### Abstract

- Top of page
- Abstract
- 1 Introduction
- 2 Basic principles of anti-Stokes refrigeration in solids
- 3 Phonon-assisted anti-Stokes photoluminescence in nanocrystal quantum dots (state of the art)
- 4 Mechanism of anti-Stokes PL in nanocrystal quantum dots
- 5 Feasibility of the anti-Stokes cooling with nanocrystal QDs
- Acknowledgements
- References
- Biographical Information
- Biographical Information
- Biographical Information
- Biographical Information

In this review, we discuss the feasibility of laser cooling of semiconductor nanocrystal quantum dots by phonon-assisted anti-Stokes photoluminescence. Taking into account recent experimental advances, in particular, the development of semiconductor nanocrystals with very high quantum yield, we analyze in detail how the various physical processes in nanocrystals might help or hinder the cooling process. Possible experimental approaches to achieve efficient optical cooling are also discussed.

### 1 Introduction

- Top of page
- Abstract
- 1 Introduction
- 2 Basic principles of anti-Stokes refrigeration in solids
- 3 Phonon-assisted anti-Stokes photoluminescence in nanocrystal quantum dots (state of the art)
- 4 Mechanism of anti-Stokes PL in nanocrystal quantum dots
- 5 Feasibility of the anti-Stokes cooling with nanocrystal QDs
- Acknowledgements
- References
- Biographical Information
- Biographical Information
- Biographical Information
- Biographical Information

It has been long recognized that interaction of a fluorescent gas with radiation can lead to cooling of the gas when the electronic energy extracted by photon emission exceeds that introduced by photon absorption 1. In contrast to the cooling of atomic ensembles by means of laser radiation pressure 2, laser cooling of condensed matter occurs in exceptional situations when the pump (absorption) frequency is lower than the mean luminescence frequency. This emission process is in violation of Stokes' law and is termed anti-Stokes photoluminescence (ASPL) and was first observed in dye solutions 3, 4. It follows from energy conservation that the additional energy of the emitted photons in ASPL is taken out of the heat reservoir (i.e., illuminated material), which loses energy and, thus, cools. The absorption of vibrational energy distinguishes ASPL from other up-conversion effects like nonlinear two-photon excited luminescence. The thermodynamic viability of the anti-Stokes process was established by Landau 5 by considering the entropy of the incident and emitted light. It was shown theoretically that the entropy lost by the sample upon cooling can be compensated by an increase in the entropy of the light as a result of the loss of monochromaticity and beam directionality. Experimentally, however, such cooling by ASPL in the condensed phase is difficult to achieve and only lately conclusive results have been reported on successful cooling in rare-earth doped glasses and a fluid solution of a laser dye 6–8. Even more recently, the cooling by more than 70 K has been attained using thulium-doped glass 9.

From a practical standpoint, however, it is far more desirable to achieve laser cooling in semiconductors, which are already being used in a variety of optoelectronic devices. The biggest potential advantage of semiconductors as compared to rare-earth doped materials is the ability to achieve optical cooling temperatures below 10 K 10. This is due to the difference of the ground state populations in the two systems. As a consequence of the Boltzmann statistics, the population at the top of the ground state manifold of energy levels in a rare-earth doped system dramatically decreases as the temperature drops below 100 K, strongly reducing the efficiency of the cooling process. On the other hand, the population of energy levels in semiconductors is governed by Fermi–Dirac statistics, which means that the lower energy valence band is populated even at temperatures close to absolute zero.

The fundamentals of laser cooling of semiconductors were considered in detail in the work of Sheik-Bahae–Epstein 10 where the range of parameters for which laser cooling is possible has been determined. Unfortunately, a number of issues make the laser cooling in semiconductors difficult to attain in practice. The main challenge is getting the cooling effects to dominate over heating. At low excitation intensities the heating is usually associated with non-radiative decay and multi-phonon emission. Therefore, high quantum efficiency is required to get net cooling. Another issue is low light extraction efficiency in bulk semiconductors and heterostructures, which is due to the high value of the refractive index. These challenges have so far prevented net cooling from being observed in semiconductor systems despite numerous experimental attempts 10–14.

