Physique de la Boite Quantique: Blocage de Coulomb et ... · Blocage de Coulomb et Effect Kondo Harold U. Baranger, Duke University . Outline-Summary OUTLINE Comp. Tech. The Kondo

1.  Coulomb Blockade in Quantum Dots • conductance through a nearly isolated system

2.  Kondo Effect: the basics, with quantum dots in mind • localized, doubly degenerate level interacting w/ continuum

3.  Kondo Effect in Quantum Dots – selected topics

Title Physique de la Boite Quantique:

Blocage de Coulomb et Effect Kondo Harold U. Baranger, Duke University

Outline-Summary OUTLINE

Comp. Tech.

The Kondo Effect in Quantum Dots

Things not covered: multi-level dots, nearly open dots, multiple dots, nonlinear regime, singlet-triplet Kondo, noise, …

  Spectroscopy of a Kondo Box • can probe in much more detail the correlated Kondo state • spin and evolution of excited states from pert. theory & QMC • tune through weak-coupling to strong-coupling cross-over

  Mesoscopic Kondo Problem• realization-dependent TK works as scaling parameter in high-T regime (from modified poor-man’s scaling) • low-T: non-universal! (theory?)

  SU(4) Kondo in Carbon Nanotubes (G. Finkelstein group)• seen with 1, 2, and 3 electrons above a closed shell • smooth shape when dot is open

Kondo Qdot Expt 1 Kondo Effect in Conductance of Quantum Dots

Comp. Tech.

[L. Kouwenhoven group, Delft]

lowest T

highest T

Behavior in “odd” valleys contradicts Coloumb blockade! [Kouwenhoven group]

Qdot Qimp? 1 Kondo Effect in Conductance of Quantum Dots: Theory

Comp. Tech.

[Glazman and Raikh; Ng and Lee]

Ng

Nonlin Expt Nonlinear I-V in the Kondo Regime

Comp. Tech.

[Kouwenhoven group,`98]

Qdot Qimp? 2 Quantum Dots Are Quantum Impurities??

Comp. Tech.

•  Can look at a single dot: real Single Impurity physics! •  Vg (gate voltage) and shape are tunable •  Random Confinement •  Energy Scales: Δ, the mean level spacing; ETh, the Thouless energy

Quantum Dot Impurity Leads Conduction Electrons

BUT Quantum Dots are not Atoms!

Previous work concerning Δ: Thimm, Kroha, von Delft (99); Simon & Affleck (02); Cornaglia & Balseiro (03)

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… ?

Tool5: non-lin. Coulomb Blockade: Non-linear Transport

Apply a bias voltage between the two leads connected to the dot:

dot L lead R lead

tunneling spectroscopy of individual quantum states

Small/Large dot Small dot / Large dot System

Comp. Tech.

No Leads.Isolated R-S system.

Kondo-couplingbetween R-S

exact one-bodystates on R

electrostaticenergy on R

Comp. Tech.

Thm: g.s. spin

Theorem: Ground State Spin

Comp. Tech.

Mattis;Marshallʼs 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’s book `94

Wilson `75

Thm: 1-body Exact Theorem: One-body basis

Comp. Tech.

Wilson `75

Thm: many-body Exact Theorem: Many-body basis

Comp. Tech.

… Marshallʼs Sign Theorem

For fixed ground state is never degenerate!

e.g. Auerbach `94

Spectr. S=1/2 (1) Finite Size Spectra: Ssmall-dot=1/2, N odd, J>0

Comp. Tech. Comp. Tech. from GS theorem, for all J>0

Construct excited state spectrum fromperturbation theory

Spectr. S=1/2 (2) Finite Size Spectra: S=1/2, N odd, J>0

weak coupling: expand in J

Δ{

Comp. Tech. ~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 UFL

(Nozières `74)

2nd excited state: Stot = 0, δΕ(2) – δΕ(1) ≈ 2VFL |φa(0)|2 |φb(0)|2

Tk » 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 UFL

For intermediate coupling, connect the two limits smoothly.

[Ribhu Kaul]

Spectr. S=1 (1) Finite Size Spectra: Ssmall-dot=1, N even, J>0

Comp. Tech.

!!!!

weak coupling: expand in J>0

strong coupling: expand in JF<0!

Nozières & Blandin `80

Scr. vs Underscr. Compare Screened vs. Under-screened Kondo

Screened Ssmall-dot=1/2, J>0, N=odd

Under-Screened Ssmall-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 Quantum Monte Carlo Results on a “Clean Box”

Clean:

Scaling:

[Ribhu Kaul]

MesoKondo2

Comp. Tech.

•  No mapping to clean problem •  New confinement energy scales

Question: How universal in mesoscopic samples (a single scale)??

Mesoscopic Kondo(R. Kaul, J. Yoo, D. Ullmo, S. Chandrasekharan, HUB)

MesoKondo1 Case 2: Fluctuations in the Fermi Sea

Comp. Tech.

What would be the effect of the mesoscopic fluctuations of the electron sea on the Kondo physics ? e.g.: renormalization scheme, scaling law, etc …

2-dot system Impurity in a dot

•  Chaotic quantum dots → RMT model (realizations = random choice of energies and

wavefunctions according to RMT statistics)

•  Integrable quantum dots –  rectangular billiard –  circular billiard (realizations = position of the impurity, chemical potential)

Quantum Dot Models

Kondo1 Magnetic Susceptibility

Comp. Tech.

QMC Data Quantum Monte Carlo Calculations

Comp. Tech.

