2D Mater. 7 (2020) 035029 https://doi.org/10.1088/2053-1583/ab93e2

The dimensionality effect on phonon localization in

RECEIVED

21 March 2020

REVISED

22 April 2020

ACCEPTED FOR PUBLICATION

18 May 2020

PUBLISHED

18 June 2020

graphene/hexagonal boron nitride superlatticesTengfei Ma1, Cheng-Te Lin2,3 and YanWang11 Department of Mechanical Engineering, University of Nevada, Reno, Reno, NV 89557, United States of America2 Key Laboratory of Marine Materials and Related Technologies, Zhejiang Key Laboratory of Marine Materials and ProtectiveTechnologies, Ningbo Institute of Materials Technology and Engineering (NIMTE), Chinese Academy of Sciences,Ningbo 315201, People’s Republic of China

3 Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100049,People’s Republic of China

E-mail: yanwang@unr.edu

Keywords: phonon localization, dimensionality, thermal conductivity, thermoelectric, coherent phonon

AbstractPhonon localization, a largely elusive phenomenon, has a great promise for improving currentapplications like thermoelectric materials and thermal barrier coatings. By freezing theout-of-plane atomic motion in graphene/hexagonal boron nitride (hBN) superlattices and randommultilayers, i.e aperiodic superlattices, we are able to completely isolate the effect of the thirddimension on phonon scattering and localization in 2D materials. In particular, we find muchmore prominent phonon heat conduction and localization when atomic motions in the thirddimension are frozen. Rigorous spectral phonon transmission and scattering phase space analysesreveal that the phase-breaking anharmonic scatterings can significantly hinder the occurrence ofphonon localization. Phonon participation ratio calculations further reveal that the flexuralmodes—arising from the degree of freedom of the third dimension—are rather extended, incontrast to the vastly localized in-plane modes in graphene/hBN random multilayers. These twofactors altogether greatly obstruct the observation of coherent phonon localization in realisticmaterials. This work will be useful for guiding the search for nanostructures possessing significantphonon localization behaviors.

1. Introduction

Anderson localization represents a peculiar statewhere the diffusion of waves is in a complete halt,which is a manifestation of destructive wave interfer-ences resulting frommultiple backscatterings in a dis-ordered medium. Since Phillip Anderson’s ground-breaking theoretical prediction of the localization ofelectron waves in 1958 [1], extensive efforts have beendevoted to observing this phenomenon for varioustypes of energy carriers like electrons [2, 3], photons[4, 5], and elastic waves [6]. Particularly, the last twodecades have witnessed the breakthrough in experi-mental observations of electron [7] and photon [8, 9]localization in various artificial materials.

Despite of its potential importance for improv-ing existing energy applications like thermoelectricmaterials and thermal barriers by reducing the latticethermal conductivity (κL), the weak localization orstrong (Anderson) localization of phonons was onlyreported for a limited number of systems [10–17].

The most investigated system is binary superlatticeconsisting of alternating layers of two differentmater-ials. The earliest studies back in the 1990 s used theGreen’s function approach to analyze the phonontransmission spectra of aperiodic superlattices andfound numerous forbidden bands [18]. Later in the2010 s, classicalmolecular dynamics simulations wereconducted to dynamically simulate lattice thermaltransport in aperiodic superlattices, such as fictitioussilicon [19], mass-mismatched Lennard-Jones [10,14, 20, 21], and more realistic Si-Ge [12, 13] andgraphene (12C)-graphene (13C) [11] systems. Not-ably, a substantial reduction (typically more than50% and up to 95%) in κL from a periodic super-lattice to its aperiodic counterpart is one straight-forward evidence of phonon localization, becausethe same interface density and material compositionof the periodic and aperiodic structures imply sim-ilar phonon scatterings, even though none of theprevious studies compared their phonon scatteringphase spaces to draw a quantitative conclusion. Less

