Electric control of spin transitions at the atomic scale

Electric control of spins has been a longstanding goal in the field of solid state physics due to the potential for increased efficiency in information processing. This efficiency can be optimized by transferring spintronics to the atomic scale. We present electric control of spin resonance transitions in single TiH molecules by employing electron spin resonance scanning tunneling microscopy (ESR-STM). We find strong bias voltage dependent shifts in the ESR signal of about ten times its line width. We attribute this to the electric field in the tunnel junction, which induces a displacement of the spin system changing the g-factor and the effective magnetic field of the tip. We demonstrate direct electric control of the spin transitions in coupled TiH dimers. Our findings open up new avenues for fast coherent control of coupled spin systems and expands on the understanding of spin electric coupling.


INTRODUCTION
Spintronics and the concept to control spin and magnetic properties using electric elds have been on the forefront of solid state research for the past several decades with the promise to increase e ciency in data processing [1][2][3][4].Different concepts have been considered such as the spin transistor [5][6][7][8], the spin Hall e ect [9,10], dopants in silicon [11][12][13], and magnetic molecules [14][15][16][17][18][19][20][21][22].Speci cally, spinelectric control allows for superior scalability and switching as electric elds are more easily contained and faster to manipulate than magnetic elds.This type of processing could be further optimized by transfering it to the atomic scale, for which scanning tunneling microscopy (STM) is an ideal platform in realizing such a goal.Speci cally, the combination of electron spin resonance spectroscopy (ESR) with STM has expanded the sensitivity of ESR to atomic scale spin systems, and has enhanced the attainable energy resolution of STM well into the neV range [23][24][25][26][27].
As the manipulation capabilities in STM are mostly based on electrical control, implementing sizeable atomic scale electrical spin control can become not only possible with ESR-STM, but also quite e ective.The applied bias voltage typically induces a very strong electric eld between the tip and sample due to the extremely small gap of only a few Ångströms [28].Moreover, ESR spectra are typically acquired by sweeping the microwave frequency or the magnetic eld, so that the bias voltage essentially becomes a free parameter to be tuned.However, so far the bias voltage in ESR-STM has not been employed for spin manipulation.
In this study, we exploit the bias voltage as an electrical means for direct manipulation of spin transitions.We use a TiH molecule on an insulating MgO layer (see Fig. 1(a)) to demonstrate a direct tuning of the -factor and the tip magnetic eld.In this system, the resonance peak shifts by many line widths within a bias voltage range of 240 mV (see Fig. 1(b)), which is much stronger than what has been predicted for this system (on a di erent adsorption site) [29] or previously measured in bulk systems [15].We explain this e ect by the strong electric eld in the tunnel junction induced by the applied bias voltage and felt by the dipolar TiH molecule.A change in the electric force shifts the equilibrium position of the TiH molecule, resulting in the -factor being modi ed and the molecule feeling a di erent magnetic eld from the spin-polarized tip.The -factor is, in part, modi ed due to a change in the crystal eld felt by the TiH [29].

