Ho Kwan

Suchetana Au-Yeung

Tim Sarkar

Oumaima Kühne

Dmitry Aiboudi

Roberto Ryndyk

Nicolas Robles

Franziska Lorente

Christian Lissel

Francesca Joachim

Moresco

Au-Yeung

Suchetana Sarkar

Tim Kühne

Oumaima Aiboudi

Dmitry A Ryndyk

Roberto Robles

Nicolas Lorente

Franziska Lissel

Christian Joachim

Francesca Moresco

A Nanocar and Rotor in One Molecule

Keywords: Molecule-rotor, Nanocar, inelastic tunneling electrons, molecular manipulation, scanning tunneling microscopy, density functional theory

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

ABSTRACT. Depending on its adsorption conformation on the Au(111) surface, a zwitterionic single molecule machine works in two different ways under bias voltage pulses. It is a unidirectional single molecule rotor while anchored on the surface. It is a fast-drivable moleculevehicle (nanocar) while physisorbed. By tuning the surface coverage, the conformation of the molecule can be selected to be either rotor or nanocar. The inelastic tunneling excitation producing the movement is investigated in the same experimental conditions for both the unidirectional rotation of the rotor and the directed movement of the nanocar.

Understanding the working principles of synthetic molecular machines is of fundamental importance to develop nanoscale mechanical molecular devices, able for example to perform calculations, [START_REF] Soe | Towards a Molecular Mechanical Calculator, Single Molecule Mechanics on a Surface[END_REF] to store energy, [START_REF] Wang | Status and Challenges for Molecular Solar Thermal Energy Storage System Based Devices[END_REF] or to produce work. [START_REF] Ohmann | Supramolecular Rotor and Translator at Work: On-Surface Movement of Single Atoms[END_REF] While the first molecular machines have been explored as ensembles in solution [START_REF] Erbas-Cakmak | Artificial Molecular Machines[END_REF][START_REF] Aprahamian | The Future of Molecular Machines[END_REF][START_REF] Kay | Synthetic Molecular Motors and Mechanical Machines[END_REF] , it is now crucial to control the mechanics of always one and the same molecule. Specific examples of controlled rotations or translations on a surface under the tip of a scanning tunneling microscope (STM) have been recently reported [START_REF] Simpson | Unidirectional Motion of Single Molecules at Surfaces, Single Molecule Mechanics on a Surface[END_REF] and it is now known that in a STM junction the tunneling electrons can release energy to specific conformational or vibrational molecular degrees of freedom, being in some cases able to trigger rotations [START_REF] Stipe | Inducing and Viewing the Rotational Motion of a Single Molecule[END_REF][START_REF] Perera | Controlled Clockwise and Anticlockwise Rotational Switching of A molecular Motor[END_REF][START_REF] Tierney | Experimental Demonstration of a Single-Molecule Electric Motor[END_REF][START_REF] Eisenhut | One-Way Rotation of a Chemically Anchored Single Molecule-Rotor[END_REF][START_REF] Stolz | Molecular Motor Crossing the Frontier of Classical to Quantum Tunneling Motion[END_REF] or translations 13- 15 . However, a general understanding of the physical mechanisms inducing controlled and directional movements on a surface is still lacking, which would allow the rational design of single molecule-machines towards specific applications.

In order to enable the rotation of a single molecule, a stable rotational axle and the restriction of the lateral diffusion on the supporting surface are the prerequisites. Different strategies of constructing molecular axle via, e.g., positioning the molecule-gear on a defect, [START_REF] Manzano | Step-by-Step Rotation of a Molecule-Gear Mounted on an Atomic-Scale Axis[END_REF] triggering onsurface reactions, [START_REF] Ren | A Unidirectional Surface-Anchored N-Heterocyclic Carbene Rotor[END_REF][START_REF] Au Yeung | Transmitting Stepwise Rotation among Three Molecule-Gear on the Au(111) Surface[END_REF] and building molecule-stators [START_REF] Zhang | A Chiral Molecular Propeller Designed for Unidirectional Rotations on a Surface[END_REF][START_REF] Jasper-Toennies | Rotation of Ethoxy and Ethyl Moieties on a Molecular Platform on Au(111)[END_REF] have been reported. On the other hand, controlling the translation on a surface is a parallel goal of research. Mobile physisorbed molecules with legs, wings or wheels have been proposed to investigate translations induced by the STM tip. [START_REF] Pawlak | Design and Characterization of an Electrically Powered Single Molecule on Gold[END_REF][START_REF] Nickel | Moving Nanostructures: Pulse-Induced Positioning of Supramolecular Assemblies[END_REF][START_REF] Shirai | Directional Control in Thermally Driven Single-Molecule Nanocars[END_REF][START_REF] Kudernac | Electrically Driven Directional Motion of a Four-Wheeled Molecule on a Metal Surface[END_REF][START_REF] Kühne | Stm Induced Manipulation of Azulene-Based Molecules and Nanostructures: The Role of the Dipole Moment[END_REF][START_REF] Grill | Rolling a Single Molecular Wheel at the Atomic Scale[END_REF][START_REF] Soe | Conformation Manipulation and Motion of a Double Paddle Molecule on an Au(111) Surface[END_REF][START_REF] Simpson | How to Control Single-Molecule Rotation[END_REF] Furthermore, the first two editions of the international Nanocar Race have provided further examples of controlled lateral motions on a surface, induced by inelastic tunneling and electric field. [START_REF] Joachim | Molecule Concept Nanocars: Chassis, Wheels, and Motors?[END_REF][28] In the STM tunnel junction, the initial excitation mechanism for rotation and translation is a priori the same. However, a rotation requires a strong local molecule-substrate interaction providing a rotational axle, while a nanocar moves fast if its adsorption is weak. It is therefore unusual that the same molecule presents both functions. Here, we present such a molecule.

Thanks to its internal charge separation and to the possibility of chemisorb or physisorb on the Au(111) surface, the 1,3-dimethyl-2-phenyl-2,3-dihydro-1H-benzoimidazol (DMBI)-based zwitterionic molecule studied in this work can rotate (without translating) in one direction or can translate (without rotating) depending on its conformation on the surface. Such dual functionality allows the comparison of inelastic tunnelling electrons excitation for a controlled translation or a unidirectional rotation on the same molecule and in the same experimental conditions.

RESULTS AND DISCUSSION

The molecule 2-(2-Methoxyphenyl)-1,3-dimethyl-1H-benzoimidazol-3-ium iodide (o-OMe-DMBI-I) [START_REF] Wei | Z. 2-(2-Methoxyphenyl)-1,3-Dimethyl-1h-Benzoimidazol-3-Ium Iodide as a New Air-Stable N-Type Dopant for Vacuum-Processed Organic Semiconductor Thin Films[END_REF] and further DMBI derivatives have been first synthetized to be employed as air-stable n-type dopants in organic electronic devices. [START_REF] Lüssem | Doped Organic Transistors[END_REF][START_REF] Wei | Use of a 1h-Benzoimidazole Derivative as an N-Type Dopant and to Enable Air-Stable Solution-Processed N-Channel Organic Thin-Film Transistors[END_REF] It was however soon observed that decomposition reactions occur during the doping process, with the halide separation or the splitting of a methyl group. [START_REF] Schwarze | Analyzing the N-Doping Mechanism of an Air-Stable Small-Molecule Precursor[END_REF] When sublimating o-OMe-DMBI-I in ultra-high vacuum (UHV), the molecule undergoes a dissociation reaction where the methoxy group is cleaved and methyl iodide is released, yielding the zwitterionic phenolate DMBI-P. [START_REF] Eisenhut | One-Way Rotation of a Chemically Anchored Single Molecule-Rotor[END_REF] . In a recent study, we showed that on Au(111) DMBI-P works as a single molecule rotor reliably converting STM voltage pulses into controllable unidirectional rotation. [START_REF] Eisenhut | One-Way Rotation of a Chemically Anchored Single Molecule-Rotor[END_REF] Here, we investigated the zwitterionic phenolate to make the methoxy cleavage superfluous, and elongated the rotating part to a naphthyl, obtaining 2-(1,3-dimethyl-1H-naphtho [2,3-d]imidazol-3-ium-2-yl)phenolate (DMNI-P) presented in Figure 1a. The chemical structure of o-OMe-DMBI-I, DMBI-P and DMNI-P are compared in Figure S1. A representative overview STM image (Figure 1b) shows a moderate coverage of about 0.1 ML. Single molecules and dimers can be observed where the majority occupies the kinks of the Au(111) reconstruction sites. At lower coverage only single molecules are present on the surface, while by increasing the coverage the relative number of dimers increases, becoming dominant at 0.2 ML (see Figure S2).

Single molecule rotors

We firstly consider the low coverage case and the isolated single adsorbed molecules (Figure 2). A typical STM image of an isolated DMNI-P molecule (Figure 2a) shows the naphtyl unit at the tail of the molecule as well as the widening of the structure roughly in the middle, as the two methyl groups of the imidazole unit appear wider in the STM image because they are not planar.

The molecule is anchored via the phenolate (purple cross). After having performed 251 pulses on different rotors, we observed that the molecule rotates in steps of 60° and in the same direction in 99% of the cases. On a given molecule rotor, the entry port for the electron tunnelling allowing a one-step rotation is not located at the anchoring point (Figure 2a), making necessary a re-position of the tip after each pulse (Figure 2c). Our moleculerotor is chiral after adsorption on the surface. We observe that DMNI-P molecules with opposite chirality rotate in opposite direction by the same voltage pulse: Isomers as the one shown in Figure 2, which we call left-DMNI-P, always rotates CCW by positive voltage pulses. Negative bias pulsse drive the rotor in the opposite direction.

