Thermometry with Sub-nanometer Resolution in the Electron

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Thermometry with Sub-nanometer Resolution in the Electron Microscope using the Principle of Detailed Balance Maureen Lagos, and Philip E Batson Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.8b01791 • Publication Date (Web): 06 Jun 2018 Downloaded from http://pubs.acs.org on June 6, 2018

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Thermometry with Sub-nanometer Resolution in the Electron Microscope using the Principle of Detailed Balance Maureen J. Lagos1,2*, Philip E. Batson1 1

Department of Physics and Astronomy;

Department of Materials and Science Engineering. Rutgers University, Piscataway, New Jersey 08854, USA. 2

Present Address: Department of Materials and Science Engineering. McMaster University, Hamilton, Ontario L8S 4L7, Canada

We measure phonon energy gain and loss down to 20 meV in a single nanostructure using an atom-wide monochromatic electron beam. We show that the bulk and surface, energy-loss and energy-gain processes obey the Principle of Detailed Balance in nanostructured systems at thermal equilibrium. Plotting the logarithm of the ratio of the loss and gain bulk/surface scattering as a function of the excitation energy we find a linear behavior, expected from detailed balance arguments. Since that universal linearity scales with the inverse of the nanosystem temperature only, we can measure the temperature of the probed object with precision down to about 1 K without reference to the nanomaterial. We also show that sub-nanometer spatial resolution (down to ~ 2 Å) can be obtained using highly-localized acoustic phonon scattering. The surface phonon polariton signal can also be used to measure the temperature near the nanostructure surfaces, but with unavoidable averaging over several nanometers. Comparison between transmission and aloof probe configurations suggests that our method exhibits noninvasive characteristics. Our work demonstrates the validity of the Principle of Detailed Balance within nanoscale materials at thermal equilibrium and, it describes a transparent method to measure nanoscale temperature, thus representing an advance in the development of a noninvasive method for measurements with angstrom resolution. Keywords: nanoscale thermometry, energy gain spectroscopy, bulk phonons, surface phonon polaritons, principle of detailed balance. The measurement of temperature in nanometer size areas with sub-nanometer resolution has been a challenge for many research areas, leading to a continuous development of thermometry 1

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techniques for nanoscale thermal characterization1. Thus, a sub-nanometer thermometry method that is applicable to a broad range of materials will have potential to impact many nanoscale research areas such as catalysis2, high temperature nanophotonics3 and solid-state microelectronic devices4. This will deepen our understanding about heat distribution in catalytic particles, thermal response of thermophotovoltaics and nanoscale heat-dissipation in transistors. Several nanoscale thermometry techniques such as scanning thermal microscopy (SThM)5, near-field optical measurements6, light reflectance7, luminescent thermometry1, and Raman spectroscopy8 among others allow probing the thermal response of objects with nanometric spatial resolution. Among all these methods, SThM is the most commonly used; however its spatial resolution is limited by the physical size of the probe. In particular, a few cases have shown the sub-10 nm spatial resolution9, but were limited to particular types of materials, signal collection, etc. Today, electron probes of the size of a hydrogen atom can be produced in a scanning transmission electron microscope (STEM)10, allowing the realization of spectroscopic studies with nanometric and atomic-scale resolution. Exploiting these remarkable spatial capabilities, an electron probe was used to map temperature gradients in semiconductor micrometer-size devices by measuring energy shifts in plasmon excitations11 through electron energy loss spectroscopy (EELS). Also, temperature measurements have been performed in crystalline12 and amorphous materials13 using electron diffraction. In spite of this significant progress in improvement of spatial resolution, there is still a need to develop a non-invasive and general method which provides local temperature measurements in smaller, single nanostructures (< 100 nm) with better spatial resolution (< 1nm) and, which allows applications in a broader range of nanomaterials. Electron microscopy is today a premier tool to image nanoobjects and to probe energy of excitations (phonons, plasmons, core-shell transitions, etc.) within and nearby nanomaterials. In particular, with new monochromators implemented in modern STEM’s one can detect and map excitation of vibrational/phonon states in a large variety of nanostructures14,15. It is well known that the inelastic vibrational scattering cross section depends on the thermal occupation of the vibrational/phonon states16,17, thus providing a straightforward path to access temperature information of objects through scattering measurements. In general, the vibrational scattering cross sections are modulated by the temperature dependent occupational statistics, a factor (n+1) for energy loss, and a factor (n) for energy gain, where n represents the Bose-Einstein statistical factor which depends on the system temperature16,17. In addition, the Principle of Detailed Balance (PDB) establishes that in a system in thermal equilibrium at temperature T, each elementary process should be balanced by its reverse process. This 2


