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Physica E 19 (2003) 221 – 224 www.elsevier.com/locate/physe

Fractal conductance uctuations in a quantum-dot molecule N. Aokia;∗ , L.-H. Linb , Y. Iwasea , T. Morimotoa , T. Sasakia , Y. Ochiaia , K. Ishibashic , J.P. Birdd a Department

of Materials Technology & Center for Frontier Electronics and Photonics, Chiba University, 1-33 Yayoi, Inage, Chiba 263-8522, Japan b Department of Applied Physics, National Chiayi University, Chiayi 600, Taiwan c Semiconductor Laboratory, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan d Department of Electrical Engineering & Center for Solid State Electronics Research, Arizona State University, Tempe, AZ 85287-5706, USA

Abstract Exactly self-similar, and statistically self-similar, features have been studied in the magneto-conductance of an open quantum-dot molecule. The fractal dimensions determined from the two dierent features are found to dier signicantly from each other. While we do not understand the detailed origins of this dierence at present, our results suggest that dierent physical processes are responsible for the two dierent types of fractal structure. ? 2003 Elsevier B.V. All rights reserved. PACS: 05.45.Df; 75.47.−m; 81.07.Ta Keywords: Magnetoconductance; Conductance uctuations; Fractal dimension; Coupled quantum dot; Low temperature transport

1. Introduction In this study, chaotic or fractal behavior in electron transport in quantum dots has been studied by means of low-temperature magnetoconductance measurements, so as to nd the relationship on quantum chaos [1–9]. Chaotic behavior has been believed to occur in the ballistic cavities and fractal behavior has previously been reported in a self-similar pattern of the magnetoconductance in quantum dots [3–5]. In this paper, we analyze the low-temperature magnetoconductance of an open quantum-dot molecule by controlling the coupling between two dots, and discuss the coupling dependence of the fractal dimension that ∗ Corresponding author. Tel.: +81-43-290-3430; fax: +81-43290-3427. E-mail address: [email protected] (N. Aoki).

is estimated by two methods for remarking the exact and statistic self-similarity. 2. Sample- and experimental-details The coupled dot was formed on the surface of a high mobility GaAs/AlGaAs wafer using the standard split-gate technique and is shown in the inset of Fig. 1. The size of the left and right dots in this gure are 1:0 × 1:0 m2 and 1:2 × 1:2 m2 , respectively. By applying a suitable gate voltage to the ve strip gates, we can control the number of modes supported by the individual quantum point contacts (QPCs). In this experiment, we modied only the gate voltage of the central QPC to control the coupling strength between the two dots, without signicantly changing the size of the individual dots. The electron

1386-9477/03/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/S1386-9477(03)00320-5

N. Aoki et al. / Physica E 19 (2003) 221 – 224

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1µm

δG1

2 1.8

δB 1 1.6 1.4

δG2

Conductance (e 2/h)

2.2

1.2 -0.4

-0.2

0

0.2

0.4

Magnetic Field (T)

δB2

δG3

Fig. 1. The magnetoconductance trace at several gate voltages (from the bottom −2:8, −2:4, −1:6 V). The inset is a SEM image of the sample.

3. Results and discussion Magnetoconductance traces, obtained with several dierent voltages applied to the gates of the coupling QPC, are shown in Fig. 1. Clear conductance uctuations are observed that are highly symmetric with respect to the central weak-localization peak at zero magnetic eld. As shown in Fig. 2, we can typically observe a three- or four-fold self-similar hierarchy in the magnetoconductance. In order to clarify those uctuations, we can estimate the fractal dimension (DF )

δB3 1

DFS = 2 − ∆G (log scale)

density and mobility of the two-dimensional electron gas (2DEG) were 3:9 × 1015 m−2 and 80 m2 /Vs, respectively. Low-temperature transport measurements were performed at 100 mK in a He3 –He4 dilution refrigerator with a low power excitation of 10 nA and using a lock-in amplier system. In order to investigate fractal transport behavior originating from the inter-dot coupling, xed voltages were applied to all gates of the structure, except for the central, coupling, QPC. The voltages applied to the entrance and the exit QPCs of the coupled dot were chosen from the individual pinch-o characteristics to allow three propagating modes. The gate voltage of the central QPC was then changed from −0:6 to −3:0 V, over which range the propagating-mode number varied from eight to essentially zero.

