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Erasable electrostatic lithography
Available online at www.sciencedirect.com
Physica E 22 (2004) 717 – 720 www.elsevier.com/locate/physe
Erasable electrostatic lithography Rolf Crook∗ , Abi C. Graham, Charles G. Smith, Ian Farrer, Harvey E. Beere, David A. Ritchie Cavendish Laboratory, Semiconductor Physics Group, Department of Physics, University of Cambridge, Madingley Road, Cambridge CB3 OHE, UK
Abstract We report a new lithography called erasable electrostatic lithography (EEL) where patterns of charge are drawn on a GaAs surface with a scanning probe. The charge locally depletes electrons from a subsurface 2D electron system to de1ne any quantum component. Crucially, EEL is performed in the same low-temperature high-vacuum environment required for measurement, so patterning, measurement, and device modi1cation are made during a single cool down. This vastly reduces the measurement-lithography cycle time compared to other lithographic techniques such as electron-beam and local oxidation by atomic force microscopy. EEL is particularly productive where device geometry is of interest, such as investigations of the 0.7 anomaly and chaotic electron trajectories in quantum billiards. ? 2003 Elsevier B.V. All rights reserved. PACS: 73.23.−b; 05.45.AC; 07.79.−v Keywords: Erasable electrostatic lithography; 1D Electron system; Quantum dot; Quantum billiard
Erasable electrostatic lithography (EEL) [1] is a new lithography where patterns of charge are drawn on a GaAs surface with a scanning probe [2], and erased with light. EEL is performed in the same environment as measurement, so device geometry can be modi1ed during an experiment. We demonstrate the unique productivity of EEL by drawing, characterising, and then erasing a series of quantum components [3,4], all during a single cool down. The wafer incorporates a 2D electron system (2DES), with electron mobility 5 × 106 cm2 V−1 s−1 anddensity3:1×1011 cm−2 ,formed at a GaAs/AlGaAs heterojunction 97 nm beneath the surface. Electronbeam fabricated metal surface electrodes are biased to
∗
Corresponding author. Tel./fax: +44-1223-337-271. E-mail address: [email protected] (R. Crook).
−1 V to de1ne a 5 m long quantum wire. EEL components are then drawn on the GaAs surface either between or adjacent to the electrodes. Components drawn between the electrodes are characterised by plotting the wire conductance G against the electrode bias Vg . When the components are drawn adjacent to the electrodes, Vg is held constant throughout the experiment. Spots of charge drawn with a probe bias less than −5 V deplete underlying 2DES electrons. The charge is erased locally with a probe bias of +3 V, or globally by illuminating the sample with a red light emitting diode. Charge patterns drawn with a negative probe bias persist unchanged for at least 1 week, whereas patterns drawn with a positive bias decay in a few hours. This electrostatic behaviour is similar to that of surface electrodes, which suggests the EEL mechanism is the charging of GaAs surface states.
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.12.107
718
R. Crook et al. / Physica E 22 (2004) 717 – 720
EEL spot 2DES
surface electrode
surface electrode
surface electrode
quantum dot quantum dot 2DES
EEL spot
surface electrode
surface electrode
(a)
(b)
surface electrode
G (2e2/h)
4
G (2e2/h)
0.4
3
G (2e2/h)
3
(a)
2
0
1 2
1 0 0 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7
-1
Vg (V)
Vg (V)
Fig. 1(a) illustrates the EEL fabrication of a point contact [5,6]. Two spots of charge are drawn laterally about the wire centre with a probe bias of −5 V. The dashed line outlines the electron depletion showing how a narrowing, or point contact, is formed [7,8]. Fig. 1(b) plots G against Vg before and after the EEL fabrication. Before fabrication, the wire potential is evidently disordered as no conductance plateaus are seen. After EEL fabrication, several plateaus are observed quantized in units of 2 e2 =h showing that a 1D electron system has been created by a point contact. A feature near 0:7 × 2 e2 =h is observed [9]. Fig. 2(a) and (b) illustrates the EEL fabrication of a quantum dot [10–12] by two methods. A small quantum dot is obtained when two spots of charge
∆G (2e2/h)
Fig. 1. (a) Point contact fabricated by EEL at the centre of a quantum wire. The dashed line outlines the 2DES electron depletion. (b) Plots of quantum wire conductance against electrode bias. The leftmost plot characterises the original wire (oDset by +0:6 V). The rightmost plot characterises the EEL fabricated point contact. A 1 kE series resistance has been subtracted from both plots.
