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Deformation mechanisms, electronic conductance and friction of metallic nanocontacts.
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Deformation mechanisms, electronic conductance and friction of metallic nanocontacts. Adrian P Sutton Close interactions between
experiments
and computer
simulations of metallic contacts have revealed role of mechanical electronic fracture.
instabilities in the evolution of the
conductance
of atomic-scale
contacts
At the same time crucial differences
scales of relaxation processes
pulled to
in the time
that occur in experiments
and in simulations have been highlighted fundamental
the central
questions about modelling
and have raised strategies.
Addresses Department of Materials, Oxford University, Parks Road, Oxford, OX1 3F’H, UK; e-mail:[email protected] Current Opinion in Solid State & Materials Science 1996, t ~627-033 Current Chemistry Ltd ISSN 1359-0266 Abbreviations AFM atomic force microscopy DLC diamond-like carbon FE free electron MCBJ mechanically controllable break junction MD molecular dynamics STM scanning tunnelling microscopy TB tight binding
Introduction Computer simulations of the formation, fracture, sliding, and lubrication of metallic contacts have been motivated by the development of a variety of proximal probe techniques ([l-3]; for reviews, see 14-61). In these experiments a tip of l-100nm radius of curvature is brought into contact with a substrate, under ultra high vacuum (UHV), or in air, or completely immersed in a liquid. These techniques have been used in studies of adhesion, friction, lubrication and fracture of single asperity nanocontacts in different environments. At the same time they have raised fundamental questions about the atomic mechanisms involved in these processes, which computer simulations have sought to address. The first simulations of the formation and fracture of metallic nanocontacts were carried out independently by me and Pethica [7] and by Landman and coworkers [8]. Both simulations modelled the interaction between a tip and a slab to gain insight into the irreversible nature of the formation and fracture of metallic contacts that were observed experimentally. Mechanisms of the sintering of the tip to the substrate and its subsequent fracture during pull-off were described. There are clear similarities between the descriptions of some of these processes [7,8]. In particular, the same mechanism by which the tip elongates during the pull-off, at small
diameters of the constriction, or ‘neck’, was found in both papers. The constriction, undergoes brief structural rearrangements during which it disorders and reorders with the introduction of an additional atomic layer. In between these structural rearrangements, which occur when the tip becomes mechanically unstable, the tip is elastically strained. There are, however, differences in the way the simulations [7] and [8] were set up, notably the boundary conditions and the manner and rate at which the strain was applied [9*]. Here differences are mentioned in [9’] in a provocative manner. These points are discussed carefully in the next section. Over the past four years increasing attention has been given to the electronic conductance of metallic contacts as they are pulled to diameters of atomic dimensions and eventual fracture. Experimentally such studies have been carried out by the mechanically controllable break junction (MCBJ) technique [lo-12,13**,14*,15”], and by stretching a metallic contact in the scanning tunnelling microscope (STM) [16-18,19’,20’*-24.*,25]. A principal goal of this work has been to determine whether the electronic conductance is quantized. The conductance is quantized if it is an integer multiple of the fundamental quantum unit, Go, where Gu=2ez/h, where e is the electronic charge and h is Planck’s constant. The contacts in these experiments are in the ballistic regime with a large splitting between transverse electron levels (0.1-l eV). This is large enough for the presence of discrete levels to be reflected in a quantization of the conductance, provided that the constriction is smooth on the scale of the Fermi wavelength, and provided that the discrete atomic structure of the contact is not such as to cause electron back-scattering. Here it is important to distinguish between two key issues. The first is whether the observed abrupt changes (‘jumps’) in the conductance are caused by structural rearrangements (mechanical instabilities) of the contact, or whether they are caused by the abrupt closing-off of conductance channels during a gradual and continuous thinning down of the contact. The second is whether the values of the conductances at the plateaux between jumps are integer multiples of Go, and thus whether the conductance is quantized. Thus, even if the jumps are caused by mechanical instabilities the conductance may or may not be quantized. The reason why molecular dynamics (MD) simulations have contributed so much to the resolution of these issues is that they produce sequences of atomic coordinates during the pull-off. These coordinates have been used to varying degrees during the pull-off to calculate the electronic conductance
828
Modelling and simulation of solids
[18,19,20”,26,27,2S**,Z9~~], about its variation.
and
to conjecture
[9*,21”]
Finally I mention briefly recent simulations of friction and lubrication of metallic nanocontacts. Some of this work was initially motivated by experimental measurements with the atomic force microscope (AFM) which revealed an atomic-scale stick-slip process [Z]. Early simulations [30-341 modelled contacts sliding at velocities of the order of lOOms-1. With the much greater computing power now available more recent simulations have reduced the sliding velocity to 1 ms-*. Although this sliding velocity is still many orders of magnitude greater than that operating in AFM experiments [Z], there are applications, such as computer disc drives, where sliding velocities are l-30 ms-1. There have also been some interesting new continuum level insights into frictional sliding. One of the main motivating questions for current work on lubrication concerns the breakdown of lubricant molecules during sliding of insulating surfaces, such as diamond and diamond-like carbon (DLC), for the computer disc industry. I briefly touch on this evolving field. Recent models and simulations of friction and lubrication of metallic nanocontacts are discussed briefly.
Simulations of the formation metallic contacts
and fracture of
In the simulations of this section I consider a computational cell containing a tip that is lowered into contact with a slab and is then pulled off. Periodic boundary conditions are applied along all three sides of the computational cell in [7,26,27,28”,29**,33]. This has the advantage that all atoms are treated dynamically, but the disadvantage that cells above and below the contact are in mechanical contact. The three atomic layers in both the slab and tip most remote from the contact are used as grips and kept static in [8,9*,21**], with periodic boundary conditions applied parallel to the slab. This has the advantage that only a two-dimensionally periodic array of contacts is modelled, but the disadvantage of introducing two new interfaces in each cell between dynamic and static atoms (which were absent in [7,26,27,28”,29”,33]). The vertical displacement of the tip, which effects its approach to and retraction from the substrate, is applied in [7,26,27,28”,29”,33] by varying the height of the periodic cell. By contrast it is effected in [8,9*,21”] by rigidly displacing the grip layers. In [7,26,27,28**,29**,33] a vertical strain of SE is imposed homogeneously on all atoms in the periodic cell every tenth time step. In between these strain increments the system evolves dynamically. The strain SE is such that it produces a constant average vertical velocity, v, of the uppermost part of the cell relative to the lower most throughout the simulation. Values of v have varied from 0.4 m s-r [29**] to 3.5 m s-r [28**] to 60 m s-1 [26,27,33] and &~-lo -5 to -4 x 104. In [8,9*,21**] the three rigid layers of the tip are moved in increments of 0.25 A over 500 time
steps, in other words at 16 m s-r towards or away from the slab. The system is allowed to evolve dynamically after each increment until no discernible variations in system properties are observed beyond natural fluctuations [8]. This method of imposing the tip displacements has been described as being adiabatic [9’]. A ‘thermostat’ is usually introduced into the MD equations of motion to prevent the whole system from heating up as a consequence of the work done on it during the pull-off. But this does not prevent local heating at the neck of the contact. A larger strain rate will generate heat more rapidly in the neck. Whether this heat injection is sufficient to melt the neck depends on the melting point, ThI, of the material. It is also expected to depend on the diameter and length of the neck because the Tht of the material in the neck is reduced significantly by the local curvature of the neck profile. Permanent melting of the neck was found [27,33] during the pull-off of a lead tip from a lead substrate at lOms-r at 300K (0.5 Thr). By contrast an iridium tip pulled off an iridium substrate at 60m s-1 at 300K (0.11 T~,I) [26] retained a well defined layered structure (except momentarily through mechanical instabilities where a new layer was introduced), until the neck was reduced to just a few atoms, when one could no longer sensibly speak of a well defined structure anymore. The strain rate also imposes a time scale on the deformation processes in the simulation. If this time scale is less than the characteristic time required for each of these processes to reach completion before the next begins, then the neck region may disorder permanently. In this case the disordering is permanent not because the neck has melted but because it has insufficient time to reorder before the next structural rearrangement begins. For a given volume of disordered material the time taken to reorder will decrease with increasing temperature up to the melting point. But, raising the temperature also increases the length of the neck, and hence the volume of disordered material, because the enhanced atomic mobility produces a smoother neck profile. These two trends with increasing temperature conflict with each other. For example, the relatively short neck of a nickel contact pulled at 3.5 m s-1 at 10K remained ordered [2800], until the neck was reduced to just one or two atoms. The same contact pulled at the same rate at 150K had a longer and smoother neck which disordered permanently once it had been reduced to a size where the conductance was 5-6G,. When the same contact was pulled at the same rate at 250K the neck was slightly longer, but there was sufficient thermal activation to reorder the neck into atomic layers after each instability and before the onset of the next. On the basis of these results, I conclude that whether the neck has an ordered (layered) structure, apart from momentary instabilities, or a permanently melted or disordered structure depends not only on the strain rate but also on the homologous temperature, and on the length and curvature of the neck.
