Robustness of a persistent spin helix against a cubic Dresselhaus field in (001) and (110) oriented two-dimensional electron gases

D. Iizasa, D. Sato, K. Morita, J. Nitta, and M. Kohda

Phys. Rev. B 98, 165112 – Published 8 October 2018

Abstract

The persistent spin helix (PSH) state in III–V semiconductor quantum wells (QWs) is a promising candidate for spin-based applications because the PSH state realizes controllable spin orientation with long spin lifetime. Although the cubic Dresselhaus spin-orbit (SO) interaction is known for breaking the PSH state in both (001)-oriented and (110)-oriented QWs, it is not well understood how the distinct symmetry of cubic Dresselhaus terms $β_{3}$ between (001) and (110) QWs affects the robustness of the PSH state. Here we investigate robustness of the PSH state between (001) and (110) QWs under various strengths of cubic Dresselhaus SO interaction based on numerical Monte Carlo approach and magnetoconductance simulation, respectively, representing optical spin excitation/detection and weak localization/weak antilocalization in magnetotransport. For electron spins initialized along $z ∥ [001]$ in a (001) QW and $x ∥ [001] and y ∥ [1 \bar{1} 0]$ in a (110) QW, where the spin distribution is developed with the helical spin mode, a (001) QW shows a more robust PSH state against the increase of $β_{3}$ than does a (110) QW. This phenomenon is contrary to numerically computed magnetoconductance, where weak localization is maintained on the variation of $β_{3}$ in a (110) QW, whereas weak antilocalization appears in a (001) QW. By deriving the spin lifetime of the PSH state in a (110) QW from a diffusion equation using a random-walk approach, we demonstrate that such a difference arises directly from the magnitude and orientation of third angular harmonics in cubic Dresselhaus fields for (001) and (110) QWs.

Received 7 August 2018

DOI:https://doi.org/10.1103/PhysRevB.98.165112

Physics Subject Headings (PhySH)

Spintronics

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

D. Iizasa¹, D. Sato¹, K. Morita², J. Nitta^1,3,4, and M. Kohda^1,3,4,*

¹Department of Materials Science, Tohoku University, Sendai 980-8579, Japan
²Graduate School of Electrical and Electronic Engineering, Chiba University, Chiba 263-8522, Japan
³Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan
⁴Center for Science and Innovation in Spintronics (Core Research Cluster), Tohoku University, Sendai 980-8577, Japan

^*makoto@material.tohoku.ac.jp

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Vol. 98, Iss. 16 — 15 October 2018

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Images

Figure 1
Schematic illustrations of momentum-dependent effective magnetic fields in a persistent spin helix state, assuming (a) $α > 0, β_{1} - β_{3} > 0$ in a (001) QW and (d) $α = 0, β_{1} - β_{3} > 0$ in a (110) QW. (b, e) Helical spin textures with spins initialized along $z ∥ [001]$ in a (001) QW and $x ∥ [001]$ in a (110) QW. Helical spin textures are developed in a $y - z$ plane in a (001) QW, whereas they are developed in a $x - y$ plane in a (110) QW. Third angular harmonics in a cubic Dresselhaus field are shown for (c) a (001) QW and for (f) a (110) QW. A (001) QW has three rotation symmetry of the effective magnetic field with constant SO field strength, whereas a (110) QW shows uniaxial alignment along the $z$ direction with modulated SO field strength.
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Figure 2
(a, c) Simulated spatial maps of spin polarization for (001) and (110) QWs at time $t = 2 ns$ after the spin excitation under different strengths of $β_{3} = 0.5, 1.0,$ and $1.5$ ( $10^{- 13} eV m$ ). Red and blue colors denote the magnitude of positive and negative spin polarizations along the $z$ ( $x$ ) direction for a (001) [(110)] QW. The PSH states in both QWs are degraded as $β_{3}$ increases. (b, d) Magnetoconductance as a function of perpendicular external magnetic field calculated using recursive Green's-function method. Magnetoconductance under various values of the normalized cubic Dresselhaus term, $Λ = β_{3} / M_{PSH}$ , is calculated for both QWs. Crossover from WL to WAL is observed clearly in a (001) QW, although WL states are preserved entirely for a variation of $Λ$ in a (110) QW.
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Figure 3
Time evolution of spin polarization at coordinate $(x, y) = (0, 0)$ with different $β_{3} = 0, 0.5, 1.0,$ and $1.5$ ( $10^{- 13} eVm$ ), respectively, in (a) a (001) QW and (b) a (110) QW. Solid lines are fitted curves using Eq. (11) for a (001) QW and Eq. (9) for a (110) QW. (c) Extracted lifetime of the PSH state under various strengths of $β_{3}$ . Filled circles represent a (001) QW with spins initialized along $z ∥ [001]$ . Filled squares and triangles, respectively, represent a (110) QW with spins initialized along $x ∥ [001] and y ∥ [1 \bar{1} 0]$ . Black solid lines are full calculations of the theoretical PSH lifetime [Eqs. (10) and (12)].
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Figure 4
(a) Time-reversal-symmetric trajectories for an electron moving clockwise (black solid arrows) and counterclockwise (pink dotted arrows) through impurity positions $R_{1}$ to $R_{8}$ . During the scattering events, spins precess around the momentum-dependent spin-orbit field with opposite directions for the clockwise and counterclockwise trajectories. (b) Three independent trajectories for electrons traveling the same distance with scattering and arriving at an identical point at time $t$ , corresponding to optically injected electrons.
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