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H. Uchiyama, [Y. Oshima](https://orcid.org/0000-0001-8293-4891), R. Patterson, S. Iwamoto, J. Shiomi, [K. Shimamura](https://orcid.org/0000-0001-6502-8731)

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[Phonon Lifetime Observation in Epitaxial ScN Film with Inelastic X-Ray Scattering Spectroscopy](https://mdr.nims.go.jp/datasets/a0cb9988-c666-4011-8256-aded90e6ffc1)

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Phonon Lifetime Observation in Epitaxial ScN Film with Inelastic X-Ray Scattering Spectroscopy Phonon Lifetime Observation in Epitaxial ScN Filmwith Inelastic X-Ray Scattering SpectroscopyH. Uchiyama,1 Y. Oshima,2 R. Patterson,3 S. Iwamoto,4 J. Shiomi,4 and K. Shimamura21Research and Utilization Division, Japan Synchrotron Radiation Research Institute (JASRI),SPring-8, 1-1-1 Koto, Sayo, Hyogo 679-5198, Japan2Optical Single Crystals Group, Environment and Energy Materials Research Division,National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan3School of Photovoltaic and Renewable Energy Engineering, University of New South Wales, Sydney 2052, Australia4Department of Mechanical Engineering, The University of Tokyo, Bunkyo, Tokyo 113-8656, Japan(Received 18 December 2017; revised manuscript received 21 March 2018; published 7 June 2018)Phonon-phonon scattering dominates the thermal properties in nonmetallic materials, and it directlyinfluences device performance in applications. The understanding of the scattering has been progressingusing computational approaches, and the direct and systematic observation of phonon modes that includemomentum dependences is desirable. We report experimental data on the phonon dispersion curves andlifetimes in an epitaxially grown ScN film using inelastic x-ray scattering measurements. The momentumdependence of the optical phonon lifetimes is estimated from the spectral width, and the highest-energyphonon mode around the zone center is found to possess a short lifetime of 0.21 ps. A comparison withfirst-principles calculations shows that our observed phonon lifetimes are quantitatively explained by three-body phonon-phonon interactions.DOI: 10.1103/PhysRevLett.120.235901Thermal management plays an important role in semi-conductordeviceswith continued advances inminiaturizationto the nanoscale. In power devices, for example, local heatingby the hot-electrons degrades the performance [1]. Anotherexample can be seen in applications for energy harvestingdevices, where heat is recycled as energy [2,3]. In the case ofpolar semiconductors, phonon properties are correlated to thethermal properties, which leads to the following conse-quences. First, the highest energy [longitudinal optical(LO)] phonon mode at the zone center dominates the heatdissipation process of hot carriers, owing to the Frölichinteraction. The thermalization rates can, hence, be reducedby the reabsorption of these high energy phonons that couplestrongly to electrons [1]. Reducing the LO phonon lifetimecan avoid local heating, where energy is returned to chargecarriers and phonondecay is inhibited, leading to deviceswithlow energy consumption and high performance. On the otherhand, prolonging theLOphonon lifetime can be helpful in thehot-carrier photovoltaic concept [3–6]. Detailed knowledgeof the available vibrational modes is central to interpretingultrafast relaxation and energy dissipation processes [4,5].Second, the velocities and lifetimes of all phonon modesdetermine the lattice thermal conductivity, which in generalleads to heat dissipation in devices [7]. This lattice thermalconductivity is an important parameter to evaluate thethermoelectric figure of merit (ZT), which determines theproperties