A mechanism is proposed to describe the occurrence of distancedependent dissipation peaks in the dynamics of an atomic force microscope tip oscillating over a surface characterized by a charge density wave state. The dissipation has its origin in the hysteretic behavior of the tip oscillations occurring at positions compatible with a localized phase slip of the charge density wave. This

model is supported through static and dynamic numerical simulations of the tip surface interaction and is in good qualitative agreement with recently performed experiments on a NbSe2 sample

Pellegrini, F.

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DOI 10.1393/ncc/i2014-11790-6Communications: SIF Congress 2013IL NUOVO CIMENTO Vol. 37 C, N. 4 Luglio-Agosto 2014Theory of charge density wave non-contact frictionF. PellegriniSISSA - Via Bonomea 265, I-34136 Trieste, ItalyCNR-IOM Democritos National Simulation CenterVia Bonomea 265, I-34136 Trieste, Italyricevuto l’ 11 Aprile 2014; approvato il 27 Maggio 2014Summary. — A mechanism is proposed to describe the occurrence of distance-dependent dissipation peaks in the dynamics of an atomic force microscope tiposcillating over a surface characterized by a charge density wave state. The dissi-pation has its origin in the hysteretic behavior of the tip oscillations occurring atpositions compatible with a localized phase slip of the charge density wave. Thismodel is supported through static and dynamic numerical simulations of the tipsurface interaction and is in good qualitative agreement with recently performedexperiments on a NbSe2 sample.PACS 68.35.Af – Atomic scale friction.PACS 73.20.Mf – Collective excitations.PACS 68.37.Ps – Atomic force microscopy.The study of the microscopic mechanisms leading to energy dissipation and frictionhas very important theoretical and practical implications. In recent years, experimentshave started to single out the effects of microscopic probes in contact or near contact withdifferent surfaces, and much theoretical effort has been devoted to the full understandingof such experiments [1]. In particular, the minimally invasive non-contact experimentsoffer a chance to investigate delicate surface properties and promise to bring new insighton localized effects and their interaction with the bulk.Recently, a non-contact atomic force microscopy (AFM) experiment [2] on a NbSe2sample has shown dissipation peaks appearing at specific heights from the surface andextending up to 2 nm far from it. These peaks were obtained with tips oscillating bothparallel and perpendicular to the surface, and in a range of temperatures compatiblewith the surface charge density wave (CDW) phase of the sample. In this paper, amodel is proposed explaining in detail the mechanism responsible for these peaks: thetip oscillations induce a charge perturbation in the surface right under the tip, but, dueto the nature of the CDW order parameter, multiple stable charge configurations existcharacterized by different “topological” properties. When the tip oscillates at distancescorresponding to the crossover of these different manifolds, the system is not allowed toc© Societa` Italiana di Fisica 6364 F. PELLEGRINIfollow the energy minimum configuration, even at the low experimental frequencies ofoscillation, and this gives rise to a hysteresis loop for the tip, leading to an increase inthe dissipation.While the idea behind this dissipation mechanism has been proposed by the authorand collaborators in the original paper [2], this article expands on the technical aspectsof the model. Further details can be found in [3].1. – The modelIn this article, the term CDW is used to indicate a periodic modulation of the chargedensity ρ, irrespective of the process behind its generation. This modulation is described,in the unperturbed system and for the simplest form of CDW, as ρ(r) = ρ0 cos(Q · r),where ρ0 is the intensity and 2πQ−1 the characteristic wavelength. Perturbations toCDWs have been studied extensively [4-7], but most studies are concerned either withuniform perturbations (e.g. an external electric field) or point-like perturbations (e.g.pinning by defects), and often consider one-dimensional models, appropriate for quasi–one-dimensional materials, where the coherence length in the perpendicular directionsis smaller than the atomic distance. What is considered here, instead, is the effectof a localized but extended perturbation of typical length scale similar to the CDWwavelength, acting on a material where the coherence length is macroscopic in morethan one dimension.Starting from the standard Fukuyama-Lee-Rice model [5, 6] for CDW, the chargemodulation is described through a Ginzburg-Landau theory as a classical elastic medium.The complex order parameter ψ(r) = A(r)eiφ(r) is considered, taking into account theamplitude degree of freedom A, as well as the phase φ. The charge density can beexpressed as ρ(r) = ρ0A(r) cos(Q · r + φ(r)), so for the unperturbed system A(r) = 1and φ(r) = φ0 are constant (at least locally), and the free energy reads:(1) F0[ψ(r)] =∫ [−2f0 |ψ(r)|2 + f0 |ψ(r)|4 + κ |∇ψ(r)|2]dr,where f0 sets the energy scale and κ represents the elastic energy contribution.Considering the external perturbation by the AFM tip, generally described as a po-tential