第二組-42.模量文獻(xiàn)綜述

文獻(xiàn)相1graphene原文的解釋?zhuān)築yapplyingsuchadifferentialpressureonagraphenemembraneheldoveracircularaperture,agrapheneballoonisformedwithabiaxialstrainappliedatthecenterofthecircular2peak3constant-N常數(shù)N驗(yàn)4razor-edge5supported6Lorentzian7suspended8narrowline9翻譯說(shuō)中文翻 楊氏模量。受壓石墨烯球面1的應(yīng)變可以從拉曼光譜G帶的峰移2直接估算出。通過(guò)比2.4±0.4TPa2±0.5TPa。［石墨烯的力學(xué)性質(zhì)是有趣的研究課題，因?yàn)槭┑臈钍夏Ａ亢蛷?qiáng)度相當(dāng)大。由于這些力學(xué)性質(zhì)，石墨烯在納米力學(xué)系統(tǒng)中的應(yīng)用很有前景。［］－［］在納米尺度明確石墨烯的這些性質(zhì)是一項(xiàng)重要的工作，不僅僅是為了它在納米力學(xué)中的應(yīng)用，也是為N起泡實(shí)驗(yàn)1Ta。［］，［］在本研究中，我們采取一種不同的方法估測(cè)石墨烯的楊氏模量：我們應(yīng)用拉曼光譜直接測(cè)定石墨烯薄膜由于受到壓力差而產(chǎn)生的應(yīng)變。因?yàn)槭┎粫?huì)讓任何氣體透過(guò)，［］左右的壓差（1巴＝1bar＝10a）。［］－［］通過(guò)對(duì)固定在圓形小孔上的石墨烯薄膜施加壓力差，就會(huì)得到中心有雙軸應(yīng)變的石墨烯球面。拉曼光譜是研究石墨烯固有性質(zhì)的有力工具。［］－［1］石墨烯的拉曼光譜對(duì)機(jī)械形變非常敏感。最近的實(shí)驗(yàn)表明，石墨烯在產(chǎn)生雙軸應(yīng)變的條件下，拉曼光譜的G2D帶都會(huì)出現(xiàn)紅移和分離成兩個(gè)峰的現(xiàn)象。［］－［］因此，石墨烯薄膜的形變可以用拉曼光譜準(zhǔn)確測(cè)量。利用拉曼光譜測(cè)量出石墨烯薄膜受壓后產(chǎn)生的形變，然后與基于有限元法的數(shù)值模擬結(jié)果進(jìn)行1原文grapheneballoons詳見(jiàn)文獻(xiàn)相關(guān)術(shù)語(yǔ)表號(hào)1內(nèi)2原文peakshift號(hào)2內(nèi)3原文constant-Nblistertest詳見(jiàn)文獻(xiàn)相關(guān)術(shù)語(yǔ)表號(hào)3內(nèi)光刻和干法刻蝕刻出的小圓孔。這些小圓孔的深度約為5??m，直徑分別為2.0??m，3.1??m，4.2??m，5.3??m，6.4??m和7.3??m。石墨烯樣本通過(guò)天然微晶石墨薄片（原料提供商：NGSNaturgraphitGmbH,Germany）：的機(jī)械剝離得到。這項(xiàng)工作直接在清潔的預(yù)制基板上進(jìn)行。好的石墨烯樣本被放在真空中，用來(lái)在石墨烯薄膜上我們采用波長(zhǎng)為514.5m（2.41e）0×顯微鏡物鏡（.0.65），n-von三軸50（1800條／毫米）分散，最后被液氮冷卻的電荷耦合器件探測(cè)器檢測(cè)。光譜分辨率大約為0.????1。聚焦光束尺寸通過(guò)刀刃法測(cè)量，空間分辨率約為1??m應(yīng)控制在0.5mW×30，步長(zhǎng)為0.3??m石墨烯樣本的掃描圖像。由于可能會(huì)破壞樣本，所以用于電子顯微鏡觀察的石薄膜向外凸出。利用電子顯微鏡的掃描圖像，我們估計(jì)直徑6.6??m圓孔上的石墨烯撓區(qū)G峰的位置與懸空區(qū)域相比，有明顯的藍(lán)移（S1），拉曼光譜2D帶和4原文razor-edgemethod號(hào)4內(nèi)5原文supportedregion詳見(jiàn)文獻(xiàn)相關(guān)術(shù)語(yǔ)表編號(hào)5內(nèi)容6原文Lorentzianshape詳見(jiàn)文獻(xiàn)相關(guān)術(shù)語(yǔ)表編號(hào)6內(nèi)容7原文suspendedregion號(hào)7內(nèi)質(zhì)。［10］，［11］眾所周知，SiO2/Si基板上的石墨烯雜質(zhì)。［12］圓孔中間，G帶圖圖1.(a)實(shí)驗(yàn)裝置示意圖(b)真空中石墨烯薄膜斜角度下的SEM圖像。由于圓孔 差，兩個(gè)不同直徑(3.6??m、6.6??m)圓孔上的

2.(a)SiO2/Si預(yù)制基板上石墨烯樣本的光(b)直徑6.4??m圓孔上，單層石墨烯球面G峰拉曼圖。(c)直徑6.4??m圓孔中心處和支光譜，在真空環(huán)境下，出現(xiàn)了紅移（2c）。圓孔中心G峰的位置移至約1568?????12b6.4??m圓孔上石墨烯樣本G峰位置圖像。在支撐區(qū)域，G1591?????1。隨著探測(cè)位置不斷移向圓孔中心，G峰位置不斷紅1568?????1。紅移現(xiàn)象的出現(xiàn)是由于石墨烯薄膜凸起產(chǎn)生了1bar時(shí)，圓孔中心G1581?????1（支持8原文narrowlinewidth詳見(jiàn)文獻(xiàn)相關(guān)術(shù)語(yǔ)表號(hào)8內(nèi)S2）。經(jīng)過(guò)不斷的抽氣和注氣，G峰的位置周期性重復(fù)出現(xiàn)。這證明了圓孔通過(guò)測(cè)量拉曼光譜G2D帶的移動(dòng)，我們可以估計(jì)出石墨烯薄膜上應(yīng)變的大γ＝（1?