Thermometry of Silicon Nanoparticles

TL;DR

This study demonstrates that silicon nanoparticles can serve as high-resolution nanothermometers by measuring their volume plasmon energy, providing a promising method for temperature mapping in nanoscale devices.

Contribution

The paper introduces a novel method for nanoscale temperature measurement using silicon nanoparticles' volume plasmon energy in STEM-EELS.

Findings

Plasmon energy varies predictably with temperature in silicon nanoparticles.

Measured nanoparticle temperatures agree within 5% of macroscopic chip thermometers.

Silicon nanoparticles can act as in situ nanothermometers in electron microscopy.

Abstract

Current thermometry techniques lack the spatial resolution required to see the temperature gradients in typical, highly-scaled modern transistors. As a step toward addressing this problem, we have measured the temperature dependence of the volume plasmon energy in silicon nanoparticles from room temperature to 1250$^\circ$C, using a chip-style heating sample holder in a scanning transmission electron microscope (STEM) equipped with electron energy loss spectroscopy (EELS). The plasmon energy changes as expected for an electron gas subject to the thermal expansion of silicon. Reversing this reasoning, we find that measurements of the plasmon energy provide an independent measure of the nanoparticle temperature consistent with that of the heater chip's macroscopic heater/thermometer to within the 5\% accuracy of the chip thermometer's calibration. Thus silicon has the potential to provide its own, high-spatial-resolution thermometric readout signal via measurements of its volume plasmon energy. Furthermore, nanoparticles in general can serve as convenient nanothermometers for \emph{in situ} electron microscopy experiments.

Figures

Equations

$$R\simeq -\frac{3}{2}\left(\frac{l-l_0}{l_0}\right)\simeq -\frac{3}{2}{\int }_{T_0}^T\alpha (T^{\prime })d{T}^{\prime }.$$

$$\begin{gathered} \int \alpha (T^{\prime })d{T}^{\prime }=(1313.41e^{-0.00588T^{\prime }} \\ +3.725T^{\prime }+0.0002774T^{\prime 2})\times 10^{-6}. \end{gathered}$$

Full text

Suggested keywords

Silicon, as the primary constituent of most semiconductor devices, is perhaps the most important and most studied material in modern technology. Silicon’s thermal properties are particularly relevant to the design of devices such as microprocessors, since heat transport is frequently a performance-limiting factor in highly-scaled and high-power density electronics2016Heiderhoff ; 2014Cahill . The current semiconductor processing node, designated with the scale label ‘10-nm’, produces devices with features that are even smaller (in the vertical direction) and multiple, non-trivial interfaces.

As such devices approach the atomic limit, classical, continuum thermal transport theory breaks down2016Heiderhoff ; 1938Casimir . Improved designs for next-generation microprocessors, memory, and opto-electronics will come with a better understanding of thermal transport at these small length scales. To gain this understanding, thermometry techniques with $\lesssim 1\,\mu$m spatial resolution are required. However, no currently available technique can resolve the thermal gradients within the smallest modern transistors.

The temperature mapping techniques of most relevance to microelectronics are generally either optical or scanning-probe2016Heiderhoff ; 2015KimMicroscale . Optical examples include micro-Raman and thermoreflectance 2014Maize ; 2008Beechem ; 1993Epperlein , both of which are diffraction-limited to 500–1000 nm spatial resolution. Mechanical scanning thermal microscopy (SThM) techniques do better by rastering a sharp tip across a sample1999Majumdar ; 2015Gomes . They extract a thermometric signal by analyzing a tip-embedded thermometer1986Williams ; 1998Mills ; 2012KimUltra , the heat transfer between the tip and sample1986Williams ; 2016MengesTemperature , or the thermal expansion of the sample1998Majumdar .

