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One-way rotation of a molecule-rotor driven by a shot noise† Jorge Echeverria,* Serge Monturet and Christian Joachim The shot noise of a tunneling current passing through a molecule-motor can sustain a one-way rotation when populating the molecular excited states by tunneling inelastic excitations. We demonstrate that a ratchet-like ground state rotation potential energy curve is not necessary for the rotation to occur. A relative shift in energy difference between the maxima of this ground state and the minima of the
Received 31st October 2013 Accepted 12th December 2013
excited states is the necessary condition to get to a unidirectional rotation. The rotor speed of rotation and its rotation direction are both controlled by this shift, indicating the necessity of a careful design of
DOI: 10.1039/c3nr05814j
both the ground and excited states of the next generation of molecule-motors to be able to generate a
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motive power at the nanoscale.
Introduction Since the rst observation of the rotation of a single molecule adsorbed on a surface,1 studies on surface-supported single molecule mechanical machineries have expanded in many directions such as molecule-gears,2,3 molecule-motors4,5 and molecule nano-vehicles.6,7 Nowadays, there are several possibilities to drive the rotation of a single molecule motor when adsorbed on a metallic surface, like pushing the rotor using an STM or AFM tip,2,8 feeding the rotor using inelastic tunneling electrons via an STM tip,9,10 applying an oscillating electric eld to the rotor to synchronise its rotation with the large oscillation period of this eld,11–13 or powering the rotor with light.4 Aside from tip pushing,2 in the other cases mentioned above the conditions for a one way rotation of a molecule-motor have not yet been claried. Currently, the literature is mainly following two directions: (a) the introduction of an intramolecular ratchet effect which is usually characterized by plotting the potential energy variation of the rotor as a function of its rotation angle14 and (b) the control of the interactions between the motor and a specic external driving force to break micro-reversibility.11 In this article, following the recent demonstration of the controlled rotation of a single molecule motor driven by a tunneling current,14 we demonstrate that the conditions for a one-way rotation of a single molecule motor are between (a) and (b). From (a), there is a need to break the symmetry of the rotational potential energy surface of the rotor as a function of its rotation angle and from (b), there is a need Nanosciences Group & MANA Satellite, CEMES/CNRS, 29 rue Jeanne Marvig, 31055 Toulouse, France. E-mail: [email protected] † Electronic supplementary information (ESI) available: Rotation energy prole of the four-ferrocene single molecule motor, NEB calculations on the rotation of the 2,20 -dibromobiphenyl model and damping behavior of the theoretical model. See DOI: 10.1039/c3nr05814j
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for the rectication of some random excitation process of the motor to drive the rotation. At the origin of this study, our molecule motor is a piano-stool organometallic complex composed of a ve-arm rotor mounted on a tripod stator as presented in Fig. 1.14–16 This molecular motor was recently operated by STM on a Au(111) surface yielding step-by-step oneway rotation, both clockwise and counter-clockwise, under a tunneling current. In order to follow the motion in the STM experiments one of the molecule arms was tagged by removing its terminal ferrocene.14 The direction of rotation is selected by positioning the STM tip on different wings of the rotor. If the tip is placed on the truncated arm, the rotation is clockwise, whereas when it is placed on a ferrocene-terminated arm the rotation is counterclockwise.14
Fig. 1 The detailed chemical structure of the molecule-motor at the basis of this article. The rotation angle q is defined along the imaginary axis passing through the Ru atom and the centroid of the central Cp ring. 4 defines a flipping dihedral angle in the arm and c is the leg deformation angle.
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In this paper, we rst build up a general model for a rotary molecule motor like the one shown in Fig. 1 to analyse the requirements to rotate one-way in a controllable manner. Then, a complete numerical analysis of the rotation dynamics is performed, paying a special attention to the average driving force and to the role played by the electronic excited states of the molecule. Finally, we demonstrate how to drive a one-way rotation using shot noise, which allows us to explain how to control the rotation direction.
Computational details The rotation potential energy prole of the molecule motor ground state was obtained by means of the ASED+ semiempirical technique.17 The ASED+ semi-empirical Hamiltonian as well as the orbital parameters used by the ASED+ soware have been extensively tested and can be found in the literature.17,18 The van der Waals parameters (vdW radius r and vdW energy coefficient E) to complete the molecular orbital ˚ E ¼ 0.110 kcal mol1), description were used for Au (r ¼ 2.50 A, 1 ˚ E ¼ 0.057 kcal mol ) and H (r ¼ 1.60 A, ˚ E ¼ 0.017 C (r ¼ 1.96 A, kcal mol1). For the general van der Waals interaction, a potential function was added to the standard ASED potential17 as in the MM4 program.19 The rotation energy prole of 2,20 dibromobiphenyl, partial geometry optimizations and calculations on this molecule to build the potential energy surfaces were done with the semi-empirical AM1 method20 as implemented in Gaussian 09.21 This method uses the AM1 Hamiltonian and the standard AM1 parameters. Numerical simulations were carried out with a home-built program that solves the dynamics equations by means of a Verlet-like algorithm based on the nite differences method (this soware can be provided upon request).
Results and discussion A simple mechanical model for a molecule-motor The electronic ground and rst excited states potential energy surfaces of the motor shown in Fig. 1 mainly depend on nine angles: the top rotor rotation angle q around the central Ru axis, the ve ipping angles 41 to 45 of the ve rotor phenyl arms and the deformation angles c1 to c3 of the three stator legs. Despite the fact that many other degrees of freedom contribute to the mechanics of this motor, their role is minor as compared to the nine principal ones. Even with only nine coordinates, the corresponding E0(q, 41, .45, c1, .c3) ground state potential energy surface is considerably difficult to master. Starting from one of the possible E0(q, 41, .45, c1, .c3) energy minima, a simple way to model the rotation mechanisms is to step by step increase q in one direction only while minimizing the molecule motor potential energy for each q. This delivers a q parametric rotation trajectory on the E0(q, 41, .45, c1, .c3) manifold. For the motor shown in Fig. 1, this trajectory was calculated using the ASED+ semi-empirical technique.17 Along this rotation trajectory, the resulting E0(q) potential energy curve is highly asymmetric, with a potential barrier height of DV ¼ 0.25 eV as presented in Fig. 2. On E0, the combination of the rotor upper C5
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Asymmetric potential energy profile for a forced clockwise rotation of the molecule motor shown in Fig. 1, obtained by means of semi-empirical calculations.