Lateral manipulation at low STM junction tunnelling resistance (e.g. 10 mV, 1 nA) has been used to confirm that the rotors are anchored on the surface (Figure S5). A STM pushing [START_REF] Moresco | Manipulation of Large Molecules by Low-Temperature Stm: Model Systems for Molecular Electronics[END_REF] does not break the Au-O bond and no lateral movement of the molecule can be observed. More interestingly, we are able to flip a molecule-rotor changing its surface chirality, therefore inverting its rotational direction without breaking the Au-O bond. An example of such flipping is reported in Figure 3. In Figure 3a one can see the STM image of a left-DMNI-P, which rotates CCW by positive voltage pulses (typically 0.4 -0.7 V). After applying a stronger voltage pulse By increasing the molecular coverage on the surface (see Methods section for details), we observe an increasing quantity of molecular dimers. Figure 4a presents the STM image of a few dimers, formed by identical DMNI-P molecules. We can separate the dimers into single DMNI-P molecules by voltage pulses. Typically, we applied a voltage pulse outside of the scan area (e.g. 5 V, 160 pA; laterally about 8 nm away from the target). An equivalent separation can be obtained by applying weaker pulses (e.g. ~2 V, 5 nA) directly above the dimer (Figure S7). After separation (Figure 4b), we observe in the STM image single adsorbed molecules, where at least one of both molecules has experienced a lateral displacement and, instead of rotating around a fixed point, can be laterally moved on the surface by voltage pulses, behaving like a nanocar.

Considering the n = 41 performed dimer separations, about 40% of them results into two separated nanocars and 60% in a nanocar and a rotor.

The schematic Figure 4c summarizes the two observed adsorptions after dimer separation:

Case I shows that the dimer forms a rotor and a nanocar by voltage pulses; case II represents that both the molecules can translate on the surface after separation, i.e. they are nanocars. The red lines illustrate the phenolate undergoing chemisorption (Au-O bond) or physisorption on the surface.

Figures 4d show the corresponding DFT calculated adsorption geometries for dimer I (rotornanocar) and dimer II (nanocar -nanocar). As on can see, the dimers are formed via hydrogen bonds between two zwitterions, where the O atoms point to the second molecule, hence contributing to the dimer stability. Our calculations show that dimers can be formed already in the gas phase with a bonding energy of 0.2 eV.

The separation of the dimers allows us to contemporary observe the voltage-pulse induced rotation and translation of DMNI-P molecules under exactly the same experimental conditions like surface preparation and tip (see also the movie in the Supporting Information).

Nanocars

In the following, we consider the driving of the nanocars on the Au(111) terraces. Figure 5a shows the STM image of a single DMNI-P nanocar after dimer separation. Its appearance is very similar to the rotor discussed above. A manipulation sequence is shown in Figure 5a and5b. We position the STM tip apex at the periphery of the molecule (see Figure S8) and apply a voltage pulse. DFT calculations of the absorbed geometry propose a second stable conformation (Figure 5c) where the O atom is rotated slightly away from the surface and the molecule is physisorbed, interacting with the Au(111) surface by van der Waals interaction. The reaction path (NEB DFT) calculations show that the energy of the car configuration is a little bit larger than the energy of the rotor configuration but the energy barrier between these configurations is about 0.4 eV making impossible switching from one to another at low temperatures. We studied the movement of the DMNI-P nanocars applying voltage pulses with the STM tip, similar to the case of the rotors. Figure 5a shows a sequence of a nanocar laterally translating on the Au(111) surface by positioning the tip near to the nanocar (red marks). By applying positive voltage pulses, the molecule is attracted by the tip. Similar to the case of the rotors (Figure 2g), a sudden jump of the tunneling current signal indicates that a hopping event occurs. In this particular sequence, the nanocar has travelled about 4 nm with four individual pulses (Figure 5b).

We performed 379 pulses on different nanocars, observing that at positive bias the molecule translates towards the tip position with very high precision, independently on its chirality. On the other hand, the molecule moves in random directions and away from the tip position at negative bias (Figure S11). Typically, the nanocars can be driven at V ≥ 0.4 V and few hundreds pA.

Comparison car-rotor

Thanks to the unique possibility of having both chemisorbed and physisorbed conformations of DMNI-P on Au(111), we have compared the unidirectional rotation and translation in the same experimental conditions, applying the same voltage pulses. We induced 251 rotation events and 379 translations on the same surface with the same STM tip and analyzed the data statistically (see Methods for details). We extracted the reaction time of each event from the I(t) plots (see Figure 2d), calculated the average rate R, and plotted the experimental yield, or action spectra, [START_REF] Kim | Action Spectroscopy for Single-Molecule Reactions -Experiments and Theory[END_REF] (Y = R/(I/e): motion probability per tunneling electron) for rotors and cars, respectively (Figure 6). [START_REF] Kim | Action Spectroscopy for Single-Molecule Reactions -Experiments and Theory[END_REF] obtaining a threshold at 370 meV. In the lower panel, the continuous line is a simulation of the action spectra of the nanocar considering a threshold voltage of 165 meV and an overall yield constant of 3.5 x 10 -10 events/electron. The fitting of the data at different currents was not possible because of the data spread.

Figure 6 shows the electronic yield for DMNI-P rotations (top plot) and translations (bottom plot) depending on the bias voltage for different currents. In both cases, the different adsorption position of the molecule on the Au(111) surface reconstruction causes a spread of the experimental data, which is obviously stronger in the case of the mobile nanocars.

At voltages higher than 400 mV, the yield for rotation and translation are very similar, and a nearly constant yield of about 5 x 10 -10 events/electron is observed in both cases. For rotations, a threshold voltage at about 400 mV is visible in the upper plot, which can be explained by the onset of the C-H stretch mode of the methyl group at the opposite side of the Au-O bond, in agreement with the case of the shorter DMBI-P. [START_REF] Eisenhut | One-Way Rotation of a Chemically Anchored Single Molecule-Rotor[END_REF] Fitting the experimental action spectra by the theory of Ref. [START_REF] Kim | Action Spectroscopy for Single-Molecule Reactions -Experiments and Theory[END_REF] (continuous lines in the upper panel of Figure 6), we obtain a threshold at 370 meV (see Methods for details).

At lower voltage, however, a translation becomes more favorable than a corresponding rotation. This indicates that low-energy vibrations are sufficient for inducing diffusion in the non-anchored physisorbed conformation, while higher energy is needed to overcome the rotational barrier of the anchored chemisorbed conformation, which can be reached by exciting the C-H stretch mode. Calculations of the vibrational modes of DMNI-P (see Figure S12), show similar modes for both rotor and nanocar and confirm the onset of the C-H stretch mode at 370 meV and a mixing of lower vibrational modes, including the C-C and C-N stretch between 155 meV and 167 meV. The continuous line in the lower panel of Figure 6, shows a simulation for a threshold voltage of 165 meV, in good agreement with the experimental data.

We therefore conclude that inelastic tunneling electrons are responsible for both rotation and translation, while for translation lower energy modes are already effective.

As already discussed in Ref. [START_REF] Eisenhut | One-Way Rotation of a Chemically Anchored Single Molecule-Rotor[END_REF] , the directionality of the movement (both as unidirectional rotation or directed translation) cannot be explained only by the inelastic excitation of vibrational modes, but is related to the virtual occupation of excited molecular electronic states during the electron tunneling through the molecule. In this specific case, the asymmetry of the molecule, its chirality on the surface, and the charge separation of the zwitterionic form are encrypted in the asymmetry of the potential energy surfaces of the excited electronic states respect to the ground state, and are therefore contributing to the observed unidirectional motion.

CONCLUSIONS

We have presented a single molecule machine showing two distinct functions depending on its adsorption conformation. Thanks to a coverage-related selectivity, we contemporarily observe rotors that are anchored by a fixed Au-O bond, and mobile nanocars that can controllably translate across the surface with atomic precision. The direction of rotation of the rotor is related to its chirality on the surface and can be inverted by manipulation, flipping the chirality without changing the anchoring position.

At bias voltages higher than 400 mV, the yields for rotation and translation are comparable. At lower voltage however, the nanocar is free to move laterally, while the rotation is blocked. This indicates that low-energy vibrational modes (i.e. C-C and C-N stretch modes at 150-170 meV) are sufficient to move the non-anchored physisorbed conformation, while higher energy modes (C-H stretch mode at 370 mV) are needed to induce the rotation of the chemisorbed anchored conformation. METHODS DMNI-P molecules were sublimated from a quartz crucible (T evap = 225 °C) on the clean Au(111) surface kept at room temperature under ultrahigh vacuum (UHV) conditions. The sublimation time was chosen between 2 -7 sec depending on the target coverage. Before evaporation, the Au(111) single crystal was cleaned by subsequent cycles of Ar + sputtering and annealing to 723 K. STM experiments were performed using a custom-built instrument operating at a low temperature of T = 5 K under ultrahigh vacuum (p ≈ 1 × 10 -10 mbar). All shown STM images were recorded in constant-current mode with the bias voltage applied to the sample.

Voltage pulse manipulations were performed by first positioning the STM tip above or near to the molecule and then by ramping up quickly the voltage bias. Both constant current and constant height modes were employed depending on the types of measurements. z(t) or I(t) curves, respectively, were recorded during the pulses, detecting the movement of the molecule.

For constant height measurements, the tip-surface distance was calibrated by recording I(z) curves. STM images were recorded before and after the application of the pulse, determining the displacement of the molecule. Non-destructive tunneling parameters were selected (e.g. V = 0.2 V, I = 5 pA) for acquiring STM images in order to lower the possibility to induce unexpected molecular movement during scanning.

All lateral manipulations were performed in constant-current mode. The lateral manipulation procedure involves three steps: (1) allowing the tip to vertically approach the molecules under a small bias and current to increase the tip-molecule interaction, (2) laterally driving the tip parallel to the surface in a precisely controlled trajectory, and (3) retracting the tip to normal scanning position. The STM captures images before and after each manipulation in all of the sequences.

The rotation or translation events were collected from the signal of tip height over time, where a sudden jump of signal can be observed. The corresponding yield (events/electron) calculations were done by the average of the population for all fixed voltage and current (n ≥ 10 for each point), and the statistical error was evaluated by standard deviation.