It was recently suggested that the indicated problems can be significantly alleviated by employing quantum confinement nanostructures 10. Due to the fact that the size of a nanostructure is smaller than the emission wavelength, the problem of light extraction is remedied. Moreover, in contrast to bulk semiconductors or heterostructures, the quantum efficiency of some nanostructures can be close to unity *at room temperature*15, 16. So far, a record efficiency of 99% has been observed in a GaAs/GaInP double heterostructure only by decreasing temperature down to 100 K 12. These advantages along with the numerous recent reports on efficient ASPL 17–26 motivated us to gain insight into the feasibility of the proposed approach and requirements for achieving net laser cooling using one particular type of semiconductor nanostructures, namely nanocrystal quantum dots (QDs) fabricated via colloidal chemistry.

These QDs represent the ultimate semiconductor-based quantum-confined system with atom-like energy levels, large optical transition dipole moment, and high photoluminescence (PL) quantum efficiency. The interest developed in nanocrystal QDs has been fueled by the high degree of reproducibility and control that is currently available in the fabrication and manipulation of these quantum-confined structures 16. It is worth-mentioning that these QDs have been used in LEDs 27, 28, photonic 29, 30 and core–shell structures 31, 32, and as biological cellular probes 33, 34.

### 2 Basic principles of anti-Stokes refrigeration in solids

- Top of page
- Abstract
- 1 Introduction
- 2 Basic principles of anti-Stokes refrigeration in solids
- 3 Phonon-assisted anti-Stokes photoluminescence in nanocrystal quantum dots (state of the art)
- 4 Mechanism of anti-Stokes PL in nanocrystal quantum dots
- 5 Feasibility of the anti-Stokes cooling with nanocrystal QDs
- Acknowledgements
- References
- Biographical Information
- Biographical Information
- Biographical Information
- Biographical Information

The optical refrigeration can be achieved only in materials that have appropriately spaced energy levels and emit light with high quantum efficiency. To illustrate this concept we consider a system with a simple energetic structure including a ground state, which we denote “0”, and an excited state, which consists of two closely spaced levels “1” and “2” (see Fig. 1).

For our aims we also assume that the spacing between the levels of the excited state, Δ*E*, could not be larger than a few *kT*, where *k* is the Boltzmann's constant and *T* is the temperature of the sample. This assumption guarantees the thermodynamical quasi-equilibrium between these two levels, which is established on a picosecond time scale after the carrier excitation. At the same time, the energy separation between the ground and exited states is supposed to be much bigger than Δ*E*. In this regime the effect of non-radiative transitions to the ground state can be neglected and carriers can only recombine from levels “1” and “2” by emitting a photon. In other words, the quantum yield (QY) of this model system is close to unity, i.e., nearly each absorbed photon results in a photon emitted by the system. Tuning the optical excitation to the transition energy “0”–“1” (Fig. 1), results in the overpopulation of the first excited level with respect to the population of the level “2” which is determined by thermal equilibrium. To restore the equilibrium, some electrons will be rapidly transferred to the highest level by absorbing phonons and, consequently, cooling the sample. This electron population will subsequently relax to the ground state by emitting photons with mean energy larger than that of the absorbed photons (Fig. 1). Evidently, luminescence from both the level “2” and “1” will occur after the equilibrium is reached. The time scale of thermal interaction in bulk solids or nanostructures is in the picosecond regime, which is several orders of magnitude shorter than the time scale for optical transitions 35. This means that the population distribution between the excited state levels reaches thermal equilibrium long before relaxing to the ground state. In the described process, the excess in the mean energy of the emitted photons over the absorbed ones is due to the thermal absorption required to equilibrate the excited levels and is carried out of the solid on the fluorescent light beam, resulting in cooling.

