Effects of different geometries on the conductance, shot noise, and tunnel magnetoresistance of double quantum dots

The spin-polarized transport through a coherent strongly coupled double quantum dot (DQD) system is analyzed theoretically in the sequential and cotunneling regimes. Using the real-time diagrammatic technique, we analyze the current, differential conductance, shot noise, and tunnel magnetoresistance (TMR) as a function of both the bias and gate voltages for double quantum dots coupled in series, in parallel, as well as for $T$-shaped systems. For DQDs coupled in series, we find a strong dependence of the TMR on the number of electrons occupying the double dot, and super-Poissonian shot noise in the Coulomb blockade regime. In addition, for asymmetric DQDs, we analyze transport in the Pauli-spin blockade regime and explain the existence of the leakage current in terms of cotunneling and spin-flip cotunneling-assisted sequential tunneling. For DQDs coupled in parallel, we show that the transport characteristics in the weak-coupling regime are qualitatively similar to those of DQDs coupled in series. On the other hand, in the case of $T$-shaped quantum dots we predict a large super-Poissonian shot noise and TMR enhanced above the Julliere value due to increased occupation of the decoupled quantum dot. We also discuss the possibility of determining the geometry of the double dot from transport characteristics. Furthermore, where possible, we compare our results with existing experimental data on nonmagnetic systems and find qualitative agreement.

Figure 1

(Color online) Schematic of a double quantum dot weakly coupled to external ferromagnetic electrodes. The magnetizations of the leads can form either parallel or antiparallel magnetic configurations. The first $\left(j=1\right)$ and second $\left(j=2\right)$ dots are coupled to each other via the hopping $t$, and to the left $\left(r=L\right)$ and right $\left(r=R\right)$ leads with the coupling strength ${\Gamma }_{rj}$. The double quantum dot is assumed to be symmetrically biased. By adjusting the couplings, the system can smoothly cross over from the serial to parallel geometry.

Figure 2

(Color online) The differential conductance in the (a) parallel $G_P$ and (b) antiparallel $G_{AP}$ magnetic configurations as a function of the bias voltage $V$ and the position of the dot levels $\epsilon \equiv {\epsilon }_1={\epsilon }_2$ for double quantum dots coupled in series. The parameters are $k_{B}T=0.15\text{meV}$, $U=2\text{meV}$, $U^{\prime }=1\text{meV}$, $t=0.25\text{meV}$, ${\Gamma }_{L2}={\Gamma }_{R1}=0$, ${\Gamma }_{L1}={\Gamma }_{R2}\equiv \Gamma /2$, with $\Gamma =0.1\text{meV}$, and $p=0.5$. Because the double quantum dot is symmetric, ${\epsilon }_1={\epsilon }_2$, there is no asymmetry associated with the bias reversal. For comparison in part (c), we also show the density plot of the differential conductance in the parallel configuration calculated using only the first-order tunneling processes, $G_P^{\left(1\right)}$.

Figure 3

(Color online) The (a) total (first and second orders) TMR and (b) the first-order TMR as a function of the bias voltage $V$ and the level position $\epsilon$ for parameters are the same as in Fig. 2. The two figures are plotted in the same scale.

Figure 4

(Color online) The (a) linear conductance in the parallel (solid line) and antiparallel (dashed line) magnetic configurations and the (b) linear TMR as a function of the level position $\epsilon$ for parameters the same as in Fig. 2. For comparison, the dotted curves show the linear conductance and the TMR calculated taking into account only the first-order processes.

Figure 5

(Color online) The current in units of [(a) and (e)] $I_0=e\Gamma /\hslash$, [(b) and (f)] differential conductance, and [(d) and (h)] the Fano factor in the parallel (solid line) and antiparallel configuration (dashed line), as well as the [(c) and (g)] TMR as a function of the bias voltage for different values of the level position $\epsilon$ as indicated in the figure. (a)–(d) $\epsilon =-2\text{meV}$ corresponds to the case when the ground state of the DQD is doubly occupied, while for (e)-(h) $\epsilon =2\text{meV}$, the double dot is empty. The parameters are the same as in Fig. 2. The dotted curves show the results obtained in the sequential tunneling approximation.

