- Email: [email protected]
Towards Coulomb drag in vertically coupled quantum wires with independent contacts
ARTICLE IN PRESS
Physica E 40 (2008) 1569–1572 www.elsevier.com/locate/physe
Towards Coulomb drag in vertically coupled quantum wires with independent contacts D. Larochea,b,, E.S. Bielejecb, J.L. Renob, G. Gervaisa, M.P. Lillyb a
Department of Physics, McGill University, Montreal, QC, Canada Sandia National Laboratories, 1515 Eubank, P.O. Box 5800, Mailstop 1314, Albuduerque, NM, USA
b
Available online 7 October 2007
Abstract We report the details of design and fabrication of independently contacted, vertically coupled quantum wires using the epoxy-bondand-stop-etch (EBASE) technique. These nanostructures are fabricated in high quality GaAs/AlGaAs parallel double quantum well heterostructures and are intended for Coulomb drag measurements of quantum wires. They will allow us to explore Coulomb drag in one-dimensional structures in a regime of small interlayer separation where the drag signal is expected to be stronger and less affected by phonon drag. r 2007 Elsevier B.V. All rights reserved. PACS: 73.21.Hb; 73.40.Gk Keywords: Quantum wires; Nanostructures; Coulomb drag; Vertical geometry
1. Introduction Over the past few years, one-dimensional structures have shown a number of striking transport properties. Apart from the universal quantized conductance in units of 2e2 =h [1,2], most of these properties arise from an enhancement of the electron–electron interactions induced by the strong confinement in one dimension. As a result of this enhancement, the Fermi-liquid (FL) theory typically used in higher dimensions becomes inadequate to describe onedimensional electronic transport and should ultimately be replaced with the Luttinger-liquid (LL) theory as a better picture to understand electronic behavior in one-dimension. This theory leads to many interesting predictions, among others the highly sought-after spin-charge separation [3,4]. Experimentally, major efforts have recently been made in the fabrication and engineering of coupled nanostructures for the investigation of non-FL characteristics in one-
dimensional systems. As an example, Coulomb drag measurements can be performed in coupled quantum wires and could provide important information on the nature of the electronic state formed inside the wire. This measurement consists of flowing an electrical current I drive into one of the wire (the drive wire) while preventing any current flow in the other wire (the drag wire). Under this condition, e–e scattering will cause the electrons in the drag wire to pile up on one side, creating a net voltage difference V drag across the wire. The quantity usually determined is the transresistance, or drag resistance, which is defined as the drag voltage divided by the drive current, RD ¼ V drag =I drive . In this paper, we report on the design and the fabrication of vertically coupled, independently contacted quantum wires and we present preliminary results on the tunnelling rate as a function of the interlayer barrier of AlGaAs/GaAs MBE-grown heterostructures.
2. Theoretical and experimental background Corresponding author. Sandia National Laboratories, 1515 Eubank, P.O. Box 5800, Mailstop 1314, Albuquerque, NM 87185, USA. Tel.: +1 505 844 0763; fax: +1 505 845 6526. E-mail address: [email protected] (D. Laroche).
