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Magnetic-field induced subband mixing in tunnel-coupled quantum point contacts: Transport spectroscopy and numerical modelling
ARTICLE IN PRESS
Physica E 40 (2008) 1855–1858 www.elsevier.com/locate/physe
Magnetic-field induced subband mixing in tunnel-coupled quantum point contacts: Transport spectroscopy and numerical modelling Andrea Bertonia,, Saskia F. Fischerb, Ulrich Kunzeb a
National Research Center S3, INFM-CNR, via Campi, 213/A, 41100 Modena, Italy Werkstoffe und Nanoelektronik, Ruhr-Universita¨t Bochum, 44780 Bochum, Germany
b
Available online 19 November 2007
Abstract We study the dependence of subband energies of a double quantum point contact on an external magnetic field applied in the transport direction. The comparison with experimental magnetotransport data allows us to identify the conditions for the modulation of anticrossing gaps and to map the subband spectra with the measured transconductance. The effect of an asymmetry of the two quantum point contacts is also investigated. r 2007 Elsevier B.V. All rights reserved. PACS: 73.23. b; 73.43.Cd; 73.63.Nm; 03.67.Lx Keywords: Magnetotransport spectroscopy; 2D Schro¨dinger equation; Quantum point contact; Coupled channels
1. Introduction The physics of coherent electron transport in coupled one-dimensional channels is of considerable interest not only for the understanding of fundamental effects in nanoelectronic systems [1], but also for possible realization of solid-state quantum logic gates [2]. Recently, high-resolution magnetotransport spectroscopy was obtained in tunnel-coupled quantum point contacts (QPCs) prepared by etched nanogrooves [3,4] from a GaAs/AlGaAs double-quantum well heterostructure, hosting two stacked two-dimensional electron gases. The transconductance was measured with different voltages applied at the top of the capping layer, close to the coupled wells, and at the bottom of the substrate. They will be indicated, in the following, as top- and back-gate voltages, respectively. The plots of the transconductance allow for a direct identification of the transverse QPCs modes responsible for the quantized conductance characteristics. Application of different cooling bias voltages makes it possible to tune the confining potential of the two Corresponding author. Tel.: +39 59 2055295.
E-mail address: [email protected] (A. Bertoni). 1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2007.08.157
QPCs, and external electric or magnetic fields change the occupation of modes. Transport occurring through a quantum channel coherently delocalized in the two QPCs is revealed. In fact this system, acting as a quantum directional coupler [5], constitutes the basic building block for possible coherent-electronic, or quantum-computing, devices [6]. 2. Numerical results In the presence of a magnetic field B, in particular when applied in the transport direction, rich variations of the mode spectra are observed. They can be explained in terms of the modulation of the subband energies, induced by the applied field [7]. In order to gain a full insight about the mode spectra in electrical and magnetic fields which are not easily accessible in experiment, and for devices whose transverse QPC wave functions do not posses a simple analytical expression, numerical modelling is essential [8]. In this work, we use a closed-boundary 2D Schro¨dinger solver to compute the transverse eigenenergies of a double QPC both in symmetric and asymmetric configurations, i.e. for devices with the same, or different QPCs geometry, respectively. In Fig. 1(a) we present the confining potential
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Fig. 1. (a) Transverse potential profile of the symmetric double QPC, with an harmonic Y confinement of 20 meV and two 15 nm wells separated by a 1 nm barrier in the Z-direction, stemming from the GaAs=Al:3 Ga:7 As band offset. A 3.7 KV/m bias is applied in the Z-direction. The transport occurs along the X-direction. (b) Energy levels of the double-QPC against the magnetic field applied in the transport direction. (c) Experimental transconductance of the double-QPC device [4], showing the transverse energy levels against the magnetic field.
