Langmuir 1996, 12, 4505-4509
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Plastic Deformation in Nanometer Scale Contacts† N. Agraı¨t,* G. Rubio, and S. Vieira Instituto Universitario de Ciencia de Materiales “Nicola´ s Cabrera”, Laboratorio de Bajas Temperaturas, C-III, Universidad Auto´ noma de Madrid, 28049 Madrid, Spain Received October 17, 1995X The combination of scanning force and tunneling microscopy techniques has made possible the study of plastic deformation in the nanometer scale with unprecedented detail.1 Retraction of the tip after indentation produces a connective neck which deforms in alternating elastic and yielding stages, in agreement with the theoretical predictions using molecular dynamics simulations. A description of the plastic deformation process in terms of a simple model based on continuum contact mechanics makes possible a quantitative analysis of the experimental results and underlines the connection with the bulk properties of the material.
1. Introduction Contact between macroscopic bodies occurs at asperities of microscopic to nanoscopic dimensions, and as a consequence understanding the processes that occur at the atomic level when two macroscopic bodies are brought into contact in fundamentally important to basic and applied problems like adhesion, contact formation, materials hardness, friction, wear, and fracture. However, knowledge of the behavior in contact and mechanical properties of asperities of such a small size is still quite limited. There is some experimental evidence that very small contacts can sustain much greater pressures than their macroscopic counterparts. Gane and Bowden2 used a 300 nm radius tip to indent a gold substrate. They found that the hardness of these very small contacts approaches ideal hardness. Even larger values for the hardness were obtained in recent indentation experiments using scanning probe microscopy techniques.3 Nanoindentation experiments in silver4 showed that the increase of hardness as the depth of indentation decreases is associated with the disappearance of dislocations, suggesting that small-scale indentation plasticity takes place by nondislocation mechanisms. From the theoretical point of view, the molecular dynamics (MD) simulations of Landman et al.5 have provided deep insight into the atomic scale processes occurring during indentation and plastic deformation. In particular, they showed that the mechanism underlying the formation and elongation of the connective neck, which forms upon separation, consists of structural transformations involving elastic and yielding stages. This stepwise behavior has also been observed in MD simulations of fracture.6 In the classical indentation experiments, as those † Presented at the Workshop on Physical and Chemical Mechanisms in Tribology, Bar Harbor, ME, August 27 to September 1, 1995. X Abstract published in Advance ACS Abstracts, Sept.15, 1996.
(1) Agraı¨t, N.; Rubio, G.; Vieira, S. Phys. Rev. Lett. 1995, 74, 3995. (2) Gane, N.; Bowden, F. P. J. Appl. Phys. 1968, 39, 1432. (3) Tangyunyong, P.; Thomas, R. C.; Houston, J. E.; Michalske, T. A.; Crooks, R. M.; Howard, A. J. Phys. Rev. Lett. 1993, 71, 3319. Thomas, R. C.; Houston, J. E.; Michalske, T. A.; Crooks, R. M. Science 1993, 259, 1883. (4) Pharr, G. M.; Oliver, W. C. J. Mater. Res. 1989, 4, 94. (5) Landman, U.; Luedtke, W. D.; Burnham, N. A.; Colton, R. J. Science 1990, 248, 454. Landman, U.; Luedtke, W. D.; Ringer, E. M. In Fundamentals of Friction: Macroscopic and Microscopic Processes; Singer, I. L., Pollock, H. M., Eds.; Kluwer: Dordrecht, 1992; p 463. (6) Lynden-Bell, R. M. Science 1994, 263, 1704. Lynden-Bell, R. M. J. Phys.: Condens. Matter 1992, 4, 2127.
