We propose a nanocavity design which is able to confine acoustic phonons by adiabatically changing the thicknesses of a GaAs/AlAs superlattice. By means of high resolution Raman scattering, we experimentally demonstrate the presence of a confined acoustic mode around 350 GHz. We observe an excellent agreement between the experimental data and numerical simulations based on a photoelastic model. We demonstrate that the spatial profile of the confined mode can be tuned by changing the magnitude of the adiabatic deformation, leading to strong variations of its mechanical quality factor and Raman scattering cross-section. The reported design could significantly improve the acoustic confinement properties of nanophononic and optomechanical devices.

Acoustic cavities confine mechanical vibrations in one or more directions of space.1 They are at the base of devices which are able to generate, manipulate, and detect high frequency acoustic phonons.2,3 Furthermore, such systems are at the core of the development of optomechanical devices.4,5 The well-established design of one-dimensional acoustic nanocavities are phononic Fabry Perot resonators capable of operating in the technologically relevant sub-THz range.6–8 They are built by enclosing a spacer between two acoustic distributed Bragg reflectors (DBRs).9,10 Most of the mechanical properties of an acoustic DBR can be described by an acoustic band diagram11 in the Bloch mode formalism.12 

In the optical domain, sophisticated cavity designs have emerged over the years, providing stronger spatial confinement, higher quality factors as well as reduced sensitivity to nanofabrication imperfections.13–15 One efficient design proposed in the optical domain consists of introducing tapered regions where the periodicity of a photonic crystal is adiabatically broken. This approach allows for reduced optical losses and hence increased optical Q-factors when going to 3D confinement. This strategy has been adopted in several optical systems, such as 2-dimensional photonic crystal membranes,16 nanobeams,17,18 waveguides,19 and micropillars.20,21

In nanophononics, with few exceptions,22–24 only the standard Fabry-Perot approach has been implemented. Taking advantage of the periodic character in nanophononic crystals, engineering of the band structure can be performed. For instance, based on the tight-binding model, phononic band structures arise from the coupling of multiple resonators and effective potentials can be introduced by changing the resonance energy of consecutive cavities.25–27 By taking a similar band engineering approach, but based on the nearly free electron model, we confine a phononic state by adiabatically perturbing the band structure of a periodic superlattice. As in the case of optics, this design could lead to the development of 3-dimensional mechanical resonators with quality factors overcoming the ones currently achieved with standard Fabry-Perot designs.4,28

We report the design and the experimental study of an adiabatic acoustic cavity operating at 350 GHz. By progressively changing the periodicity of an acoustic DBR, we adiabatically transform the acoustic band diagram of the system, leading to the generation of a confined mechanical state. We probed the presence of such a confined phononic state by high resolution Raman scattering. Furthermore, by changing the magnitude of the adiabatic transformation, we numerically demonstrate that we can significantly transform the spatial profile of the confined mode, leading to major changes in its mechanical Q-factor and Raman scattering cross-section.

We start the conception of the adiabatic cavity by designing an acoustic GaAs/AlAs DBR made of 29 layer pairs. The layer thicknesses of AlAs and GaAs are 12 nm and 3.4 nm, respectively. By choosing these thicknesses, we obtain a (λ4,3λ4) acoustic DBR, where λ corresponds to the wavelength of the acoustic phonons in GaAs and AlAs, respectively, for a frequency of 350 GHz.9 By gradually changing the period thickness [Fig. 1(a), color map], we introduce an adiabatic perturbation at the center of the structure. The envelope of the perturbation has the shape of a sin2 function and an amplitude of 7% and extends over 12.5 layer pairs. We compute the acoustic reflectivity of the system embedded in a GaAs matrix, as shown by the cyan curve in Fig. 1(b). We note the presence of a sharp dip inside the stop band at 353 GHz, corresponding to a confined mode. The dashed red line represents the simulated reflectivity spectrum of a DBR without any adiabatic perturbation. The calculated spatial profile of the confined mode is shown in Fig. 1(a) (black curve). The displacement has been normalized to the displacement amplitude obtained in the GaAs substrate. The mode is confined at the center of the structure and decays exponentially over 150 nm when we move away from the adiabatically perturbed region.

FIG. 1.