For geometry optimization and reaction path calculations, we used the DFT method as implemented in the CP2K software package (cp2k.org) with the Quickstep module [START_REF] Vandevondele | Quickstep: Fast and Accurate Density Functional Calculations Using a Mixed Gaussian and Plane Waves Approach[END_REF] . We applied the Perdew-Burke-Ernzerhof exchange-correlation functional [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] , the Goedecker-Teter-Hutter pseudo-potentials [START_REF] Goedecker | Separable Dual-Space Gaussian Pseudopotentials[END_REF] and the valence double-ζ basis sets, in combination with the DFT-D2 method of Grimme [START_REF] Grimme | A Consistent and Accurate Ab Initio Parametrization of Density Functional Dispersion Correction (Dft-D) for the 94 Elements H-Pu[END_REF] for van der Waals correction. We used 6 layers of gold, 3 upper layers allowed to be relaxed, planar supercell 29.8x19.9 Angstrom, vacuum size 40 Angstrom, maximum force 4.5x10-5 a.u. The data is analysed, and the images are made by the PyMOL Molecular Graphics System, Version 2.4 open-source build, Schrödinger, LLC.

The fitting of the action spectra of Figure 6 (rotor, upper panel) by the theory of Ref. [START_REF] Kim | Action Spectroscopy for Single-Molecule Reactions -Experiments and Theory[END_REF] was done using the function:

where K is the overall yield that controls the order of magnitude of the measured Y, the applied bias is V,  is the angular frequency of the vibrational mode mediating the reaction and  is the broadening of the reaction threshold. The fitting curves are then controlled by basically two parameters;  that fixes the threshold of a non-zero yield, and  that controls how fast the yield goes from zero to a measurable value.

The simulated curve of Figure 6 (nanocar, lower panel) was obtained fixing the threshold voltage (165 mV) and its spread (5 mV), and fitting the background yield with an overall constant of 3.5E-10 events per electron.

Figure 1 .

 1 Figure 1. (a) Chemical structure of the zwitterionic DMNI-P (color code: C = gray, H = white,

Figure 2 .

 2 Figure 2. Anchoring and rotation of an isolated DMNI-P. (a) STM image of the molecule

( 1 .

 1 5 -3.5 V) or by lateral manipulation, the molecule moves (flips) to the opposite side of the bond without changing its anchoring point (Figure3b). The STM image after flipping clearly shows that the molecule presents now the right-hand chirality. By positive voltage pulses of typically 0.4 -0.7 V, the unidirectional rotation in the opposite direction is observed, i.e. the flipped molecule rotates now clockwise (CW). The flipping is graphically represented in Figure3cafter superimposing the two images of Figure3aand b. As one can see, the two mirror images are visible, sharing the same anchoring point. Notably, the flipping is reversible and can be repeated several times. DFT calculations show that the flipping of the chirality is due to the rotation of the phenolate-head of the molecule (sequence of Figure 3d), while the naphtimidazole remains adsorbed on the surface shifting laterally. An animation of the simulated conformational change is shown in the Supporting Information.

Figure 3 .Figure 4 .

 34 Figure 3. Flipping of the chiral rotor molecule leads to the change of rotation direction (CCW 

Figure 5 .

 5 Figure 5. Manipulation of the DMNI-P nanocar. (a) Controlled lateral translation starting from

Figure 6 .

 6 Figure 6. Yield (motion probability per electron) versus bias voltage for DMNI-P rotor (top) and

ACKNOWLEDGMENTS

This work was funded by the European Union. Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union. Neither the European Union nor the granting authority can be held responsible for them. This work has received funding from then European Innovation Council (EIC) under the project ESiM (grant agreement No 101046364) and the Horizon 2020 research and innovation program under the project MEMO, grant agreement no. 766864. Support by the German Research Foundation (DFG) by the DFG Project 43234550, the Collaborative Research Centre (CRC) 1415, and the Initiative and Networking Fund of the German Helmholtz Association, Helmholtz International Research School for Nanoelectronic Networks NanoNet (VH-KO-606) is gratefully acknowledged. We thank the Center for Information Services and High Performance Computing (ZIH) at TU Dresden for computational resources. F.L. thanks the Fonds der Chemischen Industrie (FCI) for a Liebig Fellowship and C.J. the WPI MANA project for financial support.


Makoto Sakurai
email: sakurai.makoto@nims.go.jp

Tsuyoshi Ueta

Christian Joachim

Spin liquid-like behavior of weakly interacting molecular spins in nanoclusters on a graphene oxide nanosheet Weakly interacting molecular spins in on-surface synthesized nanoclusters on a graphene oxide nanosheet

Keywords: Amino-ferrocene, magnetic dipole interaction, entangled structure, nanocluster, graphene oxide, slow dynamics, 2D material

With an average diameter of about 2 nm, on-surface synthesized amino-ferrocene nanoclusters are chemisorbed onto a graphene oxide nanosheet, where their Fe ions are in an S = 5/2 high-spin state. In this two-dimensional (2D) nanomaterial, the molecular spins in a given nanocluster are weakly magnetic dipole interacting. It generates spin correlations and slow dynamics accessible by magnetic susceptibility and Mössbauer spectroscopy at a temperature where the magnetic anisotropy is negligible. Magnetic simulations show that minimizing their magnetic dipole energies produces spatially entangled structures of the spin orientations at T ≲ 15 K and that the structures behave like a spin liquid. The competition between the formation of the structures and their thermal destruction generates the slow dynamics. The ability to create emergent functionalities from dense stacks of weakly interacting magnetic molecules in a 2 nm space paves the way for new designs of an ultracompact building block and a functional component in 2D spintronic and neuromorphic devices. The data that support the findings of this study are available from the corresponding author upon reasonable request.

Introduction

Engineering the magnetic interactions between molecular spins plays an essential role in the chemical design and arrangement of molecular magnets, magnetic molecules, and their assemblies required for the integration of nanoscale spintronic and neuromorphic devices. On the one hand, molecular magnets have the ability to maintain the spin orientations through magnetic anisotropy and intramolecular exchange interactions, [1,[START_REF] Chibotaru | Molecular Nanomagnets and Related Phenomena[END_REF] and a variety of magnets have been synthesized for molecular spintronics and quantum computing, acting as information storage devices [START_REF] Ishikawa | [END_REF]4] and, for those with long spin coherence, as building blocks for quantum computation. [5,6] Some magnetic molecules have another ability to self-assemble into various nano-and micro-scale structures, [7,8] which can lead to the development of new functionalities not accessible to single molecules through structuring in combination with the intrinsic properties of the molecules themselves. The amino-ferrocene molecules used in this study (Figure 1A) can self-assemble into nanoclusters on a graphene oxide nanosheet via onsurface chemical reaction. [9] We have measured the magnetic properties dense stacked weakly interacting molecular spins into a nanocluster with a diameter of about 2 nm and discussed those properties by combining theoretical simulations and a systematic analysis of the experimental data.

The amino-ferrocene-based assemblies, referred to as nanoclusters in this study, are chemisorbed on the surface of a graphene oxide (GO) nanosheet, forming a 2D material. The electronic states of the 3d electrons of the Fe ion in ferrocene or ferrocene-derivatives are dominated by the ligand field. [START_REF] Crabtree | The Organometallic Chemistry of the Transition metals[END_REF] Inside the nanocluster, the Fe ion of a given amino-ferrocene molecule has unpaired electrons due to the through-bond electron transfer process between the Fe ion and the GO, [9] which behaves as a localized spin due to the geometry of the ferrocene unit (Figure 1A). Magnetic dipole interactions between localized spins in molecular assemblies are generally much weaker than intramolecular interactions and magnetic anisotropy in the molecule, because the dipole energy approximated as is equivalent to only 0.0006 K assuming a magnetic moment m of 1 and a distance r between them of 1 nm. However, in assemblies of small magnets with large magnetic moments, magnetic dipole interactions have been reported to play a crucial role, for example, in the magnetic ordering in high-spin molecular cluster magnet crystals [7][8] and in the modulation of magnetic relaxation in magnetic nanoparticle assemblies. [START_REF] Dormann | [END_REF]12] Here, the nearest dipole interactions dominate mainly the formation of these properties. In our amino-ferrocene-based nanoclusters, the d m = 0.28 nm distance between neighboring localized spins [9] (Figure 1A) is much shorter than the d m between neighboring magnetic moments in Fe 12 cluster magnet crystals (d m 1.4 nm) [7] and in nanoparticle assemblies (d m = 2 -10 nm) [START_REF] Dormann | [END_REF]12] , leading to the completely different situation that all molecular spins within the nanocluster are weakly connected by magnetic dipole fields of the order of 10 2 -10 3 Oe. The interactions in the nanoclusters differ from that treated in a mean-field-based theory. [START_REF] Panissod | Magnetism: Molecular to Materials IV[END_REF] In addition, the controlled on-surface selforganization of our nanoclusters can produce a uniform nanocluster distribution with a relatively large inter-cluster distance, [9] resulting in a surface structure close to an isolated magnetic nanocluster distribution on an insulating GO nanosheet. Here, we analyze the behavior of dense packed molecular spins resulting from such unique magnetic interactions in nearly zero-dimensional nanoclusters by measuring the magnetic susceptibility, magnetization, 57 Fe Mössbauer spectrum, and by simulating the distribution of the spin orientations by Markov-chain Monte Carlo methods. We also elucidate the mechanism of the slow dynamics generated in this 2D material by comparing it with that studied in spin glasses [14,15], interacting magnetic nanoparticles [11,12], and frustrated magnets [16,17].

