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1、PHYSICALREVIEWB74,0641102006Finite-temperaturequasicontinuummethodformultiscaleanalysisofsiliconnanostructuresZ.Tang,H.Zhao,G.Li,andN.R.AluruDepartmentofMechanicalScienceandEngineering,BeckmanInstituteforAdvancedScienceandTechnology,UniversityofIllinoisatU
2、rbana-Champaign,Illinois61801,USAReceived22December2005;revisedmanuscriptreceived1April2006;published23August2006Inthispaper,weextendthequasicontinuumapproachforamultiscaleanalysisofsiliconnanostructuresatÞnitetemperature.Thequasicontinuummethodusestheclas
3、sicalcontinuummechanicsframework,buttheconstitutiveresponseofthesystemisdeterminedbyemployinganatomisticdescription.ForÞnite-temperaturesolidsystemsunderisothermalconditions,theconstitutiveresponseisdeterminedbyusingtheHelmholtzfreeenergydensity.Thestaticpar
4、toftheHelmholtzfreeenergydensityisobtaineddirectlyfromtheinteratomicpotentialwhilethevibrationalpartiscalculatedbyusingthetheoryofquantum-mechanicallatticedynamics.SpeciÞcally,weinvestigatethreequasiharmonicmodels,namelytherealspacequasiharmonicmodel,theloca
5、lquasiharmonicmodel,andthereciprocalspacequasiharmonicmodel,tocomputethevibrationalfreeenergy.UsingtheÞnite-temperaturequasicontinuummethod,wecomputetheeffectofthetemperatureandstrainonthephonondensityofstates,phononGrneisenparameters,andtheelasticproperties
6、oftheTersoffsilicon.Wealsocomputethemechanicalresponseofsiliconnanostructuresforvariousexternalloadsandtheresultsarecomparedtomoleculardynamicssimulations.DOI:10.1103/PhysRevB.74.064110PACSnumbers:46.15.x,62.25.g,02.70.cI.INTRODUCTIONratelyandtheinterfa
7、ceregionsareusedtoexchangeinfor-mationbetweentheatomisticandcontinuumregions.Top-Rapidadvancesinnanotechnologyhaveledtothefabri-downapproaches,suchasthequasicontinuumQCcationofnanoscalemechanicalstructureswithapplicationsmethod,11Ð13thebridgingscalemethod,
8、14andtheheteroge-innanoelectromechanicalsystemsNEMS.1Ð3Thedesign,neousmultiscalemethod,15solvethecontinuumequationsbyoptimization,andfabricationofNEMSforvariousapplica-extractingconstitutivela
