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J. D. G. Greener, A. V. Akimov, V. E. Gusev, Z. R. Kudrynskyi, P. H. Beton, Z. D. Kovalyuk, [T. Taniguchi](https://orcid.org/0000-0002-1467-3105), [K. Watanabe](https://orcid.org/0000-0003-3701-8119), A. J. Kent, A. Patanè

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[Coherent acoustic phonons in van der Waals nanolayers and heterostructures](https://mdr.nims.go.jp/datasets/544602ba-559e-4099-bcae-26fa7b3d13e6)

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Coherent acoustic phonons in van der Waals nanolayers and heterostructuresPHYSICAL REVIEW B 98, 075408 (2018)Coherent acoustic phonons in van der Waals nanolayers and heterostructuresJ. D. G. Greener,1 A. V. Akimov,1,* V. E. Gusev,2 Z. R. Kudrynskyi,1 P. H. Beton,1 Z. D. Kovalyuk,3 T. Taniguchi,4K. Watanabe,4 A. J. Kent,1 and A. Patanè11School of Physics and Astronomy, The University of Nottingham, Nottingham NG7 2RD, United Kingdom2LAUM, UMR-CNRS 6613, Le Mans Université, Avenue O. Messiaen, 72085 Le Mans, France3Institute for Problems of Materials Science, The National Academy of Sciences of Ukraine, Chernivtsi, Ukraine4National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki305-0044, Japan(Received 22 March 2018; revised manuscript received 31 May 2018; published 9 August 2018)Terahertz (THz) and sub-THz coherent acoustic phonons have been successfully used as probes of variousquantum systems. Since their wavelength is in the nanometer range, they can probe nanostructures buried belowa surface with nanometer resolution and enable control of electrical and optical properties on a picosecondtime scale. However, coherent acoustic phonons have not yet been widely used to study van der Waals (vdW)two-dimensional (2D) materials and heterostructures. This class of 2D systems features strong covalent bondingof atoms in the layer planes and weak van der Waals attraction between the layers. The dynamical properties of theinterface between the layers or between a layer and its supporting substrate are often omitted as they are difficultto probe. On the other hand, these play a crucial role in interpreting experiments and/or designing new devicestructures. Here, we use picosecond ultrasonic techniques to investigate phonon transport in vdW InSe nanolayersand InSe/hBN heterostructures. Coherent acoustic phonons are generated and detected in these 2D systems andallow us to probe elastic parameters of different layers and their interfaces. In particular, our study of the elasticproperties of the interface between vdW layers reveals a strong coupling over a wide range of frequencies up to0.1 THz, offering prospects for high-frequency electronics and technologies that require control over the chargeand phonon transport across an interface. In contrast, we reveal a weak coupling between the InSe nanolayersand sapphire substrates, relevant in thermoelectrics and sensing applications, which can require quasi-suspendedlayers.DOI: 10.1103/PhysRevB.98.075408I. INTRODUCTIONDuring the last decade, techniques for the generation anddetection of coherent acoustic phonons with frequencies fromtens of gigahertz (GHz) to several terahertz (THz) haveenabled novel approaches to the investigation of a wide rangeof materials and nanostructures. Experiments with coherentphonons were made possible by the availability of lasers for thegeneration of ultrashort light pulses. This has led to a new fieldof research, picosecond (ps) ultrasonics, which has extendedtraditional acoustics and ultrasonics to higher frequencies andshorter wavelengths [1,2], thus stimulating and advancingtopical research areas in condensed matter physics, includingthe following.(i) Phonon transport. Of particular interest is phonontransport through the interface between two nanostructuresor between one system and its environment [3–9]. Acousticmismatch theory, which describes phonon transport throughan elastically perfect interface with strong bonds, does not*Corresponding author: andrey.akimov@nottingham.ac.ukPublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.always apply. Thus different approaches are often required tounderstand and control phonon transport in real systems.(ii) Phonon interactions. Interactions of coherent phononswith thermal phonons [10], electrons [11–15], plasmons[16–18], and magnons [19–25] are intensively studied inmetallic (for a review see Ref. [26]) and semiconductor [27,28]nanoparticles, multilayered heterostructures [13–15,29,30],and patterned surfaces [16–18,21,31]. These interactions un-derpin a wide range of phenomena, such as piezoelectricity,light scattering, phonon-assisted tunneling, etc., and the designand successful development of devices for electronics [32] andoptoelectronics [33].(iii) Phonon nanoscopy. Coherent phonons with THz fre-quency can probe interfaces and nanoobjects buried below asurface with atomic depth resolution (e.g., �1 nm). Sinceits first use to probe and control the quality of microchipcontacts [34], nanoscopy has been used to image interfaces[35,36], chemical reactions [37], and biological cells [38].Also, acoustic, optic and acousto-optic parameters of inho-mogeneous materials can be probed with a nanometer spatialdepth resolution [39].Of critical importance in the studies highlighted in (i)–(iii) is the generation and detection of quantized coherentphonons in various nanoobjects (e.g., in single nanoparticles[9,26,27,40,41] and quantum wires [42]). In this case, thephonon spectrum consists of well-defined resonances eachwith a linewidth �f that is smaller than the resonance2469-9950/2018/98(7)/075408(15) 075408-1 Published by the American Physical Societyhttp://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevB.98.075408&domain=pdf&date_stamp=2018-08-09https://doi.org/10.1103/PhysRevB.98.075408https://creativecommons.org/licenses/by/4.0/J. D. G. GREENER et al. PHYSICAL REVIEW B 98, 075408 (2018)frequency f , and finesse defined by the quality factor Q =f/�f . A number of recent works have focused on two-dimensional (2D) nanometer-thick layers where phonons arequantized in the direction perpendicular to the 2D plane.Examples include submicrometer free-standing membranesbased on Si [43], GaAs [44] and GaN [45], and phononnanocavities [46].Among 2D systems, van der Waals (vdW) crystals haveemerged as a new class of materials for several potential ap-plications. In a vdW crystal, the atoms in each layer are boundby strong covalent bonds, whereas the planes are held togetherby weak vdW interactions. Thus several physical propertiesof these materials, including elastic properties (e.g., soundvelocity) are strongly anisotropic. The extended family of vdWcrystals includes graphene, transition metal dichalchogenides(e.g., MoS2, WS2, etc), metal chalcogenides (e.g., InSe, GaSe,etc.), hexagonal boron nitride (hBN), and many others [47,48].For this new class of materials, picosecond acoustics couldhelp to address several important open questions. The interfacebetween two vdW layers or between a vdW layer and itssupporting substrate may or may not be elastically perfect,thus influencing thermal and charge transport. For example,the control of phonon transport across a vdW interface couldprovide a