Effect of chaos on two-dimensional spin transport

Chen-Rong Liu, Xian-Zhang Chen, Hong-Ya Xu, Liang Huang, and Ying-Cheng Lai

Phys. Rev. B 98, 115305 – Published 24 September 2018

Abstract

The role of classical dynamics in spin transport is an intriguing problem from the point of view of classical-quantum correspondence, as spin is a purely relativistic quantum mechanical variable with no classical counterpart. Nevertheless, due to spin-orbit coupling (generally referred to as the relativistic interaction of a particle's spin with its motion inside a potential) and because the orbital motion does have a classical correspondence, the nature of the classical dynamics can affect spin. A basic transport structure is quantum dots, whose geometrical shape can be chosen to lead to characteristically distinct classical behaviors ranging from integrable dynamics to chaos. Whether and how classical chaos can affect spin transport and if the effect can be exploited for applications in spintronics are thus issues of both fundamental and practical interest. Here we report results from systematic, full quantum computations of spin transport through quantum dots hosting different types of classical dynamics. Our main finding is that chaos can play orthogonal roles in affecting spin polarization, depending on the relative strength of the spin-orbit coupling. For weak coupling with a characteristic interaction length much larger than the system size, chaos can be beneficial to preserving spin polarization. In the strong coupling regime where the interaction length is smaller than the system dimension, chaos typically destroys spin polarization. We develop a semiclassical theory to understand these phenomena and point out their implications and potential applications in developing spintronic devices.

Received 21 May 2018
Revised 1 September 2018

DOI:https://doi.org/10.1103/PhysRevB.98.115305

Physics Subject Headings (PhySH)

Quantum chaotic transport Spin-orbit coupling Spintronics

Coherent structures Quantum dots

Semiclassical methods

Nonlinear Dynamics

Authors & Affiliations

Chen-Rong Liu¹, Xian-Zhang Chen¹, Hong-Ya Xu², Liang Huang^1,*, and Ying-Cheng Lai^2,3

¹School of Physical Science and Technology, and Key Laboratory for Magnetism and Magnetic Materials of MOE, Lanzhou University, Lanzhou, Gansu 730000, China
²School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
³Department of Physics, Arizona State University, Tempe, Arizona 85287, USA

^*huangl@lzu.edu.cn

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Vol. 98, Iss. 11 — 15 September 2018

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Images

Figure 1
Schematic illustration of spin transport in two dimensions. The Rashba spin-orbit coupling (SOC) is induced by an external electrical field $E_{Rashba}$ in the SOC region.
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Figure 2
Four cavities investigated in this paper and their spin polarization. [(a)–(d)] Cavities I to IV in the order of an increasing degree of classical chaos, respectively. All structures have the same width of $0.24 μ m$ and, for a given Fermi energy, the leads permit the same number of transmitting modes. (a) A ribbon with integrable classical dynamics, where the length of the SOC region is $1 μ m$ . (b) A cosine billiard with mixed dynamics of length $L = 1.33 μ m$ . The upper boundary is given by [21] $y = W + (M / 2) [1 - cos (2 π x / L)]$ with $M / L = 0.11$ and $W / L = 0.18$ . (c) A cosine billiard with chaotic dynamics for $L = 0.67 μ m, M / L = 0.22$ , and $W / L = 0.36$ . (d) A chaotic cavity [44] with its upper boundary being made up of an arc of radius $R = 0.38 μ m$ and another arc of radius $r = 0.2 μ m$ . The lower boundary is a section of arc of radius $r = 0.2 μ m$ . The length of the cavity is $1 μ m$ . [(e)–(h)] Contour plots of $P_{z}$ in the parameter plane of Fermi energy $E_{f}$ and SOC strength $t_{so}$ (both in units of $t_{0}$ , the hopping energy between neighboring lattice sites). The triangles mark the positions of the $t_{so}$ such that $L_{so} = π / (2 t_{so}) = \infty, 4 L, L, L / 2$ , which are investigated further in Fig. 3.
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Figure 3
Characterization of spin transport through the four different types of cavities. Total transmission $T = T^{↑ ↑} + T^{↓ ↑}$ (upper row), charge Fano factor $F \equiv F^{↑ \to ↑ ↓}$ [Eq. (10c)] for the total spin-resolved current (middle row), and the spin polarization $P_{z}$ (bottom row) vs the normalized Fermi energy. The four columns from left to right correspond to $L_{so} = \infty, 4 L, L, L / 2$ , respectively. The notions I–IV indicate the four cavities in Fig. 2. For $F$ and $P_{z}$ , a sliding window average is used with the window size $Δ = 0.0464 t_{0}$ . For $E_{f} - 0.016 t_{0} < Δ / 2$ , the average is over the interval $[0.016 t_{0}, 2 E_{f} - 0.016 t_{0}]$ .
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Figure 4
Averaged spin polarization over the Fermi energy vs the spin-orbit coupling strength. [(a)–(d)] Averaged polarization ${〈 P_{z} 〉}_{E_{f}}$ versus $t_{so}$ for four different ranges of energy averaging: $[0.016, 0.0624], [0.0632, 0.140], [0.1408, 0.2472], [0.248, 0.3816]$ , respectively, corresponding to one to four transmission modes. The spin-orbit coupling strength $t_{so}$ is normalized by ${\tilde{t}}_{so}$ , the strength value that flips the electron spin from up to down as it travels from the left to the right lead through the cavity. The ${\tilde{t}}_{so}$ values are marked by the third triangles (red) from the bottom in Fig. 2 for different cavities.
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Figure 5
Spin polarization and evolution of polarization vector for the integrable cavity. For the ribbon (cavity I), (a) $P_{z}$ vs $x$ , for spin-orbit coupling strength $L_{so} = L$ . Symbols are numerical results from Eq. (14), while the light blue oscillatory curve is from theory [Eq. (16)]. (b) A trajectory of the spin polarization vector in the $P$ space, where up and down triangles are for reflections from the upper and lower boundary, respectively.
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Figure 6
Spin polarization and evolution of polarization vector for cavities with mixed and chaotic dynamics. [(a) and (d)] Phase space trajectories, i.e., the reflection angle versus the position of reflection at the lower boundary, for the cosine cavity with mixed (cavity II) and chaotic dynamics (cavity III), respectively. Only trajectories starting from the left lead are considered. The crosses are for a typical trajectory [61]. A dominant KAM island at $(x, θ) = (0.5, π / 2)$ is present in (a). [(b) and (e)] $P_{z}$ vs $x$ for the trajectories shown in [(a) and (d)], respectively. The spin-orbit coupling strength is such that $L_{so} = 4 L$ . (c,f) $P_{z}$ versus $x$ for the same trajectories but with $L_{so} = L$ . The two curves in (c) are the envelope lines similar to that in Fig. 5. Up and down triangles are for trajectory's reflecting from the upper and lower boundary, respectively. The two solid red right triangles mark the values of the incoming and outgoing $P_{z}$ at $x = 0$ and $x = L$ , respectively.
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