# When phononic crystals fail: spatial and spectral limits of phonon interference

Roman Anufriev, Michele Diego, Sebastian Volz, Masahiro Nomura

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Roman Anufriev, Michele Diego, Sebastian Volz, Masahiro Nomura. When phononic crystals fail: spatial and spectral limits of phonon interference. Physical Review Applied, 2025, 24 (6), pp.L061001. 10.1103/4h35-f4kz. hal-05395678

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# When phononic crystals fail: spatial and spectral limits of phonon interference

Roman Anufriev\* and Sebastian Volz

LIMMS, CNRS-IIS UMI 2820, The University of Tokyo, Tokyo 153-8505, Japan

Michele Diego and Masahiro Nomura+

Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan

(Dated: December 3, 2025)

Phononic crystals are artificial periodic structures that leverage phonon wave interference to control mechanical vibrations. Their practical implementations range from evenly planted trees to periodic nanostructures covering mechanical vibrations from seismic waves to sounds and heat. However, spatial and spectral limits at which phonon interference disappears are unknown. Here, we report on experimental observations of a gradual breakdown of phonon interference in two-dimensional nanoscale phononic crystals. Measured phonon dispersion relations reveal critical dimensions and frequencies at which phonons no longer sense the in-plane periodicity of the crystal and enter an intermediate state dominated by the out-of-plane interference before ceasing to interfere altogether. These findings refine the current understanding of nanoscale phonon interference and might be relevant for phononic crystal applications in microelectronics and acoustic quantum computing.

## INTRODUCTION

In nature, periodic phononic structures evolved in multiple species helping their survival for millions of years [1–3]. Today, researchers propose artificial phononic structures that can control seismic waves, sounds, and heat alike [4, 5]. These artificial structures, called phononic crystals, consist of periodic scatterers that induce phonon wave interference and alter the mechanical, acoustic, and thermal properties of the structure. Depending on their application, phononic crystals scale inversely to their target frequency range. For example, alleys of evenly planted trees can dampen low-frequency seismic waves [6], whereas periodic microstructures can mute sounds at kilohertz and megahertz frequencies [7–9]. At even smaller scales, periodic nanostructures can manipulate hypersonic waves at gigahertz frequencies [10–13]. Yet, similar control over heat, carried by phonons at terahertz frequencies, remains a major challenge [14, 15].

Some evidence of phonon interference affecting heat conduction in two-dimensional phononic crystals was obtained at low temperatures [16–18], at which heat is still carried by phonons at gigahertz frequencies. At higher temperatures and frequencies, indirect measurements sparked conflicting interpretations. While some studies suggested that phonon interference can impact phonon transport even at terahertz frequencies [19–21], a number of experiments detected no signs of such impact [18, 22–25]. Thus, realistic limits at which phonons still sense periodic structures and experience interference remain unknown, calling for direct observations of phonon dispersion relation and its changes induced by the interference.

In this work, we investigate phonon interference in two-dimensional phononic crystals by experimentally probing their phonon dispersion relations across various scales and frequencies. We seek to reach critical dimensions and wavelengths at which phonon interference breaks down. These observations aim to uncover the nature of

the transition from coherent to incoherent phonon behaviour, which so far remained a mystery of nanoscale phonon transport.

Our phononic crystals were fabricated on a single-crystalline suspended silicon membrane with thickness of 200 nm atomically flat surfaces. Using the electron-beam lithography followed by reactive ion etching, we patterned the membrane with two-dimensional arrays of holes that formed phononic crystals. The resulting membrane had several $100 \times 70$ µm regions containing patterns with different periods ($a$) but the same 0.7 diameter-to-period ratio, which is known to be sufficient to produce a phononic bandgap [26]. Figure 1 shows six periodic patterns chosen to cover the range of phononic crystal periods found in the literature. Our $a = 150$ nm sample is close to some of the smallest fabricated nanostructures [24, 25, 27–29]. The sample with $a = 300$ nm is the widely used standard that previously showed signs of phonon interference [10, 12, 18, 30, 31]. The intermediate samples ($a = 600$ and $1100$ nm) are reminiscent of those in the pioneering works [16, 19, 32], while the microscale samples ($a = 2425$ and $4000$ nm) aim to capture the interference effects suggested by subkelvin experiments [16, 17, 33]. As all samples were fabricated simultaneously on the same membrane, they share [110] crystallographic orientation, 2 nm surface roughness of holes, and native oxidation.

