Scanning thermal microscopy and its applications for quantitative thermal measurements

In the last few decades, nanotechnology has continued to be of growing importance in many areas of science and technology. Production of structures with the length scale of several nanometers is now a standard in the electronics industry. Nowadays, technology allows the fabrication of low-dimensional structures, e.g., quantum dots, nanowires, or 2D materials like graphene, silicene, etc. Classical laws have failed in describing the transport phenomena in such structures, thus new approaches have become necessary. New opportunities and the challenges connected with heat transport at the nanoscale were described in two review papers.^1,2 Both papers contain sections that are devoted to thermal metrology at the nanoscale, and the scanning thermal microscopy is indicated as a method that has a great potential.

SThM was developed by Williams and Wickramasinghe in 1986.^3,4 It belongs to scanning probe microscopies (SPMs), in which an image shows distribution of quantity characterizing interaction between the sample and a probe scanning sample surface. The currently used SThMs utilize a working principle that was described by Majumdar et al.⁵ The idea was to combine topographical imaging with thermal imaging, alongside a use of an atomic force microscope (AFM) equipped with a thermal module. For this purpose, special AFM probes with a temperature sensor positioned close to the probe apex are used. The operational principle of the SThM is described in Sec. II. In this section, a methodology for the measurements is also analyzed. The key part of all SPMs, including the SThM, is the probe used for the imaging. The thermal probes available on the market, and those proposed in the literature, are characterized in Sec. III. Understanding of the potentialities and limitations of this measuring technique, and the proper interpretation of the experimental data, is possible with a theoretical model for the physical phenomena responsible for measurements. In the case of thermal measurements, this model is based on a description of the thermal transport in a probe–sample system. Section IV is devoted to the theory and modeling of the SThM measurements. Finally, a few examples of using the SThM for various applications are given in Sec. V. At the end of each section, an analysis on the possible directions for future development is performed. In the conclusion, the perspectives and the challenges for the SThM are summarized.

This paper is focused on the perspectives of SThM. The perspectives of any measuring method arise from the current state-of-the-art and the work on its development. Therefore, this paper is focuses on works that appear to be important for further development of the SThM equipment and methodology. The basics on SThM can be found in a previously published tutorial,⁶ and a detailed overview of the works performed in this field can be found in Refs. 7–10.

The SThM is a SPM that utilizes thermal probes suitable also for AFM imaging. In practice, the probes have a special design and differ from typical AFM probes. The thermal probe (TP) should provide the opportunity for topographical sample characterization in the contact mode, and the measurement and control of the probe tip temperature. A fulfillment of the second requirements often leads to a deterioration of the topographical image resolution, as the radius of the TP tip is typically larger than the one for the standard AFM probe.

The schematic diagram of the SThM is shown in Fig. 1. This is a standard AFM microscope that is equipped with a module for the thermal measurements. The AFM module is responsible for the mechanical part of the system. It controls the approach of the sample to achieve the probe–sample contact and maintains the probe–sample interaction force constant during the sample surface scanning. The thermal module controls “the thermal part” of the microscope that measures a signal from the temperature sensor. To increase the measurement sensitivity, the TP is usually mounted onto one leg of a measurement bridge (Wheatstone bridge or its analog). In the case of popular resistive thermal probes, the thermal module controls the sensor current and measures the sensor electrical resistance (actually, the bridge unbalanced voltage), which is related to TP temperature. A feedback loop enables to keep constant one of the following parameters: the probe current, the bridge unbalanced voltage (the probe resistance related to the sensor temperature), or the power dissipated in the sensor. The development of TPs that are employed in SThM measurements is described in Sec. III.

One more effect should also be accounted for when the thermal model of the SThM measurement is considered. Here, AFM and SThM signals are registered simultaneously. The operational principle of the AFM requires a continuous control of the deflection of the probe cantilever. For this reason, the beam deflection technique is used, which is shown in Fig. 1. However, the light beam that strikes the probe end is not only reflected but also partially absorbed. The absorption leads to the appearance of an additional heat source in the system. This effect was investigated in Ref. 13, and a dark mode measurement technique was proposed. In the “dark mode,” the optical beam is switched off when the SThM signal is measured. However, such technique is difficult (or even impossible) to implement in many SThM devices. The influence of additional light heating can be reduced by decreasing of ambient pressure or by measurements in the ac mode.¹³ The problem was also analyzed by Spiece et al. in Ref. 14. They showed that the SThM signal is influenced by the changes in the light beam alignment. This confirms that the measurements in the dark mode can improve the stability of the SThM measurements.

The structure and the operational principles of the SThM have practically not changed since 1993. The main modification observed in the last few years is a transfer of the measurements from air to vacuum. Twenty years ago, the vacuum SThMs were installed in a few laboratories in the world. Nowadays, the measurements performed in a vacuum, but most often in rough vacuum (∼10⁵–10² Pa), have become commonplace. The most important advantage of the vacuum measurements is the minimalization of the ambient conditions influence on the measured signal. Since a full control of the ambient parameters is impossible, only the measurements under vacuum can ensure a lower uncertainty in the experiment. The influence of the air that surrounds the WW probe–sample contact was investigated in Ref. 15. One of conclusions is that the influence of air is strongly reduced for pressures lower than 1 Pa at probe temperature 150 °C. It should be stated here that unambiguous definition of the pressure below which the influence of the environment on the measurements disappears is difficult. This limit depends on many factors, e.g., probe and sample materials, probe tip shape, probe temperature, and can be in mPa range or even lower. At present, commercially available SThMs microscopes enable measurements from atmospheric pressure to about 100 mPa.

