Automated plasmon peak fitting derived temperature mapping in a scanning transmission electron microscope

Over the last few years, several new approaches using TEM-based temperature measurements have been reported of free-standing materials, starting with the study of local temperature measurements in a transmission electron microscope by tracking the energy shifts of bulk plasmons as a function of temperature.^1–3 Other approaches to nanoscale thermometry utilize scanning thermal microscopy,^4–8 which is limited by the size of the cantilever tip, and Raman, fluorescence, and luminescence thermometry,^9–11 which are all limited in resolution by the optical diffraction limit.

In 2018, Xu et al. demonstrated that non-contact thermometry, based on STEM imaging and bulk plasmon electron energy-loss spectroscopy (EELS), in combination with first-principles modeling can be used to map the local temperature from the thermal expansion coefficient of low dimensional materials with a spatial resolution of better than 2 nm and a temperature accuracy of <8 K.³ The basic concept of this approach dates back to the mid-1950s,² when it was pointed out that the energy of a bulk plasmon depends on the temperature through changes in volume and the associated changes in electron density. This takes the form of the following equation for the plasmon energy in the approximation of the electron gas model:

where T is the temperature, ħ is the reduced Planck’s constant, n(T) is the temperature dependent valence electron density, e is the electron charge, m is the electron mass, and ε₀ is the vacuum permittivity.

In this article, we report that the accuracy of plasmon based thermometry can be improved significantly by using automatic peak-fitting algorithms. These serve to improve its resolution and to gather enough statistics to yield better than 6 meV precision in determining the plasmon peak position.

The experimental data were acquired using a JEOL ARM200CF equipped with a cold-field emission source and a Gatan Continuum GIF spectrometer, providing an energy resolution of better than 350 meV, as measured by the full-width at half maximum of the zero-loss peak. The Si samples were prepared by crushing a Si wafer drop-cast onto a previously calibrated Protochips heating echip as well as the Protochips Aduro double tilt holder to perform the in situ heating experiments. The probe-convergence angle was chosen to be 13.4 mrad, with a collection angle of 40.0 mrad for the spectrometer. The temperature of the Si samples was varied from room temperature to 900 °C, and at each temperature, a set of spectrum images was acquired from several Si particles.

The calibration of the plasmon peak shift as a function of temperature was performed using a Si sample as well as a Protochips heating echip that was previously calibrated using an external reference. The energy calibration of the plasmon electron energy loss spectra was done using the zero-loss peak, and the plasmon region of the low-loss spectral range was normalized using the integrated intensity in the energy range of 40–60 eV. As shown in Fig. 1, the Si bulk plasmon peak is located at ∼17 eV and the plasmon peak shift follows the thermal expansion coefficient, as previously reported by Mecklenburg et al.¹² In addition, we found that the cold-field emission electron source used in the JEOL ARM200CF provides a significantly better energy resolution compared to the plasmon EELS reported by Mecklenburg.¹² The recorded plasmon peak for Si also exhibits a significantly better signal to noise ratio compared to previously published data. We attributed this to the low-noise CMOS camera on the Gatan Continuum GIF spectrometer that has been optimized for low signal, in situ data acquisition. Finally, we find that the effect of beam-induced heating at 200 kV is significantly lower than what was previously reported.¹² We do not anticipate beam induced heating at 200 kV primary electron energy and 60 pA beam current to affect our local temperature measurements due to the high rate of heat conduction relative to the incident beam power as the electron probe is scanning across the sample.

Using the in situ heating capabilities of the Protochips Aduro double tilt holder, plasmon peak electron energy loss spectra were acquired from the Si sample every 100 °C between room temperature and 900 °C. At each sample temperature, over 500 low-loss spectra were acquired with a dispersion of 0.05 eV/channel. The dwell time per spectrum was 0.5 µs, and the full-width-half-max (FWHM) of the zero-loss peak was measured to be 350 meV. It was found that averaging the spectra of too many pixels without prior alignment of the energy loss scales causes a blurring of the plasmon peak since the zero-loss (or the entire energy scale) shifts as a function of electron beam position. Since the precise calibration of the energy scale is important, the zero-loss peak position was calibrated for each spectrum prior to determining the plasmon peak position at each pixel.

