Tunneling electron induced rotation of a copper phthalocyanine molecule on Cu(111)

J. Schaffert,1 M. C. Cottin,1 A. Sonntag,1 C. A. Bobisch,1 R. Moller,1 J.-P. Gauyacq,2 and N. Lorente3

1Faculty of Physics, Center for Nanointegration Duisburg-Essen (CENIDE),University of Duisburg-Essen, Lotharst. 1, D-47057 Duisburg, Germany

2Institut des Sciences Moleculaires d’Orsay, CNRS-Universite Paris-Sud 11,UMR 8214, Batiment. 351, Universite Paris-Sud, F-91405 ORSAY Cedex, France

3Centre d’Investigacio en Nanociencia i Nanotecnologıa (CSIC-ICN), Campus UAB, E-08193 Bellaterra, Spain(Dated: October 20, 2021)

The rates of a hindered molecular rotation induced by tunneling electrons are evaluated usingscattering theory within the sudden approximation. Our approach explains the excitation of copperphthalocyanine molecules (CuPc) on Cu(111) as revealed in a recent measurement of telegraphnoise in a scanning tunneling microscopy (STM) experiment [Schaffert et al., Nat. Mat. 12, 223(2013)]. A complete explanation of the experimental data is performed by computing the geometryof the adsorbed system, its electronic structure and the energy transfer between tunneling electronsand the molecule’s rotational degree of freedom. The results unambiguously show that tunnelingelectrons induce a frustrated rotation of the molecule. In addition, the theory determines the spatialdistribution of the frustrated rotation excitation, confirming the striking dominance of two out offour molecular lobes in the observed excitation process. This lobe selectivity is attributed to thedifferent hybridizations with the underlying substrate.

PACS numbers: 68.37.Ef,72.10.-d,79.20.Rf,74.55.+vKeywords: inelastic tunneling, rotational excitation, molecular adsorption, telegraph noise, molecular ma-nipulation

I. INTRODUCTION

Tunneling electrons permit to induce changes in atomicor molecular adsorbates at surfaces with great con-trol1–15. This breakthrough in fundamental science andin nanotechnology has stirred a lot of attention as it be-came possible to manipulate adsorbates and to induce re-actions on the atomic scale. On the one hand, these tun-neling experiments provide perfect toy systems to learnchemical rules16,17, on the other hand, promising bottom-up techniques for creating nanometer scaled devices be-come feasible18.

The first atomic manipulation experiments1,2 led toan important theoretical effort to unravel the mecha-nisms behind atomic motion induced by tunneling elec-trons. The task was complex due to the involvementof several length scales and the large number of de-grees of freedom. Most treatments have favored master-equation approaches where the atomic and electronic de-grees of freedom are perfectly separated and electronictransitions are incorporated only through atomic exci-tation and de-excitation rates19–21. These rates are ob-tained through full quantum mechanical calculations, us-ing electron-atom matrix elements fitted to or extractedfrom deformation-potential types of calculations in aGolden-rule treatment22. The atomic evolution is sup-posed to be that of a truncated harmonic potential,where the truncation indicates the rupture of an atomicbond7,19. Only a few works have treated different atomicpotential energy surfaces (PES) other than the truncatedharmonic. Avouris et al.23 used atomic wave-packetpropagations to evaluate the atomic dynamics in an ex-cited PES after electron tunneling. More recently, the an-

harmonicity of the PES as revealed by density functionaltheory (DFT) was used to explain the electron-inducedmotion of ammonia molecules on Cu(100)8,24. Despitethe simplicity of these models, much insight was gainedfor single-atom and single-molecule dynamics.