of the symmetric case, with the two channels separated by a tunnelling barrier of 1 nm. The confining potential of each QPC consists of a 15 nm well, in the Z-direction (growth direction of the GaAs=Al:3 Ga:7 As sample) and a parabola in the Y-direction (confinement induced by the etched nanogrooves). A linear bias is applied in the Z-direction to account for the different electrical biases applied to the bottom gate and the top gate. The resulting B-dependent energy levels, corresponding to the activation thresholds of the transport channels, are shown in Fig. 1(b). They mimic the experimental B-dependent trasconductance [4] reported in Fig. 1(c). The experimental data do not show the slight energy increase with the magnetic field, at high B values, shown in the simulation of Fig. 1(b). This could be due to the non linearity of the Z bias in the experiment or could indicate a different effective form of the QPCs confining potential. The two lower subbands (I), corresponding to the ground transverse states of the two QPCs, show an avoided crossing around 10 T (not shown in the figure), outside the experimental range, where the two wave functions hybridize in a bounding and an antibounding channels. Still, it is clear from Fig. 1(c) that the gap between the two levels is reduced by the magnetic field. The second (II) and third (III) couple of subbands crosses at lower values of the magnetic field in both the calculated and experimental spectra. In order to investigate the dependence of the anticrossings from the symmetry of the system, we report, in Fig. 2, the channels’ energy for the case of two QPCs with the same harmonic Y confinement but different dimensions
of the Z-direction quantum wells (asymmetric configuration). The details of the sample are given in the caption. At zero magnetic field, the energy levels (and, correspondingly, the thresholds of the transport channels) correspond to a transverse wave function completely localized in one of the two wells. This is due to the Z bias that, also in the case of QPCs with the same width, unaligns the two levels. As B is increased, the lower level of each couple increases more rapidly and (anti)crosses the second one. By changing the width of one of the two wells it is possible to modify one of the two wave functions, while leaving the other unaltered essentially: this has a strong impact on the hybridization process, when the two energies are aligned by the magnetic field. The crossing of the two first-excited states (A in Fig. 2) is turned into an anticrossing, while the anticrossing gap of the two ground states (B in Fig. 2) is strongly reduced. Finally, we compute the transconductance for a sourcedrain Fermi level offset of 1 meV, at a temperature of 4.2 K. The magnetic field dependence (not shown) essentially reproduces the transverse channels energy reported in the previous figures. On the other hand its dependence on the Z bias shows interesting features. In Fig. 3 we report the computed transconductance as a function of the voltage applied to the top (vertical axis) and bottom (horizontal axis) gates for two samples with different thickness of the substrate. A pictorial representation of the effect of the two voltages on the linear bias is also presented. The long vertical line represents the two Fermi levels, almost aligned, of the source and drain leads. A
ARTICLE IN PRESS A. Bertoni et al. / Physica E 40 (2008) 1855–1858
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channels energy (meV)
60 50 40 A
A
30
A
B
B
B
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5 10 15 magnetic field (T)
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Fig. 2. Effect of the quantum well asymmetry on the energy levels of the transport channels. In the three graphs, one of the QPCs is 15 nm wide, the other is 15 (left) 16 (middle) and 17 (right) nm. Note the opening (A) and the closing (B) of (anti)crossing gaps as a function of the magnetic field.
transport window
0.1
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Fig. 3. Pictorial representation (left) of the double-QPC potential modified by the top and back gate voltages. The two contour plots represent the computed transconductance as a function of the voltages (in Volts) applied to the top and back gates for two samples with the top gate at 10 nm from the double QPC region and the back gate at 200 nm (middle) and 400 nm (right). The transport window is 1 meV wide and the temperature is 4.2 K. As expected, an increase of the substrate thickness reduced the effect of the back gate voltage.
transconductance (anti)crossing occurs when two levels of the QPCs are aligned and, at the same time, close to the Fermi energy. The obtained trasconductance is, in both cases, quite different from the experimental one. This indicates that the Z bias in the real sample is not linear. In fact, due to the unintentional doping of the substrate layer, the effect of the top gate voltage is much stronger than the one of the back gate voltage. It is worth noting that, while the two curves corresponding to the same transverse level of the two QPCs (the two
ground states, the two first excited, and so on) present wide anticrossings, the curves corresponding to different levels crosses, since the hybridization of wave function with different symmetry is not possible. We find that this constrain is lifted by the magnetic field. 3. Conclusions We find evidence of the effects of the QPCs asymmetry on the (anti)crossing gaps in the transconductance spectrum. As
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a consequence, the switching characteristics of a directional coupler based on the double QPC device described above result to be strongly influenced by the geometry of the system, in the active region. Nevertheless it is possible, by using an external magnetic field of proper magnitude, to restore the band alignment of, at least, the lower subbands. This finding is of importance since it provides an extra degree of freedom for the external control of the quantum device operation.
References [1] [2] [3] [4] [5] [6] [7] [8]
V.I. Puller, et al., IEEE Trans. Nanotech. 4 (2005) 21. A. Bertoni, et al., Phys. Rev. Lett. 84 (2000) 5912. S. Fischer, et al., Nature Phys. 2 (2006) 91. S. Fischer, et al., Phys. Rev. B 74 (2006) 115324. A. Ramamoorthy, et al., Appl. Phys. Lett. 89 (2006) 13118. A. Marchi, et al., IEEE Trans. Nanotech. 3 (2004) 129. L.G. Mourokh, et al., Appl. Phys. Lett. 90 (2007) 132108. S. Birner, et al., IEEE Trans. Electron Dev. 54 (2007) 2137.