To measure the phonon dispersion relations at gigahertz frequencies, we used the Brillouin-Mandelstam light scattering (BMLS) spectroscopy [34], illustrated in Fig. 1g. In this method, a 2 mW laser beam ($\lambda = 532$ nm), linearly polarized along [110] direction of the sample, is focused through a microscope objective ($\times 20$, NA = 0.40) on the sample surface inclined at an angle $\theta$ to the incident beam. This angle can be changed to probe different wave numbers $q$ related to the angle as $q = 4\pi \sin(\theta)/\lambda$ [35]. The incident photons lose or gain a tiny fraction of their energy upon surface scattering by exciting or absorbing lattice vibrations at natural frequencies of the structure. The scattered photons are col-
<figure><img src="image_3.png" /><figcaption>FIG. 1. (a-f) Scanning electron microscopy images of phononic crystals with different periods. (g) Illustration of the light scattering spectroscopy techniques showing laser beam inducing acoustic waves in the crystal. (h) Scheme of the first (FBZ) and irreducible (IBZ) Brillouin zones corresponding to the microscopy images.</figcaption></figure>

lected through the same objective, equipped with a 2 mm pin hole, and directed into the tandem Fabry-Pérot interferometer measuring the energy lost or gained upon the scattering. The interferometer is equipped with a photon detector and a pair of optical cavities that simultaneously scan their resonant frequencies. Mirrors of the cavities are aligned by mechanical actuators for coarse alignment and piezoelectric transducers for fine adjustment. During the measurements, mirror parallelism is maintained by an active feedback system controlling the piezoelectric transducers. This feedback system maintains precise alignment for hundreds of hours required for these measurements. For example, at the rate of a few photons per second, the acquisition of a spectrum for one wave number takes about 24 hours in the low-gigahertz range or one week in the high-gigahertz range.

In addition, we used standard Raman spectroscopy [36] to probe the terahertz frequency range. Like in the BMLS spectroscopy, the laser beam ($\theta = 0^\circ$), linearly polarized along [110] direction of the sample, is scattered by the sample surface and collected through $\times 100$ microscope objective. The spectrum of scattered photons is captured by a Raman spectrometer equipped with a diffraction grating and a photon detector, which can resolve only relatively large changes in photon wavelength. Both methods are described in detail in our previous work [35].

The measured spectra typically consist of an elastic Rayleigh scattering peak, which was removed from the plots for clarity, and several inelastic scattering peaks. The inelastic peaks represent the natural phonon fre-

quencies of the sample, and their Stokes shifts from the Rayleigh peak give the frequencies of phonon modes. Figure 2a shows a typical BMLS spectrum with Lorentzian fits of the inelastic peaks. By measuring spectra for different wave numbers along $\Gamma - X$ direction of the first Brillouin zone, shown in Fig. 1h, we can map the phonon dispersion relation. In our experiment, the wave number range is limited to 0.05 - 0.18 $nm^{-1}$ due strong Rayleigh scattering at lower wave number and high incidence angles at higher wave numbers.

Figure 2b shows an example of dispersion relation measured on a reference 200-nm-thick membrane without holes. The color map represents the intensity of the raw experimental signal, while points and their error bars indicate the position and width of the fitted peaks. The measured dispersion reveals several phonon modes corresponding to well-known symmetric and anti-symmetric Lamb waves of a thin plate [37]. Note that phonon modes with negligible out-of-plane displacement components do not appear in the measured spectra due to the chosen polarization of the excitation and polarization selection of the interferometer. This selection rule enables us to observe separate phonon modes even in dense phonon dispersions relations. The measured modes closely resemble previous measurements [12, 38] and predictions of the theory of elasticity. Specifically, the predictions of the elastodynamic wave equation, solved for the sample geometry using finite element method (FEM) with anisotropic elastic constants for [110] crystal orientation, listed in our previous work [35]. However, shifts from the predicted modes occur for some of the measured modes
<figure><img src="image_4.png" /><figcaption>FIG. 2. (a) An example BMLS spectrum with the peaks of a multi-Lorentzian fit. (b-h) Phonon dispersion relations measured in Γ-X direction for a reference membrane and phononic crystals of different periods. Color map shows intensity of the signal, while points and their error bars indicate the positions and widths of the Lorentzian fits. Nanoscale samples (c,d) display flat modes characteristic of in-plane phonon interference, in agreement with FEM simulations. However, dispersion relations of the intermediate samples (e,f) deviate from theoretical predictions. In the microscale samples (g,h), the in-plane interference breaks down completely and dispersion relation reverts to that of the reference membrane.</figcaption></figure>

and are likely caused by imperfect alignment of the setup, slight difference between simulated and real shapes and dimensions of the structures [35], and material softening in the narrow regions [24].