Among the different evolutions of molecular dynamicson surfaces, rotations have proven to be complex3,25. Theorigin of this difficulty can be traced back to the excita-tion mechanism itself. Instead of inducing a deformationof some localized bonds due to a brief charged state ofthe molecule, rotation implies some type of partial angu-lar momentum transfer. Indeed, rotational excitation iscloser to magnetic excitations than to vibrational excita-tions.15,25–27

By studying the telegraph noise observed in the tun-neling current, Schaffert and coworkers15 have recentlyreported on the frustrated rotation of single copper ph-thalocyanine (CuPc) molecules on Cu(111), excited bythe tunneling current. Here, we present a theoreticalaccount of this rotational excitation process. The adia-batic PES for the rotation of CuPc on Cu(111) is com-puted within DFT. We characterize the molecular ad-sorption as well as its electronic structure and calculatescanning tunneling microscope (STM) constant currentimages. These predictions are compared with the experi-mental observations15 and are used to explain them (Sec-tion II). The electron-induced excitation is analyzed afterthe DFT structure calculations. A scattering theory ac-count of the rotational excitation is presented. Using thecustomary approximation of the Tersoff-Hamann treat-ment of STM images 28 we obtain a DFT-based spatiallyresolved description of the rotational excitation by tun-neling electrons (Section III). Finally, the insight gainedby our theoretical treatment is discussed and the article

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ends with some concluding remarks.

II. STUDIED SYSTEM

Our previous experimental and theoretical work15

shows that the tunneling current induces a frustratedrotational motion of CuPc molecules on Cu(111). Anin-plane molecular axis changes by ∼ 7◦ back and fortharound the central Cu ion, which remains located at ahighly symmetric bridge position on the surface. The ro-tation corresponds to transitions between the minimumenergy conformation aligned with the [−110] directionof the surface and two metastable local minima for theclockwise and anti-clockwise rotations.

A brief summary on the experimental findings readsas follows. When CuPc (four-fold symmetric in the gasphase) is adsorbed on the six-fold symmetric Cu(111)surface, the resulting STM images appear two-fold sym-metric. One pair of opposing benzopyrrole rings (lobes)appears pronounced as compared to the remaining twolobes. In the submonolayer regime, CuPc adsorbs indi-vidually, the formation of dimers or clusters is rather un-likely. In addition, the molecules appear partially fuzzy.The blurring of the molecular images is attributed toa molecular motion induced by the tunneling electrons.This is due to switching transitions between two discretelevels that correspond to two different tunneling currentsfor a fixed tip position. The switching events are ran-dom in time, with constant probabilities.15 Therefore,the definition of random telegraph noise (RTN) is ful-filled. The high-current level corresponds to nearly twicethe low-current level. The switching frequency scales lin-early with the tunneling current, hence, a one-electrondriven process is observed.

A special electronic setup was designed to analyze theRTN signal from the tunneling current in real-time, dur-ing the ongoing STM experiments. For a full character-ization of a telegraph signal, the three quantities rate,amplitude and duty cycle have to be measured. Detailson this new technique, scanning noise microscopy (SNM),are discussed in Ref. [29]. The RTN characteristics canbe obtained in spectroscopy measurements with fixed po-sition of the tunneling tip, but they can also be mappedto create noise images exhibiting e.g. spatially resolvedexcitation rates with the same resolution as the STMtopography. The study of CuPc on Cu(111) yielded ex-citation maps clearly highlighting a lobe selectivity ofthe rotation excitation mechanism. Two out of the fourmolecular lobes appear noisy in STM. In SNM only thosetwo lobes are visible with great detail. Based on the ex-perimental results the specific excitation of the two lobescould not be explained. However, the SNM amplitudemaps as well as the spectroscopy of the duty cycle15 gavea strong hint for possible in-plane rotational motion bya small angle. That was the starting point for our DFTstudies.

FIG. 1: (Color online). Minimum-energy conformationof a copper phthalocyanine (CuPc) molecule adsorbed onCu(111). The molecular Cu atom is located on a bridge siteand two of the molecular lobes lie along the dense atomic rowin [-110] or equivalent direction (dashed white line) on thesurface in agreement with the experimental findings15.

III. DENSITY FUNCTIONAL THEORYDESCRIPTION

In order to understand the induced rotation of CuPcon Cu (111), DFT studies were performed to obtain amaximum of information about the electronic and geo-metric properties of the molecule on the surface. Thecalculations were done using the vasp code30.

The calculations were performed for a 9×10 Cu-atomunit cell with 4 layers. This large unit cell is required inorder to reproduce dilute molecular densities. The largecell allows to use a single k-point. The PAW schemefor the atomic potentials31 was used, and the planewavebasis set was expanded up to a cutoff energy of 300 eV.

