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Low temperature conductance fluctuations in double-dot system
Microelectronic Engineering 63 (2002) 57–61 www.elsevier.com / locate / mee
Low temperature conductance fluctuations in double-dot system N. Aoki a,c , D. Oonishi a , Y. Iwase a , K. Ishibashi b , Y. Aoyagi,b , Y. Ochiai a,c , * a
Department of Materials Technology, Chiba University, 1 -33 Yayoi, Inage, Chiba 263 -8522, Japan b Semiconductors Laboratory, Riken, 2 -1 Hirosawa, Wako, Saitama 351 -0198, Japan c Center for Frontier Electronics and Photonics, Chiba University, 1 – 33 Yayoi, Inage, Chiba 263 -8522, Japan
Abstract We have studied the low temperature conductance fluctuations in double-dot systems. The electron wave coupling between ballistic quantum dots has been investigated in a pair of coupled quantum dots defined by several independently-tunable quantum-point-contacts (QPCs). Reproducible conductance fluctuations are observed in the low temperature magneto-conductance at low magnetic fields and a clear difference between weak and strong couplings has been identified as the gap of the QPC corresponding to the inter-dot coupling is varied. This suggests clear evidence for a specific spread of the coupling orbit over the whole dot region. 2002 Elsevier Science B.V. All rights reserved. Keywords: Quantum dot; Magnetoconductance; Conductance fluctuations; Coupled dot
1. Introduction The transport properties of quantum-dot arrays are largely affected by the quantum interference of electron waves, even in open dot systems. We have studied the low-temperature magneto-resistance (MR) in a coupled dot array defined by doubly-corrugated quantum wires or several pairs of an independently controlled quantum point contacts (QPCs). By suitable control of the gate voltage, we were able to change the coupling between the individual dots [1,2]. The trapped state of electron waves in the dot can be discussed in terms of interference of the wave function [3]. Even in ballistic transports, considerably large conductance fluctuations (CFs) have been observed and the phase coherency can be explained by a correlation function analysis for the CF in MR [1]. In this study, we compare the interference behavior of previously studied three- or four-dot quantum-dot arrays and an individually controlled two-dot systems in order to clarify the coupling effect between individual dots, by analyzing the CF at lower magnetic fields.
* Corresponding author. Tel.: 1 81-43-290-3428; fax: 1 81-43-290-3427. E-mail address: [email protected] (Y. Ochiai). 0167-9317 / 02 / $ – see front matter PII: S0167-9317( 02 )00639-1
2002 Elsevier Science B.V. All rights reserved.
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N. Aoki et al. / Microelectronic Engineering 63 (2002) 57 – 61
Fig. 1. (a) SEM photograph of the couple dot system in the 2DEG. (b) Schematic gate configuration of the coupled dot.
2. Experimental Dot arrays were formed with the standard split-gate technique on the surface of a high mobility GaAs–AlGaAs wafer. In previous measurements of three- and four-dot systems, the low temperature MR has been studied by controlling only a single gate voltage [2]. However, in the present work, we introduce a two-dot coupling system having individually controllable unit dots. The two-dot system consists of five strip gates that lie opposite to a two-open-mouth gate so that three QPCs and two plungers are constructed. The system is regarded as an almost-symmetric two-dot system. The size of each unit dot is slightly different but is of the order of 1 mm [2], and can be individually controlled by setting into a suitable gate voltage at the five strip gates. Also, we can independently control the coupling strength between the two dots by changing the gate voltage of the center QPC. The schematic configuration of the two-dot system is shown in Fig. 1. Fig. 1a shows the SEM photograph of the coupled dot sample and Fig. 1b shows the schematic gate configuration. The electron density and mobility in the two-dimensional electron gas (2DEG) are 3.88 3 10 15 m 22 and 80 m 2 / V s, respectively. Designed sizes of the left and right dots are 1.0 3 1.0 and 1.2 3 1.2 mm 2 , respectively. As shown in the figure, we can independently control all the gates consisting of the dots. The low temperature transport measurements were performed in the He 3 –He 4 dilution refrigerator with a low power excitation using a lock-in amplifier system.
