Zhiyu Wang
email: wangzhiyu23@scale.t.u-tokyo.ac.jp

J Kenji Clark
email: jkdclark@gmail.comho.ya-lun@scale.t.u-tokyo.ac.jp

Ya-Lun Ho

Sebastian Volz
email: volz@iis.u-tokyo.ac.jp

Hirofumi Daiguji
email: daiguji@thml.t.u-tokyo.ac.jp

Jean-Jacques Delaunay

Ultra-narrow and wavelength-tunable thermal emission in a hybrid metal-optical Tamm state structure

Keywords: thermal emission, ultra-narrow bandwidth, wavelength-tunable, mid-infrared, optical Tamm state

Spectral selective thermal emitters are promising technological components due to their efficiency, large range of available emission wavelengths, simplicity, and long lifetime. Despite intensive effort into narrowband thermal emitters using surface plasmon polaritons, surface phonon polaritons, and Tamm plasmons, material losses have limited the potential quality factors, with the highest reported value being 200. Here, by combining a metallic mirror and an optical Tamm state structure, we propose a hybrid structure that realizes narrowband and wavelength-tunable thermal emission. In the proposed structure, large energy can be stored between the loss-less DBRs (SiO 2 /Ge) thus drastically reducing metal losses. The metallic mirror is set at the bottom of the proposed structure to reduce the transmission losses and guarantee a large absorption/emission. This metal-optical Tamm state structure achieves experimentally a thermal emittance peak with a quality factor of over 750 at a wavelength of around 4.5 m. Moreover, taking advantage of the temperature sensitivity of Ge, the thermal emission wavelength can be tuned by adjusting the operating temperature between 100 and 150 C. Both the high-quality factor and the wavelengthtunable properties demonstrate its suitability as a practical narrowband light source.

3

The mid-infrared spectral range, where materials exhibit characteristic molecular absorption fingerprints originating from the intrinsic vibrational modes of chemical bonds, is crucial for biosensing [START_REF] Lin-Vien | The Handbook of Infrared and Raman Characteristic Frequencies of Organic Molecules[END_REF], [START_REF] Kaplan | Practical Applications of Infrared Thermal Sensing and Imaging Equipment[END_REF] and gas detection. [START_REF] Chen | Mid-infrared trace gas detection using continuous-wave difference frequency generation in periodically poled RbTiOAsO 4[END_REF] Conventional mid-infrared spectroscopy devices rely on broadband light sources with bulky gratings or interferometers in order to obtain spectrally resolved information. Miniaturized spectral selective emitters would allow for the design of spectroscopic devices without dispersive elements, drastically reducing their size and complexity.

Tailoring readily available thermal radiation into well-resolved and tunable emission at a single wavelength has been widely pursued as a means of creating low-cost and long-lifetime emitters with a large range of possible emission wavelengths. To date, wavelength-selective thermal emitters have been designed using a variety of technologies: photonic crystals utilizing band edge resonance, [START_REF] Celanovic | Two-dimensional tungsten photonic crystals as selective thermal emitters[END_REF], [START_REF] Rinnerbauer | High-temperature stability and selective thermal emission of polycrystalline tantalum photonic crystals[END_REF] surface plasmon/phonon polaritons (SPPs) excited by metallic/dielectric gratings, [START_REF] Liu | Quasi-coherent thermal emitter based on refractory plasmonic materials[END_REF], [START_REF] Dahan | Extraordinary coherent thermal emission from SiC due to coupled resonant cavities[END_REF] localized surface plasmon resonances (LSPRs) realized by metamaterials, [START_REF] Dao | Infrared perfect absorbers fabricated by colloidal mask etching of Al-Al 2 O 3 -Al trilayers[END_REF], [START_REF] Liu | Taming the blackbody with infrared metamaterials as selective thermal emitters[END_REF] and quantum-well inter-subband transition. [START_REF] De Zoysa | Conversion of broadband to narrowband thermal emission through energy recycling[END_REF], [START_REF] Inoue | Realization of dynamic thermal emission control[END_REF] However, all of the above require two-dimensional (2D) or three-dimensional (3D) nanofabrication processes that are costly, and this hinders their practical use. Therefore, to realize narrowband emission over a large-area without increasing fabrication complexity, a planar multilayer structure based on surface-state resonances offers many advantages over 2D or 3D nanostructures.

In one-dimensional (1D) structures, a certain type of surface waves known as an optical Tamm state (OTS) can be sustained at the interface between two dielectric distributed Bragg reflectors (DBRs) with overlapping stopbands. [START_REF] Kavokin | Lossless interface modes at the boundary between two periodic dielectric structures[END_REF] As the in-plane wave vector of OTSs is smaller than the free space wave vector, OTSs can be excited from free space in a 1D structure without the need for additional nanostructures to couple light. As a consequence, lithography-free fabrication of OTS-based devices is possible. A class of OTS known as Tamm plasmons (TPs) was shown to exist at the interface between a DBR and a metallic film. In contrast to the high transmission of the OTS structures, [START_REF] Goto | Optical Tamm states in one-dimensional magnetophotonic structures[END_REF] the TP structures can efficiently absorb incident light within their metal film resulting in an absorption/emission peak. [START_REF] Kaliteevski | Tamm plasmon-polaritons: Possible electromagnetic states at the interface of a metal and a dielectric Bragg mirror[END_REF] Using a TP structure configuration (i.e., a dielectric DBR on a metallic film) several narrowband thermal emitters with multilayer structures have been proposed. In 2017, a thermal emitter using a typical TP structure consisting of Si/SiO 2 DBR and Al film underneath was shown to exhibit a quality factor (Q-factor) over 30. [START_REF] Yang | Narrowband wavelength selective thermal emitters by confined Tamm plasmon polaritons[END_REF] One year later, by inserting an optical cavity between a Ge/SiO 2 /Ge DBR and an Au film, we demonstrated a TP-cavity structure that had a thermal emittance peak with a Q-factor of 88. [START_REF] Wang | Narrowband thermal emission realized through the coupling of cavity and Tamm plasmon resonances[END_REF] In 2019, deep learning method was used to optimize a Ge/SiO 2 DBR on W film and was able to achieve a Q-factor as high as 188. [START_REF] Sakurai | Ultranarrow-band wavelength-selective thermal emission with aperiodic multilayered metamaterials designed by Bayesian optimization[END_REF] Despite the efforts to improve the performance of spectrally selective thermal emitters based on TP structures, the Q-factors remain lower than the largest Qfactor of  200 reported to date [START_REF] Dahan | Extraordinary coherent thermal emission from SiC due to coupled resonant cavities[END_REF] due to the large absorption losses induced by the metallic films.

Without any metallic film, the OTS structure using two loss-less DBRs to store energy may offer a means to improve the Q value of thermal emitters in a 1D configuration.