of thermoelectric devices [2].Recent developments in first-principles density-functionaltheory (DFT) calculations now provide both dispersions andlifetimes of all phonon modes with the momentum depend-ence, and thermal conductivities in many materials can alsobe estimated precisely using these phonon properties [8–10].Experimentally, recent inelastic neutron or x-ray scatteringmeasurements reveal correlations between thermal conduc-tivities and phonon properties for several thermoelectricmaterials (such as PbTe [11], UO2 [12], AgSbTe2 [13],NaxCoO2 [14], and clathrate BaGeAu [15]). These studieshave been confined to bulk crystalline materials, mainlyowing to the long penetration depth of the neutron, and nodetailed study has been reported for epitaxial films, the mostcommon forms in device applications. Raman scattering is awell-known method, which detects the phonon energy andlifetime of the epitaxial films (for example, for wurtzite InNfilms [16–18]). However, this method can only measuresome optical phonon modes at the zone center in the first-order scattering, and it gives no informationabout thephonondispersion relations. Inelastic x-ray scattering (IXS) is aunique technique that reveals thephonondispersion curves infilms [19–22], but no detailed observation of the phononlifetime, that includes temperature and momentum depend-ence, has been investigated.ScN, a semiconductor with an electronic band gap of0.9 eV [23], is a promising thermoelectric candidate amongthe nitride semiconductors [2,24–27]. Since this material isnormally obtained as epitaxial films, the fundamentalelastic properties and the origin of the low thermalconductivity are yet unknown. This material has a rocksaltstructure, in which no phonon modes should be observed inPHYSICAL REVIEW LETTERS 120, 235901 (2018)0031-9007=18=120(23)=235901(7) 235901-1 © 2018 American Physical Societyhttps://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevLett.120.235901&domain=pdf&date_stamp=2018-06-07https://doi.org/10.1103/PhysRevLett.120.235901https://doi.org/10.1103/PhysRevLett.120.235901https://doi.org/10.1103/PhysRevLett.120.235901https://doi.org/10.1103/PhysRevLett.120.235901the first-order Raman scattering. Some high-order Ramanscattering measurements have been conducted [28–32], butthe results only show the phonon energy at some points inthe Brillouin zone. In this Letter, we present detailed anddirect observations of phonon energies and lifetimes in anepitaxial ScN film as a function of momentum, in order toelucidate both the harmonic and anharmonic contributionsof the phonon modes present in the material. Afteroptimizing the incident angle with respect to the surfaceof the film (α in Fig. 1(a). We chose α ¼ 1.1 − 1.7° in themeasurements, see Supplemental Material for the detailedoptimization [33]), we obtained information only from theScN film, without artifacts from the substrate. The exper-imental results agree with the DFT calculations; i.e., thephonon energies are described by the harmonic approxi-mation. The phonon lifetimes, estimated from the spectrallinewidth, are well explained by the lowest-order (cubic)anharmonic phonon-phonon interaction. In addition, ther-mal conductivity derived from the calculations wellreproduces that of this film, but much higher than thosein previous studies [24,26,27], suggesting that defects andimpurities also serve a vital role in the thermal conductivity.The ScN (110) film, which has a thickness of 40 μm,was grown by a hydride vapor phase epitaxy (HVPE)method on a sapphire (Al2O3) (100) (m-plane) substrate[34]. The film measured has carrier of n ¼ 1.07 ×1019 cm−3 (electron) and mobility of μ ¼ 164 cm2=V=sat room temperature (RT, T ∼ 300 K). The carrier of ScNoriginates