Nonequilibrium Kondo effect in a quantum dot coupled to ferromagnetic leads

We study the Kondo effect in the electron transport through a quantum dot coupled to ferromagnetic leads, using a real-time diagrammatic technique that provides a systematic description of the nonequilibrium dynamics of a system with strong local electron correlations. We evaluate the theory in an extension of the “resonant tunneling approximation,” introduced earlier, by introducing the self-energy of the off-diagonal component of the reduced propagator in spin space. In this way we develop a charge and spin conserving approximation that accounts not only for Kondo correlations but also for the spin splitting and spin accumulation out of equilibrium. We show that the Kondo resonances, split by the applied bias voltage, may be spin polarized. A left-right asymmetry in the coupling strength and/or spin polarization of the electrodes significantly affects both the spin accumulation and the weight of the split Kondo resonances out of equilibrium. The effects are observable in the nonlinear differential conductance. We also discuss the influence of decoherence on the Kondo resonance in the frame of the real-time formulation.

Figure 1

A single level quantum dot coupled to two ferromagnetic leads for (a) parallel and (b) antiparallel alignments of lead magnetizations. We allow for a left-right asymmetry in tunnel barriers and/or spin polarization factors.

Figure 2

Diagrams for various tunneling processes: (i) quantum charge fluctuations, (ii) the cotunneling process, and (iii) the resonant tunneling process. The horizontal lines on the Keldysh contour represent the free propagators of the quantum dot. Remaining directed lines are tunneling lines.

Figure 3

(a) Dyson equation for ${\pi }_{\overline{\sigma },\overline{\sigma }}^{\sigma ,\sigma }$ and (b) the diagrammatic expansion of the irreducible self-energy ${\sigma }_{\overline{\sigma },\overline{\sigma }}^{\sigma ,\sigma }$. Directed dotted lines carry the energy $\omega$.

Figure 4

(a) The irreducible self-energy of the reduced propagator ${\pi }_{0,0}^{\sigma ,\sigma }$. (b) Diagrammatic equations for the vertex functions ${\Lambda }_{\chi ,0}^{+\chi ,\sigma }$ and ${\Lambda }_{\chi ,0}^{-\chi ,\sigma }$. An open dotted line connects to the upper/lower horizontal line for ${\Lambda }^{\pm }$. Shaded boxes represent the off-diagonal component of the reduced propagator in the spin space ${\pi }_{\overline{\sigma },\overline{\sigma }}^{\sigma ,\sigma \left(1\right)}$.

Figure 5

(a) The nonlinear differential conductance and (b) the local magnetization $M$ for the P alignment $(P_L=P_R=0.1)$ and for symmetric coupling ${\Gamma }_L={\Gamma }_R$. Solid, dashed, dotted, and dot-dashed lines are results for various temperatures $T∕\Gamma =0.005$, 0.01, 0.025, and 0.05, respectively. Parameters are ${ϵ}_0∕\Gamma =-2$, $D∕\Gamma =50$.

Figure 6

(a) The linear conductance $G^{\mathrm{lin}}$ and the local magnetization $M$ as a function of the Zeeman energy $E_{\mathrm{Z}}$ for the P alignment $(P_L=P_R=0.2)$. In panel (a), the spin-resolved conductance is plotted by a dashed (dotted) line for up (down) spin. The equilibrium DOS for (c) $E_{\mathrm{Z}}∕\Gamma =0$ and (d) $E_{\mathrm{Z}}∕\Gamma =0.205$. Solid and dashed (dotted) lines show the total DOS and the spin resolved DOS for up (down) spin. Insets in these panels are blowups of the Kondo resonance. Parameters are $T∕\Gamma =0.005$, ${ϵ}_0∕\Gamma =-2$, $D∕\Gamma =50$.

