# A physically realizable molecular motor driven by the Landauer blowtorch
effect

Riley J. Preston  Institute of Physics, University of Freiburg, Hermann-
Herder-Strasse 3, 79104 Freiburg, Germany  Daniel S. Kosov  Authors to whom
correspondence should be addressed<EMAIL_ADDRESS>College of Science
and Engineering, James Cook University, Townsville, QLD, 4811, Australia

###### Abstract

We propose a model for a molecular motor in a molecular electronic junction
driven by a natural manifestation of Landauer’s blowtorch effect. The effect
emerges via the interplay of the electronic friction and diffusion
coefficients, each calculated quantum mechanically using nonequilibrium
Green’s functions, within a semi-classical Langevin description of the
rotational dynamics. The motor functionality is analysed through numerical
simulations where the rotations exhibit a directional preference according to
the intrinsic geometry of the molecular configuration. The proposed mechanism
for motor function is expected to be ubiquitous for a range of molecular
geometries beyond the one examined here.

##  I INTRODUCTION

Experimental demonstrations of molecular motors have used a range of external
energy sources such as light [1, 2, 3, 4, 5], chemical reactions [6, 7, 8],
thermal gradients [9], or applied electric currents [10, 11, 12, 13, 14, 15,
16, 17], the latter being of particular interest due to its conceptual
compatibility with nanoelectronics. With this in mind, this work considers a
molecular rotor subject to an applied electric current as supplied by a pair
of conducting electrodes. The molecular rotor serves as the main conducting
element in a molecular electronic junction, capable of producing mechanical
work.

There already exists a wealth of theoretical literature describing such
systems whose motor functionalities arise from a range of physical phenomena,
including, but not limited to, quantum tunneling [10] and excitation-
relaxation [18, 19] processes in asymmetric ratchet potentials, non-Markovian
behaviour of the current-induced forces leading to a bias in the
directionality [20], and non-conservative forces [21, 22]. However, previous
studies have overlooked the possible functionality which can arise due to the
inhomogeneous dissipative-excitational current-induced forces present in such
systems.

In this paper, we consider a model in which a molecular rotor is driven by the
current-induced forces imparted by electrons tunneling through it. These
forces provide the required energy to the rotational degree of freedom in
order to overcome the potential barrier for rotation. In parallel with
previous work [18, 21, 19], we model the rotational degree of freedom
classically according to a Keldysh-Langevin approach where its time-evolution
is governed by three components; an adiabatic force which sets the shape of
the ratchet potential, as well as a dissipative frictional force and a
stochastic force, the balance of which yields the steady-state temperature of
the classical rotator. Each of these forces, which arise due to the
interaction with the quantum nonequilibrium electronic environment, is
calculated self-consistently via nonequilibrium Green’s functions.

The directionality of our rotation is a result of - to our knowledge - a
hitherto unexplored contribution for motors in molecular junctions, that being
a consequence of Landauer’s blowtorch effect [23, 24] in the ratchet
potential, which emerges via the interplay of the coordinate-dependent
diffusion and viscosity coefficients. This phenomena is well-understood in the
context of chemical reaction rates [25], whereas here the scope is extended to
the study of ratchets. This is in contrast to previous research where the
directional rotation comes as a result of the non-conservativity of the
adiabatic force [18, 21] \- a phenomena which is also easily accessible with
our model via an appropriate choice of Hamiltonian, but is not the aim of this
study. Driving the motor by the blowtorch effect is of particular interest
since the dissipative and stochastic forces generally act to degrade the
device performance rather than enforce it [21]. We note that while the effect
of inhomogeneous viscosity and diffusion coefficients in ratchets has been
explored on a mathematical level [26, 27], here we propose a physically
realizable molecular electronic junction in which the effect emerges
naturally. This effect does not require an explicit time-dependence of the
Hamiltonian as it arises due to the molecular geometry. Additionally, the
directional rotation does not require an asymmetric ratchet potential,
although such asymmetric potentials can arise from our calculations via the
adiabatic force, further reinforcing motor performance. We note that the
function of our motor is reliant on the rotational dynamics being sufficiently
damped; a regime which is generally fulfilled in molecular electronic
junctions since the conducting molecule is usually embedded into an insulating
solvent or it is a part of a molecular monolayer.

It has been shown theoretically that a non-zero charge current can be pumped
through a quantum system in equilibrium via the periodic variation of two
independent parameters [28]. A particularly relevant example is described in
Ref.[29], where the coupling of a quantum system to the left and right
electrodes each assumes a periodic time-dependence. Since in our model the
coupling to each lead is implicitly time-dependent through the evolution of
the nuclear geometry, this would be equivalent to a constant, manual rotation
of our molecular motor with constant angular velocity. We use our model to
investigate the converse effect to equilibrium charge shuttling, in which an
applied charge current via the nonequilibrium electrodes _produces_ a time-
dependent variation of two independent parameters (the coupling to the left
and right electrodes) which emerges via the directed rotation of the molecular
geometry. Thus, this is an example of an adiabatic quantum motor. We do,
however, find that the operational parameter regimes of our molecular motor
differ from that of models of equilibrium charge shuttling, which we further
discuss in the results section. Our choice of Hamiltonian also mirrors an
example demonstrated in Ref. [30], where the rotation is instead considered
from a quantum perspective.

##  II MODEL

A visualisation of our proposed molecular junction configuration is shown in
Fig. 1. We have two planar electrodes bridged by a biphenyl based molecule.
The phenyl rings are prepared such that they are displaced by a dihedral angle
$\phi$ from each other - this angle is a constant for a given simulation. The
motor effect arises through the angle $\theta$, which represents the uniform
rotation of the entire molecular bridge as a rigid body. To produce an
observable directionality to the rotations, the vibrations must be adequately
damped. We find that the electronically calculated forces are generally
insufficient to achieve this regime and so we additionally include an external
equilibrium environment; for example, a solvent or molecular monolayer
surrounding the junction, which acts to further dampen the classical
vibrations. We emphasize that the proposed geometry is merely a physically
reasonable suggestion. The proposed motor effect should be ubiquitous in
molecular junction geometries provided that there is an asymmetry in the
Hamiltonian, in our case arising from the dihedral angle $\phi$.

Figure 1: Schematic of the model system. A biphenyl based molecule (green)
connects two graphene electrodes. The molecule represents a rigid rotator. The
dihedral angle between the two phenyl rings, which is critical for motor
functionality, can be adjusted by the inclusion of appropriate side groups to
atoms 1,2,3, and 4. An applied current induces a directional rotation about
the red bonds when the dihedral angle between the phenyl rings is non-zero.

The system is described by a generic tunneling Hamiltonian as per

$\hat{H}(t)=\hat{H}_{M}+\hat{H}_{L}+\hat{H}_{R}+\hat{H}_{LM}(\theta(t))+\hat{H}_{MR}(\theta(t))+H_{\text{cl}}(t).$
(1)

The total system Hamiltonian is partitioned into the following components; the
molecular Hamiltonian $\hat{H}_{M}$ for the molecular bridge, the left and
right electrodes Hamiltonians $\hat{H}_{L}$ and $\hat{H}_{R}$, the electrodes-
molecule coupling Hamiltonians $\hat{H}_{LM}(\theta(t))$ and
$\hat{H}_{MR}(\theta(t))$ which describe the coupling between the electronic
states on the rotor and the left and right electrodes, respectively, and the
classical Hamiltonian $H_{\text{cl}}(t)$ which describes the time-evolving
molecular geometry. Note that $\hat{H}_{LM}(\theta(t))$ and
$\hat{H}_{MR}(\theta(t))$ depend on time implicitly via the classical,
rotational degree of freedom $\theta$.