from the defects and impurities, and the presentcarrier concentration is much smaller than the previousliterature [25–27], indicating that the sample is of goodcrystal quality [34]. IXS measurements were performed onthe BL35XU beam line at the SPring-8 synchrotron facility[41]. An incident x-ray energy of hν ¼ 21.747 keV, whichcorresponds to Si(11 11 11) reflection, gives an energyresolution of 1.4 − 1.7 meV. In the present setup, eachanalyzer has a momentum resolution of 0.9 nm−1, or ΔQ ∼ð0.07; 0.07; 0.01Þ − ð0.05; 0.05; 0.06Þ (depending on thegeometry) for ScN. In estimating the elastic constants, themeasurements with a better resolution of 0.4 nm−1 werealso achieved. Overall uncertainty in determining the IXSpeak positions is at around �0.3 meV. The DFT calcu-lations, with a cutoff energy of 20 Hartree, were producedusing the local density approximation (LDA) based on theprojector-augmented-wave (PAW) method, as implementedin the ABINIT package [42,43]. The lattice parameters wereoptimized under the calculations at a zero pressure con-dition. In calculating the harmonic and anharmonic con-tribution, PHONOPY and PHONO3PY codes [10] were used.The quadratic and cubic interatomic potential wereobtained by a 2 × 2 × 2 (64 atoms) supercell approachwith finite atomic displacements of 0.03 Å. It should benoted that the nonanalytical term [44], which definesLO-TO (transverse optical) splitting, was adjusted toreproduce the experimentally observed LO-TO splittingat around Γ in the present calculations. The DFT calcu-lations tend to underestimate the splitting [18], perhaps dueto the underestimation of the (electronic) band gap; thepresent calculations suggest that ScN has no (electronic)band gap, though in reality ScN has an (indirect) bandgap [23,45].Figure 1(b) shows the IXS spectra on a logarithmic scalefor Q ∼ ð3þ q;−1 − q;−1Þ (left panel) and ð1; 1; 3þ qÞ(right) at RT. These momenta Q mostly correspond toq ¼ ðqq̄0Þ and (00q) in the reduced Brillouin zone(Q ¼ qþG;G is a reciprocal lattice vector) and multiplenearby q points are simultaneously sampled by the detectorarray, leading to efficient data collection. The exact q valuesare displayed as green and red points in Fig. 1(c) (also seeSupplemental Material for the detailed information of Q[33]). The peak positions in Fig. 1(b) are determined usingVoigt peak fitting and plotted in Fig. 1(d) as black solidcircles. Some of the peak intensities are too weak to beidentified, as seen in Fig. 1(b), and some of the acousticAl2O3(e)(c)(a)(f)hναsurface normal ScN (110) Al2O3 (100)1.001.0 ~(1,1,3+q) (rlu)~(3+q,-1-q,-1) (rlu)(b)(d)TOLOLATATOTOLOLOLALATATA, .pxE  Calc.Γ XΓ LX K806040200Energy (meV)Energy (meV)LXkxkykzKΓEnergy (meV)-5.5-5.0-4.5-4.0-3.5-3.0806040200ScNRTScN [110]806040200 Exp. Calc.X806040200Energy (meV)(0.50, 0.50, 3.50)Q=(1.50, 1.50, 3.50)FIG. 1. Phonon dispersion relations of ScN. (a) Experimentalconfiguration for the IXS measurements. (b) IXS spectra on alogarithmic scale, obtained for Q ∼ ð3þ q;−1 − q;−1Þ (left)and ð1; 1; 3þ qÞ (right). (c) The Brillouin zone of the ScNcrystal. The green and red solid circles represent the exactmeasured q for the left and right panels in (b), respectively.(d) Peak positions estimated from (b) (black solid circles) in thereduced Brillouin zone and corresponding phonon dispersioncurves in the DFT calculations (red lines). Black lines are guidesto eyes. Additional peak positions estimated from the spectra forQ ∼ ð4; q; 0Þ, ð3þ q;−1;−1Þ, and ð2 − q; 2 − q; 4Þ are depictedas black open circles. (e) IXS spectra (on a logarithmic scale) atthe L point. (f) Calculated phonon dispersion curve in the [qqq]direction (red). The peak positions in (e) are displayed as blackopen circles.PHYSICAL REVIEW LETTERS 120, 235901 (2018)235901-2phonon modes do not show the correct peak position, owingto the overlapping of LA (longitudinal acoustic) and TA(transverse acoustic) and a deviation from the high symmetrylines. To complete the estimates of the dispersion relations,additional measurements were performed in differentBrillouin zones [G ¼ ð400Þ and (224), not shown] andthe fitting results are depicted as black open circles inFig. 