Figure 7

The nonlinear differential conductance $G$, (a) and (c), and the local magnetization $M$, (b) and (d), for a symmetric coupling ${\Gamma }_L={\Gamma }_R$. Solid lines and dashed lines in (a) and (c) are plots for $E_{\mathrm{Z}}=0$ and $E_{\mathrm{Z}}=0.205\Gamma$, respectively. The solid lines in (a) and (c) are for the P alignment $(P_L=P_R=0.2)$ and the AP alignment $(P_L=-P_R=0.2)$, respectively. The dashed lines in (a) and (c) are for the AP alignment and the P alignment, respectively. The average occupation is $N\approx 0.89$ for all parameter regimes. The panels (e) and (f) show the corresponding DOS $\rho \left(\omega \right)$ of the solid and the dashed lines in panel (c) at $\mathrm{eV}∕\Gamma =0.205$. The dashed and dot-dashed lines are for ${\rho }_{\uparrow }\left(\omega \right)$ and ${\rho }_{\downarrow }\left(\omega \right)$, respectively. Panels (g) and (h) show plots of the DOS $\rho \left(\omega \right)$ of the dashed line in panel (a) at $\mathrm{eV}=E_{\mathrm{Z}}$ (g), and at $\mathrm{eV}=-E_{\mathrm{Z}}$ (h). The other parameters are the same as for Fig. 6.

Figure 8

The tunnel magnetoresistance TMR as a function of the bias voltage $V$. Spin polarization factors are $P_L=\mid P_R\mid =0.2$. The further parameters are the same as for Fig. 6.

Figure 9

(a) The differential conductance for the P alignment $G_{\mathrm{P}}\left(V\right)$ $(P_L=P_R=0.2)$. Solid lines, dashed lines and dotted lines, are results for ${\Gamma }_R∕{\Gamma }_L=1$, 0.5, and 0.25. (b) The differential conductance $G_{\mathrm{P}}\left(V\right)$ for ${\Gamma }_R∕{\Gamma }_L=0.5$ with the asymmetry factor $a_c=0$ (dashed line) and 0.5 (dot-dashed line). (c) The differential conductance with the AP alignment $G_{\mathrm{AP}}\left(V\right)$ $(P_L=-P_R=0.2)$ and (d) the nonequilibrium local magnetization $M$. (e), (f) The total DOS $\rho \left(\omega \right)$ in the AP alignment: (e) The solid (dashed) line is for ${\Gamma }_R∕{\Gamma }_L=0.5$ at $\mathrm{eV}∕\Gamma =-0.08$ (0.07). (f) The solid (dashed) line is for ${\Gamma }_R∕{\Gamma }_L=0.25$ at $\mathrm{eV}∕\Gamma =-0.13$ (0.11). The other parameters are the same as for Fig. 6.

Figure 10

(a) The nonlinear conductance $G\left(V\right)$ and (b) the local magnetization $M$ for various spin polarizations ($P_L$, $P_R$) and for symmetric coupling ${\Gamma }_R={\Gamma }_L$. The average value of the polarization factors in the left and right leads are fixed as $(P_L+P_R)∕2=0.1$. The solid, dashed, and dotted lines are for $P_L=0.1$, 0.4, and 0.6, respectively. The total DOS $\rho \left(\omega \right)$ at $\mathrm{eV}∕\Gamma =-0.1$ (solid line) and $\mathrm{eV}∕\Gamma =0.1$ (dashed line) for $P_L=0.4$ (c) and $P_L=0.6$ (d). The other parameters are the same as for Fig. 6.

Figure 11

The four additional diagrams of second order expansion for the irreducible self-energy ${\sigma }_{\overline{\sigma },\overline{\sigma }}^{\sigma ,\sigma }$. One bubble dresses each horizontal line. The other two diagrams, where one bubble is inside the other bubble, are depicted in Fig. 3b. Here, dashed lines carry the energy ${ϵ}_{\sigma }-{ϵ}_{\overline{\sigma }}$.

Figure 12

(a) The off-diagonal component of the reduced propagator in spin space within NCA. Double lines on the Keldysh contour are determined by Dyson equations in (b).

Figure 13

(a) The total and spin resolved DOS for parallel alignment $(P_L=P_R=0.2)$ within NCA for ${\pi }_{\overline{\sigma },\overline{\sigma }}^{\sigma ,\sigma }$ (solid lines). The panel (b) is the magnification around the Kondo resonance. Dotted lines are the results without decoherence effect and correspond to lines in Fig. 6c. Parameters are the same as those in Fig. 6.