Figure 6

(Color online) The (a) total Fano factor in the parallel and (b) antiparallel magnetic configurations as a function of the bias voltage $V$ and the position of the dot’s levels $\epsilon$ for parameters the same as in Fig. 2. Part (c) presents the Fano factor in the parallel configuration calculated in the first-order approximation. Because the Fano factor is divergent for $\left|\text{eV}\right|\lesssim k_{B}T$, this transport regime is denoted with a thick black line.

Figure 7

(Color online) The (a) current and (c) Fano factor in the parallel (solid) and antiparallel (dashed) magnetic configurations as well as the (b) TMR as a function of the bias voltage for DQD coupled in series. The parameters are $k_{B}T=0.05\text{meV}$, ${\epsilon }_1=-1\text{meV}$, ${\epsilon }_2=-2\text{meV}$, $\Delta =3\text{meV}$, $U=2\text{meV}$, $U^{\prime }=1\text{meV}$, and $t=0.05\text{meV}$, where $\Delta$ is the level spacing in each dot. The other parameters are the same as in Fig. 2. The inset in part (a) displays the current in the parallel magnetic configuration in the Pauli-spin blockade regime. The dotted lines correspond to the first-order calculation.

Figure 8

The total (sequential plus cotunneling) current (solid line) and the sequential current (dotted line) in the parallel configuration as a function of the hopping between the two dots calculated in the middle of the Pauli-spin blockade regime, $V=1\text{mV}$. The parameters are the same as in Fig. 7. The dependence of the current on $2t/\left|{\epsilon }_2-{\epsilon }_1\right|$ in the antiparallel configuration is qualitatively similar.

Figure 9

(Color online) The (a) differential conductance and (c) Fano factor in the parallel magnetic configuration, and the (b) TMR as a function of the bias voltage $V$ and the position of the dot’s levels $\epsilon \equiv {\epsilon }_1={\epsilon }_2$ for double quantum dots coupled in parallel. The parameters are the same as in Fig. 2 with ${\Gamma }_{rj}\equiv \Gamma /2$ for $r=L,R$, $j=1,2$, and $\Gamma =0.1\text{meV}$. The transport regime where the Fano factor is divergent is marked with a thick black line.

Figure 10

(Color online) (a) The bias dependence of the current and differential conductances, (b) the TMR, and (c) the Fano factor in the parallel (solid line) and antiparallel (dashed line) configu-rations for $T$-shaped double quantum dots. The parameters are ${\Gamma }_{r1}\equiv \Gamma /2$ and ${\Gamma }_{r2}=0$ for $r=L,R$, with $\Gamma =0.1\text{meV}$ and ${\epsilon }_1=1\text{meV}$, ${\epsilon }_2=0\text{meV}$, $k_{B}T=0.18\text{meV}$, $U=2\text{meV}$, $U^{\prime }=1\text{meV}$, $t=0.2\text{meV}$, and $p=0.5$. The dotted curves present the (b) TMR and (c) the Fano factor in the parallel configuration calculated for ${\Gamma }_{r2}={\Gamma }_{r1}$.

Figure 11

(Color online) The bias dependence of the current, (a) differential conductance, (b) TMR, and (c) the Fano factor in the parallel (solid line) and antiparallel (dashed line) configurations. The parameters are the same as in Fig. 10 with ${\epsilon }_2=-4\text{meV}$. The dotted curves present (b) the TMR and (c) the Fano factor in the parallel configuration calculated for ${\Gamma }_{r2}={\Gamma }_{r1}$.

Figure 12

(Color online) The (a) differential conductance in the parallel magnetic configuration, and (b) the TMR as a function of the bias voltage $V$ and the position of the second dot level $\epsilon \equiv {\epsilon }_2$ for $T$-shaped double quantum dots. The parameters are the same as in Fig. 10.