Coulomb Blockade and Non-Fermi-Liquid Behavior in a Double-Dot Device Avraham Schiller Racah Institute of Physics Eran Lebanon (Rutgers University) Frithjof B. Anders (Bremen University) Special Thanks to Yuval Oreg Outline: The single-channel Kondo effect The Kondo effect in ultra-small quantum dots The two-channel Kondo effect The Coulomb blockade in quantum boxes Two-channel Kondo physics in a lead-dot-box device From spin to charge two-channel Kondo effects Entanglement of spin and charge Discontinuity in the conductance Conclusions Resistivity minimum: The Kondo effect Fe in Cu T(K) T(K) Franck et al., 1961 De Haas & ven den Berg, 1936 Tmin c 1/5 imp Enhanced scattering at low T The Kondo Effect: Impurity moment in a metal A nonperturbative energy scale emerges TK exp 1 / J Below TK impurity spin is progressively screened All initial AFM couplings flow to a single strong-coupling fixed point A sharp resonance is formed at the Fermi energy for T<TK Local-moment formation: The Anderson model V ed + U |ed| H imp e d n Un n hybridization with conduction electrons 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 Ultra-small quantum dots as artificial atoms VL VR U e d Lead Q.D. Lead Electrostatically-defined semiconductor quantum dots Goldhaber-Gordon et al., Nature 1998 Plunger gate Quantum dot Temperature depedence Field dependence Two-channel Kondo effect Η e 1, 2 k , k c k ck JSimp s (0) 1, 2 Extra channel index T 0 r Impurity spin is overscreened by two identical channels A non-Fermi-liquid fixed point is approached for T<<TK One- versus two-channel Kondo effect Property One channel Two channel Non-Fermi-liquid 1 S (T 0) 0 log( 2) Residual entropy 2 C /T 1 / TK log( TK / T ) Diverging coefficient g (T 0) 1 / TK log( TK / T ) Diverging susceptibility Requirements for the realization of the two-channel Kondo effect Two independent conduction bands Equal coupling strength to the two bands No scattering of electrons between the bands No applied magnetic field acting on the impurity spin Is realization of the two-channel Kondo effect at all possible? The Coulomb blockade in quantum box Quantum box: Small metallic grain or large semiconductor quantum dot with sizeable Charging energy EC but dense single-particle levels D Charging energy: 2 2 Q e E (Q) VBQ EC 2C0 2C0 Energy for charging box with one electron Charging of a quantum box Two-channel Kondo effect in the charge sector (Matveev ‘91) Focus on EC>>kBT and on vicinity of a degeneracy point Introduce the charge isospin 2 z N 1 N 1 N N N 1 N H e k ck ck t ckL cqB cqB ckL z eV L , B k , k , q , Channel index Lowering and raising isospin operators Two-channel Kondo dictionary for the Charging of a quantum box Two-channel Kondo Charging of a quantum box Spin index Isospin index Channel index Physical spin Exchange interaction J Tunneling matrix element 2t Magnetic Field H Deviation from deg. point eV Bandwidth D Charging energy EC Diverging susceptibility Diverging capacitance C Spin two-channel Kondo effect in a lead-dot-box device (Oreg and Goldhaber-Gordon ‘03) In an ordinary two-lead device: Inter-lead spin-exchange spoils the two-channel Kondo effect Spin two-channel Kondo effect in a lead-dot-box device (Oreg and Goldhaber-Gordon ‘03) In an ordinary two-lead device: Inter-lead spin-exchange spoils the two-channel Kondo effect Spin two-channel Kondo effect in a lead-dot-box device (Oreg and Goldhaber-Gordon ‘03) In an ordinary two-lead device: Inter-lead spin-exchange spoils the two-channel Kondo effect Inter-lead spin-exchange is Quantum blocked in a lead-dot-box box device, for kBT < EC ! Quantum Quantum box box Quantum Quantum box box Quantum Quantum box box Tunneling is blocked by the Coulomb blockade A second screening channel is dynamically generated for temperatures below the charging energy A spin two-channel Kondo effect should develop if JBox and JLead are tuned to be equal Note: The above scenario assumes the formation of a stable local moment on the dot, and quantized charge on the box ! Our goal: A detailed quantitative theory of this scenario Extension to regimes with charge fluctuations Lead—Quantum dot—Quantum box setting (Courtesy of D. Goldhaber-Gordon) Leads Quantum box Quantum dot The model H e c c k k k e d d d Un d nd EC N Box N B ˆ 2 L , B k , t ck d d ck L, B Hybridization widths: t 2 Method of solution: Wilson’s numerical renormalization group (E. Lebanon, AS, F.B. Anders, PRB 2003) Strategy: Fix L and tune B in search of a two-channel Kondo effect ( B g )2 Hallmark of spin two-channel Kondo effect: (T ) ln(TK / T ) 20k BTK Definition of TK Symmetric dot: 2ed + U = 0 L EC 0.1D NB 0 Two-channel point is found for any U, including U = 0 Spin two-channel effect persists in the mixed-valent regime TK versus U for a symmetric dot Analytic estimate for stable moment Exponentially small TK is significantly enhanced in the mixed-valent regime Can become of the order of the charging energy EC Dependence of TK on the gate voltage NB for U = 0 Perfect Transmission for B B2CK Prediction of bosonization treatment near perfect transmission Spin 2-channel Kondo effect related to perfect transmission through dot Two-channel line and charging curve for ``realistically large’’ U/EC = 20 2-channel E line L EC 0.1D D C B A NB 0 Shift in Coulomb staircase Origin of shift in Coulomb staircase Note: shift in staircase occurs for B L , EC Diagonalize first the link between the dot and box Lead Dot Box tL tB 1 1 1 local d c1 c1 d d d 2 2 2 Site immediately coupled to the box is only half occupied Entanglement of spin and charge within the two-channel Kondo effect ˆ e2 d N C (T ) 2 EC dN B 2ed/U NB 0 Both magnetic susceptibility and charge capacitance diverge logarithmically, but with different Kondo scales (i.e., slopes) Continuous transition from spin to charge 2-channel Kondo effect Zero-temperature conductance L EC 0.1D NB 0 U D V Discontinuous jump in the conductance across the two-channel point Scaling of the conductance with distance from critical point B B CK 2 U 2e d 2 D L EC 0.1D B B CK 2 Conclusions Quantum-dot devices offer a unique opportunity to study the two-channel Kondo effect. Exploiting the Coulomb blockade, one can measure the two-channel Kondo effect in a double-dot device. Among the exotic features found: A continuous transition from a spin to a charge two-channel Kondo effect. Entanglement of spin and charge. A discontinuity in the T = 0 conductance. Enhancement of the Kondo temperature away from the local-moment regime.
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