Quantitative Characterization of Local Thermal Properties in Thermoelectric Ceramics Using “Jumping‐Mode” Scanning Thermal Microscopy

Thermoelectric conversion may take a significant share in future energy technologies. Oxide‐based thermoelectric composite ceramics attract attention for promising routes for control of electrical and thermal conductivity for enhanced thermoelectric performance. However, the variability of the composite properties responsible for the thermoelectric performance, despite nominally identical preparation routes, is significant, and this cannot be explained without detailed studies of thermal transport at the local scale. Scanning thermal microscopy (SThM) is a scanning probe microscopy method providing access to local thermal properties of materials down to length scales below 100 nm. To date, realistic quantitative SThM is shown mostly for topographically very smooth materials. Here, methods for SThM imaging of bulk ceramic samples with relatively rough surfaces are demonstrated. “Jumping mode” SThM (JM‐SThM), which serves to preserve the probe integrity while imaging rough surfaces, is developed and applied. Experiments with real thermoelectric ceramics show that the JM‐SThM can be used for meaningful quantitative imaging. Quantitative imaging is performed with the help of calibrated finite‐elements model of the SThM probe. The modeling reveals non‐negligible effects associated with the distributed nature of the resistive SThM probes used; corrections need to be made depending on probe‐sample contact thermal resistance and probe current frequency.


Introduction
Scanning thermal microscopy (SThM) is a scanning probe microscopy (SPM) method opening up access to local temperature and thermal conductivity by means of a nano-probe With the frequency of the probe-sample intermittent contact in a range of one kilohertz and higher, which is characteristic for the PeakForce mode, the probe and sample are not equilibrated thermally, which reduces fidelity of the measurements. [13] In this paper, we applied SThM to bulk ceramic samples of thermoelectric materials with the ultimate goal to elucidate the role of micro-and nano-structural features of the ceramics in the thermoelectric performance of the materials. Thermoelectric conversion is expected to play an important role in future energy technologies provided by its simplicity, reliability, and self-sufficiency to enable mobile or remote applications. [14] Ceramic materials based on transition metal oxides offer significant advantages for high-temperature waste heat harvesting due to their chemical stability, natural abundance and attractive combination of electrical, and thermal transport properties. [14][15][16] Various material design strategies guiding the search for high-performance oxides are being developed based on existing advances in understanding links between the chemical composition, structure at various length scales and thermoelectric properties. Extensive efforts toward improving the conversion efficiency are being made by considering multiphase composites comprising nano-or microscale phases. [17] Typically, the performance of composite thermoelectrics is boosted by the enhanced phonon scattering at nanoinclusions and grain boundaries, with combined effects of the grain boundary modification on electrical transport and by the charge carrier filtering effects. [14] Partial interaction between the composite components may also lead to the modification of the matrix composition and appearance of additional phases. [18][19][20] Such complex and often superimposed effects complicate the identification of various contributions to the charge and heat transport processes and the correct mechanisms behind the observed effects. The variability of the properties of ceramic materials is significant even when using nominally identical preparation routes. This fact cannot be explained without detailed studies of the material microstructure, factors influencing it, and specific microstructural features that affect the thermoelectric performance of the materials, electric and heat transport. This aspect is general for thermoelectric materials based on simple and complex oxide ceramics. Thus, investigations of thermal transport at the local scale are of key importance for the further knowledge-based design of the material microstructure and engineering of the thermoelectric performance of the composite materials.
Here, we demonstrate a complete cycle of probe calibration and quantitative mapping of the bulk thermoelectric ceramics with heated resistive SThM probes (probe active mode), and subsequent visualization of the distribution of thermal properties using a calibrated numerical model for the probe. Inspired by various synergistic effects of the chemical composition and microstructure on the thermoelectric performance observed in Ref. [19], we selected donor-doped ZnO-based composites containing ZrO 2 or CeO 2 as model materials. To extend the useful probe life with the ceramic samples, we developed and implemented a "jumping-mode" SThM (JM-SThM) imaging modality employing pixel-by-pixel measurements of the difference of out-of-and in-contact temperatures of the probe to extract information about the heat dissipation in a sample. This imaging mode is a key component for mapping thermal conductivity in ceramic samples with relatively rough surfaces. Fur-ther, we present a novel calibration method for resistive probes and employ a calibrated Finite-Elements (FE) model of the probe in quantitative measurements of sample local thermal properties. It was only recently realized that the SThM resistive probes are distributed systems, and their distributed nature, which is usually not taken into account, may strongly affect measurement results. [21] FE models naturally address this issue. In particular, we find that values of the probe-sample contact thermal resistance obtained with the use of a standard, simple lumped-element model of the probe need to be corrected due to the distributed nature of the probe, with correction depending on contact thermal resistance and probe current frequency. The analysis of the obtained JM-SThM maps shows that variation of the probe-sample boundary thermal resistance can be responsible for a major part of the contrast in the maps.

Characterization of the Composites by Scanning Electron Microscopy
First, the composite ceramic samples were characterized by a combination of the scanning electron microscopy (SEM) and energy-dispersive X-Ray spectroscopy (EDS) in order to reveal surface morphology and composition. Figure 1a,b displays high-magnification SEM images of a Zn 0.993 Al 0.007 O-20wt% ZrO 2 (ZnAlOZrO) composite ceramic sample, which was first fractured and then polished and thermally etched. The image in Figure 1b is overlaid by the EDS map revealing the distribution of Zn, Al, and Zr over the microstructural features of the sample. As can be seen, ZrO 2 inclusions in the sample are present as individual nanoparticles as well as agglomerates of particles. Figure 1c,d shows images of the surface of a Zn 0.993 Fe 0.007 O-10vol% CeO 2 (ZnFeOCeO) sample (polished and thermally etched). The SEM image in Figure 1d is overlaid by the Ce-distribution map acquired with the EDS. Brighter areas in the images in Figure 1c are CeO 2 particles, which are highlighted with red in Figure 1d. Presumably, the pits visible in Figure 1d on the grain surface are due to CeO 2 nanoparticles removed during polishing; pits with CeO 2 particles inside are seen. More details on the sample structure can be found in Section S1, Supporting Information.