Results and Discussion

The deviation from the paramagnetic Fe ions in the amino-ferrocene (AFc) molecules forming the nanoclusters on a GO nanosheet (AFc-GO sheets) was estimated by accounting static magnetization and magnetic susceptibility. On the GO nanosheets and resulting in an average inter-cluster distance of 10 nm (Figure 1A), the amino-ferrocene-based nanoclusters were synthesized for 5 hours by our on-surface chemical reaction technique (Methods). [9] The small black spots in a transmission electron microscope (TEM) image (Figure 1B) and small white spots in a scanning transmission microscope (STEM) image (Figure S1, Supporting Information) of a chemically reacted GO nanosheet correspond to amino-ferrocene-based nanoclusters on a GO nanosheet (Figure S1, Supporting Information). The diameter distribution of the AFc-GO sheets ( = 10 nm) shows an average diameter of about 2 nm (Inset of Figure 1B). As calculated from the magnetization of the AFc-GO sheets and of the pristine GO nanosheets, [9] the magnetization M AFc of the molecules on the GO nanosheets is plotted in Figure 1b as a function of , where is the Bohr magneton, H is the applied magnetic field, is the Boltzmann constant, and T is the temperature. Characterizing the paramagnetic properties of a material as a function of field and temperature, [START_REF] Kahn | Molecular Magnetism[END_REF] the Brillouin and Langevin functions are plotted as solid and dotted lines using the spin state numbers S = 5/2 or S = 1/2 (Figure 1C and Figure S2, Supporting Information), respectively. At small values, the experimental data are well fitted using S = 5/2. However, at low temperature and high field, the experimental data deviated from the lines meaning that a change in the energy levels of the 3d electrons under the applied field is not expressed simply by the Zeeman splitting, [START_REF] Kahn | Molecular Magnetism[END_REF] as increases. This is different from superparamagnetic materials for example ferromagnetic nanoparticles [START_REF] Guimarães | Principles of Nanomagnetism[END_REF] and molecular magnets. [START_REF] Bino | [END_REF]21] Note that this deviation was observed at T = 50 and 100 K, where the thermal energy is certainly larger than the magnetic interactions between the neighboring Fe ions.

Under a small magnetic field and for T > 20 K, the static susceptibility of our nanomaterial follows the Curie-Weiss law: [START_REF] Kahn | Molecular Magnetism[END_REF] (Figure 1D).

The constant [START_REF] Kahn | Molecular Magnetism[END_REF] is the slope of a linear fit to the experimental data, where g is the g-factor and W FD is the mass of the molecule.

Assuming g = 2, C gives a magnetic moment = 5.1 , confirming that the Fe ions are in the high-spin state (S = 5/2) rather than the low-spin state (S = 1/2) (Figure 1A). Note that this value was underestimated in our previous study [9] because it was determined in a magnetic field of 70 kOe and in an unsaturated M-H loop at T = 2 K. The constant was -2.9 K, indicating that there were negative interactions, as predicted by mean-field theory. [START_REF] Kahn | Molecular Magnetism[END_REF] Low-temperature zero-field 57 Fe Mössbauer spectroscopy [START_REF] Gütlich | Mössbauer spectroscopy and Transition Metal Chemistry[END_REF][START_REF] Greenwood | Mössbauer Spectroscopy[END_REF] confirms that the Fe ion in the amino-ferrocene molecules changed from a non-magnetic S = 0 state to a S = 5/2 highspin state when assembled by on-surface chemical reaction on a GO nanosheet. At 5 K, the Mössbauer spectrum of a pristine amino-ferrocene powder indicated that the Fe ions here are in the Fe 2+ S = 0 nonmagnetic state (Figure 2A). [START_REF] Dormann | [END_REF] On the other hand, for the aminoferrocene molecules assembled on a GO nanosheet surface, a magnetic hyperfine splitting was clearly seen in the spectra (Figure 2B). This is due to the change in the ionized state of the Fe ions caused by the charge transfer. [9] As the temperature increased, thermal disturbances attenuated the peaks in the spectra. The onset temperature to observe a splitting was about 15 K and the separation between the peaks did not depend on temperature (Figure 2B). This is different from the splitting temperature dependence observed for superparamagnetic particles whose magnetic properties are dominated by magnetic anisotropy. [START_REF] Mørup | [END_REF] At 5 K, the parameters characterizing the 3d electrons in the Fe ions were determined by taking the best fit of the spectrum (Figure S3, Supporting Information). As summarized in Table S1 (Supporting Information) and at 5 K, the 470 -400 kOe hyperfine magnetic fields for an AFc-GO sheet again confirm that the Fe ions are in an S = 5/2 Fe 3+ high-spin state and not in the S = 1/2 low spin state. [START_REF] Gütlich | Mössbauer spectroscopy and Transition Metal Chemistry[END_REF] At 5 K, about 82% of the Fe ions on the AFc-GO sheets have a large hyperfine field corresponding to S = 5/2. About 18% of the Fe ions on the GO nanosheets are in the S = 0 non-magnetic state. Some of the amino-ferrocene molecules have not chemically reacted with the GO surface and do not contribute to the magnetic properties of the chemically reacted sheets.

Magnetic properties of a mononuclear organometallic molecule are generally characterized by a magnetic anisotropy resulting from zero-field splitting at its metal site. [START_REF] Chibotaru | Molecular Nanomagnets and Related Phenomena[END_REF] There are few experimental reports on the magnetic anisotropy of ferrocene molecules because individual ferrocene molecules do not have unpaired 3d electrons. [START_REF] Dormann | [END_REF] Assembled on a GO nanosheet, Fe 3+ ion 3d electrons are in a high S = 5/2 spin state and the electronic cloud of the 3d electrons is spherically distributed (no orbital-angular momentum). [START_REF] Chikazumi | Physics of Magnetism[END_REF] Therefore, the spin-orbit interactions normally at the origin of magnetic anisotropy are too small to produce the hyperfine splitting observed in the Mössbauer spectra at T = 5 -15 K (Figure 2B). Firstprinciples calculations on MnCp 2 molecule, which has the same number of 3d electrons as the amino-ferrocene molecule, gives a magnetic anisotropy of about 0.2 meV in the direction perpendicular to the planes holding the cyclopentadienyl rings in MnCp 2 . [START_REF] Nawa | [END_REF] An anisotropy energy K = 0.2 meV leads to a blocking temperature T B = 0.09 K using . [START_REF] Guimarães | Principles of Nanomagnetism[END_REF] This temperature is much lower as compared to the onset temperature (≈ 15 K) in the splitting (Figure 2B). Thus, magnetic anisotropy is not at the origin of this splitting. Furthermore, from the electronic and geometric structures of the ferrocene molecule [28,29] and from the arrangement of amino-ferrocene molecules within the nanocluster (Figure 1A), there is no direct chemical coordination between neighboring Fe ions, suggesting that the bond-mediated interactions between Fe ions are weak to explain the properties generated at T ≲ 15 K.

Therefore, the splitting originates from magnetic dipole interactions between Fe ions densely arranged within a nanocluster.

We can also represent the distribution of spin orientations at Fe ion sites within the nanocluster by using an assembly of classical spins at the positions of Fe ions within a given nanocluster (Figure 3A). The lateral distance between neighboring Fe ions was taken as the experimental value (0.28 nm). [9] The vertical distance was assumed to be 0.4 nm, which is 20 % longer than the distance between two cyclopentadienyl rings of a given ferrocene molecule (0.33 nm). [29] We assumed that these localized spins have classical magnetic moments of 5 μ B . Magnetic screening [30] and magnetic interactions through spin-polarized carriers [31] are negligible due to the insulating GO nanosheet. A zero-field Markov-chain Monte Carlo simulation [START_REF] Binder | Monte Carlo Simulation in Statistical Physics[END_REF] of the assembly shows that there is a monotonic decrease in the average total dipole energy with decreasing temperature (Figure 3B). The snapshots of spin orientations and dipole energy at the sites are shown in Figure 3C-F. The vortex structure of spin orientations was formed at T = 0.01 K by minimizing the dipole energy while reducing the thermal perturbations. As the temperature increases, the structure becomes thermally perturbed, and the the low dipole energy regions (colored by red, orange, and sand in Figure 3C-F) become smaller in lateral size. The energy change is due to the formation of the entangled structures of the spin orientations in such a way that the total dipole energy is reduced. The term "entangled structure" refers here to the "entangled polymers" below the glass transition temperature. [START_REF] Porter | [END_REF] At T < 0.2 K, vortex structures of the spin orientations are formed. At 0.2 K, spatially entangled structures of the spin orientations are formed, and they are simply estimated by the size of the low dipole energy regions. Note that there is here a large change in the total dipole energy at 15 K (Figure 3B) consistent with the temperature at which splitting started in the Mössbauer spectra (Figure 2B).

The model simulation elucidated several magnetic properties generated by the dense stacking of weakly interacting molecular spins in the nanocluster. The magnetization M AFc of the molecule in AFc-GO sheets ( = 10 nm) under a static magnetic field was well fitted by an average magnetization calculated for the assembly in the field region (1 kOe 5 kOe), where the applied field is comparable to the magnetic dipole fields within the nanocluster (Figure 4A). The deviation of the data from the Brillouin curves with S = 5/2 in Figure 1C is mainly caused by the interactions between the spin orientations. In addition, this entanglement acts to resist spin alignment in the direction of the field, producing a negative in the Curie-Weiss law (Figure 1D). Simulation of an assembly (S = 5/2) with vertical distance l v of 0.20, 0.21, and 0.25 nm showed a large decrease in the average dipole energy at T ≲ 15 K (Figure 4B). It should be noted that the dipole energy of an assembly (l v of 0.2 nm) composed of magnetic moments of 1 (S = 1/2) did not change at T ≳ 2 K (Figure 4B), confirming that large magnetic moments of the molecules play an important role in the substantial decrease of the energy at T ≲ 15 K. Since the thermal fluctuations of the spin orientations are weakened by the entangled structure of the spin orientations, the temperature at which the spin fluctuations are strongly reduced is similar to the temperature at which the dipole energy is strongly reduced by the formation of the entangled structures (Figure 4C).

The model simulations also revealed features characterizing the stability of the spin orientation distribution in the assembly. The spin orientations and dipole energy at the sites at T = 0.01 K were nearly the same regardless of the Monte Carlo steps (MCS) (Figure 4D), indicating that this entangled structure, namely the vortex structure, was stable. As the temperature increased, the site distribution of spin orientations and dipole energy changed against the progress of the MCS (Figure 4E,F), keeping the total dipole energy close to the average value. The low dipole energy domain (colored by red, orange, and sand in Figure 4E,F) was formed by minimizing the energy. As the simulation progressed, the domain frequently changed its position within the nanocluster at T = 0.5 and 10 K (Figure 4E, F) and behaved like a liquid, suggesting that there are many possible spin orientation distributions in the assembly with the same energy (Figure S4, Supporting Information). This is similar to many degenerate states in spin-glass materials [START_REF] Mydosh | Spin glasses: an experimental Introduction[END_REF][START_REF] Binder | [END_REF] and spin liquid states in frustrated magnets. [16,17] As the temperature was further increased, sites with a low or high dipole energy were distributed almost in isolation and changed the positions frequently (Figure S4B, Supporting Information).

