- Email: [email protected]
Size and load dependence of nanoscale electric contact resistance
Tribology International 71 (2014) 109–113
Contents lists available at ScienceDirect
Tribology International journal homepage: www.elsevier.com/locate/triboint
Size and load dependence of nanoscale electric contact resistance Zhijiang Ye, Hyeongjoo Moon, Min Hwan Lee, Ashlie Martini n School of Engineering, University of California-Merced, CA 95343, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 20 September 2013 Received in revised form 9 November 2013 Accepted 14 November 2013 Available online 23 November 2013
Nanoscale electrical resistance between a platinum-coated atomic force microscope tip and highly oriented pyrolytic graphite surface is measured as a function of normal load and tip radius. These measurements are complemented by molecular dynamics simulations that relate load and radius to contact area. Simulation-predicted contact area and experimentally-measured resistance are used to calculate contact resistivity. The results show that the effect of load on resistance can be captured by the real contact area, while tip size, although in part captured by area, affects contact resistivity itself, potentially through interface distance. Our study provides new insight into the effect of load and geometry on nanoscale electric contact and, more significantly, highlights the role of atomic-scale contact features in determining contact resistance. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Nanoscale contact Electric resistance Atomic force microscope Molecular dynamics simulation
1. Introduction As demand for high density, high speed electronic and electromechanical devices continues to increase, implementation of true nanoscale structures has become critical for next generation electronics engineering [1–3]. However, although contact resistance is one of the major parasitic components in electronic devices, conduction at nanoscale contacts is not yet clearly understood [4]. Furthermore, electrical transport through nanoscale contacts often exhibits behavior deviating from that of conventional contacts [5,6]. This can be partially attributed to the fact that the dimension of contact is smaller than the characteristic Debye length, resulting in an enhanced electrical conduction [5,7]. Nanoscale structures also have discrete energy levels which, in turn, can result in unconventional electric behaviors. Recently, single layer graphene (SLG) and multi-layer graphene (MLG) have been the focus of extensive research due to their unique properties, such as high electrical and thermal conductivity [8–11], very low friction [1,12,13] and the quantum hall effect [10,14,15]. Many studies have focused on electrical properties due to the wide applicability of SLG and MLG to various electronic and electromechanical devices [17–21]. Since contact resistance is a major factor limiting device scalability, reliability and performance in any electronic device [22–24], it is important to understand this behavior in graphene-based materials. Although there are quite a few studies on metal-SLG contact, little is known about the electrical properties of metal-MLG contacts [29]. Of the previous metal-MLG studies, several have focused on the effect of number
n
Corresponding author. E-mail address: [email protected] (A. Martini).
0301-679X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.triboint.2013.11.012
of graphene layers on contact resistance [31,32,34]. Other studies revealed a pressure dependence of electrical conductance on a double layer graphene surface [33] and that conductance is affected by step edges observed on the surface of MLG [30,35,33]. These findings highlight at the critical role of the contact and its atomic-scale features in determining nanoscale resistance. In this paper, we explore this important relationship between a nanoscale contact and its resistance by studying the electric contact between a platinum-coated AFM tip and highly oriented pyrolytic graphite (HOPG) surface. We measure the contact resistance with different size tips under various loads using the experiments, and then investigate the contact between tip and substrate using complementary simulations. The simulations are used to predict real and apparent contact area, and then this information is used with the resistance measured experimentally to calculate contact resistivity. The results are analyzed in terms of the ability of the two contact area definitions to capture the effects of load and tip size on resistance. 2. Method An atomic force microscope (AFM; Model 5500, Agilent Technology) system was used to enable contacts between a conductive probe and a HOPG (Alfa Aesar) surface with controlled forces. HOPG with a thickness of 200 μm was used for the electrical characterization. To avoid significant contamination and/or oxidation of the surface, the surface was always cleaved using a blade razor before each measurement. The strong sp2 covalent intralayer bonds and relatively weak van der Waals interlayer bonds in HOPG made a thin layer readily exfoliated [36]. To investigate the contact
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Fig. 2. Snapshot from a molecular dynamics simulation of the apex of a 32 nm radius AFM tip placed on a seven layer graphite substrate.
