Представлены экспериментальные результаты по исследованию распространения объемных акустическихволн(10–200МГц)в синтетическихмонокристаллахалмазаподвоздействием одноосного давления и температуры. Полученные данные по упругим постоянным второго и третьего порядка были использованы для расчета характеристик анизотропии распространения акустических волн в алмазе под давлением. Обсуждаются особенности распространения волн.Experimental results ofthepropagation ofbulk acoustic waves(10–200MHz) inthe syntheticdiamond single crystal under the inﬂuence of uniaxial pressure and temperature have been presented. Obtained data of the second and the third order of elastic constants were used for calculation of the anisotropy of the acoustic waves propagation characteristics in diamond under the pressure application. The peculiarities of wave propagation have been discusse

Burkov, Sergey I.

Kuznetsov, Mikhail S.

Kvashnin, Gennady M.

Sorokin, Boris P.

Telichko, Arseny V.

Бурков, Сергей И.

Квашнин, Геннадий М.

Кузнецов, Михаил С.

Сорокин, Борис П.

Теличко, Арсений В.

Siberian Federal University Digital Repository

Journal of Siberian Federal University. Mathematics & Physics 2013, 1(6), 120–126УДК 548.534Experimental Investigation of the Linear and NonlinearElastic Properties of Synthetic Diamond Single CrystalBoris P. Sorokin∗Gennady M.KvashninMikhail S.KuznetsovTechnological Institute for Superhard and Novel Carbon Materials,Central’naya, 7a, Troitsk, Moscow, 142190RussiaArseny V.Telichko†Moscow Institute of Physics and Technology,Institutsky Lane, 9, Dolgoprudny, Moscow Region, 141700RussiaSergey I. Burkov‡Institute of Engineering Physics and Radio Electronics,Siberian Federal University,Svobodny, 79, Krasnoyarsk, 660041RussiaReceived 10.10.2012, received in revised form 20.11.2012, accepted 25.12.2012Experimental results of the propagation of bulk acoustic waves (10–200 MHz) in the synthetic diamondsingle crystal under the inﬂuence of uniaxial pressure and temperature have been presented. Obtaineddata of the second and the third order of elastic constants were used for calculation of the anisotropy of theacoustic waves propagation characteristics in diamond under the pressure application. The peculiaritiesof wave propagation have been discussedKeywords: bulk acoustic wave, uniaxial pressure, temperature inﬂuence, synthetic diamond, second- andthird-order elastic constantsIntroductionLarge good quality natural single crystals of diamond are very rare, they have the great costand therefore cannot be regarded as an important source of raw materials in such that hightechnology areas as optics and electronics. Nevertheless, diamond single crystal has a numberof unique properties and characteristics which has the interest for its potential application inacoustoelectronics: the highest among the known solids bulk (BAW) and surface (SAW) acousticwave velocities, high thermal conductivity, ionizing radiation stability, chemical resistance, andso on. Physical properties of diamond have the great importance in terms of solid state studiesand, in recent years, owing to the applied applications of diamond ﬁlms and single crystals. Themeasurements of elastic moduli (second-order elastic constants (SOEC)) and its temperature∗bpsorokin2@rambler.ru†arseny.telichko@phystech.edu‡sburkov@sfu-kras.ruc© Siberian Federal University. All rights reserved– 120 –Boris P. Sorokin ... Experimental Investigation the Linear and Nonlinear Elastic Properties ...and pressure dependences for natural diamond with a good accuracy have been executed by theauthors [1], who have developed the high-frequency ultrasound long pulse method, suitable forsamples with small geometric dimensions. But third order elastic constants (TOEC) hadn’t beendeﬁned. Indirect methods, such as X-ray diﬀuse scattering [2] and Brillouin scattering [3], havealso been used to determine the second-order elastic moduli of diamond. Recent measurement ofthe diamond’s elastic moduli was executed by the method of resonant ultrasound spectroscopy [4].To analyze the SOEC dependence of the temperature or pressure it is necessary to know thefull set of TOEC coeﬃcients. The TOEC’s for diamond have been theoretically obtained byab initio calculation, using the method of local electron density functional [5], and on the basisof linear