Supplementary Information Characterization of nanoscale temperature fields during electromigration of nanowires Wonho Jeong,, Kyeongtae Kim,, *, Youngsang Kim,, Woochul Lee,, *, Pramod Reddy Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA Department of Materials Science and Engineering, University of Michigan, Ann Arbor, MI 48109, USA These authors contributed equally to this paper. *Corresponding authors: kyokim@umich.edu, pramodr@umich.edu 1
1. DC/AC schemes Ultra-high vacuum scanning thermal microscopy (UHV-SThM) studies were performed by employing two different schemes. In the first approach (DC scheme), we measured local temperatures by placing the probe in contact with the device at a location ~100 nm away from the region where the cathode connects the nanowire. Although the temperature resolution in the DC scheme is rather poor (~2 K), it was possible to measure rapid changes in local temperatures as a larger bandwidth (~30 Hz) was employed. In the second approach, an AC scheme was employed to measure the temperature fields in micrometer-sized areas. The advantage of the AC scheme is that the temperature resolution can be significantly enhanced by narrowing the bandwidth of the measurement. However, obtaining such detailed thermal maps with the AC scheme requires a substantial amount of time (~85 minutes) for each thermal map. The details of these two schemes are provided in the following two subsections. 1.1. Local temperature measurement with the DC scheme While the nanowire was electromigrated under a cyclic bias voltage, the SThM probe was placed in contact with the device at a location ~100 nm away from the region where the cathode connects the nanowire. Further, the contact force was controlled using force feedback to maintain a constant contact force (see Fig. S1a). As the nanowire was electromigrated, the temperature of the device changes. These temperature changes were measured by monitoring the thermoelectric voltage generated from the nanoscale thermocouple integrated into the tip of the SThM probe. The thermoelectric voltage was amplified by a voltage preamplifier (gain of 1000) and low-pass filtered at 30 Hz to obtain the temperature signals. This enabled a direct in-situ measurement of the local temperature of the device. 2
Figure S1: Schematics of the DC and AC schemes. (a) In the DC scheme, the probe was placed in contact with the device (at a constant contact force using feedback) ~100 nm away from the region where the cathode meets the nanowire. The thermoelectric voltage from the thermocouple was constantly monitored while the bow-tie shaped Au nanowire was electromigrated. (b) In the AC scheme, a sinusoidal current at 1f (5 Hz) was applied, which resulted in temperature oscillations at 2f (10 Hz). As the SThM probe scanned over the bow-tie device under the constant force feedback, the temperature signals (the thermoelectric voltages) at 2f (10 Hz) were recorded using a lock-in amplifier in a bandwidth of ~0.5 Hz. The topographic information was also simultaneously obtained by monitoring the deflection signals of the cantilever. 1.2.Thermal mapping with the AC scheme In the AC scheme, a sinusoidal current was supplied to the bow-tie shaped Au nanowires. This sinusoidal current oscillating at a frequency of 1f (5 Hz) resulted in temperature oscillations at 2f (10 Hz). The SThM probe was scanned on the Au nanowire (around the bow-tie shaped region) under a constant contact force, while recording the thermoelectric voltages from the thermocouple using a lock-in amplifier (Stanford Research System, SR 830). The time constant of the lock-in amplifier was chosen to be 300 msec and the scan speed was chosen to be 40 seconds per line (128 lines in total for each thermal map). The topography of the device was simultaneously obtained from the deflection signals of the cantilever. 3
