- Email: [email protected]
Mechanical and electrical properties of metallic contacts at the nanometer scale
Pergamon
1. Phys. Chm,. Solidr Vol. 55. No. IO. pp. 116’+1174. 1994 Copyright X, 1994 Elvvicr Science Ltd Pnntcd in Great Britain. All rights resewed 0X2-3697/94 17.00 + 0.00
0022-3697(!Mpo!So
MECHANICAL AND ELECTRICAL PROPERTIES OF METALLIC CONTACTS AT THE NANOMETER SCALE ADRIAN
P. SUTTON and TCHAVDAR
Department
of Materials,
Oxford
University,
N. TODOROV OX1 3PH, U.K.
Abstract-In the first part we present computer simulations of the influence of a layer of foreign atoms on the force of adhesion between an asperity and substrate of the same metal. Regardless of whether or not wetting of the foreign layer by the tip occurs, it is found that the force of adhesion is always reduced relative to the case where no foreign layer is present. A simple liquid drop model is presented to explain this result. We then consider the electronic conductance of the system, in the absence of a foreign layer, as the asperity is brought into contact with the substrate and subsequently removed. The conductance is calculated by a quantum mechanical scheme using atomic positions generated by molecular dynamics. The conductance is found to change in jumps during contact formation and fracture. but the conductance per atom is not quantized. The jumps coincide with gross structural rearrangements resulting from mechanical instabilities in the system. Relevant experimental observations are mentioned. Keywora!~: A. interfaces, properties.
D. electrical
conductivity,
D. fracture,
brief summary of work that has been Oxford over the past six years on interactions between a metallic tip and a metallic substrate. At first sight this work may seem a long way from Bob Balluffi’s interests, whose contributions to materials science we are honouring in this Festschrift. We hope to show, however, that the overlap is strong, and indeed Bob’s interest in this work has been a great encouragement to us. The overlap stems from the interface that is created when a tip is brought into contact with a substrate. We have gained insight into the atomic mechanisms of sintering, adhesion, friction and fracture [l, 21, all of which are important interfacial phenomena. We have also explored the detailed relationship between the evolution of a metallic contact and its electronic conductance [3]: this structure-property correlation for an interface can be calculated [3] and measured [4] very precisely. Although it may be argued that the interfaces that are created between nanometer scale tips and substrates are not the same as those that exist in ordinary polycrystals, a great deal has been learnt abut interfacial phenomena in recent years both from experiments on, and simulations of, tip-substrate interactions. This is chiefly because the length scales involved in the experiment and simulations are the same, and, therefore, it is possible to establish a direct relation between the two. This close link between experiment and simulation is one of the hallmarks of Bob BalluflYs own work on interfaces. This
paper
out
properties,
D. surface
2. METHOD OF COMPUTER SIMULATION TIP-SUBSTRATE INTERACTIONS
1. INTRODUCTION
carried
D. mechanical
OF
is a at
In our dynamic simulations a paraboloidal metallic tip is brought into contact with a flat metallic substrate and is then pulled off. Figure 1 shows the geometry and periodic boundary conditions [I]. The tip in the computational cell is attached to the underside of the slab, and it interacts with the upperside of the slab in the image cell below. In this way the slabs acts as both tip holder and the substrate with which the tip interacts. The separation between the tip and the slab beneath it are controlled by varying the length of the computational cell normal to the slab. This length is decreased during formation of the contact and increased during pull-off and fracture. The tip and substrate may be of the same or different metals, and a layer of foreign metallic atoms may be introduced on the top surface of the slab, to study the influence of a foreign layer on adhesion. The slab typically contains about two thousand atoms, and the tip between two hundred and three hundred atoms. Atomic interactions are described by long-ranged, N-body potentials [5,6]. These potentials were constructed specifically to model the Van der Waals interaction at long range and metallic bonding at short range. We have carried out simulations [I] at 300 K, using a No&-Hoover thermostat, of (a) a Pb tip interacting with a clean Pb substrate, (b) an Ir tip interacting with a clean Ir substrate, (c) a Pb tip interacting with a Pb substrate covered with a layer of Ir and (d) an
1169
A. P. SUTTON and T. N. TODOROV substrate atoms by a coupling V, whose matrix elements have the functional form specified by the hopping integral. All on-site matrix elements of Y are taken to be zero. In a recent single-particle scattering theory formulation of the conductance problem [7j the zerovoltage, zero-temperature elastic conductance, g, of the final, coupled system was shown to be given by:
(1)
Fig. 1. Schematic illustration of _ the use of periodic- bound. .. aiy conditions to model tip-surface Interactions. The emboldened cell is repeated in three orthogonal directions.
