Title Interactions in Quantum Dots: Kondo and Wigner Crystal Harold U. Baranger, Duke University Correlation: caused by electron-electron interaction extreme: crystalmore typical: liquid Quantum Dots: tunable puddle of electrons connected to a Fermi sea (leads) dot sea gate (tubable)

Vertical Qdots Vertical Quantum Dots Comp. Tech. [from Tarucha, Kouwenhoven, et al. Rep. Prog. Phys. 2001]

Lateral Qdots Lateral Quantum Dots Comp. Tech. Q.Dots Comp. Tech. [C. Marcus group, Harvard] [Craig, et al. ’04 (Marcus group), Harvard]

Outline OUTLINE Comp. Tech. I. Introduction II. Toward Strong Interactions in Circular Quantum Dots low density  stronger interactions signs of “Wigner crystallization” ordering, spin effects, violation of Hund’s rules, … quite different from in bulk 2D gas III. Spectroscopy of Kondo Box Kondo correlations between a small dot and a large dot are manifest in transport spectroscopy measurements Theorem for ground state spin, scaling, etc.

Parametrize interaction strength: ratio of interaction strength to kinetic energy CircDots: Intro Toward Strong Interactions in Circular Quantum Dots ( Amit Ghosal, Devrim Güçlü, Cyrus Umrigar, Denis Ullmo, and HUB) Comp. Tech. What happens when the density of an electron gas is decreased? Electrons crystallize! (Wigner, 1934) Bulk 2D: Intermediate r s : Complex phases? Chakravarty, et al. 99; Jamei, et al. 05; Waintal 05 (Tanatar and Ceperley, 1989)

Circular qdot Circular Quantum Dot Comp. Tech. How will such physics appear in a confined system? no phase transition in a finite system, but a crossover?? broken translational invariance  smaller ? mesoscopic fluctuations? Model: parabolic confinement – circular symmetry

VMC+DMC Tool: Variational and Diffusion Quantum Monte Carlo Comp. Tech. VMC followed by DMC with fixed node approximation Slater dets taken with single particle DFT orbitals includes near-degeneracy and dynamic correlation use DMC on optimized wavefunction to project out the ground state

Density Density of Electrons: n(r) Comp. Tech. rotational symmetry – sharp rings form! – dot is bigger for larger r s 3 rings as in classical limit

Density: FPH Evolution of Rings in n(r) Comp. Tech. Characterize the strength of the ring structure by the “fractional peak height” (FPH): Development of FPH is completely smooth – no threshold [Amit Ghosal]

Pair density 1 Pair Density Comp. Tech.

Pair density 2 Development of Radial Modulation Comp. Tech. [Amit Ghosal]

Addition energy Addition Energy: Coulomb Blockade Peak Spacing Comp. Tech. Shell effects Classical structure develops for small N  localization! [Amit Ghosal]

Q.Dots: Kondo Quantum Dots: Kondo Effect Comp. Tech. [Goldhaber-Gordon webpage, Stanford] Spin interacting with a Fermi sea: low T  entropy of spin is quenched by formation of a singlet with conduction electrons high T  free spin (paramagnet) Realize using 2 quantum dots: control! interference effects! (in both dots…) tunneling spectroscopy ! [with Denis Ullmo]

Kondo1 The Kondo Problem Comp. Tech. A localized spin interacting with a conduction band

Kondo: Spin flips Kondo Resonance: Spin Flips Comp. Tech. Initial State Virtual State Final State Density of States two types of virtual states: empty and doubly occupied constructive interference between the two transmission resonance at the Fermi level Transmission resonance at E F increase in conductance!

Spectr. of Kondo Spectroscopy of Kondo State: Large dot / small dot (R. Kaul, G. Zaránd, D. Ullmo, S. Chandrasekharan, HUB) Comp. Tech. At temperatures « Δ, physical properties dominated by the (many body) ground state and low energy spectra Ground state spin? Finite-size spectra? Parametric evolution of spectrum with J ? Effect of parity, randomness… ?

