Rotational directionality via symmetry-breaking in an electrostatic motor

We consider a rotor composed of insulating arms and electronic islands A and B placed at their ends as sketched in figure 1. The rotor is immersed in a region with a homogeneous, time-independent electric field ${\bf{E}}$, directed along the horizontal axis. To specify the orientation of the rotor, we use the angle θ between the rotor arm connected to A and the horizontal line as indicated in figure 1. We allow the two arms of the rotor to be tilted with respect to each other, and denote the angle between them by β. The islands can host charges ${Q}_{{\rm{A}}}$ and ${Q}_{{\rm{B}}}$ up to a certain limit ${Q}_{{\rm{max}}}$ (chosen to be equal for both islands) which is given by their capacitances ${C}_{{\rm{A}}}={C}_{{\rm{B}}}$. As in [11], we start by considering the dynamics using a mean-field approach where charge fluctuations are neglected and the charge dynamics is purely deterministic. This mean-field approach allows one already to understand the fundamentals of the symmetry-breaking and is strictly valid for macroscopic devices (see appendix B).The hopping-process for the tunneling, which is more suitable to nanoelectromechanical devices, will be considered in section 5. The equations that describe the charge dynamics on the islands in the mean-field the are given by

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{Q}_{{\rm{A}}}}{{\rm{d}}t}={W}_{{\rm{in}}}(\theta )({Q}_{{\rm{max}}}-{Q}_{{\rm{A}}})-{W}_{{\rm{out}}}(\theta ){Q}_{{\rm{A}}},\end{eqnarray} \tag{ 1a }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{Q}_{{\rm{B}}}}{{\rm{d}}t}={W}_{{\rm{in}}}(\theta +\beta )({Q}_{{\rm{max}}}-{Q}_{{\rm{B}}})-{W}_{{\rm{out}}}(\theta +\beta ){Q}_{{\rm{B}}},\end{eqnarray} \tag{ 1b }$$

where t is time, ${W}_{{\rm{in}}}(\theta ^{\prime} )$ and ${W}_{{\rm{out}}}(\theta ^{\prime} )$ are charging and discharging rates, respectively, and their argument $\theta ^{\prime} =\theta$ for the island A and $\theta ^{\prime} =\theta +\beta$ for the island B. We assume ${W}_{{\rm{in/out}}}(\theta )$ to be positive even functions (their functional dependence on the angles θ and $\theta +\beta$ is going to be specified in section 2.3). In the following we use dimensionless quantities. We define $\tau =t/{t}_{0}$ (t₀ is a characteristic time-scale for the electronic dynamics), ${q}_{{\rm{A}}}(t)={Q}_{{\rm{A}}}(t)/{Q}_{{\rm{max}}}$ and ${q}_{{\rm{B}}}(t)={Q}_{{\rm{B}}}(t)/{Q}_{{\rm{max}}}$ and ${w}_{{\rm{in/out}}}(\theta ^{\prime} )={t}_{0}{W}_{{\rm{in/out}}}(\theta ^{\prime} )$. Then the dimensionless version of equations (1a) and (1b) is

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{q}_{{\rm{A}}}}{{\rm{d}}\tau }={w}_{{\rm{in}}}(\theta )(1-{q}_{{\rm{A}}})-{w}_{{\rm{out}}}(\theta ){q}_{{\rm{A}}},\end{eqnarray} \tag{ 2a }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{q}_{{\rm{B}}}}{{\rm{d}}\tau }={w}_{{\rm{in}}}(\theta +\beta )(1-{q}_{{\rm{B}}})-{w}_{{\rm{out}}}(\theta +\beta ){q}_{{\rm{B}}}.\end{eqnarray} \tag{ 2b }$$

Since the rotor is placed within an electric field, the islands experience a force when they are charged. Within the mean-field approach one finds the following set of coupled equations for the dynamics of the mechanical degrees of freedom (For a derivation we refer to appendix A)