(a) Displacement profile of the confined acoustic mode inside the adiabatic structure. The color map shows the adiabatic perturbation inside the structure. Green and blue regions correspond to AlAs and GaAs layers, respectively. Inset: evolution of the local acoustic minigap as a function of the position in the sample. (b) Acoustic reflectivity for the adiabatic nanocavity. The spectral position of the confined mode plotted in (a) is marked by a cyan square. The acoustic reflectivity of the unperturbed DBR is shown in red and vertically offset for better visibility.

FIG. 1.

(a) Displacement profile of the confined acoustic mode inside the adiabatic structure. The color map shows the adiabatic perturbation inside the structure. Green and blue regions correspond to AlAs and GaAs layers, respectively. Inset: evolution of the local acoustic minigap as a function of the position in the sample. (b) Acoustic reflectivity for the adiabatic nanocavity. The spectral position of the confined mode plotted in (a) is marked by a cyan square. The acoustic reflectivity of the unperturbed DBR is shown in red and vertically offset for better visibility.

Close modal

The presence of a confined state can be explained by locally applying the Bloch mode formalism in the aperiodic part of the sample and in particular, for one period of alternating AlAs/GaAs layer.20,29,30 We calculate for every pair of AlAs/GaAs layers the corresponding local acoustic band diagram. In the inset in Fig. 1(a), we show the first zone center bandgap (yellow) as a function of the position in the sample. The eigenfrequency of the confined mode is represented by the horizontal dashed line. By progressively increasing the thickness of the layers, we gradually redshift the position of the local acoustic bandgap of the system. At the center of the perturbed region, the confined mode is outside the bandgap and is therefore allowed to propagate. However, by moving away from the center, the mode enters adiabatically into the bandgap and is progressively reflected by the DBRs, leading to its confinement.

A GaAs/AlAs-based sample was fabricated by molecular beam epitaxy (MBE) on a (001) GaAs substrate, according to the design described above. The system was characterized by Raman scattering spectroscopy performed at room temperature. We used a Ti-sapphire tunable laser set at a wavelength of 913 nm, and the collected spectra were dispersed using a double HIIRD2 Jobin Yvon spectrometer equipped with a liquid N2 charge coupled device (CCD). The acoustic cavity is embedded between two optical Al0.1Ga0.9As/Al0.95Ga0.05As DBRs and constitutes the 3λ02 spacer of an optical microcavity (λ0 corresponds to the wavelength of the confined optical mode). The top (bottom) optical DBR is made of 14 (18) pairs. Due to the very small lattice mismatch between GaAs and AlAs, the residual strain induced during the fabrication process is negligible. Fabricating the acoustic structure inside an optical cavity enhances the Raman scattering signals up to a factor 105 (Refs. 31–34) and modifies the Raman scattering selection rules, allowing us to detect signals associated with the confined mode in a backscattering (BS) configuration. We collected the scattered light at normal incidence, whereas the excitation laser was incident with an angle of 10°. As the sample has been grown with a thickness gradient, we could tune the resonance frequency of the collection mode between 0.8 and 1 μm by changing the position of the laser spot. When the frequency of the scattered photons corresponded to the energy of the collection mode, we were in a condition of single optical resonance. We further enhanced the intensity of Raman signals by taking advantage of the in-plane dispersion relation of the optical cavity: by carefully changing the incidence angle of the laser, it was possible to set the incoming photons in resonance with the excitation mode. When both the spot position and the laser incidence angle were set in order to maximize a Raman signal, we were in the condition of double optical resonance (DOR).31,35

In Fig. 2(a) (Black curve), we show the simulated acoustic band diagram of the (λ4,3λ4) GaAs/AlAs acoustic DBR used for the conception of our sample, without any adiabatic defect. The measured Raman spectrum is presented in Fig. 2(b). The frequency of the inelastically scattered light is flaser+Δf, where Δf is the frequency shift introduced during the Raman process and flaser is the frequency of the incident laser. The DOR condition was optimized for maximizing Raman signals for Δf 350 GHz. We observe four clear Raman peaks in the measured spectrum. The most intense one is well located in the frequency interval of the zone-center acoustic minigap, also marked in Fig. 2(b) by a grey area. Its frequency matches well the one of the confined mechanical mode, indicated by the orange line in Fig. 2(a). This Raman peak is generated by the cavity mode (CM) confined in the adiabatic structure.

FIG. 2.

(a) Calculated acoustic band diagram for the DBR without adiabatic defects (black curve). The zone center acoustic minigap is marked by a grey area. The orange solid line corresponds to the resonance frequency of the confined mechanical mode in the adiabatic cavity. (b) Measured Raman spectrum (black curve), simulated Raman spectrum (pink curve), and spectral position of the zone center acoustic minigap (grey area). The peak corresponding to the confined mode is marked by the label CM.