Fig. 2
and the stator lower C3 concurrent symmetries22 leads to een potential wells per turn. This periodic series of barriers is due to the interaction between the ve phenyl rings of the rotor arms and the three motor legs bound to the surface. During the rotation and every time one arm interacts with one leg, the corresponding phenyl ring ips in order to avoid the leg and the latter is consequently deformed. In the curve shown in Fig. 2, this results in a ratchet-like potential energy variation. It is worth mentioning that the calculations on a motor with a truncated arm are qualitatively identical (see Fig. S1 in the ESI†). However, this asymmetry, observed along the rotation trajectory, is articial because it is created by forcing the rotation in a given direction during the step-by-step q increase. There is no asymmetry when passing from one minimum to the next on E0(q, 41, .45, c1, .c3) and, thus, no breaking of the micro-reversibility principle. Therefore, by forcing a q rotation, we are anticipating the good functioning of the molecule motor but not demonstrating it. In fact, with the initial objective of determining the conditions for a one-way rotation, an ideal oneway rotation trajectory was created on E0(q, 41, .45, c1, .c3) by such forced rotation as if the rotor was already rotating one way instead of nding the optimized excitation process to approach this ideal rotation trajectory. To illustrate how a saw-tooth like rotation trajectory can emerge from an a priori symmetric potential energy surface, let us consider the rotation of a single phenyl of the 2,20 -dibromobiphenyl molecule around its central C–C bond (Fig. 3a). The q rotation angle is here dened by the C1, C2, C3, C4 dihedral angle and the 4 torsion angle by the C2, C3, C4, Br5 dihedral angle (Fig. 3a). To build up the E0(q, 4) potential energy surface presented in Fig. 3b, the molecule potential energy was calculated as a function of (q, 4) by means of the AM1 semi-empirical technique but without optimization of the molecular geometry. This surface is periodic in (q, 4) and provides a map of one of these periods including also two energy minima A and B. There is a central energy maximum corresponding to the Br–Br repulsion. This maximum is symmetric, as expected from the original symmetry of the molecule. With no external forces applied, the molecule remains in one of the E0(q, 4) energy minima. When a rotation is imposed on q from A to B and the molecule energy is minimized by only optimizing the value of 4 at each q step of 1 , the black line
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of freedom allowed during the optimization, the rotation energy prole is then symmetric. This example illustrates the extreme care required in selecting the good degrees of freedom to describe the mechanics of a single molecule motor. Since the number of degrees of freedom is generally quite large, the ones to consider, the ones to minimize and the ones to freeze must be very well chosen to model the intramolecular physics involved. For the molecule-motor shown in Fig. 1 (and for others of the same kind), a minimum of two degrees of freedom, here the q rotation angle and a generic torsion angle 4, must be considered and not only q as used in Fig. 2. For the motor shown in Fig. 1, this torsion angle 4 can be dened by the dihedral angle between a given phenyl ring of the rotor and the central cyclopentadienyl ring. In this case, a model E0(q, 4) ground state that mimics the one calculated in Fig. 3b can be written as: E0(q, 4) ¼ V0 sin2(q/L + 4)cos2(q/L + 4)e42 + k46
Fig. 3 (a) 2,20 -Dibromobiphenyl with the significant atoms labelled. Rotation angle is the dihedral angle formed by 1-2-3-4 and torsion angle is the dihedral angle formed by 2-3-4-5. (b) Potential energy surface map built from the scan calculation of the torsion and rotation angles of 2,20 -dibromobiphenyl. The black line represents the value of the torsion angle for each rotation angle from A to B, whereas the red dashed line represents the value of the torsion angle for each rotation angle from B to A. The torsion angle was the only degree of freedom optimized during the calculation of both paths. The background potential map was obtained by means of a series of single point AM1 calculations scanning the rotation and torsion angles. Rotation and torsion angles are given in degrees. The lateral coloured bar is the scale for the energy in eV. (c) Potential energy profile for a forced rotation from A to B (black line) and from B to A (red line).
trajectory shown in Fig. 3b is obtained. If now, starting from B, the molecule is forced to return back to A, the red dashed line trajectory shown in Fig. 3b is obtained. Following both trajectories by plotting the corresponding potential energy variations as a function of q, the saw-tooth like curves shown in Fig. 3c are obtained. Of course, a unique minimum energy path is expected aer a complete search with a full optimization of the reaction coordinate of a non-forced rotation. Notice that an asymmetric energy rotation trajectory with two different pathways between A and B is also obtained when the partial optimization involves all the molecular degrees of freedom but the rotation angle. However, if the angle formed by C2, C3 and C4 is the only degree
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(1)
where L is the period along the rotation angle. This potential captures the general features of molecule motors like the one presented in Fig. 1 and also 2,20 -dibromobiphenyl presented in Fig. 3b, where the arm-leg like interactions lead to the pseudoratchet effect. An exponential term was added in (1) to break the periodicity of the function in the 4 axis and obtain a general potential energy surface similar to that found for a molecule motor. Instead of the een minima of the molecule motor shown in Fig. 1, the number of minima was limited here to four per turn (L ¼ 1) to simplify the analysis. However, potential (1) can be easily adapted to molecule motors with a different number of arms by simply modifying the value of the period L. During the rotor rotation, the steric repulsion between a rotor phenyl and one of the three stator legs is taken into account in (1) by a 46 term (with k ¼ 3.12 1016 eV). V0 models the energy barrier height and k the strength of the deformation. This E0(q, 4) periodic potential energy surface is presented in Fig. 4. When a q rotation is forced on this surface, the energy variation along the rotation trajectory minimized by using the Nudged Elastic Band (NEB) method23 is similar to the one presented in Fig. 3b (see Fig. S2 in the ESI†). The En electronic excited states involved in the rotation are certainly composed of several unoccupied molecular orbitals and the corresponding En(q, 4) potential energy surfaces associated with the rotation are rather complex.14 Since our goal here is to understand how the excited states can be involved to reach a unidirectional rotation, we can select (1) for the analytic expression of the rst excited state E1(q, 4) and include a shiing term in E1(q, 4) to change the angular position of its minima relative to E0(q, 4). The molecule-motor equation of motion Aer building a general model for the potential energy surfaces of the molecule motor, the next step is to construct its general system of equations of motion. A preparation of the molecule motor in a coherent non-stationary quantum superposition of states by mixing its ground and some molecule electronic excited states will provide the energy required to drive the motion.24 This preparation has to be cyclic in time since the
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cyclic re-preparation of the superposition. All the cn(t) for n > 0 will progressively decay in time towards the ground state with an average decay rate b. In (2) and (3), the external driving force F(t) is there to sustain a rotation and to compensate decoherence and relaxation. Considering the E0(q, 4) ground state, one E1(q, 4) excited state and the decay rate b, a new system of equations can be derived from (2) and (3) to describe the molecule motor dynamics taking into account the total decoherence of the superposition included in (2) and (3): I Fig. 4 Potential energy surface, given by eqn (1), used in numerical simulations as the ground state of the molecule motor. The lateral bar represents the scale of E0(q, 4) in eV for a value of V0 ¼ 1 eV.