Physica E 40 (2008) 1855–1858 www.elsevier.com/locate/physe
Magnetic-field induced subband mixing in tunnel-coupled quantum point contacts: Transport spectroscopy and numerical modelling Andrea Bertonia,, Saskia F. Fischerb, Ulrich Kunzeb a
National Research Center S3, INFM-CNR, via Campi, 213/A, 41100 Modena, Italy Werkstoffe und Nanoelektronik, Ruhr-Universita¨t Bochum, 44780 Bochum, Germany
b
Available online 19 November 2007
Abstract We study the dependence of subband energies of a double quantum point contact on an external magnetic field applied in the transport direction. The comparison with experimental magnetotransport data allows us to identify the conditions for the modulation of anticrossing gaps and to map the subband spectra with the measured transconductance. The effect of an asymmetry of the two quantum point contacts is also investigated. r 2007 Elsevier B.V. All rights reserved. PACS: 73.23. b; 73.43.Cd; 73.63.Nm; 03.67.Lx Keywords: Magnetotransport spectroscopy; 2D Schro¨dinger equation; Quantum point contact; Coupled channels
1. Introduction The physics of coherent electron transport in coupled one-dimensional channels is of considerable interest not only for the understanding of fundamental effects in nanoelectronic systems [1], but also for possible realization of solid-state quantum logic gates [2]. Recently, high-resolution magnetotransport spectroscopy was obtained in tunnel-coupled quantum point contacts (QPCs) prepared by etched nanogrooves [3,4] from a GaAs/AlGaAs double-quantum well heterostructure, hosting two stacked two-dimensional electron gases. The transconductance was measured with different voltages applied at the top of the capping layer, close to the coupled wells, and at the bottom of the substrate. They will be indicated, in the following, as top- and back-gate voltages, respectively. The plots of the transconductance allow for a direct identification of the transverse QPCs modes responsible for the quantized conductance characteristics. Application of different cooling bias voltages makes it possible to tune the confining potential of the two Corresponding author. Tel.: +39 59 2055295.
E-mail address: [email protected] (A. Bertoni). 1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2007.08.157
QPCs, and external electric or magnetic fields change the occupation of modes. Transport occurring through a quantum channel coherently delocalized in the two QPCs is revealed. In fact this system, acting as a quantum directional coupler [5], constitutes the basic building block for possible coherent-electronic, or quantum-computing, devices [6]. 2. Numerical results In the presence of a magnetic field B, in particular when applied in the transport direction, rich variations of the mode spectra are observed. They can be explained in terms of the modulation of the subband energies, induced by the applied field [7]. In order to gain a full insight about the mode spectra in electrical and magnetic fields which are not easily accessible in experiment, and for devices whose transverse QPC wave functions do not posses a simple analytical expression, numerical modelling is essential [8]. In this work, we use a closed-boundary 2D Schro¨dinger solver to compute the transverse eigenenergies of a double QPC both in symmetric and asymmetric configurations, i.e. for devices with the same, or different QPCs geometry, respectively. In Fig. 1(a) we present the confining potential
ARTICLE IN PRESS 1856
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Fig. 1. (a) Transverse potential profile of the symmetric double QPC, with an harmonic Y confinement of 20 meV and two 15 nm wells separated by a 1 nm barrier in the Z-direction, stemming from the GaAs=Al:3 Ga:7 As band offset. A 3.7 KV/m bias is applied in the Z-direction. The transport occurs along the X-direction. (b) Energy levels of the double-QPC against the magnetic field applied in the transport direction. (c) Experimental transconductance of the double-QPC device [4], showing the transverse energy levels against the magnetic field.
