As device length scales trend downward, small feature sizes and steep temperature gradients require thermometers with increasingly fine spatial resolution in order to capture the true peak temperature. Here, we develop analytical expressions for the true and measured temperature rises as a function of thermometer size for Gaussian, disk-shaped, and rectangular surface heat sources. We find that even a thermometer the same size as the hotspot can underestimate the true peak temperature rise by more than 15%, and this error frequently exceeds 75% and can approach 90% for certain geometries when the thermometer is ten times larger than the measured hotspot. We show that a thermometer with resolution approximately two times smaller than the hotspot size is required to measure the peak temperature rise with less than 5% error for several common hotspot geometries. We also experimentally demonstrate that a 50 × 50 × 50 nm^{3} individual upconverting NaYF_{4}:Yb^{3+},Er^{3+} nanoparticle thermometer captures the peak temperature rise due to laser heating more accurately than conventional diffraction limited optical techniques that our modeling results show would underestimate this value. In contrast to apparent self-heating effects that spuriously increase the nanoparticle thermometry signal at high excitation intensities, we measure true laser heating, as confirmed by comparing measurements on glass and diamond substrates.

## I. INTRODUCTION

The proliferation of micro- and nanotechnology has greatly increased the number of situations where heat transfer at small length scales is important. Examples include thermal management of microelectronic and optoelectronic devices, photothermal cancer therapies, laser-assisted manufacturing, and new data storage technologies.^{1,2} For some applications, any heating is undesirable, while in other cases the goal is to achieve a carefully controlled temperature rise.^{3–6} In both situations, monitoring the peak temperature during operation is crucial to either avoid or ensure crossing a critical threshold such as a melting point,^{7} a phase transition,^{8} or the onset of cell death.^{9} Although micro- and nanoscale heat sources are now common, the resulting temperature profiles are frequently probed by thermometers of characteristic dimension comparable to or larger than the hotspot, in some cases by orders of magnitude. Practical considerations often favor the use of non-contact optical techniques like infrared thermography and thermoreflectance, with diffraction limited spatial resolution on the order of 10 *μ*m and hundreds of nm, respectively.

It has been recognized previously that a coarse thermometer will underestimate the peak temperature rise in micro- or nanoscale hotspots,^{7,10,11} but neither modeling nor experiments have been used to systematically quantify these deviations. Here, we first provide analytical expressions for the true temperature profiles resulting from three representative surface heat source geometries on semi-infinite substrates as shown in Fig. 1, namely, a Gaussian heat source of 1/e^{2} radius *r*_{laser}, generalized to account for multilayer substrates in order to facilitate comparison with our experimental results, a disk-shaped heat source of radius *r*_{disk}, and a rectangular heat source of width 2*w* and length 2*l*. We then develop analytical expressions for the surface temperature profile measured using a thermometer with a Gaussian averaging profile and a 1/e^{2} radius σ for each of the three surface heat source geometries. This Gaussian averaging kernel is selected to be generically representative of diffraction limited optical methods such as the aforementioned infrared and thermoreflectance techniques. We find that the peak hotspot temperature can be severely underestimated when the characteristic thermometer dimension surpasses that of the hotspot, with errors nearing 90% for certain geometries when the thermometer is an order of magnitude larger than the hotspot. We also calculate the temperature profile measured by a thermometer with a 25 nm radius and a spatially uniform response, which represents using the temperature-dependent luminescence of an individual nanoparticle as a thermometer. In this case, the measured temperature profile is effectively indistinguishable from the true temperature profile resulting from laser heating by a diffraction limited Gaussian beam. We then experimentally verify this result by using a single 50 × 50 × 50 nm^{3} upconverting NaYF_{4}:Yb^{3+},Er^{3+} nanoparticle to measure the peak temperature in a laser heated spot. The results are consistent with our calculations for thermometer of radius *r*_{NP} = 25 nm and a reasonable range of uncertainty in our experimental parameters.