Ir tip interacting with an Ir substrate covered with a layer of Pb. Iridium is a hard metal with a melting point much greater than that of Pb. In terms of effective pair interactions the Ir-Ir bond energy is much greater (i.e. more negative) than the Pb-Pb bond energy, with the Pb-Ir bond energy between these two. This is essentially all one has to keep in mind when studying the results of the simulations on the influence of a foreign layer. For the conductance simulations [3] we calculate the electronic conductance at each time step of a molecular dynamics simulation of an Ir tip interacting with an Ir slab at 300 K, during which the tip is brought into contact with the substrate and is then pulled off to fracture. In view of the size and duration of the simulation the conductance calculation must necessarily be a model calculation employing the simplest possible basis set which allows the relationship between atomic structure and metallic conduction to be traced. For this reason the electronic structure of the entire system is described by an orthonormal, nearest-neighbour, l-s tight-binding model with zero on-site energies, a band-fi!ling of one half and hopping integral scaling calculated analytically for l-s orbitals on hydrogenic atoms. Periodic boundary conditions are not used for the conductance calculation. Instead, we consider a single tip between two semi-infinite crystals. The atomic positions within the tip are those given by the dy namic simulation. The distortion of the slabs in the simulation is negligible compared with that of the tip. Therefore, in the conductance calculation the slabs above and below the tip are replaced by semi-infinite, perfect crystals, labelled 1 and 2 respectively. We imagine that the tip atoms are initially decoupled from each other and from the substrate atoms. We then couple the tip atoms to each other and to the
where r(E) = V + VG +(E)V, EF is the Fermi energy, and p:(E) and pi(E) are the density of states operators for the respective initial substrates and are given by: p;(E) =&G;-(E)
- G;+(E)),
k = 1,2.
(2)
Here G:*(E) and G!*(E) are the Green’s operators for the respective substrates in the initial, decoupled system and G*(E) are the Green’s operators for the final, coupled system. (Superscripts + and - correspond to retarded and advanced respectively.) The trace in eqn (1) is taken in the orthonormal atomix l-s basis. The matrix elements of G!*(E) and G!*(E) are obtained by a method described in [A, and those of t(E) by solving the Dyson matrix equation [1 - VG”+(E)]t(E) = V. The matrix element, [G”+(E)lij, of Go+(E) between atomic basis states i andj is given by [Gy+(E)lij if i,j E substrate 1; [G!‘(E)lij if i,j E substrate 2; 6,/E if i,j E tip and zero otherwise. 3. THE EFFECT OF A FOREIGN LAYER ON THE FORCE OF ADHESION In this section we summarise our results for the simulations of contact formation and fracture for clean substrates and for substrates covered with a foreign layer. See Ref. [l] for further details. During the approach to contact a mechanical instability occurs [8] during which the tip collapses onto the substrate, or the substrate rises to meet the tip. Following this initial contact the contact area grows by diffusional creep. Diffusional rearrangements take place within the lower part of the tip, as the lowermost layers become incorporated in higher layers during successive mechanical instabilities. The tip is attracted to the substrate, but it is restrained from crashing into the substrate by the displacement control of the computational cell length. The tensile elastic stresses that are developed grow, and eventually lead to an instability in which the system yields
Properties at metallic contacts
im :vers ibly by merging layers in the bottom part of the tip. Following each one of these instabilities the car ttact area increases by a substantial number of ato Ims.
1171
During the pull-off the tip elongates by t-iurther mechanical instabilities during which the lower part of the tip effectively melts and a new la.yer is nucleated (see Fig. 2). At the same time a neck
Fig. 2. Snapshot of the Pb tip necking down on the Pb substrate at 300 K. Note the structural disorder in the lower part of the tip.