In addition to a narrowband emission peak, emission wavelength tunability is another requirement for a thermal emitter in real applications. Thermal emitters containing thermochromic materials such as Ge 2 Sb 2 Te 5 (GST) and VO 2 have been used to tune the emission wavelengths by controlling the operating temperatures. [START_REF] Du | Control over emissivity of zero-static-power thermal emitters based on phase-changing material GST[END_REF][START_REF] Kats | Vanadium dioxide as a natural disordered metamaterial: perfect thermal emission and large broadband negative differential thermal emittance[END_REF][START_REF] Du | Wavelength-tunable mid-infrared thermal emitters with a non-volatile phase changing material[END_REF] As these demonstrations rely on temperaturedependent phase transitions, they can only offer tunability over a specific temperature range. In addition, due to the relatively large loss introduced by these thermochromic materials, the thermal emitters usually cannot achieve high Q-factor (showing a Q < 10) emission peaks. Achieving a narrowband and wavelength-tunable thermal emission with tunability over a wide range of wavelengths and operation at an arbitrary temperature remains challenging.

In this work, we propose a 1D structure consisting of a Ge/SiO 2 OTS structure (i.e., DBR top -DBR bot ) and a Pt mirror (i.e., metal) as shown in Fig. 1a and realize narrow-band and wavelengthtunable thermal emission in the mid-infrared range. At the resonance wavelength, light is confined between the two loss-less DBRs of the hybrid metal-OTS structure leading to resonance with a high Q-factor (over 1000 is achieved through simulation) and the bottom metallic mirror blocks the light from leaking out thus providing an inelastic loss channel that guarantees a pronounced absorptance peak. Upon heating, a sharp peak is observed in the emittance spectrum with a Qfactor above 750, as shown in Fig. 1b. When this Q-factor is compared to that of other narrowband thermal emitters, it is superior to the LSPR structure by a factor of more than 55, [START_REF] Dao | Infrared perfect absorbers fabricated by colloidal mask etching of Al-Al 2 O 3 -Al trilayers[END_REF] the QW ISB-T structure by a factor of more than 10, [START_REF] Inoue | Realization of dynamic thermal emission control[END_REF] and the SPP structure and TP-based 1D structure by a factor of ~4. [START_REF] Dahan | Extraordinary coherent thermal emission from SiC due to coupled resonant cavities[END_REF], [START_REF] Sakurai | Ultranarrow-band wavelength-selective thermal emission with aperiodic multilayered metamaterials designed by Bayesian optimization[END_REF] Moreover, tuning of the emittance peak wavelength is also realized by taking advantage of the temperature sensitivity of Ge, enabling fine adjustment of the emission wavelength. The emission peak wavelength can be tuned within a range of 4.6 times the bandwidth of the emittance peak by adjusting the temperature from 100 to 150 C. The tuning range to bandwidth ratio is larger than that of any other tunable thermal emitters which have been reported. [START_REF] Du | Control over emissivity of zero-static-power thermal emitters based on phase-changing material GST[END_REF][START_REF] Kats | Vanadium dioxide as a natural disordered metamaterial: perfect thermal emission and large broadband negative differential thermal emittance[END_REF][START_REF] Du | Wavelength-tunable mid-infrared thermal emitters with a non-volatile phase changing material[END_REF] Besides the outstanding wavelength selectivity and tunability, other properties such as polarization-and angle-independent emission for near-normal angles, and large device area make this hybrid metal-OTS structure a promising candidate for spectrally selective mid-infrared light sources.

Results and discussion

Demonstrations and optimizations of the hybrid metal-OTS structure

As illustrated in Fig. 1a, the two DBRs in the hybrid metal-OTS structure are adjacent to each other with Ge layer at the interface. The combination of Ge and SiO 2 as the DBR layers provides a large refractive index contrast, resulting in the DBRs having a high reflectance in the midinfrared. The thickness of each DBR layer is set to have an optical thickness of one-quarter of the target wavelength (4.5 m). As a heat resistant material, Pt provides good thermal stability when used as the metallic layer at the bottom of the structure. The layer thicknesses of the different materials are fixed throughout this section, with t Ge = 280 nm, t SiO2 = 819 nm, and t Pt = 200 nm.

The hybrid metal-OTS structure exhibits a sharp and strong absorptance peak at normal incidence as shown in Fig. 1c (see Methods section for detailed information about simulation). The maximum of the absorptance peak is larger than 0.8, and the peak has a Q-factor of 1908, while at an off-resonance wavelength of 4.0 μm, the hybrid metal-OTS structure maintains a low absorptance of less than 0.001, a value which is much lower than the absorptance of a semi-infinite Pt mirror (about 0.043 at the same wavelength used in the calculation). According to Kirchhoff's law, the emittance of a planar surface is equal to its absorptance, [START_REF] Siegel | Thermal Radiation Heat Transfer[END_REF] and therefore, this structure is expected to show not only a sharp and strong emission peak, but also low background emission over a large range of wavelengths.

The eigenmode of the hybrid metal-OTS structure can be studied using a similar method as described in (15). As shown in the inset in Fig. 1c, we separate the hybrid structure into a top reflector and a bottom reflector along a virtual interface (i.e., the interface of two DBRs illustrated by the white dashed line), and define the reflection coefficient of an upward-propagating wave at the virtual interface as r top , and the reflection coefficient of a downward-propagating wave at the virtual interface as r bot , with the ambient being Ge in both cases. The confinement of light between the two reflectors requires (1) 𝑟 top 𝑟 bot ~1.

Here, r top r bot as a function of wavelength is plotted in Fig. 1c (in pink color). The r top r bot spectrum shows a peak with a near-unity value of 0.9992 corresponding to the same wavelength of the peak in the absorptance spectrum (in black color), suggesting that the strong and sharp absorptance peak is due to the high confinement of the eigenmode sustained at the interface between the top and the bottom reflectors. Without the bottom Pt mirror (i.e., for its corresponding OTS structure), the value of the r top r bot peak would be only 0.9977, showing that the addition of the Pt mirror enhances the confinement in the hybrid metal-OTS structure relative to a typical OTS structure.

To further compare the eigenmodes in the hybrid metal-OTS structure with the TP and OTS modes, we investigate the optical behaviors of TP, OTS, and hybrid metal-OTS structures for incident light from top (Figs. 1d and1e). As shown by the normalized electric field (|𝐸|⁄|𝐸 0 | ) distributions at the resonance wavelengths in Fig. 1d, the TP structure with the Ge/SiO 2 /Ge DBR and Pt mirror exhibits a relatively weak standing-wave pattern with a maximum normalized electric field value near the Pt mirror decaying towards the structure top, which corresponds to a relatively broad near-unity absorptance peak (the blue curve in Fig. 1e). Note that with only a three-layer DBR, this TP structure can sustain a fully developed TP mode. An increase in the number of the DBR pairs reduces its near-unity absorptance peak (see Fig. S1). Instead of using a metallic mirror to confine light, two DBRs having less material losses are employed in the OTS and hybrid metal-OTS structures. This results in improved light confinement observed as pronounced standing-wave patterns shown in the electric field distribution plots (Fig. 1d), thus revealing a much larger stored energy with sharper absorptance peaks (see the red and the black curves in Fig. 1e). Similar electric field distributions are found for the OTS and the metal-OTS structures, i.e., maximum values for the normalized electric field near the interface of the two DBRs (indicated as a white dashed line) decaying in intensity away from the interface. The nonzero electric field intensity below the bottom DBR of the OTS structure indicates that light transmits through the structure bottom, as evidenced by the transmittance peak at the resonance wavelength shown in the bottom-right plot of Fig. 1e. In contrast to the OTS structure, the metal-OTS structure possesses low transmission losses. Indeed, the metal film at the structure bottom can not only improve the light confinement (evidenced by a stronger electric field distribution in Fig. 1d), but also reduce the transmission loss and provide an additional inelastic loss channel. [START_REF] Li | Large-area, lithography-free super absorbers and color filters at visible frequencies using ultrathin metallic films[END_REF] Accordingly, the hybrid metal-OTS structure exhibits an absorptance peak with a higher Q value (Q-factors are 1908 for the hybrid metal-OTS structure and 1324 for the OTS structure) and a twofold increase in the absorption intensity compared with that of the OTS structure (see the top-right plot in Fig. 1e). The zoomed-in plots red-rectangle-marked in Fig. 1d show absorption distributions around Pt mirror in TP and metal-OTS structures (here, absorption, noted Abs., is defined as 𝜔Im(𝜀 , where is the angular frequency, and is the imaginary part of permittivity) with

