A Quantitative and Predictive Model of Electromigration-Induced

An isothermal model of electromigration breakdown of metal nanowires 80–700 nm in diameter is developed and validated using experimental data obtain...
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A Quantitative and Predictive Model of Electromigration-Induced Breakdown of Metal Nanowires Darin O. Bellisario, Zachary Ulissi, and Michael S. Strano* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States S Supporting Information *

ABSTRACT: An isothermal model of electromigration breakdown of metal nanowires 80−700 nm in diameter is developed and validated using experimental data obtained from isolated cylindrical Au nanowires. The model considers electromigration from an applied current producing a net flux of metal atoms, reducing the nanowire radius and conductivity precipitously and accounting for both mass and electronic carrier transport. The model successfully predicts the observed critical failure current, the correct scaling with nanowire radius to 3/2 power, and the impedance evolution prior to breakdown. Application to the case where feedback control is employed to limit the rate of nanowire thinning reproduces key features, including slowed necking, a threshold current and voltage after which lower bias is required to advance formation, and the dependence of these values on feedback parameters.

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electron microscopy observations.15,19,20 The current flux provides our driving force, consistent with the Black model of failure,21 and an electron-scattering cross section relates it to atomic dislocation. Experimental data in Ensinger et al.22,23 are then used to validate the model as being able to predict with exceptional accuracy not only the critical failure current but also the specific resistance path taken to failure for cylindrical NWs with a range of radii from 40 to 350 nm. After reproducing their experimental results and extracting key empirical parameters for scattering cross section and resistivity, we add feedback control algorithms used by other experimental groups, including ours, and find that all of the expected features are predicted for their NWs as well, suggesting a uniform formation mechanism across specific platforms and a general utility to our model. The resulting predictions lend insight into the formation process, providing a picture of the time evolution of the radius and the physical impact of different formation control mechanisms.

INTRODUCTION On-chip tunnel junctions enable exploration of single-molecule transistors (SMT),1−3 trace analyte detection,4 and detailed single-molecule spectroscopy.5,6 They rely on the formation of subnanometer-separated electrodes across molecules of interest, yielding measurable tunneling currents. The most successful approaches to creating sensitive and stable devices have been breaking metal nanowires mechanically7,8 or electrically.8−10 In the latter case, a current or voltage is enforced across a nanowire, inspiring self-diffusion driven by the electron wind force. When the nanowire (NW) resistance is monitored and the applied bias is adjusted, the NW can be broken such that a subnanometer gap remains.9−11 The scalability in fabrication and operation of this platform enables high throughput studies and is critical for technological application, but reproducibility in the gap formation process remains a limitation;12−14 variation in and debate over the impact of process control strategies and NW properties remain broad and unresolved in the literature.9,15,16 The body of results reflects that the geometry (e.g., cylindrical, rectangular, bow tie) and dimensions (20−400 nm in radius, width, or height) of the NW and the current−voltage path taken (e.g., specific ramp and feedback control schemes) can play crucial roles in size control of the final gap.9,12,14,16,17 Modeling the formation process is therefore critically important. Joule heating mechanisms have been investigated, suggesting a critical temperature for formation in some systems, but a connection to geometric changes in the wire as well as the isothermal current dependence, important for careful control, remains unclear.9,11,17,18 In this work, we develop a surface transport model in an anticipated, fixed “necking” region where atoms are preferentially removed by electromigration, which is supported by real-time © 2013 American Chemical Society

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METHODS Model. To describe the electromigration process, we adopted a surface transport mechanism, anticipating that translocation barriers are lower and available sites are substantially more numerous than in the bulk crystal. This presumption is later verified by deriving equivalent expressions for bulk diffusion, which prove to be incongruous with experimental results (Figure 2). For a basis, we took cylindrical metal nanowires with some time-dependent current applied, but analogous treatments could be presented for other geometries or applied bias control. An Received: April 11, 2013 Revised: May 21, 2013 Published: May 21, 2013 12373

