- Email: [email protected]

Microelectronics Journal 39 (2008) 354–358 www.elsevier.com/locate/mejo

Linear conductance through parallel coupled quantum dots R. Francoa,, J. Silva-Valenciaa, M.S. Figueirab a

Departamento de Fı´sica, Universidad Nacional de Colombia, A.A. 5997 Bogota´, Colombia Instituto de Fı´sica, Universidade Federal Fluminense (UFF), Avenida litoraˆnea s/n, CEP 24210-346, Nitero´i, Rio de Janeiro, Brazil

b

Available online 20 August 2007

Abstract We study the electronic transport through two parallel coupled quantum dots (QDs), employing the X-boson treatment for the single impurity Anderson model. We compute the linear conductance (LC) and transmission coefﬁcient for different regimes of the system, as function of the QDs energy; our results show a suppression of the linear conductance at low temperatures; when the coupling between the QDs is signiﬁcant, a drop in the transmission coefﬁcient is evident, at the energy value of the side-coupled QD. We also obtain the temperature dependence of the LC, for different hybridizations between the QDs and the energy of one of them. Our results are consistent with those obtained by other theoretical treatments and recovers what is expected when the coupling between the QDs is weak. r 2007 Elsevier Ltd. All rights reserved. PACS: 73.23.b; 73.21.La; 71.27.þa; 71.10.Ay Keywords: Quantum dot; Kondo effect; Transport; X-boson; Fano resonance

1. Introduction

2. Model and theory

Intense research in electronic transport through nanoscale systems, has been present in the last years. In particular, systems of coupled quantum dots (QDs) have been studied theoretically [1] and experimentally [2]; in these nanostructures, the presence of Kondo effect, Coulomb blockade and quantum interference process, originate rich physics, that may be used in technological applications to quantum computation and spintronics [3]. In this paper we studied the linear conductance (LC) through parallel double QDs as depicted in Fig. 1. One active QD1 is connected to a quantum wire (QW) and the other QD2 is side-connected to the active QD1; since the energy levels in two QDs (E d1 and E d2 ), the hybridization between the QD1 and the QW (V 1 ) and the quantum coupling between QDs (V 2 ) can be controlled separately, different transport regimes can be probed. We applied the X-boson method [4,5] for the impurity case, to describe the transport properties of this system.

We employ the model Hamiltonian H ¼ H L þ H R þ H D þ H T to describe the QDs system, represented in a pictorial form in Fig. 1. The Hamiltonian for the left ðLÞ and right ðRÞ leads are given by X Ha ¼ E k;a cyk;a;s ck;a;s ða ¼ L; RÞ, (1) k;a;s

where cyk;a;s ðck;a;s Þ is a creation (destruction) operator of an electron with energy E k;a , momentum k and spin s on the lead a. The interacting QDs are described by XX HD ¼ E di;s X di;ss , (2) i¼1;2 s

where we employ the Anderson impurity characterized by a localized di level E di;s , associated with the QDs (we employ the di to indicate the localized electrons at QDi ði ¼ 1; 2Þ), in the representation of the Hubbard operators [4–6]. The tunneling Hamiltonian H T is X X ðV a X yd1;0s ck;a;s þ H:c:Þ HT ¼ a¼L;R k;s

Corresponding author.

E-mail address: [email protected] (R. Franco). 0026-2692/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2007.07.061

þ

X s

ðV 2 X yd2;0s X d1;0s þ H:c:Þ.

ð3Þ

ARTICLE IN PRESS R. Franco et al. / Microelectronics Journal 39 (2008) 354–358

The tunneling amplitude V 2 is responsible for the tunneling between the two QDs, and V a for the tunneling between the QD1 and the lead a. For simplicity, we assume symmetric junctions (i.e. V L ¼ V R ¼ V 1 ) and identical leads. At low temperature and small bias voltage, the linear response conductance is given by the Landauer-type formula [1,4,6]

represents the lead, nF ðoÞ is the Fermi–Dirac distribution function, and m ¼ 0:0 is the chemical potential (Fermi energy at low temperature). The density of states (DOS), r00;s ðoÞ, is calculated from the local dressed GF Gs00 ðoÞ, and we can write this function in terms of the GF at the QDS, G sQDi ðoÞ and the GF of the QW gsc ðoÞ as Gs00 ðoÞ ¼ ½gsc ðoÞV 1 2 G sQD1 ðoÞ þ ½gsc ðoÞV 1 2 G sQD1 ðoÞV 22 G sQD2 ðoÞ.