ω0）（?ω?????）＝2.2±0.1和β＝（1?ω0）（?ω??εs）＝0.930.04［20］。其中ω0為無(wú)應(yīng)變情況下石墨烯的聲子頻率，???和εs 的G聲子移???=－2??????1%［16］雙軸應(yīng)變?yōu)?0cm-16.4??m圓孔中心處石墨烯薄膜 13?????10.19%們測(cè)量了不同直徑圓孔中心處的拉曼光譜變化。圖3展示了拉曼G2D帶都出現(xiàn)了紅移，并且紅移的3.14.2??m，5.3??m6.4??m的圓孔，G峰峰移分別5.0±3.6cm-1，8.0±2.8?????1，10.0±1.3?????1，13.0±1.0?????1。以上值分別對(duì)應(yīng)的雙軸應(yīng)變?yōu)樵龃蠖龃蟆Ｔ谥睆阶钚〉模?.1??m）3.真空中，在直徑分別為3.14.2??m，5.3??m6.4??m的圓孔上，單層石墨烯樣本拉曼光譜的比較(a)G帶光譜和(b)2D帶光譜

墨烯樣本的光譜G帶出現(xiàn)了不對(duì)稱(chēng)性。因?yàn)榧す夤獍叱叽绱蠹s為1??m，與直徑最小的圓孔相比較大，［26］－［28］，具有如下的應(yīng)力應(yīng)變關(guān)系：σ＝Eε＋Dε2。其中E是彈性模量，D9內(nèi)容原文Gruneisenparameter詳見(jiàn)文獻(xiàn)號(hào)9內(nèi)氣壓為環(huán)境大氣壓。當(dāng)小室內(nèi)氣壓被抽真空至10－5Torr以下時(shí)，石墨烯薄膜外凸，下［29］?？紤]到石墨烯薄膜的厚度，我們使用石墨的層間距0.335nm［30］膜直徑與圓孔直徑相同。另外，我們也認(rèn)為壓力均勻垂直施加在薄膜上。圓形薄膜邊緣緊緊固定在圓孔上作為邊界條件。這是合理的假定，因?yàn)樽罱麷oenigetal.表如果在數(shù)值模擬中，我們采用之前研究得出的石墨烯楊氏模量為1TPa的結(jié)論［5］，那么對(duì)于直徑為6.4??m的圓孔，模擬撓度值約為220nm,圓形薄膜中心的應(yīng)變約為0.32%。這遠(yuǎn)大于我們的測(cè)量值0.19%。如果楊氏模量作為擬合參數(shù)，并用5.3??m和6.4??m的圓孔，我們得到的楊氏模量值分別為2.7±1.2TPa，2.7±0.8TPa，2.5±0.4TPa和2.4±0.3TPa。我們估測(cè)單層石墨烯的楊氏模量值為2.4±0.4TPa4.通過(guò)數(shù)值計(jì)算得到的單層石墨烯徑向和橫向應(yīng)

心3.1??m范圍內(nèi)，兩個(gè)光譜幾乎相同。如果我們利用已知的由于雙軸應(yīng)力產(chǎn)生的G動(dòng)值0.????－1，與此相應(yīng)的直徑1??m圓孔中心處的應(yīng)變值應(yīng)為0.01，這與G峰寬度相比很小。離中心2??m處，兩個(gè)偏振方向G05。由此推得，兩個(gè)G峰位置的距離為1.????－1b中結(jié)果一致。在離中心3??m處，激光光斑（約1??m）層石墨烯樣本完了直徑7.3??m的圓孔。當(dāng)小室被抽真空后，G峰位置從1581????－11570????－1，與此對(duì)應(yīng)的雙軸應(yīng)變?yōu)?.14%（圖5.差造成的兩個(gè)正交偏振方向上，G譜的比偏振方向上的拉曼G帶光譜。

奈森參數(shù)分別為??＝2.20.1和1.06的研究結(jié)果是合變區(qū)間取值不同。比如，Leeetal.通過(guò)線性模型，得到單層石墨烯楊氏模量為1TPa，可以很好的在應(yīng)變區(qū)間0-5%擬合。Koenigetal.通過(guò)線性模型，得到1-5層石墨烯的楊氏模量，可以很好的在0.25MPa0.5MPa2.5-5atm)的壓力區(qū)間擬合。在以.上免費(fèi)獲得。：*E-mail:hcheong@sogang.ac.kr.:82-2-705-8434.Fax:82-717-我們感謝H.Kim和S.W.Lee的SEM測(cè)量工作。這項(xiàng)工作由韓國(guó)(MEST)支持的國(guó)家研究(NRF)撥款(No. 和No. )這項(xiàng)工作同也受到MEST全球前沿研究計(jì)劃下屬的高級(jí)軟電子中心提供的經(jīng)濟(jì)支持(NoBunch,J.S.;vanderZande,A.M.;Verbridge,S.S.;Frank,I.W.;Tanenbaum,M.;Parpia,J.M.;Craighead,H.G.;McEuen,P.L.Science2007,315,Garcia-Sanchez,D.;vanderZande,A.M.;Paulo,A.S.;Lassagne,B.;P.L.;Bachtold,A.NanoLett.2008,8,Chen,C.;Rosenblatt,S.;Bolotin,K.I.;Kalb,W.;Kim,P.;Kymissis,I.;H.L.;Heinz,T.F.;Hone,J.Nat.Nanotechnol.2009,4,Lee,C.;Wei,X.;Kysar,J.W.;Hone,J.Science2008,321,385?Koenig,S.P.;Boddeti,N.G.;Dunn,M.L.;Bunch,J.S.Nat.Nanotechnol.2010,6,543?546.Bunch,J.S.;Verbridge,S.S.;Alden,J.S.;vanderZande,A.M.;Parpia,J.M.;Craighead,H.G.;McEuen,P.L.NanoLett.2008,8,2458?2462.Zabel,J.;Nair,R.R.;Ott,A.;Georgiou,T.;Geim,A.K.;Novoselov,K.S.;Casiraghi,C.NanoLett.2012,12,617?621.Ferrari,A.C.;Meyer,J.C.;Scardaci,V.;Casiraghi,C.;Lazzeri,M.;Mauri,F.;Piscanec,S.;Jiang,D.;Novoselov,K.S.;Roth,S.;Geim,A.K.Phys.Rev.Lett.2006,97,187401.Malard,L.M.;Pimenta,M.A.;Dresselhaus,G.;Dresselhaus,M.S.Phys.Rep.2009,473,51?87.Pisana,S.;Lazzeri,M.;Casiraghi,C.;Novoselov,K.S.;Geim,A.K.;Ferrari,A.C.;Mauri,F.Nat.Mater.2007,6,198?201.Yan,J.;Zhang,Y.;Kim,P.;Pinczuk,A.Phys.Rev.Lett.2007,98,Casiraghi,C.;Pisana,S.;Novoselov,K.S.;Geim,A.K.;Ferrari,A.C.Appl.Phys.Lett.2007,91,233108.Das,A.;Pisana,S.;Chakraborty,B.;Piscanec,S.;Saha,S.K.;Waghmare,U.V.;Novoselov,K.S.;Krishnamurthy,H.R.;Geim,A.K.;Ferrari,A.C.;Sood,A.K.Nat.Nanotechnol.2008,3,210?215.Balandin,A.A.;Ghosh,S.;Bao,W.Z.;Calizo,I.;Teweldebrhan,D.;Miao,F.;Lau,C.N.NanoLett.2008,8,902?907.Lee,J.-U.;Yoon,D.;Kim,H.;Lee,S.W.;Cheong,H.Phys.Rev.B2011,83,Yoon,D.;Son,Y.-W.;Cheong,H.NanoLett.2011,11,3227?Calizo,I.;Balandin,A.A.;Bao,W.;Miao,F.;Lau,C.N.NanoLett.2007,7,Mohiuddin,T.M.G.;Lombardo,A.;Nair,R.R.;Bonetti,A.;Savini,G.;Jalil,Bonini,N.;Basko,D.M.;Galiotis,C.;Marzari,N.;Novoselov,A.K.;Geim,A.K.;Ferrari,A.C.Phys.Rev.B2009,79,205433.Huang,M.Y.;Yan,H.G.;Chen,C.Y.;Song,D.H.;Heinz,T.F.;Hone,J.Proc.Natl.Acad.Sci.U.S.A.2009,106,7304?7308.Yoon,D.;Son,Y.-W.;Cheong,H.Phys.Rev.Lett.2011,106,Frank,O.;Mohr,M.;Maultzsch,J.;Thomsen,C.;Riaz,I.;Jalil,R.;K.S.;Tsoukleri,G.;Parthenios,J.;Papagelis,K.;Kavan,L.;Galiotis,C.ACSNano2011,5,2231?2239.Gupta,A.;Chen,G.;Joshi,P.;Tadigadapa,S.;Eklund,P.C.NanoLett.2006,6,2667?2673.Yoon,D.;Moon,H.;Cheong,H.;Choi,J.S.;Choi,J.A.;Park,B.H.J.KoreanPhys.Soc.2009,55,1299?1303.Berciaud,S.;Ryu,S.;Brus,L.E.;Heinz,T.F.NanoLett.2009,9,Casiraghi,C.Phys.Rev.B2009,80,Liu,F.;Ming,P.B.;Li,J.Phys.Rev.B2007,76,Wei,X.;Fragneaud,B.;Marianetti,C.A.;Kysar,J.W.Phys.Rev.B2009,80,Jiang,J.;Wang,J.;Li,B.Phys.Rev.B2010,81,Blakslee,O.L.;Proctor,D.G.;Seldin,E.J.;Spence,G.B.;Weng,T.J.Appl.Phys.1970,41,3373?3382.Aljishi,R.;Dresselhaus,G.Phys.Rev.B1982,26,Hanfland,M.;Beister,H.;Syassen,K.Phys.Rev.B1989,39,英文原EstimationofYoung’sModulusofGraphenebyRamanJae-UngLee,DuheeYoon,andHyeonsikDepartmentofPhysics,SogangUniversity,Seoul121-742, TheYoung’smodulusofgrapheneisestimatedbymeasuringthestrainappliedbyapressuredifferenceacrossgraphenemembranesusingRamanspectroscopy.ThestraininducedonpressurizedgrapheneballoonscanbeestimateddirectlyfromthepeakshiftoftheRamanGband.Bycomparingthemeasuredstrainwithnumericalsimulation,weobtainedtheYoung’smodulusofgraphene.TheestimatedYoung’smodulusvaluesofsingle-andbilayergrapheneare2.4±0.4and0±0.5TPa, Graphene,Ramanspectroscopy,Young’smodulus,strain,grapheneThemechanicalpropertiesofgrapheneareinterestingresearchsubjectsbecauseitsYoung’smodulusandstrengthareknowntobeextremelyhigh.Owingtotheseproperties,grapheneisapromisingcandidateforapplicationsinnanomechanicalsystems.1?3Determinationofsuchmechanicalpropertiesofgrapheneinnanometerscaleisanimportantissuenotonlyforapplicationsinnanomechanicsbutalsoinstudyingfundamentalphysicalproperties.Valuesof～1TPafortheYoung’smodulushavebeenreportedinmechanicallyexfoliatedgraphenesamplesbythenanoindentationtechniqueusinganatomicforcemicroscopyandbytheconstant-Nblistertest.4,5Inthiswork,wetakeadifferentapproachandestimatethemodulusofgraphenebymeasuringdirectlythestrainappliedtoagraphenemembraneunderadifferentialpressureusingRamanspectroscopy.Sincegrapheneisimpermeabletoanygas,6itcansustainadifferentialpressureofseveralbars.5?7applyingsuchadifferentialpressureonagraphenemembraneheldoveracircularaperture,agrapheneballoonisformedwithabiaxialstrainappliedatthecenterofthecircularmembrane.Ramanspectroscopyisaverypowerfultoolforinvestigatingtheintrinsicpropertiesofgraphene.8?17Especially,theRamanspectrumofgrapheneverysensitivetomechanicaldeformations.RecentexperimentsdemonstratedthatboththeRamanGand2Dbandsred-shiftandsplitintotwopeakseachunderuniaxialstrain.18?21Therefore,strainonagraphenemembranecanbemeasuredbyRamanspectroscopyveryaccura y.BymeasuringthestraininducedonpressurizedgrapheneballoonsbyRamanspectroscopyandcomparingthestrainwithanumericalcalculationbasedonthefiniteelementmethod,wecoulddeducetheYoung’smodulusofgraphenelayers.Thismethodcanprovidevaluableinformationontheintrinsicpropertiesofgrapheneandcanbeappliedtoothertwo-dimensionalmaterialsforinvestigatingtheirmechanicalproperties.Graphenesampleswerepreparedonprepatternedsiliconsubstratescoveredwitha300nmthickSiO2layer.Thesubstrateswerepatternedwithroundholesphoto-lithographyanddryetching.Thedepthofaholeis～5μm,andthediametersare2.0,3.1,4.2,5.3,6.4,and7.3μm.Thesampleswereprepareddirectlyoncleanedsubstratesbymechanicalexfoliationfromnaturalmacrocrystallinegraphiteflakes(NGSNaturgraphitGmbH,Germany).Thesampleswereplacedintoavacuumchambertomakepressuredifferenceacrossthegraphenemembrane.TheRamanmeasurementswereperformedwiththe514.5nm(2.41eV)lineofanArionlaser.Thelaserbeamwasfocusedontothegraphenesamplebya40×microscopeobjectivelens(N.A.0.65)throughaquartzwindow.Thescatteredlightwascollectedandcollimatedbythesameobjective,dispersedwithaJobin-YvonTriax550spectrometer(1800grooves/mm)anddetectedwithaliquid-nitrogen-cooledcharge-coupled-devicedetector.Thespectralresolutionwasabout0.7Thefocusedbeamsizewasmeasuredwiththerazor-edgemethodandthespatialresolutionwas～1μm.Thelaserintensitywaskeptbelow0.5mWtoavoidlocalheatinginducedbythelaser.FortheRamanscanningimage,aspectrumwasmeasuredateachpositionofthesample,raster-scannedinarectanglewith30×30pointsin0.3μmsteps.Figure1ashowstheschematicdiagramoftheexperimentalsetup.Inthevacuumchamber,apressuredifferenceacrossthegraphenemembraneisappliedbypumoutthevacuumchamber.Thispressuredifferencemakesthegraphenemembranebulgeupwardlikeaballoon.Ascanningelectronmicroscope(SEM)imageofasamplemeasuredinvacuumisshowninFigure1b.Becausesamplestendtobedamagedbytheelectronbeam,thesamplesusedfortheSEMmeasure-mentswerefromadifferentbatchthanthoseforRamanmeasurements.Thisimageclearlyshowsthatthegraphenemembranebulgesupward.Thedeflectionofthegrapheneona6.6μmdiameterholeestimatedfromtheSEMimageisabout～200nm,whichisconsistentwithanumericalsimulationusingthefiniteelementmethod.Ramanspectroscopicmeasurementswereperformedonthesingle-layergraphenesheetshowninFigure2a.ThesingleLorentzianshapeofthe2DbandinFigure2cinthesupportedregionindicatedthatthesampleisclearlyasinglelayer.22,23Butinthesuspendedregion,someasymmetryofthe2Dbandisobserved.Itisconsistentwiththepreviouslyreporteddataforthesuspendedsinglelayergraphenesample.24Atatmosphericpressure,theGpeakpositioninthesupportedregiontheholeissubstantiallyblueshiftedwithrespecttothesuspendedregion(SupportingInformationFigureS1),andtheintegratedintensityratiooftheRaman2DtoGbandisabout4.TheseandthenarrowlinewidthoftheGpeakindicatethatthesupportedgrapheneishighlydoped.10,11Itiswell-knownthatgrapheneonSiO2/Sisubstratesbestronglydoped.12Atthecenterofthehole,thepeakpositionoftheGbandis1581cm?1andtheintensityofRamanGbandisverysmallrelativetothatofthe2DbandwiththeintensityratioofRaman2DtoGband～11,whichindicatesthatthesuspendedpartofthegraphenesampleisminimallydoped.24,25Furthermore,thepeakpositionindicatesthattheinitialresidualtensioninducedbyvanderWaalsforcebetweengrapheneandedgesoftheholeisverysmall.6Thisimpliesthatoursampleissuitableforinvestigatingtheintrinsicpropertiesofundopedgraphene.Whenthechamberisevacuated,therearenodistinctivechangesinthesupportedregion.Ontheotherhand,theRamanspectrumfromthecenterofthesuspendedgrapheneisredshiftedinvacuum(Figure2c).TheGpeakpositionatthecenteroftheholeisshiftedto～1568cm?1.Figure2bisanimageoftheGpositionofthegraphenesampleovera6.4μmholeinvacuum.Inthesupportedregion,thefrequencyoftheGpeakisnear1591cm?1.TheGpeakpositiongraduallyredshiftsastheprobingpositionmovestothecenteroftheholewiththeminimumvalueof～1568cm?1.Thisredshiftiscausedbythetensionduetobulgingofgraphenemembrane.Oncetheevacuationiscomplete,theRamanspectrumdidnotchangeaftermorethan2hinvacuum,andwhenthepressureisreturnedto1barbylettingairintothevacuumchamber,theGpeakpositionatthecenterwentbackto1581cm?1(SupportingInformationFigureS2).TheGpeakpositionwasaftermanycyclesofevacuationandvacuumrelease.Thisisevidencethatthereisnoappreciableleakageofthegasconfinedinthehole.ThisresultisconsistentwithpreviousBymeasuringtheshiftsoftheRamanGand2Dbands,wecanestimatethemagnitudeofthestrainonagraphenemembrane.18?21ToestimatethestrainfromtheRamanspectrum,weusethereportedvalueoftheGrüneisenparameter(γ)andthesheardeformationpotential(β).Theusedvaluesareγ=(1/ω0)(?ω/?εh)=2.2±0.1andβ=(1/ω0)(?ω/?εs)=0.93±0.0420whereω0isthephononfrequencyofunstrainedgraphene,andεhandεsarethehydrostaticandshearcomponentsofthestrain,respectively.Usingthesevalues,oneobtainstheGphononshiftduetobiaxialstrainεb,Δωb=?2ω0γεb,tobe70cm?1perbiaxialstrainof1%.16TheGpeakshift～13cm?1forthe6.4μmholecorrespondstoabiaxialstrainof～0.19%attheofthegraphenemembrane.WemeasuredthechangesoftheRamanspectrumatthecenteroftheholewithvariousdiameters.Figure3showsthatboththeRamanGand2Dbandsareredshiftedandtheshiftsaredependentonthediameterofthehole.Gpeakshiftsare5.0±3.6,8.0±2.8,10.0±1.3and13.0±1.0cm?1fortheof3.1,4.2,5.3,and6.4μm,respectively.Thesevaluesinturncorrespondtostrainvaluesof0.07±0.05,0.11±0.04,0.14±0.03and0.19±0.02%,respectively.Thebiaxialstrainonagrapheneballoonincreasesasafunctionofthesizeofthehole.SomeasymmetryisobservedinthelineshapeoftheGbandfromthesmallest3.1μmhole.Sincethelaserspotsize,～1μm,isasignificantfractionofthetotalholesize,thestrainvariationwithinthelaserspotislargerelativetotheumstrainatthecenter.Thisstrainvariationcausesasymmetriclineshapesforsmallerholes,whichcontributestolargererrorbars.Wecomparedtheobtainedstrainvalueswithanumericalsimulationbasedonthefiniteelementmethod.Thegraphenemembranewasmodeledbyaclampedcircularmembranewithalinearelasticity.Itiswell-knownthatgraphenehasanonlinearelasticity26?28withaformofσ=Eε+Dε2,whereEistheYoung’sDisthethird-orderelasticmodulus,σisthestress,andεistheuniaxialstrain.4typicalvaluesofEandDare1.0and?2.0TPa,4respectively,and