Deformation
mechanisms,
electronic conductance and friction of metallic nanocontacts Sutton
In my opinion there is no guarantee that the mechanisms of contact fracture seen in any MD simulation are the same as those operating in STM or MCBJ experiments. The STM experiments [16-18,19*,20**-24**,25] of contact fracture are usually carried out at tip retraction velocities of between lo-10 and lo-7ms-1, in other words at least 5 (possibly 6) orders of magnitude slower than the slowest MD simulations to date. In recent MCBJ experiments [15”] on gold at room temperature the contacts were initially stretched until the conductance was about lOGo. It was found that the contacts broke in a period of a few seconds without any further stretch, demonstrating that, during the pull-off, contacts of that size, were globally unstable. The absence of discernible variations in system properties beyond natural fluctuations [P] throughout 104s (or less) of an MD simulation, following a stretch of O.ZSA at 16ms-1, is no indication that the model is ‘adiabatic’ on the time scale of such experiments. In an experiment there can be a hierarchy of relaxation mechanisms, spanning a wide range of time scales, all or many of which may be accessible. These mechanisms range from rapid processes that presumably correspond to those seen in the MD simulations, to the much slower creep processes that presumably effected the fracture of the gold contacts in [lS**]. Finally, for the same total number of MD time steps, subjecting the tip-substrate system to an almost constant [7,26,27,2Soo,29”,33] rather than a jerky (8,9*,21**] velocity of pull-off is clearly a less obtrusive and better defined procedure for assessing the influence of the strain rate on the mechanisms of deformation.
Electronic conductance
of nanocontacts
Muller et a/. [lo] reported a step-wise increase in the electronic conductance of a platinum point contact at 1.2K in a vacuum, as the contact was shortened. The conductance plateaux were not at integer multiples of Go, and thus the conductance was not quantized. By adjusting the contact to coincide with one of the conductance jumps they observed two-level fluctuations where the conductance flipped back and forth between the high and low values on either side of the jump. They attributed the conductance jumps, and the two-level fluctuations, to movements of single atoms in the contact [lo]. An extremely useful, exact, single particle, multiple scattering formalism has been developed [35] which enables the conductance of an arbitrary atomic assembly to be calculated in the limit of elastic scattering only. Todorov and I [26] applied this formalism to sequences of atomic coordinates generated by MD simulations of an iridium tip brought into contact with, and subsequently pulled off, an iridium substrate at 60 ms-1 at 300K. We used an orthonormal 1 s tight binding (TB) model to describe the electron hopping through the contact. Although this is not an accurate representation of the electronic structure of iridium, it enables a sufficiently rapid calculation of the conductance to be made, in which
829
the positions of all atoms within the contact feature explicitly. By contrast, some later simulations [18,19,200*] of the electronic conductance, based on MD simulations of contact formation and fracture, replaced the internal atomic structure of the contact by a free electron (FE) jellium. Todorov and I found that there was a one-to-one correspondence between jumps in the electronic conductance and rapid structural rearrangements (mechanical instabilities) of the simulated contact. For example, during the pull-off plateaux in the conductance coincided with elastic stretching of the contact, and sudden drops in the conductance coincided with instabilities where the neck briefly disordered and reordered with the introduction of a new layer [26]. Quantized conductance plateaux have been observed just before contact fracture, in gold contacts, in both low [16] and room [17] temperature STM experiments. STM observations of quantized conductance plateaux in platinum, copper and nickel point contacts at room temperature were reported by Olesen ct al. [18]. In the same paper [18] an FE conductance calculation was presented on a highly idealized geometry; it showed conductance quantization. It was pointed out [14*] that the experimental observation of conductance quantization is far from universal, and that the detailed structure of a contact may destroy the conditions for quantization when the idealized FE model of [18] would predict it. A response to these comments appeared in [19*]. Krans et a/. [13**] displayed histograms of conductance values obtained from (more than 100) conductance traces for sodium and copper by the MCBJ technique at 4.2K. The histogram for sodium is particularly interesting because it shows conductance peaks at 1,3,5 and 6 quantum units, but no peaks at 2, 4 and 7 units. This is exactly the ‘signature’ of conductance quantization that may be expected in cylindrically symmetric constrictions, as discussed by Torres eta/. [36]. Histograms of conductances were also shown by Brandbyge eta/. [20”] for gold contacts pulled in STM. In general it is found that peaks in the conductance occur at the first few quantized values (typically up to 3Go, but occasionally to higher values, as in sodium [13”]), but that the height of the peak decreases and the width increases as the conductance increases. Taken together, these histograms [13”,20**] indicate that favourable conditions for conductance quantization appear only in the final stages of the pull-off, with conductances of just a few Go. Brandbyge et a/. [20”] concluded, on the basis of their STM experiments and simulations of the conductance, that jumps in the conductance are generally associated with mechanical instabilities, in agreement with Todorov and me [26]. Pascual et al. [9*,21”] presented conductance traces obtained at room temperature by scanning tunnelling microscopy of gold nanocontacts up to 400A in length. For contact lengths of about 50 A quantized conductances
030
Modelling
and simulation
of solids
were found. But for lengths of about 4OOA the resistance was found to increase with the exponential of the square of the length, which is consistent with electron localization in the contact, on the basis of the reasonable assumption that the volume of the contact is constant. Dips in the electronic conductance, after each abrupt drop and before the next plateau, were observed. It was conjectured that the dips correlate with the disorder accompanying atomic new layer is introduced; conductance Bratkovsky
calculations
rearrangements that occur when a however, no supporting electronic were
et a/. [28**]carried
provided. out
a comparative
study
of calculations of the electronic conductance of nickel nanocontacts pulled to fracture at lOK, 150K and 250K at a rate of 3.5 ms-1 using FE and TB models of the electronic structure applied to the same contact sequences generated by MD simulations. As in other FE calculations only the shape of the contact entered the calculation of the conductance, whereas the exact atomic structure of the contact was taken into account explicitly in the tight-binding Hamiltonian. Two types of mechanical instability were found. In the early stages of the pull-off, when the contact diameter was relatively large, the instability involved the nucleation and glide of a dislocation on an inclined 111 plane. The dislocation was dissociated into Shockley partials. The first Shockley partial glided completely across the contact, creating a stacking fault in its wake. After a further short period the second Shockley partial glided along the same path and removed the stacking fault. The second kind of instability was seen in the later stages of the pull-off when the constriction was relatively narrow, and was the same process as described in earlier simulations [7,8], involving disordering and reordering of several layers with the introduction of one more layer. For both kinds of instability the final result was that the contact lengthened by one layer, and the constriction was slightly narrower. Dips in the conductance immediately after an instability were found [28**] with the TB mode1 but not with the FE model. They are associated with a finite additional stretch required to create sufficient space for the contact to reorder after an instability. At 10K and 15OK the FE conductance displayed plateaux only at Go and 3Gu. At larger constriction widths the energy splitting between successive transverse modes is reduced, which enables more tunnelling and back-scattering to occur [20**,28**]. Thus, the conductance steps are rounded off at higher conductances. By contrast, at smaller constrictions the energy splitting between successive conductance channels is much greater, which produces sharp conductance steps. An example of resonant tunnelling was also found in the 10K simulation. The TB conductances displayed more plateaux, but they did not, in general, coincide with integer multiples of Go At 250K it was found [ZP] that both the FE and TB conductances
displayed attributed
more plateaux at quantized values. [28**] to the smoother constrictions
the higher
temperature.
This was formed at
The prediction [28”] that conductance quantization should be more evident at higher temperatures has been confirmed experimentally by Muller et a/. [15**]. The MCBJ technique was applied to 72 gold and 78 copper contacts at room temperature. Histograms of the conductances showing pronounced peaks, centred at all integer values up to .SGu, were found. By contrast, in copper at 4.2K peaks occurred only near Go and 3Gu, and smaller peaks at noninteger conductance values [ 13**]. Most recently experiments have appeared [ZP-24**,25] in which the force on the contact and the conductance are measured simultaneously, and in which the jumps in the conductance generally coincide with jumps in the force. These results support the view [26] that the conductance jumps generally occur as a result of atomic rearrangements. In response to these measurements Todorov and I [29**] have carried out simultaneous MD simulations and TB electronic conductance calculations of the pull-off of a gold nanocontact at 1K at 4 ms-1 and 0.4ms-1. We found that every drop in the conductance coincides with an abrupt reduction in the tensile force, and an elongation of the contact by one layer. Current subjects of simulation studies in this field are the temperature dependence of the hysteresis observed [37”] in the conductance in passing back and forth over a jump, and the observed [lo] two-level fluctuations at a jump. In conclusion, there is a growing consensus that jumps in the conductance are generally (but not exclusively) caused by structural changes within the contact arising from mechanical instabilities. The conductance is more likely to be an integer multiple of Go at small conductances, and not at low temperatures. Individual conductance traces, however may show plateaux at noninteger multiples of Go. At higher conductances transmission by tunnelling through the effective one-dimensional barrier of the often eliminate the and back-scattering, constriction, conductance plateaux altogether. It is evident that the interaction between experimental and simulation studies of the electronic conductance of metallic nanocontacts has been close and fertile.