Initially, non-negative matrix factorization (NNMF) was performed for each sub-pixel aligned spectrum to decompose it into three components. For this, the MALSpy package developed by Shiga et al.^13,14 was used. An example of decomposed spectra at 100 °C is shown in Fig. 2(a). For the example shown here, two peaks are identified in addition to the zero-loss peak. Specifically, we find a peak at (16.861 ± 0.004) eV, as well as an additional peak around 22 eV, potentially due to the presence of a thin layer of SiO_x or carbon on the surface of the Si samples. However, it is important to note here that the analysis of the main plasmon peak (shown as component 2) was not affected by this analysis.

The component corresponding to the plasmon peak was fitted using a Lorentzian function in the energy range of 14–20 eV to obtain the plasmon energy as a function of temperature, as shown in Fig. 2(c). The energy resolution was 4 meV, which translated into a temperature uncertainty of 28 °C. Although NNMF was able to decompose the initial spectrum into its constituent components successfully for most of the dataset and produce a roughly linear dependance of plasmon energy with temperature, it failed for some of the data [Fig. 2(b)] resulting in the outlier values of plasmon energy, forcing us to explore other methods.

Our main analysis focused on identifying the centers of the zero-loss and bulk plasmon peaks to high precision, using curve fitting and other statistical measures to determine the peak centers. Since the spectra contain multiple peaks, constraints on the peak location are needed to localize the peaks and constrain either fits or statistical analysis to a relevant range. This was performed using an iterative procedure to locate the local maxima, local minima, and peak full-width-half-max (FWHM) of the spectra for each pixel of the spectral images. The local maxima provide a measure of the mode of the distributions, which is strongly affected by Poisson noise yet can be used to determine a range of fit values and a suggestion seed parameter for curve fitting.

A study of different curve fitting and center finding methods was then performed in pursuit of a measure of the curve centers with low χ² (indicating a good quality curve fit) and high consistency across pixels of the spectral image. This included both identifying curve types and the best range to perform the fits. Center measures considered for each peak were median, mean, and mode of the distributions as well as fitting with the following curve types: Gaussian, skew Gaussian, Cauchy, Voigt, Landau, sum of two Landaus (bi-Landau), Gumbel, hyperbolic secant, logistic, Johnson’s S_U, and Fisher. Log-likelihood fits were performed using the MINUIT curve fitting package, driven by the MIGRAD¹⁵ minimization engine.¹⁶ MINUIT reports fit χ² but not log-likelihood, so χ² was used as a proxy for understanding the fit quality.

The spectral data values consist of counts of electrons detected in various energy bins, and, therefore, the bin values have Poisson uncertainties. Poisson uncertainties are approximated using the BaBar experiment’s approximation method,¹⁷ which provides an adequate approximation all the way down to n = 0. There are at least 10³ electrons per spectral bin, making this an excellent approximation that also provides a degree of insurance in the case of spectra with low statistics. From these uncertainties, the uncertainties are obtained on the fit parameters from MINUIT.

The results of various fits are discussed in detail in the supplementary material. The results were broadly consistent, with agreement and center parameter uncertainty improving among them as χ² improved. For the zero-loss peak, the best centering results were obtained from fits with a Fisher-Z distribution, closely followed by the median, which is far less computationally expensive. It is important to note that this distribution best captured the features of the zero-loss peak, which are formed from a convolution of the energy distribution of the emission source (Fowler–Nordheim), a point spread function from the spectrometer, and a narrow Gaussian representing the drift of the high voltage supply. The bulk plasmon peak was best fit using Johnson’s S_U distribution constrained to a narrow range around the mode. In this case, the distribution is best described using the Drude model, which is adequate for metals and heavily doped semiconductors, while the Drude–Lorentz model is commonly employed for Si. In testing, Cauchy and Voigt distributions gave nearly identical results but with higher χ². Notably, the sum of two Landau distributions tended to not converge or had a very poor shape fit.