strategy to probe and tune thermal conductivityfor specific applications in electronics and thermoelectrics[47–51]. The interaction of coherent phonons with charge car-riers could enable new routes for the generation of microwavefrequencies as it is done in nanoelectromechanical (NEMS)systems in the MHz-GHz frequency range [52], in piezoelectricheterostructures for the sub-THz range [53], and traditionalsemiconductor devices such as Schottky diodes [11,54] andtunneling devices [12]. Also, due to strong elastic anisotropyand phonon quantization, 2D vdW layers could provide anideal system for phonon nanoscopy and nondestructive sensi-tive imaging of molecules and cells coupled to the nanolayersby vdW forces. Although these research areas are still intheir infancy, recent experimental works have demonstrated thepossibility of generating coherent phonons in vdW nanolayers,including graphene [6], WSe2 [55], WTe2 [56], MoS2 [57], andInSe [58].Here, we use coherent phonons with frequencies from 10to 100 GHz, generated and detected using ultrafast optical(pump-probe) methods, for the nondestructive investigation ofnanomechanical oscillations in InSe nanolayers and InSe/hBNheterostructures. These studies enable us to probe the vdWbonding between the layers and their adhesion to the substrate.The GHz and sub-THz nanomechanical oscillations havefrequency, amplitude, and spectral width that depend on thestiffness of the interfacial bonds at the interface. In particular,we find that the interface between InSe and hBN can bedescribed by acoustic mismatch over a wide frequency rangef � 100 GHz.The paper is organized as follows. Section II describesour samples and the experimental techniques used for thegeneration and detection of coherent phonons. Section IIIintroduces the theoretical background used to describe theresonant phonon modes in vdW nanolayers and heterostruc-tures. Section IV presents the measured and calculated phononspectra for a variety of samples based on InSe nanolayers andInSe/hBN heterostructures. The main findings are discussedin Sec. V taking into account the properties of interfacesand phonon scattering processes. Conclusions and outlook forfuture studies and applications are presented in Sec. VI.II. EXPERIMENTAL TECHNIQUESA. VdW InSe nanolayers and heterostructuresOur experiments focus on InSe vdW crystals and InSe/hBNheterostructures. The InSe represents a relatively new additionto the family of 2D vdW crystals and is receiving an increasinginterest due to its unique and versatile electronic properties.This compound has a band gap that increases markedly withdecreasing layer thickness down to a single layer offeringfull coverage of the photonic spectrum from the violet to theinfrared range [59–61]; it has relatively light (mc ∼ 0.14me)conduction band electrons with high electron mobility (μ)even in the limit of atomically thin films (μ = 0.2 m2/Vs at300 K) [60]; furthermore, the encapsulation of InSe by hBN hasenabled the fabrication of high-quality field effect transistors(FETs) where the hBN capping layer isolates the InSe from theenvironment and also serves as dielectric layer for electrostaticgating [60]. Phonon transport and thermal conduction in 2DInSe and its heterostructures are still largely unexplored. Todate, theoretical studies have focussed on phonon propertiesand in-plane transport of the longitudinal, transverse, andflexural acoustic modes in single layers [62,63]. Experimentalwork has largely been limited to Raman studies [59,61] andelastic properties (e.g., sound velocity and elastic constants)are only known for bulk InSe [64].Our 2D InSe layers were prepared by exfoliation ofBridgman-grown crystals of rhombohedral γ -InSe [59]. Theyhave areas of about 10 − 100 μm2 and thicknesses from ∼5to ∼100 nm. After the exfoliation, the individual InSe flakeswere laid onto another layered crystal (e.g., InSe or hBN)so that the adhesive vdW force between the layers forms ahomojunction (e.g., InSe on InSe) or a heterojunction (e.g.,InSe on hBN or hBN on InSe). These samples were transferredon a c-cut sapphire (Al2O3) substrate with an atomically flatsurface (surface roughness of ∼0.1 nm). The thickness andthe surface roughness of the layers were assessed by atomicforce microscopy (AFM) in noncontact mode under ambientconditions. Representative AFM data and their discussion arepresented in Sec. V.Figure 1(a) shows the optical images for an InSe/hBNheterostructure and the individual InSe and hBN layers priorto stamping of hBN on InSe with thickness a = (81 ± 1) nmand b = (59 ± 1) nm, respectively. Room temperature (T =300 K) μ-photoluminescence (μPL) and μ-Raman maps, andrepresentative spectra are shown in Figs. 1(b)–1(e). By com-paring the spectra acquired in areas with and without the hBNlayer, we find that the encapsulation of InSe by hBN doesnot modify the energy of the Raman modes and/or of the PLemission of pristine InSe. However, it induces a significantincrease of both Raman and PL signals. This behavior, whichis not observed when the hBN layer lies below the InSelayer, is indicative of an increased absorption of light by theInSe layer due a smaller reflection of the laser light at theinterface between air and hBN (refractive index nhBN = 1.8[65]), compared to air (nair = 1) and InSe (nInSe = 2.7 [66]).075408-2COHERENT ACOUSTIC PHONONS IN VAN DER WAALS … PHYSICAL REVIEW B 98, 075408 (2018)FIG. 1. (a) Optical images of InSe and hBN layers before [(i) and(ii)] and after (iii) their stacking to form an InSe/hBN heterostructureon a SiO2 substrate. The rectangle in part iii is mapped by μPL andμRaman in part [(b) and (c)]. [(b) and (c)] Maps of the μPL (b) andμRaman (c) intensities for the sample shown in part (a-iii). Each mapis overlaid on the optical image of the sample and shows the InSe andhBN/InSe areas. [(d) and (e)] Representative μPL (d) and μRaman(e) spectra for InSe before (black) and after (red) the stamping of thehBN layer.B. Generation and detection of coherent acoustic phononsWe use femtosecond lasers and the most common pump-probe picosecond acoustics method to generate and detectcoherent phonons. This method, first proposed by Thomsonet al. [67] in the 1980s, is well developed and is describedin several reviews [1,2]. In this method, an optical pulsefrom a pump femtosecond laser excites the layers directlyor through an optoelastic transducer (e.g., a thin metal film),which absorbs light. As a result, a stress is generated in thesample due to a rapid, almost instantaneous electron and latticetemperature rise [67–69]. The stress causes dynamical strainand, correspondingly, generates a coherent wave packet oflongitudinal acoustic phonons. The spectrum of these sub-THzphonons depends on the absorption length and the thickness ofthe layer. The spatial and temporal evolutions of the coherentphonons are governed by the elastic properties of the materialsand their interfaces and can be probed by a second opticalpulse from the same pump laser or from another probe lasersynchronized with the pump one. Scanning the time delaybetween pump and probe pulses makes it possible to monitorthe temporal evolution of the wave packet of coherent acousticphonons generated in the sample.FIG. 2. (a)–(d) Schemes for pump-probe