Coupled Quantum Dots in a Graphene-Based Two-Dimensional

Publication Date (Web): July 10, 2009. Copyright © 2009 American Chemical Society. * Corresponding author: Phone number: +81-29-860-4405. Fax number:...
78 downloads 5 Views 4MB Size
Download PDF
NANO LETTERS

Coupled Quantum Dots in a Graphene-Based Two-Dimensional Semimetal

2009 Vol. 9, No. 8 2891-2896

Satoshi Moriyama,*,† Daiju Tsuya,‡ Eiichiro Watanabe,‡ Shinya Uji,§ Maki Shimizu,| Takahiro Mori,| Tomohiro Yamaguchi,| and Koji Ishibashi| International Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan, Nanotechnology InnoVation Center, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan, AdVanced Nano Materials Laboratory, National Institute for Materials Science, 3-13 Sakura, Tsukuba, Ibaraki 305-0003, Japan, and AdVanced DeVice Laboratory, RIKEN, 2-1 Hirosawa, Saitama 351-0198, Japan Received April 9, 2009; Revised Manuscript Received June 8, 2009

ABSTRACT We present an experimental demonstration of a graphene-based double quantum dot system, which exhibits single-electron transport of two lateral quantum dots coupled in series. Low-temperature transport measurements revealed honeycomb charge stability diagrams with a varied (from weak to strong) interdot tunnel-coupling regime, and we have extracted the relevant parameters associated with the double quantum dot system. These results are important for the realization of integrated quantum circuits in graphene-based electronics.

The recent discovery of novel electron-transport characteristics in graphene and few-layer graphene (FLG) demonstrates that they are attractive two-dimensional (2D) conducting materials, not only as a new subject in lowdimensional physics but also as building blocks of novel quantum nanodevices.1-3 Graphene-based nanodevices, such as quantum wires or quantum dots, can be fabricated by carving them out of graphene or FLG sheets on a substrate.4-9 In addition, the spread 2D sheet structure may open a door to realizing an integrated quantum nanodevice system. The tunnel-coupled double quantum dot (DQD) is proposed as a basic system for future solid-state quantum circuits, such as quantum bits (single charge qubit system,10 an interacting two-electron spin qubit system11,12) and quantum cellular automata.13 The DQD system is intrinsically different from a single quantum dot because of the variable interdot tunnel coupling. Therefore, it is possible that more sophisticated quantum coupling devices can be realized. Recent advances in nanoscale fabrication techniques have made it possible to realize a DQD device with tunable

interdot coupling in carbon nanotubes14-17 and semiconductor materials.18-20 In particular, as a step toward the spin-based quantum computing devices, coherent electron-spin manipulation in a DQD has been performed on the GaAs/AlGaAs 2D electron gas.21,22 However, hyperfine interactions with nonzero nuclear spins in the semiconductor material limit the coherence of electron spins. Carbon materials, with their low nuclear spin concentration arising from the ∼99% natural abundance of 12C (zero nuclear spin), in addition to their weaker spin-orbit interaction, are expected to have longer spin coherence.12

* Corresponding author: Phone number: +81-29-860-4405. Fax number: +81-29-860-4706. E-mail address: [email protected]. † International Center for Materials Nanoarchitectonics, National Institute for Materials Science. ‡ Nanotechnology Innovation Center, National Institute for Materials Science. § Advanced Nano Materials Laboratory, National Institute for Materials Science. | Advanced Device Laboratory, RIKEN.

Graphene-family samples were prepared by micromechanical cleavage of graphite crystals and deposited on the surface of a silicon substrate with a d ) 290 nm thickness of oxidized silicon. Optical microscope contrast and Raman spectroscopy measurements can be used to identify the single-, double-, triple-, and few-layer graphene flakes on a substrate.23-27 From micro-Raman spectroscopy measure-

10.1021/nl9011535 CCC: $40.75 Published on Web 07/10/2009

 2009 American Chemical Society

Here, we demonstrate DQD devices in a graphene-based 2D semimetal, which exhibits single-electron transport on two lateral quantum dots coupled in series. Experimental charge stability diagrams from weak to strong interdot coupling regimes were observed, and the energy and capacitances characterizing the DQD system were obtained. Our results suggest an important step in the realization of integrated quantum devices in graphene-based nanoelectronics.