At low temperature, magnetic dipole interactions in the dense stacking of molecular spins in the nanocluster give rise to two characteristic phenomena: spin correlations and slow dynamics. For AFc-GO sheets with = 10 nm, the imaginary component of the dynamic susceptibility of the molecule emerged at T ≲ 11 -12 K (Figure 5A). From the fluctuation-dissipation theorem, [START_REF] Kubo | Statistical Physics Ⅱ[END_REF] this suggests that spin correlations occur in this temperature range. Since there is no imaginary component for the dynamic susceptibility of pristine GO nanosheets (Figure S5, Supporting Information), these correlations occur within the nanocluster. The spin correlations are due to the magnetic interactions between the molecular spins (Figures 3B-D), which are maintained for a long time due to magnetic friction, [START_REF] Skomski | Simple Models of Magnetism[END_REF] an effect neglected in the simulation. The field-cooled (FC) and zero-field-cooled (ZFC) static susceptibility curves of the AFc-GO sheets ( = 10 nm) separate at a temperature below the blocking temperature T B of about 11 K (Figure 5B), indicating the onset of a slow magnetization relaxation analogous to that found in spin glass materials, [START_REF] Mydosh | Spin glasses: an experimental Introduction[END_REF][START_REF] Binder | [END_REF] ferromagnetic particles [12,[START_REF] Guimarães | Principles of Nanomagnetism[END_REF] and single-molecule magnets. [START_REF] Gatteschi | Molecular Nanomagnets[END_REF] The frequency-dependent (Figure 5A) is typical of a slow dynamics. [START_REF] Mydosh | Spin glasses: an experimental Introduction[END_REF] The origin of this slow dynamics in our AFc-GO sheets results from a competition between the formation of the entangled structures by magnetic dipole interactions and their destruction by thermal perturbations. Although the dipole interaction between the nearest spins in the nanocluster is equivalent to 0.7 K, the contributing dipole interactions of all spins, which are connected due to the short intermolecular distance and the magnetic moment of 5 , generate the entangled structures of spin orientations essentially at T ≲ 15 K (Figure 3B). In the Mössbauer spectra (Figure 2B), the hyperfine splitting at T ≲ 15 K also originates from these slow dynamics because the relaxation time of the magnetization at T ≲ T B is much longer than the response time of Mössbauer spectroscopy (10 -8 -10 -12 s). [START_REF] Gütlich | Mössbauer spectroscopy and Transition Metal Chemistry[END_REF] The relaxation process was further analyzed to extract the parameters characterizing nonequilibrium relaxation of weakly interacting spins in the nanocluster. The residual magnetization M(t) of AFc-GO sheets ( = 10 nm) was generated by the removal of the field after field-cooling (500 Oe) to each temperature and decayed slowly (Figure S6A, Supporting Information). Field-cooling of AFc-GO sheets forced the spin orientations to align with the direction of the field. The removal of the field caused a non-equilibrium distribution of spin orientations observed as the residual magnetization. The relaxation process was analyzed using a stretched exponential form: [START_REF] Mydosh | Spin glasses: an experimental Introduction[END_REF][START_REF] Palmer | [END_REF] , where τ is the relaxation time, is the residual magnetization, and 0 ≦ n ≦ 1. Fitting the data yielded a relaxation time τ(T) as a function of temperature and n of about 0.67. This relaxation time is assumed to follow a classical thermal activation process, i.e., an Arrhenius process: [START_REF] Mydosh | Spin glasses: an experimental Introduction[END_REF]38] , where is a prefactor characterizing the relaxation time scale and is the height of the energy barrier blocking the decay of the residual magnetization for field-cooled AFc-GO sheets. A linear fit of the plot ln(τ(T)) versus T (Figure 5C) supports this assumption, yielding = 11 K and = 1.6 × 10 4 s. The strength of the energy barrier that maintains the residual magnetization against thermal activation is equivalent to the degree of spatial entanglement of the spin orientations.

The two processes: the alignment of spins by the applied magnetic field and the subsequent removal of the field to create a non-equilibrium spin orientation distribution were applied to AFc-GO sheets ( = 10 nm) after zero-field-cooling (ZFC). By applying the field to the zero-field-cooled sheets for 2 h, the random distribution of the spin orientations after the ZFC was changed to the distribution whose spin orientations were partially aligned with the direction of the applied field. Removal of the field caused spin orientations in the nonequilibrium distribution, resulting in a residual magnetization that slowly decayed (Figure S6B, Supporting Information). Its relaxation was analyzed by the same method applied to the field-cooled sample (Figure S7, Supporting Information). The blocking energy barrier (≲ 4 K) and the residual magnetization M(0) (≲ 0.08 emu/g) generated in the zerofield-cooled AFc-GO sheets were much smaller than (= 11 K) and M(0) (∼0.15 emu/g) of the field (500 Oe)-cooled sheets (Figure 5D), indicating the difference in the number of molecules forming the entangled structures within the nanocluster. Note that depends on the field applied after ZFC (Figure 5D). A larger magnetic field applied to the zero-field-cooled AFc-GO sheets induces a larger number of spins to align their orientations with the direction of the field, resulting in a larger deviation from the random distribution and, consequently, a higher energy barrier after removal of the field. This result demonstrates that the non-equilibrium distribution and the residual magnetization, namely, the entangled structures of the spin orientations in the nanocluster can be tuned by varying the intensity of the applied field for the AFc-GO sheets whose spin orientations are randomly distributed.

Conclusions

This study shows that weakly interacting molecular spins in a nanocluster with an average diameter of about 2 nm behave like a spin liquid, which is confirmed by the results that the present system exhibits slow dynamics and spin correlations below the blocking temperature and that the spin orientation distribution has so many degenerate states. Analysis of the Mössbauer spectrum and magnetic susceptibility shows that the magnetic state of the Fe ions in the nanocluster is in the 5/2 high-spin state. The creation of unique functionalities such as slow dynamics, spin liquid-like behavior, and tunable non-equilibrium spin orientation distribution in 2 nm space on a surface provides new insights into low-dimensional magnetism and spintronics. The design and synthesis of chemically functionalized nanosheets by arranging magnetic molecules through on-surface chemistry will lead to major advances in the development of 2D materials and 2D spintronic devices.

Methods

Chemical synthesis: One mL of GO solution (4 mg/mL) was mixed with 50 mL of N,Ndimethylformamide (DMF) in a round-bottomed flask. The solution was stirred while adding amino-ferrocene powder (20 mg) and coupling agents (EDC-HCl, HOBt and trimethylpyridine) at 0 ˚C for the first 3 hours to reduce the overreaction, and then stirred at 24 ˚C for 4 hours. The on-surface chemical reaction produced amino-ferrocene-based nanoclusters with an average inter-cluster distance of 10 nm on the GO nanosheet. The as-synthesized products were separated from the supernatant liquid by using an ultrafast centrifuge after repeated washing first twice with DMF and then with ethanol to remove unreacted chemicals on the sheets, followed by drying.

Magnetization measurement:

The static magnetization M of aggregated AFc-GO sheets weighting 5-30 mg was measured using a superconducting quantum interference device (SQUID) magnetometer (MPMS-7T, Quantum Design). The static magnetic susceptibility was calculated as by using the applied static magnetic field H. The dynamic susceptibility was measured at zero static magnetic field by using a SQUID magnetometer (MPMS-1T, Quantum Design) at an alternating field where P is calculated as ;

. After repeated iterations of steps ( 2) and (3), the system was at or near equilibrium. This method is called the heatbath method. [28] The initial configurations (600 Monte Carlo steps (MCS)) in the simulation were removed for the calculation of the average value, because the configurations were out of thermodynamic equilibrium. S1.

Supporting Information

Table S1. Fitting parameters of Mössbauer spectra of AFc-GO sheets ( = 10 nm) at 5 K as shown in Figure S2.

  (amplitude = 1 or 0.5 Oe) with driving frequency f = 2 -500 Hz. The slow dynamics of AFc-GO sheets ( = 10 nm) was estimated by measuring the residual magnetization obtained by removing the field after the sheets were slowly cooled to the temperature under the field of 500 Oe. The residual magnetization of the zero-field-cooled sheets was obtained by the following three steps: 1) zero-field-cooling slowly to a target temperature; 2) applying a magnetic field for 2 hours at the temperature; 3) removing the field at each temperature. The change in the residual magnetization was then measured as a function of time at each temperature. Mössbauer spectroscopic measurements were performed using a conventional transmission geometry with the 57 Fe excitation energy of 14.4 keV to see the magnetic states of the Fe ions in the AFc-GO sheets. The source velocity was calibrated using pure α-Fe films. A sample with a diameter of about 14 mm was formed by random stacking of AFc-GO sheets, and it was dried and covered with thin Kapton sheets. For the measurements, the sample was sandwiched between thin pure Al films to keep the temperature distribution uniform. The sample with a total weight of Fe atoms of about 20 mg was placed on a stage of a low-temperature cryostat. The spectra were fitted with magnetically-split sextets with Voigt profiles (convolution of Lorentzian and Gaussian functions) and non-magnetic doublets with Lorentzian functions profiles. Markov-chain Monte Carlo simulation: The Monte Carlo algorithm generated a sequence of spin orientations as follows. (1) The initial state of the assembly (Fig. 3a) was a spin configuration where each spin orientation was random. (2) The program randomly selected a molecular site within the assembly and changed its spin orientation to the direction denoted by θ and φ in polar coordinates (0 ≦ θ ≦ π and 0 ≦ φ ≦ 2π) by using random numbers. After calculating the energy difference between the spin configurations before and after the spin orientation change, the Boltzmann factor , where k B is the Boltzmann constant and T is the temperature of the assembly, was used to estimate the probability P of the spin orientation change according to a Markov chain process. (3) The program randomly selected a fraction R with equal probability over the interval [0,1]. Then it performed the orientation change if ,

Figure 1

 1 Figure 1 (a) Schematic illustration of an amino-ferrocene molecule, approximate energy splitting of the 3d orbitals in the Fe 2+ (S = 0), Fe 3+ (S = 1/2), and Fe 3+ (S = 5/2) ions of a ferrocene unit, and schematic drawing of a nanocluster consisting of amino-ferrocene molecules with an average inter-cluster distance of 10 nm on a graphene oxide (GO) nanosheet. (b) A transmission electron microscope (TEM) image of an AFc-GO sheet ( = 10 nm). The inset shows the diameter distribution of the AFc-GO sheets.(c) Magnetization M AFc of molecules of AFc-GO sheets ( = 10 nm), Brillouin function (solid lines), and Langevin function (dotted lines) plotted against μ B H/k B T. (d) Inverse of the static susceptibility of the molecule for AFc-GO sheets ( = 10 nm) versus temperature. The solid line is from the Curie-Weiss law.