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2D Mater. 7 (2020) 035029 T Ma et al

commonly investigated systems displaying signific-ant phonon localization effects are graphene nan-oribbons [22, 23], porous graphene [24–27], andnanowires of various materials [28, 29]. However,the experimental realization of those sophisticatedsystems modeled in simulations seems to be chal-lenging, if not impossible, and the effect of scatter-ing (in addition to localization) on non-linear orreduced thermal transport was never ruled out. In2018, Luckyanova et al reported the observation ofphonon Anderson localization in periodic GaAs/AlAssuperlattices embeddedwith ErAs nanoparticles [15].Unlike the aperiodic superlattices investigated in pre-vious theoretical studies [10–14], the disorder arisesfrom randomly embedded nanoparticles rather thanrandom layer thicknesses. In particular, Luckyan-ova et al were able to observe a non-monotonicthermal conductivity-device length relation, whichwas claimed to be a signature of phonon localizationbased on their atomistic Green’s function modeling[15, 30].

The much rarer observations of phonon localiza-tion than electron and photon localization is not sur-prising. One can reasonably expect that the broad-band nature of phonon thermal transport, even atextremely low temperatures, can obscure the localiz-ation behavior when only the overall thermal trans-port behavior (e.g. κL) is observed, because not allthe phonons are localized and even for those localizedmodes, they do not necessarily follow the same trendas dictated by a single localization length. Moreover,anharmonic phonon scatterings, which eventuallylead to diffusive phonon transport, can further smearout any signature of phonon localization if one onlychecks the overall κL, thereby imposing a great chal-lenge for the experimental observation of phononlocalization. These complexities of phonon thermaltransport necessitate the significant localization ofa wide range of phonon modes to exhibit notablephonon localization behaviors [30, 31], which mayrarely exist in nature. Therefore, it is critical toidentify reliable characteristics of materials or struc-tures that could demonstrate significant phonon loc-alization behaviors. A dimensionality based scalingtheory of Anderson localization has been establishedfor electrons [32]. Specifically, and simply, the theorystates that electrons in three-dimensional (3D) sys-tems are only partially localized while those in lower-dimensional ones can almost be entirely localized.This inspires us to explore the effect of dimension-ality on phonon localization in lower-dimensionalsystems.

In this study, we choose the single-atomic-layergraphene-hexagonal boron nitride (hBN) superlat-tice as ourmodel system to investigate the dimension-ality effect on phonon localization. Both grapheneand hBN are two-dimensional (2D) materials, whichhave been extensively studied for their phonon andlattice thermal transport properties [17, 33–40].

Nonetheless, we note that they are not strictly 2Dfor phonons, since atoms in these systems can stillvibrate in all the three dimensions, particularly, in theout-of-plane direction. This, fortunately, enables usto isolate the effect of this out-of-plane degree of free-dom on phonon localization. Specifically, we conductclassical molecular dynamics simulations of ‘real-istic’ graphene-hBN superlattices, in which atoms canvibrate freely in all the three dimensions, and strictly2D ones, in which atoms can only move in-plane.In the latter case, we can effectively remove the flex-ural phonons and eliminate their effect on phononlocalization and scattering. Through the comparisonbetween 2D and 3D systems, we will be able to rig-orously understand the effect of dimensionality onphonon transport and localization behaviors.

2. Methods

2.1. Molecular dynamics simulationsWe conduct molecular dynamics simulations ongraphene/h-BN superlattices (SLs) and randommul-tilayers (RMLs), as displayed in figure 1(a) and (b),respectively. For 2D simulations, the movement ofatoms in the out-of-plane direction is completelyfrozen, rendering a truly 2D phonon system (fig-ure 1(c)). In contrast, atoms can move freely, fol-lowing the Newton’s law of motion, in all threedimensions in a 3D simulation, which is the casein real world and conventional MD simulations(figure 1(d)). The interatomic interactions betweenC, B, and N atoms are described by the optim-ized Tersoff potential [41, 42]. The system is firstrelaxed in the isothermal-isobaric ensemble at 300K and 0 pressure for 1.5 million steps (0.5 fs/step)with the Nóse-Hoover thermostat [43]. Then non-equilibrium molecular dynamics (NEMD) and equi-librium molecular dynamics (EMD) simulations areperformed respectively to obtain the thermal con-ductivity κL of the system. The LAMMPS package[44] is used for all the MD simulations in this work.

