Resonant tunneling between two-dimensional layers accounting for spin-orbit interaction

We present a theory of quantum tunneling between two-dimensional (2D) layers with Rashba and Dresselhaus spin-orbit interaction (SOI) in the layers. Accounting for SOI in the layers leads to a complex pattern in the tunneling characteristic with typical features corresponding to SOI energy. The resonant features strongly depend on the SOI parameters; for clear experimental observation the SOI characteristic energy should exceed the resonant broadening related to the particles' quantum lifetime in the layers. It appears that the experiments on hole tunneling are favorable to meet this criterion. We also consider a promising candidate for observing the effect, that is, $p$-doped SiGe strained heterostructures. As supported by our calculations, small adjustments of the parameters for experimentally studied AlGaAs/GaAs $p$-type quantum walls or designing a 2D-2D tunneling experiment for recently fabricated SiGe structures are very likely to reveal the SOI features in the 2D-2D tunneling.

Figure 1

Energy diagram of two 2D layers. (a) The $n\text{–}n$ tunneling case is realized by donor impurity doping layers, as illustrated by red circles. The electrons (dashed regions) populate QWs formed by the conduction-band profile. An external bias $U$ is applied to the QWs, which results in the tunneling current $I$. The electric field $E$ formed by the ionized donor layers has opposite directions in the two QWs. (b) The $h\text{–}h$ tunneling case occurs when the doping is $p$ type. In this case the current between the QWs formed by the valence-band potential profile is carried by holes. The electric field in this case $E$ is formed by the ionized acceptor layers with negative charge.

Figure 2

The calculated tunneling differential conductivity for the $n\text{–}n$ tunneling case. The red curve represents the calculation with the parameters relevant to the experiment [2] listed in Table 1; the blue curve corresponds to the sheet electron density being increased up to $n\sim {10}^{12}{\mathrm{cm}}^{-2}$.

Figure 3

The calculated tunneling differential conductivity for the $h\text{–}h$ tunneling case. The red curve represents the calculation with the parameters relevant to the experiment [3] listed in Table 1; the dotted curve shows the case with no SOI in the layers.

Figure 4

The calculated tunneling differential conductivity for the $h\text{–}h$ tunneling case with the sheet hole density increased by a factor of 2 (green curve) up to $p=1.4\times {10}^{11}{\text{cm}}^{-2}$ and with the QW thickness increased up to $a=20$ nm (blue curve); the dotted curve shows the case with no SOI in the layers.

Figure 5

The $h\text{–}h$ tunneling pattern for different ratios of the Rashba and Dresselhaus contributions. The green curve $\left({\delta }_R/{\delta }_D=0.83\right)$ corresponds to the parameters listed in the second column of Table 2. The other curves are obtained by varying the Dresselhaus parameter.

Figure 6

The calculated tunneling differential conductivity for the $h\text{–}h$ tunneling case using the parameters of the strained Ge QW studied in [16] listed in Table 3 (Experiment 2).