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

Transcription

1 & 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

2 Lecture plan Part I 1. Basics of Coulomb blockade and quantum conductance - Quantum tunneling and classical charging 2. From Anderson model to cotunneling - Schrieffer-Wolff transformation 3. Elastic vs. inelastic cotunneling - Bias spectroscopy 4. Exchange cotunneling and basic Kondo effect - Signatures of Kondo effect 5. Tunneling renormalization of cotunneling thresholds - Ferromagnetic leads, quasi-degenerate systems Part II 1. The nonequilibrium Kondo problem -What s the problem? 2. Poor man s scaling for nonequilibrium systems - Lineshapes for inelastic cotunneling? 3. The effect spin-orbit coupling - Source of bias-asymmetry and angular dependence of B-field Dias 2

3 Suggested literature 1. H. Bruus & K. Flensberg, Many-Body Quantum Theory in Condensed Matter Physics Oxford University Press (2004). 2. R. Hanson et al.: Spins in few electron quantum dots, Reviews of Modern Physics 79, 1217 (2007). 3. E. L. Wolf, Principles of Electron Tunneling Spectroscopy, Oxford University Press (1985). 4. J. Von Delft: Kondo effect in metals and quantum dots, lecture notes from The 4 th Windsor Summer School on Condensed Matter Theory, Available at 5. Articles cited along the way. Dias 3

4 Dias 4 J. Paaske, NBI

5 Molecular Transistor Realizations Bardeen, Brattain og Shockley, Bell Labs 1947 Present day Intel workhorse In 100 nm Single molecule (NBI/NSC) Carbon nanotube (Delft) E E C B B C 10 nm 1000 nm Dias 5

6 Basic (field effect) transistor setup? V Field Effect Transistor source gate drain V g I Current through the device (from source to drain) - turns on (Logical 1) - turns off (Logical 0) by adjusting electrical potential on the gate electrodes. Dias 6

7 Bias-spectroscopy of nanostructures s V source gate d drain V g I Dias 7 Coulomb diamonds

8 Typical nanostructures of interest and many more... Heterostructure quantum dot (GaAs/AlGaAs) Carbon nanotube Semiconductor Nano-wire C 60 Peapod Single cell Dias 8 Organic molecule Metal complex

9 Contacts of current interest Normal metal (Au) MATERIALS Ferromagnetic metal (Ni) Superconducting metal (Ti/Al, Pd/Nb) Various combinations: NDS, SDS, SDF, FDF, etc NANOGAPS Electron Beam Lithography Electromigration Mechanical break junctions New design! NSC 2 nm gap Au Nano-rods Dias 9

10 & NSC Charge conduction: Quantum tunneling + Classical charging Potential-landscape: Elctrostatic-landscape: R S R D C S C G C D source gate drain Source Drain Filled states V g V I Dias 10

11 & NSC The Harlequin diamond plot: Coulomb Blockade Chemical potential of dot or molecule: S D Plotting conductance as a function of and 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 Steady state current (sequential tunneling) 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 12 Current is flowing only when the level lies within the bias-window

13 Bias dependence & level broadening Steady state current: S D But where is the voltage in Ohm s law,? Tunneling 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 13

14 Summarizing: 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 14

15 Taking a closer look inside the diamonds Dias 15

16 Cotunneling: Lifting Coulomb blockade by quantum fluctuations Cotunneling rate (2.-order PT): Finite current: Spinful dot (odd occ.) ( -order PT): Charging cost: Kondo-effect : 0 N-1 N N+1 Dias 16

17 Inelastic Cotunneling: Bias spectroscopy Extra contribution to the current: Excited state spectroscopy!! Specific signatures: 0 spin-flip transitions (Kondo-sharpened!) vibrationally assisted transitions (sidebands!) Dias 17

18 Kondo effect ( ) Hamiltonian: J. Kondo, Prog. Theor. Phys. 32, 37 (1964) L. Glazman, M. Raikh, JETP Lett. 47, 452 (1988) T. K. Ng, P. A. Lee, Phys. Rev. Lett. 61, 1768 (1988) (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 18 Strong coupling regime: Landau Fermi Liquid Fixed Point [K.G. Wilson, 71; P. Nozières, 74]

19 Observing a Kondo peak... (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 19

20 Dot/lead-Hamiltonian (2 nd quantized many-body Hamiltonian) Single-orbital Anderson model S D Kondo-regime: Charge fluctuations are strongly suppressed! (Considered as a weak perturbation to Coulomb blockade) Dias 20

21 Projecting out charge-fluctuations ( odd) The Schrieffer-Wolff transformation Perform unitary transformation perturbatively: Construct so as to cancel the tunneling term : Satisfied with, where: Dias 21 J. R. Schrieffer, P. A. Wolff, Phys. Rev. 149, 491 (1966). P.-O. Löwdin, J. Chem. Phys. 19, 1396 (1951).

