3.1 自由电子理论(1).pdf

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1、Chapter 3Free electrons in solidsProfileTodays lectureElectronAtomBindingBorn-Oppenheimer Approximation(1/2)EnkkValence BandValence BandValence BandConduction BandConduction BandConduction BandQuantum MechanicsClassical TheoryCrystal DynamicsFree Electron ModelEnergy BandsCrystal StructureAtoms move

2、memtElectrons movementCrystal dynamicsElectron theoryFree electron theoryEnergy band theory IntroductionChapter 3 Free electrons in solids IntroductionChapter 3 Free electrons in solidsPhysics for computer science students:with emphasis on atomic and semiconductor physicsAcknowledgement:Chapter 3 Fr

3、ee electrons in solids3.1 Free electron model3.1.1 Drude Model-Classical Free Electron Model3.1.2 Sommerfeld Model-Quantum Mechanical Free Electron Model3.2 Heat capacity of free electron gas3.3 Transport properties of conductive electronsIndependent electron approximationFree electron approximation

4、Collision assumptionRelaxation time approximationDrudeClassical Free Electron Model Independent electron approximationFree electron approximationNo collisionQuantum statistics:Fermi-Dirac DistributionSommerfeldQuantum Mechanical Free Electron ModelEnergy band theoryFree Electron Model+Periodical pot

5、ential fieldDrude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEM IntroductionChapter 3 Free electrons in solids3.1 Free electron model3.1.1 Drude Model-Classical Free Electron Model3.1.2 Sommerfeld Model-Quantum Mechanical Free Electron Model3.2 Heat capacity of free electron gas3.3 Tra

6、nsport properties of conductive electrons3.4 Electron emission and contacting voltage Drude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEMUIRJOhms LawJ:Current density1:Resistivity:Conductivity:Electric field intensityDrude Model Failure of Drude ModelPaul Karl Ludwig DrudeDrude Model e

7、lectronsThe Drude model of electrical conduction was proposed in 1900 by Paul Drudeto explain the transport properties of electrons in materials(especially metals).The model,which is an application of kinetic theory,assumes that themicroscopic behavior of electrons in a solid may be treated classica

8、lly andlooks much like a pinball machine,with a sea of constantly jittering electronsbouncing and re-bouncing off heavier,relatively immobile positive ions.Drude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEMAssumption of the electron gas:(1)Independent electron approximationNo electros

9、tatic interaction and collision among free electrons(2)Free electron approximationNo electrostatic interaction between free electrons and ions(3)Collision assumptionVelocity of electrons after collision with ions only concerns with temperature,but not the velocity before collision(4)Relaxation time

10、approximationRelaxation time is independent with the position and velocity of electronsFree electron gasDrude Model Failure of Drude ModelDrude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEMWithout electric field,free e-moves randomly1/251131.2 10 m sec(120km sec)RMSBvk T mAt room tempe

11、rature,No net current With electric field,things are differentvd:drift velocity(漂移速度漂移速度),vd0KFermi-Dirac distributionFind Fermi energy EFk distributionDensity of stateFind the electron number near the EFDrude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEM Potential Well F-D Distributio

12、n Fermi Energy Density of State e-number near EF e-gas EnergyU(x,t)U(x)i/(,)()eEtx tx222d()()2d()xExxmUxU(x)0a x()0,0(),0U xxaU xxa x222d2dEmx222d20dmEx2mEk sincosAkxBkx2()2kEm(0)cos00()sincos0BBaAkaBka0sin0BAkaInfinite Potential Well 222d0dkx()0,0 xxa xDrude Model-Classical FEM Sommerfeld Model-Qua

13、ntum Mechanical FEM Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas Energysin0Aka sin0ka kan1,2,3,.n()sinnxAxa0 xa201adx2Aa2()sinnxxaa1,2,3,.n U(x)0a x 222sinnnxaaE1E1E12mEk 222d20dmExknasincosAkxBkxDrude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FE

14、M Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas EnergyU(x)0a x 2mEk 222d20dmExsin0Aka sincosAkxBkxkan1,2,3,.n kna22222nEEnma1,2,3,.n m,a，E,quantum effect strengthen nano-material22122Ema22222Ema1n 2n E1E1E1The energy of a particle in a box(black circles)and a f

15、ree particle(grey line)both depend upon wavenumber.2()2kEmDrude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEM Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas EnergyDrude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEM.00,0Ux y zLUx y zL

16、x y z 2222222(,)(,)2x y zEx y zmxyz,)()()()x y zxyz（U(x)0a x 222d20dmEx2()sinxkxa22222222nkEEnmma1,2,3,.n()0,0(),0U xxaU xxa xL222d()()()2dU xxExm x1n 2n E1E1E13D Infinite Potential Well Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas Energykna1D Infinite Potenti

17、al Well 1D Infinite Potential Well Drude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEM22222221()1()1()20()()()xyzmExxyyzzxyzkk ik jk l222kEm2222222221()1()1()()0()()()xyzxyzkkkxxyyzz2221()()xxkxx 2221()()yykyy 2221()()zzkzz L2222222(,)(,)2x y zEx y zmxyz222()()0 xxkxx222()()0yykyy222()