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial \tau }{\rm{\Lambda }}=-{\eta }_{0}\sqrt{2}(\mathrm{sin}\chi \mathrm{sin}(\theta ){q}_{{\rm{A}}}+\mathrm{cos}\chi \mathrm{sin}(\theta +\beta ){q}_{{\rm{B}}})-\gamma {\rm{\Lambda }},\end{eqnarray} \tag{ 3a }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial \tau }\theta ={\rm{\Lambda }}.\end{eqnarray} \tag{ 3b }$$

Here, we have introduced the dimensionless angular momentum ${\rm{\Lambda }}={t}_{0}L/I$, where L is the actual angular momentum and I is the moment of inertia of the rotor. The asymmetry parameter $\chi \in [0,\pi /2]$ is given by $\mathrm{tan}\chi ={l}_{{\rm{A}}}/{l}_{{\rm{B}}}$, with ${l}_{{\rm{A(B)}}}$ being the length of the arm containing island A(B). The driving strength ${\eta }_{0}\gt 0$ is

$$\begin{eqnarray}&&{\eta }_{0}=\sqrt{\displaystyle \frac{{l}_{{\rm{A}}}^{2}+{l}_{{\rm{B}}}^{2}}{2}}\left[\displaystyle \frac{{t}_{0}^{2}| {Q}_{{\rm{max}}}| E}{I}\right],\end{eqnarray} \tag{ 4 }$$

with $E=| \vec{E}|$ being the electric field strength. A surrounding environment leads to the phenomenological damping-term, whose strength is given by the dimensionless viscosity parameter γ. The dimensionless quantities can be seen in table 1.

Table 1.  Dimensionless quantities and their definitions.

Quantity	Symbol	Formula
Time	τ	$\tfrac{t}{{t}_{0}}$
Charge	${q}_{{\rm{A/B}}}$	$\tfrac{{Q}_{{\rm{A/B}}}}{{Q}_{{\rm{max}}}}$
Charging rate	${w}_{{\rm{in/out}}}(\theta )$	${t}_{0}{W}_{{\rm{in/out}}}(\theta )$
Angular momentum	Λ	$\tfrac{{t}_{0}L}{I}$
Asymmetry parameter	χ	$\mathrm{arctan}\left(\tfrac{{l}_{{\rm{A}}}}{{l}_{{\rm{B}}}}\right)$
Driving strength	${\eta }_{0}$	$\sqrt{\tfrac{{l}_{{\rm{A}}}^{2}+{l}_{{\rm{B}}}^{2}}{2}}\left[\tfrac{{t}_{0}^{2}| {Q}_{{\rm{max}}}| E}{I}\right]$

Note that ${q}_{{\rm{A}}}(t)$ and ${q}_{{\rm{B}}}(t)$ are determined by equations (2a) and (2b). Since the right-hand side of equations (2a) and (2b) depends on the angle θ we have to solve the nonlinear system of coupled equations (2a)–(3b).

We are interested in the influence of ICs on the steady-state behavior of the system. In particular we are interested in the time-average of the angular momentum (sign and magnitude) and charge current. The time-averaged angular momentum is defined as

$$\begin{eqnarray}&&\langle {\rm{\Lambda }}\rangle =\displaystyle \frac{1}{{ \mathcal T }-{\tau }_{{\rm{tran}}}}{\int }_{{\tau }_{{\rm{tran}}}}^{{ \mathcal T }}{\rm{\Lambda }}(\tau ){\rm{d}}\tau ,\end{eqnarray} \tag{ 5 }$$

where ${\tau }_{{\rm{tran}}}$ is a typical transient time³ and ${ \mathcal T }$ is sufficiently large to make $\langle {\rm{\Lambda }}\rangle$ reach its stationary value.

