Transcription

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4 1. Motivation

5 1. Motivation

6 1. Motivation L d R - V +

7 2. Bulk Transport E( k ) = 2 k 2 2m * ε µ)

8 2. Bulk Transport

9 2. Bulk Transport k d = -eeτ

10 3. Some basic concepts τ l m =vτ l ϕ λ = 2π/k

11 3. Some basic concepts ϕ ϕ ϕ

12 3. Some basic concepts ε ε E k z

13 3. Some basic concepts ϕ ϕ ϕ E k z

14 4. Coherent Transport: Landauer E µ µ µ+δµ µ L µ R = δµ = ev I = 2e bands i BZ dk n(k)v g (k,i) = 2e 2 h k V N bands E F v( k ) = 1 n( k ) = E( k ) k f( k) 2π E( k )/ k

15 4. Coherent Transport: Landauer

16 4. Coherent Transport: Landauer I ev T ε ( ) =Tr [t + t] ε ( ) I = 2e h dε f L ε ( ) f R ( ε) ( ) transmission matrix: ( ) T ε

17 4. Coherent Transport: Landauer For small voltage (linear response): G = I V = 2e2 h T(E F ) = G 0 T(E F )

18 4. Coherent Transport: Landauer M. A. Reed et al, Science 278, 252 (1997)

19 4. Coherent Transport: Landauer No Molecules/Solution Molecules in Solution Wires formed (up to 1 atom thick!) Conductance quantization Atomic short-circuit Nonlinear I/V curves Molecular levels (channels) M. A. Reed et al, Science 278, 252 (1997)

20 Brandbyge et al., PRB 52, 8499 (1995) 4. Coherent Transport: Landauer

21 5. Some Math: Green s Functions

22 5. Some Math: Green s Functions G r (E) = (E + H) 1 G r = i i E + iδ ε G r µν i i E + = lim E + iδ δ 0 + = i µ i i ν E + iδ ε i ρ(e) = 1 π Im Gr (E) ρ = 1 π Im n FD(E)G r (E) de

23 5. Some Math: Green s Functions G r (E) ρ(r) V KS (r) SCF

24 5. Some Math: Green s Functions

25 5. Some Math: Green s Functions Σ

26 5. Some Math: Green s Functions

27 5. Some Math: Green s Functions

28 6. Formulation at Equilibrium

29 6. Formulation at Equilibrium L C R Σ ε

30 6. Formulation at Equilibrium ρ ( ) = Im Σ R ( ε) t ε ( [ ]) 1/2 G ε ( [ ]) 1/2 ( ) Im Σ L ( ε) T ε ( ) =Tr [t + t] ε ( ) G = I V = 2e2 h T(E F ) = G 0 T(E F )

31 7. Away from Equilibrium L C R J. Phys. C: Solid St. Phys., , (2002) Density matrix from the incoming scattering states from left to right (with their corresponding chem. pot.)

32 7. Away from Equilibrium ˆ H = H L V L 0 + V L H C + 0 V R V R H R L C R Lippman-Schwinger Eq.: Density matrix:

33 After some algebra Away from Equilibrium Note that both the Density Matrix and the Spectral Density are more involved than in the equilibrium case (but reduce to them at equilibrium) D µν = dε ρ µν (ε) n F ( ε ε F ) ρ( ε) = 1 π Im G( ε)

34 7. Away from Equilibrium We, then, recover a Landauer-Büttiker formula for the conductance!, with explicit formulas for the transmission matrix and the density matrix (and the density in real space).

35 7. Away from Equilibrium 2 V (r) = ρ(r) ρ φ φ V L z L 2

36 8. Some Examples

37 8. Some Examples Calculation: Bandstructure of infinite chain Experiment: dzx,dyz

38 8. Some Examples

39 8. Some Examples Stokbro, Taylor, Brandbyge, Mozos, Ordejón, Computational Materials Science 27 (2003)

40 8. Some Examples Defect-free, long (5,5) tube Conductance: G = 2G 0 (around E F ) Charlier, Blase and Roche, RMP 07

41 8. Some Examples S-W defect in (5,5) CNT 10 8 Perfect tube Tube with a S-W defect Transmission Energy (ev)

42 8. Some Examples

43 9. Beyond Elastic Scattering and Independent Electrons