固體物理(黃昆) 3－10晶格的狀態(tài)方程和熱膨脹

§3－10晶格的狀態(tài)方程和熱膨脹簡諧近似：

(1)在簡諧近似的情況下，晶格原子振動(dòng)可描述為3N個(gè)線性獨(dú)立的諧振子的疊加，各振子間不發(fā)生作用，也不交換能量；一、晶體的熱膨脹和非簡諧效應(yīng)

(2)晶體中某種聲子一旦產(chǎn)生，其數(shù)目就一直保持不變，既不能把能量傳遞給其他聲子，也不能使自己處于熱平衡狀態(tài)。用簡諧近似理論不能解釋晶體的熱膨脹和熱傳導(dǎo)現(xiàn)象。晶體的非簡諧效應(yīng)：微擾項(xiàng)聲子間有相互作用能量交換系統(tǒng)達(dá)到熱平衡

兩個(gè)聲子通過非簡諧項(xiàng)的作用，而產(chǎn)生第三個(gè)聲子。這可以看成是兩個(gè)聲子的相互碰撞，最后產(chǎn)生第三個(gè)聲子。微擾項(xiàng)聲子間的相互作用遵循能量守恒和準(zhǔn)動(dòng)量守恒碰撞前后系統(tǒng)準(zhǔn)動(dòng)量不變，對熱流無影響。---正常過程(N過程)；(1)---反常過程(U過程)。(2)二、熱膨脹

熱膨脹：在不施加壓力的情況下，晶體體積隨溫度變化的現(xiàn)象稱為熱膨脹。

假設(shè)有兩個(gè)原子，一個(gè)在原點(diǎn)固定不動(dòng)，另一個(gè)在平衡位置R0附近作振動(dòng)，離開平衡位置的位移用表示，勢能在平衡位置附近展開：0R0R0（1）簡諧近似RU(r)R0兩原子間距不變，無熱膨脹現(xiàn)象（2）非簡諧效應(yīng)兩原子間距增大，有熱膨脹現(xiàn)象。由玻爾茲曼統(tǒng)計(jì)，原子離開平衡位置的平均位移(c、g均為正常數(shù)。)(1)簡諧近似：是的奇函數(shù)在簡諧近似下無熱膨脹現(xiàn)象。(2)非簡諧效應(yīng):在非簡諧效應(yīng)下，有熱膨脹現(xiàn)象。推導(dǎo)略線膨脹系數(shù)當(dāng)勢能只保留到3次方項(xiàng)時(shí)，線膨脹系數(shù)與溫度無關(guān)。若保留更高次項(xiàng)，則線膨脹系數(shù)與溫度有關(guān)。顯然，在簡諧近似下，g=0，=0。三、晶體的狀態(tài)方程和熱膨脹

由熱力學(xué)知，壓強(qiáng)P、熵S、定容比熱CV和自由能F之間的關(guān)系為：自由能F(T，V)是最基本的物理量,求出F(T，V)，其他熱力學(xué)量或性質(zhì)就可以由熱力學(xué)關(guān)系導(dǎo)出。晶格自由能F1=U(V)F2由統(tǒng)計(jì)物理知道：Z是晶格振動(dòng)的配分函數(shù)。頻率為i的格波，配分函數(shù)為：由晶格振動(dòng)決定T=0時(shí)晶格的結(jié)合能若能求出晶格振動(dòng)的配分函數(shù)，即可求得熱振動(dòng)自由能。各諧子相互獨(dú)立，忽略晶格之間的相互作用能，總配分函數(shù)為：