We also investigate the time-averaged electronic current that flows through the system, and its dependence on the ICs. The dimensionless time-averaged current is defined as

$$\begin{eqnarray}&&\langle J\rangle =\displaystyle \frac{1}{{ \mathcal T }-{\tau }_{{\rm{tran}}}}{\int }_{{\tau }_{{\rm{tran}}}}^{{ \mathcal T }}[{w}_{{\rm{out}}}(\theta (\tau )){q}_{{\rm{A}}}(\tau )+{w}_{{\rm{out}}}(\theta (\tau )+\beta ){q}_{{\rm{B}}}(\tau )]{\rm{d}}\tau ,\end{eqnarray} \tag{ 6 }$$

and the physical current is then given by $\langle J\rangle {Q}_{{\rm{max}}}/{t}_{0}$. Again, we consider the same transient-time ${\tau }_{{\rm{tran}}}$ as in equation (5) and ${ \mathcal T }$ to be large enough such that $\langle J\rangle$ reaches its stationary value after ${ \mathcal T }$. In order to calculate $\langle {\rm{\Lambda }}\rangle$ and $\langle J\rangle$ we solve the set of equations (2a)–(3b) numerically, and use the trajectories to calculate the time-integrals (5) and (6).

Although our discussion does not require any specific form of ${w}_{{\rm{in/out}}}(\theta ^{\prime} )$ we will use as an example the functional form adopted in [11] to describe a nano-scale electromechanical rotor, namely

$$\begin{eqnarray}&&{w}_{{\rm{in}}}(\theta ^{\prime} )={{\rm{e}}}^{-\xi \mathrm{cos}(\theta ^{\prime} )},\end{eqnarray} \tag{ 7a }$$

$$\begin{eqnarray}&&{w}_{{\rm{out}}}(\theta ^{\prime} )={{\rm{e}}}^{\xi \mathrm{cos}(\theta ^{\prime} )}.\end{eqnarray} \tag{ 7b }$$

In [11] this particular form of ${w}_{{\rm{in/out}}}(\theta ^{\prime} )$ stems from the exponentially decreasing (with respect to the lead-island distance) tunneling rates⁴ . In this work we will take ξ to be identical for the two islands and for source and drain⁵ .

The mean-field equations (2a)–(3b) describe a macroscopic system. For instance, the electronic part of the system can be realized using RC circuits, where the electronic islands are capacitors whose capacitance has a specific angular dependence to reproduce the (dis)charging rates from equations (7a) and (7b) (details are given in appendix B). In macroscopic capacitors the charge fluctuations can be neglected. Therefore one does not need to consider any torque-noise coming from the charge fluctuations in typical macroscopic systems and equations (2a) and (2b) are valid. In smaller scales charge fluctuations can be comparable to the charge mean value and thus charging/discharging processes must be treated stochastically. We consider this in the extreme case of Coulomb blockade limit in section 5.

The presence of the damping term $-\gamma {\rm{\Lambda }}$ leads, according to the fluctuation-dissipation theorem, to torque fluctuations [28]. We consider this by adding a Langevin noise term $\sqrt{2\gamma \tilde{T}}{\rm{\Phi }}(\tau )$ to equation (3a). Here $\tilde{T}$ is a dimensionless temperature and the relevant averages over realizations of the white noise ${\rm{\Phi }}(\tau )$ are $\langle {\rm{\Phi }}(\tau ){\rangle }_{{\rm{re}}}=0$ and $\langle {\rm{\Phi }}({\tau }_{1}){\rm{\Phi }}({\tau }_{2}){\rangle }_{{\rm{re}}}=\delta ({\tau }_{1}-{\tau }_{2})$. The relation between physical parameters and $\tilde{T}$ is given by $\tilde{T}={k}_{{\rm{B}}}{{Tt}}_{0}^{2}/I$, where k_B is the Boltzmann constant and T is the temperature. Notice that typical macroscopic physical parameters lead to a very small value of $\tilde{T}$. For instance, setting ${t}_{0}=1\;{\rm{s}}$, $I=3.4\times {10}^{-5}\;\mathrm{kg}\;{{\rm{m}}}^{2}$ and $T=300\;{\rm{K}}$ gives $\tilde{T}\approx {10}^{-16}$. Although very small, this noise will, for an infinitely long time of integration, wash out all the dependence on the ICs. Nevertheless, for finite times the question whether one can rely on the dependence on ICs depends on the time-scale one is interested in. We have recalculated all our mean-field results setting $\tilde{T}={10}^{-9}$ for times as long as a day in the experiment. Within the resolution of our figures, no differences can be observed when comparing with the the results from the deterministic equations (3a) and (3b) and therefore we can safely disregard the Langevin noise $\sqrt{2\gamma \tilde{T}}{\rm{\Phi }}(\tau )$ for macroscopic systems.