We evaluated the total energy of different high-symmetry conformations of the adsorbed molecule.Among all of them the minimum energy corresponds tothe molecular Cu atom sitting on a surface bridge siteand one of the molecular symmetry axis aligned along the[-110] direction (dense atomic row on the surface). Fig-ure 1 shows the minimum energy configuration, in perfectagreement with the experimental observations15. Thisadsorption conformation has also been found for CoPcon Cu(111) in a recent joint experimental and theoreti-cal study32, as well as in other theoretical studies33,34.

A. Adiabatic potential energy surface

The PES as a function of the rotation angle, φ, betweenone of the molecular in-plane axes and the surface [-110]direction, was computed by fixing one of the pyrrole-Natoms and the molecular Cu atom at their relaxed po-sitions and rotating the corresponding interatomic axisto the desired angle. The structure was relaxed until allforces within the molecule and the first two substrate lay-ers fell below 0.02eV/A, keeping these two atoms fixed.

3

FIG. 2: (Color online). Adiabatic potential energy surface(PES) of a CuPc on Cu (111) as the molecule rotates arounda surface normal defined by the position of its central Cuatom. Symbols correspond to computed values and lines aretwo different fits of the computed data. The two fits yield afrustrated rotation barrier between the equilibrium points atφ = 0◦ and φ = 6.9◦ of 50 meV and 70 meV, respectively.

Afterwards, the constrain on the pyrrole-N was releasedand placed on an aza-N atom, repeating the ionic conver-gence to the same thresholds. In this way, the internalmolecular and surface structure were relaxed for an ex-tensive set of ionic relaxation calculations, excluding theback-relaxation to the original equilibrium state. De-spite a careful convergence, we cannot rule out a small(∆φ ≈ ±0.1◦) uncertainty in the final positioning of themolecular axis. The PES shown in Fig. 2 reveals the ex-istence of a metastable adsorption position of the CuPcrotated by ±7◦ from the equilibrium position. This sec-ondary minimum is reminiscent of the geometry of fullCuPc monolayers on Cu(111) where the molecule appearsrotated by ±7◦ from the substrate axis.35,36

The calculation of the rotational barrier height is ofparticular importance for the determination of the rota-tional rates as shown below. Figure 2 shows the adiabaticPES calculated with the above prescriptions. In order tosmoothen the small calculation uncertainties a continu-ous function is fitted to the PES. Despite our efforts, wecould fit different types of curves with different barriers.In the present calculations we have used two differentcurves corresponding to rotational barriers of 50 meVand 70 meV in order to study how critical the chosenPES is in the switching rate determination.

All of these calculations have been performed withinthe local density approximation (LDA). The rationale be-hind this is to use the overbinding error of LDA to ob-tain a physical molecule-surface distance compensatingto some extend the neglected van der Waals interactionin LDA. It is well known, that van der Waals is a largecomponent in the binding of large organic molecules like

LDA DFT+D2z (A) 2.574 2.805

Echem (eV) 4.21 5.08

TABLE I: Computed height distances z between the Cucentral atom of the molecule and the surface plane and themolecular chemisorption energy Echem obtained with LDAand DFT-D2.

phthalocyanines on noble metal surfaces37. In order toassess the accuracy of the LDA PES, we repeated thecalculations just for the φ = 0◦-case using DFT-D2 cal-culations including van der Waals interaction38 as imple-mented in the vasp code. Table I compares the results.The adsorption energy difference is 16%, and the differ-ence in adsorption distances is 8%. As is well-known,LDA compares favorably in these values with more real-istic methods. And, particularly in the present case, theLDA PES should be a good estimation of the adiabaticPES because the missing van der Waals force is a long-range interaction which is little affected by atomic detailssuch as a small-angle in-plane rotation of the molecularaxis.