22 Effective exchange-cotunneling (Kondo) model Finishing the Schrieffer-Wolff transformation: With (exchange-)cotunneling amplitudes: (AFM exchange coupling) (Potential scattering) J. Appelbaum, Phys. Rev. Lett. 17, 91 (1966). P. W. Anderson, Phys. Rev. Lett. 17, 95 (1966). Dias 22

23 Cotunneling current (2nd order PT, finite B-field) Cotunneling-rates: for Nonequilibrium spin-occupation numbers: Dias 23

24 Cotunneling conductance (2nd, and 3rd order order PT, finite B-field) M. R. Wegewijs, Y. Nazarov, arxiv: cond-mat/ J. Paaske, A. Rosch, P. Wölfle, Phys. Rev. B 69, (2004). V. N. Golovach, D. Loss, Phys. Rev. B 69, (2004). Dias 24

25 & NSC Inelastic cotunneling (typical experiments) Goldhaber-Gordon [GaAs/AlGaAs] Zumbühl [GaAs/AlGaAs] Ralph [Charge-trap] Nygård [CNT] Cronenwet [GaAs/AlGaAs] Kogan [GaAs/AlGaAs] Babic [CNT] Schmid [GaAs/AlGaAs] Osorio [OPV5] Z z z z z z z z... z z z z z z z Z Osorio [Mn2+] Dias 25

26 J. Paaske, NBI Contacting a single molecule (Electromigration: gold wire) and a bit of chemistry... (Herre van der Zant et al., TU-Delft) Dias 26

27 The completed single-molecule transistor... Dias 27

28 OligoPhenyleneVenylene5 Chemical synthesis (Bjørnholm et al. NSC-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) The perfect void for inelastic cotunneling involving low-energy excitations! Compare: Molecule CNT-dot 100 mev 5 mev Vg (V) Dias 28

29 [Mn(terpy-O-(CH 2 ) 6 -SAc) 2 )] 2+ Chemical synthesis (Bjørnholm et al. NSC-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 2 m S=1/2 Low-Spin Dias 29

30 Spectroscopic fine-structure in carbon nanotubes: Tunneling renormalization Dias 30

31 Maria-Alm, Austria, January 2008 CNT Coulomb-blockade diamonds (bias-spectroscopy) Adding 285 electrons, one by one odd-occupied charge states with zero-bias Kondo peak. Dias 31

32 Maria-Alm, Austria, January 2008 The standard diamond Shell-filling Dias 32 H He Li Be B C N O F Ne Na Mg

33 Maria-Alm, Austria, January 2008 The standard diamond Elastic cotunneling (Kondo-peak) Inelastic cotunneling Dias 33

34 Dias 34 J. Paaske, NBI

35 Tunneling induced level-shifts in nanotube QD [Ni leads] Spin-polarized leads: N=1 (tunneling out) (tunneling in) N=2 N=0 Gate-dependent spin-splitting: ( ) (Bethe logarithms ) N=1 J. Martinek et al., Phys. Rev. Lett. 91, (2003). J. Martinek et al., Phys. Rev. Lett. 72, (R) (2005). M. Sindel et al., Phys. Rev. B 76, (2007). Dias 35

36 Dias 36 J. Paaske, NBI

37 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 37

38 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 38

39 Maria-Alm, Austria, January Gate-dependent excitation energy 7 Dias 39

40 Maria-Alm, Austria, January 2008 Tunneling-induced level shifts (2nd order PT) tunneling out tunneling in Energy of dot-state with i electrons in orbital 1 and j in orbital 2: ( V g ) Tunneling rate for orbital i=1,2 to lead =source, drain: Γ 1 Γ 2 Dias 40

41 Gate-dependent excitation energies Strong coupling sub-gap structure Unresolved?! Dias 41

42 Inelastic cotunneling in quantum dots and molecules with weakly broken degeneracies G. Begemann et al., Phys. Rev. B 82, (2010) Gate-dependent line-shapes Dias 42

43 Lecture plan Part I 1. Basics of Coulomb blockade and quantum conductance - Quantum tunneling and classical charging 2. From Anderson model to cotunneling - Schrieffer-Wolff transformation 3. Elastic vs. inelastic cotunneling - Bias spectroscopy 4. Exchange cotunneling and basic Kondo effect - Signatures of Kondo effect 5. Tunneling renormalization of cotunneling thresholds - Ferromagnetic leads, quasi-degenerate systems Part II 1. The nonequilibrium Kondo problem -What s the problem? 2. Poor man s scaling for nonequilibrium systems - Lineshapes for inelastic cotunneling? 3. The effect spin-orbit coupling - Source of bias-asymmetry and angular dependence of B-field Dias 43