All-optical single-nanoparticle ratiometric thermometry with a noise floor of 0.3 K Hz^−1/2

Two luminescence spectra of the same crystal are shown in figure 1. The striking difference between the two spectra is due to the different contributions of electrically neutral centers, NV⁰ to the spectra. These centers have a shorter zero-phonon line (ZPL) transition wavelength (575 nm in comparison to the 638 nm transition of ${\rm N}{{{\rm V}}^{-}}$). The ZPL corresponds to pure electronic optical transitions which do not change the phonon numbers of the crystal lattice and/or of local vibrations. Therefore photons at 590 nm wavelength have enough energy for electronic excitation of the NV⁰ centers only from a thermally populated vibrational level. The two spectra are scaled vertically so that they overlap at the long-wavelength side where the luminescence of NV⁰ is minimal. An advantage of the excitation at the 590 nm wavelength is significant reduction of the background under ZPL in the luminescence spectra. This background is one of the factors contributing to the detection noise [19]. The background is not reduced to zero due to a significant contribution of hot luminescence of ${\rm N}{{{\rm V}}^{-}}$. A simple way to characterize this background quantitatively is by the background ratio $r\equiv {{B}_{0}}/A$ (see figure 2), the ratio of the background level at the center of ZPL to the ZPL amplitude. The factor r has been determined for 5 crystals where it had a value of $1.8(2)$. The dispersion of r is relatively small because in the case of the 590 nm excitation r is an intrinsic property of the NV⁻ luminescence and does not depend on the concentration of NV⁰ in the crystal. Contrarily, the ratio varies between 4 and 10 in the case of 532 nm excitation. Another advantage of 590 nm excitation is that in this case the left side of the phonon band (approximately from 607 nm to 657 nm) can be very well approximated by an exponential function. Under the 532 nm excitation the background is more structured and this creates additional ambiguity in the data analysis and complicates separation of the ZPL from the phonon band.

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It can be seen in figure 3(A) that both the ZPL amplitude and the background depend on the temperature. We will discuss the background dependence later and first focus our attention on the ZPL. The temperature dependencies of the ZPL amplitudes measured on 5 crystals are shown in figure 3(B).

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The data analysis has been done by minimizing the chi-squared function defined as follows

$$\begin{eqnarray}&&{{\chi }^{2}}=\mathop{\sum }\limits_{n}{{\left( {{N}_{n}}-{{B}_{0}}{{{\rm e}}^{bn}}-\frac{A{{\Gamma }^{2}}}{{{\Gamma }^{2}}+{{(n-{{n}_{0}})}^{2}}} \right)}^{2}},\end{eqnarray} \tag{ 1 }$$

where N_n is the number of photons detected in the nth spectral bin (corresponds to one pixel of the CCD matrix in the direction of spectral dispersion). The exponential term is semi empirical and can be justified by the fact that thermal population of the vibrationally excited levels (emission of photons with a wavelength shorter than that of the ZPL comes exclusively from the vibrationally excited states) is proportional to the exponential Boltzmann factor. The ZPL has three fitting parameters, the amplitude A, the linewidth Γ and the position n₀. The exponential background is characterized by the prefactor B₀ and the exponential factor b. In principle, all the factors and their combinations such as $\pi A\Gamma$, the area under the ZPL (ZPL area divided by the total area of the spectrum gives the value called Debye–Waller factor, or DWF) are temperature dependent and can be used for temperature measurements. However, their sensitivities to the temperature are different. Although the relative errors in the estimated values of A, Γ, DWF etc (defined here as standard deviations divided by the corresponding mean values e.g. ${{\sigma }_{A}}/\bar{A}$) are all inversely proportional to the square root of the total number of the photons emitted in the ZPL, ${{N}_{{\rm ZPL}}}=\pi A\Gamma$, the proportionality constants are different in each case. Under simplifying assumption (uniform, wavelength independent background and an infinite recorded spectral interval where a single Lorentzian-shape line is located) the relative standard deviations of the ZPL parameters can be derived analytically [22]. The standard deviation of the amplitude estimate can be expressed as follows