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principle plays an important role in the process of electron inelastic scattering, requiring the scattering probability associated with a loss energy event should be balanced with a gain energy process17. This effect is also visualized in the statistical relationship between dynamical form factors for loss and gain processes, which relates spectral properties of the multi-particle system17. As a matter of fact, the concept of balance was developed and applied successfully during the last one hundred fifty years to the description of several physical and chemical phenomena18. For instance, the theory of radiation of Einstein19 and the reciprocal relations of Onsanger20 were developed using this fundamental concept, and clearly illustrate its importance in the description of nature phenomena. Through PDB an important relationship can be derived showing that energy loss P(∆q, ∆E) and energy gain P(-∆q, -∆E) probabilities are linked through the Boltzmann factor, which follows as: P (∆q, ∆E ) = e β ⋅∆E P (−∆q,− ∆E )

ln[

P (∆q, ∆E ) 1 ]=( ) ⋅ ∆E , P (− ∆q,− ∆E ) K BT

(1)

(2)

where ∆q and ∆E are the momentum and energy transferred during the scattering event (Figure 1a); and the β parameter is given by 1/KBT, where KB is the Boltzmann constant. Note that using this relationship one can determine the temperature of the system through a linear fit of the curve produced by the logarithm of the ratio between the loss and gain scattering. Experimental validation of PDB was performed on surfaces using micrometer-size electron beam scattering in the reflected geometry21,22. In this work we report the experimental verification of the PDB in a variety of nanosized systems revealing an intrinsic spectral property of materials at the nanoscale. At this point is worth discussing the localization of the phonon scattering, which depends on two different types of signals (dipole and impact scattering)16. It is well known that the dipole excitation (∆q 0) dominates the scattering process at large impact parameters, resulting mainly in the excitation of long-wavelength surface phonon modes in the aloof configuration23. The long-range effects of the scattering can be easily observed probing nanomaterials up to distances of one micron away from the specimen (Supporting Information). Because this scattering includes the excitation of ionic charges near surfaces, it can be used to determine the surface temperature of nanostructures with limited degree of localization. On the other hand, non-dipole excitations (∆q >> 0) can govern the scattering at small impact parameters, producing vibrational signals modulated by the atom columns of the crystal24,25,26. This highly-localized signal allows us to perform local temperature measurements with sub-nanometer resolution by local interaction between the fast electron and the oscillating electric

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dipoles associated with vibrational modes. In this work we show we can measure local temperatures in nanoscale objects using spatially-resolved vibrational spectroscopy (Methods), with an energy resolution of 7 - 10 meV and a spatial resolution of 1.5 - 2 Å (Supporting Information). Figure 1a shows a schematic of the STEM operation during vibrational spectroscopic studies. Typically, an atom-wide electron probe within the STEM is focused on a nanostructure with angstromlevel position accuracy. At each beam position, the impinging electrons can be scattered elastically and inelastically producing a large variety of excitations. In particular, the inelastic process involves transfer of energy (∆E) and momentum (∆q) between the electron and the material. Due to the relativistic velocities for keV electrons, most of the inelastically-scattered electrons scatter in the forward direction at different angles, are collected through an aperture and then dispersed in energy by a magnetic prism to form an excitation spectrum. Usually such a spectrum contains an intense zero-loss peak (ZLP) at zero energy, corresponding mainly to the elastically-scattered electrons. The loss (+∆E) and gain (-∆E) peaks are associated with vibrational/phonon excitations and lie adjacent to the ZLP. In this way, the generation of a vibrational spectrum which contains energy loss and gain phonon excitations provides a suitable platform to determine the temperature of nanomaterials. To analyze the implications of the statistical occupation of phonon states on the inelastic scattering we performed spectroscopic measurements for a range of temperatures (Methods). Figures 1b and 1c shows typical phonon spectroscopy results obtained in nanostructures probing MgO nanocubes in aloof and transmission geometries, respectively. The beam positions are indicated by a cross marker in the figure insets. The aloof geometry certainly provides excellent non-invasive conditions for temperature measurements because it minimizes direct electron-beam driven processes such as, heat-injection, knock-on damage and radiolisis27. The implications of these invasive conditions for the transmission geometry will be discussed below. Note in Figures 1b and 1c that several types of resonances are generated as a result of the excitation of different phonon modes of the material. Figure 1b shows the excitation of both bulk and surface modes, with the prominent scattering mainly associated with bulk modes (impact scattering). Meanwhile, Figure 1c shows only the excitation of surface phonon polariton modes (dipole scattering). A ZLP curve is also presented in each plot to illustrate clearly the appearance of the phonon scattering signal, which lies over the tails of the elastic ZLP curve. Note that the loss and gain scattering signals scale with the temperature, the higher the temperature the stronger the scattering. It is important to mention that we also performed measurements in different type of nanomaterials, such as boron nitride flakes, silicon carbide particles, and silicon carbide/silicon dioxide interfaces and found similar loss/gain vibrational response (Supporting Information). 4


