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Lectures: Condensed Matter II 1 – Electronic Transport in Quantum Dots 2 – Kondo Effect: Intro/Theory

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Lectures: Condensed Matter II 1 – Electronic Transport in Quantum Dots 2 – Kondo Effect: Intro/Theory Lectures: Condensed Matter II 1 – Electronic Transport in Quantum dots 2 – Kondo effect: Intro/theory. 3 – Kondo effect in nanostructures Luis Dias – UT/ORNL Lectures: Condensed Matter II 1 – Electronic Transport in Quantum dots 2 – Kondo effect: Intro/theory. 3 – Kondo effect in nanostructures Luis Dias – UT/ORNL Lecture 3: Outline Quantum Dots: brief review. Kondo effect: Review. Kondo effect in quantum dots. Kondo effect in Single Molecule Transistors. Kondo effect in Surfaces (STM, “quantum mirage”). Kondo effect in carbon nanotubes. “More is Different” “ The behavior of large and complex aggregates of elementary particles, it turns out, is not to be understood in terms of simple extrapolation of the properties of a few particles. Instead, at each level of complexity entirely new properties appear and the understanding of the new behaviors requires research which I think is as fundamental in its nature as any other.“ Phillip W. Anderson, “More is Different” , Science 177 393 (1972) Can you make “atoms” out of atoms? Electrostatically confine electrons within an small (nanometer-size) region. 2D Electron gas “Quantum dot” Tunnel energy barrier levels s from Charlie Marcus’ Lab website (marcuslab.harvard.edu) Making “artificial atoms”(?) out of atoms ATOM E conduction Con Many Atoms! fine i n 2D! 4p EF Band gap 3d ~eV Discrete levels valence: we might be tempted to call 4s 2D Electron gas them “artificialCRYSTAL atoms”. (3D) k EF E BUT: different energy/length scales! U (charging) Strong charging (U) → Many-body correlations density of density states 2 D ~ meV E QD Confine in 0D lowest subband 0 D Quantum dot! What are Quantum Dots ? Semiconductor Quantum Dots: Devices in which electrons are confined in nanometer size volumes. Sometimes referred to as “artificial atoms”. “Quantum dot” is a generic label: lithographic QDs , self- assembled QDs, colloidal QDs have different properties. Coulomb Blockade in Quantum Dots Ec Ec Even N Odd N conductance Vgate Coulomb Blockade in Quantum Dots Vgate Coulomb Blockade in Quantum Dots Ec Ec Ec Even N Odd N Even N conductance Vgate Coulomb Blockade in Quantum Dots Vgate “Coulomb Diamonds” (Stability Diagram) eV sd eV gate Coulomb Blockade in Quantum Dots L. P. Kouwenhoven et al. Science 278 1788 (1996). Kondo effect M.P. Sarachik et al Phys. Rev. 135 A1041 (1964). ρ/ρ 4.2K Mo .2 Nb .8 1% Fe Mo .7 Nb .3 ξξ Mo .8 Nb .2 K ~~ vv F/k/k BTTK Mo .9 Nb .1 T ( oK) CharacteristicResistivity decreasesincreases energywith withscale: the ρ(T) Kondodecreasing temperature T (usual)(KondoT Keffect) Kondo problem: s-d Hamiltonian Kondo problem: s-wave coupling with spin impurity (s-d model): Conduction band E filled EF Metal (non magnetic, s-band) Magnetic impurity (unfilled d-level) Kondo’s explanation for T min (1964) k T Perturbation theory in J3: spin∝ 2 − ρ B Rimp J1 4 J 0 log Kondo calculated the D conductivity in the linear k T response regime R() T= aT5 − c R log B tot imp imp D R D 1/5 = imp 1/5 Tmin c imp 5ak B Only one free paramenter: the Kondo temperature TK Temperature at which the perturbative expansion − ρ diverges. 1 2 J 0 kB T K ~ De A little bit of Kondo history: Early ‘30s : Resistance minimum in some metals Early ‘50s : theoretical work on impurities in metals “Virtual Bound States” (Friedel) Kenneth G. Wilson – Physics Nobel Prize in 1982 1961: Anderson"for his theory model for critical for magnetic phenomena inimpurities connection in metals with phase transitions" 1964: s-d model and Kondo solution (PT) 1970: Anderson “Poor’s man scaling” 19741974 --75:75: WilsonWilson ’’ss NumericalNumerical RenormalizationRenormalization GroupGroup (non(non PT)PT) 1980 : Andrei and Wiegmann’s exact solution Renormalization Procedure Iterative numerical solution. Renormalize by Λ1/2. Keep low energy states. γγ γγ γγξξξξξξ JJ 1 2 3N ... ... γγ ξξ ΛΛ-n/2 n~~ n HHN HHN+1 Anderson Model ed+U εF t with ed D “Quantum dot language” ed: energy level ed: position of the level (V g) U: Coulomb repulsion U: Charging energy eF: Fermi energy in the eF: Fermi energy in the metal leads t: Hybridization t: dot-lead tunneling D: bandwidth D: bandwidth NRG on Anderson model: LDOS γ γ γ ε t 1 2 3 1+U 1 ... ε γ Λ-n/2 1 n~ Single- particle peaks at ε and ε +U. d d ~T Many- body peak at K the Fermi energy: Kondo resonance Γ Γ (width ~T K). NRG: good resolution at low ω (log discretization). ε ε d d+ U History of Kondo Phenomena Observed in the ‘30s Explained in the ‘60s Numerically Calculated in the ‘70s (NRG) Exactly solved in the ‘80s (Bethe-Ansatz) So, what’s new about it? Kondo correlations observed in many different set ups: Transport in quantum dots, quantum wires, etc STM measurements