Control of vibrational states by spin-polarized transport in a carbon nanotube resonator

We study spin-dependent transport in a suspended carbon nanotube quantum dot in contact with two ferromagnetic leads and with the dot's spin coupled to the flexural mechanical modes. The spin-vibration interaction induces spin-flip processes between the two energy levels of the dot. This interaction arises from the spin-orbit coupling or a magnetic field gradient. The inelastic vibration-assisted spin flips give rise to a mechanical damping and, for an applied bias voltage, to a steady nonequilibrium occupation of the harmonic oscillator. We analyze these effects as function of the energy-level separation of the dot and the magnetic polarization of the leads. Depending on the magnetic configuration and the bias-voltage polarity, we can strongly cool a single mode or pump energy into it. In the latter case, we find that within our approximation, the system approaches eventually a regime of mechanical instability. Furthermore, owing to the sensitivity of the electron transport to the spin orientation, we find signatures of the nanomechanical motion in the current-voltage characteristic. Hence, the vibrational state can be read out in transport measurements.

Figure 1

Schematic views of a carbon nanotube quantum dot suspended between two ferromagnetic leads. (a) The spin-vibration interaction can be either induced by the intrinsic spin-orbit coupling ${\Delta }_{\text{SO}}$ or by a magnetic gradient $\partial B/\partial x$. (b) Due to the spin-vibration interaction, the dot spin's component ${\widehat{\sigma }}_x$ parallel to the mechanical displacement $u$ couples to the flexural mode. The local tangent vector is denoted by $\mathbf{t}$.

Figure 2

Spectrum of the Hamiltonian for a defect-free carbon nanotube quantum dot. The inset shows the full spectrum as a function of the magnetic field along the nanotube axis as given by the Hamiltonian (3). The circle in the inset points out the crossing point between the two levels reported in the main panel. We focus on the electron transport in which only two levels of energies ${\varepsilon }_{+}$ and ${\varepsilon }_{-}$ are involved. They have the same orbital state and opposite spin. The sketches illustrate the direction of the orbital (large green arrow) and spin (small magenta arrow) magnetic moments along the $z$ axis. The parameters are ${\Delta }_{\text{SO}}=170 \mu \mathrm{eV}$ and ${\mu }_{\mathrm{orb}}=330 \mu \mathrm{eV}/\mathrm{T}$ from Ref. [63].

Figure 3

Leading-order diagrams corresponding to the perturbation expansion of the phonon Green's function $\check{D}\left(\varepsilon \right)$ (a) and the electronic Green's function $\check{\mathcal{G}}\left(\varepsilon \right)$ (b). The plane lines indicate the electronic Green's function $\check{G}\left(\varepsilon \right)$ for vanishing spin-vibration interaction. The dashed lines represent the bare phonon Green's functions. The filled circle is the vertex for the spin-vibration interaction with coupling constant $\lambda$ which couples electronic Green's functions of opposite spin with a phonon Green's function.

Figure 4

Phonon occupation $n$ as function of the bias voltage $eV$ and gate voltage ${\varepsilon }_0$. The parameters are $p_l=-1$ and $p_r=1,{\Gamma }_l^{}={\Gamma }_r^{}=0.2\omega$, and $T=10\omega$. White color corresponds to $n_B\left(\omega \right)\approx 9.5$. Here, we assume a vanishing external damping ${\gamma }_0=0$, a large spin splitting ${\varepsilon }_z=10T=100\omega$, and the chemical potential fixed to ${\mu }_r={\varepsilon }_0-eV$ and ${\mu }_l={\varepsilon }_0$. The instability regions (in gray) correspond to $\gamma <0$ and the dashed (black) line corresponds to the analytical formula for the threshold $\gamma =0$ (see text).

Figure 5

Schematic picture of the energy levels, Fermi functions, and the spin-flip processes with rate ${\gamma }_{lr}^{s\sigma }$ for fully polarized ferromagnets. In (a)–(d), a single level contributes to the inelastic transport which is characterized by the absorption (upwards blue arrows) or the emission (downward red arrows) of a vibrational energy quantum. In (e) and (f), the resonant condition ${\varepsilon }_z=\omega$ is fulfilled. When the transport is mainly determined by the process shown in (e), optimal ground-state cooling of the oscillator is achieved. On the contrary, when the transport is dominated by the process shown in (f), a strong heating occurs which is the precursor of a mechanical instability (see also Fig. 6).