1(d). It is evident from Fig. 1(d) that all experimentalpeak positions can be assigned to ScN phonon modes. Forexample, in the [001] direction, the phonon consists of fourmodes, TA, LA, TO, and LO. The red lines in Fig. 1(d)indicate the phonon dispersion in theDFT calculations alongthe high-symmetry lines, which reproduces the measure-ments remarkably well (black). For further confirmation, wealso observed the phonon modes at Q ¼ ð1.50; 1.50; 3.50Þand (0.50, 0.50, 3.50), which correspond to the L point[Fig. 1(e)], supporting the agreement between observationand calculations, as seen in Fig. 1(f). The elastic constantsat RT were determined from spectra around Γ throughthe Christoffel equation (see Supplemental Material fordetails [33]). Experimentally obtained elastic constants areC11 ¼ 384ð5Þ, C12 ¼ 107ð5Þ, and C44 ¼ 178ð5Þ GPa, thatfairly agree with the present calculations: C11 ¼ 442,C12 ¼ 238, and C44 ¼ 160 GPa.The phonon lifetime due to phonon-phonon scattering iscorrelated to the imaginary part of the phonon self-energyΣq;jðωÞ (for the jth phonon mode at q), and in a simpleapproximation this ImΣq;jðωÞ can be regarded as a con-stant, ImΣ∘q;j ¼ ImΣq;jðωq;jÞ, which is given asImΣ∘q;j ¼18πℏ2Xq1;q2jΨð−q0;q1;q2Þj2ðn1 þ n2 þ 1Þδω0;ω1þω2− 2jΨð−q0;q1;−q2Þj2ðn1 − n2Þδω0;ω1−ω2: ð1ÞHere, ni is the Bose-Einstein distribution function for thephonon mode ωi (¼ ωq;j), andΨðq0;q1;q2Þ¼ 16ffiffiffiffiNpXffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiℏ38m0m1m2ω0ω1ω2sXe0 ⊗ e1 ⊗ e2...ϕð3Þ× exp ðiq0 · r0 þ iq1 · r1 þ iq2 · r2Þδqþq1þq2;G: ð2Þϕð3Þ is the cubic interatomic potential, and ei, mi, ri are theeigenvector, mass, and position of an atom for the phononmode ωi, respectively. Equations (1) and (2) indicate thatImΣ∘q;j has nonzero contribution only in a decay (down-conversion) process;ω0 ¼ ω1 þ ω2; q0 þG ¼ q1 þ q2; ð3aÞand in a merging (up-conversion) process;ω0 þ ω1 ¼ ω2; q0 þ q1 ¼ q2 þG: ð3bÞFollowing the fitting method described in theSupplemental Material [33], the extracted phonon line-width of the LO mode in the [001] direction, 2ImΣ∘ð00qÞ;LO,is depicted in Fig. 2(d) as black solid circles. Some of theobserved spectra (dots) and fitted results (lines) are shownin Figs. 2(a), 2(b), and 2(c). The phonon linewidthdetermined from the DFT calculations [red in Fig. 2(d)]compares excellently with the experimental results. Thecalculations indicate that the phonon linewidth in this [00q]direction is caused only by the phonon decay process. Thedetails of the phonon scattering, which contributes to thephonon linewidth, can be seen in Figs. 2(e), 2(f), and 2(g)for several q. In each Fig., decay phonons are displayedas black dots. Each momentum corresponding to a decayor product phonon is represented by the position of thedots, while the scattering intensity is indicated by thedot size.Similar reasoning applies to the TO mode in the [00q]direction, as seen in Fig. 3. Resemblance between thecalculations (red) and measurements (black) can be easilyfound in Fig. 3(d). The calculations suggest that theω0ω0XΓEnergy(meV) Energy(meV) Energy(meV)(a) (b) (c)(g)(f)(e)kzkxkykzkykzky(1.02, 1.01, 3.53) (1.02, 1.06, 4.00)Q=(1.01, 0.96, 3.08)ω1ω2ω1ω209590858075 Fit Obs.085807570075706560 Exp. Calc.2ImΣ (00 q),LO (meV) ScN LOT~300 K(d)3210ω0FIG. 2. LO phonon linewidth of ScN along the [00q] direction.