As periodic holes are introduced to the membrane, the resulting phonon interference changes the dispersion relation, as shown in Fig. 2c for the period of $a = 150$ nm. The Lamb-wave branches become flatter and open a bandgap at 18 GHz, in agreement with the FEM simulations. In addition, new branches emerge due to the Brillouin zone folding in both in-plane directions. These

effects become more apparent for the twice longer period $a = 300$ nm. Figure 2d shows multiple branches displaying flattened curvature and symmetry around the edge of the first Brillouin zone, whereas the bandgap shifts to the lower frequency. Indeed, since the wavelength of phonons experiencing Bragg diffraction in a phononic crystal is proportional[14] to its period as $\lambda_{ph} \sim 2a$, the frequency ($f_{bg} \sim 1/\lambda_{ph}$) of the bandgap caused by this diffraction becomes twice lower as the period becomes twice longer. Thus, both nanoscale phononic crystals ($a = 150$ and 300 nm) cause expected phonon interference and change
the dispersion relation in agreement with the theory of elasticity.

In the sample with a longer period of $a = 600$ nm, even more flat branches appear in the dispersion relation, as shown in Fig. 2e. However, the phonon modes were detected only below 28 GHz with no clear peaks at higher frequencies. This might be the first sign of vanishing phonon interference, as the high-frequency phonons become unable to feel pattern periodicity. Indeed, as the period exceeds one micron ($a = 1100$ nm), even the low-frequency branches weaken, leaving only a few optically active branches visible. Moreover, some modes appear similar to the membrane modes, indicated by dashed lines. Finally, the microscale structures ($a = 2425$ and 4000 nm) show no signs of the dense dispersion relation of phononic crystal predicted by the theory of elasticity (not shown in the plots for clarity). Instead, their dispersion relations revert to that of an unpatterned membrane and match the simulated membrane dispersion, of both structures shown by dashed lines in Fig. 2g,h. This surprising transition suggests that in sufficiently large phononic crystals, phonons no longer sense periodicity of the in-plane pattern yet still sense the out-of-plane confinement of the membrane.

Since the period of phononic crystals is linked to the affected phonon frequencies, a similar transition to the membrane dispersion relation may occur even in smaller phononic crystals if we measure sufficiently high frequencies. To test this hypothesis, we probed optically weak upper-gigahertz modes by recalibrating the Fabry-Pérot interferometer and performing week-long spectra acquisitions for each sample. Figure 3a,b shows measured lower and upper-gigahertz BMLS spectra. In the lower-gigahertz range, the nanoscale phononic crystals display modes predicted by the theory of elasticity for their corresponding geometry, as we observed in their dispersion relations. Inserts of Fig. 3 show simulated unit cell displacements at the frequencies of observed phonon modes.

However, in the upper-gigahertz range, the spectra blend phononic crystal modes with modes of a plain membrane at 19.6, 32, 37 GHz, and other frequencies, indicated by dashed lines. For the phononic crystals of shorter periods, the membrane modes occur at proportionally higher frequencies. Nevertheless, all the periods, except the shortest, display some membrane modes at sufficiently high frequencies, thus confirming our hypothesis of the spectral transition to the membrane dispersion relation.

Finally, to reach the frequencies where even the membrane modes disappear, we performed Raman spectroscopy. Figure 3c shows the measured terahertz spectra of all samples displaying identical peaks. These peaks correspond to first and second-order Raman scattering on bulk modes of silicon, indicated by dotted lines. Thus, terahertz phonons are incoherent as they no longer experience any detectable interference due to structure boundaries Rather, the dispersion relation in the terahertz range is defined by the atomic periodicity of silicon

crystal lattice. For example, the highest peak around 16 THz corresponds to the first-order scattering on the optical modes (LO) of bulk silicon, while the peaks at 9 and 29 THz are caused by the second-order scattering on transverse acoustic (2TA) and transverse optical (2TO) modes. Likewise, all other weaker peaks in the terahertz range correspond to well-known scattering processes of bulk silicon [39].

Let us now discuss the implications of these results. Measured dispersion relations show that nanoscale phononic crystals ($a \le 300$ nm) cause phonon interference in agreement with the theory of elasticity up to at least 30 GHz, consistent with previous spectroscopic measurements [12, 30, 31, 35]. We will refer to this behaviour as “in-plane coherent”, to contrast it with the “out-of-plane coherent” behaviour observed at larger scales and higher frequencies. In this out-of-plane coherent regime, phonons still experience confinement caused by the top and bottom surfaces but become insensitive to the periodic holes. For example, samples with the periods of 600 and 1100 nm transitioned to the out-of-plane coherence roughly above 24 GHz, whereas in microscale samples ($a = 2425$ and 4000 nm), the out-of-plane coherence defines dispersion relation at least above 6 GHz. Thus, the transition from in-plane to out-of-plane coherence, albeit gradual, occurs at frequencies inversely proportional to the period of phononic crystal.