Friction and lubrication
of metallic contacts
Useful reviews on nanotribology have appeared recently [38’,39]. Sorensen et a/. [40*] have extended earlier atomistic simulations [31] of frictional sliding of clean copper contacts significantly. In the sliding of a (111) tip over a (111) surface, of the same orientation and in a [liO] direction, a stick-slip process was observed in which Shockley partial dislocations were nucleated in one corner of the contact and glided across it. Energy was dissipated as heat during each slip phase of the stick-slip process,
Deformation mechanisms, electronic conductance and friction of metallic nanocontaets
and the heat was adsorbed by thermostats positioned as remotely as possible from the sliding interface. Sliding tips with different crystal orientations from the substrate were also modelled, and some were found to slide with an average force of zero in the sliding direction, although this was dependent on the area of the contact as well as the relative orientation of the crystal axes in the tip and substrate. A finite average sliding force always arose when the sliding took place by a stick-slip mechanism. This is consistent with the view that energy dissipation, in the absence of wear, is associated with a stick-slip process. This mechanism of energy dissipation has been compared by Tabor [41] to running a finger nail over the teeth of a comb. For a (100) tip sliding on a (100) substrate, along a close-packed direction, with parallel orientations of the crystal axes in the tip and substrate, it was found [40*] that sliding took place inside the tip on inclined 111 planes, and resulted in transfer of material from the tip to the substrate. This mechanism of junction growth differs somewhat from that found in [31]. It is tempting to think that frictional sliding and grain boundary sliding [42**] are related processes. Indeed Sorensen et al. [40*] compared their results on tips sliding over misoriented substrates with further simulations of the sliding of grain boundaries on the same planes and crystal lattice misorientations. But there is an essential difference, which was seen clearly in the simulations performed by Sprrensen et a/. [40’]. A stress concentration, which is a lIJr singularity in elastic continuum theory, is generated around the periphery of a pinned contact that is subjected to a shear stress. Such a stress concentration cannot arise at a flat grain boundary unless there is some pinning mechanism, such as a triple junction or a second phase particle. (In reality grain boundaries are almost never flat, and stress concentrations do occur.) Dislocations are nucleated at the stress concentration and enter the contact, causing ‘microslip’. Johnson [43**] has used this picture to develop a continuum model for a single asperity contact of the transition from static to kinetic friction, and of the interaction between adhesion and friction. Dislocations at interfaces across which there is a change of crystal structure or orientation, may have quite different Burgers vectors from either crystal lattice [42”]. Such interfacial dislocations have been observed [42’*] at many static homophase and heterophase interfaces, and at some sliding interfaces. But it is unclear whether they play a significant role in frictional sliding. They have been observed [42”] directly in the electron microscope at the contact between a sphere and a substrate, as the sphere rotates about an axis normal to the substrate to minimize the interfacial energy of the contact. But, in frictional sliding the requirements that the dislocations are glissile into the interface (i.e. their motion does not involve diffusion, as in climb), and that they are able to glide out of the interface to avoid obstacles, such as inclusions, leads one to the conclusion that the predominant agents of
frictional sliding crystal lattice.
are probably
screw dislocations
Sutton
831
of either
Gao et a/. [44’] have performed MD simulations of the sliding of two gold (111) surfaces, exposing flat-top pyramidal asperities, in the presence of n-CroH34 (hexadecane) lubricant molecules. The relative sliding velocity was lOms-r, which is comparable with the velocity of a magnetic head above a computer hard disc. All atomic interactions were described by simple empirical models. It was found that as the asperities were brought closer together a layering of the lubricant developed between the asperities, which resulted in an oscillatory frictional force. Severe deformations of the asperities were found even before direct contact between them took place. These deformations were caused by stresses as large as 4 GPa that were transmitted by the lubricant molecules. The mechanisms of degradation of the lubricant remain one of the primary concerns of the magnetic disc industry. Degradation is a process involving chemical change of the lubricant molecules. It can be modelled only by a procedure that provides an adequate description of bond breaking and bond formation processes, and the energy barriers associated with transition states. This is the world of ‘tribochemistry’ and it is where some of the most challenging problems in modelling tribological processes lie. Some first steps in this direction have been taken in [34,45,46].
Conclusions In this review I have given a critical discussion of the manner and rate at which strains are applied to contacts in MD simulations in order to stretch them to fracture. The gulf in time scales of relaxation processes in MD simulations, compared with those in the recent experiments of Muller et a/. [1.5**], raises serious questions about the suitability of MD to model such processes. Nevertheless, there have been some remarkable successes of MD simulations of contact fracture, coupled to electronic conductance calculations. In particular, the result from simulations that jumps in the electronic conductance are generally caused by mechanical instabilities has received strong support from simultaneous experimental measurements of forces and conductances. Also, the prediction by simulations, that the conditions for conductance quantization are more favourable at higher temperatures, has also been confirmed experimentally. Finally, both simulations and experiments find that conditions for quantization of the conductance are more favourable in the final stages of the pull-off than in thicker contacts. In my view, the major challenge in this area will be the development of new modelling techniques that can describe structural rearrangement mechanisms lasting seconds rather than nanoseconds. Recent modelling of frictional sliding of clean metallic contacts and of lubricated metallic junctions was reviewed briefly. In my view there are two major challenges here.
832
Modelling
and simulation
of solids
The first is to relate the work on single asperity contacts to modelling of friction and wear of real engineering surfaces at which discussed
there are multi-asperity contacts. This has been by Stoneham eta/. [47]. The second is to model
chemical changes of surfaces and lubricants during friction. This will involve a modelling strategy in which chemical reactions as well as atomic dynamics can be described satisfactorily.
This paper provides experimental confirmation of the prediction [28**] that conditions for conductance quantization are more favourable at room temperature than at liquid helium temperatures. It also demonstrates the global instability of metallic nanocontacts of less than a certain cross-sectional area. 16.
Agrdit N, Rodrigo JG, Vieira S: Conductance steps and quantization in atomic-size contacts. Phys Rev B 1993, 47:12345-l 2348.
1 7.
Pascual JI, MBndez J, G6mez-Herrero J, Bar6 AM, Garcia N, Binh VT: Quantum contact in gold nanostructures by scanning tunneling microscopy. Phys Rev Left 1993, 71:1852-l 855.
18.
Olesen L, Lsegsgaard E, Stensgaard I, Besenbacher F, Schiotz J, Stoltze P, Jacobsen KW, Narskov JK: Quantized conductance in an atom-sized point contact Phys Rev Lett 1994, 72:2251-2254.
Acknowledgements I am very grateful to Tchavdar Todorov for valuable comments on an earlier draft of this review, and to him and John Pethica for innumerable discussions.
References
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reading
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Blijchl PE, Joachim C, Fisher AJ (Eds.): Computations for the Nanoscale. Dordrecht: Kluwer; 1993. [NATO ASI Series E, vol 240.1
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Pascual JI, Mbndez J, G6mez-Herrero J, Barb AM, Garcia N, Landman U, Luedtke WD, Bogachek E, Cheng HP: Electrical and mechanical properties of metallic nanowires - conductance quantization and localization. J Vat Sci TechnolB 1995, 13:1280-1284. This paper contams provocative remarks about MD simulations, some of which led to the response in this review. 10.
Muller CJ, Van Ruitenbeek JM, De Jongh U: Conductance and supercurrent discontinuities in atomic scale metallic constrictions of variable width. Phys Rev Lett 1992, 69:140-l 43.
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Muller CJ, Van Ruitenbeek JM, De Jongh U: Experimental observation of the transition from weak link to tunnel junction. Physica C 1992,191:485-504.
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13. ..
Krans JM, van Ruitenbeek JM, Fisun W, Yanson IK, de Jongh LJ: The signature of conductance quantization in metallic point contacts. Nature 1995, 375:767-769. This paper presents experimental data on conductance quantization in sodium and copper at 4.2K in the form of histograms, and shows that conditions for quantization are more favourable at lower conductances and in a FE metal. 14. .
