Contactless Scanning Near‐Field Optical Dilatometry Imaging at the Nanoscale

To date, there are very few experimental techniques, if any, that are suitable for the purpose of acquiring quantitative maps of the thermal expansivity of 2D materials and nanostructured thin films with nanoscale lateral resolution in spite of huge demand for nanoscale thermal management, for example in designing integrated circuitry for power electronics. Besides, contactless analytical tools for determining the thermal expansion coefficient (TEC) are highly desirable because probes in contact with the sample significantly perturb any thermal measurements. Here, ω‐2ω near‐field thermoreflectance imaging is presented as a novel, all‐optical, and contactless technique to map the TEC at the nanoscale with precision. Testing of this technique is performed on nanogranular films of gold and multilayer graphene (ML‐G) platelets. With ω‐2ω near‐field thermoreflectance, it is demonstrated that the TEC of Au is higher at the metal‐insulator interface, with an average of (17.12 ± 2.30) ×10−6 K−1 in agreement with macroscopic techniques. For ML‐G, the average TEC is (−5.77 ± 3.79) x10−6 K−1 and is assigned to in‐plane vibrational bending modes. A vibrational‐thermal transition from graphene to graphite is observed, where the TEC becomes positive as the ML thickness increases. The nanoscale method here reported demonstrates results in excellent agreement with its macroscopic counterparts, as well as superior capabilities to probe 2D materials and interfaces.


Introduction
With critical advances in integrated power electronics, [1] there has been a tremendous demand for nanoscale thermal management in the pursuit of miniaturization and integration of submicrometric components preserving high power density, for DOI: 10.1002/admi.2023008064] The need for heat dissipation is essential in many of such applications [5] as it assists in preserving the efficiency and lifespan of the circuitry components.Heat spreaders and heat sinks are typically integrated for the purpose of discharging local thermal buildup from long device operation times. [6]With tremendous demand for miniaturization, the understanding of heat dissipation in low-dimensional systems and thin film structures beyond the macroscopic level is rapidly becoming essential toward the development of next-generation information technology.However, a major cause of power electronics failure is due to thermal stress induced by a difference in thermal expansion coefficients of different materials in contact upon heat exposure. [7]In spite of such a demand, there are few experimental techniques suitable for the purpose of providing thermal expansion information beyond the macroscopic level.Thermal expansion in inhomogeneous thin films is customarily investigated macroscopically with dilatometers, [8][9][10] thus only measuring their average expansivity from bulk materials with known geometry through an external heat source. [8] microscope-based pump-probe dilatometer has also been proposed, [11] but mesoscopic and nanoscopic thermal expansion information is generally not provided, not even by microscopic dilatometry techniques.[14] These methods are also macroscopic in nature, however, and they require crystalline materials.
To date, guesses on the local thermal properties of surfaces and near-surface features at the mesoscopic level chiefly rely on scanning thermal microscopy (SThM) [15][16][17][18][19] a form of contactmode scanning probe microscopy.In SThM, heat is generated in the sample due to an external AC electrical source, while detection is measured through periodical oscillations of the probe cantilever to determine the thermal expansion coefficient of the material.However, heat detection in SThM requires the cantilever to be in contact with the sample surface, which acts as a huge thermal sink that is also irreproducible due to unpredictable sample-probe distance.[22] Alternative methods for nanoscale thermal detection have been proposed, [23][24][25][26] and they involve combinations of conventional SThM with other techniques, or they use synchrotron-based methods assisted with high-resolution transmission electron microscopy. [27]Such methods were also occasionally proposed for acquiring thermal expansion information with nanoscale resolution. [28,29]However, the configurations of the proposed systems are highly specialized and require a combination of multiple techniques, some of which are available only at large-scale synchrotron facilities.