d2 q X ¼ 1 ebt Gðt ti ÞVq E0 ðq; 4Þ dt2 i X dq ebt Gðt ti ÞVq E1 ðq; 4Þ g þ dt i X þ FaGðt ti Þ
(4)
i
rotor will lose rapidly the coherence of the initial preparation due to the large number of internal degrees of freedom of the molecule-motor and to the nite lifetime of the excited states. For the two angles used in our molecular motor model, the system of semi-classical equations of motion can be written as:25 X d2 hqi dhqi 2 þ F ðtÞa I ¼ Vq jcn ðtÞj En ðq; 4Þ g 2 dt dt
(2)
X d h4i dh4i 2 I0 þ FðtÞa jc ¼ V ðtÞj E ðq; 4Þ g 4 n n dt dt2
(3)
2
where the molecular rotor is described as a disk of mass m ¼ 1024 kg and of radius a ¼ 1 nm, with an extremely small I ¼ 1042 kg m2 momentum of inertia with respect to q. In (3), the momentum of inertia I0 of the rotation along 4 is certainly smaller (around 1044 kg m2) than in (2). However, we have observed that this difference has very little effect on the dynamic behavior of the system, with a small decrease in the nal speed of about 2%. Accordingly, the same value of I ¼ 1042 kg m2 was used both in (2) and (3). A friction coefficient g ¼ 5 1031 kg m2 s1 was chosen to run (2) and (3) in an over-damped regime aer analyzing the damping behavior of the system (see Fig. S3 in the ESI†). This friction is smaller than the value calculated for a Xe atom sliding on a silver surface.26 Critically damped and slightly under-damped scenarios have been also tested with similar results. The change in the nal speed when the value g is decreased to 5 1032 is only of 5%. However, smaller values of g involving under-damped behavior of the system lead to a loss of the unidirectional rotation. Notice also that we have not taken into account the presence of thermal noise in our model since the actual experiments exploring single molecule motors on a surface are usually performed at very low temperature.27 The coherent non-stationary preparation is described in (2) and (3) by the cn(t) summation over the En(q, 4) ground and electronic excited states potential energy surfaces. If the superposition coherence is step-by-step preserved along the rotation time, a semi-classical motion will result based on the multiple potential energy surfaces En(q, 4) employed for the dynamics and according to the cn(t) amplitudes of the superposition.25 However, it is difficult to keep such coherence for a long time even by a
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I
d2 4 X ¼ 1 ebt Gðt ti ÞV4 E0 ðq; 4Þ dt2 i X ebt Gðt ti ÞV4 E1 ðq; 4Þ þ i
d4 X þ g FaGðt ti Þ dt i
(5)
where G(t ti) is a time shied step function that permits the force F to act at a given time series ti and to reset the excited state preparation with a total depopulation of the ground state. Therefore, there is no more coherence in (4) and (5) between the ground and the excited states as compared to (2) and (3). Before discussing the F(t) optimization to drive the rotor one way, let us comment on what happens when this rotor is driven only in its ground state by an external random force F(t). In this case, the system (4) and (5) simply reduces to the system: I
d2 q dq ¼ Vq E0 ðq; 4Þ g þ F ðtÞa dt2 dt
(6)
I
d2 4 d4 þ F ðtÞa ¼ V4 E0 ðq; 4Þ g dt2 dt
(7)
Here, we have tested whether a one-way rotation trajectory can be stabilized on E0(q, 4) using a simple Gaussian distributed F(t) driving force. This force was applied by means of a random generator half of the times on (6) and half of the times on (7). The system (6) and (7) was solved using a Verlet-like algorithm based on the nite differences method. Due to the extremely small I value, much care has been taken to ensure the stability of the numerical solutions. For a given V0 value in E0 and exploring different friction parameters relative to the average F(t) strength, it was not possible to construct a q trajectory on E0(q, 4) corresponding to a one-way rotation of the rotor. This fact is in good agreement with the general behavior of Brownian motors, which cannot undergo directed motion even in an anisotropic medium.28 Driving a one-way rotation with shot noise In a molecular tunneling junction, the tunneling current intensity is the result of billions of electron transfers per second
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happening at random time. The temperature independent statistical time distribution of this random process is a Poisson distribution.29 Its rst momentum gives the tunneling current intensity through the molecule depending on the bias voltage used whereas its second momentum measures the time uctuations of this intensity and is characteristic of a shot noise. The force F during each excitation in (4) and (5) can be constant for all random distributed ti or also be random in strength following, for example, a Gaussian distribution. The rst case is equivalent to consider that each electron transfer event through the molecule is inelastically active. This is certainly not the case in a molecular tunnel junction.29 Therefore, a Gaussian distribution for F along the random ti represents better the low efficiency of an inelastic tunneling excitation. By selecting a Poisson time-dependent excitation distribution the energy is provided to the molecule-motor by the shot noise. At each time ti, a zero mean Gaussian distributed F is applied to the molecule. Moreover, F must act on q or 4 with an equal probability over time. Unlike in (6) and (7), which were based only on a ground state excitation, a one-way rotation trajectory is now obtained with (4) and (5) using the above described Poisson noise and with both E0(q, 4) and E1(q, 4) being fully symmetric in q and 4. As presented in Fig. 5, the necessary condition for such a rotation is a shi of the E1(q, 4) minima relative to the E0(q, 4) ones. The rotation stops when there is a coincidence between a E1(q, 4) minimum and a E0(q, 4) maximum or between the E1(q, 4) and E0(q, 4) minima. Notice that such behavior was previously reported in another context for the directed transport of Brownian particles in a double symmetric potential.30 According to Fig. 5, a change in the rotation direction is obtained by changing the min–max relative shi between E0(q, 4) and E1(q, 4). This was observed experimentally with the molecule-motor shown in Fig. 1 by locating the tip apex of the STM on different rotor arms to be coupled to different excited states of the molecule at the same positive bias voltage range.14 This gives a design rule for new molecule-motors with a possible reversible choice of the rotation direction since excited states with different min–max relative shis with respect to the
Fig. 5 Effect of the electronic states shift on the rotation efficiency and direction of a model with a symmetric ground state and a symmetric excited state. The average time between hits is 3.5 ns for a 10 ms simulation with a barrier height of 0.25 eV both in E0 and E1.