Krans JM, Muller CJ, van der Post N, Postma FR, Sutton AP, Todorov TN, van Ruitenbeek JM: Comment on quantized conductance in an atom-sized point contact Phys Rev Lett 1995, 7412146. This comment and the reply [lg.] summarize the controversy over the origin of the conductance jumps and whether the conductance is quantized. 15. ..
Muller CJ, Krans JM, Todorov TN, Reed MA: Quantization effects in the conductance of metallic contacts at room temperature. Phys Rev B 1996, 531022-l 025.
19. .
Olesen L, Laegsgaard E, Stensgaard I, Besenbacher F, Schiatz J, Stoltze P, Jacobsen KW, Nerskov JK: Olesen et a/.. reply. Phys Rev Left 1995, 74:2147. See I41 above. 20. ..
Brandbyge M, Schiatz J, Sorensen MR, Stoltze P, Jacobsen KW, Nerskov JK, Olesen L, Laegsgaard E, Stensgaard I, Besenbacher F: Quantized conductance in atom-sized wires between 2 metals. Phys Rev B 1995, 52:8499-8514. As well as containing experimental data in the form of histograms on conductances in nanowires, this paper gives a full account of FE calculations based on MD simulations, which address the observed form of the experimental histograms. 21. ..
Pascual JI, Mend& J, Gbmez-Herrero J, Bar6 AM, Garcia N, Landman U, Luedtke WD, Bogachek EN, Cheng HP: Properties of metallic nanowires - from conductance quantization to localization. Science 1995, 267:1793-l 795. This paper describes experimental results on electron localization in long nanocontacts. Experimental conductance traces for shorter nanocontacts show quantization and dips after each conductance drop. Similar dips were predicted by TB conductance simulations in [28**]. 22. ..
Agrdit N, Rubio G, Vieira S: Plastic deformation of nanometrescale gold connective necks. Phys Rev Left 1995, 74:3995-3998. The first experimental evidence of the correlation between jumps in the tensile force and jumps in the electronic conductance. The contact diameter was relatively large. 23. ..
Rubio G, Agra’it N, Vieira S: Atomic-sized metallic contacts - mechanical properties and electronic transport, Phys Rev Len 1996, 76:2302-2305. Simultaneous measurements of force and conductance of an atomic-sired gold contact in air, where jumps in the conductance were always due to atomic rearrangements in the contact. 24.
Staider A, Diirig U: Study of yielding mechanics in nanometersized Au contacts. Appl Phys Let! 1996, 68:637-639. gmultaneous measurements of force and conductance of atomic-sized gold contacts in UHV showing correlations between variations in conductance and tensile force. 25.
Staider A, Diirig U: Study of plastic flow in ultrasmall Au contacts. J Vat Sci Technol B 1996, 1411259-l 263.
26.
Todorov TN, Sutton AP: Jumps in electronic conductance due to mechanical instabilities. Phys Rev Leti 1993, 70:2138-2141,
27.
Sutton AP, Todorov TN: Mechanical and electrical properties of metallic contacts at the nanometre scale. J Phys Chem Solids 1994, 55:1169-l 174.
28. ..
Bratkovsky AM, Sutton AP, Todorov TN: Conditions for conductance quantization in realistic models of atomic-scale metallic contacts. Phys Rev B 1995, 525036-5051. The only comparative study of FE and TB calculations of the conductance evolution based on the same MD simulations of nano contact pulling. It is also the only simulation study of the temperature dependence of the mechanical response and conductance of metallic nanocontacts. It explains several experimentally observed features abbut the conductance and predicts others, which have since been confirmed. 29.
Todorov TN, Sutton AP: Force and conductance jumps in atomic-scale metallic contacts. Phys Rev B 1996, in press. ysimulation study which reproduces the experimentally observed correlation between jumps in the tensile force and in the conductance. It also discusses time scales of relaxation processes in nanocontacts. 30.
Landman U, Luedtke WD, Nitzan A: Dynamics of tip-surface interactions in atomic force microscopy. Surf Sci 1989, 2lO:L177-L184.
Deformation
mechanisms,
electronic
conductance
and friction
of metallic
nanocontacts
Sutton
833
31.
Nieminen JA, Sutton AP, Pethica JB: Static junction growth during frictional sliding of metals. Acfa Metal/ Mater 1992, 40:2503-2509.
A careful simulation study of frictional sliding between clean metal surfaces, considering the effect of a crystal lattice misorientation between the tip and substrate.
32.
Nieminen JA. Sutton AP, Pethica JB, Kaeki K: Mechanism of lubrication by a thin solid film on a metal surface. Model Simul Mater Sci Eng 1992, 1:63-90.
41.
33.
Sutton AP, Pethica JB, Rafii-Tabar H, Nieminen JA: Mechanical properties of metals at the nanometre scale. In Electron Theory in Alloy Design. Edited by P&for DG, Cottrell AH. London: The Institute of Materials; 1992:191-233.
42.
Sutton AP, Bafluffi RW: Interfaces in crystalline materials. New York: Oxford University Press; 1995. L integrated texl on the structure, thermodynamics, kinetics and mechanical and electrical properties of interfaces in crystalline materials.
34.
Harrison JA, White CT, Cotton RJ, Brenner DW: Atomistic simulations of friction at sliding diamond interfaces. MRS Bull May 1993, 18:50-53.
43. ..
35.
Todorov TN, Briggs GAD, Sutton AP: Elastic quantum transport through small structures. J Phys-Condens Matter 1993, 52389-2406.
36.
Torres JA, Pascual JI, Saenz JJ: Theory of conduction through narrow constricttons in a three-dimensional electron gas. Phys Rev B 1994,49:16581-l 6584.
37. ..
Kraos JM: Size effects in atomic-scale point contacts [Phd Thesis]. Rijksuniversiteit Leiden 1 QQ6. Den Haag: Cip-data Koninklijke Bibliotheek. . . ..--. (ISBN 90-9008989-6). Rich In Information about MCBJ experiments and results, some of whtch are not published elsewhere.
36. .
Bhushan B, lsraelachvili JN, Landman U: Nanotribology: friction, wear and lubrtcatton at the atomic scale. Nature 1995, 374:607-616. A useful review of some of the experimental and simulation work in this field. 39.
Nanotrfbology.
40. .
Serensen MR, Jacobsen KW, Stoltxe P: Simulations of atomicscale sliding friction. Phys Rev B 1996, 53:2101-2113.
MRS Bull 1993,
18, No. 5.
Tabor D: Friction as a dissipative process. In Fundamentals of Friction: Macroscopic and Microscopic Processes. Edited by Singer IL, Pollock HM. Dordrecht: Kluwer; 1992:3-24.
Johnson KL: A continuum mechanics model of adhesion and friction in a single asperity contact In Micro/nanotribo/ogy and Its Applications. Edited by B Bhushan. Dordrecht: Ktuwer; 1997, in press. A continuum theory of a single asperity contact in which the transition from static to kinetic friction is treated, together with the interaction between friction and adhesion. 44. .
Gao JP, Luedtke WD, Landmao U: Nano-elastohydrodynamics - structure, dynamics, and flow in nonuntfonrr lubricatad junctions. Science 1995, 270:605-608. Using simple empirical potentials these simulations reveal interesting structural changes within an aikane lubricant. The changes have considerable consequences for the friction and wear of soft metallic (gold) asperities. 45.
Ramos MMD: Theory of processes at surfaces and interfacas [PhD Thesisl. oxford: University of Oxford; 1 Q92.
46.
Godwin PD, Horafield AP, Stoneham AM, Bull SJ, Ford U, Harker AH, Pettifor DG, Sutton AP: Computational materials synthesis Ill: synthesis of hydrogenated amorphous carbon from molecular precursors. Phys Rev 6 1 gQ6, in press.
47
Stoneham AM, Ramos MMD, Sutton AP: How do they stick together. The statics and dynamics of interfaces. Phil Mag A 1993, 67:797-811.
Deformation mechanisms, electronic conductance and friction of metallic nanocontacts. Adrian P Sutton Close interactions between
experiments
and computer
simulations of metallic contacts have revealed role of mechanical electronic fracture.
instabilities in the evolution of the
conductance
of atomic-scale
contacts
At the same time crucial differences
scales of relaxation processes
pulled to
in the time
that occur in experiments
and in simulations have been highlighted fundamental
the central
questions about modelling
and have raised strategies.
