CORRIGENDUM: Characterization of nanoscale temperature fields during electromigration of nanowires

Quantitative studies of nanoscale heat dissipation (Joule heating) are essential for advancing nano-science and technology. Joule heating is widely expected to play a critical role in accelerating electromigration induced device failure. However, limitations in quantitatively probing temperature fields—with nanoscale resolution—have hindered elucidation of the role of Joule heating in electromigration. In this work, we use ultra-high vacuum scanning thermal microscopy to directly quantify thermal fields in nanowires during electromigration. Our results unambiguously illustrate that electromigration begins at temperatures significantly lower than the melting temperature of gold. Further, we show that during electromigration voids predominantly accumulate at the cathode resulting in both local hot spots and asymmetric temperature distributions. These results provide novel insights into the microscopic details of hot spot evolution during electromigration and are expected to guide the design of reliable nanoscale functional devices.

with a 500 nm thick thermally grown SiO 2 layer. Subsequently, source/drain electrodes were defined by photolithography and evaporation (Ti/Au, 3/70 nm thick) in order to ensure good electrical contact with the Au nanowires. The detailed fabrication process of SThM probes, with integrated gold-chromium thermocouples, can be found elsewhere 21 . In order to carry out thermal measurements during electromigration the bow-tie samples were placed in a UHV chamber, and all experiments were performed under UHV conditions (,10 29 torr) at room temperature (,300 K) or at a low temperature (,100 K).
To perform electromigration we adopted a process originally developed by others 8,11,27 . Briefly, we applied a cyclic voltage to the Au nanowires of the bow-tie structure (Figs. 1c, d) while the electrical resistance was continuously monitored. The applied voltage was linearly ramped up in steps of 5 mV every 100 ms until the resistance increased by 1 , 5%. The threshold value was manually controlled depending on the monitored resistance of the nanowire samples. Upon detection of the desired increase in the resistance the voltage was rapidly ramped down. We note that the resistance of the device increased irreversibly as electromigration proceeded; this is in contrast to reversible changes in resistance due to temperature variations. This process was continued until the electromigration process was completed resulting in a nanoscale gap in the bow-tie devices as shown in Fig. 1b. The formation of a nanoscale gap is readily recognized by a rapid increase in the electrical resistance.
To measure the local temperature rise of the Au nanowires during the electromigration process, we first placed the SThM probe in mechanical contact with the bow-tie shaped devices at a contact force of 150 6 25 nN. The contact location was chosen to be on the cathode side (i.e. the electrode from which electrons enter the nanowire) at a point that is ,100 nm from where the cathode meets the nanowire as illustrated in Fig. 1a. This location is well suited because: 1) it is sufficiently close to the nanowire to record temperatures that are representative of the local temperature rise of the nanowire, 2) this is the region where the temperature rise is the largest after electromigration (as explained in detail later), and 3) the measured temperature signal is not affected by the electromigration induced topographical changes of the devices. The temperature rise of the nanowires was recorded via the thermocouple integrated into the probe while a cyclical voltage was applied to the bow-tie devices to initiate electromigration. The voltage output of the integrated thermocouple (DV) is related to the local temperature rise (DT) by DT|b~DV j j=S T , where S T 5 16.3 6 0.1 mV/K is the Seebeck coefficient of the thermocouple and b 5 0.083 6 0.006 K/K is the sensitivity of the probe (see SI) 21 . Fig. 1c shows a two-dimensional color plot of the measured local temperature rise and evolution of the electrical resistance as the applied voltage is cycled at room temperature. Data obtained in a similar experiment performed at a lower temperature (,100 K) are shown in Fig. 1d. Since the thermal time constant of the SThM probe is sufficiently small (,100 ms) 21 , it tracks the variations in the local temperatures with high fidelity. It can be seen from Figs. 1c, d that the maximum temperature attained during electromigration in both the high and low temperature measurements occurs when the resistance changes by ,10%. Specifically, the maximum local temperature is ,660 K for the experiment performed at room temperature and ,400 K for the experiment performed at a low temperature (,100 K), which were measured at the location aforementioned. From the data shown in Figs. 1c, d, it is clear that the overall trends of temperature rise during electromigration are relatively independent of the ambient temperature. However, the maximum temperature rise observed during electromigration is significantly smaller in low temperature measurements-a fact that can be leveraged in creating molecule-based devices where a large increase in the local absolute temperature is detrimental to the stability of molecules 19,20 .