0.5
G (2e2/h)
(b)
-1.5
(c)
0.2
0 0
(d)
1 Vt (V)
2 0
(e)
Fig. 2. (a) Small quantum dot fabricated by EEL local erasure of a barrier. (b) Large quantum dot fabricated between two adjacent EEL point contacts. (c) Conductance of the quantum wire after EEL fabrication of a small quantum dot (solid) and a large quantum dot (dashed). (d) Conductance of the quantum wire as a function of the probe bias. Coulomb blockade oscillations are highlighted. (e) Scanned gate microscope (SGM) image of the small quantum dot. A 1 kE series resistance has been subtracted from all plots. Scale bars are 1 m.
are drawn to create a barrier and a spot of charge then erased from the centre by applying +3 V to the tip. The solid line in Fig. 2(c) characterises the small quantum dot. In addition to poorly de1ned conductance plateau, three small oscillations are seen below 2 e2 =h in the tunnelling regime. These oscillations are due to Coulomb blockade where electrons
R. Crook et al. / Physica E 22 (2004) 717 – 720
surface electrode quantum billiard 2DES
surface electrode 2DES
EEL lines
(a)
0.54
G (2e2/h)
0.52 0.50 0.48 0.46 0.44
0 (b)
10
20
30
40 50 B (mT)
70
80
90 100
39.5 mT
31.5 mT
(c)
60
(d)
Fig. 3. (a) Quantum billiard fabricated by EEL adjacent to a quantum wire. The dashed line outlines the 2DES electron depletion. (b) Plot of device conductance against perpendicular magnetic 1eld. (c and d) High-resolution conductance images of the quantum billiard. A 1 kE series resistance has been subtracted from the plot. Scale bars are 1 m.
can only tunnel through the dot barriers when the dot energy level matches the reservoir Fermi energy [10,11]. The dot conductance is periodic in units of e when plotted against Cg Vg , where Cg is the constant
719
capacitance between the gates and the dot. The oscillation period measured from Fig. 2(c) is Vg =11 mV, so Cg = e=Vg = 1:5 × 10−17 F. The total dot capacitance, including an approximate 2 × 10−17 F between the 2DES and the dot, is therefore Ct =3:5×10−17 F. The charging energy, which is the energy required to add an electron to the dot, is equal to Ec =e2 =2Ct =2 meV. By comparison with similar dots [11], the dot radius is approximately 50 nm and contains about 20 electrons. Only three oscillations are seen because changing Vg soon takes the dot barriers beyond their tunnelling regime. With the gates biased to observe Coulomb blockade and the probe positioned 100 nm above the dot, the probe bias Vt was swept and the resulting plot shown in Fig. 2(d). Five Coulomb blockade oscillations are seen with a period of about 300 mV which corresponds to a probe to dot capacitance of 5 × 10−19 F. To draw larger EEL quantum dots, a diDerent charge pattern is used. Two adjacent narrow point contacts are drawn, as shown in Fig. 2(b), to de1ne a central quantum dot with an approximate radius of 250 nm. The dashed line in Fig. 2(c) characterises the large quantum dot. Conductance plateaus are lost or obscured, and irregular large-period oscillations are seen above 2 e2 =h. These oscillations are interpreted as length resonances where electrons are scattered back and forth between the dot barriers, and interfere either constructively or destructively depending on their path length [10,12]. The oscillations are observed as a function of Vg because the dot size, and therefore the path length, is weakly dependent on Vg . Scanned gate microscopy (SGM) [13–17] images are generated by scanning a biased probe over a quantum component, while the device conductance is recorded to determine the colour of the associated pixel. SGM images highlight regions which determine the device conductance such as microconstrictions [7,8] or quantum dots [14,15]. Therefore, SGM is a useful tool to spatially characterise EEL de1ned quantum components and conveniently requires no modi1cation to the apparatus. Fig. 2(e) presents an SGM image made after a small quantum dot was drawn using the pattern of Fig. 2(a). As the probe passes the dot, the dot barriers are reduced because the scanning probe is biased more positive than the probe bias used to draw the EEL barrier spots, and the wire conductance increases. Halos seen in the SGM
