1. INTRODUCTION

The photon-induced annihilation of thermal quanta from matter, i.e., laser cooling, is of key importance in ultracold science and has been instrumental in the creation of gas phase quantum condensates, matter-wave interferometry, and macroscopic tests of quantum mechanics. The laser cooling of solids, in particular, is enabled through photon upconversion processes in which phonons are annihilated from the material [1–3]. On this principle, bulk cooling of solids has been achieved through photoluminescence on specific electronic transitions in Yb-doped solids [4], with demonstrations even reaching cryogenic temperatures [5–8]. Photoluminescence cooling, enhanced through exciton–phonon coupling, was also employed for the first demonstration of the laser cooling of an undoped direct band gap semiconductor [9]. However, this method is not effective with indirect band gap semiconductors like silicon and germanium due to the need of a phonon for momentum conservation and the consequent competition with nonradiative recombination. In parallel efforts, efficient narrowband cooling of individual phonon modes in solids has also been achieved by means of optical forces in geometry-engineered microstructures [10–12] and through Brillouin scattering [13], albeit without net cooling. The search still continues for a universal laser refrigeration mechanism that could be applied to any transparent material, with any geometry, at any optical wavelength.

In this context, Raman scattering of light from optical phonons in solids has been suggested as a promising alternative technology, especially since Raman scattering is available in all materials and also exhibits the requisite photon upconversion, i.e., anti-Stokes scattering [3,13,14]. The key distinction is that Raman cooling involves only virtual excited states, whereas photoluminescence mechanisms involve real excitations of the electronic states of a system. Photoluminescence thus relies on the existence of suitable natural materials, whereas Raman scattering can be potentially engineered. However, two key challenges must be resolved before the Raman cooling of a bulk solid can be achieved. First, spontaneous Stokes Raman scattering always dominates anti-Stokes scattering in bulk materials due to the event probabilities being scaled by $n_0+1$ and $n_0$, respectively, where $n_0={(e^{\hslash {\omega }_0/k_{B}T}-1)}^{-1}$ is the Bose–Einstein distribution function. It is essential to invert this imbalance to have any possibility of achieving cooling. Second, the anti-Stokes scattering efficiency needs to be enhanced so as to overcome the absorption of the pump light by the material. This is critical since each anti-Stokes Raman scattering event annihilates $\hslash {\omega }_0$ energy from the material (${\omega }_0$ is phonon frequency), whereas each absorption event adds $\hslash {\omega }_{\text{photon}}(\gg \hslash {\omega }_0)$ energy to the material.

Here we show that both the above requirements can be met in nearly any semiconductor by engineering the optical density of states (DoS) to a suitable form. Accordingly, we develop expressions for the spontaneous Raman scattering efficiency in a material with a modified photonic DoS, and derive the necessary and minimum condition to achieve cooling by this method. As a specific case, we demonstrate the surprising possibility of cooling all Raman-active phonon modes in undoped crystalline silicon using a single telecom wavelength pump.

Before we proceed, we must recognize the encouraging experimental efforts of past researchers also moving in the direction of this result. It has been reported that resonant Raman scattering (using excitonic enhancement) in certain semiconductors [3,9] can be used to tilt the Stokes versus anti-Stokes balance in favor of cooling. Ding and Khurgin [3] showed that by pumping GaN at 375.8 nm, the anti-Stokes-to-Stokes ratio of 0.38 can be reached naturally. More recently, Zhang et al. reported [9] a ratio of 1.43 in CdS nanobelts by means of exciton resonance, firmly establishing that anti-Stokes dominance can indeed be achieved. Since this method relies on interaction with an exciton, it can only allow the narrowband cooling of one or a few phonon modes that lie within the resonance, a constraint similar to optomechanical cooling [10–13]. Additional challenges to this method are the large optical absorption that occurs due to strong exciton–photon interaction and its impracticality for cooling indirect band gap semiconductors. A different perspective on Raman cooling has been offered by Rand [14], using a multipump method on specific electronic transitions of rare-earth elements like ${\mathrm{Ce}}^{3+}$ and ${\mathrm{Yb}}^{3+}$. However, as in other efforts, each photon pumping configuration chosen would target only one phonon mode.