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ground state can be reached depending on the location of the excitation. Knowing the detailed topology of the excited electronic states potential energy surface of a molecule-motor becomes now an important part of its design. For instance, an elegant example of a molecule able to shi its min–max using hydrogen bonds was recently reported by Garc´ıa-Iriepa et al.31 With the same barrier height (0.25 eV) in E0(q, 4) and E1(q, 4), a molecule-motor of the type presented in Fig. 1 is able to rotate one-way in a controllable manner for an average Gaussian F larger than 0.2 pN. The necessary condition is that the E0(q, 4) and E1(q, 4) min–max must be shied as discussed above and presented in Fig. 5. The average time interval between two ti that forges the Poisson distribution in (4) and (5) must also be tuned in order to get a stable unidirectional rotation. For example, according to the solution of (4) and (5) and for an excited state decay time b1of 1 ns, the rotor of the molecule-motor rotates one-way for an average time separation between two ti larger than 0.35 ns, reaching a maximum rotation speed at a 1.25 ns ti interval. Aer this threshold, the average speed decays exponentially due to the decrease in the energy supplied to the molecule per unit of time as presented in Fig. 6a. Therefore, when b decreases, that is for very long E1(q, 4) life times, it is necessary to increase the average excitation time further in order to keep control on the rotation. The rotation is better controlled when the average time interval between two ti is long enough as compared to the lifetime of the excited state (here longer than 2 ns). For example, the same rotation velocity is reached (7.5 106 turns per s) when the system is driven with an average interval of 0.35 ns or of 7 ns between two consecutive ti leads. This corresponds to a tunneling current intensity below 1 nA. However, notice that the energy provided in the rst case to the molecule-motor is twenty times larger than in the second and, thus, the rotation efficiency will be signicantly different for both cases. A representation of the two rotation behaviors is presented in Fig. 6b. To evaluate how a motor following (4) and (5) can move a load, this system of equations was solved again when applying a constant antagonist force to the rotation direction in (4). As shown in Fig. 7 and at its maximum speed, the motor is able to work against forces up to 0.65 pN. This value is in good agreement with the universal performance of motors proposed by Marden et al.32 as a function of the motor intrinsic mass. According to this roadmap, our 1024 kg mass molecule-motor is expected to bring a maximum output force of about 0.1 pN. The motor efficiency is h ¼ W/Ein, where Ein is the energy delivered to the motor and W the maximum work produced by it. When driving the motor with a tunneling current, it is not the bias voltage V applied to the molecular tunnel junction which must be considered as the energy provided to the motor for its functioning. It is the inelastic tunneling effect that populates the excited states and the spectral power of the corresponding shot noise. The shot noise applied is not a constant excitation, neither in amplitude nor in time. Therefore, the provided energy can only be estimated here considering the maximum of the random force in (4) and (5) with a Gaussian distribution ranging between 300 pN and 300 pN, which has been calculated to lead to the motor maximum rotation speed. For the
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the external force applied. This gives h ¼ 0.002 for our shot noise driven motor. This is very close to h ¼ 0.001 for a Brownian motor driven by correlated Gaussian white noise33 but less than for a motor driven by a temporally asymmetric unbiased external force34 (h ¼ 0.01) or for a ashing ratchet35 (h ¼ 0.05). On the other hand, the barrier height along their respective potential energy surfaces of the states controls the rotor average velocity. We have employed for the simulations the value previously calculated with the ASED+ semi-empirical method, which is 0.25 eV for the ground state, leading to a calculated average velocity of 3.43 107 turns per s. Increasing the energy barrier of both states up to 0.44 eV leads to a slightly slower rotation (3.28 107 turns per s). However, when decreasing the barrier in E0(q, 4) to 0.25 eV while keeping E1(q, 4) at 0.44 eV, a smaller average velocity is calculated (2.25 107 turns per s). In contrast, a barrier of 0.44 eV in E0(q, 4) and of 0.25 eV in E1(q, 4) leads to a large average velocity of 4.11 107 turns per s. Further modications of the energy barriers must be accompanied by a new optimization of the average time interval between two ti and of the average Gaussian force in order to keep the one-way rotation.
Conclusions
Fig. 6 (a) Average velocity as a function of the average time between two hits, for a model with a shift of 0.6 rad between electronic states. (b) Angle of rotation as a function of time. The red line is for an average hitting time of 0.35 ns whereas the black line is for an average hitting time of 7 ns.
Fig. 7 Average rotation velocity of the molecule-motor as a function of the load. The load is modelled as an external applied force against the rotor rotation.
optimized one-way rotation conditions discussed above, this leads to Ein ¼ 3 1019 J. According to Fig. 7 and under those conditions, the motor is able to develop a maximum work of 6.5 1022 J before stopping and going backward constrained by
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Formally equivalent to a particle in solution, which is subject to a thermal noise due to random collisions with solvent molecules,28 a surface-mounted molecule in ultra-high vacuum can be exposed to a shot noise coming from the STM tip. This noise has been recently used to drive the motion of a single molecule motor.14 Inspired by this recent achievement, we have carried out a systematic study of the dynamics of a single molecule motor model driven by a shot noise. The use of an excited state for driving the rotation of the molecule motor allows the conversion of a Poisson time-distributed applied force to a controlled unidirectional motion. For a barrier height of 0.25 eV, the motor reaches a maximum speed of 3.43 107 turns per s. Furthermore, the direction of rotation of a molecule-motor can be reversed depending on the characteristics of the excited state involved in the process. We have demonstrated that a molecule without an intrinsic asymmetry can reconstruct a one-way rotation by using its internal characteristics. These ndings will help to optimize the chemical structure of new molecule-motors, not only to rotate one-way but also to develop a true motive power able to move atoms, small molecules or to drive cooperatively a train of solid state nano-gears.
Acknowledgements The authors acknowledge nancial support from the AUTOMOL project (ANR 09-NANO-040) and CNRS. J. E. thanks the Generalitat de Catalunya and the European Union for a Beatriu de Pin´ os scholarship. The authors also thank C. Collard for technical assistance with numerical simulations and CALMIP for computing time at Hyperion supercomputer.