(a)–(c) IXS spectra at q ∼ ð00qÞmeasured at RT. The lines are fitsto a function described in the Supplemental Material. (d) Exper-imentally obtained phonon linewidth (black solid circles) andlinewidth in the DFT calculations (at T ¼ 300 K, red line).(e)–(g) Plot of the calculated decay phonon [ω1 andω2 in Eq. (3a)].Momenta of decay phonons (q1 and q2) are indicated by theposition of the black dots. The blue arrows in (e) and (f) show onepair of decay phonons from the original (00q) phonon (ω0,green arrow).PHYSICAL REVIEW LETTERS 120, 235901 (2018)235901-3merging process dominates the decay process in TO, incontrast to LO. Technically speaking, as seen in Fig. 3(e),calculations predict that the decay process rarely happens atthe zone center (0.1% of the total scattering), but itincreases for a higher q [10.5% in Fig. 3(f), 22.1% inFig. 3(g)].When rapid reabsorption of LO phonons by chargecarriers takes place before the LO phonon decay at the zonecenter, phonon bottleneck effects can be enhanced.Figure 2(e) shows that the decay phonons predominantlyappear around theL points. This decay process is somethingsimilar to the Ridley channel [46], where LO phonons decayinto acoustic andTOphononmodes (LO → LA=TAþ TO).However, theprocess obtainedhere ismore complicated.Thedecay phonons no longer have pure transverse or longi-tudinal polarization, because they are not on the high-symmetry lines. Furthermore, the decay process can be alsoconsidered to be Klemens-like (LO → 2LA=TA) [47],because the LA and TO modes have similar energy at L[Fig. 1(f)]. This LO linewidth increases with temperature[Fig. 4(a), and the derived phonon linewidth is depicted inblack solid circles in Fig. 4(b)].Again, the observation iswellexplained by the calculations [red line in Fig. 4(b)]. A similartemperature dependence of linewidth can be found, forexample, in the Raman scattering measurements of wurtziteInN for the highestE1ðLOÞ phononmode [18]. InRef. [18], alarge temperature-dependent peak shift of thismodewas alsoobserved (∼2.5 meV shift between T ¼ 80 and 600 K), incontrast to ScN (0.2� 0.3 meV between T ¼ 20 and 600 K[Fig. 4(a)]). This difference may be caused by the structuraldifferences [wurtzite (InN) and rocksalt (ScN)]. In fact, inMgO, another rocksalt structure, the peak position of the LOmode is mostly unchanged in this temperature range [48].The lattice thermal conductivity, κð2Þ (rank 2 tensor), isestimated by solving the linearized phonon Boltzmannequation using the single-mode relaxation time, τ∘q;jð¼ ℏ=2ImΣ∘q;j);κð2Þ ¼ 1NV0Xq;jCq;jvq;j ⊗ vq;jτ∘q;j: ð4ÞHere,NV0 is the volume of the solid, ½vq;j�α ¼ ∂ωq;j=∂qα isthe group velocity, andCq;j is the heat capacity. Specifically,κð2Þ of ScN is isotropic due to the crystal symmetry, and thescalar thermal conductivity κ [¼ trðκð2ÞÞ=3] can be defined.The calculated lattice thermal conductivity of ScN is κScN ¼43 W=m=K at T ¼ 300 K and 21 W=m=K at T ¼ 600 K,which shows a good agreement with the experimentallyobtained value, κScN ¼ 36.4 W=m=K at RT (seeSupplemental Material for the detailed information [33]).Thesevalues are close to those ofMgO (κMgO ¼ 51 W=m=Kat T ¼ 300 K and 23 W=m=K at T ¼ 600 K [49,50]),which has the same structure and similar molar mass toScN. Note that, the isotope effect seen inGaN [9] is expectedto be negligible in ScN, because without any intentionalincorporation of Sc and N isotopes, the content of these willbe extremely small.The previous literature reports significantly lower ther-mal conductivity of 10 − 20 W=m=K [24,26,27] than thepresent study, and this deviation is presumed to arise fromthe imperfectness of the crystals. Table I, whichEnergy(meV) Energy(meV) Energy(meV)XΓω2ω1ω2ω1ω1ω2kxx5x5x5Q=(1.01, 