We propose that engineering the photonic DoS [15,16] at high-transparency wavelengths can be used to invert the imbalance of Stokes versus anti-Stokes spontaneous Raman scattering efficiencies, and can also be used to enhance anti-Stokes Raman scattering through engineered photonic resonances. This proposal is outlined in Fig. 1. This method circumvents the optical absorption issue confronted by techniques that use excitons [3,9], can be applied to indirect band gap semiconductors, and also avoids band-tail absorption confronted in photoluminescence approaches [17]. Such DoS engineering has been employed previously in experiments with W1 silicon waveguides [18] to enhance spontaneous Stokes Raman scattering by a factor of up to 13. It has also been shown that the naturally occurring photonic DoS of a high-$Q$ resonator can be used to enhance anti-Stokes Brillouin scattering by orders of magnitude while completely suppressing Stokes scattering, thus enabling Brillouin cooling [13]. The possibility of by-design DoS engineering to achieve spontaneous anti-Stokes Raman cooling is clear.

 

Fig. 1. Concept of DoS engineering for Raman cooling of an arbitrary number of phonon modes. (a) Stokes scattering (red) dominates anti-Stokes scattering (blue) from any chosen phonon mode ${\omega }_{0,i}$ in bulk media. (b) It is proposed that an engineered photonic DoS with a complete band gap can be used to (c) suppress Stokes scattering while simultaneously enhancing anti-Stokes scattering intensity, as demonstrated previously with Brillouin cooling [13].

Download Full Size | PDF

2. RAMAN SCATTERING EFFICIENCY WITH MODIFIED DOS

Consider a spontaneous Raman scattering process with an incident photon frequency of ${\omega }_i$ and a scattered photon frequency of $\omega$. The scattering probability is given by [16,19]

To quantitatively evaluate the effects of arbitrary photonic DoS on the Raman scattering efficiency, we use the formulas found in [20,21] for the following calculations. The Raman scattering efficiency per unit length and solid angle is given in [20] as

Note that the above equations hold only for linear dispersion relations. Following Eq. (1), if the photonic DoS is modified, then Eqs. (6) and (7) become

Since the typical phonon energies are small compared with the photon energy, the Raman tensor for both processes is symmetrical and has approximately the same values. From Eqs. (8) and (9), we see that it is also possible to adjust the scattering efficiency in the Stokes and anti-Stokes directions by aligning the material’s crystal (Raman selection rules) suitably with respect to the anisotropic photonic DoS.

3. APPLICATION TO LASER COOLING

Now that we have obtained the DoS-modified Raman scattering efficiencies, we can calculate whether this can be used to achieve Raman cooling in solids. We consider a highly transparent medium in which Raman scattering occurs over a volume of cross-sectional area $A$ and effective length $L$. We can define the net scattering efficiencies by ${\eta }_S=L{S}_S$ and ${\eta }_{\mathrm{AS}}=L{S}_{\mathrm{AS}}$, such that $I_S={\eta }_S{I}_{\text{pump}}$ and $I_{\mathrm{AS}}={\eta }_{\mathrm{AS}}{I}_{\text{pump}}$ are the respective scattered intensities, with $I_{\text{pump}}$ being the incident light intensity. The experiment should be performed at high-transparency wavelengths far away from any electronic resonances, exciton resonances, or two-photon absorption in the material. High-order scattering is neglected since the probabilities are much smaller [19]. It is then fair to assume that the only energy exchange mechanisms between the medium and the pump laser are residual absorption and Raman scattering. Under these assumptions, the net power transferred into the medium is given by

The absorbed power can be calculated by $P_{\text{abs}}=A{I}_{\text{pump}}(1-e^{-\alpha L})$, where $\alpha$ is the absorption coefficient at the incident frequency. Since the absorption coefficient is small for a transparent material, the absorbed power is approximately $P_{\text{abs}}=AL\alpha I_{\text{pump}}$. Presently, we consider only one phonon mode for which the scattered optical power in the Stokes sideband is $P_S=A{\eta }_S{I}_{\text{pump}}=AL{S}_S{I}_{\text{pump}}$, and, similarly, $P_{\mathrm{AS}}=AL{S}_{\mathrm{AS}}{I}_{\text{pump}}$ for the anti-Stokes sideband. For each Raman scattered photon, one phonon is added or removed from the system. The heating (cooling) power due to scattering is thus obtained by multiplying the creation (annihilation) rate by the phonon energy $\hslash {\omega }_0$:

Upon substituting these relations into Eq. (10), we have the cooling condition

Through Eqs. (15), (8), and (9), one can determine that the cooling efficiency is directly proportional to the phonon occupation number $n_0$. Thus, materials with lower phonon energies are preferred. For instance, even though diamond is highly transparent, its longitudinal optical (LO) and transverse optical (TO) phonons have an occupation of $n_0=0.00168$ owing to their large $1332.5{\mathrm{cm}}^{-1}$ wavenumber [23]. On the other hand, cooling of LO, TO phonons in silicon ($519{\mathrm{cm}}^{-1}$) would be more efficient since $n_0=0.09$ [24]. Other promising materials include Ge (TO, LO, $300.7{\mathrm{cm}}^{-1}$, $n_0=0.31$) [25] and GaAs (TO, $268{\mathrm{cm}}^{-1}$, $n_0=0.38$, and LO, $285{\mathrm{cm}}^{-1}$, $n_0=0.34$) [26]. However, material transparency is of critical importance and will typically dominate material selection decisions.

4. RAMAN COOLING OF SILICON

Among common semiconductor materials, intrinsic crystalline silicon is extremely transparent in the telecom range [27–29] and has a Raman scattering efficiency comparable to the optical absorption coefficient [21]. It is thus a promising material with which to demonstrate Raman cooling. In this section, we show that the cooling of undoped silicon is practical using a telecom pump when a diamond-structure three-dimensional (3D) photonic crystal is used to generate a complete photonic band gap.

First, we consider a simplification to Eqs. (8) and (9) for cases where an isotropic or nearly isotropic photonic DoS is available, which will be used later. The ratio of the DoSs in these equations can then be pulled out from the integral as a simple prefactor $D(\omega )/(2\times \frac{{\omega }^2}{2{\pi }^2{c}^3})$, where the numerator is the photonic DoS of the crystal and the denominator is the vacuum DoS. The factor 2 in the denominator accounts for both polarizations. We then simply need to integrate the variation of the Raman tensor element $({\int }_{\mathrm{\Omega }}{|R|}^2\mathrm{d}\mathrm{\Omega })$ over the solid angle $\mathrm{\Omega }$ over which scattering takes place. In the case of crystalline silicon (point group $O_h$), we can define $\widehat{x}$, $\widehat{y}$, and $\widehat{z}$ as the [100], [010], and [001] directions of the material crystal, respectively. The Raman tensor for phonon polarization in each of these directions is [22,30]

The net cooling condition [Eq. (14)] demands minimization of the Stokes efficiency, and thus, a photonic DoS design that can fully suppress Stokes scattering is strongly desired. This is available in various 3D photonic crystal structures, including diamond lattice [31], Yablonovite [32], woodpile crystal [33,34], inverse opals [35], and two-dimensional crystal stacks [36]. As an example to illustrate the method of photonic DoS engineering, we select spherical air inclusions in silicon structured as a diamond lattice (inset, Fig. 2).

 

Fig. 2. Band structure and DoS of a diamond-structure photonic crystal consisting of air spheres in silicon. The refractive index of silicon is $n=3.44$ in the calculation; the radii of the air spheres are $r=0.3a$, where $a$ is the lattice constant of the photonic crystal. The yellow-shaded region denotes the range of the photonic band gap. The frequency ($\omega a/2\pi c$) is in nondimensional units.

Download Full Size | PDF

The fabrication of this 3D photonic crystal structure can be achieved through colloidal self-assembly [37] and holographic lithography [38,39]. In particular, single-crystal photonic crystals can be produced by epitaxial growth of the material in a polymer template. This was demonstrated for single-crystal GaAs with a template patterned by holographic lithography [40], and is possible to extend for silicon.