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21 M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr, J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, ¨ Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski and O. D. J. Fox, Gaussian 09, Revision A.1, Gaussian, Inc., Wallingford CT, 2009. 22 J. Echeverr´ıa, A. Carreras, D. Casanova, P. Alemany and S. Alvarez, Chem. – Eur. J., 2011, 17, 359–367. 23 D. Sheppard, R. Terrell and G. Henkelman, J. Chem. Phys., 2008, 128, 134106. 24 C. Joachim, J. Phys.: Condens. Matter, 2006, 18, S1935–S1942. 25 C. Joachim, Mol. Phys., 1984, 52, 1191–1208. 26 B. N. J. Persson and A. Nitzan, Surf. Sci., 1996, 367, 261–275. 27 D. Lensen and J. A. A. W. Elemans, So Matter, 2012, 8, 9053– 9063. 28 R. Astumian, Science, 1997, 276, 917–922. 29 C. Joachim and M. A. Ratner, Proc. Natl. Acad. Sci. U. S. A., 2005, 102, 8801–8808. 30 L. Sanchez-Palencia, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2004, 70, 011102. 31 C. Garc´ıa-Iriepa, M. Marazzi, F. Zapata, A. Valentini, D. Sampedro and L. M. Frutos, J. Phys. Chem. Lett., 2013, 4, 1389–1396. 32 J. H. Marden and L. R. Allen, Proc. Natl. Acad. Sci. U. S. A., 2002, 99, 4161–4166. 33 L.-R. Nie and D.-C. Mei, Phys. Lett. A, 2009, 373, 3816–3821. 34 B. Ai, H. Xie, H. Liao and L. Liu, J. Stat. Mech.: Theory Exp., 2006, 2006, P09016. 35 J. M. R. Parrondo and B. J. de Cisneros, Appl. Phys. A: Mater. Sci. Process., 2002, 75, 179–191.
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One-way rotation of a molecule-rotor driven by a shot noise† Jorge Echeverria,* Serge Monturet and Christian Joachim The shot noise of a tunneling current passing through a molecule-motor can sustain a one-way rotation when populating the molecular excited states by tunneling inelastic excitations. We demonstrate that a ratchet-like ground state rotation potential energy curve is not necessary for the rotation to occur. A relative shift in energy difference between the maxima of this ground state and the minima of the
Received 31st October 2013 Accepted 12th December 2013
excited states is the necessary condition to get to a unidirectional rotation. The rotor speed of rotation and its rotation direction are both controlled by this shift, indicating the necessity of a careful design of
DOI: 10.1039/c3nr05814j
both the ground and excited states of the next generation of molecule-motors to be able to generate a
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motive power at the nanoscale.
Introduction Since the rst observation of the rotation of a single molecule adsorbed on a surface,1 studies on surface-supported single molecule mechanical machineries have expanded in many directions such as molecule-gears,2,3 molecule-motors4,5 and molecule nano-vehicles.6,7 Nowadays, there are several possibilities to drive the rotation of a single molecule motor when adsorbed on a metallic surface, like pushing the rotor using an STM or AFM tip,2,8 feeding the rotor using inelastic tunneling electrons via an STM tip,9,10 applying an oscillating electric eld to the rotor to synchronise its rotation with the large oscillation period of this eld,11–13 or powering the rotor with light.4 Aside from tip pushing,2 in the other cases mentioned above the conditions for a one way rotation of a molecule-motor have not yet been claried. Currently, the literature is mainly following two directions: (a) the introduction of an intramolecular ratchet effect which is usually characterized by plotting the potential energy variation of the rotor as a function of its rotation angle14 and (b) the control of the interactions between the motor and a specic external driving force to break micro-reversibility.11 In this article, following the recent demonstration of the controlled rotation of a single molecule motor driven by a tunneling current,14 we demonstrate that the conditions for a one-way rotation of a single molecule motor are between (a) and (b). From (a), there is a need to break the symmetry of the rotational potential energy surface of the rotor as a function of its rotation angle and from (b), there is a need Nanosciences Group & MANA Satellite, CEMES/CNRS, 29 rue Jeanne Marvig, 31055 Toulouse, France. E-mail: [email protected] † Electronic supplementary information (ESI) available: Rotation energy prole of the four-ferrocene single molecule motor, NEB calculations on the rotation of the 2,20 -dibromobiphenyl model and damping behavior of the theoretical model. See DOI: 10.1039/c3nr05814j
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for the rectication of some random excitation process of the motor to drive the rotation. At the origin of this study, our molecule motor is a piano-stool organometallic complex composed of a ve-arm rotor mounted on a tripod stator as presented in Fig. 1.14–16 This molecular motor was recently operated by STM on a Au(111) surface yielding step-by-step oneway rotation, both clockwise and counter-clockwise, under a tunneling current. In order to follow the motion in the STM experiments one of the molecule arms was tagged by removing its terminal ferrocene.14 The direction of rotation is selected by positioning the STM tip on different wings of the rotor. If the tip is placed on the truncated arm, the rotation is clockwise, whereas when it is placed on a ferrocene-terminated arm the rotation is counterclockwise.14
Fig. 1 The detailed chemical structure of the molecule-motor at the basis of this article. The rotation angle q is defined along the imaginary axis passing through the Ru atom and the centroid of the central Cp ring. 4 defines a flipping dihedral angle in the arm and c is the leg deformation angle.
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In this paper, we rst build up a general model for a rotary molecule motor like the one shown in Fig. 1 to analyse the requirements to rotate one-way in a controllable manner. Then, a complete numerical analysis of the rotation dynamics is performed, paying a special attention to the average driving force and to the role played by the electronic excited states of the molecule. Finally, we demonstrate how to drive a one-way rotation using shot noise, which allows us to explain how to control the rotation direction.
Computational details The rotation potential energy prole of the molecule motor ground state was obtained by means of the ASED+ semiempirical technique.17 The ASED+ semi-empirical Hamiltonian as well as the orbital parameters used by the ASED+ soware have been extensively tested and can be found in the literature.17,18 The van der Waals parameters (vdW radius r and vdW energy coefficient E) to complete the molecular orbital ˚ E ¼ 0.110 kcal mol1), description were used for Au (r ¼ 2.50 A, 1 ˚ E ¼ 0.057 kcal mol ) and H (r ¼ 1.60 A, ˚ E ¼ 0.017 C (r ¼ 1.96 A, kcal mol1). For the general van der Waals interaction, a potential function was added to the standard ASED potential17 as in the MM4 program.19 The rotation energy prole of 2,20 dibromobiphenyl, partial geometry optimizations and calculations on this molecule to build the potential energy surfaces were done with the semi-empirical AM1 method20 as implemented in Gaussian 09.21 This method uses the AM1 Hamiltonian and the standard AM1 parameters. Numerical simulations were carried out with a home-built program that solves the dynamics equations by means of a Verlet-like algorithm based on the nite differences method (this soware can be provided upon request).
Results and discussion A simple mechanical model for a molecule-motor The electronic ground and rst excited states potential energy surfaces of the motor shown in Fig. 1 mainly depend on nine angles: the top rotor rotation angle q around the central Ru axis, the ve ipping angles 41 to 45 of the ve rotor phenyl arms and the deformation angles c1 to c3 of the three stator legs. Despite the fact that many other degrees of freedom contribute to the mechanics of this motor, their role is minor as compared to the nine principal ones. Even with only nine coordinates, the corresponding E0(q, 41, .45, c1, .c3) ground state potential energy surface is considerably difficult to master. Starting from one of the possible E0(q, 41, .45, c1, .c3) energy minima, a simple way to model the rotation mechanisms is to step by step increase q in one direction only while minimizing the molecule motor potential energy for each q. This delivers a q parametric rotation trajectory on the E0(q, 41, .45, c1, .c3) manifold. For the motor shown in Fig. 1, this trajectory was calculated using the ASED+ semi-empirical technique.17 Along this rotation trajectory, the resulting E0(q) potential energy curve is highly asymmetric, with a potential barrier height of DV ¼ 0.25 eV as presented in Fig. 2. On E0, the combination of the rotor upper C5
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Asymmetric potential energy profile for a forced clockwise rotation of the molecule motor shown in Fig. 1, obtained by means of semi-empirical calculations.