experiments in trans-mission (a) and reflection [(b), (c), and (d)] geometries. The stress inthe nanolayers is excited directly [(a)–(c)] or through an Al-transducer(d). (e) A generic pump-probe signal and its analysis to deduce thephonon spectrum: (top) temporal signal; (middle) temporal signalafter subtraction of a slow decaying background; and (bottom) phononspectrum obtained by fast Fourier transform (FFT) of the temporalsignal after subtraction of the background. In this example, the FFTspectrum shows two phonon resonances.For our studies, we use two 120-fs pulsed Ti:sapphirelasers with wavelength λ ≈ 800 nm and repetition rate of 80MHz. The experiments were carried out using an asynchronousoptical sampling (ASOPS) technique [70] and also the tra-ditional method in which the laser output is split into pumpand probe beams. The pump beam passes through an acousto-optic modulator and frequency doubler. The pump pulse withwavelength λ ≈ 400 nm is focused onto the sample to an areaof ∼ 20 × 20 μm2 for normal incidence and ∼ 20 × 30 μm2for 45◦ incidence. The probe beam with λ ≈ 800 nm is focusedon the sample surface to a diameter of several microns, over-lapping spatially with the pump spot. The power for pump andprobe were kept below 20 and 0.5 mW, respectively, to avoidnonlinear effects and photooxidation of the InSe layers [71].We used four experimental schemes shown in Figs. 2(a)–2(d) inwhich the intensities of the transmitted [Fig. 2(a)] or reflected[Figs. 2(b)–2(d)] probe beams are measured. The pump beamis directed at an angle of about 45◦ [Fig. 2(a) and 2(b)] or iscollinear and in the opposite direction to the incident probebeam [Figs. 2(c) and 2(d)]. For coherent phonon generation inthin (<70 nm) layers, we used Al transducers [Fig. 2(d)]. TheAl films were deposited on the vdW layers by electron beamevaporation and their thicknesses (∼30 nm) were measured byAFM.Figure 2(e) demonstrates the procedure for obtaining thespectrum of coherent phonons from the temporal evolutionof an arbitrary pump-probe signal. The top panel showsschematically a typical pump-probe temporal trace, which075408-3J. D. G. GREENER et al. PHYSICAL REVIEW B 98, 075408 (2018)includes an instantaneous rise and a slow decay modulatedby periodic oscillations. In a picosecond acoustics experiment,these oscillations arise from coherent acoustic phonons and canbe revealed more clearly by subtracting to the signal a slowdecay background, as fitted by a polynomial or exponentialfunction [see the middle panel in Fig. 2(e)]. The spectrumof coherent phonons is then obtained by performing a fastFourier transform (FFT) of the subtracted curve [bottom panelof Fig. 2(e)]. The procedure of subtracting a slow-decayingbackground may cause artifacts showing low-frequency lineswith Q∼1 in the FFT spectrum. Thus the frequency rangefrom 0 to 10 GHz is not considered in the analysis of theFFT spectra. The relative intensities of the lines in the phononspectrum are generally difficult to model as they are determinedby photoelastic parameters [72], which are not known forthe studied materials. On the other hand, we can analyze thefrequencies and linewidths of the phonon resonances. Theseare governed by the main elastic properties of the layers andcan be compared with the values obtained by solving elasticequations (see Sec. III and Appendix).III. THEORETICAL BACKGROUNDWe describe coherent phonons as classical wave packetsof acoustic waves with displacement vector u, wavevector kand given strain, ε, and stress, σ , tensors. For propagationof longitudinal acoustic phonons (u||k) along the x direction,perpendicular to the plane of the vdW layers, the solution ofthe elastic equations for the Fourier component ω in the ithlayer isũ(ω,x) = Ai cos (kix) + Bi sin (kix), (1)where ki = ω/si is the phonon wavevector and si is the soundvelocity in the ith layer (for details see Appendix). The solutionfor the elastic wave in a semi-infinitive substrate isũS(ω,x) = AS exp (−ikix). (2)The frequency dependent coefficients Ai , Bi , and AS definethe coherent phonon spectrum and can be found by applyingboundary conditions to the displacement and stress at eachinterface. The simplest, most common case for the boundaryconditions is the acoustic mismatch (AM) model, i.e., thedisplacement and stress are continuous at each interface [73].This approximation works for relatively thick layers andperfect, atomically flat interfaces. An example of AM is shownschematically in Fig. 3(a) for an InSe/hBN heterostructure ona substrate. Here, the elastic coupling between the boundaryatoms in the two materials is assumed to be absolutely rigid.For the wave propagating in a medium (i) and reaching theinterface with another medium (j ), the amplitude reflectioncoefficient Rij for the strain isRij = Zj − ZiZj + Zi, (3)where Zi and Zj are the acoustic impedances (Z = ρs, whereρ is the density and s is the sound velocity) of the materials. Forphonons propagating from a high to a low impedance material(Zi > Zj ), Rij < 0 and the phase for the strain wave of thereflected beam is changed by π . In the opposite case (Zi < Zj ),FIG. 3. Schematics showing different strengths of elastic cou-pling between van der Waals layers (e.g., InSe and hBN) andbetween a van der Waals layer and a sapphire substrate. (a) Rigidinterface between InSe and hBN, and between InSe and sapphire.The elastic properties of the rigid interfaces are described by theacoustic mismatch model. (b) Rigid interface between InSe and hBN,but weak elastic coupling between InSe and sapphire. (c) Weak elasticcoupling for both InSe/hBN and InSe/sapphire interfaces, as describedby springs with stiffnesses ηH and ηS , respectively.Rij > 0 and the phase does not change on reflection. The caseZi = Zj corresponds to Rij = 0.When Zi �= Zj , the interference of waves reflected at aninterface results in phonon quantization and nanomechanicalresonances. For a single layer of thickness a on a substratewith acoustic impedance ZS , the phonon frequencies aregiven byfn = (2n − 1)s4a, for Zi < ZS, (4)andfn = ns2a, for Zi > ZS, (5)where n is an integer number.In general, the AM model may not describe well the phonontransport in thin layers. Also, it may not be valid when thephonon dispersion plays a role, e.g., at high phonon frequenciesor when the interface between the layers is not elasticallyperfect. Here we consider layers with thicknesses a > 5 nm,larger than the vdW interlayer separation, i.e., ∼0.8 nm and0.7 nm in InSe and hBN, respectively (see Table I). Also, themaximum phonon frequency studied in this work is 120 GHz,which is much smaller than the frequency at which phonondispersion effects become important (∼1 THz). Thus the mainlimitation to the validity of the AM model to our structures isTABLE I. Elastic parameters of InSe, hBN, and Al used in thecalculations.InSe hBN Al SapphireDensity, kg/m3 5500 2180 2700 4000LA sound velocity, m/s 2500a 3440b 6420 11 000Acoustic impedance, MPa s/m 14 7.5 17 44Elastic constant, C33, GPa 36 26vdW interlayer period, nm 0.8 0.7Spring stiffness of vdW bond, 3.7 3.41019 N/m3aReference [64].bReference [75].075408-4COHERENT ACOUSTIC PHONONS IN VAN DER WAALS … PHYSICAL REVIEW B 98, 075408 (2018)the imperfectness of the