VOLTAGE DEPENDENT ESR-STM
The measurements were done on TiH molecules that adsorb on the bridge-site between two O atoms of the MgO double layer.They are labelled as TiH OO in Fig. 1(a).Varying the bias voltage continuously, we observe the evolution of the ESR peak as a function of both bias voltage and external magnetic eld at a constant microwave radiation frequency of 61.545 GHz and a microwave amplitude of 20 mV.This is shown for two di erent setpoint currents of  sp = 100 pA and  sp = 250 pA in Fig. 2(a) and (b), respectively.Unless otherwise noted, the corresponding setpoint voltage is  sp = 100 mV.The horizontal features in Fig. 2(a) and (b) are due to the interaction of the microwaves with the background density of states and are not related to the ESR signal [30][31][32][33].Comparing the slope of the ESR peak in Fig. 2(a) and (b), we directly see that the change in the resonance condition is more pronounced for the higher setpoint current, which points towards an in uence of the electric eld rather than the bias voltage.We have obtained similar results for TiH molecules adsorbed on top of an O atom of the MgO layer (labeled TiH O in Fig. 1(a)), which are presented in the Supplementary Information [34].
For a more quantitative analysis of the evolution of the ESR peak, we exploit the linear dependence of the ESR resonance on the magnetic eld as where  Z is the Zeeman energy,  res is the resonance frequency,  is the -factor, and  ext,tip are the external magnetic eld and the eld of the tip felt by the spin system (henceforth the tip eld), respectively.Furthermore, we assume the spin to be  = 1 /2 [35], ℎ is Planck's constant, and  B is the Bohr magneton.Both the tip eld  tip and the -factor will be a function of the applied bias voltage.Analyzing the data at di erent frequencies, we extract the -factor and the tip eld  tip dependency on the bias voltage at four di erent setpoint currents, which is shown in Fig. 2(c) and (d) (for details on the curve tting, see the Supplementary Information [34]).We can clearly see that both the -factor and the tip eld  tip monotonically increase with increasing bias voltage.This indicates that both quantities are sensitive to the changing electric eld.In addition, the change is stronger at a larger setpoint current, which is consistent with our interpretation as a smaller tip-sample distance will lead to a stronger adjustment of the electric eld with respect to bias voltage.
One notable di erence in the behavior of the -factor and the tip eld  tip is around zero bias voltage, where the effects of the electric eld vanish.. Interestingly, near zero bias voltage the tip eld is relatively stagnant as a function of the set point current, while the increase in the -factor is comparable to non-zero bias voltages.Calculations in the literature show that the -factor increases as the moleculesubstrate distance decreases for TiH O [29,36] (we expect a similar behavior for TiH OO ).We have measured approach curves demonstrating that the molecule-substrate coupling increases as the tip-sample distance is reduced.This indicates a decrease in the molecule-substrate distance, which provides an overall consistent behavior for the increasing -factor for larger set point currents (see Supplementary Information for details [34]).Our ndings show that adjusting the tip-sample distance results in changes to both the tip eld and the factor.The changes due to the tip-sample distance have pre- viously been attributed to the tip eld [35,37,38], while theoretical considerations of an electric eld dependence have not taken changes in the tip eld into account [29].However, as we show here, the two e ects cannot be easily separated.
To compare our results with literature, we calculate an effective frequency shift as a function of applied bias voltage of 0.83 GHz/V and 4.3 GHz/V for the -factor and the tip eld, respectively, at a setpoint current of 250 pA.These values are orders of magnitude larger than what has recently been reported for the ESR peak shift of 5.7 kHz/V in a bulk matrix of HoW 10 nanomagnets [15].We can reach these values because the electric eld becomes extremely large between the tip and sample.Comparing the spin-electric coupling (SEC) constants, which relate the frequency shift to the applied electric eld, the situation looks a bit di erent.For the HoW 10 nanomagnets [15], a value of 11.4 Hz/(V/m) was reported, while we estimate values of 0.4 Hz/(V/m) and 2.2 Hz/(V/m) for the -factor and the tip eld  tip , respectively, assuming a tipsample distance of about 5 Å.While this indicates a more e cient coupling mechanism for the HoW 10 nanomagnets, the particular TiH system was not optimized a priori for high SEC, so we anticipate spin systems with superior SEC to be identi ed in the future.
Furthermore, the response of the Zeeman splitting to an electric eld has been previously calculated speci cally for the TiH molecule on MgO, albeit on an oxygen site TiH rather than on bridge site TiH [29].The calculated frequency shift is estimated to be about 0.2 GHz/V, which is smaller than what we have observed experimentally.Neglecting the e ect of the tip eld, which was not considered in the calculations, we nd a four times stronger change in the frequency shift for the -factor in the experiment.We surmise that additional changes other than the crystal eld gradient and the equilibrium position of the whole TiH molecule, such as a change in the Ti-H bond or simply the di erent adsorption site, contribute to this di erence.The sensitivity of the TiH molecule to the local environment is already illustrated by changing the spin state from 3 ⁄ 2 in the gas phase to 1 ⁄ 2 upon adsorption on the surface, as well as changing the -factor from about 2 to 0.6 by moving to a di erent binding site on the MgO [36,39].The ability now to tune the -factor and the tip eld  tip by means of the bias voltage opens up an entirely new degree of freedom for in situ electrical manipulation of the spin transitions.