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1 8 γαΔt 2 + k 0 = −Δ3 ln(2r − Δ) − 2Δ2 r − 2Δr 2 − r 3 2 3 8 k 0 ≡ −Δ3 ln(2r0 − Δ) − 2Δ2 r0 − 2Δr0 2 − r0 3 3

electron-scattering treatment where surface atoms interact with the current flux through an empirical scattering cross section provides a convenient representation of the wind force scattering rate FS FS = F0(r , t ) σcs N (r )

(1)

(4)

where F0(r,t) is the incident current flux in the surface region as a function of time t in a wire of radius r, σCS the atomic scattering cross section, N(r) the number of atoms in the surface region, and FS the resulting number of scattering events per time. It has been observed experimentally that well before breakage “necking” occurs, where some region of the NW is thinned, often with a downstream extrusion forming from the lost material.15,19,20 We assume that the specific shape of the necking region can be reduced to some equivalent symmetric representation (cylindrical in this case) and that the length of the hypothetical region δ is constant (Figure 1).

Choosing representative parameters, we can qualitatively observe a precipitously accelerating reduction in radius with time, as we expect from experimental results (Figure 1B).9,10 Experimental Data Set. To validate our proposed model, we used the experimental results of Ensinger et al.23 We chose this work because they report experimental data from a prototypical process (increasing the current flux at a fixed rate through a cylindrical NW until breakage) and because they provide a broad data set including several important vectors for deriving model parameters, including initial resistance, current and voltage with time, and failure current information, all as functions of a range of well-characterized initial radii.22,23 They used ion track-etched polycarbonate membranes as templates to form cylindrical nanowires using electrochemical Au deposition. The resulting NWs were left in the membrane and were 30 μm in length and 40−360 nm in radius. Their radii were verified by scanning electron microscopy and conductometry. Single NWs were contacted with an upper Cu electrode at room temperature, and contact resistances were estimated to be O(10 Ω). They then applied increasing current to the NWs at a fixed rate of 0.1 mA/s until breakage occurred, measuring the applied voltage over this time.22,23

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RESULTS Predicting Critical Failure Current. Ensinger et al. ramped a current applied to their NWs at a fixed rate of 0.1 mA/s until breakage occurred, observing a trend in the critical failure current density (relative to the initial radius).23 We evaluated our model by applying it to these data, optimizing the scattering cross section for fit.24 Solving eq 4 for current and substituting rc (critical failure radius) for r results in

Figure 1. (A) Diagram of the model geometry. (B) Predicted nanowire radius vs time trace for a NW with an initial radius of 40 nm (using eq 4).

Icrit ⎡ 2α ⎤1/2 =⎢ ( −Δ3 ln(2rc − Δ) − 2Δ2rc − 2Δrc 2 − 8rc 3 − k 0)⎥ ⎣ γΔ ⎦

For a fixed length of the necking region δ, the number of Au atoms in the surface region is N = πδnAu(2rΔ − Δ2), where r is the radius of the neck, Δ the depth of the surface region under consideration (one atomic layer), and nAu the number density of bulk gold. In this surface region we have a resulting electron flux of I/eπr2. If we define a new empirical cross section ϕCS as the product of the electron-scattering cross section σCS and the fraction of scattering events that yield atomic loss, we have a simple expression for atomic loss in the neck region

(5)

The result (Figure 2) shows excellent agreement with the experimental data. We can benchmark the model by comparison to a pure regression, finding them to agree well within experimental error. The log−log slope (which is the order of the r dependence of the rate of radius change) we predict is −0.499; this corresponds to a 3/2 power scaling between the critical current and initial radius (eq 6). The fit yields a ϕCS of 8.86 × 10−27 cm2.