The molecular Hamiltonian consists of two conducting electronic levels, each
localised on one of the phenyl rings. It then takes the form

$\hat{H}_{M}=E_{1}\hat{d}^{{\dagger}}_{1}\hat{d}_{1}+E_{2}\hat{d}^{{\dagger}}_{2}\hat{d}_{2}+v(\hat{d}^{{\dagger}}_{1}\hat{d}_{2}+\hat{d}^{{\dagger}}_{2}\hat{d}_{1}).$
(2)

$E_{1}$ and $E_{2}$ are the energies of the first and second electronic
levels, respectively, while $v$ is the hopping amplitude.

The electrodes are described as non-interacting fermionic baths and the
Hamiltonian is taken in the standard form,

$\hat{H}_{L}+\hat{H}_{R}=\sum_{k\alpha}\epsilon_{k\alpha}\hat{d}^{\dagger}_{k\alpha}\hat{d}_{k\alpha},$
(3)

where we use a subscript $k\alpha$ to denote an operator acting on state $k$
in the $\alpha$ electrode which has energy $\epsilon_{k\alpha}$.

The molecule-electrode coupling, $\hat{H}_{LM}$ and $\hat{H}_{MR}$, are
defined according to

$\displaystyle\hat{H}_{LM}$ $\displaystyle=\sum_{k\in
L}\Big{(}t_{k1}(\theta(t))\hat{d}^{\dagger}_{k}\hat{d}_{1}+\text{h.c.}\Big{)},$
(4) $\displaystyle\hat{H}_{RM}$ $\displaystyle=\sum_{k\in
R}\Big{(}t_{k2}(\theta(t)+\phi)\hat{d}^{\dagger}_{k}\hat{d}_{2}+\text{h.c.}\Big{)}.$
(5)

The matrix elements $t_{ki}$ (and their conjugates) describe the tunneling
amplitudes between electrode states $k$ and the molecular bridge states $i$,
where state $1$ is only coupled to the left electrode and state $2$ is only
coupled to the right electrode. Note that $t_{ki}$ depends explicitly on the
classical rotational coordinate $\theta$. We choose to express
$t_{k\alpha,i}(\theta)=t_{k\alpha,i}s_{\alpha}(\theta)$, where the classical
dependence emerges through $s_{\alpha}(\theta)$, which takes the following
forms for the left and right electrodes:

$\displaystyle s_{L}$
$\displaystyle=1+\frac{A}{2}\left(\cos(2\theta)-1\right),$ (6) $\displaystyle
s_{R}$ $\displaystyle=1+\frac{A}{2}\left(\cos(2(\theta+\phi))-1\right).$ (7)

With this dependence, the coupling amplitude is maximised when a phenyl ring
is coplanar with its corresponding electrode and minimised when the phenyl
ring is orthogonal to the electrode with a magnitude of $1-A$ times the
maximum value. This dependence of the tunneling amplitudes on the rotational
angle can be realized physically using graphene electrodes, where the rotation
of the molecular bridge out of the electrode plane lowers $\pi$-conjugation,
reducing the corresponding tunneling amplitude.

Finally, the classical Hamiltonian is given by a rigid rotator expression,

$H_{\text{cl}}(t)=\frac{L^{2}}{2I}+U_{\text{cl}}(\theta),$ (8)

where $L$ is the angular momentum of the molecular geometry, $I$ is the moment
of inertia and $U_{\text{cl}}(\theta)$ is the classical potential for the
rotation. In our calculations we set $U_{\text{cl}}(\theta)=0$, such that the
rotational potential results entirely from the interaction with the electronic
environment, calculated quantum mechanically. In any case, the inclusion of a
non-zero classical potential will not have a qualitative difference on the
observed motor effect.

##  III Current-induced torque and ”blowtorch” temperature

The operator for the torque acting on the classical rotational coordinate due
to the quantum, electronic environment is given by

$\hat{\tau}=-\partial_{\theta}\hat{H}(t)=-\sum_{k\alpha,i}\left[\partial_{\theta}t_{k\alpha
i}(\theta)\hat{d}_{k\alpha}^{\dagger}\hat{d}_{i}+h.c.\right],$ (9)

where $\partial_{\theta}$ is the partial derivative with respect to $\theta$.
The summation in the above runs over both electrodes, $\alpha\in\\{L,R\\}$,
and both molecular electronic states, $i\in\\{1,2\\}$. The torque operator is
then expressed in terms of a mean term and a deviation from the mean,

$\hat{\tau}=\langle\hat{\tau}\rangle+\delta\hat{\tau},$ (10)

where each can be quantified in terms of nonequilibrium Green’s functions. As
is covered in detail in the appendix, a time-scale separation between the slow
classical rotation of the rotor and the fast electron tunneling allows for a
perturbative expansion of the mean torque in terms of the small parameter -
the derivative with respect to central time in the molecular bridge Green’s
functions. The perturbative expansion is

$\langle\hat{\tau}\rangle=\tau_{(0)}(\theta)+\tau_{(1)}(\theta,\dot{\theta})+...,$
(11)

where $\tau_{(n)}$ is of $n^{\text{th}}$ order in the central time
derivatives. We truncate the expansion after the first order. We calculate a
conservative potential according to

$U=-\int^{\theta}_{\theta_{0}}d\theta^{\prime}\tau_{(0)}(\theta^{\prime}),$
(12)

where the choice of $\theta_{0}$ is arbitrary. Equation (12) entirely defines
the ratchet potential for our rotational coordinate due to the electronic
environment. Finally, the torque operator is then mapped onto a classical
torque such that we obtain a classical equation of motion for the rotational
coordinate. It takes the form of a Langevin equation,

$I\ddot{\theta}=\tau_{(0)}(\theta)-(\xi_{\text{solv}}+\xi(\theta))\dot{\theta}+\delta\tau(t),$
(13)

where $\xi(\theta)$, calculated via $\tau_{(1)}$, is the electronic friction
coefficient while $\xi_{\text{solv}}$ is the friction due to the interaction
with an external solvent. $\delta\tau(t)$ is a classical stochastic force
quantified according to a diffusion coefficient,
$D_{\text{tot}}=D(\theta)+D_{\text{solv}}$, where the electronic part is
defined according to

$\langle\delta\tau(t)\delta\tau(t^{\prime})\rangle=D(\theta)\delta(t-t^{\prime}).$
(14)

Each of the electronic forces, $\tau_{(0)}(\theta)$, $\xi(\theta)$, and
$D(\theta)$, are calculated quantum mechanically via nonequilibrium Green’s
functions while the forces due to interaction with the external solvent,
$\xi_{\text{solv}}$ and $D_{\text{solv}}$, are input parameters to the model
which allow us to artificially increase the damping of the dynamics. We have
applied the white-noise approximation in calculating the electronic part of
the diffusion coefficient which is justified due to the clear separation of
time-scales between the electronic and classical dynamics [31]. The same
cannot be said for the external damping, whose dynamics may occur on similar
time-scales to the classical rotations. However, the operation of our motor is
governed chiefly by the behaviour of the electronic component and as such, we
predict that a more accurate approach to the modelling of the external solvent
is not important for the observed motor effect. In analogy with the
fluctuation-dissipation theorem, we can define an effective ”blowtorch”
temperature for the classical rotation according to [32, 33, 25, 34]

$k_{B}T_{\text{eff}}(\theta)=\frac{D(\theta)+D_{\text{solv}}}{2(\xi(\theta)+\xi_{\text{solv}})}.$
(15)

Expressions for the diffusion coefficient, viscosity and average torque in
terms of nonequilibrium Green’s functions along with relevant derivations are
given in the appendix.