1172
A. P. SUlTON
and T. N. TODOROV
develops in the tip and the strain becomes localized in this region. These are the basic mechanisms of contact formation and fracture in these nanometer scale contacts, and they are irreversible. Their irreversibility is further evidenced by the hysteresis in the forcedisplacement curves. The effect of a foreign layer on the substrate is to lead to either increased or decreased wetting of the substrate by the tip. For the Ir layer on the Pb substrate it is found that the Pb tip is strongly attracted to the layer. Through spreading out over the Ir layer, the Pb tip atoms replace Ir dangling bonds on the surface with relatively strong Ir-Pb bonds. Thus the Pb tip wets the Ir layer, and a larger pile of Pb tip atoms remains on the surface after pull-off to fracture. On the other hand, for the Pb layer on the Ir substrate, wetting by the Ir tip is not favourable becuase it would involve breaking stronger Ir-Ir bonds within the tip to form weaker Ir-Pb bonds on the surface. In this case the Ir tip leaves the substrate intact after the pull-off, although many Pb atoms are pulled out of the surface layer to cover the Ir tip. The most surprising result is that the force required to pull the tip off the surface (which is called the force of adhesion) is reduced when a foreign layer is present regardless of whether wetting is favourable or unfavourable. This reduction is relative to the case where no foreign layer is present. When wetting is not favourable the result is not surprising, since the energy required to break the bonds across the interface between the tip and substrate is reduced. But when wetting is favourable, as in the case of the Ir layer between the Pb tip and Pb substrate, one might have expected to see the force of adhesion to be increased. But in fact the force of adhesion is reduced
between two plates (see Fig. 3). On one plate the contact angle is maintained at a neutral value of 90” and the contact area there is held fixed. At the other plate the contact angle is fixed by the wetting condition and the contact area is variable. The separation of the plates, d, is increased, until the liquid drop breaks. This model may be solved analytically [I] and it is found that the force requited to break the drop is always reduced when the contact angle at the lower plate is not 9O”, i.e. when the wetting is either favourable or unfavourable. This liquid drop model reproduces the results of the simulations precisely. It is in agreement with a large body of experimental data (see references in [l]).
4. CHANGES IN ELECTRONIC CONDUCTANCE DURING CONTACT FORMATION AND FRACTURE Figure 4 shows the conductance (in units of 2e2/h) as a function of the iteration (time step) number, N, of the dynamic simulation for an Ir tip interacting with an Ir substrate, following the initial equilibration which occurs for 1 6 N < 1050 [3]. For 1050 < N < 3200 the computational cell length is being decreased, whereas for 3200 < N Q 6200 it is being increased. Owing to the truncation of the hopping integral the conductance is zero until substrate atoms come within hopping range of tip atoms. This happens at N = 1784. After that the conductance rises as the bonds between the single atom at the bottom of the tip and its three neighbours in the
PI. The explanation [l] for this result is quite simple. When wetting is strongly favourable fracture does not take place between the foreign layer and tip, which is an energetically stable interface, but within the tip. The Pb atoms adjacent to the Ir layer remain in fixed positions throughout the pull-off to fracture. The contact angles also remain more or less constant. The role of the Ir layer is to constrain the necking-down to fracture that occurs higher up in the tip. It is this constraint on the plastic deformation within the tip that reduces the force of adhesion. By contrast, when the Ir layer is not present, necking-down to fracture still takes place within the Pb tip, but the contact angle remains at about 90” as the contact area on the Pb substrate shrinks a little under the tensile load. These observations suggest a simple “liquid-drop” model [I] of the effect of a foreign layer on the force of adhesion. The tip is modelled as a liquid drop
Fig. 3. Liquid drop between two plates at x = 0 and x = d. ‘6 co&t angl; at x = 0 is &intained at 90” and the contact radius is maintained at R, during the pull-off. The contact angle at x = d is maintained at JlO. but the contact area at x = d is variable. The volume of the drop is conserved as it is stretch& The shape of the drop is described by the function y = y(x).
1173
Properties at metallic contacts
1750
2550
3350 batktn
4150
4950
5750
number N
Fig. 4. The conductance, in units of 2e2/~, vs the iteration number, N, of the dynamic simulation throughout the For formation and breaking of the contact. 1050 < N < 3200 the length of the computational cell normal to the substrate is being decreased, whereas for 3200 < N < 6200 it is being increased.
