Temperature mapping of operating nanoscale devices by scanning probe thermometry

Imaging temperature fields at the nanoscale is a central challenge in various areas of science and technology. Nanoscopic hotspots, such as those observed in integrated circuits or plasmonic nanostructures, can be used to modify the local properties of matter, govern physical processes, activate chemical reactions and trigger biological mechanisms in living organisms. The development of high-resolution thermometry techniques is essential for understanding local thermal non-equilibrium processes during the operation of numerous nanoscale devices. Here we present a technique to map temperature fields using a scanning thermal microscope. Our method permits the elimination of tip–sample contact-related artefacts, a major hurdle that so far has limited the use of scanning probe microscopy for nanoscale thermometry. We map local Peltier effects at the metal–semiconductor contacts to an indium arsenide nanowire and self-heating of a metal interconnect with 7 mK and sub-10 nm spatial temperature resolution.

Unlike macroscopic contact thermometers, they typically not infer temperature by measuring only one physical property, for example, the electrical resistance, of a calibrated sensor in equilibrium with the system of interest. Instead, they detect a heat-flux-related signal across a tip-sample contact _ Q ts ¼ T sensor À T sample À Á =R ts À Á that is not only proportional to the temperature difference between a probe sensor (T sensor ) and a sample (T sample ), but also influenced by an unknown thermal contact resistance (R ts (T)) 29,30 . The well-established concept of equilibrium contact thermometry cannot be easily adapted to the nanoscale, as it would require R ts to be small compared with the resistance between the sensor and its thermal reservoir (R cl ), a condition that practically cannot be achieved for high-resolution scanning probes forming nanoscopic contacts 18,19,[22][23][24][25] .
To measure the temperature in such a non-equilibrium situation, one has to account for the thermal coupling strength (given by the value of R ts ) between sample and sensor. This demand is comparable to that for an infrared thermometer, requiring emissivity corrections to quantify the temperature of a non-ideal black body as potential that governs a (radiative) heat flux. Previous attempts addressing this issue on the nanoscale including a two-pass method 23 and dynamic adjustment of the probe sensor temperature 24,31 are complicated to implement and could not demonstrate reliable temperature mapping in those cases where the surface topography is inhomogeneous. This is mainly because R ts of a nanoscopic contact is not only difficult to predict 13,14,29,30 but a position-dependent property that varies dynamically during the imaging of a surface.
Here we first elucidate the challenge of using thermal scanning probes for nanoscale thermometry, before we present an approach that enables the elimination of contact-related artefacts. We illustrate how the perturbing variations of R ts can be separated from sample temperature variations by simultaneously probing a time-dependent and a time-averaged heat flux signal between a self-heated scanning probe sensor and a temperaturemodulated sample.

Results
Self-heating of a metal interconnect. To demonstrate the technique, we first characterize the Joule heating of a gold interconnect structure shown schematically in Fig. 1 and as an atomic force micrograph in Fig. 2a. An alternating voltage bias ( ± 3 V) is applied to the interconnect, inducing an AC current of ±2.4 mA (see Supplementary Fig. 1 for a detailed schematic of the data acquisition). When using sufficiently small frequency (f ¼ o/(2p) ¼ 10 kHz), the applied voltage bias leads to a steady-state temperature increase and a continuous modulation of the interconnect temperature (T sample ¼ T RT þ DT sample (1 þ sin(2ot))) at twice the excitation frequency, with T RT denoting room temperature. The change in sample temperature (DT sample ) affects the tip-sample heat flux ð _ Q ts ¼ _ Q ts;DC þ _ Q ts;AC sin 2ot ð ÞÞ, which is related to the measured electrical power (P el ) dissipated in the probe sensor and also by the sensor temperature (T sensor ¼ T sensor,DC þ T sensor,AC sin(2ot)), which is proportional to the electrical resistance (R) of the sensor (see the Supplementary Note 1 for the derivation of the relation between power, resistance and temperature of the probe). By simultaneously measuring the temperature-dependent steady-state (DC) and the alternating (AC) response of the probe sensor, the sample temperature increase can be derived as  interconnect (B100 nm wide) on a silicon (111) substrate wafer covered with 150 nm of amorphous silicon nitride (SiN x ). An alternating voltage bias (V cos(ot)) is applied to modulate the sample temperature. The sample temperature field (red-blue colour scheme indicates the temperature from hot to cold) is inferred by simultaneously probing a time-dependent and a time-averaged heat flux signal between the heatable (red coloured) sensor and the temperature-modulated sample. (b) The sensor is decoupled from its thermal reservoir by the thermal resistance of the cantilever (R cl ). The tip-sample heat flux _ Q ts À Á is not only a function of the temperature difference between the sensor (T sensor ) and sample (T sample ), but also of the tip-sample contact thermal resistance (R ts ). Changes of the tip-sample contact due to (c-d) the material in contact with the tip, (d-e) the tipsample contact area and tribological effects cause variations of R ts .