## II. MODELING THE TRUE AND MEASURED SURFACE TEMPERATURE RISE FOR REPRESENTATIVE SURFACE HEAT SOURCE GEOMETRIES

### A. Gaussian surface heat source

We begin by considering a surface heat source with a Gaussian intensity distribution [Fig. 1(a)]. Experimentally, this situation corresponds to laser heating of a sample with small optical penetration depth such that the incident laser light is effectively absorbed at the sample surface. The results presented here are thus inapplicable for larger optical penetration depths, but the analysis can be modified to account for such effects.^{12,13} A low optical penetration depth of ∼10 nm can be achieved by depositing a thin, optically absorbing metal film onto the sample surface, which is common in time- and frequency-domain thermoreflectance (TDTR and FDTR) measurements and is also the approach we later employ in our experiments. We therefore generalize our Gaussian heat source calculations for a multilayer system. Because our experiments use a continuous wave laser, we consider only the steady-state temperature rise, but the transient thermal response due to pulsed and modulated laser heating has been studied extensively in the context of TDTR and FDTR.^{14–16} Here, we adapt solutions for heat conduction in a layered system including radial effects that have been described thoroughly in the TDTR and FDTR literature.^{16–18} This class of axisymmetric problems can be solved using Hankel transforms.^{14,16} We solve for the steady-state temperature rise $\theta (r,z)=T(r,z)\u2212T\u221e$, where $T\u221e$ is the far-field sample temperature.

If the sample surface is surrounded by an insulating medium like air, the Hankel transforms of the temperature rise and heat flux at the top surface of an arbitrary layer *i* [$\theta ~surface,i(q)$ and $Q~surface.i(q)$, respectively, where *q* is the Hankel transform variable] are related to the Hankel transforms of the temperature and heat flux at the bottom of that same layer [$\theta ~bottom,i(q)$ and $Q~bottom,i(q)$, respectively] by

where *k*_{i} is the thermal conductivity (assumed to be isotropic) and *d*_{i} is the thickness of layer *i*. Considering Gaussian heating at the topmost surface of the sample, $Q~surface(q)$ can be expressed as

where $\alpha $ is absorptivity, *P* is laser power, and the product $\alpha P$ thus represents the absorbed laser power. The solution for multiple layers is obtained by multiplying the matrices for individual layers together,^{14} giving

where $Mn$ is the matrix corresponding to the bottom layer. Here, we neglect the effects of thermal boundary resistance between the layers, but such effects can be accounted for by including additional matrices that relate the temperature and heat flux at bottom surface of a given layer to the same quantities at the top surface of the layer below via a thermal boundary conductance. While the steady-state temperature rise resulting from continuous wave laser heating at the scales considered here is insensitive to thermal interface resistance between the metal transducer layer and the semi-infinite substrate, in the context of transient or modulated measurement techniques like TDTR and FDTR this parameter can be critical at higher modulation frequencies.^{19} To determine the unknown surface temperature, here we take the *n*th layer to be semi-infinite, in which case Eq. (3) yields $C~\theta ~surface(q)+D~Q~surface(q)=0$ and $\theta ~surface(q)$ is given by

By substituting the expression for $Q~surface$ given in Eq. (2) into Eq. (4) and taking the inverse Hankel transform, the surface temperature rise $\theta surface(r)$ can be expressed as

This integral can be evaluated numerically, and in practice, the upper bound can be set to $2/rlaser$ without loss of accuracy.^{14}

Next, we consider the temperature profile that will be measured by a Gaussian thermometer with a 1/e^{2} radius $\sigma $ given the true surface temperature profile $\theta surface(r)$. Mathematically, this situation is very similar to that of beam offset TDTR measurements used to measure anisotropic thermal properties, in which the temperature sensing (probe) laser beam is offset from the heating (pump) beam by a distance $x0$.^{16,20} In this case, some of the axial symmetry of the problem is lost, and the measured temperature rise as a function of offset distance $x0$ is given by

