Spin dynamics and fluctuations in the streaming regime

Spin dynamics of two-dimensional electrons in moderate in-plane electric fields is studied theoretically. The streaming regime is considered, where each electron accelerates until reaching the optical phonon energy, then it emits an optical phonon, and a new period of acceleration starts. Spin-orbit interaction and elastic scattering result in anisotropic relaxation of electron-spin polarization. The overall spin dynamics is described by a superposition of spin modes in the system. The relaxation time of the most long-living mode depends quasiperiodically on the inverse electric field. The spin modes can be conveniently revealed by means of spin noise spectroscopy. It is demonstrated that the spectrum of spin fluctuations consists of peaks with the low-frequency peak much narrower than satellite ones, and the widths of the peaks are determined by the decay times of the modes.

Figure 1

Illustration of the streaming regime in the momentum space. During a ballistic motion inside the needle (red solid line), an electron spin rotates with the frequency $\mathbf{\Omega }\left(\mathit{p}\right)$ (blue arrows). The electron can be elastically scattered by an impurity (star and dashed arrow), then it reaches the active area (red arrow), instantaneously emits an optical phonon (wavy arrow), and returns to $\mathit{p}=0$ (dotted arrow).

Figure 2

(a) The spin distribution in the logarithmic scale for ${\tau }_p=3t_{\mathrm{tr}}, {\tau }_s^z=10{\tau }_p$ at the times $t=0$ (black solid curve), $1.8{\tau }_p$ (red dashed curve), $10{\tau }_p$ (blue dotted curve), and $20{\tau }_p$ (green dash-dotted curve). (b) The decay rates of the first three $z$-spin modes for $n=0$ (black solid curve), $n=1$ (red dashed curve), and $n=2$ (blue dotted curve) as functions of $\mathrm{\Delta }\varphi$, Eq. (21). The green dash-dotted line denote the spin splitting corresponding to panel (a). The inset shows the decay rates of the $x$ modes (dots) and the analytical approximation Eq. (20) (solid line).

Figure 3

(a) The decay rates of the first four spin modes, $-\mathrm{Im}\left({\delta }_n^{\left(1\right)}{\tau }_p\right)$ for $n=0$ (red solid curve), $n=1$ (blue dashed curve), $n=2$ (green dotted curve), and $n=3$ (magenta dash-dotted curve). (b) Spin-relaxation rate of the most long-living mode as a function of electric field (black solid curve) and its approximation Eq. (27) (red dashed curve). The spin-relaxation rate in the presence of both components of the effective field is shown by the blue dotted curve at ${\beta }_{xy}={\beta }_{yx}/3$.

Figure 4

Average spin $|s_{n^{*}}^{\left(1\right)}|$ in the most long-living mode as a function of ${\mathrm{\Omega }}_{\mathrm{dr}}{t}_{\mathrm{tr}}$ (black solid curve) and its asymptotic, Eq. (28) (blue dotted curve). Inset: Average spin in the $n\mathrm{th}$ mode, $|s_n^{\left(1\right)}|$, for $n=0$ (red solid curve), 1 (blue dashed curve), 2 (green dotted curve), and 3 (magenta dash-dotted curve).

Figure 5

Spin noise spectra in the streaming regime calculated after Eq. (34) for ${\tau }_p=3t_{\mathrm{tr}}$ and ${\beta }_{xy}=0$.