Transcript Slide 1
Introduction to the Kondo Effect in Mesoscopic Systems Resistivity minimum: The Kondo effect Fe in Cu T(K) T(K) Franck et al., 1961 Tmin c 1/5 imp De Haas & ven den Berg, 1936 Enhanced scattering at low T 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 Photoemission spectra CeCu2Si2 CeSi2 Energy (meV) Reinert et al., PRL 2001 DOS A(e) Patthey et al., PRL 1987 Occupied DOS A(e)f(e) Planner tunnel junction with magnetic impurities T(K) DG(0)/G0(0) Wyatt, PRL (1964) DG(0)/G0(0) log(T) enhancement of the conductance V (mV) Zero-bias anomaly 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 Kondo-assisted tunneling in ultrasmall quantum dots Goldhaber-Gordon et al., Nature 1998 Plunger gate Quantum dot 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 TK exp( 1 / J ) 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 Local-moment formation: The Anderson model V ed + U |ed| H imp e d n Un n hybridization with conduction electrons Energy scales: Inter-configurational energies ed and U+ed Hybridization width G = pV2 Condition for formation of local moment: G e d ,U e d J Kondo screening 2G 1 1 p e d U e d Free local moment Schrieffer & Wolff 1966 Charge fluctuations T 0 TK TLM e d The Anderson model: spectral properties Kondo resonance ed EF ed+U A sharp resonance of width TK develops at EF for T<TK Unitary scattering for T=0 and <n>=1 p n 1 2 A(e 0, T 0) sin ( ) , pG 2 Bulk versus tunnel junction geometry Tunnel-junction geometry: Bulk geometry: Tunneling through impurity opens a new channel for conductance Impurity blocks ballistic motion of conduction electrons Ultrasmall quantum dots as artificial atoms VL VR U e d Lead Q.D. Lead Anderson-model description of quantum dot Ingredient Magnetic impurity Quantum dot 1. Discrete singleparticle 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 Tunneling through a quantum dot Kondo resonance increases tunneling DOS, enhances conductance For GL=GR , unitary limit corresponds to perfect transmission G=2e2/h e2 4GLGR I (V , T ) 2 A(e , T ;V ) f (e L ) f (e R )de 2 h (GL GR ) Zeeman splitting with magnetic field H H eV Resonance condition for spin-flip-assisted tunneling: BgH = eV Resonance in dI/dV for eV = BgH eV Electrostatically-defined semiconductor quantum dots Goldhaber-Gordon et al., Nature 1998 Plunger gate Quantum dot Temperature depedence Field dependence More semiconductor quantum dots… van der Wiel et al., Science 2000 Conductance vs gate voltage dI/dV (e2/h) Differential conductance vs bias T varies in the range 15-800mK Carbon nano-tube quantum dots Nygard et al., Nature 2000 Lead Nano-tube Conductance vs gate voltage T varies in the range 75-780mK Lead Carbon nano-tube quantum dots Nygard et al., Nature 2000 BgH Physical mechanism: tuning of Zeeman energy to level spacing Pustilnik et al., PRL 2000 Magnetic-field-induced Kondo effect! Carbon nano-tube quantum dots Nygard et al., Nature 2000 BgH Physical mechanism: tuning of Zeeman energy to level spacing Pustilnik et al., PRL 2000 Magnetic-field-induced Kondo effect! Phase-shift measurement in Kondo regime Ji et al., Science 2000 F Vp Relative transmission phase Two-slit formula: T tu td ei 2 tu td 2 tu td cos( ) 2 2 Aharonov-Bohm phase 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 and transmission phase Entin-Wohlman et al., 2002 Magnitude of oscillations & phase evolution Nonequilibrium splitting of the Kondo resonance The Kondo resonance in the dot DOS splits with an applied [Meir & Wingreen, 1994] bias into two peaks at L and R 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 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 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-ofequilibrium Kondo effect and field-driven Kondo effect