2. SThM probe calibration The sensitivity of the SThM probe is defined as the temperature rise of the tip (when in contact with the sample), as measured by the integrated thermocouple, per unit temperature rise of the sample S1. In order to determine the sensitivity of the SThM probe, we employed a Au line (40 nm thick and 4 µm wide) that was deposited on a Si substrate with a thermally grown 500 nm thick SiO 2 layer. The Au line used in the experiment was patterned in a four-probe configuration as shown in Fig. S2a. In order to heat the Au line, an AC current at a known frequency 1f (5 Hz) and amplitude (I 0 ) was supplied through it. The temperature rise of the Au line was measured using the 3 method S2. To elaborate, the electrical current at a frequency of 1f results in Joule heating and temperature oscillations at a frequency of 2f. The temperature oscillations in turn result in the oscillations of the line resistance at a frequency of 2f. Therefore, the voltage across the line oscillates at 3f, and it is related to the temperature oscillations by T2f 2( dt / dr) V3f / Io, where ΔT 2f is the amplitude of the temperature oscillations at 2f, R is the resistance of the line and ΔV 3f is the amplitude of the voltage oscillations across the line at 3f. In order to characterize the sensitivity of the SThM probe (heat transfer between the probe and the heated line), we also measured the temperature rise of the thermocouple from the thermoelectric voltage output of the thermocouple (ΔV TE,2f ). The measured thermoelectric voltage is related to the temperature rise of the thermocouple by DT TC,2 f = DV TE,2 f / S TC, where ΔT TC,2f is the temperature rise of the thermocouple and S TC is the Seebeck coefficient of the thermocouple, which was determined to be 16.3 ± 0.1 V/K S1. The measured temperature rise of the thermocouple per unit temperature rise of the sample is shown in Fig. S2b. The slope of the graph gives the sensitivity of the SThM probe, 4
which is estimated to be 0.083 ± 0.006 K/K. The obtained sensitivity is ~2.5 times larger than that was used in our previous work S1. We believe that this increased sensitivity is due to the presence of an additional insulating layer of Al 2 O 3 on the outermost surface of the SThM probe, which results in an increase of the contact area. This additional Al 2 O 3 layer (~10 nm thick) was coated by atomic layer deposition to prevent possible electrical short between the sample and the probe during extensive scanning measurements. Finally, we note that our frequency response measurements, which are similar to those reported in our previous work S1, show that the response of the probe is invariant until 10 Hz. Therefore, the sensitivity of the SThM probe is the same as that reported above even in the case of DC measurements. Figure S2: Estimation of the sensitivity of the SThM probe. (a) Schematic diagram of the experimental setup used to measure the sensitivity of the SThM probe. (b) The measured temperature rise of the thermocouple as a function of the temperature rise of the heated Au line. 3. Estimating the maximum temperature of the device using finite element modeling calculation In the manuscript, we reported that the maximum temperature (before electromigration) is ~20 % larger than what we obtained in the local temperature measurements using the DC scheme. To quantify this deviation, we computed the temperature profile in an as-fabricated device using a 5
finite element modeling software (COMSOL). The result of this calculation is shown in Fig. S3 in a normalized form by dividing the temperatures by the maximum temperature, which is found to occur in the middle of the nanowire. It is clear from Fig. S3 that the temperature of the point at which the local temperature measurements were performed is smaller (~0.8 times) than the maximum temperature of the device. Figure S3: Finite element modeling of the temperature field in an as-fabricated device. The temperature profile of an as-fabricated device shows the temperature at which the local temperature measurements performed (indicated by the black dot) is ~20 % lower than the maximum temperature of the device. 4. Finite difference scheme to calculate the time derivative of the number density of atoms We note that in all the devices studied in our work the nanogap is preferentially created in the cathode after electromigration. This behavior can be qualitatively understood using a relatively simple model. We begin by noting that the conservation of mass during electromigration implies that S3, S4 : 6
