- Email: [email protected]
Tuning of tunneling rates in quantum dots using a quantum point contact
ARTICLE IN PRESS
Physica E 34 (2006) 500–503 www.elsevier.com/locate/physe
Tuning of tunneling rates in quantum dots using a quantum point contact M.C. Roggea,, C. Frickea, B. Harkea, F. Hohlsa,b, R.J. Hauga, W. Wegscheiderc a
Institut fu¨r Festko¨rperphysik, Universita¨t Hannover, 30167 Hannover, Germany Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, Great Britain c Angewandte und Experimentelle Physik, Universita¨t Regensburg, 93040 Regensburg, Germany
b
Available online 17 April 2006
Abstract We study the influence of asymmetric tunneling rates of a lateral quantum dot connected to source and drain leads. We use a quantum point contact (QPC) to detect charging events on the dot. Since the mean charge depends on the ratio of the tunneling rates the QPC can be used to detect symmetric configurations where both rates are equal. Thus, the transport properties of the dot can be investigated concerning this symmetry. We concentrate on the visibility of features that correspond to transitions with excited states. We interpret the results with the number of channels used for transport, the electron flow direction and the rate symmetry. r 2006 Elsevier B.V. All rights reserved. PACS: 73.63.Kv; 73.23.Hk; 72.20.My Keywords: Quantum dots; Quantum point contact; Charge detection; Tunneling rates; Excited states
During the last decades the research on semiconductor nanostructures has dealt with quantum point contacts (QPCs) and quantum dots (QDs) separately. QPCs were used as a realization of a quasi 1D system showing a quantized conductance [1]. Other effects like the 0.7 anomaly were observed [2]. QDs received great interest, because as a 0D system they were proposed as a realization of a qubit [3]. Many effects have been studied so far [4] but only in recent years both systems were combined to gain further information. The idea was to use a QPC as a charge detector for QDs. This was experimentally implemented in lateral structures using electron beam lithography [5–7] as well as local anodic oxidation with atomic force microscopes [8–10]. In this paper we demonstrate how the investigation of QD systems can benefit from the additional data measured with a QPC. The parameter space can be expanded beyond the limit of measurable transport, because charge detection Corresponding author. Tel.: +49 511 762 19002; fax: +49 511 762 2904. E-mail address: [email protected] (M.C. Rogge).
1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2006.03.022
does not rely on high tunneling rates. As long as charging takes place on the dot this can be measured. Apart from this quantitative enhancement the QPC can yield results which are new in quality. We found, that the charging of the dot depends not on the strength of the tunneling rates but on the ratio of the rates to source and drain leads. Thus, the implementation of the QPC enables us to detect the symmetry of these rates. This is of big relevance to understand effects that occur in transport. With the knowledge of the symmetry we investigate features of excited dot states and interpret the results in terms of tunneling rates influenced by the number of transport channels. The coupled device with a QD and a QPC is embedded in the inversion layer of a GaAs/AlGaAs heterostructure. The two-dimensional electron system (2DES) is located 34 nm below the surface. The sheet density is n ¼ 4:59 1015 m2 and the mobility is m ¼ 64:3 m2 =V s. The technique of local anodic oxidation (LAO) [11] is used to create the desired potential within the 2DES. A negative voltage is applied to the tip of an atomic force microscope (AFM) that touches the sample surface. With a thin water film as
ARTICLE IN PRESS M.C. Rogge et al. / Physica E 34 (2006) 500–503