In this paper, we present and utilize for the first time contactless near-field thermo-reflectance imaging as a novel tool for precisely mapping the thermal expansion coefficient at the nanoscale.Near-field thermoreflectance [30,31] is a class of pumpprobe techniques based on super-resolution, aperture-type, scanning near-field optical microscopes (SNOMs) -optical instruments capable of lateral resolution beyond the diffraction limit through the use of evanescent light waves. [32,33][36][37][38] In pump-probe near-field thermoreflectance experiments, samples are periodically heated by a pump laser beam from an inverted optical microscope, while periodic changes in the sample's local reflectivity caused by the resulting heat waves are measured, in phase and amplitude, through an upright aperture-type SNOM operating in reflection mode from a low-intensity direct-current (DC) laser ("probe") of wavelength different from the pump.Near-field thermoreflectance tools can be operated in non-contact mode, [30,31] thus making them contactless and offering significant advantages, in terms of little invasiveness, over methods requiring electrical currents to heat the sample or nanosized thermocouples to measure their thermal properties.
The use of scanning near-field thermoreflectance methods to map the thermal conductivity and heat capacity of specific samples by locking the probe beam detection at the same frequency  as the pump beam in an - pump-probe technique is well assessed. [30,31]Here, we demonstrate that one-of-a-kind -2 near-field thermoreflectance experiments, in which the probe is locked in for detection at double frequency (2) as the pump, are spectacularly sensitive to the local expansivity of surfaces, and can thus be successfully used to satisfy the longstanding quest for a contactless tool capable of imaging the thermal expansion coefficient (TEC) of composite materials at the nanoscale.This novel technique, which can be termed scanning near-field optical dilatometry (SNOD), is here demonstrated to quantitatively estimate and successfully image the nanoscale thermal expansion coefficient of samples with both positive and negative thermal expansivity.SNOD is anticipated to be uniquely suited for thermal expansion coefficient imaging in a variety of thin films and devices, thus solving fundamental and technological problems in nanoscale thermal management.

Operation Principles of SNOD
Panels a and b in Figure 1 elucidate the principle of operation of thermoreflectance experiments and the differences between - and -2 operation modes, where this second mode of operation is specifically pioneered in this work.Samples include a transparent substrate with an optically absorbing and heterogeneous thin film on top.As these thin films are heterogeneous and of different material properties, their TEC is expected to vary from point to point (x,y) of the probed surface.Due to the high transparency of the sample, the pump beam ( 0 = 405 nm, purple, in Figure 1a,b) pulses at a frequency  through an optical chopper and periodically heats the film, from which heat dissipates both along the surface or through the substrate and, to a limited extent, air at the top.Air experiences a stronger change in density than the solid sample, which results in oscillations of the refractive index of air and the surface reflectivity [Δ  (x,y)].Such oscillations are probed by an upright SNOM in reflection mode ( = 532 nm, green, in Figure 1a,b).This allows to locally probe the oscillations in the film's surface temperature [ΔT f, (x,y)] which are linear in the reflectivity: where h is a constant that only depends on , on the tip material, the tip aperture radius and length, as well as the focal distance from the detector. [30]he interpretation of - measurements summarized by Equation (1) assumes that the near-field thermoreflectance entirely originates from the sample surface, which on its turn assumes the evanescent field to be spatially localized at lengths much shorter than the spatial fluctuations of the sample surface along the z-axis. [30]This is normally not the case, as the near-field decays over lengths of the order as the tip radius a (i.e. a few tens nm) and these fluctuations can be either timeindependent [i.e., from position dependent z = z(x,y)] or timedependent, from z = z(t) at a given scanned point (x,y).Performing SNOM measurements in AFM noncontact mode eliminates the contribution from time-independent fluctuations, but not from time-dependent ones, as the tip resonant frequency is normally much larger than .As seen in Figure 1b, the thermal expansivity [ɛ(x,y)] of the sample surface under sinusoidal heating induces time-dependent oscillations of the z-axis position of the sample surface at the same frequency: z(x, y, t) = z 0 + (x, y) cos(t) (2) Because the reflected evanescent field is not point-like but can be approximated to a Gaussian profile peaked at a constant distance z 0 from the sample and of full width at half maximum of about the tip aperture radius, a, the near-field reflectivity from temperature-induced, time-dependent, fluctuations of the underlying surface is given by where a Taylor expansion of the Gaussian was used to work out the right-hand term of Equation (3) in conjunction with Equation ( 2) and the half-angle trigonometry identity 2cos 2 (t) = 1+cos(2t).Thus, the amplitude of the time-dependent SNOM signal at double-frequency (2) is given by: Then, ɛ(x,y) can be derived from Δ 2 (x,y) by means of Equation (4) if Δ  (x,y) is known from an independent - experiment, and the lateral resolution of this method will be of the order of a.The z-axis resolution will be more accurate, however, as the use of a lock-in method enables the detection of quantities ɛ(x,y)/a ≈ 10 −2 and, therefore ɛ(x,y) ≈ 0.5 nm with a ≈ 50 nm tip aperture, while the phase of the -2 experiments allows for the discrimination between negative and positive ɛ.As the two experiments (- and -2) are carried out simultaneously on the same instrument, this opens a robust pathway towards contactless near-field based scanning optical nano-dilatometry.