18、()0zzkzz2222222()()()()()()2xyzExyzmxyz.00,0.,.Ux y zLUx y zL x y z Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas Energy2222()2xyzEkkkm0,()0;,()00,()0;,()00,()0;,()0 xxxLxyyyLyzzzLz()sinxxxAk xxxnkL()sinyyyAk yyynkL()sinzzzAk zzznkL,)()()()x y zxyz（1,2,3,.xn 1,

19、2,3,.yn 1,2,3,.zn 222222222222()222()xxyzzyEkkkkmmmLnnn,1,2,3,.xyzn n n L2222222(,)(,)2x y zEx y zmxyzDrude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEM Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas Energy222()()0 xxkxx222()()0yykyy222()()0zzkzzB

20、oundary conditionFind the characteristics of motion of the free electronsQuantized EnergyHow the energy levels are occupied by free electronsFermi-Dirac distributionFind Fermi energy EFFind the electron number near the EF1n 2n E1E1E10K0KDrude Model-Classical FEM Sommerfeld Model-Quantum Mechanical F

21、EM Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas EnergyFermi-Dirac Distribution()()/()/111111llBFBllEEk TE Ek TlFeee Drude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEM Potential Well F-D Distribution Fermi Energy Density of State e-number near EF

22、 e-gas EnergyDrude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEM Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas EnergyOccupancy Condition at T=0K-Classical theory222222()2xyzEnnnmL,1,2,3,.xyzn n n L2220()xyzE nnn22022EmLDrude Model-Classical FEM So

23、mmerfeld Model-Quantum Mechanical FEM Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas EnergyOccupancy Condition at T=0K-Quantum mechanicallyPaulis Exclusion PrincipleDrude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEM Potential Well F-D Distribution

24、 Fermi Energy Density of State e-number near EF e-gas EnergyOccupancy Condition at T=0K-Quantum mechanicallyThe Fermi energy：the highest energy a fermion can take atabsolute zero temperature.Fermi-Dirac Distribution()()/()/111111llBFBllEEk TE Ek TlFeee Drude Model-Classical FEM Sommerfeld Model-Quan

25、tum Mechanical FEM Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas EnergyDrude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEM Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas Energy()/1()1FBE Ek TF EeOccupancy Cond

26、ition at T0K-Quantum mechanicallyDrude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEM Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas EnergyOccupancy Condition at T=0K and T0K-Quantum mechanically1.Fermi Energy value？2.e-number near EF？Explain Find t

27、he characteristics of motion of the free electronsQuantized EnergyHow the energy levels are occupied by free electronsFermi-Dirac distributionFind Fermi energy EFFind the electron number near the EF1n 2n E1E1E1k distributionDensity of state0K0KDrude Model-Classical FEM Sommerfeld Model-Quantum Mecha

28、nical FEM Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas EnergykzkykxkFThe radius of the Fermi sphere is called the Fermi wave vector kF222,xxyyzzknknknLLL 2()2FFkEmAt T=0K,Inside Fermi sphere,all orbits are occupied;Outside Fermi sphere,all orbits are empty.Dru

29、de Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEM22022EmLxyzkk ik jk l Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas EnergyCalculation of Fermi Energy(EF)at T=0K222200()xyzEnnnEn E,1,2,3,.xyzn n n 222222()22xyzkEkkkmmStates on the same sphere are d

30、egenerate statesFermi wave vector kFFermi sphere 2()2FFkEmDrude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEM Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas EnergyCalculation of Fermi Energy(EF)at T=0KDistribution density of k:(k)Fermi sphere()kVN

31、knumberddddkdofd dnumber of electronsEach k2 electrons 222(2)ksLLS 214kSks222,xxyyzzknknknLLL ky=kx=0-112-23-31-12-2-33L 2L 22xnL2ynL2D3222(2)kvLLLV 318kVkv3DFV 33324386FFFkkVVkV3226FkNV323FkNnV22022/3()(3)22FFkEnmmEF0=5.84*10-38n2/3J 38Vk2()2FFkEm21/3(3)FknFermi-energy EF:The valance electron densi

32、ty n=N/V is 1.4021028m-322/3232.122FEneVm1/31230.746FNkV42.46 10FFBETKkComparison:For a conductor of a=4cm4112.3 10eVEDrude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEMFermi wave vector kF:Fermi temperature TF:Potential Well F-D Distribution Fermi Energy Density of State e-number near

33、 EF e-gas EnergyCalculation of Fermi Energy(EF)at T=0KNumber of kNumber of e-Density of e-E.g.Find the characteristics of motion of the free electronsQuantized EnergyHow the energy levels are occupied by free electronsFermi-Dirac distributionFind Fermi energy EFFind the electron number near the EF1n

34、 2n E1E1E1k distributionDensity of state0K0KDrude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEM Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas EnergyFind the characteristics of motion of the free electrons by quantum mechanicsQuantized EnergySchrod

35、inger EquationHow the energy levels are occupied by free electronsFermi-Dirac distributionFind Fermi energy EFk distributionDensity of stateFind the electron number near the EF电子气能量eVCTEDrude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEM Potential Well F-D Distribution Fermi Energy Den

