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Physical Review B

Material metrics for laser cooling of solids

Jedo Kim, Ankur Kapoor, Massoud Kaviany

Published Mar 19, 2008• 12 citations
Publisher

Abstract

The material metrics for optimal laser cooling of ion-doped solids are derived using atomic and molecular dynamics properties of the constituents. The anti-Stokes process is modeled as an optical phonon coupling of the bound electron, followed by a photon absorption. The transition dipole moment is estimated using a simplified charge-displacement model and the Judd--Ofelt theory of rare-earth ions both suggesting that transitions with high-energy gaps and similar angular momentum states should be used. The electron-phonon coupling is interpreted as a derivative of electronic energy with respect to displacement of the nearest neighboring ligands whose stretching mode frequency is approximated using the molecular data. The Debye-Gaussian model is used for the phonon density of states of diatomic crystal. Then, the Fermi golden rule is used for photon-induced, phonon-assisted electronic transition probability and applied to the cooling rate equation by defining a phonon-assisted transition dipole moment. Based on the material metrics, an example blend is investigated for its cooling performance and a general guide is proposed for selection of better performing laser cooling hosts. Furthermore, the cooling rate limits are discussed and three distinct characteristic times are identified with the photon-induced, phonon-assisted transition time controlling the rate. The metrics guide the selection of host materials for optimal cooling, and predict a noticeable increase in the absorption rate when using a blend of cation atoms.