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Once we obtain the spectroscopic measurements, we determine the temperature of the nanostructure following the procedure presented below. As a descriptive example we use experimental data obtained from a 100 nm-size suspended MgO nanocube (black curve) probed in the intersecting geometry at room temperature (Figure 2a). The spectrum was acquired from a cube, which is attached to a neighboring larger nanostructure supported on an amorphous carbon (a-C) film, as shown in the inset of Figure 2a. The physical origins of the phonon resonances shown in the spectrum (black curve) are well understood15,26 and will be used here to describe our results. Briefly, the two resonances at around 35 and 50 meV correspond to the excitation of the short-wavelength acoustic bulk modes and the excitation at around 90 meV corresponds to the excitation of the longitudinal optical bulk mode. Note that the acoustic loss excitations (< 60 meV) have their counterpart gain excitations which are clearly revealed in the data. Also, notice that small scattering signal lies in the Restrahlend band of the material, which corresponds to the excitation of surface optical phonon modes23, but there are not corresponding gain excitations above ~ 60 meV because the gain signal there is smaller than the noise. As a rule of thumb, one should expect to observe gain excitations at a certain energy if the amplitudes of loss excitation peaks are at least four times larger than the product of the Boltzman factor and the background noise. The next step in the process is to remove the elastic contributions from the vibrational scattering signal. In order to do that, we assume that the tails of a ZLP curve acquired in a vacuum region are a good approximation to the elastic components above 20 meV (Supporting Information). To obtain a spectrum which does not include excitations from any part of the sample we acquired a ZLP curve by locating the electron probe in vacuum at about 400 µm from the specimen. We achieved this goal by cutting in half the grid which supports the MgO nanoparticles (Methods) and, we thus produced vacuum regions of ~ 0.5 mm in size. The curve is shown in the Figure 2a and is labeled ZLP. A fast electron can couple to surface modes through the long-range coulomb field during aloof excitation14,28. We illustrate this behavior for vibrational excitations in Figure 2b, where a swift electron traveling with an impact parameter of about 1 µm (inset Figure 2b) can excite surface modes of the a-C film due to the long-range phononic response. This scattering appears as broadening of the tails, becoming more prominent at high temperatures (Figure 2b). Thus, in order to perform a truly local temperature measurement it is imperative to remove the contribution of surface phonon excitations from surrounding nanostructures; otherwise the measurement would contain non-local information. A more extensive discussion of substrate effects on the vibrational scattering and their removal from the experimental data is presented in the supporting information. In the particular case we are analyzing (Figure 2a), we obtained the vibrational contributions from the a-C substrate for an 5 ACS Paragon Plus Environment