where $\theta surface(x2+y2)=\theta surface(r)$ is given by Eq. (5). While Eq. (6) can be evaluated numerically, doing so is computationally costly due to the required triple integral. Feser and Cahill^{20} showed that the axial symmetry of the problem can be preserved by recognizing that the intensity profile of a temperature sensing beam offset from the heating beam by a distance $x0$ can be equivalently represented as an axisymmetric ring-shaped profile concentrically aligned with the heating beam, with an intensity distribution given by

where $I0$ is the zeroth-order modified Bessel function of the first kind and *P*_{sense} is the absorbed power from the sensing laser beam. We assume that *P*_{sense} ≪ *αP* and thus any additional heating caused by the sensing beam is negligible. In thermoreflectance experiments, this assumption is typically valid because the probe wavelength and power are selected such that optical absorption is minimized and the resulting temperature rise is negligible. However, in cases where the probe-induced temperature rise cannot be ignored, this additional temperature rise can be calculated in the same manner used to determine the temperature rise caused by the pump. The total temperature rise is simply the sum of the pump- and probe-induced temperature rises because of the principle of superposition.^{19,21} Using a Taylor series expansion for $I0$, the Hankel transform of Eq. (7) can ultimately be expressed as

where the polynomial $ln(x)$ is defined recursively as

and the first term is given by $l0=\pi $. In practice, the first 40 terms of the summation are sufficient to calculate the Hankel transform if $x0\u22644\sigma $.^{20} $\theta measured(x0)$ can thus be expressed as

### B. Disk-shaped surface heat source

We now turn to our second surface heat source geometry, a disk-shaped heat source of radius *r*_{disk} with a constant heat flux $Qdisk$ [Fig. 1(b)]. Carslaw and Jaeger^{22} give the surface temperature rise in this situation as

where *k* is substrate thermal conductivity, $J0$ is the zeroth-order Bessel function of the first kind, and $J1$ is the first-order Bessel function of the first kind. The temperature measured by a Gaussian beam offset from the disk by a distance $x0$ can be calculated by substituting Eq. (11) into Eq. (6); similarly, we can again follow the method of Feser and Cahill^{20} and preserve the axial symmetry of the problem by expressing the offset Gaussian sensing beam as an equivalent axisymmetric ring-shaped profile. The Hankel transform of the surface temperature rise, $\theta ~surface$, for a disk-shaped surface heat source is given by

### C. Rectangular surface heat source

The final geometry we consider is a rectangular surface heat source of width 2*w* and length 2*l* on a semi-infinite substrate with a constant heat flux $Qrectangle$ [Fig. 1(c)]. The surface temperature rise $\theta surface(x,y)$ is obtained by solving the heat conduction equation in the Cartesian coordinate system assuming isotropic properties. In brief, this problem can be solved using a Green's function approach by combining Green's functions for a plane source in an infinite solid and a plane source in a semi-infinite solid and then taking the integral over times *t* ranging from $t=0$ to $t=\u221e$ to find the steady-state solution.^{23} The resulting expression for $\theta surface(x,y)$ is

where *k* is substrate thermal conductivity. Because the rectangular heat source itself is not axisymmetric, in this case, we can no longer preserve axial symmetry by representing the offset Gaussian probe as a ring-shaped beam concentrically aligned with the heat source. Therefore, the surface temperature rise measured by a Gaussian beam offset from the center of the rectangle by a distance $x0$ is given by

and the double integral in Eq. (14) must be evaluated numerically. We consider an offset only in the *x* direction, meaning that Eq. (14) represents the temperature measured along the line *y* = 0 as a function of the *x* direction offset $x0$.