Template Small dot / Large dot System Comp. Tech. No Leads. Isolated R-S system. Kondo-coupling between R-S exact one-body states on R electrostatic energy on R

Template Exact Theorem: One-body basis Comp. Tech. Wilson `75

Template Exact Theorem: Many-body basis Comp. Tech.

Template Exact Theorem: Ground State Spin Comp. Tech. … Marshall’s Sign Theorem For fixed ground state is never degenerate! Ground state spin fixed in parametric evolution: Calculate ground state spin in perturbation theory! e.g. Auerbach `94

Spectr. S=1/2 (1) Finite Size Spectra: S small-dot =1/2, N odd, J>0 Comp. Tech. from GS theorem, for all J>0 Construct excited state spectrum from perturbation theory

Spectr. S=1/2 (2) Finite Size Spectra: S=1/2, N odd, J>0 Comp. Tech. weak coupling: expand in J Δ{ ~Δ Triplet: simply flip spin Higher states: a single-particle excitation is necessary

Spectr. S=1/2 (3) Finite Size Spectra: S=1/2, N odd, J>0 Comp. Tech. weak coupling: expand in J strong coupling: expand in U FL (Nozières `74) 2 nd excited state: S tot = 0, δΕ(2) – δΕ(1) ≈ 2V FL |φ a (0)| 2 |φ b (0)| 2 T k » Δ → Fermi liquid description - one e locked into a singlet with the impurity - local interactions between e :

Spectr. S=1/2 (4) Finite Size Spectra: S=1/2, N odd, J>0 Comp. Tech. weak coupling: expand in J strong coupling: expand in U FL For intermediate coupling, connect the two limits smoothly. [Ribhu Kaul]

Spectr. S=1 (1) Finite Size Spectra: S small-dot =1, N even, J>0 Comp. Tech. !!!! weak coupling: expand in J>0 strong coupling: expand in J F <0! Nozières & Blandin `80

Scr. vs Underscr. Compare Screened vs. Under-screened Kondo Screened S small-dot =1/2, J>0, N=odd Under-Screened S small-dot =1, J>0, N=even

QMC method QMC Method Comp. Tech. 1D fermions = XY spin chain efficient simulation in continous time: simulate spin chains using“directed loop” Wilson 75; Evertz et al. 93; Beard & Wiese 96; Syljuasen & Sandvik 02

QMC clean box QMC Results on a “Clean Box” Clean: P QMC (S z 2 =1) fit to simple singlet-triplet form: Scaling: [Ribhu Kaul]

Qdot Qimp? 2 Quantum Dots Are Quantum Impurities? Comp. Tech. Real Single Impurity Physics! V g (gate voltage) and Shape are tunable Random Confinement Energy Scales:  the mean level spacing E Th, the Thouless energy Quantum DotImpurity LeadsConduction Electrons BUT Quantum Dots are not Atoms!

MesoKondo2 Mesoscopic Kondo Comp. Tech. No mapping to clean problem New confinement energy scales Question: How universal in mesoscopic samples (a single scale)??

Conclusions CONCLUSIONS Quantum dots can be used to study e-e interactions in several interesting contexts  Circular Quantum Dots low density  stronger interactions interactions act on pre-existing modulation caused by interference smooth increase in modulation up to r s =20  Spectroscopy of Kondo Box can probe in much more detail the correlated Kondo state ground state spin fixed during any parametric evolution sketch spin and evolution of excited states from pert. theory confirmed by QMC calculation (clean box) some results for mesoscopic Kondo problem [R. Kaul, G. Zaránd, D. Ullmo, S. Chandrasekharan, HUB] PRL 96, (2006) [A. Ghosal, D. Güçlü, C. Umrigar, D. Ullmo, and HUB] Nature Physics 2, 336 (2006)

THE END

Template Title Comp. Tech.

Spectr. S=0 Finite Size Spectra: S=1/2, N even, J>0 Comp. Tech.

Angular plots Pair Density in Angular Direction Comp. Tech. Pair density along outer ring, with 1 electron fixed on outer ring r s = 0.41 r s = 4.7 r s = 14.7 angular modulation is weak Friedel oscillations for small to moderate r s intriguing “bump” at θ=π