)|𝐸(𝑧)| 2 𝜔 Im(𝜀)
logarithmic-scale. In the TP structure, most of the incident light is absorbed by the Pt mirror. This contrasts with the hybrid metal-OTS structure where most of the light is absorbed by the dielectric DBRs (see Fig. S2). The strong electric field of the DBRs results in two-order weaker light absorption in the Pt mirror. Interestingly, this weak metal absorption property of the metal-OTS structure is associated with a robust optical behavior (i.e., absorptance peak) independent of the metal used in the hybrid metal-OTS structure. Therefore, the metal is not limited to commonly used low-loss metals such as Ag, Au, and Al, and a variety of metals could be chosen, such as Ti for good adhesion, Pt for good thermal stability, or Pd for catalysis, without any significant change in the optical behavior (see Fig. S3 and Table S1). Note that this is a consequence of the non-zero loss of the dielectric materials, as evidenced by the fact that when the dielectric loss is decreased to zero in simulation, the metal-dependent absorption will become obvious (see Fig. S3).

To optimize the optical behavior of the hybrid metal-OTS structure, we examine the absorptance spectra when the number of DBR layers is varied (Fig. 2a). Here, as demonstrated on the right part, the configuration of the DBR bot and the metallic film are fixed, and the layer number of DBR top is varied (N top = 5, 7, and 9). From N top = 5 to N top = 7, both the absorptance peak intensity and Q-factor improved. By adding two more layers (i.e., N top = 9), the peak intensity decreases drastically. As a result, the structure with the 7-layer DBR exhibits the best optical behavior when considering both absorptance intensity and Q value. The optical behaviors change can be attributed to different optical "impedance matching" conditions for N top = 5, 7 and 9. [START_REF] Yang | Narrowband wavelength selective thermal emitters by confined Tamm plasmon polaritons[END_REF] In the case of N top = 7, it is found that the structure achieves the best impedance matching condition (the transmission loss through the top reflector matches the absorption loss of the lower reflector), thus generating a minimum reflection from the structure and a maximum for the absorptance peak.

The angular and polarization dependences are characterized for the hybrid metal-OTS structure with 7 layers in the DBR top . The simulated absorptance spectra for different incident angles of the transverse electric (TE-) and transverse magnetic (TM-) polarized light are shown in the left part of Fig. 2b. For both TE-and TM-polarized light, when the incident angle is small (e.g., within 5), the wavelength deviation is relatively small (less than 3 nm). However, for a larger incident angle, the resonance wavelength shows a larger angular dependence (e.g., the wavelength difference is as large as 47 nm between the case of normal incidence and the 20 TM-polarized incidence). As the incident angle increases, TE-polarized spectra exhibit weaker absorptance peaks, while the TM-polarized spectra show stronger peaks. This intensity change can be explained by the difference in the impedance matching conditions occurring for different polarizations and incident angles. The right part of Fig. 2b shows the wavelength difference (Δλ) between the absorptance peaks of the TM and the TE polarizations for varying incident angles. The wavelength difference increases with the incident angle, but the difference caused by the different polarizations is relatively small compared with the peak resonance bandwidths (for the incident angle of 20, the TM and TE show a wavelength difference of 1.2 nm while the bandwidth of the peak is around 2.5 nm). In summary, the resonance wavelength of the hybrid metal-OTS structure exhibits a smaller angular and polarization dependence at small incident angles, so for the structure applied as thermal emitters, the emission near normal angle should be utilized to avoid a broadening of the emission spectrum.

Tunability of the absorptance peak wavelength

Figure 3a shows absorptance as a function of incident wavelength and the thickness of the middle Ge layer t mid (the absorptance is shown on a logarithmic-scale for a clear demonstration of the ultra-narrow absorptance peaks). High-absorptance lines represent resonance modes, and by varying t mid , the resonance wavelengths can be adjusted from 3.5 μm to a wavelength longer than 5.5 μm. Therefore, the narrow absorptance peak can be artificially designed within a wide range of wavelengths by simply adjusting t mid . The inset of Fig. 3a shows the absorptance spectra for three different t mid values (i.e., t mid = 0, 560, and 1120 nm) as marked by the circles referring to different orders of the resonance modes. It is observed that higher-order modes have narrower bandwidths, allowing narrower bandwidth resonances to be designed without altering the DBR layer structure.

In addition to the ability to design the absorptance peak wavelength by adjusting t mid , the wavelength can also be actively tuned effect as a result of the temperature-dependent material properties. In Fig. 3b, we simulated the absorptance spectra of the hybrid structure with (t mid = 1120 nm) and without (t mid = 0 nm) the middle Ge layer at different temperatures by considering the temperature dependence of the SiO 2 and Ge refractive indices as well as the thermal expansion coefficients of Ge and SiO 2 . Using a linear fitting procedure, the thermo-optic coefficients of Ge and SiO 2 are estimated to be 4.77  10 -4 /C and 1.23 10 -5 /C, respectively (see Fig. S4), [START_REF] Li | Refractive index of silicon and germanium and its wavelength and temperature derivatives[END_REF], [START_REF] Tan | Temperature dependence of refractive index of glassy SiO2 in the infrared wavelength range[END_REF] and the thermal expansion coefficients of Ge and SiO 2 are taken directly from literatures as 5.90

 10 -6 /C and 5.60  10 -7 /C, respectively. ( 25), [START_REF] El-Kareh | Fundamentals of Semiconductor Processing Technologies[END_REF] Without the Ge middle layer, the simulated absorptance peak shows little variation with temperature, while for the structure with a thick Ge middle layer, a strong wavelength-tuning behavior is observed. As the temperature changes from 100 to 150 C, the absorptance peak red-shifts by 23 nm. The electric field distributions, shown to the right of the absorptance spectra, can be used to explain these different behaviors. For the structure without Ge middle layer, the maximum electric field exists in the SiO 2 layer, while for the structure with a thick Ge middle layer, the maximum electric field is within the Ge layer.

Because Ge has both a more than one-order larger thermo-optic coefficient and thermal expansion coefficient than SiO 2 , the structure with a Ge middle layer has a larger sensitivity to temperature.

The middle Ge layer does not only store a large amount of energy, allowing for a high Q-factor resonance, but also acts as a wavelength tuning material. This structure with a thick middle Ge layer (t mid around 1120 nm) was therefore chosen for fabrication.