substrate below strengthen. For 1940 f N < 2090, when, apart from thermal fluctuations, there is a stable single atom contact, the conductance settles at a value of 0.93 + 0.05. Between N x 2100 and N z 2230 we see the first mechanical instability in which the single atom at the base of the tip is incorporated into the layer above, and the number of layers in the tip is reduced by one. The system becomes unstable in the sense that, once induced, the structural rearrangement persists even if the change in cell length is suspended. Now there are 13 atoms of the tip in direct contact with the slab. During the rearrangement the conductance undergoes a sharp increase and for 2300 < N < 2600 it settles at a value of 9.0 f 0.3, giving a conductance per atom of 0.69 + 0.02. The next instability occurs between N ~2650 and N ~2750. The number of layers in the tip is again reduced by one, and the number of tip atoms in direct contact with the slab becomes 25. Once again the conductance undergoes a sharp increase and for 2800 < N < 3000 it settles at a value of 15.2 & 0.5, yielding a conductance per atom of 0.61 + 0.02. We see the onset of one further reduction in the number of tip layers just before the cell length starts being increased at N = 3200. Between successive sharp increases, the conductance exhibits some small variation. This variation is due to two factors: (i) thermal vibrations giving rise to random fluctuations in the conductance; and (ii) the growing compressive force promoting improved registry between the tip and substrate atoms, which leads to a further slight increase in the conductance between jumps. The above observations indicate that the main factor governing the conductance is the number of tip atoms in direct contact with the substrate.
However the conductance does not vary linearly with this number. The variation in the conductance per atom is due to two independent factors. Firstly, even with a perfect geometry (i.e. when all bond lengths are equal to that in the perfect crystal), the conductance for a given contact area depends on the shape of the rest of the tip. Secondly, there is interference between the single atom contacts that make up a multi-atom contact. This phenomenon is best illustrated by a calculation discussed in [A in which, using the same tight binding model as here, we showed that in the limit of an infinite ideal contact between two f.c.c. (111) semi-infinite perfect crystals, the conductance per atom is 0.81 (in units of 2e*/h), whereas the conductance of an ideal single atom contact between these semi-infinite crystals is unity. Between N = 3200 and N = 6200 the tip is being pulled off the slab. As in the simulations discussed in Section 3, fracture does not take place between the lowermost layer of the tip and the slab, but through the formation of a neck within the tip. Mechanical instabilities are induced during the elongation of the tip, each of which results in the formation of a new layer in the neck region. This is reflected in sharp decreases in the conductance at N x 3500, 3900 and 4300. Beyond N NN4500 the neck becomes highly disordered and without a well defined layer structure. Just before fracture, between N x 5300 and N x 5600, the neck reduces to a single atom width. Finally, when fracture occurs, a pile of tip atoms is left behind on the slab. We emphasize that the mechanical instabilities, which manifest themselves as the generation or loss of tip layers, and the resulting discontinuities in the conductance, are a general feature of atomic scale contacts, even though the particular abrupt rearrangements and conductance jumps depend on the initial structure of the tip. Our results are in agreement with experimental observations [4] of jumps in electronic conductance during contact formation.
5. CONCLUSIONS Contact formation (sintering) at the nanometer scale proceeds by a sequence of mechanical instabilities in which layers of atoms merge. Fracture of nanometer scale contacts also proceeds by another sequence of mechanical instabilities in which the asperity lengthens by nucleating new whole layers in a neck. The introduction of a foreign layer, which is wetted by the tip, results in a reduction of the force of adhesion because the constraint of the fixed contact angle reduces the plastic flow within the neck of
I174
A. P. SUTTON and T. N. TODGROV
the tip. This can be modelled by regarding the tip as a liquid drop. The abrupt structural changes within the tip during contact formation and fracture lead to abrupt changes in its conductance. The main factor controlling the conductance is the area of the contact while perturbations of the internal structure of the tip are of secondary importance. Finally, the conductance does not vary linearly with the number of atoms in the contact. Acknowledgement-Some of this research was carried out in the Materials Modelling Laboratory of the Department of Materials of Oxford University, which is funded in part by SERC Grant number GR/H58278.
REFERENCES 1. Sutton A. P., Pethica J. B., Rat&Tabar H. and Nieminen J. A., Electron Theory in Alloy Design (Edited by D. G. Pettifor and A. H. Cottrell), Ch. 7, p. 191. The Institute of Materials (1992). 2. Nieminen J. A., Sutton A. P.. Pethica J. B. and Kaski K., Mod. Sitn. Mater. Sci. &g. 1, 83 (1992). 3. Todorov T. N. and Sutton A. P.., P/tvs. &a. fett. 70. 2138 (1993). 4. Muller C. J., van Ruitenbeek J. M. and de Jongh L. J., P/W. Rev. L&t. 69, 140 (1992). 5. Sutton A. P. and khen J., Phil. Mug. Len. 61, 139 (1990). 6. Rafii-Tabar H. and Sutton A. P., Phil. Mug. Lett. 63, 217 (1991). 7. Todorov T. N., Brings G. A. D. and Sutton A. P., J. Phys. Condenr. &&&et S, 2389 (1993). 8. Pethica J. B. and Sutton A. P.. J. Vat. Sci. Technol. A6. 2498 (1988). I
1. Phys. Chm,. Solidr Vol. 55. No. IO. pp. 116’+1174. 1994 Copyright X, 1994 Elvvicr Science Ltd Pnntcd in Great Britain. All rights resewed 0X2-3697/94 17.00 + 0.00
