>1mm. Macroscopic. Mesoscopic. < 1nm. Atomic-size. Ohm s law UCF. Strong Quantum limit

Transcripción

1 W >1mm Macroscopic I L V G W 2 Ohm s law / L L,W < 1mm Mesoscopic G UCF e 2 / h < 1nm Atomic-size G 2e 2 / h Strong Quantum limit

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3 Electrons mesoscopic metallic wires l + e L e-ph e-e e-mag. Mesoscopic regime L, W L ~ 1 T ~ 1K L

4 Diffusive and ballistic mesoscopic conductors lf L, W L L diffusive l F << << W,L W ballistic l F << W,L < ~ 30 nm (metallic wires made by nanolitography)

5 G/G ( ) Aharonov-Bohm effect B r P A A transm. 2 P P 2Re A A B (h/e)/( r 2 ) B (T) e A exp i kfl A. dx e A A exp i kf L L A. dx G L r exp 2 e / h r L B r F. Pierre and N.O. Birge, PRL 86, (2002) 2

6 Original experiment: Webb et al T=10mK h/e (AB) diameter=0.8µm Webb et al., PRL 54, 2696 (1985)

7 Heteroestructuras semiconductoras: Resonant tunneling 10 nm GaAs AlxGa1-x As Prediction: Tsu & Esaki, 1973 Experimental test: Sollner et al. 1983

8 Gas de electrones bidimensional (2DEG)

9 Gas de electrones bidimensional (2DEG) Silicon MOSFET (Metal Oxide Semiconductor Field Effect Transistor) 13 2 n 10 cm Heterouniones semiconductoras 11 2 n 10 cm e 1 10 mm L ( T 1K ) lf nm Superficie de Helio líquido 9 2 n 10 cm

10 Van Wees et al PRL 1988 Quantum point contact V SD I V cpq AlGaAs GaAs 2DEG V cpq [V]

11 Imaging current flow in Quantum point contact M. A. Topinka et al. Science (2000)

12 Aharonov-Bohm effect in 2DEG W. G. van der Wiel et al. 2003

13 El efecto Hall cuántico (entero) h H R H ie i 1,2,3,... 2 Resistencia Hall von Klitzing et al., 1980

14 Puntos cuánticos GaAs 10 nm Al xga1-x As

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16 Coulomb blockade Kondo effect ~ E C Goldhaber-Gordon et al Nature (1998) Van der Wiel et al Nature (2000)

17 Atomic-size contacts Contact formation with STM Nanofabricated break-junctions Mechanically controlable break-junctions (MCBJ) Review: N. Agraït, ALY and J.M. van Ruitenbeek Phys. Rep. 2003

18 Conductance steps one atom limit atomic chains

19 Curious properties macroscopic 1 10 A/ mm 2 maximal current densities atomic-sized 10 8 A/ mm 2! poor conductors become better conductors: macroscopic scale Au is better conductor than Pb atomic-size scale Pb is better conductor than Au

20 Gold Conductance histograms

21 Transport through a single molecule Smit et al., Nature 419, 906 (2002)

22 Otros fenómenos de interés: - Fluctuaciones de corriente - Superconductividad

23 Fluctuaciones de corriente V mesoscopic conductor I(t) I(t) noise Caracterización básica: espectro de fluctuaciones de corriente i t S(, V ) e I ( t) I (0) I (0) I ( t) I ( t) I( t) I( t) t T 0 S(, V) 0 Shot noise!

24 Full counting statistics P t (N) probability of transferring N electrons in t (measuring time) Example : tunnel junction P (N) asymmetric P t (N) N red: exact (Poisson) blue: Gaussian fit Prediction theory FCS : non gaussian character of noise depends on conductor and on interactions

25 Superconductividad Kamerlingh Onnes (Leiden ) Gran avance en bajas temperaturas: He líquido (T = 4.2 K) Descubrimiento de la superconductividad.

26 1.- Resistividad cero Experimento original: mercurio (Hg) de gran pureza Temperatura Crítica

27 2.- Efecto Meissner

28 3.- Calor específico

29 4.- Cuantización del flujo h 0 0 2e h e

30 Basic microscopic theory BCS (1957) Electrons in a lattice Cooper Pair

31 Energy spectrum in a superconductor ground state Cooper pair condensate low-energy excitations breaking a Cooper pair E min =2 normal empty states supercond. Fermi level 2 Gap occupied states

32 Superconducting tunnel junctions * Unión tunel * S-I-N S-I-S

33 Mesoscopic SNS junctions Josephson effect Dubos et al., PRL 86, (2001)

34 di/dv (normalized) Unión N-I-S mediante STM I W V 3 data fridge 40mK theory T=60mK theory T=65mK theory T=70mK 3 Al T eff ~ 65 mk on this curve Resolution ~ 15 µev V (µv) best STM resolution to date H. Le Sueur et al. (2007) 0

35 Transition from tunnel to contact Pb (sample) Pt-Rh (tip) N. Agraït et al. (1992)

36 G(V) Andreev Reflection ev/ Normal metal Superconductor S <Ψ h (r)ψ i (r)> N

37 I [ na ] Transporte superconductor en contactos atómicos Superconductivity in Al one-atom 2 /3 2 Al Pb Nb etc Tc(Al) = 1.14 K Tc(Pb) = 7.22 K Tc(Nb) = 9.3 K 30 T = 17 mk 20 I S V [ µv ]

38 Trends : graphene and carbon nanostructures

39 Novoselov et al., Nature 438, 197 (2005)

40 Graphene: peculiar band structure + K K Dirac s equation Linear dispersion relation

41 Superconductors Proximity effect Andreev transport Josephson effect Dirac fermions, Klein tunneling Quantum dot Physics Spin-orbit Interactions

42 V SD (mv) G Diff (e 2 /h) Hybrid nanostructures based on graphene and CNTs SWCNT + S leads Graphene + S leads V 4.64 V V V BG (V) 0.0 Eichler et al. Phys. Rev. Lett. (2007) Heersche et al. Nature (2007)

43 Andreev bound states in supercurrent-carrying carbon nanotubes revealed J.D. Pillet, C. H. L. Quay, C. Bena, A. Levy Yeyati and P. Joyez Nature Phys. 6 (2010)

44 Fitting the experimental data: gate voltage dependence Exp. Model

45 Programa Clase I: Introducción: Sistemas y fenómenes de interés. Imagen de Landauer del transporte cuántico. Clase II: Formalismo de scattering. Caso multicanal y multiterminal. Fluctuaciones de corriente en formalismo de scattering. Clase III: Formalismo de scattering en sistemas superconductores. Reflexiones de Andreev y estados ligados en uniones SNS. Clase IV: Introducción a las técnicas de funciones de Green. Funciones de Green en modelos continuos y discretos. Clase V: Técnicas de funciones de Green de no-equilibrio Clase VI: Aplicación a diversos ejemplos: el contacto puntual normal Clase VII: Técnicas de funciones de Green en sistemas superconductores Clase VIII: Temas avanzados: reflexión de Andreev en grafeno y/o FCS. Problemas: página web

46 Bibliografía S. Datta, Electronic transport in mesoscopic systems, Cambridge New York, Y. Nazarov & Y. Blanter, Quantum transport: introduction to nanoscience, Cambridge University Press, H. Bruus & K. Flensberg, Many-body Quantum theory in Condensed Matter Physics, Oxford University Press, A. M. Zagoskin, Quantum theory of many-body systems, Springer-Verlag, New York, N. Agraït, A. Levy Yeyati & J.M. van Ruitenbeek, Quantum properties of atomic-size conductors, Physics Reports 377, 81 (2003).