Spin-dependent tunneling into an empty lateral quantum dot

Motivated by the recent experiments of Amasha et al. [Phys. Rev. B 78, 041306(R) (2008)], we investigate single electron tunneling into an empty quantum dot in presence of a magnetic field. We numerically calculate the tunneling rate from a laterally confined, few-channel external lead into the lowest orbital state of a spin-orbit coupled quantum dot. We find two mechanisms leading to a spin-dependent tunneling rate. The first originates from different electronic $g$ factors in the lead and in the dot, and favors the tunneling into the spin ground (excited) state when the $g$ factor magnitude is larger (smaller) in the lead. The second is triggered by spin-orbit interactions via the opening of off-diagonal spin-tunneling channels. It systematically favors the spin-excited state. For physical parameters corresponding to lateral GaAs/AlGaAs heterostructures and the experimentally reported tunneling rates, both mechanisms lead to a discrepancy of $\sim 10\mathrm{\%}$ in the spin-up vs spin-down tunneling rates. We conjecture that the significantly larger discrepancy observed experimentally originates from the enhancement of the $g$ factor in laterally confined lead.

Figure 1

(Color online) (a) Potential profile of the dot and the semi-infinite lead when considered separately. (b) Top view of the dot-lead structure. (c)–(e) Schematics of the tunneling mechanisms. (c) Symmetric tunneling. (d) Asymmetric tunneling: a larger (smaller) $g$ factor in the lead enhances the tunneling into the ground (excited) state. (e) Asymmetric tunneling: spin-orbit interactions open the off-diagonal spin-tunneling channels, which always favors tunneling into the excited state.

Figure 2

Spin-tunneling asymmetry due to the $g$ factor inhomogeneity varying (a) the $g$ factor difference, (b) the length $L$ of the barrier ramp, (c) the barrier height $V_0$, and (d) the dot asymmetry $\left(l_x-l_y\right)$. When not varied, the parameters are $g_{\text{lead}}=-0.39$, $g_{\text{dot}}=-0.32$, $V_0=12\text{meV}$, $L=240\text{nm}$, and $l_x-l_y=0$. In all panels, the tunneling rate into the ground state has been kept fixed at 200 Hz.

Figure 3

Spin-tunneling asymmetry due to the enhancement of the $g$ factor in the lead as a function of (a) the barrier length for $g$ factor values similar to those reported in Refs. 20, 21 and (b) the lead $g$ factor for two fixed barrier lengths.

Figure 4

Description of extended and evanescent spinors. (a) In-plane direction of the external and spin-orbit magnetic field. At a downward/upward arrow the fields are (anti)parallel/perpendicular. (b) Azimuthal $\left(\varphi \right)$ and inclination $\left(\theta \right)$ angles for the spin direction of an evanescent spinor. [(c) and (d)] External and spin-orbit magnetic fields, and Zeeman energy rescaled by $\mu =\left(g/2\right){\mu }_B$ for (c) extended and (d) evanescent spinor.

Figure 5

Influence of the magnetic field strength and orientation on the spin-orbit-induced spin-tunneling asymmetry. (a) At a downward/upward arrow the external and spin-orbit magnetic fields are (anti)parallel/perpendicular. (b) The asymmetry if only linear (lin), cubic (D3), or both (dashed) spin-orbit interactions are present. (c) Asymmetry for varied out-of-plane field orientation. This is the only case in this manuscript, where the vector potential $\mathbf{A}$ contributes to the kinematic momentum operator, $\hslash \mathbf{k}=-i\hslash \mathbf{\nabla }+e\mathbf{A}$. (d) Asymmetry as a function of the in-plane field strength.

Figure 6

Dependence of the spin-orbit-induced spin-tunneling asymmetry on (a) the barrier height $V_0$, (b) the barrier ramp $L$ [see Fig. 1a], (c) the dot-lead offset $y_{\text{offset}}$ [see Figs. 1a, 1b], (d) the lead width $w_{\text{width}}$. The widths at which the lowest three transverse channels become open are indicated. In all panels, the tunneling rate into the ground state has been kept fixed at 200 Hz.

Figure 7

(a) Full scattering eigenstate (thick line) and the confinement potential (thin solid line) of the dot-barrier-lead system. (b) Potential of an isolated dot and its bound eigenstate and the perturbation $\delta V$. (c) Extended eigenstate for the same problem. (d) Potential profile and an extended eigenstate of an isolated lead.

Figure 8

Comparison of two scattering approaches. (a) Wigner method: the reflection phase with a magnified resonance. (b) The flux method: lifetime according to Eq. (A10). The methods’ schematics are on the right. The model parameters [depicted in Fig. 7a] are $V_g=7\text{meV}$, $V_0=13\text{meV}$, $L=50\text{nm}$, and $l_0=100\text{nm}$.

Figure 9

Comparison of tunneling formulas varying the width of the potential well $l_0$. Parameters are as in Fig. 8. Tunneling rate for the dot bound states computed using (a) Eq. (A5) and (b) Eq. (A4), where only two states are shown for clarity. (c) Comparison of Eqs. (A5, A9). (d) Comparison of Eqs. (A5, A9), with the latter evaluated at the energy of the isolated dot state. In (c) and (d), the relative difference $\delta {\Gamma }_{ij}=\left|{\Gamma }_i-{\Gamma }_j\right|/{\Gamma }_i$ is plotted.