Figure 2 shows the true normalized surface temperature profile $\theta surface(x)/\theta surface,max$ and the measured profiles for different values of $\sigma dchar$ for a Gaussian [Fig. 2(a)], disk [Fig. 2(b)], square [Fig. 2(c)], or rectangular [Fig. 2(d)] heat source, where $\sigma $ is the characteristic thermometer dimension (i.e., the 1/e^{2} Gaussian laser beam radius $\sigma $) and $dchar$ is that of the hotspot (*r*_{laser}, *r*_{disk}, or *w*, the latter being half the width of the square or rectangular heat sources). Because a rectangular heat source of width 2*w* and length 2*l* has two characteristic dimensions, Fig. 2(d) shows $\theta surface(x)/\theta surface,max$ for different values of $\sigma dchar=\sigma w$ and a fixed ratio of $lw=100$. The Gaussian heat source calculations are performed for a two-layer system consisting of a 50 nm optically absorbing layer on top of a semi-infinite substrate. The ratio of the thermal conductivity of the absorbing top layer to that of the substrate is $kabsksub=25$, typical of a metal film deposited on a low thermal conductivity substrate. Because the first 40 terms of the summation in Eq. (9) are sufficient to calculate the Hankel transform only if $x0\u22644\sigma $, when $|xdchar|>2$ for the $\sigma dchar=0.5$ cases shown in Figs. 2(a) and 2(b), we can no longer represent the offset sensing beam as an equivalent axisymmetric ring-shaped profile concentrically aligned with the heating beam. Instead, we use Eq. (6) to calculate the measured temperature profile. We also verify that Eqs. (6) and (10) give identical results when $x0\u22644\sigma $, and both methods can be applied (see supplementary material).

### D. Error in the measured temperature rise

That quantity that we are ultimately interested in is the percent error in the measured temperature rise, $\eta (x0)$, defined as

Although our analytical framework allows us to calculate the percent error $\eta $ at any spatial location, we now focus on the hotspot center [i.e., $\eta (x0=0)$] since that is where the maximum error will occur, which can be observed in Fig. 2. $\eta (x0=0)$, which we refer to as $\eta $ from here on, thus represents the percent error in the measured peak temperature rise. Because both the Gaussian and disk-shaped heat source geometries can be characterized with only a single geometric parameter, i.e., the 1/e^{2} Gaussian laser beam radius *r*_{laser} and the disk radius *r*_{disk}, respectively, in these cases $\eta $ can be expressed as a function of a single dimensionless parameter, $\sigma dchar$. Figure 3(a) shows $\eta (\sigma dchar)$ for both Gaussian and disk-shaped heat sources. The rectangular heat source of width 2*w* and length 2*l* has two characteristic dimensions and $\eta $ can therefore no longer be expressed as a function of a single dimensionless parameter. Figure 3(b) thus shows $\eta (\sigma dchar=\sigma w)$, with contours representing different values of the aspect ratio $lw$. The results in Figs. 3(a) and 3(b) demonstrate that even when $\sigma dchar=1$, $\eta $ can be as large as 16%, and such errors increase sharply as the thermometer becomes larger. For example, when $\sigma dchar=10$, the peak temperature rise is underestimated by 77% and 88% for the Gaussian and disk-shaped heat sources, respectively. For more complicated geometries or irregular hotspot shapes, we would again expect large values of $\eta $ for $\sigma dchar>1$. The shape of the full temperature profile will be unknown in such cases, highlighting the need for experimental thermometry techniques with nanoscale spatial resolution.

## III. EXPERIMENTAL MEASUREMENT OF THE PEAK TEMPERATURE RISE IN A LASER HEATED SPOT USING AN INDIVIDUAL UPCONVERTING NANOPARTICLE

### A. Upconverting nanoparticle (UCNP) background

To experimentally demonstrate how a thermometer with $\sigma dchar\u226a1$ can accurately measure the peak temperature of nanoscale hotspots, we use a thermometer with a 25 nm radius to measure the peak temperature rise in a laser heated spot, which corresponds to the Gaussian surface heat source of Fig. 1(a). Laser heating has wide-ranging applications, including additive manufacturing,^{6} photothermal therapies,^{3} and catalysis.^{24} Laser heating can also occur inadvertently in other scenarios, such as biological imaging^{25} and optical pumping,^{26} where it is considered a parasitic side effect. Accurately measuring the peak temperature rise caused by laser heating is thus important for improving thermal design and minimizing thermal damage. An attractive method for probing the peak temperature rise in micro- and nanoscale hotspots is to employ the temperature-dependent luminescence of an individual nanoparticle.^{27–29} By placing a single nanoparticle at the location of interest and measuring the temperature-dependent luminescence it emits, a far-field optical temperature readout can be obtained with minimal sample perturbation. While the diameter of the exciting laser beam remains diffraction limited, the thermal information comes only from the luminescence emitted by the nanoparticle. The spatial resolution of the single-point temperature measurement is thus governed by the nanoparticle size, which can be far smaller than the diffraction limited laser beam diameter.^{30} Lanthanide-doped upconverting nanoparticles (UCNPs), such as the popular NaYF_{4}:Yb^{3+},Er^{3+} composition, are frequently used for thermometry because of their excellent photostability, uniformity, tunability, and biocompatibility.^{31,32}

