Atomic rheology of gold nanojunctions
Jean Comtet, Antoine Lainé, Antoine Niguès, Lydéric Bocquet, Alessandro Siria
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Atomic rheology of gold nanojunctions
Jean Comtet, Antoine Lain´e, Antoine Nigu`es, Lyd´eric Bocquet, Alessandro Siria*
Laboratoire de Physique Statistique de l’Ecole Normale Sup´erieure, UMR CNRS 8550,PSL Research University, 24 Rue Lhomond 75005 Paris, France * [email protected]
Despite extensive investigations of dissipation and deformation processes inmicro- and nano- sized metallic samples [1–7], the mechanisms at play duringdeformation of systems with ultimate, molecular size remain elusive. Whilemetallic nanojunctions, obtained by stretching metallic wires down to theatomic level are a system of choice to explore atomic scale contacts [5, 8–11],it has not been possible up to now to extract the full equilibrium and out ofequilibrium rheological flow properties of matter at such scales. Here, by usinga quartz-tuning fork based Atomic Force Microscope (TF-AFM), we combineelectrical and rheological measurement on angstr¨om-size gold junctions tostudy the non linear rheology of this model atomic system. By submitting thejunction to increasing sub-nanometric deformations we uncover a transitionfrom a purely elastic regime to a plastic, and eventually to a viscous-likefluidized regime, akin to the rheology of soft yielding materials [12–14], thoughorders of magnitude difference in length scale. The fluidized state furthermorehighlights capillary attraction, as expected for liquid capillary bridges. Thisshear fluidization cannot be captured by classical models of friction betweenatomic planes [15, 16], pointing to unexpected dissipative behavior of defect-free metallic junctions at the ultimate scales. Atomic rheology is therefore apowerful tool to probe the structural reorganization of atomic contacts.
Solid metallic materials in the micron range and below have been shown to exhibit dras-tically different mechanical behavior as compared to their macroscopic counterparts. Suchsize effects take origin in the decreasing density of defect-mediated plastic events [1], suchas occurring during dislocation gliding [2] or twinning [3], as well as an increasing surface tovolume ratio [4]. However, extending measurements of the mechanical response to the ∼ G of the junction, whichvaries approximately in multiples of G = 2 e /h ≈ µ S. Quantization of the conductanceoccurs as the molecular lateral size of the junction leads to ballistic electronic transport, forwhich conductance G is simply proportional to the number N of conductance channels, akato the number of atoms in the cross-section, with G ≈ N · G . This quantization allows adirect readout of the transverse dimension of the junction at the single atom level. Fig. 1cshows the successive thinning of the junction from N = 7, down to N = 1 atoms.To simultaneously probe the mechanical properties of the junction, we further excite theTF-AFM via a piezo dither, with a periodic force F ∗ = F exp( iωt ) [N], leading to sub-nanometric oscillations a ∗ = a exp( iωt + φ ) of the tuning fork and upper gold electrode.The viscoelastic behavior of the gold nanojunction can be characterized by the compleximpedance Z ∗ = F ∗ /a ∗ = F/a · exp( − iφ ) [N/m], where the real part Z ′ = ℜ [ Z ∗ ] and imag-inary part Z ′′ = ℑ [ Z ∗ ] characterize respectively the conservative (elastic) and dissipativeresponse of the junction (See Fig. 1a and Methods section ’Measurement of viscoelasticproperties’).We show in Fig. 1d, the variations of Z ′ and Z ′′ measured with a fixed oscillation am-plitude a = 70 pm, along with the variations of junction conductance during elongation2 A d Gold SubstrateTuning ForkPiezo Dither: F.e iωt
Gold ElectrodeGold Nanojunction a, ω
PLL, ωPID, E ~ a.e i(ωt+φ) IncreasingSeparation N = G / G Z * ( N / m ) N=7 N=4 N=1Z’Z’’
200 pm~ a b c -40 -20 400102030405060 0 20 a ( p m ) δω/2π (Hz)∆Vi Piezoscanner FIG. 1:
FIGURE 1: Experimental Set-Up ( a ) Schematic of the experimental set-up.A quantum point contact, consisting of a nanojunction of gold (red dashed box) is formedbetween a gold electrode, attached to the tuning fork and a gold substrate. A bias ∆ V [V]is applied between the electrode and the substrate, allowing measurements of the junctionconductance G [S]. Gold substrate position is controlled with a piezoscanner. A piezodither excites the tuning fork with an oscillatory force F ∗ = F · e iωt , leading to theoscillation a ∗ = a · e i ( ωt + φ ) of the gold electrode and tuning fork. Two feedback loops tunethe excitation frequency ω [Hz] (Phase-Locked Loop PLL) and excitation force F [N](PID) to systematically excite the tuning fork at resonance and maintain a set oscillationamplitude, allowing direct measurement of the gold nanojunction viscoelastic impedance Z ∗ = F ∗ /a ∗ (see Methods section ’Measurement of viscoelastic properties’). Inset:resonance curve of the free tuning fork in air. ( b ) Schematic representation of the idealizedjunction geometry, for N = G/G ≈
4. The junction is assumed to have a rod-like shapewith height h and surface area A ≈ N πd /
4. ( c ) Typical conductance trace for increasingseparation between the two surfaces. Conductance varies stepwise, in multiples of G . ( d )Simultaneous measurement of the conservative (Z’, black) and dissipative (Z”, red) part ofthe mechanical impedance Z ∗ of the gold junction, here in the low amplitude (elastic)regime. Red arrows indicate transient decrease in stiffness Z ′ and increase in dissipation Z ′′ interpreted as a signature of plastic reorganization of the junction.(Fig. 1c). On each plateau in conductance, the stiffness Z ′ is approximately constant, indi-cating a constant mechanical structure of the junction, while the dissipative modulus Z ′′ isvanishingly small pointing out to the absence of intrinsic dissipation in the structure for suchdeformations. At each change in conductance, we observe transient decrease in stiffness Z ′ and slight increase in dissipation Z ′′ (red arrows, Fig. 1d), which can be interpreted as signa-3ures of plastic reorganization during the transformation between states of different contactsize. Those measurements are consistent with previous measurements on metallic junctions,reporting that elongation is caused by successive elastic and yielding events, concomitantwith changes in the electrical conductance [5, 10, 11, 20].Going beyond this simple picture requires a distinct strategy to probe the full out-of-equilibrium rheological flow properties and dissipative mechanisms at play in the junction.Our methodology consists in maintaining a fixed number of atoms in the cross-sectionthrough a feed-back control of the junction conductance G , and submitting the junctionto increasing shear to probe its viscoelastic flow properties under a wide range of shear rateand shear stress (with shear amplitude ranging from 20 pm up to 1 nm, see Extended DataFig. 5). This approach is similar in spirit, although at radically different length scales, tothe exploration of the mechanical response of foams and emulsions [12, 13], and allows for aquantitative measurement of the rheological response at the atomic scale.As shown in Fig. 2a for a contact number of N = 11 atoms, the fluctuations in thedimensionless conductance trace N = G/G increase at large oscillation amplitude, butthe mean contact conductance and thus the averaged junction geometry remains fixed. InFig. 2b, simultaneous measurements of the elastic ( Z ′ = ℜ ( Z ∗ ), black) and dissipative( Z ′′ = ℑ ( Z ∗ ), red) part of the mechanical impedance Z ∗ [N.m − ] allows to extract the fulllinear and non-linear rheological flow properties of the junction. Three consecutive regimesare highlighted as a function of the oscillation amplitude, from elastic to plastic, to liquid-like behavior. Remarkably, in spite of orders of magnitude of difference in scales, this overallphenomenology echoes similar trends for plastic flows in soft yielding materials [12, 13].At low oscillation amplitude, the gold junction is unperturbed, and we observe a purelyelastic response, characterized by the absence of dissipation in the junction ( Z ′′ ≃
0) and afinite positive stiffness Z ′ > S N = 0 .
