材料物理化学 第三篇习题.docx
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材料物理化学 第三篇习题.docx
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材料物理化学第三篇习题
Chap11、Semiconductors
11.1(a)Forasemiconductor,showthatnpproductobtainedfromEq.(11.27)isproportionaltoexp(-βEg)andthusisindependentofthepositionofthechemicalpotential
inthebandgap.
Eq.(11.27):
(b)Thelawofmassactioninsemiconductorsforreactioncreatingpairsofelectronsandholes[e.g.,Eq.(11.28)]hastheformn(T)P(T)∝exp(-βEg).Explainthesignificanceofthislaw.(Hint:
ThelawofmassactionisdescribedinSection4.6)
Eq.(11.28):
(c)EvaluatethenpproductatT=300KforSiwithEg=1.11evandmeds﹡=1.05mandmhds﹡=0.58m.
11.2UsingEq.(11.30)andmeds﹡=1.05mandmhds﹡=0.58mforSi,calculatethechangeinthepositionofthechemicalpotentialµintheenergygapofintrinsicSibetweenT=0and300K.
Eq.(11.30):
11.3CalculatethevaluesofNcandNvasdefinedinEq.(11.27)forSiatT=300K.Theappropriateofdensity-of-stateseffectivemassesforSiaremeds﹡=1.05mandmhds﹡=0.58m.
11.4ConsiderasemiconductorwithabulkenergygapEg=1.5evandwithme﹡=mh﹡=0.1m.Calculatetheincreaseintheenergygapofthissemiconductorwhenitisincorporatedintothefollowingstructures:
(a)Aquantumwell(d=2)withLx=10nm.
(b)Aquantumwire(d=1)withLx=Ly=10nm
(c)Aquantumdot(d=0)withLx=Ly=Lz=10nm
11.5AHalleffectmeasurementiscarriedoutonarectangularbarofasemiconductorwithdimensionsLx=0.04m(thedirectionofcurrentflow)andLy=Lz=0.002m.WhenacurrentIx=5mAflowsinthe+xdirectionandamagneticfieldBz=0.2Tisappliedinthe+zdirection,thefollowingvoltagesaremeasured:
Vx=6VandVy=+0.3mV(i.e.,increasinginthe+ydirection).Determinethefollowingpropertiesofthesemiconductorbarfromthesedata:
(a)Thesignofthedominantchargecarriers.
(b)Theconcentrationofthedominantchargecarriers.
(c)Theelectricalconductivityσ.
(d)Themobilityµofthedominantchargecarriers.
11.6UsingEq.(11.59),estimatetheincrease△nintheelectronconcentrationinann-typesemiconductorduetotheuniformabsorptionoflightwithα=105m-1,I0=1W/m2,andhω=1eV,aquantumefficiencyη=1,andaminority-carrierlifetimeτp=10-3s.
Eq.(11.59):
11.7UsingthedefinitionoftheHallmobilityµH=︱σRH︱andtheexpressionforRHforanintrinsicsemiconductorgiveninEq.(11.49),showthatµH=︱µh-µe︱.
Eq.(11.49):
11.8ConsiderthestructuraltransformationofabinarycrystalABfromthehexagonalwurtzitecrystalstructuretothecubiczincblendecrystalstructureinwhichthedensityoftheatomsremainsconstant.Findthelatticeconstantoftheresultingcubiccrystalifthelatticeconstantsoftheinitialwurtzitecrystalarea=0.3400nmandc=0.5552nm.
11.11Listallofthelocaltetrahedralbondingunits,A-B4,whicharepresentintheternarysemiconductingcompoundsCu2SiTe3,Cu3PS4,andCuSi2P3.Notethateachtetrahedronmustcontainanaverageoffourbondingelectronsperatom.
11.13Derivetheexpressionfortheshift△Eoftheelectronenergybandsfromonesideofap-njunctiontotheotherunderzerobiasasgiveninEq.(11.93).Calculatethemagnitddeofthebuilt-inelectricpotentialVB=△E/eforSiatT=300KforNd=Na=2×1024m-3.Usingthesesameparameters,calculatethedepletionwidthdandthemaximumelectricfieldQ/∈AforaSip-njunctionatT=300K.