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mentioned above, a hard tip of known geometry is used to indent the sample. The size of the indentation marks on the surface and the force necessary to produce them are used to obtain the hardness of the sample. If the force during indentation is recorded, the elastic properties can also be obtained. In the study of very small contacts this method has a serious drawback: it is very difficult to have a tip of a few nanometers of well-characterized geometry. We have adopted a different approach. Using a tip of the same material as the sample, it is possible to form a connective neck by cohesive bonding of tip and sample. The geometry of these necks can be obtained from their electrical conductance, and a nanometer scale contact of known and controlled dimensions can be formed. In previous works we studied nanometer-scale contacts in lead7 and with much better resolution in gold1 in both cases at low temperature (4.2 K). (The same approach has been recently used by Stadler et al.8) In this article we present results for gold contacts measured at room temperature, which are essentially identical to the low temperature results, and explore the applicability of continuum mechanics to the determination and prediction of the properties of such small contacts. 2. Experimental Section 2.1. Experimental Setup. The experimental setup consists of a tip and a cantilever beam on which the substrate is mounted. The force exerted on the substrate by the tip causes the deflection of the beam and is measured using a scanning tunneling microscope (STM) which provides subnanometer resolution (see inset of Figure 1). This STM tip works in the constant current mode, maintaining a constant tunneling gap and consequently the force exerted by this tip on the beam is constant. The tip, cantilever beam, and STM tip are mounted on independent piezoelectric tubes. The cantilever beam was made of phosphorus bronze and its elastic constant was 380 N/m at room temperature. The force resolution was better than 5 nN. In addition to the bias voltage applied between the beam and the STM tip, a small voltage (10 mV) is applied between the beam and the tip in order to measured the conductance of the contact. The conductance of a macroscopic contact is proportional to the contact radius and inversely proportional to the resistivity of the material. However, if the contact dimensions are smaller than the mean free path of the electrons (of the order of 40 nm at room temperature and much larger at low temperatures) the conductance is proportional to the contact area and independent (7) Agraı¨t, N.; Rodrigo, J. G.; Rubio, G.; Sirvent, C.; Vieira, S. Thin Solid Films 1994, 253, 199. (8) Stalder, A.; Du¨rig, U. J. Vac. Sci. Technol., in press.
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Figure 2. Formation of the asperities: (a) tip and substrate; (b) indentation; (c) on retraction, a connective neck forms; (d) rupture of the connective neck produces two opposing asperities.
Figure 1. Conductance and force evolution during retraction and approach measured at room temperature. The cycle starts with the asperities in contact (right-hand side of the figure). Inset: experimental setup. of the resistivity of the material.9 Using the corrected Sharvin’s equation,10 the conductance GS of a contact of small radius a is given by
GS )
(
)
2e2 Aπ P h λF2 2λF
(1)
where e is the electron charge, h is Planck’s constant, λF is the Fermi wavelength (for Au, λF ) 5.19 Å), A is the contact area, and P is the perimeter of the contact. Note that for circular contacts the term that depends on the perimeter is less than 10% for a contact radius larger than about 2 nm. We have not taken into account other effects as backscattering in the contact, which according to recent numerical simulations could reduce the conductance of a given contact by as much as 30%, at least for very small contacts.11 Henceforward the conductance will be implicitly given in units of 2e2/h. For Au a conductance unit equals 0.086 nm2 (when the perimeter term can be neglected). When tip and substrate are not in contact, there is still a measurable tunneling current; however, the corresponding conductance is at least 1 order of magnitude smaller than the smallest contact whose conductance is of the order of one unit.12 The onset of mechanical contact is always signaled by an abrupt jump in conductance. The substrate is a thin Au foil, and the tip is a thin Au wire. The substrate was cleaned by immersing in a 3:1 concentrated H2SO4/30% H2O2 solution, and the tip was clipped. This is the same setup used in previous experiments performed at liquid helium temperature (4.2 K).1,7 Here we will present the results of measurements at room temperature in high vacuum (10-6 Torr) conditions. 2.2. Experimental Procedure. The conductance between the tip and substrate and the deflection of the cantilever beam are measured simultaneously as the tip is advanced or retracted (9) Jansen, A. G. M.; van Gelder, A. P.; Wyder, P. J. Phys. C: Solid State Phys. 1980, 13, 6073. (10) Torres, J. A.; Pascual, J. I.; Sa´enz, J. J. Phys. Rev. B 1994, 49, 16581. We are not taking quantum size effects into account because the necks are wide enough to comprise many quantum channels, and the jumps observed in the conductance are always several quantum units high. (11) Bratkovsky, A. M.; Sutton, A. P.; Todorov, T. N. Phys. Rev. B, in press. (12) Agraı¨t, N.; Rubio, J. G.; Vieira, S. Phys. Rev. B 1993, 47, 12345.
at a constant speed. Typically the initial position of the tip (either in contact or out of contact) and the amplitude of the cycle are set and a series of cycles (10) is acquired. In order to obtain reproducible results, it is necessary to condition the contacting surfaces. This is accomplished by pressing the tip against the substrate repeatedly. As a result the adsorbates on the contact area are displaced and a cohesive bonding between tip and substrate is formed. As the tip is retracted a connective neck forms and eventually breaks, leaving a protrusion on the substrate. Since tip and substrate are of the same material, a similar protrusion must form on the tip (see Figure 2). We can image the protrusion on the substrate using the tip as an STM tip.1,7 Note that the STM image gives a convolution of both. The surface of these in situ formed asperities is clean as shown in the exponential dependence of the conductance with separation before contact, and the lack of repulsive force before electrical contact is established. The experimental data are taken on subsequent shallower contacts formed between these in situ prepared asperities. Initial excessive contamination on the surface of either tip or substrate would prevent cohesive bonding and the formation of the asperities. In this case the indentation curves show large repulsive forces with insignificant conductance and are quite irreproducible.