ments, we confirmed that the measured devices were fabricated on a triple-layer graphene (TLG) sheet. With electron-beam lithography, a DQD device structure was patterned into a thin ZEP resist (80 nm thickness) that protected chosen areas during the O2 reactive ion etching process. Next, Cr/Au (10/50 nm) metal was deposited on the TLG for source-drain and side-gate contact formation, again using electron-beam lithography. Graphene, a singleatomic layer of graphite, is a zero-gap semiconductor 2D system where electrons behave as Dirac fermions, i.e., massless relativistic particles. In contrast, the FLG is a semimetallic 2D system with an overlap between valence and conduction bands. Double-layer graphene has a very small overlap (a few millielectronvolts), but increasing the number of graphene layers increases the band overlap, so that theoretically they may behave as a semimetal.28 Indeed, our TLG appears to be a metallic (or narrow-gap semiconducting) 2D system, as shown below. Electrical transport measurements were carried out in a pumped He cryostat at 4 and 1.5 K and in a dilution refrigerator with a base temperature of 30 mK and electron temperature of ∼240 mK, which allowed us to use the results of one as a reference for the other under the same measurement conditions.29 We measured the dc current (I) through the TLG double-dot device by applying a dc bias voltage (Vsd). Figure 1a shows a scanning electron microscope (SEM) image of the fabricated TLG device. A highly p-doped Si substrate was used as a back gate (Vbg) and three lateral TLG side gates, labeled the center gate (Vcg) and Vg1(2), which were expected to couple to the individual dot 1(2). Quantum dots were designed in a triangular shape (dot 1, S1 ≈ 0.004 µm2; dot 2, S2 ≈ 0.005 µm2), and the narrow constriction was about 20 nm in length and 15 nm in width, as shown in Figure 1b. These constrictions are expected to act as tunnel barriers due to the quantum size effect. Figure 1c shows the current through the DQD device as a function of Vbg, measured at 4 K. The current curve shows a smooth background with a minimum around the charge neutrality point at Vbg ∼ 4 V and has small structures with fluctuations (black arrows) that appear to indicate Coulomb oscillations. In the inset of Figure 1c, Coulomb oscillations are observable at 1.5 K on the smooth background. However, the current pinch-off that has usually been observed in singlelayer graphene nanostructures4-7 is not observed in the present sample. At the base temperature in a dilution refrigerator (Te ) 240 mK), Coulomb oscillations are again observed, and the current pinch-off was again not observed. These results do not provide clear evidence of the energy gap that causes complete current suppression; therefore, the present TLG could be a metallic (or narrow-gap semiconducting) 2D system. Besides, no clear evidence for a single carrier (electron or hole) or mixed carriers (electron and hole) was observed. Hereafter, unless otherwise mentioned, Vbg is fixed at 4.0 V, near the charge neutrality point. Figure 2a shows current as a function of center-gate voltage (Vcg); over 150 oscillations are observed. We find paired peaks in some Vcg regions, as shown in the inset of 2892

Figure 1. Graphene-based double dot device. (a) SEM image of the measured device with electrode assignment. Bright areas show etched triple-layer graphene. The two isolated islands (quantum dots) are connected via two narrow constrictions to wide source and drain regions. Three lateral TLG side gates are fabricated close to the quantum dots. (b) Schematic picture of the device with dimensions of relevant structures. (c) Back-gate voltage (Vbg) dependence of the device for Vsd ) 1.4 mV at T ) 4 K and for Vsd ) 0.35 mV at 1.5 K (inset). Current minimum is observed around Vbg ∼ 4 V at 4 K. Other gate voltages are grounded. At lower temperature (inset), Coulomb blockade oscillations are observed near the current minimum region.

Figure 2a. The peak splitting of a paired peak was defined as ∆Vs, and the spacing to the nearest-neighbor paired peak was defined as ∆Vp. In the inset of Figure 2a, ∆Vs ∼ 40 mV and ∆Vp ∼ 100 mV, and the deviation around the averaged value is smaller than 5 mV in both cases. These characteristics show evidence of the electrostatic coupling of the two dots, which is similar to the data reported on the GaAs/ Nano Lett., Vol. 9, No. 8, 2009

Figure 2. Formation of the series-coupled double-dot system. (a) Center-gate voltage dependence of the current at fixed Vbg ) 4.0 V, Vg1 ) Vg2 ) 0 V, and Vsd ) 25 µV at base temperature (Te ) 240 mK). Inset: Magnified view of the oscillations in some gate voltage ranges show paired peaks, indicating the formation of a coupled double-dot system. (b) Histograms of the nearest-neighbor peak spacings in Figure 2a. (c, d) Current as a function of the side gates of each dot for Vg1 (c) and Vg2 (d) at Vbg ) 4.0 V, Vcg ) 9.0 V, and Vsd ) 25 µV.