1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2007.09.175
Coulomb drag is caused by e–e scattering via Coulomb interactions which conserve both energy and momentum. At low one-dimensional electron density, Coulomb drag is
ARTICLE IN PRESS 1570
D. Laroche et al. / Physica E 40 (2008) 1569–1572
expected to be stronger due to the enhancement of Coulomb interactions and the reduction of screening coming from one-dimensional confinement [5]. The drag can either occur when electrons are scattered along the drive current direction (positive drag) or opposite to it (negative drag). The negative drag is expected at really low electron densities whereas positive drag is expected at higher densities [6]. Coulomb drag has been thoughtfully studied in bilayer systems of two-dimensional electrons and was found fully consistent with FL theory as it showed a quadratic dependence with temperature [7–10]. In one dimension, LL theory predicts the Coulomb drag to exhibit a distinct temperature dependence from that of a FL. The main contribution to the transresistance comes from interwire backscattering yielding four main temperature regions over which the drag behavior changes drastically [5]. These regions are defined, in a regime where the two wires are strongly coupled together, by the temperature T L at which the thermal wavelength of the electrons becomes of order of the system size and by the energy (or mass) M of a soliton. In the low temperature regime, where ToT L , Coulomb drag decreases linearly with temperature due to soliton tunnelling.pffiffiffiffiffiffiffiffiffiffiffiffiffiffi If theffi temperature is even lower such that To ðT L MÞ expðM=T L Þ, the Coulomb drag is expected to decrease as T 2 . On the other hand, if the thermal wavelength is bigger than the system size, but ToM, thermally activated solitons should cause an activated behavior of the drag resistivity described by rD ðTÞ r~0 eM=T ,
(1)
where r~0 is a constant. In the high-temperature regime, when T4T L ; M the Coulomb drag resistivity should then depend on the strength of the e–e interactions and be described by a power-law of the form 4K3 T rD ðTÞ r~0 , (2) E0 where E 0 is a constant of the order of the Fermi energy and K is a parameter accounting for the strength of the e–e interactions. If there are no interactions, K is unity and one expects a linear increase of the drag with temperature, as predicted by FL theory. Exploring the different regimes of the Coulomb drag should therefore provides significant insights on the e–e interaction-strength in one-dimension, as well as possibly and unambiguously determining the correct underlying theory for one-dimensional conduction. Coulomb drag measurements between parallel quantum wires in a horizontal geometry have already been performed in several experiments [6,11]. In this geometry, the two wires, which are separated by a soft electrostatic barrier, must be relatively far apart so that interwire tunnelling is minimized. This separation leads to a weaker Coulomb drag signal as it decreases exponentially with the interwire separation. In these experiments, there is also an additional contribution coming from phonon drag which does not decay exponentially and remains strong at
interwire distance above 100 nm [12]. Therefore, these soft barriers cause the drag signal to be significantly influenced by phonon drag. Coupling the wires in a vertical geometry allows us to explore the Coulomb drag in the regime of short interwire separation. This minimizes the phonon-induced drag since, in this geometry, the wires have a hard MBE-defined barrier allowing for much smaller tunneling conductance at a given interwire distance. 3. Device fabrication The vertically coupled quantum wires reported here are fabricated in a GaAs/AlGaAs double quantum well heterostructure with electrostatic TiAu split gates defining the wires. A sketch of the device and the effect of applying a negative bias on the gates are shown in Fig. 1(a). The device consists of 2 pinch off gates and 2 plunger gates, one set on each side of the device, forming two wires that are 2 mm long and 0:5 mm wide. These dimensions were chosen as a balance between the requirements to observe ballistic quantized transport, usually sharper in short wires, and that of observing a strong enough drag signal, larger in longer wires. In order to deposit the metallic gates on both side of the double quantum well heterostructure, we employ the epoxy-bond-and-stop-etch (EBASE) technique [13]. This technique allows the bottom and the top split gates to be approximately 150 nm from their respective quantum well, which is of sufficient proximity to form wellconfined wires. A scanning electron microscope picture of the device made on a GaAs/AlGaAs heterostructure with a 12 nm high barrier is shown in Fig. 1(b). The gates are patterned such that, when an appropriate negative bias is applied on them, only regions C&D and E&F are electrically connected through a quantum wire. With such design, one can probe both wires independently and an arbitrary number of conduction sub-bands can be occupied depending on the bias applied on the gates. In order to determine which barrier height should be used in the double quantum well heterostructure, the tunnelling rate was measured as a function of the barrier height. The results are shown in Fig. 2. The data show the expected exponential decay behavior for tunnelling and follow the relation C ¼ 0:88e1:63d ,
(3)
where C is the tunnelling conductance and d is the barrier height. The wires described previously have a total surface of 1 mm2 or smaller, depending on how heavily the 2DEGs are depleted by the gates. Assuming that the onedimensional tunnelling rate is at most as big as the twodimensional tunnelling rate, we conclude that a 15 nm height barrier would contribute at most to 109 Siemens to the interwire conduction, which is three orders of magnitude smaller than the drag conductance measured by Debray et al. [11].