At thermal equilibrium, the population of the levels “1” and “2” follows the Boltzmann distribution. As the cooling process is driven by thermal energy extraction via the luminescence up-conversion, the carriers' recombination dynamics in the system presented in Fig. 1a can be described by the following set of coupled rate equations:

- ((1))

where *n*_{1} and *n*_{2} are the population densities of levels “1” and “2”, respectively, *γ*_{1}(*γ*_{2}) is the recombination rate for transition “1” → “0” (“2” → “0”), and *γ*_{r} is the rate of relaxation “2” → “1”.

Solving Eqs. (1) for continuous-wave excitation one obtains equations for both the resonant (*I*_{1} = *γ*_{1}*n*_{1}) and up-converted emission intensity (*I*_{2} = *γ*_{2}*n*_{2})

- ((2))

where the coefficients *c*_{1} = *γ*_{1}/*γ*_{r} and *A* = *γ*_{2}/(*γ*_{r} + *γ*_{2}) are the ratios of the recombination rates. Note that these parameters can be estimated from the analysis of experimental PL decay curves 16, 36.

The first conclusion of this analysis is that the intensity of the up-converted emission is a linear function of the excitation power. Also resonant emission (transition “1” → “0”) shows conventional thermal quenching (Fig. 2). However, the most pronounced feature of this model is that the intensity of the up-converted emission (“2” → “0”) increases with temperature (Fig. 1b) by gaining energy from the thermal bath in contrast to the conventional quenching of resonant or Stokes-shifted photoluminescence (SSPL) with increasing temperature. As follows from Eq. (2), at high excitation intensity and/or at elevated temperatures (150–300 K), the temperature dependence of the ASPL intensity can be modeled by a single exponential growth function if *c*_{1} ≫ *A*. It is noteworthy that the energy gap value (Δ*E* = 100 meV) used in our model (Fig. 2) is larger than the highest phonon energy in any semiconductor material. This fact along with the exponential functional form of the probability for ASPL and SSPL excitations with respect to Δ*E*, means that the anti-Stokes cooling process is provided by the absorption of multiple phonons. This “anomalous” temperature behavior of the ASPL intensity may, therefore, be used as an indicator of phonon-assisted processes while analyzing the mechanisms of up-converted luminescence in materials. The growth of the ASPL intensity (and, therefore, the cooling efficiency) is clearly stronger at elevated temperatures (Fig. 2). Below 100 K, a high excitation intensity would be required in order to provide an appropriate level of cooling.

It is noteworthy that the energy level structure presented in Fig. 1a is just one of a number of possible schematics for cooling. Generally, a cooling system consists of both a ground state and an excited state, well separated from each other in energy, with at least one of the states split into two or more levels 37.

The thermodynamical processes in this kind of refrigerator have been considered in great detail by Mungan and Gosnell 37. An optical cooling device can be analyzed in terms of a general refrigeration process similar to that shown in Fig. 1b. The energy and entropy flow from the pump source (e.g., a laser). Heat is extracted by coolant from the cold reservoir. In the case of laser cooling of solids, the coolant and cold reservoir are the same object, i.e., the sample with levels “1” and “2” coupled through phonon-mediated transitions (Fig. 1a). Energy and entropy are output to the hot reservoir (vacuum at *T* = 0 K), with the coolant experiencing changes in energy and entropy. The maximum energetic efficiency (*η*_{max}) of this cooling process can be estimated considering a steady-state reversible operation of the coolant. As is typical for a reversible process, the equation for the efficiency can be simplified by substituting for the pump and hot reservoir temperatures in terms of level population, which gives 38

- ((3))

where *E*_{abs} is the energy of the pump transition from level “0” to “1”. In other words, we can consider the process as a cooling cycle where the absorption of energy *E*_{0–1} causes the extraction of energy Δ*E*.