SThM Measurement Principle
Two types of SThM probes are usually used for local measurements of thermal conductivity: thermocouple-based probes [11,22] and resistive probes. [4,23] The resistive probes are commercially available, while thermocouple-based probes are usually custommade. For the experiments, we used Kelvin Nanotechnology KNT-SThM-2an probes with built-in 100 Ω in-series resistors.
The KNT probes are resistive with sensor and heater functions combined in one ca. 40-nm-thick and 1-1.5 µm-wide stripe of Pd deposited near the sharp apex of a Γ-shaped SiN x cantilever. The sensor is heated by passing an electric current, which creates a heat source localized at the probe tip apex. The resistance of the sensor R is a function of temperature, which is close to linear dependence R = R 0 (1 + α · ΔT), where R 0 is the sensor resistance at room temperature and ΔT is the sensor temperature change. Pd has a considerable thermal coefficient of electrical resistance (TCR) α (α ≈ 3.4 × 10 −3 1/K at 300 K in bulk pure Pd [24] ), and the resistance change of the heated Pd stripe sensor is used to determine variations in the amount of heat escaping through the tip apex into a sample, which allows mapping of sample thermal conductivity with a high spatial resolution of about 50-100 nm. As illustrated by the schematic in Figure 2a, the probe resistance was monitored with the use of a Wheatstone bridge and a home-made pre-amplifier assembled on the basis of low-noise instrumentation amplifier integrated circuits. SThM was performed in two modes: with a DC current (DC mode) and with an AC current (AC mode) through the probe. In the latter case, the voltage oscillations through the probe were detected at the 3rd harmonic of the AC current (ω-3ω method, or 3ω SThM [4] ). The details of the experimental setup can be found in the Experimental section.
The KNT resistive probes are widely used for SThM, and different aspects of their static and dynamic behavior have been addressed in a number of reports with experimental and numerical methods. [4,7,12,21,[25][26][27][28] The measurements can be quantitative, [2][3][4] and the typical approach to quantification of the sample thermal properties can be illustrated with a simplified lumped-element equivalent electrical circuit shown in Figure 2b. We conduct experiments in a vacuum at a pressure of about 10 −3 Torr, when the heat transfer between the probe and environment by air can be neglected. Apparently, the situation is more complex if the SThM measurements are performed in air, where heat flow through air [25,29,30] and water meniscus [31] is possible. The measurements in vacuum significantly simplify extracting the thermal conductivity from the SThM data, because the thermal "bridge" between probe and sample shrinks to probe-sample solid-solid contact with the air pressure reduction. [25,30] In vacuum, the heat from the Pd stripe is transferred to the environment via the thermal resistance of the probe, p th R , (that is, via the sensing tip and lever) and the thermal resistance of the probe-sample contact, c th R . With known p th R and the power Q, measuring temperature T yields the value of c th R through the resistance of the c th p th  R R circuit in Figure 2b. The temperature T is determined by measuring the electrical resistance of the probe. For quantitative measurements, probes need to be calibrated, namely, the sensitivity of the probe resistance  to temperature changes of the sensor-heater need to be determined and probe thermal resistance p th R has to be measured. The contact resistance c th R in the model in Figure 2b consists of three components, one of which, the spreading thermal resistance of the sample, s th R , contains information about the thermal conductivity of the sample material. Two other components are in series with it: the thermal resistance of the portion of the sensing tip between the heater element and the probesample contact, t th R , and the thermal resistance of the boundary between the sample and the probe tip apex, b th R , which may include the resistance of the interface between dissimilar materials (Kapiza resistance) as well as the resistance of contamination layers on the sample surface.

SThM Imaging Mode: JM-SThM
SThM measurements necessarily involve measurements of the probe temperature in contact with the surface. Typically, these measurements are realized in the "contact mode" regime when the probe is moved along the surface with a constant applied force against the surface. Generally, it has to be expected that there is a time delay between the change of the probe location and the moment when the heated probe and colder surface come to the steady state at the new probe position. In another aspect, when dealing with ceramic samples in SPM, one needs to take into account a relatively high surface roughness remaining after polishing, a fact that may create a hindrance for meaningful SThM measurements, especially when compared with the results obtained on smooth surfaces of Si wafers usually used to demonstrate and characterize the performance of SThM (see, for example, Refs. [23,32]). The high surface roughness not only creates uncertainty in the probe-sample contact properties, but also damages the gentle probe during the rasterscanning motion in contact with the surface. The problem is aggravated by the presence of pores in nearly all ceramics.
To accommodate SThM to rough surfaces of ceramics, we have modified the way the SThM is performed. First, we have modified the standard sample holder of the microscope, which is equipped with an auxiliary piezo stack for excitation of probe oscillations employed for semi-contact measurements (tapping mode) of topography, so that the holder can accept SThM probes, and added springy electrical leads/contacts to interface the SThM probes with the measurement circuit. With this feature, the sample topography was inspected before starting SThM to find a smooth area of interest using a higher eigenmode oscillation of the SThM probe cantilever, similar to the standard semi-contact imaging. A lower Q-factor higher eigenmode was used in order to reduce self-excitation of the vibrations in vacuum.
Second, the SThM mapping was conducted retracting the probe from the surface to a distance of about 4 µm and then engaging the probe into contact with the sample at the next pixel with the probe moving from one spatial point to another out of contact with the sample. Thereby, scanning is performed with the probe "jumping" from one point to another. We call this  mode "jumping-mode" SThM (JM-SThM). The probe deflection and thermal signals-bridge balance voltage for the DC SThM and amplitude of the 3rd harmonic for AC (ω-3ω) SThM-were collected with the probe in and out of contact at each pixel of a map, and quantification of sample properties was made based on the difference of in-and out-of-contact signals. In particular, such an approach allowed the elimination of the signal drift, which was present in the measurements at DC, from the final data (see Section S2, Supporting Information, for details). Typical time dependences of the one-pixel thermal signals recorded during approach half-cycles of the probe moves for the ZnAlOZrO sample are presented in Figure 3. On contact, thermal signals jump down which is caused by the heat dissipation into the sample. Importantly, as can be seen, the relaxation of the thermal signals in contact takes place with a characteristic time of around 200 ms. In DC JM-SThM mode (Figure 3a), it occurs due to: i) stabilization of the local probesample temperature difference; and ii) stabilization of the probe-sample contact force (probe deflection) by the SPM feedback system. The in-contact signal inside the ZnO matrix is different from the signal in the ZrO 2 inclusions (particles), which is attributed to the difference in the contact thermal resistance in the different regions of the sample. In the AC JM-SThM (Figure 3b), a similar trend is observed, however, the difference between the matrix and inclusion is relatively smaller at the signal level scale. Since the 3ω signal is insensitive to the average probe-sample temperature difference, the relaxation of the signal in the AC JM-SThM is attributed only to the stabilization of the probe-sample contact force by the SPM feedback system.