##  IV RESULTS

We now present results for our model system. Results are acquired via
computational simulations of the Langevin dynamics produced by our model
according to (13). From long Langevin trajectories in time, we calculate the
average rotation rate for the classical rotational degree of freedom for a
chosen set of parameters. The common parameters for all calculations, unless
otherwise specified, are as follows. The electrode temperatures are set such
that $k_{B}T_{\alpha}\approx 2.72\times 10^{-2}\text{eV}$, and the solvent is
in thermal equilibrium with the electrodes with a corresponding viscosity
coefficient of $\xi_{\text{solv}}=5$ a.u.. The moment of inertia of the
classical rotational coordinate is approximated according to two phenyl rings
as $I=4.5\times 10^{5}$ a.u.. We use the wide-band approximation, and express
the level broadening as

$\Gamma_{\alpha}=\Gamma_{\alpha}^{\text{max}}s_{\alpha}^{2}(\theta),$ (16)

where the maximum level broadenings $\Gamma_{\alpha}^{\text{max}}$ are input
parameters in our calculations. We take the maximum level broadenings due to
the left and right electrodes, respectively, as
$\Gamma^{\text{max}}_{L}\approx 0.272\text{eV}$ and
$\Gamma^{\text{max}}_{R}=\Gamma^{\text{max}}_{L}/2$. For the molecular
Hamiltonian, we take $E_{1}=E_{2}=0$ while the hopping amplitude is given by
$v=1.25\text{eV}$. We apply the voltage, $\mathcal{V}$, symmetrically in all
cases such that $\mu_{L}=-\mu_{R}$. Finally, we take $A=0.95$, such that the
Hamiltonian coupling element when the phenyl ring is perpendicular to the
electrode is $5\%$ of the corresponding coplanar value.

Figure 2: $U$ calculated according to (12) with $k_{B}T_{\text{eff}}$ overlaid
on top for different voltages. The nonhomogeneous temperature with local hot
spots is the manifestation of the Landauer blowtorch effect. Dihedral angle
between phenyl rings: $\phi=-\pi/4$. Figure 3: $U$ calculated according to
(12) with $k_{B}T_{\text{eff}}$ overlaid on top for different dihedral angle
between phenyl rings $\phi$. Voltage $\mathcal{V}=5V$.

In Fig. 2, we observe the periodic ratchet potentials generated for a range of
voltages along with the corresponding inhomogeneous effective temperatures
overlaid on top. At equilibrium, the rotational coordinate is in thermal
equilibrium with the electrodes and solvent. Of principal importance are the
energies of the molecular orbitals which are $\pm 1.25eV$ for our parameters,
which are off-resonant when $\mathcal{V}<2.5V$. Increasing the voltage in the
off-resonant regime - exemplified by the $\mathcal{V}=2V$ case - increases the
height of the energy barrier for rotation while the temperature of the
rotational coordinate differs only slightly from equilibrium. Conversely, in
the resonant regime when $\mathcal{V}>2.5V$, the inhomogeneous temperature as
a function of $\theta$ yields clear periodic hot-spots which we refer to as
the blowtorch. Further increasing the voltage magnifies these hotspots while
decreasing the energy barrier for rotation. The value of $\phi=-\pi/4$ was
chosen specifically here to illustrate a situation in which a periodic
blowtorch increases the probability for the forwards rotation (increasing
$\theta$). This is because the effects of the potential gradient are nullified
in the region where the blowtorch is applied, resulting in an effective
decrease to the barrier for rotation in the forwards direction [25]. We also
observe numerically that our Langevin coefficients are independent of the sign
of the voltage. Thus, the rotational direction must also be independent of the
sign of the voltage. In other words, our mechanism for the rotation of the
molecular structure is independent of the direction of electron tunneling
through the junction. If we take this to be true, this then justifies our
decision to have $\Gamma^{\text{max}}_{L}\neq\Gamma^{\text{max}}_{R}$, since
otherwise the symmetry of the system would prevent any non-zero average
rotation.

Figure 4: The rotation rate, $R$, as a function of dihedral angle between
phenyl rings $\phi$. Each $R$ point is calculated via averaging over a
trajectory with a length of $\approx 1.6\times 10^{6}\;\mathrm{ns}$. Voltage
$\mathcal{V}=5V$.

We additionally observe the dependence on $\phi$ in Fig. 3. When $\phi=0$ and
the two phenyl rings are coplanar, the ratchet potential and corresponding
effective temperature distribution are symmetric, ruling out any possible
rotation as is to be expected. Upon comparing $\phi=\pi/4$ with $\phi=-\pi/4$,
corresponding to opposite chiralities of the molecular bridge, we observe the
dependence on $\theta$ to be flipped such that we should observe equal and
opposite average rotation rates - a result which we observe directly in Fig.
4. The case of $\phi=1.3$ was chosen to highlight the possibility of
deformation to the potential which can have a significant effect on the
rotation rate.

We now turn to numerical simulations of the dynamics. In Fig. 4, we observe
the average rotation rate, $R$, over a trajectory as a function of $\phi$. The
rotations go to zero when $\phi=0$ and $\pm\frac{\pi}{2}$, as is to be
expected from symmetry arguments. We find that $R(-\phi)=-R(\phi)$ as expected
from the previous discussion. Short example trajectories of the rotational
coordinate as a function of time are plotted for different values of $\phi$ in
Fig. 5, for the readers intuition.

In Ref. [29], equilibrium charge shuttling was shown to be maximised when
$\phi=\pm\pi/4$; a result which we can readily reproduce by applying a manual
rotation to $\theta$ such that it increases or decreases linearly with time.
We find here that the rotation rate due to an applied voltage follows a
similar trend, reaching a minimum/maximum at $\phi=\pm\pi/4$. However, we
observe a deviation from this behaviour around $\phi=\pm 1.3$ due to the rapid
current-induced deformation of the ratchet potential. We also note that
equilibrium charge shuttling can even be observed even when
$\Gamma^{\text{max}}_{L}=\Gamma^{\text{max}}_{R}$; a regime in which we do not
observe a net rotation by applying a voltage since our mechanism for rotation
is independent of the direction of the current. In contrast, models for
equilibrium charge pumping show that the produced current is _reversed_ upon
reversing the rotation of the molecular configuration [29]. We find that the
direction of rotation in our model is determined by the choice of $\phi$ as
well as the choices of $\Gamma_{L}^{\text{max}}$ and
$\Gamma_{R}^{\text{max}}$. We have arbitrarily chosen
$\Gamma_{L}^{\text{max}}>\Gamma_{R}^{\text{max}}$ to produce the displayed
results. If we instead choose
$\Gamma_{R}^{\text{max}}>\Gamma_{L}^{\text{max}}$, the observed rotational
directions are reversed - a result we have observed numerically but not shown
here.