AlGaAs DQD system.30,31 In other gate voltage ranges, regular Coulomb oscillations are observed, but the distribution of the peak spacings is larger than that in the pairedpeak regions. Figure 2b shows the histogram of the nearestneighbor peak spacings, obtained from Figure 2a. The observed large distribution could be explained by a change in the interdot tunnel coupling tuned by Vcg. More details are provided below using a charge stability diagram of the DQD system. Note that the pinch-off characteristic was not observed over a wide Vcg range. After this, Vcg was changed to 9.0 V, near the current suppression region in Figure 2a. Then, we measured the current as a function of each gate voltage Vg1(2) for dot 1(2); the results are shown in parts c and d of Figure 2, respectively. In each case, the current oscillations are strongly suppressed in similar gate-voltage regions, ∼0 V < Vg1, Vg2 < ∼1.7 V. Further, irregular peak spacings are observed. This feature is interpreted as stochastic Coulomb blockade through a DQD.32,33 These results suggest that a DQD is formed in TLG islands. Now, we focus on measurements in the range of 0 V < Vg1, Vg2 < 1.7 V. Figure 3a shows an experimental charge stability diagram with a small source-drain bias (Vsd ) 50 µV < 4 kBTe). Honeycomb structures, which are characteristic of coupled quantum dots, are clearly observed in the transconductance (dI/dVg1) as a function of Vg1 and Vg2. Large dI/dVg1 values are observed near the vertices of the hexagonal lattice. These results indicate that Vg1 and Vg2 predominantly couple to dots 1 and 2, respectively, and the two dots are weakly coupled. In this case, the number of charges (n, m) on the two dots is constant in each honeycomb cell. In a series-coupled DQD system, the resonant current flows at the vertices where the three charge states in a DQD (e.g., (n, m), (n + 1, m), and (n, m + 1)) are degenerated, so-called triple points. The offresonant tunneling process may occur when the energy levels of one of the two dots align with its neighboring leads. This is called a cotunneling process and occurs at the edge of the Nano Lett., Vol. 9, No. 8, 2009

honeycomb cells.34 In Figure 3a, the cotunneling current along the edges of the honeycomb cells is suppressed. However, the transconductance near the triple points is slightly blurred, connecting to the curved edges. If electrostatic coupling is completely dominant, the triple points become a “sharpened point” in the linear transport regime. In contrast, when the effect of quantum tunnel coupling is not negligible, they become blurred along the edges of the honeycomb cells. Therefore, the results indicate that the DQD system is in a predominantly electrostatic-coupling regime with weak tunnel coupling. At large source-drain bias, the triple points should change to a triangular shape, and flow of resonant current should be possible only along the one side of the triangle that connects to the original triple points.34 Figure 3b focuses on one of the vertices of the honeycomb structure in the weak tunnel-coupling region, measured at Vsd ) 800 µV (>4kBTe). Regions where the current flows are of the same order in size as the triangular shapes are indicated by solid lines. However, the boundary of the triangle is not completely obvious in the data, and the regions are broadened around the original triple point. These results are due to the effect of quantum tunnel coupling16 and the increase in the cotunneling current. The cotunneling current that flows along the edge of the honeycomb cells increases rapidly as Vsd is increased. Therefore, the triangular shape is not clearly identified. For the same reason, the excitation spectra could not be investigated in the present sample. Recent studies of the transport through quantum dots in single-layer graphene exhibit similar behavior, in which the conductivity through graphene constrictions increases rapidly under a higher applied bias.8 From the dimensions of the triangles δVg1 and δVg2, we can deduce the conversion factors R1(2) for the dot1(2) using the relation R1(2) ) |eVsd|/δVg1(2). The conversion factors are 2893

Figure 3. Charge stability diagrams for series-coupled quantum dots. (a) Color scale plot of the transconductance (dI/dVg1) calculated from dc current (I) as a function of Vg1 and Vg2 at Vsd ) 50 µV. We measured the current as a function of Vg1 with stepwise variation of Vg2. White dashed lines show the honeycomb structure. Inset: A schematic charge stability diagram of the coupled quantum dots to obtain the relevant parameters, based on ref 34. (b) One of the vertices of the honeycomb structure at Vsd ) 800 µV. Solid red lines show the triangular shape. Dashed lines, which are extended from the triangles, are used to define δVg1 and δVg2. They are designated by using the conversion factor and the corresponding Vsd. For details, see the text. (c, d) Source-drain bias dependence of the current peak width of the cotunneling along the edges of the honeycomb structure, as a function of Vg1 (c) and Vg2 (d). (e) Capacitance model of a DQD system.

