Dissipation-relaxation dynamics of a spin-$\frac{1}{2}$ particle with a Rashba-type spin-orbit coupling in an ohmic heat bath

Spin-orbit coupling (SOC), which is inherent in a Dirac particle that moves under the influence of electromagnetic fields, manifests itself in a variety of physical systems, including nonrelativistic ones. For instance, it plays an essential role in spintronics developed in the past few decades, particularly by controlling spin current generation and relaxation. In the present work, by using an extended Caldeira-Leggett model, we elucidate how the interplay between spin relaxation and momentum dissipation of an open system of a single spin-$\frac{1}{2}$ particle with Rashba-type SOC is induced by the interactions with a spinless, three-dimensional environment. Staring from the path-integral formulation for the reduced density matrix of the system, we have derived a set of coupled nonlinear equations that consists of a quasiclassical Langevin equation for the momentum with a frictional term and a spin precession equation. The spin precesses around the effective magnetic field generated by both the SOC and the frictional term. It is found from analytical and numerical solutions to these equations that a spin torque effect included in the effective magnetic field causes a spin relaxation and that the spin and momentum orientations after a long time evolution are largely controlled by the Rashba coupling strength. Such a spin relaxation mechanism is qualitatively different from, e.g., the one encountered in semiconductors in which essentially no momentum dissipation occurs due to the Pauli blocking.

Figure 1

Single particle energies $E_{\pm 1}\left(\mathit{p}\right)$ as functions of $p_x$ for $B=0$ (solid lines), $B<B_{\mathrm{c}}$ (dashed lines), and $B>B_{\mathrm{c}}$ (dotted lines). The upper (lower) lines correspond to $E_{+1}$ ($E_{-1}$). Note that these energies have the rotation symmetry in the $p_x\text{−}{p}_y$ plane, while the plotted values of the momenta and energies are normalized, respectively, by the asymptotic value (44) at $B=0$ and the absolute value of $E_{\min }=-m{\alpha }^2/8$ corresponding to $E_{-1}(\mathit{p}\left(\infty \right))$ at $B=0$.

Figure 2

Schematic view of the three-dimensional vectors that represent the spin $\mathit{n}\left(t\right)$, the momentum $\mathit{p}\left(t\right)$, and the effective and external magnetic fields ${\mathit{b}}_{\mathrm{eff}}\left(t\right)$ and $\mathit{B}$. The zenith and azimuth angles ${\theta }_n$ and ${\varphi }_n$ of the spin direction with respect to the total magnetic field ${\mathit{b}}_{\mathrm{eff}}+\mathit{B}$ are indicated. The angle between $\mathit{p}\left(t\right)$ and $\mathit{p}\left(0\right)$ is denoted by ${\varphi }_p$. The Rashba coupling $\mathit{\alpha }$ and initial momentum $\mathit{p}\left(0\right)$ are set in the $z$ and $x$ directions, respectively, while both $\mathit{p}$ and ${\mathit{b}}_{\mathrm{eff}}$ always lie in the $x\text{−}y$ plane.

Figure 3

Asymptotic values of the momentum angle $|{\varphi }_p\left(\infty \right)|$ as functions of the initial values of the spin angles ${\theta }_n\left(0\right)$ and ${\varphi }_n\left(0\right)$, plotted for (a) ${\tau }_{\mathrm{prec}}\ll {\tau }_{\mathrm{damp}}$ and (b) ${\tau }_{\mathrm{prec}}\gg {\tau }_{\mathrm{damp}}$ at $B=0$. The parameters are set as follows: (a) $\alpha p_0/\gamma =167.50$ and (b) $\alpha p_0/\gamma =3.6750$, with $m\gamma /p_0^2=0.00081633$ in both cases.