ELECTRIC CONTROL OF MULTIPLE SPINS
We demonstrate direct manipulation through SEC on two di erent types of dimers with di erent distances between the TiH molecules [35,40].In the rst example, the two bridge site TiH molecules (TiH OO ) are 644 pm apart (see in-set in Fig. 3(a)), such that the coupling is relatively strong ( ≈ 61.1 GHz).We identify three transitions in this dimer in Fig. 3(a).These transitions (labeled I, II, and III) are well separated near zero bias voltage and subsequently broaden as well as intersect as we increase the bias voltage [40].The white dashed lines are ts to a dimer spin Hamiltonian assuming a linear dependence of the -factors and the tip eld  tip on the bias voltage (for details see the Supplementary Information [34]).The corresponding energy levels at a constant external eld of 2.2 T are plotted in Fig. 3(b) with the transitions being indicated.We identify transition III as a clock transition that would not be visible if the two -factors in the dimer were equal [40].Therefore, we know that the two  factors are not equal even at zero bias voltage.Furthermore, as shown in Fig. 3(a) we can tune transitions II and III such that they are located at the same external magnetic eld value, which demonstrates that we can manipulate the spin transitions in a dimer by means of SEC.
If the two TiH molecules are 1.04 nm apart (see inset in Fig. 3(d)), the interaction between them is reduced ( ≈ 0.67 GHz), which shifts the energy of the singlet state | close to the triplet state | 0 as shown in Fig. 3(c) [40,41].The singlet state | and the triplet state | 0 undergo an avoided crossing (see inset in Fig. 3(c)), which can be observed experimentally [41].We have tuned the tip-sample distance such that we can observe this avoided crossing in a bias voltage range between 0 mV and 200 mV as shown in Fig. 3(d).The four transitions that are visible in Fig. 3(d) are labelled I through IV corresponding to the transitions indicated in Fig. 3(c).We can clearly see how the two pairs of transitions associated with each TiH molecule in the dimer approach the avoided crossing and separate again.The white dashed lines are ts to the same dimer spin Hamiltonian as before, just with a weaker exchange interaction, which corroborates the experimental observations (for details on the parameters see the Supplementary Information [34]).For smaller magnetic elds below the avoided crossing, the transitions I and II are strongly in uenced by the SEC, which indicates that the wave functions of the corresponding energy levels are located on the TiH molecule under the tip.As transitions III and IV are much less in uenced by the applied bias voltage, we conclude that the corresponding wave functions are located on the TiH molecule next to the tip.The slope is not vertical, so we expect some in uence of the electric eld on the TiH molecule next to the tip about 1 nm away.For higher magnetic elds above the avoided crossing, the situation is reversed, such that the wave functions for transitions III and IV are in the TiH below the tip and the wave functions for transitions I and II are in the TiH next to the tip.

OPTIMIZING COHERENCE IN COUPLED SPIN STATES
The ability to manipulate spin interactions in dimers through SEC clearly demonstrates the versatility of voltage dependent ESR-STM.However, the tunneling current itself is the biggest source of decoherence in the ESR excitation [42].As a nal proof-of-principle, we exploit both the bias voltage and the tip-sample distance as two degrees of freedom to move the avoided crossing to zero bias voltage, where the tunneling current is minimized and correspondingly the coherence time is maximized.This should enhance and maximize the coherent evolution of entangled states in a TiH dimer that has recently been demonstrated [41].
In order to move the avoided crossing of the second TiH dimer in Fig. 3(d) to zero bias voltage, we increase the tipsample distance such that the setpoint reduces from  sp = 100 mV and  sp = 400 pA to  sp = 50 mV and  sp = 112 pA.Here, the avoided crossing shifts in bias voltage when adjusting the tip-sample distance, but essentially remains at the same position in external magnetic eld.Fig. 4(a) shows the corresponding measurement, where the avoided crossing is now moved close to zero bias voltage.At zero bias voltage only the homodyne detection scheme allows to observe the ESR peaks, which typically appear as asymmetric peaks [40].This can be seen in Fig. 4(b) for three di erent current setpoints, where the avoided crossing is above zero voltage (blue), near zero voltage (red), and below zero voltage (yellow).The shifts of the resonances corresponding to the movement of the avoided crossing in bias voltage is clearly visible.This demonstrates that by considering the bias voltage in ESR-STM, we can manipulate spin structures in a more complex manner than previously possible.