In Fig. 2, we present the well-known photonic band structure and DoS of a diamond-structure silicon–air photonic crystal, with radii of air spheres of $r=0.3a$, where $a$ is the lattice constant. These calculations are performed using the MIT Photonic-Bands package [41]. Since the first Brillouin zone (FBZ) of a diamond-structure (face-centered cubic) crystal is a truncated octahedron, which is nearly spherical [36], we can make the simplifying assumption that the photonic DoS is approximately isotropic. We can then compute the photonic DoS through [42]

We now impose this DoS onto the Raman efficiency formulas and assume that the escape efficiency for the scattered photons is 1, i.e., all the scattered light escapes the medium [43]. In Fig. 3(a), we plot the total scattering efficiency per unit length of the Stokes ($S_S$) and anti-Stokes ($S_{\mathrm{AS}}$) processes as a function of pump frequency for the selected photonic crystal. Note that we include the contribution of the triply degenerate LO/TO phonon modes of silicon into a single $S_S$ and $S_{\mathrm{AS}}$. We invoke the known properties of silicon [21], $M=28.09$ amu, $N=2.5\times {10}^{22}{\mathrm{cm}}^{-3}$, $|d|=1.9\times {10}^{-15}{\mathrm{cm}}^2$, and $a=850\mathrm{nm}$ as the lattice constant of the diamond-structure photonic crystal, to complete our calculations. Several observations can be made from this computation. The pump light lies within the photonic band gap in the 160–203 THz (1476–1873 nm) range. Therefore, little to no scattering will occur. Pumping in the 203–220 THz (1362–1476 nm) range, however, completely suppresses the Stokes process. Simultaneously, the anti-Stokes process is enhanced by a factor of $\sim 15$.

 

Fig. 3. Cooling all Raman-active phonons in silicon. (a) Raman efficiencies per unit length $S_{\mathrm{AS}}$ and $S_S$ (in intrinsic crystalline silicon patterned with the Fig. 2 diamond photonic crystal) as functions of pump frequency. The contributions from all three Raman-active phonon modes are included. The yellow-shaded region denotes the band gap for the pump. Band-edge absorption, not included, will become significant at pump frequencies higher than 260 THz (approximately the band gap energy). (b) Calculated cooling and heating efficiency per unit length. The Stokes process is suppressed over a wide range, resulting in net phonon energy removal and the simultaneous cooling of all Raman-active phonons.

Download Full Size | PDF

In Fig. 3(b), we plot the cooling and heating efficiencies per unit length, i.e., $\frac{{\omega }_0}{{\omega }_{\mathrm{AS}}}{S}_{\mathrm{AS}}$ and $\frac{{\omega }_0}{{\omega }_S}{S}_S$, as functions of pump frequency. We observe that net cooling can occur within the Stokes suppression range (203–220 THz, 1362–1476 nm) as long as Eq. (14) can be satisfied. The majority of optical absorption in crystalline silicon (band gap energy of $\approx 1.12\mathrm{eV}$) over this frequency range comes from three- and four-phonon assisted absorption [29], and is very small. To evaluate the possibility of cooling, we invoke experimentally measured absorption coefficients from [27,28] and compare them against the net cooling efficiency per unit length $\frac{{\omega }_0}{{\omega }_{\mathrm{AS}}}{S}_{\mathrm{AS}}-\frac{{\omega }_0}{{\omega }_S}{S}_S$ (presented in Fig. 4). As expected, the cooling condition is satisfied for optical pumping around 203–210 THz (1427–1476 nm). This demonstrates that the total phonon energy removal rate can exceed the energy absorbed, leading to net energy removal from the system.

 

Fig. 4. Net Raman cooling is achievable in silicon: the net cooling efficiency per unit length ($\frac{{\omega }_0}{{\omega }_{\mathrm{AS}}}{S}_{\mathrm{AS}}-\frac{{\omega }_0}{{\omega }_S}{S}_S$) can exceed the absorption coefficient for the design presented. The absorption coefficient for silicon is extracted from [27,28]. Near 210 THz (1427 nm), the net cooling overcomes the absorption.