and nonlinear force constants, calculated according to the Keating’s theory [6]. Bymeasurement of Raman shifts of optical phonons with uniaxial stress authors [7] have obtainedthe Keating’s anharmonic parameters of the valence-force-ﬁeld model as well as the diamond’sTOEC values. In addition, the experimental TOEC values were found by the method of shock-wave loading [8].During the last 3–4 years Technological Institute for Superhard and Novel Carbon Materials(Russia) began the production of suﬃciently large samples of synthetic dielectric diamond withlinear dimensions up to 10 mm in reproducible quality. The aims of this study were the measure-ments of the BAW velocities, including the investigations of uniaxial pressure and temperatureinﬂuences, calculation of the second and third order elastic constants and anisotropy of BAWpropagation parameters in synthetic dielectric diamond single crystal (type IIa with low nitrogencontent < 2 · 1016 cm−3) under the action of uniaxial pressure.ExperimentDiamond single crystal specimens with the [110] and [001] crystalline faces oriented to within5′ were prepared with a nonparallelism of opposite faces up to 1 µm/cm and a ﬂatness betterthan ±100 nm/сm. BAW phase velocities were measured by the interferometer pulse-phasemethod like [1], realized by Ritec Advanced System RAM-5000 in the frequency range 10–200MHz. Piezoelectric quartz transducers with fundamental frequencies of 12 MHz for longitudinal(L) and 18 MHz for shear (S) waves have been used. The measurements were performed on theharmonics transducers up to 200 MHz. Epoxy resin’s acoustic contact between the sample andfused silica buﬀer has been used. Since there was diﬃcult to determine the layer thickness ofan acoustic contact, a special method to minimize its inﬂuence has been used. High frequencymeasurements should also to minimize the dispersion errors due to diﬀraction eﬀects and theacoustic wave’s reﬂections on the side walls of the sample. Taking into account the phase errorand sample preparation’s inaccuracy, the phase velocity experimental error has been estimatedas ±0.05 %. Pure acoustic modes have been investigated only.When the measurements of BAW velocities dependences on the uniaxial pressure P have beenperformed, the scheme of echo pulse measurement without buﬀers has been used. Pressure up to30 MPa was produced by the Instron 5965 automatic electromechanical system with inaccuracy ofthe measurement up to 5%. The BAW’s phase velocity variations, arising under the inﬂuence ofuniaxial pressure on diamond single crystal, were measured by the time shifting of one of reﬂectedacoustic pulses. More accurate and reproducible results can be obtained only if the median regionof pulse was taken into account to measure the required time shift. The limiting time resolutionwas close to 20 picosecond. The pulse has been chosen as far as from the exciting one to obtain thebest sensitivity. All the v = f(P ) dependences were the linear ones. Temperature dependences,– 121 –Boris P. Sorokin ... Experimental Investigation the Linear and Nonlinear Elastic Properties ...investigated within the -60. . . +70◦C, were observed the linear ones too.Calculation of Second- and Third-order Elastic Constantsof DiamondWell-known equations obtaining for the 3 independent second-order elastic coeﬃcients C11,C12, and C44 of cubic symmetry crystals with BAW phase velocities are given in the Tab. 1.Under the calculation of the Cij coeﬃcients the density value ρ0=3516 kg/m3, obtained bymeasurement of unit cell parameter [9], has been used. The SOEC values of synthetic diamondat room temperature are presented in the Tab. 2 in comparison with known data. Takinginto account the eﬀect linear thermal dilatation [10] and v = f(T ) dependences, the SOECtemperature coeﬃcients have been calculated too (Tab. 2). Since the phase velocity variationsare proportional to the pressure it is convenient to introduce controlling coeﬃcientsαv =1v(0)(∆v∆P)∆P→0. (1)Table 1. BAW velocities of diamond single crystal (type IIA) at room temperaturePropagation direction Polarization BAW’s ρ0v2 Phase velocity,N U type m/s[100] [100] L C11 17542±1[100] ⊥(100) S C44 12828±1[110] [110] L12(C11 + C12 + 2C44) 18333±1[110] [001] S1 C44 12829±1[110] [11¯0] S212(C11 − C12) 11659±1Table 2. Elastic moduli CIJ (GPa) of diamond