CONCLUSIONS
The ability to tune spin transitions at the nanoscale by means of an electric eld opens up many new and interesting possibilities in the atomic manipulation capabilities of STM far beyond the proof-of-principle presented here.It adds the otherwise unconsidered bias voltage to the degrees of free-dom for customizing spin systems to speci c needs.In this regard, the tip-sample distance, which has previously been used, and the bias voltage present ideal tuning parameters for manipulating complex spin structures.Furthermore, the bias voltage opens avenues towards a more complete understanding of the ESR mechanism in the STM and its dynamics as well as its sources of decoherence and dissipation.This becomes particularly interesting for future applications in timeresolved experiments as it enables fast switching schemes for the bias voltage, which is not possible for magnetic elds or the tip-sample distance (e.g.coherent evolution [41], qubit operations [43,44]).Looking on a broader perspective, we have established SEC in ESR-STM, which connects to the well established eld of spintronics on an atomic scale.Moreover, studying the in uence of the electric eld within ESR-STM opens new possibilities and a better understanding for optimizing SEC in bulk materials.

Supplementary Material for Electric Control of Spin Transitions at the Atomic Scale TIP AND SAMPLE PREPARATION
We cleaned Ag(100) in UHV by repeated cycles of Ar + ion sputtering at 5 kV and annealing at 820 K. MgO was grown on the clean Ag by simultaneous evaporation of Mg onto the sample surface, leaking of O 2 into the UHV space, and heating of the Ag substrate.Deposition times varied from 15 to 20 minutes with the Mg Knudsen cell being heated to 500 K, the O 2 being leaked to 10 −6 mbar and heating of the Ag to 520 K.After the MgO growth, we deposited Fe and Ti on the surface using e-beam evaporators by applying an emission voltage of 850 V and an emission current of 8.5 mA for Fe and 19 mA for Ti.Furthermore, the sample was kept below 16 K during Fe and Ti deposition to ensure that the atomic species did not form clusters on the surface.The Ti species naturally hydrate due to the residual hydrogen gas found in the UHV space [1].To create ESR sensitive tips we picked up between one and ten Fe atoms [2].Dimers studied in this letter were either found naturally occurring on the sample or were created via atom manipulation [3].

MAGNETIC FIELD/BIAS VOLTAGE SWEEPS
We performed magnetic eld/bias voltage sweeps on TiH molecules found on islands of MgO with a height of two monolayers (ML).Measurements were done by irradiating the junction at one frequency, and taking bias voltage dependent sweeps as a function of magnetic eld.To minimize artifacts due to drift, we waited at least for two hours after approaching the tip and applying the microwave radiation prior to starting a sweep.To ensure that we do not drift o the molecular species under investigation, we performed atom tracking between bias sweeps while the magnetic eld was ramping to the next value.In addition, we set the ramp rate of the magnet to relatively low values (≈ 2.5 mT/s), ensuring minimal heating and slow adjustment of the STM junction.During each bias sweep, atom tracking was turned o and the tip position was set to hold.Lastly, we modulated the radiation at a chopping frequency of 107 Hz and set the demodulation frequency of our lock-in ampli er to the same frequency.This way we can pick up the ESR signal of the system in the lock-in ampli er and increase our signal to noise ratio [4].These sweeps took anywhere from four to twelve hours depending on the number of points being measured.

EXTRACTING 𝑔-FACTORS AND TIP FIELDS
To extract the bias voltage dependency of the -factor and the magnetic eld of the tip presented in the main text in Fig. 2, we measured magnetic eld/bias voltage sweeps on a bridge site TiH molecule (TiH OO ) at four di erent microwave frequencies (i.e.Zeeman energies) and four di erent current set points.Fig. S1 shows magnetic eld/bias voltage sweeps at four di erent current set points.We keep the -axis scaling the same in all panels to more clearly show the e ect of the tip-sample distance on the bias voltage shift of the ESR signal.Already there is a clear indication that the spin-electric coupling (SEC) is stronger at smaller tip-sample distances.Fig. S2 shows magnetic eld/bias voltage sweeps measured at four di erent microwave frequencies.The horizontal features in all panels of Figs.S2 and S9 are due to the interaction of the microwaves with the background density of states and not related to the ESR signal (cf.[5,6]).
We can extract the dependencies of the g-factor and the tip eld on the bias voltage at a speci c current set point by the procedure outlined in Fig. S3.We extract the magnetic eld positions of the ESR signal maxima at each bias voltage from a magnetic eld/bias voltage sweep and do a spline t of the bias voltage vs. magnetic eld points as shown in Fig. S3(a).In practice, bias voltages smaller than ±20 mV do not show a clear ESR signal, which we attribute to too low currents close to zero bias voltage.To bridge this gap, we interpolate the missing data points with a spline t.We then use the spline ts at four di erent microwave frequencies and perform a  To demonstrate the overall consistency of this multidimensional t, we plot the extracted ESR peak positions along with the ESR resonance positions calculated from the tted values, which are shown in Fig. S4 for all microwave frequencies and current setpoints.We see an overall good agreement and a continuous evolution.To illustrate the agreement quantitatively, we calculate the di erence between the experimental  data and the modeled resonances, which is shown in Fig. S5 for the corresponding data in Fig. S4.We see that the deviations are generally small and never exceeding 2 mT.Therefore, we conclude that we have an overall consistent model.

