Controlling spin without magnetic fields: The Bloch-Rashba rotator

C. E. Creffield

Phys. Rev. B 102, 085436 – Published 31 August 2020

Abstract

We consider the dynamics of a quantum particle held in a lattice potential and subjected to a time-dependent spin-orbit coupling. Tilting the lattice causes the particle to perform Bloch oscillations, and by suitably changing the Rashba interaction during its motion, the spin of the particle can be gradually rotated. Even if the Rashba coupling can only be altered by a small amount, large spin rotations can be obtained by accumulating the rotation from successive oscillations. We show how the time dependence of the spin-orbit coupling can be chosen to maximize the rotation per cycle, and thus how this method can be used to produce a precise and controllable spin rotator, which we term the Bloch-Rashba rotator, without requiring an applied magnetic field.

Received 12 June 2020
Revised 11 August 2020
Accepted 17 August 2020

DOI:https://doi.org/10.1103/PhysRevB.102.085436

Physics Subject Headings (PhySH)

Rashba coupling Spintronics

Nanowires

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

C. E. Creffield

Departamento de Física de Materiales, Universidad Complutense de Madrid, E-28040 Madrid, Spain

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Issue

Vol. 102, Iss. 8 — 15 August 2020

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Images

Figure 1
(a) Creutz ladder representation of the lattice Hamiltonian (1). The index $j$ labels the sites of the lattice. Black lines along the edges of the ladder represent standard single-particle hopping between neighboring sites which conserves the spin-orientation. Diagonal hopping terms (shown in red) represent processes in which a particle hops by one lattice site and flips its spin, which arise from the Rashba interaction. The red dotted/solid lines have amplitudes of $- J_{so} / J_{so}$ due to the $σ_{y}$ term in $H_{latt}$ . (b) Dispersion relation of the lattice Hamiltonian. For $J_{so} = 0$ the system exhibits the standard single-band dispersion relation $E_{k} = - 2 J cos k$ . As $J_{so}$ is increased, the spectrum splits into two cosinusoidal bands, displaced from the origin by an amount proportional to the spin-orbit coupling.
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Figure 2
Schematic form of the Bloch-Rashba rotator. A particle is placed in a lattice which is subjected to a small tilt causing the particle to undergo an oscillatory motion through the lattice (Bloch oscillation). An external electric field regulates the Rashba spin-orbit coupling, causing the spin of the particle to rotate about an axis mutually perpendicular to its motion and the Rashba field. We consider the particle motion to be in the $x$ -direction, while the Rashba field is aligned with the $z$ -axis; consequently the particle spin will rotate about the $y$ -axis, in the $S_{x} - S_{z}$ plane.
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Figure 3
A Gaussian wave packet placed in a tilted lattice potential makes an oscillatory motion along the lattice. The amplitude of the oscillation and its frequency are governed by the size of the tilt, Eq. (3). The lattice tilt used here, $V_{0} = 0.01 J$ , gives a Bloch period of $T_{B} = 200 π / J$ .
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Figure 4
(a) Bloch oscillation of the Gaussian wave packet. The center of mass of the wave packet makes a sinusoidal oscillation along the lattice, the period of which is determined by the lattice tilt, $T_{B} = 2 π / V_{0}$ [see Eq. (3)]. (b) Solid lines denote $〈 S_{z} 〉$ , the dashed lines denote $〈 S_{x} 〉$ . The $x$ and $z$ components of the particle's spin oscillate sinusoidally in time, with the same period as the Bloch oscillation. When the SOC is increased from $J_{so} = 0.0005 J$ (black lines) to $J_{so} = 0.001 J$ (red lines), the amplitude of the oscillations in $〈 S_{x} 〉$ and $〈 S_{z} 〉$ correspondingly increases.
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Figure 5
The spin projections, $〈 S_{x} 〉$ and $〈 S_{z} 〉$ as a function of time. When the Rashba tunneling $J_{so}$ is held constant, the electron spin rotates at a constant rate while the electron wave packet propagates in one direction through the lattice. However, when the wave packet reverses to complete a cycle of Bloch oscillation, the spin retraces its trajectory to its original configuration, and so no spin-rotation is acquired. When $J_{so}$ is quenched to zero during the second half of the Bloch cycle, the electron spin is frozen, and so does not retrace its trajectory. Accordingly the rotation angle changes in steps as the Bloch oscillation continues. Flipping the sign of the Rashba tunneling, $J_{so} \to - J_{so}$ in the second half-cycle causes the spin to continue rotating at the same rate during the entire Bloch oscillation. $J_{so}$ can also be varied continuously, $J_{so} = J_{0} sin ω_{B} t$ , with the same frequency as the Bloch oscillation, to achieve this effect. Inset: When $J_{so}$ is held constant, the Bloch vector oscillates over a small range (black symbols). By making $J_{so}$ time dependent, the Bloch vector can now progressively step around a great circle in the $S_{x} - S_{z}$ plane.
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Figure 6
Area enclosed in the parameter plane ( $x - J_{so}$ ) for the different protocols shown in Fig. 5. The displacement $x$ is measured in units of the amplitude of the Bloch oscillation. (a) Constant $J_{so}$ . The trajectory is a straight line; as it does not enclose an area, the net spin-rotation is zero. (b) Quenched driving. $J_{so}$ takes two values, $J_{so} = 1$ in the first half-period, and $J_{so} = 0$ in the second, so the trajectory encloses a rectangle. (c) Flipped driving. $J_{so}$ again takes two values, but is now negative ( $J_{so} = - 1$ ) in the second half-period. The area again is again rectangular, but encloses twice the area obtained for quenched driving. As a consequence the spin rotates more rapidly. (d) Sinusoidal driving. The trajectory now encloses a circle.
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