In the case of the symmetric rotor, solutions of the coupled equations (2a)–(3b) possess the following invariance: Substituting $\tilde{{\rm{\Lambda }}}=-{\rm{\Lambda }}$ and $\tilde{\theta }=-\theta$ simultaneously in equations (2a)–(3b), i.e. changing the direction of rotation and reflecting around $\theta =0$, one obtains the same equation. This invariance (with the rotation inversion) occurs also for reflections around $\theta =\pi /2$ and around $\theta =3\pi /2$. Due to these invariances, for every counter-clockwise (positive Λ) rotating solution with IC $({{\rm{\Lambda }}}_{0},{\theta }_{0})$ there is another clockwise (negative Λ) rotating one with IC $(-{{\rm{\Lambda }}}_{0},-{\theta }_{0})$. This invariance will be of fundamental importance when we consider the time-averaged angular momentum $\langle {\rm{\Lambda }}\rangle$, which is a function of the IC, averaged over ${\theta }_{0}$ for a rotor initially discharged (${q}_{{\rm{A}}}(\tau =0)={q}_{{\rm{B}}}(\tau =0)=0$) and at rest (${{\rm{\Lambda }}}_{0}=0$)

$$\begin{eqnarray}&&\overline{\langle {\rm{\Lambda }}\rangle }=\displaystyle \frac{1}{2\pi }{\int }_{0}^{2\pi }{\rm{d}}{\theta }_{0}\langle {\rm{\Lambda }}\rangle ,\end{eqnarray} \tag{ 8 }$$

which becomes exactly zero in the presence of this invariance.

It turns out that the steady-state dynamics depends sensitively on the ratio ${\eta }_{0}/\gamma$ [11]. For a given ξ one has three dynamical regimes: for very small ${\eta }_{0}/\gamma$ there is no motion at all and the rotor rests in the position $\theta =\pi /2$. For larger ${\eta }_{0}/\gamma$ there is an interval where oscillations are found. Above a certain threshold of ${\eta }_{0}/\gamma$ rotational motion sets in where the angular momentum increases approximately as the square root of ${\eta }_{0}/\gamma$. As said before, here we focus on parameters where rotational motion occurs and we will consider the case $\xi =2$ as an example.

Figure 2 shows $\langle {\rm{\Lambda }}\rangle$ calculated in the IC-space for two values of ${\eta }_{0}/\gamma$. Red stands for clockwise (negative $\langle {\rm{\Lambda }}\rangle$) and blue for counter-clockwise (positive $\langle {\rm{\Lambda }}\rangle$) rotation direction. Throughout the IC-space, for both negative and positive $\langle {\rm{\Lambda }}\rangle$, one has the same $| \langle {\rm{\Lambda }}\rangle |$. In both plots one observes intertwined patterns spiraling towards $({\theta }_{0}=\pi /2,{{\rm{\Lambda }}}_{0}=0)$ and $({\theta }_{0}=3\pi /2,{{\rm{\Lambda }}}_{0}=0)$. These points correspond to the standstill situation (${\rm{\Lambda }}=0$) with the rotor in a vertical orientation. Both locations are fixed points in phase space: once the system reaches those states, it stays there forever. The spiraling pattern around these fixed points can be understood as follows: the rotor transiently oscillates and as it swings it gains energy until it reaches the rotatory steady-state. In each half oscillation the rotor has the chance to set into the rotatory steady-state with a certain rotation direction. If it does not have enough energy, it performs another half oscillation gaining more energy and having the opportunity to start rotating into the opposite direction. This process is repeated until the stationary rotatory state is reached, whose direction depends on the IC of the rotor. For ICs closer to the fixed points a larger number of oscillations is required to reach the steady-state⁶ and small changes of the IC can lead to a different direction of rotation. This fact causes the complex spiraling pattern seen in the figures. As one decreases the driving strength ${\eta }_{0}$ the rotor gains energy more slowly and therefore more and more oscillations are required in order to reach the rotatory regime, which in turn leads also to finer spiraling patterns, as one sees in figure 2(b).

