Evidence of phonon Anderson localization on the thermal properties of disordered atomic systems

Anderson localization originates from coherent backscattering and destructive interference between multiple wave scatterings in strongly disordered media.^1–3 It is a captivating wave phenomenon that strongly hinders the wave diffusion, which was first observed in electrons since the pioneering work of Anderson.⁴ However, the notion of Anderson localization is not limited to quantum electrons models, but also applies to classical waves in disordered media including acoustic waves,^5–7 electromagnetic waves,⁸ and hydrodynamical waves.⁹ 

Thermal phonons, as quasiparticles, have both particle and wave natures. While the particle pictures have been extensively studied using the phonon Boltzmann transport equation, the wave picture is much less investigated.¹⁰ The nature of phonon wave transport is fundamentally dependent on the magnitude of the phonon mean free path (MFP), relative to the system size. If the MFP is comparatively shorter than the system size, phonon propagates in the diffusive regime, similar to a classical Brownian particle.¹¹ If the system size is smaller than MFP, ballistic propagation occurs. In the ballistic regime, long-wavelength phonons dominate thermal transport, and those phonons are coherent phonons that are especially pronounced at lower temperatures. However, coherent phonons are often disrupted by disorder entities such as vacancies and impurities. If there is a high density of scatterers in the medium, exponential localization of eigenmodes can be induced, i.e., the amplitudes decay exponentially along the medium with a rate of decay inversely proportional to a certain localization length.^12,13 Anderson localization is an unusual phenomenon where sufficient disorder on the lattice can break the very concept of band structures in a yet elastic regime. For phonons, this is due to either disordered bond strengths or disordered masses or both.¹⁴ 

Phonon Anderson localization could potentially offer a new way to obtain materials with low thermal conductivity. The previous works on phonon localization in a disordered medium are mainly featured by an exponential decay of phonon transmission to zero with increasing system length, which leads to a thermal conductivity maximum with length.^15,16 In this Perspective, we will summarize the evidences and characteristics of phonon localization in disordered atomic systems from both experimental and theoretical aspects.

A number of experiments have directly observed the localization of electrons,¹⁷ photons,¹⁸ matter waves,¹⁹ and acoustic waves.²⁰ However, for thermal phonons, since heat conduction is a broadband phenomenon, the experimental evidences of phonon localization still remains ambiguous and have been reported only in the past few years.^14,21–23 There are several ways to probe the Anderson localization in phonons experimentally, by directly measuring the localization-induced phonon behavior and thermal property change. These measurements include Raman spectroscopy, femtosecond-pump–probe reflectivity measurements, inelastic neutron scattering and x-ray scattering, Fourier transform infrared spectroscopy (FTIR) and dielectric spectroscopy, thermal conductivity measurements, etc. The obtained information serves as certain signals of phonon localization.

In general, phonon Anderson localization in disordered systems can be probed by directly measuring the phonon behavior change. For example, vibrational spectroscopy, especially Raman scattering, is sensitive to the nanoscale order and can give information on the coherence range.²⁴ Howie et al.¹⁴ demonstrated the mass-induced localization effect in dense hydrogen–deuterium mixtures of various concentrations. Within a band of modes, the Raman activity is still in the in-phase modes that are typically with lower frequencies. Thus, the effect of broadening the entire band is to lower the frequencies of the Raman active modes, which is an indication of increased localization. The comparison of calculated and measured Raman spectra is demonstrated in Fig. 1. For another example, Hofmann et al.²⁵ utilized spatially resolved Raman spectroscopy to reveal a continuous change of the silicon Raman peak position and peak width along the nanotip, which is attributed to a smooth change between bulk properties at the base to size-induced phonon confinement in the apex of the nanotip. This approach allows to exclude heating effects that normally overwhelm the phonon localization signature.

For nm-thick films, localized surface modes lie in the range of several 10 to a few 100 GHz, and the coherent surface reflectivity oscillations are on a picosecond timescale, which is in the observable range of pump–probe experiments.²⁶ Döring et al. used the femtosecond-pump–probe reflectivity measurements to probe the phonon localization in multilayer systems (metals, semiconductors, oxides, and polymers). The standing phonon waves that are reflected by adjacent interfaces and localized within specific layers as well as strongly damped surface modes were observed at GHz frequencies.²⁶ 

Several other measurements can also be used to observe the phonon behavior change due to localization. Bermudez et al.²⁷ used a minimally perturbing measurement of the resonance fluorescence to study phonon Anderson localization. The cationic disorder and phonon localization in (Ni_1/3Nb_2/3)_xTi_1−xO₂ ceramic solid solutions were uncovered with combined Raman, FTIR, and dielectric spectroscopies.²⁸ With increasing (Ni + Nb)-substitution levels, the Raman and infrared modes gradually broaden, suggesting the increased disorder strength in (Ni_1/3Nb_2/3)_xTi_1−xO₂, which implies the possible phonon localization. By using inelastic neutron and x-ray scattering, the phonon localization and related changes in the lattice dynamics in a PbSe crystal was observed.²⁹ The results reveal that localization occurs at temperatures close to predicted but involves more spectral weight than expected. It drives unanticipated changes in the lattice dynamics including an unexpected sharpening of the longitudinal acoustic (LA) phonon at high temperatures. By using inelastic neutron scattering, Manley et al.²² show that ferroelectric phonon localization drives polar nanoregions (PNRs) in relaxor ferroelectric PMN [Pb(Mg_1/3Nb_2/3)O₃]-30%PT. At the frequency of a pre-existing resonance mode, nanoregions of standing ferroelectric phonons develop with a coherence length equal to one wavelength and the PNR size, which was attributed to Anderson localization of ferroelectric phonons by resonance modes.