01 Hz − / , definedin terms of the distribution of N as S N = σ/ √ ν , with ν [Hz] the sampling frequency).The constant elastic stiffness Z ′ in the elastic regime allows a coarse characterization of thecontact height h (Fig. 1b) which is found typically in the range h ≈ − a for a fixed contact size N , we evidence an abrupt decrease in the stiffness Z ′ and a corresponding increase in the4 * ( N / m ) Oscillation Amplitude a (pm) Elastic
Plastic
Liquid strain ε = a/h ε Y Z ∞ ab
10 11 1210 11 12
150 200 250 300 350100101112N 05101520 150 200 250 300 350100’’Z ’2.5% 5% 7.5% 10% D e n s i t y ( a . u . ) D e n s i t y ( a . u . ) N N
Oscillation Amplitude a (pm) elastic plastic a Y FIG. 2:
FIGURE 2: Atomic Rheology. (a)
Dimensionless conductance trace N = G/G as a function of the oscillation amplitude of the TF-AFM for sampling rate of50 Hz. At the transition between the elastic and plastic regimes of deformation, thecurrent noise increases, while retaining a fixed mean value. Histograms show currentdistribution for the elastic (green) and plastic (red) regimes and the correspondingdistribution width σ . (b) Measurement of the viscoelastic properties of the junction with afixed cross-sectional area. Variation of storage ( Z ′ , black) and loss ( Z ′′ , red) impedance ofthe junction as a function of oscillation amplitude a of the TF-AFM. With increasingoscillation amplitude, we observe a successive transition from elastic (green, constant Z ′ = Z ′ , Z ′′ = 0) to plastic (red, decrease in Z ′ , increase an plateau in Z ′′ = Z ′′∞ ) up to aliquid-like regime showing capillary adhesion (blue, Z ′ < Z ′′ (going from green to red zone in Fig. 2b; note that rheological curves5re completely reversible upon decrease of the shear rate, see Extended Data Fig. 6). Thisdramatic change points to a dissipative reorganisation of the junction under shear, whichis further evidenced by the increase in current fluctuations (Fig. 2a, red histogram withspectral density S N = 0 .
04 Hz − / ). This mechanical response is in striking analogy withthe (non-linear) rheology of soft yielding materials, such as foams and emulsions [12, 13]where it is a signature of the onset of yielding and plasticity. Here this regime occursabove a threshold oscillation amplitude a Y , which we define as the point where Z ′ decreasesand Z ′′∞ increases to half their asymptotic values ( Z ′ ( a Y ) ≈ Z ′ /
2, roughly concomitant to Z ′′ ( a Y ) ≈ Z ′′∞ /
2, alternative definitions giving similar results).Using the quantized variation of the junction conductance with lateral number of atoms( G = N · G ), we now vary – atom by atom – the lateral size of the junction, from sizesof N ≈
30 atoms down to
N ≈
A ≈ N πd / ≈ − . (Extended data Fig. 5). As shown in Fig. 3a, the yieldforce, defined as F Y = Z ′ · a Y , is roughly proportional to the mean number N of atoms ina cross-section (lower axis) or equivalently the cross-sectional area A (upper axis).The corresponding yield stress σ Y = F Y / A is accordingly found to be roughly independentof contact area and of the order of 5 GPa (Fig. 3b, left axis), while the yield strain ǫ Y = a Y /h = σ Y /E gold is found of the order 5-10% (Fig. 3b, right axis).Importantly, those values are much larger than in macroscopic gold samples, for which σ Y ranges from 55 to 200 MPa [21] or in single crystal gold nano-pillars, where σ Y is of theorder of 500 MPa [22]. Such high values of yield stress and yield strain are consistent withthe expected absence of defects in atomic size junctions.As a first modeling approach, one may compare the values of the experimental yield stressto a Frenkel-type of estimate [11, 15, 16], based on the slippage of a perfect crystal (Fig.3c). For slip along the { } plane of a fcc crystal, the resolved shear stress is given by τ max ≈ G/
