1. Coulomb Blockade and Non-Fermi-Liquid Behavior in a Double-Dot Device Avraham Schiller Racah Institute of Physics Eran Lebanon (Rutgers University) Special Thanks to Yuval Oreg Frithjof B. Anders (Bremen University)

2. Outline: The single-channel Kondo effect The Kondo effect in ultra-small quantum dots Two-channel Kondo physics in a lead-dot-box device Conclusions The two-channel Kondo effect The Coulomb blockade in quantum boxes From spin to charge two-channel Kondo effects Entanglement of spin and charge Discontinuity in the conductance

3. 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

4. The Kondo Effect: Impurity moment in a metal A nonperturbative energy scale emerges Below T K 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<T K

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

6. 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

7. dd U VLVL Q.D.Lead VRVR Ultra-small quantum dots as artificial atoms

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

9. Two-channel Kondo effect Impurity spin is overscreened by two identical channels A non-Fermi-liquid fixed point is approached for T<<T K Extra channel index

10. One- versus two-channel Kondo effect PropertyOne channelTwo channelNon-Fermi-liquid Residual entropy Diverging coefficient  Diverging susceptibility

11. Requirements for the realization of the two-channel Kondo effect No scattering of electrons between the bands Two independent conduction bands Equal coupling strength to the two bands No applied magnetic field acting on the impurity spin Is realization of the two-channel Kondo effect at all possible?

12. The Coulomb blockade in quantum box Quantum box: Small metallic grain or large semiconductor quantum dot with sizeable Charging energy E C but dense single-particle levels  Charging energy: Energy for charging box with one electron

13. Charging of a quantum box

14. Two-channel Kondo effect in the charge sector (Matveev ‘91) Focus on E C >>k B T and on vicinity of a degeneracy point Introduce the charge isospin Lowering and raising isospin operatorsChannel index

15. Two-channel Kondo dictionary for the Charging of a quantum box Two-channel KondoCharging of a quantum box Spin index Channel index Exchange interaction Magnetic Field Bandwidth   H D Isospin index Physical spin Tunneling matrix element Deviation from deg. point Charging energy   2t2t eV ECEC Diverging susceptibility  Diverging capacitance C

16. Spin two-channel Kondo effect in a lead-dot-box device In an ordinary two-lead device: Inter-lead spin-exchange spoils the two-channel Kondo effect (Oreg and Goldhaber-Gordon ‘03)

17. 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)

18. In an ordinary two-lead device: Inter-lead spin-exchange spoils the two-channel Kondo effect Quantum box Inter-lead spin-exchange is blocked in a lead-dot-box device, for k B T < E C ! Spin two-channel Kondo effect in a lead-dot-box device (Oreg and Goldhaber-Gordon ‘03)

19. Quantum box Quantum box Quantum box Quantum box Quantum box Quantum box Tunneling is blocked by the Coulomb blockade

20. A second screening channel is dynamically generated for temperatures below the charging energy A spin two-channel Kondo effect should develop if J Box and J Lead are tuned to be equal Our goal: A detailed quantitative theory of this scenario Note: The above scenario assumes the formation of a stable local moment on the dot, and quantized charge on the box ! Extension to regimes with charge fluctuations

21. Lead—Quantum dot—Quantum box setting (Courtesy of D. Goldhaber-Gordon) Leads Quantum box Quantum dot

22. The model Hybridization widths:

23. 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 Hallmark of spin two-channel Kondo effect: Definition of T K

24. Symmetric dot: 2  d + U = 0 Two-channel point is found for any U, including U = 0 Spin two-channel effect persists in the mixed-valent regime

25. T K versus U for a symmetric dot Exponentially small T K is significantly enhanced in the mixed-valent regime Analytic estimate for stable moment Can become of the order of the charging energy E C

26. Perfect Transmission for     2CK Dependence of T K on the gate voltage N B for U = 0 Prediction of bosonization treatment near perfect transmission Spin 2-channel Kondo effect related to perfect transmission through dot

27. Shift in Coulomb staircase B D C E A Two-channel line and charging curve for ``realistically large’’ U/E C = 20 2-channel line

28. Lead BoxDot tLtL Origin of shift in Coulomb staircase tBtB Note: shift in staircase occurs for Diagonalize first the link between the dot and box Site immediately coupled to the box is only half occupied

29. 2  d /U Entanglement of spin and charge within the two-channel Kondo effect 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

30. Zero-temperature conductance V Discontinuous jump in the conductance across the two-channel point

31. Scaling of the conductance with distance from critical point

32. 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.