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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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Publié par

larec

Nombre de lectures

Scattering cross-section Electric field

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 2d123τLϕϕT−2/3−1/3TT−1T−1/2T−3/2T−3/4201Electron-phononscattering111=+ltotlelinelasticinelastic105TLϕ−1T−3/2T−2Tτϕ−2T−3T−4Td123λFτmomentumrelaxation timelele=νFτmeanfreepathelastichtapeerfna

cStaetimpurityirnθscatteringcross section σσgrcoLsss-ceitnom/niaenrfeeapdensityofimpuritieshtmeannumberofimpuritiesin thetubeσLni=NDistance betweenimpurities1Lle=N=niσanisotropicscatteringl*ele→le=1−cosθNanosciences-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=2mεF⇒F2mFh2k=2mεFF2hTypical valuessqiuzaen1tu00mn bmo-x1 μ2mD GaAsΔ2D≈10mK−1K(γ<Δ2D)D=2All states inside disk of⎛kFL⎞2kF2Sradius kFLare occupiedπ⎜⎟==N⎝2π⎠4πεmS2F2=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=2mεF⇒F2mFh2k=2mεFF2hTypical valuesnearεF3D metallic grains size 10nm-100nmΔ3D≈100nK−100μK(γ>Δ2D)Δ<Ec(except in 1D)=Δ=nVmDDFεπh128333/223)(Δ=h823233/2πεmVD

FromDrude to Einstein formula33Numberofelectrons4⎛kFL⎞1kF1perunit ofvolumen=23π⎝⎜2π⎠⎟L3=6π2=3νhkFvFνD22Drude Formulaσ=neτ=e2ndvFτ=e2νDmpFvFdGraphicilustration: wirelengthLV voltage diff.εFεVJ=eNτL=eNDL2L=νe2DVL=σE=νeNVVeNanosciences-M1-chapter 2Einstein relation7

Δ ,cE ,seigrenecitsiretcarahcowtfo oitaRecnatcudnoc otytivitcudnocmorFcEh1Δnoitaler nietsniEmorf ecnatcudnoCLσσν1-dL−Cube ofsideL−Nanosciences-M1-chapter 2=8=G independantofEF, τOnlydependson theratio Ec/Δ=2dimensionlessG=eNTConductancehThoulessnumberDependencewithL11−d2Δ∝dEc∝2NT∝LLLd=1⇒insulator⎛⎝⎜⎞⎠⎟=Δ=122222GLLLeLDLehEehNdddcT

nIetpreratitnohtdiwfohToh=h2D=EcτL#li noefwstiadttehs withinlue21e=Quantum de conductance=h25800ΩsNT<1insulatorN(2L)=N(L)2NT>1NT∝Ld−2d=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.μ2LCkR−DkRnwiresection w, lengthL=waveguide2ε(k)=hk⊥2+k2m2LTransmissionmatrixJmAkL−BkLhCurrentvelocity2j=eNtvFcharge# ofstatesTransmissionIn slice eVcoefficientLReservoirchem. pot.μ1=μ2+eVvF)(1=Δ=NVeVevhFVkw⊥=πkmwslennahc =sedoM

CkR−BkLAkLxirtamnoissimsnarTJtrtTTTT−=++2**122111***ABTCDtrtrttCD−⎛⎝⎜⎞⎠⎟=⎛⎝⎜⎞⎠⎟=⎛⎝⎜⎜⎜⎜⎞⎠⎟⎟⎟⎟⎛⎝⎜⎞⎠⎟***11CDTABtrtrttAB⎛⎝⎜⎞⎠⎟=⎛⎝⎜⎞⎠⎟=−−⎛⎝⎜⎜⎜⎜⎞⎠⎟⎟⎟⎟⎛⎝⎜⎞⎠⎟xirtam/feoc noissimsnarT11MultichannelLandauerformulaλα⎛eλα/20⎞TT+=⎛⎜e0⎞⎟⎜⎟=Tλ⎝0eλβ/2⎠⎝0eβ⎠λ<1→1α22+trtt=2α∑eλα+e−λα+2=α∑cosh(λα)+1λα>1→0Nanosciences-M1-chapter 2Input (L)ijOutput (L)tiiitjijtjjMore than1 channel(mode)2G=2etrtt+h=22GehNffe)1<αλ(/lennahcfo #=ffeNTsezilanogaidhcihw sisab :sedom noisuffiDarbeglaevitacilpitlum T−DkR

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