Selective suppression of Dresselhaus or Rashba spin-orbit coupling effects by the Zeeman interaction in quantum dots

We study single- and two-electron parabolic quantum dots in the presence of linear Dresselhaus and Rashba spin-orbit interactions. Contributions of both types of spin-orbit coupling are investigated in the context of the spin polarization of the system at high magnetic fields. We demonstrate that for negative Landé factors the effect of the Dresselhaus coupling is suppressed at high magnetic field, which for structures without inversion asymmetry leads to a completely spin-polarized system and a strict antisymmetry of the wave functions with respect to the interchange of spatial-electron coordinates. For negative Landé factor the Rashba coupling is preserved at high field and consequently the spin polarization of the systems as well as the spatial antisymmetry of the two-electron wave function remain approximate.

Figure 1

(Color online) Solid curves show the lowest-energy levels corresponding to the total angular momentum quantum numbers $J=1/2,-1/2,-3/2,\dots$ calculated for $g=0$ and with only a single type of SO coupling present with coupling constant of $c=10.8\text{meV}\text{nm}$ ($\alpha =c,\beta =0$ or equivalently $\alpha =0,\beta =c$). The energy levels are calculated with respect to the lowest-energy FD level. For $J=-1/2$ we plot also the second-energy level (dashed curve). The dotted curves show the FD energy levels for $l=0,-1$, and $-2$ and $n=0$.

Figure 2

(Color online) Zoom of the low-energy part of Fig. 1 where we additionally plotted (dotted curves) the lowest-energy $J=\pm 1/2$ levels calculated in a basis where only the lowest Landau levels ($n=0$ and $l\le 0$) are included. Main contributions of the FD states to the lowest-energy $J=\pm 1/2$ states of Fig. 1 are given when only the Dresselhaus coupling is present. We use $\left(n,l,s\right)$ notation, where $n$ is the main quantum number, $l$ is the orbital momentum, and $s=\uparrow \downarrow$ is the orientation of the spin. The size of labels $\left(n,l,s\right)$ for each energy level indicates the importance of the FD contributions.

Figure 3

(Color online) Average $z$ component of the spin for $J=\pm 1/2$ energy levels in the presence of Dresselhaus coupling. Applied parameters correspond to Figs. 1, 2. In the inset square of the average $z$ component of the spin is plotted (the values are identical for Rashba and Dresselhaus types of coupling).

Figure 4

(Color online) Solid curves show the lowest-energy levels corresponding to the total angular momentum quantum numbers $J=1/2,-1/2,-3/2,\dots$ calculated when only Dresselhaus coupling with $\beta =10.8\text{meV}\text{nm}$ is present together with the Zeeman effect with $g=-0.44$. The energy levels are calculated with respect to the lowest-energy FD level. For $J=-1/2$ we also show the second-energy level (dashed curve). The dotted curves show the FD energy levels for a given value of $l=2,1,0,-1,-2,\dots$. The direction of the spin is indicated for $\left|l\right|\le 2$ FD energy levels. Dominating FD contributions to $J=1/2$ energy level as well as to $J=-1/2$ and $J=-3/2$ at high field are also given.

Figure 5

(Color online) Same as Fig. 4 but now for pure Rashba coupling.

Figure 6

(Color online) Solid lines show the low-energy spectrum for $\alpha =\beta =10.8\text{meV}\text{nm}$ and $g=-0.44$ with respect to the lowest-energy FD level in the absence of SO coupling. The red solid (gray solid in print version) and black solid lines correspond to $-1$ and 1 eigenvalues of ${\sigma }_{z}P$ operator, respectively. The dotted curves represent the FD energy levels shifted down on the energy scale by $-0.19\text{meV}$. The dashed curves show the spectrum for pure Rashba coupling.

Figure 7

(Color online) Low-energy spectrum for two-electron states in the presence of a single type of spin-orbit coupling (Rashba or Dresselhaus) with coupling constant $c=2.7\text{meV}\text{nm}$ ($\alpha =c$, $\beta =0$ or $\alpha =0$, $\beta =c$) and $g=0$. The stepwise curves, referred to the right axis, show the ground-state angular momentum. The solid (dotted) line corresponds to the case without (with) SO coupling. Energy levels are calculated with respect to twice the FD ground-state energy.