Here, we use individual 50 × 50 × 50 nm^{3} hexagonal NaYF_{4} nanorods doped with 20% Yb^{3+} and 2% Er^{3+} (Mesolight Inc.) to quantify the true peak temperature rise in a laser induced hotspot (see supplementary material for nanoparticle TEM images). NaYF_{4}:Yb^{3+},Er^{3+} is an upconverting system in which the Yb^{3+} ions absorb two 980 nm photons and then transfer this energy to a single Er^{3+} ion, exciting Er^{3+} to its ^{4}F_{7/2} state.^{33} Er^{3+} then non-radiatively decays to its ^{2}H_{11/2} and ^{4}S_{3/2} manifolds, whose close spacing gives rise to the temperature-dependent luminescence.^{27} From these manifolds, Er^{3+} returns to the ground state by emitting a single photon in the green wavelength range (∼515–565 nm). The temperature-dependent luminescence can be calibrated using the following relationship:^{34}

where *I*(*λ*) is the emission spectrum, Δ*E* is the energy difference between the ^{2}H_{11/2} and ^{4}S_{3/2} energy levels, *k*_{B} is the Boltzmann constant, *T* is temperature, and *A* is a constant based on the radiative transition rates from ^{2}H_{11/2} and ^{4}S_{3/2} to ^{4}I_{15/2}. The temperature-dependent luminescence intensity ratio *r* represents the emission intensity from ^{2}H_{11/2} relative to that from ^{4}S_{3/2}. This commonly employed approach is known as ratiometric thermometry.^{31,32}Fig. 4(a) shows *r*(*T*) for five individual 50 × 50 × 50 nm^{3} particles (see supplementary material for details). While the temperature range of our calibration extends only to 400 K, others have shown that *r*(*T*) data for NaYF_{4}:Yb^{3+},Er^{3+} UCNPs can be described by Eq. (16), specifically a linear fit to the transformed form $ln\u2061(r)=ln(A)\u2212(\Delta EkBT)$, with *r*^{2} > 0.99 up to temperatures as high as 900 K.^{35} We follow the convention of our previous work^{36} and some other single particle thermometry work^{28,37} by choosing *λ*_{1} = 513 nm, *λ*_{2} = 535 nm, and *λ*_{3} = 548 nm to exclude the peak centered at ∼556 nm originating from a non-thermal excited state-to-excited state transition known to occur at high excitation intensities. The inset of Fig. 4(a) shows representative emission spectra at two different temperatures. We image the nanoparticles using a custom-built scanning confocal microscope, and we follow the sample preparation and single particle identification protocols described in our previous work.^{27,36}