The vast majority of the SThM measurements are performed in the contact mode. However, non-contact mode experiments can be performed in fluids with a controlled gap between the probe tip and the sample surface. This mode is used in the double-scan technique for an actual surface temperature measurement (TCM). The method is described in Sec. II A. The possibility of utilizing the non-contact SThM for thermal conductivity measurements (CCM) was proposed by Zhang et al. and it was analyzed in Ref. 16. The lack of a poorly defined tip–sample contact simplifies the description of the heat transfer between the TP and the sample. The lateral resolution of the non-contact measurements is inadequate as it is defined by the heat spreading in the fluid not by the probe tip radius. Nevertheless, the quantitative thermal measurements based on the analysis of the force and thermal approach-retraction curves^14,17,18 are worthy of consideration.

New application fields for the SThM can be opened by the SThM operating in a liquid environment for both the TCM¹⁹ and CCM.²⁰ Experiments performed in dodecane proved the sensitivity to the local thermal conductivity and a good lateral resolution of the measurements. Since the heat flows to the sample predominantly through the liquid rather than through the weakly defined solid–solid, the in-liquid measurements can provide a lower uncertainty. The measurements in the liquid environment provide the possible applications of SThM in the investigation of the thermal phenomena in biological systems.

The methodologies of the SThM measurements differ depending on the operating mode. The main challenge in the TCM is the determination of the actual temperature at the sample surface. The problem is that, during measurements, the temperature sensor is not in its thermal equilibrium with the sample. Let us assume that the local surface temperature is higher than the ambient one. In this case, the TP is heated. The heat flux flows through the tip–surface contact (and its surroundings for measurements in air). The probe–sample system cannot reach the thermal equilibrium because the TP cantilever is mounted onto a holder at the ambient temperature. The heat flux requires a temperature gradient, so the sensor temperature must be lower than the temperature of the sample surface.

Both the double-scan and the null-point methods are dedicated to the thermocouple TPs, in which the thermal sensor (the metal–metal junction) is in contact with the sample. In the case of the resistive TP, this situation is more complex as the thermal senor is distributed and only the mean sensor temperature can be measured. This scenario is especially pronounced for the batch-fabricated TP from Kelvin Nanotechnology (KNT). In the case of the doped silicon (DS) probes, the sensor is a part of cantilever from which the probe tip protrudes.²⁸ Therefore, the sensor is relatively distant from the sample surface.

DS TPs were used by the Gotsmann group for temperature measurements using high-vacuum SThM.²⁹ To measure the temperature of a sample surface, a few steps are required. First, the cantilever temperature must be calibrated to obtain the relationship between the probe temperature and the probe electrical resistance. In the next step, the tip–sample thermal resistance is determined. The whole thermal resistance $R th=1/ G th$ can be calculated using Eq. (2). Its value for the TP, being far from the sample, gives the cantilever a thermal resistance $R c th$⁠. From the change in $R th$ during the approach toward the sample, the tip–sample thermal resistance $R s th$ can be obtained. This measurement is performed at room temperature $T 0$ and repeated for each tip position on the sample (i.e., each pixel of an image). Finally, the same procedure is completed for the heated sample. As $R s th$ is known, the heat flux to the sample depends on sample temperature, the local temperature change can be calculated.²⁵ This method acts as a variant of the double-scan technique with an idle and active investigated device. Another method for determining $T s$ requires Joule heating with the ac current. The temperature of the heated element has dc (steady state) and ac (alternating) components, which can be measured simultaneously. Based on these measurements, the sample temperature increase is calculated.²⁶ The most sophisticated method is a dual-sensing technique, in which the total dc heat flux through the tip–sample contact is simultaneously monitored with the ac heat flux caused by the periodic sample heating.²⁷ The temperature resolution achievable by this technique was estimated to be 190 μK.

All the described methods for the sample temperature measurements are attempts to overcome the problem that the thermal equilibrium cannot be reached in the TP–sample system. It is difficult to define which of them are suitable for standard measurement, i.e., those that can be performed on a typical SThM for real samples (rather than those specifically prepared for the measurements). For the imaging of the temperature distribution, the double-scan technique seems to be a good choice. For the local temperature measurements, the null-point method is the most promising. For the quantitative temperature measurements, the TP needs to be calibrated. The calibration procedure requires a well-defined temperature of the sample surface. The simplest solution is to use a Peltier module with an attached calibrated temperature sensor (e.g., a thermocouple).³⁰ However, special devices have most often been manufactured to provide a temperature uniformity at the surface.^31–35 They were used not only for temperature sensors calibration but also for the overall characterization of the probe–sample contact.

When in the CCM mode the TP is heated, its temperature depends on the effectiveness of the heat extraction from the heated region into the surroundings. In the steady state, this process is described by Eq. (2), for which, $G th$ is the thermal conductance for the total heat flux from the heated region to the surroundings. Its determination requires the knowledge of the dissipated power P and the mean probe temperature rise $θ$⁠. However, a determination of $G th$ is not a goal of the measurements since the aim is to discover the sample thermal conductivity. In SThM systems that operate in ambient conditions, there are three channels for the heat extraction from the heated TP to the sample, along the cantilever to the probe holder, and to the surroundings via air. These channels are described by the three thermal conductances $G s th,$ $G c th$⁠, and $G a th$⁠, respectively; the sum of which gives $G th$ and the thermal conductivity of the sample affects $G s th$⁠. However, the heat transfer to the sample is fairly complex and it depends on many factors. The heat flows through the contact but it also passes through the medium surrounding the contact (i.e., the air and the water meniscus). The energy is also transferred by the thermal radiation, while some mechanisms are excluded for the vacuum SThM. The heat transport through the contact is dependent upon the sample surface roughness, the constriction resistance, and the boundary resistance. A more detailed analysis of the heat fluxes in the TP–sample system is presented in Sec. IV. However, it should be stated here that the analysis of the share of $G s th,$ $G c th$⁠, and $G a th$ in $G th$ show that, for the Wollaston wire (WW) and the KNT TPs, $G s th$ is of around 30% of $G th$ for samples with $k>100 W m − 1 K − 1$⁠.³⁶ Similar values for the WW probes were obtained in other analyses.^37,38 The $G s th$ share in $G th$ provided here is taken from the analysis completed for two specific probe types. This value can significantly differ when comparing the probe types. Moreover, it can also vary from probe to probe for the same probe type. Therefore, each probe requires separate calibration.