An immediate way to picture the underlying principle of contactless SNOD through -2 experiments is depicted in Figure 1b.Consider the sample surface to be, on average, at a temperature and height T Mid and z Mid , respectively.At a time t Hi = /2 during a pump-beam cycle, the sample is heated up to T Hi = T Mid +ΔT f, , which expands it locally and places its surface at z Hi = z Mid -ɛ(x,y)/2 below the tip.Cooling the sample at a time t Lo = 3/2 during the same cycle, will bring the temper-ature down to T Lo = T Mid -ΔT f, , thus locally placing its surface at a larger distance, z Hi = z Mid +ɛ(x,y)/2, from the tip.Hence, the temperature and position of the sample surface relative to the contactless tip oscillate at .However, the sample surface sits away from the Gaussian peak of the reflected evanescent field at both t Hi and t Lo , where the sample reflectivity reaches equal minima,  Lo = Δ  -Δ 2 = (1-ɛ 2 /2a 2 )Δ  because of the symmetry of the Gaussian-shaped profile of the reflected evanescent field.Because the reflectivity reaches two minima, at /2 and 3/2, for each pump-beam period, then the reflectivity signal from thermal expansivity oscillations occurs at a double frequency as the excitation.In addition, the amplitude of the reflected evanescent field will reach maxima at t Mid = 0, /, and 2/, also twice as frequently as the pump beam, thus confirming the 2 oscillations of the signal associated with the sample's thermal expansion.
After measuring ɛ(x 0 ,y 0 ) via Equation ( 4), the next critical step is represented by properly taking into account the contributions from the substrate toward the determination of the local TEC [ f (x 0 ,y 0 )] of the film, in which the former can be substantial (up to 50%) even for thermally insulating substrates like glass.The two quantities are linked by the relationship: [8][9][10]

𝜀
where ΔT f (x 0 ,y 0 ) can be obtained through - thermoreflectance measurements via Equation(1), and d(x 0 ,y 0 ) is the local thin-film ) measured by - pump-probe near-field scanning thermoreflectance imaging. [30]e) Thermal expansion maps at 45 Hz; f) 60 Hz; g) 80 Hz; and h) 100 Hz obtained from -2 pump-probe SNOD measurements.i) Pictorial representation of the measured nanogranular gold thin film, for which j) the linear thermal expansion coefficient (TEC) is observed to be highest at the thin-film edge, corresponding to the metal-insulator interface, while it is relatively constant (and in good agreement with the macroscopic measurements by Frenkel et al. [54] ) at other areas of the film.