V (r) coupling to the charge density, adds an interaction term to the free energy:(2) FV [ψ(r)] =∫ [V (r)Re(ψ(r)eiQ·r)]dr.A well-studied case is the impurity one [7,8], where V (r) =∑i δ(r−ri). In that case phaseoscillations are often enough to describe the ground state of the system, which comesfrom the balance of elastic and potential energy. These point-like perturbations, however,only impose a likewise point-like constraint on the order parameter, and therefore cannotlead to a phase slip in the absence of an external driver [7]. I will therefore consider thecase where V (r) has a finite width of the order of the wavelength 2πQ−1, and minimizethe total free energy F = F0 +FV given a specific shape of V (r). Following in the stepsof the impurity model and considering only phase perturbations, the functional would be(3) F [φ(r)] =∫ [κ |∇φ(r)|2 + V (r)ρ0 cos(Q · r + φ(r))]dr.THEORY OF CHARGE DENSITY WAVE NON-CONTACT FRICTION 65This model, however, is inadequate to describe this specific effect. In fact, considering apurely one-dimensional model would lead to a linear behavior of φ where the potentialis zero. Since we expect a decay far from the perturbation, this is a clearly unphysicalresult. Moreover, due to the nature of the phase, which is defined modulo 2π, given someboundary conditions the solution is not univocally defined unless the total variation ofφ is also specified. Assuming the phase to have the unperturbed value φ0 far from theperturbation, we can define the integer winding number N of a solution as the integral(4) N =12π∫∇φ(x)dx,taken along the CDW direction Q (with N = 0 typically representing the unperturbedcase). Since any change in the winding number along the Q direction would extend to thewhole sample and unnaturally raise the energy of such a solution, to recover a physicalresult one needs to take into account the amplitude degree of freedom, which will allowfor the presence of dislocations and local changes in the winding number.For these reasons, the minimization will be performed in two dimensions, with thecomplete complex order parameter and in a subspace with a defined winding number. Thefinal result is expected to be similar to what previously considered in the wider contextof phase slip [7] and more specifically in the case of localized phase slip centers [9, 10].Namely, the local strain induced by the perturbation on the phase will reduce the orderparameter amplitude, to the point where a local phase slip event becomes possible. Inmore than one dimension, the boundary between areas with different winding numberwill be marked by structures such as vortices.From this preliminary analysis, the mechanism responsible for the dissipation peakscan be understood: as the tip approaches the surface, it encounters points where theenergies of solutions with different winding number undergo a crossover. At these pointsthe transition between manifolds is not straightforward, due to the mechanism requiredto create the vortices, therefore the oscillations lead to jumps between different manifolds,resulting in hysteresis for the tip and ultimately dissipation.2. – SimulationsTo asses the validity of the proposed mechanism, numerical simulations of the tip-surface interaction were performed. A two-dimensional model is considered, since ittakes into account the relevant effects while keeping the simulation simpler; moreover,the experimental substrate has a quasi–two-dimensional structure, so that volume effectscan be expected to be negligible. Differently from the experimental system [11], theCDW is characterized by a single wave vector Q, leading to a simpler order parameterand a clearer effect. To represent the effect of the tip, the shape of a van der Waalspotential C/r6 is integrated over a conical tip at distance d from the surface. The resultcan be reasonably approximated in the main area under the tip by a Lorentzian curve(5) V (r; d) =V0(d)r2 + σ(d)2,where r is the distance in the plane from the point right below the tip and the parametersare found to scale like V0(d) = V¯ /d and σ(d) = σ¯d2.66 F. PELLEGRINI0.6 0.8 1 1.2 1.4d [nm]-80-60-40-200E [eV] -101-101-101N=0N=1N=2Fig. 1. – Free energy F as a function of tip distance d for subspaces with different winding numberN . Results from simulations on a 201 × 201 grid with parameters (see text) f0 = 10 eV/nm,κ = 1 eV, Q = 2.5 nm−1, V¯ = 9.4 eV · nm, σ = 1.2 nm−1 and boundary conditions ψ0 = i.Insets: charge density ρ (full lines) and phase φ (dashed lines) along the line passing right belowthe tip (indicated by the vertical dashed line) for different winding number N , at the positionsindicated by the dots on the energy curves.Knowing the shape of the perturbation, the total free energy F = F0+FV is minimizednumerically on a square grid of points with spacing much smaller than the character-istic wavelength of the CDW, imposing a constant boundary condition ψ0 on the sidesperpendicular to Q, while setting periodic boundary conditions in the other directionto allow for possible phase jumps. The minimization is carried out through a standardconjugated gradients algorithm [12]. The parameters employed are order of magnitudeestimates of the real parameters, which reproduce the relevant experimental effects in aqualitative fashion.The