change. Interestingly this shift is not constant for the whole measurement. For low gate voltages the steps shift to the right while for more positive voltages the steps appear shifted to the left compared to the solid line. This effect is caused by the ratio of the tunneling rates to source and drain GS and GD . With a strong asymmetry the mean charge that is related to a dot state is governed by the side with the stronger tunneling rate. Imagine GS 4GD . If the chemical potential of a dot state mN is above the chemical potential of the source mS the mean charge of this state is 0 because electrons leave the dot immediately after they have entered it. If mN omS the state is occupied with a full electron, because each electron that leaves the dot via drain is immediately replaced. Thus, the mean charge placed on the dot is altered only for resonance condition with the side that has a stronger tunneling rate (which is source in this case). Since for finite V SD the resonance conditions for source and drain are split in energy and shifted compared with V SD ¼ 0, the charging appears for either lower or higher gate voltages. This is exactly what happens in Fig. 2. For lower gate voltages the step is shifted to the right which corresponds to resonance with
0.25 VSD=0 mV S
GQPC (2e2/h)
electrolyte an oxide is formed below the tip. This oxide depletes the 2DES below and thus insulating lines can be written. Fig. 1(a) shows an AFM image of the device. The oxide lines are colored bright, while the remaining 2DES is kept dark. The QD is connected via tunnel barriers to leads source (S) and drain (D). The tunneling rates are influenced by the voltage applied to two inplane gates G1 and G2. The QPC is placed next to the QD with its own leads S0 and D0 . A third gate G3 is used to tune the QPC. With the device cooled to 40 mK the differential conductance G QD of the QD is measured using standard lock in technique. At the same time the DC current through the QPC is recorded and the conductance G QPC is derived. A first measurement that demonstrates the working principle is shown in Fig. 1(b). G QD is plotted as a function of the voltage applied to gate G2 as well as G QPC . The differential conductance exhibits several Coulomb peaks that correspond to ground state transitions of the dot. In between the electron number is constant. For each peak in GQD the QPC conductance shows a steplike decrease. These changes of the QPC conductance reflect a potential change for the QPC induced by the capacitive coupling of the additional electron on the dot. Thus, the steps are correlated with charging events on the dot. As the tunneling barrier to the drain is influenced by V G2 the Coulomb peaks vanish for lower voltages. In contrast, the QPC signal is not influenced. The typical steps are still visible reflecting the presence of the dot. This measurement demonstrates how the parameter space is expanded [5]. Thus, we found that the charge detection is not influenced by the strength of the tunneling rates as long as tunneling is possible. The next measurement will show the effect of the ratio of the rates. This is presented in Fig. 2. Again the QPC conductance is plotted as a function of V G1 . The solid line is taken in the linear regime with V SD ¼ 0. The regular steps are again visible marking charge transitions. When a positive voltage V SD ¼ 1 mV is applied (dashed line) the steps stay, but their positions
501
VSD=1 mV
D
S
0.05 -310
-290
-270 VG1 (mV)
-250
D
-230
Fig. 2. GQPC as a function of V G1 for V SD ¼ 0 mV (solid) and V SD ¼ 1 mV (dashed). For lower voltages the dashed line appears shifted to the right, for higher voltages to the left. This is due to asymmetric tunneling rates as indicated in the sketches.
6
S’
D’ QPC
S
QD
G1
D
4
GQPC (2e2/h)
G3
2
GQD (10-2e2/h)
0.25
G2 0.1 500 nm
0 0
(a)
(b)
50
100
VG2 (mV)
Fig. 1. (a) AFM image of the device; bright, oxide lines; dark, remaining 2DES. (b) Coulomb blockade measurement. GQD and GQPC as a function of V G2 . Each Coulomb peak in GQD corresponds to a step in GQPC . For vanishing GQD the steps in GQPC are still visible.
