Negative tunnel magnetoresistance and differential conductance in transport through double quantum dots

Spin-dependent transport through two coupled single-level quantum dots weakly connected to ferromagnetic leads with collinear magnetizations is considered theoretically. Transport characteristics, including the current, linear and nonlinear conductances, and tunnel magnetoresistance are calculated using the real-time diagrammatic technique in the parallel, serial, and intermediate geometries. The effects due to virtual tunneling processes between the two dots via the leads, associated with off-diagonal coupling matrix elements, are also considered. Negative differential conductance and negative tunnel magnetoresistance have been found in the case of serial and intermediate geometries, while no such behavior has been observed for double quantum dots coupled in parallel. It is also shown that transport characteristics strongly depend on the magnitude of the off-diagonal coupling matrix elements.

Figure 1

(Color online) Schematic picture of the DQD system coupled to ferromagnetic leads. The parameter ${\gamma }_{\beta }^{\sigma }$ (for $\beta =L,R,\sigma =\uparrow ,\downarrow$) describes here a spin-dependent dot-lead couplings, whereas $\alpha$ takes into account difference in the coupling of a given electrode to the two dots $\left(\alpha ∊⟨0,1⟩\right)$. In particular, for $\alpha =0$ double quantum dots are in the serial geometry, while for $\alpha =1$ the system is in the parallel geometry.

Figure 2

(Color online) Current (a) and differential conductance (b) in the parallel (P, solid line) and antiparallel (AP, dashed line) magnetic configurations, as well as tunnel magnetoresistance (c), calculated as a function of the bias voltage for the parameters: $E=10k_{B}T$, $\Delta E=0$, $U_0=20k_{B}T$, $U=100k_{B}T$, $p=0.4$, $t=0.25\Gamma$, $\Gamma =5\mu \text{eV}$, $\alpha =0$, and $I_0=e\Gamma /\hslash \approx 1.215\text{nA}$.

Figure 3

(Color online) Current (a) and differential conductance (b) in the parallel (solid line) and antiparallel (dashed line) magnetic configurations, and tunnel magnetoresistance (c), calculated as a function of the bias voltage for $E=-10k_{B}T$. The other parameters are the same as in Fig. 2.

Figure 4

(Color online) Renormalization of the dots’ levels spacing for both spin orientations in the parallel and antiparallel magnetic configurations. The parameters are the same as in Fig. 3.

Figure 5

(Color online) Tunnel magnetoresistance as a function of the bias voltage and the average level position. The other parameters as in Fig. 3.

Figure 6

(Color online) The current (a), differential conductance (b) in the parallel (solid line) and antiparallel (dashed line) configurations, and tunnel magnetoresistance (c) as a function of the bias voltage obtained for the parameters: $\alpha =1$, $q=0.25$, $E=10k_{B}T$, while the other parameters are as in Fig. 2.

Figure 7

(Color online) Current (a) and differential conductance (b) in the parallel (solid line) and antiparallel (dashed line) magnetic configurations, and tunnel magnetoresistance (c), calculated as a function of the bias voltage for $\alpha =0.25$. The other parameters are the same as in Fig. 6.

Figure 8

(Color online) Tunnel magnetoresistance as a function of the bias voltage and the average level position, calculated for $\alpha =0.25$. The other parameters are the same as in Fig. 7.

Figure 9

(Color online) Bias voltage dependence of TMR for indicated values of the asymmetry parameter $\alpha$ and for $q=0$ (a) and $q=0.25$ (b), calculated for $E=10k_{B}T$. The other parameters are the same as in Fig. 2.

Figure 10

(Color online) (a) Tunnel magnetoresistance as a function of the bias voltage and parameter $q$, calculated for $\alpha =0.25$. (b) Cross sections of TMR for several values of $q$ as indicated in the figure. The other parameters are the same as in Fig. 6.

Figure 11

(Color online) Bias voltage dependence of TMR calculated for $q=1$ and for different values of the asymmetry parameter $\alpha$, as indicated in the figure. The other parameters are the same as in Fig. 2.

Figure 12

Graphical representation of the self-energy ${\Sigma }_{|0,\sigma ⟩|0,\sigma ⟩}^{|\sigma ,0⟩|\sigma ,0⟩}$ (a). The summations are over lead and spin degrees of freedom $\alpha =L,R$ and ${\sigma }^{\prime }=\uparrow ,\downarrow$. Each tunneling line carries lead index $\alpha$, spin $\sigma$, and frequency $\omega$.

Figure 13

Graphical equation for the self-energy ${\Sigma }_{|0,\sigma ⟩|\sigma ,0⟩}^{|\sigma ,0⟩|\sigma ,0⟩}$.