thickness.Being scanning-probe techniques, SNOM and SNOD are uniquely suited to determine d(x 0 ,y 0 ) through a simultaneous AFM scan of the sample.Also, in Equation ( 5),  s is the TEC of the substrate, which, for most of the cases of practical interest, is known and is expected to be homogeneous.More complicated is the determination of ΔT s (x 0 ,y 0 ,z 0 ), the change in temperature at a point of the substrate at a level z 0 below the measured point of the surface.Multiple neighboring substrate points (x,y) may affect the thermal expansion around (x 0 ,y 0 ).In an ideal case, where the substrate is perfectly insulating and the film is enough conducting for heat to be overwhelmingly dissipated inplane along the surface, the integral from the second addend of Equation ( 5) yields a negligible value, but this is normally not the case. [30]o facilitate the determination of Equation (5), ΔT s (x 0 ,y 0 ,z 0 ) can be divided in two components, as depicted in Figure 1c: i) an (x-y)-independent component [T s,av (z 0 )] originating from the film's temperature oscillation T f,av averaged over the entire scanned region (panel c, top), and ii) the local deviation from the average [T s, (x 0 ,y 0 ,z 0 )] which originates from the fluctuations of the measured film's temperature T f (x 0 y 0 ) (panel c, bottom).Due to the linearity of the equation of heat, this can be solved independently for cases i) and ii), by using T s,av (0) = T f,av and T s (x 0 ,y 0 ,0) = T f (x 0 y 0 ), respectively, as boundary conditions, and the solutions can be superimposed as where and A is the cut-off area of the sample.ΔT s (x 0 ,y 0 ,z 0 ), obtained from Equation ( 6), can then be integrated over z 0 in Equation ( 5) to yield  f (x 0 ,y 0 ).Specifically, component i) offers a substrate temperature uniformly decaying over z irrespective of (x,y) (Figure 1c, top), while component ii) results in a local temperature oscillation of the form Θ s (x-x 0 ,y-y 0 ,z 0 ), which is the product of spherical harmonics that only depend on r = [(x-x 0 ) 2 + (y-y 0 ) 2 + z 0 2 ] 1/2 and azimuthal components only depending on  = atan{z 0 /[(xx 0 ) 2 +(y-y 0 ) 2 ] 1/2 } (Figure 1c, bottom, see also Supporting Information).Under the additional assumption that multiple Θ s (x-x 0 ,yy 0 ,z 0 ) components from neighboring substrate points have random signs, and therefore tend to cancel out due to the randomness of thermal fluctuations in disordered media, the integration in Equation 5is, at the present stage, carried out by neglecting the contribution from any component at×≠ x 0 and y ≠ y 0 (i.e. with a cut-off area A corresponding to 1 pixel, so T s (x 0 ,y 0 ,z 0 ) ≈ Θ s (x 0 ,y 0 ,z 0 )).Nonetheless, it is anticipated that this simplified model of the substrate's contributions to the thermal expansivity is sufficient for optically transparent substrates at low enough thermal conductivity.
In the following section, we will explore the use of SNOD for investigating specific systems of practical interest, with both positive and negative TEC.2]

Nanogranular and Continuous Thin Films of Gold on Glass
Nanogranular and continuous thin films of gold on glass are ideal testing samples for SNOD as their data interpretation is simplified by heat being predominantly dissipated along the Au thinfilm, which is very thermally conducting, while the relative uniformity of the thin film minimizes T f (x 0 ,y 0 ) from Figure 1c.For this system, the substrate's contribution to the sample's expansivity from Equation ( 5) is overwhelmingly represented by T s,av (z 0 ) in Equation ( 6), which is independent of (x,y) and, consequently, (x 0, y 0 ).
Figure 2 shows the SNOD images and the corresponding calculated thermal expansion maps from an 80-nm thick Au film, where the granularity of the sample is noticeable from the AFM micrograph in panel a, while SNOM measurements in transmission and reflection mode are shown in panels b and c, respectively.The corresponding - thermoreflectance map is shown in panel d, with its -2 thermal expansion counterparts at four different  presented in panels e-h, respectively.The TEC is an intensive property that will only depend on the crystal structure in the bulk of a solid, even though it can assume different values at surfaces and interfaces. [39]Thus, it cannot depend on measuring parameters, such as .To validate this, thermal expansion maps are measured for  = 45, 60, 80, and 100 Hz in the same region as the AFM micrograph in panel a.We can thus observe that the thermal expansion maps are independent of , which is an indication of their genuineness.It can also be observed that the TEC is highest in the proximity of Au-glass interfaces (panel i) and relatively constant elsewhere.From panels e-h, we estimate the average volumetric TEC of Au to be (51.35± 6.89)×10 −6 K −1 with an average linear TEC of (17.12 ± 2.30)×10 −6 K −1 , and uncertainties determined by the standard deviation from all the measured .Comparison of these values with the TEC for Au recorded by macroscopic dilatometry techniques from the literature [40][41][42][43][44][45][46][47][48][49][50][51][52][53] is reported in Table 1.In addition, the TEC of the same nanogranular Au thin film probed by SNOD was also measured macroscopically using an optical dilatometer (see Supporting Information), and comparable TEC estimates were obtained.We can thus infer that SNOD provides quantitatively correct measurements of TEC of Au thin films, with additional information at the grain boundaries, and at the interface with the glass substrate.