Importantly, a part of the DC sensor temperature component (T sensor,DC ) is set by the sensing voltage bias applied to the scanning probe, which typically heats the sensor to about 20-200 K above T RT . As a result, T sensor,DC and T sensor,AC are not the same, and the two unknown DT sample and R ts can be extracted simultaneously from the two measurable quantities T sensor,DC and T sensor,AC via equations (1) and (2) for each pixel of an image under the assumption that, for a given scan position, R ts is approximately constant within the range of sample temperature modulation. Under this assumption, R ts can be directly expressed as a ratio between the measured heat fluxes and sensor temperatures, while previously R ts had to be either estimated by modelling simulations 14,15,18,19,22,25 or be determined from reference measurements 23 . The fact that we can avoid this step reduces systematic and propagated measurement errors dramatically. Figure 2b shows the thermal resistance (R ts ) map recorded simultaneously to the AC tip-sample heat flux _ Q ts;AC À Á , while raster scanning the probe in contact with the surface. Measurements were conducted in a high-vacuum environment (o10 À 6 mbar) that diminishes parasitic heat conduction between the probe sensor and the sample via air-mediated conduction/convection. Large variations in R ts can be observed in correlation with topography features that change the tip-sample contact area. These features perturb _ Q ts , as evident from the signal variations at the interconnect edges in Fig. 2c.
Previous approaches for inferring temperature maps using scanning thermal microscopy (SThM) did not fully account for these contact-related effects, but typically assumed a position-independent value for R ts (refs 14,15,18,19,22,25) to transform a heat-flux-related signal to a temperature field map. Such a procedure, however, can lead to significant systematic errors as shown in the 'apparent' temperature map in Fig. 2d. The section of the interconnect that is heated most even shows a reduced temperature in the centre of the metal line with respect to its edges, which is highly implausible. By accounting for position-dependent variations of R ts , however, we can obtain a corrected temperature map (see Fig. 2e). The rather homogenous temperature field and the absence of nanoscopic, contact-related features in Fig. 2e, are important results of our work and highlighting the benefits of the scanning probe thermometry approach presented here. As this technique is applicable to various kind of material surfaces, it also enables to map the temperature field extending around the heated interconnect. To illustrate this, Fig. 2f shows the temperature distribution of Fig. 2e encoded in terms of both height and colour. Not only the self-heating of the thinnest interconnect segment exhibiting the typical, nearly parabolic temperature profile due to Joule heating but also the temperature decay into the floating interconnect segment and the spreading of heat into the substrate are resolved. This real-space temperature field information can be particularly valuable when comparing with models and simulations 18,23 , and potentially provides further insight into the thermal coupling between the interconnect and the substrate.
Self-heating of an InAs nanowire. Having established a reliable approach to image nanoscopic temperature fields, we use it to study self-heating of an indium arsenide (InAs) nanowire 32 . Again, a bipolar AC voltage bias is applied (see Fig. 3a), inducing not only Joule heating in the nanowire, but also Peltier heating/cooling at the metal-semiconductor interfaces. In contrast to previous SThM approaches mapping only overall temperature fields, our method can be applied to study both effects separately, an important option that facilitates direct characterization of thermoelectric effects in real space that might find relevance for the characterization of operating logic, memory-and energy-harvesting devices. For a linear device (for example, an ohmic resistor), the two effects can be distinguished in a single measurement scan (see ref. 33 and ref. 26), because the Peltier effect has a linear dependence _ Q / I À Á on an electrical current, matching the frequency of the applied voltage bias (see Fig. 3a), whereas Joule heating has a . Previous approaches inferred local temperature fields using a constant, position-independent (x,y) tipsample thermal resistance (for example, the mean of R ts (x, y)). This leads to significant artefacts, illustrated as (d) an apparent temperature distribution. By combining position-dependent thermal resistance variations with the local heat flux maps, we can calculate (e) the true sample temperature rise (DT sample ). This relative temperature increase can be encoded in terms of both height and colour as shown (f), to illustrate the characteristic (parabolic) Joule heating along the interconnect segment, as well as heat spreading (indicated by the white dotted arrows and lines) into the SiN x /Si substrate and the electrically floating interconnect segment. It is noteworthy that the colourbar applies also to d,e.