of magnetic structures on metallic surfaces (e.g., single atoms, molecules. “Quantum mirage”) ... History of Kondo Phenomena Observed in the ‘30s Explained in the ‘60s Numerically Calculated in the ‘70s (NRG) Exactly solved in the ‘80s (Bethe-Ansatz) So, what’s new about it? Kondo correlations observed in many different set ups: Transport in quantum dots , quantum wires, etc STM measurements of magnetic structures on metallic surfaces (e.g., single atoms, molecules. “Quantum mirage”) ... Kondo Effect in Quantum Dots Kowenhoven and Glazman Physics World – Jan. 2001. Coulomb Blockade in Quantum Dots Ec Ec Even N Odd N conductance Vg Coulomb Blockade in Quantum Dots Vg Coulomb Blockade in Quantum Dots Ec Ec Ec Even N Odd N Even N conductance Vg Coulomb Blockade in Quantum Dots Vg Y. Alhassid Rev. Mod. Phys. 72 895 (2000). Kondo Effect in Quantum Dots D. Goldhaber-Gordon et al ~E c ~E c Nature 391 156 (1998) Even N Odd N ΓΓΓ 2 ETcK=e DOS/2C Conductance (G) Conductance Vg •T>T K: Coulomb blockade ( low G ) •T<T K: Kondo singlet formation •Kondo resonance at E F (width T K). •New conduction channel at E F: Zero-bias enhancement of G Vg Kondo effect in Quantum Dots D. Goldhaber-Gordon et al. Nature 391 156 (1998) Semiconductor Quantum Dots: Allow for systematic and controllable investigations of the Kondo effect. QD in N odd Coulomb Blockade 25mk valley: realization of the Kondo regime of the Anderson impurity problem. 1K Also in: Kowenhoven and Glazman Physics World , (2001). Kondo Effect in CB-QDs 25mk 1K NODD valley: Conductance rises for low T (Kondo effect) Kondo Temperature T k : only scaling parameter (~0.5K, depends on V g) Kowenhoven and Glazman Physics World – Jan. 2001. From: Goldhaber-Gordon et al. Nature 391 156 (1998) Examples of “fundamental” physics (in the “More is Different” sense) we can learn from quantum dots: Strongly correlated effects : Charging effects due to electron-electron interactions are dominant in QDs. Quantum effects (spin, tunneling, discreet energy levels, interference) probed in a very controllable way. Quantum Many-body physics: Kondo effect , Quantum phase transitions, … Kondo Effect in Double QDs “Side dot” configuration “Parallel” configuration Chen, Chang, Melloch, PRL 92 176801 (2004) R. Potok et al. Nature 446 167 (2007). Tunability of intradot and interdot parameters (couplings, gate voltage). Prospects for experimental probe of many-body phenomena, e.g: - SU(4) Kondo, RKKY interactions,... - Non-Fermi liquid physics (2-ch Kondo) - Quantum phase transitions. Craig et al., Science 304 565 (2004) Two-channel Kondo effect. Spin 1/2 coupled to two J1 … independent bands: 2- channel Kondo model … (“overscreened”). J2 CJ Bolech,R. Potok N etAndrei al. Nature PRL 88446237206167 (2007). (2002) Non- Fermi- liquid (NFL) behavior for J1=J2. Impurity entropy (NFL): √ Simp = k B log( 2) (NRG, Bethe ansatz). NFL fixed point Recent expts in q- dots. Kondo effect in Single Molecule Transistors C60 Single molecule transistors: C 60 molecules “caught” between electrodes (break junction). Zero-bias peak as a function of gate voltage: correct Kondo scaling. Correct behavior vs. Bias. T >50K . T=18K T=50K K Yu, Natelson, NanoLett. 4 79 (2004). Transport in molecular junctions. Coulomb blockade effects . Features consistent with (mv) sd vibrational modes in dI/dV. V Kondo signatures. S D (mv) sd V ε d=-U/2 (phs) Yu & Natelson, Nano. Lett. 4 79 (2003) Vg (V) Park et al. Nature 407 57 (2000) Connection of vibrational modes with 2-channel Kondo: LDS, E. Dagotto – PRB 79 155302 (2009) ; arXiv:0902.3225. Kondo effect in Single Molecule Transistors Pasupathy et al., Science 306 86 (2004). C60 Similar expts (D. Ralph’s group). Supression of the Kondo resonance in the presence of a magnetic field (top left, black curve, B=10T) and magnetic leads (top right, parallel [green] and antiparallel From Dan Ralph’s webpage: [blue] magnetizations). http://people.ccmr.cornell.edu/~ralph/ Kondo effect in surfaces (STM images). Manoharan et al., Nature 403 512 (2000). t´c td Magnetic (Co, Fe) atoms on metallic surfaces ! Right ingredients for Kondo. In this case, Kondo is marked by a dip at zero-bias conductance (dI/dV at V=0). Kondo effect surfaces: STM measurements. t´c td STM atomic manipulation: can build local structures (“quantum corrals ”). Elliptical shape: imaging (top) and dI/dV measurements (bottom). Cobalt atoms on Cu(111) shown. Manoharan et al., Nature 403 512 (2000). Kondo effect surfaces: STM measurements. Manoharan et al., Nature 403 512 (2000). t´c td One extra atom placed in one foci: a peak in the dIdV appears in the other focus athough NO ATOM is there! (“quantum mirage”). Theory: “focusing” of Kondo-scattered surface electrons*. *Schiller and Agam, PRL 86 484 (2001).. Magnetic impurities on metallic surfaces: Kondo + STM Kondo/Fano phenomenology Co quite generic: Early experiments: Co on Au(111) Madhavan et al., Science 280 567 (1998).
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