Addresses Department of Materials, Oxford University, Parks Road, Oxford, OX1 3F’H, UK; e-mail:[email protected] Current Opinion in Solid State & Materials Science 1996, t ~627-033 Current Chemistry Ltd ISSN 1359-0266 Abbreviations AFM atomic force microscopy DLC diamond-like carbon FE free electron MCBJ mechanically controllable break junction MD molecular dynamics STM scanning tunnelling microscopy TB tight binding
Introduction Computer simulations of the formation, fracture, sliding, and lubrication of metallic contacts have been motivated by the development of a variety of proximal probe techniques ([l-3]; for reviews, see 14-61). In these experiments a tip of l-100nm radius of curvature is brought into contact with a substrate, under ultra high vacuum (UHV), or in air, or completely immersed in a liquid. These techniques have been used in studies of adhesion, friction, lubrication and fracture of single asperity nanocontacts in different environments. At the same time they have raised fundamental questions about the atomic mechanisms involved in these processes, which computer simulations have sought to address. The first simulations of the formation and fracture of metallic nanocontacts were carried out independently by me and Pethica [7] and by Landman and coworkers [8]. Both simulations modelled the interaction between a tip and a slab to gain insight into the irreversible nature of the formation and fracture of metallic contacts that were observed experimentally. Mechanisms of the sintering of the tip to the substrate and its subsequent fracture during pull-off were described. There are clear similarities between the descriptions of some of these processes [7,8]. In particular, the same mechanism by which the tip elongates during the pull-off, at small
diameters of the constriction, or ‘neck’, was found in both papers. The constriction, undergoes brief structural rearrangements during which it disorders and reorders with the introduction of an additional atomic layer. In between these structural rearrangements, which occur when the tip becomes mechanically unstable, the tip is elastically strained. There are, however, differences in the way the simulations [7] and [8] were set up, notably the boundary conditions and the manner and rate at which the strain was applied [9*]. Here differences are mentioned in [9’] in a provocative manner. These points are discussed carefully in the next section. Over the past four years increasing attention has been given to the electronic conductance of metallic contacts as they are pulled to diameters of atomic dimensions and eventual fracture. Experimentally such studies have been carried out by the mechanically controllable break junction (MCBJ) technique [lo-12,13**,14*,15”], and by stretching a metallic contact in the scanning tunnelling microscope (STM) [16-18,19’,20’*-24.*,25]. A principal goal of this work has been to determine whether the electronic conductance is quantized. The conductance is quantized if it is an integer multiple of the fundamental quantum unit, Go, where Gu=2ez/h, where e is the electronic charge and h is Planck’s constant. The contacts in these experiments are in the ballistic regime with a large splitting between transverse electron levels (0.1-l eV). This is large enough for the presence of discrete levels to be reflected in a quantization of the conductance, provided that the constriction is smooth on the scale of the Fermi wavelength, and provided that the discrete atomic structure of the contact is not such as to cause electron back-scattering. Here it is important to distinguish between two key issues. The first is whether the observed abrupt changes (‘jumps’) in the conductance are caused by structural rearrangements (mechanical instabilities) of the contact, or whether they are caused by the abrupt closing-off of conductance channels during a gradual and continuous thinning down of the contact. The second is whether the values of the conductances at the plateaux between jumps are integer multiples of Go, and thus whether the conductance is quantized. Thus, even if the jumps are caused by mechanical instabilities the conductance may or may not be quantized. The reason why molecular dynamics (MD) simulations have contributed so much to the resolution of these issues is that they produce sequences of atomic coordinates during the pull-off. These coordinates have been used to varying degrees during the pull-off to calculate the electronic conductance
828
Modelling and simulation of solids
[18,19,20”,26,27,2S**,Z9~~], about its variation.
and
to conjecture
[9*,21”]
Finally I mention briefly recent simulations of friction and lubrication of metallic nanocontacts. Some of this work was initially motivated by experimental measurements with the atomic force microscope (AFM) which revealed an atomic-scale stick-slip process [Z]. Early simulations [30-341 modelled contacts sliding at velocities of the order of lOOms-1. With the much greater computing power now available more recent simulations have reduced the sliding velocity to 1 ms-*. Although this sliding velocity is still many orders of magnitude greater than that operating in AFM experiments [Z], there are applications, such as computer disc drives, where sliding velocities are l-30 ms-1. There have also been some interesting new continuum level insights into frictional sliding. One of the main motivating questions for current work on lubrication concerns the breakdown of lubricant molecules during sliding of insulating surfaces, such as diamond and diamond-like carbon (DLC), for the computer disc industry. I briefly touch on this evolving field. Recent models and simulations of friction and lubrication of metallic nanocontacts are discussed briefly.
Simulations of the formation metallic contacts
and fracture of
In the simulations of this section I consider a computational cell containing a tip that is lowered into contact with a slab and is then pulled off. Periodic boundary conditions are applied along all three sides of the computational cell in [7,26,27,28”,29**,33]. This has the advantage that all atoms are treated dynamically, but the disadvantage that cells above and below the contact are in mechanical contact. The three atomic layers in both the slab and tip most remote from the contact are used as grips and kept static in [8,9*,21**], with periodic boundary conditions applied parallel to the slab. This has the advantage that only a two-dimensionally periodic array of contacts is modelled, but the disadvantage of introducing two new interfaces in each cell between dynamic and static atoms (which were absent in [7,26,27,28”,29”,33]). The vertical displacement of the tip, which effects its approach to and retraction from the substrate, is applied in [7,26,27,28”,29”,33] by varying the height of the periodic cell. By contrast it is effected in [8,9*,21”] by rigidly displacing the grip layers. In [7,26,27,28**,29**,33] a vertical strain of SE is imposed homogeneously on all atoms in the periodic cell every tenth time step. In between these strain increments the system evolves dynamically. The strain SE is such that it produces a constant average vertical velocity, v, of the uppermost part of the cell relative to the lower most throughout the simulation. Values of v have varied from 0.4 m s-r [29**] to 3.5 m s-r [28**] to 60 m s-1 [26,27,33] and &~-lo -5 to -4 x 104. In [8,9*,21**] the three rigid layers of the tip are moved in increments of 0.25 A over 500 time
steps, in other words at 16 m s-r towards or away from the slab. The system is allowed to evolve dynamically after each increment until no discernible variations in system properties are observed beyond natural fluctuations [8]. This method of imposing the tip displacements has been described as being adiabatic [9’]. A ‘thermostat’ is usually introduced into the MD equations of motion to prevent the whole system from heating up as a consequence of the work done on it during the pull-off. But this does not prevent local heating at the neck of the contact. A larger strain rate will generate heat more rapidly in the neck. Whether this heat injection is sufficient to melt the neck depends on the melting point, ThI, of the material. It is also expected to depend on the diameter and length of the neck because the Tht of the material in the neck is reduced significantly by the local curvature of the neck profile. Permanent melting of the neck was found [27,33] during the pull-off of a lead tip from a lead substrate at lOms-r at 300K (0.5 Thr). By contrast an iridium tip pulled off an iridium substrate at 60m s-1 at 300K (0.11 T~,I) [26] retained a well defined layered structure (except momentarily through mechanical instabilities where a new layer was introduced), until the neck was reduced to just a few atoms, when one could no longer sensibly speak of a well defined structure anymore. The strain rate also imposes a time scale on the deformation processes in the simulation. If this time scale is less than the characteristic time required for each of these processes to reach completion before the next begins, then the neck region may disorder permanently. In this case the disordering is permanent not because the neck has melted but because it has insufficient time to reorder before the next structural rearrangement begins. For a given volume of disordered material the time taken to reorder will decrease with increasing temperature up to the melting point. But, raising the temperature also increases the length of the neck, and hence the volume of disordered material, because the enhanced atomic mobility produces a smoother neck profile. These two trends with increasing temperature conflict with each other. For example, the relatively short neck of a nickel contact pulled at 3.5 m s-1 at 10K remained ordered [2800], until the neck was reduced to just one or two atoms. The same contact pulled at the same rate at 150K had a longer and smoother neck which disordered permanently once it had been reduced to a size where the conductance was 5-6G,. When the same contact was pulled at the same rate at 250K the neck was slightly longer, but there was sufficient thermal activation to reorder the neck into atomic layers after each instability and before the onset of the next. On the basis of these results, I conclude that whether the neck has an ordered (layered) structure, apart from momentary instabilities, or a permanently melted or disordered structure depends not only on the strain rate but also on the homologous temperature, and on the length and curvature of the neck.