We are developing a temperature mapping technique, plasmon energy expansion thermometry (PEET) 2015MecklenburgScience , with the capability for $\lesssim 10$ nm spatial resolution inside a thermometric material. The technique is scanning, but, unlike most scanning techniques, it is non-contact in the sense that the heat transfer between the probe and the sample is negligible. PEET infers a material’s temperature from measurements of its volume plasmon energy. The plasmon energy, $E_{p}=\hbar\sqrt{e^{2}n/\epsilon_{0}m}$ in the electron gas model (where $e$ and $m$ are the electronic charge and mass respectively), gives the valence electron density $n$. The electron density in turn indicates the temperature via the material’s coefficient of thermal expansion (CTE), which is determined separately. In a scanning transmission electron microscope (STEM) equipped with electron energy loss spectroscopy (EELS), $E_{p}$ can be mapped with sufficiently high spatial resolution to observe the density changes at grain boundaries2015MecklenburgScience . Thus temperature mapping with resolution approaching the atomic limit can be achieved.

In this communication we share two main results. First, we have measured the temperature dependence of silicon’s bulk plasmon energy, which has not been reported previously. This measurement is a necessary step toward the goal of applying PEET to determine the temperature gradients within an operating transistor, using the transistor’s own silicon as the thermometric read-out material.

Second, we show how nanoparticles can serve as fiducial thermometers for in situ TEM experiments. A compact PEET thermometer in or near the TEM field of view (FOV) can provide an improved temperature determination without the complications of external wiring or additional thermal loading. Nanoparticles are small and can be easily dispersed. With a variety of nanoparticles commercially available (e.g. silicon, aluminum, indium, and tungsten), the specific type can be chosen to best meet the experiment’s requirements (e.g. operating temperature range and chemical compability). Similar ideas for fiducial thermometers have been implemented previously in an optical context, for instance with nitrogen-vacancy centers in diamond 2013Kucsko or lanthanide ion-doped nanocrystals2010Vetrone . The PEET approach allows implementation in a TEM, and without requiring any additional hardware more exotic than a standard EELS spectrometer. In a sense each nanoparticle serves as an expansion thermometer in the style of Fahrenheit’s mercury-in-glass design, but with a construction that is much simpler, cheaper, and smaller (vs., for example, the approach of Ref. 2002Gao, ).

To accomplish these two goals we measured the plasmon energy in silicon nanoparticles as a function of temperature using a chip-style TEM-sample heating holder (DENS Solutions Wildfire S3, Fig. 1 top). Relative to furnace-type heating holders, this type of holder equilibrates faster, drifts less, consumes less power, and provides more accurate temperature read-out2016MeleMEMS . As shown in Fig. 1 (top), each chip had a $300\,\mu$m$\times 300\,\mu$m, SiNx-encapsulated, spiral Joule heater/thermometer atop a silicon nitride membrane with nearby $<20$ nm-thick, $100\,\mu$m2 electron-transparent windows 2016MeleMEMS ; 2012MeleMolybdenum . The specifications for these chips list a guaranteed temperature range of room temperature to 1,300∘C, a maximum temperature of 1,500∘C, achievable temperature change rates of 200∘C/ms, and settling times of $<2$ s. At 1250∘C (1523 K) the heater drew 6.5 mA at 2.7 V, dissipating 18 mW.

The window temperature was determined via a four-wire measurement of the heater resistance, which had been calibrated vs. temperature by the manufacturer to an accuracy of $5\%$. By design the chip featured a temperature gradient, with the temperatures of different windows varying by more than $15$% relative to the difference from ambient at a given heater power. The temperature calibration was only accurate for the windows nearest the center of the heater.

Samples were prepared by dropcasting silicon nanoparticles from $1\mu$l of an ethanol solution onto a chip (Figure 1). According to the vendor (SkySpring Nanomaterials), the nanoparticles were manufactured by chemical vapor deposition (CVD), had 99% purity, and a 100 nm average particle size.

Generally speaking, 100 nm is roughly one mean-free path for plasmon production, so nanoparticles of this thickness are preferred for PEET. Particle size-dependent effects are a potential source of systematic errors, but these only appear in much smaller particles. For instance, the bulk plasmon resonance has been observed to change in silicon nanoparticles with diameters $\lesssim 10$ nm1992Mitome . Similarly, size-dependent melting effects, which likely would have concomitant effects on the CTE, are only seen in particles with diameters $\lesssim 15$ nm2006Hirasawa .