Fig. 1. Scanning electron microscope images of AFM tips with radii (a) 32 nm, (b) 77 nm, and (c) a simplified schematic diagram of the experimental setup.
area dependence, tips with two different apex radii (32 nm and 77 nm, shown in Fig. 1(a) and (b)) were prepared by depositing Pt on commercially available probes (Team Nanotec HSC Probe, NanoScience Instruments Inc.). The commercial probe was originally coated with tungsten carbide and its apex was in a hemispherical cone shape. Their nominal tip radii (the values specified by the vendor) were 20 nm and 40 nm. A dense Pt film was coated on the probes by a direct current sputter deposition scheme in an Ar pressure of 10 mTorr. The electrical characterization was conducted under ambient conditions at room temperature ( 300 K). The forces applied by the cantilever (with a spring constant of 3.0 N/m) were in the range of 100–800 nN during the electrical measurement. The tip–substrate contact resistances were obtained using a semiconductor parameter analyzer (HP4145B). An illustration of the experiential setup is shown in Fig. 1(c). In this setup, electrical current was measured during a potentiodynamic sweep performed from 0 to 50 mV when the tip is in contact with the HOPG surface. We observed a consistent linear relationship between the bias and the current (data not shown) such that the contact resistance could be acquired by applying the Ohms law (R ¼ V=I) with currents measured under any given bias; we used a
bias of 50 mV for this purpose. All sources of resistance except for that from the Pt–HOPG contact were found to be negligible ( o15 Ω) compared to the measured resistance which was in the range of kΩ. A capillary condensation of water from the ambient, which is usually called a meniscus, can be a medium for an alternative current path between the tip and the sample [37,38]. Considering a significantly higher resistance of the water meniscus than the Pt and HOPG contact, we also neglected this meniscus effect [31]. After the electrical characterization, the tip apex shape was examined using a field emission scanning electron microscopy as shown in Fig. 1. The tip was found to have maintained its apex shape throughout the measurement despite the mechanical and electrical stresses. A complementary atomistic model described the near-contact region of the experimental system. As shown in Fig. 2, the model includes the apex of a Pt AFM tip and an adjacent graphite substrate. Seven graphene sheets were ABA-stacked, with an interlayer distance of 0.335 nm. Here we used 7 layers of graphene to represent the top layers of the HOPG thin film. The assumption is reasonable in light of a recent report showing that the transition to a bulk-like resistivity appears at 7–8 layers of graphene [34]. The atoms in the bottommost layer of graphite were fixed to model a supporting substrate and the atoms at one end of all the layers were fixed as well to prevent relative sliding. The boundaries were periodic in the x y plane (in the plane of the graphene sheets), and the boundary in the z-direction was formed by the fixed bottom layer of graphite and the rigid body of atoms at the top of the tip. The model tips had a hemispherical geometry and consisted of platinum atoms in a FCC structure, with the same radii as the experimental tips (32 and 77 nm). A constant external normal load was maintained on the atoms treated as a rigid body at the top of the tip. A Langevin thermostat was applied to the free atoms in the system to maintain a temperature of 300 K. The interatomic interactions within the tip and substrate were described via the Embedded Atom Method (EAM) potential and the Adaptive Intermolecular Reactive Empirical Bond Order (AIREBO) potential [39] respectively, and the long range interactions between tip and substrate were modeled using the Lennard-Jones (LJ) potential (energy minimum 0.022 eV, zero-crossing distance 0.295 nm). The simulations were performed using LAMMPS simulation software [40].
