Spin filtering effect generated by the inter-subband spin-orbit coupling in the bilayer nanowire with the quantum point contact

The spin filtering effect in the bilayer nanowire with quantum point contact is investigated theoretically. We demonstrate the new mechanism of the spin filtering based on the lateral inter-subband spin-orbit coupling, which for the bilayer nanowires has been reported to be strong. The proposed spin filtering effect is explained as the joint effect of the Landau-Zener intersubband transitions caused by the hybridization of states with opposite spin (due to the lateral Rashba SO interaction) and the confinement of carriers in the quantum point contact region.


Effective 2D Hamitonian in the two-subband model with the intra-and inter-subband SO interaction
The 3D Hamiltonian of the conduction band electron in the nanowire has the form where m * is the conduction band effective mass and the potential energy U(x, y, z) = U QPC (x, y) + |e|F y y +U con f (z) (2) is the sum of the electron energy in the QPC (U QPC ), in the lateral electric field (second term) and the confinement potential energy along the grown direction z, which includes the band potential energy profile and the Hartree energy that describes the electron-electron interaction. H SO in (1) is the Hamiltonian of the Rashba SO interaction, which is given by where σ σ σ = (σ x , σ y , σ z ) is the Pauli matrices vector, β is the Rashba spin-orbit coupling constant and k is the wave vector. Now, we derive the effective 2D Hamiltonian in the two-subband model starting from its 3D form given by Eq. (1). For this purpose we define the four element basis {|φ 1 ⟩| ↑⟩, |φ 1 ⟩| ↓⟩, |φ 2 ⟩| ↑⟩, |φ 2 ⟩| ↓⟩} which consists of the spin-degenerate ground (φ 1 ) and first excited (φ 2 ) eigenstates of the Hamiltonian The matrix elements, including spin, are given by , , , If we define the intra-subband SO coupling constants β 11 = ⟨φ 1 |dU con f (z)/dz|φ 1 ⟩ and β 22 = ⟨φ 2 |dU con f (z)/dz|φ 2 ⟩, the intersubband SO coupling constant β 12 = ⟨φ 1 |dU con f (z)/dz|φ 2 ⟩, the inter-subband coupling constant δ = ⟨φ 1 |d/dz|φ 2 ⟩ , and we 1/3 neglect the electric field of the QPC, as negligibly small compared with the lateral electric field (∂U QPC /∂ x, ∂U QPC /∂ y ≪ F y ) we obtain where Assuming β 11 = β 22 = 0 we obtain the effective Hamiltonian used in the present paper When deriving Hamiltonian (7) we have made the four assumptions, which require a more detailed justification.
(i) We have neglected the electric field of the QPC in Hamiltonian H SO . In order to justify this assumption, we have performed additional calculations taking into account the electric field of the QPC i.e., ∂U QPC /∂ x, ∂U QPC /∂ y and we have not observed any influence of this field on the results. This can be understood if we realize that the electric field of the QPC is a few orders of magnitude smaller than the lateral electric field F y .
(ii) We have assumed the presence of the lateral electric field F y . This assumption is based on the recent paper, in which Gvozdic and Ekenberg 1 pointed out that in the modulation-doped wide quantum wells, used for the bilayer nanowires fabrication, the large intrinsic F y exists.
(iii) We have assumed that the intra-subband SO coupling constants β 11 = β 22 = 0. This assumption is based on the results of our recent paper 2 , in which we have studied the realistic model of the nanowire with the gated Al 0.48 In 0.52 As/Ga 0.47 In 0.53 As double quantum well. For this nanowire we have determined the SO coupling constants by the self-consistent Schrödinger-Poisson approach. The exemplary results are presented in Fig. 1. Figure 1 shows that the inter-subband SO coupling constant β 12 is an even function of the gate voltage and exhibits the "resonant behavior" at V g = 0 corresponding to the symmetric geometry of the heterostructure. Simultaneously, for V g = 0, the intra-subband SO coupling constants β 11 and β 22 change the sign and for V g = 0 are equal to β 11 = β 22 = 0. The similar behavior has been recently reported by Calsaverini et. al. for InSb/Al 0.12 In 0.88 Sb double quantum well 3 . Therefore, the assumption β 11 = β 22 = 0 corresponds to the symmetric geometry of the heterostructure, which is convenient for the studied effect because then the inter-subband SO interaction, which generates the spin filtering effect, is maximal.

2/3
(iv) We have neglected the Dresselhaus SO interaction. In our paper, we consider the bilayer nanowire, which can be experimentally realized in the system built from double quantum wells with the central barrier or a wide quantum well, in which the Coulomb repulsion gives raise to the soft barrier in the middle of the quantum well. For such a wide quantum wells needed to fabricate the bilayer nanowires, the strength of the Dresselhaus SO interaction is a few order of magnitudes weaker than the Rashba SO coupling. This results from the fact that the Dresselhaus coupling parameter γ D ∼ ⟨k 2 z ⟩, and for the infinite quantum well takes the form γ D ∼ 1/d 2 (d-is the quantum well width in the z direction). To give some concrete example, based of the formula 4 we have calculated the linear Dresselhaus SO coupling parameters for the realistic model of the bilayer nanowire with the gated Al 0.48 In 0.52 As/Ga 0.47 In 0.53 As double quantum well 2 (the same structure has been considered for the calculations of the Rashba parameters presented in Fig. 1). Results for γ 3D D = 0.0237 meVnm 35 are presented in Fig. 2. If we compare these results with the Rashba SO parameters presented in Fig. 1 we see that for the considered double quantum well, the Dresselhaus SO coupling constants are four orders of magnitudes smaller than the Rashba coupling constants. This means that the Dresselhaus SO interaction does not affect the spin filtering presented in the paper. In order to support this argument in more quantitative manner we performed the additional calculations with the inclusion of the Dresselhaus term and obtained that P(E F ) presented in Fig. 2(b) in the main paper did not change.