ARTICLE IN PRESS M.C. Rogge et al. / Physica E 34 (2006) 500–503
502
source. For higher voltages the drain resonance is responsible for charging which results in a shift to the left. Thus we find the following asymmetries: on the left we have GS 4GD and on the right GS oGD . This is illustrated by the sketches in Fig. 2. In between we find smoothed steps that correspond to a symmetric configuration (GS ¼ GD ). These experiments demonstrate that we are able to introduce asymmetry of tunneling rates in a controlled manner. Thus, we can use this method to investigate how the transport of the dot depends on the symmetry of tunneling rates. This is done in the two following measurements which were taken in a perpendicular magnetic field of B ¼ 3:7 T. Fig. 3 shows the differential conductance G QD as a function of gate voltages and V SD . Bright colors represent high values of G QD . Several Coulomb diamonds are visible that are typical for single QDs in the Coulomb blockade regime. The enclosing lines of these diamonds appear due to ground state transitions. Lines with positive slope correspond to resonance with source, lines with negative slope to resonance with drain. In addition to the ground state transitions a few more lines are visible. They correspond to transitions with excited states involved. Interestingly they appear almost only for positive V SD . This is due to the asymmetric ratio of the tunneling rates. As indicated with the sketch in Fig. 3 the tunneling rate to the source is higher (GS 4GD ). The differential conductance G QD is the derivative of the current I QD through the dot with respect to V SD . Thus, to understand the measurement we have to think about how the current depends on the tunneling rates and the number of transport channels. Assuming one transport channel due
to a ground state transition N2N þ 1 we get I QD /
GS GD . GS þ GD
For strong asymmetries GS 5GD this formula can be simplified to I QD /
GS GD ¼ GS . GD
If GS bGD we get I QD / GD . Thus the weak tunneling rate controls the current. If a second channel due to a transition N2N þ 1 with N þ 1 an excited state is available for transport electrons have two ways to enter the dot. Thus the incoming tunneling rate is doubled. Whether this is GS or GD depends on V SD . If the electron has decided which way to enter the outgoing rate stays the same as for one channel because now there is still only one way out. In Fig. 3, we have the configuration GS bGD (see sketch). Thus the current I QD is proportional to GD . For a positive voltage V SD electrons enter the dot via the drain lead. Thus GD is the incoming rate. As explained this rate is doubled if a second channel is available. Thus, also the current is doubled and we get a peak in differential conductance when the second channel enters the transport window. This is why we observe additional lines on the right side of the measurement. For negative V SD electrons traverse the dot from source to drain. Therefore, GS is doubled for two channels but GD stays the same. Thus, also the current stays nearly as it is and almost no additional lines are observed on the left side. For GS 5GD the situation is just the opposite. This is shown in Fig. 4. Here we have GS 5GD as indicated by the 120
-258
0.1
190
1 E-4
GQD (e2/h)
VG2 (mV)
GQD (e2/h)
VG1 (mV)
VG2 (mV)
0.2
1 E-4
150 -268 -1.0 -0.5
0.0 0.5 VSD (mV) S
1.0
1.5
60 -1.5
-1.0
-0.5 0.0 0.5 VSD (mV)
1.0
1.5
D S
Fig. 3. GQD of the QD as a function of gate voltages and V SD for asymmetric tunneling rates (see sketch). Several Coulomb diamonds are visible along with excited states. They appear much stronger for positive V SD .
D
Fig. 4. Measurement as in Fig. 3 but with the opposite asymmetry (see sketch). Now excited states are visible much better on the left.
ARTICLE IN PRESS M.C. Rogge et al. / Physica E 34 (2006) 500–503
sketch. Thus, I QD is proportional to GS . An increase in GD for positive V SD has no notable effect on the current and almost no additional lines are observed on the right side of the measurement. If instead GS is increased due to an additional channel for negative V SD also the current is increased and lines reflecting excited states appear on the left of the measurement. In summary, we have studied the visibility of excited states in transport of a single QD under different tunnel configurations. We used a QPC to detect the charge of the dot. Since charging is influenced by the symmetry of the tunneling rates of the dot we were able to detect the symmetry of tunneling rates. Therefore, asymmetry could be introduced deliberately. With this the transport of the dot was investigated. We found that for strong asymmetry the current depends not on the number of transport channels only but also on the direction of the electron flow. This work was supported by BMBF. References [1] C.W.J. Beenaaker, H. van Houten, in: Quantum Transport in Semiconductor Nanostructures Solid State Physics, vol. 44, Academic Press, New York, 1991.