Eq.(11.93):
.
Chap12、MetalsandAlloys
12.1ReferringtoSection12.5,showthattheconditionforthetangencyoftheFermispheretotheBrillouinzoneboundaryfortheFCClatticeisN=1.36.
12.2DeriveEq.:
12.3DeriveEq.:
Chap13、Ceramics
13.1ForthesiliconoxynitridecompoundSi2N2O,assumethatSi,N,andOatomshavetheirusualvalences(4,3,and2)andthattheNandOatomsdonotformcovalentbondswitheachother.
(a)GivenalocalbondingunitSi-NxOyforSiwithx+y=4,determinex(andy)forthiscrystalstructure.
(b)WhatarethelocalbondingunitsforNandO?
13.2FortheSixNyOzternaryphasediagram,locatethefollowingcompounds:
SiO2,Si3N4,Si2N2O,andSi3N2O3.
13.3Findtheaveragenumberofbridgingoxygens,b,andnonbridgingoxygens,n,forthefollowingglasses:
(a)CaO•SiO2,and
(b)soda-lime(i.e.,2CaO•3Na2O•15SiO2)
Chap14、Polymers
14.1ApolymerwhoseviscoelasticpropertiesaredescribedbyEq.(14.40)(i.e.,theMaxwellmodel)issubjectedtoatime-dependentstressσ=σ0exp(-it).Findthesteady-statestrain.ComparethisresulttothatofapolymerthatobeystheVoigtmodel,givenbyEq.(14.37).
Eq.(14.40):
Eq.(14.37):
14.2Consideranelastomerconsistingofmonomersthatareopticallyanisotropic[i.e.,theyhaveapolarizabilityα11(ω)forlightparalleltothechainaxisandα┴(ω)forlightpolarizedperpendiculartothechainaxis].AssumethatthereareNchainsperunitvolume.Let
bethemeanindexofrefractionofthematerial.Theelastomerisstretchedwithastechingparameters,asdefinedinSection14.5.Showthattheelastomerwillhaveabirefringencegivenby
Obtainanexpressionforthestressopticalcoefficient.C≡δn(ω)/σ,whereσistheappliedstress.
Chap15、DielectricandFerroelectricMaterials
15.1GiventheLandaufree-energydensityforaferroelectricoftheform
Whereb>c.Leta=a0(T-TC)andassumethatbandcareconstant.FindPzandχasafunctionofTforthestateofthermalequilibrium.
15.2DesignapiezoelectricactuatorthatcanbeusedtosweepanSTMheadoverthesurfaceofasolid.Whatistheareathatcanpracticallybecovered?
15.3AdaptWeissmolecularfieldtheory(seeChapter9)todescribeaferroelectric.Assumethattherearejusttwoorientationsfortheelectric-dipolemomentofaunitcellandthatNNcellsinteractviaanexchangeinteraction.ObtainthehysteresiscurveandvaluesforthecoercivefieldEc,saturationpolarizationPsat,andremanentpolarizationPrem.
15.4BaTiO3isaparaelectricforT>TC=130℃andhasaCurieconstantC=76,000K.
(a)IfthelatticeconstantforthecubicunitcellofBaTiO3isa=0.401nm,calculatetheelectric-dipolemomentµofthisunitcell.
(b)WhatwouldthecorrespondingpolarizationP=µnbeatT=0K?
Chap16、Superconductors
16.1(a)Deriveexpressionforthedifferenceinentropy△S(T)=Sn(T)-Ss(T)andthedifferenceinspecificheat△C(T)=Cn(T)-Cs(T)betweenthenormalandsuperconductingstatesintermsofthecriticalmagneticfieldHc(T)anditsfirstderivativedHc/dTandsecondderivatived2Hc/dT2.[Hint:
UseEq.(16.3)andstandardthermodynamicrelationships.]