interfaces. To account for the elasticproperties of an imperfect interface, we use the spring model[5,74]: the adjacent layers of two materials, i and j , are coupledby effective “springs” with stiffness ηij . Figure 3(b) shows thescenario in which an InSe/hBN heterostructure on sapphire hasAM between the layers, but is weakly coupled to the substrateby an effective spring of stiffnesses ηS . Figure 3(c) shows thescenario in which both interfaces are imperfect, i.e. there isalso a spring of stiffness ηH at the InSe/hBN interface.The boundary conditions in the spring model include thecontinuity of the stress at the interfaces with springs, fromwhich the coefficients Ai , Bi , and AS can be derived (see Ap-pendix). Similarly to the AM model, the spring model revealsnanomechanical resonances in the heterostructure. However,the resonance frequencies fn differ from those predicted by theAM model; also, due to the weaker coupling at the interfaces,these resonances can be narrower.For acoustic phonons propagating from material i to mate-rial j and back from j to i, it is convenient to use parametersGij and Gji , respectively, related to ηij = ηji :Gij = Gji = ηij(Zi + Zj)2πZiZj. (6)Since in our work we use no more than two vdW nanolayers,we consider two parameters GH and GS , which characterizethe coupling between the vdW nanolayers and between theinner nanolayer with the substrate, respectively,GH = ηH (Z1 + Z2)2πZ1Z2(7)andGS = ηS(ZS + Z2)2πZSZ2, (8)where Z1, Z2, and ZS are the acoustic impedances of the outer,inner nanolayers, and substrate, respectively. The parametersGH and GS have the dimension of frequency. For a single layerweakly coupled to the substrate, e.g., GS � fn, the resonancefrequencies are given by Eq. (5).IV. MEASURED AND MODELLED PHONON SPECTRAThis section presents the main experimental findings. Thefirst section, Sec. IV A, focuses on nanomechanical reso-nances in InSe nanolayers and explores their dependenceon the InSe layer thickness and elastic coupling with thesubstrate. Section IV B concentrates on vdW heterostructuresand demonstrates that the AM model is well suited to describea vdW interface, indicating an elastically ideal interface andstrong interlayer coupling for both homo (e.g., InSe/InSe)and hetero (e.g., InSe/hBN) interfaces over a wide range ofsub-THz frequencies.One of the main aims of the present work is to obtain thevalues of GS or GH , which are governed by the coupling pa-rameters ηS and ηH for single nanolayers and heterostructures,respectively. To obtain these parameters, first, we obtain thefrequencies fn of the phonon resonances from the measuredphonon spectra and compare them with the theoretical valuesof fn obtained from the calculated spectra for various GS (GH ).FIG. 4. Phonon spectra for different InSe nanolayers on a sapphiresubstrate. The insets show the temporal evolution of the pump-probe signals after background subtraction. Vertical bars indicatethe calculated phonon frequencies for an InSe nanolayer (thicknessa = 100 nm) coupled to the sapphire substrate in the limit of strong(blue bars, GS � fn) and weak (red bars, GS � fn) elastic coupling.The values of GS are deduced from the best fits of the spring modelto the data.To make this comparison, we calculate the modeled strainspectrum asS(f ) ∼ f√∑i(|Ai |2 + |Bi |2), (9)with the bulk elastic parameters from Table I and the layerthicknesses, as measured by AFM. We find that the couplingparameters GS (GH ) do not depend directly on the thicknessof the nanolayers and are governed by the roughness of thesurface at the interface with coupling parameter ηS (ηH ).A. InSe nanolayersExperiments on relatively thick (a ∼ 100 nm) InSe layerswere carried out using the scheme shown in Fig. 2(a). Figure 4shows the measured coherent phonon spectra for three separateInSe layers with approximately the same thicknessa ∼ 100 nmon a sapphire substrate. The background free temporal evolu-tions are also shown in the insets of Fig. 4. It can be seen thatall spectra consist of a distinct spectral line between 10 and20 GHz. Despite the layers having similar thickness, the linesare centered at different frequencies and the spectral width istwice larger for the lowest curve. Blue and red vertical barsindicate the phonon frequencies calculated from Eqs. (4) and(5), which correspond to AM and weak elastic coupling of thelayers with the substrate (GS � f ), respectively. The valuesof GS obtained from the comparison of the measured and075408-5J. D. G. GREENER et al. PHYSICAL REVIEW B 98, 075408 (2018)FIG. 5. Measured (solid lines) and calculated (dotted lines)phonon spectra for InSe nanolayers with thicknesses a = 41 nm on asapphire substrate. Red and blue curves correspond to acoustic match-ing and weak coupling of the InSe nanolayer to the sapphire substrate,respectively. Phonons are excited via a 33-nm-thick Al transducer.Vertical dashed-dotted lines indicate the central frequencies of themeasured phonon resonances. The inset shows the background freetemporal evolution of the measured signal.calculated spectra are shown in the same graph. The resonanceat the top of Fig. 4 indicates a regime of weak coupling of InSeto the substrate; the one at the bottom has a broad spectralline centered at a frequency equal to that expected from AM;the middle spectrum shows instead two lines that cannot bedescribed by a single coupling parameter GS : the spectrallines centered at 13.4 and 18.0 GHz are reproduced by thespring model with GS = 1 and 20 GHz, respectively. Thus,from this study, we conclude that InSe layers on sapphire canexhibit different mechanical properties and elastic coupling tothe substrate.Experiments with thinner InSe layers were performed usingthe scheme shown in Fig. 2(d) using an Al transducer. Thedeposition of the Al film by electron beam evaporation onInSe produced uniform high-quality thin (33 nm) film. Thuswe expect strong elastic coupling with GijRAMij � f for theAl/InSe interface. Figure 5 shows the measured and calculatedphonon spectra for InSe layers with thickness a = 41 nm onsapphire. Four lines can be seen in the measured spectrumof the 41-nm-thick InSe layer (Fig. 5). The assumption ofAM with the substrate (GS � f ) reproduces the three high-frequency spectral lines. The low frequency line centered atf = 26 GHz agrees with the calculated spectrum by assuminga lower GS-value (GS = 8 GHz).Figure 6 shows the measured spectra for three InSe layerswith thicknesses from a = 5.4 nm to a = 12 nm. We can com-FIG. 6. Measured phonon spectra for InSe nanolayers with dif-ferent thickness on a sapphire substrate. Phonons are excited via a38-nm-thick Al-transducer. Vertical arrows indicate the frequenciesof the phonon resonances obtained from the calculated spectra withcorresponding values ofGS . The shaded area marks the low-frequencyspectrum. The inset shows the temporal evolution of the pump-probesignals after background subtraction for the 6.5-nm-thick layer.pare the measured resonances with the calculated ones only forf > 40 GHz. The low frequency resonances with f ∼ 20 GHz(see shaded square in Fig. 6) have a very low quality factorQ < 1. For these resonance modes, we may conclude onlythat GS > 10 GHz and that the phonon resonance frequencyhas weak dependence