Sparse Multilayer Graphene (ML-G) Platelets on Glass
To further substantiate the capabilities of SNOD, we have performed TEC imaging measurements on a system of complementary characteristics, represented by sparse multi-layer graphene (ML-G) platelets deposited on optically transparent and thermally insulating glass, [54] as summarized in Figure 3.As far as this system is concerned, the substrate's contribution to the sample's expansivity from Equation ( 5) is overwhelmingly represented by T s (x-x 0 ,y-y 0 ,z 0 ) in Equation (6), which, at a certain subsurface level z 0 , may depend on a relatively large number of substrate points (x,y) around (x 0, y 0 ).However, because of the sparseness of the ML-G platelets, heat is here overwhelmingly dissipated through the substrate, orthogonally from the surface, and therefore a small cut-off area A in Equation ( 7) appears to be adequate.
From the AFM micrograph in Figure 3a, it can be observed that the ML-G platelets are dispersed on the substrate surface with little overlap and some folding, which results in a broad range of number of stacking graphene layers.The SNOM transmission and reflection-mode maps over the same region are presented in Figure 3b,c respectively, while panel d shows the temperature profile of the sample.A local optical absorption profile can be determined using the near-field transmission and reflectance signals to determine the local temperature profile and be used to calculate the local thermal expansion maps from  f (x 0 ,y 0 ) in Equation (5).TEC maps of ML-G platelets are presented in Figure 3e-i for frequencies  = 100, 200, 400, and 700 Hz, respectively.Because the TEC is an intensive, material-specific,parameter, the results in Figure 3e-i should not depend on  and any The negative TEC can be associated to the in-plane bending mode of graphene, whereas the positive TEC (layer-number independent at > 200 layers) can be assigned to the vibrational modes along the crystallographic z-axis of graphite.
differences are the result of measuring uncertainties.In fact, the differences between the TEC determined from such independent measurements are marginal and consistent in magnitude with normal fluctuations during sequential AFM and SNOM scans.
From the summary in Figure 3h, which correlates average TEC results from panels e-i and the ML-G thickness from AFM (panel a), we can observe that the thinner the ML-G the more negative the TEC.A transition point can be observed at ≈ 175 layers.For platelets thicker than 175 layers, the TEC transitions to positive, as it happens in bulk graphite. [56]Furthermore, the linear TEC above 175 layers undergo a plateau at (33.75 ± 3.24)×10 −6 K −1 which is consistent with the out-of-plane thermal expansion of graphite along the crystallographic c-axis, [55][56][57][58] which corroborates the validity of SNOD as an accurate and quantitative method for thermal expansivity measurements.Conversely, a value of (−5.77 ± 3.79)×10 −6 K −1 is observed via SNOD for the thinnest ML-G platelets, below ≈ 90 layers, which can be assigned to the in-plane bending modes of graphene, [55,56] and is consistent with previous macroscopic measurements, [56][57][58][59][60][61][62][63][64][65][66][67][68][69] as shown in Table 2.A significant advantage of SNOD over all of such techniques rests in that it provides site-specific thermal expansion information, while other probing methods only yield an ensemble aver-age of the thermally induced expansion and offer minimal information on the distribution of the thermal expansion throughout the whole sample.In addition, through the systematic use of this newly invented SNOD technique, we have been able to prove that the transition between graphene and graphite occurs, from a thermal expansion point of view, at ≈ 175 layers and, therefore, at a higher thickness than the transition of electronic properties between few-layer graphene and graphite, which typically occurs at ≈ 10 layers or below. [70]

Conclusion
In this article, we have described the invention and construction of the first contactless nano-dilatometry apparatus capable of imaging the TEC at the nanoscale in nanostructured thin-film materials, 2D materials, and related electronic nanodevices.Our apparatus, a scanning near-field dilatometer, is all-optical in nature and, therefore, does not require any external heater or electrical contact to vary the sample's temperature, nor it requires any physical thermometer or nano-thermocouple to probe it.Our method is a -2 pump-probe method, where the probe is an aperture-type reflection-mode SNOM, which makes it uniquely suited for probing optically absorbing and thermally conducting See References Graphite 0-40 [57-59]   Nano EELS Graphene/ Cu Mesh Mono−2.14 ± 0.79 Bilayer−1.09± 0.25 Trilayer−0.87 ± 0.17 Bulk−0.07 ± 0.01 [60]   Low Temperature