The thermal conductivity measurements using the SThM are based on the initial calibration of the TP–sample thermal exchange. For this purpose, a set of reference samples are employed. There are two basic strategies for the TP calibration: the implicit procedure, which is the simplest one, and the explicit method based on the thermal exchange calibration.³⁹ The SThM signals are measured for the reference samples, and experimental points are fitted with a theoretically predicted dependence to obtain the calibration curve. An exemplary calibration curve is shown in Fig. 2;⁴⁰ more examples can be found in the literature, for example, in Refs. 41–44.

An analogous formula for the electron transport was first derived by Maxwell.⁴⁹ This model was used for the interpretation of the experimental data obtained in various experiments.^16,37,50,51 To determine the sample thermal conductivity k, three parameters of the model must be known: $h eff$⁠, b, and $R bnd th$⁠. According to Eq. (2), the $θ(P)$ dependence allows the $R th=1/ G th$ determination (here, $θ$ is the mean probe temperature). For the probe in air, $R th$ is correlated to $h eff$ while the parameter b can be obtained from the independent topographical measurements.^52,53 It is the determination of $R bnd th$ that is the most challenging task. First, $R th$ is measured for two or more reference samples with a known k. Then, $R th$ is calculated using Eqs. (4) and (5) with $J s$ as a parameter; the latter is adjusted until $R th | measured= R th | calculated$⁠. Finally, $R bnd th$ is calculated using Eqs. (6) and (7).⁴⁶ Here, it should be remembered that $R bnd th$ depends on the sample, hence its values may considerably change for different samples.

At this point, attention should be paid to a simplification of the adopted model. First, this is one-dimensional model, in which the radial heat transport to the surroundings is described by $h eff$⁠. The influence of the sample on the temperature distribution in the probe is described by the boundary condition provided by Eqs. (6) and (7). However, when measurements are performed in a fluid, the heat is transferred not only through the contact but also through its surroundings. The existence of a parallel channel for the TP–sample heat transport lowers $R s th$⁠. This effect can be considered through the effective thermal contact radius $b eff>b$⁠; it was shown that $b eff$ is proportional to $1/k$⁠.⁴⁰ Moreover, $R bnd th$ depends on the materials that are in contact and their surface roughness.

The modeling of the heat transport in the TP–sample system for TPs, other than WW probes, is a more challenging task. This is because the complex geometry used by the simple one-dimensional model is unjustified. Therefore, the models based on electro-thermal analogies or the finite element method, described in Sec. IV, are employed.

The calibration method assumes that for all the reference sample conditions of the thermal transport are the same, except the k values. However, the thermal transport from the TP to the sample is influenced by numerous factors: the local sample topography,^54,55 the sample roughness,⁵⁶ the contact environment (air or water meniscus),^15,57,58 the thermal radiation,⁵⁹ and the thermal boundary resistance (Kapitza resistance).⁶⁰ The influence of the contact environments can be excluded for measurements taken under vacuum. The topographical effect can be removed from the measured signal by numerical data correction procedures.⁵⁵ The influence of the radiative heat transfer is often considered negligible, especially for ambient SThM, and for the probe–sample tip distances larger than 1 μm.^9,61,62 However, a huge enhancement was observed for the radiative heat transfer in sub-30 nm gaps.⁶³ Therefore, this heat transport mechanism could also be considered, especially in vacuum SThM.⁶⁴ In practice, it is difficult to collect a set of reference samples with the same surface roughness, which do not exceed a few nanometers. Moreover, to avoid a complex data analysis, the reference samples should be thermally isotropic, and their thermal conductivities should cover a range for which a measured k is expected. This problem can be solved by the manufacturing of reference samples with the same surface and required k. To achieve this goal, a sample that consists of steps of SiO₂ on a Si substrate was fabricated.⁶⁵ The steps thicknesses range from 7 nm to 0.95 μm; the sample is shown in Fig. 3.

The probe is the main element for any SPM. It defines the capabilities and the resolution of a particular microscope when investigating physical quantities at the nanoscale. The commonly available TPs are resistive probes, which utilize the dependence of the electrical resistance on temperature. In the case of metal resistors, the linear approximation described by Eq. (1) is commonly used for the estimation of this dependence. The historic pioneering WW TPs were proposed in 1994.⁶⁹ The active part of this probe is a ∼200 μm long and 5 μm diameter V-shape bended Pt90/Rd10 wire. The tip radius is ∼1 μm and its electrical resistance is ∼2 Ω. As the spatial resolution of measurements is comparable with the probe tip radius, the WW TP can be used for microscale but not nanoscale measurements. Moreover, handmade TPs do not have repeatable shapes. In 1998, batch-fabricated TPs were introduced (KNT probes). Initially, they were thermocouple probes,⁷⁰ but commercially available ones are resistive.⁷¹ The basic parameters of these probes are as follows: a tip radius smaller than 100 nm, an active element in a form of a thin-film Pd resistor with the resistance of around 100 Ω, and the maximum probe temperature is 200 °C. The Pd resistor is connected in series with two ballast resistors, each with around 100 Ω resistance. The whole probe electrical resistance is between 275 and 425 Ω (typically 325 Ω). The spatial resolution of the KNT probes is around one order of magnitude greater than the one for the WW TPs. As these are batch-fabricated probes, their parameters are more repeatable. However, their electrical resistance can still vary by 30%, and they are not suitable for measurements that require nanometer resolution. TPs that enable real nanoscale measurements are doped silicon (DS) probes with the tip radius ∼10 nm. They were originally designed for data storage systems based on thermomechanical writing.⁷² Similar probes were also used for material softening measurements.⁷³ Their temperature sensor is a low-doped Si resistor that is connected to two highly doped silicon cantilevers. The shape of the probe is similar to that of a typical AFM probe. Commercially available DS probes have a tip radius that is smaller than 20 nm and enables measurements up till 350 °C;⁷⁴ their electrical resistance is ∼1300 Ω. A weak point of the DS probes is the strong nonlinearity of their resistance, which is dependent on the temperature.²⁷ However, in the temperature range from 20 to 100 °C, the linear approximation is valid.¹¹ TPs with Pt nanowire resistive sensor are offered by Nanonics Imaging Ltd.⁷⁵ This is a glass cantilever that incorporates double Pt wire of ∼100 nm diameter. The reported tip radius is ∼50 nm⁷⁶ and, therefore, it is comparable with the KNT probe. They can operate up to 400 °C and their electrical resistance is ∼40 Ω.