To construct maps, the signals were averaged over 100 ms time intervals within out-of-contact, R p,out , and in-contact, R p,in parts of the signal. To take into account the relaxation, the difference (in-out) signals for mapping were calculated for in-contact time points corresponding to 0.7-1 s after the probe-sample contact was established. The algorithm used for data collection is based on a fixed time for data acquisition at each pixel. While the algorithms for the choice of the in-contact data points for mapping can be different, for example, based on the time after the probe-sample contact is established, in this work, we used a simplified approach, where the in-contact data points were selected based on the time passed after the start of the probe motion toward sample from the highest position in the out-ofcontact state. The exact time interval of 100 ms duration where the in-contact data points were used for mapping is indicated with the hatched rectangles in the plots in Figure 3. The position of this time interval in respect to the start of the approach move is constant for all pixels. The probe vertical motion is controlled by a probe scanner with position referenced to the microscope, not to the sample surface. Therefore, the time-tocontact varied across a map due to sample topography within an interval of about 200 ms. We neglected these variations in this work since the changes of the signals on the relaxation tail at the end of the data acquisition, that is, about 0.7-1 s after the probe-sample contact is established, were smaller than the signal noise and much smaller that the random pixel-to-pixel signal variations, which is seen in the plots in Figure 3. For this reason, we considered such an approach as a reasonable compromise between accuracy and mapping time. In order to reduce the mapping time to manageable limits, the number of pixels in the maps was relatively small in this work. Typically, 50 × 50-pixel maps were recorded, which took about 2 h to complete for each. The multi-dimensional array data postprocessing was performed using a Python script. Details of the JM-SThM thermal map processing can be found in Section S2, Supporting Information.
Here, a principal difference between our "jumping mode" and the "peak-force" mode of Ref. [13] should be noted, which is in the time spent by the cantilever at a point and the value of the applied force at the probe-sample contact. While in the "peak-force" mode, the force is constantly changing at a microsecond time scale, the force in the "jumping mode" is kept constant by the feedback system of SPM, which, in turn, keeps the thermal contact constant.

Calibration of the Probe and of the Finite-Elements Probe Model
As can be deduced from the lumped element model in Figure 2b, the temperature rise of the probe sensor stripe when Small Methods 2023, 7, 2201516 the probe is away from a sample is proportional to the Joule power generated in the sensor: where I is the electric current flowing through the probe, R 0 is the room temperature resistance of the sensor, and ΔR is the probe resistance change on heating.
The values of the sensor TCR, α, and the probe thermal resistance, p th R , are a priori unknown because of the limited reproducibility of the probe manufacturing process and need to be calibrated. In a typical TCR calibration sequence, α or the product α × R 0 is measured first, and then p th R is determined with the help of the last equality of Equation (1) assuming a known R 0 . To determine α, a commonly reported procedure is to heat the probe in an oven, temperature chamber, or in a liquid bath [33] where the probe is completely submerged into a uniform temperature field and equilibrated with it to ensure uniform heating of the sensor stripe during R versus T measurements. This method, while yielding a quite accurate value of the α for the Pd stripe for the use in the lumped-element model, is a step that may affect the probe. Also, in contrast to the lumped-element description, the probe sensor is a distributed system with a non-uniform temperature and, correspondingly, resistivity distribution in the course of SThM experiments. This fact is usually not taken into account in calibrations, as noted in Ref. [21]. Other calibration protocols were proposed; some of them need complicated calibration samples, multi-step procedures, or include numerical simulations. [7,27,[34][35][36] In this work, we developed and implemented a different approach based on the determination of the probe tip apex temperature as a function of the probe current. All calibration measurements are performed in the same setup as the SThM experiments. The results of the calibration measurements are used to calibrate the probe numerical finite-elements (FE) model, which self-consistently accounts for the distributed nature of the probe and which is used for signal interpretation and quantification of sample thermal properties in post-processing of the experimental data and images. The details of the FE model can be found in Section S3, Supporting Information.
Our calibration procedure was derived from the idea of the so-called "null-point" temperature measurement, when the difference in temperature of the probe tip apex and sample is nullified together with the tip-sample heat flow, which allows exclusion of the tip sample thermal resistance from influencing measurements of sample temperature. [30] We used this approach, but instead of measuring the sample temperature with a probe, we measured the temperature of the probe tip apex (T apex ) by setting the sample temperature. As a sample, we employed here a miniature PT100 thermistor heated by self-heating with an electric current. The SThM probe current is set and after that, we change the current through the PT100 thermistor and search for the thermistor current value when no response is detected by moving the probe in and out of contact with the thermistor (Figure 4a). The temperature of the thermistor is monitored via its electrical resistance. At the zeroresponse ("null") point, the probe current and the Wheatstone bridge balance voltage, reflecting the resistance of the probe, can be related to the temperature of the PT100 heater. The dependence of the bridge balance voltage change versus PT100 temperature is plotted in the inset in Figure 4a. The dependence is strictly linear, and thereby the "null-point" can be found easily by measurements at just two temperature points. By varying the probe current and searching for the null-point with the thermistor, calibration curves for the probe can be plotted (Figure 4a, inset), which allows extracting information about probe temperature and resistance as functions of current. We note from Equation (1)   . SThM probe calibration using a PT100 resistive heater. a) Time dependence of the heater temperature and respective changes of the SThM signal. This is an extract of the typical "null-point search" experimental run when the probe current is set, and the sample (PT100) temperature is adjusted so that the probe response is zero or near zero when moving the probe from in-contact to out-of-contact (to a distance of about 4 µm from the sample surface) and back. Inset: dependence of the bridge balance voltage for the probe in contact with the surface versus heater temperature. b) A summary of several measurements of the "null-point" that are based on null-point conditions for several probe current values. The plots show probe calibration graphs as described in the text.
the sensor element, the values of p th R and α obtained this way cannot be used for lumped-element-based calculations. Still, the data can be used for calibration of the thermal conductivity of the cantilever materials (SiN x and gold leads) and TCR of the sensor's Pd in the numerical model.