An important question to address in this context is: What is the local temperature of hot spots at which electromigration is actually initiated? In order to answer this question it is instructive to present the data shown in Figs. 1c, d in a different form that enables visualization of the relationship between the resistance change and the temperature rise. Therefore, in Figs. 2a-d we present the measured temperature rise as a function of the resistance of the device during electromigration. In Fig. 2a, it can be seen that before electromigration is initiated the temperature increases approximately linearly with the resistance. In contrast, for the devices where the resistance had irreversibly changed by 2.5% or 10% (Figs. 2b, c), the relationship between temperature and resistance as well as that between power dissipation and resistance were found to be non-linear. The onset of the nonlinearity in the relationship between temperature and resistance is highlighted by a red arrow in Fig. 2b and indicates the point at which electromigration is initiated when the local temperature of the nanowires reached ,430 K. Similar analysis of the data obtained in low temperature measurements indicates that electromigration is initiated at ,315 K. In addition, the temperature and power dissipation traces which are scaled to overlap at lower powers do not continue to overlap with each other in the shaded regions in Fig. 2b, c. This is due to the fact that upon additional electromigration Joule heating is more localized to regions where voids accumulate thus changing the relationship between the temperature measured by the probe and the total power dissipation. Further, from FEM simulations we estimated that the temperature rise (before electromigration) at the middle of the nanowire is ,20% larger than what we measured at the location shown in Fig. 1a (see SI). This implies that the local temperature in the hot spots when electromigration starts is ,520 K in room temperature measurements and ,380 K in low temperature measurements. These values are larger than those reported in previous studies 10,20,28 . This deviation can be attributed to the indirect way in which previous temperature measurements were performed that resulted in a spatial average of the device temperature, which is lower than the local temperature in the hot spots. Our results clearly show that electromigration of Au nanowires is indeed initiated at temperatures much lower than the melting temperature of Au (,1337 K) 29 , which indicates that the electric field contributes dominantly to electromigration.
In order to obtain further insights, we performed additional experiments where we first terminated the electromigration process at certain representative states (the resistance of the devices had increased by 2.5%, 10% and 40%, respectively). Subsequently, we mapped the topography and temperature fields of the devices under   a small sinusoidal bias voltage with an amplitude ,0.3 V and a frequency of 5 Hz. The applied voltage bias was sufficiently small that no further electromigration occurs during thermal characterization. The topography was obtained in contact mode at a contact force of 150 6 25 nN while the temperature fields were mapped by locking in to the second harmonic component (10 Hz) of the voltage output from the integrated thermocouple (see ref 21 for more details).
The obtained topography for Au nanowires when the resistance of the device had changed by 0%, 2.5%, 10% and 40% are shown in Figs. 3a-d, respectively. The measured topography unambiguously shows the formation of hillocks and depressions, which mostly emerged at an early stage of the electromigration process where the resistance increase was ,10%. The corresponding temperature fields are shown in Figs. 3e-h, which highlight an important feature: an increase in the asymmetry of the temperature fields upon electromigration. Specifically, before electromigration the temperature fields are symmetrical about the center line (the dotted line) with the maxima of the temperature fields being located in the middle of the nanowire. However, as electromigration progresses and results in a 40% resistance change, the temperature fields become increasingly asymmetric with the temperature rise being larger at the intersection of the nanowire and the cathode. The color codes used in Figs. 3e-h and 3m-p represent normalized temperature rises with red being hotter than blue. The increase in the asymmetry of the temperature fields can be qualitatively understood by hypothesizing that voids/vacancies are created and accumulated primarily in the cathode of the device (see SI). This hypothesis is consistent with previous studies that have observed the accumulation of voids in nanoscale devices 9,10,13,14,30 .
Further, thermal imaging measurements performed at a low temperature (,100 K) show similar trends (Figs. 3m-p). However, in this case the asymmetry in the temperature fields is less prominent than the measurements performed at room temperature. These results can be partly understood by noting that the mobility of Au atoms is lower at 100 K than room temperature, which requires a larger current density for initiating electromigration 10 (as experimentally observed, ,1.9 3 10 12 A/m 2 for low temperature measurements versus ,1.3 3 10 12 A/m 2 for room temperature measurements). We believe that this large current density creates voids both in the cathode and in the nanowire leading to a less asymmetrical temperature field. In spite of this reduced asymmetry in the temperature fields, we note that the location of nanogap formation still showed the same bias-polarity dependence i.e. the voids accumulated primarily in the cathode. The fact that there are similar trends in the low and high temperature measurements is consistent with the view point that electromigration is primarily driven by the electric field and the ambient temperature plays a less significant role.
Finally, we note that we have limited our topography and thermal imaging studies to the devices whose resistances have changed by 40% or less because the voltage required for obtaining appreciable temperature changes increases with increasing resistance. In general, large voltages tend to induce further electromigration preventing thermal imaging of devices before the resistance increases further.