on GS . The red vertical arrows in Fig. 6and the values above them show the measured resonancefrequencies and the corresponding GS values obtained fromthe comparison with the calculated spectra. For a = 12 nm,two spectral lines are observed corresponding to GS = 10and 90 GHz. For thinner layers, one spectral line is detectedwith values of GS from 13 GHz (a = 6.5 nm) to 210 GHz(a = 8 nm).From the data and analysis (Figs. 4 –6) of InSe layers witha wide range of thicknesses, from 5 to 100 nm, we concludethat the coupling parameter GS of an InSe layer to sapphirecan have values from ∼1 GHz to more than 100 GHz, withoutany direct correlation with the InSe layer thickness. Our resultsalso show that the measured phonon spectrum cannot always bedescribed by a single parameter GS . We have observed severalcases when a good fit in the whole frequency range requirestwo or more values of GS .B. Van der Waals heterostructuresFigure 7 shows the results for two vdW heterostructures.The first heterostructure is a homojunction obtained by exfo-liation and mechanical contact of two InSe layers of thickness075408-6COHERENT ACOUSTIC PHONONS IN VAN DER WAALS … PHYSICAL REVIEW B 98, 075408 (2018)FIG. 7. (a) Optical images of an InSe/InSe homojunction (left)and of an InSe/hBN heterojunction (right). For the homojunction,B, T, and H correspond to the base, top and overlapping areasof the InSe layers, respectively. For the heterojunction, H corre-sponds to the InSe layer that is overlapped with hBN. Measured(solid lines) and calculated (dotted lines) phonon spectra for theInSe(a = 77 nm)/InSe(a = 50 nm) homojunction (b) and the InSe(a = 50 nm)/hBN (b = 81 nm) heterojunction (c). The upper andlower panels in (b) correspond to the phonon spectra in areas H and T,respectively. The theory curves are shown for different values of GSand/or GH . In (b) and (c), the red dotted curves are the best theoreticalfits to the measured spectra. The insets are temporal signals afterbackground subtraction.a = 50 nm and b = 77 nm on a sapphire substrate. The opticalimage of the heterostructure [see left image in Fig. 7(a)] showsthree distinct areas: the area H of the heterostructure wherethe layers overlap and two regions where the top (T) andbase (B) InSe layers do not overlap, thus they can be probedseparately. In the experiments, we used the geometry presentedin Fig. 2(c). The lower blue solid curve in Fig. 7(b) showsthe phonon spectrum for the nonoverlapping area T. It revealsa peak at f = 21 GHz in agreement with that simulated bythe spring model with GS = 10 GHz [see lower blue dottedcurve in Fig. 7(b)]. The red solid curve in Fig. 7(b) for theoverlapping area H reveals a different spectrum: it consists ofseveral lines at a regular frequency interval of 8.8 GHz. Theindividual frequencies agree well with those calculated for asingle InSe layer with a total thickness equal to the sum ofthe individual layer thicknesses (e.g., a + b = 127 nm) underthe assumption of GH � 50 GHz and GS = 26 GHz. Thuswe conclude that the interface between the InSe layers canbe considered as elastically “ideal” for transport of coherentphonons with f up to at least 50 GHz.FIG. 8. Measured (solid line) and calculated (dotted line) phononspectra for an hBN/InSe heterostructure on a sapphire substrate. Theinset shows the temporal evolution of the pump-probe signals afterbackground subtraction. The hBN and InSe layers have thickness ofa = 14 and 10 nm, respectively. Phonons are excited from a 35-nm-thick Al transducer. The theory curves are shown for different valuesof GS and GH . The red dotted curve is the best theoretical fit to theexperimental spectrum.Figure 7(c) shows the results for an InSe/hBN heterostruc-ture on a sapphire substrate measured using the transmissiongeometry shown in Fig. 2(a). In this heterostructure, an indi-vidual InSe nanolayer (a = 50 nm) is stamped onto an hBNlayer (b = 81 nm) to form a heterojunction. Two resonances atf = 18 and 30 GHz are clearly seen in the measured spectrumindicating efficient elastic coupling between the layers. If therewas no good elastic contact between the layers, the spectrumwould reveal only the nanomechanical resonances excited inthe top InSe layer. In fact, the underlying hBN layer cannotbe excited directly as it is transparent at the wavelength of thepump probe. As for the case of the InSe/InSe homojunction,also in this case the calculated spectrum indicates acousticmismatch at the InSe/hBN vdW heterointerface [see dottedlines in Fig. 7(c)]. The best agreement between our modelwith the experiment is obtained for GH � 30 GHz and GS =10 GHz [Fig. 7(c), red dotted line].Similar results were obtained for other vdW heterostruc-tures and experimental geometries. For example, Fig. 8 showsthe results for an InSe/hBN heterostructure on sapphire withthicknesses of the hBN and InSe layers of a = 14 nm andb = 10 nm, respectively. Coherent phonons were injected intothe heterostructure from an Al-film (thickness of 35 nm)evaporated on top of the hBN layer [experimental schemeFig. 2(d)]. From the observation of four distinct spectral linesin the frequency range up to 150 GHz and the comparisonwith the calculated phonon spectra (dotted lines in Fig. 8), we075408-7J. D. G. GREENER et al. PHYSICAL REVIEW B 98, 075408 (2018)conclude that there is a strong coupling at the heterointerface.The best agreement is obtained assuming GH � 100 GHz andGS = 10 GHz. If we assume that the coupling at the vdWheterointerface drops to GH = 30 GHz (see the lowest dottedcurve in Fig. 8), the spectrum would show only two resonances.In summary, the measured spectra of coherent phononsreveal nanomechanical resonances with frequency, amplitude,and spectral width that depend on the strength of elastic cou-pling between the different layers and/or with the supportingsubstrate. Our study of the acoustic mismatch between vdWlayers in InSe/InSe homojunctions and InSe/hBN heterojunc-tions reveal an ideal interface and strong coupling over a widerange of frequencies up to ∼100 GHz.V. DISCUSSIONIn this section, we examine how the frequency and thefinesse of the phonon resonances depend on the elastic proper-ties of the nanolayers and vdW heterostructures, and on phononscattering at interfaces, i.e., the interface between two vdWlayers or the interface between a vdW layer and its supportingsubstrate.A. Elastic properties of InSe and hBNThe layered crystal structure of InSe and hBN causesanisotropic elastics properties, e.g., elastic constant tensor cijand sound velocity s. For our experiments with propagation ofcoherent phonons in the direction x perpendicular to the vdWlayer planes, the sound velocity is calculated as s = √c33/ρ(for values see Table I) using elastic parameters measuredin bulk crystals by standard ultrasonics [64,75] and Brillouinscattering [76].To assess whether the use of elastic parameters for bulkcrystals is a valid assumption to interpret our findings,we derive elastic parameters from an empirical Lennard-Jones potential U(x) for the vdW interaction between thelayers:U (x) = EB[(x0x)12− 2(x0x)6], (10)where x0 is the distance between the edges of the atomicmonolayers and EB is their binding energy. In hBN, x0corresponds to the interlayer period, e.g., x0 = c ∼ 0.7 nm.In InSe, each vdW monolayer consists of four covalently ionicbonded atomic sheets (Se-In-In-Se) of thickness 0.53 nm. Thedistance between Se-atoms at the boundaries of each vdWlayer is x0 ∼ 0.38 nm [77,78]. The value of EB for hBN isEB = 13 meV/Å2[79]; also, vdW interactions in InSe werefound to be as weak as in graphite and other vdW crystals [78],for which EB is in the range 13 to 21 meV/Å2[79].For a small change �x in the equilibrium separation of thevdW layers, i.e., �x = x − x0 � x0, Eq. (10) can be writtenasU (�x) = 36EBx−20 (�x)2, (11)from which we derive the specific stiffness of the vdWcoupling:η0 = 72EBx−20 . (12)Alternatively, we can estimate the value of η0 from theelastic constant of the bulk crystals:η0 = c33/c. (13)Using Eqs. (12) and (13) for hBN, we derive η0 = 3.1 ×1019 N/m3 and η0 = 3.7 × 1019 N/m3, respectively. Thus dif-ferent models give similar stiffnesses for hBN. Also, usingEq. (13) and the values for c33 and x0 for InSe (see Table I),we derive η0 = 4.5 × 1019 N/m3. From this value of η0 andEq. (12), we estimate that EB = 15 meV/Å2for InSe, withinthe range of values of EB reported for other vdW crystals [80].We use the same “spring” model approach to describethe elastic coupling between two materials (Sec. III B, andAppendix) and calculate the coupling parameter G0 betweentwo vdW layers. Using the estimated values of η0, we obtainG0 ∼ 1.0 and 1.5 THz for InSe and hBN, respectively. Thesevalues are smaller than the zone edge phonon frequencies forpropagation perpendicular to the layers, i.e., f = 1.2 and 2.4THz for InSe [81] and hBN [75], respectively, but much higherthan the maximum phonon frequency f ∼ 100 GHz measuredin our experiments. Thus we conclude that our use of thecontinuous approximation with bulk elastic parameters andsound velocity of InSe and hBN is valid in our analysis of thephonon spectra.B. Elastic properties of interfacesIn the analysis of phonon transport, the elastic propertiesof interfaces are as important as the elastic properties of thenanolayers themselves. The interface defines the absolute value|R| and phase shift �ϕ of the specular phonon reflection at theinterface. The value of �ϕ varies from 0 to π depending onthe elastic coupling. This phase shift governs the interferenceof coherent phonons and hence defines the frequency fn of thephonon resonances.We use a phenomenological spring model to describe theinterface between two materials (see Sec. III and Appendix).In this model, the interface is described as a weightless spring(Fig. 3) with a frequency independent specific stiffness η. Asshown in Sec. IV, the value of fn measured in several layersand heterostructures agrees well with acoustic mismatch (G �fn), thus indicating a good elastic contact at the interface overa wide frequency range. However, in some cases, we have toassume a weak coupling between a layer and its supportingsubstrate (GS ∼ fn) and/or different values of GS . To interpretthese results, we examine the morphology of the interfaces.The elastic coupling between the exfoliated nanolayersdepends on the quality of the surfaces, which can be nonuni-form over the area (5 × 5 μm2) of the probing spot. Figure 9shows the AFM images and x profiles for an InSe layer on asapphire substrate. The AFM data show that the roughness ofthe InSe surface (up to 1 nm) is significantly larger than forthe sapphire substrate (∼ ±0.1 nm). Also, it is larger than thatof hBN, which has an abrupt and atomically smooth surface[82,83]. Thus the most likely reason for the imperfectness075408-8COHERENT ACOUSTIC PHONONS IN VAN DER WAALS … PHYSICAL REVIEW B 98, 075408 (2018)FIG. 9. (a) Atomic force microscopy (AFM) image of a 5 μm x5 μm region of InSe (scale bar = 1 μm). Inset: AFM profile alongthe white dotted line. (b) AFM image of a 1 μm x 1 μm region ofInSe (scale bar = 0.2 μm). (c) AFM image of an InSe nanolayer on asapphire substrate. (Inset) AFM profile along a line through regionsmarked by squares 1-2-3. These areas show regions with differentsurface roughness (scale bar = 5 μm).of the interface revealed in our experiments is the surfaceroughness of the InSe layer. This also implies inhomogeneitiesin the coupling parameter ηS and ηH . Two atomically flat vdWlayers interact elastically via vdW forces and have a bindingenergy EB ∼ 10 meV/Å2, which does not depend stronglyon the specific material [78,79]. The value of EB gives astiffness η0 ∼ 1019 − 1020 N/m3 and corresponding couplingparameter G0 ∼ 103 GHz. Due to the strong dependence of thevdW potential energy on distance [Eq. (10)], the increase of xby only one layer of atoms due an imperfect interface results inthe decrease of η0 and G0 by almost two orders of magnitude(G∼10 GHz), leading to changes in the complex phononreflectivity R [Eq. (A9)] and phonon resonance frequency fn.Thus, for sub-THz phonons, the areas of atomically perfectinterface behave as in the case of AM while the increase of thedistance at the interface of just one atomic layer results in thereflection of sub-THz phonons, as for the case of a free surface.To characterize the coherent phonon resonances, we con-sider two types of regions. The first type, labeled “AM,” hasan atomically flat interface: the distance x between the layersis uniform with a coupling parameter GAM ∼ 103 GHz. Thesecond type corresponds to a “free surface” (FS): the distancebetween the layers is x � 2x0 with a coupling parameterGFS � f . It is important to compare the average size of the AMand FS regions, LAM and LFS, respectively, with the phononwavelength �. For InSe, the value of � = s/f varies from250 nm down to 25 nm for f in the range 10–100 GHz. Inthe assumption that the correlation length is small comparedwith � we may consider LAM, and LFS � �, and then for thephonon resonances observed in our experiments the interfaceroughness is averaged over �. In this case, we can characterizethe interface using a single coupling parameter:G = G0SAMSAM + SFS, (14)where SAM and SFS are the areas of the AM and FS regionswithin the probe spot. We refer to this type of interface as“mono.” For SAM � SFS, we have G = G0 ∼ 1 THz and theinterface may be considered as elastically perfect and the AMmodel valid for all acoustic phonons, provided that the phonondispersion is not important. An alternative case is that of a “bro-ken” interface, i.e., SAM � SFS and hence G � G0. In general,the interface may have a more complicated morphology. Forinstance, in the area of the probe spot, there may be areas withdifferent mean values of G. Examples of areas of InSe withdifferent surface roughness and hence different G are shownin Fig. 9(c) by squares labelled 2 and 3. Their typical in planesize (∼1 μm) is larger than �. The corresponding profilesin Fig. 9(c) of these areas demonstrate that the roughness ofthe surface in area 2 is higher than in area 3, thus leading toG3 > G2 and a “dual” interface. In general, we can identifyboth “mono” and “dual” interfaces for InSe layers on a sapphiresubstrate.In contrast, we find that the interface between two vdWlayers (e.g., the interface in InSe/InSe and InSe/hBN) iselastically perfect within the measured frequency range. Thecomparison between the experimental and calculated spectrain Figs. 7 and 8 indicates