$$\begin{eqnarray}&&\frac{{{\sigma }_{A}}}{A}=\frac{1}{N_{{\rm ZPL}}^{1/2}}{{\left[ {{a}_{1}}+{{a}_{2}}r+{{a}_{3}}{{({{r}^{2}}+r)}^{1/2}} \right]}^{1/2}},\end{eqnarray} \tag{ 2 }$$

where $[{{a}_{1}},{{a}_{2}},{{a}_{3}}]=[3,3,1]$ are analytically derived assuming an infinite spectral interval. For a finite spectral interval, comparable to the ZPL linewidth the analysis is more complicated but can be done numerically. This is shown in figure 2. The inset in figure 2 demonstrates how $N_{{\rm ZPL}}^{1/2}{{\sigma }_{A}}/\bar{A}$ (and similar for Γ and N_ZPL) depends on the background when the linewidth is about 12 times narrower than the recorded spectral interval. Surprisingly, the results for the amplitude A are the least dependent on the details of the model and the theoretical curve (equation 2) overlaps well with Monte Carlo simulations. The convergence of the data obtained by numerical simulations to the curves derived for the infinite range is very slow for ${{N}_{{\rm ZPL}}}$ and Γ for which $[{{a}_{1}},{{a}_{2}},{{a}_{3}}]=[1,3,1]$ and $[{{a}_{1}},{{a}_{2}},{{a}_{3}}]=[2,4,4]$ respectively (these curves are not shown in the figure).

Based on the experiments performed on a single NV center and theoretical analysis [23, 24], it is expected that the linewidth increases with temperature while the DWF decreases with increasing temperature. It is then apparent that the amplitude (divided by the total luminescence intensity to account for the changes in the laser light intensity etc) has a larger temperature sensitivity than both DWF and Γ. It is clear from the noise analysis and the sensitivity argument that the amplitude has a significant advantage in the accuracy over Γ and DWF.

A simple Debye model for the density of the phonon modes and scaling laws for the electron–phonon coupling can be used to predict how the DWF and the width of the ZPL depend on the temperature. In particular, at room temperature and above the ZPL width is proportional to T² [25] while the decrease of the DWF is described by an exponential dependence [19, 26]. These two factors combined together result in the following expression for the temperature dependence of the ZPL amplitude normalized by A_R, its value at $T={{T}_{R}}=295\;{\rm K}$ which was the temperature in the laboratory

$$\begin{eqnarray}&&\frac{A}{{{A}_{R}}}=\frac{T_{R}^{2}}{{{T}^{2}}}{\rm exp} \left( -\frac{2{{\pi }^{2}}\;S({{T}^{2}}-T_{R}^{2})}{3T_{D}^{2}} \right),\end{eqnarray} \tag{ 3 }$$

where T_D is the Debye temperature of diamond and S is the parameter determining the strength of coupling between the electronic degrees of freedom in the NV-center and the phonon bath of the crystal. Equation (3) effectively depends on a single parameter ${{T}_{{\rm D}}}/\sqrt{S}$. For the sample of 5 crystals which have been studied, ${{T}_{{\rm D}}}/\sqrt{S}=1.0(1)\times {{10}^{3}}$ K. The theoretical sensitivity of the ZPL amplitude according to equation (3) reads

$$\begin{eqnarray}&&\Phi _{A}^{-1}\equiv \frac{1}{A}\frac{{\rm d}A}{{\rm d}T}=-\frac{2}{T}-\frac{4{{\pi }^{2}}ST}{3T_{{\rm D}}^{2}}.\end{eqnarray} \tag{ 4 }$$

Given the mean and the variance of ${{T}_{{\rm D}}}/\sqrt{S}$, the value of inverse sensitivity ${{\Phi }_{A}}=94(10)\;{\rm K}$ at T = 295 K. The sensitivity can also be estimated by fitting a cubic-polynomial to the experimental data points in figure 3(B). The polynomials fit the data visibly better than the single-parameter equation (3) but the value of ${{\Phi }_{A}}=96(10)$ K at 295 K estimated using the derivatives of the polynomials agrees well with the estimate obtained from equation (4). Given a small dispersion of the temperature sensitivity we will use the value of ${{\Phi }_{A}}\approx 95$ K also for crystals which have not been calibrated in the oven.