Resonance Au/Graphene/SiO 2 /Si −7.4 [61]   Raman Spectroscopy Graphene/LiNbO 3 −10 to -5 [62]   Au/Graphene/ SiO 2 /Si Mono−3.2 ± 0.2 [63]   Bilayer−3.6± 0.4 Trilayer−3.8 ± 0.6 Graphene/ODTS −0.6 ± 0.5 [64]   Graphene/SiO 2 /Si −3.68 ± 0.49 [65]   Graphene/Ge −(9.4 ± 7.7) [66]   Stress-Strain Curve 3D/Graphene −3.69 ± 0.12 to [67]   Pris-Graphene/SiO Using SNOD, we have also found that the TEC of nanogranular Au thin films is higher at the lateral glass/Au interface.Such a finding would have been impossible with any other techniques.It would have been impossible with macroscopic contactless techniques because of poor resolution, which would be insufficient to resolve them by several orders of magnitude.To the best of our knowledge, the same effect has never been observed before using SThM, arguably because the strong interaction with the sample of supposed SThM "nano-thermocouples" (that are, in fact, rather voluminous) may produce too large lateral heat spreads for the thermal expansion gradients to be observed over few microns.
As far as 2D materials are concerned, SNOD has led us to observe for the first time that, from a thermomechanical point of view, the graphene-graphite transition occurs at ≈175 layers.Such a finding has so far been elusive with other techniques because of the difficulty of reproducibly heating up highly thermally conducting ML-G flakes on thermally insulating samples without a contactless nanoscale dilatometer.The discovery that such a transition occurs over a hundred layers is particularly remarkable because, as far as the electronic structure is concerned, few-layer graphene is known to transition to graphite at ≈ 10 layers or less. [70]Therefore, the discovery that "graphene is thicker than we thought" from a thermomechanical standpoint may have critical implications including, for example, biodevices and bioimplants, in addition to nano-electronic devices and heat spreaders.Collectively, our work heralds the significance of the development of a near-field-based super-resolution optical technique for nanodilatometry imaging applications.Furthermore, our method can be employed with electrically driven samples, for example in operando, where an electronic circuit with electrical contacts can be biased at the frequency  to cause Joule heating and a SNOD signal at 2 in alternative to, or in conjunction with, the optical pump.Owning to the unique properties of evanescent waves, we can probe thermal events beyond the limits of diffraction which will be invaluable for studying a large variety of 2D materials and nanostructured thin-film systems where thermomechanics is critical for their applications.

Experimental Section
SNOM Apparatus and Transmission/Reflection SNOM Scans: Experiments described in this study were recorded on a WiTec Alpha 300S aperture type SNOM/AFM consisting of an AFM oscillator that can be equipped with hollow cantilevers (SNOM-NC, NT-MDT) with 20 × 20× 13 μm LxWxH pyramidal tips, at the end of which a subwavelength aperture (typical radius: a = 50-60 nm) was obtained by focussed ion beam (FIB) milling.The pyramidal shape of the tips ensures that any back-reflected light from the sample, which is also reflected by the inclined tip surface, does not hit the sample again, thus making stray light a technical impossibility.The tip apex was situated at the focal plane of an upright laser confocal optical microscope, from which different lasers could be focused to produce an evanescent field next to the tip aperture.The WiTec Alpha 300S can acquire both transmission and reflection-mode SNOM images while simultaneously measuring the surface morphology via AFM (panels a in Figures 2 and 3).Transmission-mode SNOM images (panel b in Figures 2 and 3) were obtained by illuminating the sample with light focused through the SNOM cantilever with a 532 nm diode laser (50 mW Excelsior, Spectra Physics) that was passed through a single-mode optical fiber (OZ Optics, NA = 0.12).Evanescent waves thus generated at the apex of the cantilever were used to probe the sample, where scattered photons that emerge from the substrate were collected with an inverted optical microscope from below and fed into a photomultiplier tube (U64000 Hamamatsu).The typical 532 nm probe power at the tip apex was ≈ 1.2 μW.Reflection-mode SNOM images (panel c in Figures 2 and 3) were obtained from the same apparatus as the transmission-mode scans, but with scattered photons collected at a grazing angle in the near-field by a subminiature accessory (SMA, WiTec) and, from there, fed into the photomultiplier tube.It is essential to be able to acquire both transmission and reflection-mode SNOM images as they provide information on the local sample absorbance and, from there, determine the sample heat profile in conjunction with knowledge of the pump-beam illumination.