Download Full Size | PDF

5. PRACTICAL CONSIDERATIONS

There are a few experimental notes that we must discuss in this context. Geometric imperfections always exist in engineered photonic systems, leading to band-tail states, which can increase optical absorption near the band edge, and possibly in the midgap localization states. Since we tune the optical pump just blue of the band edge, the band-tail states will reduce the cooling efficiency since they increase $S_S$. However, since the LO and TO phonon energies in silicon are quite high ($519{\mathrm{cm}}^{-1}$), the scattered light can be placed well within the photonic band gap, away from the band tails. Ultimately, the photonic system design can also be revised to compensate for lost efficiency by engineering the anti-Stokes DoS.

The large surface area of an air-inclusion photonic crystal can also lead to an increase in overall absorption due to the electronic surface states of the material crystal, and must be controlled. Surface passivation through hydrogen termination and monolayer/dielectric deposition has been proven to be effective in reducing these contributions [44,45]. Crystal impurities, dopants, and defects in silicon can also increase optical absorption substantially for photons lower than the band gap energy [46,47]. For instance, to minimize free-carrier absorption at the wavelength of interest, the doping concentration for either p- or n-type dopants should be lower than ${10}^9{\mathrm{cm}}^{-3}$ [47] to reach the cooling condition in the present example.

The escape efficiency of the scattered photons, which is the escape probability of anti-Stokes photons from the material after scattering, must be engineered to approach 100%. Since the radiation pattern for scattered anti-Stokes light depends on the polarization of the pump light relative to both the material crystal and photonic crystal orientations, it is possible to engineer the emission of anti-Stokes light toward surfaces at angles smaller than the critical angle for total internal reflection. Other approaches for improving the escape efficiency may include a gradient index region at the surface of the cooling crystal for increasing the critical angle, and LED-inspired geometries [48].

Several different photonic crystal architectures are known to offer the requisite 3D band gaps, including woodpile structure [33,34], inverse opals [35], and two-dimensional crystal stacks [36]. The selection of a particular crystal design can be made based on our ability to control the aforementioned parasitic effects in each structure.

Cooling efficiency, or heat lift per watt of pump power, is a key metric of interest. For photoluminescence-based refrigeration [5], the single-pass cooling efficiency is typically around 0.1% per interacting (i.e., fully absorbed) photon. In comparison, Raman cooling efficiency is much higher, at 5%–10% per interacting (i.e., scattered) photon, calculated via the ratio of phonon energy to photon energy. However, since the interaction probability in Raman scattering is small, the single-pass net cooling efficiency is very low, around ${10}^{-7}{\mathrm{cm}}^{-1}$ (Fig. 4). To boost efficiency, photoluminescence coolers employ pump photon recycling through a multipass cavity [5]. However, due to the high pump absorption that is required for efficient fluorescence, one can only reach a photon recycling factor of about 10, bringing the net cooling efficiency into the 1%–3% regime. In the case of Raman cooling, the cooling efficiency could be boosted up by several orders of magnitude into the 1% regime by adding a high-finesse (finesse of $\approx {10}^5-{10}^6$) resonator around the engineered crystal. This is an advantage of working in the highly transparent regime of the material. Even further efficiency enhancement can come from more careful design of the photonic DoS to express strategically placed anti-Stokes resonances.

6. CONCLUSIONS

In this study, we have demonstrated that spontaneous Raman cooling of transparent semiconductors is achievable when the photonic DoS of the material is engineered. Cooling is reached by significantly enhancing anti-Stokes Raman scattering over the optical absorption probability, along with simultaneous rejection of Stokes scattering events. As a particular example, we show that our method can be applied to the net laser cooling of undoped silicon, which is currently unattainable by any other laser-cooling method. Furthermore, the cooling of all three triply degenerate modes (1 LO and 2 TO) in silicon is possible simultaneously using a single telecom wavelength pump.

In a broader context, Raman scattering and Brillouin scattering are similar processes differing primarily by the phonon populations they influence. The natural occurrence of suitable photonic DoS has already been proven to allow Brillouin cooling [13]; thus, Raman cooling remains an exciting prospect. The ability to exert such control on elementary photon–phonon scattering processes through photonic DoS engineering allows us to operate in highly transparent regimes far away from any material absorption, band edges, excitons, or atomic transitions in nearly any semiconductor. Such a broadly applicable method for laser cooling based on Raman scattering can greatly impact our ability to control the thermal states of matter.