The final step in the data analysis is to eliminate fluctuations of the ZPL amplitude due to the fluctuations in the excitation intensity. For this purpose, the amplitude has been divided by the integral intensity of the spectrum in the range from 610 to 660 nm.

The change of temperature $\Delta T$ can be determined using $\Delta A$, the change of the ZPL amplitude as follows

$$\begin{eqnarray}&&\Delta T=\frac{\Delta A}{A}{{\Phi }_{A}},\end{eqnarray} \tag{ 5 }$$

while the noise floor of the temperature measurements reads

$$\begin{eqnarray}&&{{\sigma }_{T}}=\frac{{{\sigma }_{A}}}{A}{{\Phi }_{A}}.\end{eqnarray} \tag{ 6 }$$

The value of ${{\eta }_{T}}\equiv {{\sigma }_{T}}\sqrt{{{t}_{{\rm m}}}}$, where t_m is the measurement time is called the noise floor of the temperature measurements and is expressed in units of K ${\rm H}{{{\rm z}}^{-1/2}}$. To obtain the experimental value for the noise floor, luminescence of one crystal has been monitored for about 22 min at room temperature of 295 K. 900 spectra have been collected with integration time of 1 s per spectrum and then analysed as explained above. The difference between the ZPL amplitude at the beginning of the series and its values at later times has been multiplied by 95 K for conversion to the corresponding temperature change and then plotted in figure 4(A). Note also how well a simple exponent fits the left side of the phonon band as shown in figure 4(B). The standard deviation of the temperature fluctuations estimated from the data equals 0.30 K and given the 1 s integration time the noise floor equals 0.3 K ${\rm H}{{{\rm z}}^{-1/2}}$. Below we estimate the noise floor using Poisson statistics of photon counting.

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The number of photons detected in ZPL in one second ${{N}_{{\rm ZPL}}}\approx 9.0\times {{10}^{5}}$ and the relative background factor r = 1.65 for this crystal. Substitution of these numbers into right hand side of equation (2) leads to a value of ${{\sigma }_{A}}/\bar{A}$ in perfect agreement with the experimental number of ${{\sigma }_{A}}/\bar{A}\approx 0.0032$. Moreover, the relative standard deviations of Γ and DWF determined experimentally for the 900 data points in figure 4 are 0.0044 and 0.0037, in agreement with the simulations shown in figure 2. Thus, the observed fluctuations confirm Poisson statistics of the photon counting in these experiments. This conclusion is not trivial as it indicates negligable luminescence intermittency (blinking) in these crystals as compared to quantum dots (QDs) [27].

The long time drift which can be identified in the data by fourth-order polynomial fits to the three segments shown in figure 4(A) indicate possible slow fluctuations of the room temperature. But 0.6 K, the peak-to-peak range of these fluctuations is within the 68% confidence interval $(\pm \sigma )$ defined by the short-time fluctuations. The cause of the apparent slow drift has not been further investigated. The overall detected photon count rate observed for this crystal is 52 MHz. Using the numbers for the photon detection efficiency reported previously [19] we estimate that there are about 100 NV centers in the crystal under consideration (the exact estimate depends on the population of the metastable signet state). Panel C in figure 4 shows a change of 8 K detected by a nano-diamond of a similar detected brightness ($\approx 50$ MHz count rate). For this measurement, the slide with nano-diamonds has been placed in the heating chamber, close to its optical window. The spectra have been measured first at a room temperature and then a temperature of 8 K higher. A smaller numerical aperture of the microscope objective (NA 0.55 with a long working distance) and the window of the heating chamber have reduced the photon collection efficiency by a factor of about 3. The standard deviation of the data at the room temperature is 0.33 K and agrees with the estimate obtained for Panel A. The peak-to peak range of 1.6 K and the standard deviation of 0.5 K at the higher temperature are slightly larger. The two distinct spikes and the apparent drift in the temperature data are explained by imperfect temperature stabilization.