quadratic response _ Q / I 2 À Á that can be detected in the higher harmonic response (2f) of the probe sensor. Applying the same procedure as described for the metal interconnect, we can simultaneously map variations in the thermal resistance (see Fig. 3b) and variations in the tip-sample heat flux (see Fig. 3c), now separated for the two frequencies ( _ Q ts;1f and _ Q ts;2f ) corresponding to Peltier and Joule effects, respectively. The assignment of the (1f and 2f) signals to the heat generation due to Peltier and Joule effects, respectively, is supported by an analysis of the observed effects as a function of voltage bias amplitude (see Supplementary Note 2 and Supplementary Fig. 3). By combining the thermal resistance and heat flux measurements, we can successfully eliminate tip-sample contact-related artefacts and calculate the sample's temperature field according to equations (1) and (2) expanded for two frequencies.
The instantaneous Peltier temperature field, shown in Fig. 3d, induced by net heat generation or extraction at the nanowire-metal contacts exhibits a symmetrical, almost linear temperature profile along the nanowire length with a vanishing temperature change in the centre (see Supplementary Note 3 and Supplementary Fig. 4 for the related phase signals). It is noteworthy that the Peltier temperature at a certain phase is shown; in time average, there is no heating or cooling of the nanowire in response to the bipolar AC excitation. Conversely, the Joule-heating-related temperature increase of the nanowire (see Fig. 3d) does not show a nearly parabolic temperature distribution as observed for the self-heated metal interconnect, but indicates a temperature increase towards the electrical contacts (see also Supplementary Fig. 2). This could stem from either a very non-homogenous thermal coupling of the respective nanowire sections to the substrate or enhanced Joule heating near the Au/Ni-InAs contacts. As there is no indication of a non-uniform thermal coupling to the substrate in the Peltier measurement, we conclude that the Joule heating along this particular nanowire must be unevenly distributed, favouring the nanowire-metal contact regions.

Discussion
To discuss the method presented here, we first consider the sample-temperature resolution of our measurements. As argued above, the sample-temperature resolution is directly correlated to the sensitivity for variations in heat flux. We characterized the voltage noise-limited temperature resolution (dT res À probe ) of our thermal probes, which range from 1 to 100 mK ffiffiffiffiffiffi Hz p À Á À 1 as a function of sensor temperature and operation frequency. For the measurements presented, we calculate a temperature resolution of dT res À probe ¼ 2 mK ffiffiffiffiffiffi Hz p À Á À 1 and 20 mK ffiffiffiffiffiffi Hz p À Á À 1 for the interconnect and nanowire experiments, respectively, which is two to three orders of magnitude larger than in a recent nanoscale thermometry approach of similar spatial resolution based on plasmon energy expansion thermometry 12 . With a probe sensor thermal resistance R cl on the order of 2 Â 10 5 K W À 1 , this translates into a sub-100 pW heat flux resolution d _ Q res ¼T res ÂR À1 cl À Á at 20 kHz operation frequency. A comparable heat flux resolution has previously only been achieved by MEMS nanocalorimeters 34 , which, however, do not provide local imaging functionality. Achieving this high sensitivity is important, as it allows the total heat flux to be limited to values that are small enough not to perturb the heat generation in the sample volume beneath the tip (o100 nW for the nanowire experiment). The heat flux used in our experiments is comparable to that used in typical non-contact, radiative heat flux experiments based on Raman or fluorescence thermometry, thus ruling out one of the main concerns against using contactbased methods for nanoscale thermometry 3,12 .
The sample temperature resolution is limited by the heat flux resolution of the thermal scanning probe system and could be estimated based on the intrinsic resolution of the probe sensor and the tip-sample thermal resistance dT res;sample ¼ _ Q res ÂR ts À Á . Typically, either the intrinsic noise in the sensor circuitry 22 or the noise extracted from nominally flat regions in the image data 23,28 are used to quantify the resolution. The latter is a more conservative approach, as it includes additional noise sources affecting the measured sample temperature field, such as mechanical noise of the tip-sample contact or changes at the tip apex. Therefore, we prefer to directly extract the sample temperature resolution as root mean square noises from the experimental data of Figs 2e and 3f, reaching 7 and 85 mK, respectively. These sample temperature resolutions have to be considered in the context of lateral resolution, which is directly linked to the value of R ts via the size of the tip-sample contact. The sample temperature resolution of the nanowire experiments is less than those of the metal interconnect studies, whereas spatial resolution is higher. In the nanowire experiment, we observe a tip-sample thermal resistance (up to R ts ¼ 6 Â 10 8 K W À 1 ) similar to that of a previous SThM resolution demonstration of 6 nm 35 , which also matches the smallest features observable in the nanowire scans. Therefore, we conclude that the method is suitable for achieving a sub-10 mK temperature resolution at sub-10 nm lateral temperature resolution with the microscope used.