(b)Evaluatetheseexpressionsfor△S(T)and△C(T)forthecasewhereHc(T)canbeapproximatedbyHc0[1-(T/Tc)2]andshowthat:
(ⅰ)△S(Tc)=△S(0)=0
(ⅱ)△S(T)>0for0 (ⅲ)△C(Tc)=-4µ0Hc02/Tc.Calculate△C(Tc)fromthisexpressionusingTc=1.175KandHc0=105Oe=8360A/mforAlandcomparewiththemeasuredresult-225Jm-3K-1forAl. (ⅳ)△C=0forT=Tc/ andT=0K. Eq.(16.3): 16.2(a)UsingEq.(16.5),calculatethecondensationenergyinJ/m3andineVperelectronatT=0KforthesuperconductorPbforwhichHc0=6.39×104A/m. (b)Compareyourresultfrompart(a)withtheexpression(0)((0)/EF)wherethesuperconductingenergygap2(0)=2.6meVforPb.Here(0)/EFisthefractionofconductionelectionswhoseenergiesareactuallyaffectedbythecondensation. Eq.(16.5): 16.3ConsidertheLondonpenetrationdepthλLdefinedinEq.(16.10). (a)CalculateλL(0K)forthesuperconductingAl,Pb,andNb. (b)IfasuperconductorhasaLondonpenetrationdepthλL(0K)=200nm,whatistheconcentrationnsofsuperconductingelectronsatT=0.5Tc. Eq.(16.10): 16.4WhentransportcurrentiflowsthroughasuperconductingwireofradiusR,itspathisconfinedtoaregionofthicknessλ,thepenetrationdepth,justinsidethesurfaceofthewire. (a)InthiscaseshowthatthecriticalcurrentdensityJc=ic/AeffisindependentofRandcanbeexpressedintermsofthecriticalfieldHcbyJc=Hc/λ.HereAeffistheeffectiveareathroughwhichthecurrentflows,withAeff< (b)CalculateJcforsuperconductingPbatT=0K.[Note: Hc0=803Oe=63919A/mandλ(0)=39nm] (c)SketchJc(T)/Jc(0)fromT=0KtoTcusingthetemperaturedependenciesofHcandλgiveninEqs.(16.6)and(16.11),respectively. Eq.(16.6): Eq.(16.11): 16.5AtypeⅡsuperconductorhasTc=125k,ΘD=250K,and(Tc)=50.Onthebasisofstandardtheories[free-electronmodel,Debyemodel,BCStheory,G-Ltheory,PaulilimitforHc2giveninEq.(16.33)],estimatethefollowing: (a)Thesuperconductingenergygap2ε(0). (b)TheuppercriticalfieldHc2(0)=Hp. (c)Thecoherencelengthξ(0)andthepenetrationdepthλ(0). (d)ThethermodynamiccriticalfieldHc0=Hc(0). (e)ThecoefficientsγandAoftheelectronicandphononcontributionstothespecificheat,γTandAT3,respectively. Eq.(16.33): 16.7UseEq.(16.20)tofindthelimitingvaluesofλ(l)andξ(l)(a)inthecleanlimitwheretheelectronmeanfreepathl>>ξ0,and(b)inthedirtylimitwherel<<ξ0. Eq.(16.20): and 16.8(a)CalculatethedensityofvorticesperunitareaB/Φ0forthefollowingvaluesofB,theaveragefluxdensitypresentinthemixedstateofasuperconductor.TakeHc2=1.6MA/m. (i)B=μ0Hc2/2.(ii)BBc2=μ0Hc2. (b)Calculatetheaverageseparationdbetweenthevorticesfromyouranswersinpart(a)andcompareyouranswerswiththeconherencelengthξ.[Hint: YoucanobtainξwiththehelpofEq.(16.22).] Eq.(16.22): 16.10CalculatethenumberofholesNholeperCuionintheCuO2copper-oxygenlayersinthesuperconductorYBa2Cu3O7-xforthecasesofx=0,0.25,and0.5.Assumethefollowingionicchargestatesfortheionsinthisstructure: Y3+,Ba2+,Cu2+,andO2-. 16.11ForthecompoundwiththechemicalformulaLa1.7Sr0.3CuO3.9: (a)whatisthetotalnumberofelectronsperformulaunitoutsideclosedshells? (b
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