Cooling a Magnetic Nanoisland by Spin-Polarized Currents

We investigate cooling of a vibrational mode of a magnetic quantum dot by a spin-polarized tunneling charge current exploiting the magnetomechanical coupling. The spin-polarized current polarizes the magnetic nanoisland, thereby lowering its magnetic energy. At the same time, Ohmic heating increases the vibrational energy. A small magnetomechanical coupling then permits us to remove energy from the vibrational motion and cooling is possible. We find a reduction of the vibrational energy below 50% of its equilibrium value. The lowest vibration temperature is achieved for a weak electron-vibration coupling and a comparable magnetomechanical coupling. The cooling rate increases at first with the magnetomechanical coupling and then saturates.

Figure 1

Schematic consisting of ferromagnetic leads, a quantum dot with a localized magnetic moment, and a single vibrational mode.

Figure 2

(a) Polarization $⟨J_z⟩/\hslash$ of the localized magnetic moment along the $z$ direction, (b) charge current $⟨I\hslash /e\mathrm{\Gamma }⟩$, (c) vibrational energy $⟨H_{\text{ph}}⟩/{\epsilon }_0$, and (d) probability $P_1$ to find one electron on the quantum dot as a function of time for the three different lead setups. The remaining parameters are $p=0.9$, $\hslash \mathrm{\Gamma }=0.01{\epsilon }_0$, $B=0.7{\epsilon }_0/(g{\mu }_B)$, $\hslash \omega =0.75{\epsilon }_0$, $k_{B}T=2{\epsilon }_0$, $q=0.4{\epsilon }_0$, $\xi =0.06{\epsilon }_0$, $\lambda =0.2{\epsilon }_0$, and $eV=1.2{\epsilon }_0$.

Figure 3

Effective vibrational temperature $T_{\text{ph}}$ in the stationary limit versus electron-vibration coupling $\lambda$ and magnetization-vibration coupling $\xi$. Inset: Vibrational energy $⟨H_{\text{ph}}⟩$ versus time for different electron-vibration coupling strengths $\lambda$. The remaining parameters are as in Fig. 2.

Figure 4

Effective vibrational temperature $T_{\text{ph}}$ in the stationary limit versus magnetic field $g{\mu }_{B}B$ and spin-magnetization coupling $q$. The used parameters are $\xi =0.12{\epsilon }_0$, $\lambda =0.1{\epsilon }_0$, with the remaining parameters as in Fig. 2.

Figure 5

(a) Vibrational cooling rate ${\mathrm{\Gamma }}_{\text{cool}}$ versus the magnetomechanical coupling $\xi$ for two electron-vibration couplings $\lambda =0.1{\epsilon }_0$ and $\lambda =0.2{\epsilon }_0$. (b) Effective vibrational temperature in the initial, thermalized state ($T_{\text{init}}$, $V=0$, red dashed line) and in the final steady state ($T_{\text{ph}}$, $V>0$, blue solid line) as a function of the lead temperature $T$ for $\xi =0.12{\epsilon }_0$ and $\lambda =0.1{\epsilon }_0$. The remaining parameters are as in Fig. 2.