The probe thermal resistance p th R can be determined alternatively with the probe lifted away from the unheated sample in the vacuum chamber and with the cantilever deflection detection laser switched off. Such measurements are more accurate since they avoid parasitic heating of the sensor from the sample (as seen in the drift of the off-contact signals in Figure 4a with increasing PT100 temperature) and by the laser. We have made such measurements with our probes. For comparison, the p th R value calculated from the fit of the ΔT versus I 2 dependence in Figure 4b is ca. 8.8 × 10 5 K W −1 , while the second approach yielded (6.6 ± 0.3) × 10 5 K W −1 for the same probe. This difference arises almost solely due to the temperature base used to calculate the p th R , that is, T apex or the average temperature of the sensor element. We confirmed this fact in the numerical model.
Similarly, owing to the distributed nature of the sensor, the α determined from the temperature of the probe apex cannot be used directly both for sample property quantification and in the numerical model of the probe as a material property value for the Pd stripe. However, the α of the probe sensor material still can be determined approximately with the use of the probe apex temperature instead of the average temperature of the Pd sensor. We have performed numerical modeling of the probe response, looking for a relationship between the average probe sensor temperature and the apex temperature. As expected, the difference is systematic, and with different values of p th R determined based on the average sensor temperature, α apex calculated with the use of the apex temperature is 0-15% smaller than α Pd set in the model for the sensor material ( Figure S5, Supporting Information), which suggests the direction and the amount of correction of the α value determined from the fits of the calibration procedure in order to obtain an approximate value of α Pd . The latter value can be further refined with the help of the FE model by fitting the experimental dependencies T apex (I 2 ) and R(I 2 ) with the corresponding dependencies generated by the numerical model. For illustration, a result of such a procedure for one of our probes is shown in Figure 5a.
Altogether, four parameters were set in the FE model relying on the calibration measurement: Pd and Au electrical resistivity (and thermal conductivity as follows from the Wiedemann-Franz law), Pd α, and thermal conductivity of the cantilever material (SiN x ). As a result, to get the experimental value of R 0 = 100 ± 4 Ω, the Pd resistivity was reduced from ρ = 10.8 × 10 −8 Ω m at 300 K in pure bulk Pd to 2.9 × 10 −8 Ω m in the calibrated model, which is about 27% of the bulk value. It is known that the resistance of Pd thin films is a strong function of film thickness and microstructure and widely varies depending on deposition conditions. [37] Therefore, such reduction is reasonable for the thin-film Pd sensors of the probe. The parameter α was determined to be equal to 1.85 × 10 −3 1/K. Its reduction in comparison with the bulk value of 3.4 × 10 −3 1/K is smaller than that expected from the Matthiessen rule ρ × α = const, [38] which can be attributed to the deviation of the thickness of the Pd film from its nominal value of 40 nm. This result generally points to a need to have direct access to the resistance of the Pd stripe, which is hindered by the presence of the built-in NiCr resistors in the standard configuration of KNT probes. The Pd thermal conductivity in the model was reduced to 27% of its bulk value following the Wiedemann-Franz law. To reproduce the experimentally obtained value of p th R , we set the thermal conductivity of the cantilever materials k SiN = 2.5 W m −1 K −1 in the FE model. This value is close to 2.8 W m −1 K −1 reported in Ref. [39] for room temperature. However, k SiN varies widely in different reports and, for example, is about 12 W m −1 K −1 at room temperature in SiN x membranes of 100 nm thickness in another report by the same group. [40] Additionally, the cantilever was modeled as a 400 nm-thick rectangular slab, while the shape of the Small Methods 2023, 7, 2201516 cross-section of the real cantilever is more complex than rectangular. This needs to be kept in mind when comparing the model material parameter values with literature ones. The gold leads along the cantilever significantly contribute to the cantilever thermal conduction, strongly affecting the probe thermal resistance. The electrical and thermal conductivities of gold in the calibrated model with the nominal lead thickness of 145 nm (as in the manufacturer's specifications) were set to 35% of the bulk values; such a reduction is possible in thin films of gold as a result of grain boundary scattering of electrons. [41]

Accounting for the Thermal Resistance of the Probe Tip
After calibrating the probe parameters, the FE model was used to calculate the response of the probe to the presence of a finite c th R at its apex (refer to the schematic in Figure 2b for the meaning of c th R ), both for DC and AC ω-3ω measurements. As illustrated in Figure 2b, the contact resistance c th R arises due to three contributions: from the probe tip, from the probe-sample interface, and from the sample. The latter-the spreading thermal resistance of the sample, s th R -contains information about the thermal conductivity of the sample material and is the ultimate goal of the measurements. Two other components in series with it-the thermal resistance of the sensing tip, t th R , and the thermal resistance of the contact boundary, b th R -are essentially parasitic in the SThM measurement, and possibly need to be reduced to achieve higher sensitivity to the material properties embedded into s th R . In any case, R . In turn, in Ref. [23], the temperature dependence of the material thermal conductivity was used for the same goal. While these approaches were successful, the first one cannot be implemented in the case of ceramic materials of interest in our studies, and the second one requires a low-temperature microscope, which is not readily available. In our approach, we can separate the tip resistance from other contributions into c th R with the use of the calibrated FE model of the probe. After that, the boundary resistance can be determined in the measurements of materials with known thermal conductivity that are present in the ceramics of interest as inclusions. Because of the distributed nature of the electrical resistance of the Pd sensor of the probe and uneven current and temperature distribution over the senor, t th R can be dependent on the thermal resistance present to the probe by a sample at the tip apex, that is, on the sum bs th b th s th R R R = + . Moreover, t th R generally should be a function of the current frequency, which directly affects the results of the measurement with use of the 3rd harmonic of the probe current frequency (i.e., ω-3ω technique).