EELS spectrum images of silicon nanoparticles at different temperatures were acquired in a JEOL JEM-2100F TEM equipped with a Gatan Quantum SE GIF. The microscope was operated at 80 kV with a beam current of 100 pA, a 0.5 nm probe, and a convergence semi-angle of 12 mrads. (The 80 kV accelerating voltage enhances the plasmon production rate by roughly a factor of two relative to the rate at 200 kV.) The spectrometer collected 64 spectra per second with a semi-collection angle of 20 mrad, a 2.5 mm entrance aperture, a dispersion of 25 meV/channel, and $26\times$ vertical binning.

In each spectrum the silicon plasmon energy was determined by fitting the zero loss peak (ZLP), the silicon nitride plasmon peak, and the silicon plasmon peak, as shown in Fig. 1. Fitting the ZLP with a Gaussian function in a fit window of full-width 0.85 $e$V centered around the spectrum maximum returned a full-width at half-maximum (FWHM) of $0.76\pm 0.01$ $e$V. In a region of interest (ROI) bare of any material but the electron-transparent membrane, the silicon nitride plasmon peak was fit with a Lorentzian function in a fit window extending from 19.5 to 26.5 $e$V relative to the ZLP center. The peak center and width from this fit were then fixed, and a two-Lorentzian fit in the window 13.5–25.0 $e$V was performed over the entire FOV. This fit had four free parameters: the amplitude of the silicon nitride peak, and the amplitude, center, and width of the silicon peak. The difference between the silicon peak center and the ZLP center is taken to be the silicon plasmon energy2015MecklenburgScience .

Typical data extracted from a 75 nm-diameter silicon nanoparticle are shown in Figure 2. The TEM image with its diffraction contrast reveals the most detailed structural information, showing the nanoparticle’s 8–10 nm-thick oxide coating and two distinct crystal grains. The high-angle annular dark field STEM image shows the grains only, while the plasmon energy maps show none of these features and are basically uniform. Including the fit of the silicon nitride peak in the data analysis is necessary to achieve this uniformity; without it, the plasmon energies within 10 nm of the nanoparticle edge appear to be systematically higher than those in the interior (the low-amplitude silicon plasmon gets pulled higher by the slope in the silicon nitride background). Histograms of the silicon plasmon energies are well-fit by Gaussian distributions.

Converting these plasmon energy differences into temperature differences requires integrating silicon’s linear CTE $\alpha(T)\equiv(1/l)(dl/dT)$, where $l$ is a length in the material1975Slack . The plasmon energies $E_{p}$ at an unknown temperature $T$ and the known reference temperature $T_{0}$ are related to the CTE by the ratio $R\equiv[{E_{p}(T)-E_{p}(T_{0})}]/{E_{p}(T_{0})}$, where

[TABLE]

Okada and Tokumaru1984Okada provide an empirical formula for the CTE, valid between 120 and 1500 K, which integrated gives (for $T^{\prime}$ in kelvin)

[TABLE]

At $T=300$, 600, 900, 1200, and 1500 K, this expression gives the CTE $\alpha=2.57$, 3.83, 4.19, 4.38, and 4.56 (all $\times 10^{-6}$) respectively, which is to say that silicon’s CTE is consistently increasing with temperature, though more slowly after a shoulder in the neighborhood of 700 K. (Regarding PEET’s sensitivity in silicon, it is unfortunate that, compared to that of other materials, silicon’s high-temperature CTE is small, smaller even than that of diamond1975Slack .) In the range 298 to 1500 K the integrated CTE $f(T)\equiv\int_{T_{0}}^{T}\alpha(T^{\prime})dT^{\prime}\simeq\alpha_{1}\Delta T+\alpha_{2}\Delta T^{2}$ ranges from 0 to $4.85\times 10^{-3}$ and is approximated with the coefficients $\alpha_{1}=3.25\times 10^{-6}$ K*-1* and $\alpha_{2}=6.84\times 10^{-10}$ K*-2*, where $\Delta T\equiv T-T_{0}$ and $T_{0}=298$ K. (For comparison, in aluminum the corresponding numbers2015MecklenburgScience are $\alpha_{1}=23.5\times 10^{-6}$ K*-1* and $\alpha_{2}=89\times 10^{-10}$ K*-2* in the range 25 to 650∘C.) However, while the quadratic approximation to Eq. 2 is good to better than $5\times 10^{-5}$ through the whole range, the relative errors are as large as 27% near room temperature where $f(T)$ is small. Since for many applications the lower end of the range will be the most interesting region, we invert $f(T)$ numerically to find temperatures.