Zero-loss and bulk plasmon peaks were fitted for every pixel in the spectral images, resulting in maps of the fitted energy difference (taken from different regions of the same particle), as well as distributions of the fitted energy differences [see Figs. 3(a) and 3(b), respectively]. These maps demonstrated that the fitting procedure could be performed rapidly and reliably (see the supplementary material). They also revealed variability in the energy difference, around 50–60 meV (0.3%), as the results of Poisson noise of the spectra due to the limited luminosity of the electron beam on the sample. Thus, adequate temperature determination requires locating the centers of the peaks with a precision of about 300 times better than what is achievable using the spectrum of a single pixel with the present luminosity.

Peak centering can be improved using greater statistics in two ways: first, by increasing the exposure time at each pixel and second, by combining the spectra of all the pixels in a spectral image. Spectral images in the current dataset include up to 43 000 pixels, which should naively offer a factor of 200 improvement, and roughly achieving the desired temperature resolution. Since an increased pixel exposure time can lead to increased beam damage or beam induced heating for samples other than Si, the second approach is preferred for most experimental setups.

Due to drift in the energy scale, the spectra cannot be simply added. Rather, the zero-loss peak is used to calibrate the energy scale for each spectrum. The spectrum is then shifted accordingly to keep the median zero-loss peak energy at zero. We shift spectra in energy in order to correct for calibration drift and manifest by a non-zero center of the zero-loss peak. Spectral data come in the form of histograms of electron counts with a set energy binning. Adding spectral histograms with sub-bin horizontal shifts cannot be done simply due to binning mismatches. To overcome this, the spectral histograms are re-binned, subdividing each bin into 50 bins, so that the new bin energy resolution is 1 meV. This eases the potential binning mismatch to an acceptable level. Distributing the original bin content uniformly among the sub-bins tends to produce spectra with a stair-step behavior. To solve this, the original bin content is then distributed into the new smaller bins using quadratic interpolation. This interpolated the new bin content using a quadratic curve, defined using the old bin’s integral and intersecting two points at either edge of the original bin, which are the midpoints between the bin content of that bin and the corresponding neighboring bin. The new bin content is the integral of this quadratic in the new bin’s region. This resulted in a smoothed stacked spectrum. The point of stacking spectra is to gain statistical power that would not be gained by fitting the spectra of individual pixels and using that to form numerical corrections. Variations in the estimate of the zero-loss peak center slightly widened the peaks in the summed spectra but did not significantly influence the center peak energy since they average out. Our quadratic sub-bin interpolation method provided a means of shifting the spectra energy, which implements the energy correction so that the spectra can be stacked smoothly and in a way that leverages the combined statistics. Any systematic error from the zero-loss peak center finding and resulting shifting was calibrated out at a later step.

Spectra used for the combined spectrum were hand-selected to exclude edge effects of the sample, which were revealed as increased variance on the energy difference maps. An example stacked spectrum is shown in Fig. 4.

The resulting stacked spectra have the same major features as the spectra of the individual pixels, but with greatly reduced noise. The same iterative characterization procedure was used on the stacked spectra as on the individual spectra. Then, the zero-loss peaks were fitted with a Fisher-Z distribution and the bulk plasmon peaks were fitted with Johnson’s S_U distribution. Merging every spectrum in the image removes our ability to understand the variability in the resulting energy difference by examining the population of results. For the sake of examining the effectiveness of high statistics stacked histograms, a spectral image is divided into four strips. A stacked spectrum was made for each strip (each with about ¼ of the statistics of the whole image), and the standard deviation of these four results is then considered to understand the variability of the calculations. A workflow diagram is included in the supplementary material indicating the entire process to arrive at this result. Data processing would commence immediately after a scan is performed and processed on board the microscope’s computer. Once a spectral image is acquired, an operator would review the map of plasmon peak positions to select a rectangular region for use in a stacked-spectrum analysis. This ensures the temperature measurement is not skewed by sample edge effects and provides an opportunity to detect sample contamination. Then, a second software script stacks the spectra and analyzes the plasmon peak position of the stacked spectra, reporting a high precision measure of the sample temperature.