3. Results and discussion Fig. 3 shows the resistance measured with 32 and 77 nm radius tips under loads between 100 and 800 nN. For the 77 nm radius tip in Fig. 3(b), there are three sets of data. All resistances measured with the 77 nm tip are smaller than those measured with the 32 nm tip. Further, for both tips, resistance decreases monotonically with increasing load. The potential sources of the measured resistance are the Pt-coated tip, the HOPG and the contact itself. The resistivity of
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it has been shown that geometry and loading conditions may affect real and apparent area very differently [46–48]. The contrast between real and apparent contact area in our simulations is illustrated in Fig. 4 where the contour map shows that the distribution of the tip–substrate atom distances and the black spheres represent tip atoms in contact with the substrate. Tip atoms in contact are defined as those within 0.3 nm of a substrate atom. The threshold value of 0.3 nm is used simply because it is the average of the Pt interatomic distance (0.27 nm) and HOPG interlayer distance (0.33 nm). As expected, the real contact area appears to be only a subset of the apparent contact area. Using the simulation data, we can determine the real and apparent areas of contact as functions of load for both model tips. We quantify real contact area, Ar, as the number of atoms in the contact multiplied by the average surface area per atom [47]. The apparent contact area, Aa, is calculated as the circle having a radius equal to half the distance between the contact atoms furthest from each other in the plane of the contact. Variation of contact size with load and tip radius calculated from MD simulation using these two definitions of contact area is shown in Fig. 5(a). The size of the contact quantified by either definition of area increases with load and tip size. We next use the contact area results from simulation and the experimentally-measured resistance to calculate the contact resistivity where we assume R Rc ; the results are shown in Fig. 5(b). Note that the contact resistance from experiment shown in Fig. 3 was obtained at slightly different loads that the contact areas from simulation shown in Fig. 5(a). Therefore, power-law interpolation
Fig. 3. Contact resistance as a function of load measured by experiment for (a) 32 nm and (b) 77 nm tips; EXP1 - EXP3 correspond to three different measurements taken using the same tip.
Pt is very small (ρPt 10 7 Ω m [41]), so the tip can be neglected as a major factor. The resistance of the HOPG film in this system can be estimated as Rg ¼ ρg =kt, where k is a geometric constant and t is a thickness [42]. For our experiments, t ¼0.2 mm and the geometric constant can be estimated to be on the order of 1. The maximum resistivity of HOPG is in the out-of-plane direction and has been found to be up to ρg 10 3 Ω m [43]. This means that the largest possible resistance contribution of the HOPG is likely to be on the order of 10 Ω. This is orders of magnitude smaller than the measured resistivity reported in Fig. 3 which suggests that the observed resistance is predominantly due to the contact itself. Contact resistance can be defined as Rc ¼ ρc =A where ρc is the contact resistivity with units of Ω m2 and A is the contact area [44]. This Ohms law-based relationship was derived for larger scale contacts. However, although alternatives to Ohms law have been developed for cases where continuum assumptions may break down [50,51], we apply the simplest form of the contact resistance equation here. This approximation is justified by recent work demonstrating ohmic scaling to the atomic limit [52]. The challenge in evaluating contact resistance (Rc ¼ ρc =A) is that contact area is difficult to define at nanoscale. There are two possible definitions of contact area, real and apparent. On larger length scales, the apparent contact is the area between two solid surfaces defined by the boundaries of their macroscopic interface and the real contact includes the effect of the roughness and can be considered as the sum of the contact areas of many micro-scale asperities [45]. These concepts have been applied to smaller-scale contacts, such as those considered here, with the apparent area defined in terms of the boundaries of the nanoscale interface and the real contact area as the sum of atomic-scale “asperities” [46–48]. For nanoscale contacts,
Fig. 4. Illustration of the real (black spheres) and apparent (color contour) contact areas for the (a) 32 nm and (b) 77 nm tips from an MD simulation of contact with a 200 nN load. The colors in the contour plot reflect local distances between the tip and substrate; legend units are nm. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
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4. Conclusion To summarize, we study the nanoscale electric contact between an AFM tip and a HOPG surface. The resistance of different size tips (32 and 77 nm radii) under various loads (100–800 nN) is measured by AFM with voltage applied between tip and substrate. Analysis of the various potential contributions to the measured resistance indicates that the contact itself is the dominant factor. Contact resistance is related to contact resistivity and contact size. Atomistic simulations are used to quantify contact size in terms of real and apparent contact area. Contact resistivity is then calculated using measured resistance and both definitions of area from simulation. We observe that contact resistivity is independent of load only if the real contact area is used in the calculation which indicates real contact area is a better measure of contact size. The effect of tip size on contact resistance, however, cannot be fully captured by either definition of contact which indicates that contact resistivity is somewhat affected by tip size. Analysis suggests that one mechanism by which this might happen is through the interface distance. Overall, this study reveals that, although load and tip size affect the resistance of nanoscale electrical contacts, these effects can be predominantly captured by correctly defining contact size as the real contact area. Since real contact area can be readily calculated from simulations, this study encourages further collaboration between experiments and simulations for investigating nanoscale electrical contacts. Most significantly, this study emphasizes the role of atomic-scale contact features in determining contact resistance at the nanoscale.