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R. Crook et al. / Physica E 22 (2004) 717 – 720
image are Coulomb blockade oscillations observed as a function of probe to dot separation and therefore probe to dot capacitance which is a predominantly radial function [14,15]. A row of closely spaced EEL spots de1nes a linear barrier, or line, in the 2DES. We demonstrate EEL lines by drawing the 1:4 m × 2:9 m quantum billiard [18–24] illustrated in Fig. 3(a). The EEL spots are separated by 100 nm and drawn with a probe bias of −5 V. Further EEL spots tune the billiard entrance and exit to each transmit one degenerate 1D subband (n = 2) which maximises the fractal dimension [20]. For the duration of the experiment, the surface electrode bias is taken to −1 V to deplete underlying electrons and so separate the billiard source and drain 2DES regions. Fig. 3(b) plots billiard conductance against perpendicular magnetic 1eld. Note that the structure is reproducible and is not noise. The structure is symmetric about 0 mT where weak localisation suppresses conductance. A box counting algorithm con1rms that the data is fractal with dimension 1.44, which indicates that this quantum billiard is a chaotic system [21–23]. Fourier analysis reveals magnetic periodicities from 3 to 6 mT, depending upon the magnetic 1eld. Such periodicities are understood to be the signature of classical orbits with associated theoretical scarred wave functions [25–27]. The classical orbits have a well-de1ned area, which is the origin of the observed AB-like magnetic periodicity. For example, a strong 3 mT periodicity is observed at low 1elds, which corresponds to an AB-area A=fh=e =1:4 m2 . Figs. 3(c) and (d) present SGM conductance images at 31.5 and 39:5 mT, made by scanning the probe 100 nm above the sample surface while the billiard conductance is recorded to determine the colour of the associated image pixel. During imaging the probe bias is small to ensure the shape of the billiard does not change. Structure in the images, which is seen on many
length scales and turns out to be fractal, is interpreted as electron interference [3] due to wavelength-scale modi1cations to electron trajectories. References [1] R. Crook, et al., Nature 424 (2003) 751. [2] S.H. Tessmer, P.I. Glicofridis, R.C. Ashoori, L.S. Levitov, M.R. Melloch, Nature 392 (1998) 51. [3] C.W.J. Beenakker, H. van Houten, in: H. Ehrenreich, D. Turnbull (Eds.), Solid State Physics, Vol. 44, Academic Press, New York, 1991. [4] C.G. Smith, Rep. Progress Phys. 59 (1996) 235. [5] B.J. van Wees, et al., Phys. Rev. Lett. 60 (1988) 848. [6] D.A. Wharam, et al., J. Phys. C 21 (1988) L209. [7] R. Crook, C.G. Smith, M.Y. Simmons, D.A. Ritchie, Physica E 12 (2002) 695. [8] A.A. Starikov, et al., arxiv.org/abs/cond-mat/0206013 (2002). [9] K.J. Thomas, et al., Phys. Rev. B 58 (1998) 4846. [10] U. Meirav, M.A. Kastner, S.J. Wind, Phys. Rev. Lett. 65 (1990) 771. [11] M. Field, et al., Phys. Rev. Lett. 70 (1993) 1311. [12] C.G. Smith, et al., J. Phys. C 21 (1988) L893. [13] M.A. Eriksson, et al., Appl. Phys. Lett. 69 (1996) 671. [14] M.T. Woodside, P.L. McEuen, Science 296 (2002) 1098. [15] R. Crook, et al., Phys. Rev. B 66 (2002) 121301. [16] T. Ihn, et al., Physica E 12 (2002) 691. [17] M.A. Topinka, et al., Nature 410 (2001) 183. [18] R. Crook et al., Phys. Rev. Lett. 91 (2003) 246803. [19] C.M. Marcus, A.J. Rimberg, R.M. Westervelt, P.F. Hopkins, A.C. Gossard, Phys. Rev. Lett. 69 (1992) 506. [20] A.P. Micolich, et al., Phys. Rev. Lett. 87 (2001) 036802. [21] A.S. Sachrajda, et al., Phys. Rev. Lett. 80 (1998) 1948. [22] R. Ketzmerick, Phys. Rev. B 54 (1996) 10841. [23] R.A. Jalabert, H.U. Baranger, A.D. Stone, Phys. Rev. Lett. 65 (1990) 2442. [24] R.P. Taylor, et al., Phys. Rev. Lett. 78 (1997) 1952. [25] R. Akis, D.K. Ferry, J.P. Bird, Phys. Rev. Lett. 79 (1997) 123. [26] J.P. Bird, et al., Phys. Rev. Lett. 82 (1999) 4691. [27] I.V. Zozoulenko, R. Schuster, K.F. Berggren, K. Ensslin, Phys. Rev. B 55 (1997) R10209.