In the calibrated FE model, bs th R was modeled as a resistance of a thin thermally resistive layer between the tip apex and a surface at room temperature (equal to the temperature of the chip end of the cantilever). The probe resistance was calculated for the DC, and the amplitude of the 3rd harmonic of the current frequency was calculated for the ω-3ω measurements. Figure 5b presents the result of two calculation runs with varying interface layer resistance at the probe apex obtained at DC and at a current frequency of 1 kHz. As seen, the calculated "measured" contact resistance seen by the probe is not equal to the interface layer resistance, which could be explained by a non-zero value of t th R . It is also apparent that t th R and b th R do not simply add up. It could be assumed that t th R can be determined by "short-circuiting" the b th R in the model. However, the results in Figure 5b indicate that t th R should be treated as dependent on the sum ( bs th s th R R + ) as well as the current frequency and increases with the resistance of the sample. The dependencies shown in Figure 5b are linear. As seen, the t th R is relatively large, amounting to about 1.4 × 10 7 K W −1 at DC for the probe, based on the value at 0 b th R = , and drops by a factor of 6 for the 1 kHz AC current. The slope of the line for the AC response is also smaller by a factor of about 5. We have not performed a detailed analysis of the reasons for such behavior leaving it outside the scope of this work; such an analysis will be performed in the near future. In turn, the effect of the total thermal resistance presented to the tip on the value of t th R can be readily interpreted by examination of the calculated maps of the temperature and current density distribution in Figures 6a,b, as well as values changes of the current density along the surface of the Pd stripe displayed in Figure 6c. The differences are shown between the largest contact resistance used and the "short-circuit condition" (a small value of the resistance of the resistive layer, such that any further reduction of its resistance yields negligibly small changes of the "measured" c th R ). First, it should be noted that the current density in the sensing stripe falls toward the apex. In the shown map in Figure 6b, the current density near the apex is about 15 times smaller than in the middle along the stripe. From inspection of the map in Figure 6c, it is seen that the cooling of the apex side of the sensor stripe leads to the current redistribution toward the apex resulting in a shift of the effective "center of gravity of the current" toward the apex. Hence, the density of heat generation is also shifted toward the apex, reducing the distance that heat needs to travel to reach the sample surface, which results in the reduction of the effective thermal resistance of the tip. Based on this picture drawn for DC, we may speculate that at AC, a similar effect leads to the dependence seen in Figure 5b for 1 kHz. However, at AC, the probe response is determined by temperature oscillations propagating in the form of evanescent thermal waves. This results in different and frequencydependent distributions of resistivity and AC current density over the sensor stripe of the probe, which, in turn, yields the frequency dependence of the effect value.
The calculated calibration curves in Figure 5b were applied to the experimental SThM maps to obtain thermal resistance maps of the sub bs th R presented and discussed in the next section. for the two samples are presented in Figure 7. All the maps in Figure 7 were recorded with one and the same probe. Altogether, the useful life of the probe lasted 7-8 JM-SThM maps; after that the probe degradation led to nearly contrastless images. The numbers of the JM-SThM maps in the imaging sequence are indicated in Table 1. The maps show agglomerated particles of zirconia and ceria as inclusions embedded into the matrixes of doped ZnO (see also SEM images of the samples in Figure 1). The bs th R maps were acquired in both DC and AC modes, and as seen comparing maps in Figure 7b,c,e, the results are consistent.

Analysis and Quantification of the Thermal Resistance Maps
For in-detail quantification of the maps in Figure 7, we adopt the approach used in the SThM measurements, where the thermal size of probe-sample contact is taken equal to the radius of the probe tip apex instead of the size calculated based on one of the contact mechanics models (Hertzian or, more usually, Derjaguin-Muller-Toporov). The criterium for the choice of the contact size is the correct range of the probe-sample interface thermal resistance, i th ρ , between 10 −9 and 10 −7 K m 2 W −1 , where interfacial resistances for different material pairs fall. [26,33] The lower interface resistance bound, 10 −9 K m 2 W −1 , corresponds to clean metal-metal interfaces where primarily free electrons are responsible for the heat transfer. [42] The higher value bound of the interface thermal resistance (close to 10 −7 K m 2 W −1 ) corresponds to phononmediated thermal transport through clean interfaces between materials with very different crystal lattice properties, such as bismuth and diamond [43] as well as between amorphous materials. Following this approach, the values of the contact radii for the KNT probes are 50-60 nm, that is, larger by over one order of magnitude than with the use of the contact mechanics models. [33] More direct measurements of the thermal contact size also yielded values in the range of 50-80 nm for the KNT probes. [32,44] The failure of the contact mechanics models to correctly predict the contact size, when applied directly based on a nominal radius of the probe apex, can be explained by the flattening of the apex as a result of imaging. [33] With the radius of the probe-sample contact r = 50 nm and the thermal conductivity of the ZnO matrix k ZnO = 40 W K −1 m −1 , we can estimate the spreading thermal resistance s th R of ZnO under the probe as: This value is close to that obtained in the work of Spièce et al. [32] , who were able to measure the interface resistance between a KNT probe and a thermally grown SiO x , and found a value of i th ρ = 8.9 × 10 −8 K m 2 W −1 . However, the range of the values varies between the maps in Figure 7 is not accurately reproducible; in our measurements, we also register a few times lower as well as much larger values varying between maps. The interface resistance can be determined by several factors. For clean and structurally perfect interfaces, it is a function of the phonon and electron structure of the materials, [45,46] and may vary between inclusions and the matrix due to these factors.  Large and varying values of the interface resistance in our experiments can be attributed to the sample surface condition and varying properties of the probe apex surface that change as a result of the probe sample interaction during imaging. The surface of our ceramic samples is not perfect; it most likely has a disordered layer and contaminations left after polishing. Structural disorder at the interface of crystalline contacting bodies can significantly increase the interface thermal resistance, [47] enhancing the scattering of phonons and electrons, and reducing the energy transfer across the interface. Apparently, the same effects are produced by "weakening" the interfaces with, for example, contaminations or simply by reduction of the bonding strength. [48,49] Therefore, the upper bound for the interface thermal resistance in the SThM experiments can be above 10 −7 K m 2 W −1 , a value that is valid for clean, structurally perfect interfaces with a strong bonding between contacting materials. Besides, the surface roughness of both the sample and probe surfaces may reduce the contact area with the heat transfer taking place only through asperities that are in contact.