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electron beam probing the material in a similar configuration as probed the MgO cube. We thus performed a spectrum acquisition in a circular hole of an empty Quantifoil grid, using a fast electron travelling with an impact parameter of about 100 nm from the hole edge. The curve is plotted in Figure 2a and we clearly see that the electron excites vibrational modes of the a-C substrate in the region of interest (< 60 meV). We performed these measurements multiple times and acquired ZLP curves before and after each measurement, thus ruling out ZLP broadening due to the instabilities in the microscope optics or electrical system. Figure 2c shows the subtracted data including only the signal from the excitation of modes of the MgO cube. The black curve shows the subtracted raw data and the blue curve shows the deconvoluted scattering signal using a Richardson-Lucy deconvolution method29,30. It is possible to observe that the deconvoluted signal exhibits the same scattering peaks with slightly narrower resonances and the surface phonon excitations are better resolved, suggesting that our experimental data contains minor deviations from the single scattering signal produced by the instrument response. Our deconvolution analysis showed that convergence can be achieved after a few iterations (~ 5). We also estimate that multiple scattering events for the phonon excitations are about two orders of magnitude smaller than the single scattering signal, thus lying within the signal noise. Once we derived the phonon scattering signal from the nanostructure we calculate the ratio between the loss and gain scattering, and we then plot the logarithm of the ratio as a function of excitation energy (Figure 2d). Figure 2d shows plots generated from three different spectra acquired in the same MgO nanocube, for acquisition points that were spaced by about 2 nm among them. To render a better visualization of the curves, the upper and lower curves were shifted upwards/downwards, respectively. Note that such a ratio between scattering signals exhibits a linear behavior, thus verifying that the PDB holds for electron scattering from lattice vibrations. We will show below that PDB also holds for other nanoscale systems at thermal equilibrium. The temperature of the nanostructure can be derived from the slope of the curve. Our analysis indicates that the nanocube temperature is about 290 K with experimental errors within 7 - 10 K. Our results agree with the temperature values accepted for conditions of room temperature (293 – 298 K) indicating a good accuracy in our measurements. The experimental errors associated with the temperature measurements shown here are expressed with confidence interval of about 99% (three standard deviations) establishing a good precision in our measurements. Noise reduction in the vibrational scattering allows us to obtain deviation standards down to 1 K. Also, because we used acoustic phonon scattering highly localized near the atom columns24,25,26, which in our study correspond mainly to the excitation of short-wavelength acoustic modes of MgO lying near the Brillouin zone, we obtained temperature measurements with sub6 ACS Paragon Plus Environment

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nanometer resolution (down to values of the order of the nearest neighbor distances, being about 2 Å for MgO). We also determined the temperature using the deconvoluted scattering signal and we found similar results, indicating that our scattering signal does not appear to be drastically influenced by the instrument response. However, deconvoluted processed data could be used to treat data acquired with less good energy resolution conditions (> 10 meV), where the instrument response can modify drastically the shape of the scattering amplitudes resulting in deviations of the linear behavior of PDB (Supplementary Information), or systems which lead to prominent multiple scattering signals. In addition, we performed measurement of the nanostructure temperature including the substrate scattering signal (gray curve of Figure 2a). In this way we would expect to obtain an average measurement of the cube and substrate. Our results showed that measured temperature values can shift a few degrees (2 - 4 K) within our precision. In this case the substrate effects are not critical because the temperature of the substrate and the cube are apparently the same. Certainly, composite systems (i.e. nanostructure + substrate) which exhibit temperature gradients between the nanostructure and substrate will produce curves (Figure 2d) with different slopes resulting in an averaged curve that might deviate strongly from linearity. It is well known that the incoming electron can deposit energy into the object which can be dissipated gradually through time. The heating process usually starts with a plasmon excitation dumping its energy into the creation of electron-hole pairs (non-radiative process). Plasmons can also decay via radiative process through photon emission. Both energy-transfer processes take place in the scale of femtoseconds (see for instance Reference (31), which shows the temporal evolution of plasmonic fields in a nanosphere induced by a relativistic electron). The formed hot carriers can subsequently relax by interactions with lattice phonons within picosecond times resulting in a higher local lattice temperature. The final relaxation occurs through heat diffusion to the surrounding areas in the time scale of hundreds of picoseconds to nanoseconds. It is important to bear on mind that there is one single electron at the time in the microscope column due to low beam current. Thus, in these experiments we have about one incoming electron about each nanosecond, implying that most incoming electrons experience a fully or almost-fully relaxed system. In order to explore possible temperature increases due to heating by the fast electron we measured the temperature in a large variety of nanostructures at room temperature. In particular, we explored arrangements of nanoparticles which might contain fewer available channels for heat dissipation by thermal diffusion or thermal radiation, that might result in significant temperature 7