Figure 5: Short time trajectories of the rotational angle $\theta$ (expressed
here in terms of the number of revolutions) for different values of the
dihedral angle $\phi$. Voltage $\mathcal{V}=5V$.

Fig. 6 demonstrates the voltage dependence of the rotational rate. We observe
negligible rotation in the off-resonant regime when $\mathcal{V}<2.5V$. In the
resonant regime, the average rotation rate increases approximately linearly
due to the increasing magnitude of the applied blowtorch with increasing
voltage along with the lowering of the energy barrier required for rotation.
For even higher voltages, we expect that the rotation rate would begin
decreasing back towards zero since the large effective temperatures will
overwhelm the potential entirely, removing any directional preference. This,
however, would occur beyond the realms of physically achievable voltages for
our model.

Figure 6: The rotation rate, $R$, as a function of voltage $\mathcal{V}$. Each
$R$ point is calculated via averaging over a trajectory with a length of
$\approx 1.6\times 10^{6}\;\mathrm{ns}$. Dihedral angle $\phi=-\pi/4$. Figure
7: The rotation rate $R$, and directionality, $Dir$, as a function of the
moment of inertia of the classical rotational coordinate. The final point on
each plot corresponds to our usual choice of $I$ for two phenyl rings. The
trajectory length was chosen for each value of $I$ to ensure convergence of
the results. Dihedral angle $\phi=-\pi/4$, voltage $\mathcal{V}=5V$.

The function of our molecular motor requires sufficient damping - a regime we
achieve via the inclusion of an external solvent to the system. In Fig. 7, we
observe the dependence of the rotation rate on the moment of inertia of the
molecular configuration, where $I\approx 1.15\times 10^{-45}\text{kgm}^{2}$ is
the physically reasonable value corresponding to our chosen molecular
configuration. In the overdamped case where $I$ is unrealistically small, the
rotation rate is orders of magnitude larger than for realistic values for $I$.
The rotation rate asymptotically decreases towards zero with increasing moment
of inertia, where in the underdamped case, the preference of a given direction
will become vanishingly small. As an additional insight, we define the
directionality according to

$Dir=\frac{n_{\text{forw}}}{n_{\text{forw}}+n_{\text{back}}},$ (17)

where $n_{\text{forw}}$ and $n_{\text{back}}$ are the number of forward and
backward rotations over the full length of the trajectory. $Dir=1$ would
correspond to a trajectory in which the molecular motor rotates
unidirectionally forwards. For the physically realistic value of $I$,
$50.68\%$ of all rotations are forwards. This directionality is far smaller
than what has been demonstrated for motors governed chiefly by quantum effects
[10].

##  V CONCLUSIONS

In this paper, we have proposed an experimentally realizable model for a
molecular motor in a molecular electronic junction whose operation is governed
by Landauer’s blowtorch effect. This contrasts with other theoretical models
for molecular motors which generally disregard the inhomogeneous temperature
of the electronic environment induced by the nonequilibrium electrodes. We
have demonstrated that directional rotations can be produced entirely as a
result of the behaviour of the viscosity and diffusion coefficients - these
are exerted by tunneling quantum electrons on the classical rotator and
calculated exactly via nonequilibrium Green’s functions - while the rotational
potential is periodic and subsequently introduces no intrinsic directionality
of its own. This effect is, however, limited to regimes where the rotations
are sufficiently damped and we anticipate that the small electronic friction
alone will not be enough to produce a non-negligible rotational preference,
hence our choice to additionally include an external solvent which increases
the dampening of the rotation.

DATA AVAILABILITY

The data that supports the findings of this study are available within the
article.

##  Appendix A Torque, Rotational Viscosity and Diffusion Coefficient in Terms
of Nonequlibrium Green’s functions

We use the standard definitions for the lesser $G_{ij}^{<}(t,t^{\prime})$,
greater $G_{ij}^{>}(t,t^{\prime})$, retarded $G_{ij}^{R}(t,t^{\prime})$ and
advanced $G_{ij}^{A}(t,t^{\prime})$ components of the electronic Green’s
functions in our derivations. Expressing the torque operator in the Heisenberg
picture, we compute the average torque as

$\langle\hat{\tau}\rangle=i\sum_{k\alpha i}\left[\partial_{\theta}t_{k\alpha
i}(\theta)G_{ik\alpha}^{<}(t,t)+\partial_{\theta}t_{ik\alpha}(\theta)G_{k\alpha
i}^{<}(t,t)\right].$ (18)

This torque is computed for the exact, nonadiabatic Green’s functions.

We now perform a perturbative expansion of the mean torque given in (18). It
is a mathematical convenience to perform this expansion under a Wigner
transformation of the time since it allows for the easy recognition of
different time-scales within the system. The Wigner time coordinates are
defined according to

$T=\frac{t+t^{\prime}}{2},\;\;\;\;\;\;\;\tau=t-t^{\prime},$ (19)

where $T$ is the central time, associated with the long time-scales of
classical vibration and $\tau$ is the relative time, related to electronic
tunneling. Thus, in our theory the small parameter naturally emerges via
derivatives with respect to $T$. We introduce an auxilliary two-time function,

$\mathcal{T}(t,t^{\prime})=i\sum_{k\alpha i}\left[\partial_{\theta}t_{k\alpha
i}(\theta(t^{\prime}))G_{ik\alpha}^{<}(t,t^{\prime})+\partial_{\theta}t_{ik\alpha}(\theta(t))G_{k\alpha
i}^{<}(t,t^{\prime})\right],$ (20)

where $\mathcal{T}(t,t)=\langle\hat{\tau}(t)\rangle$. Next, the Green’s
functions spanning both the electrode and molecular space can be decomposed
via the Dyson equation

$G^{<}_{k\alpha
i}(t,t^{\prime})=\int_{-\infty}^{\infty}dt_{1}\sum_{j}\left[g_{k\alpha}^{<}(t,t_{1})t_{k\alpha
j}(t_{1})G^{A}_{ji}(t_{1},t^{\prime})\right.+\left.g_{k\alpha}^{R}(t,t_{1})t_{k\alpha
j}(t_{1})G^{<}_{ji}(t_{1},t^{\prime})\right],$ (21)

where $g_{k\alpha}(t,t_{1})$ is the free Green’s function for electrode
$\alpha$. The resultant equation for $\mathcal{T}(t,t^{\prime})$ is

$\mathcal{T}(t,t^{\prime})=i\sum_{ij}\int_{-\infty}^{\infty}dt_{1}\Big{[}G_{ij}^{<}(t,t_{1})\Phi_{ji}^{A}(t_{1},t^{\prime})+G_{ij}^{R}(t,t_{1})\Phi_{ji}^{<}(t_{1},t^{\prime})+\Psi_{ij}^{<}(t,t_{1})G_{ji}^{A}(t_{1},t^{\prime})+\Psi_{ij}^{R}(t,t_{1})G_{ji}^{<}(t_{1},t^{\prime})\Big{]}.$
(22)

Here we have introduced the self-energy-like terms, $\Psi$ and $\Phi$, which
contain any information about the coupling to the electrodes. These are
defined as ($c=<,>,R,A$)