Figure 2

 2 Figure 2 Mössbauer spectra of (a) pristine amino-ferrocene powder at 5 K. (b) aminoferrocene-based nanoclusters on a graphene oxide nanosheet ( = 10 nm) at T = 5 -15 K.

Figure 3

 3 Figure 3 (a) Schematic illustration of the fundamental unit of Fe ions of amino-ferrocene molecules (left panel) and the positions of Fe ions within the nanocluster (right panel). (b) Plot of the average total dipole energy of the spin assembly, calculated using a zero-field Markov-chain Monte Carlo method, versus temperature. Snapshots of spin orientations at sites of the assembly (top), spin orientations (left), and dipole energy (right) at sites of the layers forming the assembly calculated at (c) T = 0.01 K, (d) 0.5 K, (e) 10 K, and (f) 200 K under a zero field.

Figure 4 .

 4 Figure 4. (a) Calculated magnetization M AFc of the molecule plotted versus applied field at T = 10 K, 20 K, 50 K (open circles) compared to experimental values (solid line). (b) Average dipole energy of assemblies (l c = 0.20, 0.21, 0.25 nm, S = 5/2) and assembly (l c = 0.20 nm, S = 1/2) versus temperature. (c) Standard deviation (STD) of the dipole energy of the assemblies (l c = 0.20 nm, S = 1/2, 5/2) versus temperature. Snapshots of spin orientations and dipole energy at sites of the bottom layer of the assembly (l c = 0.20 nm, S = 5/2) versus each MCS at temperature T of (d) 0.01, (e) 0.5, (f) 10 K.

Figure 5 .

 5 Figure 5. (a) Dynamic susceptibility (real component: , imaginary component: ) of the molecules of AFc-GO sheets ( = 10 nm) as a function of temperature T. (b) Fieldcooled (FC) and zero-field-cooled (ZFC) static susceptibilities of the molecules plotted against temperature, showing a separation between the curves at the blocking temperature T B . (c) Logarithmic plot of the relaxation time τ(T) of AFc-GO sheets ( = 10 nm) in nonequilibrium spin orientations obtained after field-cooling (red open circle) and zero-fieldcooling (orange and blue open circles) against the inverse of temperature. (d) Blocking energy barrier in the relaxation of the residual magnetization by applying the static field H to zero-field-cooled AFc-GO sheets (〈d〉 = 10 nm) for 2 hours and removing the field.

Figure S2 .

 S2 Figure S2. Magnetization M AFc of the molecule within amino-ferrocene-based nanoclusters on a graphene oxide sheet (AFc-GO sheets ( = 10 nm)) at each temperature as a function of . Solid and dashed lines are Brillouin and Langevin functions, respectively, with spin quantum number S = 5/2. Note that the deviation of the data from the line occurred at larger onset fields with increasing temperature.

Figure S3 .

 S3 Figure S3. Mössbauer spectrum of AFc-GO sheets ( = 10 nm) at 5 K and fitting to the spectrum by using two magnetically split sextets with Voigt profiles (convolution of Lorentzian and Gaussian functions) and two non-magnetic doublets with Lorentzian profiles. The parameters obtained from the fit are listed in TableS1.

Figure S4 .

 S4 Figure S4. (a) Dipole field of the assembly versus MCS. The blue line shows the average value. (b) Snapshots of spin orientations and dipole energy at sites of the bottom layer of the assembly (l c = 0.20 nm, S = 5/2) versus each MCS at temperature T of 200 K.

Figure S5 .

 S5 Figure S5. (a) Real and (b) imaginary components of dynamic susceptibility of pristine GO at a driving frequency (f = 2, 10, 100, and 500 Hz) of the AC field ( = 0.5 Oe).

Figure S6 .

 S6 Figure S6. (a) Magnetization relaxation of AFc-GO sheets ( = 10 nm) after field-cooling (H = 500 Oe) and subsequent field removal. (b) Magnetization relaxation of AFc-GO sheets ( = 10 nm) in non-equilibrium spin orientations by the application of a static field of 500 Oe for 2 hours after zero-field-cooling to the target temperature and subsequent removal of the field at each temperature. The solid lines in (a), (b) show curves fitted to the data using the stretched exponential form.

Figure S7 .

 S7 Figure S7. Logarithmic plot of relaxation time τ(T) against the inverse of temperature of AFc-GO sheets ( = 10 nm) in non-equilibrium spin orientations produced by the application of magnetic fields (500 -2000 Oe) for 2 hours after the zero-field-cooling and subsequent removal of the field at each temperature.

Acknowledgements

We thank Prof. K. Mibu of Nagoya Institute of Technology and Prof. E. Kita of Tsukuba university for the insightful discussion. The Mössbauer spectroscopic measurements were performed under the Nanotechnology Platform Program of MEXT, Japan. This work was supported by JSPS KAKENHI Grant Number JP21K0482, Grand-in-Aid for User Facility Service in NIMS, and by the International Center of Materials Nanoarchitectonics (WPI-MANA), MEXT, Japan.

Data Availability Statement

Supporting Information is available from the Wiley Online Library or from the author.

Conflict of Interest

The authors declare no conflict of interest.

Supporting Information

Spin liquid-like behavior of weakly interacting molecular spins in nanoclusters on a graphene oxide nanosheet

Makoto Sakurai,* 1 Tsuyoshi Ueta [START_REF] Chibotaru | Molecular Nanomagnets and Related Phenomena[END_REF] and Christian Joachim 1,[START_REF] Ishikawa | [END_REF] 1 International Center for Materials Nanoarchitectonics (WPI-MANA), National Institute for Material Sciences (NIMS), 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan 2 Jikei University School of Medicine, 8-3-1 Kokuryo, Chofu, 182-0022, Japan.


Ho Kwan

Au-Yeung

Suchetana Sarkar

Tim Kühne

Oumaima Aiboudi

Dmitry A Ryndyk

Roberto Robles

Franziska Lissel

Nicolas Lorente

Christian Joachim

Francesca Moresco

Thermal with Electronic Excitation for the Unidirectional Rotation of a Molecule on Surface

Keywords: Single-molecule rotor, microscopic reversibility principle, thermal excitation, electron tunneling, scanning tunneling microscopy

ABSTRACT. Exploring the limits of the microscopic reversibility principle, we investigated the interplay between thermal and electron tunneling excitations for the unidirectional rotation of a molecule-rotor on the Au(111) surface. We identified a range of moderate voltages and temperatures where heating the surface enhances the unidirectional rotational rate of a chemisorbed DMNI-P rotor. At higher voltage, inelastic tunneling effects dominate while at higher temperature the process becomes stochastic. At each electron transfer event during tunneling, the quantum mixing of ground and excited electronic states brings part of the surface thermal energy in the excited electronic states of the molecule-rotor. Thermal energy contributes therefore to the semi-classical unidirectional rotation without contradicting the microscopic reversibility principle.

The unidirectional rotation of a single-molecule rotor adsorbed on a surface cannot be induced thermally without breaking the microscopic reversibility principle. [START_REF] Tolman | The Principle of Microscopic Reversibility[END_REF] According to the energy equipartition theorem, the thermal energy coming from the surface is equally distributed to all accessible mechanical degrees of freedom of the rotor, leading to random movements. [START_REF] Eisenhut | One-Way Rotation of a Chemically Anchored Single Molecule-Rotor[END_REF][START_REF] Stolz | Molecular Motor Crossing the Frontier of Classical to Quantum Tunneling Motion[END_REF] To make unidirectional rotation possible, one should include the electronic excited states of the molecule (completely or in a quantum mixing), [START_REF] Joachim | The Driving Power of the Quantum Superposition Principle for Molecule-Machines[END_REF] a possibility which is absent for a classical rotor. This is for example the case of the Feringa et al. molecular motors, [START_REF] Koumura | Light-Driven Monodirectional Molecular Rotor[END_REF] which rotate always in the same direction when excited by UV light in solution, bringing the molecules from their electronic ground state (S 0 ) to their excited state (S 1 ). [START_REF] Koumura | Light-Driven Monodirectional Molecular Rotor[END_REF] The intramolecular mechanics of an isolated molecule is quantized. A simple example is a single NH 3 molecule in a cold molecular trap, where rotation and vibration spectra can be recorded. [START_REF] Bethlem | Electrostatic Trapping of Ammonia Molecules[END_REF][START_REF] Van Veldhoven | Decelerated Molecular Beams for High-Resolution Spectroscopy[END_REF] Instead, using PF 3 molecules on a solid surface, leads to gearing effects on the electronic ground state. This was first detected by electron stimulated desorption spectroscopy. [START_REF] Zwanzig | Rotational Relaxation in a Gear Network[END_REF][START_REF] Alvey | The Direct Observation of Hindered Rotation of a Chemisorbed Molecule: Pf3 on Ni(111)[END_REF] On a surface, however, the soft quantum mechanical degrees of freedom of both NH 3 and PF 3 rapidly decohere, and the rotation around their umbrella axis becomes random and semiclassical. [START_REF] Klauber | Evidence for Chemisorption Site Selection Based on an Electron-Donor Mechanism[END_REF] A similar mechanical decoherence occurs for the rotation of larger adsorbed molecules like terbutyl-decacyclene on Cu(100). [START_REF] Gimzewski | Rotation of a Single Molecule within a Supramolecular Bearing[END_REF] At room temperature, this molecule was the first to show a classical random rotation after manipulation with the tip of a scanning tunneling microscope (STM). This random rotation occurs in the ground electronic state respecting the microscopic reversibility principle.

