Spin-orbit effects in single-electron states in coupled quantum dots

Spin-orbit effects in single-electron states in laterally coupled quantum dots in the presence of a perpendicular magnetic field are studied by exact numerical diagonalization. Dresselhaus (linear and cubic) and Bychkov-Rashba spin-orbit couplings are included in a realistic model of confined dots based on GaAs. Group theoretical classification of quantum states with and without spin-orbit coupling is provided. Spin-orbit effects on the $g$ factor are rather weak. It is shown that the frequency of coherent oscillations (tunneling amplitude) in coupled dots is largely unaffected by spin-orbit effects due to symmetry requirements. The leading contributions to the frequency involves the cubic term of the Dresselhaus coupling. Spin-orbit coupling in the presence of a magnetic field leads to a spin-dependent tunneling amplitude, and thus to the possibility of spin to charge conversion, namely, spatial separation of spin by coherent oscillations in a uniform magnetic field. It is also shown that spin hot spots exist in coupled GaAs dots already at moderate magnetic fields, and that spin hot spots at zero magnetic field are due to the cubic Dresselhaus term only.

Figure 1

Energy spectrum of a single dot in magnetic field. (a) The Fock-Darwin spectrum, Eq. (7). (b) Lowest orbital excited levels ($n=0$, $\mid l\mid =1$) without (dashed) and with (solid) spin-orbit coupling. Arrows indicate the spin states. For clarity the energy’s origin here is shifted relative to case (a). Both the shift in energy levels as well as the splitting at $B=0$ grow as ${\alpha }^2$. (c) Anticrossing at the critical magnetic field (here about $13\mathrm{T}$). For clarity, a linear trend was subtracted from the data.

Figure 2

(Color online) Calculated corrections to the effective $g$ factor by spin-orbit interactions. Formulas (18) scaled by the values at $B=0$ (and thus independent on ${\alpha }_{\mathit{SO}}$) are plotted. The actual numerical values of $\delta g$ at $B=0$ are $\delta g_{D-D}\left(0\right)=1.0\times {10}^{-2}$, $\delta g_{\mathit{BR}-\mathit{BR}}\left(0\right)=8.6\times {10}^{-4}$, $\delta g_{D-D3}\left(0\right)=9.4\times {10}^{-4}$, and $\delta g_{D3-D3}\left(0\right)=2.5\times {10}^{-5}$. At the anticrossing $\delta g_{D-D}\left(B_{\mathrm{acr}}\right)=2.4\times {10}^{-3}$, $\delta g_{\mathit{BR}-\mathit{BR}}\left(B_{\mathrm{acr}}\right)=1.0\times {10}^{-4}$, $\delta g_{D-D3}\left(B_{\mathrm{acr}}\right)=1.8\times {10}^{-3}$, and $\delta g_{D3-D3}\left(B_{\mathrm{acr}}\right)=3.4\times {10}^{-4}$.

Figure 3

Single electron spectrum of a symmetric $\left({\mathrm{C}}_{2v}\right)$ lateral double dot structure as a function of the interdot separation, at $B=0$, derived by applying group theoretical considerations. Single dot states at $d=0$ and $d=\infty$ are labeled by the principal $\left(n\right)$ and orbital $\left(l\right)$ quantum numbers, while the double dot states are labeled according to the four irreducible representations ${\Gamma }_i$ of ${\mathrm{C}}_{2v}$. The lowest double dot states have explicitly excitation level of the $d=0$ and $d=\infty$ states they connect. Every state is doubly (spin) degenerate, and the spin index is not given.

Figure 4

(Color online) Calculated energy spectrum of a double quantum dot at $B=0$, as a function of interdot distance. Spin-dependent terms are not included in the Hamiltonian. Vertical bars indicate couplings due to spin-orbit interactions. Group theoretical symbols are shown with the lines on the left. Single dot levels are denoted by the highest orbital momentum (0, 1, 2,…) present in the degenerate set. This labeling is on the right. Every state is doubly degenerate, and the spin index is not given.

Figure 5

(Color online) Calculated corrections to the tunneling energy $T$ from spin-orbit terms at $B=0$. The labels indicate which spin-orbit terms are involved. Only $D\text{−}D3$ and $D3\text{−}D3$ are of second order. The remaining contributions are of fourth order.

Figure 6

(Color online) Calculated corrections to selected lowest energy levels due to $H_D$. All states have spin $\sigma =+1$. The graph demonstrates a transition between the symmetry group of the double dot $H_0$ (states $\Gamma$) and that of single dots (states $\Psi$). The transition is induced by $l_z$ which by symmetry couples states ${\Gamma }_1\leftrightarrow {\Gamma }_3$ and ${\Gamma }_2\leftrightarrow {\Gamma }_4$. Thus the anticrossing mechanism will induce transition ${\Gamma }_{1\left(3\right)}\leftrightarrow {\Gamma }_1\pm {\Gamma }_3$ and ${\Gamma }_{2\left(4\right)}\leftrightarrow {\Gamma }_2\pm {\Gamma }_4$. These linear combinations are equal to a single dot solution ${\Psi }_{n,l,\sigma }$ in the case $d\to 0$ and a combination ${\Psi }_{n,l,\sigma }^{\pm }\equiv {\Psi }_{n,l,\sigma }^d\pm {\Psi }_{n,l,\sigma }^{-d}$ of functions with the same orbital momenta in the case $d\to \infty$.

Figure 7

Computed energy spectrum of the double dot Hamiltonian without $H_{\mathit{SO}}$, as a function of magnetic field. The quantum dot separation is $50\mathrm{nm}$ (single dot confinement length is $20\mathrm{nm}$). The energy levels are labeled according to the symmetry of the states at $B=0$. Two crossings (one between ${\Gamma }_2$ and ${\Gamma }_3$, the other between ${\Gamma }_1$ and ${\Gamma }_2$) and one anticrossing (between ${\Gamma }_2$ and ${\Gamma }_4$) are indicated. In the limit $B\to \infty$ the states merge to Landau levels.

Figure 8

(Color online) Calculated energies of the four lowest states of Hamiltonian $H_0+H_Z$ at $B=1\mathrm{T}$. The vertical dashed lines indicated doublets in the effective Hamiltonian. The inset displays the anticrossing at $89\mathrm{nm}$ due to $H_D$. Dashed lines are energies of $H_0+H_Z$, solid lines of $H_0+H_Z+H_D$.

Figure 9

Calculated corrections to the energies of the four lowest states due to the linear Dresselhaus term $H_D$, at $B=1\mathrm{T}$. Solid lines are numerical data; dashed lines are analytical expressions computed by Eq. (35) using the LCSDO approximation for the states.

Figure 10

Calculated spin-orbit corrections to the effective $g$ factor (relative to the conduction band value $g^{*}$) as functions of magnetic field. The distance between the dots is $50\mathrm{nm}$. Solid lines are numerical data; dashed lines are analytical values computed using Eq. (35). Contributions come from linear spin-orbit terms (a, b), the mixed Dresselhaus correction from $H_D$ and $H_{D3}$ (c) and the cubic Dresselhaus $H_{D3}$ correction (d).

Figure 11

Calculated spin-orbit corrections to the tunneling energy $T$ as a function of magnetic field. The interdot distance is $50\mathrm{nm}$. Solid lines are numerical data; dashed lines are analytical expressions computed by Eq. (35). (a) Tunneling without spin-orbit interactions present. (b) Contribution to the tunneling from linear Dresselhaus for both spin up and down. (c) The mixed linear-cubic and (d) cubic-cubic Dresselhaus contributions.