Two-stage Kondo effect in T-shaped double quantum dots with ferromagnetic leads

The linear-response transport properties of a T-shaped double quantum dot strongly coupled to external ferromagnetic leads are studied theoretically by using the numerical renormalization group method. It is shown that when each dot is occupied by a single electron, for antiparallel alignment of leads' magnetizations, the system exhibits the two-stage Kondo effect. For parallel alignment, however, the second stage of the Kondo effect becomes suppressed due to the presence of ferromagnetic-contact-induced exchange field. The difference between the two magnetic configurations results in highly nontrivial behavior of the tunnel magnetoresistance, which for some parameters can take giant values. In addition, the dependence of the linear conductance and tunnel magnetoresistance on external magnetic field, the double-dot levels' position, and the spin polarization of the leads is thoroughly analyzed. It is shown that the second stage of the Kondo effect can be restored by fine-tuning of the magnetic field or the dots' levels. The effect of spin-dependent tunneling on the low-temperature transition from the high to low conducting state of the system, which occurs when changing the hopping between the dots, is also discussed.

Figure 1

Schematic of a T-shaped double quantum dot coupled to ferromagnetic leads. The first dot is directly coupled to the leads with the coupling strengths ${\Gamma }_{L\sigma }$ and ${\Gamma }_{R\sigma }$, while the second dot is coupled to the first one via the hopping matrix element $t$. The magnetizations of the leads can form either parallel or antiparallel magnetic configuration.

Figure 2

The linear response conductance $G$ as a function of temperature $T$ for different values of the hopping $t$ between the two dots in the case of nonmagnetic leads. The parameters are $U=0.5,\Gamma =U/5,{\varepsilon }_1={\varepsilon }_2=-U/2$, and $p=0$.

Figure 3

The linear conductance $G$ in the parallel (a) and antiparallel (b) magnetic configuration and the resulting TMR (c) as a function of temperature. The parameters are $U=0.5,\Gamma =0.1,{\varepsilon }_1={\varepsilon }_2=-U/2$, and $p=0.4$. The vertical lines correspond to temperatures $30T^{*},T^{*}$, and $T^{*}/30$ in the case of $t=\Gamma /3$, indicated for further reference.

Figure 4

The same as in Fig. 3 calculated in the absence of particle-hole symmetry. The left column corresponds to ${\varepsilon }_1=-U/2$ and ${\varepsilon }_2=-U/3$, while the right column is calculated for the case of ${\varepsilon }_1=-U/3$ and ${\varepsilon }_2=-U/2$.

Figure 5

The linear conductance $G$ in parallel (a) and antiparallel (b) magnetic configuration and the TMR (c) plotted as a function of temperature in the case of ${\varepsilon }_1={\varepsilon }_2=-U/2$ for different spin polarizations of the leads. The parameters are the same as in Fig. 3, except for $t=\Gamma /3$ and $p$ indicated in the figure.

Figure 6

The same as in Fig. 5 calculated for ${\varepsilon }_1=-U/2$ and ${\varepsilon }_2=-U/3$ (left column), and ${\varepsilon }_1=-U/3$ and ${\varepsilon }_2=-U/2$ (right column).

Figure 7

The dependence of the low-temperature value of the linear conductance in the parallel configuration, $G^{\mathrm{P}}$, on the spin polarization of the leads $p$ for different DQD level positions, as indicated, and for $t=\Gamma /3$ and $T={10}^{-8}$. The other parameters are the same as in Fig. 3.

Figure 8

The low-temperature linear conductance in the antiparallel (a) and parallel (b)–(d) configurations as a function of hopping between the dots $t$ for different spin polarization $p$, as indicated. In the case of (a) and (b) ${\varepsilon }_1={\varepsilon }_2=-U/2$, (c) ${\varepsilon }_1=-U/2$ and ${\varepsilon }_2=-U/3$, and (d) ${\varepsilon }_1=-U/3$ and ${\varepsilon }_2=-U/2$. The parameters are the same as in Fig. 3 with $T={10}^{-8}$. The vertical line indicates the value of $t=\Gamma /3$ that was used in Figs. 5 and 6.

Figure 9

The linear conductance in the parallel (a) and antiparallel (b) magnetic configuration and the TMR (c) as a function of magnetic field for different values of the hopping between the dots, as indicated, and for ${\varepsilon }_1={\varepsilon }_2=-U/2$. The other parameters are the same as in Fig. 3 with $T={10}^{-8}$.

Figure 10

The same as in Fig. 9 calculated for ${\varepsilon }_1=-U/2$ and ${\varepsilon }_2=-U/3$. Note the logarithmic scale for positive and negative magnetic fields.

Figure 11

The same as in Fig. 9 calculated for ${\varepsilon }_1=-U/3$ and ${\varepsilon }_2=-U/2$. Note the logarithmic scale for positive and negative magnetic fields.

Figure 12

The dependence of the linear conductance in the parallel configuration on magnetic field $\tilde{B}\equiv B-B_0$ around the point where $G^{\mathrm{P}}$ becomes suppressed in the case of ${\varepsilon }_1=-U/2,{\varepsilon }_2=-U/3$ (solid line), and ${\varepsilon }_1=-U/3,{\varepsilon }_2=-U/2$ (dashed line). Here, $B_0$ is the magnetic field at which $G^{\mathrm{P}}$ has local minimum. The inset presents the quadratic scaling of $G^{\mathrm{P}}$ as a function of $\tilde{B}$ for ${\varepsilon }_1=-U/2$ and ${\varepsilon }_2=-U/3$. The other parameters are the same as in Fig. 3 with $T={10}^{-8}$.

Figure 13

The linear conductance in the parallel (left column) and antiparallel (right column) magnetic configuration as a function of DQD levels calculated at (a)–(b) $T=T^{*}/30$, (c)–(d) $T=T^{*}$, (e)–(f) $T=30T^{*}$, with $T^{*}$ being the second-stage Kondo temperature for $t=\Gamma /3,T^{*}\approx 6\times {10}^{-5}$. The parameters are the same as in Fig. 3 with $t=\Gamma /3$.

Figure 14

The TMR as a function of DQD levels calculated at (a) $T=T^{*}/30$, (b) $T=T^{*}$, (c) $T=30T^{*}$, with $T^{*}\approx 6\times {10}^{-5}$. Parameters are the same as in Fig. 3 with $t=\Gamma /3$. The black areas in insets mark the regions where TMR is negative.