Physica E 22 (2004) 717 – 720 www.elsevier.com/locate/physe
Erasable electrostatic lithography Rolf Crook∗ , Abi C. Graham, Charles G. Smith, Ian Farrer, Harvey E. Beere, David A. Ritchie Cavendish Laboratory, Semiconductor Physics Group, Department of Physics, University of Cambridge, Madingley Road, Cambridge CB3 OHE, UK
Abstract We report a new lithography called erasable electrostatic lithography (EEL) where patterns of charge are drawn on a GaAs surface with a scanning probe. The charge locally depletes electrons from a subsurface 2D electron system to de1ne any quantum component. Crucially, EEL is performed in the same low-temperature high-vacuum environment required for measurement, so patterning, measurement, and device modi1cation are made during a single cool down. This vastly reduces the measurement-lithography cycle time compared to other lithographic techniques such as electron-beam and local oxidation by atomic force microscopy. EEL is particularly productive where device geometry is of interest, such as investigations of the 0.7 anomaly and chaotic electron trajectories in quantum billiards. ? 2003 Elsevier B.V. All rights reserved. PACS: 73.23.−b; 05.45.AC; 07.79.−v Keywords: Erasable electrostatic lithography; 1D Electron system; Quantum dot; Quantum billiard
Erasable electrostatic lithography (EEL) [1] is a new lithography where patterns of charge are drawn on a GaAs surface with a scanning probe [2], and erased with light. EEL is performed in the same environment as measurement, so device geometry can be modi1ed during an experiment. We demonstrate the unique productivity of EEL by drawing, characterising, and then erasing a series of quantum components [3,4], all during a single cool down. The wafer incorporates a 2D electron system (2DES), with electron mobility 5 × 106 cm2 V−1 s−1 anddensity3:1×1011 cm−2 ,formed at a GaAs/AlGaAs heterojunction 97 nm beneath the surface. Electronbeam fabricated metal surface electrodes are biased to
∗
Corresponding author. Tel./fax: +44-1223-337-271. E-mail address: [email protected] (R. Crook).
−1 V to de1ne a 5 m long quantum wire. EEL components are then drawn on the GaAs surface either between or adjacent to the electrodes. Components drawn between the electrodes are characterised by plotting the wire conductance G against the electrode bias Vg . When the components are drawn adjacent to the electrodes, Vg is held constant throughout the experiment. Spots of charge drawn with a probe bias less than −5 V deplete underlying 2DES electrons. The charge is erased locally with a probe bias of +3 V, or globally by illuminating the sample with a red light emitting diode. Charge patterns drawn with a negative probe bias persist unchanged for at least 1 week, whereas patterns drawn with a positive bias decay in a few hours. This electrostatic behaviour is similar to that of surface electrodes, which suggests the EEL mechanism is the charging of GaAs surface states.
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.12.107
718
R. Crook et al. / Physica E 22 (2004) 717 – 720
EEL spot 2DES
surface electrode
surface electrode
surface electrode
quantum dot quantum dot 2DES
EEL spot
surface electrode
surface electrode
(a)
(b)
surface electrode
G (2e2/h)
4
G (2e2/h)
0.4
3
G (2e2/h)
3
(a)
2
0
1 2
1 0 0 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7
-1
Vg (V)
Vg (V)
Fig. 1(a) illustrates the EEL fabrication of a point contact [5,6]. Two spots of charge are drawn laterally about the wire centre with a probe bias of −5 V. The dashed line outlines the electron depletion showing how a narrowing, or point contact, is formed [7,8]. Fig. 1(b) plots G against Vg before and after the EEL fabrication. Before fabrication, the wire potential is evidently disordered as no conductance plateaus are seen. After EEL fabrication, several plateaus are observed quantized in units of 2 e2 =h showing that a 1D electron system has been created by a point contact. A feature near 0:7 × 2 e2 =h is observed [9]. Fig. 2(a) and (b) illustrates the EEL fabrication of a quantum dot [10–12] by two methods. A small quantum dot is obtained when two spots of charge
∆G (2e2/h)
Fig. 1. (a) Point contact fabricated by EEL at the centre of a quantum wire. The dashed line outlines the 2DES electron depletion. (b) Plots of quantum wire conductance against electrode bias. The leftmost plot characterises the original wire (oDset by +0:6 V). The rightmost plot characterises the EEL fabricated point contact. A 1 kE series resistance has been subtracted from both plots.