Since the ceramic sample surface roughness can be a major contributor to the boundary resistance, we have assessed the surface roughness and its variation between materials in the maps of Figure 7. For that, we obtained high-resolution images     of the sample surface topography in the semi-contact (tapping mode) with the use of a sharp silicon cantilever (Nanosensors PPP-NHCR, nominal tip apex radius <10 nm). For the examination of the roughness and its potential effect on the variation of the contact thermal resistance, we applied Fourier analysis of the high-resolution topographic maps comparing Fourier spectra of the topography in the spatial frequency range above the inverse of the probe-sample contact diameter 1/(2r), which corresponds to topography wavelengths, λ sp , "seen" by the probe apex (such that λ sp ≤ 2r). The high-resolution topographic images are shown in Figure S6a,b, Supporting Information, and the corresponding map of 2D DFT transform of the selected areas from the samples are shown in Figure S6c, Supporting Information. As could be concluded from the spectra, the difference between the ZrO 2 and CeO 2 inclusions and the ZnO matrix is small in the spatial frequency range of interest. We have also performed roughness analysis using the Gwyddion software.
[50] The software allows calculating roughness parameters along selected cross-section line profiles in the images with the decomposition of the profiles into the long-range (low-frequency component) waviness and short-range (high-frequency component) roughness; the roughness parameters are calculated based on the high-frequency component only. With the cut-off frequency set to 1/130 nm −1 , the RMS roughness along the scan lines is in the range 0.1-1 nm, a value below the lattice unit cell. We conclude that the roughness is not the reason for the boundary resistance variation in the bs th R map in Figure 7. It is important to note also that the high-resolution topography images do not show the inclusions of ZrO 2 as agglomerates of nanoparticles; there are no depressions and grooves, as seen in the SEM images in Figure 2, which could appear between nanoparticles. This can be explained by the efficient integration of the nanoparticles in the inclusions due to sintering. The grainy structure of the inclusions can be revealed only after the thermal etching, which was performed for the SEM images but was not made for the SPM experiments. It should also be noted that in contrast to ZrO 2 , CeO 2 inclusions in the ZnFeOCeO sample are integral entities in the SEM images in Figure 1.

Analysis of the Local Thermal Conductivities in the Composite Ceramics
As was estimated above, the thermal resistance maps in Figure 7 are overwhelmingly dominated by the probe-sample boundary resistance, and the signal variations due to changes in the material thermal conductivity are on the order of 1% of the total contact thermal resistance or less. Under such conditions, the value of the thermal conductivity of the inclusion still can be evaluated differentially using the thermal conductivity of the doped ZnO matrix as a reference value and assuming that the boundary resistance does not vary between materials. From another perspective, it may be assumed that the knowledge of the thermal conductivity of the ZnO matrix and ZrO 2 inclusion in the ZnAlOZrO samples can be used for probe calibration and, in particular, to determine the probe-sample thermal contact radius, how it is frequently done in SThM experiments.
Following this route, we have measured the thermal conductivities of the ZnAlO and ZnFeO matrixes with the use of bulk ZnO ceramic samples doped with Al and Fe, respectively, but without nanoparticles of ZrO 2 and CeO 2 , in the temperature range from 900 °C down to 150 °C. The temperature dependence for both the compositions showed a linear dependence on 1/T ( Figure S1, Supporting Information) down to the lowest temperature of the measurements (150 °C), indicating that the thermal conductivity is controlled by the Umklapp processes (phonon-phonon scattering) without a noticeable contribution of grain boundaries in the heat transport. [51] Therefore, we determined the thermal conductivity of the bulk sample at room temperature by extrapolating the 1/T dependence down to room temperature, and because of the negligible contribution of the grain boundaries, the room temperature values obtained this way can be applied to individual grains of the matrix in samples doped with nanoparticles. The room temperature thermal conductance of ZnAlO and ZnFeO matrixes was found to be about 30 and 33 K W −1 , respectively. These values were used as reference values in the further interpretation of the maps in Figure 7.
In turn, according to the most recent literature data, room temperature thermal conductivity of the monoclinic ZrO 2 is 6-7 W m −1 K −1 (with a significant temperature dependence), [52,53] which is in agreement with earlier modeling results of Ref. [54]. The material studied in Ref. [52] was a ceramic of pure, monoclinic zirconia with a grain size of 63 nm and 98% of the theoretical density, which is similar to the apparent characteristics of zirconia inclusions of our sample (Figure 1, Ref. [19]). In particular, the authors of Ref. [52] stressed that the temperature dependence of thermal conductivity of the monoclinic ceramic samples is linear versus 1/T, indicating the lack of any significant contribution of the grain boundary phonon scattering even with the grain size as small as 63 nm. Therefore, similar to the ZnO matrix, we can apply the values of the thermal conductivity obtained with bulk samples in Ref. [52] to the nanoscale-sized ZrO 2 inclusions in our samples.