Discussion
The above results raise an important question: What is the source of the observed asymmetry in the temperature fields after electromigration? Is it due to the formation of a crack/notch in the electrode or caused by a local change in the electrical resistivity resulting from the accumulation of voids, which enhance electron scattering and increase heat dissipation? In order to answer this question, we performed finite element modeling (FEM) simulations using COMSOL (Joule heating module). In these simulations, we first considered the case where a notch is developed in the nanowire without a change in the local electrical resistivity (see Methods for details). The computed temperature field when a current density of 1.3 3 10 11 A/m 2 , which is comparable to that used in our scanning measurements, is applied across such a notched nanowire is shown in Fig. 4a. It can be clearly seen that the temperature field is largely symmetric highlighting that structural effects alone-i.e. notch formation as the breakdown proceeds-cannot explain the asymmetrical temperature fields observed in the experiments. For comparison, we performed a second FEM simulation (Fig. 4b) where in addition to the notch, the local electrical resistivity of the cathode was increased by a factor of ten to 2.2 3 10 27 V?m-to mimic the effect of accumulation of voids-in the region of the cathode highlighted by the dotted line in Fig. 4b. It can be seen that the modeled temperature field is asymmetric in correspondence with experimental observations providing strong support for our hypothesis of voids accumulation in the cathode during electromigration. This hypothesis can also be understood using a simple model that is discussed in the SI.
Additional insights into the breakdown mechanism of Au nanowires during electromigration can be obtained by a more detailed analysis of the data in Figs. 2a-d. Before the initiation of electromigration (Fig. 2a), it is clear that the resistance and the temperature change are linearly related. This should be expected because for a metallic wire the relationship between the temperature rise (DT) and the resistance (R) is given by, where R 0 is the wire resistance when DT is zero, and a is the temperature coefficient of resistance (TCR) of the metallic wire. This implies that, therefore, if aR 0 changes, the slope (dT/dR) becomes non-linear. As mentioned above, in Fig. 2a  In addition, from the data shown in Figs. 2b, c, it is clear that immediately after the initiation of electromigration both the resistance and the slope (dT/dR) increase implying that the TCR decreases (indicated as region ii)). This observed decrease in TCR can be attributed to our hypothesis of voids creation that results in a reduced electron mean free path, which is known to suppress TCR 31,32 . It can be also seen in Figs. 2b, c that at larger temperature rises, which  correspond to higher voltage biases (marked as region iii)), the slope of the temperature (dT/dR) and power dissipation (dP/dR) curves decrease simultaneously. This can be understood by noting that an increased electrical resistance due to voids accumulation reduces the overall Joule heating in the device. This behavior was also observed in Fig. 2d for the device where the resistance had changed by 40%.
In conclusion, by employing UHV-SThM we successfully studied temperature fields in Au nanowires during electromigration at both room temperature (,300 K) and a low temperature (,100 K). From these measurements, we were able to obtain information regarding both the local temperature rise and spatial inhomogeneity of electrical resistivity in the nanowires during electromigration. Specifically, the later information was obtained by thermally imaging the nanowires under a small sinusoidal bias to elucidate asymmetries in temperature fields that are directly related to the local electrical resistivity. We also found that the maximum absolute temperature at the onset of electromigration can be substantially suppressed by lowering the ambient temperature. Further, our results provide strong evidence for accumulation of voids in the cathode of the devices during electromigration, which results in the preferential nanogap formation on the cathode. The experimental approach reported here enables deeper insights into the role of Joule heating (heat dissipation) in the failure of nanoscale devices. This approach can be applied to investigate a variety of interesting heat related phenomena relevant to many areas of nano-science and technology.

Methods
Finite element modeling (FEM) simulations. To compute temperature fields of the nanowire devices, we used commercial FEM software COMSOL (Joule heating module). An electrical current was simulated through the Au nanowire from the source electrode to the drain electrode, while all other surfaces were electrically insulated. In addition, the temperature of the bottom surface (of the 500 mm thick Si wafer on which the nanowire is located) was modeled to be at 300 K, while all other surfaces were modeled to be thermally insulating. The thermal conductivity and electrical resistivity of materials were chosen to be 320 W/m?K and 2.2 3 10 28 V?m respectively for bulk Au and 160 W/m?K and 4.4 3 10 28 V?m respectively for thin film Au in the bow-tie region 33 . Further, the thermal conductivity of Si and SiO 2 were assigned to be 150 W/m?K and 1.4 W/m?K, respectively 34 .