also deduced by the Vsd dependence of the cotunneling peak width with respect to each gate-voltage Vg1(2), which is shown in parts c and d of Figure 3. From the observed linear relation, we deduce conversion factors of R1 ≈ 0.07 and R2 ≈ 0.05 (meV/mV). Having understood the qualitative behavior of the sample, we estimate the energy parameters and capacitances associated with the DQD. The equivalent circuit and relevant parameters are shown in Figure 3e. All capacitances are obtained from the size and separation of the honeycomb cells and from the details of the triple point (Figure 3a inset) with the conversion factors obtained above. On the basis of the formula in ref 34 (eqs 2, 8, and 10 in section II) and the typical experimental honeycomb cells in Figure 3a, the capacitances are obtained as Cg1 ∼ 2.2 aF, Cg2 ∼ 2.9 aF, and Cm ∼ 10 aF, where Cm is the coupling capacitance between the dots. The total capacitances of dots 1 and 2 are C1 ∼ 30 aF and C2 ∼ 60 aF, respectively. The back-gate and center-gate capacitances (Cbg and Ccg) are estimated from the Coulomb peak spacing as a function of each voltage (∆Vbg and ∆Vcg), given as Cbg(cg) ) e/∆Vbg(cg). Both values are not determined precisely but are in a range of 1-4 aF because of the large distribution in peak spacing. However, the values are smaller by an order than the total 2894

capacitance, which means that the capacitances between the leads and the dot are the main contributors to the total capacitance. The back-gate capacitance to each dot can be estimated with the simple formula as Cbg ) πε0(ε + 1)D/ tan-1(4d(ε0 + 1)/(εD)), where ε is dielectric constant and D the 2D disk diameter.35 For the left and right dots with the effective D1 ∼ 70 nm, D2 ∼ 80 nm, respectively, Cbg for the left and right dots are calculated to be 6 and 7 aF. These values are in reasonable agreement with the experimental values given before. From the obtained capacitances, the charging energy EC1(2) and coupling energy Em are obtained as EC1 ∼ 6 meV, EC2 ∼ 3 meV, and Em ∼ 0.9 meV, respectively. Calculated values by the simple isolated 2D island model, EC ) e2/(4ε0εeffD), where εeff is the effective dielectric constant,36 EC1(2), becomes ∼20 meV. This might be a bit larger than the experimental values but could be explained by including the effects of surrounding contacts and the interdot coupling. Similar measurements have been carried out in different gate-voltage ranges within 0 V < Vg1, Vg2 < 1.7 V, and varied interdot couplings were observed. Panels a and b of Figure 4 show the charge stability diagrams with a fixed Vg2 range and different Vg1 ranges. The interdot coupling could be changed also by Vg1 and Vg2, because the side gates Nano Lett., Vol. 9, No. 8, 2009

controlled by the lateral TLG side gates that work on each dot independently to control their electrostatic potential. In addition to that, it turned out that tunnel coupling of both lead-to-dot and dot-to-dot could be tuned. In fact, charge stability diagrams from weak to strong interdot coupling regimes were demonstrated, and the charging energies and capacitances that characterize the DQD system were obtained. This new material system may become a promising candidate for possible integrated quantum computing devices with the expected absence of nuclear spins.

Figure 4. Demonstration of variable interdot coupling in the coupled quantum dots. Color scale plot of the transconductance (dI/dVg1) is calculated from dc current (I) as a function of Vg1 and Vg2 at Vsd ) 50 µV in different side-gate voltage ranges. The transconductance scale bar is the same as in Figure 3. We measured the current as a function of Vg1 with stepwise variation of Vg2. Stability diagram is shown for intermediate coupling (a), for a strong coupling regime (b), and finally, the diagram becomes an array of lines where a single dot is formed (c).

