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Scattering

µ 1. µ 2. H + U. Scattering. Numbers ( e , g ,U)  Matrices (H, S , U) Rate equations  NEGF formalism. SOURCE. DRAIN. H + U. INSULATOR. V G. V D. I. Unified approach. [H]: Energy levels [U]: Channel potential [ S 1,2 ]: Injection from contacts

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Scattering

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  1. µ1 µ2 H + U Scattering Numbers (e,g,U)  Matrices (H, S, U) Rate equations  NEGF formalism

  2. SOURCE DRAIN H + U INSULATOR VG VD I Unified approach [H]: Energy levels [U]: Channel potential [S1,2]: Injection from contacts [Ss]: Scattering Unified approach to quantum transport Details of ingredients vary

  3. H + U Physics of Ss Scattering other than contacts (Responsible for Ohm’s Law)

  4. H + U Phase BreakingSP ihdy/dt –Hy – SPy = SP Environment changes during conduction through interactions (eg. Atoms vibrating, spins flipping, light absorption) Many-electron theory needed S, S function of occupancy (Pauli exclusion)

  5. E.g. Electron-phonon H + U Sin(E) ~ G.F = n(E-Eph) x Pabs(Eph) + n(E+Eph) x Pem(Eph)  We’ll revisit this later

  6. H + U Enforces Irreversibility ihdy/dt –Hy – SPy = SP Environment changes during conduction through interactions Brings system back to equilibrium PA->B/PB->A = exp[-(EB-EA)/kT]

  7. Signatures of Scattering Depending on DOS, scattering can increase or decrease G Phonon assisted Tunneling  Sidebands

  8. Signatures of Scattering Current picks up signatures of these vibrations (Inelastic Electron Tunneling Spectroscopy) Expt. Mark Reed (Yale) Electron can lose energy by setting molecule vibrating

  9. T T1 T2 Scattering leads to Ohm’s Law T T1 T2 What’s the net transmission if we know T1 and T2 ? Simplest guess: T = T1T2 Correction: T = T1T2 + T1T2R1R2 + ..

  10. Since R1,2 = 1 – T1,2 T = T1T2/(T1+T2-T1T2) Bottom-Up treatment of Ohm’s Law T T1 T2 Net result: T = T1T2 + T1T2R1R2 + T1T2(R1R2)2 + … = T1T2/(1-R1R2)  1/T = 1/T1 + 1/T2 -1

  11. L: Scattering length The limits L 0, ∞ make sense There is an additive component ! T TL TL+dL Let r = (1-T)/T = R/T Then, r(L+dL) = r(L)+ r(dL) = r(L) + dL/L • dr/dL = 1/L • r(L) = L/L • T(L) = L/(L+L)

  12. Contact Resistance Intrinsic Device Resistance OHM’s LAW Bottom-Up treatment of Ohm’s Law G = (2q2/h)MT (Landauer Theory) R = h/2q2MT = h/2q2M + (h/2q2M)R/T Rdevice = (h/2q2M)L/L #modes depends on how many half wavelengths are fitted in M = A/(lF/2)2 = AkF2/p2 In 3-D, kF3 = 3p2n, and kF = mvF/ħ = mL/tħ Thus Rdevice = rL/A, where conductivity s = 1/r ~ nq2t/m

  13. t t1 t2 But we’re adding probabilities here !! What if we want to include quantum effects? t t1 t2 Correct way: Deal with t1, t2, r1, r2 (complex #s) Simplest guess: t = t1t2 Correction: t = t1t2 + t1t2r1r2 + ..

  14. Including phases then… t t1 t2 Net result: t = t1t2/(1-r1r2) T = |t|2 = |t1t2/(1-r1r2)|2 = T1T2/[1+R1R2-2(R1R2)cosDj] Dj = 2kDx • Resistances may not add !! • Resistances may be tunable by altering phase (e.g. path-length, temperature, magnetic field)

  15. Interference with electrons Goldman group, Stonybrook Chandrasekhar et al, PRL ‘85

  16. When do we simply add resistances? When interferences are unimportant. e.g. Impurities, temperature All cosine phase terms drop out in numerator (“Dephasing”) Ironically, we know how to solve the quantum problem!! So how would the classical equations come out of NEGF? (Dephasing or Incoherence)

  17. ‘s’ H + U T12 = [g1g2/(g1+g2)]D(E) T1s = [g1gs/(g1+gs)]D(E) Ts2 = [gsg2/(gs+g2)]D(E) Simple Model for Scattering Think of scattering center as a ‘virtual’ contact in equilibrium that extracts and reinjects charge after randomizing its phase Property of regular contacts: In thermal equilibrium determined by f1,2

  18. ‘s’ = 0 H + U     Solve for fs and calculate (f1 - fs (f1 - f2 (f1 - fs (f2 - fs +.T2s(E) +.T12(E) I1=dE T1s(E) Is=dE T1s(E) ) ) ) ) Buttiker Probe

  19. ‘s’ H + U   (f1 - f2 I1=dE T(E) ) 1/T1s 1/Ts2 1/T12 Elastic Scattering Elastic scattering (No energy exchange) Set integrand of Is = 0 fs = (T1sf1+Ts2f2)/(T1s+Ts2) T = T12 + T1sTs2/(T1s+Ts2) Like resistor network!

  20. ‘s’ H + U Inelastic Scattering A lot of energy exchange (‘Thermalization’) fs(E) = 1/[1+exp(E-ms)/kT] Adjust ms for zero Is Here we assume all energy relaxation processes allowed Many parallel ‘channels’ Scattering exchanges energy among them (‘Vertical Flow’) A single vibration (phonon) can only cause change in energy of ħw0

  21. 2q I1 = dETr[S1inA-G1Gn] h Beyond Buttiker Probes Instead of gs assume Ss has some energy structure More importantly, can’t calculate transmission between scatterer and channel very easily T1s ≠Tr(g1GgsG+), since Ssin ≠ gsfs Use instead

  22. El-El S(E): complicated function of f We get G(E) = 2p|t|2f(E)D(E) Instead of G(E) = 2p|t|2D(E) Coulomb Blockade Kondo effect Self-energy for Interacting systems

  23. Self-energy for dephasing scattering Ss = hGh+ Vibrations/Spins (“Bosons”) Ssin(E) = D[Gn(E-ħw)Nw+ Gn(E+ħw)(Nw+1)] for 1 mode (Slide 5) E E ħw ħw E-ħw E+ħw Ssout(E) = D[Gp(E-ħw)Nw + Gp(E+ħw)(Nw+1)] Phonon Absorption Phonon Emission

  24. Ss = hGh+ Phonon emission Polaronic shift Phonon absorption Self-energy for dephasing scattering Vibrations/Spins (“Bosons”) Gn = GSsinG+ Gs = Ssin + Ssout Ss = H(Gs)

  25. ‘s’ H + U Going beyond self-energy A more accurate way is to directly solve the transport problem in ‘Fock’ space like in chapter 3 Beyond a certain point, calculating S(E) becomes hard, and may need summation of various infinite (possibly) non-convergent series

  26. Summary Classical equations come out of quantum if we lose phase information (‘memory’) through dephasing While the consequence of dephasing is simple to implement (Newton’s Laws!), it’s not easy to get out of NEGF, especially if we have intermediate degrees of dephasing (ie, some incoherence, but not enough to make it classical) Needs careful physical considerations, but could have really important roles to play (e.g. spectroscopy)

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