The cooling can also be quantified in terms of quantum efficiency (*η*_{q}) and the cooling power *P*_{cool}. In systems with a near-unity QY the cooling energy per photon is determined by the difference between the average energy of an emitted photon (*E*_{PL}) and the value of *E*_{abs}, with the corresponding wavelengths *λ*_{PL} = *hc*/*E*_{PL} and *λ*_{abs} = *hc*/*E*_{abs}. According to an empirical rule known as Vavilov's law, the *λ*_{PL} and QY are independent 39 (or almost independent 40) of the excitation wavelength in the optical spectral region of luminescence. For the excitation in the anti-Stokes PL regime, the emitted photons have higher energy than the absorbed ones. In this case the average cooling energy per photon in one cycle is given by

- ((4))

From Eq. (4) we can obtain an expression for the cooling efficiency (*η*) defined as the ratio of the cooling power to the absorbed power

- ((5))

where = *λ*_{PL}/*η*_{q}.

Equation (5) enables one to calculate the cooling efficiency for any given material using just experimental spectroscopic parameters. Also, Eqs. (4) and (5) help the understanding of why the anti-Stokes cooling is so difficult to obtain in practice. Indeed, the requirement for positive cooling efficiency (*η* > 0) implies *η*_{q} > *λ*_{PL}/*λ*_{abs} ≈ 1−*kT*/*E*_{abs}. On the other hand, the value of *E*_{abs} must be sufficiently high (∼20*kT*) in order to minimize the multi-phonon relaxation 37. This imposes strong restrictions on the QY value, which must be close to unity (desirably at room temperature where the ASPL efficiency is higher). Among all semiconductor materials, only colloidal QDs satisfy this condition 16, 41.

From Eq. (5), we can conclude that the larger the difference between the ASPL maximum and the excitation wavelengths, the higher will be the efficiency of the emitter used in the cooling system. For dye molecules or rare-earth elements there are limitations for increasing the excitation wavelength because it leads to a strong decrease of the absorption coefficient. We will discuss this point for the case of QDs in Section 3.4.

### 5 Feasibility of the anti-Stokes cooling with nanocrystal QDs

- Top of page
- Abstract
- 1 Introduction
- 2 Basic principles of anti-Stokes refrigeration in solids
- 3 Phonon-assisted anti-Stokes photoluminescence in nanocrystal quantum dots (state of the art)
- 4 Mechanism of anti-Stokes PL in nanocrystal quantum dots
- 5 Feasibility of the anti-Stokes cooling with nanocrystal QDs
- Acknowledgements
- References
- Biographical Information
- Biographical Information
- Biographical Information
- Biographical Information

Most of the practical applications of semiconductor colloidal QDs are based on nanocrystal ensembles incorporated into a variety of host materials, such as organic or inorganic solvents, glass, or polymer films. However, the highest QY value that has been reported to date is for QDs colloidal solutions in organic solvents and we will focus on this case using some of the experimental data discussed above. The presence of the host material implies that, in terms of the anti-Stokes optical cooling, the absorption of excitation light by both QDs themselves and by the solvent must be taken into account. Also, the distribution of QD sizes results in the corresponding distribution of the emission wavelength, *λ*_{PL}. The cooling model described in Section 2 can be modified in order to take these effects into account 85 by rewriting Eq. (4) in terms of cooling power as

- ((10))

where *P*_{ex} is the PL excitation power, *λ*_{abs} corresponds to the wavelength of excitation (*λ*_{exc}), *A*_{QD} is the absorbance of QDs themselves, *A*_{comp} is the absorbance of the composite QDs and host material (both at *λ*_{exc}), and *λ*^{*} is the mean wavelength of the fluorescence given by

- ((11))

where is the PL energy spectrum.

In Eq. (10) the host absorbance (*A*_{comp}) also takes into account both the intrinsic solvent absorption and the absorption due to impurities. The absorption due to such non-luminescent components can be described by a quantity *A*_{nl}. For small values of *A*_{QD} and *A*_{nl}, we have *A*_{comp} = *A*_{QD} + *A*_{nl} and one obtains from Eq. (10) the following expression for the net cooling power

- ((12))

Obviously, the anti-Stokes cooling take place if *P*_{cool} > 0. Then the threshold QY value (above which the cooling is possible) can be derived from Eq. (12)

- ((13))

As discussed in Sections 3 and 4, the ASPL efficiency in QDs strongly depends on excitation wavelength (Figs. 4b and 5). Using these experimental data and Eq. (13), we can calculate the spectral distribution of *η*_{th} values for our CdSe/ZnS QDs in toluene. As one can see from Fig. 4, *λ*^{*} of the ASPL band of QDs depends on the excitation wavelength and has to be estimated using Eq. (11).