0022-3697(!Mpo!So
MECHANICAL AND ELECTRICAL PROPERTIES OF METALLIC CONTACTS AT THE NANOMETER SCALE ADRIAN
P. SUTTON and TCHAVDAR
Department
of Materials,
Oxford
University,
N. TODOROV OX1 3PH, U.K.
Abstract-In the first part we present computer simulations of the influence of a layer of foreign atoms on the force of adhesion between an asperity and substrate of the same metal. Regardless of whether or not wetting of the foreign layer by the tip occurs, it is found that the force of adhesion is always reduced relative to the case where no foreign layer is present. A simple liquid drop model is presented to explain this result. We then consider the electronic conductance of the system, in the absence of a foreign layer, as the asperity is brought into contact with the substrate and subsequently removed. The conductance is calculated by a quantum mechanical scheme using atomic positions generated by molecular dynamics. The conductance is found to change in jumps during contact formation and fracture. but the conductance per atom is not quantized. The jumps coincide with gross structural rearrangements resulting from mechanical instabilities in the system. Relevant experimental observations are mentioned. Keywora!~: A. interfaces, properties.
D. electrical
conductivity,
D. fracture,
brief summary of work that has been Oxford over the past six years on interactions between a metallic tip and a metallic substrate. At first sight this work may seem a long way from Bob Balluffi’s interests, whose contributions to materials science we are honouring in this Festschrift. We hope to show, however, that the overlap is strong, and indeed Bob’s interest in this work has been a great encouragement to us. The overlap stems from the interface that is created when a tip is brought into contact with a substrate. We have gained insight into the atomic mechanisms of sintering, adhesion, friction and fracture [l, 21, all of which are important interfacial phenomena. We have also explored the detailed relationship between the evolution of a metallic contact and its electronic conductance [3]: this structure-property correlation for an interface can be calculated [3] and measured [4] very precisely. Although it may be argued that the interfaces that are created between nanometer scale tips and substrates are not the same as those that exist in ordinary polycrystals, a great deal has been learnt abut interfacial phenomena in recent years both from experiments on, and simulations of, tip-substrate interactions. This is chiefly because the length scales involved in the experiment and simulations are the same, and, therefore, it is possible to establish a direct relation between the two. This close link between experiment and simulation is one of the hallmarks of Bob BalluflYs own work on interfaces. This
paper
out
properties,
D. surface
2. METHOD OF COMPUTER SIMULATION TIP-SUBSTRATE INTERACTIONS
1. INTRODUCTION
carried
D. mechanical
OF
is a at
In our dynamic simulations a paraboloidal metallic tip is brought into contact with a flat metallic substrate and is then pulled off. Figure 1 shows the geometry and periodic boundary conditions [I]. The tip in the computational cell is attached to the underside of the slab, and it interacts with the upperside of the slab in the image cell below. In this way the slabs acts as both tip holder and the substrate with which the tip interacts. The separation between the tip and the slab beneath it are controlled by varying the length of the computational cell normal to the slab. This length is decreased during formation of the contact and increased during pull-off and fracture. The tip and substrate may be of the same or different metals, and a layer of foreign metallic atoms may be introduced on the top surface of the slab, to study the influence of a foreign layer on adhesion. The slab typically contains about two thousand atoms, and the tip between two hundred and three hundred atoms. Atomic interactions are described by long-ranged, N-body potentials [5,6]. These potentials were constructed specifically to model the Van der Waals interaction at long range and metallic bonding at short range. We have carried out simulations [I] at 300 K, using a No&-Hoover thermostat, of (a) a Pb tip interacting with a clean Pb substrate, (b) an Ir tip interacting with a clean Ir substrate, (c) a Pb tip interacting with a Pb substrate covered with a layer of Ir and (d) an
1169
A. P. SUTTON and T. N. TODOROV substrate atoms by a coupling V, whose matrix elements have the functional form specified by the hopping integral. All on-site matrix elements of Y are taken to be zero. In a recent single-particle scattering theory formulation of the conductance problem [7j the zerovoltage, zero-temperature elastic conductance, g, of the final, coupled system was shown to be given by:
(1)
Fig. 1. Schematic illustration of _ the use of periodic- bound. .. aiy conditions to model tip-surface Interactions. The emboldened cell is repeated in three orthogonal directions.
Ir tip interacting with an Ir substrate covered with a layer of Pb. Iridium is a hard metal with a melting point much greater than that of Pb. In terms of effective pair interactions the Ir-Ir bond energy is much greater (i.e. more negative) than the Pb-Pb bond energy, with the Pb-Ir bond energy between these two. This is essentially all one has to keep in mind when studying the results of the simulations on the influence of a foreign layer. For the conductance simulations [3] we calculate the electronic conductance at each time step of a molecular dynamics simulation of an Ir tip interacting with an Ir slab at 300 K, during which the tip is brought into contact with the substrate and is then pulled off to fracture. In view of the size and duration of the simulation the conductance calculation must necessarily be a model calculation employing the simplest possible basis set which allows the relationship between atomic structure and metallic conduction to be traced. For this reason the electronic structure of the entire system is described by an orthonormal, nearest-neighbour, l-s tight-binding model with zero on-site energies, a band-fi!ling of one half and hopping integral scaling calculated analytically for l-s orbitals on hydrogenic atoms. Periodic boundary conditions are not used for the conductance calculation. Instead, we consider a single tip between two semi-infinite crystals. The atomic positions within the tip are those given by the dy namic simulation. The distortion of the slabs in the simulation is negligible compared with that of the tip. Therefore, in the conductance calculation the slabs above and below the tip are replaced by semi-infinite, perfect crystals, labelled 1 and 2 respectively. We imagine that the tip atoms are initially decoupled from each other and from the substrate atoms. We then couple the tip atoms to each other and to the
where r(E) = V + VG +(E)V, EF is the Fermi energy, and p:(E) and pi(E) are the density of states operators for the respective initial substrates and are given by: p;(E) =&G;-(E)
- G;+(E)),
k = 1,2.
(2)
Here G:*(E) and G!*(E) are the Green’s operators for the respective substrates in the initial, decoupled system and G*(E) are the Green’s operators for the final, coupled system. (Superscripts + and - correspond to retarded and advanced respectively.) The trace in eqn (1) is taken in the orthonormal atomix l-s basis. The matrix elements of G!*(E) and G!*(E) are obtained by a method described in [A, and those of t(E) by solving the Dyson matrix equation [1 - VG”+(E)]t(E) = V. The matrix element, [G”+(E)lij, of Go+(E) between atomic basis states i andj is given by [Gy+(E)lij if i,j E substrate 1; [G!‘(E)lij if i,j E substrate 2; 6,/E if i,j E tip and zero otherwise. 3. THE EFFECT OF A FOREIGN LAYER ON THE FORCE OF ADHESION In this section we summarise our results for the simulations of contact formation and fracture for clean substrates and for substrates covered with a foreign layer. See Ref. [l] for further details. During the approach to contact a mechanical instability occurs [8] during which the tip collapses onto the substrate, or the substrate rises to meet the tip. Following this initial contact the contact area grows by diffusional creep. Diffusional rearrangements take place within the lower part of the tip, as the lowermost layers become incorporated in higher layers during successive mechanical instabilities. The tip is attracted to the substrate, but it is restrained from crashing into the substrate by the displacement control of the computational cell length. The tensile elastic stresses that are developed grow, and eventually lead to an instability in which the system yields
Properties at metallic contacts
im :vers ibly by merging layers in the bottom part of the tip. Following each one of these instabilities the car ttact area increases by a substantial number of ato Ims.
1171
During the pull-off the tip elongates by t-iurther mechanical instabilities during which the lower part of the tip effectively melts and a new la.yer is nucleated (see Fig. 2). At the same time a neck
Fig. 2. Snapshot of the Pb tip necking down on the Pb substrate at 300 K. Note the structural disorder in the lower part of the tip.