### B. Self-heating considerations

To study laser heating, we designed samples consisting of a 50 nm platinum (Pt) film evaporated on a portion of a 130 *μ*m thick borosilicate glass cover slip. These parameters are chosen to decouple the roles of optical absorption and thermal dissipation so as to achieve the highest temperature rise possible. Even a 50 nm Pt film is optically thick;^{38} consequently, the higher absorptivity of the Pt film (*α* ≈ 0.3^{39}) will govern the absorption of the incident 980 nm laser light, which also serves as the excitation source for the UCNP thermometer. Placing a UCNP on a metal vs glass surface may change the overall UCNP emission intensity but is not expected to change the ratiometric thermometry signal apart from surface heating that results due to the higher absorptivity metal. Conversely, borosilicate glass is highly transparent at 980 nm (∼1% absorptance for a standard 130 *μ*m cover slip). The low thermal conductivity glass substrate (*k*_{glass} ≈ 1 W m^{−1} K^{−1}) will govern heat dissipation. We spin coat dilute nanoparticle-containing solutions onto the samples, resulting in a sparse distribution of primarily single nanoparticles on the sample surface.^{27,36} In the context of our experimental system, using a laser to generate a hotspot with a known temperature profile is advantageous because our piezo-controlled nanopositioning stage allows us to precisely position the sample with respect to the laser heat source. This capability contrasts with other common heater structures such as microfabricated metal lines where placing a single nanoparticle at the precise location of interest is challenging. Furthermore, many metal structures will absorb the laser used to excite the nanoparticle luminescence, resulting in a non-negligible laser-induced temperature rise even if the laser is not intentionally used as a heat source.

One challenge for studying laser heating with UCNPs is the apparent self-heating effect that we recently reported.^{36} At high laser excitation intensities, radiative and non-radiative relaxation from highly excited states in Er^{3+} distorts the Boltzmann population distribution such that the luminescence intensity ratio *r* increases, even though the true nanoparticle temperature remains unchanged. Here, we briefly illustrate this phenomenon by considering a nanoparticle on an uncoated glass substrate; extensive details are given in Ref. 36. First, we estimate the temperature rise of both the nanoparticle and the glass substrate due to the incident laser light. To make a highly conservative estimate for the nanoparticle temperature rise, i.e., an upper bound on $\theta NP$, we neglect all heat dissipation into the substrate and consider only conduction heat loss through the air surrounding the nanoparticle. Even for a high excitation intensity of 5 × 10^{5} W cm^{−2}, near the limit of our experimentally accessible range, we estimate an upper bound of $\theta NP\u22484K$ (see supplementary material for details). Next, we confirm that direct laser absorption by the glass substrate also results in a negligible temperature rise. We estimate an upper bound on the substrate temperature rise $(\theta substrate)$ by approximating heat dissipation in the glass as radial conduction through a cylindrical wall, since the low absorptance of glass (∼1% at 980 nm for a standard 130 *μ*m glass cover slip) implies that the incident light will be absorbed throughout the thickness of the glass, rather than primarily at the surface. This calculation predicts an upper bound of $\theta substrate\u22481K$ (see supplementary material for details). Thus, neither the substrate nor the nanoparticle is expected to heat beyond the few-K level even at our highest *I*_{exc}. However, in our experiments, we measure an increase in *r* for a nanoparticle on a bare portion of the glass substrate equivalent to a temperature rise of nearly 50 K at *I*_{exc} = 3 × 10^{5} W cm^{−2} based on the calibration of Fig. 4(a). This apparent self-heating effect is illustrated by the blue points in Fig. 4(b), and the origins of this non-thermal effect are discussed extensively in Ref. 36.

