AC Kondo eect in quantum dots Ram on Aguado a,b, * , Rosa L opez b , Gloria Platero b , Carlos Tejedor c a Department of Applied Physics, Delft University of Technology, 2628 CJ Delft, The Netherlands b Instituto de Ciencia de Materiales (CSIC), Madrid, Spain c Universidad Aut onoma de Madrid, Spain Abstract Correlation eects in the transport properties of con®ned electrons coupled by quantum-mechanical tunneling to delocalized electrons in the leads ± quantum dots ± are discussed in the presence of arbitrary intense ac potentials. Using a time dependent Anderson model we propose a consistent procedure that, for the ®rst time, allows one to study the full range of parameters (from Kondo physics to Coulomb blockade). Our method takes care of charge conservation by means of a generalized Friedel±Langreth sum rule. We analyze the ac Kondo problem at zero temperature in a wide range of parameters. The eect of the ac potential on the linear conductance for dierent values of the external ®eld is discussed. Ó 1998 Elsevier Science B.V. All rights reserved. Keywords: Quantum dots; Kondo eect; Photon-assisted tunneling It has been predicted [1±6] and recently ob- served experimentally [7,8] that the transport at low temperatures through a quantum dot (QD) coupled by tunneling barriers to two leads contain- ing Fermi liquids, is governed by Kondo-like phys- ics [9]. For a dot symmetrically coupled to the leads at zero-temperature, a Kondo (or Abriko- sov±Suhl) singularity exist in the quasiparticle den- sity of states (DOS) at the Fermi level ( F ) of the leads. This produces a perfect transparency with conductance G  2e 2 =h [1,2] in the symmetric con- ®guration  0 U =2,  0 being the QD level and U the on-site interaction. The interest of the Kondo physics in QDs resides in the possibility of exter- nally controlling parameters as U and  0 or C (cou- pling with the leads) which determine the Kondo temperature T K   U C p exppj F   0 jU   0 =CU  [7,8]. Another interesting possibility is to apply a dc or ac bias in order to study non-equilibrium eects [10,11]. This last case can be actually made by means of an ac gate voltage. The problem becomes very interesting because the frequency range for the observation of photon-assisted tunneling (PAT) [12] is roughly that of T K in QDs going from few mK to 1 K, i.e. frequencies between 100 MHz and 20 GHz [8]. The aim of our work is to take care of charge neutrality that is crucial for the existence of the Kondo singularity. This re- quires to generalize the Friedel±Langreth (FL) sum rule [13] to the case of a time dependent situ- ation. Our method is an extension of those [6,14] developed in the static case where an eective self- energy eliminates consistently all the problems of the perturbative approaches. The time dependent Anderson±Hamiltonian reads Physica B 256±258 (1998) 165±168 * Corresponding author. Fax: 31 15 2783251; e-mail: ra- mon@qt.tn.tudelft.nl 0921-4526/98/$ ± see front matter Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 5 8 8 - 2