503
[2] K.J. Thomas, J.T. Nicholls, M.Y. Simmons, M. Pepper, D.R. Mace, D.A. Ritchie, Phys. Rev. Lett. 77 (1996) 135. [3] D. Loss, D.P. DiVincenzo, Phys. Rev. A 57 (1998) 120. [4] L.P. Kouwenhoven, C.M. Marcus, P.L. McEuen, S. Tarucha, R.M. Westerveld, N.S. Wingreen, in: L.L. Sohn, L.P. Kouwenhoven, G. Scho¨n (Eds.), Mesoscopic Electron Transport, vol. 345 of Series E, Kluwer, Dordrecht, 1997, pp. 105–214. [5] M. Field, C.G. Smith, M. Pepper, D.A. Richie, J.E.F. Frost, G.A.C. Jones, D.G. Hasko, Phys. Rev. Lett. 70 (1993) 1311. [6] J.M. Elzerman, R. Hanson, J.S. Greidanus, L.H.W. van Beveren, S. De Franceschi, L.M.K. Vandersypen, S. Tarucha, L.P. Kouwenhoven, Phys. Rev. B 67 (2003) 161308. [7] A.C. Johnson, C.M. Marcus, M.P. Hanson, A.C. Gossard, Phys. Rev. B 71 (2005) 115333. [8] R. Nemutudi, M. Kataoka, C.J.B. Ford, N.J. Appleyard, M. Pepper, D.A. Ritchie, G.A.C. Jones, J. Appl. Phys. 95 (2004) 2557. [9] R. Schleser, E. Ruh, T. Ihn, K. Ensslin, D.C. Driscoll, A.C. Gossard, Appl. Phys. Lett. 85 (2004) 2005. [10] C. Fricke, M.C. Rogge, B. Harke, M. Reinwald, W. Wegscheider, F. Hohls, R.J. Haug, PRB 72 (2005) 193302; M.C. Rogge, B. Harke, C. Fricke, F. Hohls, M. Reinwald, W. Wegscheider, R.J. Haug, PRB 72 (2005) 233402. [11] U.F. Keyser, H.W. Schumacher, U. Zeitler, R.J. Haug, K. Eberl, Appl. Phys. Lett. 76 (2000) 457.
Physica E 34 (2006) 500–503 www.elsevier.com/locate/physe
Tuning of tunneling rates in quantum dots using a quantum point contact M.C. Roggea,, C. Frickea, B. Harkea, F. Hohlsa,b, R.J. Hauga, W. Wegscheiderc a
Institut fu¨r Festko¨rperphysik, Universita¨t Hannover, 30167 Hannover, Germany Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, Great Britain c Angewandte und Experimentelle Physik, Universita¨t Regensburg, 93040 Regensburg, Germany
b
Available online 17 April 2006
Abstract We study the influence of asymmetric tunneling rates of a lateral quantum dot connected to source and drain leads. We use a quantum point contact (QPC) to detect charging events on the dot. Since the mean charge depends on the ratio of the tunneling rates the QPC can be used to detect symmetric configurations where both rates are equal. Thus, the transport properties of the dot can be investigated concerning this symmetry. We concentrate on the visibility of features that correspond to transitions with excited states. We interpret the results with the number of channels used for transport, the electron flow direction and the rate symmetry. r 2006 Elsevier B.V. All rights reserved. PACS: 73.63.Kv; 73.23.Hk; 72.20.My Keywords: Quantum dots; Quantum point contact; Charge detection; Tunneling rates; Excited states
During the last decades the research on semiconductor nanostructures has dealt with quantum point contacts (QPCs) and quantum dots (QDs) separately. QPCs were used as a realization of a quasi 1D system showing a quantized conductance [1]. Other effects like the 0.7 anomaly were observed [2]. QDs received great interest, because as a 0D system they were proposed as a realization of a qubit [3]. Many effects have been studied so far [4] but only in recent years both systems were combined to gain further information. The idea was to use a QPC as a charge detector for QDs. This was experimentally implemented in lateral structures using electron beam lithography [5–7] as well as local anodic oxidation with atomic force microscopes [8–10]. In this paper we demonstrate how the investigation of QD systems can benefit from the additional data measured with a QPC. The parameter space can be expanded beyond the limit of measurable transport, because charge detection Corresponding author. Tel.: +49 511 762 19002; fax: +49 511 762 2904. E-mail address: [email protected] (M.C. Rogge).