and its temperature coeﬃcients at room temper-ature in comparison with previous data[1] [3] [4] This workC11 1079 1080.4 1078.16 1081.9±1.0C12 124 127.0 126.63 125.2±0.8C44 578 576.6 577.56 578.6±0.2Experiment TheoryTC11TCij =1Cij(∆Cij∆T)∆T→0, 10−5K−1-1.71 -0.96TC12 -4.65 -5.9TC44 -1.29 -1.5Using the solutions of Green-Christoﬀel equations, earlier derived for explanation of small am-plitude acoustic wave propagation in crystals, superposed with bias pressure [11], one can obtain– 122 –Boris P. Sorokin ... Experimental Investigation the Linear and Nonlinear Elastic Properties ...the relations between αv coeﬃcients (1) and third-order elastic constants (Tab. 3). The proce-dure of the TOEC’s calculation was executed by the Wolfram Mathematica 8 software includingthe least-square method option. Our results on the TOEC’s complete set of diamond singlecrystal as well as known calculated and experimental data, are shown in the Tab. IV. Takinginto account the results on second- and third-order elastic constants the theoretical calculationof the TCij coeﬃcients by the relations [11] has been executed (Tab. 2).As one can see from the Tab. 4, our results are close to the published earlier. The distinctionbetween C456 constants can be explained by the relatively small eﬀect concerned with such kindof constants, because in our experiments and calculations only the uniaxial pressure has beenused. More essential eﬀect would be expected if the shear stress was applied. The problem ofmore accurate deﬁnition of the C456 constant was known for other cubic crystals and hasn’tsolved for the present.Table 3. Controlling coeﬃcients for the some BAW modes of diamondN U P λ0 = ρ0v2 αvi =1vi (0)(∆vi∆P)∆P→0αvi ,10−13Pa−1[001][001][110]C1112λ0[2S12C11 + S12C111 + (S11 + S12)C112] -6.2±0.5[11¯0] C4414λ0[(2S11+2S12−S44)C44+ (S11++S12)C144+ (S11 + 3S12)C155 − S44C456]-13.8±0.5[110] C4414λ0[(2S11+2S12+S44)C44+(S11++S12)C144 + (S11 + 3S12)C155 + S44C456]-15.6±0.5[110][11¯0][001]12(C11− C12)14λ0[4S12λ0 + S12C111 + (S11 − S12)C112−−S11C123]-22.7±0.5[110]12(C11++C12+ 2C44)12λ0[4S12λ0 + S12C111 + (S11 + 3S12)C112++2S11C144 + 4S12C155 + S11C123]-10.8±0.5[001] C4412λ0[2S11C44 + S12C144 + (S11 + S12)C155] -17.7±0.5[11¯0][11¯0]12(C11− C12)18λ0[2 (2S11+2S12 + S44)λ0+(S11++S12)C111 − (S11 − S12)C112 − 2S12C123]-2.4±0.5[110]12(C11++C12+ 2C44)18λ0[2 (2S11 + 2S12−S44)λ0+(S11++S12)C111 + (3S11 + 5S12)C112 + 4 (S11++S12 − S44)C155 + 4S12C144 + 2S12C123]-10.9±0.5[001] C4414λ0[4S12C44 + (S11 + S12)C144++ (S11 + 3S12)C155 − S44C456]-14.2±0.5– 123 –Boris P. Sorokin ... Experimental Investigation the Linear and Nonlinear Elastic Properties ...Table 4. Third order elastic constants CIJK (GPa) of diamond at room temperature in compar-ison with previous data[5] [6] [7] [8] This workC111 -6300±300 -6475 -7367 -7603±600 -7660±500C112 -800±100 -1947 -2136 -1909±554 -1550±500C123 0±400 982 1040 835±1447 3470±500C144 0±300 115 186 1438±853 -3130±300C155 -2600±100 -2998 -3292 -3938±375 -2630±300C456 -1300±100 -135 76 -2316±743 -700±300Analysis of BAW Propagation in Diamond under the Uniax-ial Stress InﬂuenceUsing the own software and data one obtained the anisotropy of BAW propagation parametersin the diamond under the uniaxial stress inﬂuence. As an example the results of anisotropicbehavior of phase velocities and αv coeﬃcients at diﬀerent directions of pressure application forthe waves, which can propagate along all the directions in the (001) crystallographic plane, aredisplayed on the Fig. 1.When pressure is applied along [001] direction (Fig. 1b), αv coeﬃcients for shear waves havethe diﬀerent values, and as a result the eﬀect of removing of degeneracy for shear phase velocitieshas to be taken into account. Thus the acoustic axes along the [100] and [010] directions ofundisturbed crystal should disappear. Actually, in this case the analysis of symmetry changingby Curie principle shows that the point symmetry of stressed crystal will be the tetragonal 4mm,and the X or Y axis of such crystal will not have be the acoustic axis property. The insertion onthe Fig. 1c shows that the acoustic axes, initially coinciding with [100] or [010] directions splitson two acoustic axis, inclined to the initial direction [11].The greatest eﬀect of the uniaxial pressure inﬂuence is observed for the longitudinal pressureapplication