II. THE MODEL OF THE EDGE AND SPIN-ORBIT COUPLING 2 Let us consider a sample with 2DEG created within a GaAs-based quantum well. The electrical current in the plane of the 2DEG will be driven along the y direction. We will be interested in the physics close to one of the two edges of the sample along which the current flows. The coordinate in the direction perpendicular to the edge will be x , x > 0 corre- sponding to the region where the 2DEG is present (see Fig. 1). We will describe the electronic states for electrons in the 2DEG within the effective mass approximation, treat- ing the electrons as noninteracting quasiparticles. Qualita- tive changes in our results introduced within a self-consistent mean-field treatment will be studied afterwards. The length scale characterizing the electrons is given by their Fermi wavelength. It typically attains values 33 F = √ 2 / n$_{2}$$_{D}$ ∼ 2 . 5 a ∗ $_{B}$, where n$_{2}$$_{D}$ is the two-dimensional electronic density and a ∗ B = 9 . 79nm is the effective Bohr radius in GaAs. Since this is about two orders of magnitude larger than the inter- atomic distances, we can employ the effective mass theory and approximate the form of the confining potential with a simple functional form. In our work will assume that the confining edge potential, V ( x ) , is independent of the coordinate y , directed along the edge. Within our analytical derivations we model V ( x ) as an abrupt step of height V , V ( x ) = V ( x ) = V ( − x ) , where ( x ) is the unit step function. The actual value of the step is much larger than the Fermi energy of the 2DEG. This is frequently used to set the potential step to infinity. Howeve r, here it is essential that the step is finite as the whole spin-o rbit coupling is nonzero only in the region of the nonzero gradient of the confining potential. Since this value should be of the order of the work function of the electrons in the 2DEG, we fix this value to V = 2 . 7eV ∼ 230Ha ∗ . One of the results of our work is the demonstration that the current induced spin polarization is rather independent of this value so that we do not need to be concerned with its precise numerical value, as long as it is large ( >> E$_{F}$ ) but finite value. Once we establish the analytical result for this model of abrupt potential step, we will also consider more general forms: step with a linear slope on a distance d , V$_{d}$ ( x ) (used in Fig. 1), and a partially self-consistent, density-dependent model with a small triangular-shaped dip, V$_{n}$ ( x ) , close to the edge. In the absence of this dip, the decrease in the density to zero at the edge in the 2DEG results in a slight local deple- tion of electrons there. A small negative value of the depth of the dip, − V$_{0}$ , that is found self-consistently results in a charge neutral edge which is the physically expected situation. The essential part of our model is the spin-orbit (SO) cou- pling term in the electrons’ Hamiltonian. In general, several different types of SO interactions can be similar in magni- tude: the Rashba-Bytchkov, the Dresselhaus or the impurity- induced SO coupling. However, since they are all small per- turbations to the effective Hamiltonian, we can consider them separably and superpose their outcomes in the sense of first order perturbation theory in terms of SO parameter appearing in the SO interaction. In our paper we will consider a special where r q , ( k ) is the reflection amplitude of the eigenstate with energy E = ( 1 / 2 )( k $^{2}$+ q $^{2}$) . For large negative times only the incoming part (moving from the right in the Fig. 1) of this In our previous work$^{22}$, we have given estimates for the total induced spin per unit length of the edge using arguments based on wavepacket propagation. We considered a single- electron wavepacket state, q , k , $_{k}$( t ) , for a specific momentum in the y direction q , spin , with momentum k of uncertainty k , with a well defined quantum number for the projection of the spin in the z direction, $_{z}$, attaining values ± 1, and q is the y component of the electron’s momentum (Fig. 1). Hence, the spin-orbit term is a potential energy which, for states with pos- itive momentum in the + y direction (the current), is smaller (larger) in the spatial region of negative derivative of the edge potential, V ′ ( x ) < 0, for electrons with their spin down (up). Since close to x = 0 we have V ′ ( x ) = − V ( x ) < 0, we expect that close to the edge we will find majority of spin-down elec- trons. Of course, in equilibrium for zero total current density there will be equal number of electron with q > 0 and q < 0 so that in this special case zero total spin polarization will be observed. Within the framework of this model it is very easy to under- stand the origin of the current-induced spin polarization. The gradient of the potential energy is directed only in the x direc- tion, the momentum is confined only within the xy plane so that the only nonzero component of the spin operator comes from its z component due to the mixed product form in Eq. 2. Specifically, the SO interaction is simply a spin-dependentpo- tential where E characterizes the strength of the SO coupling, are Pauli matrices, ⃗ k is the operator of electron’s momen- tum and V ( x ) is the effective one-electron potential energy. Within our study the latter corresponds to various models of the edge potential: V ( x ) , V$_{d}$ ( x ) or V$_{n}$ ( x ) . The strength of the SO coupling in GaAs is known to be approximately E ≈ 5 . 3 ˚ A 2 = 5 . 53 × 10 − $^{4}$a ∗ B which is 10 6 times larger than the strength of the SO coupling in vacuum$^{26}$. case of the impurity-induced SO coupling for the special case when the “impurity” is the edge potential only. In general, t he SO contribution to the Hamiltonian takes the form 27 ˆ V SO = $_{E}$⃗ · ( ⃗ k × V ( x ) , (2) ˆ V SO = − E $_{z}$qV ′ ( x ) (3) q , k , $_{k}$( t ) = ∫ k + k / 2 k − k / 2 dk √$_{2}$ k ( e − $^{ikx}$+ r q , ( k ) e$^{ikx}$e iqy ) × e i ( k $^{2}$/ 2 + q $^{2}$/ 2 ) $^{t}$, (4) III. SPIN POLARIZATION AND THE SPIN-DEPENDENT PHASE-SHIFT