Deformation
mechanisms,
electronic conductance and friction of metallic nanocontacts Sutton
In my opinion there is no guarantee that the mechanisms of contact fracture seen in any MD simulation are the same as those operating in STM or MCBJ experiments. The STM experiments [16-18,19*,20**-24**,25] of contact fracture are usually carried out at tip retraction velocities of between lo-10 and lo-7ms-1, in other words at least 5 (possibly 6) orders of magnitude slower than the slowest MD simulations to date. In recent MCBJ experiments [15”] on gold at room temperature the contacts were initially stretched until the conductance was about lOGo. It was found that the contacts broke in a period of a few seconds without any further stretch, demonstrating that, during the pull-off, contacts of that size, were globally unstable. The absence of discernible variations in system properties beyond natural fluctuations [P] throughout 104s (or less) of an MD simulation, following a stretch of O.ZSA at 16ms-1, is no indication that the model is ‘adiabatic’ on the time scale of such experiments. In an experiment there can be a hierarchy of relaxation mechanisms, spanning a wide range of time scales, all or many of which may be accessible. These mechanisms range from rapid processes that presumably correspond to those seen in the MD simulations, to the much slower creep processes that presumably effected the fracture of the gold contacts in [lS**]. Finally, for the same total number of MD time steps, subjecting the tip-substrate system to an almost constant [7,26,27,2Soo,29”,33] rather than a jerky (8,9*,21**] velocity of pull-off is clearly a less obtrusive and better defined procedure for assessing the influence of the strain rate on the mechanisms of deformation.
Electronic conductance
of nanocontacts
Muller et a/. [lo] reported a step-wise increase in the electronic conductance of a platinum point contact at 1.2K in a vacuum, as the contact was shortened. The conductance plateaux were not at integer multiples of Go, and thus the conductance was not quantized. By adjusting the contact to coincide with one of the conductance jumps they observed two-level fluctuations where the conductance flipped back and forth between the high and low values on either side of the jump. They attributed the conductance jumps, and the two-level fluctuations, to movements of single atoms in the contact [lo]. An extremely useful, exact, single particle, multiple scattering formalism has been developed [35] which enables the conductance of an arbitrary atomic assembly to be calculated in the limit of elastic scattering only. Todorov and I [26] applied this formalism to sequences of atomic coordinates generated by MD simulations of an iridium tip brought into contact with, and subsequently pulled off, an iridium substrate at 60 ms-1 at 300K. We used an orthonormal 1 s tight binding (TB) model to describe the electron hopping through the contact. Although this is not an accurate representation of the electronic structure of iridium, it enables a sufficiently rapid calculation of the conductance to be made, in which
829
the positions of all atoms within the contact feature explicitly. By contrast, some later simulations [18,19,200*] of the electronic conductance, based on MD simulations of contact formation and fracture, replaced the internal atomic structure of the contact by a free electron (FE) jellium. Todorov and I found that there was a one-to-one correspondence between jumps in the electronic conductance and rapid structural rearrangements (mechanical instabilities) of the simulated contact. For example, during the pull-off plateaux in the conductance coincided with elastic stretching of the contact, and sudden drops in the conductance coincided with instabilities where the neck briefly disordered and reordered with the introduction of a new layer [26]. Quantized conductance plateaux have been observed just before contact fracture, in gold contacts, in both low [16] and room [17] temperature STM experiments. STM observations of quantized conductance plateaux in platinum, copper and nickel point contacts at room temperature were reported by Olesen ct al. [18]. In the same paper [18] an FE conductance calculation was presented on a highly idealized geometry; it showed conductance quantization. It was pointed out [14*] that the experimental observation of conductance quantization is far from universal, and that the detailed structure of a contact may destroy the conditions for quantization when the idealized FE model of [18] would predict it. A response to these comments appeared in [19*]. Krans et a/. [13**] displayed histograms of conductance values obtained from (more than 100) conductance traces for sodium and copper by the MCBJ technique at 4.2K. The histogram for sodium is particularly interesting because it shows conductance peaks at 1,3,5 and 6 quantum units, but no peaks at 2, 4 and 7 units. This is exactly the ‘signature’ of conductance quantization that may be expected in cylindrically symmetric constrictions, as discussed by Torres eta/. [36]. Histograms of conductances were also shown by Brandbyge eta/. [20”] for gold contacts pulled in STM. In general it is found that peaks in the conductance occur at the first few quantized values (typically up to 3Go, but occasionally to higher values, as in sodium [13”]), but that the height of the peak decreases and the width increases as the conductance increases. Taken together, these histograms [13”,20**] indicate that favourable conditions for conductance quantization appear only in the final stages of the pull-off, with conductances of just a few Go. Brandbyge et a/. [20”] concluded, on the basis of their STM experiments and simulations of the conductance, that jumps in the conductance are generally associated with mechanical instabilities, in agreement with Todorov and me [26]. Pascual et al. [9*,21”] presented conductance traces obtained at room temperature by scanning tunnelling microscopy of gold nanocontacts up to 400A in length. For contact lengths of about 50 A quantized conductances
030
Modelling
and simulation
of solids
were found. But for lengths of about 4OOA the resistance was found to increase with the exponential of the square of the length, which is consistent with electron localization in the contact, on the basis of the reasonable assumption that the volume of the contact is constant. Dips in the electronic conductance, after each abrupt drop and before the next plateau, were observed. It was conjectured that the dips correlate with the disorder accompanying atomic new layer is introduced; conductance Bratkovsky
calculations
rearrangements that occur when a however, no supporting electronic were
et a/. [28**]carried
provided. out
a comparative
study
of calculations of the electronic conductance of nickel nanocontacts pulled to fracture at lOK, 150K and 250K at a rate of 3.5 ms-1 using FE and TB models of the electronic structure applied to the same contact sequences generated by MD simulations. As in other FE calculations only the shape of the contact entered the calculation of the conductance, whereas the exact atomic structure of the contact was taken into account explicitly in the tight-binding Hamiltonian. Two types of mechanical instability were found. In the early stages of the pull-off, when the contact diameter was relatively large, the instability involved the nucleation and glide of a dislocation on an inclined 111 plane. The dislocation was dissociated into Shockley partials. The first Shockley partial glided completely across the contact, creating a stacking fault in its wake. After a further short period the second Shockley partial glided along the same path and removed the stacking fault. The second kind of instability was seen in the later stages of the pull-off when the constriction was relatively narrow, and was the same process as described in earlier simulations [7,8], involving disordering and reordering of several layers with the introduction of one more layer. For both kinds of instability the final result was that the contact lengthened by one layer, and the constriction was slightly narrower. Dips in the conductance immediately after an instability were found [28**] with the TB mode1 but not with the FE model. They are associated with a finite additional stretch required to create sufficient space for the contact to reorder after an instability. At 10K and 15OK the FE conductance displayed plateaux only at Go and 3Gu. At larger constriction widths the energy splitting between successive transverse modes is reduced, which enables more tunnelling and back-scattering to occur [20**,28**]. Thus, the conductance steps are rounded off at higher conductances. By contrast, at smaller constrictions the energy splitting between successive conductance channels is much greater, which produces sharp conductance steps. An example of resonant tunnelling was also found in the 10K simulation. The TB conductances displayed more plateaux, but they did not, in general, coincide with integer multiples of Go At 250K it was found [ZP] that both the FE and TB conductances
displayed attributed
more plateaux at quantized values. [28**] to the smoother constrictions
the higher
temperature.
This was formed at
The prediction [28”] that conductance quantization should be more evident at higher temperatures has been confirmed experimentally by Muller et a/. [15**]. The MCBJ technique was applied to 72 gold and 78 copper contacts at room temperature. Histograms of the conductances showing pronounced peaks, centred at all integer values up to .SGu, were found. By contrast, in copper at 4.2K peaks occurred only near Go and 3Gu, and smaller peaks at noninteger conductance values [ 13**]. Most recently experiments have appeared [ZP-24**,25] in which the force on the contact and the conductance are measured simultaneously, and in which the jumps in the conductance generally coincide with jumps in the force. These results support the view [26] that the conductance jumps generally occur as a result of atomic rearrangements. In response to these measurements Todorov and I [29**] have carried out simultaneous MD simulations and TB electronic conductance calculations of the pull-off of a gold nanocontact at 1K at 4 ms-1 and 0.4ms-1. We found that every drop in the conductance coincides with an abrupt reduction in the tensile force, and an elongation of the contact by one layer. Current subjects of simulation studies in this field are the temperature dependence of the hysteresis observed [37”] in the conductance in passing back and forth over a jump, and the observed [lo] two-level fluctuations at a jump. In conclusion, there is a growing consensus that jumps in the conductance are generally (but not exclusively) caused by structural changes within the contact arising from mechanical instabilities. The conductance is more likely to be an integer multiple of Go at small conductances, and not at low temperatures. Individual conductance traces, however may show plateaux at noninteger multiples of Go. At higher conductances transmission by tunnelling through the effective one-dimensional barrier of the often eliminate the and back-scattering, constriction, conductance plateaux altogether. It is evident that the interaction between experimental and simulation studies of the electronic conductance of metallic nanocontacts has been close and fertile.