For example, with the 700 °C sample (as shown in Fig. 3), the calculated energy differences for the four strips were 15.3733, 15.3701, 15.3723, and 15.3725 eV. The standard deviation of these results was 1.3 meV, a 50-fold improvement over results using the spectra of individual pixels. Fitting the stack of all four strips at once should yield an additional twofold improvement in resolution. The sample used was a particularly small spectral image. Using larger area images can easily give an additional 4× greater statistics, for an additional 2× greater precision.

The complete stacked spectra for each temperature point for two different datasets (both taken from a Si sample) were computed and fitted (see Table I). The energy differences vs temperature trends are shown in Fig. 5. It is important to note here that the precision with which the plasmon peak position is determined far exceeds the energy resolution of the cold-field emission source, which is measured to be 350 meV. Linear least-squares fits of the temperature dependence in Fig. 5 were performed using weights inversely proportional to the square root of the energy uncertainty.

As can be seen in Fig. 5, the two datasets demonstrate similar slopes of the plasmon peak shift as a function of temperature. However, there appears to be a constant offset in the plasmon peak energy regardless of sample temperature. This could be due to surface layers on the Si particles (such as silicon oxide), which could create a secondary plasmon peak that could weigh in on the bulk Si plasmon peak or slight variations in the dispersion calibration between the two experiments, caused by a change in focus. It is, therefore, important to determine the plasmon peak position of the sample at a known temperature prior to performing any in situ heating experiment. This initial calibration step where the plasmon peak energy is measured at a known temperature will ensure that the plasmon peak shift approach can accurately determine the sample temperature using the slope determined here. It is interesting to note that this effect was not observed previously, potentially due to the significantly lower accuracy in determining the plasmon peak position in earlier studies. Future studies will investigate the effects of improving the energy resolution, using, for example, a monochromated electron source, on the accuracy of the plasmon peak position measurements, as well as the reproducibility of the absolute plasmon peak energy using other sample materials.

In conclusion, using stacked spectra with interpolated sub-binning provided a greatly improved bulk plasmon peak energy resolution of 6 meV, which corresponds to a temperature precision of 50 °C. The zero-loss peak was best approximated by a Fisher-Z distribution but can be well centered using its median. The bulk plasmon peak was best approximated and centered by Johnson’s S_U distribution. While it was not directly demonstrated that these measurements are performed with a spatial resolution of 2 nm, previous work on MoS₂ by Hu et al. has shown that the localization of the plasmon peak signal in the energy range considered here is as low as 1.8 nm. Although variations of internal field due to impurities and defects in the semiconductor have an effect on the local electron density in Si and the plasmon peak energy, since the sample is a stable particle without these defects, on average such effects should be negligible. Considering that the electron probe size in these experiments is of the order of 1 Å, the spatial resolution of the reported temperature measurements will follow the delocalization of the plasmon peak signal and has been measured at the edge of a Si sample.

See the supplementary material for additional peak fitting and temperature mapping approaches.

B.S. and R.K. acknowledge support from the National Science Foundation (Grant No. DMR-1831406). The work at Sivananthan Laboratories was supported by the U.S. Department of Defense under Contract No. DMEA-0006. The acquisition of the UIC JEOL JEM-ARM200CF was supported by the NSF MRI-R² Grant (No. DMR-0959470). The acquisition of the Gatan Continuum spectrometer was supported by a grant from the National Science Foundation MRI program (Grant No. DMR-1626065). The support from the UIC Research Resources Center (RRC), in particular F. Shi, is acknowledged.