1172
A. P. SUlTON
and T. N. TODOROV
develops in the tip and the strain becomes localized in this region. These are the basic mechanisms of contact formation and fracture in these nanometer scale contacts, and they are irreversible. Their irreversibility is further evidenced by the hysteresis in the forcedisplacement curves. The effect of a foreign layer on the substrate is to lead to either increased or decreased wetting of the substrate by the tip. For the Ir layer on the Pb substrate it is found that the Pb tip is strongly attracted to the layer. Through spreading out over the Ir layer, the Pb tip atoms replace Ir dangling bonds on the surface with relatively strong Ir-Pb bonds. Thus the Pb tip wets the Ir layer, and a larger pile of Pb tip atoms remains on the surface after pull-off to fracture. On the other hand, for the Pb layer on the Ir substrate, wetting by the Ir tip is not favourable becuase it would involve breaking stronger Ir-Ir bonds within the tip to form weaker Ir-Pb bonds on the surface. In this case the Ir tip leaves the substrate intact after the pull-off, although many Pb atoms are pulled out of the surface layer to cover the Ir tip. The most surprising result is that the force required to pull the tip off the surface (which is called the force of adhesion) is reduced when a foreign layer is present regardless of whether wetting is favourable or unfavourable. This reduction is relative to the case where no foreign layer is present. When wetting is not favourable the result is not surprising, since the energy required to break the bonds across the interface between the tip and substrate is reduced. But when wetting is favourable, as in the case of the Ir layer between the Pb tip and Pb substrate, one might have expected to see the force of adhesion to be increased. But in fact the force of adhesion is reduced
between two plates (see Fig. 3). On one plate the contact angle is maintained at a neutral value of 90” and the contact area there is held fixed. At the other plate the contact angle is fixed by the wetting condition and the contact area is variable. The separation of the plates, d, is increased, until the liquid drop breaks. This model may be solved analytically [I] and it is found that the force requited to break the drop is always reduced when the contact angle at the lower plate is not 9O”, i.e. when the wetting is either favourable or unfavourable. This liquid drop model reproduces the results of the simulations precisely. It is in agreement with a large body of experimental data (see references in [l]).
4. CHANGES IN ELECTRONIC CONDUCTANCE DURING CONTACT FORMATION AND FRACTURE Figure 4 shows the conductance (in units of 2e2/h) as a function of the iteration (time step) number, N, of the dynamic simulation for an Ir tip interacting with an Ir substrate, following the initial equilibration which occurs for 1 6 N < 1050 [3]. For 1050 < N < 3200 the computational cell length is being decreased, whereas for 3200 < N Q 6200 it is being increased. Owing to the truncation of the hopping integral the conductance is zero until substrate atoms come within hopping range of tip atoms. This happens at N = 1784. After that the conductance rises as the bonds between the single atom at the bottom of the tip and its three neighbours in the
PI. The explanation [l] for this result is quite simple. When wetting is strongly favourable fracture does not take place between the foreign layer and tip, which is an energetically stable interface, but within the tip. The Pb atoms adjacent to the Ir layer remain in fixed positions throughout the pull-off to fracture. The contact angles also remain more or less constant. The role of the Ir layer is to constrain the necking-down to fracture that occurs higher up in the tip. It is this constraint on the plastic deformation within the tip that reduces the force of adhesion. By contrast, when the Ir layer is not present, necking-down to fracture still takes place within the Pb tip, but the contact angle remains at about 90” as the contact area on the Pb substrate shrinks a little under the tensile load. These observations suggest a simple “liquid-drop” model [I] of the effect of a foreign layer on the force of adhesion. The tip is modelled as a liquid drop
Fig. 3. Liquid drop between two plates at x = 0 and x = d. ‘6 co&t angl; at x = 0 is &intained at 90” and the contact radius is maintained at R, during the pull-off. The contact angle at x = d is maintained at JlO. but the contact area at x = d is variable. The volume of the drop is conserved as it is stretch& The shape of the drop is described by the function y = y(x).
1173
Properties at metallic contacts
1750
2550
3350 batktn
4150
4950
5750
number N
Fig. 4. The conductance, in units of 2e2/~, vs the iteration number, N, of the dynamic simulation throughout the For formation and breaking of the contact. 1050 < N < 3200 the length of the computational cell normal to the substrate is being decreased, whereas for 3200 < N < 6200 it is being increased.
