Spin relaxation in a zinc-blende (110) symmetric quantum well with $\delta$ doping

The spin relaxation of a two-dimensional electron system (2DES) formed in a symmetric quantum well is studied theoretically when the quantum well is parallel to the (110) plane of the zinc-blende structure, the spin polarization is perpendicular to the well, and electrons occupy only the ground subband. The spin-relaxation rate is calculated as a function of the distribution of donor impurities which are placed in the well layer. Considered processes of the spin relaxation are (1) the intrasubband process by impurity-potential-induced spin-orbit interaction (SOI), which is the Elliott-Yafet mechanism in the 2DES, and (2) virtual intersubband processes consisting of a spin flip by (2a) well-potential-induced SOI or (2b) the Dresselhaus SOI, and a scattering from an impurity. It is shown that all of the above processes disappear when all impurities are located on the center plane of the well. Even if impurities are distributed over three (110) atomic layers, the spin-relaxation rate is two orders of magnitude lower than that for the uniform distribution over the well width of 7.5 nm. In GaAs/AlGaAs type-I quantum wells, the processes (1) and (2a) interfere constructively, being dominant over (2b) for the well width of $\sim$10 nm, while in some type-II quantum wells, they can interfere destructively.

Figure 1

(a) $\delta$ doping at the well center ($z=0$). Filled circles ($•$) represent impurities. (b) $\delta$ dopings at $z=\pm z_{\mathrm{d}}$. (c) The well potential $V_{\mathrm{well}}\left(z\right)$ with height $V_0$ and width $W$.

Figure 2

Each electron state has a parity in addition to a spin ($z$ component) since the well potential $V_{\mathrm{well}}$ is even in $z$. When $z_{\mathrm{d}}=0$ [Fig. 1] and the impurity potential $V_{\mathrm{imp}}$ is also even in $z$, some perturbation terms change the parity and the spin at the same time, while all others change neither the parity nor the spin. $H_{\mathrm{imp}}^{\mathrm{so},xy}$ and $H_{\mathrm{imp}}^{\mathrm{so},z}$ ($H_{\mathrm{D}}^{\mathrm{so},xy}$ and $H_{\mathrm{D}}^{\mathrm{so},z}$) denote terms with ${\sigma }_x$ or ${\sigma }_y$ and terms with ${\sigma }_z$, respectively, of $H_{\mathrm{imp}}^{\mathrm{so}}$ ($H_{\mathrm{D}}^{\mathrm{so}}$).

Figure 3

(a) Intrasubband process by impurity-potential-induced SOI, $H_{\mathrm{imp}}^{\mathrm{so}}$. (b) Intersubband processes by well-potential-induced SOI, $H_{\mathrm{well}}^{\mathrm{so}}$, combined with impurity potential, $V_{\mathrm{imp}}$. (c) Intersubband processes by the Dresselhaus SOI, $H_{\mathrm{D}}^{\mathrm{so}}$, combined with $V_{\mathrm{imp}}$. In (b) and (c), the summation is taken over excited subbands with odd parity $(n=\mathrm{odd})$.

Figure 4

Spin-flip scattering rate ${\overline{P}}^{\mathrm{sf}}\left(\varepsilon \right)$ divided by $P_0$ [Eq. (39)]. $k=\left|\mathit{k}\right|$ with $\mathit{k}=(k_x,k_y)$ and $k_{\mathrm{s}}$ is the inverse of the screening length. (a) Dependence on the position of $\delta$ doping, $z_{\mathrm{d}}$. Contributions from each spin-flip scattering process in Fig. 3 are also shown. (b) Dependence on the width of the doped layer, $w_{\mathrm{d}}$, with all three processes in Fig. 3 considered.

Figure 5

Spin-flip scattering rate ${\overline{P}}^{\mathrm{sf}}\left(\varepsilon \right)$ divided by $P_0$ [Eq. (39)], as a function of the position of $\delta$ doping, $z_{\mathrm{d}}$. (a) Dependence on $k=\left|\mathit{k}\right|$. (b) Dependence on $k_{\mathrm{s}}$, the inverse of the screening length.