n t + ( Ñ J ) = 0 (S1) where n is the local number density of atoms and J = J E + J S + J T is the local atomic flux. Here, J E, J S, and J T are the atomic fluxes induced by the electric field, back stress and temperature gradients, respectively S3, S4. Equation S1 can be simplified further by neglecting the small variations in the current density along the y-direction (see Fig. S4b), to give: n( x, t) J x t x 0 (S2) where n(x, t) is the local number density of atoms at a position x and time t, and J x is the x- component of the local atomic flux. It is known that after the onset of electromigration J E dominates over all other components ( J S and J T ) S3, S4. The relative effect of J E and J T on n(x, t)/ t at t = 0 along the centerline of the device is shown in Fig. S4c. Thus, under these approximations. Further, J x can be related to the x-component of the local electric current density j x by: J x ~ J E, x = ez * n(x,t)dr(x,t) k B T j x (S3) where ez * is the effective charge of atoms, (x, t) is the local resistivity of the device at t, T is the absolute temperature, and D is the local diffusion constant given by: a/ B D D0e E k T (S4) 7
where E a is the activation energy for atomic diffusion of Au and is ~0.12 ev (Ref. S4) and D 0 is a constant. In order to model the atomic flux, we first computed the temperature profile in an asfabricated device (before electromigration) using a finite element modeling software (COMSOL). The result of the computed temperature field is shown in Fig. S4a. This temperature profile was used in conjunction with our simplified one-dimensional model, described by equations S2, S3 and S4, to compute the time derivative of the local number density of atoms ( n(x, t)/ t) at t = 0 along the centerline of the device (Fig. S4) using a finite difference scheme. The obtained temperature profile was used to compute J E, x at each location of x using equations S3 and S4. In this calculation, we assume that (x, t = 0) = 4.4 10-8 Ω m. Subsequently, we computed J x / x by the following approximation J x / x» [J x (x + Dx)- J x (x)]/ Dx, where Dx= 10 nm. We note that this one-dimensional model is a good approximation in the nanowire and in close proximity to the nanowire as both the electric current density and temperature fields have a very weak y- dependence. We also note that the values of n(x, t = 0)/ t presented in Fig. S4b are normalized such that the maximum value of n( x,t = 0) / t is equal to 1 for the highest electric current density, therefore, the results are independent of the values of n(x, t = 0) and D 0 in equations S3 and S4, respectively. These results are shown in Fig. S4b where it can be seen that n(x, t = 0)/ t < 0 in the cathode signifying a depletion of the number density. In contrast, n(x, t = 0)/ t > 0 in the anode implying an increase in the number density. Further, it can be shown that this asymmetry in the time derivative of the local number density of atoms is ultimately related to the presence of temperature gradients in the device, in the absence of which equations S2, S3 and S4 would not predict any accumulation or depletion of atoms in either the anode or the cathode. 8
Figure S4: Temperature profile of a nanowire device and time derivative of the local number density of atoms at t = 0 along the centerline of the device. (a) The calculated temperature field (top panel) and temperature profiles (bottom panel) along the centerline (shown by the dotted line) of the bow-tie structure for the two different electric current densities supplied through the device (current densities were estimated using the cross-sectional area of the nanowire (~10-14 m 2 )). (b) The time derivative of the local number density of atoms at t = 0 (bottom panel) along the centerline (the dotted line) of a bow-tie shaped nanowire (top panel) obtained for the two different electric current density conditions using a finite difference scheme. The depletion of atoms is seen in the converging region of the cathode, while the accumulation of atoms is observed in the converging region of the anode. (The electron flow direction is indicated by the black arrow.) The region of the graph from -0.225 µm to 0.225 µm corresponds to the nanowire of the device and the origin corresponds to the center of the device. (c) The relative effect of J E and J T on n(x, t)/ t at t = 0 along the centerline calculated for a current density of j = 1.5 10 12 (A/m 2 ) and the corresponding temperature profile. The atomic fluxes induced by the electric field are much larger than those induced by the temperature gradients. References S1. Kim, K.; Jeong, W. H.; Lee, W. C.; Reddy, P. ACS Nano 2012, 6, (5), 4248-4257. S2. Cahill, D. G. Rev Sci Instrum 1990, 61, (2), 802-808. S3. Korhonen, M. A.; Borgesen, P.; Tu, K. N.; Li, C. Y. J. Appl. Phys. 1993, 73, (8), 3790. S4. Trouwborst, M. L.; van der Molen, S. J.; van Wees, B. J. J. Appl. Phys. 2006, 99, (11), 114316. 9