TIP APPROACH
The coupling of the TiH molecule to the substrate can be inferred from the evolution of the tunnel junction transmission as a function of the tip sample distance.A similar situation has been analyzed previously in a di erent context [7].Assuming that the TiH molecule is coupled to the substrate by the molecule-substrate coupling  s and to the tip by the molecule-tip coupling  t , the junction transmission  can be written as [7,8] The transmission  describes the junction conductance in units of the quantum of conductance  0 = 2 2 /ℎ, where  is the electron charge and ℎ is Planck's constant.Since our junction is in the tunneling limit, i.e.  t  s , we can easily see that a change in the molecule-substrate coupling  s has a direct impact on the evolution of the junction transmission.We can reasonably assume that in the tunneling regime, the molecule-tip coupling  t increases exponentially with decreasing tip-sample distance.If the moleculesubstrate coupling  s increases/decreases as the tip-sample distance decreases, the junction transmission  will evolve less/more than exponentially, respectively.The tip approach for the tunnel junction measured in the main text in Fig. 2 is shown in Fig. S6(a).The blue line represents the data, while the red line represents an exponential t to the data points at -positions > 60 pm.A small but clear subexponential deviation of the data can be seen.The relative difference between data and t is also plotted in Fig. S5(b) indicating that the junction transmission evolves below the tted exponential evolution.From this behavior, we conclude that the molecule-substrate coupling  s increases as the tip approaches the molecule.Therefore, it is likely that the molecule is pushed towards the surface in this approach range.This provides an overall consistent picture of an increasing -factor as the molecule-substrate distance decreases [9] and explains the evolution of the -factor at zero bias voltage for decreasing tip-sample distance.