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increase of the probed nanostructure (for instance see Figure S10a). Although there is no complete theory which fully accounts for the complex phenomenon associated with heat generation induced by an electron beam and the corresponding heat transport at nanoscale, very simple models predict quite minor temperature increases (∆T < 1 K) in small nanostructures (> 5 nm in size)27,32. With our approach we cannot access sub-kelvin range. Our experimental results show that within our precision there is no significant increase of the temperature of the nanostructures, as shown by the histogram in Figure S10b. Note that the average temperature of the system is about 295 K with a very narrow temperature distribution. Although a few measurements indicated higher values, those shifts can be considered within our error precision. Our experimental results seem to suggest that for the studied cases here, each incoming swift electron probes a fully or almost-fully relaxed nanosystem with phonon states at thermal equilibrium in each scattering event. Following the same method presented above we measured the temperature of nanostructures at higher temperatures than 300 K. Figure 3a shows the results of temperature measurements for the MgO cube of Figure 1b. The measured values are slightly smaller (~ 20 K) than the target values provided by the heating holder unit, but are still within our measurement error bars. We also verified that the calibration of the heating e-chip is reasonable by performing measurements in empty grids using the surface phonon scattering of the ceramic heater material (Methods). The small differences shown in our work might be associated with the heat dissipation of the suspended nanostructure driven by thermal conduction and radiative processes33. In addition, note that the errors for the high-temperature conditions are usually larger and are typically given by the propagation error theory, as ∆T = KB T2 ∆β, where ∆β represents the error in the measurement of curve slope. This indicates that at high temperatures we could approach conditions that limit our abilities to detect small temperature gradients in nanosized systems, but those effects could be minimized, obtaining small errors associated with the slope measurement. We also measured nanostructure temperatures using surface phonon scattering excitations. As is well known, those excitations usually involve lattice displacements within the outermost surface layers, producing stronger response fields in regions near the surface. Thus, one should expect that the temperature measurements correspond to vibrational behavior in the structure surface. We then compared these surface measurements with the bulk (Figure 3b) finding that the measurements performed using the surface scattering (black curves) match the trend established by the bulk scattering (gray curves), confirming that the temperature on the surface and inside the nanostructure are quite the same. In terms of heat injection, we can say that experiments probing in an intersecting-geometry might be considered as non-invasive as well. 8


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We thus show above the validity of the Principle of Detailed Balance in MgO nanostructures. We verify PDB through the linearity of curves shown in Figure 2d, which include a range of inelastic scattering by different type of phonons. We further tested the validity of this fundamental law in other materials. Figure 4 shows the experimental verification of the PDB in several nanomaterials at different temperatures. Measurements were conducted in a single SiC nanoparticle, MgO nanosphere and BN flake at room temperature, ~ 600 K and ~ 1200 K, respectively. The measurements associated with the SiC particle only included the bulk phonon scattering of the SiC. The measurements for the BN flakes considered the surface phonon polariton signal of both Reststralhen bands (only the lower band is shown in the plot). We also conducted measurements in a SiC/SiO2 interface, which represents the classical system for a solid-state device (transistor), at room temperature. For this case the beam intersected the SiC region. Note in Figure 4 that all of the curves exhibit the typical linear behavior, with slope scaling like T-1, as predicted by the PDB (Equation 2). This universal linear behavior represents an important spectral property of nanoscale structures which does not depend on the experimental conditions or instrument characteristics, revealing an intrinsic property of inelastic bulk and surface phonon scattering. In conclusion, we present a transparent and general method to measure the temperature of a single nanostructure using acoustic and optical phonon scattering signal induced by an electron probe. The spatial resolution will depend on the type of scattering signal used to determine the temperature: short wavelength bulk phonons usually provide highly-localized scattering signal down to the atomlevel and, surface phonon excitations can yield average temperatures values over a region of interest. We also verified that the Principle of Detailed Balance holds for inelastic electron scattering from bulk and surface lattice vibrations in nanoscaled objects. In addition, we think this general principle can aid in the determination of inelastic scattering signal at ultra-low energies (< 10 meV). Our method can be also utilized to determine effective temperatures of out-of-equilibrium systems like glasses, which are known for violating the Fluctuation-Dissipation Theorem. And we also think that our results will stimulate investigation towards atomic-resolution temperature spectroscopy in nanostructures exploring different collection conditions. We foresee that our method can be implemented for temperature measurements in monochromatic STEM in a near future, and think that it will be useful for characterization studies which require spatially-resolved thermal imaging capabilities and nanoscale thermal response, such as catalysis, nanoscale energy transport, photonic applications, among others.