The data for *A*_{nl} = 0 in Fig. 8 represent an ideal case where no excitation energy is dissipated by the host absorption. Even in this case, QY as high as 0.965 is required to provide anti-Stokes cooling with excitation at 620 nm when the difference between the excitation and emission wavelengths (and, therefore, the cooling efficiency is a maximum (see Fig. 4a and Eq. (5)). In practice, the absorption by the host material cannot be neglected. Even the very small absorption of toluene (spectroscopy grade) of 5 × 10^{−4} raises the value of *η*_{th} (Fig. 8). Such a high value of QDs QY is still a challenge at present 68. Extrapolating the data in Fig. 8 by a linear function we can predict that the excitation wavelength has to be at least 700 nm to provide efficient cooling with QDs of 0.85 QY, which can be more or less routinely achieved 64, 65. It is noteworthy, however, that at this wavelength the excitation of ASPL in CdSe/ZnS QDs samples discussed here, is highly inefficient (see Fig. 4). Since the ASPL efficiency depends linearly on the pump power, a rather obvious solution is to increase the excitation level. However, at high excitation intensity a number of undesirable side effects come into play, such as photodegradation of the sample. Also, radiative or non-radiative nonlinear effects like two-photon absorption or Auger recombination, respectively, can strongly reduce the efficiency of ASPL and therefore, the cooling efficiency.

Several means can be suggested in order to enhance the QY of the QDs or to eliminate unwanted nonlinear effects. One of the possible approaches is metal-enhanced fluorescence, a newly recognized technology where the interactions of the light emitter with metallic colloids or surfaces provides efficient PL enhancement. In the coupling process, the energy of the light emitter can be transferred into surface plasmon modes of the metal nanostructures and then radiated in such a way that the effective emission efficiency is enhanced. A significant (up to fivefold) enhancement in the PL intensity was recently reported for the CdSe-ZnS/polyelectrolyte/Au nanostructure 86 and for CdTe QDs deposited on the silver nano-island films 87. Theoretical analysis shows that the threshold and the efficiency of laser cooling can be significantly improved due to energy transfer from semiconductor to metal 88.

It has been recently realized that careful band-gap engineering of semiconductor QDs can provide core–shell nanostructures for which nonlinear Auger recombination is simply inactive 89. One such approach, which involved the use of type-II core/shell hetero-QDs, was recently demonstrated in Ref. 90. Spatial separation of electrons and holes in these nanostructures produces a significant imbalance between negative and positive charges, which results in a strong local electric field. This field spectrally displaces the absorbing transition in excited QDs with respect to the emission line via the carrier-induced Stark effect. Moreover, this also should increase the exciton–phonon coupling and therefore the ASPL efficiency 84. Although there are no reports to date on ASPL in such QDs, these studies might open the way for anti-Stokes cooling at high pump powers.

Finally, perhaps the simplest solution to the problem of insufficient QY of QDs is to decrease the dot temperature by one of the conventional methods. Cooling the sample down to ∼100 K results in the QY close to unity (Section 3.4). To this end, the most feasible concept for QDs anti-Stokes cooling is combined in a Peltier and optical cooling device, when first the sample is cooled down to the threshold QY by thermoelectric cooling and then phonon-assisted optical cooling takes over.

To conclude, recent experimental results demonstrate that all the requirements for optical cooling using semiconductor colloidal QDs, outlined in Section 3.3 can be met. The appropriate electronic structure, strong electron–phonon interaction, high quantum efficiency of PL, and the efficient thermally stimulated anti-Stokes PL in QDs are the conditions that are highly favorable for the first observation of anti-Stokes cooling. The rapid development of new nano-technological tools and techniques in this field ensures that answering the question about the feasibility of optical cooling with QDs one can say “when” rather than “if”.