Introduction to the Kondo Effect in Mesoscopic Systems

1. Introduction to the Kondo Effect in Mesoscopic Systems

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

3. Intermediate-valence and heavy fermion systems: Enhancement of thermodynamic and dynamic properties CexLa1-xCu6 [from Onuki &Komatsubara, 1987] On-set of lattice coherence at high concentration of Ce Strongly enhanced thermodynamics Single-ion scaling up to x=0.5

4. Photoemission spectra CeCu2Si2 CeSi2 Energy (meV) Patthey et al., PRL 1987 Reinert et al., PRL 2001 DOS A(e) Occupied DOS A(e)f(e)

5. Planner tunnel junction with magnetic impurities Wyatt, PRL (1964) DG(0)/G0(0) T(K) V (mV) DG(0)/G0(0) Zero-bias anomaly log(T) enhancement of the conductance

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

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

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

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

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

11. Local-moment formation:The Anderson model ed + U V |ed| hybridization with conduction electrons

12. Energy scales: Inter-configurational energies edand U+ed Hybridization width G= prV2 Condition for formation of local moment: Schrieffer & Wolff 1966 Kondo screening Free local moment Charge fluctuations T TK 0

13. The Anderson model: spectral properties Kondo resonance ed ed+U EF A sharp resonance of width TK develops at EF for T<TK Unitary scattering for T=0 and <n>=1

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

15. Ultrasmall quantum dots as artificial atoms VL VR U d Lead Q.D. Lead

16. Anderson-model description of quantum dot Ingredient Magnetic impurity Quantum dot 1. Discrete single- particle levels Atomic orbitals Level quantization 2. On-site repulsion Direct Coulomb repulsion Charging energy EC=e2/C 3. Hybridization With underlying band Tunneling to leads

17. Tunneling through a quantum dot Kondo resonance increases tunneling DOS, enhances conductance For GL=GR , unitary limit corresponds to perfect transmission G=2e2/h

18. Zeeman splitting with magnetic field H H eV eV Resonance condition for spin-flip-assisted tunneling: mBgH = eV Resonance in dI/dV for eV = mBgH

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

20. More semiconductor quantum dots… van der Wiel et al., Science 2000 Differential conductance vs bias Conductance vs gate voltage dI/dV (e2/h) Tvaries in the range 15-800mK

21. Carbon nano-tube quantum dots Nygard et al., Nature 2000 Lead Lead Nano-tube Conductance vs gate voltage Tvaries in the range 75-780mK

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

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

24. Phase-shift measurement in Kondo regime Ji et al., Science 2000 F Vp Relative transmission phase Two-slit formula: Aharonov-Bohm phase

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

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

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

28. 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

29. 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