9. Taking gold shear modulus G = 27 GPa and a Schmid factor m ≈ .
5, we finda yield stress σ Y ≈ τ max /m ≈ σ Y ( G P a ) F Y ( n N ) ε Y
0% 5%15%10%20%0 1 2cross-sectional area A (nm ) 0 1 210 -1 cross-sectional area A (nm )0.5 1.5 0.5 1.5GoldJunction piezo dither+ substrate (excitation) tuning fork (probe) a TF = 60 pmω TF /2π = 31 kHzω S /2π = 1 - 500 kHza S = 0 - 500 pm ττ c FIG. 3:
FIGURE 3: Yielding Threshold. ( a ) Yield force F Y , ( b ) yield stress σ Y (leftaxis) and yield strain ǫ Y (right axis) as a function of cross-sectional atom number N (andsurface area A of the junction, see upper axis). ( c ) Slip in a perfect crystal between twoatomic planes under shear stress τ . ( d ) Schematic of the set-up. An additional piezodither is placed below the substrate, allowing to shear the junction at an additionalfrequency ω s / π , while the junction properties are simultaneously probed by the TF-AFM(see Methods section ’Effect of excitation frequency on the plastic transition’). ( e ) yieldstrain ǫ Y as a function of the additional excitation frequency ω s (respective to theTF-AFM frequency ω TF ) for N = 15. Error bars represent standard deviation and aretaken to 10 % for single valued points.is here induced by this external shear while junction properties are simultaneously probed σ Y ( N ) and ǫ Y ( N ) curves, one may remark that the yieldstrain and yield stress appear to exhibit local maxima for a number of specific values of theconductance channel number N , typically N ≈ , , ,
23. Interestingly these values areclose to the (so-called) “magical” numbers for the more stable gold wires predicted theoret-ically for
N ≈ , , − ,
23 in [24] and observed in Transmission Electron Microscope[25]. This would suggest that the more stable gold wires presents a larger yield strain toenter in a plastic regime. This observation however deserves a dedicated exploration whichwe leave for future work.We now turn to the regime of large deformations (Fig. 2b, red zone). Surprisingly,the dissipative modulus exhibits a plateau in this regime and thus becomes independent ofthe imposed deformation a , while the junction stiffness decreases steadily (Fig. 2b). Thecorresponding dissipative force is accordingly expected to increase linearly with deformation,as F D ≈ Z ′′∞ × a , suggesting a “viscous-like” dissipation, for which dissipation is proportionalto the imposed velocity with a linear response between driving and dissipation.One would accordingly expect F D = η ˙ γ A , with ˙ γ ∼ aω/h ∼ s − the typical shear-rate, A the contact area and η a material viscosity.To further assess these views, we plot in Fig. 4a the friction coefficient F D / ˙ γ – defined as F D / ˙ γ = Z ′′∞ · h/ω – as a function of the contact lateral size N (lower axis) or cross section A (upper axis). This plot confirms that the dissipative force is proportional to the contactarea, allowing us to infer a viscosity η (Fig. 4b), which is found to be roughly constant andof the order of η ∼ · Pa.s at the TF-AFM excitation frequency ω TF / π ≈
31 kHz.This viscous-like regime is completely unexpected for defect-free crystalline systems (Fig.2c), for which plastic flow should occur at constant stress [15, 16] (see Methods section’Comparison with Prandtl-Tomlinson model’ and Extended Data Fig. 7).To get further insights in this viscous-like behavior, one may define a Maxwell time-scale8 M characterizing relaxation of the system as η = G gold · τ M with G gold = 27 GPa the shearmodulus of gold [13]. The corresponding value τ M ∼ µ s is huge as compared to microscopictime scales (typically in the picosecond range), and very close to the excitation time scale ∼ /ω TF = 5 µ s. This therefore suggests that the excitation does fix the relaxation time-scale of the junction under strong deformation, in direct line with the behavior of yieldingmaterials – emulsions, foams or granular materials – where the fluidity (inverse viscosity) isfixed by the excitation time-scale itself [14, 26, 27]. cross-sectional area A (nm )0 5 10 15 20 25 30