ARTICLE IN PRESS D. Laroche et al. / Physica E 40 (2008) 1569–1572
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Fig. 1. (a) Sketch of the design used for the double quantum wires fabrication. The left-hand side shows the whole structure without bias on the gates. The depleting effect of applying a suitable negative bias on the gates is shown on the right-hand side. For clarity, only the 2DEGs are shown on the right-hand side, but the same gates and ohmic contacts than the ones on the left-hand side are present. (b) A SEM picture of the double quantum wires. Note that the gate on the other side of the wafer (top side) appears as a faint white arm.
rather nonuniform. Our current design should avoid these difficulties. In the future, we will make measurements on newly fabricated vertically coupled quantum wires. An important milestone will be to map out completely the occupied subbands of each quantum wire as a function of different gate voltages. We expect this to be nonetheless non-trivial since all gates are capacitively coupled to each other; yet it should be achievable. Subsequently, we will perform Coulomb drag measurements over wide range of temperatures, densities and magnetic fields. 4. Conclusion
Fig. 2. Log plot of the tunnelling conductance as a function of the interlayer distance of double quantum well heterostructures. The data follow an exponential decay with the barrier height best described by C ¼ 0:88e1:63d .
Tunnelling measurements have already been performed in vertically coupled quantum wires with small interlayer barrier [14] and also for cleaved edge overgrowth wires [15]. In the former case, separate gates were used to achieve independent contacts. These gates were close to the wires and were thus affecting their density profile, making it
In conclusion, we have fabricated vertically coupled quantum wires with independent contacts using split-gate and EBASE techniques. The vertical geometry and the rigid barrier between the two layers will allow us to make Coulomb drag measurements in a regime that has never been explored before. This will provide important information on the many-body interactions that are occurring in one-dimensional quantum wires. Acknowledgements We acknowledge the outstanding technical assistance of D. Tibbets and C. Morath. This work has been supported by the Natural Sciences and Engineering Research Council
ARTICLE IN PRESS D. Laroche et al. / Physica E 40 (2008) 1569–1572
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of Canada (NSERC) and the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, US Department of Energy. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000. One of the authors (G.G.) acknowledge receipt of support from the Alfred P. Sloan Foundation under their fellowship program. References [1] [2] [3] [4]
B.J. van Wees, et al., Phys. Rev. Lett. 60 (1988) 848. E.G. Haanappel, D. van der Marel, Phys. Rev. B. 39 (1989) 5484. S.-I. Tomonaga, Prog. Theor. Phys. 5 (1950) 544. J.M.J. Luttinger, Math. Phys. 4 (1963) 1154.
[5] P. Debray, V.N. Zverev, V. Gurevich, R. Klesse, R.S. Newrock, Semicond. Sci. Technol. 17 (2002). [6] M. Yamamoto, M. Stopa, Y. Tokura, Y. Hirayama, S. Tarucha, Science 313 (2006) 204. [7] T.J. Gramila, J.P. Eisenstein, A.H. MacDonald, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 66 (1991) 1216. [8] M.C. Bønsager, K. Flensberg, B.Y.-K. Hu, A. Jauho, Phys. Rev. Lett. 77 (1996) 1366. [9] M.P. Lilly, J.P. Eisenstein, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 80 (1998) 1714. [10] M. Kellogg, J.P. Eisenstein, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 93 (2004) 036801. [11] P. Debray, et al., J. Phys. Condens. Matter 13 (2001) 3389. [12] A.J.J. Rojo, Phys.: Condens. Matter 11 (1999). [13] M.V. Weckwerth, et al., Supperlatt. Microstruct. 20 (1996) 561. [14] E.S. Bielejec, J.A. Seamons, J.L. Reno, M.P. Lilly, Appl. Phys. Lett. 86 (2005) 83101. [15] O.M. Auslaender, et al., Science 303 (2005) 308.