由于非線性振動(dòng)，格波頻率i也是宏觀量V的函數(shù)，所以NeutralclusteranionclusterDopantCsIhC5VCsIhC5VCSiGeSn0.000.951.230.590.000.560.410.000.440.000.060.140.000.351.090.530.001.080.350.000.800.000.060.37StructuralandelectronicpropertiesoftheneutralandtheanionclustersonIn12X(X=C,Si,Ge,Sn)ThegeometricandelectroniccharactersofIn12X(X=C,Si,Ge,Sn)clustersinitsneutralandanionicchargestateshavebeeninvestigatedusingthedensityfunctionalcalculationswiththegeneralizedgradientapproximation.ThegroundstateofIn12X(X=C,Sn)andtheanionclustershavealowsymmetryCsstructureinsteadofanicosahedron.However,theneutralandanionclustersontheSiandGeatomdopedfavoranicosahedralstructure.FortheneutralIn12X(X=C,Si,Ge,Sn)clusters,ourcalculationdemonstratethattheenergygapsbetweenthehighestoccupiedmolecularorbitalandthelowestunoccupiedmolecularorbitalare0.6ev,1.16eV,1.15eV,0.88eVrespectively,andthemagneticmomentsarezero.However,theenergygapsforitsanionclusterare<0.3eV.Theseresultscanbeexplainedbyelectronicshellclosingof2pshell(40e)forneutralclusters,andindicatethatInatomsinclustersaretrivalent.I.INTRODUCTIONSincethediscovery1andmacroscopicproduction2offullerenes,therehavebeenextensiveintereststosearchforothersimilarstableclusters.Thesestableclustersnamed“magic”clusterscouldactascandidatesfornovelcluster-assembledmaterials.Thesphericaljelliummodelpredictsthattheclusterswith2,8,20,40,...valenceelectronshavehigherstabilityduetotheclosureofelectronicshells.3,4Forexample,theAl13having39valenceelectronsbydopingasinglemonovalentatomorbysubstitutinganAlwithatetravalentatomhas40valenceelectronsandcouldgainelectronicstabilitybyclosingthe1s21p61d102s21f142p6shell,asystematicstudyofbinaryaluminum(Al)clustershasbeenperformedbothexperimentallyandtheoretically5-11recently.DopingasingleC,Si,Ge,Sn,orPbatomintoanAl12cluster,thebinaryneutralclusterscouldgaintheelectronicstabilitywithperfecticosahedrastructureand40valenceelectronsclosingtheelectronicshell.Atthesametime,thetheoreticalstudiesfoundaddingorremovingelectronsfromtheneutralclustermightalteritsstructuralandelectronicproperties8,11.Thoughthealuminum(Al)andindium(In)atombelongtothesamegroupintheperiodictableandtheyhavetwos-valenceelectronsandonep-valenceelectron,thereareonlyafewstudiesconcerningindiumclusters.Intheirclusters,theelectronicfeatureischaracterizedonlybyonepelectroninasmallclusterbecauseanenergygapexistsbetweenthesandpbands.Astheclustersizeincreases,thesandpbandshybridizationandthecomposedatomsarecharacterizedastrivalentatoms.Foraluminumclusters,ithasbeenrevealedthatthes/phybridizationoccursaroundn~5bymeasuringtheirionizationpotential(IPs)[12,13].However,forindiumclusters,ithasbeenreportedtheInatomsaremonovalentupton~15bytheirIPsandphotoelectronspectroscopy[14].AtsushiNakajimaetal[15]foundIPsofIn7NaandIn13NaclusterswerehigherthanthoseofIn7andIn13,andtheIPincrementscouldbeexplainedbyelectronicshellclosingofthe1p(8e)and2pshell(40e),whereInatomsintheclustersweremonovalentandtrivalentrespectively.Meanwhile,theIn12Na3-wasfoundbestablegeometricallyorelectronically.Itsstabilitywasexplainedbytheelectronicshellmodelcorrespondingtothe2pshellclosing(40e),whenindiumatomsweretrivalent.ButnoelectronicclosingshellstructurewasobservedinpureInnclustersaroundn=13.TheythoughttheNaatomadditioninducess/phybridizationintheInnclusters.Howeverrecently,MinoruAkutsuetal16studiedtheelectronicpropertiesofSiandGeatomsdopedInclusters,InnSimandInnGem,byphotoionizationspectroscopyoftheneutralsandphotoelectronspectroscopyoftheanions.Theyfoundthereactivityoftheclustershaslocalminimaatn=10and12forInnSiandatn=9,10,and12forInnGeindicatingthattheIn10Si(In10Ge)andIn12Si(In12Ge)werestableelectronicallyorgeometricallyascomparedtoothersizeclusters.AndthechangingsizefrommonovalenttotrivalentforInnclustersdidnotshowobviousevidence,sotheoreticalcalculationstillwasneeded.Toourbestknowledge,thereisnostudyreportedonthegeometricalandelectronicpropertiesofthetetravalentatomdopingIn12clustersbyfar.Therefore,weperformedthedensityfunctionaltheory(DFT)calculationstoexplorethestructuralandtheelectronicpropertiesofIn12XandIn12X-(X=C,Si,Ge,Sn)clusters.II.COMPUTATIONALDETAILSAllthecalculationshavebeenperformedbasingupontheall-electrondensityfunctiontheory(DFT)17withthegeneralizedgradientapproximation(GGA)