Figure 4

Time evolution of the spin and momentum for two sets of the initial spin angles, namely, $\{{\varphi }_n\left(0\right),{\theta }_n\left(0\right)\}=\{\pi /4,\pi /12\}$ and $\{{\varphi }_n\left(0\right),{\theta }_n\left(0\right)\}=\{\pi /4,2\pi /3\}$, as plotted for (a)–(c) ${\tau }_{\mathrm{prec}}\ll {\tau }_{\mathrm{damp}}$ and (d)–(f) ${\tau }_{\mathrm{prec}}\gg {\tau }_{\mathrm{damp}}$. The definition of each angle is illustrated in Fig. 2, while the parameter values are the same as in Fig. 3. For other sets of the initial angles, see Appendix pp2.

Figure 5

Time evolution of the spin and momentum at $B=0$, as plotted for ${\tau }_{\mathrm{prec}}\ll {\tau }_{\mathrm{damp}}$ under the initial conditions (a)–(d) $\{{\varphi }_n\left(0\right),{\theta }_n\left(0\right)\}=\{0,\{\pi /12,\pi /4,2\pi /3\}\}$ and (e)–(h) $\{{\varphi }_n\left(0\right),{\theta }_n\left(0\right)\}=\{\pi /2,\{\pi /12,\pi /4,2\pi /3\}\}$.

Figure 6

Time evolution of the spin and momentum at $B=0$, as plotted for ${\tau }_{\mathrm{prec}}\gg {\tau }_{\mathrm{damp}}$ under the initial conditions (a)–(d) $\{{\varphi }_n\left(0\right),{\theta }_n\left(0\right)\}=\{0,\{\pi /12,\pi /4,2\pi /3\}\}$ and (e)–(h) $\{{\varphi }_n\left(0\right),{\theta }_n\left(0\right)\}=\{\pi /2,\{\pi /12,\pi /4,2\pi /3\}$.

Figure 7

Asymptotic values of the momentum angle $|{\varphi }_p\left(\infty \right)|$ as a function of the initial values of the spin angles ${\theta }_n\left(0\right)$ and ${\varphi }_n\left(0\right)$ as plotted for (a) ${\tau }_{\mathrm{prec}}\ll {\tau }_{\mathrm{damp}}$ and (b) ${\tau }_{\mathrm{prec}}\gg {\tau }_{\mathrm{damp}}$ at $B=0.027563\gamma <B_{\mathrm{c}}$.

Figure 8

Time evolution of the spin and momentum at $B=0.027563\gamma <B_{\mathrm{c}}$, calculated under the initial conditions $\{{\varphi }_n\left(0\right),{\theta }_n\left(0\right)\}=\{\pi /4,\{\pi /12,2\pi /3\}\}$. Results for (a)–(c) ${\tau }_{\mathrm{prec}}\ll {\tau }_{\mathrm{damp}}$ and (d)–(f) ${\tau }_{\mathrm{prec}}\gg {\tau }_{\mathrm{damp}}$.

Figure 9

Time evolution of the spin and momentum at $B=137.42\gamma >B_{\mathrm{c}}$, calculated under the initial conditions $\{{\varphi }_n\left(0\right),{\theta }_n\left(0\right)\}=\{\pi /4,\{\pi /12,2\pi /3\}\}$. Results for (a)–(d) ${\tau }_{\mathrm{prec}}\ll {\tau }_{\mathrm{damp}}$ and (e)–(h) ${\tau }_{\mathrm{prec}}\gg {\tau }_{\mathrm{damp}}$.

Figure 10

Time evolution of the velocity, $\mathit{v}=\dot{\mathit{x}}$, at $B=0$, as plotted for (a) and (b) ${\tau }_{\mathrm{prec}}\ll {\tau }_{\mathrm{damp}}$ and (c) and (d) ${\tau }_{\mathrm{prec}}\gg {\tau }_{\mathrm{damp}}$ under the initial conditions $\{{\varphi }_n\left(0\right),{\theta }_n\left(0\right)\}=\{\pi /4,\pi /12\}$. ${\varphi }_v$ denotes the angle between $\mathit{v}\left(0\right)$ and $\mathit{v}\left(t\right)$ in the $x\text{−}y$ plane. The values $p_0$ and $\alpha$ are the same as in Fig. 3.