2.2. Spectral phonon transmission analysis inNEMDThe spectrally decomposed heat flux in our NEMDsimulations is firstly calculated with equation (1) toquantify the phonon transmission [45, 46]:

Q(ω) =∑i∈L̃

∑j∈R̃

(− 2

tsimuω

∑α,β

Im⟨v̂αi (ω)∗Kαβij v̂βj (ω)⟩

),

(1)

in which Q(ω) is the spectral heat current; ω is thefrequency; i and j are respectively the atom indicesin the left(L̃) and right(R̃) part of a chosen cross sec-tion; tsimu is the total NEMD simulation time;α and βare Cartesian coordinates; v̂ is the Fourier transformof atom velocity and* denotes the complex conjugate;and Kαβij is the force constant matrix.

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2D Mater. 7 (2020) 035029 T Ma et al

Figure 1. (a) and (b) Atomic structures of representativeSLs and RMLs. (c) and (d) Atomic structures in the 2D and3D simulations. In 2D simulations (c), the out-of-planemotion of atoms is completely frozen, while in 3Dsimulations (d), atoms can move freely in all the threedimensions. (e) Thermal conductivity as a function ofdevice length for various systems obtained from NEMDsimulations. The two solid lines represent the differencebetween the thermal conductivity of RMLs and theircorresponding SLs. (f) Thermal conductivity as a functionof average layer thickness for various systems obtained fromEMD simulations. Panels e and f share the same legend.

3. Results and discussions

NEMD simulations allow us to investigate latticethermal transport in finite-sized systems conveni-ently. As shown in figure 1(e), the thermal conduct-ivity of both 2D and 3D SLs increases quickly as thedevice length increases all the way to 500 nm, thelongest length studied in this work. As revealed in pre-vious experimental and theoretical studies [10, 47],this results from the ballistic transport of coher-ent phonons. In particular, coherent phonons arethe modes originating from the large unit cell (i.e.one period) of the superlattice, instead of each indi-vidual basematerial constituting the SL, and thereforethey are not scattered at the interfaces. Furthermore,figure 1(e) demonstrates much lower thermal con-ductivity of RMLs than the corresponding SLs, eventhough they have exactly the same interface dens-ity and thus presumably the same level of phonon-interface scatterings. The reason for the reduction isnow well understood as a result of phonon localiza-tion, which severely quenches coherent phonons andtheir contribution to thermal transport [10, 48].

Since the localization mechanism hinders coher-ent phonon transport exponentially along its path,it is reasonable without losing the essential phys-ics to assume that the coherent phonon contribu-tion to thermal transport is largely eliminated inlong RMLs. In this regard, the difference between thethermal conductivity of a RML and its correspond-ing SL (i.e. κL,SL −κL,RML) approximately represents

the coherent phonon contribution suppressed by loc-alization (indicated in the legend as ‘κL,SL −κL,RML,2D’ and ‘κL,SL −κL,RML, 3D’ in figures 1(e) and (f).As shown in figure 1(e), phonon localization causes amuch greater reduction in κL for 2D systems than 3Dones. Moreover, this difference becomes more prom-inent in longer structures, where coherent phononsare supposed to play amore important role in thermaltransport because of their long mean-free-path [10].

To exclude the influence of finite device length(system boundaries) on thermal transport in NEMDsimulations, we further conduct EMD (which sup-posedly predicts bulk-limit thermal conductivity)simulations on SLs and RMLs with various averagelayer thicknesses. As shown in figure 1(f), phononsaremore significantly localized in 2D systems than 3Dones, agreeing with our NEMD results in figure 1(e).Particularly, κL is reduced by 53% to 89% in 2D sys-tems, which is much greater than the reduction in 3Dones (7% to 57%), as shown in figure 1(f).