Experimental results

The hybrid metal-OTS structure was fabricated and its reflectance and emittance spectra were measured as shown in Fig. 4 (see Methods section for the detailed information about the fabrication and the measurement). As shown in Fig. 4a, the measured reflectance spectrum under normal incidence (0) shows a sharp dip at 4.516 μm with a bandwidth of 4.7 nm (corresponding to a Qfactor of 961). The reflectance dip blue-shifts by only 1.4 nm when the incident angle is varied from 0° to 5° under transverse-electric (TE) polarized incidence. An additional 5.4 nm shift is observed when the incident angle is further increased from 5° to 10°. This angular dependence reveals a relatively robust optical behavior for near-normal incidence (e.g., within 5). More information about the angular and polarization dependence is provided in the Supporting Information (Fig. S5). In Fig. 4(b), the simulations are performed for different incident angles (TEpolarized) with the actual layer thicknesses determined by cross-section scanning electron microscopy (inset in Fig. 4a). The wavelength of the reflectance dips matches well with the measured spectra of the fabricated structure. The deviations in the modulation depth and bandwidth may be due to imperfections in the deposited dielectric layers introducing a larger optical loss (see Fig. S6).

Figure 4c shows the measured emission spectra of the structure for varying temperatures between 100 and 150 °C. As temperature rises, in addition to the intensity increasing, the emission peak shifts to longer wavelengths. After normalizing by the emission of a blackbody, the emittance spectra of the structure for different temperatures are plotted in Fig. 4d. The room temperature emittance (grey colored, calculated as 1 -reflectance) is shown as a reference. The red-shift of the emittance peak wavelength with increasing temperature is observed. For every 10 C increase in temperature, the wavelength of the emittance peak red-shifts by about 5.4 nm, with a total change from 100 to 150 C of 27.1 nm. The shift of the emittance peak is 4.6 times the average bandwidth (5.9 nm) of the measured emittance peaks. Based on the data of thermo-optic coefficients of Ge and SiO 2 and their thermal expansion coefficients, [START_REF] Li | Refractive index of silicon and germanium and its wavelength and temperature derivatives[END_REF][START_REF] Tan | Temperature dependence of refractive index of glassy SiO2 in the infrared wavelength range[END_REF][START_REF] Kagaya | Temperature dependence of the linear thermal expansion coefficient for Si and Ge[END_REF][START_REF] El-Kareh | Fundamentals of Semiconductor Processing Technologies[END_REF] the trend of the emittance wavelength shift can be reproduced by simulation (see Fig. S7). Note that this wavelength tuning behavior is not caused by the thermal degradation of the sample, as can be confirmed by the reflectance spectra taken before and after thermal emission measurements (see Fig. S8). The emittance spectra for different temperatures show similar maximum emittance values about 0.35, and the Q-factors of these emittance peaks are 789, 974, 762, 775, 703, and 684 (average Q-factor of 780) for temperatures from 100 to 150 C. Compared with the room temperature emittance, the emittance peaks at elevated temperatures exhibit smaller Q-factors and modulations, which should be attributed to increased losses in each layer material due to the elevated temperatures. Nonetheless, the Q-factor and tunable range vs. bandwidth ratio are superior to those of any other reported thermal emitters (see the list in Table 1). This temperature sensitive narrowband thermal emitter is expected to be used as a finely tunable monochromatic light source for the analysis of characteristic molecular absorptions in applications such as volatile vapor identification and biosensing.

Conclusions

A hybrid metal-OTS structure with optimized layer thicknesses was investigated as a wavelengthselective and wavelength-tunable thermal emitter in the mid-infrared range. The OTS features allow the structure to store a large amount of energy, resulting in a high-Q-factor resonance, while the metallic mirror helps to avoid light leakage, guaranteeing a large absorption at the resonance wavelength. The structure is optimized for a narrow bandwidth and highly tunable absorptance peak under normal incidence. The hybrid structure was fabricated, and ultra-narrow emission peaks with an average Q-factor of 780 were observed, representing a 4-fold increase over the best experimental results reported so far. Additionally, by taking advantage of the large thermo-optic and thermal expansion coefficients of Ge, the sharp emittance peak was tuned over a range as large as 4.6 times the emittance peak bandwidth. This work demonstrates the possibility of achieving narrowband and tunable thermal emitters in the mid-infrared for use as light sources in chemicaland bio-sensing.

Materials and methods

Simulation details

The simulated reflectance, absorptance, and transmittance spectra, and the near-field electric field magnitude distributions were computed using rigorous coupled-wave analysis (DiffractMOD, Rsoft Design Group, Ossining, USA). The results for r top r bot as a function of wavelength were calculated using the transfer-matrix method. The complex permittivities of Pt and Ti are described by the Drude-Lorentz dispersion model. [START_REF] Rakić | Optical properties of metallic films for vertical-cavity optoelectronic devices[END_REF] The refractive indices of Ge [START_REF] Palik | Handbook of Optical Constants of Solids[END_REF] and SiO 2 (31) are modeled according to literature values.

Fabrication details

The thin films were deposited on a fused quartz substrate using an RF-magnetron sputtering system (CFS-4ES, Shibaura Engineering Works Co., Ltd., Yokohama, Japan). First, a 200 nm Pt layer is deposited with a 10 nm Ti layer underneath as an adhesion layer. Then, a SiO 2 layer is deposited with a 5 nm Ti adhesion layer. Afterward, the rest Ge/SiO 2 layers were deposited. Finally, an additional SiO 2 layer was deposited on the top of the structure as a protection layer (see Fig. S9 for detailed information about optical behavior with and without the protection layer). All the RF sputtering processes were done with Ar at a flowing rate of 13 sccm and a pressure of 0.6-0.7 Pa.

After the deposition, the sample was annealed in air at 150 C for 6 hours to reduce the internal stress.

Measurement details

The reflectance spectra of the fabricated structures were measured with incident angle of 0, 5, and 10 using an FT-IR spectrometer (VIR-300, JASCO Co., Tokyo, Japan) with a spectral data spacing of 0.241 cm -1 , and calibration was done with an Au mirror (see Fig. S10). To measure the thermal emission of the fabricated hybrid structure, the sample was heated up to 150 °C using a heating stage (UH200CV, SPLEAD Co., Tokyo, Japan). Emission from the sample was measured through the external port of a Thermo Nicolet iS50 FT-IR (THERMO FISHER SCIENTIFIC Co., Waltham, MA, USA) with a spectral data spacing of 0.241 cm -1 . A carbon soot film was deposited on a Si substrate to act as a blackbody reference. The surface temperature of the fabricated structure was measured using a K-type thermocouple equipped with a surface temperature probe (IK-500, AS ONE Corp., Osaka, Japan); the temperature of the carbon soot surface was determined using an infrared thermometer (IT-545S, HORIBA Ltd., Kyoto, Japan). Emittance spectra of the fabricated hybrid metal-OTS structure at different temperatures including the room temperature (marked as "RT" in grey color) shown as a reference.

Tables

Figure 1 .