3. Results and Discussion 3.1. Room Temperature Results. All the results that we will discuss here correspond to a reproducible evolution of the contact, that is, subsequent cycles of equal starting position and amplitude result in very similar conductance and force curves. Figure 1 shows the evolution of conductance and force during a retraction/approach cycle. Starting at an initial conductance of about 100 units (about 9 nm2) the tip is retracted until the contact breaks. As the tip approaches the substrate, contact is reestablished about 1 nm beyond the point of rupture. The conductance after rupture is nonzero (tunneling regime) but cannot be appreciated in the scale of the figure. In the noncontact portion of the force curve a small long-range attractive force is detected (the curve bends slightly downward). This long-range attractive force is sometimes observed and highly variable from one experiment to the other. Note that while the noncontact part of the curve is reversible, the contact part is highly hysteretical. Figure 3 shows the evolution of a nonbreaking neck. In this case we can clearly see that the conductance varies in steps that are correlated to sudden relaxations of the force as predicted by MD simulations5 and observed experimentally1 at low temperature. These correlated features can be also observed in Figure 1 on closer inspection. 3.2. The Rigid-Perfectly-Plastic Neck. The overal behavior of the force and current curves depicted in Figures 1 and 3 can be described using a rigid-perfectly-plastic model.13 If elastic strains are negligible compared to plastic strains, elastic deformation may be neglected and (13) Johnson, K. L. Contact Mechanics; Cambridge University Press: Cambridge, 1985.
Plastic Deformation
Figure 3. Elongation-contraction cycle without breaking the connective neck at room temperature. Note the steps in the conductance correlated to force relaxations.
Figure 4. Conductance and force for the simple purely plastic deformation model: (a) rigid setup; (b) effect of elasticity, hard spring (black) and soft spring (gray). Note that the retracting branch for the soft spring is unstable.
the material may be considered a rigid solid that flows plastically at a constant stress. Figure 4a shows the deformation of a rigid-perfectly-plastic neck. We have assumed that the evolution of the contact area (i.e., conductance) is reversed for compressive and tensile stresses, and since the contact deforms at a constant apparent pressure, the force is proportional to the contact area. Note that the force changes from compressive (positive) to tensile (negative) at the right-hand side of the cycle. The figure represents a cycle in which the neck is broken. A force cycle with no rupture will also have a change from tensile to compressive at the left-hand side. The area of the cycle gives the work spent in the plastic deformation cycle (crushing and re-forming of the asperities). In Figure 4b we have introduced the effect of the cantilever beam (a soft and a hard beam are depicted). Note that the change from compressive to tensile forces takes place along a straight line whose slope is equal to the spring constant of the beam. During this change the conductance is constant since the pressure is smaller than that necessary to produce plastic deformation. Note that
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Figure 5. Discrete elastic-configuration model of plastic deformation. The dotted lines represent the different configurations of the neck (denoted by greek letters). The solid lines show the actual evolution of conductance and force during plastic deformation. The slope of the linear segments of the force is given by (1/keff + 1/kbeam)-1.
