 10, 11]. The central result of the drift diffusion approach is a differentialequation that describes the spatial and temporal evolution of carriers with a certain spinpolarization  n σ . Ref. [3] derived this equation for a number of special cases starting fromthe Wigner distribution function. In a coordinate system where the x-axis coincides withthe direction of electric field driving transport, this equation is of the form: ∂n σ ∂t  − D ∂  2 n σ ∂x 2  − A ∂n σ ∂x  + B n σ  = 0 (1)where D  =  D  0 00  D  00 0  D  ,  (2)(3) D  is the diffusion coefficient, and  A  and  B  are dyadics (9-component tensors) that dependon  D , the mobility  µ  and the spin orbit interaction strength in the material.Solutions of Equation (1), with appropriate boundary conditions, predict that the ensem-ble averaged spin  | S | ( x ) =    S  x  2   ( x ) +   S  y  2 ( x ) +   S  z  2 ( x ) should decay exponentiallywith  x  according to: | S | ( x ) =  | S | (0) e − x/L (4)where1 L  =  µE  2 D  +   µE  2 D  2 +  C  2 .  (5)Here  E   is the strength of the driving electric field and  C   is a parameter related to the spinorbit interaction strength.The quantity  L  is the characteristic length over which  | S |  decays to 1 /e  times its srcinalvalue. Therefore, it is defined as the “spin diffusion length”. Equation (5) clearly shows that  spin diffusion length depends on the  sign   of the electric field  E  . It is smaller for upstreamtransport (when  E   is positive) than for downstream transport (when  E   is negative).This difference assumes importance in the context of spin injection from a metallic fer-romagnet into a semiconducting paramagnet. Ref. [1] pointed out that the spin injectionefficiency across the interface between these materials depends on the difference between thequantities  L s /σ s  and  L m /σ m  where  L s  is the spin diffusion length in the semiconductor,  σ s is the conductivity of the semiconductor,  σ m  is the conductivity of the metallic ferromagnet,and  L m  is the spin diffusion length in the metallic ferromagnet. Generally,  σ m  >> σ s . How-ever, at sufficiently high retarding electric field,  L s  << L m , so that  L s /σ s  ≈  L m /σ m . Whenthis equality is established, the spin injection efficiency is maximized. Thus, ref. [1] claimedthat it is possible to circumvent the infamous “conductivity mismatch” problem [13], whichinhibits efficient spin injection across a metal-semiconductor interface, by applying a highretarding electric field in the semiconductor close to the interface. A tunnel barrier betweenthe ferromagnet and semiconductor [14] or a Schottky barrier [15, 16] at the interface doesessentially this and therefore improves spin injection.The result of ref. [1] depends on the validity of the drift diffusion model and Equation (4)which predicts an exponential decay of spin polarization in space. Without the exponentialdecay, one cannot even define a “spin diffusion length”  L . The question then is whetherone expects to see the exponential decay under all circumstances, particularly in quantumconfined structures such as quantum wires. The answer to this question is in the  negative  .Equation (1), and similar equations derived within the drift diffusion model, do not accountfor energy quantization in quantum confined systems and neglect the influence of subbandstructure on spin depolarization. This is a serious shortcoming since in a semiconductorquantum wire, the spin orbit interaction strength is  different   in different subbands. It isthis difference that results in D’yakonov-Perel’ (D-P) spin relaxation in quantum wires.Without this difference, the D-P relaxation will be completely absent in quantum wires andthe corresponding spin diffusion length will be always infinite [17]. The suband structure istherefore vital to spin relaxation.  