Friction and lubrication
of metallic contacts
Useful reviews on nanotribology have appeared recently [38’,39]. Sorensen et a/. [40*] have extended earlier atomistic simulations [31] of frictional sliding of clean copper contacts significantly. In the sliding of a (111) tip over a (111) surface, of the same orientation and in a [liO] direction, a stick-slip process was observed in which Shockley partial dislocations were nucleated in one corner of the contact and glided across it. Energy was dissipated as heat during each slip phase of the stick-slip process,
Deformation mechanisms, electronic conductance and friction of metallic nanocontaets
and the heat was adsorbed by thermostats positioned as remotely as possible from the sliding interface. Sliding tips with different crystal orientations from the substrate were also modelled, and some were found to slide with an average force of zero in the sliding direction, although this was dependent on the area of the contact as well as the relative orientation of the crystal axes in the tip and substrate. A finite average sliding force always arose when the sliding took place by a stick-slip mechanism. This is consistent with the view that energy dissipation, in the absence of wear, is associated with a stick-slip process. This mechanism of energy dissipation has been compared by Tabor [41] to running a finger nail over the teeth of a comb. For a (100) tip sliding on a (100) substrate, along a close-packed direction, with parallel orientations of the crystal axes in the tip and substrate, it was found [40*] that sliding took place inside the tip on inclined 111 planes, and resulted in transfer of material from the tip to the substrate. This mechanism of junction growth differs somewhat from that found in [31]. It is tempting to think that frictional sliding and grain boundary sliding [42**] are related processes. Indeed Sorensen et al. [40*] compared their results on tips sliding over misoriented substrates with further simulations of the sliding of grain boundaries on the same planes and crystal lattice misorientations. But there is an essential difference, which was seen clearly in the simulations performed by Sprrensen et a/. [40’]. A stress concentration, which is a lIJr singularity in elastic continuum theory, is generated around the periphery of a pinned contact that is subjected to a shear stress. Such a stress concentration cannot arise at a flat grain boundary unless there is some pinning mechanism, such as a triple junction or a second phase particle. (In reality grain boundaries are almost never flat, and stress concentrations do occur.) Dislocations are nucleated at the stress concentration and enter the contact, causing ‘microslip’. Johnson [43**] has used this picture to develop a continuum model for a single asperity contact of the transition from static to kinetic friction, and of the interaction between adhesion and friction. Dislocations at interfaces across which there is a change of crystal structure or orientation, may have quite different Burgers vectors from either crystal lattice [42”]. Such interfacial dislocations have been observed [42’*] at many static homophase and heterophase interfaces, and at some sliding interfaces. But it is unclear whether they play a significant role in frictional sliding. They have been observed [42”] directly in the electron microscope at the contact between a sphere and a substrate, as the sphere rotates about an axis normal to the substrate to minimize the interfacial energy of the contact. But, in frictional sliding the requirements that the dislocations are glissile into the interface (i.e. their motion does not involve diffusion, as in climb), and that they are able to glide out of the interface to avoid obstacles, such as inclusions, leads one to the conclusion that the predominant agents of
frictional sliding crystal lattice.
are probably
screw dislocations
Sutton
831
of either
Gao et a/. [44’] have performed MD simulations of the sliding of two gold (111) surfaces, exposing flat-top pyramidal asperities, in the presence of n-CroH34 (hexadecane) lubricant molecules. The relative sliding velocity was lOms-r, which is comparable with the velocity of a magnetic head above a computer hard disc. All atomic interactions were described by simple empirical models. It was found that as the asperities were brought closer together a layering of the lubricant developed between the asperities, which resulted in an oscillatory frictional force. Severe deformations of the asperities were found even before direct contact between them took place. These deformations were caused by stresses as large as 4 GPa that were transmitted by the lubricant molecules. The mechanisms of degradation of the lubricant remain one of the primary concerns of the magnetic disc industry. Degradation is a process involving chemical change of the lubricant molecules. It can be modelled only by a procedure that provides an adequate description of bond breaking and bond formation processes, and the energy barriers associated with transition states. This is the world of ‘tribochemistry’ and it is where some of the most challenging problems in modelling tribological processes lie. Some first steps in this direction have been taken in [34,45,46].
Conclusions In this review I have given a critical discussion of the manner and rate at which strains are applied to contacts in MD simulations in order to stretch them to fracture. The gulf in time scales of relaxation processes in MD simulations, compared with those in the recent experiments of Muller et a/. [1.5**], raises serious questions about the suitability of MD to model such processes. Nevertheless, there have been some remarkable successes of MD simulations of contact fracture, coupled to electronic conductance calculations. In particular, the result from simulations that jumps in the electronic conductance are generally caused by mechanical instabilities has received strong support from simultaneous experimental measurements of forces and conductances. Also, the prediction by simulations, that the conditions for conductance quantization are more favourable at higher temperatures, has also been confirmed experimentally. Finally, both simulations and experiments find that conditions for quantization of the conductance are more favourable in the final stages of the pull-off than in thicker contacts. In my view, the major challenge in this area will be the development of new modelling techniques that can describe structural rearrangement mechanisms lasting seconds rather than nanoseconds. Recent modelling of frictional sliding of clean metallic contacts and of lubricated metallic junctions was reviewed briefly. In my view there are two major challenges here.
832
Modelling
and simulation
of solids
The first is to relate the work on single asperity contacts to modelling of friction and wear of real engineering surfaces at which discussed
there are multi-asperity contacts. This has been by Stoneham eta/. [47]. The second is to model
chemical changes of surfaces and lubricants during friction. This will involve a modelling strategy in which chemical reactions as well as atomic dynamics can be described satisfactorily.
This paper provides experimental confirmation of the prediction [28**] that conditions for conductance quantization are more favourable at room temperature than at liquid helium temperatures. It also demonstrates the global instability of metallic nanocontacts of less than a certain cross-sectional area. 16.
Agrdit N, Rodrigo JG, Vieira S: Conductance steps and quantization in atomic-size contacts. Phys Rev B 1993, 47:12345-l 2348.
1 7.
Pascual JI, MBndez J, G6mez-Herrero J, Bar6 AM, Garcia N, Binh VT: Quantum contact in gold nanostructures by scanning tunneling microscopy. Phys Rev Left 1993, 71:1852-l 855.
18.
Olesen L, Lsegsgaard E, Stensgaard I, Besenbacher F, Schiotz J, Stoltze P, Jacobsen KW, Narskov JK: Quantized conductance in an atom-sized point contact Phys Rev Lett 1994, 72:2251-2254.
Acknowledgements I am very grateful to Tchavdar Todorov for valuable comments on an earlier draft of this review, and to him and John Pethica for innumerable discussions.
References
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Mate CM, McClelland GM, Erlandsson R, Chiang S: Atomic scale friction of a tungsten tip on a graphite surface. Phys Rev Lett 1987, 59:1942-l 945.
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Diirig U. Ziiger 0, Pohl DW: Observation of metallic adhesion using the scanning tunneling microscope. Phys Rev Left 1990, 65:349-352.
4.
Gijntherodt H-J, Anselmetti D, Meyer E (Eds): Forces in Scanning Probe Methods. Dordrecht: Kluwer; 1995. [NATO ASI Series E, vol 286.)
5.
Gewirth AA, Siegenthaler H (Eds.]: Nanoscale Probes of the Solid/Liquid Interface. Dordrecht: Kluwer; 1995.[NATO ASI Series E, vol 288.1
6.
Blijchl PE, Joachim C, Fisher AJ (Eds.): Computations for the Nanoscale. Dordrecht: Kluwer; 1993. [NATO ASI Series E, vol 240.1
7.
Sutton AP, Pethica JB: Inelastic flow processes in nanometre volumes of solids. J Phys-Condens Matier 1990, 25317-5326.
8.
Landman U, Luedtke WD, Burnham NA, Colton RJ: Atomistic mechanisms and dynamics of adhesion, nanoindentation and fracture. Science 1990, 248:454-461.
9. .
Pascual JI, Mbndez J, G6mez-Herrero J, Barb AM, Garcia N, Landman U, Luedtke WD, Bogachek E, Cheng HP: Electrical and mechanical properties of metallic nanowires - conductance quantization and localization. J Vat Sci TechnolB 1995, 13:1280-1284. This paper contams provocative remarks about MD simulations, some of which led to the response in this review. 10.
Muller CJ, Van Ruitenbeek JM, De Jongh U: Conductance and supercurrent discontinuities in atomic scale metallic constrictions of variable width. Phys Rev Lett 1992, 69:140-l 43.