(Ref.18)usingtheDMol3package19.TheBecke-Lee-Yang-Parr(BLYP)correlationexchangefunctionisadoptedwithDNPbasisset.BLYPfunctionisthecombinationoftheexchangefunctionexploitedbyBecke20andthecorrelationfunctionexploitedbyLee,Yang,andParr21.DNPbasisfunctions,comparabletoGaussian6-31G**sets,arethedoublenumericalatomicorbitalaugumentedbypolarizationfunctions:i.e.,functionswithangularmomentumonehigherthanthatofthehighestoccupiedorbitalinthefreeatom.TheelectronicstructureisobtainedbysolvingtheKohn-Sham(KS)equations22self-consistentlyusingthespinunrestrictedscheme.Self-consistentfieldproceduresweredonewithaconvergencecriterionof10?6a.u.ontheenergyandelectrondensity.GeometryoptimizationswereperformedusingtheBroyden-Fletcher-Goldfarb-Shanno(BFGS)algorithm23withaconvergencecriterionof10-3.onthedisplacementand10-5.ontheenergy.III.RESULTSANDDISCUSSIONSOneofthemostimportantthingsinstudyingclustersistodetermineitsequilibriumgeometry.FortheneutralandanionIn12X(X=C,Si,Ge,Sn)clusters,threepossiblegeometricconfigurationswithIh,C5v,Cssymmetryshowninfig.1areconsidered.ThethreeconfigurationsarethefavorablegeometriesoftheneutralandthechargedAl12X(X=C,Si,Ge,Sn,Pb)clusters.Theicosahedronisthetetravalentatomsittingatthecenter,whiletheC5vstructureisthetetravalentatomadsorbedonthevertex.Intable.1,wetabulatedtherelativevalueofthecalculatedbindingenergiesforthestructuresasshowninFig.1intheneutralandanionstates.ForIn12X(X=C,Sn)anditsanions,wefoundtheCsgeometryisground-statestructures,whiletheicosahedrongeometryforIn12Snanditsanionisnearlyenergeticallydegenerate(ΔE=0.06eV).ThestructurewithIhsymmetrywaslowest-energystructureofAl12Snchargedcluster8,10.Meanwhile,ourcalculationshowsthattheicosahedraconfigurationisthegroundstatesofIn12X(X=Si,Ge)anditsanionclusters.ThoughthedifferenceofthebindingenergybetweentheIhandCsstructurefortheIn12Snanditsanionclusterisverylitter,theCsgeometryhasthehigherbindingenergy(0.35eV~0.60eV)fortheSiandGeatomdopingIn12clusterseitherintheneutralorintheanionsthantheIhgeometry.TheaverageIn-SnandIn-InbondlengthsintheCsstructureforIn12Snare3.507?and3.295???forIn12Sn-respectively.WhiletheaverageIn-CandIn-InbondlengthsintheCsstructureforIn12Care2.464?and3.315???forIn12C-.Fortheseclusters,therearetwelveX(Si,Ge,Sn)-InbondsintheIhstructures,whiletheCsstructureshaveonlyfiveX-Inbonds.ThesingleSiandGeatomssitatthecenteroftheicosahedron.IntheseclusterswithIh??forIn12SiandIn12????forIn12SiandIn12??foritsanion.FromtheSitoGe,thesizeofclustersisincreasingfortheIn12SiandIn12Geclusters,whichmayattributetothesizeofatomdopedsittingatthecenteroftheclusters,andsuggestsalsothebondingstrengthisweakerandweaker.Andthesizeoftheclustersforneutralclusterissmallerthanitsanion,whichindicatesthenetchargehasanobviouseffectonthestructureofSiandGedopingIn12clusters.InFig.2,Weshowthedensityofstates(DOS)ofthegroundstatesstructures.Thedensityofstatesiscalculatedbybroadeningthebandenergyofeachorbitalwiththeline-shapeLorentzianfunction.TheDOSforthegroundstatesofneutralclustersandtheiranionclustersareshownforadirectcomparison.Fromtheresults,wecouldseethatthegroundstatesstructurewiththesamesymmetryhadthesimilarstatesdensity(DOS).Forexample,theIn12SiandIn12GeclusterswithIhsymmetry,thelocationsandwidthsoftheirpeaksarealmostsame.FortheIn12CandIn12SnwiththesameCssymmetry,theirDOSarealmostsimilar,exceptthatthepeakwidthoftheIn12CislargerthantheclusterdopedSn,whichmayattributetothelocationoftheatomsdopedinPeriodicTable.AsonemovedowntothePeriodicTable,theelementbecomesincreasinglymoremetallicandlesselectronegative,andthesizeislargemoreandmore.ComparingtheDOSoftheanionclusterswiththeirneutralclusters’,wecouldseetheenergylevelsofelectroncouldbealteredbyaddinganelectron.Forexample,thefirsthighpeakfromtheleft,itislocalizedataround-3.8eVintheneutralstateforIn12SiandIn12GehavingthesameIhstructure.However,itisshiftedtoaround-4.4eVintheanionclusters.Takingawayanelectronincreasestheenergylevelofthevalenceelectron.XiLietal10hasputforwardthattheelectronegativityandthesizeofatomsbetwoimportantfactorsinthestructureevolutionfortheAl12X-speciesfromX=CtoPb.TheythoughtthattheAl12C-favoredtheCsstructurebecauseofthehighelectronegativity,andtheCsstructurewasthegroundstateofAl12Snforitslargesize.Inourcomputation,thegroundstatesofIn12CandIn12SnfavoringCsstructuresatisfyapparentlythereason.Thetotalelectrondensityoftheneutralclustersisshowninfig.3.WecouldseetheelectrondistributionofCinIn12CislitterthanotherIndiumcluster,whilethedifferenceexistshardlyinIn12Sn.InthePeriodicTable,theSnantInatomisproximateandthustheirelectronegativityaresamenearly.ButtheelectronegativityofCatomishigherthanthem,whichresultsinthedifferenceofthetotalelectrondensity.Infig.3,weshowalsothetotalelectrondensityoftheneutralIn12SiandIn12GeclusterswithsameIhstructure.TheelectrondistributionofInatomsinIn12SiandIn12Geclustersissameduetotheirhighsymmetry,WealsocarriedouttheHOMO-LUMObandgap,andfoundtheneutralclusterduetotheclosedelectronicshellpredictedbythejelliummodelhadalargerbandgapthantheiranionclusters.Theyare0.6,1.16,1.15,0.88eVforC,Si,Ge,Sndopedclusters,respectively.Fromtheseresults,wecouldseethestructuraleffectsonthebandgap.ThestablestructuresforIn12SiandIn12Gearetheicosahedron,andtheirbandgapsarearound1.15eV.However,thebandgapoftheCsstructuresthatarethegroundstatesofIn12CandIn12Sn,aresmallerthantheclusterswithIhstructure,whichindicatestheIhstructurewithhighsymmetrywillbekineticallymorestablethantheCsstructure.Comparingtheneutralclusters,thebandgapoftheiranionisallsmaller(<0.3eV).Sotheneutralclustersareverystablekinetically,andconsistentwithcompletelythejelliummodelfor40electrons.AndtheHOMO-LUMOgapsoftheneutralIn12X(X=Si,Ge)havingclosedelectronicstructureswereabout0.8eVreportedbyMinoruAkutsu20.Forhavingclosedelectronicstructure,wealsoobtainfromthespinsthatarezeroforallneutralclusters.However,thespinoftheIn12clusteris2,whichindicateitstwoelectrons(2p2)areunpaired.DuetothespinoftheneutralclustersofIn12X(X=C,Si,Ge,Sn),theyshouldhavetheclosed1S21p61d102S21f142P6electronicshell.SowesuggestthattheIndiumatombetrivalentintheIn12X,whichisconsistentwiththeresultsobtained