DOI
10.1103/physrevb.77.115127

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The material metrics for optimal laser cooling of ion-doped solids are derived using atomic and molecular dynamics properties of the constituents. The anti-Stokes process is modeled as an optical phonon coupling of the bound electron, followed by a photon absorption. The transition dipole moment is estimated using a simplified charge-displacement model and the Judd--Ofelt theory of rare-earth ions both suggesting that transitions with high-energy gaps and similar angular momentum states should be used. The electron-phonon coupling is interpreted as a derivative of electronic energy with respect to displacement of the nearest neighboring ligands whose stretching mode frequency is approximated using the molecular data. The Debye-Gaussian model is used for the phonon density of states of diatomic crystal. Then, the Fermi golden rule is used for photon-induced, phonon-assisted electronic transition probability and applied to the cooling rate equation by defining a phonon-assisted transition dipole moment. Based on the material metrics, an example blend is investigated for its cooling performance and a general guide is proposed for selection of better performing laser cooling hosts. Furthermore, the cooling rate limits are discussed and three distinct characteristic times are identified with the photon-induced, phonon-assisted transition time controlling the rate. The metrics guide the selection of host materials for optimal cooling, and predict a noticeable increase in the absorption rate when using a blend of cation atoms.
The enhanced laser cooling performance of rare-earth-ion-doped nanocrystalline powders is predicted, compared to the bulk material, using Yb3+:Y2O3{\mathrm{Yb}}^{3+}:{\mathrm{Y}}_{2}{\mathrm{O}}_{3}Yb3+:Y2​O3​ as the model material. This is achieved by enhancing the off-resonance, phonon-assisted absorption, which is proportional to the three factors considered in this paper: the dopant concentration, the pumping field energy, and the excitation coefficient. Using the energy transfer theory for concentration quenching, the optimum concentration corresponding to the maximum cooling power is found to be considerably larger than the currently used value, suggesting noticeable enhancement effects for laser cooling. The pumping field energy is enhanced in random nanopowders compared with bulk crystals under the same irradiation, due to the multiple scattering of photons. Photons are thus localized in the medium and do not propagate through, increasing the photon absorption of the pumping beam (and it is shown that the reabsorption of the fluorescence is negligible). Using molecular dynamics simulations, the phonon density of states (DOS) of the nanopowder is calculated, and found to have broadened modes, and extended small tails at low and high frequencies. The second-order electronic transition rate for the anti-Stokes luminescence is calculated using the Fermi golden rule, which includes the influence of this phonon DOS, and is shown to have enhancement effects on the laser cooling efficiency using nanopowders. It is finally concluded that these three enhancement mechanisms are essentially to increase the population of the three participating carriers (electron, photon, and phonon) in the interacting volume, and this also points out directions for enhancing laser cooling performance in bulk materials.
Laser cooling of rare-earth doped solids has been demonstrated across a wide range of material platforms, inspiring the development of simple phenomenological models such as the four-level model to elucidate the universal properties of laser cooling under various operating conditions. However, these models usually require the input of full absorption spectra that must be provided experimentally or by additional complicated atomic modeling. In this Letter, we propose that a four-level model, when extended to admit effective energy levels adaptive to the pumping photon energy, can accurately predict the cooling efficiency as a function of temperature and pumping frequency using only few inputs such as the absorption coefficient measured at a single frequency and temperature. Our model exploits the quasi-equilibrium properties of the excitation of rare-earth ions for the determination of the effective four energy levels. The model is validated against published experimental results for a span of materials including ytterbium/thulium-doped glass and crystals. With the verified model, we derive explicit expressions for the optimal frequency and the operating bandwidth of pumping lasers. Our model significantly simplifies the modeling process of laser cooling and is expected to stimulate further development of optical refrigeration.
A mechanism for laser cooling of crystals doped with rare-earth ions is proposed. The mechanism involves two-photon Raman scattering through the dipole-allowed 5d ion level, electron–phonon transitions between the Stark-split sublevels, and fluorescence from the excited ion level. It is shown that, under the conditions of coherent pumping, the total population of the ground level is transferred to the excited level of the rare-earth ion, thus increasing the cooling power and accelerating the cooling process. The involvement of dipole-allowed transitions yields an increase in the cooling efficiency by 14% compared to the efficiency of direct pumping at transitions between the 4f levels. Estimations show that the temperature of the sample cooled by the proposed mechanism can reach 82.9 K for current materials with a background absorption coefficient of 4.0×10−4 cm−1.
Publisher Summary This chapter focuses on the concept of laser cooling of a solid, which may occur when the average energy of the photons emitted by the solid is larger than the energy of the ones it absorbs. More formally, the anti-Stokes emission, which occurs at frequencies larger than that of the pump laser, must dominate the Stokes emission that occurs at smaller frequencies. A crucial additional requirement is that the nonradiative decay rates of the laser-pumped states be negligible in comparison to their radiative decay rates. For the sake of specificity, these states will be taken to be those of a set of isolated ions embedded within an insulating host, although in general they could equally well be those of gas-, liquid-, or solid-phase neutral atoms or molecules, or even those of the energy bands of intrinsic semiconductors. Outside the realm of impurity-doped insulators, recent experiments with semiconductors make net anti-Stokes fluorescence cooling of these materials appear achievable. A long-term idea for improving laser cooling of solids takes advantage of photonic band gap materials. The idea is to fabricate a refractive-index structure within the cooling element that would establish a three-dimensional photonic gap immediately below the pump frequency. In this way, emission at Stokes frequencies falling within the gap would be inhibited, thereby enhancing the cooling power derived from the anti-Stokes emission.
We present a theoretical scheme for laser cooling of rare-earth-doped solids with optical superradiance (SR), which is the coherent, sharply directed spontaneous emission of photons by a system of laser-excited rare-earth ions in the solid-state host (glass or crystal). We consider an Yb+{}^{+}+-doped ZnF4{}_{4}4​-BaF2{}_{2}2​-LaF3{}_{3}3​-AlF3{}_{3}3​-NaF (ZBLAN) sample pumped at a wavelength 1015 nm, with a rectangular pulsed source with a power of \ensuremath{\sim}433 W and a duration of 10 ns. The intensity of the SR is proportional to the square of the number of excited ions. This unique feature of SR permits an increase in the rate of the cooling process in comparison with the traditional laser cooling of the rare-earth-doped solids with anti-Stokes spontaneous incoherent radiation (fluorescence). This scheme overcomes the limitation of using only low phonon energy glasses for laser cooling.
The cooling of solids via laser-excited anti-Stokes luminescence has reached the cryogenic regime and is now being considered for several applications that benefit from vibration-free cryogenic refrigeration. 4f-4f transitions in rare-earth-doped crystals with low phonon energies currently offer the best laser-cooling performance. These transitions, however, are electric-dipole (ED) forbidden and thus have low cross sections, limiting the minimum achievable temperatures in practical materials to the 50–100 K regime. In contrast, ED-allowed transitions may offer greater cross sections and potentially enable lower minimum temperatures or more compact devices. The search for laser cooling with ED-allowed transitions has been an active field of research over the past decades with a wide variety of materials having been studied. This paper reviews the fundamentals of solid-state laser cooling, provides a preliminary theoretical assessment showing that laser-cooling with ED-allowed transitions is in principle possible, and reviews the experimental results reported to date. While cooling with ED-allowed transitions has not yet been realized in the solid state, the existing experimental body of work along with the theoretical considerations presented here suggest that further research on this topic may be fruitful.