At low temperature (T = 5 K), a single O 2 molecule can rotate step-by-step, but still randomly. [START_REF] Stipe | Inducing and Viewing the Rotational Motion of a Single Molecule[END_REF] W. Ho and co-workers were the first using STM inelastic tunneling electronic excitation to trigger the rotation of a single isolated molecule. Also in that case, the O 2 rotational motion is semi-classical, but the energy for the rotation is delivered by the tunneling current producing multiple electron transfer events per second through O 2 . At each electron transfer event, a quantum mixing of the virtual cationic ground state and the anionic excited electronic states occurs, with a very short femtosecond quantum occupation time. [START_REF] Stipe | Inducing and Viewing the Rotational Motion of a Single Molecule[END_REF] This experiment was followed by several others, where chiral or asymmetric molecules adsorbed on a surface were investigated aiming at reaching unidirectional rotation by inelastic electron tunneling. [START_REF] Eisenhut | One-Way Rotation of a Chemically Anchored Single Molecule-Rotor[END_REF][START_REF] Stolz | Molecular Motor Crossing the Frontier of Classical to Quantum Tunneling Motion[END_REF][START_REF] Tierney | Experimental Demonstration of a Single-Molecule Electric Motor[END_REF][START_REF] Perera | Controlled Clockwise and Anticlockwise Rotational Switching of a Molecular Motor[END_REF][START_REF] Au-Yeung | A Nanocar and Rotor in One Molecule[END_REF][START_REF] Jasper-Toennies | Rotation of Ethoxy and Ethyl Moieties on a Molecular Platform on Au(111)[END_REF] Keeping the STM bias voltage and current as small as possible to avoid any inelastic tunneling effects, molecule-rotors in the ground state were further investigated by increasing the surface temperature, thus observing random rotations. [START_REF] Eisenhut | One-Way Rotation of a Chemically Anchored Single Molecule-Rotor[END_REF] Recently, the unidirectional rotation induced by inelastic tunneling at low temperature was shown to be maintained also at a temperature higher than 5 K. [START_REF] Stolz | Molecular Motor Crossing the Frontier of Classical to Quantum Tunneling Motion[END_REF] In that case, the structural asymmetry was not provided by the small C 2 H 2 molecule itself but by the chiral supporting surface.

Understanding the interplay between thermal and electronic excitations for the controlled rotation of a single-molecule rotor is then important for the design of single-molecule machines able to store thermal energy or to produce work. In terms of quantum engineering, this opens fascinating perspectives in the direction of mono-thermal motors. [START_REF] Scully | Extracting Work from a Single Heat Bath Via Vanishing Quantum Coherence[END_REF] In this article, we studied the role of thermal energy and electronic excitations for the controlled rotation of a single-molecule rotor on a metal surface. We first separately investigated the starting point of random thermal rotations (by increasing the temperature from T = 5 K) and of unidirectional rotations induced by inelastic tunneling electrons (by STM bias voltage pulses small in voltage and very long in time keeping T = 5 K). Then, we combined tunneling electrons with moderate heating to understand how thermal energy contributes to the one-way rotation.

RESULTS AND DISCUSSION

We selected the DMNI-P molecule-rotor chemisorbed on Au(111) (Figure 1). Compared to other single-molecule rotors, [START_REF] Stolz | Molecular Motor Crossing the Frontier of Classical to Quantum Tunneling Motion[END_REF][START_REF] Stipe | Inducing and Viewing the Rotational Motion of a Single Molecule[END_REF][START_REF] Simpson | How to Control Single-Molecule Rotation[END_REF] this molecule strongly chemisorbs at low molecular coverage on Au(111), making it impossible to induce any lateral movement on the surface. [START_REF] Au-Yeung | A Nanocar and Rotor in One Molecule[END_REF] In the STM tunnel junction, the needed structural asymmetry is carried internally by the molecule-rotor itself.

At a temperature of 5 K, the isolated DMNI-P molecules in known to reliably rotate in one direction by a bias voltage of V ≥ 350 mV applied for a few seconds. [START_REF] Au-Yeung | A Nanocar and Rotor in One Molecule[END_REF] This specific moleculerotor therefore represents a well-suitable model system to investigate the onset of the unidirectional rotation by combining thermal and low-voltage tunneling electronic excitations. Each voltage pulses were applied under I = 250 pA and V = 0.5 V for 5 s in constant current mode. Images size: 3 nm x 3 nm, I = 5 pA and V = 0.2 V.

Thermal excitation

We first consider the thermal heating of the Au(111) surface. We stepwise increased the surface temperature T by the steps of 0.5 -1 K in order to determine the minimum T triggering a random rotation of the molecule-rotor. For each temperature step, we let the system relax reaching a thermal equilibrium in the STM junction. For STM images we used very low voltage and tunneling current (typically I = 5 pA and V = 10 mV). At these conditions, inelastic tunneling effects involving electronic excited states and tip interactions with the molecule-rotor are negligible. The first rotation event was detected at T = 12 K after waiting 17 hours (See Supplementary Movie 1). When increasing T further, we observed more frequent random rotary motions (Figure 3a and 3b recorded at T = 17 K and T = 23 K respectively, and Supplementary Movie 2 for T = 15 K). At T = 29 K, the STM image recorded during the rotation of the molecule shows a hexagonal shape (Figure 3c) indicating that the molecule-rotor randomly visits the six stable rotational stations during the scanning. In agreement with the microscopic reversibility principle, the experiment confirms that a purely thermal heating of the DMNI-P molecule-rotor does not induce a unidirectional rotation, but only a random one. It is also interesting to note that the anchoring of the molecule-rotor on the surface survives at least up to 77 K, where random and very fast oscillations are observed (Supplementary Figure S4). Even at this relatively high surface temperature, the rotational dynamics remains in the electronic ground state, where only 6 stable stations can be observed on Au (111).

Electronic excitation at low temperature

In the second part of the experiment, we kept the temperature constant at T = 5 K, aiming at determining the smallest possible electronic excitation inducing a directional one-step rotation of 60°. By progressively reducing the STM bias voltage, we measured how the time needed to observe a one-step unidirectional rotation event extends from a few seconds to many hours (Figure 4a and4b). The experiment was systematically repeated on the same molecule (shown in the inset of Figure 4a) after decreasing the bias voltage and increasing the pulsing time duration (always keeping the same tunneling current I = 700 pA). At 150 mV, the first one-step directed rotation event was observed while keeping the STM tip apex at the same position on the molecule-rotor for 35 950 s (around 10 h) in a 12 h pulse (applied at the position of the red cross in the inset of Figure 4a). The lateral thermal drift is negligible in our experimental setup.

After 139 pulses, we plotted the response time as a function of the bias voltage (Figure 4a).

Each response time is measured on the corresponding z(t) curve (like in Figure 4b). The obtained experimental yield, or action spectrum [START_REF] Kim | Action Spectroscopy for Single-Molecule Reactions -Experiments and Theory[END_REF] (Y = R(I/e)), is plotted in Figure 4c. For V > 400 mV, we confirm the nearly constant yield of about 5 x 10 -10 rotations/electron previously reported, [START_REF] Au-Yeung | A Nanocar and Rotor in One Molecule[END_REF] which is assigned to the onset of the C-H stretch vibrational mode at 370 meV. This vibration is the principal energy entry port to trigger a one-step rotation event in the higher bias voltage range. At lower bias voltage, we observe a sharp drop of the yield between 330 and 400 mV, while another nearly constant region appears at 200 -300 mV. For V < 200 mV, a rapid decrease of yield is again observed, reaching the low yield of 6 x 10 -15 rotations/electron at 150 mV. By fitting the data (continuous line in Figure 4c), there are two thresholds, one at 165 meV and the other at 370 meV, corresponding to a large density of DMNI-P deformation mode (C-C stretch manifold) and the above-mentioned C-H stretch mode of DMNI-P, respectively. Those modes contribute to the unidirectional rotation, providing the rotational energy inelastically.

Thermal with inelastic electronic excitation

To investigate the interplay between thermal energy and tunneling electronic excitations, we determined the range of temperatures and STM bias voltage pulses where both thermal heating and electronic excitation contribute to the rotation. We therefore repeated the above rotation experiments by applying STM voltage pulses (50 mV < V < 500 mV) increasing progressively the temperature in the interval 5 K < T ≤ 17 K. In such conditions, we observed a faster rate of unidirectional rotation for T > 5 K while applying voltage pulses. In Figure 5, the rate of the unidirectional rotation (1/reaction time) depending on the voltage is plotted for three different temperatures (all data points measured at I = 700 pA). Increasing the temperature leads to a faster response time for unidirectional rotation. This effect is diminished by increasing the voltage, becoming negligible for V > 500 mV. At this energy, the C-H vibrational mode is excited and the inelastic electronic excitation becomes the dominating energy source for a one-way rotation, thus overcoming the STM junction temperature contribution.

Action spectra taken at T = 12 K and 15 K are shown in Supplementary Figure S5.

In Figure 6, we summarize the mechanical response of a single DMNI-P molecule-rotor by the combination of thermal and tunneling electronic contributions. On the x,y base plane, the applied voltage and temperature are indicated. The blue dots correspond to observed unidirectional rotation, and the orange dots to the (T, V) values where we observed random rotations. On the x-z vertical curve, the yield curve is sketched from Figure 4c In Figure 6, we can identify four different (T, V) rotation regimes, indicated with different colors in Figure 6, and discussed in the following:

(1) The small grey area indicates the very low voltages and surface temperature regimes, where no rotation events have been observed, or experiments were not performed. In those conditions we do not expect the creation of a rotation minimum energy path (random or not) on the ground state potential energy surface.