3. Results and discussion The low field CFs, DG, are obtained from the low temperature MR near zero field are shown in Fig. 2b. The gate voltage of the upper gate is 2 2.0 V, that of the lower left and right are 2 0.8 V and 2 0.92 V, respectively, in order to form three quantized channel width. The center QPC is swept from 2 0.6 to 2 3.0 V to vary the coupling strength, where the propagating-channel number is decreased from eight to essentially zero. The amplitude of the CF is almost 0.1 3 e 2 /h for the whole data of the
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Fig. 2. (a) Gate configuration of the coupled dot is shown and both outermost QPCs are kept with three quantized channel width. (b) Typical CF in the low temperature magneto-conductance (MC) at T 5 0.1 K.
MR. In order to obtain real CF components, we subtract a suitable averaged MC at high temperatures from the raw data at low temperatures. The CF has been analyzed using a suitable FFT program using a conventional PC. Each amplitude of DG at 100 mK is almost similar for various gate voltages of the center QPC between the dots, as shown in Fig. 2b. The phase-breaking of electron waves in the coupled-dot system can be discussed using auto-correlation function analysis for the low field CF. In the case of a longer lifetime of the phase breaking, as shown in the previous study in the single dot system, many higher harmonics in the FFT power spectrum for the oscillating part of correlation function calculated from low field CFs exist [1]. Therefore, the spread behavior of the wave function over the two dots, as illustrated in Fig. 2a, must be analyzed by the FFT power spectrum of the CF depending upon the gate voltages. In this sense, the higher harmonics should be significant in the case of the wide spread or the gate opening. This indicates that FFT power analysis of the correlation function provides a powerful tool for investigating the coupling between dots in the array systems. The interference area, Sc , in the coupled dot is estimated from the correlation field DBc which is obtained from the auto-correlation function of the CF. The results are shown in the left of Fig. 3. This figure shows the dependence of Sc and DBc on the center-QPC gate voltage, which is varied from 2 0.6 to 2 3.0 V. The other QPCs of the dots are configured to support three quantized channels during these measurements. DBc decreases and Sc increases with opening the center QPC (lowering the gate voltage). We can determine another interference area, S, using a half width, DB, of the localization peak at zero magnetic field. In the right of Fig. 3, a similar dependence is seen based on analysis using localization peak at zero field. S also increases as the gate voltage decreases. These results may be related to an enhancement of the coupling between electron waves due to an increase of the spread of interference area over the two dots. FFT power spectrum indicates the coupling behavior more clearly. In order to compare above two results with FFT power spectrum, we plot above results together with FFT powers. Fig. 4 shows the dependence of the FFT power on the center-QPC gate voltage, which is varied from 2 0.6 to 2 3.0 V. Also the other QPCs of the dots are kept to three quantized channels. A large spread of the FFT power extends towards higher frequencies with lowering gate voltage at the center QPC. Interference areas of Sc and S from correlation function and weak localization, respectively, are almost similar as well as
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N. Aoki et al. / Microelectronic Engineering 63 (2002) 57 – 61
Fig. 3. (Left) Result from auto-correlation function and corresponding interference area; (right) from localization peak at zero magnetic fields.
the FFT power dependences on the gate voltage. However, S is slightly larger than Sc as shown in the figure. Although this difference is not clear at present, it may be related to the difference of electron wave dynamic between classical trajectory process and wave function scar in the coupled dots. Nevertheless, S and Sc spread widely and also the coupling become stronger with decreasing of the center gate voltage. A similar dependence has been observed in a previous study in the dot array systems [2]. Therefore, the above results also show a clear evidence of a specific spread of the coupling orbit over the whole dot region. Also, it shows that we can control a real coupling of coherent electron waves in the coupled dots system. Here, however, we try to analyze such coupling behavior in a slightly different size in such double dot configuration in order to compare with previous measurements. The behavior of this double dot system seems to be similar in the case of three- or four-dot configurations even at a slightly different sized dot. As shown in Fig. 4, a narrow frequency region at the lower frequencies around 20 T 21 is almost independent on the gate voltage of the center QPC and is a slightly different
Fig. 4. FFT power result together with the two interference areas of S and Sc .