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In figure 2 one can also see a consequence of the invariance discussed above in section 3.1, namely an anti-symmetry around ${\theta }_{0}=0$ (${\theta }_{0}=\pi$) and ${\theta }_{0}=\pi /2$(${\theta }_{0}=3\pi /2$) and ${{\rm{\Lambda }}}_{0}=0$. This anti-symmetry implies in particular that $\overline{\langle {\rm{\Lambda }}\rangle }=0$. In the following section we will show how one can induce a preferred rotation direction, i.e. to make $\overline{\langle {\rm{\Lambda }}\rangle }\ne 0$, even in the case when the angular momentum is initially zero.

Now we turn our attention to the second quantity of interest defined in section 2.2, namely the time-averaged current $\langle J\rangle$. The time-averaged current only depends on the magnitude (and not on the sign) of the angular momentum. Therefore all ICs lead to the same time-averaged current and we do not present a plot of $\langle J\rangle$ as a function of the ICs $({{\rm{\Lambda }}}_{0},{\theta }_{0})$.

Individual alterations of the parameters β or χ (e.g. just ${l}_{{\rm{A}}}\ne {l}_{{\rm{B}}}$) do not lead to a preferred rotational direction, since for every solution $(\theta (t),{\rm{\Lambda }}(t))$ of equations (3a) and (3b) there is another solution $(\tilde{\theta }=-\theta (t)+\phi ,\tilde{{\rm{\Lambda }}}=-{\rm{\Lambda }}(t))$, where ϕ is a fixed angle. Although those changes do not introduce a preferred rotational direction, it is instructive to understand the effects of changing β and χ in equations (3a) and (3b). Examples can be seen in figure 3, where $\langle {\rm{\Lambda }}\rangle$ is plotted as a function of the ICs $({{\rm{\Lambda }}}_{0},{\theta }_{0})$ again. Red stands for clockwise (negative Λ) and blue for counter-clockwise (positive Λ) rotational direction. Notice that although $\langle {\rm{\Lambda }}\rangle$ changes sign across the IC-space, its absolute value is constant. In figure 3(a) one sees the case ${l}_{{\rm{A}}}\ne {l}_{{\rm{B}}}$ ($\chi \ne \pi /4$) and $\beta =\pi$, while in b) the case ${l}_{{\rm{A}}}={l}_{{\rm{B}}}$ ($\chi =\pi /4$) and $\beta =2$ is shown. The figure 3(a) looks qualitatively similar to figures 2(a) and (b). Again, one has fixed points (the same as for the symmetric rotor) and a patterned structure that spirals towards them. Nevertheless, the point anti-symmetry around the fixed points $\theta =\pi /2$ and $\theta =3\pi /2$ (${\rm{\Lambda }}=0$) is no longer present. This can be seen by comparing the lengths of the line segments 1 and 2 (3 and 4) in figure 3(a), which are the widths of regions with the same $\langle {\rm{\Lambda }}\rangle$ measured symmetrically with respect to the aforementioned fixed points. The point anti-symmetry around the fixed points $\theta =0$ and $\theta =\pi$ is preserved though. Figure 3(b) presents other differences when compared to figures 2(a) and (b): two fixed points disappear and other two are localized at different positions, namely $\theta =2\pi -\beta /2$ and $\theta =\pi -\beta /2$. Nevertheless, the IC-space is still point anti-symmetric with respect to the fixed point positions, and therefore again one has $\overline{\langle {\rm{\Lambda }}\rangle }=0$. The complete breakage of these point anti-symmetries is just possible by setting both $\chi \ne \pi /4$ and $\beta \ne \pi$, and it is exactly this symmetry-breaking that will be exploited to generate directed rotational motion in the next section.