$\Psi^{c}_{ij}(t,t^{\prime})=\sum_{k\alpha}\partial_{\theta}t_{ik\alpha}(\theta(t))g_{k\alpha}^{c}(t,t^{\prime})t_{k\alpha
j}(\theta(t^{\prime})),$ (23)
$\Phi^{c}_{ij}(t,t^{\prime})=\sum_{k\alpha}t_{ik\alpha}(\theta(t))g_{k\alpha}^{c}(t,t^{\prime})\partial_{\theta}t_{k\alpha
j}(\theta(t^{\prime})).$ (24)

Application of the Wigner transform to (22) results in

$\int d\tau
e^{i\omega\tau}\mathcal{T}(t,t^{\prime})=\text{Tr}\Big{\\{}ie^{\frac{1}{2i}\lambda(\partial_{T}^{G}\partial_{\omega}^{\Phi}-\partial_{\omega}^{G}\partial_{T}^{\Phi})}\left(\tilde{G}^{<}\tilde{\Phi}^{A}+\tilde{G}^{R}\tilde{\Phi}^{<}\right)+ie^{\frac{1}{2i}\lambda(\partial_{T}^{\Psi}\partial_{\omega}^{G}-\partial_{\omega}^{\Psi}\partial_{T}^{G})}\left(\tilde{\Psi}^{<}\tilde{G}^{A}+\tilde{\Psi}^{R}\tilde{G}^{<}\right)\Big{\\}},$
(25)

where we use $\tilde{G}$ to denote the Wigner transform of $G$, defined as

$\tilde{G}(T,\omega)=\int d\tau e^{i\omega\tau}G(T,\tau),$ (26)

and the same applies for the self-energy-like terms. Functions in the Wigner
space carry dependence on $T$ and $\omega$ which we subdue for brevity. We now
propose the ansatzes,

$\tilde{G}=\tilde{G}_{(0)}+\lambda\tilde{G}_{(1)}+\lambda^{2}\tilde{G}_{(2)}+...,$
(27)
$\tilde{\Psi}=\tilde{\Psi}_{(0)}+\lambda\tilde{\Psi}_{(1)}+\lambda^{2}\tilde{\Psi}_{(2)}+...,$
(28)
$\tilde{\Phi}=\tilde{\Phi}_{(0)}+\lambda\tilde{\Phi}_{(1)}+\lambda^{2}\tilde{\Phi}_{(2)}+...,$
(29)

in which $\tilde{G}_{(n)}$ is of $n^{th}$ order in our small parameter, and
the same applies to $\tilde{\Psi}$ and $\tilde{\Phi}$. Terms with $n=0$
correspond to the adiabatic approximation, while the higher order terms go
beyond this and account for the dynamical corrections due to molecular
rotations. We use $\lambda$ in the above as a book-keeping term which makes
clear the ”smallness” of the term in question. For example, a term
proportional to $\lambda$ will be first order in our small parameter, and so
on. We let $\lambda=1$ at the end of the derivation.

We substitute these expansions into (25) and consider each order of $\lambda$
separately. In the adiabatic case, we retain only the $n=0$ terms from
(27)-(29) while the exponentials in (25) disappear, resulting in

$\int d\tau
e^{i\omega\tau}\mathcal{T}_{(0)}(t,t^{\prime})=i\text{Tr}\Big{\\{}\tilde{G}_{(0)}^{<}\tilde{\Phi}_{(0)}^{A}+\tilde{G}_{(0)}^{R}\tilde{\Phi}_{(0)}^{<}+\tilde{\Psi}_{(0)}^{<}\tilde{G}_{(0)}^{A}+\tilde{\Psi}_{(0)}^{R}\tilde{G}_{(0)}^{<}\Big{\\}},$
(30)

where we have let $\lambda=1$. We then apply the inverse Wigner transform and
let $\tau=0$ which yields

$\tau_{(0)}=-\int\frac{d\omega}{\pi}\text{ImTr}\Big{\\{}\tilde{\Psi}_{(0)}^{<}\tilde{G}_{(0)}^{A}+\tilde{\Psi}_{(0)}^{R}\tilde{G}_{(0)}^{<}\Big{\\}}.$
(31)

We use ImTr to denote the imaginary part of the trace, where we have used the
fact that $(X^{<})^{\dagger}=-X^{<}$ and $(X^{A})^{\dagger}=X^{R}$ for an
arbitrary term $X$. (31) specifies the adiabatic torque.

We now consider the first-order non-adiabatic correction to the average
torque. This is found by retaining the first-order terms in (25), which are
linear in $\lambda$. With some work, we find

$\tau_{(1)}=-\frac{1}{\pi}\int
d\omega\text{ImTr}\left\\{\tilde{\Psi}_{(0)}^{R}\tilde{G}_{(1)}^{<}+\tilde{\Psi}_{(1)}^{<}\tilde{G}_{(0)}^{A}\right.\left.+\tilde{\Psi}_{(1)}^{R}\tilde{G}_{(0)}^{<}+\tilde{\Psi}_{(0)}^{<}\tilde{G}_{(1)}^{A}\right\\}\\\
+\frac{1}{2\pi}\int
d\omega\text{ReTr}\left\\{\partial_{T}\tilde{\Psi}_{(0)}^{<}\partial_{\omega}\tilde{G}_{(0)}^{A}+\partial_{T}\tilde{\Psi}_{(0)}^{R}\partial_{\omega}\tilde{G}_{(0)}^{<}\right.\left.-\partial_{\omega}\tilde{\Psi}_{(0)}^{<}\partial_{T}\tilde{G}_{(0)}^{A}-\partial_{\omega}\tilde{\Psi}_{(0)}^{R}\partial_{T}\tilde{G}_{(0)}^{<}\right\\},$
(32)

where ReTr denotes the real part of the trace. We find that $\tau_{(1)}$ is
proportional to $\dot{\theta}$ and as a result, it can be alternately
expressed as

$\tau_{(1)}=-\xi(\theta)\dot{\theta},$ (33)

where $\xi$ is the electronic viscosity coefficient. Thus, (32) denotes the
dissipative frictional torque.

The fluctuations about the average torque are treated as a Gaussian stochastic
variable which is quantified entirely by its first two moments:

$\langle\delta\hat{\tau}(t)\rangle=0,\;\;\;\;\;\;\;\langle\delta\hat{\tau}(t)\delta\hat{\tau}(t^{\prime})\rangle=D\delta(t-t^{\prime}),$
(34)

where $D$ is the electronic diffusion coefficient which we aim to find an
expression for. Note that we have taken the white-noise approximation such
that the stochastic force is delta-correlated.