### C. Determining the measured peak temperature rise

In sharp contrast with the particle on bare glass, we measure a much steeper increase in *r* with *I*_{exc} using a particle on a portion of the cover slip coated with 50 nm of Pt. The results are shown by the red points in Fig. 4(b). Additionally, above a certain threshold excitation intensity value (*I*_{threshold} ≈ 2.5 × 10^{5} W cm^{−2}), the luminescent signal from the UCNP is lost and we can no longer perform measurements, consistent with reports that bare UCNPs degrade above ∼600 K.^{35} To convert this increase in *r* with *I*_{exc} to a temperature rise, we first subtract the apparent self-heating contribution, which is determined using the data shown in Fig. 4(b) (see supplementary material for details). We can then isolate $\theta NP(Iexc)$. We approximate $\theta NP(Iexc)$ as measuring $\theta substrate(Iexc)$ averaged over a circular area of radius *r*_{NP} = 25 nm, which is reasonable since the small nanoparticle size and lack of a conduction pathway to $T\u221e$ minimize parasitic heat sinking (i.e., heat flow from the sample to the nanoparticle thermometer will not alter the sample temperature). This situation contrasts with AFM-based thermometry techniques like scanning thermal microscopy, where heat can flow upward from the sample into the AFM tip and such parasitic heat sinking is thus a serious concern. We have also previously demonstrated that the thermal contact resistance between the nanoparticle and substrate is negligible.^{36} Additionally, because the nanoparticles are hexagonally faceted, it is reasonable to assume facet-plane contact and take *r*_{NP} = 25 nm as the contact radius. We also assume that the emitted nanoparticle luminescence is spatially uniform since the Er^{3+} ions, from which the luminescence originates, are distributed evenly throughout the nanoparticle volume with approximately 0.3 ions/nm^{3} for a 2% Er^{3+} concentration; prior photon antibunching measurements have also confirmed the presence of many Er^{3+} emitters even in small single UCNPs.^{33} Thus, to quantify the nanoparticle thermometer's spatial averaging, we represent the nanoparticle luminescence intensity profile as uniform over a circular area of radius *r*_{NP} and express the measured temperature rise as

where the expression for $\theta surface$ is given in Eq. (5) and $\varphi =atan(yx)$ is the polar angle. For a nanoparticle thermometer with *r*_{NP} = 25 nm and a Gaussian heat source with an experimentally determined 1/e^{2} radius of *r*_{NP} = 584 nm (see supplementary material), the percent error $\eta $ calculated using Eqs. (17) and (15) is 0.01% and thus the nanoparticle thermometer effectively measures the true peak temperature rise.

## IV. DISCUSSION

### A. Comparison of experimental and modeling results for the measured peak temperature rise

The red points in Fig. 4(c) show $\theta NP(Iexc)$ obtained from converting the measured *r*(*I*_{exc},*T*) data after eliminating the apparent self-heating contribution. In our experiments, the laser is positioned to maximize the emitted luminescence intensity, implying that the nanoparticle is centered within the laser spot and thus records the peak hotspot temperature. We measure a maximum temperature rise of *θ*_{NP} = *θ*_{substrate} ≈ 167 K at *I*_{exc} = 2.2 × 10^{5} W cm^{−2}. To compare this experimental result with the analytical result of Eq. (17), we consider a range of Pt thermal conductivity (*k*_{Pt}) and absorptivity (*α*_{Pt}) values. Because grain boundaries and film surfaces will increase scattering in thin films relative to the bulk, we expect the thermal conductivity of the 50 nm Pt film to be reduced below the bulk value of *k*_{Pt,bulk} ≈ 70 W m^{−1} K^{−1}.^{40} Specifically, we vary *k*_{Pt} between 25 and 60 W m^{−1} K^{−1} and *α*_{Pt} between 0.25 and 0.35. These ranges capture the spectrum of possible *k*_{Pt}^{40,41} and *α*_{Pt}^{39} values based on literature data since no direct characterization was available for the Pt films used in this work.

Using Eq. (17), we calculate the corresponding range of the peak temperature rise within the laser heated spot recorded by a nanoparticle thermometer with *r*_{NP} = 25 nm, which is represented by the red shaded band in Fig. 4(c). Conversely, the blue shaded band in Fig. 4(c) represents the range of possible peak temperatures that would be measured by a thermometer with a conventional diffraction limited Gaussian thermometer of 1/e^{2} radius σ = 1 *μ*m calculated using Eq. (10) and the same range of *k*_{Pt} and *α*_{Pt}. Our experimental data are consistent with the portion of the red shaded band that is distinct from the blue shaded band and incompatible with the blue shaded band. In other words, the data agree only with modeled values for thermometers with $\sigma \u2264$ ∼400 nm (see supplementary material for details of this estimate), including a thermometer with $\sigma =rNP=25nm$. This experimental result thus demonstrates that an individual UCNP thermometer with $rNPrlaser\u226a1$ measures a temperature rise consistent with the range of modeled values for the true peak temperature rise in a laser heated spot. In situations where the spatial distribution of the thermometer averaging kernel is well known, deconvolution may provide another approach for determining the true peak temperature.