The temperature sensitivities of the resistive TPs are 0.0033, 0.22, 2.8, and 0.38 Ω/K for WW, KNT, DS, and Nanonics TPs, respectively. Nanonics TPs possess the highest effective coefficient of resistance at around $0.01 K − 1$⁠, which is calculated as the ratio of the temperature sensitivity to the probe resistance at room temperature. The lowest $∼0.0007 K − 1$ is for KNT TPs. The temperature sensitivity is important for TCM. For CCM, the most important issue is the sensitivity to k changes. Generally, it strongly depends on $k$⁠. However, it is difficult to find data to compare the sensitivity of different types of probes. Sensitivities of WW and KNT TPs can be compared based on the experimental results published in Ref. 65. For WW TP, a change in the effective thermal conductivity of sample from $1.28$ to $50 W m − 1 K − 1$ causes a change in the probe–sample thermal conductance from $4.0$ to $4.8μ W K − 1$⁠, i.e., by a factor of 20%. For KNT TP, a change in the effective thermal conductivity of sample from $2$ to $50 W m − 1 K − 1$ creates a modification to the probe–sample thermal conductance from $145$ to $158μ W K − 1$⁠, i.e., by 9%. This leads to the conclusion that the k sensitivity of the WW TPs is more than two times higher. The upper limits of the sensitivity ranges are $8 W⋅ m − 1⋅ K − 1$ for WW TP and greater than $30 W⋅ m − 1⋅ K − 1$ for KNT TP. Relative uncertainties read from graphs do not exceed ∼2.0% for WW TP and 4.8% for KNT TP.⁶⁵ It proves that WW TP is more suitable for CCM quantitative measurements.

The analysis based on the 3D finite element models are completed for all the mentioned resistive probes, operating in the non-contact mode that showed the highest sensitivity of the DS probes.⁴⁵ This is almost three times higher than the one for the KNT TPs. The lowest sensitivity is revealed to be the Nanonics probes. Calculations were performed for probes in air and the analysis showed that the highest spatial resolution of the k measurement can be achieved with Nanonics TPs. However, it should be noticed here that the parameters determined for the non-contact measurements are defined by the heat transfer through the air and differ from these obtained for the contact mode. For instance, the resolution in high-vacuum conditions differs by a few orders of magnitude. For the DS TPs, the one determined in Ref. 45 was 16.6 μm, while the resolution achieved experimentally was ∼10 nm.⁷³ 

The second type of TP, which is quite popular, is the thermocouple TP. A wire thermocouple TP was used in the first SThM measurements by Williams and Wickramasinghe.⁴ Nowadays, the thermocouple TPs are fabricated by batch technology.⁷⁰ These probes are often used in TCM because of two reasons. The thermocouple junction is highly localized and positioned in the vicinity of the probe–sample contact; it can measure temperature without a current flowing through the junction, i.e., in a truly passive regime. Recently, thermocouple TPs became commercially available.^77,78 Especially interesting are VERTISENSE^TM TPs with hollow SiO₂ tip. Thermal insulation of thermocouple junction results in high sensitivity, and TCM and CCM modes can be switched by changing the position of laser beam used for the detection of probe bending.⁷⁸ 

There are many proposals for TPs that utilize other phenomena: the piezoresistive effect,⁷⁹ properties of the metal–semiconductor junction (Schottky diode),⁸⁰ the fluorescence,⁸¹ etc. However, mainly because of their complexity, these can be treated as examples of possible alternative solutions. The same applies to the proposal to improve the sensitivity and the resolution of TPs by the modification of the probe tip. Examples are a wire thermocouple TP with a small diamond tip⁸² or a KNT probe with an attached high-thermal conductivity nanowire.⁸³ 

An interesting approach to the thermal measurements using the AFM is to use a typical silicon AFM probe as the TP.⁸⁴ The idea is to utilize the temperature dependence of the reflectance (thermoreflectance) for the temperature measurements. The thermoreflectance thermometry was previously used, for example, for the thermal imaging of active semiconductor devices.⁸⁵ In the AFM-thermoreflectance imaging reported in Ref. 84 the temperature distribution on the surface of test sample with a tungsten line heater was measured. To determine the temperature of the silicon probe, the intensity of the light reflected from the probe cantilever was measured. A light beam used for the detection of the cantilever bending was simultaneously used for the thermoreflectance measurements. Therefore, any modifications of the AFM system were not needed. The thermal image was obtained using a conventional AFM, applying typical probe, without extra accessories for the data acquisition. This imaging was performed in the TCM; for the CCM, the probe should be heated. However, the thermal conductance contrast can also be obtained by heating the sample not the TP.⁸⁶ 