Fig. 2
and the stator lower C3 concurrent symmetries22 leads to een potential wells per turn. This periodic series of barriers is due to the interaction between the ve phenyl rings of the rotor arms and the three motor legs bound to the surface. During the rotation and every time one arm interacts with one leg, the corresponding phenyl ring ips in order to avoid the leg and the latter is consequently deformed. In the curve shown in Fig. 2, this results in a ratchet-like potential energy variation. It is worth mentioning that the calculations on a motor with a truncated arm are qualitatively identical (see Fig. S1 in the ESI†). However, this asymmetry, observed along the rotation trajectory, is articial because it is created by forcing the rotation in a given direction during the step-by-step q increase. There is no asymmetry when passing from one minimum to the next on E0(q, 41, .45, c1, .c3) and, thus, no breaking of the micro-reversibility principle. Therefore, by forcing a q rotation, we are anticipating the good functioning of the molecule motor but not demonstrating it. In fact, with the initial objective of determining the conditions for a one-way rotation, an ideal oneway rotation trajectory was created on E0(q, 41, .45, c1, .c3) by such forced rotation as if the rotor was already rotating one way instead of nding the optimized excitation process to approach this ideal rotation trajectory. To illustrate how a saw-tooth like rotation trajectory can emerge from an a priori symmetric potential energy surface, let us consider the rotation of a single phenyl of the 2,20 -dibromobiphenyl molecule around its central C–C bond (Fig. 3a). The q rotation angle is here dened by the C1, C2, C3, C4 dihedral angle and the 4 torsion angle by the C2, C3, C4, Br5 dihedral angle (Fig. 3a). To build up the E0(q, 4) potential energy surface presented in Fig. 3b, the molecule potential energy was calculated as a function of (q, 4) by means of the AM1 semi-empirical technique but without optimization of the molecular geometry. This surface is periodic in (q, 4) and provides a map of one of these periods including also two energy minima A and B. There is a central energy maximum corresponding to the Br–Br repulsion. This maximum is symmetric, as expected from the original symmetry of the molecule. With no external forces applied, the molecule remains in one of the E0(q, 4) energy minima. When a rotation is imposed on q from A to B and the molecule energy is minimized by only optimizing the value of 4 at each q step of 1 , the black line
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of freedom allowed during the optimization, the rotation energy prole is then symmetric. This example illustrates the extreme care required in selecting the good degrees of freedom to describe the mechanics of a single molecule motor. Since the number of degrees of freedom is generally quite large, the ones to consider, the ones to minimize and the ones to freeze must be very well chosen to model the intramolecular physics involved. For the molecule-motor shown in Fig. 1 (and for others of the same kind), a minimum of two degrees of freedom, here the q rotation angle and a generic torsion angle 4, must be considered and not only q as used in Fig. 2. For the motor shown in Fig. 1, this torsion angle 4 can be dened by the dihedral angle between a given phenyl ring of the rotor and the central cyclopentadienyl ring. In this case, a model E0(q, 4) ground state that mimics the one calculated in Fig. 3b can be written as: E0(q, 4) ¼ V0 sin2(q/L + 4)cos2(q/L + 4)e42 + k46
Fig. 3 (a) 2,20 -Dibromobiphenyl with the significant atoms labelled. Rotation angle is the dihedral angle formed by 1-2-3-4 and torsion angle is the dihedral angle formed by 2-3-4-5. (b) Potential energy surface map built from the scan calculation of the torsion and rotation angles of 2,20 -dibromobiphenyl. The black line represents the value of the torsion angle for each rotation angle from A to B, whereas the red dashed line represents the value of the torsion angle for each rotation angle from B to A. The torsion angle was the only degree of freedom optimized during the calculation of both paths. The background potential map was obtained by means of a series of single point AM1 calculations scanning the rotation and torsion angles. Rotation and torsion angles are given in degrees. The lateral coloured bar is the scale for the energy in eV. (c) Potential energy profile for a forced rotation from A to B (black line) and from B to A (red line).
trajectory shown in Fig. 3b is obtained. If now, starting from B, the molecule is forced to return back to A, the red dashed line trajectory shown in Fig. 3b is obtained. Following both trajectories by plotting the corresponding potential energy variations as a function of q, the saw-tooth like curves shown in Fig. 3c are obtained. Of course, a unique minimum energy path is expected aer a complete search with a full optimization of the reaction coordinate of a non-forced rotation. Notice that an asymmetric energy rotation trajectory with two different pathways between A and B is also obtained when the partial optimization involves all the molecular degrees of freedom but the rotation angle. However, if the angle formed by C2, C3 and C4 is the only degree
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(1)
where L is the period along the rotation angle. This potential captures the general features of molecule motors like the one presented in Fig. 1 and also 2,20 -dibromobiphenyl presented in Fig. 3b, where the arm-leg like interactions lead to the pseudoratchet effect. An exponential term was added in (1) to break the periodicity of the function in the 4 axis and obtain a general potential energy surface similar to that found for a molecule motor. Instead of the een minima of the molecule motor shown in Fig. 1, the number of minima was limited here to four per turn (L ¼ 1) to simplify the analysis. However, potential (1) can be easily adapted to molecule motors with a different number of arms by simply modifying the value of the period L. During the rotor rotation, the steric repulsion between a rotor phenyl and one of the three stator legs is taken into account in (1) by a 46 term (with k ¼ 3.12 1016 eV). V0 models the energy barrier height and k the strength of the deformation. This E0(q, 4) periodic potential energy surface is presented in Fig. 4. When a q rotation is forced on this surface, the energy variation along the rotation trajectory minimized by using the Nudged Elastic Band (NEB) method23 is similar to the one presented in Fig. 3b (see Fig. S2 in the ESI†). The En electronic excited states involved in the rotation are certainly composed of several unoccupied molecular orbitals and the corresponding En(q, 4) potential energy surfaces associated with the rotation are rather complex.14 Since our goal here is to understand how the excited states can be involved to reach a unidirectional rotation, we can select (1) for the analytic expression of the rst excited state E1(q, 4) and include a shiing term in E1(q, 4) to change the angular position of its minima relative to E0(q, 4). The molecule-motor equation of motion Aer building a general model for the potential energy surfaces of the molecule motor, the next step is to construct its general system of equations of motion. A preparation of the molecule motor in a coherent non-stationary quantum superposition of states by mixing its ground and some molecule electronic excited states will provide the energy required to drive the motion.24 This preparation has to be cyclic in time since the
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cyclic re-preparation of the superposition. All the cn(t) for n > 0 will progressively decay in time towards the ground state with an average decay rate b. In (2) and (3), the external driving force F(t) is there to sustain a rotation and to compensate decoherence and relaxation. Considering the E0(q, 4) ground state, one E1(q, 4) excited state and the decay rate b, a new system of equations can be derived from (2) and (3) to describe the molecule motor dynamics taking into account the total decoherence of the superposition included in (2) and (3): I Fig. 4 Potential energy surface, given by eqn (1), used in numerical simulations as the ground state of the molecule motor. The lateral bar represents the scale of E0(q, 4) in eV for a value of V0 ¼ 1 eV.
