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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