Methods Sample Preparation 9


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We obtained SiC nanoparticles and BN flakes from powder materials. We prepared the microscopy specimen following the traditional procedure of mixing powder with ethanol; then the solution was sonicated with ultrasonic vibration for 30 minutes and it was finally drop casted in a standard amorphous carbon Quantifoil grid or in a heating MEM-based chip. This last sample support allows us to increase the specimen temperature (see heating holder section). The SiC particles and BN flakes range between 100 to 1000 nm in size. The MgO nanostructures were produced by burning a magnesium wire in air and collecting the combustion smoke in a Quantifoil grid or a in a heating chip unit. With this method we obtained nanostructures ranging between 30 - 500 nm in size. Most of the nanostructures consist of nanocubes, but in some instances when the atmosphere composition was not well controlled nanosphere, nanorods, and plates where also produced. It is important to mention that after the drop casting on the standard Quantifoil was finished, the grids were cut in half producing in this way vacuum region of about 500 nm in size. Finally, a section of MOSFET transistor was used to prepare the SiC/SiO2 interface system. The whole sample preparation was done using the standard focus-ion beam procedure.

Heating Holder Nanostructures were heated to different temperatures using a heating MEM-based chip (Protochips model) which fits into a Nion-designed holder. This device allows us to uniformly change the temperature of the specimen between room temperature and about 1500 K. The chip unit is composed of a ceramic material which contains a 3x3 array of nine circular holes. The surface temperature of the ceramic material can be calibrated using the method proposed in this work. Inside each hole there is an amorphous carbon membrane which contains an arrangement of 2 µm holes (see inset Figure 2b). The samples were loaded into the chip following the procedure described in the Sample preparation section.

Scanning Transmission Electron Microscopy (STEM) The experimental vibrational data were obtained using an UltraSTEM 100 microscope equipped with a monochromator operating at 60 kV. Three main optical elements allow the experimental realization of spatially-resolved spectroscopic studies in nanomaterials: (i) An aberration corrector allows the formation of an atom-size electron probe, producing about 1 Å of spatial resolution (Supporting Information). (ii) A monochromator allows the reduction of the energy spread of the field-emitted electrons (250 meV) to about 7.5 meV with 125 ms of acquisition time and 6.5 meV with 8 ms of acquisition time (Supporting Information). (iii) A post-column spectrometer designed for highdispersion conditions, (0.7 – 1 meV per channel) and current throughput of about 40% of the incident beam current at optimum resolution, and (iv) a CMOS-based Hamamatsu camera to record a spectrum of 2D images with 2048x500 pixels. 10


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STEM Imaging The whole imaging work was done using a probe with convergence semi-angle of about 30 mrad. This probe produces about of 1 Å resolution with a non-monochromated beam. The transmitted high angle scattered electrons where collected using a high angle annular detector with inner/outer collection angle of 80 and 200 mrad, respectively.

Vibrational Data Acquisition Spectroscopy was performed using a probe of 30 mrad convergence semi-angle with a beam current of about 50 - 75 pA (0.4 µA emission current), producing a probe size of about 1.5 Å with 250 meV wide energy distribution. Monochromated operation produces a 7.5 - 10 meV wide beam (acquisition time 62 ms – 1 s) and beam current between 5 – 10 pA. Under these optical conditions, the spatial resolution remains between 1.5 – 2.0 Å (Supporting Information). The spectrometer entrance aperture subtends about 20 mrad half angle at 60 kV. With these collection conditions, one is able to collect scattering signal from phonon excitations across the Brillouin Zone of most nanomaterials. The spectra were acquired as a function of probe position. For each position, 30 spectra were collected and then the beam was blanked for 15 s and additional 30 background (dark current) spectra were collected. The spectra were acquired using 1 and 2 s of acquisition time. Once, the spectra is acquired we applied a dark image subtraction to each image, then aligned multiple exposures using cross correlation, and finally summed them to produce a single spectrum associated with a single point (Figure 1b, 1c and 2a). Zero Loss (ZLP) background curves are also acquired in vacuum following the similar procedure as described above. The ZLP curves were acquired in vacuum regions which were very distant from the specimen aiming to avoid long-range coupling effects with surface excitations (Supporting Information). In most cases, we acquired spectra which contain surface scattering contributions from substrates, which were also subtracted in order to perform a proper local temperature measurement (Supporting Information). Each spectrum was acquired after locating the ZLP position in the middle of the EELS detector. Intensity variations in the central region of the detector, which are induced by the set of light apertures located inside the EELS camera, are neglected within an energy range of about 100 meV. This is equivalent to an energy window of 100 – 200 energy channels of the 2048 channel camera. Most of the data presented in this work consists of phonon excitations spaced apart in energy by less than 150 meV. To properly treat intensity variations generated by the microscope/detector configuration, a gain normalization procedure was performed in the EELS camera before collecting the data.