(2) The light blue area corresponds to the (T, V) values between 5 K < T < 12 K where a stepby-step unidirectional rotation takes place by applying a bias voltage (150 mV < V < 700 mV).

In this area, the inelastic tunneling electrons provide energy to the molecule-rotor via vibrational modes. At each electron transfer event, such tunneling electrons virtually occupy both cationic ground S 0 +() and anionic excited S 1 -() electronic states, where  is the angle of rotation. [START_REF] Eisenhut | One-Way Rotation of a Chemically Anchored Single Molecule-Rotor[END_REF] The occupation is virtual because the S 0 + and S 1 -tunneling resonances are more than 2 eV far from the bias voltage of the C-H stretching excitation (see Supplementary Figure S6). In these conditions, the vibrational energy is able to trigger a dynamical path on the potential energy surface S 1 -() to pass over the rotation barrier, without contradicting the microscopic reversibility principle. The needed asymmetry of the S 1 -() surface respect to the S 0 +() is coming from the chirality of the molecule and by its charge separation in the electric field of the STM junction. At small bias voltage, the long response time can be explained by the competition between the intramolecular vibration energy redistribution and the energy relaxation towards the Au(111) surface.

(3) In the yellow area, both blue and orange points are present, i.e., both unidirectional and random-direction rotations are possible. When reducing the bias voltage to nearly 0 V, the molecule rotates randomly. Interestingly, increasing the surface temperature (12 K < T < 20 K) with a bias voltage 50 mV ≤ V ≤ 250 mV keeps the rotation unidirectional, thus significantly reducing the rotation response time (Figure 5). In this narrow (T, V) range, the inelastic tunneling effect is weak with a low probability of exciting low-energy vibrational modes at 165 meV and below (Figure 4c). As already mentioned, the tunneling current can be described by a quantum mixing of the S 0 +() and S 1 -() virtual electronic configurations. Therefore, at each electron transfer event, part of the thermal energy coming from the surface is redistributed intramolecularly and trigger a dynamical path on the S 1 -() potential energy surface. This means that the rotational barrier on S 1 -() can be overcome, and a one-step of unidirectional rotation happens similarly as discussed above for inelastic effects. In other words, each electron transfer event "projects" part of the thermal energy captured by the molecule-rotor on the S 1 -() state, practically sharing it between S 0 +() and S 1 -(). This is a quantum motor as presently discussed in quantum information theory. [START_REF] Scully | Extracting Work from a Single Heat Bath Via Vanishing Quantum Coherence[END_REF] The thermal contribution to the unidirectional rotation diminishes at higher voltages (Figure 5), indicating that the opening of the inelastic C-H stretch channel is taking the lead to project energy to the S 1 -() state, making the thermal contribution negligible.

(4) In the orange area (T > 20 K and independently of the bias voltage 0 < V < 700 mV) the rotation is non-directional. The surface temperature is now too large to take advantage from the topology of the S 1 -() potential energy surface relative to S 0 +( described above, and the rotation becomes a stochastic process.

CONCLUSIONS

In conclusion, we have investigated the unidirectional rotation of chemisorbed DMNI-P molecule-rotors on the Au(111) surface induced by thermal and tunneling electronic excitations.

While in a well identified voltage and thermal range the induced rotation can be clearly separated as due to thermal effect (random rotation) or electronic inelastic excitations (unidirectional rotation), there is a narrow area of voltages and temperatures where the surface thermal excitation enhances the unidirectional rotational rate. This effect can be rationalized describing the electron transfer events during tunneling by a quantum mixing of virtual electronic states. During each of those events, part of the thermal energy captured from the surface is distributed among the electronic states of the molecule-rotor, increasing the probability of overcoming the rotational energy barrier. In this specific voltage range, the probability of inelastic excitation of vibrational modes is small, and a unidirectional rotation of the moleculerotor is observed even without involving inelastic tunneling effect, opening new perspectives for the development of quantum motors.

METHODS

The Au(111) single crystal was cleaned by subsequent cycles of Ar+ sputtering and annealing

to 450 °C. 2-(1,3-dimethyl-1H-naphtho[2,3-d]imidazol-3-ium-2-yl) phenolate (DMNI-P)
molecules were sublimated from a quartz crucible at T = 225 °C on a clean Au(111) surface held at room temperature under ultra-high vacuum (UHV) conditions (p ≈ 1 × 10 -10 mbar). STM experiments were performed by using a custom-built instrument operating at low temperature under UHV conditions. For thermal excitation measurement, the surface temperature was controlled by a Zener diode (I, V) with a temperature measurement resolution of 0.1 K. The system was stabilized for at least 1 hour in order to achieve thermal equilibrium for each target temperature step. All shown STM images were recorded in constant-current mode with the bias voltage applied to the sample.

Voltage pulses manipulations were performed by positioning the STM tip above the molecule and subsequently ramping up the voltage bias. Both constant current and constant height modes were employed depending on the types of measurements. z(t) or I(t) curves, respectively, were recorded during the pulses, detecting the movement of the molecule. For constant height measurements, the tip-surface distance was calibrated by recording I(z) curves. STM images were recorded before and after the application of the pulse, determining the displacement of the molecule. STM images were taken under non-destructive parameters (e.g., V = 0.2 V, I = 5 pA).

The rotation events were collected from the signal of tip height over time, where a sudden jump of signal can be observed. The corresponding yield (events/electron) calculations were done by the average of the population for all fixed voltages and currents (typically n ≥ 10 for each point unless specifically described), and the statistical error was evaluated by standard deviation.

For geometry optimization and reaction path calculations, we used the DFT method as implemented in the CP2K software package (cp2k.org) with the Quickstep module [START_REF] Vandevondele | Quickstep: Fast and Accurate Density Functional Calculations Using a Mixed Gaussian and Plane Waves Approach[END_REF] . The Perdew-Burke-Ernzerhof exchange-correlation functional [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] , the Goedecker-Teter-Hutter pseudopotentials [START_REF] Goedecker | Separable Dual-Space Gaussian Pseudopotentials[END_REF] and the valence double-ζ basis sets, in combination with the DFT-D2 method of Grimme [START_REF] Grimme | A Consistent and Accurate Ab Initio Parametrization of Density Functional Dispersion Correction (Dft-D) for the 94 Elements H-Pu[END_REF] for van der Waals (vdW) correction were applied. We used 6 layers of gold, where the 3 upper layers were allowed to be relaxed (planar supercell 29.8 x 19.9 Å, vacuum size 40 Å, maximum force 4.5 x 10 -5 a.u.). The data was analysed, and the images were generated by the PyMOL Molecular Graphics System, Version 2.4 open-source build, Schrödinger, LLC.

The fitting of the action spectrum of Fig. 4c by the theory of Ref. [START_REF] Kim | Action Spectroscopy for Single-Molecule Reactions -Experiments and Theory[END_REF] was calculated by: where K is the overall yield that controls the order of magnitude of the measured yield Y, V is the applied bias,  is the angular frequency of the vibrational mode mediating the reaction, and  is the broadening of the reaction threshold. The fitting curves are controlled by basically two parameters;  fixes the threshold of a non-zero yield, and  controls how fast the yield goes from zero to a measurable value.

ASSOCIATED CONTENT Supporting Information. The following files are available free of charge:  A supporting information PDF file is containing further experimental results and supplementary calculations.

Figure 1 .

 1 Figure 1. (a) Chemical structure and (b) adsorption geometry of the DMNI-P molecule-rotor on

Figure 2 .

 2 Figure 2. STM images sequence of the tip-induced unidirectional rotation of DMNI-P on

Figure 3 .

 3 Figure 3. Thermal activation of DMNI-P rotation. (a) The molecule starts rotating at T = 17 K,

Figure 4 .

 4 Figure 4. (a) Response time for a unidirectional rotation over the applied bias voltage. Inset

Figure 5 .

 5 Figure 5. Interplay between thermal and inelastic electronic excitations for unidirectional

  for the inelastic electronic excitation at T = 5 K, illustrating the two main thresholds. According to Figure 4c, three regions are determined by the excitation energy threshold of soft mechanical and the C-H stretch modes at 165 meV and 370 meV, respectively.

Figure 6 .

 6 Figure 6. Summary of combining thermal and electronic excitations. Plot of yield over bias



  A mp4 movie (movie1.mp4) showing the random movement at T = 12 K. The STM images are repeated every 15 minutes. All STM images were taken under I = 5 pA and V = 100 mV, image size 30 nm x 30 nm.  A mp4 movie (movie2.mp4) showing the random movement at T = 15 K. The STM images are repeated every 15 minutes. All STM images were taken under I = 5 pA and V = 50 mV, image size 30 nm x 30 nm.

ACKNOWLEDGMENT This work was funded by the European Union. Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union. Neither the European Union nor the granting authority can be held responsible for them. This work has received funding from then European Innovation Council (EIC) under the project ESiM (grant agreement No 101046364) and the Horizon 2020 research and innovation program under the project MEMO, grant agreement no. 766864. Support by the German Research Foundation (DFG) by the DFG Project 43234550, the Collaborative Research Centre (CRC) 1415, and the Initiative and Networking Fund of the German Helmholtz Association, Helmholtz International Research School for Nanoelectronic Networks NanoNet (VH-KO-606) is gratefully acknowledged. F.L. thanks the Fonds der Chemischen Industrie (FCI) for a Liebig Fellowship and C.J. the WPI MANA project for financial support. R.R. and N.L. acknowledge financial support from the Spanish State Research Agency Grant PID2021-127917NB-I00 funded by MCIN/AEI/10.13039/501100011033 and by "ERDF A way of making Europe".

AUTHOR INFORMATION

Corresponding Authors. *Email: francesca.moresco@tu-dresden.de Author Contributions. The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.