 1 Figure 1. (a) Illustration of the hybrid metal-OTS structure as a wavelength-tunable thermal emitter. (b) Measured emittance spectrum at 130 C. (c) Simulated absorptance spectrum (in black color) for the hybrid metal-OTS structure under normal incident light with t Ge = 280 nm, t SiO2 = 819 nm, and t Pt = 200 nm (here, t Ge , t SiO2 , and t Pt are the thicknesses of the Ge, SiO 2 , and Pt layers, respectively); the calculated r top r bot as a function of wavelength (in pink color) (r top and r bot are the reflection coefficients of a wave incident on the virtual interface, indicated as the white dashed line, with different directions as shown in the inset). (d) The simulated normalized electric field magnitude ( ) distributions of the TP, the OTS, and the hybrid metal-OTS structures at the |𝐸| |𝐸 0 | wavelengths of the absorptance peaks ( is the incident electric field magnitude). The zoomed-|𝐸 0 | in plots with their absorption distributions along the Pt mirror are shown. (e) The absorptance spectra of the TP, OTS, and hybrid metal-OTS structures for normal incidence. The zoom-in plots for the absorptance spectrum of the hybrid metal-OTS structure, and the absorptance and transmittance spectra of the OTS structure are shown on the right side.

Figure 2 .

 2 Figure 2. (a) Simulated absorptance spectra for the hybrid metal-OTS structures with different layer numbers (N top ) in the DBR top under normal incidence, with schematic diagrams shown on the right side. (b) Left: Simulated absorptance spectra for the hybrid metal-OTS structures with N top = 7 for different polarizations and incident angles of light; right: the wavelength differences between the TE-and TM-polarizations calculated for different incident angles.

Figure 3 .

 3 Figure 3. (a) The simulated absorptance varying with light wavelength and thickness of the middle Ge layer t mid under normal incident light on a logarithmic-scale. The absorptance spectra for the three t mid thicknesses are exhibited in the inset on the right side (the structure for t mid = 560 nm is identical to the hybrid metal-OTS structure discussed in Fig. 1 and Structure 4 in Fig. 2. (b) The simulated absorptance spectra of the structures with t mid = 0 and 1120 nm at temperatures from 100 to 150 C, together with their normalized electric field magnitude distributions at the wavelengths of their absorptance peaks at room temperature.

Figure 4 .

 4 Figure 4. Measured (a) and simulated (b) reflectance spectra of the fabricated hybrid metal-OTS structure for three different incident angles (0, 5, and 10) under TE-polarized incident light. The actual thicknesses of the fabricated device were determined by scanning electron microscopy cross-sectional view in the inset, where the red curve reveals the normalized electric field distribution according to the actual dimension. (c) Emission spectrum of the fabricated hybrid metal-OTS structure at different temperatures of 100, 110, 120, 130, 140, and 150 °C. (d)

TOC

  Graphic. 3D schematic diagram of a portable spectroscopy device realized with the proposed thermal emitter that enables narrowband and wavelength-tunable emission in the midinfrared (MIR) range. The thermal emitter consisting of different functional components indicated in white (i.e., DBR and Pt layer) is shown with the superimposed electric field distribution (red curve) at the resonance wavelength. Taking advantage of the large field confinement and temperature-sensitive middle Ge layer, a narrowband and wavelength-tunable thermal emission is realized by controlling the operating temperature. The "normalized emission" spectra are based on the measurement results. By measuring the transmission through the formaldehyde sample at different operating temperatures, the fingerprint of formaldehyde (e.g., C=O stretch as shown in the TOC) can be detected. Since the emission peak is more than one order sharper than the fingerprint of C=O stretch in formaldehyde (the quality factor of emission peak is about 780, while the quality factor of the fingerprint is less than 70), the fingerprint can be detected without the need of optical filters or gratings, thus realizing a portable spectroscopy device. Moreover, the proposed thermal emitter is fabricated by conventional layer deposition methods (e.g., sputtering), enabling a large area and low-cost monochromatic light source in the MIR range.

Table 1 . A summary of recently reported thermal emitters

 1

								Page 26 of 27
	Structures	𝜆	(m)*	ε**		Q	Δλ tune /Δλ band ***	Refs.
	Al-MIM structure		~4.2	0.95		~14	-	[8]
	Quantum-well structure		~10	0.74		72	-	[11]
	Hybrid TP-cavity structure	4.731	0.90		88	0.1	[16]
	Optimized TP structure	6.534	0.82		188	-	[17]
	SiC grating		11.5	0.46		~200	-	[7]
	GST metasurface		~3	0.66~0.94	5.4	0.5	[27]
	VO 2 -sapphire 1D structure		~11	0.3~1.0	4.1	0.6	[19]
	Cr-GST-Au trilayer		~10	~0.7		3.3	0.9	[20]
	GST-Au 1D structure		~10	0.2~1.0	~3.6	1.1	[18]
	GST-TP structure		~4.2	0.43~0.70 47~172	~2	[28]
	Hybrid metal-OTS structure	4.565	0.35		780	4.6	this work
	* (m) is the central wavelength of the emission peak. 𝜆
	**ε indicates peak emittance.
	*** 𝛥𝜆 tune	is the tunable wavelength range;	𝛥𝜆 band	is the bandwidth of the emittance peak.

ACS Paragon Plus Environment ACS Photonics

Acknowledgments

This work was supported through Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Numbers 17H03229 and 19J13668, JSPS CREST (JPMJCR17I3), and Leadership Development Program for Ph.D. (LDPP), The University of Tokyo. This work was conducted at Takeda Sentanchi Supercleanroom, The University of Tokyo, supported by "Nanotechnology Platform Program" of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan, Grant Number JPMXP09F-19-UT-0129. We would like to extend our grateful appreciation to Ms. Ayako Mizushima and Dr. Eric Lebrasseur from The University of Tokyo for important technical support.

Supporting Information: Absorptance spectra of the TP structure with additional DBR pairs, the electric field and absorption distributions of the hybrid metal-OTS structure at the absorptance peak wavelength for normal incidence, the hybrid metal-OTS structure with different metal layers, the fitting procedure of the thermo-optic coefficients for Ge and SiO 2 , the simulated absorptance dispersion diagrams of the fabricated structure, the simulated spectrum with increased dielectric material losses, the simulated trend of the emittance wavelength shift, the stability test showing the reflectance spectrum after the emission measurements, the reflectance spectra with and without the top SiO 2 protection layer, and the calibration of the measured reflectance spectrum.

Conflict of interest


Zhongwei Zhang

Shiqian Hu

Qing Xi

Tsuneyoshi Nakayama

Sebastian Volz

Jie Chen

Baowen Li

Rapid Communications

The ability to control monochromatic and coherent phonons is crucial for many phonon-related applications but remains very challenging. Here, we demonstrate a promising phonon nanocapacitor for storing and emitting phonons, built by the carbon schwarzite based host-guest system (HGS). This phonon nanocapacitor takes advantage of the inherently strong phonon confinement of the hybridized modes in HGS. The monochromaticity and coherence of the stored phonons are well demonstrated via the ultralong phonon lifetime and coherent time. More interestingly, the frequency of the nanocapacitor is widely tunable from gigahertz to several terahertz by engineering the host-guest interaction. Finally, the stored phonons with different polarizations can be emitted separately with the application of uniaxial strain along a particular direction. This work may provide new opportunities for studying the coherent wave effect of phonons.