Acknowledgments Fig. 5. (a) Variation of real and apparent contact area with load and tip radius from MD simulation. (b) Contact resistivity calculated from the experimentally measured resistance and the simulation-predicted contact areas. Real contact area Ar represented by hollow symbols and apparent contact area Aa by solid symbols for the 32 nm (squares) and 77 nm (triangles) radius tips.
AM and ZY authors would like to thank the U.S. National Science Foundation for its support through Grant no. 1068552CMMI. ML and HM acknowledge support from UC Merced faculty start-up fund. References
was used to estimate contact area from the simulation data at the experimentally-measured loads. We observe that the contact resistivity calculated using apparent contact area increases with load, but is nearly constant when calculated using the real contact area. Ideally, the effect of load on resistance can be captured entirely by the contact area such that the contact resistivity is a constant. Therefore, the trends in Fig. 5 (b) indicate that real contact area is a better measure of contact size than apparent contact for nanoscale electrical resistance studies. We also observe that the contact resistivity in Fig. 5 (b) calculated using either measure of contact area is larger for the smaller tip. This suggests that there is some other factor (in addition to area) that changes with tip size and affects contact resistance. A previous study on the resistivity of few-layer graphene showed the resistivity is linearly related to the distance between adjacent graphene layers [49]. We assume a similar relationship is applicable to contact resistance. Our simulations reveal that the average distance between the bottommost atoms of the tip and the topmost atoms of the substrate is 7% smaller with the 77 nm radius tip as compared to the 32 nm radius tip due to greater deformation. This is very consistent with the observation that the contact resistivity calculated from the real contact area of the 77 nm radius tip is 9% smaller than that calculated from the real contact area of the 32 nm tip. Therefore, tip size appears to affect resistance not only through contact area but also through near-contact deformation. Further study is needed to quantify the effect of deformation and determine if there are other factors involved.
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Contents lists available at ScienceDirect
Tribology International journal homepage: www.elsevier.com/locate/triboint
Size and load dependence of nanoscale electric contact resistance Zhijiang Ye, Hyeongjoo Moon, Min Hwan Lee, Ashlie Martini n School of Engineering, University of California-Merced, CA 95343, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 20 September 2013 Received in revised form 9 November 2013 Accepted 14 November 2013 Available online 23 November 2013
Nanoscale electrical resistance between a platinum-coated atomic force microscope tip and highly oriented pyrolytic graphite surface is measured as a function of normal load and tip radius. These measurements are complemented by molecular dynamics simulations that relate load and radius to contact area. Simulation-predicted contact area and experimentally-measured resistance are used to calculate contact resistivity. The results show that the effect of load on resistance can be captured by the real contact area, while tip size, although in part captured by area, affects contact resistivity itself, potentially through interface distance. Our study provides new insight into the effect of load and geometry on nanoscale electric contact and, more significantly, highlights the role of atomic-scale contact features in determining contact resistance. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Nanoscale contact Electric resistance Atomic force microscope Molecular dynamics simulation
1. Introduction As demand for high density, high speed electronic and electromechanical devices continues to increase, implementation of true nanoscale structures has become critical for next generation electronics engineering [1–3]. However, although contact resistance is one of the major parasitic components in electronic devices, conduction at nanoscale contacts is not yet clearly understood [4]. Furthermore, electrical transport through nanoscale contacts often exhibits behavior deviating from that of conventional contacts [5,6]. This can be partially attributed to the fact that the dimension of contact is smaller than the characteristic Debye length, resulting in an enhanced electrical conduction [5,7]. Nanoscale structures also have discrete energy levels which, in turn, can result in unconventional electric behaviors. Recently, single layer graphene (SLG) and multi-layer graphene (MLG) have been the focus of extensive research due to their unique properties, such as high electrical and thermal conductivity [8–11], very low friction [1,12,13] and the quantum hall effect [10,14,15]. Many studies have focused on electrical properties due to the wide applicability of SLG and MLG to various electronic and electromechanical devices [17–21]. Since contact resistance is a major factor limiting device scalability, reliability and performance in any electronic device [22–24], it is important to understand this behavior in graphene-based materials. Although there are quite a few studies on metal-SLG contact, little is known about the electrical properties of metal-MLG contacts [29]. Of the previous metal-MLG studies, several have focused on the effect of number
n
Corresponding author. E-mail address: [email protected] (A. Martini).
