Spin vibronics in interacting nonmagnetic molecular nanojunctions

We show that in the presence of ferromagnetic electronic reservoirs and spin-dependent tunnel couplings, molecular vibrations in nonmagnetic single molecular transistors induce an effective intramolecular exchange magnetic field. It generates a finite spin accumulation and precession for the electrons confined on the molecular bridge and occurs under (non)equilibrium conditions. The effective exchange magnetic field is calculated here to lowest order in the tunnel coupling for a nonequilibrium transport setup. Coulomb interaction between electrons is taken into account as well as a finite electron-phonon coupling. We show that for realistic physical parameters, an effective spin-phonon coupling emerges. It is induced by quantum many-body interactions, which are either of electron-phonon or Coulomb type. We investigate the precession and accumulation of the confined spins as function of bias and gate voltages as well as their dependence on the angle enclosed by the magnetizations between the left and right reservoir.

Figure 1

(a) Sketch of the Anderson-Holstein spin-valve geometry. The spinful electronic level with energy ${\varepsilon }_{\sigma }$ is coupled to a single mechanical harmonic mode with frequency $\mathrm{\Omega }$ with a coupling constant $\lambda$. The electrons are able to tunnel from the spin-polarized leads to the nonmagnetic device due to finite tunneling couplings ${\mathrm{\Gamma }}_{L/R\sigma }$. The left and right ferromagnets have different magnetizations, which are characterized by an angle ${\varphi }_{\alpha },\alpha =L/R$. (b) The underlying coordinate system where $\varphi$ is the relative angle between the two directions ${\mathbf{n}}_{L/R}$ of the left and the right magnetizations.

Figure 2

Effective exchange magnetic field in units of $\mathrm{\Gamma }$ induced by the electron-phonon and Coulomb interaction. The color scale shows the sum $B_{\mathrm{ex}}=B_L+B_R$. Panel (a) shows the result for pure Coulomb interaction $U=30k_{B}T$ without electron-phonon coupling $\lambda =0$. In (b), the electron-phonon coupling is set to $\lambda =10k_{B}T,\mathrm{\Omega }=5k_{B}T$, while $U=0$. The finite $\lambda$ broadens the resonances, shifts it to larger values of the gate voltage, and reverses its direction as compared to panel (a). In (c), both interactions are present. The remaining parameters are $k_{B}T=\mathrm{\Gamma }$, $\varphi =\pi /2$, and $p=0.9$.

Figure 3

Interaction-induced spin accumulation illustrated by the spin expectation values $\langle {\sigma }_x\rangle$ (left column), $\langle {\sigma }_y\rangle$ (middle column), and $\langle {\sigma }_z\rangle$ (right column) for varying bias and gate voltages and for $U=30k_{B}T$. The upper row shows the results for pure Coulomb interaction without electron-phonon coupling $\lambda =0$. The lower row refers to the case with an additional electron-phonon interaction $\lambda =10k_{B}T,\mathrm{\Omega }=5k_{B}T$. The remaining parameters are as in Fig. 2.

Figure 4

Same as in Fig. 3, but as a function of the enclosed angle $\varphi$ for a single-particle energy ${\epsilon }_0=30k_{B}T$ and for $U=30k_{B}T$. The upper row shows the results for $\lambda =0$, while the lower row shows the case with an additional electron-phonon interaction $\lambda =10k_{B}T,\mathrm{\Omega }=5k_{B}T$.

Figure 5

Differential conductance of the Anderson-Holstein spin valve as a function of the bias and the gate voltage in units of $G_{\mathrm{tunn}}=(e^2/h)\mathrm{\Gamma }/\left(2\pi k_{B}T\right)$. In (a), the noninteracting case is shown: $U=\lambda =0$, while (b) refers to $U=30k_{B}T,\lambda =0$. In panel (c), we set $U=0,\lambda =10k_{B}T$, and in (d), we have chosen $U=30k_{B}T,\lambda =10k_{B}T,\mathrm{\Omega }=5k_{B}T$. All calculations have been performed with $\mathrm{\Omega }=5k_{B}T$, $p=0.9$, and $\varphi =\pi /2$.

Figure 6

Differential conductance in the crossover regime (see text) for $U=30k_{B}T,\lambda =\mathrm{\Omega }=10k_{B}T$. The remaining parameters are as in Fig. 5.

Figure 7

Angular dependence of the differential conductance for $U=30k_{B}T$ without electron-phonon coupling $\lambda =0$ in (a), and with $\lambda =10k_{B}T,\mathrm{\Omega }=5k_{B}T$ in (b). The remaining parameters are as in Fig. 5.

Figure 8

(a) Comparison of the differential conductance for equilibrated and nonequilibrated phonon modes. Within a voltage range $\left|eV\right|\le 20\mathrm{\Gamma }$ we do not find significant deviations from the calculation where the phonon is assumed to be in equilibrium. (b) Same as in (a) but for the nontrivial spin projection along the $z$-axis. We note that $\langle {\sigma }_z\rangle \ne 0$ reflects the finite exchange magnetic field from the leads. Other parameters are chosen as $\lambda =2.5\mathrm{\Gamma },k_{B}T=\mathrm{\Gamma },\varphi =\pi /2,\mathrm{\Omega }=5\mathrm{\Gamma },\epsilon =-40\mathrm{\Gamma }$.