Herein, as the core of this work, we will con-duct spectral phonon transmission analyses compar-ing 2D and 3D systems to confirm that the lower κLof RMLs than corresponding SLs is indeed causedby phonon localization, identify the critical phononmodes that are responsible for the localization beha-viors, and unveil the reason for the significantlystronger phonon localization in 2D systems.

First, to rule out the possibility that the reducedκL in RMLs compared to SLs is caused by increasedphonon scatterings, we calculate the volume of three-phonon scattering phase space (P) for RMLs andtheir corresponding SLs. P quantifies the number ofphonon triplets satisfying necessary selection rules for3-phonon scatterings, which can provide a reliableestimation of the strength of phonon scattering. Thescattering phase space can be calculated from latticedynamics as:

P±(qj) =1

Nq

∑q1,q2,j1,j2

δ(ωqj ±ωq1j1 −ωq2j2)δq±q1,q2+G

P(qj) =1

D(2P(+)(qj)+ P(−)(qj)),

(2)

where Nq is the number of q points, D is a nor-malization factor that is related to the number ofphonon branches and determined by the dimension-ality, P(+) is the phase space for phonon combina-tion processes (i.e. two lower energy phonons com-bine into a single, higher energy phonon), and P(−) isfor phonon splitting processes in which one phononsplits into two lower energy phonons. Only three-phonon processes are considered in the above equa-tion, because higher order scattering process can beignored in most materials [49].

As shown in figure 2, SLs and RMLs of the samedimensionality have similar volume of 3-phononscattering phase space, of which the difference is

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2D Mater. 7 (2020) 035029 T Ma et al

Figure 2. The overall scattering phase space of acoustic phonon modes for (a) 3D SL, (b) 3D RML, (c) 2D SL, and (d) 2D RML.The value of scattering phase space is denoted by the color scale. Red means larger scattering phase space.

much smaller than that between the κL of the twostructures as shown in figure 1. In particular, thesummed value of P for the 2D SL with an aver-age layer thickness of d= 4 UC is 4.82× 10−5 versus5.02× 10−5 for the corresponding 2D RML, and thesummed P for the same 3D SL is 8.29× 10−4 versus2.86× 10−4 for 3D RML. The drastically smaller 3-phonon scattering phase space of the 2D SL than its3D SL counterpart is due to the removed flexuralphonon modes, which, at the same time, eliminatesall the 3-phonon triplets involving at least one flex-ural modes. As a result, the κL of 2D SLs is muchhigher than the corresponding 3D SLs, as shown infigure 1(e) and (f).More importantly, the fact that 2DSL and 2D RML have similar scattering phase spaceand that the 3D RML has even lower P than the cor-responding 3D SL (due to the opening of bandgapsin RMLs) indicate that phonon scattering is not thereason, at least not a significant one, for the muchlower κL of RMLs than their corresponding SLs. Theonly reason, as far as we can see, is the phonon localiz-ation induced by the random layer thickness in RMLs,as will be discussed in detail next.

Here, we calculate the phonon transmission spec-tra Γ(ω) from the same set of NEMD simulationsreported in figure 1. Notably, Sääskilahti et al ’sapproach [45, 46] enables us to compute the Γ(ω) insteady-state NEMD simulations, similar to the atom-istic Green’s function approach [49] but is advantage-ous in its capability of dealing with much larger sys-tems and the natural inclusion of anharmonic effects.Our transmission results for SLs and RMLs of various

device length (number of periods, NP) are presentedin figures 3(a), (c), (e), and (g). As expected, phonontransmission decreases as device length increases. For3D SLs and RMLs, we are able to decompose thetransmission into that contributed by in-planemodesand flexural modes. As shown in figures 3(a) and (c),the in-plane phonons dominate heat conduction inshort SLs and RMLs, while the relative significanceof flexural phonons increases as the device becomeslonger. This trend can be comprehended more easilyin figures 3(b) and (d), where we plot transmission asa function of device length, i.e. the number of peri-ods. Evidently, the transmission function of flexuralphonons decreases muchmore slowly than that of in-plane phonon modes, clearly demonstrating a ratherballistic transport behavior in short 3D SLs and 3DRMLs.