1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2006.03.022
does not rely on high tunneling rates. As long as charging takes place on the dot this can be measured. Apart from this quantitative enhancement the QPC can yield results which are new in quality. We found, that the charging of the dot depends not on the strength of the tunneling rates but on the ratio of the rates to source and drain leads. Thus, the implementation of the QPC enables us to detect the symmetry of these rates. This is of big relevance to understand effects that occur in transport. With the knowledge of the symmetry we investigate features of excited dot states and interpret the results in terms of tunneling rates influenced by the number of transport channels. The coupled device with a QD and a QPC is embedded in the inversion layer of a GaAs/AlGaAs heterostructure. The two-dimensional electron system (2DES) is located 34 nm below the surface. The sheet density is n ¼ 4:59 1015 m2 and the mobility is m ¼ 64:3 m2 =V s. The technique of local anodic oxidation (LAO) [11] is used to create the desired potential within the 2DES. A negative voltage is applied to the tip of an atomic force microscope (AFM) that touches the sample surface. With a thin water film as
ARTICLE IN PRESS M.C. Rogge et al. / Physica E 34 (2006) 500–503
change. Interestingly this shift is not constant for the whole measurement. For low gate voltages the steps shift to the right while for more positive voltages the steps appear shifted to the left compared to the solid line. This effect is caused by the ratio of the tunneling rates to source and drain GS and GD . With a strong asymmetry the mean charge that is related to a dot state is governed by the side with the stronger tunneling rate. Imagine GS 4GD . If the chemical potential of a dot state mN is above the chemical potential of the source mS the mean charge of this state is 0 because electrons leave the dot immediately after they have entered it. If mN omS the state is occupied with a full electron, because each electron that leaves the dot via drain is immediately replaced. Thus, the mean charge placed on the dot is altered only for resonance condition with the side that has a stronger tunneling rate (which is source in this case). Since for finite V SD the resonance conditions for source and drain are split in energy and shifted compared with V SD ¼ 0, the charging appears for either lower or higher gate voltages. This is exactly what happens in Fig. 2. For lower gate voltages the step is shifted to the right which corresponds to resonance with
0.25 VSD=0 mV S
GQPC (2e2/h)
electrolyte an oxide is formed below the tip. This oxide depletes the 2DES below and thus insulating lines can be written. Fig. 1(a) shows an AFM image of the device. The oxide lines are colored bright, while the remaining 2DES is kept dark. The QD is connected via tunnel barriers to leads source (S) and drain (D). The tunneling rates are influenced by the voltage applied to two inplane gates G1 and G2. The QPC is placed next to the QD with its own leads S0 and D0 . A third gate G3 is used to tune the QPC. With the device cooled to 40 mK the differential conductance G QD of the QD is measured using standard lock in technique. At the same time the DC current through the QPC is recorded and the conductance G QPC is derived. A first measurement that demonstrates the working principle is shown in Fig. 1(b). G QD is plotted as a function of the voltage applied to gate G2 as well as G QPC . The differential conductance exhibits several Coulomb peaks that correspond to ground state transitions of the dot. In between the electron number is constant. For each peak in GQD the QPC conductance shows a steplike decrease. These changes of the QPC conductance reflect a potential change for the QPC induced by the capacitive coupling of the additional electron on the dot. Thus, the steps are correlated with charging events on the dot. As the tunneling barrier to the drain is influenced by V G2 the Coulomb peaks vanish for lower voltages. In contrast, the QPC signal is not influenced. The typical steps are still visible reflecting the presence of the dot. This measurement demonstrates how the parameter space is expanded [5]. Thus, we found that the charge detection is not influenced by the strength of the tunneling rates as long as tunneling is possible. The next measurement will show the effect of the ratio of the rates. This is presented in Fig. 2. Again the QPC conductance is plotted as a function of V G1 . The solid line is taken in the linear regime with V SD ¼ 0. The regular steps are again visible marking charge transitions. When a positive voltage V SD ¼ 1 mV is applied (dashed line) the steps stay, but their positions
501
VSD=1 mV
D
S
0.05 -310
-290
-270 VG1 (mV)
-250
D
-230
Fig. 2. GQPC as a function of V G1 for V SD ¼ 0 mV (solid) and V SD ¼ 1 mV (dashed). For lower voltages the dashed line appears shifted to the right, for higher voltages to the left. This is due to asymmetric tunneling rates as indicated in the sketches.