the tensile branch for the soft beam is unstable and both the conductance and the force show an abrupt jump (not necessarily conducive to rupture). This instability would take place when the spring constant of the beam is smaller than the derivative the force with respect to the displacement. As a consequence a soft beam can only be used to study small contacts. This model implies that the apparent pressure (force/ contact area) is constant during the deformation process. This was shown explicitly in ref 7 by dividing the force cycle by the conductance cycle to obtain a pressure cycle. In the cases in which a long-range attractive force is present (the force cycle is bent downward), it is always possible to subtract a position-dependent force from the whole cycle that makes the apparent pressure roughly constant in both branches simultaneously.1 This force is assumed to vary smoothly with tip displacement. In practice for nonbreaking cycles (like the one in Figure 3) we use a straight line. The fact that this procedure makes possible a roughly constant pressure with the same average value for cycles of very different amplitudes and starting points and that this force is different for different runs of the experiment indicates that this often observed long range attractive force plays no role in the process of plastic deformation,1 i.e., it acts on the cantilever but not on the contact. In Figures 1 and 3 we show the measured forces, but for the quantitative analysis we use the mechanical force, that is, the one obtained from the measured cycle by subtracting the attractive force. 3.3. The Discrete Elastic-Configuration Model. The detailed features of the experimental curves are completely ignored by the model described in the preceding section. The following model is based on continuum contact mechanics and follows naturally from the results of the molecular dynamics simulations.5 Consider a neck of elastic material with a given geometry (or configuration). For example R in Figure 5. Application of a force to this neck will result in purely elastic deformation if the force is below a certain threshold. As
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the threshold is reached (point B), the neck simply changes discontinuously to a new configuration (β). If the force is compressive, the new configuration will be shorter and its force curve will be displaced to the right-hand side of the figure. Note that the difference in length between the two configurations is given by the separation of the intercepts of the force vs displacement of both configuration with the x-axis. In the abrupt change from configuration R to configuration β, the force decreases (relaxation) but does not become zero. Since the elastic deformations do not have a great effect on the minimal cross section of the neck (contact area), the conductance for each configuration is constant. Since configuration β is shorter, mass conservation requires that its section be larger (larger conductance). Note that the length of the conductance steps does not directly give the contraction of the neck. For example, the difference in length between the different configurations depicted in the figure is constant, but the length of the conductance steps is quite irregular due to the difference in force thresholds. At point H the direction of the tip movement changes and the stresses change from compressive to tensile. At point I the tensile threshold is reached and the neck jumps to the longer configuration γ, etc. If we call the spring constant of a configuration keff, the slope in the force cycle will be given by (1/keff + 1/kbeam)-1, where kbeam is the spring constant of the beam. As pointed in the previous section we can eliminate the effect of the long-range attractive force by requiring that the threshold pressure be equal for both branches of the force curve. This defines the zero point for the mechanical forces. Thus from the experimental curves we can easily obtain keff, the pressure threshold, and the elongation/ contraction length for the different configurations. This model shows the basic features in the experimental force curves in Figure 3 and those in ref 1, namely, correlation between the force relaxations and the conductance plateaus, and linear dependence of the force between relaxations. In order to compare the mechanical properties of the contacts with the bulk properties of the material, we will approximate the properties of a neck (according to continuum elasticity) by those of a contact. According to contact mechanics13 the effective spring constant of a contact depends on the contact radius, a, and the elastic properties of the material
keff ) BaE/(1 - ν2)
(2)
where E is Young’s modulus and ν is Poisson’s ratio (0.42 for Au) and B is a factor that depends on the pressure distribution in the contact but is approximately 1 (B ) 1, corresponds to uniform displacement of the contact area, in which case the pressure is minimum at the center of the contact and diverges on the perimeter (flat punch); B ) 2/3, corresponds to Hertzian pressure distribution, in which case the pressure is maximum at the center of the contact and zero on the perimeter). The pressure distribution in the neck (typically large stresses on the perimeter) will be closer to that of a flat punch. This equation overestimates the spring constant of the neck, nevertheless it is a good approximation if l < aπ, where l is the length of the neck. A simple estimate of the shape of the neck obtained from the conductance curves shows that the necks are not very slender and the approximation is good. In Figure 6 we show the effective spring constant and maximum apparent pressure as a function of contact radius for the different configurations of several different necks at room temperature and at 4.2 K. Taking into
Figure 6. Experimental values of the effective spring constant keff of the necks and the maximum pressures as a function of contact radius for many different necks at room temperature (triangles) and liquid helium temperature (circles).