⎛ F ⎞ dN = − ϕCS⎜ 2 ⎟[πδnAu(2r Δ − Δ2)] ⎝ eπ r ⎠ dt I (2r Δ − Δ2 ) = −δϕcsnAu e r2

Icrit = Aπr0 3/2

Our extracted value for A in this system is 1029.17 A/m2. We can validate our surface transport premise by comparison to a bulk transport model derived and optimized in the same fashion. In Figure 2 (inset), we can see that the resulting −1/r necking rate dependence is unsuitable for describing the data. Note that while the data has a clear overall trend, there remains considerable variability in the observed failure current, as high as O(10 mA), well beyond the reported measurement error bars. Uncontrolled experimental variables (e.g., organic impurities at

(2)

Substituting in the number of atoms in the necking region N = πr2δ × nAu we have (2r Δ − Δ2 ) dr = −γI dt r3

γ≡

ϕcs 4π e

(6)

(3)

For a constant current ramp at rate α A/s (I = αt), we can integrate analytically 12374

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r ≫ le ρAu ρS

=1−

3 3 (1 − p) + 3 (1 − p)2 4k 8k

∞

∑ v=1

pv − 1 v2

(8)

r ≪ le 2 ⎞ 1+p 3k 2 ⎡⎢ 1 + 4p + p ⎛ ⎜⎛ 1 ⎟⎞ ⎜ln + 1.059⎟ k− 2 ⎝ ⎝ ⎠ ⎠ ρs 1−p 8 ⎢⎣ (1 − p) k ∞ 2 3 2k3 1 + 11p + p ⎤⎥ − (1 − p)2 ∑ v 3pv − 1 ln v − 15 (1 − p)3 ⎥⎦ v=1

ρAu

=

(9)

where k ≡ r/le and v is the electron velocity.27 In all cases, we take the infinite series to convergence. For radii between the extreme limits, we approximated the resistivity by calculating the value at that radius from both equations and averaging those two values weighted by the proximity to each, choosing demarcations of 60 and 20 nm. Note that while FS scattering depends continuously on the NW radius, the MS grain scattering is connected to the wire dimensions only indirectly through the crystallization kinetics. The grain size and specularity have been shown to vary between experimental systems for the same material because of differences in grain, surface facets, and impurity levels resulting from fabrication methods.22,25,26 For Au, values from 0.4 to 0.9 for RG and 0 to 0.6 for p have been inferred in different experimental setups.22,25 To extract the crystal properties of our experimental data set, we calculated the initial resistance for a NW of a given radius using Matthiessen’s rule and optimized p and αG given Karim et al.’s measurements. As a first approximation, verified by the quality of the fit, we took the grain size to be constant across the entire diameter range. This yields an αG of 0.22 and a specularity of 0.36 with a root-mean-square deviation of 8.7 Ω. Figure 3

Figure 2. Observed critical failure current from Ensinger et al. compared to surface diffusion model predictions (eq 5). The data is linearized on a log−log plot (inset), showing that the model prediction is as good as a pure linear regression. Comparison to an equivalent bulk model validates the surface diffusion treatment.

the NW surface, surface charging at the polymer interface, and variation in contact resistance) may provide this experimental scatter. This observation is important later when we reproduce the traces from single experiments. Predicting Resistance Evolution during Necking. In order to relate our r(t) predictions to time-dependent observables, we require valid expressions for resistivity. It has been broadly demonstrated that a bulk Au resistivity, or even a modified but radius-independent resistivity, is insufficient to describe NW resistance.22,25−30 In this size regime, two effects are considered critical: the change in grain boundary scattering behavior from bulk due to smaller grain sizes and the increased importance of carrier scattering off surface states. Mayadas and Shatkes (MS) pioneered the commonly accepted behavior in the former case, expressed here for a cylinder:22,25,26,28 ⎡1 α ⎛ 1 ⎞⎤ = 3⎢ − G + αG 2 − αG 3 ln⎜1 + ⎟⎥ ⎝ ⎣3 ρG α ⎠⎦ 2

ρAu

αG ≡

le R G Dg 1 − R G

(7)

Figure 3. Nanowire resistance vs radius data from Karim et al. with a fit using FS and MS models of resistivity.

where ρG is the resistivity due to enhanced grain boundary scattering, ρAu the bulk Au resistivity, le the mean free path of electrons (we take to be 40 nm in Au at 298 K), RG the reflectivity of the grain boundaries, and DG the mean grain diameter. Fuchs and Sondheimer (FS), with later refinements for cylinders by Dingle, described the effect of surface scattering as dependent on the specularity of the material, p, with p = 0 representing a completely diffuse surface scattering behavior and p = 1 representing a totally reflective surface.27,29,30 Two regimes are considered: when the radius is sufficiently above or below the electron mean free path, expressed in eq 8 or 9, respectively, for a cylinder.