0.96, 3.08)(a) (b) (c)(e) (f) (g)kxkzkykz kzky ky(1.02, 1.01, 3.53) (3.99, 1.01, 0.03)ω0ω0 ω00555045400555045402ImΣ (00q),TO (meV)ScN TO(d) Exp. Calc.1.51.00.50.0T~300 K050454035 Fit Obs.FIG. 3. TO phonon linewidth of ScN in [00q]. (a)–(c) IXSspectra at q ∼ ð00qÞmeasured at RT. (d) Experimentally obtainedphonon linewidth (black solid circles) and linewidth in the DFTcalculations (at T ¼ 300 K, red line). (e)–(g) Plot of the calcu-lated decay phonons (black dots) and phonons involved in themerging process (red dots). Momenta of the phonons involved inthe scattering processes (�q1 and q2 for the decay or mergingprocess) are indicated by the position, and the scattering intensityis indicated by the size, which was enlarged by a factor of 5compared to that in Fig. 2. The red arrows display one pair ofphonons involved in the merging process [ω1 and ω2 in Eq. (3b)],and the green arrow shows the original phonon (ω0).  20 K 150 KRT (~300 K)450 K600 KT=Energy  (meV)(a)9288848076(b)2ImΣ q,LO (meV)τ q,LO (ps)Temperature (K) Exp. Calc.Q=(1.01, 0.96, 3.08) 765432160040020000.60.40.20.1q=(0, 0, 0.08) FIG. 4. Temperature dependence of LO phonon linewidth ofScN at Q ¼ ð1.01; 0.98 3.08Þ. (a) Temperature dependent IXSspectra. (b) Comparison of the calculations (red line) andmeasurements (black circles) in the reduced Brillouin zone.PHYSICAL REVIEW LETTERS 120, 235901 (2018)235901-4summarizes the results of the carrier and thermal conduc-tivity measurements, including the present study, shows anegative correlation. Given the fact that the defects (such asnitrogen vacancies) and impurities (such as oxygen andchlorine) enhance the carrier concentration of ScN[25–27,34], and reduce the thermal conductivity in general[51], the negative correlation in Table I should be the resultsfrom the defects and impurities. This also indicates thatenriching impurities and/or defects is effective to reducethermal conductivity for thermoelectric applications.Furthermore, the good agreement between the presentmeasurements and calculations testifies that the presentsample is close to the ideal (perfect) crystal, that issupported by the lowest carrier concentration in Table I.When the obtained results are compared to other nitrides,the LO phonon lifetime of ScN at RT is 0.21 ps near Γ[Fig. 4(b)], which is much shorter than the E1ðLOÞ phononlifetimes of GaN (0.56 ps [52]) and InN (0.45 ps [18]). Therapid heat dissipation resulting from the short LO phononlifetime may be useful in nitride power devices. Moreover,the ideal thermal conductivity for ScN (43 W=m=K atT ¼ 300 K) is smaller than those for other nitrides(176 W=m=K for InN [53] and ∼240 W=m=K forGaN [9]). Utilizing the low thermal conductivity, ScN issuitable for thermoelectric application as indicated inRefs. [2,24–27]. As discussed above, further reductionof the thermal conductivity may be possible with intro-ducing more defects and impurities.In summary, direct observations of the phonondispersion relations and lifetimes of ScN have been made,with no additional (i.e., surface and substrate) contribu-tions. All observed properties are consistent with the DFTcalculations, and the optical phonon lifetime is wellexplained by the three-body phonon-phonon interaction.The experiments demonstrate the possibility of investigat-ing thermal properties through phonon observation withmomentum dependence. Eventually, this technique can bein general applied to epitaxial films, to help to reveal thecorrelation between the phonon and thermal properties, thatprovide guidelines in fabricating the next-generation devi-ces with layered structures, such as thermoelectric, highpower, and photovoltaic devices.H. U. thanks I. 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