It is noteworthy that a meaningful demonstration of a spatial temperature resolution in SThM measurements is challenging. Ideally, it requires a step-like temperature variation in the absence of any other variation influence the heat flux-related measurement signal such as a material-or topography-related change in the tip-sample thermal resistance. However, using contact model estimations to calculate the size of the mechanical contact between the tip and sample are oftentimes inherently inaccurate and can be in contradiction to the heat flux resolution, which is a strong function of contact size. Therefore, a meaningful resolution definition for SThM measurements should be based on the tip-sample heat flux resolution, as heat flux-related signals are measured and directly reflecting the tradeoff between sample temperature resolution and spatial resolution via their common dependence on the tip-sample contact size. In our experiments, we can extract a heat flux resolution of 140 pW in a 100-Hz bandwidth, which is a factor of 25 or more improvement compared with previous SThM measurements 22,23,28 .
Finally, we may also discuss limitations of the scanning probe thermometry approach presented. Although the technique described can be easily implemented, it is limited to cases where the sample temperature can be modulated, for example, by electrical or optical means. Steady heat sources may not be fully characterized by the presented approach and alternative techniques, for example, based on multiple measurements 23,24 might be applied to those cases. Although the presented modulation-based thermometry approach simplifies the temperature field characterization as all signals can be characterized in a single measurement scan, it comes on the additional experimental effort to create a temperature modulation of known frequency in the first step. As the modulation frequency can be flexible chosen, even the transient regime of local heat sources might be characterized to extract further information, for example, about the thermal diffusivity of the system, beyond the direct imaging of a steady-state temperature field itself (see Supplementary Discussion and Supplementary Fig. 5). Furthermore, care is needed not to perturb the sample heat source by the measurement heat flux between the scanning probe sensor and the sample. It is noteworthy that this is a common challenge to any thermometry method, even for non-contactbased approaches where optical or electron beam-based sensing fluxes are used for detection.
To summarize, we have introduced a versatile scanning probe thermometry technique to enable quantitative, spatially resolved temperature measurements down to few-nanometre spatial resolution and sub-10 mK temperature resolution. The presented approach accounts for contact-related artefacts, so far limiting the reliability of scanning probe microscopy for nanoscale thermometry, and may promote further understanding of local thermal processes in various areas of nanoscience. It can be used to characterize surfaces of almost arbitrary materials and is expected to have a direct impact on understanding of nanoscale devices dealing with local thermophysical effects such as the spin-Peltier effect 5 , the magnetocaloric effect or the thermoplasmonic effect 9 .

Methods
Experimental setup. Experiments where conducted using a custom-built high-vacuum SThM 35 situated in an electromagnetically shielded, temperaturestabilized laboratory 36 . Our typical cantilevers have a nominal spring constant of 0.15-0.20 N m À 1 and a resonance frequency in the range of 50 kHz, making them well suited for contact-mode operation. The cantilever is highly phosphorus doped (10 20 cm À 3 ) for good electrical conductivity. The heater region of (4 Â 6) mm below the tip has a nominal dopant density of 5 Â 10 17 cm À 3 . The thermal resistance of the cantilever when situated in vacuum is typically 2 Â 10 5 K W À 1 . The thermal resistance is mainly defined by the heat flux from the heater region along the anchor beams of the cantilever into the chip body. The thermal isolation of the heater region is maximized in a tradeoff between the mechanical and thermal properties of the lever. For scanning probe operation, we require a critical thickness of the cantilever, which is already a thin beam of only 300 nm in the region of the hinge. More than 90% of the total power dissipation in the cantilever occurs in the small region of the heater element below the tip. The heater equilibrates in a few microseconds. The electrical resistance is on the order of 1.2 kO at room temperature and increases to about 3.3 kO at 550°C. Overall, the heater can be heated up to 1,000°C. Further details on the cantilever, its design and fabrication are reported by Drechsler et al. 37 .
Sample fabrication. The metal interconnect test structure consists of nanoscale gold interconnects on a silicon (111) substrate waver coated with 150 nm of amorphous silicon nitride (SiN x ). The narrow gold lines (B100 nm wide) were fabricated via electron-beam lithography (Vistec EBPG5200) and patterned by an etching process.
InAs nanowires were grown by metal-organic vapour-phase epitaxy on a silicon(111) wafer 38 . The nanowire (B120 nm diameter) was separated from the growth substrate and transferred to a silicon substrate covered with a B100-nm-thick silicon oxide. Electrical contacts (Au/Ni) were fabricated by e-beam lithography.