36、sity of State e-number near EF e-gas Energy 3333244383223VVkZ Ekkkddd()dddZZkg EEkE3/21/222dd2()dd2ZkVmg EEkE3/2222C2VmkzkykxkF1/2()g EC EDrude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEMDOS:the number of electronic energy states(orbits)per unit energy.-g(E)Density of state(DOS)Remem

37、ber:In crystal dynamics,the density of states g()is defined as the number of oscillators(or k)per unit frequency interval.Each k state represents two possible electron states:one for spin up,the other for spin down,thus the total number of electronic states in a sphere of diameter k is:Density of st

38、ate(DOS)222kEm Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas Energyd()dZg EE2ddEkkm22ddZVkk2mEk 1/2dd2km EE1/22322d2()4d24d(2)2VmEm EZkkkE3/21/222d2()d2ZVmg EEEDrude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEMAnother Way to Calculate DOS3/2222C2

39、Vm1/2()g EC E Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas EnergyFor a free electron gas in 3D,the density of states for free electrons increases with the increase of energy E.1Dd2()2 d4d2LZkkk 222d2d22()dd2ZLkL mLmg EEEkmE1/2()g EC E2D2d2()2d4d4SZkkkkk22dd()d

40、dZkSkkS mmSg EEEkg(E)has no relationship with E or k3D3D22232d2()4d24d8VVkZkkkkkdk2222222222dd2()ddZVkkVkmmVkmVmEg EEEkDrude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEM Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas EnergyDrude Model-Classical FE

41、M Sommerfeld Model-Quantum Mechanical FEMSituation of Orbits Occupied by Electrons Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas Energy1/2()g EC Eg(E)()()()N Ef E g E()f E()N E0()()dNg E f EETotal number of e-Define N(E)as Electron energy distribution functionN

42、(E)dE=number of e-with energies between E and E+dEg(E)f(E)dE=number of e-with energies between E and E+dE()()()N Eg E f EDOS g(E):the number of electronic energy states(orbits)per unit energy.f(E):probability that an electronic energy state be occupied.T0KT=0KEFEFDrude Model-Classical FEM Sommerfeld

43、 Model-Quantum Mechanical FEM1/2()g EC Eg(E)()()()N Ef E g E0()()dNg E f EETotal number of e-3/21/2222()2Vmg EEAnother method of getting Fermi energyT=0K003/21/203/2232300()()d(2)d(2)23FFEEFVVNf E g EEmEEmE2323220223322FNEnmVm222FFekEm1/321/3233FNknV Potential Well F-D Distribution Fermi Energy Dens

44、ity of State e-number near EF e-gas EnergyDrude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEMFind the characteristics of motion of the free electrons by quantum mechanicsQuantized EnergySchrodinger EquationHow the energy levels are occupied by free electrons0K0KFermi-Dirac distribution

45、Find Fermi energy EFk distributionDensity of stateFind the electron number near the EF电子气能量eVCTE Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas Energyd()()dtEE NE g Ef EE005/21/23/20002dd=5FFEEFCC EE ECEEE5/200235.5FFCEEEN2/320232FNEmV322322VmCDrude Model-Classi

46、cal FEM Sommerfeld Model-Quantum Mechanical FEMTotal energy for electron gas T=0KAverage energy for free electronsAt T=0K，the average energy of a free electron is 60%of the Femi energy.Q:Is the result similar to that of classical theory?1/2()g EC Eg(E)()()()N Ef E g E0()()dNg E f EE00()dFEg EE03/223

47、(2)3FVmE Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas Energy035tFENE0()()dNg E f EE()()dtEE g Ef EE()/01()d1FBE Ek Tg EEe3/2()/01de1FBE Ek TCEE0220351()512BFFk TNEEDrude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEMTotal energy for electron gas T

48、0K1/2()g EC Eg(E)()()()N Ef E g E22220000111212BFFFFFFk TTEEEEETFermi energyEF0 Potential Well F-D Distribution Fermi Energy Density of State e-number near EF e-gas EnergyDrude Model-Classical FEM Sommerfeld Model-Quantum Mechanical FEMFind the characteristics of motion of the free electrons by quan

49、tum mechanicsQuantized EnergySchrodinger EquationHow the energy levels are occupied by free electrons0K0KFermi-Dirac distributionFind Fermi energy EFk distributionDensity of stateFind the electron number near the EF电子气能量 Potential Well F-D Distribution Fermi Energy Density of State e-number near EF

50、e-gas EnergyA.Piccard,E.Henriot,P.Ehrenfest,E.Herzen,Th.De Donder,E.Schrdinger,J.E.Verschaffelt,W.Pauli,W.Heisenberg,R.H.Fowler,L.Brillouin;B.P.Debye,M.Knudsen,W.L.Bragg,H.A.Kramers,P.A.M.Dirac,A.H.Compton,L.de Broglie,M.Born,N.Bohr;I.Langmuir,M.Planck,M.Skodowska-Curie,H.A.Lorentz,A.Einstein,P.Lang