Introduction to the Kondo Effect in Mesoscopic Systems

T(K) Resistivity minimum: The Kondo effect De Haas & ven den Berg, 1936 Franck et al., 1961 Fe in Cu Enhanced scattering at low T

Intermediate-valence and heavy fermion systems: Enhancement of thermodynamic and dynamic properties Strongly enhanced thermodynamics Single-ion scaling up to x=0.5 On-set of lattice coherence at high concentration of Ce Ce x La 1-x Cu 6 [from Onuki &Komatsubara, 1987]

Photoemission spectra Energy (meV) CeSi 2 CeCu 2 Si 2 Patthey et al., PRL 1987 Reinert et al., PRL 2001 Occupied DOS A(  )f  DOS A(  )

Planner tunnel junction with magnetic impurities V (mV) T(K)  G(0)/G 0 (0) Wyatt, PRL (1964) log(T) enhancement of the conductance Zero-bias anomaly

Kondo-assisted tunneling through a single charge trap Ralph & Buhrman, PRL 1994 dI/dV has image of Anderson impurity spectrum Zero-bias anomaly splits with magnetic field

Kondo-assisted tunneling in ultrasmall quantum dots Goldhaber-Gordon et al., Nature 1998 Quantum dot Plunger gate Temperature depedence Field dependence

Cobalt atoms deposited onto Au(111) at 4K (400A x 400A) Madhavan et al., Science 280 (1998)

STM spectroscopy on and off a Co atom Madhavan et al., Science 280 (1998)

The Kondo Effect: Impurity moment in a metal A nonperturbative energy scale emerges Below T K impurity spin is progressively screened Universal scaling with T/T K for T<T K Conduction electrons acquire a  /2 phase shift at the Fermi level All initial AFM couplings flow to a single strong-coupling fixed point

Local-moment formation: The Anderson model  d |  d + U hybridization with conduction electrons V

Energy scales: Inter-configurational energies  d and U+  d Hybridization width  =  V 2 Condition for formation of local moment: T TKTK 0 Charge fluctuations Free local moment Kondo screening Schrieffer & Wolff 1966

The Anderson model: spectral properties EFEF dd  d +U Kondo resonance A sharp resonance of width T K develops at E F for T<T K Unitary scattering for T=0 and =1

Bulk versus tunnel junction geometry Bulk geometry: Impurity blocks ballistic motion of conduction electrons Tunnel-junction geometry: Tunneling through impurity opens a new channel for conductance

dd U VLVL Q.D.Lead VRVR Ultrasmall quantum dots as artificial atoms

Anderson-model description of quantum dot IngredientMagnetic impurityQuantum dot Discrete single- particle levels 1.Atomic orbitalsLevel quantization On-site repulsion2.Direct Coulomb repulsion Charging energy E C =e 2 /C Hybridization3.With underlying band Tunneling to leads

Kondo resonance increases tunneling DOS, enhances conductance For  L =  R, unitary limit corresponds to perfect transmission G=2e 2 /h Tunneling through a quantum dot

Zeeman splitting with magnetic field HH eV Resonance condition for spin-flip-assisted tunneling:  B gH = eV eV Resonance in dI/dV for eV =  B gH

Electrostatically-defined semiconductor quantum dots Goldhaber-Gordon et al., Nature 1998 Quantum dot Plunger gate Temperature depedence Field dependence

More semiconductor quantum dots… van der Wiel et al., Science 2000 Conductance vs gate voltage Differential conductance vs bias dI/dV (e 2 /h) T varies in the range mK

Carbon nano-tube quantum dots Nygard et al., Nature 2000 Conductance vs gate voltage Nano-tube Lead T varies in the range mK

Magnetic-field-induced Kondo effect! Carbon nano-tube quantum dots Nygard et al., Nature 2000 Physical mechanism: tuning of Zeeman energy to level spacing  B gH Pustilnik et al., PRL 2000

Magnetic-field-induced Kondo effect! Carbon nano-tube quantum dots Nygard et al., Nature 2000 Physical mechanism: tuning of Zeeman energy to level spacing  B gH Pustilnik et al., PRL 2000

Phase-shift measurement in Kondo regime Ji et al., Science 2000  Two-slit formula: Relative transmission phase Aharonov-Bohm phase VpVp

Conductance of dot vs gate voltage Aharonov-Bohm oscillatory part Magnitude of oscillations & phase evolution Kondo valley Plateau in measured phase in Kondo valley ! Change in phase differs from  /2 But, no simple relation between  and transmission phase Entin-Wohlman et al., 2002

Nonequilibrium splitting of the Kondo resonance The Kondo resonance in the dot DOS splits with an applied bias into two peaks at  L and   R [Meir & Wingreen, 1994] Is this splitting measurable? Use a three-terminal device, with a probe terminal weakly connected to the dot YES! Sun & Guo, 2001; Lebanon & AS 2002

Measuring the splitting of the Kondo resonance de Franceschi et al., PRL (2002) Quantum wire Third lead Quantum dot Kondo peak splits and diminishes with bias Varying  V

Nonequilibrium DOS for asymmetric coupling to the leads de Franceschi et al., PRL (2002) Relative strength of coupling to left-and right-moving electrons is controlled by perpendicular magnetic field

Conclusions Mesoscopic systems offer an outstanding opportunity for controlled study of the Kondo effect In contrast to bulk systems, one can study an individual impurity instead of an ensemble of them New aspects of the Kondo effect emerge, e.g., the out-of- equilibrium Kondo effect and field-driven Kondo effect