We note that the decreasing trend of the trans-mission (Γ)-device length (NP) data of the in-planemodes at 13 THz, 20 THz, and 25 THz in the 3D and2DSLs in figures 3(a), (b), (e), and (f) is not caused byphonon localization; instead, it arises from the anhar-monic scattering of phonons in the periodic SL struc-tures. This conclusion is first evidenced by the almostperfect fittings (solid lines in figures 3(b) and (f)to the data points in figures 3(b) and (f) using theΓ(L) = Γ0λ/(L+λ) relation, which generally applieswell to diffusive transport of phonons [10]. Strictlyspeaking, the quality of curve fittings is a straight-forward but may not be a conclusive evidence of thenature (localized, diffusive, or ballistic) of phonontransport. Thus, to further support our assertion that

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2D Mater. 7 (2020) 035029 T Ma et al

Figure 3. Spectral transmission function Γ obtained from NEMD simulations. Panels (a), (c), (e), and (g) are Γ(ω) of 3D SLs, 3DRMLs, 2D SLs, and 2D RMLs of various device length (number of periods, NP), respectively, and panels (b), (d), (f), and (h)show the length-dependency of Γ for each system at different phonon frequencies. Solid curves in panels (a), (c), (e), and (g)represent the transmission function for in-plane phonons while dashed curves are for out-of-plane (flexural) phonon modes. Thesolid lines in (b),(d),(f), and (h) are fitted curves with Γ(L) = Γ0λ/(L+λ), while the dashed lines are fitted curves withΓ(L) = Γ0exp(−L/Lloc)/(L+λ).

the aforementioned in-plane phonon modes are notlocalized, we can compare between the Γ(ω) curvesin figure 3(a) (3D SLs) and e (2D SLs). As shown infigure 3(e), all the transmission spectra curves signi-ficantly overlap in the 0-15 THz range, indicating thatthose in-plane phonons are strongly ballistic. In con-trast, the same range of phonons in the 3D SLs dis-play a notable decrease in their transmission as thedevice length increases, as shown in figure 3(a). Thiscan be captured more straightforwardly by compar-ing the Γ-NP (device length) data for the in-planemodes at 13THz in figure 3(b) for 3DSLs and those infigure 3(f) for 2D SLs, in which the former decreasesrather significantly with device length while the lat-ter is almost constant, suggesting the former is dif-fusive while the latter is ballistic. This reaffirms ourassertion that there should be no or, at least, negli-gible localization for the 13 THz in-plane modes inboth 2D and 3D SLs due to their same atomic struc-tures and that any significant localization should leadto a decreasing Γ-NP trend in figure 3(f). By compar-ing the phonon scattering phase space in figures 2(a)and (c), it is obvious that the low-frequency acous-tic phonons in the 3D SL are scattered much more(by one order of magnitude) severely than those inthe corresponding 2D SL, leading to a more diffusivetransport behavior in 3D SLs.

It is worth noting that the in-plane phononmodes at 40 THz behave rather differently fromother in-plane modes in both 3D and 2D SLs, asshown in figures 3(b) and (f). Specifically, we can-not achieve reasonable fitting to the data with Γ(L) =Γ0λ/(L+λ) (solid lines), but a modified rela-tion Γ(L) = Γ0exp(−L/Lloc)/(L+λ) (dashed lines),which includes both the scattering and localizationeffects (with Lloc being the localization length), worksmuch better for those phonon modes. This trace of

phonon localization is not surprising, even thoughthe SL structure is periodic. In fact, 40 THz corres-ponds to a band edge, as shown in the phonon disper-sion relations of the SLs (figures 4(a) and (c), and ithas been established that it is generally easier for local-ization to occur at band edges [50, 51]. Similar obser-vation of enhanced phonon localization near bandedges has been reported in GaAs/AlAs superlattices[15].