[Ribhu Kaul]

Exact density of states:

Finite temperature formalism: introduces smoothed local density of states

[Zaránd and Uvardi]

Poor Manʼs Scaling (1-loop RG) with Mesoscopic Flucts.

Realization-Dependent Kondo Temperature

€

δTKTK0 ≈

1ρ0

δρsm (ω)dωωTK

0

D

∫

•  Careful: even in the bulk is a poor approximation quantitatively

•  However, we only wish to describe fluctuations, so

symbols → QMC

lines → f(T/T )

T = T [ρ (ω)]

k

sm k k

Comparison with QMC: RMT Model

Shows that the realization-dependent TK is the scaling parameter in the universal functions for T>TK

0

(strong sense of Kondo temperature)

Mesoscopic fluctuations of the local density of states related to closed classical trajectories:

Relation to Classical Dynamics

→ “analytical continuation” of classical dynamics (basically dS/dE = t)

P (t) = classical probability of return cl

sum rule (conservation of classical probability) :

Relation to Classical Dynamics: Kondo Temperature

(diagonal approximation)

Square Circle

Comparison with QMC: Integrable Quantum Dots

QMC Data Quantum Monte Carlo Calculations

Comp. Tech.

[Ribhu Kaul]

SU(4) Kondo Effect in Carbon Nanotube Quantum Dots: Kondo Effect without Charge Quantization

Gleb Finkelstein

Duke University Experiment: Alex Makarovski

NRG: F. Anders, M. Galpin and D. Logan

Thanks: H. Baranger, L. Glazman, K. LeHur, J. Liu, G. Martins, K. Matveev, E. Novais, M. Pustilnik, D. Ullmo

Support: NSF DMR-0239748

Nanotubes: metallic and semiconducting

E

k k- quasi-momentum along the length of the nanotube

metal

• Metallic or semiconducting • Two degenerate bands

semiconductor

k

C= 2πR

E

k

k

Degenerate orbitals: ‘shells’

Two degenerate subbands

k

E

=> degenerate orbitals

k

E

shell

W.J. Liang, M. Bockrath, and H. Park, PRL (2002) M. R. Buitelaar et al., PRL (2002)

Quantization along length

Samples

doped Si

SiO2

nanotube

Vgate Vsource-drain

A

Single-wall CNT ~2 nm in diameter

Measurement: differential conductance = dI/dV (Vgate)

Groups of 4 peaks: orbital degeneracy

Energy gap

Tunneling grows with Vgate

Quantum Dot

Leads and island formed within the same nanotube

4 degenerate levels (↑, ↓, ↑, ↓) in the dot are coupled one-to-one to 4 modes (↑, ↓, ↑, ↓) in the leads

The modes are not mixed by tunneling; t amplitude does not depend on α or σ Hamiltonian has SU(4) symmetry

two leads ν = L, R

spin σ, orbital α N number of electrons

SU(4) tunneling Hamiltonian

Kondo effect in Nanotubes

G

Vgate 1e 2e 3e

SU(4) theories for 1 electron: Double dots, dots with symmetries: D. Boese et al., PRB (2002) L. Borda et al., PRL (2003) K. Le Hur and P. Simon, PRB (2003) G. Zarand et al., SSC (2003) W. Izumida et al., J.P.Soc.Jpn. (1998) A. Levy Yeyati et al., PRL (1999) Nanotubes: M.S. Choi et al., PRL (2005)

1 electron SU(4) Kondo experiment: Quantum dots: S. Sasaki et al., PRL (2004) Nanotubes: P. Jarillo-Herrero et al., Nature (2005)

Kondo effect in Nanotubes

G

Vgate 1e 2e 3e

2 electron SU(4) theory M.R. Galpin, D.E. Logan and H.R. Krishnamurthy, PRL (2005) C.A. Busser and G.B. Martins, PRB (2007)

2-e Kondo in nanotubes – triplet or SU(4) W.J. Liang, M. Bockrath and H. Park, PRL (2002) B. Babic, T. Kontos and C. Schonenberger, PRB (2004)

Interactions ↑↓ = ↑↓ = ↑↓ Exchange ↑↑ is small

Kondo effect in Nanotubes

G

Vgate 1e 2e 3e

Friedel sum rule: Σ δi = πN Kondo singlet: δ↑= δ↓= δ↑= δ↓

1e δi = π/4 2e δi = π/2

G(2e) = 2G(1e)

G=G0 Σ sin2(δi )

Temperature dependence of conductance

Growth of the signal in the valleys due to the Kondo effect

Ec, Δ ~ 100K

1 2

3 1

2 3

Spectroscopy at finite bias

E

Vgate

ΓL >> ΓR

Tunneling from the weakly coupled lead probes the density of states in the Kondo / Mixed Valence system 1e 2e 3e

Tunneling density of states

EF

2.25e2/h 0 Width of the resonance: 10 K ~ T0

T = 3.3 K

Conclusions CONCLUSIONS

Quantum dots can be used to probe Kondo-type correlations in several interesting contexts

  Spectroscopy of a Kondo Box • can probe in much more detail the correlated Kondo state • spin and evolution of excited states from pert. theory & QMC • tune through weak-coupling to strong-coupling cross-over

  Mesoscopic Kondo Problem• realization-dependent TK works as scaling parameter in high-T regime (from modified poor-man’s scaling) • low-T: non-universal! (theory?)

  SU(4) Kondo in Carbon Nanotubes (G. Finkelstein group)• seen with 1, 2, and 3 electrons above a closed shell • smooth shape when dot is open

Credits: Ribhu Kaul, Jaebeom Yoo, G. Zaránd, D. Ullmo, S. Chandrasekharan, HUB