shows that with these parameter values the FS and MS models describe the data exceptionally well over the wide diameter range. At radii above 110 nm, one can observe a slight systematic deviation, which could be attributed to a slight change in grain size resulting from fabrication of these larger NWs. If desired, this effect could be accounted for by fitting a different αG for initial radii in this range. Note again that, just as in the case of the critical failure point, undetermined variables are providing variation in the initial resistance data for specific data points. Using these resistivity parameters, we modeled the time traces of individual experiments and used those results to empirically 12375

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determine the necking region length, δ, for each radius (Figure 4). As earlier noted, the critical failure current and the initial

Figure 4. (A) Observed (circles) and model-predicted (lines) resistance vs current traces for different initial radii. Because it is ramped at a fixed rate, current is a proxy for time in this system. (B) The resulting necking region lengths optimized to fit the curves in (A).

resistance data both show the influence of uncontrolled variables, leading to variation in the data that is impossible to simulate. In modeling the time evolution of single NW experiments, this provides an unpredictable offset in the initial NW resistance and the final breakage current. As we are interested in prototypical cases, we rectify this experimental variation by proportionally shifting the current data and the resistance data from each experiment such that the failure current and initial resistance collapse to their respective trend lines. Despite the simplicity of the model assumptions on evolution of the necking region dimensions, excellent prediction of timedependent behavior is achieved. With the validity of the model’s prediction established through experimental observables, it can be used to examine unmeasured quantities, the most relevant of which is the effective neck radius, shown in Figure 1 for the 40 nm radius NW. Predicting Nanowire Response to a Control Scheme. The value of our predictions is in devising optimal strategies for gap formation. In particular, there has been debate in the literature over the value and parametrization of using a feedback loop strategy for controlling the applied bias during formation.9,11,15 In a method pioneered by Johnson et al., the resistance of the NW is monitored as the current or voltage is increased, as in Karim’s system, but when the resistance increases by some threshold percentage, the applied current or bias is rapidly reduced by a percentage to arrest the formation; then the process repeats.9 We used a finite-difference method with time steps of 1 ms to predict the result of using this control method for Karim et al.’s NWs, with a variety of parametrizations, given our model of electromigration (Figure 5). The experiments of Ensinger et al. did not employ a feedback control mechanism during electromigration breakdown. However, our model can be used to predict the behavior under these conditions despite substantial differences associated with

Figure 5. (A) Simulated current versus time trace in a 50 nm initial radius nanowire for cases of no feedback and feedback with different parameters. Discrete feedback control evolution with a percentage reduction in current (ΔJ/J) in response to a percentage reduction in resistance (ΔR/R) is overlaid on the equivalent continuous feedback behavior (see text). (B) Simulated evolution of radius and resistance with time. (C) Evolution of the conductivity with the applied current. Curves are included (in pink) for x = −0.25 (solid), −0.4 (dashed), and −0.7 (dotted) to show trends in deviation from the uncontrolled scenario. (D) Experimental results of a feedback loop in our own electron beam lithography-defined nanowires showing qualitative agreement (see Supporting Information for experiment details). (E) The effect of the feedback parameter on the turnaround radius.

electron beam lithography-based fabrication in those systems such as longitudinal contact with an oxide substrate; thin, rectangular NW dimensions; and electron beam evaporation for crystal formation.9 The time for formation is lengthened, and the reduction in radius is made more gradual. Most distinguishing, once a critical current is reached, successively lower and lower currents are required to provide the same resistance increases, providing a maximal current at some turnaround radius. These observations are completely consistent with experimental results in other systems (Figure 5D) and are captured by our model as a result of a 1/r2 dependence of radius reduction rate on current. We can understand this point by considering the continuous analog of this discrete feedback process

dI Ix dR =α+ (10) dt R dt which, with eq 3, defines the smooth curves in Figure 5. x is the negative, unitless control parameter corresponding to the 12376

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the key behaviors observed experimentally.9 This work provides greater insight into the nature of electromigration failure, which must be carefully controlled for use in single-molecule transistors and sensing architectures taking advantage of the electron tunneling phenomenon.