Figure 8

(Color online) Mean value of the $z$ component of the total spin for the ground state at various coupling constants $c$ (given in meV nm) for a single type of SO coupling present and $g=0$. The dotted curves are for the ground state without SO coupling. The lower panel corresponds to pure Dresselhaus coupling and upper one to pure Rashba coupling.

Figure 9

(Color online) Solid red and dashed curves show the odd and even $s$-parity energy levels for the two-electron states in the presence of both types of SO coupling with $\alpha =\beta =10.8\text{meV}\text{nm}$ and in the absence of the Zeeman contribution. Energy levels are calculated with respect to twice the FD ground-state energy. The gray dotted lines (referred to the right axis) show the spectrum without the SO coupling. Some of these energy levels are labeled by the angular momentum $L$ quantum number. The inset shows the zoom of the avoided crossing marked by the dashed circle.

Figure 10

(Color online) Real parts of the components of the two-electron wave function [see Eq. (11)]. The contours were plotted for $y_1=y_2=0$ in the $\left(x_1,x_2\right)$ plane for $\alpha =\beta =10.8\text{meV}\text{nm}$ at $B=0$. Blue (red) regions correspond to positive (negative) values of the wave function. The rows correspond to states with increasing energy from the bottom to the top and the columns to components of the two-electron wave function indicated by arrows at the top of the plot. The stars mark the states that enter into the avoided crossing shown in the inset of Fig. 9.

Figure 11

(Color online) Solid curves show the ground-state expectation values of the interchange of spatial coordinates of electrons for $\alpha =\beta$ and $g=0$. The dotted line shows the values for no SO coupling and the dashed line corresponds to $\alpha =4\beta =10.8\text{meV}\text{nm}$ (same result is obtained for $\beta =4\alpha =10.8\text{meV}\text{nm}$).

Figure 12

(Color online) Mean value of the $z$ component of the total spin for the ground state calculated for $g=0$ and pure Dresselhaus SO coupling (dotted curve, $\beta =10.8\text{meV}\text{nm}$, $\alpha =0$) and dominant Dresselhaus coupling (solid curve, for $\alpha =\beta /2$).

Figure 13

(Color online) Low-energy spectrum for two-electron states in the presence of pure Dresselhaus coupling for $\beta =10.8\text{meV}\text{nm}$ including the Zeeman effect $g=-0.44$. Energy levels are calculated with respect to twice the FD ground-state energy. The dotted lines show the energy levels obtained without SO coupling. At the bottom of the plot the average values of the $R$ operator and $⟨S_z⟩$ for the SO ground state are plotted with respect to the right axis.

Figure 14

(Color online) Same as Fig. 13 but now for pure Rashba coupling with $\alpha =10.8\text{meV}\text{nm}$.

Figure 15

(Color online) Same as Fig. 13 but for equal Rashba and Dresselhaus SO coupling constants $\alpha =\beta =10.8\text{meV}\text{nm}$. Energy levels of odd (even) $s$ parity are plotted with solid red (black dashed) lines. Dotted lines show the energy levels obtained for pure Rashba coupling.

Figure 16

(Color online) $z$ component of the spin for pure Rashba coupling (blue curve), for pure Dresselhaus coupling (red curve), and for equal Rashba and Dresselhaus coupling constants $\alpha =\beta =10.8\text{meV}\text{nm}$ (dotted curve) in the presence of the Zeeman effect $g=-0.44$. Dashed line shows the result for no SO coupling. The inset shows the zoom of the higher magnetic-field region.

Figure 17

(Color online) Expectation values of the $R$ operator for pure Rashba coupling (blue line), for pure Dresselhaus coupling (red line), and for equal Rashba and Dresselhaus coupling constants $\alpha =\beta =10.8\text{meV}\text{nm}$ in the presence of the Zeeman effect $g=-0.44$. The inset shows the zoom of the higher magnetic-field region.