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point from previous corrugation dot systems. This may suggest a classical behavior due to backscattering orbit [4] or an electron wave trapping in the inside of the individual dots.
4. Summary We have studied electron wave transport in a coupled open-dot system. Reproducible CFs are observed in the low temperature magneto-conductance at lower magnetic fields and a clear difference between previous three- or four-dot and double-dot systems has not been appeared as the strength of the inter-dot coupling is varied. FFT analysis and interference area from two analysis provides clear evidence of a specific spread of the coupling orbit over the whole two-dot region.
Acknowledgements The authors are very much indebted to Professors D.K. Ferry and J.P. Bird for valuable discussions on this study and for important comments and suggestions on the coupling of the dot array system. This work was supported by a JSPS-NSF (US) cooperative science program and also supported by CREST.
References [1] Y. Okubo, N. Sasaki, Y. Ochiai, J.P. Bird, K. Ishibashi, Y. Aoyagi, T. Sugano, D. Vasileska, R. Akis, D.K. Ferry, Physica B 246 / 247 (1998) 266. [2] M. Elhassan, J.P. Bird, A. Shailos, C. Prasad, R. Akis, D.K. Ferry, Y. Takagaki, L.-H. Lin, N. Aoki, Y. Ochiai, K. Ishibashi, Y. Aoyagi, Phys. Rev. B 64 (2001) 085325. [3] R. Akis, J.P. Bird, D.K. Ferry, D. Vasileska, Physica E 7 (2000) 745. [4] L.-H. Lin, N. Aoki, K. Nakao, A. Andresen, C. Prasad, F. Ge, J.P. Bird, D.K. Ferry, Y. Ochiai, K. Ishibashi, Y. Aoyagi, T. Sugano, Phys. Rev. B 60 (1999) R16299.
Low temperature conductance fluctuations in double-dot system N. Aoki a,c , D. Oonishi a , Y. Iwase a , K. Ishibashi b , Y. Aoyagi,b , Y. Ochiai a,c , * a
Department of Materials Technology, Chiba University, 1 -33 Yayoi, Inage, Chiba 263 -8522, Japan b Semiconductors Laboratory, Riken, 2 -1 Hirosawa, Wako, Saitama 351 -0198, Japan c Center for Frontier Electronics and Photonics, Chiba University, 1 – 33 Yayoi, Inage, Chiba 263 -8522, Japan
Abstract We have studied the low temperature conductance fluctuations in double-dot systems. The electron wave coupling between ballistic quantum dots has been investigated in a pair of coupled quantum dots defined by several independently-tunable quantum-point-contacts (QPCs). Reproducible conductance fluctuations are observed in the low temperature magneto-conductance at low magnetic fields and a clear difference between weak and strong couplings has been identified as the gap of the QPC corresponding to the inter-dot coupling is varied. This suggests clear evidence for a specific spread of the coupling orbit over the whole dot region. 2002 Elsevier Science B.V. All rights reserved. Keywords: Quantum dot; Magnetoconductance; Conductance fluctuations; Coupled dot
1. Introduction The transport properties of quantum-dot arrays are largely affected by the quantum interference of electron waves, even in open dot systems. We have studied the low-temperature magneto-resistance (MR) in a coupled dot array defined by doubly-corrugated quantum wires or several pairs of an independently controlled quantum point contacts (QPCs). By suitable control of the gate voltage, we were able to change the coupling between the individual dots [1,2]. The trapped state of electron waves in the dot can be discussed in terms of interference of the wave function [3]. Even in ballistic transports, considerably large conductance fluctuations (CFs) have been observed and the phase coherency can be explained by a correlation function analysis for the CF in MR [1]. In this study, we compare the interference behavior of previously studied three- or four-dot quantum-dot arrays and an individually controlled two-dot systems in order to clarify the coupling effect between individual dots, by analyzing the CF at lower magnetic fields.