B. Electronic structure

Transition metal phthalocyanines capture charge fromnoble metal surfaces37. Similarly, in the case of CuPc,a full electron is captured into the first empty orbital ofπ character (e2g in D4h notation). This has clearly beenseen for CuPc on Ag(100)37. Copper surfaces are morereactive than silver surfaces, hence more charge is cap-tured. Our DFT results indicate a shift of the two e2gorbitals below the Fermi energy, and the overall trans-ferred charge approaches two electrons. Although theDFT Kohn-Sham orbitals strictly do not possess phys-ical meaning, they are commonly used to evaluate themolecule’s charge population. The results can be under-stood as a qualitative prediction that can be easily putto test e.g. when computed constant current STM im-ages are compared to the experiment. Indeed, as shownin Ref. 15 there is qualitative agreement between theTersoff-Hamann simulated image28 and the experimen-tal data. Here, we present calculations done within theTersoff-Hamann theory but integrating over a given en-ergy window:

I ∝∑ν

|ψν(~r)|2F (E − εν), (1)

where E is the energy window from the Fermi energy,ψν(~r) is an eigenstate of the full system’s Hamiltonianwith its eigenvalue εν . The window function F (E − εν)reads

F (E − εν) =

{1 if|εν − EF | ≤ |E − EF |0 otherwise.

4

Figure 3 shows the evaluated constant current imageusing Eq. (1) for various values of E. For energy windows(equivalent to the applied STM bias) between -800 meVand 800 meV,the resulting images slightly vary. The im-ages are dominated by the contribution of the extendedπ-like e2g orbitals. Indeed, the density of states pro-jected on these orbitals15 show broad features spanningthe above energy range. Due to a lack of wave functionamplitude on-top of the molecular Cu atom, the STMimage shows a depression at the molecular center. Thecharacteristic four lobes resembling the two e2g orbitalsappear as protrusions.

Figure 3 also showcases a comparison between the cal-culated images for the molecule aligned with the [-110]axis (the ground state or minimum energy configuration)and the molecule rotated by ∼ 7◦ (metastable state).Only small variations can be seen. There are some slightasymmetries due to inaccuracies during the coordinaterelaxation, but more significantly the rotated moleculesshow sharper images (smaller sized spots in the lobe re-gion), indicative of a lower hybridization with the sub-strate. The simulated images also show a difference be-tween the lobes aligned with the [-110] axis and the per-pendicular ones. The latter show a larger local densityof states (see for example the ground state image at 200meV, and also at -200 meV). This is due to the strongerinteraction of the molecule with the surface along the[-110] axis, and has connections with the difference be-tween excitation probabilities as revealed in our excita-tion rate study below. Indeed, this finding relates tothe experimentally observed factor of nearly two, sepa-rating the high-current and the low-current level of theRTN tunneling current signal. The experimental factorof about two was reproduced by our DFT study whencomparing cross-sections within the calculated constant-height images.

IV. TUNNELING-ELECTRON INDUCEDROTATION

Energy transfer from a tunneling electron to an adsor-bate has been studied in detail for various systems.39,40

In the case of vibrational excitation by electron colli-sion, the resonant process associated with the transientcapture of the electron by the target molecule has earlybeen recognized to be very efficient for transferring en-ergy from an electron to a molecule. Resonant processesare very active in electron collisions on free molecules (seee.g.41) or adsorbed molecules42, as well as in tunnelingconditions43,44. For vibrational excitation the large massratio between electron and atoms makes a recoil mecha-nism only weakly efficient and any trapping phenomenon(a resonance) which significantly increases the electron-molecule interaction time can boost the vibrational exci-tation efficiency. The situation is completely different inthe case of rotational excitation. The rotational motion isslow, so that an electron-molecule interaction can be seen

as a sudden process with the fast electron colliding witha fixed molecule. The excitation process is then relatedto a recoil phenomenon in which the scattered electrontransfers recoil angular momentum to the target. Thespecificity of the rotational excitation compared to thevibrational excitation comes from the change in the rel-ative orders of magnitude of electronic and nuclear mo-menta when going from linear to angular momenta. Inthe case of a free molecule, quantization of the angularmomentum renders the angular momentum of a scatteredelectron at the same order of magnitude as that of thelow rotational levels of the molecule, so that exchangecan be very efficient and lead to significant rotational ex-citation45. In the present case, the non-spherical symme-try of the tunneling electrons and the frustrated rotationimplies that neither the electron nor the molecule has awell-defined angular momentum. However, distributionsof angular momentum can be associated with such sys-tems, and the efficiency of the recoil in rotational excita-tion remains very high25. One can stress that magneticexcitation by tunneling electrons also involves angularmomentum transfer. Essentially, spin excitation and ro-tational excitation can be described along the same linesand both can be very efficient26,27.

