Noncollinear magnetoconductance of a quantum dot

We study theoretically the linear conductance of a quantum dot connected to ferromagnetic leads. The dot level is split due to a noncollinear magnetic field or intrinsic magnetization. The system is studied in the noninteracting approximation, where an exact solution is given, and, furthermore, with Coulomb correlations in the weak tunneling limit. For the noninteracting case, we find an antiresonance for a particular direction of the applied field, noncollinear to the parallel magnetization directions of the leads. The antiresonance is destroyed by the correlations, giving rise to an interaction induced enhancement of the conductance. The angular dependence of the conductance is thus distinctly different for the interacting and noninteracting cases when the magnetizations of the leads are parallel. However, for antiparallel lead magnetizations, the interactions do not alter the angle dependence significantly.

Figure 1

Schematic drawing of the quantum dot system, connected to two ferromagnetic leads. In addition, there is an applied magnetic that spin polarizes the quantum dot in a direction noncollinear to the magnetization of the leads. In this paper, we focus on how the conductance depends on the angle $\varphi$ and the interplay between interactions and spin interference.

Figure 2

(Color online) The conductance for the noninteracting case in both the parallel (left panel) and the antiparallel (right panel) cases for different values of the magnetic strength and at the symmetric point ${\xi }_0=0$. For the parallel case, destructive interference leads to an antiresonance for $\varphi =\pi ∕2$. The antiresonance sharpens as $B∕{\Gamma }^0$ decreases. The conductance is maximal when the applied field is parallel to magnetizations in the leads ($\varphi =0$ or $\pi$). In contrast, for the antiparallel case the current is maximal for $\varphi =\pi ∕2$ and vanishes at $\varphi =0$ and $\pi$ because of the spin-valve effect.

Figure 3

Contour plots of the conductance in the noninteracting case for the parallel (top panel) and antiparallel (lower panel) geometry for different values of the angle $\varphi$ and the orbital dot quantum dot energy ${\xi }_0∕{\Gamma }_0$ ($B∕{\Gamma }_0=1$ in both figures). For the antiparallel geometry the same angular dependence is observed for all ${\xi }_0$, whereas a change is seen for the parallel geometry as explained in the text.

Figure 4

The three different cotunneling regimes considered $({\epsilon }_2={\epsilon }_{\uparrow }+{\epsilon }_{\downarrow }+U)$.

Figure 5

The two possible second order processes contributing to the tunneling amplitude, here shown for the parallel configuration in the first regime. Because of interactions, the path to the right is suppressed when the lower energy spin-state is occupied by one electron. Hence the interactions tend to destroy the interference effects seen in Fig. 2, valid for $U=0$, and instead one gets the angular dependence shown in Fig. 6.

Figure 6

(Color online) The conductance for the interacting case in both the parallel (left panel) and the antiparallel (right panel) cases for different values of the Coulomb interaction and for the symmetric point ${\xi }_0=0$. For the parallel case, we see how the antiresonance at $\varphi =\pi ∕2$ is destroyed by interactions, due to the blocking of one of the paths in Fig. 5. For large $U$ one instead has a crossover to a spin-valve effect at $\varphi =\pi$, where the dot is occupied by an electron with spin opposite to the lead electrons. The angle dependence thus dramatically depends on the interaction. This is not so for the antiparallel case shown in the right panel, where merely an overall suppressing is seen.

Figure 7

(Color online) The cotunneling result for the parallel geometry in the three different regimes $[j=\mathrm{i},\mathrm{ii},\mathrm{iii}]$ with (i) ${\xi }_0=0$, (ii) ${\xi }_0∕B=2$, and (iii) ${\xi }_0∕B=-4$.