FIGURE 4: Liquid-like behavior. ( a ) Dissipative hydrodynamic force F D / ˙ γ ,with ˙ γ = aω/h as a function of cross-sectional atom number N (and surface area of thejunction, see upper axis). ( b ) Viscosity η as a function of N . ( c ) Measured viscosity as afunction of oscillation frequency ω s /ω TF for N = 15. Error bars are standard deviation andare taken to 10 % for single valued points. ( d ) Liquid-like response of the junction at largedeformation, with signatures of both viscous-like dissipation (viscosity η ) and capillaryadhesion (surface tension γ ).9o assess this dependence of the viscous behavior on excitation frequency, we measureas shown in Fig. 3c the viscosity η of the liquefied gold junction induced by an additionalexcitation frequency ω s (see Fig. 3c, Methods section ’Effect of excitation frequency onthe plastic transition’ and Extended Data Fig. 2). As shown in Fig. 4c, the measuredviscosity is found to be roughly independent of excitation frequency for ω s /ω TF <
1, as theresponse of the junction is accordingly fixed by the lowest excitation time 1 /ω TF . For largerexcitation frequency ω s /ω TF >
1, we observe a thinning behavior with η ∼ η TF ( ω s /ω TF ) − α with α ≈ .
28 and η TF ≈ · Pa.s. This thinning behavior further highlights the dominantrole of excitation frequency in the liquid-like dissipative behavior of the junction.Our experiments thus point to unexpected dissipation channels during the deformationof the junction under large strain. Keeping in mind the yielding process in macroscopicsoft materials [13, 14], such fluidization under stress might be due to the collective atomicreconstruction process, potentially favored by the rapid surface diffusion of gold atoms [6,7, 28, 29]. Such additional dissipation channels, completely unexpected for dislocation-free ordered crystalline systems open exciting perspective for the fundamental modeling ofdissipative processes at the atomic scale. Finally, the slight deviation from linearity observedfor
N >
20 in Figs. 3a and 4a suggests a possible transition to more traditional dislocation-based mechanisms as sample volumes becomes sufficiently large.Surprisingly, the liquid-like character of the gold junction is also recovered in the conser-vative elastic response in the form of a negative stiffness under large strain (Fig. 2b, bluezone).This regime is therefore associated with an attractive adhesive response of the junction,which is reminiscent of capillary adhesion of macroscopic capillary bridges [30] (Fig. 4d).This additional signature of the liquid-like behavior of the junction at large oscillation am-plitudes, whose conservative mechanical response becomes dominated by surface effects isfurther confirmed by additional force spectroscopy measurements in vacuum, showing ”jump-to-contact” of the liquified gold (Methods section ’Capillary attraction at large oscillationamplitude’ and Extended Data Fig. 3). Effect of local junction heating on the observed ad-hesive behavior can be discarded by order of magnitude estimates (Methods section ’Energybalance for the shear induced fluidization of the junction’ and Extended Data Fig. 4).Those observations allow us to estimate the surface stress of the liquefied gold meniscus.Using the expression for the adhesive force induced by a perfectly wetting liquid between two10pherical contacts, one get a stiffness Z ′ ≈ − πγ ( R/h ) (Fig. 4d) [30]. Identifying the radius R with that of the cross-sectional area of the bridge (Fig. 2b), we find for the experimentof Fig. 2, R/h ∼ .
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