Compensation effect in carbon nanotube quantum dots coupled to polarized electrodes in the presence of spin-orbit coupling

We study theoretically the Kondo effect in a carbon nanotube quantum dot attached to polarized electrodes. Since both spin and orbit degrees of freedom are involved in such a system, the electrode polarization contains the spin and orbit polarizations as well as the Kramers polarization in the presence of the spin-orbit coupling. In this paper we focus on the compensation effect of the effective fields induced by different polarizations by applying magnetic field. The main results are (i) while the effective fields induced by the spin and orbit polarizations remove the degeneracy in the Kondo effect, the effective field induced by the Kramers polarization enhances the degeneracy through suppressing the spin-orbit coupling; (ii) while the effective field induced by the spin polarization cannot be compensated for by applying a magnetic field, the effective field induced by the orbit polarization can be compensated for; and (iii) the presence of the spin-orbit coupling does not change the compensation behavior observed in the case without the spin-orbit coupling. These results are observable in an ultraclean carbon-nanotube quantum dot attached to ferromagnetic contacts under a parallel applied magnetic field along the tube axis and it would deepen our understanding on the Kondo physics of the carbon nanotube quantum dot.

Figure 1

Schematic diagram of the cotunneling process. (a) The configuration before the tunneling and (b) the configuration after the tunneling. There are six possible tunneling processes involving different spin and orbit flips in carbon nanotube quantum dot ($m\iff m^{\prime }$).

Figure 2

The spin-polarization effect and its compensation behavior in the absence of the spin-orbit coupling. (a) The Kondo peak splitting due to the spin polarization and (b) the Kondo subpeaks evolution under the external magnetic field applied. For clarity, here and hereafter the curves have been shifted perpendicularly to show the change of the subpeak position with the polarization in (a) and the field in (b). The parameters used are $P^o=P^k=0$ and ${\Delta }_{\text{so}}=0$. In (b) the magnetic fields applied are $0,0.0025\Gamma ,0.005\Gamma$, and $0.0075\Gamma$ (from top to bottom).

Figure 3

The orbit-polarization effect and its compensation behavior. (a) The Kondo subpeaks splitting due to the orbit polarization and (b) the Kondo subpeaks evolution under the external magnetic field applied. The parameters used are $P^k=0$ and ${\Delta }_{\text{so}}=0$. In (b) the magnetic fields applied are $0,0.0025\Gamma ,0.005\Gamma ,0.006\Gamma ,0.0075\Gamma$ (from top to bottom). The dotted line denotes the complete compensation effect at $B=0.006\Gamma$.

Figure 4

The spin-polarization effect and its compensation behavior in the presence of the spin-orbit coupling. (a) The Kondo peak splitting due to the spin polarization at zero field and (b) the Kondo subpeaks evolution under the external magnetic field applied when $P^s=0.5$. The parameters used are $P^o=P^k=0$ and ${\Delta }_{\text{so}}=0.0075\Gamma$. In (b) the magnetic fields applied are $0,0.0025\Gamma ,0.005\Gamma ,0.0075\Gamma$ (from top to bottom).

Figure 5

The orbit-polarization effect in the presence of the spin-orbit coupling and its compensation behavior. (a) The Kondo subpeaks splitting due to the orbit polarization at zero field and (b) the Kondo subpeaks evolution under the external magnetic fields applied when $P^o=0.3$. The parameters used are $P^k=0$ and ${\Delta }_{\text{so}}=0.0075\Gamma$. In (b) the magnetic fields applied are $0,0.0025\Gamma ,0.005\Gamma ,0.006\Gamma ,0.0075\Gamma$ (from top to bottom). The dotted line denotes the compensation effect at $B=0.006\Gamma$.

Figure 6

The Kramers polarization effect in the presence of the spin-orbit coupling and its compensation behavior. (a) The Kondo subpeaks merging due to the Kramers polarization at zero field and (b) the Kond subpeaks evolution under the external magnetic fields applied with $P^k=0.2$. The parameter used are ${\Delta }_{\text{so}}=0.0075\Gamma$. In (b) the magnetic fields applied are $0,0.0025\Gamma ,0.005\Gamma ,0.006\Gamma ,0.0075\Gamma$ (from top to bottom). The dotted line denotes the compensation effect at $B=0.006\Gamma$.