As expected, the phonon localization behavior ismore prominent in RMLs than SLs. As shown in fig-ures 3(d) and (h), we can obtain notably improved fit-ting by Γ0exp(−L/Lloc)/(L+λ) (dashed lines) thanΓ(L) = Γ0λ/(L+λ) (solid lines) for all the four fre-quencies (13 THz, 20 THz, 25 THz, and 40 THz)in 3D and 2D RMLs, where localization is primar-ily caused by the disordered layer thicknesses inRMLs. The stronger anharmonic phonon scatter-ings in 3D systems, however, can break the phaseof phonons more severely, rendering them harder tobe localized—which is a manifestation of destruct-ive coherent wave interference. As a result, thesemore strongly scattered phonons show a rather dif-fusive transport behavior, for which the fitting bythe scattering-limited relation Γ0λ/(L+λ) alone canwork reasonably well, which is the case for the 13THz in-plane phonons in 3D RMLs, as shown infigure 3(d). In contrast, the Γ-device length rela-tion for 2D RMLs shown in figure 3(h), particularlyfor those at 13 THz, shows a strong deviation fromthe Γ0λ/(L+λ) trend (solid yellow line), necessitat-ing the consideration of phonon localization (dashedyellow line). In addition, the high-frequency phon-ons, which are subjected to rather strong anharmonicphonon scatterings even in 2D RMLs, also demon-strate quite diffusive (scattering-limited) transportbehavior, as shown in figure 3(h) for the in-plane

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2D Mater. 7 (2020) 035029 T Ma et al

Figure 4. Phonon dispersion relations and participation ratios (color scale) for (a) 3D SL (a), (b) 3D RML, (c) 2D SL, and (d) 2DRML. A smaller value (redder colors) of the participation ratio indicates that the mode is more strongly localized.

phonons at 20 THz and 25 THz. Thus, we can con-clude that even for 2D RMLs, in which phononscatterings are generally suppressed because of thereduced scattering phase space, it is still challenging toobserve strong phonon localization behaviors for cer-tain phonon modes, particularly the high-frequencyones, because the anharmonic phonon scatteringsare already strong enough to smear out the localiz-ation behavior (i.e. the exponential decay of trans-mission of conductance). As one can naturally derive,it is even harder for one to capture signatures ofphonon localization if one merely observe the κL-device length relation [15, 30], because of the exist-ence of numerous diffusive modes that obscure thelocalization behavior of others. In contrast, the com-parison between a disordered system and its periodiccounterpart, such as RMLs versus SLs, can serve as amore reliable and feasible, though not as straightfor-ward, system for demonstrating phonon localizationeffect on thermal transport, as long as the effect ofphonon scattering can be ruled out.

Finally, to reaffirm that the observed greatlyreduced thermal conductivity of RMLs results fromphonon localization, we calculate the phonon parti-cipation ratio—a reliable parameter to quantify thelevel of localization property of each phononmode—to rigorously understand the mode-wise localizationof phonons in our systems. The participation ratio ofthe phonon mode at the jth branch and q point is cal-culated by:

PRqj = (N∑i

|e(i,qj)|2

Mi)2/N

N∑i

|e(i,qj)|4

M2i. (3)

In the above equation, e(i,qj) is the polarizationvector of the corresponding phonon mode for the ithatom, N is the total number of atoms in the super-cell, and Mi is the mass of atom i. Figures 4(a), (b),(c), and (d) show the phonon dispersion relationsand participation ratios (denoted by the color scale)

of a 3D RML (NP = 16 and d = 4UC) and the cor-responding 3D SL, 2D RML, and 2D SL, respect-ively. The primary difference between the 2D struc-tures and their 3D counterparts is the missing of flex-ural branches, which correspond to the out-of-planemotions that are frozen in the 2D systems. The com-parison between figure 4(a) and (b), and betweenfigure 4(c) and (d), clearly demonstrates strongerphonon localization in RMLs, which have disorderedlayer thickness. Moreover, the comparison betweenfigure 4(b) and (d) reveals that the low-frequency(e.g. 5 THz) flexural phonons in RMLs are mostlyextended, i.e. non-localized, which is in stark contrastwith the in-plane phonons, which have much smal-ler participation ratio, i.e. more localized. This agreeswith the observations in figure 3(b) and (d), whichshows that the transmission of the flexural phononsat 5 THz does not decrease as much as that of thein-plane phonons, when one switches from a 3D SL(the blue circles in figure 3(b)) to a 3D RML (the bluecircles in figure 3(d)). Therefore, we can concludethat the flexural phonons, which are rather exten-ded and ballistic, impose an additional challenge toobserve significant phonon localization behavior in3D systems.

