Kondo, Fano and Dicke effects in side quantum dots

1. Kondo, Fano and Dicke effects in side quantum dots Pedro Orellana UCN-Antofagasta

2. Collaborators Gustavo Lara,Universidad de Antofagasta Enrique V. Anda P. Universidade Católica de Rio de Janeiro

3. Outline • Kondo effect in quantum dots • Fano effect in side attached quantum dots • Model • Results • Conclusions

5. Kondo effect-Phenomenology:Behavior of the resistance as a functionof the temperature in macroscopic systems.

6. Kondo Effect – More of 40 Years after the Discovery Jun Kondo's paper "Resistance Minimum in Dilute Magnetic Alloys" was published in Progress of Theoretical Physics 32 (1964) 37. Although more forty years have passed since then, the importance of this work has not diminished, but continues to increase. Kondo solved the long-standing mystery of resistance minimum phenomenon in his study, thereby opening a door to fundamental and universal physics; this is now known as the Kondo effect. 

7. Kondo effect in quantum dots Conductance as a function of the temperature in quantum dots

8. Kondo effect in quantum dots

9. Ng, T. K. & Lee, P. A. On-site Coulomb repulsion and resonant tunneling. Phys. Rev. Lett. 61, 1768–1771 (1988).  • Glazman, L. I. & Raikh, M. E. Resonant Kondo transparency of a barrier with quasilocal impurity states. JETP Lett. 47, 452–455 (1988). 

12. Fano effect in a single quantum dot The condition for the Fano resonance is the existence of two scattering channels: a discrete level and a broad continuum band.

13. Noninteracting pictureFano antiresonance Destructive interference between two paths

18. Kondo effect in a quantum wire with a side coupled quantum-dot

20. Two side attached quantum dots

21. We adopt the two-fold Anderson Hamiltonian. Each dot has a single level l (with l = 1, 2), and intra-dot Coulomb repulsion U. The two side attached dots are coupled to the QW with coupling t0 . We use the finite-U slave boson mean-field approach in which all the boson operators are replaced by their expectation value.

22. To fin the solution of this correlated fermions system, we use the finite-U slave boson mean-field approach. We appeal to this semi analytical approach where, generalizing the infinite-U slave-boson approximation the Hilbert space is enlarged at each site, to contain in addition to the original fermions a set of four bosons represented by the creation (annihilation) operators They act as projectors onto empty, single occupied (with spin up and down) and doubly occupied electron states, respectively. Then, each creation (annihilation) operator of an electron with spin  in the l-th QD, is substituted by

25. As we work at zero temperature, the bosons operators expectation values and the Lagrande multipliers are determined by minimizing the energy <H> with respect to these quantities. It is obtained in this way, a set of nonlinear equations for each quantum dot, relating the expectation values for four bosonics operator, the three Lagrange multipliers, and the electronic expectation values,

26. The operator Zi, ‘s are chosen to reproduce the correct limit when U tends to zero in the mean field approximation,

28. To obtain the electronic expectation values <…>, the Hamiltonian, He is diagonalized. Their stationary states can be written as where ajk and blk are the probabilities amplitudes to find the electron at the site j and the l-th QD respectively, with energy

29. The amplitudes a’s and b’s obey the following linear difference equations

31. CONTACTS

32. Transmission

34. For U sufficiently large the transmission can be written approximately as a superposition of Fano and Briet-Wigner line shapes

35. Density of states

37. This phenomenon is in analogy to the Dicke effect in optics, that take place in the spontaneous emission of two closely lying atoms radiating a photon, with a wavelength larger than the separation between them, in the same environment

38. The luminescence spectrum is characterized by a narrow and a broad peak, associated with long-and-short lived states, respectively. The former state, coupled weakly to the electromagnetic field, called subradiant, and latter, strongly coupled, superradiant state

39. In the electronic case, however the level broadening are produced by indirect coupling of the up-down QDs through the QW. Two Kondo temperatures

40. Non equilibrium transport

41. Current(solid line, black) and differential conductance (dashed line,red) for Vg=-3, U=6 for a) δV=0.1 and b) δV=0.5

42. Current (solid line, black) and differential conductance (dashed line) for Vg=-3, on site energy, U=6, for a) δV=0.1 and b) δV=0.5

44. CONCLUSIONS • We have studied the transport through two single side coupled quantum dots using the finite-U slave boson mean field approach at T=0. • We have found that the transmission spectrum shows a structure with two antiresonances localized at the renormalized energies of the quantum dots. • The DOS of the system shows that when the Kondo correlations are dominant there are two Kondo regimes each with its own Kondo temperature. • The above behavior of the DOS is due to quantum interference in the transmission through the two different resonance states of the quantum dots coupled to common leads. • This result is analogous to the Dicke effect in optics.

45. Side attach double quantum dot molecule

46. Transmission probability

47. Artificial Molecule Coupled quantum dot system Series connection Science 274 5291 (1996)

48. Energetically the double quantum dot molecule can be modeled as two wells potential connected by a barrier

49. Transmission spectrum

50. Transmission spectrum for 1=2=Vg=-3, tc = 0.5  (solid line) and tc =  (dashed line), for on site energy, a) U = 2 ,b) U = 4 , c) U = 8  and d) U =16 . The figure allows to study the interplay between the Kondo effect and the inter-dot anti-ferromagnetic correlation. Increasing U, a sharp feature develops close to the Fermi energy, indicating the appearance of a Kondo resonance. We can see that this process is more rapidly defined for the case where tc =0.5 than for tc =