Introduction. Manipulations of phonons are of primary importance for the exploration of the wave nature of phonons, atomic imaging, and other phononic related fields [1][2][3][4]. For example, the phonon is demonstrated as a new carrier to control and exchange information [2,5,6], where the monochromaticity and coherence of phonons are crucial factors. To achieve this, various optomechanical and electromechanical systems have been developed, including the isolated trapped ions system [7], two-level systems [8], and other mechanical oscillators [1,9].

The phonon interference has been used for phononic control, for instance, the subterahertz (THz) acoustic nanocavities based on GaAs/AlAs [10] and the phonon nanocapacitor based on defective silicon for specific THz phonons [11]. However, the state-of-the-art studies indicate that the manipulation of phonons still encounters notable challenges: (i) Few approaches allow for controlling monochromatic phonons, especially for the ultrahigh THz frequency. (ii) Only specific phonons with fixed frequency can be controlled in various systems [11,12]. (iii) The coherence is difficult to maintain over long distances [9,13]. Therefore, the development of a tunable phonon capacitor is highly desirable.

Host-guest systems (HGSs), such as clathrates, perovskites, and skutterudites [14][15][16], are usually favorable for thermoelectrics and have received extensive attention. Due to the weak interactions with the host cages, the guest atoms in HGSs possess isolated rattling, which results in the hybridized * jie@tongji.edu.cn phonon modes with strong confinement and flat band structure. As an inherent feature of HGSs, the unique phonon confinement of the hybridized mode provides a novel platform to control monochromatic phonons [14,17]. More interestingly, the frequency of hybridized modes (ω h ) generally depends on the mass of the guest atoms and the coupling strength with host cages, exhibiting a strong tunability [18][19][20].

In this work, by utilizing a carbon schwarzite based HGS, we demonstrate a tunable phonon nanocapacitor for storing and emitting monochromatic and coherent phonons. The coherence of the stored phonon is discussed based on the classical coherence function. By considering the rattling motions of guest atoms, we also emphasize the importance of phonon anharmonicity for the phonon nanocapacitor. Our work may provide new opportunities to study thermal transport in nanophononic crystals [5,21,22].

The atomic structure of the phonon nanocapacitor is shown in Fig. 1(a). It is composed of two homogeneous parts: the host cage at the center and HGS layers at two sides. The host cage is a carbon schwarzite structure with a large void space [23], analogous to the empty clathrates [24]. We use classical molecular dynamics (MD) simulations to study the dynamics for the phonon storage and emission processes. MD simulations have been widely used in literature studies on the coherent phonon transport in various superlattice structures [13,[25][26][27]. The covalent C-C bond in the host cage is modeled by the optimized Tersoff potential [28], and the nonbonded interaction between the host cage and guest atom is modeled by the Lennard-Jones (LJ) potential (see Table S1 in Ref. [29]), which can well capture the "on-center" and "off-center" rattling behaviors in HGSs [20,30]. All MD simulations are performed by using the LAMMPS package [31] with a timestep of 0.35 fs.

Phonon storage. We first consider the HGS with on-center rattling motion by using the case I LJ parameters (Table S1 in Ref. [29]) and fixed atom mass (Table S2 in Ref. [29]). Under this condition, the highest frequency for acoustic phonon in the host cage is ω max ≈ 7.8 THz, and the hybridized mode ω h ≈ 2 THz in HGS (see Fig. S1 in Ref. [29]). The whole system is initially equilibrated with a Langevin thermostat at a high temperature of 600 K (> 2π hω max /k B ∼ 375 K), ensuring the full excitation of acoustic phonons in the cage. Afterwards, we remove the high-temperature thermostat and apply the low-temperature heat sinks at two ends to simulate the cooling process [11], as shown in Fig. 1(a). During the cooling process, the temperature difference between cavity and heat sinks leads to the directional propagation of phonon energy, and the phonon energy spectrum in the cavity E (ω) is recorded (see Sec. IV in Ref. [29]). Compared to the wide-frequency spectrum in the initial state [dotted line in In other words, except for the phonons with frequency close to ω h , the majority of phonons escaped from the cavity after the cooling process. We have further verified that the successful storage of monochromatic phonons in the nanocapacitor is quite robust for various initial temperatures and system sizes. The increase of the initial temperature or the cavity/HGS length results in a longer escape time for the majority of phonons, which is typically within 100 ns for various cases (see Sec. V in Ref. [29]).

It should be noted that the storage of monochomatic phonons in the cavity is different from the Bose-Einstein condensation phenomenon where the boson particles condensate to the ground state (with the lowest eigenfrequency) when cooling the system below a critical transition temperature T c . In our study, however, the stored single-frequency mode in the cavity is not the ground state, but is the hybridized mode whose frequency is higher than the ground-state frequency and is highly tunable via the host-guest interaction. Furthermore, the estimated T c in our system is 2.96 K (see Sec. VIII in Ref. [29], and also Ref. [START_REF] Pitaevskii | Bose-Einstein Condensation and Superfluidity[END_REF] therein), which is on the same order of magnitude compared to the literature result [START_REF] Misochko | [END_REF]. The recorded energy spectrum in the cavity reveals that an obvious peak around ω h already appears at t = 20 ps during the cooling process (Fig. S8 in Ref. [29]), for which the averaged temperature of the cavity is 165.9 K, much higher than the estimated T c . With further cooling, the averaged temperature of the cavity drops and the peak around ω h becomes more distinct. Therefore, the absolute value of low-temperature heat sink used in our simulation will only change the monochromaticity of the stored phonon, but will not change the working mechanism of our proposed phonon nanocapacitor.

This phonon storage process can be understood based on the interfacial phonon transmission ( ) between the mirror and cavity parts as described by the acoustic mismatch model [34] (

ω) = 4ρ m υ m g (ω)cosθ m ρ c υ c g (ω)cosθ c ρ m υ m g (ω)cosθ m + ρ c υ c g (ω)cosθ c 2 , ( 1
)
where υ g is the phonon group velocity, ρ is the mass density, cosθ is the directional cosine (see Fig. S2 in Ref. [29]), and the superscripts m and c indicate the mirror and cavity parts, respectively. As depicted in Fig. 1(a), two types of phonon transport events take place: (1) For phonons with ω = ω h , since the group velocity is nonzero in both the mirror and cavity parts, this leads to a nonzero transmission coefficient at the interface according to Eq. ( 1), so that these phonons in the cavity can propagate into HGS given enough time;

(2) For phonons with ω = ω h , the flat band of hybridized modes (zero group velocity) results in the completely forbidden phonon transmission [ (ω h ) = 0] from the cavity to HGS. Therefore, HGS acts as a mirror (total reflection) for the hybridized modes. At the end of the process, only phonons with hybridized modes frequency are confined in the cavity [Fig. 1(b)], achieving the purpose of storing phonons with a particular frequency.