that the phonon frequency would bevery different if the layers were not elastically coupled. Thuswe may exclude the case of weak coupling (GH � f ) at theinterface between two vdW layers in all our heterostructures.Also, we can explain the phonon spectra using the acousticmismatch model over a wide frequency range up to 120 GHz(Fig. 8).At a first glance, it may seem surprising that the interface ofInSe/InSe and InSe/hBN heterostructures has a stronger elasticcoupling compared to the InSe/sapphire interface since thesurface roughness of the InSe layer is the same in both cases.To explain this behavior, we note that there is a difference inacoustic impedances between InSe and sapphire: Zsapphire =3.2ZInSe. Thus the value of G0 for the interface of InSe withsapphire is noticeably smaller than for InSe/InSe and InSe/hBNinterfaces [see Eqs. (7) and (8)]. Also, the more rigid sapphiresubstrate may not be able to adapt to the InSe surface comparedto the case of two van der Waals layers.In summary, we have shown that the frequencies of thephonon resonances are dependent on both the thickness of thelayer and elastic coupling at the interface. The latter factor wasnot considered in earlier works [55–57] where it was assumedthat the nanolayer has either a perfect elastic contact with thesubstrate [57] or behaves as a suspended film [55]. In ourwork, we show that the elastic properties of vdW interfaces canbe probed by sub-THz coherent phonons. None of the otherexisting measuring techniques can access this information.In particular, some of the available microscopy techniques,such as TEM, examine only static surface properties, whereas075408-9J. D. G. GREENER et al. PHYSICAL REVIEW B 98, 075408 (2018)others, such as AFM, can only probe surfaces and/or low-frequency elastic properties.C. Scattering and decay of coherent phononsAn important characteristic of a phonon resonance is itslinewidth �f ∼ (2τ ∗)−1, where τ ∗ is the phonon decay time.The measured linewidths indicate τ ∗ � 50 ps. Different mech-anisms can contribute to τ*, e.g.,(τ ∗)−1 = (τS)−1 + (τA)−1 + (τI )−1 + (τD)−1 + (τR)−1,(15)where τS is the time of the diffusive scattering at interfacesand free surfaces; τA is the anharmonic decay time; τI is theeffective time related to inhomogeneities in film thickness; τDdescribes the decay of the laterally localized resonance dueto acoustic diffraction, and τR describes the phonon decaytime due to energy emission into the substrate. Only the latterone is included in our previous theoretical analysis and can beexpressed asτR = − 2as ln |R| , (16)where s is the sound velocity of the nanolayer and a is thelayer thickness. Using Eq. (3) for AM (GS � f ) and for anInSe nanolayer layer with a ∼ 100 nm on a sapphire substrate,we estimate τR ∼ 100 ps, which is about three times longerthan the value τ*∼30 ps derived from the measured phononspectra (see lower spectrum in Fig. 4). The calculated valueof τR increases for weak coupling and correspondingly smallvalues of GS . Thus the difference between the measured andcalculated τR is due to other decay mechanisms, which arediscussed below.The value of τA is governed by third-order elastic constants,which are not known for the studied vdW layers. We mayassume that τA is not much different from the values found inother crystalline materials for which τA > 1 ns at frequenciesbelow 100 GHz [84]. This time is significantly longer than themeasured time and thus it will be not considered further.The value of τD is governed by the lateral dimension LG ofthe local AM and FS resonators characterized by an effectiverigidity G at the interfaces in Eq. (14). The value of τD canbe estimated as the time taken for an acoustic wave to travela diffraction length LD , e.g., τD ≈ LD/s ≈ (L2G/�)/s, whereLG is the diameter of the acoustic beam. Since LG � �, τD ismuch longer than the measured time. For example, for phononswith �∼100 nm in an InSe film strongly coupled to sapphire,τD becomes equal to the measured time when LG ∼ �. Thusthe losses due to diffraction are not important only for thelocal resonators with LG significantly exceeding the acousticwavelength (LG � �).The decay time τI could be caused by inhomogeneousbroadening of the resonance spectral line due to the long-scale(∼1 μm in the plane of the nanolayers) thickness variationsof the InSe and hBN nanolayers. The AFM profiles of ourlayers show variations in the thickness of ±10%. Thus τI canrepresent a significant contribution to the measured lifetime.For example, the measured lifetime τ*∼80 ps for 13 GHzphonons in an InSe nanolayer weakly coupled to sapphire(upper spectrum in Fig. 4) could be assigned to ±15% varia-tions in the InSe film thickness.Finally, the decay time τS in Eq. (15) accounts for thelosses caused by diffusive scattering at an interface due to theirroughness, point defects, monolayer steps and/or boundariesbetween regions with different G. The AFM image in Fig. 9shows variations of the InSe layer thickness over length scales>100 nm. We examine their effect on acoustic wave scatteringusing the small slope approximation (SSA) [85,86]. For ourestimate of τS , we use the dependence of the specular scatteringprobability (SSP) on the root-mean square roughness σrms ofthe surface [87]. The data presented in Ref. [87] for 117-GHzphonons in GaN can be applied to estimate τS for ∼20 GHzphonons in InSe films as both systems have similar phononwavelengths (∼70 nm); also, the correlation length of thesurface roughness of InSe (∼ 10–40 nm measured by AFM)and in GaN (∼ 24–38 nm in Ref. [87]) are similar. For valuesof τS equal to the measured ones, the SSP is ≈ 0.15, implyingσrms ≈ 8 nm [87]. This value exceeds by an order of magnitudethe value σrms ∼ 1 nm for the InSe surface (see AFM data inFig. 9). Thus we may conclude that diffusive scattering of ∼20GHz phonons at the interfaces with σrms � 1 nm is negligibleand, therefore, its contribution to τS can be ignored.In summary, from the comparison of the measured andcalculated spectra, we find that the measured spectral widths�fn � 10 GHz exceed those calculated by acoustic mismatchby a factor of 2. In accordance with the estimates presentedabove, this difference should be mostly attributed to theinhomogeneous broadening of the spectral line.VI. CONCLUSIONS AND OUTLOOKWe have used picosecond ultrasonics to measure the coher-ent acoustic phonon spectra of InSe layers with thicknessesfrom 5 to ∼100 nm and their heterostructures. The spectraconsist of several well separated lines in the frequency rangefrom ∼10 to ∼100 GHz, revealing the existence of nanome-chanical resonances in all layers due to phonon quantization inthe direction perpendicular to the layer plane. The frequenciesof quantized phonons depend not only on the layer thickness,but also on the elastic coupling parameters at the interfaces.The values of the resonance frequencies measured in thepresent work agree well with those derived from the continuouselastic theory using bulk elastic constants and effective springswith frequency independent stiffnesses. In some cases, theinhomogeneities of the elastic coupling at the interfaces shouldalso be taken into the account to explain the phonon spectrum.Our data and analysis indicate that the vdW