We have made an attempt to determine the probesample thermal contact radius with k ZnO = 30 W m −1 K −1 , k ZrO2 = 6.5 W m −1 K −1 , and the data in Figure 7 for the ZnAlOZrO sample as summarized in Table 1, for both DC and AC measurements, assuming that the probe-sample boundary thermal resistance does not vary between materials. The contact radius values obtained this way turn out to be too small for the KNT probes: about 7 and 10 nm, respectively. The corresponding values for the probe-matrix interface resistance, i th ρ , are 1 × 10 −9 and 9 × 10 −9 K m 2 W −1 , respectively, with the former value being significantly out of the expected range. Account of the measurement uncertainty does not bring the values of the radius and i th ρ for the DC measurements in the expected range. It can be done only with k ZrO2 reduced to 1-2 W m −1 K −1 , which is too small. Such values are characteristic of cubic stabilized zirconia, where the thermal conductivity is determined by a significant number of oxygen vacancies playing the dominant role in phonon scattering, which is not the case for the monoclinic phase. [52] Apparently, with the boundary resistance dominating, we have to include in the analysis of the maps the possible variation of the boundary resistance b th R between materials. Indeed, Small Methods 2023, 7, 2201516 the data of Table 1 become consistent with expectations if we introduce into the analysis the material-dependent interface thermal resistance i th ρ . For that, we set k ZrO2 = 6.5 W m −1 K −1 with a contact-sample thermal radius of r = 50 nm as was determined multiple times for the KNT probes [26,27,32] (keeping in mind that this size can change as a result of the sampleprobe interaction in the course of the probe usage) and evaluate the difference between bs th R for the inclusion and matrix,  Table 1 for the DC ZnAlOZrO map, we find that the data can be explained by a larger i th ρ for zirconia inclusions with a difference i th ρ ∆ = (3.1 ± 1.6) × 10 −8 K m 2 W −1 relative to the ZnO matrix (which yields an about order of magnitude larger variation of the contact resistance than due to the spreading resistance of the materials). In turn, with this value of i th ρ ∆ , the AC data in Figure 7c ( bs th R ∆ = 2.9 × 10 6 K W −1 between the inclusion and matrix) can be matched with r = 63 nm. The value of i th ρ ∆ = 3.1 × 10 −8 K m 2 W −1 is reasonable taking into account the difference of lattice structures of the monoclinic ZrO 2 and hcp ZnO, and hence, of the phonon spectra of the materials. Different disordered layers on the surface can also contribute differently to the contact resistance. The increase of the contact thermal radius from 50 to 63 nm between maps #2 and #4 (Figure 7b,c) can be ascribed to the probe wear and flattening of the probe tip apex during mapping. It is worth noting also that, bs th R ∆ = 4.5 × 10 6 K W −1 of the DC of map (map #2, Figure 7b) is within the range of the data from the AC map with an account of uncertainty. It is also important to note that assuming equal interface resistance of the matrix and inclusion, a difference of bs th R ∆ = 1.35 × 10 6 K W −1 in the contact resistances, which is less than 30% of the measured bs th R ∆ in the DC map, would yield k ZrO2 = 6.5 W m −1 K −1 with the DC map data, r = 50 nm and k ZnO = 30 W m −1 K −1 . This means that most of the contrast between the matrix and the zirconia inclusions in the maps of the ZnAlOZrO sample is a result of the varying contact interface resistance, while the sensitivity of the probe to the material thermal conductivity is low. Now, we apply the same approach to the map for CeO 2 in Figure 7e (map #6) obtained with a DC current through the SThM probe with the use of the same probe. The thermal conductivity of undoped, pure CeO 2 at room temperature is k CeO2 = 14 W m −1 K −1 , [55] and the thermal conductivity of the matrix in the ZnAlOZrO sample is k ZnO = 33 W m −1 K −1 as mentioned above. Assuming that the b th R values are equal for the matrix and CeO 2 inclusion in the map in Figure 7e and applying the data of Table 1 with r = 63 nm for the probe-sample contact radius, we find k CeO2 = 11.3 W m −1 K −1 , which is very close to 14 W m −1 K −1 of Ref. [55] (r = 50 nm results in an even better match k CeO2 = 13.1 W m −1 K −1 ). However, the contrast between the inclusion and the matrix in the map is weak, and, as a result, the measurement uncertainty is large. It should be noted that the value of k CeO2 = 14 W m −1 K −1 corresponds to a difference between the inclusion and matrix contact resistance, bs th R ∆ , of about 30% less than the value in Table 1, which is already quite small compared to the difference for the ZnAlOZrO sample. Apparently, with this sample, the b th R of the matrix and inclusion are nearly equal. This indirectly verifies the conclusion of the previous paragraph.
For completeness, Figure 7f shows the map for the ZnFeOCeO samples obtained with AC current through a probe (map #10 in Table 1), which shows a very weak contrast and is unsuitable for quantification. The contact resistance in this map is substantially larger-evidencing probe degradation, and a trace of probe damage is clearly seen at the end of the scan at the bottom of the map. The probe degradation modalities leading to such a behavior are discussed in Section S6, Supporting Information. Based on the observed probe degradation, the probe sensitivity is mainly reduced due to the adsorption of impurities and picking up debris from the sample surface.

Conclusions
To summarize, we have developed a novel SThM imaging mode, "jumping mode", JM-SThM, to preserve the probe integrity while imaging rough surfaces of ceramic samples.
We have also described and tested a SThM probe calibration technique, with probe calibration requiring no other setup but the setup used for imaging. This calibration technique is used to calibrate a finite elements model of the probe, which further allows quantitative imaging of sample thermal properties. The FE modeling reveals non-negligible effects associated with the distributed nature of the KNT resistive probes, which was not accounted for before. Corrections should be applied to the contact resistance values obtained with the standard lumped-element model of the probe to account for the distributed effects in quantitative imaging. Our experiments with real thermoelectric ceramics showed that the JM-SThM can be used for reliable imaging of ceramic samples. Quantitative analysis of a series of images showed a large probe-sample boundary thermal resistance, which reduces the sensitivity of the measurements of thermal conductivity in the range of interest close to 10 W K −1 m −1 and above. Moreover, we found in the case of the ZnAlOZrO composite ceramics that most of the contrast between the ZnO matrix and ZrO 2 inclusions is due to varying probe-sample boundary resistance. Apparently, such material dependence of the probe-sample boundary thermal resistance generally needs to be accounted in the qualitative and quantitative interpretation of SThM images. The measurement sensitivity can be improved in part by increasing the thermal conductivity of the material of the probe tip apex. Based on the observed gradual probe degradation mode, it is concluded that the probe sensitivity is reduced in the course of imaging due to the adsorption of impurities and debris from the sample surface, which calls for improved techniques of sample surface preparation. For instance, delicate surface treatment using controllable plasma action or sample annealing immediately in the camera of the microscope can be used to improve the sensitivity of the resistive probes to the bulk thermal properties.

