Cotunneling through quantum dots coupled to magnetic leads: Zero-bias anomaly for noncollinear magnetic configurations

Cotunneling transport through quantum dots weakly coupled to noncollinearly magnetized leads is analyzed theoretically by means of the real-time diagrammatic technique. The electric current, dot occupations, and dot spin are calculated in the Coulomb blockade regime and for the arbitrary magnetic configuration of the system. It is shown that an effective exchange field exerted on the dot by ferromagnetic leads can significantly modify the transport characteristics in noncollinear magnetic configurations, in particular the zero-bias anomaly found recently [I. Weymann et al., Phys. Rev. B 72, 113301 (2005)] for antiparallel configuration. For the asymmetric Anderson model, the exchange field gives rise to precession of the dot spin, which leads to a nonmonotonic dependence of the differential conductance and tunnel magnetoresistance on the angle between magnetic moments of the leads. An enhanced differential conductance and negative tunnel magnetoresistance were found for certain noncollinear configurations.

Figure 1

(Color online) Schematic of a quantum dot coupled to ferromagnetic leads with noncollinearly aligned magnetizations. The net spin moments of the left ${\mathbf{S}}_L$ and right ${\mathbf{S}}_R$ leads form an angle $\phi$. There is a symmetric bias voltage applied to the system.

Figure 2

(Color online) (a) Differential conductance $G=dI∕dV$ as a function of the bias voltage $V$ and the angle $\phi$ for the symmetric Anderson model. The parameters are $k_{\mathrm{B}}T=0.5\Gamma$, $\epsilon =-15\Gamma$, $U=30\Gamma$, and $p_L=p_R\equiv p=0.5$. (b) Bias dependence of differential conductance for several angles.

Figure 3

(Color online) (a) Tunnel magnetoresistance as a function of the bias voltage $V$ and the angle $\phi$ for the symmetric Anderson model. The parameters are the same as in Fig. 2. (b) TMR as a function of the bias voltage for several values of $\phi$.

Figure 4

(Color online) (a) $S_x$ and (b) $S_z$ components of the dot spin as a function of the bias voltage for several values of the angle $\phi$ as indicated in the figure. The other parameters are the same as in Fig. 2. In (a) the curves for $\phi =0$ and $\phi =\pi$ coincide.

Figure 5

(Color online) (a) The differential conductance $G=dI∕dV$ as a function of the bias voltage $V$ and the angle $\phi$ for the asymmetric Anderson model, $\epsilon =-12\Gamma$ and $U=30\Gamma$. The other parameters are the same as in Fig. 2. (b) Bias dependence of differential conductance for several values of the angle $\phi$.

Figure 6

(Color online) (a) Tunnel magnetoresistance as a function of the bias voltage $V$ and the angle $\phi$ for the asymmetric Anderson model. The parameters are the same as in Fig. 5. (b) TMR as a function of the bias voltage for several values of $\phi$.

Figure 7

(Color online) (a) Differential conductance $G$ and (b) tunnel magnetoresistance as a function of the angle $\phi$ for several values of $x=(2\epsilon +U)∕\mid eV\mid$ and for $\epsilon =-12\Gamma$. The other parameters are the same as in Fig. 2.

Figure 8

(Color online) (a) Differential conductance and (b) tunnel magnetoresistance as a function of angle between the leads’ magnetizations for several values of the level position $\epsilon$, as indicated in the figure, and for $eV=\Gamma$. The other parameters are the same as in Fig. 2.

Figure 9

(Color online) The (a) $S_x$, (b) $S_y$, and (c) $S_z$, components of the dot spin as a function of angle $\phi$ for several values of $\epsilon$ and for $eV=\Gamma$. The other parameters are the same as in Fig. 2.