substrate below strengthen. For 1940 f N < 2090, when, apart from thermal fluctuations, there is a stable single atom contact, the conductance settles at a value of 0.93 + 0.05. Between N x 2100 and N z 2230 we see the first mechanical instability in which the single atom at the base of the tip is incorporated into the layer above, and the number of layers in the tip is reduced by one. The system becomes unstable in the sense that, once induced, the structural rearrangement persists even if the change in cell length is suspended. Now there are 13 atoms of the tip in direct contact with the slab. During the rearrangement the conductance undergoes a sharp increase and for 2300 < N < 2600 it settles at a value of 9.0 f 0.3, giving a conductance per atom of 0.69 + 0.02. The next instability occurs between N ~2650 and N ~2750. The number of layers in the tip is again reduced by one, and the number of tip atoms in direct contact with the slab becomes 25. Once again the conductance undergoes a sharp increase and for 2800 < N < 3000 it settles at a value of 15.2 & 0.5, yielding a conductance per atom of 0.61 + 0.02. We see the onset of one further reduction in the number of tip layers just before the cell length starts being increased at N = 3200. Between successive sharp increases, the conductance exhibits some small variation. This variation is due to two factors: (i) thermal vibrations giving rise to random fluctuations in the conductance; and (ii) the growing compressive force promoting improved registry between the tip and substrate atoms, which leads to a further slight increase in the conductance between jumps. The above observations indicate that the main factor governing the conductance is the number of tip atoms in direct contact with the substrate.
However the conductance does not vary linearly with this number. The variation in the conductance per atom is due to two independent factors. Firstly, even with a perfect geometry (i.e. when all bond lengths are equal to that in the perfect crystal), the conductance for a given contact area depends on the shape of the rest of the tip. Secondly, there is interference between the single atom contacts that make up a multi-atom contact. This phenomenon is best illustrated by a calculation discussed in [A in which, using the same tight binding model as here, we showed that in the limit of an infinite ideal contact between two f.c.c. (111) semi-infinite perfect crystals, the conductance per atom is 0.81 (in units of 2e*/h), whereas the conductance of an ideal single atom contact between these semi-infinite crystals is unity. Between N = 3200 and N = 6200 the tip is being pulled off the slab. As in the simulations discussed in Section 3, fracture does not take place between the lowermost layer of the tip and the slab, but through the formation of a neck within the tip. Mechanical instabilities are induced during the elongation of the tip, each of which results in the formation of a new layer in the neck region. This is reflected in sharp decreases in the conductance at N x 3500, 3900 and 4300. Beyond N NN4500 the neck becomes highly disordered and without a well defined layer structure. Just before fracture, between N x 5300 and N x 5600, the neck reduces to a single atom width. Finally, when fracture occurs, a pile of tip atoms is left behind on the slab. We emphasize that the mechanical instabilities, which manifest themselves as the generation or loss of tip layers, and the resulting discontinuities in the conductance, are a general feature of atomic scale contacts, even though the particular abrupt rearrangements and conductance jumps depend on the initial structure of the tip. Our results are in agreement with experimental observations [4] of jumps in electronic conductance during contact formation.
5. CONCLUSIONS Contact formation (sintering) at the nanometer scale proceeds by a sequence of mechanical instabilities in which layers of atoms merge. Fracture of nanometer scale contacts also proceeds by another sequence of mechanical instabilities in which the asperity lengthens by nucleating new whole layers in a neck. The introduction of a foreign layer, which is wetted by the tip, results in a reduction of the force of adhesion because the constraint of the fixed contact angle reduces the plastic flow within the neck of
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the tip. This can be modelled by regarding the tip as a liquid drop. The abrupt structural changes within the tip during contact formation and fracture lead to abrupt changes in its conductance. The main factor controlling the conductance is the area of the contact while perturbations of the internal structure of the tip are of secondary importance. Finally, the conductance does not vary linearly with the number of atoms in the contact. Acknowledgement-Some of this research was carried out in the Materials Modelling Laboratory of the Department of Materials of Oxford University, which is funded in part by SERC Grant number GR/H58278.
REFERENCES 1. Sutton A. P., Pethica J. B., Rat&Tabar H. and Nieminen J. A., Electron Theory in Alloy Design (Edited by D. G. Pettifor and A. H. Cottrell), Ch. 7, p. 191. The Institute of Materials (1992). 2. Nieminen J. A., Sutton A. P.. Pethica J. B. and Kaski K., Mod. Sitn. Mater. Sci. &g. 1, 83 (1992). 3. Todorov T. N. and Sutton A. P.., P/tvs. &a. fett. 70. 2138 (1993). 4. Muller C. J., van Ruitenbeek J. M. and de Jongh L. J., P/W. Rev. L&t. 69, 140 (1992). 5. Sutton A. P. and khen J., Phil. Mug. Len. 61, 139 (1990). 6. Rafii-Tabar H. and Sutton A. P., Phil. Mug. Lett. 63, 217 (1991). 7. Todorov T. N., Brings G. A. D. and Sutton A. P., J. Phys. Condenr. &&&et S, 2389 (1993). 8. Pethica J. B. and Sutton A. P.. J. Vat. Sci. Technol. A6. 2498 (1988). I