Moreover, even for the 2D RMLs, there are stillnumerous low-frequency-long-wavelength phononsthat are largely extended, i.e. the blue-colored regionaround 13 THz and the Γ point in figure 4(d). Fortu-nately, the group velocities (i.e. the slope of the dis-persion curves) of these extended in-plane modes aremuch smaller than the shorter-wavelengthmodes, i.e.the red-color region around 13 THz but away fromthe Γ point. As a result, we are still able to observe aprominent localization-like Γ-device length relationfor the 13 THz-in-plane phonons in 2D RMLs, asdepicted in figure 3(h) by the yellow dashed line.

4. Conclusions

In summary, we have found much more prominentphonon localization behavior in 2D graphene/h-BN

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2D Mater. 7 (2020) 035029 T Ma et al

systems, in which the out-of-plane motion of atomsare frozen, than in 3D systems. Spectral phonontransmission analysis and participation ratio calcu-lations reveal that a significant number of flexuralmodes are rather extended in 3D RMLs, in con-trast to the vastly localized in-plane modes. Thismakes it more challenging to observe significantphonon localization in realistic 2D materials, inwhich atoms can move freely in all the three dimen-sions. Moreover, the more intense phase-breakinganharmonic phonon scatterings in 3D systems—dueto the much larger phonon scattering phase spacethan 2D ones—further make it harder for destructivecoherent phonon interference, i.e. phonon localiza-tion, to occur. Specifically, many phonon modes in3D systems, though they might have a low particip-ation ratio that suggests strong localization, demon-strate a diffusive behavior than localized. Thus, thesefactors altogether obstruct the direct observation ofthe coherent phonon localization phenomenon in 3Dsystems, particularly when one only checks the overallthermal transport behaviors. Instead, the observationof reduced thermal conductivity from an ordered sys-tem to its disordered counterpart, such as the super-lattices and random multilayers studied in this work,can be used as an indirect but feasible criterion forjudging the occurrence of phonon localization, whichmust be accompanied by ruling out the effect ofphonon scattering.

Finally, we note that even though the perfectly 2Dphonon systems as studied in our theoretical work arenot accessible in experiments, at least for now, it ispossible to approach them by applying certain tensilestrains to 2Dmaterials, which can effectively suppressthe out-of-planemotion of atoms and lead to reducedanharmonic phonon scatterings [52–54]. Currently,graphene/hBN superlattices can be fabricated withcontrolled layer thickness [55–57]. Another phononicsystem, graphene nanomesh, of which the location,size, and shape of the holes can bewell controlledwithvarious experimental techniques [58–60], may alsoserve as a feasible material system for experimentalvalidation of our predictions.

Acknowledgments

The authors would like to thank Dr Sebastian Volzand Dr Kimmo Sääskilahti for fruitful discussions onthe spectral phonon transmission calculations. Theauthors would like to thank the University of Nevada,Reno (YW’s startup fund) for providing support forthis study and acknowledge the support of Research& Innovation and the Office of Information Techno-logy at the University of Nevada, Reno for computingtime on the Pronghorn High-Performance Comput-ing Cluster.

ORCID iDs

Tengfei Ma https://orcid.org/0000-0001-6220-2706Cheng-Te Lin https://orcid.org/0000-0002-7090-9610Yan Wang https://orcid.org/0000-0001-9474-6396

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The dimensionality effect on phonon localization in graphene/hexagonal boron nitride superlattices1. Introduction2. Methods2.1. Molecular dynamics simulations2.2. Spectral phonon transmission analysis in NEMD

3. Results and discussions4. ConclusionsAcknowledgmentsReferences