This loss of in-plane coherence does not appear to be caused by diffuse surface scattering due to mostly elastic scattering at these wavelengths. Indeed, the probability of elastic surfaces scattering can be estimated [40] as $\exp(16\pi^2\sigma^2/\lambda_{ph}^2)$, which for the surface roughness $\sigma = 2$ nm and phonon wavelength $\lambda_{ph} = 100$ nm ($k = 0.01$ nm$^{-1}$) gives the probability of 94%. Likewise, various phonon-phonon processes, such as Umklapp scattering or Akhiezer damping, are unlikely to cause observed loss of coherence since phonon mean free path at gigahertz frequencies is still estimated at tens of microns despite these processes [41–44]. Therefore, the observed transition might be a wave phenomenon likely originating from the finite phonon coherence length and competing influences of the in-plane interference and the out-of-plane membrane confinement.

These findings challenge one prevailing assumption in the field. Current literature presumes the existence of a size [17, 29] or frequency [18, 30] limit, below which the phonon transport is controlled by dispersion relation of phononic crystal and remains coherent. Above this limit, phonons are presumed to cease interfering, phonon dispersion relation returns to that of bulk material, and phonon transport becomes incoherent. Our data suggest that this picture is incomplete. Although the dispersion relation of phononic crystals indeed disappears above certain size and frequency limits, only the in-plane coherence is lost. The out-of-plane coherence can persist up to much higher, yet unknown, limits. For instance, unpatterned membranes can support Lamb waves up to a few hundred gigahertz [30, 45, 46]. Hence, at room tem-
<figure><img src="image_5.png" /><figcaption>FIG. 3. At gigahertz frequencies, (a,b) BMLS spectra of most phononic crystals, measured at $q = 0.01$ nm$^{-1}$, start displaying membrane modes at sufficiently high frequencies, which signifies a transition from in-plane to out-of-plane coherent regime. Inserts show displacements predicted by FEM simulations for some of the prominent phonon modes. (c) At terahertz frequencies, Raman spectroscopy reveals only bulk silicon modes related to atomic vibrations, depicted in the inserts, which implies the incoherent regime.</figcaption></figure>

perature, the true incoherent regime with bulk phonon properties is likely reached around several hundred gigahertz, and certainly above 1 THz, as evidenced by our high-frequency data.

In the context of thermal transport, our findings imply that in-plane interference affects heat conduction only at low temperatures. For example, at subkelvin temperatures, phonon spectra span only a few tens of gigahertz, and thus experimental works [16, 17, 33] assumed coherent subkelvin thermal conductance, which breaks as temperature or period is increased. Our measurements on similar microscale phononic crystals align with these results. The $a = 1100$ nm sample indeed displays some coherent phonon modes in the low-gigahertz range, which is consistent with the measurements and interpretations on the $a = 970$ nm sample by Zen et al. [16]. Our microscale samples ($a = 2425$ and $4000$ nm) display mainly phonon modes of a membrane above 6 GHz, which might explain apparent weakening of coherent conduction in thermal measurements on samples with the same periods [17, 33]. At higher temperatures, the phonon spectrum extends to the terahertz range, of which only a small gigahertz part is affected by the in-plane interference. This observation is consistent with reports [18, 25] of either minimal or undetectable impact of phonon interference

on heat conduction at temperatures above 4 K. Thus, at room temperature, the impact of phonon interference in such structures must be negligible, contrary to optimistic suggestions found throughout literature [19, 21, 32].

In conclusion, we investigated phonon interference in phononic crystals and observed a transition from the in-plane to the out-of-plane coherent phonon transport, followed by the final transition into the incoherent regime. The theory of elasticity failed to capture this behaviour and deviated from the experiment at relatively low frequencies. This discrepancy likely occurs due to the assumption of infinite phonon coherence length and calls for improved theoretical frameworks that would account for the limited length of phonon coherence. This refined picture of phonon interference at nanoscale might be relevant for applications of phononic crystals in microelectronics [33, 47] and acoustic quantum computing [48, 49]. As acoustic quantum systems rely on phononic bandgaps at increasingly higher temperatures and gigahertz frequencies [49], understanding the limits of phonon interference forming the bandgaps becomes important for achieving low-noise and high-coherence systems. Likewise, phonon interference in the gigahertz range is essential for controlling phonons in optomechanics [50, 51] and heat in low-temperature sensors [33]. Our data sug-
gest that coherent control over frequencies up to 25 GHz in silicon require phononic crystals with periods below a micron, while higher gigahertz frequencies require even shorter periods at the scale of hundred nanometers.

Authors thank Ryoto Yanagisawa and Laurent Jalabert for advice on fabrication and measurements. This work has been partially funded by the CNRS Energy unit (Cellule Energie) through the project SEGMENT.

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