The noise floor of the temperature measurements can be further reduced by exploiting the temperature dependence of the exponential fit to the short-wavelength side of the phonon band. In particular, the exponential factor b (see equation (1) and the caption to figure 2) demonstrates inverse sensitivity $\Phi \approx 300$ K. Although this makes b three times less sensitive to the temperature change than the ZPL amplitude, the short-time relative fluctuations of b are more than three times smaller. Weighted average calculation of $\Delta T$ using the values of $\Delta T={{\Phi }_{A}}\Delta A/A$ and $\Delta T={{\Phi }_{b}}\Delta b/b$ reduces the short-term noise floor to 0.23 K ${\rm H}{{{\rm z}}^{-1/2}}$. Unfortunately, it has been observed that the long-term instability divided by its short-time variance is larger for the exponential factor than the same ratio determined for the ZPL amplitude. The origin of the long-term instability needs further investigations.

Conventionally, the sensitivity of a probe $\Phi _{F}^{-1}\equiv {\rm d}F/(F{\rm d}T)$ is defined as a relative change of the temperature sensitive factor F per one degree of temperature increment. Although this is an important parameter (see, for example, a review article [7] where the data on sensitivity is reported for many sensors), the detection of the temperature change by luminescent nano-sensors (unlike in bulk measurements) is fundamentally limited also by the number of detected photons which cannot be increased by increasing the concentration of the nano-particles. If a probe responds to a temperature change by changing its total luminescence intensity (see, for example, a paper by Wang et al [28]), the minimum change of the temperature detectable by one particle reads

$$\begin{eqnarray}&&{{\sigma }_{T}}={{\Phi }_{I}}\frac{{{\sigma }_{N}}}{N}=\frac{{{\Phi }_{I}}}{\sqrt{DI{{t}_{{\rm m}}}}},\end{eqnarray} \tag{ 7 }$$

where ${{\sigma }_{T}}$ is the temperature change which can be detected with 68% confidence, t_m is the measurement time, I is the photon emission rate per particle, D is the photon detection efficiency, and $N=DI{{t}_{{\rm m}}}$ is the number of detected photons. The expression is based on the assumption that Poisson statistics of photon counting is the dominating source of noise. This is frequently the case in single nano-particle measurements where the number of detected photons is so small that ${{\sigma }_{N}}/N={{N}^{-1/2}}$ is much larger than, for example, the relative fluctuation of the excitation laser intensity. The temperature resolution reported in many publication has units of K and the term originates from bulk measurement where it is likely to be determined by the stability of the experimental setup, which may have a complicated dependence on the measuring time.

Using equation 7 we can evaluate the noise floor of the CdTe QDs [28]. The reported sensitivity of this sensor is 1.1% K⁻¹ and radiative decay rate of their excitonic states is about 50 MHz [29, 30]. This rate sets the maximum photon emission rate at 50 MHz (to achieve this number the optical transition must be saturated by a high intensity of the exciting laser). The rate of detected photons depends on the detection efficiency but in the case of QDs may be reduced by luminescence intermittency effect [31]. The maximum detected count rate observed for similar QDs is about 500 kHz [32]. The corresponding noise floor is about 0.13 K ${\rm H}{{{\rm z}}^{-1/2}}$. The paper reports temperature resolution of 0.02 K estimated from the signal-to-noise characteristics of the fluorescence spectrometer used in balk measurements and therefore the Poisson fluctuations will dominate single-particle measurements. This is an impressive value of the noise floor but CdTe QDs suffer from thermal instability [28]. In addition, it can be problematic to ensure high stability of the excitation intensity (better than 0.1%) on the level of a single particle which can exhibit, for example, significant diffusion. Therefore so called ratiometric optical thermometry appears much more suitable for single-particle applications because ratiometric probes display a change in the relative intensities of different regions of the detected spectra [33].