11.
Muller CJ, Van Ruitenbeek JM, De Jongh U: Experimental observation of the transition from weak link to tunnel junction. Physica C 1992,191:485-504.
12.
Krans JM, Muller CJ, Yanson IK, Govaert Th CM, Hesper R, van Ruitenbeek JM: One atom point contacts. Phys Rev B 1993, 48:14721-l 4724.
13. ..
Krans JM, van Ruitenbeek JM, Fisun W, Yanson IK, de Jongh LJ: The signature of conductance quantization in metallic point contacts. Nature 1995, 375:767-769. This paper presents experimental data on conductance quantization in sodium and copper at 4.2K in the form of histograms, and shows that conditions for quantization are more favourable at lower conductances and in a FE metal. 14. .
Krans JM, Muller CJ, van der Post N, Postma FR, Sutton AP, Todorov TN, van Ruitenbeek JM: Comment on quantized conductance in an atom-sized point contact Phys Rev Lett 1995, 7412146. This comment and the reply [lg.] summarize the controversy over the origin of the conductance jumps and whether the conductance is quantized. 15. ..
Muller CJ, Krans JM, Todorov TN, Reed MA: Quantization effects in the conductance of metallic contacts at room temperature. Phys Rev B 1996, 531022-l 025.
19. .
Olesen L, Laegsgaard E, Stensgaard I, Besenbacher F, Schiatz J, Stoltze P, Jacobsen KW, Nerskov JK: Olesen et a/.. reply. Phys Rev Left 1995, 74:2147. See I41 above. 20. ..
Brandbyge M, Schiatz J, Sorensen MR, Stoltze P, Jacobsen KW, Nerskov JK, Olesen L, Laegsgaard E, Stensgaard I, Besenbacher F: Quantized conductance in atom-sized wires between 2 metals. Phys Rev B 1995, 52:8499-8514. As well as containing experimental data in the form of histograms on conductances in nanowires, this paper gives a full account of FE calculations based on MD simulations, which address the observed form of the experimental histograms. 21. ..
Pascual JI, Mend& J, Gbmez-Herrero J, Bar6 AM, Garcia N, Landman U, Luedtke WD, Bogachek EN, Cheng HP: Properties of metallic nanowires - from conductance quantization to localization. Science 1995, 267:1793-l 795. This paper describes experimental results on electron localization in long nanocontacts. Experimental conductance traces for shorter nanocontacts show quantization and dips after each conductance drop. Similar dips were predicted by TB conductance simulations in [28**]. 22. ..
Agrdit N, Rubio G, Vieira S: Plastic deformation of nanometrescale gold connective necks. Phys Rev Left 1995, 74:3995-3998. The first experimental evidence of the correlation between jumps in the tensile force and jumps in the electronic conductance. The contact diameter was relatively large. 23. ..
Rubio G, Agra’it N, Vieira S: Atomic-sized metallic contacts - mechanical properties and electronic transport, Phys Rev Len 1996, 76:2302-2305. Simultaneous measurements of force and conductance of an atomic-sired gold contact in air, where jumps in the conductance were always due to atomic rearrangements in the contact. 24.
Staider A, Diirig U: Study of yielding mechanics in nanometersized Au contacts. Appl Phys Let! 1996, 68:637-639. gmultaneous measurements of force and conductance of atomic-sized gold contacts in UHV showing correlations between variations in conductance and tensile force. 25.
Staider A, Diirig U: Study of plastic flow in ultrasmall Au contacts. J Vat Sci Technol B 1996, 1411259-l 263.
26.
Todorov TN, Sutton AP: Jumps in electronic conductance due to mechanical instabilities. Phys Rev Leti 1993, 70:2138-2141,
27.
Sutton AP, Todorov TN: Mechanical and electrical properties of metallic contacts at the nanometre scale. J Phys Chem Solids 1994, 55:1169-l 174.
28. ..
Bratkovsky AM, Sutton AP, Todorov TN: Conditions for conductance quantization in realistic models of atomic-scale metallic contacts. Phys Rev B 1995, 525036-5051. The only comparative study of FE and TB calculations of the conductance evolution based on the same MD simulations of nano contact pulling. It is also the only simulation study of the temperature dependence of the mechanical response and conductance of metallic nanocontacts. It explains several experimentally observed features abbut the conductance and predicts others, which have since been confirmed. 29.
Todorov TN, Sutton AP: Force and conductance jumps in atomic-scale metallic contacts. Phys Rev B 1996, in press. ysimulation study which reproduces the experimentally observed correlation between jumps in the tensile force and in the conductance. It also discusses time scales of relaxation processes in nanocontacts. 30.
Landman U, Luedtke WD, Nitzan A: Dynamics of tip-surface interactions in atomic force microscopy. Surf Sci 1989, 2lO:L177-L184.
Deformation
mechanisms,
electronic
conductance
and friction
of metallic
nanocontacts
Sutton
833
31.
Nieminen JA, Sutton AP, Pethica JB: Static junction growth during frictional sliding of metals. Acfa Metal/ Mater 1992, 40:2503-2509.
A careful simulation study of frictional sliding between clean metal surfaces, considering the effect of a crystal lattice misorientation between the tip and substrate.
32.
Nieminen JA. Sutton AP, Pethica JB, Kaeki K: Mechanism of lubrication by a thin solid film on a metal surface. Model Simul Mater Sci Eng 1992, 1:63-90.
41.
33.
Sutton AP, Pethica JB, Rafii-Tabar H, Nieminen JA: Mechanical properties of metals at the nanometre scale. In Electron Theory in Alloy Design. Edited by P&for DG, Cottrell AH. London: The Institute of Materials; 1992:191-233.
42.
Sutton AP, Bafluffi RW: Interfaces in crystalline materials. New York: Oxford University Press; 1995. L integrated texl on the structure, thermodynamics, kinetics and mechanical and electrical properties of interfaces in crystalline materials.
34.
Harrison JA, White CT, Cotton RJ, Brenner DW: Atomistic simulations of friction at sliding diamond interfaces. MRS Bull May 1993, 18:50-53.
43. ..
35.
Todorov TN, Briggs GAD, Sutton AP: Elastic quantum transport through small structures. J Phys-Condens Matter 1993, 52389-2406.
36.
Torres JA, Pascual JI, Saenz JJ: Theory of conduction through narrow constricttons in a three-dimensional electron gas. Phys Rev B 1994,49:16581-l 6584.
37. ..
Kraos JM: Size effects in atomic-scale point contacts [Phd Thesis]. Rijksuniversiteit Leiden 1 QQ6. Den Haag: Cip-data Koninklijke Bibliotheek. . . ..--. (ISBN 90-9008989-6). Rich In Information about MCBJ experiments and results, some of whtch are not published elsewhere.
36. .
Bhushan B, lsraelachvili JN, Landman U: Nanotribology: friction, wear and lubrtcatton at the atomic scale. Nature 1995, 374:607-616. A useful review of some of the experimental and simulation work in this field. 39.
Nanotrfbology.
40. .
Serensen MR, Jacobsen KW, Stoltxe P: Simulations of atomicscale sliding friction. Phys Rev B 1996, 53:2101-2113.
MRS Bull 1993,
18, No. 5.
Tabor D: Friction as a dissipative process. In Fundamentals of Friction: Macroscopic and Microscopic Processes. Edited by Singer IL, Pollock HM. Dordrecht: Kluwer; 1992:3-24.
Johnson KL: A continuum mechanics model of adhesion and friction in a single asperity contact In Micro/nanotribo/ogy and Its Applications. Edited by B Bhushan. Dordrecht: Ktuwer; 1997, in press. A continuum theory of a single asperity contact in which the transition from static to kinetic friction is treated, together with the interaction between friction and adhesion. 44. .
Gao JP, Luedtke WD, Landmao U: Nano-elastohydrodynamics - structure, dynamics, and flow in nonuntfonrr lubricatad junctions. Science 1995, 270:605-608. Using simple empirical potentials these simulations reveal interesting structural changes within an aikane lubricant. The changes have considerable consequences for the friction and wear of soft metallic (gold) asperities. 45.
Ramos MMD: Theory of processes at surfaces and interfacas [PhD Thesisl. oxford: University of Oxford; 1 Q92.
46.
Godwin PD, Horafield AP, Stoneham AM, Bull SJ, Ford U, Harker AH, Pettifor DG, Sutton AP: Computational materials synthesis Ill: synthesis of hydrogenated amorphous carbon from molecular precursors. Phys Rev 6 1 gQ6, in press.
47
Stoneham AM, Ramos MMD, Sutton AP: How do they stick together. The statics and dynamics of interfaces. Phil Mag A 1993, 67:797-811.