II. SEMICLASSICAL MODEL OF SPIN RELAXATION In this paper, we have studied spin relaxation using a microscopic semiclassical modelthat is derived from the Liouville equation for the spin density matrix [18, 22]. Our modelhas been described in ref. [10] and wil not be repeated here. This model allows us to studyD’yakonov-Perel’ spin relaxation taking into account the detailed subband structure in thesystem being studied.In technologically important semiconductors, such as GaAs, spin relaxation is dominatedby the D’yakonov-Perel’ (D-P) mechanism [12]. This mechanism arises from the Dresselhaus[19] and Rashba [20] spin orbit interactions that act as momentum dependent effective mag-netic fields  B ( k ). An electron’s spin polarization vector  S  precesses about  B ( k ) accordingto the equation d S dt  =  Ω ( k )  × S  (6)where Ω( k ) is the angular frequency of spin precession and is related to  B ( k ) as  Ω ( k ) =( e/m ∗ ) B ( k ), where  m ∗ is the electron’s effective mass. If the direction of   B(k)  changesrandomly due to carrier scattering which changes  k , then ensemble averaging over the spinsof a large number of electrons will lead to a decay of the ensemble averaged spin in space andtime. This is the physics of the D-P relaxation in bulk and quantum wells. In a quantumwire, the direction of   k  never changes (it is always along the axis of the wire) in spite of scattering. Nevertheless, there can be D-P relaxation in a  multi-subband   quantum wire, aswe explain in the next paragraphs.We will consider a quantum wire of rectangular cross-section with its axis along the [100]crystallographic orientation (which we label the x-axis), and a symmetry breaking electricfield  E  y  is applied along the y-axis to induce the Rashba interaction (refer Fig. 1). Then,the components of the vector  Ω ( k ) due to the Dresselhaus and Rashba interactions are givenby Ω D ( k ) = 2 a 42    nπW  y  2 −  mπW  z  2  k x ˆ x ; ( W  z  > W  y ) Ω R ( k ) = 2 a 46    E  y k x ˆ z   (7)where  a 42  and  a 46  are material constants, ( m,n ) are the transverse subband indices,  k x  is thewavevector along the axis of the quantum wire, and  W  z ,W  y  are the transverse dimensions  of the quantum wire along the z- and y-directions. Therefore, B ( k ) =  m ∗ e  [ Ω D ( k ) + Ω R ( k )]= 2 m ∗ a 42 e    −  mπW  z  2 +  nπW  y  2  k x ˆ x  + 2 m ∗ a 46 e    E  y k x ˆ z   (8)Thus,  B (k)  lies in the x-z plane and subtends an angle  θ  with the wire axis (x-axis)given by θ  = arctan  a 46 E  y a 42  mπW  z +  nπW  y  nπW  y −  mπW  z   (9)Note from the above that in any given subband in a quantum wire, the  direction   of   B ( k )is fixed, irrespective of the magnitude of the wavevector  k x , since  θ  is independent of   k x . Asa result, there is no D-P relaxation in any given subband, even in the presence of scattering.However,  θ  is different in different subbands because the Dresselhaus interaction is differ-ent in different subbands. Consequently, as electrons transition between subbands becauseof inter-subband scattering, the angle  θ , and therefore the direction of the effective magneticfield  B ( k ), changes. This causes D-P relaxation in a  multi-subband   quantum wire. Sincespins precess about different axes in different subbands, ensemble averaging over electronsin all subbands results in a gradual decay of the net spin polarization. Thus, there is no D-Pspin relaxation in a quantum wire if a single subband is occupied, but it is present if multiplesubbands are occupied and inter-subband scattering occurs. This was shown rigorously inref. [17].The subband structure is therefore critical to D-P spin relaxation in a quantum wire. Infact, if a situation arises whereby all electrons transition to a single subband and remainthere, further spin relaxation due to the D-P mechanism will cease thereafter. In this case,spin no longer decays, let alone decay exponentially with distance. Hence, spin depolariza-tion (or spin relaxation) cannot be parameterized by a constant spin diffusion length.The rest of this paper is organized as follows. In the next section, we describe our modelsystem, followed by results and discussions in section IV. Finally, we conclude in section V. III. MODEL OF UPSTREAM SPIN TRANSPORT We consider a non centro-symmetric (e.g. GaAs) quantum wire with axis along [100]crystallographic direction. We choose a three dimensional Cartesian coordinate system with
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