6
S’
D’ QPC
S
QD
G1
D
4
GQPC (2e2/h)
G3
2
GQD (10-2e2/h)
0.25
G2 0.1 500 nm
0 0
(a)
(b)
50
100
VG2 (mV)
Fig. 1. (a) AFM image of the device; bright, oxide lines; dark, remaining 2DES. (b) Coulomb blockade measurement. GQD and GQPC as a function of V G2 . Each Coulomb peak in GQD corresponds to a step in GQPC . For vanishing GQD the steps in GQPC are still visible.
ARTICLE IN PRESS M.C. Rogge et al. / Physica E 34 (2006) 500–503
502
source. For higher voltages the drain resonance is responsible for charging which results in a shift to the left. Thus we find the following asymmetries: on the left we have GS 4GD and on the right GS oGD . This is illustrated by the sketches in Fig. 2. In between we find smoothed steps that correspond to a symmetric configuration (GS ¼ GD ). These experiments demonstrate that we are able to introduce asymmetry of tunneling rates in a controlled manner. Thus, we can use this method to investigate how the transport of the dot depends on the symmetry of tunneling rates. This is done in the two following measurements which were taken in a perpendicular magnetic field of B ¼ 3:7 T. Fig. 3 shows the differential conductance G QD as a function of gate voltages and V SD . Bright colors represent high values of G QD . Several Coulomb diamonds are visible that are typical for single QDs in the Coulomb blockade regime. The enclosing lines of these diamonds appear due to ground state transitions. Lines with positive slope correspond to resonance with source, lines with negative slope to resonance with drain. In addition to the ground state transitions a few more lines are visible. They correspond to transitions with excited states involved. Interestingly they appear almost only for positive V SD . This is due to the asymmetric ratio of the tunneling rates. As indicated with the sketch in Fig. 3 the tunneling rate to the source is higher (GS 4GD ). The differential conductance G QD is the derivative of the current I QD through the dot with respect to V SD . Thus, to understand the measurement we have to think about how the current depends on the tunneling rates and the number of transport channels. Assuming one transport channel due
to a ground state transition N2N þ 1 we get I QD /
GS GD . GS þ GD
For strong asymmetries GS 5GD this formula can be simplified to I QD /
GS GD ¼ GS . GD
If GS bGD we get I QD / GD . Thus the weak tunneling rate controls the current. If a second channel due to a transition N2N þ 1 with N þ 1 an excited state is available for transport electrons have two ways to enter the dot. Thus the incoming tunneling rate is doubled. Whether this is GS or GD depends on V SD . If the electron has decided which way to enter the outgoing rate stays the same as for one channel because now there is still only one way out. In Fig. 3, we have the configuration GS bGD (see sketch). Thus the current I QD is proportional to GD . For a positive voltage V SD electrons enter the dot via the drain lead. Thus GD is the incoming rate. As explained this rate is doubled if a second channel is available. Thus, also the current is doubled and we get a peak in differential conductance when the second channel enters the transport window. This is why we observe additional lines on the right side of the measurement. For negative V SD electrons traverse the dot from source to drain. Therefore, GS is doubled for two channels but GD stays the same. Thus, also the current stays nearly as it is and almost no additional lines are observed on the left side. For GS 5GD the situation is just the opposite. This is shown in Fig. 4. Here we have GS 5GD as indicated by the 120
-258
0.1
190
1 E-4
GQD (e2/h)
VG2 (mV)
GQD (e2/h)
VG1 (mV)
VG2 (mV)
0.2
1 E-4
150 -268 -1.0 -0.5
0.0 0.5 VSD (mV) S
1.0
1.5
60 -1.5
-1.0
-0.5 0.0 0.5 VSD (mV)
1.0
1.5
D S
Fig. 3. GQD of the QD as a function of gate voltages and V SD for asymmetric tunneling rates (see sketch). Several Coulomb diamonds are visible along with excited states. They appear much stronger for positive V SD .
