Confinement of spin-orbit induced Dirac states in quantum point contacts

Tommy Li

Phys. Rev. B 92, 085302 – Published 10 August 2015

Abstract

The quantum transmission problem for a particle moving in a quantum point contact in the presence of a Rashba spin-orbit interaction and applied magnetic field is solved semiclassically. A strong Rashba interaction and parallel magnetic field form emergent Dirac states at the center of the constriction, leading to the appearance of resonances which carry spin current and become bound at high magnetic fields. These states can be controlled in situ by modulation of external electric and magnetic fields, and can be used to turn the channel into a spin pump which operates at zero bias. It is shown that this effect is currently experimentally accessible in $p$ -type quantum point contacts.

Received 2 February 2015

DOI:https://doi.org/10.1103/PhysRevB.92.085302

Authors & Affiliations

Tommy Li

School of Physics, University of New South Wales, Sydney 2052, Australia

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Issue

Vol. 92, Iss. 8 — 15 August 2015

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Images

Figure 1
Confinement of a Dirac fermion in a QPC, which occurs when the parallel magnetic field is below the critical value $β = \frac{1}{2} g μ_{B} B_{x} < β_{c}$ . (a) The 1D dispersion near the anticrossing, with the bold lines showing the region in which Dirac fermion behavior is expected to appear. The two spin branches $ε_{k}^{+}, ε_{k}^{-}$ are identified as the positive and negative energy Dirac states, respectively. Arrows indicate the direction of spin polarization. (b) The potential barrier $U (x)$ (lower curve), and the situation in momentum space (three upper figures). There are three regions which form the scattering wave function. In the asymptotic region $| x | > b$ , the kinetic energy lies in the upper (positive energy) branch. Inside the barrier, $| x | < a$ , the kinetic energy lies in the lower (negative energy) branch. Quantization of motion in the region bounded by vertical lines gives rise to a subbarrier resonance. The energy is indicated by the horizontal line. The tunneling regions are shown with dashed lines.
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Figure 2
Predicted number of quasibound states for parameters $η, ζ$ , given by Eq. (12).
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Figure 3
The QPC conductance in units of $\frac{e^{2}}{h}$ for a Gaussian barrier $U (x) = U_{0} e^{- \frac{x^{2}}{w^{2}}}$ with parameters $w = 50 nm, E_{F} = 0.6 meV, \frac{m α}{ℏ k_{F}} = 0.6$ , and values of magnetic field $g B_{x} = 3 T$ (solid, red), 2 T (dashed, blue), 1 T (dotted, green). (a) The conductance obtained from the analytical formula for the resonant transmission probability (10). (b) The conductance obtained by brute-force numerical integration of the Schrödinger equation corresponding to the Hamiltonian (3).
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Figure 4
Spin current and spin pumping via the bound state. (a) The current densities (arb. units) in the spin-up and spin-down components of the wave function at resonance, $J_{x +} = ψ_{+}^{*} J_{x} ψ_{+}$ (red, positive curve) and $J_{x -} = ψ_{-}^{*} J_{x} ψ_{-}$ (blue, negative curve), where $ψ = ψ_{+} {| + 〉}_{y} + ψ_{-} {| - 〉}_{y}$ . The barrier is shown by the dashed line. Since the state is localized, the total current everywhere is zero; however the state possesses a spin current $J_{x y} = \frac{J_{x +} - J_{x -}}{2}$ . (b) Upon adiabatically switching on the magnetic field, a particle is captured inside the barrier from the left lead, initially in a spin-down state. (c) When the magnetic field is switched off instantaneously, the resonance decays, either returning the particle to the left lead in a spin-up state, or transferring it to the right lead in a spin-down state. The mirror process occurs for a particle injected from the right lead. This process increases the number of spin-up carriers in the left lead and the number of spin-down carriers in the right lead; i.e., it pumps spin across the channel in the absence of dc current.
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Figure 5
The resonant spectrum calculated in the Hartree approximation with $N = 0$ (a) and $N = 1$ (b) particles captured in the channel. The barrier $U (x)$ and effective potential $U_{eff} (x)$ which account for the Hartree potential are shown in the background. The resonant energies are indicated by solid horizontal lines, and the chemical potential $μ$ is shown by dashed horizontal lines.
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