Thermoreflectance (-) and Nano-Dilatometry (-2, SNOD) Scans: - and -2 thermoreflectance images (corresponding, respectively, to panel d and panels e-h in Figures 2 and 3) were recorded using the abovementioned WiTec instrument in the configuration reported in Figure 1, with the cantilever in noncontact (liftoff) mode.In this configuration, the sample is illuminated ("pumped") with a 405-nm laser beam (500 mW, Apinex) from below the sample through the same inverted optical microscope previously used to collect the transmission-mode SNOM signal (see section above).The typical pump laser power at the sample was ≈ 50 mW.The 405 nm laser was modulated with a mechanical chopper (SciTec Instruments, 300 CD/HRG), and the chopper angular velocity, , was varied to record different scans (as in panels e-h of Figures 2,3).During the pulsed illumination and heating of the sample from the 405-nm pump beam, the sample surface was scanned ("probed") in the near field with the upright SNOM microscope in reflection mode.These probing SNOM measurements were carried out by focussing the same 532-nm laser mentioned above (50 mW Excelsior, Spectra-Physics) at the FIB-milled aperture of the scanning SNOM tip (SNOM-NC, NT-MDT, see above) through the upright confocal optical microscope.It is worthwhile noting that the power at the sample of this 532-nm probe beam is significantly lower than the 405-nm pump.The evanescent light scattered off the surface of the sample was again collected at a grazing angle by the WiTec SMA reflection mode accessory.A 405 nm notch filter (Thorlabs Inc., NF405-13) and 530 nm long pass filter (Melles-Griot) were positioned in series with the reflection mode SMA to ensure the detected optical signal at the photomultiplier purely originated from the 532-nm probe-beam light, not from the 405-nm pump.The signal from each point on the sample was obtained by connecting the photomultiplier tube with a dual-phase lock-in amplifier (830, Stanford Research Systems) with the reference signal originating from the mechanical chopper, either at  [30] (panels d of Figures 2 and 3) or 2 (panels e-h of Figures 2 and 3) frequency.Raw image data were saved in matrix format and processed in ad hoc Matlab routines implementing the model presented by Equations 1 and 4-7 to obtain the TEC images.
Testing Samples-Nanogranular and Continuous Thin Film of Gold on Glass: Nanogranular gold thin films were selected due to the chemical inertness of Au as a noble metal.Au thin films were thermally evaporated onto 1×1-in glass substrates (Corning 7101).These substrates were precleaned in a bath sonicator with multiple 15' baths (soap water, deionized water, reagent-grade acetone, and methanol).The substrates were then blown dry with medical-grade nitrogen.The cleaned substrates were loaded into a custom-built thermal evaporator, [39] and Au growth was monitored through a Sycom STM-2 thickness monitor controlling an Inficon EasyRate water-cooled sensor equipped with a Kurt-Lesker 5-MHz quartz oscillator.A growth rate of 2 nm min −1 up to 80 nm thickness was used.80 nm was expected to be the optimal trade-off for SNOD imaging between film thinness (required for homogeneous optical absorption and heat generation along the z-axis) and film thickness (required for substantial heat dissipation in the x-y direction to use these as test samples).The quality of the films according to specifications was tested by AFM prior to SNOD measurements.
