Quantum transport in nanoscale solids

Quantum transport in nanoscale solids The Landauer approach Dietmar Weinmann Institut de Physique et Chimie des Matériaux de Strasbourg Strasbourg, ESC 2012 p. 1

Quantum effects in electron transport R. A. Webb, S. Washburn, Physics Today, Dec. 1988 Strasbourg, ESC 2012 p. 2

Quantum effects in electron transport R. A. Webb, S. Washburn, Physics Today, Dec. 1988 far from classical quantum effects in electronic transport Strasbourg, ESC 2012 p. 2

Outline Basic concepts Quantum conductance, Landauer s approach Strasbourg, ESC 2012 p. 3

Basic concepts Strasbourg, ESC 2012 p. 4

Quantum versus classical behavior microscopic macroscopic atoms, molecules quantum large pieces of matter classical Strasbourg, ESC 2012 p. 5

Quantum versus classical behavior microscopic macroscopic atoms, molecules quantum large pieces of matter classical intermediate: Nanoscale systems Strasbourg, ESC 2012 p. 5

Quantum versus classical behavior microscopic macroscopic atoms, molecules quantum large pieces of matter classical intermediate: Nanoscale systems quantum effects and large number of atoms quantum and statistical physics "mesoscopic regime" [B. Reulet, friday] Strasbourg, ESC 2012 p. 5

Quantum behavior isolated system A: i t ψ A(t) = H A ψ A (t) Strasbourg, ESC 2012 p. 6

Quantum behavior isolated system A: i t ψ A(t) = H A ψ A (t) time evolution ψ A (t) = exp( ih A t/ ) ψ A (0) Strasbourg, ESC 2012 p. 6

Quantum behavior isolated system A: i t ψ A(t) = H A ψ A (t) time evolution ψ A (t) = exp( ih A t/ ) ψ A (0) How can classical behavior emerge? Strasbourg, ESC 2012 p. 6

Coupling to an environment realistic, A coupled to environment E: A E combined system H = H A + H E + H AE ψ(t) = exp( iht/ ) ψ(0) Strasbourg, ESC 2012 p. 7

Coupling to an environment realistic, A coupled to environment E: A E combined system H = H A + H E + H AE ψ(t) = exp( iht/ ) ψ(0) Do we have ψ A (t) = exp( ih A t/ ) ψ A (0)? Strasbourg, ESC 2012 p. 7

Coupling to an environment realistic, A coupled to environment E: A E combined system H = H A + H E + H AE ψ(t) = exp( iht/ ) ψ(0) Do we have ψ A (t) = exp( ih A t/ ) ψ A (0)? Coupling leads to decoherence on a timescale τ φ Strasbourg, ESC 2012 p. 7

Phase coherence time/length Quantum effects are suppressed due to decoherence phase coherence time τ φ Electrons move phase coherence length L φ Strasbourg, ESC 2012 p. 8

Phase coherence time/length Quantum effects are suppressed due to decoherence phase coherence time τ φ Electrons move phase coherence length L φ ballistic motion (v F Fermi velocity): L φ = v F τ φ diffusive motion (D diffusion constant): L φ = Dτ φ Strasbourg, ESC 2012 p. 8

Phase coherence time Experiment in GaAs [Choi et al., Phys. Rev. B 87] L φ 1 10 µm at T 1 K Strasbourg, ESC 2012 p. 9

Spin coherence Magnetic impurities Other charges Magnons Spin Orbit Phonons Other spins Photons etc... Spin is often weakly coupled to other degrees of freedom Strasbourg, ESC 2012 p. 10

Spin coherence Magnetic impurities Other charges Magnons Spin Orbit Phonons Other spins Photons etc... Spin is often weakly coupled to other degrees of freedom Spin coherence time τ φ,s τ φ,o Example: τ φ,s > 100 ns in GaAs (Kikkawa&Awschalom, PRL 98) Quantum computing, Spin electronics,... [Coupling to mechanical degrees of freedom: R. Leturq, thursday] Strasbourg, ESC 2012 p. 10

Quantum conductance Landauer s approach Strasbourg, ESC 2012 p. 11

Conductance quantization Point contact D. A. Wharam et al. J. Phys. C 21, L209 (1988) B. J. van Wees et al. Phys. Rev. Lett. 60, 848 (1988) www.ilorentz.org Strasbourg, ESC 2012 p. 12

Conductance quantization Point contact D. A. Wharam et al. J. Phys. C 21, L209 (1988) B. J. van Wees et al. Phys. Rev. Lett. 60, 848 (1988) www.ilorentz.org steps of 2e 2 /h in the conductance! Strasbourg, ESC 2012 p. 12