A. Scattering theory

Let us consider the transition of an electron from aninitial state |ψi〉 of energy εi in the STM tip into a finalstate in the sample |ψf 〉 of energy εf . The process maybe inelastic such that εi 6= εf . The excess energy maythen be transferred to a frustrated rotational state ofthe molecule, defined as an eigenstate of the potentialenergy curve in Figure 2. The molecule is excited fromits frustrated rotation ground state |R0〉 to an excitedstate |Rn〉, gaining an energy E = En − E0 from theelectron (E0 and En are the energies of the initial andfinal rotational states).

We can then say that the global system is initially inthe state |i〉 = |R0〉 ⊗ |ψi〉 at energy Ei = E0 + εi andthe final state and energy are |f〉 = |Rn〉 ⊗ |ψf 〉 andEf = En + εf , respectively.

The transition rate per unit time for an inelastic elec-tron transfer from tip to the substrate is given by theT -matrix:

1

τine=

2π

~∑i,f

|Ti,f |2δ(Ei − Ef ). (2)

In general, it is difficult to determine the connecting po-tential, and hence the T -matrix, in particular its inelasticpart. However, in the present case, the molecular rota-tion is a slow motion compared to the electron trans-mission from the tip into the substrate. Therefore, onecan use a sudden approximation in which the T-matrix isevaluated for fixed positions of the molecule with respectto the substrate (fixed φ-angle) and then used to com-

5

FIG. 3: (Color online). Constant current simulated images for different energy windows, Eq. (1). Both, the moleculealigned with the [-110] surface direction and the molecule rotated by ∼ 7◦ are shown. An image variation due to the differenthybridization of the molecule with the substrate is apparent between the two types of molecules.

pute the T-matrix element between initial and final ro-tational states. This approach has been introduced veryearly in the treatment of rotational excitation of electron-free molecule collisions45 and later also used for adsorbedmolecules25. Expression (2) can be further simplified inthe Tersoff-Hamann approximation28: the T -matrix for afixed position of the molecule is taken proportional to theelectronic wavefunction evaluated at the tip apex (~r0):

Ti,f = 〈ψi|T |ψf 〉 ≈ Cψf (φ,~r0), (3)

where ψf (φ,~r0) is the molecule+substrate wavefunction.It is computed for an STM tip located at ~r0, a fixedposition with respect to the substrate and for an angle φof the molecule rotated from its equilibrium position. Cis the proportionality constant. We explicitly write thedependence of the wavefunction on the molecular angle φ(relative position of the molecule and substrate), for itslater use in the evaluation of the rotational excitation.

We can now replace each quantity in expression (2) by

its value:

1

τine=

2π

~∑i,f

∑n>NR

ftip(εi)[1− fsub(εf )]

× |〈R0|〈ψi|T |ψf 〉|Rn〉|2δ(E0 + εi − En − εf ).(4)

In the summation we explicitly defined electron flowfrom the tip to the substrate by using the Fermi factors ofthe tip ftip and the substrate fsub. The index NR is thatof the Rn level above which the molecule actually rotates.Indeed, excitation from the ground state R0 to low lyingrotational excited states Rn is not sufficient to lift themolecule out of its equilibrium potential well (Fig. 2).The rotational excitation must be strong enough for themolecule to overcome the potential barrier separating theequilibrium well from the two metastable ±7◦ side wells(see Fig. 2). This limit corresponds to the level NR inthe formulas. For the PES with a rotational barrier of 70meV, NR is equal to 25 . This emphasizes, that jumpsfrom one well to the other have to involve highly excitedstates of the frustrated rotation and thus, a very efficient

6

FIG. 4: Probability of de-excitation into the right well fromthe level n of the rotational PES, Eq. (8). For levels belowthe 70 meV barrier there is an alternation of levels among leftand right and central wells. Above the barrier, the rotationalstates become increasingly delocalized about all wells and theprobability to find the molecule in the right well approaches1/3.