Apart from the localization-induced change in vibrational spectra, the heat conduction in certain disordered systems exhibits some unique features that can be regarded as the proof of phonon Anderson localization. Luckyanova et al.²³ reported localization behavior in GaAs/AlAs superlattices (SLs) with ErAs nanodots randomly distributed at the interfaces [Figs. 2(a)–2(e)]. With an increasing number of superlattice periods, the measured thermal conductivities near room temperature increased and eventually saturated, indicating a transition from ballistic to diffusive transport. In contrast, at cryogenic temperatures, the thermal conductivities first increased but then decreased, signaling phonon wave localization.²³ 

The observation of the peaks in samples at low temperatures with two different concentrations of ErAs nanodots at the interfaces [Figs. 2(f)–2(h)] suggests that, rather than being caused by small variations in the arrangement of the dots across different samples, a new heat conduction mechanism, which reduces the transport of long-wavelength, low-frequency (terahertz range) phonons, is unfolding with an increasing sample thickness. This unexpected trend strongly points to the presence of phonon localization in these SLs.

Compared with experimental investigation, atomistic modelling provides alternatives to study phonon localization in a more intuitive and comprehensive way. In this section, we summarize the evidence of phonon localization from the perspective of atomistic modellings.

Localization is characterized by a phonon frequency-dependent localization length, $ξ$⁠. Phonon transmission decays exponentially for localized modes as $e−L/ξ$⁠, where L is the system length. As mentioned previously, this exponential decay is a signature of strong localization caused by random disorder. Thus, investigation on system length-dependent phonon transmission is imperative in searching for the evidence of phonon localization, which is more accessible from atomistic modellings than experimental measurements. To obtain the phonon transmission, one can make resource to molecular dynamics (MD) simulations^30,31 and atomistic Green's function method.^32,33

The exponential decay of phonon transmission with increasing system size has been observed previously in a series of random systems, including GaAs superlattices with randomly distributed ErAs particles,¹⁵ graphene phononic crystal structures,³⁴ graphene/h-BN superlattices,³⁵ heterogenous interfaces with atomic inter-diffusion,³⁶ etc. Recently, Hu et al.¹⁰ provided the direct observation of phonon Anderson localization at the mode level using the exponential decay of mode-resolved transmission in Si/Ge aperiodic superlattices (ap-SLs). At the extremely low-frequency region (<0.7 THz), ballistic modes were observed [Figs. 3(a) and 3(d)], which do not sense much of the disorder. For propagating modes, the transmission shows irregular, oscillating features [Figs. 3(b) and 3(e)]. For localized modes, the exponential decay of transmission with respect to the central region length [Figs. 3(c) and 3(f)] serves as direct evidence of phonon Anderson localization. Also, Juntunen et al.³⁷ proposed a phenomenological model accounting for the interplay between phonon localization and phase breaking processes in ap-SLs, which accurately reproduce the simulated thermal conductivities.

The phonon localization length can be extracted from the exponential decay of phonon transmission $T(ω)$ against the system length from the relation $T(ω)=T0(ω)⋅e−L/ξ$⁠, where $T0(ω)$ is the transmission of the pristine structure, L is the system length, and $ξ$ is the localization length. Ni et al.³⁶ computed the frequency-resolved phonon localization length of graphene/h-BN heterostructure with atomic interface diffusion. The coefficient of determination, R², was used to evaluate the goodness of the exponential fitting. For phonon modes with certain frequency, the fitting can achieve very high values of R², which provides the evidence of Anderson localization for the modes with these frequencies. From the well-fitted curves (⁠$R2>0.96$⁠), the frequency-dependent phonon localization length can be extracted, as plotted in Fig. 4(a). It can be clearly seen that more phonon modes (especially with frequencies below 10 THz) are localized in the models with composition-graded interfaces, with respect to the one with uniform intermixing section. It also shows that larger scatter size (larger heterogeneous particle) traps more phonons.

Guo et al.¹⁶ indicate that phonon localization is mainly contributed by modes in the moderate- and/or high-frequency range. To ensure the population of these localized phonon modes, the system temperature cannot be too low. On the other hand, the system temperature shall be sufficiently low to suppress the anharmonic phonon–phonon scattering, which will destroy the phonon coherence. This is also confirmed in Fig. 4(b), in which the population of localized modes obviously decreases with temperature.

Phonon wave packet allows for intuitive observation of phonon wave propagation in the coordinate space, which can be used to view the behavior of localized modes.