Physica E 40 (2008) 1569–1572 www.elsevier.com/locate/physe
Towards Coulomb drag in vertically coupled quantum wires with independent contacts D. Larochea,b,, E.S. Bielejecb, J.L. Renob, G. Gervaisa, M.P. Lillyb a
Department of Physics, McGill University, Montreal, QC, Canada Sandia National Laboratories, 1515 Eubank, P.O. Box 5800, Mailstop 1314, Albuduerque, NM, USA
b
Available online 7 October 2007
Abstract We report the details of design and fabrication of independently contacted, vertically coupled quantum wires using the epoxy-bondand-stop-etch (EBASE) technique. These nanostructures are fabricated in high quality GaAs/AlGaAs parallel double quantum well heterostructures and are intended for Coulomb drag measurements of quantum wires. They will allow us to explore Coulomb drag in one-dimensional structures in a regime of small interlayer separation where the drag signal is expected to be stronger and less affected by phonon drag. r 2007 Elsevier B.V. All rights reserved. PACS: 73.21.Hb; 73.40.Gk Keywords: Quantum wires; Nanostructures; Coulomb drag; Vertical geometry
1. Introduction Over the past few years, one-dimensional structures have shown a number of striking transport properties. Apart from the universal quantized conductance in units of 2e2 =h [1,2], most of these properties arise from an enhancement of the electron–electron interactions induced by the strong confinement in one dimension. As a result of this enhancement, the Fermi-liquid (FL) theory typically used in higher dimensions becomes inadequate to describe onedimensional electronic transport and should ultimately be replaced with the Luttinger-liquid (LL) theory as a better picture to understand electronic behavior in one-dimension. This theory leads to many interesting predictions, among others the highly sought-after spin-charge separation [3,4]. Experimentally, major efforts have recently been made in the fabrication and engineering of coupled nanostructures for the investigation of non-FL characteristics in one-
dimensional systems. As an example, Coulomb drag measurements can be performed in coupled quantum wires and could provide important information on the nature of the electronic state formed inside the wire. This measurement consists of flowing an electrical current I drive into one of the wire (the drive wire) while preventing any current flow in the other wire (the drag wire). Under this condition, e–e scattering will cause the electrons in the drag wire to pile up on one side, creating a net voltage difference V drag across the wire. The quantity usually determined is the transresistance, or drag resistance, which is defined as the drag voltage divided by the drive current, RD ¼ V drag =I drive . In this paper, we report on the design and the fabrication of vertically coupled, independently contacted quantum wires and we present preliminary results on the tunnelling rate as a function of the interlayer barrier of AlGaAs/GaAs MBE-grown heterostructures.
2. Theoretical and experimental background Corresponding author. Sandia National Laboratories, 1515 Eubank, P.O. Box 5800, Mailstop 1314, Albuquerque, NM 87185, USA. Tel.: +1 505 844 0763; fax: +1 505 845 6526. E-mail address: [email protected] (D. Laroche).