account systematic errors due to the calibration of the cantilever beam, the error introduced in the subtraction of the long-range attractive force correction and the possible effects of scattering on the conductance, we can consider these values accurate within about 40%. The maximum pressure for the different configurations typically ranges from 2 to 6 GPa, these differences are likely to be due to the detailed atomic structure. The effective spring constant increases rather linearly and the values are between the solid lines that represents the effective spring constant for a flat punch for values of E between 40 and 120 GPa, values which are adequate for Au and depend on the crystalline orientation.14 Since the contact area in our experiments is extremely small, the experiment is actually conducted between two single crystals of unknown orientation. We must also note that the features of the experimental force curves are much sharper than those in MD simulations and that the elastic stages show no loss of linearity (cf. refs 5 and 6). 3.4. Nanometer Scale Mechanical Properties. The yield strength, σy, gives the stress necessary to produce plastic deformation under uniaxial compressive or tensile stresses. At contact the situation is more complicated because the yielding material is contained by the surrounding material which behaves elastically,13 full plasticity is reached for pressures about 3σy, and, as a consequence, a typical indentation test gives a hardness equal to 3σy. However, even in the case of contacts, first yield occurs at a pressure of about σy.13 In our experiment we are measuring the first yield of each configuration and consequently σy,nano ≈ maximum pressure ≈ 4 GPa. Thus atomic-scale contacts are more than 1 order of magnitude stronger than macroscopic contacts (for Au, σy ) 0.21 GPa). Theoretically, the ideal shear strength (no dislocations nor defects), τy,theo, of a metal is considered to be of the order of G/30,15 where G is the shear modulus. For Au a more detailed calculation by Kelly16 gave τy,theo ) 0.74 (14) Young’s modulus values for Au are 117 and 43 GPa in the (111) and (100) directions at room temperature and 126 and 46.5 GPa at 0 K. (15) Kittel, C. Introduction to Solid States Physics, 6th ed.; Wiley: New York, 1986; p 559. (16) Kelly, A. Strong Solids; Clarendon: Oxford, 1973; p 28.
Plastic Deformation
GPa, and σy,theo ) 2τy,theo ) 1.5 GPa. This value is the same order of magnitude as our experimental value. We conclude that the strength of atomic scale contacts behave as ideally hard solids. 3.5. Energetic Considerations. The energy dissipated in each force relaxation, that is, the energy necessary to bring about the configurational change, can be directly obtained from the force cycle. Referring to Figure 5, the energy to pass from configuration R to configuration β is simply the hatched area between R and β. The dependence of this energy with the size of the contact area is easy to estimate. If the contact radius of both configurations is very similar (which is normally true except for very small necks) the energy to change configurations is given approximately by σy,nanoπa2∆x, where a is the contact radius, and ∆x is the change in length of the neck. If we assume that this energy is of the same order of magnitude as the heat of fusion (0.13 eV/ atom), we find that the zone around the narrowest part of the neck where the atomic rearrangements take place is about 4-5 atomic layers thick.8 3.6. Surface Energy Effects. The effect of surface energy on the mechanical properties of the contacts could be important in the case of very small contacts.17 Since a long neck with a small radius has a larger area than a shorter neck with a larger radius, there will be an additional attractive force acting on the contact due to surface energy. An estimate of the importance of this force can be easily obtained by comparing the changes in energy due to plastic deformation and to surface change, which are, respectively, ∆Eplas ≈ πr2σy∆x, and ∆Esurf ≈ πrγ∆x, for a cylinder of radius r whose length varies by ∆x. When the radius equals γ/σy, both effects will be of the same order. For a nanocontact we must use σy,nano, (17) Rabinowicz, E. Friction and Wear of Materials; Wiley: New York, 1965.
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and for Au, γ ) 1.12 J/m2, consequently γ/σy ≈ 0.25 nm. That is, surface energy effects will only be important for necks of atomic dimensions. 4. Conclusions Plastic deformation of the connective necks formed in nanometer-scale gold contacts can be described accurately using a continuum contact mechanics model in which the neck changes abruptly from one configuration to another at a certain pressure threshold. Each configuration deforms purely elastically with an elastic constant determined by contact mechanics and the bulk material properties. The transition from one configuration to the next takes place when the elastic stresses reach a value of 2-6 GPa. In compression (tension) this transition is to a shorter (longer) configuration with a larger (smaller) cross section. The length difference is always of the order of atomic separation (about 0.2 nm). The high value of the hardness (more than 1 order of magnitude larger than the bulk value) which is of the order of the theoretical value for ideal perfect crystals, suggests that plastic deformation at the nanoscale is by a nondislocation mechanism, that is, by atomic rearrangements in the immediate vicinity of the narrowest section as predicted by molecular dynamics simulations. This interpretation is also supported by direct measurement of the amount of energy spent in the change of configuration. We are currently studying the mechanical properties of even smaller necks consisting of one or a few atoms. Another unresolved interesting question of practical importance is for which contact radii the crossover to bulk behavior takes place. Acknowledgment. This work has been supported by the CICYT under Contract MAT92-0170. LA950888I