rotational excitation process is needed.Making use of i) the Tersoff-Hamann approximation,

ii) Eq. (3) and iii) the zero Kelvin expression for theFermi distribution functions Θ(εF − ε), we can furthersimplify the expression by assuming that the tip is madeof a material where the details of the band structurecan be summarized in a density of states (DOS) func-tion Dtip(ε). Finally, the rate for the excitation over therotational barrier 1/τine, is

1

τine=

2π

~∑

n>NR,f

Dtip(εf + E)|C|2

× |〈R0|ψf (φ,~r0)|Rn〉|2

Θ (εF + eV − εf − E)Θ(εf − εF ). (5)

The total electron current is equal to the electron tran-sition rate from tip to substrate times the electron charge:I = e× 1

τTot, where 1

τTotis similar to Eq. (5) except that

the summation over the rotation index n runs over the

entire rotational spectrum. In the case of the total cur-rent, there is no restriction in the sum over n, while in theinelastic rate, Eq. (5), the summation is restricted to theover-barrier levels n > NR. We can define the fraction ofthe current that induces rotation from one potential wellto another, also known as the inelastic electron fraction,by the ratio of the inelastic current to the total current:

η(~r0, V ) =1

τine/

1

τTot. (6)

The benefit from the implementation of the inelastic frac-tion is that the unknown constant C in Eq. (3) and the

FIG. 5: General excitation and de-excitation scheme in be-tween rotational levels within the potential energy surfaceleading to the molecular frustrated rotation. A direct de-excitation from the highly excited level (upper red solid line)to the left side well is indicated by an arrow. The de-excitationcan also take place via many intermediate states dependingon the degree of excitation. Here we choose one intermedi-ate state (dotted line) to exemplify the mediated two-stepde-excitation.

tip’s DOS factor cancel out under the assumption of aroughly constant DOS of the tip over the energy rangeof interest.

Using the above expressions, the inelastic electron frac-tion becomes:

η(~r0, V ) =

∑n>NR,f

|〈R0|ψf (φ,~r0)|Rn〉|2Θ(εF + eV − εf − E)Θ(εf − εF )∑n,f |〈R0|ψf (φ,~r0)|Rn〉|2Θ(εF + eV − εf − E)Θ(εf − εF )

. (7)

The inelastic electron fraction η does not evaluate thenumber of molecules that actually do rotate, but ratherthe fraction of electrons that excite the molecule over a

certain rotational level NR. Indeed, on the time scale ofthe experiment, a molecule excited by a tunneling elec-tron from the rotational ground state R0 to a rotational

7

level Rn will quickly relax to a lower state, i.e. it willend up localized in one of the three potential wells andthe experimental transition rate concerns transitions be-tween these potential wells. Relaxation from level Rn caninvolve collisions with substrate electrons (electron-holepair creation in a process very similar to the one dis-cussed here) or the transfer of the rotational excitation(coordinate φ) to other degrees of freedom of the heavyparticles in the molecule-substrate system (intramolecu-lar relaxation or transfer to phonons). Figure 5 shows asimplified scheme of the excitation/de-excitation processleading to the transfer from the equilibrium potential wellto another. For simplicity only one intermediate level inthe relaxation is displayed in the figure.

We did not try to evaluate in detail this very com-plex relaxation process but resort to a geometrical sta-tistical approximation. We evaluate the branching ratiofor the de-excitation from the level Rn towards a givenmetastable well in terms of the weight of the rotationalwavefunction Rn in the φ-range of the metastable well,approximated as the φ-region beyond a critical angle φc.By inspection of the PES, we choose φc = 5◦. The frac-tion of molecules excited to a given level n that are even-tually trapped in the right well is:

pR(n) =

∫ 15◦

φc

|Rn(φ)|2dφ/∫ 15◦

−15◦|Rn(φ)|2dφ, (8)

and we can assume that the same probability rules thetrapping in the left well. The ±15◦-limits reflect that themolecule is not freely rotating above the surface, but it isconfined to a 30◦-sector due to the molecule+substratejoint symmetry and to the high barrier separating theequivalent φ regions (see Fig. 2). Figure 4 shows theprobability of de-excitation into the right well from thelevel n of the rotational PES, pR(n). For low n, the ex-cited states are localized in the equilibrium well so thatthe branching ratio is equal to zero. Above the energy ofthe bottom of the metastable wells, some of the states arelocalized in the side wells. Owing to our approach whichconsiders even and odd functions, the branching ratio isequal to 0.5 for these states. Above the frustrated ro-tation barrier (70 meV for the shown case), i.e. for thestates included in the summation in Eq. (7), the proba-bility steadily increases to 1/3, consistent with a roughlyequal partitioning among the wells.