We then discuss the coherent feature of the stored phonons in the cavity. A previous theoretical study by Latour and Chalopin [35] suggests that the phonon coherence length (L c ) is very close to the phonon mean free path ( ) for various frequencies. Therefore, we estimate the coherent length of the stored phonons based on this approximation L c ≈ = τ υ g . To compute the phonon lifetimes τ for different polarizations, we apply the frequency match method (see Sec. IV in Ref. [29]) and Lorentzian fitting to the energy spectrum as

E (k, ω) = 1 + ω(k)-ω h γ 2 , ( 2
)
in which k is the wave vector, is the peak height, and γ is the linewidth obtained from Lorentzian fitting. Due to the small linewidth [Fig. 1(c)], ultralong lifetimes (τ -1 = 2γ ) are obtained for the stored phonons (τ LA ≈ 3644 ns and τ TA ≈ 1568 ns). This observed lifetime is several orders of magnitude higher than the intrinsic phonon lifetimes in typical materials, such as graphene and silicon [36], and is also much longer than the cooling time during the storing process (∼100 ns). Moreover, such ultralong lifetimes have been verified in the nanocapacitor with another size (see Fig. S4 in Ref. [29]). Combined with the group velocity (υ LA g = 12.71 km/s and υ TA g = 8.19 km/s) from the phonon dispersion (Fig. S3 in Ref. [29]), we obtain the coherent length for the stored phonons as L LA c ≈ 46.28 mm and L TA c ≈ 12.84 mm. This length scale is much longer than the length of the cavity (∼20 nm), indicating that the phonon coherence is well preserved in our system.

Analogous to the coherence in classical optics, the phonon coherence can be described by the coherence function C(t ) as [37] C(t ) = q k s ; t + t q k s ; t q k s ; 0 q k s ; 0

, ( 3
)
where q( k s ; t ) is the time-dependent phonon normal mode velocity for the stored phonons (k, s) [see Eq. (S4) in Ref. [29]], and • denotes the ensemble average. Here, we obtain q( k s ; t ) after 100 ns of cooling process, and then calculate the coherence function for LA phonons. As demonstrated by Latour and Chalopin [35], the spatial coherence function follows cos( 2π λ l )e -l/l c , where l is the spatial length, λ is the wavelength, and l c is the coherence length. Similarly, the calculated coherence function shown in Fig. 2(a) follows cos(2πω h t )e -t/τ c , where τ c is the coherence time. Due to the limited calculation time, the envelope of C(t ) is nearly not decayed within 4500 ps. The almost flat envelope of C(t ) close to unity indicates the ultrastrong coherence for the stored phonon mode ω h , which agrees very well with the predicted ultralong coherent length.

To further understand how the coherence is established during the cooling process, we record the envelopes of coherence functions for phonon mode ω h at various time snapshots during the cooling process, as shown in Fig. 2(b). Before the cooling process (t 0 = 0), the whole system is fully thermalized so that the coherence for the phonon mode is quite weak. Correspondingly, the envelope of the coherence function quickly decays to zero at the time t 0 = 0. During the cooling process, the degree of coherence for the phonon mode gradually increases over time, as manifested by an increase of the plateau value for the envelope. This plateau value finally saturates to unity after cooling for 20 ns. This observation agrees very well with the convergent monochromatic peak in energy spectrum in the long-time limit, as shown in Fig. 1

(b).

As the other phonons (ω = ω h ) continuously escape from the cavity during the cooling process, this essentially reduces the numbers of phonon scattering channels that involve the phonon mode ω h , leading to the increasing degree of coherence for the stored phonon mode ω h shown in Fig. 2(b).

Different rattling motions of guest atoms, i.e., on-center or off-center, have a critical impact on the phonon dynamics in HGSs [18,38]. So far, we have only considered the phonon storage in on-center HGSs [case-I in Fig. 3(a)]. When repeating the same simulations for the off-center HGSs with strong anharmonicity [case-II and case-III in Fig. 3(a)], we find that the capability of storing monochromatic phonons is immediately destroyed by the off-center motions [Fig. 3(b)]. To understand such difference, we compute the phonon dispersion at finite temperature based on the spectral energy density method [39]. As shown in Fig. 3(c), there is very limited frequency shift in ω h and negligible band broadening for the hybridized modes at high temperature for the on-center case, due to the weak anharmonicity. As a result, the hybridized mode ω h can be successfully stored in the cavity during the adiabatic cooling. In contrast, a remarkable blueshift in ω h and obvious band broadening are observed at elevated temperature for the off-center cases. These changes eventually lead to the dissipation and escape of phonons with frequency ω h from the cavity via anharmonic phonon scatterings, causing the failure of the phonon storage in the off-center HGS. In real materials, the HGSs with case-I potential well can be achieved by carefully selecting the types of host cages and guest atoms [19,38].

From above discussion, we find that on-center rattling motion and the minimized broadening of hybridized modes are favorable for the storage of monochromatic and coherent phonons in HGS based nanocapacitors. In a typical HGS, the upper and lower bound frequencies of the hybridized modes, ω ub and ω lb , are given by [18]

ω ub = 1 2π (m + M )K e mM ; ω lb = 1 2π 4m f + (m + M )K e -[4m f + (m + M )K e ] 2 -16mMK e f 2mM . (4)
Herein, m and M are the mass of the guest atom and the whole host cage, respectively. K e and f are the effective force constants between guest atom and host cage, and between nearest-neighbor cages, respectively. The detailed derivation of this equation is included in Ref. [29]. By plugging in the simulation parameters (Table S2 in Ref. [29]), Eq. ( 4) predicts ω ub = 2.05 THz and ω lb = 1.98 THz, which are quantitatively in good agreement with the results from lattice dynamics calculations ω ub = 2.12 THz and ω lb = 1.98 THz [Fig. S1(b) in Ref. [29]].

Tunability of phonon nanocapacitor. The feature of minimized broadening in the hybridized mode requires ω ub ≈ ω lb . Such condition can be satisfied in the carbon schwarzite based HGS, as the large cage structure always ensures M m. In this case, the frequency of hybridized modes ω h can be simplified as

ω h ≈ 1 2π K e m . (5)
In a LJ model system, we further introduce a dimensionless scaling factor η to unify the change of m or K e , i.e., m = m 0 /η or K e = ηK 0 . The feasibility of phonon storage for different frequencies is first studied by tuning K e with η. The well preserved highintensity peaks [Fig. 4(a)] and quite small linewidth [inset in Fig. 4(b)] demonstrate the successful storage of monochromatic phonons in a wide range of frequencies. Furthermore, we carry out first-principles calculations to explore the tunability in real materials where m and K e are simultaneously varied [40] (see Sec. VII in Ref. [29], and also Refs. [41][42][43] therein). By substituting the guest atoms in HGS, ω h can be gradually tuned from gigahertz to several terahertz [Fig. 4(b)]. Therefore, highly tunable phonon storage can be achieved in such HGS based nanocapacitor by engineering the interactions between the host cage and guest atoms.