0.1

0.01 0.01

0.1

1

∆B (log scale)

Fig. 2. Self-similar hierarchy structure in the low temperature conductance uctuations at Vg = −2:4 V and the fractal dimension estimated by the slope. Solid lines are smoothed ts that show the background of the magneto-resistance.

from the relation DF = 2 − , where 1 ¡ DF ¡ 2 and is determined from the ratio of G to B. The gate voltage dependence of the fractal dimension from the self-similarity (DFS ) is shown in Fig. 3a. The value of DFS increases with increasing negative gate voltage. On the other hand, alternative estimation for the fractal dimension is the box counting technique, where

N. Aoki et al. / Physica E 19 (2003) 221 – 224

D FS

(a)

(b)

1.6

1.3

1.6

D

FB

δB >0.03 T 1.3

δB<0.01 T 1 -3

-2.5

-2

. -1.5

-1

-0.5

Gate Voltage lt (V)

Fig. 3. Gate voltage dependence of the fractal dimensions obtained from self-similar (a) and box counting (b) determinations. The closed circles and the closed squares in (b) are obtained from the magnetic eld region at B ¿ 0:03 T and B ¡ 0:01 T, respectively.

the G(B) trace is covered with a mesh of identical squares. Then, the fractal dimension for the box counting (DFB ) is obtained by calculating the number of occupied squares in the mesh, N (B), as a function of the square size, B. Finally, DFB is dened as the ratio of log N (B) to log B and the gate voltage dependence of DFB is shown in Fig. 3b. The upper slope (lled circles) was obtained from the ratio at B ¿ 0:03 T, and the lower slope (lled squares) was obtained from that at B ¡ 0:01 T. Before discussing the coupling-dependent evolution of the fractal dimensions, let us consider the role of the central QPC and the inter-dot coupling. In our previous study of three-dot arrays [7], the high frequency components observed in an FFT analysis of the magnetoconductance were found to be reduced by decreasing the inter-dot coupling. This observation was argued to arise from the fact that, in dots that are multiply connected by phase-coherent electrons, the resulting hybridization of the density of states should increase

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the complexity of the density of states, giving rise to more complicated structure in the magneto-resistance. Reducing the inter-dot coupling, however, will suppress the hybridization eect, thereby inducing a transition to uctuations with suppressed high-frequency content. A similar suppression of the high frequency components of the uctuations, due to the variation of the conductance of the coupling QPC, has also been observed in the structure studied here [8]. Since such a high frequency components should predominantly inuence box-counting analysis of the fractal behavior in the region of small B, the lower slope in Fig. 3b appears to reect this evolution of hybridization eects. As can be seen in Fig. 3, we obtained dierent gate-voltage dependences of the fractal dimension from the geometrical and the statistical fractal analyses of the coupled-dot magnetoconductance. At present, we do not have a clear understanding of the origin of this dierence, although it seems possible that the dierent fractal analyses probe dierent aspects of electron transport. Micolich et al. have shown universal scaling of the statistical self-similarity of the conductance uctuations by introducing the parameter Q, which is dened as the ratio of average energy spacing to the broadening of the energy levels in the dot [4]. In future work we plan to consider how our results compare with this predicted universal behavior. In conclusion, we have shown that the two dierent fractal dimensions are determined by based on exactly or statistically self-similar structure in the magnetoconductance of an open quantum-dot molecule. While the origin of the dierence is not clear yet, it suggests that dierent physical origins are responsible for the two dierent fractal structure. Acknowledgements This work was supported by JSPS-NSF (US) cooperative science program. Work at RIKEN is supported by CREST. References [1] C.M. Marcus, A.J. Rimberg, R.M. Westervelt, P.F. Hopkins, A.C. Gossard, Phys. Rev. Lett. 69 (1992) 506.

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