Z 2e qnF G¼ s r ðoÞ do, hrc ðmÞ qo 00;s 2

(4)

V1=VR

GsQD1 ðoÞ ¼

Ds1 , o E~d1 þ V 21 Ds1 gsc ðoÞ

(6)

GsQD2 ðoÞ ¼

Ds2 , 2 ~ o E d2 þ V 2 Ds2 ðV 21 gsc ðoÞ2 G sQD1 ðoÞÞ

(7)

QD 1 QW

QW

V2

ð5Þ

Using the X-boson method, GF are [4,5]

where rc;s ðoÞ ¼ ð1=pÞIm gsc ðoÞ with gsc ðoÞ ¼ ð1=2DÞ ln½ðo þ DÞ=ðo DÞ (o ¼ o þ iZ in this case, with Z ! 0þ ), being Green’s function (GF) of the QW that V1=VL

355

where again o ¼ o þ iZ, with Z ! 0þ ; the quantity Dsi ¼ hX di;00 i þ nQDi;s is responsible for the correlation in the cumulant X-boson approach [4,5] and E~di ¼ E di þ Li, where Li renormalizes the localized level and is obtained by the minimization of the free energy, computed by the GFs [4–6].

QD 2 Fig. 1. Schematic display of parallel double QDs.

1.1 0.9 0.8 G (2e2/h)

EQD2=Δ, V2=10-5V1, D=100Δ, Δ=(πV1/2D)=1

T=10-3 Δ T=8 10-2Δ T=5 10-1Δ T=Δ

1.0

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -10 -9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

G (2e2/h)

EQD1/Δ

1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 10-3

V2=10-5V1, EQD2=Δ, D=100Δ, Δ=(πV12/2D)=1

10-2

10-1

EQD1=2.0Δ EQD1=Δ EQD1=-0.5Δ EQD1=-Δ EQD1=-1.7Δ

100

101

T/Δ Fig. 2. Linear conductance (LC) vs energy of the QD1 (G vs E QD1 ¼ E d1 ), for different temperature values (up). LC G vs temperature T, for several regimes of the system (QD1) (down). The energy of the QD2 E QD2 ¼ E d2 ¼ D is ﬁxed and the coupling between the QDs is practically zero (V 2 ¼ 105 V 1 ).

ARTICLE IN PRESS R. Franco et al. / Microelectronics Journal 39 (2008) 354–358

356

a splitting in two peaks, with a minimum between them, this minimum is characteristic of the LC for a side-coupled QD. Our results for the lower temperatures, are in qualitative agreement with the one obtained by Vernek et al. [1], using the ﬁnite-U slave boson approach (USBMFT), for a more general conﬁguration of two parallel QDs, and zero temperature ðT ¼ 0Þ; a non-zero coupling between the QDs originate a suppression (a plateau) of the LC. The lower ﬁgure shows a decrease of the LC at low temperature, if compared with the result of Fig. 2, this is in qualitative agreement with the reported by the same system for Kim et al. [1] (non-crossing approach (NCA)), and with the LC result obtained by Chung et al. [1] (numerical renormalization group (NRG)), in a more general conﬁguration of parallel-coupled QDs, that include a J coupling between spins of the localized electrons in the QDs; when the J coupling is important, there is a drop of the LC. In our case we have a coupling between the QDs by hybridization V 2 , that originate a JðoÞ ¼ 2V 2 = ðE d2 D2 ðoÞÞ coupling by the Schrieffer–Wolff transformation for the Anderson impurity, the D2 ðoÞ is the effective hybridization between the QDs, and is given by D2 ðoÞ ¼ V 22 Imð½gsc ðoÞV 1 2 G sQD1 ðoÞ þ gsc ðoÞÞ, where o ¼ o þ iZ, with Z ! 0þ .

3. Results and discussion In our calculations we choose the following parameters D ¼ ðpV 21 =2DÞ ¼ 1, D ¼ 100D, the temperature T and the QDs energies (E d1 and E d2 ) are in D units. In Fig. 2 (up) we present the results for the LC G vs E d1 (energy for the QD1) for different temperature values, Fig. 2 (down) shows the LC G vs T (temperature), for several regimes, linked with E d1 values; the coupling between the QDs V 2 ¼ 105 V 1 , is very weak, in this case the results reproduce the expected ‘‘behavior’’ for a QD embedded in a QW, at the limit of inﬁnity electronic repulsion inside the QD [7]. In Fig. 3, we show the same results as in Fig. 2, but we consider a ﬁnite hybridization between the QDs (V 2 ¼ V 1 ), for the same ﬁxed energy value of the QD2 (E d2 ¼ E QD2 ¼ D), used in Fig. 2. Presence of two peaks (upper ﬁgure) is evident, the minimum between the peaks is a manifestation of the QDs coupling; a situation similar to the one obtained for the conductance in a QW with a sidecoupled QD [4,6], that shows a minimum for G. In this case we have a combination of the ‘‘behaviors’’ for an embedded QD and a side-coupled QD, a peak similar to the obtained for an embedded QD (see Fig. 2), that suffers