Testing Samples-Sparse ML-G Platelets on Glass: ML-G nanoplatelets on glass substrates (Corning 7101) were prepared according to the method by Sharifi et al. [54] Briefly ML-G was obtained from surfactantassisted exfoliation of nanocrystalline graphite powder (average flake di-mension: d av ≈ 700 nm) using RNA VI from torula utilis (Sigma-Aldrich), thus forming stable ML-G suspensions in water.These suspensions were filtrated on nitrocellulose membranes (EMD, Millipore) which were transferred on the requisite glass substrate and subsequently etched in multiple acetone and methanol baths, leaving behind an ensemble of surfactantfree nanoplatelets on their substrate.These samples were thoroughly washed in subsequent acetone, methanol and water baths to eliminate any traces of surfactant RNA and filtration membrane, as ascertained by AFM, [54] prior to being submitted for SNOD measurements.

Figure 1 .
Figure1.a) Schematic of - pump-probe scanning near-field thermoreflectance imaging experiments for contactless nano-optical probing of temperature oscillations (ΔT f ) at the sample surface via local changes of the air's refractive index,[30] b) -2 pump-probe experiments for contactless nano-optical probing the local thermal expansion of inhomogeneous thin films on transparent substrates (sample).Both experiments can be implemented in a commercial apparatus mounting aperture-type SNOM tips in a scanning probe microscope (AFM) with upright-confocal (532 nm) and inverted (405 nm) optical microscopes for sample illumination, and a submillimeter accessory (SMA) for 532-nm evanescent wave detection in reflection mode.With thermal expansion, the Gaussian evanescent wave is off-peak at both maxima (Hi) and minima (Lo) of the sample temperature, so thermal-expansion related oscillations occur at angular frequency 2; and c) Model for determining the substrate temperature used in Equation (5) from the linear superposition of average (T f,av , top) and local (T f , bottom) film temperature oscillations of ΔT f as detailed in the Supporting Information.

Figure 2 .
Figure 2. a) AFM micrograph of a nanogranular and continuous gold thin film on glass at the metal-glass interface, along with b) transmission-mode and c) reflection-mode SNOM images as well as d) nanoscale temperature oscillation profile [ΔT f,(x,y)) measured by - pump-probe near-field scanning thermoreflectance imaging.[30]e) Thermal expansion maps at 45 Hz; f) 60 Hz; g) 80 Hz; and h) 100 Hz obtained from -2 pump-probe SNOD measurements.i) Pictorial representation of the measured nanogranular gold thin film, for which j) the linear thermal expansion coefficient (TEC) is observed to be highest at the thin-film edge, corresponding to the metal-insulator interface, while it is relatively constant (and in good agreement with the macroscopic measurements by Frenkel et al.[54] ) at other areas of the film.

Figure 3 .
Figure 3. a) AFM micrograph of sparse ML-G platelets on glass at the metal-glass interface, along with b) transmission-mode and c) reflection-mode SNOM images as well as d) Nanoscale temperature oscillation profile [ΔT f, (x,y)) measured by - pump-probe near-field scanning thermoreflectance imaging.e) Thermal expansion maps at 100 Hz; f) 200 Hz; g) 400 Hz; and h) 700 Hz obtained from -2 pump-probe measurements.i) Schematic representation of the sparse ML-G platelets, for which j) a negative linear thermal expansion coefficient (TEC) is observed up to 90-layer thickness.A thermomechanical transition occurs at ≈175 layers, where the TEC becomes positive, thus demonstrating the appearance of graphite-like characteristics.The negative TEC can be associated to the in-plane bending mode of graphene, whereas the positive TEC (layer-number independent at > 200 layers) can be assigned to the vibrational modes along the crystallographic z-axis of graphite.

Table 1 .
Comparison of the thermal expansion coefficient (TEC) of Au measured by SNOD in this work with existing probing methods from the literature (where XAS indicates X-ray absorption spectroscopy and EXAFS indicates extended X-ray absorption fine structure).

Table 2 .
Comparison of TEC from SNOD experiments on multi-layer graphene/graphite nanoplatelets with those achieved from macroscopic probing methods available from the literature (where EELS indicates Electron Energy Loss Spectroscopy).