To summarize, the SThM probes evolve and become similar to those used in the conventional AFM. The WW TPs first used for SThM imaging are not commercially available nowadays. They were replaced by the KNT probes, and the popularity of DS probes is growing. The main disadvantage of the KNT TPs is the relatively low sensitivity in CCM. This is mainly caused by a high heat flux in the probe holder via the gold contact pads. In air, this heat flux is ∼2/3 of the dissipated power. This effect takes place in all TPs and can be minimized by the thermal separation of the heated part and the probe holder. This can be achieved through the replacement of the probe cantilever by two thin cantilevers, like in DS TP or, more generally, U shape probes. The latter are very similar to the conventional AFM probes and provide a similar resolution in the topographical imaging. However, because of the complex fabrication technology, they are much more expensive. An interesting idea is to use the standard probes with a thermoreflectance temperature detection and light heating. The thermocouple TPs also become commercially available. Especially interesting are probes providing thermal insulation of temperature sensor.

The tendency for reducing the probe tip radius to improve the spatial resolution results also in increase of the thermal resistance to the heat flow through the probe–sample contact. A comparison of the WW and KNT TPs sensitivities with respect to k shows that the probes with larger tip radii are better for the thermal measurements, when the lateral resolution is of less importance.

To correctly interpret the measurements, an understanding of the physical phenomena that occur in the measuring system is required. In the case of the SThM measurements, the analysis of the thermal transport phenomena is of key importance. This analysis is often based on highly simplified models, electro-thermal analogies, or the finite element method. This is because of the complex geometry of the TP–sample system since an analytical description of the thermal transport is not possible. Simplified analytical models, based on the thermal fin equation with a Joule dissipation term, were used for both the WW^87,88 and KNT⁵¹ TPs. The main advantages of these model are the possibility of determining the temperature distribution along the TP⁸⁹ and its frequency characteristics.^51,90 However, it should be noticed that the thermal fin model only corresponds to the geometry of the WW probe. Its use when describing other probes is difficult to justify.

A well-known tool used in heat transfer modeling is the thermal quadrupole method.⁹¹ This method is based on electro-thermal analogies. Heat fluxes are caused by the temperature differences, analogous to electric currents deriving from the differences of electric potentials (voltage) at separate positions of a circuit. The connection between the heat flux and the temperature difference is given by Eq. (2), which is an analog of Ohm's law. The thermal quadrupole method was used for modeling of the TP–sample system with a KNT probe for dc + ac and transient modes.⁹² First, the probe–sample system was modeled using a finite element method. Based on a calculated heat fluxes distribution, three zones were distinguished in the system: the contact zone that embraces the vicinity of the probe–sample contact, the active zone encompassing the heat sources, and the coupling zone connecting the active zone and the probe mounting. The characteristic lengths of these zones were: ∼100 nm, ∼10 μm, and ∼100 μm, respectively. The thermal quadrupoles model of the system, which is slightly modified relatively to the one described in Ref. 92, is shown in Fig. 5. Three resistors, connected in series in the contact zone, model the heat transfer through the probe–sample contact. $R cns _ p th$ and $R cns _ s th$ are constriction resistances of the probe and the sample, respectively, and $R bnd th$ is the boundary thermal resistance. They are connected in parallel with $R surr th$⁠, which represents the heat transfer through the contact surroundings; the heat capacity of this zone was omitted. The resistor $R a th$ models the heat losses from the active zone to the ambient, and the capacitor $C a th$ represents the heat capacity of this zone. The chain of quadrupoles, in the coupling zone, models the heat transfer along the TP cantilever. $R C _ i th$ resistors enable heat flux along the cantilever, $R c _ i th$ resistors model the heat losses to ambient, and $C c _ i th$ capacitors correspond to the heat capacities of the consecutive cantilever elements. The cantilever is connected to the probe holder through $R C th$⁠. The current source $J Q$ represents the power dissipated in the active zone.

The case of the non-contact SThM results in $R solid th→∞$⁠. In the vacuum SThM, $R surr th$ is defined by $R rad th$⁠, and the term $1 R surr th$ can be neglected if the other mechanisms are present. The thermal resistance $R surr th$ strongly depends on the tip–sample distance. It should be stated here that Eq. (9) is valid if all thermal resistances in the right side are in parallel and connect two objects (the sample and the probe), which temperatures are T₀ and T, respectively [Eq. (2)]. In reality, the situation is more complicated; however, the model described above is commonly used as a reasonable simplification.

Derivation of formula for the constriction resistance in intermediate regime for the diaphragm separated two different materials is not obvious. The diffusive term is additive, so Eq. (10) can be used. However, the ballistic term is not additive and cannot be divided into sum of two components. Wexler pointed out to this problem.⁹⁴ 

The above considerations based on $R cns th$ and $R bnd th$ apply to an ideal interface. However, real tip–sample interfaces are never ideal. The sample surface is rough, covered by oxide and often contaminated. This leads to a weakening of the mechanical coupling.⁹⁹ In the case of the solid–solid contact, the roughness can be accounted for by multiplying $R b th$ by the scaling factor $A 0/A$⁠, where $A 0$ and A are the apparent and the real contact areas, respectively.⁹⁹ However, in general, any attempts to describe the actual contact lead to highly complex theoretical models, for which its application toward experimental data analysis is far from straightforward.¹⁰⁰ An interesting fact, which can be useful for the analysis of the heat transport through the solid–solid contact, is a correlation between the shear forces and the thermal conductance $1/ R solid th$ in the nanoscale junction. In the case of the phonon heat transport, the junction thermal conductance is correlated to the normalized shear force through the heat capacity and the phonons group velocity.¹⁰¹ 

The next channel for the probe–sample heat transfer is the heat transfer through the air that surrounds the contact. This is very efficient channel and can be observed at relatively long distances, up to ∼10 μm for the WW probe. At an ambient pressure, it is found that around 40% of the WW probe–sample heat flux flows through the air; for the KNT probe, this share can exceed 65%.³⁶ These values were obtained for a specific sample, a gold-coated pyroelectric sensor, and can considerably differ for samples of different thermal conductivities. They also strongly depend on the air pressure, which is clearly represented in Fig. 6.^15,102 So, provided values should not be treated as reference ones, they simply illustrate that the heat transfer through the air surrounding the contact is effective.

