Oscillating spin-orbit interaction in two-dimensional superlattices: Sharp transmission resonances and time-dependent spin-polarized currents

Viktor Szaszkó-Bogár, F. M. Peeters, and Péter Földi

Phys. Rev. B 91, 235311 – Published 10 June 2015

Abstract

We consider ballistic transport through a lateral, two-dimensional superlattice with experimentally realizable, sinusoidally oscillating, Rashba-type spin-orbit interaction (SOI). The periodic structure of the rectangular lattice produces a spin-dependent miniband structure for static SOI. Using Floquet theory, transmission peaks are shown to appear in the mini-bandgaps as a consequence of the additional, time-dependent SOI. A detailed analysis shows that this effect is due to the generation of harmonics of the driving frequency, via which, e.g., resonances that cannot be excited in the case of static SOI become available. Additionally, the transmitted current shows space- and time-dependent partial spin polarization, in other words, polarization waves propagate through the superlattice.

Received 16 February 2015
Revised 19 May 2015

DOI:https://doi.org/10.1103/PhysRevB.91.235311

Authors & Affiliations

Viktor Szaszkó-Bogár^1,2,*, F. M. Peeters², and Péter Földi^1,3

¹Department of Theoretical Physics, University of Szeged, Tisza Lajos körút 84, H-6720 Szeged, Hungary
²Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium
³ELI-ALPS, ELI-HU Non-Profit Ltd., Dugonics tér 13, H-6720 Szeged, Hungary

^*vszaszko@physx.u-szeged.hu

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Vol. 91, Iss. 23 — 15 June 2015

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Images

Figure 1
Panel (a): Schematic view of a lateral, rectangular two-dimensional superlattice that can be realized in semiconductor heterostructures (e.g., InAlAs/InGaAs- or GaSb/AlSb-based systems). The gate electrode (on the top of the structure, yellow) can be used to induce time-dependent spin-orbit interaction. Panel (b): Our one-dimensional model. Quantum wires represented by gray lines are subjected to oscillating Rashba-type SOI. Black arrows show the input and output leads in which no spin-orbit coupling is present.
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Figure 2
Conductance (in units of $G_{0} = 2 e^{2} / h$ ) of a $7 \times 7$ array in the presence of oscillating and stationary Rashba-type spin-orbit interaction [see Eq. (2)]. Panel (a) focuses on an energy range that corresponds to a mini-bandgap without the oscillating part of the SOI (see the inset). In this panel, $G$ is plotted for different oscillating SOI amplitudes $ω_{1},$ with $ω_{0}$ being fixed. According to panel (b), when $ω_{0} / Ω$ is considerably lower than $ω_{1} / Ω$ , the mini-bandgap disappears. Parameters are $ω_{0} / Ω = 1.0, ν_{α} = 3.0$ (a) and $ω_{1} / Ω = 0.4, ν_{α} = 3.0$ (b).
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Figure 3
Conductance (in units of $G_{0} = 2 e^{2} / h$ ) of a $7 \times 7$ array as a function of the input energy $E_{0}$ and the frequency $ν_{α}$ of the oscillating part of the SOI. Additional parameters: $ω_{0} / Ω = 1.0, ω_{0} / Ω = 0.3 .$
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Figure 4
The weights $| t_{1 m} |^{2} + {| t_{1 m} |}^{2}$ of the frequency components $E_{m}$ [see Eqs. (17) and (19)] in the transmitted spinor-valued wave function as a function of the harmonic order (the integer $m$ ). Figures (a), (b)…(f) correspond to the six peaks that can be identified in the front curve ( $ν_{α} = 3$ ) of Fig. 3—from left (lowest energy) to the right (highest energy), respectively. For a clear identification of the peaks, the same symbols were used as in Fig. 3. Additional parameters: $ω_{0} / Ω = 1.0, ω_{0} / Ω = 0.3 .$
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Figure 5
Snapshots of the (un-normalized) electron density along a $7 \times 7$ array for time instants indicated in the labels of the vertical axes. [As a reference, the value of $n = 1$ corresponds to the input plane wave (in this case with unpolarized spin state).] Panels (a)–(e) correspond to the first, broad peak seen in Fig. 3 [panel (a) of Fig. 4], while panel (e) corresponds to the parameters of the second, narrow peak in Fig. 3 [panel (b) of Fig. 4].
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Figure 6
Panel (a): The degree of spin polarization $p$ and the particle density $n$ in the output arm of a $7 \times 7$ array at $τ = 0 .$ The spin orientation corresponding to “spin-up” and “spin-down” (in the $z$ direction) input spinors are shown in panel (b). Parameters: $ω_{0} / Ω = 1.0, ω_{0} / Ω = 0.3, E_{0} / ℏ Ω = 153.5 .$
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