1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2007.09.175
Coulomb drag is caused by e–e scattering via Coulomb interactions which conserve both energy and momentum. At low one-dimensional electron density, Coulomb drag is
ARTICLE IN PRESS 1570
D. Laroche et al. / Physica E 40 (2008) 1569–1572
expected to be stronger due to the enhancement of Coulomb interactions and the reduction of screening coming from one-dimensional confinement [5]. The drag can either occur when electrons are scattered along the drive current direction (positive drag) or opposite to it (negative drag). The negative drag is expected at really low electron densities whereas positive drag is expected at higher densities [6]. Coulomb drag has been thoughtfully studied in bilayer systems of two-dimensional electrons and was found fully consistent with FL theory as it showed a quadratic dependence with temperature [7–10]. In one dimension, LL theory predicts the Coulomb drag to exhibit a distinct temperature dependence from that of a FL. The main contribution to the transresistance comes from interwire backscattering yielding four main temperature regions over which the drag behavior changes drastically [5]. These regions are defined, in a regime where the two wires are strongly coupled together, by the temperature T L at which the thermal wavelength of the electrons becomes of order of the system size and by the energy (or mass) M of a soliton. In the low temperature regime, where ToT L , Coulomb drag decreases linearly with temperature due to soliton tunnelling.pffiffiffiffiffiffiffiffiffiffiffiffiffiffi If theffi temperature is even lower such that To ðT L MÞ expðM=T L Þ, the Coulomb drag is expected to decrease as T 2 . On the other hand, if the thermal wavelength is bigger than the system size, but ToM, thermally activated solitons should cause an activated behavior of the drag resistivity described by rD ðTÞ r~0 eM=T ,
(1)
where r~0 is a constant. In the high-temperature regime, when T4T L ; M the Coulomb drag resistivity should then depend on the strength of the e–e interactions and be described by a power-law of the form 4K3 T rD ðTÞ r~0 , (2) E0 where E 0 is a constant of the order of the Fermi energy and K is a parameter accounting for the strength of the e–e interactions. If there are no interactions, K is unity and one expects a linear increase of the drag with temperature, as predicted by FL theory. Exploring the different regimes of the Coulomb drag should therefore provides significant insights on the e–e interaction-strength in one-dimension, as well as possibly and unambiguously determining the correct underlying theory for one-dimensional conduction. Coulomb drag measurements between parallel quantum wires in a horizontal geometry have already been performed in several experiments [6,11]. In this geometry, the two wires, which are separated by a soft electrostatic barrier, must be relatively far apart so that interwire tunnelling is minimized. This separation leads to a weaker Coulomb drag signal as it decreases exponentially with the interwire separation. In these experiments, there is also an additional contribution coming from phonon drag which does not decay exponentially and remains strong at
interwire distance above 100 nm [12]. Therefore, these soft barriers cause the drag signal to be significantly influenced by phonon drag. Coupling the wires in a vertical geometry allows us to explore the Coulomb drag in the regime of short interwire separation. This minimizes the phonon-induced drag since, in this geometry, the wires have a hard MBE-defined barrier allowing for much smaller tunneling conductance at a given interwire distance. 3. Device fabrication The vertically coupled quantum wires reported here are fabricated in a GaAs/AlGaAs double quantum well heterostructure with electrostatic TiAu split gates defining the wires. A sketch of the device and the effect of applying a negative bias on the gates are shown in Fig. 1(a). The device consists of 2 pinch off gates and 2 plunger gates, one set on each side of the device, forming two wires that are 2 mm long and 0:5 mm wide. These dimensions were chosen as a balance between the requirements to observe ballistic quantized transport, usually sharper in short wires, and that of observing a strong enough drag signal, larger in longer wires. In order to deposit the metallic gates on both side of the double quantum well heterostructure, we employ the epoxy-bond-and-stop-etch (EBASE) technique [13]. This technique allows the bottom and the top split gates to be approximately 150 nm from their respective quantum well, which is of sufficient proximity to form wellconfined wires. A scanning electron microscope picture of the device made on a GaAs/AlGaAs heterostructure with a 12 nm high barrier is shown in Fig. 1(b). The gates are patterned such that, when an appropriate negative bias is applied on them, only regions C&D and E&F are electrically connected through a quantum wire. With such design, one can probe both wires independently and an arbitrary number of conduction sub-bands can be occupied depending on the bias applied on the gates. In order to determine which barrier height should be used in the double quantum well heterostructure, the tunnelling rate was measured as a function of the barrier height. The results are shown in Fig. 2. The data show the expected exponential decay behavior for tunnelling and follow the relation C ¼ 0:88e1:63d ,
(3)
where C is the tunnelling conductance and d is the barrier height. The wires described previously have a total surface of 1 mm2 or smaller, depending on how heavily the 2DEGs are depleted by the gates. Assuming that the onedimensional tunnelling rate is at most as big as the twodimensional tunnelling rate, we conclude that a 15 nm height barrier would contribute at most to 109 Siemens to the interwire conduction, which is three orders of magnitude smaller than the drag conductance measured by Debray et al. [11].