Here, we provide a final expression for $D$, while the derivation follows Ref.
[33]:

$\displaystyle D(\theta)$ $\displaystyle=\frac{1}{2\pi}\int
d\omega\text{Tr}\left\\{\tilde{G}_{(0)}^{>}\tilde{\Phi}_{(0)}^{A}\tilde{G}_{(0)}^{<}\tilde{\Phi}_{(0)}^{A}+\tilde{G}_{(0)}^{R}\tilde{\Phi}_{(0)}^{>}\tilde{G}_{(0)}^{<}\tilde{\Phi}_{(0)}^{A}\right.+\tilde{G}_{(0)}^{>}\tilde{\Phi}_{(0)}^{A}\tilde{G}_{(0)}^{R}\tilde{\Phi}_{(0)}^{<}+\tilde{G}_{(0)}^{R}\tilde{\Phi}_{(0)}^{>}\tilde{G}_{(0)}^{R}\tilde{\Phi}_{(0)}^{<}+\tilde{\Psi}_{(0)}^{>}\tilde{G}_{(0)}^{A}\tilde{\Psi}_{(0)}^{<}\tilde{G}_{(0)}^{A}$
$\displaystyle+\tilde{\Psi}_{(0)}^{R}\tilde{G}_{(0)}^{>}\tilde{\Psi}_{(0)}^{<}\tilde{G}_{(0)}^{A}\left.+\tilde{\Psi}_{(0)}^{>}\tilde{G}_{(0)}^{A}\tilde{\Psi}_{(0)}^{R}\tilde{G}_{(0)}^{<}+\tilde{\Psi}_{(0)}^{R}\tilde{G}_{(0)}^{>}\tilde{\Psi}_{(0)}^{R}\tilde{G}_{(0)}^{<}+\tilde{G}_{(0)}^{>}\tilde{\zeta}_{(0)}^{<}+\tilde{\zeta}_{(0)}^{>}\tilde{G}_{(0)}^{<}\right.+\tilde{G}_{(0)}^{>}\tilde{\Psi}_{(0)}^{<}\tilde{G}_{(0)}^{A}\tilde{\Phi}_{(0)}^{A}$
$\displaystyle+\tilde{\Psi}_{(0)}^{>}\tilde{G}_{(0)}^{A}\tilde{\Phi}_{(0)}^{A}\tilde{G}_{(0)}^{<}+\tilde{G}_{(0)}^{>}\tilde{\Psi}_{(0)}^{R}\tilde{G}_{(0)}^{<}\tilde{\Phi}_{(0)}^{A}\left.+\tilde{\Psi}_{(0)}^{R}\tilde{G}_{(0)}^{>}\tilde{\Phi}_{(0)}^{A}\tilde{G}_{(0)}^{<}+\tilde{G}_{(0)}^{>}\tilde{\Psi}_{(0)}^{R}\tilde{G}_{(0)}^{R}\tilde{\Phi}_{(0)}^{<}+\tilde{\Psi}_{(0)}^{R}\tilde{G}_{(0)}^{R}\tilde{\Phi}_{(0)}^{>}\tilde{G}_{(0)}^{<}\right\\},$
(35)

where we have introduced an additional self-energy-like term, defined as
($c=<,>,R,A$)

$\zeta^{c}_{ij}(t,t^{\prime})=\sum_{k\alpha}\partial_{\theta}t_{ik\alpha}(\theta(t))g_{k\alpha}^{c}(t,t^{\prime})\partial_{\theta}t_{k\alpha
j}(\theta(t^{\prime})),$ (36)

whose perturbative expansion is defined in the usual way. The diffusion
coefficient according to (35) then gives a means of quantifying the stochastic
force in numerical simulations.

##  Appendix B Solving for the Adiabatic and First Order Green’s Functions

What remains is to calculate explicit expressions for both the adiabatic and
first order Green’s functions, as well as the self-energy-like terms, in the
frequency domain. The Green’s functions evolve according to the Keldysh-
Kadanoff-Baym equations, given in the Wigner space as [33, 25, 31]

$\Big{(}\omega+\frac{i}{2}\partial_{T}-e^{\frac{1}{2i}\lambda\partial_{\omega}^{G}\partial_{T}^{h}}h(T)\Big{)}\tilde{G}^{R/A}=I+e^{\frac{1}{2i}\lambda\left(\partial_{T}^{\Sigma}\partial_{\omega}^{G}-\partial_{\omega}^{\Sigma}\partial_{T}^{G}\right)}\tilde{\Sigma}^{R/A}\tilde{G}^{R/A},$
(37)
$\Big{(}\omega+\frac{i}{2}\partial_{T}-e^{\frac{1}{2i}\lambda\partial_{\omega}^{G}\partial_{T}^{h}}h(T)\Big{)}\tilde{G}^{</>}=e^{\frac{1}{2i}\lambda\left(\partial_{T}^{\Sigma}\partial_{\omega}^{G}-\partial_{\omega}^{\Sigma}\partial_{T}^{G}\right)}\Big{(}\tilde{\Sigma}^{R}\tilde{G}^{</>}+\tilde{\Sigma}^{</>}\tilde{G}^{A}\Big{)},$
(38)

where we have shown the retarded/advanced and the lesser/greater terms
collectively. Here, we adopt the convenient notation for derivatives,
$\partial_{T}^{G}$, which denotes a partial derivative acting on the $G$ term
with respect to $T$, and so on. We have once again introduced the book-keeping
parameter, $\lambda$, for clarity in our perturbative expansions. The self-
energies take the conventional form ($c=<,>,R,A$):

$\Sigma^{c}_{ij}(t,t^{\prime})=\sum_{k\alpha}t_{ik\alpha}(\theta(t))g_{k\alpha}^{c}(t,t^{\prime})t_{k\alpha
j}(\theta(t^{\prime})),$ (39)

and we apply our usual ansatz to the self-energies,

$\tilde{\Sigma}=\tilde{\Sigma}_{(0)}+\lambda\tilde{\Sigma}_{(1)}+\lambda^{2}\tilde{\Sigma}_{(2)}+....$
(40)

To solve for the form of the adiabatic and first-order Green’s functions, we
take a perturbative expansion of the exponentials in (37) and (38) as well as
substituting in our perturbative ansatzes, (27) and (40). Truncating after the
zeroth order and solving for $\tilde{G}_{(0)}$ yields the standard adiabatic
Green’s functions as follows:

$\tilde{G}_{(0)}^{R/A}=\Big{(}\omega
I-h-\tilde{\Sigma}_{(0)}^{R/A}\Big{)}^{-1},$ (41)
$\tilde{G}_{(0)}^{</>}=\tilde{G}_{(0)}^{R}\tilde{\Sigma}_{(0)}^{</>}\tilde{G}_{(0)}^{A}.$
(42)

For the first-order, we consider terms linear in $\lambda$ such that we obtain

$\tilde{G}_{(1)}^{R/A}=\frac{1}{2i}\tilde{G}_{(0)}^{R/A}\Big{[}\tilde{G}_{(0)}^{R/A},\partial_{T}h\Big{]}\tilde{G}_{(0)}^{R/A},$
(43)
$\tilde{G}_{(1)}^{</>}=\tilde{G}_{(0)}^{R}\tilde{\Sigma}_{(0)}^{</>}\tilde{G}_{(1)}^{A}+\tilde{G}_{(1)}^{R}\tilde{\Sigma}_{(0)}^{</>}\tilde{G}_{(0)}^{A}+\frac{1}{2i}\tilde{G}_{(0)}^{R}\Big{(}\partial_{T}h\tilde{G}_{(0)}^{R}\partial_{\omega}\tilde{\Sigma}^{</>}+\tilde{G}_{(0)}^{</>}\partial_{T}h+h.c\Big{)}\tilde{G}^{A}_{(0)}.$
(44)

We now solve for the adiabatic and first-order components of the self-energy-
like terms. Rather than considering each variant of self-energy individually,
we will instead consider the following more general expression ($c=<,>,R,A$)