MODELLING COUPLED SPINS
The models presented in Fig. 3(b) and (c) of the main text are based on a coupled spin Hamiltonian found in literature [1,10,11]:  This Hamiltonian works on the spin operators of the coupled spins, where  1 and  2 are the -factors of the TiH molecule beneath the tip and beside the tip, respectively,  tip is the tip eld that is only considered to a ect the TiH molecule beneath the tip,  ext is the external magnetic eld,  is the Heisenberg interaction energy between the two spins, and  is the dipole interaction between the two spins.For the dipole interaction, we estimate  = 13.3MHz for the more distant dimer and  = 50 MHz for the closer dimer, which is a signi cantly smaller contribution than the other interactions.
Modelling of the experimentally observed transitions is done by considering the energy di erence between two eigenvalues of the spin Hamiltonian.Modeling the TiH molecules as spin-1 ⁄ 2 systems, the spin Hamiltonian in Eq. (S2) can be diagonalized to analytically nd the four eigenstates, three triplet states (|T + , |T 0 and |T -) and one singlet state (|S ).In the case of the avoided crossing, we observe four transitions with energies: [10,12].We then equate these energy di erences to the energy ℎ of the microwave radiation, which leads to the following set of equations: To incorporate the e ect of the bias voltage in the modelling we assume a linearly dependence of  1 ,  2 , and  tip on the bias voltage in the range from 0 mV to 200 mV.This is based on the results presented in Fig. 2(c) and (d) of the main text.We nd that to get accurate results,  2 also has to shift with the bias voltage, which implies that the electric eld of the tip still a ects the TiH molecule next to the tip apex.This is to be expected as the tip and sample can be approximated as a plate capacitor close to the tip apex.Furthermore, we assume that  and  are not a ected by the bias voltage.We choose the values for  according to the exponential distance dependence between the molecules in the dimer that has been established previously [1,3,11].The comparison is shown in Fig. S7, where the red line is given by  =  0 exp (−( −  0 )/) with  0 = 0.72 nm,  = 94 pm, and  0 = 27.7 GHz [3].We then input a constant  and  into our set of equations and solve for the external magnetic eld values  ext constituting the positions of the ESR peaks as described above.For each set of  1 ,  2 and  tip , we solve for   0  + ,   0  + ,   + and   0 for the case of the dimer with the avoided crossing, and   0  + ,   0  + and   0 for the case of the dimer with a larger interaction energy.Finally, we superimpose the calculated ESR transitions over the data to nd the parameters with the best t.The t parameters  1 ,  2 ,  tip ,  and  for the two di erent dimers are presented in Table SI.
To plot the modelled eigenergies in Fig. 3(b) and (c) of the main text, we simply input our estimated values for  and linearly changing  1 ,  2 and  tip into the diagonalized eigenergies of Eq. (S2).Here, the -axis in Fig. 3(b) and (c) is an "e ective" bias voltage that we model with linearly shifting values for  1 ,  2 and  tip , but for a constant external magnetic eld  ext .Therefore, the evolution of the energy levels in Fig. 3(b) and (c) and the experimental data in Fig. 3 Magnetic eld/bias voltage sweeps were also performed on on-site TiH molecules (TiH O ).Fig. S8(a) shows such a sweep where the ESR signal can be clearly seen.We see a linear shift of the ESR peak at positive bias voltages and no signal at negative bias voltages.We found that increasing the set point current of the magnetic eld/bias voltage sweeps on TiH O increased the linear shift of the ESR signal with respect to the bias voltage, which is consistent with our observations on TiH OO molecules.We found that the shift of the TiH O is much stronger than on the TiH OO for similar setpoint currents.We attribute this to the tip being closer to the sample when measuring on TiH O than when measuring on TiH OO .This is due to the smaller local density of states on the TiH O molecule, which leads to the tip sample distance being smaller on the TiH O than on the TiH OO for comparable set points.This is supported by the di erent appearance of the TiH O compared to the TiH OO as the TiH OO molecules appear brighter than the TiH O molecules (cf.

SPIN-ELECTRIC COUPLING FOR DIFFERENT TIPS AND TIH OO MOLECULES
As a consistency check we performed magnetic eld/bias voltage sweeps on various TiH OO molecules found on the sample with the same tip.Fig. S9 shows three such sweeps on three di erent TiH OO molecules from which we conclude that the measurements are consistent and reproducible.
Furthermore, the data presented in the main text was measured using two di erent ESR-functionalized tips.The rst tip was used for the measurements shown in Fig. 1, Fig. 2 and Fig. 3

Figure 1 :
Figure 1: ESR on TiH molecules (a) STM topography of 2 ML MgO on a Ag(100) substrate decorated with individual TiH molecules and Fe atoms (  = 100 mV,   = 20 pA).The di erent species are labelled and circled accordingly.(b) Schematic of the tunnel junction during the ESR experiment.Force vectors representing the electric force induced by the bias voltage and elastic force of the Ti-MgO bond are shown.Additionally, the electric forces may act on the Ti-H bond.(c) Magnetic eld sweeps performed at di erent STM junction bias voltages  bias ( sp = 100 mV,  sp = 250 pA,  rf = 61.545GHz,  rf = 20 mV).The ESR peak positions are labelled with black arrows.

Figure 2 :
Figure 2: Voltage dependence of the ESR signal (a)-(b) Magnetic eld/bias voltage sweeps performed at two di erent current set points ( sp = 100 mV,  rf = 61.545GHz,  rf = 20 mV, (a)  sp = 100 pA, (b)  sp = 250 pA).White dashed lines show a spline t to the ESR peak positions as a function of bias voltage.(c)-(d) Extracted -factor and tip eld vs. bias voltage at four current set points.