ARTICLE IN PRESS D. Laroche et al. / Physica E 40 (2008) 1569–1572
1571
Fig. 1. (a) Sketch of the design used for the double quantum wires fabrication. The left-hand side shows the whole structure without bias on the gates. The depleting effect of applying a suitable negative bias on the gates is shown on the right-hand side. For clarity, only the 2DEGs are shown on the right-hand side, but the same gates and ohmic contacts than the ones on the left-hand side are present. (b) A SEM picture of the double quantum wires. Note that the gate on the other side of the wafer (top side) appears as a faint white arm.
rather nonuniform. Our current design should avoid these difficulties. In the future, we will make measurements on newly fabricated vertically coupled quantum wires. An important milestone will be to map out completely the occupied subbands of each quantum wire as a function of different gate voltages. We expect this to be nonetheless non-trivial since all gates are capacitively coupled to each other; yet it should be achievable. Subsequently, we will perform Coulomb drag measurements over wide range of temperatures, densities and magnetic fields. 4. Conclusion
Fig. 2. Log plot of the tunnelling conductance as a function of the interlayer distance of double quantum well heterostructures. The data follow an exponential decay with the barrier height best described by C ¼ 0:88e1:63d .
Tunnelling measurements have already been performed in vertically coupled quantum wires with small interlayer barrier [14] and also for cleaved edge overgrowth wires [15]. In the former case, separate gates were used to achieve independent contacts. These gates were close to the wires and were thus affecting their density profile, making it
In conclusion, we have fabricated vertically coupled quantum wires with independent contacts using split-gate and EBASE techniques. The vertical geometry and the rigid barrier between the two layers will allow us to make Coulomb drag measurements in a regime that has never been explored before. This will provide important information on the many-body interactions that are occurring in one-dimensional quantum wires. Acknowledgements We acknowledge the outstanding technical assistance of D. Tibbets and C. Morath. This work has been supported by the Natural Sciences and Engineering Research Council
ARTICLE IN PRESS D. Laroche et al. / Physica E 40 (2008) 1569–1572
1572
of Canada (NSERC) and the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, US Department of Energy. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000. One of the authors (G.G.) acknowledge receipt of support from the Alfred P. Sloan Foundation under their fellowship program. References [1] [2] [3] [4]
B.J. van Wees, et al., Phys. Rev. Lett. 60 (1988) 848. E.G. Haanappel, D. van der Marel, Phys. Rev. B. 39 (1989) 5484. S.-I. Tomonaga, Prog. Theor. Phys. 5 (1950) 544. J.M.J. Luttinger, Math. Phys. 4 (1963) 1154.
[5] P. Debray, V.N. Zverev, V. Gurevich, R. Klesse, R.S. Newrock, Semicond. Sci. Technol. 17 (2002). [6] M. Yamamoto, M. Stopa, Y. Tokura, Y. Hirayama, S. Tarucha, Science 313 (2006) 204. [7] T.J. Gramila, J.P. Eisenstein, A.H. MacDonald, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 66 (1991) 1216. [8] M.C. Bønsager, K. Flensberg, B.Y.-K. Hu, A. Jauho, Phys. Rev. Lett. 77 (1996) 1366. [9] M.P. Lilly, J.P. Eisenstein, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 80 (1998) 1714. [10] M. Kellogg, J.P. Eisenstein, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 93 (2004) 036801. [11] P. Debray, et al., J. Phys. Condens. Matter 13 (2001) 3389. [12] A.J.J. Rojo, Phys.: Condens. Matter 11 (1999). [13] M.V. Weckwerth, et al., Supperlatt. Microstruct. 20 (1996) 561. [14] E.S. Bielejec, J.A. Seamons, J.L. Reno, M.P. Lilly, Appl. Phys. Lett. 86 (2005) 83101. [15] O.M. Auslaender, et al., Science 303 (2005) 308.