Conductance and conductivity conductance conductivity Ohm s law I = GV j = σe sample material global local Strasbourg, ESC 2012 p. 13

Conductance and conductivity conductance conductivity Ohm s law I = GV j = σe sample material global local classical: L G = σ A L A Strasbourg, ESC 2012 p. 13

Scaling of the conductance G = σ A L A L d 1 1d G 1/L 2d G 3d G L Strasbourg, ESC 2012 p. 14

Non-locality of quantum transport R. A. Webb, S. Washburn, Physics Today, Dec. 1988 Strasbourg, ESC 2012 p. 15

Non-locality of quantum transport R. A. Webb, S. Washburn, Physics Today, Dec. 1988 quantum transport is non-local G σ A L Conductance has to be considered Strasbourg, ESC 2012 p. 15

Electrons in a perfect 1d wire L µ 1 µ 2 1d wavefunctions ψ k (x) exp(ikx) k > 0: forward propagation with v = E/ k Current carried by a state: I k = ev/l Strasbourg, ESC 2012 p. 16

Electrons in a perfect 1d wire L µ 1 µ 2 1d wavefunctions ψ k (x) exp(ikx) k > 0: forward propagation with v = E/ k Current carried by a state: I k = ev/l Density of states: ρ + = L/hv Total current carried by k > 0 states: I + = 0 de f µ1 I k ρ + Strasbourg, ESC 2012 p. 16

Quantum conductance of a perfect 1d wire L µ 1 µ 2 I + = e h 0 de f µ1 Strasbourg, ESC 2012 p. 17

Quantum conductance of a perfect 1d wire L µ 1 µ 2 I + = e h 0 de f µ1 I = e h 0 de f µ2 Strasbourg, ESC 2012 p. 17

Quantum conductance of a perfect 1d wire L µ 1 µ 2 I + = e h 0 de f µ1 I = e h 0 de f µ2 Net current: I = I + + I = e h 0 de (f µ1 f µ2 ) Strasbourg, ESC 2012 p. 17

Quantum conductance of a perfect 1d wire L µ 1 µ 2 I + = e h 0 de f µ1 I = e h 0 de f µ2 Net current: I = I + + I = e h 0 Low temperature f µ Θ(µ E): de (f µ1 f µ2 ) I = e h (µ 1 µ 2 ) }{{} ev = e2 h V E µ 1 µ 2 G = I V = e2 h k Strasbourg, ESC 2012 p. 17

Landauer: Conductance from scattering I 1 TI 1 RI1 RI 2 TI 2 µ 1 µ 2 I 2 Strasbourg, ESC 2012 p. 18

Landauer: Conductance from scattering I 1 TI 1 RI1 RI 2 TI 2 µ 1 µ 2 I 2 I = TI 1 TI 2 = e h T (µ 1 µ 2 ) }{{} ev Strasbourg, ESC 2012 p. 18

Landauer: Conductance from scattering I 1 TI 1 RI1 RI 2 TI 2 µ 1 µ 2 I 2 I = TI 1 TI 2 = e h T (µ 1 µ 2 ) }{{} ev two terminal conductance G 2t = e2 h T finite for T = 1! Strasbourg, ESC 2012 p. 18

Four terminal conductance [H.-L. Engquist, P. W. Anderson, Phys. Rev. B 24, 1151 (1981)] I 1 µ 1 µ 2 I 2 I A I B µ A voltage probes, no net current µ B Strasbourg, ESC 2012 p. 19

Four terminal conductance [H.-L. Engquist, P. W. Anderson, Phys. Rev. B 24, 1151 (1981)] I 1 µ 1 µ 2 I 2 I A I B µ A voltage probes, no net current four terminal conductance G 4t = µ B ei µ A µ B G 4t = e2 h T 1 T T 1: infinite conductance Strasbourg, ESC 2012 p. 19

Contact resistance difference between two and four terminal resistance: contact resistance ( 1 1 ) = h ( 1 G 2t G 4t e 2 T 1 T ) = h T e 2 Strasbourg, ESC 2012 p. 20

Contact resistance difference between two and four terminal resistance: contact resistance ( 1 1 ) = h ( 1 G 2t G 4t e 2 T 1 T ) = h T e 2 µ µ1 µ A µ B equilibration in the reservoirs leads to dissipation µ 2 Strasbourg, ESC 2012 p. 20

Electrons in a quasi-1d clean quantum wire H = 2 2m ( 2 x 2 + 2 y 2 ) +V (y) y x Strasbourg, ESC 2012 p. 21