0.5
G (2e2/h)
(b)
-1.5
(c)
0.2
0 0
(d)
1 Vt (V)
2 0
(e)
Fig. 2. (a) Small quantum dot fabricated by EEL local erasure of a barrier. (b) Large quantum dot fabricated between two adjacent EEL point contacts. (c) Conductance of the quantum wire after EEL fabrication of a small quantum dot (solid) and a large quantum dot (dashed). (d) Conductance of the quantum wire as a function of the probe bias. Coulomb blockade oscillations are highlighted. (e) Scanned gate microscope (SGM) image of the small quantum dot. A 1 kE series resistance has been subtracted from all plots. Scale bars are 1 m.
are drawn to create a barrier and a spot of charge then erased from the centre by applying +3 V to the tip. The solid line in Fig. 2(c) characterises the small quantum dot. In addition to poorly de1ned conductance plateau, three small oscillations are seen below 2 e2 =h in the tunnelling regime. These oscillations are due to Coulomb blockade where electrons
R. Crook et al. / Physica E 22 (2004) 717 – 720
surface electrode quantum billiard 2DES
surface electrode 2DES
EEL lines
(a)
0.54
G (2e2/h)
0.52 0.50 0.48 0.46 0.44
0 (b)
10
20
30
40 50 B (mT)
70
80
90 100
39.5 mT
31.5 mT
(c)
60
(d)
Fig. 3. (a) Quantum billiard fabricated by EEL adjacent to a quantum wire. The dashed line outlines the 2DES electron depletion. (b) Plot of device conductance against perpendicular magnetic 1eld. (c and d) High-resolution conductance images of the quantum billiard. A 1 kE series resistance has been subtracted from the plot. Scale bars are 1 m.
can only tunnel through the dot barriers when the dot energy level matches the reservoir Fermi energy [10,11]. The dot conductance is periodic in units of e when plotted against Cg Vg , where Cg is the constant
719
capacitance between the gates and the dot. The oscillation period measured from Fig. 2(c) is Vg =11 mV, so Cg = e=Vg = 1:5 × 10−17 F. The total dot capacitance, including an approximate 2 × 10−17 F between the 2DES and the dot, is therefore Ct =3:5×10−17 F. The charging energy, which is the energy required to add an electron to the dot, is equal to Ec =e2 =2Ct =2 meV. By comparison with similar dots [11], the dot radius is approximately 50 nm and contains about 20 electrons. Only three oscillations are seen because changing Vg soon takes the dot barriers beyond their tunnelling regime. With the gates biased to observe Coulomb blockade and the probe positioned 100 nm above the dot, the probe bias Vt was swept and the resulting plot shown in Fig. 2(d). Five Coulomb blockade oscillations are seen with a period of about 300 mV which corresponds to a probe to dot capacitance of 5 × 10−19 F. To draw larger EEL quantum dots, a diDerent charge pattern is used. Two adjacent narrow point contacts are drawn, as shown in Fig. 2(b), to de1ne a central quantum dot with an approximate radius of 250 nm. The dashed line in Fig. 2(c) characterises the large quantum dot. Conductance plateaus are lost or obscured, and irregular large-period oscillations are seen above 2 e2 =h. These oscillations are interpreted as length resonances where electrons are scattered back and forth between the dot barriers, and interfere either constructively or destructively depending on their path length [10,12]. The oscillations are observed as a function of Vg because the dot size, and therefore the path length, is weakly dependent on Vg . Scanned gate microscopy (SGM) [13–17] images are generated by scanning a biased probe over a quantum component, while the device conductance is recorded to determine the colour of the associated pixel. SGM images highlight regions which determine the device conductance such as microconstrictions [7,8] or quantum dots [14,15]. Therefore, SGM is a useful tool to spatially characterise EEL de1ned quantum components and conveniently requires no modi1cation to the apparatus. Fig. 2(e) presents an SGM image made after a small quantum dot was drawn using the pattern of Fig. 2(a). As the probe passes the dot, the dot barriers are reduced because the scanning probe is biased more positive than the probe bias used to draw the EEL barrier spots, and the wire conductance increases. Halos seen in the SGM