The inelastic fraction of electrons that make themolecule jump from the equilibrium well into the rightwell, ηR, is then obtained by including the branching ra-tio pR(n) into the expression (7):

ηR(~r0, V ) =

∑n>NR,f

pR(n)|〈R0|ψf (φ,~r0)|Rn〉|2Θ(εF + eV − εf − E)Θ(εf − εF )∑n,f |〈R0|ψf (φ,~r0)|Rn〉|2Θ(εF + eV − εf − E)Θ(εf − εF )

. (9)

At this point, one can stress that the quantity ηR canbe directly compared with the equivalent quantity in theexperiment, Ref. 15. Furthermore, the excitation proba-bility ηR(~r0, V ) depends on where the tip apex is located,~r0, i.e. expression (9) provides a geometrical map of theexcitation process.

As was experimentally shown15, the derivative of theefficiency with respect to the bias gives density of statesinformation. This can easily be comprehended, as thederivative changes one Θ function in expression (9) intoa delta function. The obtained quantity strongly resem-bles the local density of states with information on theinvolved rotational states.

The above treatment parallels that used to computethe changes in conductance due to magnetic excitations(see a review in 27). Indeed, both cases deal with anangular momentum transfer, in the present case, the ro-tational angular momentum of an adsorbed molecule, inthe magnetic case the spin of a magnetic molecule. Thetheoretical treatments just evaluate the sharing of theelectron flux among the different rotational or magneticchannels, respectively.

B. Results

Figure 6 shows the spatial distribution of the inelasticfraction of electrons producing the molecular rotation.The maximum corresponds to ∼ 2.5× 10−5 for a bias of0.5 V. The equivalent experimental number15 is rather∼ 1.2 × 10−7 at 0.6 V. The calculations have been per-formed for a rotational barrier of 70 meV. As we showin Fig. 2, there is some uncertainty in the determinationof the rotational barrier and 50 meV is also consistentwith our computed PES. As expected, a change in thebarrier height strongly influences the rate value. Indeed,the rotational rate, for a tip located at the maximumefficiency spot, changes by a factor of ∼ 3 between the70 and the 50-meV barriers. This is probably one ofthe largest sources of uncertainty in the present calcu-lation. Another uncertainty is hidden inside the PEScalculations of section IIA. It corresponds to the PES asa function of the angle φ when all the other coordinateshave been relaxed, so that it includes an implicit adia-batic assumption for these coordinates, justified by theslowness of the rotational motion. In addition, in the

8

FIG. 6: Inelastic fraction of electrons inducing the molecularrotation. The probabilities of all states above the rotationalbarrier are added and a factor 1/3 is used for the branchingratio into the right well.

presence of distortions of the molecule geometry as theangle φ is varied, a more complex Schroedinger equationshould be used than the pure rotation one used above todefine the Rn wave-functions. In the present case, thedistortion of the molecular frame are below 1 pm duringthe rotation, hence justifying the use of a single variablein the Schroedinger equation.

The calculation involves the integration over angle, φ,the PES rotational levels and the electronic wavefunc-tion, ψf (φ,~r0). Since each angular value of ψf (φ,~r0) in-volves a full self-consistent calculation of the molecule onthe surface for that angle, it is a very costly calculationto perform. However, high Rn rotational states exhibit alarge number of nodes and thus require a large number ofφ-values for the integrals in Eq. (9). We balanced accu-racy and computation costs by discretizing the integra-tion with nine φ-values at which full DFT -calculationswere performed. We then used a linear interpolation ofψf between the ab initio points. In this way we can havea double scale of integration points for the ψf and Rnparts of the integral.

