Nanosciences M1 chapter

Cnrs/Lcmi

Description

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

Sujets

• Scattering cross-section
• Electric field

Informations

• Publié par: larec
• Nombre de lectures: 26

Extrait

τ = vEe m steadystate d d v t = 0 Nanosciences-M1-chapter 2 ircsedlacissalCwal edurD :noitpC 2 λ F Electricfield E level imreFtanortcelemorfsemoc tropsnarT rewsna tcerroc tubsuouibudyllacisyhP = 2 σ τ en m alumrof edurD 2 === τ σ r rr jenv en m EE ytisnedtnerruC seitirupmi yb  noitaxaleR ztneroL r =− m vd td eEm v τ stiucricdnaseriw-onanfo ecnatcudnoC dleifcirtcele yb noitareleccA 1

em :noitaxalermutnemoM l e mμ1 mm1 Electron- electron scattering Nanosciences-M1-chapter 2 d 1 2 3 τ L ϕϕ T − 2/3 − 1/3 T T − 1 T − 1/2 T − 3/2 T − 3/4 2 01 Electron- phonon scattering 111 =+ l tot l e l inelastic inelastic 105T L ϕ − 1 T − 3/2 T − 2 T τ ϕ − 2 T − 3 T − 4 T d 1 2 3 λ F τ momentumrelaxation time l e l e =ν F τ meanfreepath elastic htapeerfna

cStaet impurity irn θ scattering cross section σ σ grco L sss-ceitnom/ n i aenrfeeap densityofimpurities ht meannumberofimpuritiesin thetube σLn i =N Distance betweenimpurities 1L l e = N = n i σ anisotropicscattering l * el e → l e = 1 − cos θ Nanosciences-M1-chapter 23

hSfi tniF reims ruafec v τ = Ee m Electricfieldshiftsmomentum Relaxation drives shiftedFermi Surface backto equilibrium Nanosciences-M1-chapter 24

eDn 2=D isytfo m2= sat Volume availablein phase space? States belowE F are occupied ets 22 ε= h k F,k 2 = 2m ε F ⇒ F 2m F h 2 k = 2m ε F F 2 h Typical values s q i u z a e n 1 tu 00 m n  b m o - x 1    μ 2 m D GaAs Δ 2 D ≈ 10mK − 1K ( γ<Δ 2D ) D=2 All states inside disk of ⎛ k F L ⎞ 2 k F 2 S radius k F Lare occupied π ⎜⎟ == N ⎝ 2 π ⎠ 4 π ε mS 2 F 2 = N ε F = 2 π h N = N Δ π 2 h Sm 2 2D density of Δ 2D = 2 π h n 2D = 1 = m 2 = c s o ta n t s e t s ant mS S Δ 2 π h Nanosciences-M1-chapter 25

F )( h επ ε ∂ ∂ == ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = Δ 1 8 21 3 22 3/2 1/2 3 N nV m V DF D h = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ π ε 1 6 2 22 3/2 3/2 N m V π ππ 4 32 1 6 3 2 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ == F F kL kVN L F k suidar fo erehps fo emulov3=D )D3( setatsfoytisneD 62 retpahc-1M-secneicsonaN 22 ε= h k F ,k 2 = 2m ε F ⇒ F 2m F h 2 k = 2m ε F F 2 h Typical values near ε F 3D metallic grains size 10nm-100nm Δ 3D ≈ 100nK − 100 μ K ( γ>Δ 2D ) Δ< E c (except in 1D) = Δ = n V m D D F ε π h 1 2 8 3 3 3/2 23 )( Δ= h 8 2 3 23 3/2 π ε mV D

FromDrude to Einstein formula 3 3 Numberofelectrons 4 ⎛ k F L ⎞ 1 k F 1 perunit ofvolume n = 2 3 π ⎝ ⎜ 2 π ⎠ ⎟ L 3 = 6 π 2 = 3 ν h k F v F ν D 22 Drude Formula σ= ne τ = e 2 ndv F τ = e 2 ν D m p F v F d Graphicilustration: wirelengthL V voltage diff. ε F ε V J = eN τ L = eN D L 2 L =ν e 2 D V L =σ E =ν eNV Ve Nanosciences-M1-chapter 2 Einstein relation 7

Δ , c E ,seigrene citsiretcarahcowtfo oitaR ecnatcudnoc otytivitcudnocmorF c E h 1 Δ noitaler nietsniE morf ecnatcudnoC L σσν 1-d L − Cube ofsideL − Nanosciences-M1-chapter 2 = 8 = G independantofE F , τ Onlydependson theratio E c / Δ = 2 dimensionless G = e N T Conductance h Thoulessnumber DependencewithL 1 1 − d2 Δ∝ d E c ∝ 2 N T ∝ L L L d=1 ⇒ insulator ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = Δ = 1 22 2 22 G L L LeL D L e h E e h N d dd c T

nIetpreratitno htdiw fohTo h = h 2 D = E c τ L # li   n o e f w st i a d t t e h s within lue 2 1 e = Quantum de conductance = h 25800 Ω s N T <1insulator N(2L) = N(L) 2 N T >1 N T ∝ L d − 2 d=1, L →∞ insulator sunbmre Dwelltime=residence timeτ=L 2 /D mlate Nanosciences-M1-chapter 2 2 e = h N(2L) = N(L) 2 Ohm’s law 9

hhh πππΔ =−−=== 22 2 2 2 22 2 2 1 22LmL nn m 01 n Single channel Nanosciences-M1-chapter 2 Landauerformula L 2 j = e t 2 V = GV h h Reservoir chem. pot. μ 2 L Ck R − Dk R n wiresection w, lengthL =waveguide 2 ε (k) = h k ⊥ 2 + k 2 m2 L Transmission matrix J m Ak L − Bk L h Current velocity 2 j = eNtv F charge # ofstatesTransmission In slice eVcoefficient L Reservoir chem. pot. μ 1 =μ 2 +eV v F )( 1 = Δ = NVe Ve vh F V k w ⊥ = π k m w slennahc =sedoM

Ck R − Bk L Ak L xirtam noissimsnarT J trt TTTT − = ++ 2 ** 1 2 2 1 1 1 * * * A B T C D t r t r tt C D − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ * * * 1 1 C D T A B t r t r tt A B ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = − − ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ xirtam/feoc noissimsnarT 11 Multichannel Landauerformula λ α ⎛ e λ α /2 0 ⎞ TT + = ⎛ ⎜ e0 ⎞ ⎟ ⎜⎟ = T λ ⎝ 0e λ β /2 ⎠ ⎝ 0e β ⎠ λ< 1 → 1 α 22 + trtt = 2 α ∑ e λ α + e −λ α + 2 = α ∑ cosh ( λ α ) + 1 λ α > 1 → 0 Nanosciences-M1-chapter 2 Input (L) i j Output (L) t ii i t ji j t jj More than1 channel(mode) 2 G = 2 e trtt + h = 2 2 G e h N ffe )1< α λ( /lennahcfo #= ffe N Tsezilanogaidhcihw  sisab :sedom noisuffiD arbegla evitacilpitlum T − Dk R