of the symmetric case, with the two channels separated by a tunnelling barrier of 1 nm. The confining potential of each QPC consists of a 15 nm well, in the Z-direction (growth direction of the GaAs=Al:3 Ga:7 As sample) and a parabola in the Y-direction (confinement induced by the etched nanogrooves). A linear bias is applied in the Z-direction to account for the different electrical biases applied to the bottom gate and the top gate. The resulting B-dependent energy levels, corresponding to the activation thresholds of the transport channels, are shown in Fig. 1(b). They mimic the experimental B-dependent trasconductance [4] reported in Fig. 1(c). The experimental data do not show the slight energy increase with the magnetic field, at high B values, shown in the simulation of Fig. 1(b). This could be due to the non linearity of the Z bias in the experiment or could indicate a different effective form of the QPCs confining potential. The two lower subbands (I), corresponding to the ground transverse states of the two QPCs, show an avoided crossing around 10 T (not shown in the figure), outside the experimental range, where the two wave functions hybridize in a bounding and an antibounding channels. Still, it is clear from Fig. 1(c) that the gap between the two levels is reduced by the magnetic field. The second (II) and third (III) couple of subbands crosses at lower values of the magnetic field in both the calculated and experimental spectra. In order to investigate the dependence of the anticrossings from the symmetry of the system, we report, in Fig. 2, the channels’ energy for the case of two QPCs with the same harmonic Y confinement but different dimensions
of the Z-direction quantum wells (asymmetric configuration). The details of the sample are given in the caption. At zero magnetic field, the energy levels (and, correspondingly, the thresholds of the transport channels) correspond to a transverse wave function completely localized in one of the two wells. This is due to the Z bias that, also in the case of QPCs with the same width, unaligns the two levels. As B is increased, the lower level of each couple increases more rapidly and (anti)crosses the second one. By changing the width of one of the two wells it is possible to modify one of the two wave functions, while leaving the other unaltered essentially: this has a strong impact on the hybridization process, when the two energies are aligned by the magnetic field. The crossing of the two first-excited states (A in Fig. 2) is turned into an anticrossing, while the anticrossing gap of the two ground states (B in Fig. 2) is strongly reduced. Finally, we compute the transconductance for a sourcedrain Fermi level offset of 1 meV, at a temperature of 4.2 K. The magnetic field dependence (not shown) essentially reproduces the transverse channels energy reported in the previous figures. On the other hand its dependence on the Z bias shows interesting features. In Fig. 3 we report the computed transconductance as a function of the voltage applied to the top (vertical axis) and bottom (horizontal axis) gates for two samples with different thickness of the substrate. A pictorial representation of the effect of the two voltages on the linear bias is also presented. The long vertical line represents the two Fermi levels, almost aligned, of the source and drain leads. A
ARTICLE IN PRESS A. Bertoni et al. / Physica E 40 (2008) 1855–1858
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channels energy (meV)
60 50 40 A
A
30
A
B
B
B
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15
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5 10 15 magnetic field (T)
0
5
10
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Fig. 2. Effect of the quantum well asymmetry on the energy levels of the transport channels. In the three graphs, one of the QPCs is 15 nm wide, the other is 15 (left) 16 (middle) and 17 (right) nm. Note the opening (A) and the closing (B) of (anti)crossing gaps as a function of the magnetic field.
transport window
0.1
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Fig. 3. Pictorial representation (left) of the double-QPC potential modified by the top and back gate voltages. The two contour plots represent the computed transconductance as a function of the voltages (in Volts) applied to the top and back gates for two samples with the top gate at 10 nm from the double QPC region and the back gate at 200 nm (middle) and 400 nm (right). The transport window is 1 meV wide and the temperature is 4.2 K. As expected, an increase of the substrate thickness reduced the effect of the back gate voltage.
transconductance (anti)crossing occurs when two levels of the QPCs are aligned and, at the same time, close to the Fermi energy. The obtained trasconductance is, in both cases, quite different from the experimental one. This indicates that the Z bias in the real sample is not linear. In fact, due to the unintentional doping of the substrate layer, the effect of the top gate voltage is much stronger than the one of the back gate voltage. It is worth noting that, while the two curves corresponding to the same transverse level of the two QPCs (the two
ground states, the two first excited, and so on) present wide anticrossings, the curves corresponding to different levels crosses, since the hybridization of wave function with different symmetry is not possible. We find that this constrain is lifted by the magnetic field. 3. Conclusions We find evidence of the effects of the QPCs asymmetry on the (anti)crossing gaps in the transconductance spectrum. As
ARTICLE IN PRESS 1858
A. Bertoni et al. / Physica E 40 (2008) 1855–1858
a consequence, the switching characteristics of a directional coupler based on the double QPC device described above result to be strongly influenced by the geometry of the system, in the active region. Nevertheless it is possible, by using an external magnetic field of proper magnitude, to restore the band alignment of, at least, the lower subbands. This finding is of importance since it provides an extra degree of freedom for the external control of the quantum device operation.
References [1] [2] [3] [4] [5] [6] [7] [8]
V.I. Puller, et al., IEEE Trans. Nanotech. 4 (2005) 21. A. Bertoni, et al., Phys. Rev. Lett. 84 (2000) 5912. S. Fischer, et al., Nature Phys. 2 (2006) 91. S. Fischer, et al., Phys. Rev. B 74 (2006) 115324. A. Ramamoorthy, et al., Appl. Phys. Lett. 89 (2006) 13118. A. Marchi, et al., IEEE Trans. Nanotech. 3 (2004) 129. L.G. Mourokh, et al., Appl. Phys. Lett. 90 (2007) 132108. S. Birner, et al., IEEE Trans. Electron Dev. 54 (2007) 2137.