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R. Crook et al. / Physica E 22 (2004) 717 – 720
image are Coulomb blockade oscillations observed as a function of probe to dot separation and therefore probe to dot capacitance which is a predominantly radial function [14,15]. A row of closely spaced EEL spots de1nes a linear barrier, or line, in the 2DES. We demonstrate EEL lines by drawing the 1:4 m × 2:9 m quantum billiard [18–24] illustrated in Fig. 3(a). The EEL spots are separated by 100 nm and drawn with a probe bias of −5 V. Further EEL spots tune the billiard entrance and exit to each transmit one degenerate 1D subband (n = 2) which maximises the fractal dimension [20]. For the duration of the experiment, the surface electrode bias is taken to −1 V to deplete underlying electrons and so separate the billiard source and drain 2DES regions. Fig. 3(b) plots billiard conductance against perpendicular magnetic 1eld. Note that the structure is reproducible and is not noise. The structure is symmetric about 0 mT where weak localisation suppresses conductance. A box counting algorithm con1rms that the data is fractal with dimension 1.44, which indicates that this quantum billiard is a chaotic system [21–23]. Fourier analysis reveals magnetic periodicities from 3 to 6 mT, depending upon the magnetic 1eld. Such periodicities are understood to be the signature of classical orbits with associated theoretical scarred wave functions [25–27]. The classical orbits have a well-de1ned area, which is the origin of the observed AB-like magnetic periodicity. For example, a strong 3 mT periodicity is observed at low 1elds, which corresponds to an AB-area A=fh=e =1:4 m2 . Figs. 3(c) and (d) present SGM conductance images at 31.5 and 39:5 mT, made by scanning the probe 100 nm above the sample surface while the billiard conductance is recorded to determine the colour of the associated image pixel. During imaging the probe bias is small to ensure the shape of the billiard does not change. Structure in the images, which is seen on many
length scales and turns out to be fractal, is interpreted as electron interference [3] due to wavelength-scale modi1cations to electron trajectories. References [1] R. Crook, et al., Nature 424 (2003) 751. [2] S.H. Tessmer, P.I. Glicofridis, R.C. Ashoori, L.S. Levitov, M.R. Melloch, Nature 392 (1998) 51. [3] C.W.J. Beenakker, H. van Houten, in: H. Ehrenreich, D. Turnbull (Eds.), Solid State Physics, Vol. 44, Academic Press, New York, 1991. [4] C.G. Smith, Rep. Progress Phys. 59 (1996) 235. [5] B.J. van Wees, et al., Phys. Rev. Lett. 60 (1988) 848. [6] D.A. Wharam, et al., J. Phys. C 21 (1988) L209. [7] R. Crook, C.G. Smith, M.Y. Simmons, D.A. Ritchie, Physica E 12 (2002) 695. [8] A.A. Starikov, et al., arxiv.org/abs/cond-mat/0206013 (2002). [9] K.J. Thomas, et al., Phys. Rev. B 58 (1998) 4846. [10] U. Meirav, M.A. Kastner, S.J. Wind, Phys. Rev. Lett. 65 (1990) 771. [11] M. Field, et al., Phys. Rev. Lett. 70 (1993) 1311. [12] C.G. Smith, et al., J. Phys. C 21 (1988) L893. [13] M.A. Eriksson, et al., Appl. Phys. Lett. 69 (1996) 671. [14] M.T. Woodside, P.L. McEuen, Science 296 (2002) 1098. [15] R. Crook, et al., Phys. Rev. B 66 (2002) 121301. [16] T. Ihn, et al., Physica E 12 (2002) 691. [17] M.A. Topinka, et al., Nature 410 (2001) 183. [18] R. Crook et al., Phys. Rev. Lett. 91 (2003) 246803. [19] C.M. Marcus, A.J. Rimberg, R.M. Westervelt, P.F. Hopkins, A.C. Gossard, Phys. Rev. Lett. 69 (1992) 506. [20] A.P. Micolich, et al., Phys. Rev. Lett. 87 (2001) 036802. [21] A.S. Sachrajda, et al., Phys. Rev. Lett. 80 (1998) 1948. [22] R. Ketzmerick, Phys. Rev. B 54 (1996) 10841. [23] R.A. Jalabert, H.U. Baranger, A.D. Stone, Phys. Rev. Lett. 65 (1990) 2442. [24] R.P. Taylor, et al., Phys. Rev. Lett. 78 (1997) 1952. [25] R. Akis, D.K. Ferry, J.P. Bird, Phys. Rev. Lett. 79 (1997) 123. [26] J.P. Bird, et al., Phys. Rev. Lett. 82 (1999) 4691. [27] I.V. Zozoulenko, R. Schuster, K.F. Berggren, K. Ensslin, Phys. Rev. B 55 (1997) R10209.