d2 q X ¼ 1 ebt Gðt ti ÞVq E0 ðq; 4Þ dt2 i X dq ebt Gðt ti ÞVq E1 ðq; 4Þ g þ dt i X þ FaGðt ti Þ
(4)
i
rotor will lose rapidly the coherence of the initial preparation due to the large number of internal degrees of freedom of the molecule-motor and to the nite lifetime of the excited states. For the two angles used in our molecular motor model, the system of semi-classical equations of motion can be written as:25 X d2 hqi dhqi 2 þ F ðtÞa I ¼ Vq jcn ðtÞj En ðq; 4Þ g 2 dt dt
(2)
X d h4i dh4i 2 I0 þ FðtÞa jc ¼ V ðtÞj E ðq; 4Þ g 4 n n dt dt2
(3)
2
where the molecular rotor is described as a disk of mass m ¼ 1024 kg and of radius a ¼ 1 nm, with an extremely small I ¼ 1042 kg m2 momentum of inertia with respect to q. In (3), the momentum of inertia I0 of the rotation along 4 is certainly smaller (around 1044 kg m2) than in (2). However, we have observed that this difference has very little effect on the dynamic behavior of the system, with a small decrease in the nal speed of about 2%. Accordingly, the same value of I ¼ 1042 kg m2 was used both in (2) and (3). A friction coefficient g ¼ 5 1031 kg m2 s1 was chosen to run (2) and (3) in an over-damped regime aer analyzing the damping behavior of the system (see Fig. S3 in the ESI†). This friction is smaller than the value calculated for a Xe atom sliding on a silver surface.26 Critically damped and slightly under-damped scenarios have been also tested with similar results. The change in the nal speed when the value g is decreased to 5 1032 is only of 5%. However, smaller values of g involving under-damped behavior of the system lead to a loss of the unidirectional rotation. Notice also that we have not taken into account the presence of thermal noise in our model since the actual experiments exploring single molecule motors on a surface are usually performed at very low temperature.27 The coherent non-stationary preparation is described in (2) and (3) by the cn(t) summation over the En(q, 4) ground and electronic excited states potential energy surfaces. If the superposition coherence is step-by-step preserved along the rotation time, a semi-classical motion will result based on the multiple potential energy surfaces En(q, 4) employed for the dynamics and according to the cn(t) amplitudes of the superposition.25 However, it is difficult to keep such coherence for a long time even by a
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I
d2 4 X ¼ 1 ebt Gðt ti ÞV4 E0 ðq; 4Þ dt2 i X ebt Gðt ti ÞV4 E1 ðq; 4Þ þ i
d4 X þ g FaGðt ti Þ dt i
(5)
where G(t ti) is a time shied step function that permits the force F to act at a given time series ti and to reset the excited state preparation with a total depopulation of the ground state. Therefore, there is no more coherence in (4) and (5) between the ground and the excited states as compared to (2) and (3). Before discussing the F(t) optimization to drive the rotor one way, let us comment on what happens when this rotor is driven only in its ground state by an external random force F(t). In this case, the system (4) and (5) simply reduces to the system: I
d2 q dq ¼ Vq E0 ðq; 4Þ g þ F ðtÞa dt2 dt
(6)
I
d2 4 d4 þ F ðtÞa ¼ V4 E0 ðq; 4Þ g dt2 dt
(7)
Here, we have tested whether a one-way rotation trajectory can be stabilized on E0(q, 4) using a simple Gaussian distributed F(t) driving force. This force was applied by means of a random generator half of the times on (6) and half of the times on (7). The system (6) and (7) was solved using a Verlet-like algorithm based on the nite differences method. Due to the extremely small I value, much care has been taken to ensure the stability of the numerical solutions. For a given V0 value in E0 and exploring different friction parameters relative to the average F(t) strength, it was not possible to construct a q trajectory on E0(q, 4) corresponding to a one-way rotation of the rotor. This fact is in good agreement with the general behavior of Brownian motors, which cannot undergo directed motion even in an anisotropic medium.28 Driving a one-way rotation with shot noise In a molecular tunneling junction, the tunneling current intensity is the result of billions of electron transfers per second
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happening at random time. The temperature independent statistical time distribution of this random process is a Poisson distribution.29 Its rst momentum gives the tunneling current intensity through the molecule depending on the bias voltage used whereas its second momentum measures the time uctuations of this intensity and is characteristic of a shot noise. The force F during each excitation in (4) and (5) can be constant for all random distributed ti or also be random in strength following, for example, a Gaussian distribution. The rst case is equivalent to consider that each electron transfer event through the molecule is inelastically active. This is certainly not the case in a molecular tunnel junction.29 Therefore, a Gaussian distribution for F along the random ti represents better the low efficiency of an inelastic tunneling excitation. By selecting a Poisson time-dependent excitation distribution the energy is provided to the molecule-motor by the shot noise. At each time ti, a zero mean Gaussian distributed F is applied to the molecule. Moreover, F must act on q or 4 with an equal probability over time. Unlike in (6) and (7), which were based only on a ground state excitation, a one-way rotation trajectory is now obtained with (4) and (5) using the above described Poisson noise and with both E0(q, 4) and E1(q, 4) being fully symmetric in q and 4. As presented in Fig. 5, the necessary condition for such a rotation is a shi of the E1(q, 4) minima relative to the E0(q, 4) ones. The rotation stops when there is a coincidence between a E1(q, 4) minimum and a E0(q, 4) maximum or between the E1(q, 4) and E0(q, 4) minima. Notice that such behavior was previously reported in another context for the directed transport of Brownian particles in a double symmetric potential.30 According to Fig. 5, a change in the rotation direction is obtained by changing the min–max relative shi between E0(q, 4) and E1(q, 4). This was observed experimentally with the molecule-motor shown in Fig. 1 by locating the tip apex of the STM on different rotor arms to be coupled to different excited states of the molecule at the same positive bias voltage range.14 This gives a design rule for new molecule-motors with a possible reversible choice of the rotation direction since excited states with different min–max relative shis with respect to the
Fig. 5 Effect of the electronic states shift on the rotation efficiency and direction of a model with a symmetric ground state and a symmetric excited state. The average time between hits is 3.5 ns for a 10 ms simulation with a barrier height of 0.25 eV both in E0 and E1.