* Corresponding author. Tel.: 1 81-43-290-3428; fax: 1 81-43-290-3427. E-mail address: [email protected] (Y. Ochiai). 0167-9317 / 02 / $ – see front matter PII: S0167-9317( 02 )00639-1
2002 Elsevier Science B.V. All rights reserved.
58
N. Aoki et al. / Microelectronic Engineering 63 (2002) 57 – 61
Fig. 1. (a) SEM photograph of the couple dot system in the 2DEG. (b) Schematic gate configuration of the coupled dot.
2. Experimental Dot arrays were formed with the standard split-gate technique on the surface of a high mobility GaAs–AlGaAs wafer. In previous measurements of three- and four-dot systems, the low temperature MR has been studied by controlling only a single gate voltage [2]. However, in the present work, we introduce a two-dot coupling system having individually controllable unit dots. The two-dot system consists of five strip gates that lie opposite to a two-open-mouth gate so that three QPCs and two plungers are constructed. The system is regarded as an almost-symmetric two-dot system. The size of each unit dot is slightly different but is of the order of 1 mm [2], and can be individually controlled by setting into a suitable gate voltage at the five strip gates. Also, we can independently control the coupling strength between the two dots by changing the gate voltage of the center QPC. The schematic configuration of the two-dot system is shown in Fig. 1. Fig. 1a shows the SEM photograph of the coupled dot sample and Fig. 1b shows the schematic gate configuration. The electron density and mobility in the two-dimensional electron gas (2DEG) are 3.88 3 10 15 m 22 and 80 m 2 / V s, respectively. Designed sizes of the left and right dots are 1.0 3 1.0 and 1.2 3 1.2 mm 2 , respectively. As shown in the figure, we can independently control all the gates consisting of the dots. The low temperature transport measurements were performed in the He 3 –He 4 dilution refrigerator with a low power excitation using a lock-in amplifier system.
3. Results and discussion The low field CFs, DG, are obtained from the low temperature MR near zero field are shown in Fig. 2b. The gate voltage of the upper gate is 2 2.0 V, that of the lower left and right are 2 0.8 V and 2 0.92 V, respectively, in order to form three quantized channel width. The center QPC is swept from 2 0.6 to 2 3.0 V to vary the coupling strength, where the propagating-channel number is decreased from eight to essentially zero. The amplitude of the CF is almost 0.1 3 e 2 /h for the whole data of the
N. Aoki et al. / Microelectronic Engineering 63 (2002) 57 – 61
59
Fig. 2. (a) Gate configuration of the coupled dot is shown and both outermost QPCs are kept with three quantized channel width. (b) Typical CF in the low temperature magneto-conductance (MC) at T 5 0.1 K.
MR. In order to obtain real CF components, we subtract a suitable averaged MC at high temperatures from the raw data at low temperatures. The CF has been analyzed using a suitable FFT program using a conventional PC. Each amplitude of DG at 100 mK is almost similar for various gate voltages of the center QPC between the dots, as shown in Fig. 2b. The phase-breaking of electron waves in the coupled-dot system can be discussed using auto-correlation function analysis for the low field CF. In the case of a longer lifetime of the phase breaking, as shown in the previous study in the single dot system, many higher harmonics in the FFT power spectrum for the oscillating part of correlation function calculated from low field CFs exist [1]. Therefore, the spread behavior of the wave function over the two dots, as illustrated in Fig. 2a, must be analyzed by the FFT power spectrum of the CF depending upon the gate voltages. In this sense, the higher harmonics should be significant in the case of the wide spread or the gate opening. This indicates that FFT power analysis of the correlation function provides a powerful tool for investigating the coupling between dots in the array systems. The interference area, Sc , in the coupled dot is estimated from the correlation field DBc which is obtained from the auto-correlation function of the CF. The results are shown in the left of Fig. 3. This figure shows the dependence of Sc and DBc on the center-QPC gate voltage, which is varied from 2 0.6 to 2 3.0 V. The other QPCs of the dots are configured to support three quantized channels during these measurements. DBc decreases and Sc increases with opening the center QPC (lowering the gate voltage). We can determine another interference area, S, using a half width, DB, of the localization peak at zero magnetic field. In the right of Fig. 3, a similar dependence is seen based on analysis using localization peak at zero field. S also increases as the gate voltage decreases. These results may be related to an enhancement of the coupling between electron waves due to an increase of the spread of interference area over the two dots. FFT power spectrum indicates the coupling behavior more clearly. In order to compare above two results with FFT power spectrum, we plot above results together with FFT powers. Fig. 4 shows the dependence of the FFT power on the center-QPC gate voltage, which is varied from 2 0.6 to 2 3.0 V. Also the other QPCs of the dots are kept to three quantized channels. A large spread of the FFT power extends towards higher frequencies with lowering gate voltage at the center QPC. Interference areas of Sc and S from correlation function and weak localization, respectively, are almost similar as well as
60
N. Aoki et al. / Microelectronic Engineering 63 (2002) 57 – 61
Fig. 3. (Left) Result from auto-correlation function and corresponding interference area; (right) from localization peak at zero magnetic fields.
