Photoluminescence (PL) is a quantum process governed by conservation of photon rate [1]. In optical refrigeration, endothermic PL is generated by a narrow-line pump at the absorption tail of a photoluminescent material, followed by emission at shorter wavelengths [2,3]. Recent studies have demonstrated cryogenic temperatures with doped glasses of quantum efficiency (QE) approaching unity [4]. Thus far, endothermic PL has been studied under thermal population that is far weaker than the photonic excitation. Under this condition, the photon rate conservation of the PL process is undisturbed by the thermal emission, obscuring the generalization of PL and thermal emissions. In this Letter we theoretically and experimentally study endothermic PL at high temperatures, where thermal and PL excitations compete in populating states above the bandgap. The fundamental physics that governs the interplay between PL and thermal emissions is expressed by the generalized Planck law, describing the spontaneous emission rate of a photoluminescent material [1,5] (similar formulations for the spontaneous emission rate of luminescent systems can be found in Refs. [6–11]):

We start with simulating the PL evolution of an ideal material, under constant optical pump rate and temperature increase. Physically, as will be shown, this corresponds to a situation where the material is optically pumped at its absorption resonance, generating constant PL, with additional variable heat flux that controls the temperature. For the sake of generality, the material is chosen to have a band-like emissivity function, as shown in Fig. 1(a) (inset). This emissivity function can describe both materials with discrete energy gaps, such as small molecules, and semiconductors (by expending the emissivity into the high energy spectrum). As an example, the emissivity function is chosen to be unity between 1.3 and 1.7 eV and zero elsewhere. In addition, for simplicity the PL is assumed to have unity QE and only radiative heat transfer is accounted for. In line with our hypothesis, we solve Eq. (1) by balancing the incoming and outgoing photonic and energy rates, at steady state. For a given incoming photon and energy rate the solution uniquely defines the thermodynamic state of the PL absorber, which is characterized by its quantities $T$ and $\mu$. The only way to conserve both the PL and energy rates is if each emitted photon is blueshifted with the increase in pumped heat. Figure 1(a) presents the evolution of emission spectrum and chemical potential (inset) as a function of temperature. Figure 1(b) presents the total emitted photon rate (inset) and the rate of photons with energy above 1.45 eV in the case of endothermic PL (blue line) and thermal emission (red line). The thermal emission is calculated by setting $\mu =0$, and applying only the energy balance. As evident, at low temperatures, the emission’s line shape at the band edge is narrow, and is blueshifted with temperature increase [Fig. 1(a)], while the total emitted photon rate is conserved [Fig. 1(b) inset]. Remarkably, this process is characterized by the reduction of photon rate near the band edge, where electrons are being thermally pumped to the high-energy regime as long as $\mu >0$. The red portion of the emission in Fig. 1(a) represents the thermal population, $R_0$. At low temperatures the PL photon rate is far above the rate of thermal emission, while $R_0$ increases and becomes significant at high temperatures. The temperature rise leads to the reduction in the chemical potential, according to the relation

 

Fig. 1. (a) Emission evolution of PL material with temperature. Insets: the emissivity function and the chemical potential temperature dependence. (b) Emission rates of energetic photons and total photon rate (inset) for PL (blue line) and thermal emission (red line) at various temperatures.

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Moving to the experimental part, the need for high-QE PL at high temperatures limits the use of solid-state semiconductors, due to reduction in their QE by temperature-dependent nonradiative recombination mechanisms [14]. On the other hand, rare-earth ions such as neodymium and ytterbium are an excellent choice of materials as their electrons are localized and insulated from interactions. This results in the conservation of their high QE at extremely high temperatures [15]. Due to the lack of interaction between electrons, each energy gap can be populated at different $\mu$ values, but thermalization equalizes $\mu$ within each spectral band. We experimentally study the transition between the PL and thermal emission at the 905 nm fluorescence band of a neodymium (${\mathrm{Nd}}^{+3}$)-doped silica fiber tip under 532 nm PL excitation of a few milliwatts (mW). The 905 nm emission corresponds to the transition between the $4{\mathrm{F}}_{3/2}$ and $4{\mathrm{I}}_{9/2}$ energy levels of the ${\mathrm{Nd}}^{3+}$ system, the latter being the ground state, which is essential for maximizing the thermal radiation signal. To control the heat load separately from the PL excitation, we use a ${\mathrm{CO}}_2$ laser operating at 10.6 μm wavelength with power levels up to 150 mW. At this wavelength, the photons are efficiently absorbed by the silica matrix [16] and converted to a constant heat flow. Although the 532 nm pump is exothermic in the sense that the absorbed energy is higher than the emitted PL energy, the heat associated with the few-mW pump thermalization is negligible, in comparison with the heat load imposed by the ${\mathrm{CO}}_2$ laser. Therefore, the PL harvests the thermal energy pumped by the ${\mathrm{CO}}_2$ laser, and, thus, it is endothermic. The temperature and the chemical potential are uniquely defined by these two pumps. The experimental setup is sketched in Fig. 2(a). The power spectrum is measured by a calibrated spectrometer. The weak PL excitation at 532 nm is kept constant at 1 mW, while the ${\mathrm{CO}}_2$ laser power varies between 0 and 150 mW. We monitor the temperature by fluorescence intensity ratio thermometry [17] (see Supplement 1). The spectral results are shown in Fig. 2(b). As we increase the thermal load, the temperature rises, and the PL exhibits a blueshift evolution. This is shown by the reduction in the emission of the low photon-energy peak at 905 nm, and enhancement of the high photon-energy peak at 820 nm. This trend continues until it reaches the transition temperature of $\sim 1500\mathrm{K}$. As we increase the temperature further, the emitted photon rate increases sharply at all wavelengths [dotted lines in Fig. 2(b)]. Figure 2(c) shows the total number of emitted photons at wavelengths between 600 and 1000 nm (blue line). To compare PL to thermal emission under equivalent conditions, we turn off the weak PL pump while the thermal current is unchanged (red line). The thermal emission at temperatures lower than 1150 K is below our detection limit and we extrapolate the experimental values down to 300 K. The inset of Fig. 2(c) shows that the measured chemical potential values (blue dots) are in good agreement with the theory (gray line) (see detailed description in Supplement 1). As demonstrated, the total photon rate is conserved at various temperatures, as long as $\mu >0$, and increases sharply at $\mu =0$, while converging with the thermal emission.