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ground state can be reached depending on the location of the excitation. Knowing the detailed topology of the excited electronic states potential energy surface of a molecule-motor becomes now an important part of its design. For instance, an elegant example of a molecule able to shi its min–max using hydrogen bonds was recently reported by Garc´ıa-Iriepa et al.31 With the same barrier height (0.25 eV) in E0(q, 4) and E1(q, 4), a molecule-motor of the type presented in Fig. 1 is able to rotate one-way in a controllable manner for an average Gaussian F larger than 0.2 pN. The necessary condition is that the E0(q, 4) and E1(q, 4) min–max must be shied as discussed above and presented in Fig. 5. The average time interval between two ti that forges the Poisson distribution in (4) and (5) must also be tuned in order to get a stable unidirectional rotation. For example, according to the solution of (4) and (5) and for an excited state decay time b1of 1 ns, the rotor of the molecule-motor rotates one-way for an average time separation between two ti larger than 0.35 ns, reaching a maximum rotation speed at a 1.25 ns ti interval. Aer this threshold, the average speed decays exponentially due to the decrease in the energy supplied to the molecule per unit of time as presented in Fig. 6a. Therefore, when b decreases, that is for very long E1(q, 4) life times, it is necessary to increase the average excitation time further in order to keep control on the rotation. The rotation is better controlled when the average time interval between two ti is long enough as compared to the lifetime of the excited state (here longer than 2 ns). For example, the same rotation velocity is reached (7.5 106 turns per s) when the system is driven with an average interval of 0.35 ns or of 7 ns between two consecutive ti leads. This corresponds to a tunneling current intensity below 1 nA. However, notice that the energy provided in the rst case to the molecule-motor is twenty times larger than in the second and, thus, the rotation efficiency will be signicantly different for both cases. A representation of the two rotation behaviors is presented in Fig. 6b. To evaluate how a motor following (4) and (5) can move a load, this system of equations was solved again when applying a constant antagonist force to the rotation direction in (4). As shown in Fig. 7 and at its maximum speed, the motor is able to work against forces up to 0.65 pN. This value is in good agreement with the universal performance of motors proposed by Marden et al.32 as a function of the motor intrinsic mass. According to this roadmap, our 1024 kg mass molecule-motor is expected to bring a maximum output force of about 0.1 pN. The motor efficiency is h ¼ W/Ein, where Ein is the energy delivered to the motor and W the maximum work produced by it. When driving the motor with a tunneling current, it is not the bias voltage V applied to the molecular tunnel junction which must be considered as the energy provided to the motor for its functioning. It is the inelastic tunneling effect that populates the excited states and the spectral power of the corresponding shot noise. The shot noise applied is not a constant excitation, neither in amplitude nor in time. Therefore, the provided energy can only be estimated here considering the maximum of the random force in (4) and (5) with a Gaussian distribution ranging between 300 pN and 300 pN, which has been calculated to lead to the motor maximum rotation speed. For the
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the external force applied. This gives h ¼ 0.002 for our shot noise driven motor. This is very close to h ¼ 0.001 for a Brownian motor driven by correlated Gaussian white noise33 but less than for a motor driven by a temporally asymmetric unbiased external force34 (h ¼ 0.01) or for a ashing ratchet35 (h ¼ 0.05). On the other hand, the barrier height along their respective potential energy surfaces of the states controls the rotor average velocity. We have employed for the simulations the value previously calculated with the ASED+ semi-empirical method, which is 0.25 eV for the ground state, leading to a calculated average velocity of 3.43 107 turns per s. Increasing the energy barrier of both states up to 0.44 eV leads to a slightly slower rotation (3.28 107 turns per s). However, when decreasing the barrier in E0(q, 4) to 0.25 eV while keeping E1(q, 4) at 0.44 eV, a smaller average velocity is calculated (2.25 107 turns per s). In contrast, a barrier of 0.44 eV in E0(q, 4) and of 0.25 eV in E1(q, 4) leads to a large average velocity of 4.11 107 turns per s. Further modications of the energy barriers must be accompanied by a new optimization of the average time interval between two ti and of the average Gaussian force in order to keep the one-way rotation.
Conclusions
Fig. 6 (a) Average velocity as a function of the average time between two hits, for a model with a shift of 0.6 rad between electronic states. (b) Angle of rotation as a function of time. The red line is for an average hitting time of 0.35 ns whereas the black line is for an average hitting time of 7 ns.
Fig. 7 Average rotation velocity of the molecule-motor as a function of the load. The load is modelled as an external applied force against the rotor rotation.
optimized one-way rotation conditions discussed above, this leads to Ein ¼ 3 1019 J. According to Fig. 7 and under those conditions, the motor is able to develop a maximum work of 6.5 1022 J before stopping and going backward constrained by
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Formally equivalent to a particle in solution, which is subject to a thermal noise due to random collisions with solvent molecules,28 a surface-mounted molecule in ultra-high vacuum can be exposed to a shot noise coming from the STM tip. This noise has been recently used to drive the motion of a single molecule motor.14 Inspired by this recent achievement, we have carried out a systematic study of the dynamics of a single molecule motor model driven by a shot noise. The use of an excited state for driving the rotation of the molecule motor allows the conversion of a Poisson time-distributed applied force to a controlled unidirectional motion. For a barrier height of 0.25 eV, the motor reaches a maximum speed of 3.43 107 turns per s. Furthermore, the direction of rotation of a molecule-motor can be reversed depending on the characteristics of the excited state involved in the process. We have demonstrated that a molecule without an intrinsic asymmetry can reconstruct a one-way rotation by using its internal characteristics. These ndings will help to optimize the chemical structure of new molecule-motors, not only to rotate one-way but also to develop a true motive power able to move atoms, small molecules or to drive cooperatively a train of solid state nano-gears.
Acknowledgements The authors acknowledge nancial support from the AUTOMOL project (ANR 09-NANO-040) and CNRS. J. E. thanks the Generalitat de Catalunya and the European Union for a Beatriu de Pin´ os scholarship. The authors also thank C. Collard for technical assistance with numerical simulations and CALMIP for computing time at Hyperion supercomputer.
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