Electrons in a quasi-1d clean quantum wire H = 2 2m ( 2 x 2 + 2 y 2 ) +V (y) y x separable! H = H x + H y ψ(x,y) χ n (y) exp(ikx) E n,k = E n + 2 k 2 2m Strasbourg, ESC 2012 p. 21

Electrons in a quasi-1d clean quantum wire H = 2 2m ( 2 x 2 + 2 y 2 ) +V (y) y x separable! H = H x + H y ψ(x,y) χ n (y) exp(ikx) E n,k = E n + 2 k 2 2m conduction channels conductance per open channel: e 2 /h E E F k Strasbourg, ESC 2012 p. 21

Conductance of a many-channel scatterer Generalization of the 2-terminal conductance: G 2t = e2 h n,m T n,m n T n,m m sum runs over occupied channels E n,e m < E F Strasbourg, ESC 2012 p. 22

Conductance of a many-channel scatterer Generalization of the 2-terminal conductance: G 2t = e2 h n,m T n,m n T n,m m sum runs over occupied channels E n,e m < E F Generalizations to finite voltage, temperature textbooks Strasbourg, ESC 2012 p. 22

Conductance quantization Point contacts: T n,m δ n,m G 2t e2 h N c N c : # of occupied channels van Wees et al. Phys. Rev. Lett 88 www.ilorentz.org Strasbourg, ESC 2012 p. 23

Summary Quantum transport: - quantum effects: low temperatures; small lengthscales relevant in nanoscale systems Strasbourg, ESC 2012 p. 24

Summary Quantum transport: - quantum effects: low temperatures; small lengthscales relevant in nanoscale systems - Landauer formalism: Quantum conductance transmission Strasbourg, ESC 2012 p. 24

Hall Effect Classical: Hall 1879 V x = IR V H = B en s I R H = V H I = B en s independent of B linear in B Drude Strasbourg, ESC 2012 p. 25

Hall Effect Classical: Hall 1879 V x = IR V H = B en s I strong magnetic field, clean sample: R H = V H I = B en s cyclotron orbits r c = mv F /eb; ω c = eb/m independent of B linear in B Drude Strasbourg, ESC 2012 p. 25

Hall Effect Classical: Hall 1879 V x = IR V H = B en s I strong magnetic field, clean sample: R H = V H I = B en s cyclotron orbits r c = mv F /eb; ω c = eb/m independent of B linear in B Drude Quantum mechanics (L φ > r c ): quantized motion Landau levels E = E 0 + (n + 1/2) ω c Strasbourg, ESC 2012 p. 25

Hall Effect Classical: Hall 1879 V x = IR independent of B V H = B en s I R H = V H I = B en s linear in B Drude strong magnetic field, clean sample: cyclotron orbits r c = mv F /eb; ω c = eb/m Quantum mechanics (L φ > r c ): quantized motion Landau levels E = E 0 + (n + 1/2) ω c DOS hω c degenerate N D = 2SB/φ 0 E Strasbourg, ESC 2012 p. 25

Quantum Hall Effect v. Klitzing et al. Phys. Rev. Lett. 80 more recent measurement from www.ptb.de Strasbourg, ESC 2012 p. 26

Quantum Hall Effect v. Klitzing et al. Phys. Rev. Lett. 80 E F between Landau levels: R x = 0 more recent measurement from www.ptb.de Strasbourg, ESC 2012 p. 26

Quantum Hall Effect v. Klitzing et al. Phys. Rev. Lett. 80 E F between Landau levels: R x = 0 more recent measurement from www.ptb.de R H = h e 2 1 n quantized! Application: Resistance standard Strasbourg, ESC 2012 p. 26

Quantum Hall Effect Strasbourg, ESC 2012 p. 27

Edge channels 2d, strong magnetic field skipping orbits edge channels (1 per Landau level) Strasbourg, ESC 2012 p. 28

Quantum Hall Effect C D A B Strasbourg, ESC 2012 p. 29

Quantum Hall Effect C D A B T n,m = δ n,m spatial separation of forward and backward channels Strasbourg, ESC 2012 p. 29

Quantum Hall Effect C A D B I = 2 e2 h N µ 1 µ 2 LL e V x = V A V B = 0 R x = V x /I = 0 T n,m = δ n,m spatial separation of forward and backward channels Strasbourg, ESC 2012 p. 29

Quantum Hall Effect C A D B I = 2 e2 h N µ 1 µ 2 LL e V x = V A V B = 0 R x = V x /I = 0 T n,m = δ n,m spatial separation of forward and backward channels V H = V A V C = (µ 1 µ 2 )/e R H = V H /I = h 1 2e 2 N LL Strasbourg, ESC 2012 p. 29