Phonon emission. The phonon emission is another important process for the application of phonon nanocapacitors. In HGSs, the phonon confinement of hybridized modes can be manipulated by the external strains, which allows for the emission of phonons stored in the cavity. To quantify the change in the degree of confinement, we compute the participation ratio (PR) for each phonon mode λ as [44] PR

λ = N i α ε * iα,λ ε iα,λ M i 2 N N i α ε * iα,λ ε iα,λ 2 M 2 i , ( 6
)
where N is the total number of atoms, and ε iα,λ and M i are the αth (x, y, z) eigenvector component of the eigen-mode λ and the mass for the ith atom, respectively. In the pristine (strain-free) HGS, the phonons close to ω h are completely localized (PR = 0) [inset in Fig. 5(a)], which are triplet degenerated and behave as a "locon" [45]. Therefore, three phonon polarizations are simultaneously stored in the cavity. Interestingly, each polarization can be emitted separately by applying a small amount of uniaxial strain along a particular direction. For instance, when applying 0.5% of uniaxial tensile strain along the [100] direction, the LA branch shifts to a notably lower frequency, while the other two branches are unchanged [Fig. 5(a)]. This frequency shift causes a nonzero PR value for LA phonons at the originally stored frequency (ω h ≈ 2 THz), which results in a nonzero (ω h ) and thus the emission of a LA phonon in the cavity [Fig. 5(c)]. Meanwhile, two TA phonons remain stored in the cavity as their frequencies are unchanged. This procedure is also valid for emitting the transverse mode [Figs. 5(b) and 5(d)]. Such phonon emission process can be realized by periodically pumping the stress using a piezoelectric transducer [9].

The carbon schwarzite structure has only one type of cage structure with tunable Gaussian curvature [23], and relatively weak anharmonicity, making it possible to store monochromatic phonons. Moreover, the insertion of the guest atom into the schwarzite cage only leads to negligible size change of the cage structure, due to the strong stiffness of the covalently bonded carbon schwarzite cage and the weak van der Waals interaction between the cage and the guest atom. Therefore, there is almost no strain between the HGS and central cavity. These unique advantages make the carbon schwarzite an ideal candidate for building the phonon nanocapacitor. For other HGSs, such as clathrates [46,47], they may have multiple flat modes and strong phonon anharmonicity, due to the complex crystal symmetry and different types of cage structures, which may pose challenges for the storage of monochromatic phonons. In addition, the local strain generally can exist in real experiments, due to the lattice mismatch between the HGS and cavity, which will cause a frequency shift of the eventually stored phonons compared to the ideal strain-free case.

Summary. We have proposed a phonon nanocapacitor for storing and emitting monochromatic and coherent phonons, by utilizing the inherent hybridized modes in the carbon schwarzite based HGS. Our proposed nanocapacitor shows a wide-frequency tunability, from gigahertz to several terahertz. The ultralong coherent time of the stored phonon is demonstrated via the classical coherence function. The phonon nanocapacitor can be synthesized via the solid-state reaction process [48], and the nanocapacitor with different guest atoms would also be achieved by varying the stoichiometric mixtures during the reaction [49].

FIG. 1 .

 1 FIG. 1. HGS based phonon nanocapacitor for phonon storage. (a) Atomistic structure for the phonon nanocapacitor composed of full HGS (mirror layers) at the two sides and the carbon schwarzite cavity in the center. At both ends, the phonon nanocapacitor is coupled to heat sinks for applying adiabatic cooling along the length direction. The gray and blue atoms are C atoms in the host cage and guest atoms, respectively. The upper plane figures show the perspective views of the unit cell for each part. (b) E (ω) in the cavity for various time snapshots after applying adiabatic cooling from 600 K. (c) E (k, ω) for the stored longitudinal acoustic (LA) and transverse acoustic (TA) phonons (symbols), and the Lorentzian fitting (solid line) after 100 ns of adiabatic cooling. Due to the degeneracy, only one TA mode is shown. The dimensions of the nanocapacitor are L x = 19.6 nm, and L y = L z = 3.9 nm, with three unit cells of mirror layers on each side.

Fig. 1 (

 1 Fig. 1(b)], only one dominant peak around 2 THz persists even after a long time (100 ns) of continuous cooling [solid line in Fig. 1(b)], demonstrating the feasibility of storing a monochromatic phonon in the cavity.In other words, except for the phonons with frequency close to ω h , the majority of phonons escaped from the cavity after the cooling process. We have further verified that the successful storage of monochromatic phonons in the nanocapacitor is quite robust for various initial temperatures and system sizes. The increase of the initial temperature or the cavity/HGS length results in a longer escape time for the majority of phonons, which is typically within 100 ns for various cases (see Sec. V in Ref.[29]).It should be noted that the storage of monochomatic phonons in the cavity is different from the Bose-Einstein condensation phenomenon where the boson particles condensate to the ground state (with the lowest eigenfrequency) when cooling the system below a critical transition temperature T c . In our study, however, the stored single-frequency mode in the cavity is not the ground state, but is the hybridized mode whose frequency is higher than the ground-state frequency and is highly tunable via the host-guest interaction. Furthermore, the estimated T c in our system is 2.96 K (see Sec. VIII in Ref.[29], and also Ref.[START_REF] Pitaevskii | Bose-Einstein Condensation and Superfluidity[END_REF] therein), which is on the same order of magnitude compared to the literature result[START_REF] Misochko | [END_REF]. The recorded energy spectrum in the cavity reveals that an obvious peak around ω h already appears at t = 20 ps during the cooling process (Fig.S8in Ref.[29]), for which the averaged temperature of the cavity is 165.9 K, much higher than the estimated T c . With further cooling, the averaged temperature of the cavity drops and the peak around ω h becomes more distinct. Therefore, the absolute value of low-temperature heat sink used in our simulation will only change the monochromaticity of the stored phonon, but will not change the working mechanism of our proposed phonon nanocapacitor.This phonon storage process can be understood based on the interfacial phonon transmission ( ) between the mirror and cavity parts as described by the acoustic mismatch model[34]

FIG. 2 .

 2 FIG. 2. (a) The coherence function C(t ) for the stored phonon mode ω h after 100 ns of cooling process. (b) The envelopes of C(t ) for the phonon mode ω h at various time snapshots during the cooling process. Here the LA polarization is used in the calculations for both plots.

FIG. 3 .

 3 FIG. 3. The impacts of rattling motions on phonon storage. (a) Three types of potential well. case-I: steep potential well for oncenter rattling; case-II and case-III: gentle and asymmetric potential well for off-center rattling, respectively. Here the gentle potential well has strong phonon anharmonicity, and exhibits off-center rattling at high temperature. (b) Storage of phonons after cooling. (c) Phonon dispersion at 0.5 and 600 K for three types of systems.

FIG. 4 .

 4 FIG. 4. Frequency tunability of the phonon nanocapacitor. (a) The stored E (ω) peaks at different frequencies after 100 ns of cooling. (b) Frequencies of stored phonons as a function of scaling factor η in the model system from MD simulations and in real materials from first-principles calculations, respectively. The solid line shows the ω h ∼ √ η law. The inset in (b) is the linewidth γ for the stored phonons in (a).

FIG. 5 .

 5 FIG. 5. Phonon emission process. (a) and (b) show the participation ratio (PR) of three acoustic branches in HGS under 0.5% of uniaxial tensile strain along the [100] and [010] directions, respectively. The insets in (a) and (b) are the PR for the pristine (strain-free) and strained HGS along the [001] direction, respectively. (c) and (d) show the energy spectrum in the cavity for the LA and TA 1 phonons, respectively, before (solid line) and after (dashed line) applying strain. The insets in (c) and (d) show the other two polarizations are almost unaffected by the strain.

Acknowledgments. This project is supported in part by the grants from the National Natural Science Foundation of China (Grant No. 11890703), and the Science and Technology Commission of Shanghai Municipality (Grants No. 19ZR1478600, No. 18JC1410900, and No. 17ZR1448000). J. C. acknowledges support from the National Youth 1000 Talents Program in China.