Coulomb-promoted spintromechanics in magnetic shuttle devices

Exchange forces on the movable dot (“shuttle”) in a magnetic shuttle device depend on the parity of the number of shuttling electrons. The performance of such a device can therefore be tuned by changing the strength $U$ of Coulomb correlations to block or unblock parity fluctuations. We show that by increasing $U$ the spintromechanics of the device crosses over, at $U=U_c\left(T\right)$, from a mechanically stable regime to a regime of spin-induced shuttle instabilities (neglecting electric forces). This is due to enhanced spin-dependent mechanical forces as parity fluctuations are reduced by a Coulomb blockade of tunneling and demonstrates that single-electron manipulation of single-spin controlled nanomechanics is possible.

Figure 1

Mechanism of Coulomb-promoted spintromechanics: A spin-up electron ($\uparrow$) loaded to the shuttle (lower left [red] dot) from the fully polarized source electrode ($S$) interacts with the magnetizations in the leads to create a magnetic exchange force that attracts the shuttle to the source. An external magnetic field “rotates” the spin into a spin-down state ($\downarrow$), thereby reversing the sign of the exchange force so that the shuttle (lower right [blue] dot) is pushed towards the drain ($D$). In the case of a full Coulomb blockade ($U^{*}\to \infty$, see text), which prevents double occupation, the shuttle becomes mechanically active and energy is pumped into the mechanical shuttle vibrations. With a partial Coulomb blockade double occupation of the shuttle occurs with probability $exp(-U^{*}/T)$, where $T$ is the temperature, leading to electron parity fluctuations, which are detrimental to the energy pumping mechanism.

Figure 2

Characteristics of the mechanical dot vibrations as determined by Eqs. (2) and (A24) for the NEM device sketched in Fig. 1: (a) The envelope of the dot vibrations (in units of the tunneling length $\lambda$) as a function of time $t$ (in units of $\hslash /{\mathrm{\Gamma }}_N$, where ${\mathrm{\Gamma }}_N={\mathrm{\Gamma }}_L{\mathrm{\Gamma }}_R/({\mathrm{\Gamma }}_L+{\mathrm{\Gamma }}_R)$ is the width of the energy level on the dot, ${\mathrm{\Gamma }}_{L,R}$ being partial widths due to tunneling to the left and right electrodes) plotted for three different values of the Coulomb blockade energy $U$ at $T=5{\mathrm{\Gamma }}_N$ and $eV=100{\mathrm{\Gamma }}_N$. For small values of $U$ spontaneous vibrations are damped out (left panel), while for large $U$ they are amplified and develop into sustained finite-amplitude vibrations (right panel). A crossover between the two regimes occurs at $U=U_c$, when the vibration amplitude stays constant over time (middle panel). (b) Temperature dependence of the critical value of the Coulomb blockade energy $U_c\left(T\right)$ for $eV=10{\mathrm{\Gamma }}_N$ and symmetric tunnel junction $\gamma =1$ (left panel), for $eV=100{\mathrm{\Gamma }}_N$ and different values of the asymmetry parameter $\gamma ={\mathrm{\Gamma }}_R/{\mathrm{\Gamma }}_L$ (right panel). The results were obtained for the following parameters: vibration frequency $\omega =1{\mathrm{\Gamma }}_N/\hslash$, external magnetic field $H=0.5{\mathrm{\Gamma }}_N/{\mu }_B$, magnetic exchange energy $J=1.5{\mathrm{\Gamma }}_N$, dot level detuning energy $({\varepsilon }_0-{\varepsilon }_F)=2{\mathrm{\Gamma }}_N$, and the electromechanical coupling constant $\kappa ={\hslash }^{2}J/\left(ml\lambda {\mathrm{\Gamma }}_N^2\right)=0.09$, where $\kappa$ is the coefficient multiplying r.h.s of Eq. (2) written in terms of dimensionless variables $x$ and $t$. Such a value of $\kappa$ is in reasonable agreement with characteristic masses of macromolecules.

Figure 3

Temperature dependence of the critical value of the Coulomb blockade energy $U_c\left(T\right)$ for a device with a strongly biased drain electrode ${\varepsilon }_F-{\mu }_R\gg T$, for ${\mu }_L-{\varepsilon }_F=50{\mathrm{\Gamma }}_N$ and different values of the asymmetry parameter $\gamma$. Other parameters are the same as in Fig. 2.