Interplay of Coulomb interaction and spin-orbit effects in multilevel quantum dots

We study electron transport through a multilevel quantum dot with Rashba spin-orbit interaction in the presence of local Coulomb repulsion. We focus on the parameter regime in which the level spacing is larger than the level broadening. Motivated by recent experiments, we compute the level splitting induced by the spin-orbit interaction at finite Zeeman fields $B$, which provides a measure of the renormalized spin-orbit energy. This level splitting is responsible for the suppression of the Kondo ridges at finite $B$ characteristic for the multilevel structure. In addition, the dependence of renormalized $g$ factors on the relative orientation of the applied $B$ field and the spin-orbit direction following two different protocols used in experiments is investigated.

Figure 1

The considered setup consists of two parallel quantum dots with energies ${\epsilon }_{1/2}=V_{\mathrm{G}}\pm \delta$ coupled by a hopping amplitude $t$ and a SOI of strength $\alpha$. The levels are split by an external Zeeman field $B$. The local Coulomb interaction is $U$ and the interaction between electrons on the two dots is $U^{\prime }$. The system is coupled to noninteracting leads by hopping amplitudes $t_{\beta ,j}$ with $\beta =L,R$ and $j=1,2$.

Figure 2

Schematic representation of the noninteracting level structure of the isolated dot as relevant in Sec. 3b.

Figure 3

Qualitative reproduction of the conduction region of Fig. 1(c) in Ref. 23. The chosen system parameters are $t=\alpha =1$, $\delta =0.9$, $\phi =0.46\pi$, $U=U^{\prime }=1.5$, $t_{L,1}=-0.5$, $t_{L,2}=0.45$, $t_{R,1}=0.4$, $t_{R,2}=-0.3$ (Ref. 45).

Figure 4

Bare spin-orbit parameter $\alpha$ dependence of the generated effective single-particle parameters ${\tilde{B}}_S$ [upper (blue) curves] and $\tilde{\beta }$ [lower (red) curves] for different orientations of the bare magnetic field (full lines $\phi =\frac{\pi }{6}$, dashed lines $\phi =\frac{\pi }{4}$, dotted lines $\phi =\frac{\pi }{3}$). The remaining parameters are $t=1$, $\delta =0$, $B=1.2$, $U=U^{\prime }=1$, $t_{\mathrm{Coup}}=0.4$.

Figure 5

Bare Zeeman field dependence of the effective single-particle energy spectrum (full lines) as well as of the corresponding noninteracting ones (dashed lines). As a guide to the eye, the chemical potential and the crossing position of the renormalized levels are given as straight black lines. The parameters are $V_{\mathrm{G}}=1.25$, $t=1$, $\alpha =0.6$, $\phi =0.25\pi$, $U=U^{\prime }=1$, $t_{\mathrm{Coup}}=0.4$.

Figure 6

Renormalization-induced $\phi$ dependence of the effective parameters. Here, we compensated for the bare angular dependence of the Zeeman fields by dividing by the corresponding trigonometric function. The initial parameters are $t=1$, $\alpha =0.6$, $B=1.2$, $U=U^{\prime }=1$, $t_{\mathrm{Coup}}=0.4$.

Figure 7

Spin-orbit energy versus orientation of the Zeeman field for several values of the interaction $U=U^{\prime }$ (full lines). The dashed lines show the spin-orbit energies normalized to the angular dependence of the noninteracting case. The initial conditions are given by $t=1,\alpha =0.6,t_L=t_R=0.3$.

Figure 8

Determination of the $g$ factor $g_{\mathrm{Cond}}$ from the gate-voltage dependence of the Coulomb blockade peaks in the linear conductance. Bare parameters: $t=1$, $\alpha =1$, $\phi =0.5\pi$, $U=U^{\prime }=0.5$, $t_{\mathrm{Coup}}=0.3$. The inset shows a linear fit to the extracted gate-voltage difference $\Delta$.

Figure 9

Zeeman field orientation dependence of $g_{\mathrm{Cond}}$ as extracted from the gate-voltage dependence (see Fig. 8) for different values of the Coulomb interaction $U=U^{\prime }=0,0.25,0.5,1$ and parameters as in Fig. 8.

Figure 10

Angular dependence of $g_{\mathrm{Levels}}$ as extracted from the level splitting (solid lines) for the same parameters as in Fig. 8 and different values of the Coulomb interaction $U=U^{\prime }=0,0.25,0.5,1$. In addition, the angular dependence normalized to the noninteracting case is shown (dashed lines).

Figure 11

Angular dependence of the spin-orbit energy for the parameters of Fig. 3 (full lines). For reference, the angular dependence of the noninteracting case is shown (dashed lines). The inset shows a zoom-in of the $\phi >0.4\pi$ range (note the logarithmic scale on the $y$ axis).

Figure 12

Angular dependence of $g_{\mathrm{Cond}}^{\pm }$ for the parameters of Fig. 3 (full lines). For reference, the dependence of the noninteracting case is shown by the dashed lines.

Figure 13

Angular dependence of $g_{\mathrm{Levels}}^{\pm }$ for the parameters of Fig. 3 (full lines). Normalization with respect to the bare angular dependence of the noninteracting case is shown by the dashed lines.