umappliedinourexperimentswas～0.2%,thefirsttermismuchlargerthanthesecondterm,andsotheelasticbehaviorofgraphenemaybeassumedtobelinearinourexperimentalconditions.Theconfinedairinsidetheholeunderthegraphenemembraneisinitiallyatatmosphericpressure.Whenthevacuumchamberisevacuatedtoapressurebelow10?5Torr,thegraphenemembranebulgesupwardasaresult,thevolumeoftheconfinedgasincreases.Becauseofthiseffect,thepressuredifferenceacrossthemembranedropsslightly.Ifoneusesthedeflectionof～200nmestimatedfromtheSEMimage(Figure1b),thepressuredifferenceis～0.96atm.ThisvaluewasusedastheinitialestimateofthepressuredifferenceintheiterativeproceduretofindtheYoung’smodulus.Withtheiterativefittingprocedure,thedeflectionconvergesto～160nmandthepressuredifferenceto0.97atm.WeassumedthatthePoisson’sratioofgrapheneisthesameasthatofgraphite,0.16.29Forthethicknessofthegraphenemembrane,weusedtheinterlayerspacingofgraphite,0.335nm,30andignoredthesmallthicknesschangebythedeflection.Numericalsimulationswereperformedusingacommercialfiniteelementprogram,ABAQUS.Thediameterofthecircularmembraneistakentobethesameasthediameterofthehole.Auniformpressurewasappliedperpendiculartothemembrane.Asaboundarycondition,theedgesofthemembranewereclampedtothecircularedgeofthehole.ThisisreasonablesinceKoenigetal.recentlyreportedthatthegraphenemembranesfirmlyadheretothesubstrateuptoapressuredifferenceof>2.5MPa(25IfweusethepreviouslyreportedYoung’smodulusvalue4,5of1TPainsimulation,thedeflectionis～220nmandthestrainatthecenterofthecircularmembraneis～0.32%forthe6.4μmhole.Thisismuchlargerthanthemeasuredvalueof0.19%.IfweinsteadusetheYoung’smodulusasafittingparametertoreproducethemeasuredstrainatthecenter,weobtainaYoung’smodulusvalueof2.4TPa.Tofurtherconfirmtheconsistencyofourysis,werepeatedforthedifferentholediameters.WeobtainedYoung’smodulusvaluesof2.7±1.2,2.7±0.8,2.5±0.4and2.4±0.3TPafordiametersof3.1,4.2,5.3,and6.4μm,respectively.OurestimationoftheYoung’smodulusofsingle-layergrapheneis2.4±0.4TPa.Figure4showsthecalculatedstraindistributiononthecircularmembraneusingtheYoung’smodulusvalueof2.4TPa.Thestraininthetransversedirectiondoesnotvarymuchwhereastheradialstrainvariesalot.Atthecenter,abiaxialstrainof0.19%isreproduced.Neartheedgeofthemembrane,thetransverseandradialstrainsaresignificantlydifferent,resultinginashearcomponent.WeattemptedtodetectthisshearcomponentusingpolarizedRamanspectroscopy.Figure5ashowsthechangeoftheGpeakpositionalongthediameterofthegraphenemembrane,measuredintwoorthogonalpolarizations.Theincidentlaserispolarizedintheverticaldirectionandthelaserspotpositionismovedhorizontally. yzerforthescatteredsignalwasseteithervertically(0°)orhorizontally(90°).TheGpeakpositionisindependentofthepolarizationnearthecenterwhereasasmallshiftisseenneartheedges.Figure5bcomparestheRamanspectraforthetwopolarizationsatdifferentpositions.Atthecenterorat1μmfromthecenter,thespectraarealmostidentical.IfweusetheknownsplittingoftheGpeakduetouniaxialstrain,thesplittingwouldbe0.2cm?1correspondingtoauniaxialstrainof0.01%at1μmfromthecenter,whichismuchsmallerthanthewidthoftheGAtthepositionof2μmawayfromthecenter,asmallrelativeshiftoftheGpeakforthetwopolarizationsisobserved.Theuniaxialcomponentofthestrainatthispositionis0.05%fromoursimulation.ThiswouldgiveaGpeaksplittingof1.0cm?1,whichisconsistentwithwhatisseeninFigure5b.At3μmawayfromthecenter,thelaserspot(～1μm)coversboththesuspendedandthesupportedregions,resultinginlinewidthsandasymmetriclineWealsoperformedthesameexperimentonabilayergraphenesample.bilayergraphenesamplefullycovereda7.3μmdiameterhole.Whenthechamberisevacuated,theGpeakpositionisshiftedfrom1581to1570cm?1,correspondstoabiaxialstrainof0.14%(Figure6).Usingthesameprocedureasforthesinglelayersample,wefoundthattheYoung’smodulusvalueforbilayergrapheneis2.0±0.5.TheYoung’smodulusvaluesforsingle-andbilayer2.4±0.4and2.0±0.5TPa,respectively,canbecomparedwiththatofgraphite1.02TPa.29(Thedataforafour-layergraphenesampleisalsoincludedinSupportingInformation.)ThisseemsreasonableifoneconsidersthefactthattheGrüneisenparametersforgrapheneandgraphiteareγ=2.2±0.1and1.06,respectively.20,31SincetheGrüneisenparameteriscloselyrelatedtoelasticproperties,abetweentheGrüneisenparameterandtheYoung’smodulusisOurestimationsfortheYoung’smodulusaresignificantlylargerthanthereportedvaluesintheliterature,～1TPa.4,5AprobableexplanationisthattheYoung’smodulusmaynotbeconstantindifferentstrainranges.Forexample,Leeetal.fittedtheirdatamostlyintherangeof0?5%ofstrainwithamodelassumingalinearbehaviorobtaintheYoung’smodulusvalueof1.0TPaforsinglelayergraphene.Koenigetontheotherhand,fittedtheirdataintherangeof0.25to0.5MPa(2.5to5atm)withalinearmodeltoobtaintheYoung’smodulusfor1?5layergraphene.Inbothcases,ifgraphenehasasignificantsofteningathigherstrainranges,theestimatedYoung’smoduluscouldbesmallerthanthevalueestimatedinthesmallstrainrange.Inourwork,theumstrainwasonly0.19%,atleastanorderofmagnitudesmallerthantheumstraininpreviouswork.Thebehavioratevenlowerstrainwasinspectedbyrepeatingthemeasurementsasafunctionofthepressureinthevacuumchamberbetween0and1atm.(SeeFiguresS4andS5inSupportingInformatio