FIG. 2.

(a) Calculated acoustic band diagram for the DBR without adiabatic defects (black curve). The zone center acoustic minigap is marked by a grey area. The orange solid line corresponds to the resonance frequency of the confined mechanical mode in the adiabatic cavity. (b) Measured Raman spectrum (black curve), simulated Raman spectrum (pink curve), and spectral position of the zone center acoustic minigap (grey area). The peak corresponding to the confined mode is marked by the label CM.

Close modal

We implemented a photoelastic model to calculate the Raman spectrum of the structure.9,36–39 The simulated Raman spectrum is shown in Fig. 2(b) (pink curve), after convoluting it with a Gaussian curve (2σ=7GHz) to account for the experimental resolution. We note that the simulated spectrum reproduces all the features of the experimental data very well, accounting for the good quality of the sample growth.40 The peaks located around 310 GHz and 380 GHz would be normally observable in a backscattering (BS) geometry in structures with no optical confinement. These mechanical modes are localized in the DBRs (propagative modes). The mode at 369 GHz, in contrast, is usually active in forward scattering (FS) geometry. Small differences between the experiments and simulations in the relative intensities can be related to the DOR condition since different frequency intervals of the Raman spectrum present different optical enhancements. In addition, there are variations in the photoelastic constants along the structure related to electronic resonance effects in layered systems. Such effects have not been taken into account in our simulations. The vertical dashed lines in Figs. 2(a) and 2(b) correspond to the condition q=2klaser, where klaser is the wavevector of the incident laser. They indicate the modes usually Raman active in a BS geometry for a superlattice.11,36 We observe that in the measured Raman spectrum the peaks associated with backscattering are red shifted with respect to these frequencies. The introduction of an adiabatic defect in a superlattice also affects the spectral position of the Raman peaks associated with propagative modes, as it increases the average thicknesses of the layers at the center of the structure.

We numerically investigated the confinement properties of the adiabatic cavity by exploring the effect of the adiabatic sin2 transformation. We define α as the maximum amplitude of the adiabatic transformation introduced in the system. In Fig. 3(a), the color map shows the evolution of reflectivity spectra of the adiabatic cavity as a function of α. The white region corresponds to the acoustic stop band located around 350 GHz. Starting from a perfect DBR, we observe that when we increase the magnitude of α, a sharp dip appears and gradually red-shifts inside the stop band. This dip corresponds to the confined acoustic mode and reaches the center of the stop band for α=7% (dashed cyan line), as also shown in Fig. 1(b). By further increasing α, a second sharp dip in the stop band appears, evidencing the presence of a second confined mode. Both dips are clearly visible for α=11% (dashed green line). Eventually, by raising α up to 15% (dashed red line), the first mode disappears in the Bragg oscillations and the second mode reaches the center of the acoustic stop band.

FIG. 3.

(a) Color map showing the evolution of the acoustic reflectivity spectra for the adiabatic cavity as a function of α. The dashed cyan, green, and red lines mark the values of α corresponding to 7%, 11%, and 15% adiabatic changes, respectively. The labels “mode 1” and “mode 2” indicate the evolution of the reflectivity dips for the first and second confined mode, respectively. (b) Evolution of the mechanical quality factor of the first (black curve) and second (brown curve) confined modes as a function of the adiabatic transformation. The points corresponding to α=7% and α=15% are marked by a cyan square and red triangle, respectively. Insets: spatial profiles of the first (cyan curve) and second (red curve) confined mode for α=7% and α=15%, respectively. (c) Simulated Raman spectra for α=7% (cyan curve), α=11% (green curve), and α=15% (red curve). An offset between the spectra has been introduced for clarity. The square and triangle symbols indicate the resonance frequencies of the first and second mode, respectively.

FIG. 3.

(a) Color map showing the evolution of the acoustic reflectivity spectra for the adiabatic cavity as a function of α. The dashed cyan, green, and red lines mark the values of α corresponding to 7%, 11%, and 15% adiabatic changes, respectively. The labels “mode 1” and “mode 2” indicate the evolution of the reflectivity dips for the first and second confined mode, respectively. (b) Evolution of the mechanical quality factor of the first (black curve) and second (brown curve) confined modes as a function of the adiabatic transformation. The points corresponding to α=7% and α=15% are marked by a cyan square and red triangle, respectively. Insets: spatial profiles of the first (cyan curve) and second (red curve) confined mode for α=7% and α=15%, respectively. (c) Simulated Raman spectra for α=7% (cyan curve), α=11% (green curve), and α=15% (red curve). An offset between the spectra has been introduced for clarity. The square and triangle symbols indicate the resonance frequencies of the first and second mode, respectively.