### B. Measurements on a diamond substrate

To further confirm that the steeper increase in $r(Iexc)$ for a particle on Pt-coated glass relative to uncoated glass truly represents heating rather than excitation intensity artifacts,^{36} we repeated the same experiments on a CVD diamond substrate (Element 6, vendor-specified *k*_{diamond} > 1000 W m^{−1} K^{−1} at 293 K). With *k*_{diamond} at least 1000× higher than *k*_{glass}, from Eq. (17), we find that the temperature rise measured by a nanoparticle of *r*_{NP} = 25 nm on a Pt-coated diamond substrate should be nearly 60× smaller than that measured by a particle on Pt-coated glass. Because *r*(*T*) can be well approximated as linear over the *T* range of interest, the additional increase in *r* beyond the apparent self-heating contribution should similarly be close to 60× smaller for Pt-coated diamond. At the maximum *I*_{exc} used for the glass substrate, Δ*r* for a particle on Pt-coated glass compared to a particle on bare glass is ∼1. Thus for a diamond substrate this same Δ*r* should be <0.02, comparable to our detection limit,^{27} and we expect *r*(*I*_{exc}) for a particle on Pt-coated diamond vs bare diamond to be indistinguishable. Figure 4(d) shows that we indeed observe this result. The turquoise diamond-shaped points show $r(Iexc)$ for a particle on bare diamond and the red diamond-shaped points show $r(Iexc)$ for a particle on Pt-coated diamond. As expected, these two datasets are essentially indistinguishable, since the measured increase in *r* in both cases is caused only by the apparent self-heating effect. We also replot the data from Fig. 4(b) for uncoated glass (blue circular points). Despite the >1000× substrate thermal conductivity contrast between diamond and glass, the increase in *r* with *I*_{exc} is nearly identical whether the particle is on bare glass, bare diamond, or Pt-coated diamond, further emphasizing that the apparent self-heating is a non-thermal effect.

## V. CONCLUSIONS

In summary, we have shown that a thermometer with resolution σ ∼ 2× smaller than the characteristic heater size $dchar$ is necessary to quantify the peak temperature rise with less than 5% error in many cases. This conclusion holds true for heat sources and thermometers of any size, although the strongest practical implications are for micro- and nanoscale heat sources where satisfying this constraint is traditionally challenging due to the paucity of thermometers with nanoscale spatial resolution. We provide analytical expressions for the error in the measured temperature rise as a function of $\sigma dchar$ for several common heat source geometries and a Gaussian thermometer representative of conventional diffraction limited optical thermometry techniques. These expressions can be readily adapted for other heat source geometries or thermometer averaging kernels. We have also experimentally demonstrated that individual UCNPs of characteristic dimension *σ* = *r*_{NP} = 25 nm can serve as far-field, non-contact optical thermometers that measure a temperature rise consistent with the range of modeled values for the true peak temperature rise in laser heated spots with characteristic dimensions of ∼1 *μ*m. This experimental capability can easily be extended to other hotspot morphologies. We also find that, in contrast to the non-thermal apparent self-heating effect we previously reported,^{36} the laser induced heating we measure on Pt-coated glass can be effectively eliminated by replacing the glass with a higher thermal conductivity diamond substrate, further confirming the non-thermal nature of apparent self-heating. Single-UCNP thermometry thus offers a practical, minimally invasive approach to accurately characterize hotspots at the micro- and nanoscale.

## SUPPLEMENTARY MATERIAL

See the supplementary material for additional information on the numerical methods used to calculate the modeled temperature rise data, the upconverting nanoparticles used in the experimental measurements, and the determination of the experimentally measured temperature rise and associated error.

## ACKNOWLEDGMENTS

We thank Jason Wu for providing the evaporated Pt films. This work was supported in part by a National Science Foundation (NSF) GOALI (Grant No. 1512796) with Seagate Technology LLC. A.D.P. acknowledges support from an NSF Graduate Research Fellowship under Grant No. DGE 1752814.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.