The calculation shows that the two molecular lobesaligned perpendicular to the dense atomic row direction[-110] yield the largest rotational rate, in excellent agree-ment with the experiment15. The ratio of rates betweenthe lobes perpendicular to [-110] and the parallel onesis roughly 5, Fig. 6, in very good agreement with theexperimental one. By studying the electronic states ofmolecule and surface and their relaxed geometry, we con-clude that it is the local binding of the molecule to thesurface that determines the efficiency of the angular mo-mentum transfer, i.e. the excitation mechanism is a localelectronic effect.

V. CONCLUSIONS

The DFT study on CuPc on Cu(111) yields a molecu-lar adsorption conformation in good agreement with theexperimental data15. The molecule is adsorbed flat onthe surface with the central Cu atom sitting on a highlysymmetric bridge site with two out of the four molecu-lar lobes aligned along the densely packed [-110], [0-11]or [10-1] directions. No buckled or bent conformationis found. In addition, we calculated the evolution ofthe total energy of the system when the molecule is ro-tated around a surface normal, going through the centralatom of the molecule. The resulting PES reveals the ex-istence of two local minima for symmetrical rotations by±7◦ clockwise and anti-clockwise from the equilibriumposition. The molecule can undergo a frustrated rota-tional motion around its equilibrium position as soon asit gains an energy higher than the rotational barrier (70meV) by flipping from the equilibrium well into one ofthe side wells of the PES as depicted in Fig. 2. Thesecomputational results confirm the experimental interpre-tation that the blurred STM images of one opposite pairof benzopyrrole rings of CuPc are caused by jumps ofthe molecule between three adsorption wells induced bytunneling electrons15.

The simulated STM images show that the four CuPclobes are not equivalent but split into two groups (lobesalong the dense atomic rows and perpendicular to it).Their position relative to the underlying Cu lattice aredifferent and consequently they hybridize differently withthe substrate. The strong hybridization of the π-like or-bitals located on the molecular lobes leads to a sizablecharge transfer of 2 electrons from the surface to themolecule. As a result, the simulated STM image is dom-inated by the contribution of the e2g (π-like) orbitals ofthe molecule. This contribution is slightly different onthe two types of lobes in good agreement with the sym-metry reduction observed in the experiment.

By analyzing the telegraph noise in the tunneling cur-rent, which causes a fuzzy appearance of CuPc in STMimages, Ref. [15] showed that the molecular rotation rateis maximal on the two lobes perpendicular to the denseatomic row [-110] direction, whereas rotation cannot ef-fectively be induced on the remaining lobes. We haveperformed calculations that revealed the excitation mech-anism of tunneling electrons inducing the frustrated rota-tional dynamics and leading to fluctuations in the STMtunneling current signals. Our approach is based on thesudden approximation for the rotation and the Tersoff-Hamann approximation for the T -matrix in the tunnelingregime. The main result of the theory is that the tunnel-ing electron has a finite probability of exciting the molec-ular rotation: the electron is briefly in contact with themolecule and departs leaving it in one of the rotationallevels; when the excited rotational level is above the po-tential energy barrier separating the three frustrated ro-tation wells, then the jump of the molecule from one wellto another becomes possible. This process is analogous

9

to the one found in magnetic excitation by tunneling elec-trons, where the adsorbate spin changes depending on theweight of each magnetic state in the tunneling symme-try.26,27 Our calculations show a clear difference in theexcitation rates over the two types of molecular lobes,identical to the selectivity observed experimentally. Weconnect this selectivity with the slight differences in theSTM images of the molecular lobes, attributed to thedifferent hybridization between the e2g orbitals (mainlylocated on the molecular lobes) and the substrate. Thesedifferent hybridizations lead to different rotational rates.The lobe selectivity observed in the telegraph noise in-duced by tunneling electrons thus appears to be of elec-

tronic origin.

Acknowledgments

J.S., A.S., C.A.B. and R.M. gratefully acknowledge fi-nancial support by the Deutsche Forschungsgemeinschaftthrough the SFB616 ’Energy Dissipation at Surfaces’.N.L. is supported by the ICT-FET Integrated ProjectAtMol (http://www.atmol.eu). M.C.C. thanks the Stu-dienstiftung des deutschen Volkes.

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