In humid air, a water meniscus can be formed between the tip and the sample. This meniscus acts as an additional channel for the heat flow. The influence of the water meniscus was demonstrated in Ref. 103. A theoretical model for the thermal transport through the meniscus was proposed in Ref. 30, but it overestimates the influence of this mechanism on the overall heat flow to the sample. Detailed analysis of the heat conduction through a water meniscus was based on a consideration of the capillary forces between the nanoscale contacts;¹⁰⁴ it was shown that this influence is relatively weak. The share of the meniscus thermal conductance with respect to the whole thermal conductance of the sample was estimated to be between 1% and 6%.^9,104 The thermal conductance of the water meniscus depends on numerous factors, including the probe tip radius,¹⁰⁴ the relative humidity of the surrounding air,¹⁰⁵ and the hydrophobicity/hydrophilicity of the sample surface and its roughness.^9,106 All these issues means that the theoretical prediction of the water meniscus thermal conductance is practically impossible, and complete exclusion of the humidity influence on the probe–sample heat transfer can be difficult, even in high-vacuum systems. However, if not completely excluded, this heat transport channel can be considerably reduced by measurements under vacuum below ∼10⁻³ Pa or by measurements with the TP, for which the temperature is sufficiently high.^58,103

The last heat transfer channel that is mentioned above is the radiation transfer. This is because the temperatures of the sample surface and the probe tip are relatively low, and the radiative thermal conductance in the air SThM is negligible.¹⁰⁵ Such an approach is justified when the tip–surface gap exceeds Wien's wavelength (∼10 μm at ambient temperature). For smaller gaps, the near field radiative heat transfer occurs, and the energy transfer is dominated by evanescent waves that are not considered in Planck's theory of the thermal radiation.¹⁰⁷ It has been experimentally demonstrated that changing the gap from 50 nm to a few nanometers can increase the radiative thermal conductance by more than an order of magnitude for selected pairs of materials.⁵⁹ The largest increase was measured for the SiO₂–SiO₂ pair, with the highest conductance seen at around 2 nW K⁻¹; the tip radius was 225 nm. The theoretical thermal conductance for the heat flux through the contact can be calculated using Eqs. (7) and (14). Assuming that $R b th=50 m 2⋅ K⋅ n W − 1$ and $k Si O 2=1.46 W⋅ m − 1⋅ K − 1$⁠, $G solid th=1/ R solid th$ is ∼300 nW K⁻¹ for the SiO₂–SiO₂ contact. Therefore, the radiation share relative to the thermal transport is still negligible. The radiative heat transfer was also analyzed using the boundary element method.⁶⁴ The radiative thermal conductance that is obtained for the 4 nm gap and a tip similar to that used in the described experiment was 1.26 nW K⁻¹; this is in agreement with the experimental data. The same analysis for the conductance through air provided the result $G air th=15 nW⋅ K − 1$⁠. This value is obtained for the heat flux from only the rounded part of the tip, and the real conductance for the whole tip is larger.

The distributed character of the sensor also causes the distribution of heat sources in the active mode. These effects were analyzed for both the WW³⁶ and the KNT¹⁴ TPs. The temperature and heat sources distributions along the sensor depend on the sample thermal conductivity. Therefore, they affect the thermal resistances defined in this section and, additionally, they complicate the heat transfer model. These thermal resistances also differ for the passive (TCM) and the active (CCM) modes. The influence of temperature distribution along WW TP on the thermal transport in the TP–sample system can be, at least partially, taken into account through the function describing the normalized distribution of temperature along the probe $F(x)=θ(x)/ θ 0$⁠, where $θ 0$ is the normalizing temperature. The function $F(x)$ is called the form factor and in the case of ac experiments depends also on the frequency. The explicit formula for $F(x)$ and its use for calculation of the thermal resistances is described in Ref. 36.

This model was used for the analysis of the $U dc$⁠, $U n(ω)$⁠, and $φ n(ω)$ sensitivities on $G s th$⁠.¹² 

The analysis for the signal formation in a thermocouple TP, which operates in the active mode driven by an ac current at frequency $ω$⁠, was shown in Ref. 109. The resulting temperature disturbance contains the dc component, and two ac components at $ω$ and $2ω$⁠. The $ω$ component arises due to the Peltier effect, while the $2ω$ component is due to both the Peltier and Joule effects. However, the contribution of the Joule effect to the $2ω$ temperature component is dominant. Based on the results of the theoretical analysis, a $2ω$ measuring method was proposed; its usefulness was experimentally proven elsewhere.¹⁸ As already mentioned, due to its complex geometry, an analytical description of the physical processes that are responsible for the SThM measurements is not possible. Therefore, numerical modeling is often used. It is useful for modeling either nanoscale processes or processes in the whole TP–sample system. An example of the nanoscale processes analysis is the modeling of the tip–sample radiative heat transport.⁶⁴ The geometry of the model is shown in Fig. 7, and the results of the calculations of the spectral and total exchanged power between the SiO₂ tip and the SiO₂ surface are shown in Fig. 8.

