J. Paaske, NBI. What s the problem? Jens Paaske, NBI Dias 1

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1 Nonequilibrium Quantum Transport What s the problem? Jens Paaske, NBI Dias 1

te ga dra ain Most important property: V g I Adjusting the potential on B, say, changes

2 Basic 3-terminal setup sou urce Three metallic electrodes:? V 1. Emitter (Source) 2. Base (Gate) 3. Collector (Drain)?te ga dra ain Most important property: V g I Adjusting the potential on B, say, changes the current from E to C. Field Effect Transistor Central question: What happens to the I/V characteristics when the central region becomes very small? Dias 2

3 Molecular Transistor Realizations Bardeen, Brattain og Shockley, Bell Labs 1947 Intels latest... In 100 nm One of our latest (NBI/NSC) Carbon nanotube (Delft) E E C B B C 10 nm 1000 nm Dias 3

or Resistance: Conductance: Conductance relates to material

4 Conductance of very small objects? Macroscopic metals, semiconductors, insulators Ohm s law: or Resistance: Conductance: Conductance relates to material details via conductivity : L A REDUCING SIZE Micro, or nano-scopic Quantum Dots Dias 4

5 Small objects? Three important lessons from quantum mechanics: 1. Discrete (quantized) energy levels (e.g. hydrogen atom, potential well, ) 2. Tunneling through barrier (Heisenberg uncertainty principle, ) 3. Electrons are Fermions Pauli exclusion principle: p There are no fermion states in which two or more particles share the same quantum numbers Example: Infinite square well Schrödinger equation for single particle in given potential: Eigenvalues: Smaller box Larger level-spacing! Dias 5

6 Charge conduction: Quantum Tunneling + Classical Charging Quantum tunneling: Rate of tunneling through potential barrier found from transition matrix element, i.e. Hamiltonian-overlap: Potential-landscape Reminder: Transmission through potential step L Source Drain Filled states (Pauli) 0 L x Dias 6

Free Fermi-gas model of metal electrodes For many practical purposes a metal is merely a

A metal ground state is a filled Fermisea, with electron energies distributed according

a continuum with a density of states DOS: From this, the number of particles is:

7 Free Fermi-gas model of metal electrodes For many practical purposes a metal is merely a system of free electrons. A metal ground state is a filled Fermisea, with electron energies distributed according to the Fermi-function: Energy of single electron in level N: For cm, : : Joule Much like a continuum with a density of states DOS: From this, the number of particles is: Pauli-principle Relating to the chemical potential,, via the Fermi-function. N electrons condensed in a Fermi-sea Real metals most often have constant DOS for Dias 7

8 Steady state current Consider a single quantum level of energy Occupations: S Source: D Drain: Dot: Tunneling rates: Steady state (nonequilibrium) occupation number of the level: Source and drain currents: Steady state current: Dias 8 Current is flowing only when the level lies within the bias-window

9 Bias dependence & level broadening Steady state current: S D But where is the voltage-drop appearing? [ ] Tunneling also broadens the quantum level and smears energy conservation:, Heisenberg! This changes the current to:, Conductance through a single level cannot exceed the conductance quantum: Dias 9

10 Charge conduction: Quantum Tunneling + Classical Charging Classical charging: Q: What is the energy cost for adding an electron to the dot? A: Consider the system as four conductors whose surfaces are equipotentials. The electrostatic energy of the dot is then: Elctrostatic-landscape R S R D With total capacitance: so ource C S gate CG C D dr rain The total charge on the dot is that at zero-potentials plus the induced charge: V g V I This results in a total electrostatic energy for dot with electrons: Background charge for neutral dot Dias 10

11 The diamond plot: Coulomb Blockade Chemical potential of dot: S Plotting conductance as a function of and D I 0 N-1, N N, N+1 Current thresholds: 0 I=0 N-1 N N+1 gives the slopes: for. C.B. C.B. C.B. C.B. C.B. Addition energy: Dias 11

12 Summary so far: I. Conductance through a single quantum level is limited by e 2 /h II. Current is blocked by Coulomb-repulsion except for special resonant values of V and V g. III. Varying V and V g leads to characteristic Coulomb-diamonds for the conductance: Single-electron transistor. InAs-wire based Quantum Dot, T. Sand Jespersen, NBI Dias 12

13 Nanostructures of current interest Heterostructure quantum dot (GaAs/AlGaAs) Carbon nanotube Semiconductor Nano-wire C 60 Peapod Single cell Dias 13 Organic molecule Metal complex

14 Contacts of current interest Normal metal (Au) MATERIAL LS F Ferromagnetic metal (Ni) Superconducting metal (Ti/Al, Pd/Nb) Various combinations: NDS, SDS, SDF, FDF, etc NANOGAPS New design! NSC 2 nm gap Electron Beam Lithography h Electromigration Mechanical break junctions Au Nano-rods Dias 14

Schönenberger) Single molecule devices NBI (J.