$\Xi^{c}_{\alpha,ii^{\prime}}=\sum_{k}A_{ik\alpha}(t)g^{c}_{k\alpha}(t,t^{\prime})B_{k\alpha
i^{\prime}}(t^{\prime}),$ (45)

where $A$ and $B$ are arbitrary functions of time. Obviously, when $A_{k\alpha
i}=B_{k\alpha i}=t_{k\alpha i}$, we obtain $\Sigma^{c}$, while different
choices allow us to obtain $\Psi$, $\Phi$ and $\zeta$. We apply the Wigner
transform to the above while making use of the shift operator, defined
according to $f(x+h)=e^{hd^{f}_{x}}f(x)$ where we use $d^{f}_{x}$ to denote
the derivative with respect to $x$ which acts on $f$ (to avoid ambiguity) , to
obtain

$\displaystyle\tilde{\Xi}^{c}_{\alpha,ii^{\prime}}$
$\displaystyle=\sum_{k}\int^{\infty}_{-\infty}d\tau
e^{i\omega\tau}e^{\frac{\tau}{2}d^{A}_{T}}A_{ik\alpha}(T)g^{c}_{k\alpha}(t,t^{\prime})e^{\frac{-\tau}{2}d^{B}_{T}}B_{k\alpha
i^{\prime}}(T)$ (46) $\displaystyle=\sum_{k}\int^{\infty}_{-\infty}d\tau
e^{i\omega\tau}e^{\frac{1}{2i}\overleftarrow{\partial_{\omega}^{e}}(d^{A}_{T}-d^{B}_{T})}A_{ik\alpha}(T)g^{c}_{k\alpha}(t,t^{\prime})B_{k\alpha
i^{\prime}}(T),$ (47)

where the $\overleftarrow{\partial_{\omega}^{e}}$ notation denotes the
derivative operator acting to the left on the exponential. Now we take all the
terms that are independent of $\tau$ outside of the integral, leaving us with

$\displaystyle\tilde{\Xi}^{c}_{\alpha,ii^{\prime}}$
$\displaystyle=\sum_{k}e^{\frac{1}{2i}\overrightarrow{\partial_{\omega}^{e}}(d^{A}_{T}-d^{B}_{T})}A_{ik\alpha}(T)B_{k\alpha
i^{\prime}}(T)\int^{\infty}_{-\infty}d\tau
e^{i\omega\tau}g^{c}_{k\alpha}(t,t^{\prime})$ (48)
$\displaystyle=\sum_{k}e^{\frac{1}{2i}\overrightarrow{\partial_{\omega}^{G}}(d^{A}_{T}-d^{B}_{T})}A_{ik\alpha}(T)B_{k\alpha
i^{\prime}}(T)\tilde{g}^{c}_{k\alpha}(T,\omega).$ (49)

Finally, we take a power series expansion of the exponential to find

$\displaystyle\tilde{\Xi}^{c}_{\alpha,ii^{\prime}}$
$\displaystyle=\sum_{k}A_{ik\alpha}\tilde{g}^{c}_{k\alpha}B_{k\alpha
i^{\prime}}+\frac{1}{2i}\sum_{k}\frac{\partial\tilde{g}^{c}_{k\alpha}}{\partial\omega}\left(\frac{dA_{ik\alpha}}{dT}B_{k\alpha
i^{\prime}}-A_{ik\alpha}\frac{dB_{k\alpha
i^{\prime}}}{dT}\right)+...=\tilde{\Xi}^{c}_{(0),\alpha,ii^{\prime}}+\tilde{\Xi}^{c}_{(1),\alpha,ii^{\prime}}+...,$
(50)

where the functional dependencies are clear from the context. Thus, (50)
allows us to calculate each of the required orders of self-energy-like terms.
If we consider $A_{k\alpha i}=B_{k\alpha i}=t_{k\alpha i}$, the adiabatic
component corresponds to the standard self-energy. We make the wide-band
approximation for the electrodes. The retarded/advanced component is given by

$\tilde{\Sigma}_{(0),\alpha,ii^{\prime}}^{R/A}=\mp\frac{i}{2}\Gamma_{\alpha,ii^{\prime}},$
(51)

where the level-broadening takes the form

$\Gamma_{\alpha,ii^{\prime}}=2\pi t_{\alpha i}^{*}t_{\alpha
i^{\prime}}\rho_{\alpha},$ (52)

where density of states $\rho$ is a constant and $t_{\alpha ki}=t_{\alpha i}$
under the wide-band approximation. The equation for the lesser case takes the
form

$\tilde{\Sigma}_{(0),\alpha,ii^{\prime}}^{<}(\omega,T)=if_{\alpha}(\omega)\Gamma_{\alpha,ii^{\prime}}(T),$
(53)

where $f_{\alpha}(\omega)$ is the Fermi-Dirac distribution;

$f_{\alpha}(\omega)=\frac{1}{e^{\frac{\omega-\mu_{\alpha}}{k_{B}T_{\alpha}}}+1}.$
(54)

Here, $\mu_{\alpha}$ is the chemical potential for the $\alpha$ lead while
$T_{\alpha}$ is the macroscopic temperature and $k_{B}$ is Boltzmann’s
constant. The form of $\Psi$, $\Phi$ and $\zeta$ can be found equivalently by
replacing $\Gamma$ in the above equations with, $\Gamma^{\Psi}$,
$\Gamma^{\Phi}$ and $\Gamma^{\zeta}$, respectively, as given by

$\Gamma^{\Psi}_{\alpha,ii^{\prime}}=2\pi\partial_{\theta}t_{\alpha
i}^{*}t_{\alpha i^{\prime}}\rho_{\alpha},$ (55)
$\Gamma^{\Phi}_{\alpha,ii^{\prime}}=2\pi t_{\alpha
i}^{*}\partial_{\theta}t_{\alpha i^{\prime}}\rho_{\alpha},$ (56)
$\Gamma^{\zeta}_{\alpha,ii^{\prime}}=2\pi\partial_{\theta}t_{\alpha
i}^{*}\partial_{\theta}t_{\alpha i^{\prime}}\rho_{\alpha}.$ (57)

Under the wide-band approximation, $\tilde{\Xi}_{(1)}^{R/A}=0$, and so we need
only consider the lesser case.