 

Fig. 2. (a) Experimental setup. (b) PL spectra evolution with temperature. (c) Total photon rate of PL and thermal emissions (inset: chemical potential versus temperature). (d) Energetic photon ($\lambda <850\mathrm{nm}$) rate for PL and thermal emissions.

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In Fig. 2(d) we compare the emission rate of energetic photons at wavelengths between 600 and 850 nm under PL excitation of 1 mW (blue line) and 4 mW (purple line) to thermal emission (red line). As can be seen, at temperatures below 1300 K the energetic-photon rates generated by endothermic PL excitation exceed that of thermal emission at similar temperatures by orders of magnitude.

The experimental part clearly validates our hypothesis on the nature of PL rate conservation with temperature rise. This can be simply seen by the convergence of the PL and thermal lines in Fig. 2(c): if the photon conservation law is not applied, the two separated curves could not converge. Also in Fig. 2(d), the order of magnitude enhancement in energetic photons emitted by endothermic PL between 400 and 1100 K (blue and purple curves) cannot be explained by the contribution of thermal emission (red curve). Last, in agreement with our theoretical model, the reduction of the low-energy PL photon rate (near 905 nm), is completely compensated by the increase of PL photon rate at 820 nm. Therefore, we demonstrated enhanced spectral blueshift, which is the outcome of energy balance in a number-conserved photonic system.

Another discussion should be addressed to the experimental evolution of endothermic PL near the critical temperature where the transition from PL to thermal emission is abrupt but not as sharp as expected from the simulation. We note that for PL, photons are pumped at a certain energy and emitted at a different one, which requires directional energy transfer. This is in contrast to the meaning of complete thermal equilibrium between modes through thermalization. In other words, thermalization only approximately equilibrates between adjacent frequencies, and small variation in the chemical potential is inevitable. We postulate that at the critical temperature these small differences in chemical potential lead to gradual transition into thermal emission. Other smearing effects of the transition could be unequal distribution of excitation and temperature across the sample.

Before concluding, we note that such a “thermally enhanced PL” (TEPL) can be used as an efficient heat pump. This can be considered for conversion of heat to electricity in a similar way to the solar thermal photovoltaics (STPV) concept [18,19], in which the energetic portion of thermal radiation is harvested by a low bandgap photovoltaic cell. While STPV efficiency depends on the thermal radiation, $R_0$, TEPL efficiency depends on the enhanced emission $R$. This advantage is related to the reduced entropy in TEPL in comparison with STPV when emitting the same energy flux. While a thermal emitter is always characterized by maximal entropy per energy, the TEPL is characterized by reduced entropy due to $\mu >0$. This can be seen from the entropy term in the Gibbs free energy equation:

To conclude, we theoretically and experimentally study endothermic PL at high temperatures. Our findings demonstrate how endothermic PL, in contrast to thermal emission, conserves the photon rate while the emission is blueshifted due to temperature rise. We also demonstrate how, at a critical temperature, the PL abruptly transforms into thermal emission where the photon rate increases sharply with temperature. Finally, we demonstrate how such a number-conserved photonic system is more efficient in emitting energetic photons than thermal radiation. These new findings show that endothermic PL is an ideal optical heat pump. We envision a novel TEPL-based device that harvests thermal losses in photovoltaics with record efficiency.