the FFT power dependences on the gate voltage. However, S is slightly larger than Sc as shown in the figure. Although this difference is not clear at present, it may be related to the difference of electron wave dynamic between classical trajectory process and wave function scar in the coupled dots. Nevertheless, S and Sc spread widely and also the coupling become stronger with decreasing of the center gate voltage. A similar dependence has been observed in a previous study in the dot array systems [2]. Therefore, the above results also show a clear evidence of a specific spread of the coupling orbit over the whole dot region. Also, it shows that we can control a real coupling of coherent electron waves in the coupled dots system. Here, however, we try to analyze such coupling behavior in a slightly different size in such double dot configuration in order to compare with previous measurements. The behavior of this double dot system seems to be similar in the case of three- or four-dot configurations even at a slightly different sized dot. As shown in Fig. 4, a narrow frequency region at the lower frequencies around 20 T 21 is almost independent on the gate voltage of the center QPC and is a slightly different
Fig. 4. FFT power result together with the two interference areas of S and Sc .
N. Aoki et al. / Microelectronic Engineering 63 (2002) 57 – 61
61
point from previous corrugation dot systems. This may suggest a classical behavior due to backscattering orbit [4] or an electron wave trapping in the inside of the individual dots.
4. Summary We have studied electron wave transport in a coupled open-dot system. Reproducible CFs are observed in the low temperature magneto-conductance at lower magnetic fields and a clear difference between previous three- or four-dot and double-dot systems has not been appeared as the strength of the inter-dot coupling is varied. FFT analysis and interference area from two analysis provides clear evidence of a specific spread of the coupling orbit over the whole two-dot region.
Acknowledgements The authors are very much indebted to Professors D.K. Ferry and J.P. Bird for valuable discussions on this study and for important comments and suggestions on the coupling of the dot array system. This work was supported by a JSPS-NSF (US) cooperative science program and also supported by CREST.
References [1] Y. Okubo, N. Sasaki, Y. Ochiai, J.P. Bird, K. Ishibashi, Y. Aoyagi, T. Sugano, D. Vasileska, R. Akis, D.K. Ferry, Physica B 246 / 247 (1998) 266. [2] M. Elhassan, J.P. Bird, A. Shailos, C. Prasad, R. Akis, D.K. Ferry, Y. Takagaki, L.-H. Lin, N. Aoki, Y. Ochiai, K. Ishibashi, Y. Aoyagi, Phys. Rev. B 64 (2001) 085325. [3] R. Akis, J.P. Bird, D.K. Ferry, D. Vasileska, Physica E 7 (2000) 745. [4] L.-H. Lin, N. Aoki, K. Nakao, A. Andresen, C. Prasad, F. Ge, J.P. Bird, D.K. Ferry, Y. Ochiai, K. Ishibashi, Y. Aoyagi, T. Sugano, Phys. Rev. B 60 (1999) R16299.