15 Central experimental collaborations Nano Junctions Quantum dots NBI (J. Nygård) Harvard/NBI (C. Marcus) Basel (C. Schönenberger) Single molecule devices NBI (J. Nygård) NSC (T. Bjørnholm) Delft (H. van der Zant) IBM Zürich (H. Riel) Dias 15

16 Actual physical systems involved Thomas Sand Jespersen Jesper Nygård Jonas Ralph Hauptmann Kasper Grove Rasmussen NBI Au-rods Thomas Bjørnholm & Co. (NSC) Nanotubes Wires Molecules Delft GaAs Harvard Basel Herre Van der Zant Charles M. Marcus Christian Schönenberger Dias 16

17 Bias-spectroscopy of nanostructures μ s V source gate μ d drain Coulomb diamonds V g I Dias 17

18 Taking a closer look inside the diamonds Dias 18

19 Cotunneling: Lifting Coulomb blockade by quantum fluctuations Cotunneling rate (2.-order PT): Finite current: Spinful dot (odd occ.) ( -order PT): Strong correlations! Charging cost: Kondo-effect : Universality! 0 N-1 N N+1 Dias 19

20 Inelastic Cotunneling: Bias spectroscopy Extra contribution to the current: Excited state spectroscopy!! Specific signatures: 0 spin-flip transitions (Kondo-sharpened!) vibrationally assisted transitions Dias 20

21 Cotunneling Live Tunneling amplitude from source to molecule: Electrostatic charging cost for 2 electrons: Tunneling amplitude from molecule to drain: Dias

22 Kondo Resonance: Basic Notions (Liang et al., Nature 2002) Nucleon Strong coupling: - Singlet (S=0) Binding energy T K ~4K Weak coupling: - Doublet (S=1/2) Quark Spin is screened when lowering temperature! Dias 22

(conduction (lead) electron s) Transition probability in 3 rd

(Poor man s scaling [PWAnderson, 64]): Universal scaling

Wiel, Science 2000) Interaction induced energy-scale!

23 Kondo effect: what s under the hood? Hamiltonian: (exchange amplitude J) (localized (dot) spin S) (conduction (lead) electron s) Transition probability in 3 rd order perturbation-theory: Perturbative Renormalization Group (Poor man s scaling [PWAnderson, 64]): Universal scaling curve: Integrate down to relevant energy-scale: ( Van der Wiel, Science 2000) Interaction induced energy-scale! Dias 23 Strong coupling regime: Landau Fermi Liquid Fixed Point [K.G. Wilson, 71; P. Nozières, 74]

The Nonequilibrium Kondo problem Current with

Paaske, J. Kroha, P. Wölfle, PRL90, 076804 (2003).

24 The Nonequilibrium Kondo problem Current with leading logarithmic singularities: A. Rosch, J. Paaske, J. Kroha, P. Wölfle, PRL90, (2003). Summing log-singularities requires RG with multiple energy-scales? Poor man s scaling generalized to flow of coupling-functions with multiple peaks! How to generalize Wilson s approach what is low energy? How to incorporate decoherence and relaxation? Dias 24

25 Overview of the quantum impurity problem Equilibrium systems Impurities in bulk materials (e.g. Fe impurities in Au, ) Thermalized to a single reservoir Nonequilibrium systems Impurities in tunnel-junctions Single atom/molecule-transistor Coupled to two reservoirs! Strong Coupling Numerical Renormalization Group (Wilson '74 - NOBEL PRIZE) Exact solution: Bethe Ansatz (Andrei, Tsvelik, ) Bosonization (CFT) Strong Coupling? Unsolved! Attempts Numerical Renormalization Group (F. Anders) Scattering State Bethe Ansatz (N. Andrei) Weak coupling Weak coupling Dias 25 Perturbation theory (Kondo, Appelbaum) Renormalized perturbation theory Poor man s scaling (Anderson, Zawadowski,Wölfle) Wegner Flow Equations [Kehrein] Functional Renormalization Group [Schoeller; JP, HS, PW] Renormalized perturbation theory Poor man's scaling [JP, AR, JK, PW]