Zhang et al.³⁸ reported that the composition-graded SiGe NWs possess much lower thermal conductivity than that of the corresponding models with an abrupt interface. Figure 5 demonstrates the wave packet in composition-graded SiGe (red) and SiGe with abrupt interface (gray) at t = 0, 7.5, 11.5, 20, and 60 ps, respectively. In the system with the abrupt interface, the wave packet (gray) scattered into two waves at 11.5 ps when interacting with the interface: a transmitted wave and a reflected one. While for the system with the composition gradient, the wave packet (red) started to scatter once it meets the heterogeneous particles (around the region of the first gray dashed line). At 7.5 ps, a large portion of the wave transmitted into the composition-graded region, and it is trapped in this region until the end of the simulation (60 ps). The extremely low propagation speed of the trapped waves indicates strong localization in the disordered region.

Hu et al. performed phonon wave packet simulations for the C₃N/disordered C₃N phononic crystal heterostructure.³⁹ The transmitted wave was trapped near the interface and showed an exponential decay behavior with an increase in the travelling distance for the wave packet, presenting a localized nature. This unusual behavior confirmed that the low-frequency phonons in D-C₃N were localized by the disorder.

Phonon participation ratio (P) is a measure of the degree of spatial localization of a vibrational mode. For a propagative mode, all atoms participate in the vibration and the participation ratio is close to unity, while for a highly localized mode, the participation ratio is on the order of O(1/N). As a result, the phonon participation ratio allows to gain insights into which mode is localized and how severely it is localized. This approach is widely used to reveal the phonon transport behavior in terms of localization.^13,40–46 For example, Yang et al.⁴⁷ studied the thermal transport of a nanoscale three-dimensional (3D) Si phononic crystal (PnC) with spherical pores [Fig. 6(a)]. Obviously, the values of P in 3D PnCs are much smaller than those of bulk Si [Fig. 6(b)], which means that phonon modes in PnCs are likely localized. They further define the number of phonon modes whose value of P is less than 0.5 divided by the total number of phonon modes as localization ratio (LR). It was concluded that there are more localized phonon modes with the increase in porosity. To view the spatial localization in 3D PnCs, the energy distribution of the 3D PnCs was computed, as illustrated in Fig. 6(a). Only the strongly localized modes (P < 0.2) are included in the summation of energy. It is obvious that the intensity of localized modes is much higher at boundaries, indicating that most of phonons are localized at internal boundaries of PnCs.

In this Perspective, we summarized the evidence of phonon Anderson localization in disordered systems from both experimental and theoretical aspects. As phonon Anderson localization is a phenomenon occurring for coherent phonons, its observation and detections require experimental or modelling techniques that can extract information on the coherent phonon behavior. Although for nanoscale systems, the range of spatial coherence is well reflected by the vibrational spectroscopy such as Raman spectroscopy,²⁴ the existing measurable evidence of phonon localization still remains ambiguous in disordered systems. Most measured quantities suggest the possible phonon Anderson localization. The wave behavior of thermal phonons is strongly mode dependent, for example, ballistic modes, propagating modes, and localized modes are simultaneously observed in the same structure of Si/Ge aperiodic superlattices.¹⁰ Even for the localized modes, they do not necessarily follow the same trend as dictated by a single localization length.³⁵ Besides, coherent phonon transport can easily be destroyed by increased anharmonicity at elevated temperature,³⁶ which further smears out any signature of phonon localization. These facts make the experimental observation of localized modes more complicated.

The wave-related characteristic scales, i.e., wavelength or coherence length, mostly belong to the nanoscale, making the phonon localization in nano-phononic crystals extensively explored by atomistic simulation methods via either non-equilibrium Green's function formalism (NEGF) or MD simulations.⁴⁸ Compared with experimental measurements, atomistic simulations can easily create proper conditions for the observation of coherent transport and phonon localization, such as highly disordered systems and low temperature. Meanwhile, atomistic simulations can conduct mode analyses, which efficiently and intuitively bring a whole picture of phonon localization from different aspects. The signature of phonon Anderson localization, i.e., the exponential decay of phonon transmission with length, can be easily observed from the phonon transmission calculations. Besides, the phonon participation ratio, phonon wave packet, and energy distribution analyses offer different aspects for intuitively comprehending the phonon Anderson localization.

This Perspective only discussed disordered atomic systems, which are the most studied media for phonon Anderson localization. We have to mention that phonon localization was also found in other systems, such as one-dimensional harmonic and anharmonic chain,⁴⁹ isotopically disordered harmonic lattices,⁵⁰ and topologically nontrivial intralayer heterojunction structure^51,. Besides atomistic modelling, other approaches including numerical exact diagonalization,⁵² localization theory and kinetic theory,⁵⁰ and analytic investigation⁵⁰ are also useful tools to analyze phonon localization. The structural design of disordered systems can be guided by the machine learning method to maximize the Anderson localization of phonons.⁵³ These approaches provide essential and helpful complement to investigate phonon localization and advance our understanding of this fascinating phenomenon.

Y.N. acknowledges the support of National Natural Science Foundation of China (NNSFC) under Grant No. 11774294.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

The authors have no conflicts to disclose.