Spin relaxation in a quantum dot due to Nyquist noise

We calculate electron and nuclear spin relaxation rates in a quantum dot due to the combined action of Nyquist noise and electron-nuclei hyperfine or spin-orbit interactions. The relaxation rate is linear in the resistance of the gate circuit and, in the case of spin-orbit interaction, it depends essentially on the orientations of both the static magnetic field and the fluctuating electric field, as well as on the ratio between Rashba and Dresselhaus interaction constants. We provide numerical estimates of the relaxation rate for typical system parameters, compare our results with other, previously discussed mechanisms, and show that the Nyquist mechanism can have an appreciable effect for experimentally relevant systems.

Figure 1

(Color online) Typical gated lateral quantum dot structure considered in the text, with gates connected to an electrical circuit of impedance matrix $Z$. The fluctuating electric field produced by one particular gate is shown schematically.

Figure 2

(a) Equivalent circuit scheme used in our model; (b) energy diagram, showing the second-order relaxation process releasing the Zeeman energy $\hslash \omega$ into the electromagnetic environment.

Figure 3

(Color online) Angular dependence of the spin relaxation rate ${\Gamma }_{\mathrm{SO}}$ on magnetic field direction $(\theta ,\phi )$, according to Eq. (15), for ${\phi }^{*}=-\pi ∕4$. The shape will be rotated around the $z$ axis for different values of ${\phi }^{*}$, which is determined by the electric field direction $\zeta$ and the spin-orbit coupling constant ratio expressed via $\gamma$.

Figure 4

Electron spin relaxation rate due to Nyquist noise, Eqs. (10, 16), at $\hslash {\Omega }_0=1\mathrm{meV}$, $R∕R_Q={10}^{-2}$, $d=0.5\mu \mathrm{m}$, ${\lambda }_{\mathrm{SO}}=1\mu \mathrm{m}$, $\phi =0$, $\theta =\pi ∕2$, $\zeta =0$, and $\gamma =-\pi ∕4$. At low fields, the hyperfine rate (HF) dominates over the spin orbit rate (SO).