Figure 3 :
Figure 3: Interaction tuning in dimers (a) Magnetic eld/bias voltage sweep on a strongly coupled dimer ( sp = 150 mV,  sp = 1 nA,  rf = 61.545GHz,  rf = 20 mV).The inset shows the topography of the dimer with the red dot indicating the position of the tip during measurements ( sp = 100 mV,  sp = 20 pA).The white dashed lines are ts to the ESR peak positions corresponding to the transitions in panel (b).(b) Modelled behaviour of the spin states at a constant external magnetic eld that result in the ESR transitions measured by the experiment in (a).The colored arrows show the observed transitions.(c) Modelled behaviour of the spin states at a constant external magnetic eld that result in the ESR transitions measured by the experiment in (d).The colored arrows show the observed transitions.(d) Magnetic eld/bias voltage sweep showing the avoided crossing of two coupled TiH molecules ( sp = 100 mV,  sp = 400 pA,  rf = 61.545GHz,  rf = 20 mV).The inset shows the topography of the dimer with the red dot indicating the position of the tip during measurements ( sp = 100 mV,  sp = 20 pA).The white dashed lines are ts to the ESR peak positions corresponding to the transitions in panel (c).

Figure 4 :
Figure 4: Tuning the avoided crossing (a) Magnetic eld/bias voltage sweep showing the avoided crossing of the TiH dimer in Fig. 3(c) near zero bias voltage ( sp = 50 mV,  sp = 112 pA,  rf = 61.545GHz,  rf = 20 mV).(b) ESR sweeps measured at zero bias showing how the resonances shift with respect to current set point ( sp = 50 mV,  rf = 61.545GHz,  rf = 20 mV).This shows that the tip-sample distance can be adjusted to bring the avoided crossing exactly to zero bias.

Figure S3 :
Figure S3: (a) Magnetic eld/bias voltage sweep with a dashed line indicating the spline t performed over the full bias voltage range.(b) Representation of the linear ts at each bias voltage to extract the bias voltage dependencies of the -factor and tip eld  tip .Two linear ts are shown at 150 mV and −150 mV represented by the dashed and dotted lines, respectively.

Figure S4 :
Figure S4: Comparison of the extracted ESR peak positions with the peak positions calculated from the tted -factors and tip elds  tip .The di erent panels show the di erent microwave frequencies.We nd generally good agreement for all frequencies and current setpoints.

Figure S5 :
Figure S5: Di erences between the extracted and the calculated ESR peak positions shown in Fig. S4.The deviations are always less than 2 mT indicating good overall agreement.

Figure
Figure S6: (a) Junction transmission as a function of -displacement (tip-sample distance).The exponential t is done in the low transmission regime revealing the sub-exponential evolution of the data.(b) Di erence between data and t normalized to the t showing the sub-exponential evolution of the tip-approach.

Figure S7 :
FigureS7: Comparison of the t parameters for the exchange coupling  in the two dimers with the exponential dependence reported in literature[11].
S6) Using this set of equations we can solve for four di erent external magnetic elds  ext (  0  + ,   0  + ,   + and   − ) numerically using input values for  , ,  1 ,  2 and  tip .The microwave frequency  is a known input quantity.The resulting external magnetic eld values  ext represent the positions of the ESR peaks on the external magnetic eld axis for the given transitions.In the case of the dimer with the stronger interaction energy, we model the positions of the ESR peaks by considering the following transition energies: |  0 −   − |, |  0 −   + | and |  −   0 | [11].
Figure S8: (a) Magnetic eld/bias voltage sweep measured on a TiH O molecule ( sp = 100 mV,  sp = 75 pA,  rf = 19 GHz,  rf = 20 mV).(b) ESR sweeps measured on TiH O at di erent bias voltages.The evolution of the ESR peak shows a similar voltage dependence as for TiH OO , but the spin-electric coupling is stronger.
Fig. 1(a) of the main text).
(a).The second tip was used in the avoided crossing measurements shown in Fig.3(d) and Fig.4.In addition, over the course of this study we have observed spin-electric coupling in the ESR signal for ve tips.Lastly, during our experiments we never encountered an ESR tip nor a TiH molecule that did not show spin-electric coupling.

Table SI :
S2) Fit parameters for the two dimers presented in the main text.The bias voltage values in the table are the extremal values at the edge of the interval.The parameters for the bias voltage values in between are linearly interpolated.The corresponding -factors and tip elds di er between the dimers because they were measured with di erent tips and at di erent current and voltage set points  sp and  sp .