when pressure coincides with the wave normal (Fig. 1d) for all BAW types.ConclusionFor the ﬁrst time the third-order elastic constants of synthetic diamond single crystal wereobtained by direct ultrasonic measurement of BAW phase velocity variations under the actionof uniaxial pressure. The results obtained are helpful for solid state physics in the ﬁeld ofanalysis of anharmonic behavior of acoustic phonons, as well as for estimation of the diamond’sprospective applications in the acoustoelectronic sensors of pressure or acceleration. Now thereis the possibility of the systematic search for more eﬀective crystalline directions and cuts to beused in such kind of devices.This work has supported by the Russian Ministry of Education under the Federal Program”Research and Development on Priority Directions of Scientiﬁc and Technological Complex ofRussia for 2007–2013” (State Contract №16.513.12.3025).– 124 –Boris P. Sorokin ... Experimental Investigation the Linear and Nonlinear Elastic Properties ...Fig. 1. Anisotropic behavior of phase velocities (a) and αv coeﬃcients at the pressure applicationalong [001] direction (b), perpendicularly to the wave normal (c), and along the wave normal (d)for the waves, which can propagate along all the directions in the (001) crystallographic plane.References[1] H.J.McSkimin, P.Andreatch, The elastic stiﬀness moduli of diamond as a function of pres-sure and temperature, J. Appl. Phys., 43(1972), no. 7, 2944–2948.[2] E.Prince, W.A.Wooster, Determination of elastic constants of crystals from diﬀuse reﬂexionsof X-rays. III. Diamond, Acta Cryst., 6(1953), 450–454.[3] E.S.Zouboulis, M.Grimsditch, A.K.Ramdas, S.Rodriguez, Temperature dependence of the– 125 –Boris P. Sorokin ... Experimental Investigation the Linear and Nonlinear Elastic Properties ...elastic moduli of diamond: A Brillouin-scattering study, Phys. Rev. B, 57(1998), no. 5,2889–2896.[4] A.Migliori, H.Ledbetter, R.G.Leisure, C.Pantea, J.B.Betts, Diamond’s elastic stiﬀnessesfrom 322 K to 10 K, J. Appl. Phys., 104(2008), no. 5, 053512(4).[5] O.H.Nielsen, Optical phonons and elasticity of diamond at megabar stresses, Phys. Rev. B,34(1986), no. 8, 5808–5819.[6] C.S.G.Cousins, Elasticity of carbon allotropes. I. Optimization, and subsequent modiﬁca-tion, of an anharmonic Keating model for cubic diamond, Phys. Rev. B, 67(2003), no. 2,024107(13).[7] E.Anastassakis, A.Cantarero, M.Cardona, Piezo-Raman measurements and anharmonicparameters in silicon and diamond, Phys. Rev. B, 41(1990), no. 11, 7529–7535.[8] J.M.Jr. Lang, , Y.M.Gupta, Experimental determination of third-order elastic constants ofdiamond, Phys. Rev. Lett., 106(2011), no. 12, 125502(4).[9] K.Haruna, H.Maeta, K.Ohashi, T.Koike, Thermal expansion coeﬃcient of synthetic dia-mond single crystal at low temperatures, Jpn. J. Appl. Phys. (Pt. 1), 31(1992), 2527–2529.[10] T.Sato, K.Ohashi, T.Sudoh, K.Haruna, H.Maeta, Thermal expansion of a high purity syn-thetic diamond single crystal at low temperatures, Phys. Rev. B, 65(2002), no. 9, 092102(4).[11] K.S.Aleksandrov, B.P.Sorokin, S.I.Burkov, Eﬀective Piezoelectric Crystals for Acoustoelec-tronics, Piezotechnics and Sensors, vol. 2, Novosibirsk: SB RAS Publishing House (2008).(in Russian).Экспериментальное исследование линейныхи нелинейных упругих свойств синтетическогомонокристалла алмазаБорис П.СорокинГеннадий М.КвашнинМихаил С.КузнецовАрсений В.ТеличкоСергей И.БурковПредставлены экспериментальные результаты по исследованию распространения объемных аку-стических волн (10–200 МГц) в синтетических монокристаллах алмаза под воздействием одноос-ного давления и температуры. Полученные данные по упругим постоянным второго и третьегопорядка были использованы для расчета характеристик анизотропии распространения акусти-ческих волн в алмазе под давлением. Обсуждаются особенности распространения волн.Ключевые слова: объемная акустическая волна, одноосное давление, влияние температуры, син-тетический алмаз, упругие постоянные второго и третьего порядка.– 126 –

Experimental Investigation of the Linear and Nonlinear Elastic Properties of Synthetic Diamond Single Crystal

Experimental Investigation of the Linear and Nonlinear Elastic Properties of Synthetic Diamond Single Crystal

Abstract

Similar works

Full text

Available Versions

Siberian Federal University Digital Repository