Experimental Section
Sample Synthesis and Preparation: The samples selected for the experiments were ceramic composites based on ZnO matrix doped with Al or Fe and containing ZrO 2 or CeO 2 nano-and microparticles, respectively, for the expected reduction of thermal conductivity, potentially combined with other favorable effects on the electrical properties. [19] Zn 0.993 Al 0.007 O-20wt% ZrO 2 (ZnAlOZrO) and Zn 0.993 Fe 0.007 O-10vol% CeO 2 (ZnFeOCeO) composite ceramic samples for SThM were prepared using a solid-state route. Details on experimental procedures for processing Zn 0.993 Al 0.007 O-20wt% ZrO 2 (ZnAlOZrO) composites can be found elsewhere. [19] For the preparation of the nanocomposite ceramic, ZrO 2 nanopowder provided by Sigma-Aldrich with a particle size below 100 nm was used. The matrix with the Zn 0.993 Al 0.007 O composition was pre-synthesized by mixing the corresponding amounts of ZnO (Alfa-Aesar, 99.99%) and Al 2 O 3 (Sigma-Aldrich, 99.7%) precursors. Multiple annealing steps at 900-1100 °C for 5-15 h, with intermediate regrindings, were conducted. Thus prepared Zn 0.993 Al 0.007 O powder was dispersed in alcohol and subjected to ultrasonication with the simultaneous gradual addition of the corresponding ZrO 2 powder in order to improve homogenization. The final mixture was dried by delicate heating using ultrasound; diskshaped samples were isostatically compacted and sintered at 1500 °C for 5 h in air. The densities of the sintered nanocomposite samples were within 94-99% of the theoretical density.
In the case of Zn 0.993 Fe 0.007 O-10vol% CeO 2 (ZnFeOCeO) composite, the Zn 0.993 Fe 0.007 O matrix was prepared by mixing the corresponding amounts of ZnO (Alfa-Aesar, 99.99%) and FeC 2 O 4 ·2H 2 O (99%, Merck). Thus obtained precursor powder was annealed three times at 950 °C for 6 h, with multiple intermediate grindings, followed by the addition of nanosized CeO 2 (<25 nm particle size BET, 99.95%) powder under ultrasonication conditions. The obtained powder was uniaxially compacted and sintered at 1500 °C for 10 h to produce ceramic samples for further characterization.
The samples were fractured and polished for SEM investigations. To reveal grain boundaries in the SEM images, the samples for SEM were thermally etched after polishing.
In turn, for SPM, extra-fine polishing of samples was performed without thermal etching. The surface of the fractured ceramic composite samples was step-by-step polished by the silicon carbide grinding papers and then abrasive diamond paste with particle size decreasing down to 0.25 µm. Fine polishing with an alkali-based solution of 60 nm-size colloidal silica (SF1 Polishing Suspension, Logitech, United Kingdom) was performed for around 30 min, to remove the mechanically strained region from the sample surface that could appear after previous polishing steps. After that, the samples surfaces were washed with acetone and isopropanol. The last step was repeated right before installing samples into the SPM for measurements in vacuum.
Scanning Electron Microscopy: Both polished then thermally etched and freshly-fractured ceramics were characterized by SEM combined with energy dispersive spectroscopy (SEM/EDS). The structural and microstructural characterization was performed using a Rigaku D/Max-B diffractometer (Cu Kα) and a Hitachi SU-70 SEM equipped with a Bruker Quantax 400 EDS detector.
Scanning Thermal Microscopy: SThM was performed on a SPM NTEGRA Aura (NT-MDT, Russia) in a medium vacuum at a pressure of 10 −3 Torr. KNT-SThM-2an (Kelvin Nanotechnology, UK) resistive probes (obtained from NanoAndMore, GmbH, Germany) with a nominal tip radius below 100 nm and a spring constant of 0.45 N m −1 were used. The probe-sample force was set to 3 nN during all SThM experiments. As illustrated by the schematic in Figure 2, the probe resistance was monitored with the use of a Wheatstone bridge and a home-made pre-amplifier assembled on the basis of low-noise instrumentation amplifier integrated circuits. SThM was performed in two modes: with a DC current (DC mode) and with an AC current (AC mode) through the probe. In the latter case, the voltage oscillations through the probe were detected at the 3rd harmonic of the AC current (ω-3ω method, or 3ω SThM [4] ). The AC measurements were performed with the help of a Zurich Instruments HF2LI lock-in amplifier.
FE Simulations: Numerical modeling of the probe was carried out with the use of the Joule Heating module of the COMSOL Multiphysics finite elements analysis package (COMSOL, AB). The dimensions and thickness of the KNT probe cantilever and the gold electrical leads were taken from manufacturer's specifications, and the dimensions of the Pd sensor stripe were determined from the authors' own SEM images. The thickness of the Pd stripe was set equal to 40 nm as indicated in manufacturer's specifications. The material properties were a subject of the model calibration process, as described above in the paper. A more detailed description of the numerical model can be found in Section S3, Supporting Information.
Statistical Analysis: The reproductivity and consistency of the reported measurements were proved with the same samples that are described in the manuscript and three different KNT probes from one and the same probe set as delivered by the probe manufacturer. Processing of the raw JM-SThM map data was performed with the help of homemade Python scripts as well as Gwyddion v.2.61 software [50] using the algorithms described in the main text and Supporting Information. The values in Table 1 were obtained over pixels in respective spatial regions of the JM-SThM maps in Figure 7 (corresponding to a matrix or to an inclusion) with more than 50 pixels for each map region. Average, avr, and standard deviation (RMS), sd, values were calculated for each data subset and shown as avr ± sd in Table 1. The data for Table 1 were extracted and calculated from the JM-SThM maps with the use of the Gwyddion v.2.61 software. [50]

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.