Nanosciences M1 chapter
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Nanosciences M1 chapter

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Niveau: Supérieur, Master, Bac+4
Nanosciences-M1-chapter 2 1 Acceleration by electric field Conductance of nano-wires and circuits 2 F? ? v mEe dt dv m ?= r 0= dt dv EElectric field Classical description: Drude law Lorentz Relaxation by impurities steady state m eEv ?= Current density EE m ne vnej rrr ?? === 2 Drude formula m ne ?? 2 = Physically dubiuous but correct answer Transport comes from electron at Fermi level

  • electron- phonon scattering

  • scattering cross-section

  • nanosciences-m1-chapter

  • ?ie nn

  • section ?

  • electric field

  • all states

  • momentum relaxation


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Nombre de lectures 26
τ=vEemsteadystateddvt=0Nanosciences-M1-chapter 2ircsedlacissalCwal edurD :noitpC2λFElectricfieldElevel imreFtanortcelemorfsemoc tropsnarTrewsna tcerroc tubsuouibudyllacisyhP=2στenmalumrof edurD2===τσrrrjenvenmEEytisnedtnerruCseitirupmi yb noitaxaleRztneroLr=mvdtdeEmvτstiucricdnaseriw-onanfo ecnatcudnoCdleifcirtcele ybnoitareleccA1
em :noitaxalermutnemoMlemμ1mm1Electron-electronscatteringNanosciences-M1-chapter 2d123τLϕϕT2/31/3TT1T1/2T3/2T3/4201Electron-phononscattering111=+ltotlelinelasticinelastic105TLϕ1T3/2T2Tτϕ2T3T4Td123λFτmomentumrelaxation timeleleFτmeanfreepathelastichtapeerfna
cStaetimpurityirnθscatteringcross section σσgrcoLsss-ceitnom/niaenrfeeapdensityofimpuritieshtmeannumberofimpuritiesin thetubeσLni=NDistance betweenimpurities1Lle=N=niσanisotropicscatteringl*elele=1cosθNanosciences-M1-chapter 23
hSfi tniF reims ruafecvτ=EemElectricfieldshiftsmomentumRelaxation drives shiftedFermiSurface backto equilibriumNanosciences-M1-chapter 24
eDn2=Disytfom2=satVolume availablein phase space?States belowEFare occupiedets22ε=hkF,k2=2mεFF2mFh2k=2mεFF2hTypical valuessqiuzaen1tu00mn bmo-x1  μ2mD GaAsΔ2D10mK1K(γ<Δ2D)D=2All states inside disk ofkFL2kF2Sradius kFLare occupiedπ==N2π4πεmS2F2=NεF=2πhN=NΔπ2hSm22D density of Δ2D=2πhn2D=1=m2= csotantsetsantmSSΔ2πhNanosciences-M1-chapter 25
F)(hεπε===Δ18213223/21/23NnVmVDFDh=πε162223/23/2NmVπππ43216323==FFkLkVNLFk suidar foerehps fo emulov3=D)D3( setatsfoytisneD62 retpahc-1M-secneicsonaN22ε=hkF,k2=2mεFF2mFh2k=2mεFF2hTypical valuesnearεF3D metallic grains size 10nm-100nmΔ3D100nK100μK(γ>Δ2D)Δ<Ec(except in 1D)=Δ=nVmDDFεπh128333/223)(Δ=h823233/2πεmVD
FromDrude to Einstein formula33Numberofelectrons4kFL1kF1perunit ofvolumen=23π2πL3=6π2=3νhkFvFνD22Drude Formulaσ=neτ=e2ndvFτ=e2νDmpFvFdGraphicilustration: wirelengthLV voltage diff.εFεVJ=eNτL=eNDL2Le2DVLE=νeNVVeNanosciences-M1-chapter 2Einstein relation7
Δ ,cE ,seigrenecitsiretcarahcowtfo oitaRecnatcudnoc otytivitcudnocmorFcEh1Δnoitaler nietsniEmorf ecnatcudnoCLσσν1-dLCube ofsideLNanosciences-M1-chapter 2=8=G independantofEF, τOnlydependson theratio Ec/Δ=2dimensionlessG=eNTConductancehThoulessnumberDependencewithL11d2Δ∝dEc2NTLLLd=1insulator=Δ=122222GLLLeLDLehEehNdddcT
nIetpreratitnohtdiwfohToh=h2D=EcτL#li noefwstiadttehs withinlue21e=Quantum de conductance=h25800ΩsNT<1insulatorN(2L)=N(L)2NT>1NTLd2d=1, L→∞insulatorsunbmreDwelltime=residencetimeτ=L2/DmlateNanosciences-M1-chapter 22e=hN(2L)=N(L)2Ohm’s law9
hhhπππΔ====222222222122LmLnnm01nSingle channelNanosciences-M1-chapter 2LandauerformulaL2j=et2V=GVhhReservoirchem. pot.μ2LCkRDkRnwiresection w, lengthL=waveguide2ε(k)=hk2+k2m2LTransmissionmatrixJmAkLBkLhCurrentvelocity2j=eNtvFcharge# ofstatesTransmissionIn slice eVcoefficientLReservoirchem. pot.μ1=μ2+eVvF)(1=Δ=NVeVevhFVkw=πkmwslennahc =sedoM
CkRBkLAkLxirtamnoissimsnarTJtrtTTTT=++2**122111***ABTCDtrtrttCD==***11CDTABtrtrttAB==xirtam/feoc noissimsnarT11MultichannelLandauerformulaλαeλα/20TT+=e0=Tλ0eλβ/20eβλ<11α22+trtt=2αeλα+eλα+2=αcosh(λα)+1λα>10Nanosciences-M1-chapter 2Input (L)ijOutput (L)tiiitjijtjjMore than1 channel(mode)2G=2etrtt+h=22GehNffe)1<αλ(/lennahcfo #=ffeNTsezilanogaidhcihw sisab :sedom noisuffiDarbeglaevitacilpitlum TDkR