affected the central barrier through the existing capacitances between the side gates and the central barrier. In Figure 4a, the triple points are much more blurred than those in Figure 3a, indicating stronger coupling. In this case, the two dots are interacting through the larger quantum mechanical tunnel coupling, which resembles a two-atom molecule with covalent bonding.16,17 The separation of vertices becomes larger in Figure 4b, and finally, in Figure 4c, an array of lines is observed. In this case, the DQD behaves as a single dot. To evaluate the amount of interdot tunnel coupling, we can use the fractional peak splitting, F ) 2∆Vs/∆Vp, first introduced in refs,30,31 and.38 In the charge stability diagram, ∆Vs is defined as the diagonal splitting between vertices, and ∆Vp is defined as the distance between vertex pairs. F ) 1 corresponds to the strong coupling limit where the DQD behaves as a single dot, whereas F ) 0 corresponds to the weak coupling limit where the DQD behaves as two isolated dots. We find the value F ∼ 0.3 for Figure 3a, and F ∼ 0.6 and F ∼ 0.7 for Figure 4a and 4b, respectively. The values of ∆Vs and ∆Vp for Figure 3 and 4 are similar to those in the inset of Figure 2a. Therefore, the observed distribution of nearest-neighbor peak spacings (Figure 2b) could be explained by the change in the interdot coupling from F ) 0 to 1, so that, in the inset of Figure 2a, both charge states of the two dots are swept simultaneously with Vcg. In conclusion, we have demonstrated a series-coupled DQD device in a graphene-based 2D semimetal. The measured TLG appears to have no energy gap; at least, the energy gap might be smaller than the energy scale (∼10 meV) of the DQD system in the Coulomb blockade regime. The charge states in the DQD system can be Nano Lett., Vol. 9, No. 8, 2009

Acknowledgment. We thank Dr. Keiji Ono of RIKEN for useful discussions and suggestions. This work was supported by Grant-in-Aid for Young Scientists (A) (20686008), the Nanotechnology Network Program, the World Premier International Research Center (WPI) Initiative on Materials Nanoarchitectonics from the Ministry of Education, Culture, Sports, Science and Technology, and a research grant from the Murata Science Foundation. References (1) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Zhang, Y.; Dubonos, S. V.; Grigorieva, I. V.; Firsov, A. A. Science 2004, 306, 666–669. (2) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Katsnelson, M. I.; Grigorieva, I. V.; Dubonos, S. V.; Firsov, A. A. Nature 2005, 438, 197–200. (3) Zhang, Y.; Tan, Y.-W.; Stormer, H. L.; Kim, P. Nature 2005, 438, 201–204. ¨ zyilmaz, B.; Zhang, Y.; Kim, P. Phys. ReV. Lett. 2007, (4) Han, M. Y.; O 98, 206805. (5) Sols, F.; Guinea, F.; Castro Neto, A. H. Phys. ReV. Lett. 2007, 99, 166803. (6) Stampfer, C.; Gu¨ttinger, J.; Molitor, F.; Graf, D.; Ihn, T.; Ensslin, K. Appl. Phys. Lett. 2008, 92, 012102. (7) Stampfer, C.; Schurtenberger, E.; Molitor, F.; Gu¨ttinger, J.; Ihn, T.; Ensslin, K. Nano Lett. 2008, 8, 2378–2383. (8) Ponomarenko, L. A.; Schedin, F.; Katsnelson, M. I.; Yang, R.; Hill, E. W.; Novoselov, K. S.; Geim, A. K. Science 2008, 320, 356–358. (9) Gu¨ttinger, J.; Stampfer, C.; Molitor, F.; Graf, D.; Ihn, T.; Ensslin, K. New J. Phys. 2008, 10, 125029. (10) Barence, A.; Deutsch, D.; Ekert, A.; Jozsa, R. Phys. ReV. Lett. 1995, 74, 4083–4086. (11) Loss, D.; DiVincenzo, D. P. Phys. ReV. A 1998, 57, 120–126. (12) Trauzettel, B.; Bulaev, D. V.; Loss, D.; Burkard, D. Nat. Phys. 2007, 3, 192–196. (13) Lent, C. S.; Tougaw, P. D.; Porod, W. Appl. Phys. Lett. 1993, 62, 714–716. (14) Mason, N.; Biercuk, M. J.; Marcus, C. M. Science 2004, 303, 655– 658. (15) Sapmaz, S.; Meyer, C.; Beliczynski, P.; Jarillo-Herrero, P.; Kouwenhoven, L. P. Nano Lett. 2006, 6, 1350–1355. (16) Gra¨ber, M. R.; Coish, W. A.; Hoffmann, C.; Weiss, M.; Furer, J.; Oberholzer, S.; Loss, D.; Scho¨nenberger, C. Phys. ReV. B 2006, 74, 075427. (17) Jørgensen, H. I.; Grove-Rasmussen, K.; Wang, K.-Y.; Blackburn, A. M.; Flensberg, K.; Lindelof, P. E.; Williams, D. A. Nat. Phys. 2008, 4, 536–539. (18) Fasth, C.; Fuhrer, A.; Bjo¨rk, M. T.; Samuelson, L. Nano Lett. 2005, 5, 1487–1490. (19) Hu, Y.; Churchill, H. O. H.; Reilly, D. J.; Xiang, J.; Lieber, C. M.; Marcus, C. M. Nat. Nanotechnol. 2007, 2, 622–625. (20) Chan, V. C.; Buehler, T. M.; Ferguson, A. J.; McCamey, D. R.; Reilly, D. J.; Dzurak, A. S.; Clark, R. G. J. Appl. Phys. 2006, 100, 106104. (21) Koppens, F. H. L.; Buizert, C; Tielrooij, K. J.; Vink, I. T.; Nowack, K. C.; Meunier, T.; Kouwenhoven, L. P.; Vandersypen, L. M. K. Nature 2005, 442, 766–771. (22) Petta, J. R.; Johnson, A. C.; Taylor, J. M.; Laird, E. A.; Yacoby, A.; Lukin, M. D.; Marcus, C. M.; Hanson, M. P.; Gossard, A. C. Science 2005, 309, 2180–2184. 2895