0301-679X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.triboint.2013.11.012
of graphene layers on contact resistance [31,32,34]. Other studies revealed a pressure dependence of electrical conductance on a double layer graphene surface [33] and that conductance is affected by step edges observed on the surface of MLG [30,35,33]. These findings highlight at the critical role of the contact and its atomic-scale features in determining nanoscale resistance. In this paper, we explore this important relationship between a nanoscale contact and its resistance by studying the electric contact between a platinum-coated AFM tip and highly oriented pyrolytic graphite (HOPG) surface. We measure the contact resistance with different size tips under various loads using the experiments, and then investigate the contact between tip and substrate using complementary simulations. The simulations are used to predict real and apparent contact area, and then this information is used with the resistance measured experimentally to calculate contact resistivity. The results are analyzed in terms of the ability of the two contact area definitions to capture the effects of load and tip size on resistance. 2. Method An atomic force microscope (AFM; Model 5500, Agilent Technology) system was used to enable contacts between a conductive probe and a HOPG (Alfa Aesar) surface with controlled forces. HOPG with a thickness of 200 μm was used for the electrical characterization. To avoid significant contamination and/or oxidation of the surface, the surface was always cleaved using a blade razor before each measurement. The strong sp2 covalent intralayer bonds and relatively weak van der Waals interlayer bonds in HOPG made a thin layer readily exfoliated [36]. To investigate the contact
110
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Fig. 2. Snapshot from a molecular dynamics simulation of the apex of a 32 nm radius AFM tip placed on a seven layer graphite substrate.
Fig. 1. Scanning electron microscope images of AFM tips with radii (a) 32 nm, (b) 77 nm, and (c) a simplified schematic diagram of the experimental setup.
area dependence, tips with two different apex radii (32 nm and 77 nm, shown in Fig. 1(a) and (b)) were prepared by depositing Pt on commercially available probes (Team Nanotec HSC Probe, NanoScience Instruments Inc.). The commercial probe was originally coated with tungsten carbide and its apex was in a hemispherical cone shape. Their nominal tip radii (the values specified by the vendor) were 20 nm and 40 nm. A dense Pt film was coated on the probes by a direct current sputter deposition scheme in an Ar pressure of 10 mTorr. The electrical characterization was conducted under ambient conditions at room temperature ( 300 K). The forces applied by the cantilever (with a spring constant of 3.0 N/m) were in the range of 100–800 nN during the electrical measurement. The tip–substrate contact resistances were obtained using a semiconductor parameter analyzer (HP4145B). An illustration of the experiential setup is shown in Fig. 1(c). In this setup, electrical current was measured during a potentiodynamic sweep performed from 0 to 50 mV when the tip is in contact with the HOPG surface. We observed a consistent linear relationship between the bias and the current (data not shown) such that the contact resistance could be acquired by applying the Ohms law (R ¼ V=I) with currents measured under any given bias; we used a
bias of 50 mV for this purpose. All sources of resistance except for that from the Pt–HOPG contact were found to be negligible ( o15 Ω) compared to the measured resistance which was in the range of kΩ. A capillary condensation of water from the ambient, which is usually called a meniscus, can be a medium for an alternative current path between the tip and the sample [37,38]. Considering a significantly higher resistance of the water meniscus than the Pt and HOPG contact, we also neglected this meniscus effect [31]. After the electrical characterization, the tip apex shape was examined using a field emission scanning electron microscopy as shown in Fig. 1. The tip was found to have maintained its apex shape throughout the measurement despite the mechanical and electrical