Roughly speaking, silicon’s plasmon shifts $-0.1$ m$e$V/K. Even a 1200 K temperature change produces a peak shift that is barely discernible by eye (see Supplementary Information). For the data in Fig. 2 the measured standard deviation of the single-pixel plasmon energies is 20 m$e$V, which corresponds to a 200 K shift. With such uncertainties, meaningful temperatures cannot be calculated at the single-pixel level; the integrated CTE $f(T)$ is valid over only a limited temperature range. Furthermore, $f(T)$ is non-linear. Thus the operations of computing the temperature from the plasmon energies and averaging over some ROI do not commute — the averaging must be done first. To suppress systematics arising from a weak silicon plasmon signal, we compute the mean plasmon energy $\overline{E_{p}}(T)$ for an ROI in the interior of the nanoparticle at the unknown temperature $T$. Finding the corresponding mean energy $\overline{E_{p}}(T_{0})$ in a similar ROI in a map acquired at the reference temperature $T_{0}$, we calculate $-2\overline{R}/3=f(T)$ and then invert to find the temperature.

The nanoparticle plasmon energy maps shown in Fig. 2 represent two data points in a temperature scan designed to demonstrate the utility of such nanoparticles as nanothermometers. (For a more comprehensive view of the entire dataset see the Supplementary Information.) This particular scan consisted of two room temperature data points, followed by two ramps down from high temperature to room temperature in 100∘C steps (according to temperature as determined by the holder), with the first ramp beginning at 1200∘C and the second at 1250∘C. Interleaving two ramps with 100∘C steps, as opposed performing a single ramp with 50∘C steps, gives an important indication of the stability of the nanoparticles with respect to thermal cycling and repeated STEM imaging. For maximum utility as nanothermometers, the nanoparticles should be robust to both perturbations.

The results of this scan are shown in Fig. 3. The plasmon energy versus temperature plot shows a total shift in the plasmon energy of 120 m$e$V — a mere 3% of the peak’s 3.7 $e$V FWHM — across the entire measured range between room temperature and 1523 K, highlighting the necessity of using curve-fitting to extract the thermometric signal. The plasmon energies determined in the two interleaved temperature ramps are themselves gratifyingly interleaved, showing no significant systematic shift between the first and second ramps. To estimate the error in the PEET determination, we require that the $\chi^{2}$ per degree of freedom in the linear fit of Fig. 3 be unity, which gives a PEET error of 30 K. (Standard error propagation applied to the invertible, quadratic approximation to $f(T)$ gives errors that are too small by a factor of 8 for reasons that are not presently understood.) The four separate room-temperature plasmon energy measurements have a standard deviation of 2 m$e$V, an energy shift which is equivalent to 20 K. This value gives an additional measure of the error in PEET’s temperature determination that is of the same order as the first. Comparing the temperatures derived from resistance measurements of the chip’s 300 $\mu$m heater/thermometer to those derived from PEET applied to the 75 nm silicon nanoparticle, we find that they agree at the 5% level, the stated accuracy of the chip’s temperature calibration.

While applying PEET to nanoparticles we encountered various pitfalls, but the problems were usually easily recognized and even quantifiable. A change in the experimental parameters between the first and the last ambient-temperature measurements warns of a possible systematic. (Of course taking both measurements, and more within an experiment if possible, is a necessary part of a sound experimental protocol.) In cases with independent thermometers, like the one described here, this warning might be unrelated to PEET and concern the other thermometer instead. For instance, a change in the zero-power resistance of a heater/thermometer indicates that it has been damaged, either through use or through processing (e.g. plasma cleaning), and that its temperature calibration can no longer be considered reliable. In other cases the problem concerns PEET: the nanoparticle might change, either in its morphology, its plasmon energy, or both. We have seen evidence of alloying or doping within a heating experiment, and also signs of beam-induced damage. Aberration-corrected microscopes are particularly hazardous in the latter regard, for a total beam current that is harmless in an uncorrected probe can, concentrated, radically transform a nanoparticle, making it useless for thermometry. Whatever the source of the change, the shift in a nanoparticle’s plasmon energy under nominally identical conditions gives a quantitative measure of the magnitude of a potential systematic.