D
Fig. 4. Measurement as in Fig. 3 but with the opposite asymmetry (see sketch). Now excited states are visible much better on the left.
ARTICLE IN PRESS M.C. Rogge et al. / Physica E 34 (2006) 500–503
sketch. Thus, I QD is proportional to GS . An increase in GD for positive V SD has no notable effect on the current and almost no additional lines are observed on the right side of the measurement. If instead GS is increased due to an additional channel for negative V SD also the current is increased and lines reflecting excited states appear on the left of the measurement. In summary, we have studied the visibility of excited states in transport of a single QD under different tunnel configurations. We used a QPC to detect the charge of the dot. Since charging is influenced by the symmetry of the tunneling rates of the dot we were able to detect the symmetry of tunneling rates. Therefore, asymmetry could be introduced deliberately. With this the transport of the dot was investigated. We found that for strong asymmetry the current depends not on the number of transport channels only but also on the direction of the electron flow. This work was supported by BMBF. References [1] C.W.J. Beenaaker, H. van Houten, in: Quantum Transport in Semiconductor Nanostructures Solid State Physics, vol. 44, Academic Press, New York, 1991.
503
[2] K.J. Thomas, J.T. Nicholls, M.Y. Simmons, M. Pepper, D.R. Mace, D.A. Ritchie, Phys. Rev. Lett. 77 (1996) 135. [3] D. Loss, D.P. DiVincenzo, Phys. Rev. A 57 (1998) 120. [4] L.P. Kouwenhoven, C.M. Marcus, P.L. McEuen, S. Tarucha, R.M. Westerveld, N.S. Wingreen, in: L.L. Sohn, L.P. Kouwenhoven, G. Scho¨n (Eds.), Mesoscopic Electron Transport, vol. 345 of Series E, Kluwer, Dordrecht, 1997, pp. 105–214. [5] M. Field, C.G. Smith, M. Pepper, D.A. Richie, J.E.F. Frost, G.A.C. Jones, D.G. Hasko, Phys. Rev. Lett. 70 (1993) 1311. [6] J.M. Elzerman, R. Hanson, J.S. Greidanus, L.H.W. van Beveren, S. De Franceschi, L.M.K. Vandersypen, S. Tarucha, L.P. Kouwenhoven, Phys. Rev. B 67 (2003) 161308. [7] A.C. Johnson, C.M. Marcus, M.P. Hanson, A.C. Gossard, Phys. Rev. B 71 (2005) 115333. [8] R. Nemutudi, M. Kataoka, C.J.B. Ford, N.J. Appleyard, M. Pepper, D.A. Ritchie, G.A.C. Jones, J. Appl. Phys. 95 (2004) 2557. [9] R. Schleser, E. Ruh, T. Ihn, K. Ensslin, D.C. Driscoll, A.C. Gossard, Appl. Phys. Lett. 85 (2004) 2005. [10] C. Fricke, M.C. Rogge, B. Harke, M. Reinwald, W. Wegscheider, F. Hohls, R.J. Haug, PRB 72 (2005) 193302; M.C. Rogge, B. Harke, C. Fricke, F. Hohls, M. Reinwald, W. Wegscheider, R.J. Haug, PRB 72 (2005) 233402. [11] U.F. Keyser, H.W. Schumacher, U. Zeitler, R.J. Haug, K. Eberl, Appl. Phys. Lett. 76 (2000) 457.