stresses. A complementary atomistic model described the near-contact region of the experimental system. As shown in Fig. 2, the model includes the apex of a Pt AFM tip and an adjacent graphite substrate. Seven graphene sheets were ABA-stacked, with an interlayer distance of 0.335 nm. Here we used 7 layers of graphene to represent the top layers of the HOPG thin film. The assumption is reasonable in light of a recent report showing that the transition to a bulk-like resistivity appears at 7–8 layers of graphene [34]. The atoms in the bottommost layer of graphite were fixed to model a supporting substrate and the atoms at one end of all the layers were fixed as well to prevent relative sliding. The boundaries were periodic in the x y plane (in the plane of the graphene sheets), and the boundary in the z-direction was formed by the fixed bottom layer of graphite and the rigid body of atoms at the top of the tip. The model tips had a hemispherical geometry and consisted of platinum atoms in a FCC structure, with the same radii as the experimental tips (32 and 77 nm). A constant external normal load was maintained on the atoms treated as a rigid body at the top of the tip. A Langevin thermostat was applied to the free atoms in the system to maintain a temperature of 300 K. The interatomic interactions within the tip and substrate were described via the Embedded Atom Method (EAM) potential and the Adaptive Intermolecular Reactive Empirical Bond Order (AIREBO) potential [39] respectively, and the long range interactions between tip and substrate were modeled using the Lennard-Jones (LJ) potential (energy minimum 0.022 eV, zero-crossing distance 0.295 nm). The simulations were performed using LAMMPS simulation software [40].
3. Results and discussion Fig. 3 shows the resistance measured with 32 and 77 nm radius tips under loads between 100 and 800 nN. For the 77 nm radius tip in Fig. 3(b), there are three sets of data. All resistances measured with the 77 nm tip are smaller than those measured with the 32 nm tip. Further, for both tips, resistance decreases monotonically with increasing load. The potential sources of the measured resistance are the Pt-coated tip, the HOPG and the contact itself. The resistivity of
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it has been shown that geometry and loading conditions may affect real and apparent area very differently [46–48]. The contrast between real and apparent contact area in our simulations is illustrated in Fig. 4 where the contour map shows that the distribution of the tip–substrate atom distances and the black spheres represent tip atoms in contact with the substrate. Tip atoms in contact are defined as those within 0.3 nm of a substrate atom. The threshold value of 0.3 nm is used simply because it is the average of the Pt interatomic distance (0.27 nm) and HOPG interlayer distance (0.33 nm). As expected, the real contact area appears to be only a subset of the apparent contact area. Using the simulation data, we can determine the real and apparent areas of contact as functions of load for both model tips. We quantify real contact area, Ar, as the number of atoms in the contact multiplied by the average surface area per atom [47]. The apparent contact area, Aa, is calculated as the circle having a radius equal to half the distance between the contact atoms furthest from each other in the plane of the contact. Variation of contact size with load and tip radius calculated from MD simulation using these two definitions of contact area is shown in Fig. 5(a). The size of the contact quantified by either definition of area increases with load and tip size. We next use the contact area results from simulation and the experimentally-measured resistance to calculate the contact resistivity where we assume R Rc ; the results are shown in Fig. 5(b). Note that the contact resistance from experiment shown in Fig. 3 was obtained at slightly different loads that the contact areas from simulation shown in Fig. 5(a). Therefore, power-law interpolation
Fig. 3. Contact resistance as a function of load measured by experiment for (a) 32 nm and (b) 77 nm tips; EXP1 - EXP3 correspond to three different measurements taken using the same tip.