0.6

T=Δ T=3 10-1 Δ T=4 10-1Δ T=5 10-1Δ

G (2e2/h)

0.5

EQD2=Δ, V1=V2, D=100Δ, Δ=πV12/2D=1

0.4 0.3 0.2 0.1 0.0 -10 -9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

EQD1/Δ 0.4

G (2e2/h)

0.3

EQD2=Δ, V1=V2=1, D=100Δ, Δ =πV1

2/2D=1

10-2

10-1

EQD1=2Δ EQD1=Δ EQD1=0.0 EQD1=-0.5Δ EQD1=-Δ

0.2

0.1

0.0 10-3

100

101

T/Δ Fig. 3. The same results of Fig. 2, but considering a ﬁnite coupling between the QDs V 2 ¼ V 1 .

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1.0 V2=0.2V1 V2=V1 V2=0.4V1

EQD1=0.0, EQD2=Δ, T=Δ, D=100Δ

T (ω)

0.8 0.6 0.4 0.2 0.0 -5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

ω/Δ

G (2e2/h), RQD1, nQD1, RQD2, nQD2

1.0 0.9

EQD2=Δ, T=Δ, V1=V2, D=100Δ , Δ = πV12/2D=1

RQD2 nQD2 RQD1 nQD1 G

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

EQD1/Δ Fig. 4. The transmission coefﬁcient TðoÞ vs o at different couplings between the QDs (up). Conductance G, occupation numbers of the QDs nQD1 , nQD2 , and number of holes in the QDs RQD1 and RQD2 vs E QD1 ¼ E d1 , the energy of the QD1, at temperature T ¼ D (down).

Fig. 4 (up), shows the transmission coefﬁcient TðoÞ ¼ r00;s ðoÞ=rsc ðmÞ vs o at different couplings between the QDs, there is a suppression of TðoÞ, at oE QD2 , that is the consequence of a destructive quantum interference process between the ‘‘direct’’ channel transmission through the QD1, and the ‘‘resonant’’ channel through the QD2. This result is in agreement with the one obtained by Kim et al. [1], by NCA in the same system; another theoretical treatments (NRG), in a more general conﬁguration for the QDs reported a similar situation (Chung et al. and Dias da Silva et al. [1]), TðoÞ drop at oE QD2 ¼ E d2 , for signiﬁcative couplings between the QDs. In the Dias da Silva et al. work [1], they compute the DOS for the local site of the QD1 r11 ðoÞ and not the transmission coefﬁcient TðoÞ, but it is known that r11 ðoÞ / TðoÞ [1,4]. Fig. 4 (down) displays the LC G, occupation numbers of the QDs nQD1 , nQD2 , and the number of holes in the QDs RQD1 and RQD2 vs E QD1 ¼ E d1 , the energy of the QD1, at temperature T ¼ D; is evident that the QD2 is in an intermediate valence regime, and that the minimum in G is linked to the variation of the occupation in the QD2.

4. Conclusions Suppressed LC G can be understood in terms of a Fanoquantum interference between the channels associated with a direct path (QW to QD1 to QW) and a Kondo resonant path (QW to QD1 to QD2 to QD1 to QW). Acknowledgments We acknowledge the ﬁnancial support of DINAIN:20601003550, DIB:8003060 and COLCIENCIAS: 1101-333-18707 (Colombia); CNPq and FAPERJ‘‘Primeiros Projetos’’ (Brazil). References [1] L.G.G.V. Dias da Silva, et al., Phys. Rev. Lett. 97 (2006) 96603; T.-S. Kim, et al., Phys. Rev. B 63 (2001) 245326; E. Vernek, et al., Physica E 34 (2006) 608; C.-H. Chung, et al., preprint cond-mat 0607772, 2006. [2] A.W. Holleitner, et al., Science 297 (2002) 70; D.M. Schro¨er, et al., preprint cond-mat 06070441, 2006. [3] D. Loss, et al., Phys. Rev. A 57 (1998) 120; A.A. Aligia, et al. Phys. Rev. B 70 (2004) 075307.

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