Another example is the analysis of the heat transfer performed for the KNT probe with an attached carbon nanotube tip.⁸³ 

A systematic analysis of the sensitivity and the spatial resolution of the commercially available resistive TPs, operating in the non-contact mode, was recently published.⁴⁵ This represents a good example of the whole TP–sample system analysis. Temperature distributions that are obtained for the four probe types are shown in Fig. 9. The modeling possibilities for the whole system can be found in many papers, for example, in Refs. 11 and 110–112.

Theoretical and numerical models of the SThM measurements are necessary for identifying the potentialities of the measuring methods and for the proper interpretation of the experimental data. The currently used models meet the first of these requirements. They enable analysis of the thermal transport, including phenomena that is not described by classical theory, i.e., Fourier law and the heat equation (namely, Fourier–Kirchhoff's differential equation of conduction). They also provide possibilities for the analysis of the SThM signal sensitivity and the probe–sample thermal conductance $G s th$⁠. However, the quantitative measurements of the local temperature or the local thermal conductivity are still challenging tasks. For example, it is well-known that the sample roughness influences $G s th$⁠, but there is no method that considers this fact when analyzing the raw experimental data. In general, the complexity of the models is a problem. As shown earlier in this section, some effects can be omitted without a significant deterioration of the measurement accuracy. The parameters used in the models are often difficult to define because ambient conditions are not well-defined. A clear definition of the measurement conditions and the development of a possible simple model for the measurements are prerequisites for a wider use of the SThM in material research.

Since the SThM invention, the usefulness of this scanning probe microscopy in a variety of applications has been proven. An extensive, systematic overview of the experimental studies performed in both the TCM and CCM can be found in Refs. 8 and 9. The quantitative SThM measurements can be divided into two groups. The first one encompasses the experiments for the quantitative characterization of the heat exchange in the TP–sample system, which enables a determination of the thermal boundary resistance,^15,60,113 the mechanical contact radius,^52,53 the effective thermal exchange radius,^34,46,52 and the shares of the various heat transport channels relative to the total heat flux from the TP to its surrounding.^36–38 Most of these issues were already analyzed in Sec. IV. The second group is the research on the thermal transport, temperature distribution, and the determination of the thermal conductivity of various structures. The SThM was used, sometimes in combination with other methods (e.g., Raman spectroscopy), for an analysis of the heat dissipation from heated nanowires⁵⁴ and carbon nanotubes.^114,115 The cited works were based on the quantitative analyses of the SThM images of deliberately fabricated structures. The observed thermal conductivity of an individual multiwall carbon nanotube was 3000 W m⁻¹ K⁻¹.¹¹⁴ The next example is the application of the SThM toward an analysis of the thermal transport into graphene through nanoscopic contacts.⁶⁸ The thermal transport was dominated by the thermal interface resistance, which decreased with an increasing number of graphene layers. The usefulness of the SThM is not limited to an investigation of the thermal transport. It was also successfully used in combination with an angular dependent ferromagnetic resonance for analysis of the spin precession modes in Co stripes.¹¹⁶ 

The main applications of the quantitative SThM are the temperature and the thermal conductivity measurements. The high transverse spatial resolution of the SThM allows a temperature mapping of the operating nanoscale electronic devices²⁶ and the nanostructures.^{52,54,114,115} The SThM was also used for an investigation into the hot spots that arise near a junction in silicon nanowire diode.²⁵ The in-depth (longitudinal) spatial resolution is important for the thermal conductivity measurements in thin films. As follows from Eq. (8), the longitudinal resolution is defined by the tip–sample contact radius. The high-resolution vacuum SThM enables a determination of the thermal conductivity of the 3 nm thick HfO₂ layer deposited on Si;²⁹ the 2 nm thick SiO₂ layer was employed as a reference. The ambient SThM was used for the investigation on ZnO films deposited onto Si.⁴⁴ The layer thickness was set in the range of 12–118 nm, and the obtained values of k were correlated with the morphology, grain structure, and defect balance in the layers. A variant of the null-point method was used for the determination of local k for a ∼117 μm thick carbonaceous layer.¹⁷ The experiment revealed heterogeneous surface structures with different k. The SThM was also employed for k measurement of individual nanowires embedded in a matrix.^51,117 The use of this technique for k measurements of biological samples has also been suggested.¹¹⁸ 

The papers cited in this section provide some idea of the SThM potentialities for quantitative measurements. However, it should be clearly stated here that only a portion of the possible examples is presented since many others can be found, it is far from a comprehensive review. Despite all the limitations, the SThM opens opportunities for investigations into nanoscale thermal phenomena, which has not been provided by any other technique. It should be possible to use SThM techniques to investigate many other nanoscale effects accompanying thermal phenomena.

Since its development in 1986, the SThM became a recognized technique for micro- and nanoscale thermal measurements with a well-established theoretical background and dedicated equipment. The potentialities and the limitations of the SThM are now clearly defined. The method allows qualitative (imaging) and quantitative temperature and thermal conductance measurements. The lateral spatial resolution is found to be strongly dependent on the TP used and whether the measurements are taken in air or in vacuum conditions. For sharp tips in a vacuum, the spatial resolution can be greater than 10 nm. The resolution of the temperature measurements of 0.2 mK can also be reached. The thermal conductance measurements are, generally, less accurate because of the relatively narrow dynamic range for the measured signal and the complex heat transport mechanism in the TP–sample system. In addition, the temperature sensor is distributed in the most popular resistive KNT TPs, which makes it difficult to model the system. Similar to other SPMs, the SThM is useful for the investigation of samples with high-quality, flat, and smooth surfaces.