Pt is very small (ρPt 10 7 Ω m [41]), so the tip can be neglected as a major factor. The resistance of the HOPG film in this system can be estimated as Rg ¼ ρg =kt, where k is a geometric constant and t is a thickness [42]. For our experiments, t ¼0.2 mm and the geometric constant can be estimated to be on the order of 1. The maximum resistivity of HOPG is in the out-of-plane direction and has been found to be up to ρg 10 3 Ω m [43]. This means that the largest possible resistance contribution of the HOPG is likely to be on the order of 10 Ω. This is orders of magnitude smaller than the measured resistivity reported in Fig. 3 which suggests that the observed resistance is predominantly due to the contact itself. Contact resistance can be defined as Rc ¼ ρc =A where ρc is the contact resistivity with units of Ω m2 and A is the contact area [44]. This Ohms law-based relationship was derived for larger scale contacts. However, although alternatives to Ohms law have been developed for cases where continuum assumptions may break down [50,51], we apply the simplest form of the contact resistance equation here. This approximation is justified by recent work demonstrating ohmic scaling to the atomic limit [52]. The challenge in evaluating contact resistance (Rc ¼ ρc =A) is that contact area is difficult to define at nanoscale. There are two possible definitions of contact area, real and apparent. On larger length scales, the apparent contact is the area between two solid surfaces defined by the boundaries of their macroscopic interface and the real contact includes the effect of the roughness and can be considered as the sum of the contact areas of many micro-scale asperities [45]. These concepts have been applied to smaller-scale contacts, such as those considered here, with the apparent area defined in terms of the boundaries of the nanoscale interface and the real contact area as the sum of atomic-scale “asperities” [46–48]. For nanoscale contacts,
Fig. 4. Illustration of the real (black spheres) and apparent (color contour) contact areas for the (a) 32 nm and (b) 77 nm tips from an MD simulation of contact with a 200 nN load. The colors in the contour plot reflect local distances between the tip and substrate; legend units are nm. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
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4. Conclusion To summarize, we study the nanoscale electric contact between an AFM tip and a HOPG surface. The resistance of different size tips (32 and 77 nm radii) under various loads (100–800 nN) is measured by AFM with voltage applied between tip and substrate. Analysis of the various potential contributions to the measured resistance indicates that the contact itself is the dominant factor. Contact resistance is related to contact resistivity and contact size. Atomistic simulations are used to quantify contact size in terms of real and apparent contact area. Contact resistivity is then calculated using measured resistance and both definitions of area from simulation. We observe that contact resistivity is independent of load only if the real contact area is used in the calculation which indicates real contact area is a better measure of contact size. The effect of tip size on contact resistance, however, cannot be fully captured by either definition of contact which indicates that contact resistivity is somewhat affected by tip size. Analysis suggests that one mechanism by which this might happen is through the interface distance. Overall, this study reveals that, although load and tip size affect the resistance of nanoscale electrical contacts, these effects can be predominantly captured by correctly defining contact size as the real contact area. Since real contact area can be readily calculated from simulations, this study encourages further collaboration between experiments and simulations for investigating nanoscale electrical contacts. Most significantly, this study emphasizes the role of atomic-scale contact features in determining contact resistance at the nanoscale.
Acknowledgments Fig. 5. (a) Variation of real and apparent contact area with load and tip radius from MD simulation. (b) Contact resistivity calculated from the experimentally measured resistance and the simulation-predicted contact areas. Real contact area Ar represented by hollow symbols and apparent contact area Aa by solid symbols for the 32 nm (squares) and 77 nm (triangles) radius tips.
AM and ZY authors would like to thank the U.S. National Science Foundation for its support through Grant no. 1068552CMMI. ML and HM acknowledge support from UC Merced faculty start-up fund. References
was used to estimate contact area from the simulation data at the experimentally-measured loads. We observe that the contact resistivity calculated using apparent contact area increases with load, but is nearly constant when calculated using the real contact area. Ideally, the effect of load on resistance can be captured entirely by the contact area such that the contact resistivity is a constant. Therefore, the trends in Fig. 5 (b) indicate that real contact area is a better measure of contact size than apparent contact for nanoscale electrical resistance studies. We also observe that the contact resistivity in Fig. 5 (b) calculated using either measure of contact area is larger for the smaller tip. This suggests that there is some other factor (in addition to area) that changes with tip size and affects contact resistance. A previous study on the resistivity of few-layer graphene showed the resistivity is linearly related to the distance between adjacent graphene layers [49]. We assume a similar relationship is applicable to contact resistance. Our simulations reveal that the average distance between the bottommost atoms of the tip and the topmost atoms of the substrate is 7% smaller with the 77 nm radius tip as compared to the 32 nm radius tip due to greater deformation. This is very consistent with the observation that the contact resistivity calculated from the real contact area of the 77 nm radius tip is 9% smaller than that calculated from the real contact area of the 32 nm tip. Therefore, tip size appears to affect resistance not only through contact area but also through near-contact deformation. Further study is needed to quantify the effect of deformation and determine if there are other factors involved.
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