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ASSOCIATED CONTENT Supporting Information is available in the online version of the paper.

AUTHOR INFORMATION Corresponding Author * Email: [email protected]

Author Contributions M.J.L initiated the project and conceived the experiments. M.J.L prepared the electron microscopy samples and conducted the EELS-STEM experiments. M.J.L performed the data analysis. M.J.L and P.E.B. discussed the results and wrote the manuscript. All the authors read and commented on the manuscript.

Notes The authors declare no competing financial interest.

ACKNOWLEDGEMENTS M. J. Lagos and P. E. Batson acknowledge the financial support of U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award # DE-SC0005132. The authors acknowledge R. Haber, V. Amarasinghe and L. C. Feldman from Rutgers University and, C. Bittencourt from University of Mons for providing samples to perform the spectroscopic studies. D. Schlom and D. A. Muller from Cornell University are also acknowledged for providing microscopy samples to characterize the microscope performance. The authors are also grateful to H. Yang and. E. Garfunkel (Rutgers University) for providing assistance during the preparation of MgO samples. We also thank O. Krivanek, N. Dellby and T. Lovejoy for discussions regarding the optical configuration of the microscope.

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Figure 1. STEM vibrational spectroscopy and energy loss/gain scattering signal of nanostructures. a, Schematic of the operation a STEM microscope performing spatially-resolved vibrational spectroscopy, which allows the detection of vibrational loss/gain scattering events. b, c, Experimental phonon spectra (raw data) collected from MgO nanocubes at different temperatures (300, 600 and 800 K) in transmission and aloof configurations, respectively. The figure insets show STEM images of the MgO particles with a cross indicating the probe position. Note that depending on the position of the probe different type of phonon modes can be excited. Also, note the drastic variation of phonon loss/gain scattering as function of temperature. A spectrum acquired in a vacuum region is also plotted and labelled ZLP.

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Figure 2. Temperature measurement of a single nanostructure using highly-localized acoustic phonon signal. a, Phonon spectrum (raw data) of a 100 nm suspended MgO cube (inset figure) at room temperature (RT). The probe position is indicated by the cross mark in the inset. The spectrum exhibits two main loss peaks at around 35 and 50 meV and their corresponding gains excitations are also present. The spectra acquired in a vacuum region and near to amorphous carbon substrate are also displayed and labelled ZLP and substrate, respectively. b, Spectra acquired in the vacuum as a function of temperature for a probe located 1 µm away from the circular edge (figure inset). Excitations of surface modes induce broadening of the tails. c, Subtracted loss/gain spectrum (black curve) and deconvoluted spectrum (blue curve) using Richarson-Lucy (RL) method. d, Linear plots of the logarithm of the ratio between loss and gain scattering as a function of excitation energy. Note that linear behaviour verifies the Principle of Detailed Balance. The upper and lower linear curves were shifted about ± 0.5 to render a better visualization. The temperature measurement results agree with values of RT conditions.

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Figure 3. Temperature measurements using bulk and surface phonon scattering as a function of temperature. a, Linear plots of the logarithm of the ratio between loss and gain scattering as a function of excitation energy for the same MgO nanocube heated up to about 600 and 800 K. The experimental measurements agree well with the values provided by the heating holder unit. Note that when the temperature is increased the slope of the linear curves reduces imposing a limit to discriminate between high temperature values. b, Linear plots generated using bulk and surface scattering signal. The plots using bulk signal (gray curves) were reproduced from part (a) and the plots using surface scattering (black curves) were superposed over the bulk scattering. Both data follow the same trend revealing that temperature inside and on the surface of the nanostructure are similar.

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Figure 4. Experimental verification of the Principle of Detailed Balance in nanomaterials using bulk and surface phonon scattering. Plot obtained for a single SiC nanoparticle and a SiO2/SiC interface sample at room temperature, and for a single MgO nanosphere and a BN flake at high temperatures. The curve for the SiC flake was shifted vertically by 0.5 for visualization purposes. Note the linear behaviour of the plots, as predicted by the Principle of Detailed Balance. The measured temperature values for each nanosystem are shown over the dotted lines.

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TOC Graphic

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Thermometry with Sub-nanometer Resolution in the Electron

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