byAtsushiNakajimaetal23.ConclusionsWehaveperformedadensityfunctionalcalculationsonthestructuresandelectronicpropertiesofIn12X(X=C,Si,Ge,Sn)anditsanionclusterswiththegeneralizedgradientapproximation.TheCsstructurehasahighestbindingenergyfortheIn12X(X=C,Sn)andtheiranionclustersinsteadofanicosahedron.However,thehighestbindingenergystructuresaretheicosahedronfortheIn12X(X=Si,Ge)andtheiranionclusters.Foralltheneutralandanionclusters,theC5vstructuresoptimizedfromanicosahedronwithavertexcappedbyatetravalentatomhavethelowestbindingenergy.Thedifferenceofthebindingenergyisonly0.06eVbetweentheIhstructureandtheCsstructureforIn12Snanditsanion.FromtheDosoftheneutralandtheanionclusters,theadditionalchargeandthestructureaffecttheelectronicpropertiesoftheseclusters.Duetotheclosedelectronicshellfortheneutralcluster,thebandgapsbetweenthehighestoccupiedmolecularorbitalandthelowestunoccupiedmolecularorbitalare0.6ev,1.16eV,1.15eV,0.88eVfortheIn12X(X=C,Si,Ge,Sn)respectively.However,thebandgapsfortheanionclusterare<0.3eV.Moreover,wefindthespinsoftheneutralclustersarezero,whichfavorsmuchmorethe2pshellclosing.TheresultofthespinsandthebandgapsshowstheIndiumatominneutralclustersistrivalent.Fig.1TheschematicfiguresforthestateconsideredofIn12X(X=C,Si,Ge,Sn)anditsanionclusters.ThetetravalentatomsitsatthecenterofIhstructure,whiletheC5vstructuresarethetetravalentatomadsorbedonthevertex.ThetetravalentatomontheoutsideofIn12clustersformsCsstructure.Table1.ThecalculatedbindingenergiesineVforIn12X(X=C,Si,Ge,Sn)andtheiranionclusters.Fig.2ThedensityofstatesofthegroundstructuresforIn12XandIn12X-(X=C,Sn,Si,Ge)clusters.The(a)isfortheneutralclusters;the(b)isforthenegativelychargedclusters.ThedensityiscalculatedbybroadendingthebandenergyoftheoccupiedorbitalandtheunoccupiedorbitalseparatelywiththeLorentzianfunction.(a)(b)Fig.3theelectrontotaldensityof(a)theneutralIn12CandIn12SnclusterswithsameCsstructure,(b)theelectrontotaldensityoftheneutralIn12SiandIn12GeclusterswithsameIhstructure.TheatomsdopedarelocatedattheleftsideoftheCsstructures.TheSiandGeatomsarelocatedthecenterofthecage.ReferenceH.W.Kroto,J.R.Heath,S.C.O’Brien,R.F.Curl,andR.E,Smalley,Nature(London)318,162(1985).W.Kratschmer,L.D.Lamb,K.Fostiropoulos,andD.R.Huffman,Nature(London)347,354(1990).3.W.D.Knight,K.Clemenger,W.A.deHeer,W.A.Saunders,M.Y.Chou,andM.L.Cohen,Phys.Rev.Lett.52,2141(1984).W.A.deHeer,Rev.Mod.Phys.65,611(1993).,L.,Phys.Rev.B2002,65,153404.H.Kawamata,;Y.Negishi,;A.Nakajima,;,Chem.Phys.Lett.2001,337,255.D.E.Bergeron,A.W.Castleman,N.,S.N.Kahanna,Chem.Phys.Lett.2005,415,230.G.ChenandY.Kawazoe,J.Chem.Phys.2007,126,014703.K.H.YoungandJ.Jaehoon,J.Chem.Phys.2004,121(17),8500.X.Li,andL.S.Wang,Phys.Rev.B65,153404(2002).S.F.LiandX.G.Gong,Phys.Rev.B70,075404(2004).12.K.E.Schriver,,E.C.Honea,R.L.Whetten,.64,1990,2539.13.J.L.Persson,,H.P.Cheng,R.S.Berry,Chem.Phys.Lett.186,1991,215.14.M.Gausa,G.Gantef?r,H.O.Lutz,K.H.Meiwes-Broer,Int.J.Massspectrum.IonProcesses,102,227,1990.15.N.Atsushi,H.Kuniyoshi,S.Tsuneyoshi,N.Takashi,T.Tetsuya,etal,J.Phys.Chem.1993,97,86-90.16.A.Minoru,K.Kiichirou,A.Junko,M.Ken,M.Masaaki,andN.Atsushi,J.Phys.Chem.A2007,111,573-577.17.,D.Sánch