Close modal

The spatial profiles of the first and second confined modes are plotted in the insets in Fig. 3(b), calculated for α=7% and α=15%, respectively. Both modes are confined at the center of the structure. However, the second mode presents two maxima in its displacement pattern. It is therefore possible to select the desired spatial profile of the mode which is optimally confined by changing the amplitude of the adiabatic deformation and to finely tune its mechanical resonance frequency.

To characterize the mechanical resonator performance, we studied the evolution of the confinement properties of the two considered modes as function of the parameter α [Fig. 3(b)]. The values of the mechanical quality factors (Q-factors) increase when the resonance frequencies approach the center of the acoustic minigap. Maximal values for the mechanical Q-factors of the first and second modes are reached for α=7% (Qmechanical = 1520, cyan square) and α=15% (Qmechanical = 1220, red triangle), respectively. To compare this design to a standard Fabry-Perot cavity, we have simulated the Q-factor of an acoustic Fabry-Perot resonator composed of 14 (λ4,3λ4) GaAs/AlAs layer pairs for each DBR and one λ2 AlAs spacer. This structure contains the same number of layers as the adiabatic system. The Q-factor reached is 1570, very close to the value reached for α=7%. For the acoustic Fabry-Perot resonator, the effective length of the confined mode is LeffFP=92nm. For an adiabatic cavity with α=7%, the effective length is LeffAdiab=133nm. Such a difference can be explained by considering that for an adiabatic cavity, the Bragg condition is reached in a smooth way when moving away from the center of the structure, leading to a spatially extended mode. This is in strong contrast to a standard Fabry-Perot resonator, where a localized mode is generated by abruptly introducing a defect.

In Fig. 3(c), we plot the simulated Raman spectra around 350 GHz for cavities with α=7% (cyan curve), α=11% (green curve), and α=15% (red curve). The resonance frequencies of the first and second modes are marked in the three curves by a square and a triangle, respectively. For α=7%, the first confined mode is clearly Raman active [as already shown in Fig. 2(c)]. For α=11%, both the first and the second modes are present. However, as shown by the green curve, the only Raman active mode is the first mode. For α=15%, the only existing confined mode is the second one. Also, in this case, no particular feature in the Raman spectrum indicates its presence. The second mode presents a different spatial symmetry in strain with respect to the first one, resulting in a Raman inactive mode. By tuning the parameter α, it is thus possible to tailor the spatial profile of the adiabatic confined mode and select its symmetry.

In conclusion, we demonstrated the adiabatic confinement of longitudinal acoustic phonons at a resonance frequency of 350 GHz by progressively breaking the periodicity of an acoustic superlattice. We probed the presence of a confined mode by performing Raman scattering spectroscopy experiments in a DOR configuration. The presented results were obtained on a sample where the DBR periodicity was adiabatically broken with α=7%, which is technologically challenging to fabricate even by MBE, due to the nanometric thicknesses of the layers. Numerical simulations based on transfer matrix calculations and a photoelastic model well reproduce our experimental data, accounting for the high quality of the MBE grown sample and showing the feasibility of actually fabricating these systems. We investigated the impact of the adiabatic transformation magnitude on the spatial profile of the confined modes and on their mechanical quality factors. We demonstrated that in the presented adiabatic cavity, the energy and spatial profile of the confined modes can be tailored by controlling the artificial phononic potential. As it has already been shown for standard acoustic Fabry-Perot designs,4,28 it is possible to fabricate out of these planar structures 3-dimensional optomechanical microresonators operating at extremely high mechanical frequencies, and for which the confined mechanical and optical modes strongly interact. Combining the simultaneous localization of photons and phonons, the reported system has the potential of being at the heart of a future generation of optomechanical resonators based on DBR structures.

This work was partially supported by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d'Avenir” program (LabexNanoSaclay, reference: ANR-10-LABX-0035), the ERC Starting Grant No. 715939 NanoPhennec, the French Agence Nationale pour la Recherche (grant ANR QDOM), and the French RENATECH network.

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