A good example is given by ${\rm M}{{{\rm n}}^{+2}}$-doped semiconductor nanocrystals [34, 35] where the relative intensities of two peaks in the spectra, one associated with direct emission from an excitonic state of the QD and the other from the state of ${\rm M}{{{\rm n}}^{+2}}$ localized in the band gap, depend on the temperature. The temperature is assessed using the ratio $R\equiv {{I}_{1}}/({{I}_{1}}+{{I}_{2}})$, where I₁ and I₂ are the intensities emitted in the two bands. Using elementary error propagation calculations, the temperature noise floor can be estimated as

$$\begin{eqnarray}&&{{\eta }_{T}}=\frac{{{\Phi }_{R}}}{{{D}^{1/2}}}{{\left( \frac{{{I}_{2}}/{{I}_{1}}}{{{I}_{1}}+{{I}_{2}}} \right)}^{1/2}}.\end{eqnarray} \tag{ 8 }$$

The sample with the highest precision (DPA-MnO5ZnS10) has the following characteristics: sensitivity $\Phi _{R}^{-1}=(3.1\pm 0.1)\times {{10}^{-3}}$ K⁻¹, exciton life time ${{\tau }_{2}}=18.1\pm 0.1\;{\rm ns}$ and a very long lifetime of the ${\rm M}{{{\rm n}}^{+2}}$ level which mounts ${{\tau }_{1}}\approx 2$ ms. The intensities of the two bands are about equal at room temperatures because the ${\rm M}{{{\rm n}}^{+2}}$ level has several orders of magnitude higher population than the exciton state. Given ${{I}_{1}}\approx {{I}_{2}}$ the expression above simplifies to ${{\eta }_{T}}={{(2D{{I}_{1}})}^{-1/2}}$. The publications on these sensors do not state the actual count rates. In the most favourable for the detection case, ${{\tau }_{1}}$ is the radiative lifetime and the corresponding emission photon rate under saturation is 0.5 kHz and then ${{\eta }_{T}}\approx 10/\sqrt{D}$ K ${\rm H}{{{\rm z}}^{-1/2}}$, much higher than the value obtained for CdTe DQs.

For completeness, we also consider ODMR measurements carried in a 150 nm crystal containing about 500 NV centers as presented in [16]. The results were much less impressive than with a bulk diamond crystal obtained by the same authors and others [17]. In [16], the luminescence intensity was measured at two pairs of RF frequencies positioned symmetrically on both sides near the points of the largest gradient of the approximately Lorenzian ODMR line to find the change of the zero-field splitting Δ. The noise floor for the nano-diamond measurements has not been reported but the following theoretical expression can be derived (for the procedure used in [16] and briefly discribed above) using Poisson statistics of the photon counting

$$\begin{eqnarray}&&{{\eta }_{T}}=\frac{4\sqrt{3}{{\Gamma }_{{\rm ODMR}}}}{9{{C}_{{\rm ODMR}}}}{{\left( \frac{{\rm d}\Delta }{{\rm d}T} \right)}^{-1}}\frac{1}{\sqrt{DI}},\end{eqnarray} \tag{ 9 }$$

where ${{\Gamma }_{{\rm ODMR}}}\approx 10\;{\rm MHz}$ is the ODMR width, ${{C}_{{\rm ODMR}}}\approx 0.1$ is the contrast of the ODMR, and ${\rm d}\Delta /{\rm d}T\approx 77$ kHz K⁻¹ is the sensitivity. The nano-diamond used in the intracellular experiments had a photon count rate detected during the ODMR measurement of about 10 MHz [36]. We have used these data and have estimated ${{\eta }_{T}}\approx 0.32$ K ${\rm H}{{{\rm z}}^{-1/2}}$, a value practically identical to the noise floor demonstrated in this work with a nano diamond of a similar brightness.