proportional change in J to be induced by a proportional change in R, or in the language of a discrete feedback loop, x ≡ ΔI/I × I/ ΔI . More negative x values therefore correspond to more aggressive controla longer time to failure and lower maximal current. As x increases, the turnaround occurs at an earlier time and higher radius despite a longer process, leading to more thinning occurring in the “tail” portion of the current−time trace (Figure 5, parts A and C). Furthermore, the radius reduction at the turnaround radius can be extracted from x, providing physical intuition on observed current traces. Limitations of the Model. The model predicts many aspects of electromigration behavior despite its simplicity. However, there are several limitations of note. First, our model is isothermal, that is, we have neglected Joule heating. Temperature increases of 100−300 K have been suggested,15,17,18 yielding higher mobility of Au atoms, higher resistivity of the NW, and potential temperature-gradient diffusion. In some systems this alone can be used to account for the initial failure of the nanowire.17,18 A more general model should include thermal and surface transport effects. Regardless, in this experimental system we have demonstrably been able to reproduce the observed results without consideration of a change in temperature. Second, our reduced representation of the necking region as symmetric and fixed in length is not, and is not meant to be, precisely observed. The actual necking region could be asymmetric (a dependence of r on position), for example bowing in the middle, providing a higher current density over a smaller region. Additionally, the necking region length could change considerably over the course of the process. In fact, asymmetric behavior could be approximated as a reduction in the length of effective neck, δ, as the highest current density region shows the dominant kinetics. Our simplified reduction of these effects has proven sufficient to describe the current−voltage behavior of the formation process, but a finer examination of δ and ϕ should take them into account. As is, our model does not predict actual wire morphology. This could provide a clearer physical representation but does not impact the demonstrated predictive ability of our model. Finally, we expect the actual failure at the end of the process to be stochastic, as only a handful of atoms will generally form the tunnel gap. As a result, conductivity behavior with time at the end of the process cannot be well-described. For a cutoff of ∼100 atoms/nm, this corresponds to a maximum valid radius of ∼1.5 nm for Au. By the same token, the model cannot account for the shape and size of the gap formed. Nevertheless, our deterministic model describes the necking process leading up to the final break, providing anticipation and adjustment of the speed at which the radius is reduced and, critically, the voltage applied at failure.

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ASSOCIATED CONTENT

S Supporting Information *

Evolution of observables (resistance, current, and voltage) at the turnaround radius with the feedback parameter x and experimental methods for the trace in Figure 5D. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139. Phone: 617-324-4323. Email: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS D.O.B. thanks the Department of Defense and American Society for Engineering Education for funding through the National Defense Science and Engineering Graduate program.

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VARIABLES N Number of Au atoms in the surface region δ Length of the necking region r Radius of the necking region r0 Radius of the wire and initial radius of the necking region Δ Depth of the Au surface region I Current flowing through the wire α Fixed rate at which current is ramped ϕCS Empirical scattering cross section t Time nAu Number density of Au atoms in an fcc lattice γ Constant for convenience defined in eq 3 Icrit Current at which precipitous wire failure occurs A Empirical constant for power law relationship in eq 6 ρAu Bulk resistivity of Au at 298 K ρG Mayadas and Shatkes grain-boundary resistivity le Electron mean free path in Au Dg Mean grain diameter RG Reflectivity of the Au crystal grains ρS Fuchs−Sondheimer surface electron scattering p Specularity of the nanowire surface region x Continuous feedback control parameter

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CONCLUSIONS We have developed a model of electromigration failure in metal nanowires based on a surface diffusion mechanism driven by the current flux and resulting electronic wind force. Taking a representative control volume over which transport is kinetically favored, consistent with ubiquitous experimental observation, precipitously more rapid thinning of the nanowire is observed. Application of our model to the experimental data set on cylindrical Au nanowire failure from Ensinger et al.22,23 predicts, including strong diameter dependences, the critical failure current and the time-dependent resistance pathway taken to failure. Finally, we used the model to investigate the effect that a feedback control scheme would have, yielding reproduction of

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