## References

  * Wilcken  _et al._ [2018] R. Wilcken, M. Schildhauer, F. Rott, L. A. Huber, M. Guentner, S. Thumser, K. Hoffmann, S. Oesterling, R. de Vivie-Riedle, E. Riedle, and H. Dube, Complete Mechanism of Hemithioindigo Motor Rotation, Journal of the American Chemical Society 140, 5311 (2018).
  * Kistemaker  _et al._ [2015] J. C. M. Kistemaker, P. Štacko, J. Visser, and B. L. Feringa, Unidirectional rotary motion in achiral molecular motors, Nature Chemistry 7, 890 (2015).
  * Balzani  _et al._ [2006] V. Balzani, M. Clemente-León, A. Credi, B. Ferrer, M. Venturi, A. H. Flood, and J. F. Stoddart, Autonomous artificial nanomotor powered by sunlight, Proceedings of the National Academy of Sciences 103, 1178 (2006).
  * van Delden  _et al._ [2005] R. A. van Delden, M. K. J. ter Wiel, M. M. Pollard, J. Vicario, N. Koumura, and B. L. Feringa, Unidirectional molecular motor on a gold surface, Nature 437, 1337 (2005).
  * Koumura  _et al._ [1999] N. Koumura, R. W. Zijistra, R. A. Van Delden, N. Harada, and B. L. Feringa, Light-driven monodirectional molecular rotor, Nature 401, 152 (1999).
  * Kinosita  _et al._ [2000] K. Kinosita, R. Yasuda, H. Noji, and K. Adachi, A rotary molecular motor that can work at near 100efficiency, Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences 355, 473 (2000).
  * Leigh  _et al._ [2003] D. A. Leigh, J. K. Y. Wong, F. Dehez, and F. Zerbetto, Unidirectional rotation in a mechanically interlocked molecular rotor, Nature 424, 174 (2003).
  * Juluri  _et al._ [2009] B. K. Juluri, A. S. Kumar, Y. Liu, T. Ye, Y.-W. Yang, A. H. Flood, L. Fang, J. F. Stoddart, P. S. Weiss, and T. J. Huang, A Mechanical Actuator Driven Electrochemically by Artificial Molecular Muscles, ACS Nano 3, 291 (2009).
  * Barreiro  _et al._ [2008] A. Barreiro, R. Rurali, E. R. Hernández, J. Moser, T. Pichler, L. Forró, and A. Bachtold, Subnanometer Motion of Cargoes Driven by Thermal Gradients Along Carbon Nanotubes, Science 320, 775 (2008).
  * Stolz  _et al._ [2020] S. Stolz, O. Gröning, J. Prinz, H. Brune, and R. Widmer, Molecular motor crossing the frontier of classical to quantum tunneling motion, Proceedings of the National Academy of Sciences 117, 14838 (2020).
  * Ren  _et al._ [2020] J. Ren, M. Freitag, C. Schwermann, A. Bakker, S. Amirjalayer, A. Rühling, H.-Y. Gao, N. L. Doltsinis, F. Glorius, and H. Fuchs, A Unidirectional Surface-Anchored N-Heterocyclic Carbene Rotor, Nano Letters 20, 5922 (2020).
  * Zhang  _et al._ [2019] Y. Zhang, J. P. Calupitan, T. Rojas, R. Tumbleson, G. Erbland, C. Kammerer, T. M. Ajayi, S. Wang, L. A. Curtiss, A. T. Ngo, S. E. Ulloa, G. Rapenne, and S. W. Hla, A chiral molecular propeller designed for unidirectional rotations on a surface, Nature Communications 10, 3742 (2019).
  * Eisenhut  _et al._ [2018] F. Eisenhut, J. Meyer, J. Krüger, R. Ohmann, G. Cuniberti, and F. Moresco, Inducing the controlled rotation of single o-MeO-DMBI molecules anchored on Au(111), Surface Science 678, 177 (2018).
  * Mishra  _et al._ [2015] P. Mishra, J. P. Hill, S. Vijayaraghavan, W. V. Rossom, S. Yoshizawa, M. Grisolia, J. Echeverria, T. Ono, K. Ariga, T. Nakayama, C. Joachim, and T. Uchihashi, Current-Driven Supramolecular Motor with In Situ Surface Chiral Directionality Switching, Nano Letters 15, 4793 (2015).
  * Perera  _et al._ [2013] U. G. E. Perera, F. Ample, H. Kersell, Y. Zhang, G. Vives, J. Echeverria, M. Grisolia, G. Rapenne, C. Joachim, and S.-W. Hla, Controlled clockwise and anticlockwise rotational switching of a molecular motor, Nature Nanotechnology 8, 46 (2013).
  * Tierney  _et al._ [2011] H. L. Tierney, C. J. Murphy, A. D. Jewell, A. E. Baber, E. V. Iski, H. Y. Khodaverdian, A. F. McGuire, N. Klebanov, and E. C. H. Sykes, Experimental demonstration of a single-molecule electric motor, Nature Nanotechnology 6, 625 (2011).
  * Kudernac  _et al._ [2011] T. Kudernac, N. Ruangsupapichat, M. Parschau, B. Maciá, N. Katsonis, S. R. Harutyunyan, K.-H. Ernst, and B. L. Feringa, Electrically driven directional motion of a four-wheeled molecule on a metal surface, Nature 479, 208 (2011).
  * Ribetto  _et al._ [2022] F. D. Ribetto, S. E. Deghi, H. L. Calvo, and R. A. Bustos-Marún, A dynamical model for Brownian molecular motors driven by inelastic electron tunneling, The Journal of Chemical Physics 157, 164102 (2022).
  * Echeverria  _et al._ [2014] J. Echeverria, S. Monturet, and C. Joachim, One-way rotation of a molecule-rotor driven by a shot noise, Nanoscale 6, 2793 (2014).
  * Nie and Mei [2009] L.-R. Nie and D.-C. Mei, Effect of correlated noises on Brownian motor, Physics Letters A 373, 3816 (2009).
  * Calvo  _et al._ [2017] H. L. Calvo, F. D. Ribetto, and R. A. Bustos-Marún, Real-time diagrammatic approach to current-induced forces: Application to quantum-dot based nanomotors, Physical Review B 96, 165309 (2017).
  * Dundas  _et al._ [2009] D. Dundas, E. J. McEniry, and T. N. Todorov, Current-driven atomic waterwheels, Nature Nanotechnology 4, 99 (2009).
  * Landauer [1975] R. Landauer, Inadequacy of entropy and entropy derivatives in characterizing the steady state, Physical Review A 12, 636 (1975).
  * Landauer [1993] R. Landauer, Statistical physics of machinery: forgotten middle-ground, Physica A: Statistical Mechanics and its Applications 194, 551 (1993).
  * Preston  _et al._ [2021] R. J. Preston, M. F. Gelin, and D. S. Kosov, First-passage time theory of activated rate chemical processes in electronic molecular junctions, J. Chem. Phys. 154, 114108 (2021).
  * Buttiker [1987] M. Buttiker, Transport as a consequence of state-dependent diffusion, Zeitschrift fur Physik B Condensed Matter 68, 161 (1987).
  * Luchsinger [2000] R. H. Luchsinger, Transport in nonequilibrium systems with position-dependent mobility, Physical Review E 62, 272 (2000).
  * Brouwer [1998] P. W. Brouwer, Scattering approach to parametric pumping, Physical Review B 58, R10135 (1998).
  * Yadalam and Harbola [2016] H. K. Yadalam and U. Harbola, Statistics of an adiabatic charge pump, Physical Review B 93, 035312 (2016).
  * Pshenichnyuk and Čížek [2011] I. A. Pshenichnyuk and M. Čížek, Motor effect in electron transport through a molecular junction with torsional vibrations, Physical Review B 83, 165446 (2011).
  * Preston  _et al._ [2022] R. J. Preston, T. D. Honeychurch, and D. S. Kosov, Emergence of negative viscosities and colored noise under current-driven Ehrenfest molecular dynamics, Physical Review B 106, 195406 (2022).
  * Zwanzig [2001] R. Zwanzig,  _Nonequilibrium Statistical Mechanics_ (Oxford University Press, 2001).
  * Preston  _et al._ [2020a] R. J. Preston, V. F. Kershaw, and D. S. Kosov, Current-induced atomic motion, structural instabilities, and negative temperatures on molecule-electrode interfaces in electronic junctions, Phys. Rev. B 101, 155415 (2020a).
  * Preston  _et al._ [2020b] R. J. Preston, T. D. Honeychurch, and D. S. Kosov, Cooling molecular electronic junctions by AC current, J. Chem. Phys. 153, 121102 (2020b).