The basic arrangement of SThM is well-established and it will probably not change considerably. A useful modification could be a possibility of independent control of the sample and the probe holders. The main changes will concern the control of the environment in which the measurements are carried out. Nowadays, many measurements are performed in the rough vacuum to avoid heat transport through the air and the formation of a thin water film on the sample surface. This trend could be continued in the future. It must be remembered that the complete exclusion of the influence of the ambient air could require a vacuum no worse than a few mPa. Such a low pressures may induce changes in the SThM equipment. In addition, not all samples can be placed in vacuum. Thus, some limitations on tested samples can be expected. The opposite trend has also become noticeable. Measurements in well-controlled ambient conditions (e.g., inert gas or liquid) are also good option. Combined with the non-contact mode, they enable an avoidance of the influence of the probe–sample (solid–solid) contact on the measured signal. The thermal resistance of this contact depends on many factors; therefore, it is difficult to estimate. The price paid for the non-contact measurements is the deterioration of the lateral spatial resolution. Measurements in liquid environment give opportunities for SThM use for investigation of biological samples, possibly living cells.

A separate problem is TP heating by the light beam used for the measurement of the cantilever deflection. As shown for KNT TPs, the SThM signal changes when the beam is switched off, but it also depends on the position of the light spot on the cantilever. Since the dark mode in a typical AFM is not possible, a simple solution to this problem is also not possible. This effect could be minimized by a reduction of the light absorption in the TP or by moving the light spot away from the heated zone, like for the WW TP. Other methods for the cantilever deflection measurement can also be considered.

The TPs used for the SThM became similar to those used for the AFM. This results in an improved lateral spatial resolution but also causes problems for the thermal conductance measurements. For the WW probes, a significant share of the total heat transfer in the tip–sample system arises from the solid–solid conduction due to the large contact area, while this share is very small in the case of the KNT probes where conduction through the surroundings is dominant. The tip radius of the DS probes is approximately ten times smaller than the one for the KNT probes. The heat flux that passes through the contact depends on the constriction $R cns th$ [Eq. (11)] and the boundary $R bnd th$ [Eq. (14)] thermal resistances, which are proportional to $b − 1$ and $b − 2$⁠, where b is the contact radius. These resistances are connected in parallel and define the heat flux through the tip–sample contact, $J s$ [Eq. (6)]. Among these two resistances, only $R cns th$ depends on the sample thermal conductivity k. With a decreasing b, $R cns th$ and $R thb$ increase, which results in a decrease in the $J s$ share relative to the total heat flux from the TP to the surroundings. Moreover, $R bnd th$ increases much quicker than $R cns th$ and the sensitivity of the SThM signal to k deteriorates with a decrease in b. These facts are confirmed by the analysis of the heat fluxes in the TP–sample system performed for the WW and KNT probes.³⁶ Therefore, a compromise between the spatial resolution and the sensitivity to k should be found. A possibly batch-fabricated successor to the WW TP could be useful for quantitative thermal conductance measurements with a ∼1 μm spatial resolution. Of course, “classical” WW TPs can be still fabricated by “do it yourself” method, but such a solution is against the standardization of SThM measurements.

TPs with a high $R c th$ thermal resistance (the resistance to the heat flux traveling to the probe holder) could significantly improve the sensitivity of the k measurements. A possible solution is to replace the probe cantilever by using two prongs, similar to DS TP. Generally, a thermal transport engineering which improves TP sensitivity to k is desirable. A good example that meets these requirements is a thermocouple TP with hollow tip.⁷⁸ The full potential of the thermoelectric probes remains unused, but it seems to change. They offer a real passive mode without a current flowing through the sensor although they can also operate in the active mode. The localization of the temperature sensor for the thermocouple TP is much greater than for the resistive one. It appears that the only problem is that the commercially available microscopes are mainly dedicated to resistive TPs.

According to the methodology of SThM measurements, the possibility of controlling heat fluxes via the temperatures of the probe and sample holders for thermal conductance measurements is still vastly underestimated. The possibility to control the temperature of the sample holder is provided in a typical AFM. Such possibility for probe holder could be an interesting option for improvement of the k measurements sensitivity.

The problem which was not practically considered yet is TP wearing during measurements. On the one hand, systematic inspection of probe parameters should be a standard during measurements. On the other hand, measuring methods which minimize TP wearing should be considered and developed.

A question that remains open is the standardization of the quantitative SThM measurements. The reliability of these measurements can only be proved through interlaboratory comparisons. These standard measuring procedures must include the TP calibration and the determination of the calibration curve for the thermal conductance measurements, if necessary. Standard methodologies must also account for the ambient conditions, the sample roughness, TP wearing, the influence of the substrate on the measurements performed on layered samples, etc.

Such an intercomparison between three laboratories was carried out for nano-thermomechanical analysis.¹¹⁹ The glass transition temperature $T g$ of five polymeric materials was measured. The maximum temperature difference between results obtained in various laboratories was about 20 K. However, the difference of $T g$ between SThM data and those obtained by differential scanning calorimetry (DSC) reached 50 K. As $T g$ of investigated samples was between 50 and 200 °C, the difference between SThM data and DSC was significant.

The scanning thermal microscopy has passed from the development stage to the point where it is used in the study of materials and processes that occur in material structures. As a consequence, it has become a tool used in the research analyses not the object of the research. However, such a complex measurement tool is only as useful as the accompanying operation manual that describes how to perform the measurements. For SThM, this manual is still incomplete, and its preparation remains a task for the coming years.

These perspectives represent the authors' point of view on the future of the SThM. One can, of course, agree with the presented conclusions or have different opinions. The near future will verify the accuracy of the forecasts.

The authors acknowledge Silesian University of Technology for financial support through the statutory funds—Project Nos. 14/030/BK-21/0011 and 14/030/BK-22/0015.

The authors have no conflicts to disclose.

Jerzy Bodzenta: Conceptualization (lead); Investigation (equal); Methodology (equal); Writing – original draft (lead); Writing – review & editing (equal). Anna Kaźmierczak-Bałata Conceptualization (supporting); Investigation (equal); Methodology (equal); Writing – original draft (supporting); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.