26 Dias 26 J. Paaske, NBI

27 Different system, same effect Finite-bias Singlet-Triplet Kondo effect Dias 27

28 Spectroscopic fine-structure in carbon nanotubes Dias 28

29 Dias 29 J. Paaske, NBI

30 Tunneling induced level-shifts in nanotube QD [Ni leads] Spin-polarized leads: N=1 (tunneling out) (tunneling in) Gate-dependent spin-splitting: ( ) N=2 N=0 (Bethe logarithms ) Fingerprint for devices with broken symmetries! N=1 Dias 30

31 Dias 31 J. Paaske, NBI

32 Gate-dependent exchange-field (tunneling induced Lamb-shift ) Findings and prospects: Electrical spin-control (not via induction fields!) Allows for much faster switching (Spintronics) Extremely localized magnetic field of order 1T (even 70 Tesla!!!) Single electron spin control (Qubit initialization) Dias 32

33 Tunneling induced level-shifts in nanotube QD [Au leads] Different tunneling-amplitudes to different orbitals: N=1 N=1 N=2 N=0 N=2 N=0 N=1 N=1 Dias 33

34 17 21 Gate-dependent excitation energy 7 Dias 34

35 Gate-dependent excitation energies Strong coupling sub-gap structure Unresolved?! Dias 35

36 Charting unknown territory Discovering the Ω -, Brookhaven 1964 Dias 36

OligoPhenyleneVenylene5 Chemical synthesis (Bjørnholm et al. NSC-Copenhagen) Copenhagen) Low temperature bias-spectroscopy in electromigrated gold-junction (van der Zant et al.

37 OligoPhenyleneVenylene5 Chemical synthesis (Bjørnholm et al. NSC-Copenhagen) Copenhagen) Low temperature bias-spectroscopy in electromigrated gold-junction (van der Zant et al., TU-Delft) 17_megah_lockin.dat di/dv (ns) Vb (mv V) The perfect void for inelastic cotunneling involving low-energy excitations! Compare: Molecule CNT-dot 100 mev 5 mev Vg (V) Dias 37

38 [Mn(terpy-O-(CH 2 ) 6 -SAc) 2 )] 2+ Chemical synthesis (Bjørnholm et al. NSC-Copenhagen) Copenhagen) Low temperature bias-spectroscopy in electromigrated gold-junction (van der Zant et al., TU-Delft) S=1/2 S=1 Al 2 O 3 gate S=5/2 S=0 AuPd Au SiO 2 Electrical Spin Control! N=5 N=6 S=5/2 High-Spin AuPd S=1/2 2 μm Low-Spin Dias 38

39 Doing the calculations Be very careful about forrmalism! In phisics, find small parameter and expiand Lev Gor kov, 1996 Consider a spherical cow of radius R NBI (K. Flensberg) NBI (B.M. Andersen) NBI (P. Hedegård) NBIA (V. Körting) NBI (K.G.L. Pedersen) Karlsruhe (P. Wölfle) Köln (A. Rosch) Bonn (J. Kroha) Jülich (T. Costi) Jülich (M. Wegewijs) Dias 39

40 Possible links with other NBI activities? (A Nano Theory viewpoint) J. Paaske, NBI Dynamical renormalization group Nonlinear structure formation in dark matter Dark Cosmology Center Boltzmann equation with collisionless dynamics, Dynamical friction, i Violent relaxation Kramers-Chandrasekhar equation Dynamics in reaction coordinate, Dynamical fluctuations in solvent Quark gluon plasma dynamics High Energy Heavy Ion Nonequilibrium quantum transport Nano Theory Molecular reaction dynamics Center for Molecular Movies Quantum information storage and manipulation Quantum optics Coherent spin-control Transport in optical lattices Dias 40

Cotunneling and Kondo effect in quantum dots. Part I/II

& NSC Cotunneling and Kondo effect in quantum dots Part I/II Jens Paaske The Niels Bohr Institute & Nano-Science Center Bad Honnef, September, 2010 Dias 1 Lecture plan Part I 1. Basics of Coulomb blockade