Figure 6

Phonon occupation $\overline{n}$ as function of the bias voltage $eV$ and gate voltage ${\varepsilon }_0$. We consider the resonant regime ${\varepsilon }_z=\omega$ with ${\gamma }_0={10}^{-5}\omega ,\lambda /\omega =0.01$, and ${\mu }_{l,r}={\varepsilon }_0\pm eV/2$. The other parameters are $p_l=-1$ and $p_r=1,{\Gamma }_l^{}={\Gamma }_r^{}=0.2\omega$, and $T=10\omega$. White color corresponds to $n_B\left(\omega \right)\approx 9.5$. The instability region (in gray) corresponds to ${\gamma }_{\mathrm{tot}}<0$ and the black dashed line shows the analytic formula for the threshold ${\gamma }_{\mathrm{tot}}=0$ (see text). The upper and lower sketches indicate the schematic behavior of the energy levels and the inelastic vibration-assisted spin-flip processes which lead to cooling for $eV>0$ and heating for $eV<0$, respectively. Absorption (emission) of a vibrational energy quantum occurring in resonance is shown as blue (red) bold wiggled arrows.

Figure 7

Minimal phonon occupation on the surface $({\varepsilon }_0,eV)$ for a polarized left lead with $-1\le p_l<0$ and a normal right lead as a function of ${\varepsilon }_z/\omega$. The parameters are ${\Gamma }_l={\Gamma }_r=0.2\omega ,T=10\omega ,Q={10}^4$, and $\lambda =0.05\omega$. The minimum of ${\overline{n}}_{\min }$ is approached at resonance ${\varepsilon }_z=\omega$ with ${\overline{n}}_{\min }\simeq 0.2$ for a fully polarized left lead $p_l=-1$.

Figure 8

Elastic correction to the linear conductance at $T=0$, symmetric coupling ${\Gamma }_l={\Gamma }_r=\Gamma ,\omega =5\Gamma$, and zero chemical potentials. (a) Parallel magnetization configuration $\left(p_r=p_l=0.8\right)$. (b) Antiparallel configuration $\left(p_r=-p_l=0.8\right)$. The different magnetization configurations in (a) and (b) lead to $G_{\mathrm{ec}}\left({\varepsilon }_0\right)\ne G_{\mathrm{ec}}(-{\varepsilon }_0)$ in (a) whereas the correction is symmetric $G_{\mathrm{ec}}\left({\varepsilon }_0\right)=G_{\mathrm{ec}}(-{\varepsilon }_0)$ in (b). In the range ${\varepsilon }_{+}{\varepsilon }_{-}<0$ the correction to the conductance is negative (see text).

Figure 9

Inelastic contribution to differential conductance at zero temperature and ${\varepsilon }_0=2\omega ,{\Gamma }_l={\Gamma }_r=0.2\omega$, and symmetrically applied voltage. In (a), the polarization of the ferromagnetic leads are aligned parallel $p_l=p_r=0.4$. In (b), we show the antiparallel configuration with $p_r=-p_l=0.4$. The peaks of the inelastic differential conductance appear at voltages $eV/2={\varepsilon }_{\pm }$ and $eV/2={\varepsilon }_{\pm }+\omega$ (see text).

Figure 10

Inelastic current for fully polarized antiparallel ferromagnets $\left(p_r=-p_l=1\right)$ at resonance ${\varepsilon }_z=\omega ,T=10\omega$, and $\Gamma =0.2\omega$. (a) Equilibrated vibration with $\overline{n}=n_B\left(\omega \right)$. (b) Unequilibrated vibration with a coupling constant $\lambda =0.01\omega$ and an intrinsic damping of ${\gamma }_0={10}^{-5}\omega$. The nonequilibrium phonon occupation $\overline{n}$ corresponding to the inelastic current in (b) is shown in Fig. 6. For $eV>0$, the oscillator is strongly cooled $\overline{n}\ll n_B\left(\omega \right)$, leading to a suppression of the inelastic current in (b) compared to the case of equilibrated vibration in (a).

Figure 11

Current for equilibrated (a) and unequilibrated vibration (b) for fully antiparallel polarized ferromagnets $p_r=-p_l=1,T=10\omega ,{\varepsilon }_0=0,\Gamma =0.2\omega ,\lambda =0.01\omega$, and ${\gamma }_0={10}^{-5}\omega$. For $eV>0$, the current in (b) is suppressed compared to the current in (a). At negative voltages in (b), the oscillator approaches the mechanical instability and sharply decreases. In (c) and (d), we show the differential conductance $d{I}_{\mathrm{in}}/dV$ corresponding to (a) and (b), respectively.

Figure 12

Current and differential conductance for equilibrated [(a) and (c)] and unequilibrated vibration [(b) and (d)] at polarization $p_r=-p_l=0.5,T=10\omega ,{\varepsilon }_0=0,\Gamma =0.2\omega ,\lambda =0.2\omega$, and ${\gamma }_0={10}^{-5}\omega$. In (b) the current at $eV<0$ decreases since the oscillator approaches the mechanical instability and the phonon occupation strongly increases. At $eV>0$, the current is suppressed compared to the current in (a) since $\overline{n}<n_B\left(\omega \right)$. In (c) and (d), we show the differential conductance $dI/dV$ corresponding to (a) and (b), respectively.