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Physica B 284}288 (2000) 1764}1765

Kondo e!ect in double quantum dot systems Wataru Izumida *, Osamu Sakai Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan Department of Physics, Tohoku University, Sendai 980-8578, Japan

Abstract The tunneling conductance of the double dot systems is calculated by using numerical renormalization group method. In low temperature, the following states continuously appear as increasing the coupling between two dots: (1) Kondo singlet state, (2) local singlet state between spins on the dots, and (3) double occupancy in even orbital. The conductance shows peaks at the transition regions between these states. Especially, the peak at the boundary between (1) and (2) shows anomalous narrowing of the width in the strongly correlated cases. We notice this is related to the quantum transition suggested previously for the two impurity Kondo model. 2000 Elsevier Science B.V. All rights reserved. Keywords: Double quantum dot; Kondo singlet state; Local singlet state; Two impurity Kondo model

The Kondo e!ect has been observed in recent experimental works for the tunneling conductance through quantum dot [1}3]. In double dot systems, split gate between dots enables one to control the coupling between the dots. How the Kondo e!ect appears in the double dot systems [4}6]. We investigate the following model Hamiltonian for the double dot connected to leads in series, as lead}dot}dot}lead: H"H #H #H , \

(1)

H " e cR c # e cR c , I *IN *IN I 0IN 0IN IN IN

(2)

H " e (n #n )# t(dR d #h.c.) *N 0N *N 0N N N # ;(n n #n n ), *t *s *t *s

(3)

H " <(dR c #dR c )#h.c. (4) \ *N *IN 0N 0IN IN We consider the case symmetric with respect to the interchange of the left and right. cR is the creation operator of the electron in the leads, dR is that in the dots, and * Corresponding author. Fax: #81-11-716-6175. E-mail address: [email protected] (W. Izumida) Present address: Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397, Japan.

n is the number operator of the electron in the dots. The su$ces L and R mean the &left' and &right', respectively. e is the energy of dot's orbital, and can be changed by the gate voltage on the dots. t is the hopping between left and right dots, and it can be controlled by split gate between dot. < is the matrix element between dot and lead, and can also be controlled by split gate between lead and dot. ; is the Coulomb interaction between electrons in the dot. We investigate from weakly to strongly correlated cases by decreasing D/;, where D"p"<"o is the hybridization strength between dot and lead. (o is the density of state of the conduction electron in leads.) Here we consider the situation that the total occupation number in the dot is "xed at 2, 1n 2#1n 2"2, * 0 by adjusting the gate voltage on the dot. At zero temperature, the linear response conductance for the applied bias voltage is written as follows by using the e!ective parameters t and D of the "xed point non-interacting Hamiltonian [4,5]: 4(t/D) 2e G" . h +1#(t/D),

(5)

The calculated conductance by using numerical renormalization group method is shown in Fig. 1. We show the conductance from weakly to strongly correlated cases, 1.5;10\)D/p;)6.0;10\. The broken line shows

0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 9 3 5 - X

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W. Izumida, O. Sakai / Physica B 284}288 (2000) 1764}1765

Fig. 1. Tunneling conductance at zero temperature.

the conductance for the non-interacting case. The horizontal axis gives the hopping normalized by hybridization, t/D. The conductance almost overlaps on the non-interacting curve in the region t;D and t

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boundary between (1) and (2). The width of the peak becomes narrow in the strongly correlated cases. We note this narrowing is closely related to the quantum transition between the Kondo and the local singlet states suggested for the two impurity Kondo model [7,8]. We have another small peak (or shoulder) structure for the strongly correlated cases (D/p;:2.0;10\) at larger t side of the main peak. For the larger t with t9;/4, the local spins are not formed, but `(3) two electrons occupy the even orbital of the dotsa. The small peak appears around the boundary between (2) and (3). For the weakly correlated cases, the small peak could not be recognized. In the double dot systems, we can observe the phenomena related to the quantum transition between the Kondo and the local spin singlet states by controlling the gate voltages.

References [1] D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav, M.A. Kastner, Nature 391 (1998) 156. [2] D. Goldhaber-Gordon, J. GoK res, M.A. Kastner, H. Shtrikman, D. Mahalu, U. Meirav, Phys. Rev. Lett. 81 (1998) 5225. [3] S.M. Cronenwett, T.H. Oosterkamp, L.P. Kouwenhoven, Science 281 (1998) 540. [4] T. Aono, M. Eto, K. Kawamura, J. Phys. Soc. Japan 67 (1998) 1860. [5] A. Georges, Y. Meir, Phys. Rev. Lett. 82 (1999) 3508. [6] W. Izumida, O. Sakai, Y. Shimizu, Physica B 259}261 (1999) 215. [7] B.A. Jones, C.M. Varma, J.W. Wilkins, Phys. Rev. Lett. 61 (1988) 125. [8] O. Sakai, Y. Shimizu, J. Phys. Soc. Japan 61 (1992) 2333.

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