(23) Ferrari, A. C.; Meyer, J. C.; Scardaci, V.; Casiraghi, C.; Lazzeri, M.; Mauri, F.; Piscanec, S.; Jiang, D.; Novoselov, K. S.; Roth, S.; Geim, A. K. Phys. ReV. Lett. 2006, 97, 187401. (24) Gupta, A.; Chen, G.; Joshi, P.; Tadigadapa, S.; Eklund, P. C. Nano Lett. 2006, 6, 2667–2673. (25) Graf, D.; Molitor, F.; Ensslin, K.; Stampfer, C.; Jungen, A.; Hierold, C.; Wirtz, L. Nano Lett. 2007, 7, 238–242. (26) Ni, Z. H.; Wang, H. M.; Kasim, J.; Fan, H. M.; Yu, T.; Wu, Y. H.; Feng, Y. P.; Shen, Z. X. Nano Lett. 2007, 7, 2758–2763. (27) Moriyama S.; Morita Y.; Watanabe E.; Tsuya D.; Uji S. Submitted . (28) Partoens, B.; Peeters, F. M. Phys. ReV. B 2006, 74, 075404. (29) Moriyama, S.; Fuse, T.; Suzuki, M.; Aoyagi, Y.; Ishibashi, K. Phys. ReV. Lett. 2005, 94, 186806. (30) Waugh, F. R.; Berry, M. J.; Mar, D. J.; Westervert, R. M. Phys. ReV. Lett. 1995, 75, 705–708. (31) Golden, J. M.; Halperin, B. I. Phys. ReV. B 1996, 53, 3893–3900.

2896

(32) Ruzin, I. M.; Chandrasekhar, V.; Levin, E. I.; Glazman, L. I. Phys. ReV. B 1992, 45, 13469–13478. (33) Kemerink, M.; Molenkamp, L. W. Appl. Phys. Lett. 1994, 65, 1012– 1014. (34) Van der Wiel, W. G.; Fanceschi, S. De.; Elzerman, J. M.; Fujisawa, T.; Tarucha, S.; Kouwenhoven, L. P. ReV. Mod. Phys. 2003, 75, 1– 22. (35) Gelmont, B.; Shur, M. S.; Mattauch, R. J. Solid-State Electron. 1995, 38, 731–734. (36) We used εeff ) (1(vacuum) + 4(SiO2))/2 ) 2.5 by taking into account the geometrical configuration.37 (37) Schnez, S.; Molitor, F.; Stampfer, C.; Gu¨ttinger, J.; Shorubalko, I.; Ihn, T.; Ensslin, K. Appl. Phys. Lett. 2009, 94, 012107. (38) Livermore, C.; Crouch, C. H.; Westervelt, R. M.; Campman, K. L.; Gossard, A. C. Science 1996, 274, 1332–1335.

NL9011535

Nano Lett., Vol. 9, No. 8, 2009