interface ofInSe/InSe and InSe/hBN heterostructures can be described bythe acoustic mismatch approach and considered as elasticallyideal for phonon frequencies up to at least 150 GHz. In contrast,the elastic coupling of InSe layers to a substrate may notalways be ideal and the phonon resonances may not alwaysbe described by a single elastic coupling parameter.The present work focuses on elastic properties of interfacesrather than phonon properties of the layers, as reported in ear-lier works [55–58]. Furthermore, our conclusions are broaderthan those in the earlier work by Beardsley et al. [58], whichreported InSe layers weakly coupled to a substrate. In thepresent work, we show that the elastic coupling of a vdW075408-10COHERENT ACOUSTIC PHONONS IN VAN DER WAALS … PHYSICAL REVIEW B 98, 075408 (2018)nanolayer to its supporting substrate may vary and can bedescribed by the spring model; also, in both InSe/InSe andInSe/hBN vdW heterostructures, we observe a strong elasticcoupling, well described by the acoustic mismatch model.Means of probing phonon transport across an interface canprovide the foundation for designing and exploiting new ma-terials and quantum systems: the strength of coupling betweenthe layers is critical to both charge and phonon transport andthus pivotal to the future development of functional devices,including new device concepts for high-frequency electronicsand thermoelectrics. The spectrum of nanomechanical reso-nances changes significantly when the distance between thelayers changes by only one layer of atoms. Thus it could beused to probe and manipulate interfaces with an unprecedentedspatial resolution and over a wide frequency range. To date,the only technique that could offer such spatial resolution isTEM, but this is time-consuming and requires special samplepreparation to expose interfaces to the probing electron beam.On the other hand, it should be possible to use a focusedlaser beam to generate coherent phonons as it scans over thesample to measure the spectrum and lifetime of the phononsin the structures or the spectrum of phonons emitted into thesubstrate.Finally, of particular interest is the possibility of usingstrain to modulate the electronic properties for high-frequencyacoustoelectric and acousto-optic devices. In particular, 2DInSe exhibits high mechanical flexibility, can sustain highmechanical strain (>20%) and its electronic properties canchange under both uniaxial and biaxial compressive strains[88]. These properties, which have been only recently predictedby theory [88], are highly desirable for new “straintronic”devices and sub-THz nano-electro-mechanical and optome-chanical devices. In particular, thin layers and heterostructuresweakly coupled to the substrate may have high finesse (>10)and used as sensors for single molecules adsorbed at surfacesand interfaces. Further studies may involve the study of singleatomic layers and correspondingly THz phonons. In this case,the approximation of continuous media and acoustic mismatchapproach may not be valid. There are no principle limitationsto these studies, which will contribute to important advancesin the physics of phonons and their exploitation in novel 2Dstructures.ACKNOWLEDGMENTSThis work was supported by the Engineering and PhysicalSciences Research Council [Grant No. EP/M012700/1] (EP-SRC); the EU FP7 Graphene Flagship Project 604391; TheLeverhulme Trust; and the National Academy of Sciencesof Ukraine. K.W. and T.T. acknowledge support from theElemental Strategy Initiative conducted by the MEXT, Japan.APPENDIXFor propagation of longitudinal acoustic phonons along thex direction, perpendicular to the plane of the vdW layers, theelastic equation isρi∂2ui∂2t= ci∂2ui∂2x+ ∂σ0∂x, (A1)where the index (i) defines the vdW layer or the substrate (s)with density ρi and elastic constant ci , and σ0 is the stressgenerated by the optical pump pulse. In our experiments, thepump pulse is absorbed in a thin InSe layer or Al transducerand with good approximation we may consider the excitationas spatially homogeneous due to fast transport of photoexcitedcarriers and heat in the layer [69]. Then σ0 does not dependon x except at the interfaces at which the second term in theright part of Eq. (A1) is equal to zero. For comparison with theexperiment, it is convenient to present the results in a spectraldomain for the Fourier components of displacement ũ(ω,x)and strain ε̃(ω,x) = dũ/dx. In the spectral domain, Eq. (A1)may be written asd2ũi(ω,x)dx2+ ω2si2ũi(ω,x) = 0, (A2)where si = √ci/ρi is the sound velocity in the ith layer orsubstrate. The general solution of Eq. (A2) for phonons invdW layers is given by Eqs. (1) and (2) in Sec. III.In the acoustic mismatch model, the displacement and stressare continuous at each interface. An example for an InSe/hBNheterostructure on a substrate is schematically shown inFig. 3(a). There are two vdW nanolayers with thicknesses a(InSe, layer 1) and b (hBN, layer 2) deposited on a sapphiresubstrate (s). There are three interfaces at x = −a, x = 0, andx = b. The stress σ0 induced by the optical pump pulse isgenerated only in the InSe nanolayer. In contrast, σ0 = 0 inthe hBN layer and the sapphire substrate, which do not absorbthe pump light at λ = 400 nm. The boundary conditions maybe written asA1k1 sin (kix) + B1k1 cos (kix) + σ̃0 = 0 at x = −a, (A3)A1 = A2B1c1k1 + σ̃0 = B2c2k2}at x = 0, (A4)andA2 cos (k1b) + B2 sin (k2b) = As exp (−iksb)−A2c2k2 sin (k2b) + B2c2k2 cos (k2b)= −iAscSks exp (−iksb)⎫⎬⎭ at x = b.(A5)Solving Eqs. (A3), (A4), and (A5), we find five complexamplitudes A1, A2, B1, B2, and AS , and substituting theminto Eqs. (1) and (2), we get the Fourier spectrum of coherentphonons at the coordinate x.The spring model describes the elastic properties of aninterface that is not ideal. In this model, the neighboringboundaries of two materials are considered to be connected bysprings with specific stiffness η [see Figs. 3(b) and 3(c)]. Theparameter η characterizes the force �F per unit area, whichappears during the modulation of the distance �x betweenthe boundaries of the neighboring materials. In the linearapproximation, according to Hooke’s law, we can write�F = η�x. (A6)075408-11J. D. G. GREENER et al. PHYSICAL REVIEW B 98, 075408 (2018)The boundary conditions for stress at the free surface, x =−a, are the same as Eq. (A3). For the interface at x = 0 andx = b, the conditions for the continuity of the stress can bewritten, respectively, asB1c1k1 + σ̃0 = ηH (A2 − A1)(A7)B2c2k2 = ηH (A2 − A1),and−A2c2k2 sin (k2b) + B2c2k2 cos (k2b)= ηS[As exp (−iksb) − A2 cos (k2b) − B2 sin (k2b)]−Asic3k3 exp (−ik3b)= ηS[As exp (−iksb) − A2 cos (k2b) − B2 sin (k2b)]⎫⎪⎪⎬⎪⎪⎭.(A8)Equations (A7) and (A8) give the solution for coefficientsAi and Bi .For strain waves incident from i to j , the equation for thereflection coefficient at the interface isRij = ηij (Zj − Zi) − iZjZiωηij (Zj + Zi) + iZjZiω= GijRAMij − ifGij + if, (A9)where RAMij is the reflection coefficient given by Eq. (3). ForGij = 0, Rij = −1 as for a free surface. For GijRAMij � f ,Rij ≈ RAMij . In contrast to the scenario of AM, in this case Rijis complex, thus implying that the phase change at the interfacedepends on η. 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