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In order to obtain a preferred rotational direction one needs to choose ${l}_{{\rm{A}}}\ne {l}_{{\rm{B}}}$ ($\chi \ne \pi /4$) and $\beta \ne \pi$ simultaneously. The resulting set of equations does not feature pairs of solutions $(\theta (t),{\rm{\Lambda }}(t))$ and $(\theta ^{\prime} (t),{\rm{\Lambda }}^{\prime} (t))$ such that $\theta ^{\prime} (t)=-\theta (t)+\phi$ and ${\rm{\Lambda }}^{\prime} (t)=-{\rm{\Lambda }}(t)$.

To quantify the resulting directionality of the rotatory motion we define a directionality measure in the IC-space

$$\begin{eqnarray}&&M=\displaystyle \frac{\overline{\langle {\rm{\Lambda }}\rangle }}{\sqrt{\bar{\langle {\rm{\Lambda }}{\rangle }^{2}}}},\end{eqnarray} \tag{ 9 }$$

where the overbar was defined in equation (8). Note that M is zero when $\overline{\langle {\rm{\Lambda }}\rangle }=0$. For the case where all the ${\theta }_{0}$ lead to rotation in the same direction (and with the same angular velocity) one has $M=\pm 1$. By changing from negative to positive values of M (or vice versa), one just changes the direction of rotation.

The figure 4 shows M as a function of the parameters β and χ. White indicates no rotational directionality, while from white to blue (red) an increasing value of the directionality measure is found, with a positive(negative) $\overline{\langle {\rm{\Lambda }}\rangle }$. In figure 4 we set the parameter $\gamma =1$ and vary ${\eta }_{0}$ (in (a)–(c)). Along the two lines $\beta =\pi$ and $\chi =\pi /4$ we have M = 0, as expected. The M values are also anti-symmetric about those lines and, consequently, point-symmetric with respect to $(\chi =\pi /4,\beta =\pi )$. This point-symmetry steams from the analytic form in which the parameters χ and β appear into equation (3a), and therefore it is present independently of the chosen γ and ${\eta }_{0}$ (and ξ). In figure 4 (a) (${\eta }_{0}/\gamma =5$) and (b) (${\eta }_{0}/\gamma =10$) there are regions with $| M| =1$, in which a total rotational directionality is achieved. For instance, the point C in figure 4(b) has M = 1. In this case all the ICs lead to the same $\langle {\rm{\Lambda }}\rangle$ (even for large $| {{\rm{\Lambda }}}_{0}|$). However, in figure 4(a) almost all the parameter space presents M = 0, since for such small ratio ${\eta }_{0}/\gamma$ the rotor has $\langle {\rm{\Lambda }}\rangle =0$ for most of the $(\chi ,\beta )$ values. Regions where $| M| \lt 1$ are also displayed in figures 4(b) and (c) (${\eta }_{0}/\gamma =30$). Interestingly, figure 4(c) presents no $| M| =1$ regions, although larger extensions of the parameter space have $| M| \gt 0$. No significant changes in these figures are verified when $\xi =1$ or $\xi =4$.

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Examples for the cases $1\gt | M| \gt 0$ are the points A and B. For these points we generate the IC-space of figure 5. In figures 5(a) and (b) we plot $\langle {\rm{\Lambda }}\rangle$ in the IC-space for the same parameters as in the point A(B) of figure 4. There, we find $M=0.47$($M=0.88$). The white line is drawn in figure 5 just with the purpose of indicating the locus where the IC-average is performed. By comparing the segments of the white line which are inside regions with positive and negative $\langle {\rm{\Lambda }}\rangle$ in both figures 5(a) and (b), and noting that $| \langle {\rm{\Lambda }}\rangle |$ is the same for both red and blue regions, one can understand why the point B presents a stronger rotational directionality than point A in figure 4(b). We do not show a figure with $\langle {\rm{\Lambda }}\rangle$ as a function of the ICs for the same parameters as on the point C of figure 4(b) because all the ICs lead to the same $\langle {\rm{\Lambda }}\rangle$ value (even for ${{\rm{\Lambda }}}_{0}\ne 0$).

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