charge density ρ and phase φ profile along the line passing right below the tip isshown in the insets of fig. 1 for solutions with different winding number N . The positionof the attractive tip is indicated by the vertical dashed line, and the phase slip occurredunder it, leading to and increase in charge density, is clearly visible. Parallel lines farfrom the tip always revert to the N = 0 manifold, through the creation of vortices.The main curves in fig. 1 represent the minimum energy at defined N as a function ofthe distance d. Since the solution with a given winding number lies in a local minimum, itis possible to use the minimization algorithm to find solutions in a certain subspace, evenTHEORY OF CHARGE DENSITY WAVE NON-CONTACT FRICTION 670.8 1 1.2 1.4d [nm]102030405060F [eV/nm]2 1051055 1042.5 1041.25 1046.7 103104105[Hz]5678W [eV/cycle][Hz]Fig. 2. – Force as a function of distance for evolutions with d0 = 1nm, d¯ = 0.4 nm and differentvalues of ω with Γ = 10−7 eV · s. Inset: total work W as a function of oscillation frequency ω.when this is not the global minimum for a specific distance, by starting from a reasonableconfiguration (namely, the minimum at close distance). Two different crossing points canbe seen, which would give rise to two peaks in the experimental dissipation trace. Ofcourse a more complex CDW configuration or different parameters would give rise tomore peaks.To completely justify the validity of the dissipation mechanism, we need to look intothe dynamics of the CDW, to guarantee that the evolution through a crossing pointdoes not lead to immediate relaxation between different manifolds. To do this, the timeevolution of the system was simulated, following the time-dependent Ginzburg-Landauequation [10]:(6) −Γ∂ψ∂t=δFδψ∗.This equation can be interpreted as an overdamped relaxation, with a coefficient Γ, ofthe order parameter towards the equilibrium position. Integrating this equation (througha standard Runge-Kutta algorithm [12]), the force as a function of the distance can becomputed for a tip performing a full oscillation perpendicular to the surface according tothe law d(t) = d0 + d¯ cos(ωt). Figure 2 shows the force evolution during such oscillations68 F. PELLEGRINIat different frequencies. As we can see the tip suffers a hysteresis even at low frequencies,since the decay from one manifold to the other happens far from the crossing point. Thearea of the loops represents directly the dissipated energy per cycle W , as reported inthe inset.3. – ConclusionBased on these simulations, the theory hereby presented is shown to be compatiblewith the experimental findings: it accounts for the existence of multiple peaks, their ap-pearance far away from the surface and their nature being related to the CDW structureof the sample.It is interesting to notice that the vortices appearing in our simulations have beendescribed before in the context of CDW conduction noise [13], where their creation andmovement justifies the phase slip near the CDW boundaries. In this sense, this theorylies in between these macroscopic effects and the simple one-dimensional model of defectpinning and phase slip [7], as is appropriate for a localized but extended perturbation.To conclude, I have presented a mechanism to explain peaks in the dissipation of atip oscillating at specific distances above a CDW surface: these occur around instabilitypoints corresponding to the crossing of energy levels characterized by different windingnumbers. Numerical simulations in a system of reduced complexity support the validityof this mechanism.It would be interesting to investigate the same effect in other systems displaying CDWor even spin density waves, as well as systems where the origin of charge modulation isrelated to the Fermi surface and not to other effects.∗ ∗ ∗The author thanks his collaborators G. E. Santoro and E. Tosatti. He acknowledgesresearch support by SNSF, through SINERGIA Project CRSII2 136287/1, by ERC Ad-vanced Research Grant N. 320796 MODPHYSFRICT, and by MIUR, through PRIN-2010LLKJBX 001,REFERENCES[1] Vanossi A., Manini N., Urbakh M., Zapperi S. and Tosatti E., Rev. Mod. Phys., 85(2013) 529.[2] Langer M., Kisiel M., Pawlak R., Pellegrini F., Santoro G. E., Buzio R., GerbiA., Balakrishnan G., Baratoff A., Tosatti E. and Meyer E., Nat. Mater., 13(2014) 173177.[3] Pellegrini F., Santoro G. E. and Tosatti E., Phys. Rev. B, 89 (2014) 245416.[4] Gru¨ner G., Rev. Mod. Phys., 60 (1988) 1129.[5] Fukuyama H. and Lee P. A., Phys. Rev. B, 17 (1978) 535.[6] Lee P. A. and Rice T. M., Phys. Rev. B, 19 (1979) 3970.[7] Tucker J. R., Phys. Rev. B, 40 (1989) 5447.[8] Tu¨tto˝ I. and Zawadowski A., Phys. Rev. B, 32 (1985) 2449.[9] Maki K., Phys. Lett. A, 202 (1995) 313.[10] Gor’kov L. P., Zh. Eksp. Teor. Fiz., 86 (1984) 1818.[11] McMillan W. L., Phys. Rev. B, 12 (1975) 1187.[12] Press W. H., Teukolsky S. A., Vetterling W. T. and Flannery B. P., in NumericalRecipes: The Art of Scientific Computing, 3rd edition (Cambridge University Press) 2007.[13] Ong N. P., Verma G. and Maki K., Phys. Rev. Lett., 52 (1984) 663.

Theory of charge density wave non-contact friction

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Theory of charge density wave non-contact friction

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