Tunnel magnetoresistance of quantum dots coupled to ferromagnetic leads in the sequential and cotunneling regimes

We study electronic transport through quantum dots weakly coupled to ferromagnetic leads with collinear magnetization directions. Tunneling contributions of first and second order in the tunnel-coupling strength are taken into account. We analyze the tunnel magnetoresistance (TMR) for all combinations of linear and nonlinear response, at or off resonance, with an even or odd dot-electron number. Different mechanisms for transport and spin accumulation of the various regimes give rise to different TMR behavior.

Figure 1

Single-level quantum dot coupled to ferromagnetic leads. The magnetic moments of the electrodes are either parallel or antiparallel to each other.

Figure 2

An example for the time evolution of the reduced density matrix. The grey regions define irreducible diagrams of first and second order in tunneling, respectively. The direction of each tunneling line indicates whether an electron of respective spin leaves or enters the dot, thus, leading to a change of the dot state, as indicated on the forward and backward Keldysh propagators.

Figure 3

Linear conductance for nonmagnetic leads $(p=0)$ as a function of the level position. The dashed line corresponds to the first-order contribution $G^{\left(1\right)}$, the dotted line represents the second-order conductance $G^{\left(2\right)}$ and the solid line presents the sum $G^{\left(1\right)}+G^{\left(2\right)}$. The parameters are $k_{B}T=\Gamma$ and $U=20\Gamma$. The figure was generated using the scheme for the perturbation expansion in the presence of sequential tunneling.

Figure 4

A sketch presenting different transport regimes. The respective regimes are separated by solid lines.

Figure 5

The first-order tunnel magnetoresistance as a function of the bias and gate voltages. The parameters are: $k_{B}T=1.5\Gamma$, $U=40\Gamma$, and $p=0.5$.

Figure 6

The first-plus-second-order tunnel magnetoresistance as a function of bias and gate voltage. The parameters are the same as in Fig. 5. The figure was generated using the crossover scheme.

Figure 7

(Color online) The total linear conductance (a) in the parallel (solid line) and antiparallel (dashed line) configuration and the resulting tunnel magnetoresistance [solid line in (b)] as a function of the level position. The dashed line in part (b) represents the first-order tunnel magnetoresistance. The dotted-dashed curve presents the TMR calculated using the approximation Eq. (24). The parameters are $k_{B}T=1.5\Gamma$, $U=40\Gamma$, and $p=0.5$. The figure was generated using the scheme for the perturbation expansion in the presence of sequential tunneling.

Figure 8

The total currents (a) in the parallel (solid line) and antiparallel (dashed line) magnetic configurations as a function of level position for $\mathrm{eV}=20\Gamma$. Part (b) shows the first-order contribution to the TMR (dashed line) and the total TMR (solid line). The inset in part (b) shows the total TMR at lower temperature, $k_{B}T=0.5\Gamma$. The other parameters are the same as in Fig. 7. The figure was generated using the crossover scheme.

Figure 9

(Color online) The total current (a) in the parallel (solid line) and antiparallel (dashed line) magnetic configurations as a function of the bias voltage. Part (b) shows the first-order contribution to the TMR (dashed line) and the total TMR (solid line). The dotted-dashed curve presents the TMR calculated using the approximation Eq. (26). The parameters are $k_{B}T=1.5\Gamma$, $\epsilon =-U∕2$, $U=40\Gamma$, and $p=0.5$. The figure was generated using the crossover scheme.

Figure 10

The differential conductance (a) for parallel and antiparallel configurations and the tunnel magnetoresistance (b) as a function of the bias voltage for different values of temperature. The maximum in conductance for antiparallel configuration at zero bias is clearly demonstrated. The other parameters are the same as in Fig. 9. The figure was generated using the scheme for the perturbation expansion in the Coulomb blockade regime.

Figure 11

The total current (a) in the parallel (solid line) and antiparallel (dashed line) magnetic configurations as a function of the bias voltage. Part (b) shows the first-order contribution to TMR (dashed line) and the total TMR (solid line). The parameters are $k_{B}T=1.5\Gamma$, $\epsilon =20\Gamma$, $U=40\Gamma$, and $p=0.5$. The figure was generated using the perturbation expansion in the presence of sequential tunneling.

Figure 12

The total current (a) in the parallel (solid line) and antiparallel (dashed line) magnetic configuration as a function of the bias voltage. Part (b) shows the first-order contribution to the TMR (dashed line) and the total TMR (solid line). The parameters are $k_{B}T=1.5\Gamma$, $\epsilon =-10\Gamma$, $U=40\Gamma$, and $p=0.5$. The figure was generated using the perturbation expansion in the presence of sequential tunneling.

Figure 13

The occupation probabilities of the spin-up and spin-down dot levels as a function of the level position in the parallel (a) and antiparallel (b) configuration. The zeroth-order occupation probabilities for the spin-up and spin-down levels are equal in both magnetic configurations, and are represented by the dotted lines. The total occupation probability of the spin-up (spin-down) level is presented by the solid (dashed) line. In the antiparallel configuration, the dashed and solid lines coincide. The parameters are $k_{B}T=1.5\Gamma$, $U=40\Gamma$, and $p=0.5$. The figure was generated using the scheme for the perturbation expansion in the presence of sequential tunneling.