WO2021154351A2

WO2021154351A2 - Guiding electrons in graphene with a carbon nanotube

Info

Publication number: WO2021154351A2
Application number: PCT/US2020/057042
Authority: WO
Inventors: Jean-Damien PILLET; Austin Cheng; Philip Kim
Original assignee: President And Fellows Of Harvard College; École Polytechnique
Priority date: 2019-10-25
Filing date: 2020-10-23
Publication date: 2021-08-05
Also published as: WO2021154351A3

Abstract

Systems, methods of manufacturing, and methods of operation provide for electron waveguides. An upper insulating layer has an upper surface and a lower surface opposite the upper surface; a lower insulating layer; at least one carbon nanotube ("CNT") on the upper surface of the upper insulating layer. A CNT axis extends in a first direction. A conductive layer is disposed between the upper insulating layer and the lower insulating layer and on the lower surface of the upper insulating layer. The conductive layer is conductively connected to a conductive layer electrode. At least two CNT electrodes are conductively connected to the CNT to produce a voltage across the CNT to form a potential well in the conductive layer. Electrons within the conductive layer are confined in a direction perpendicular to the first direction and parallel to the lower surface of the upper insulating layer.

Description

GUIDING ELECTRONS IN GRAPHENE WITH A CARBON NANOTUBE

CROSS-REFERENCE TO RELATED APPLICATIONS [0001] This application claims the benefit of priority to U.S. Provisional Application No. 62/926,101, entitled “Guiding Electrons in Graphene with a Carbon Nanotube,” filed on October 25, 2019, the disclosure of which is hereby incorporated by reference in its entirety.

STATEMENT OF GOVERNMENTAL INTEREST [0002] This invention was made with government support under Grant No. N00014-16-1- 2921 awarded by Office of Naval Research; and under Grant No. DE-SC0012260 awarded by Department of Energy. The government has certain rights in the invention.

COPYRIGHT NOTICE

[0003] This patent disclosure may contain material that is subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent document or the patent disclosure as it appears in the U.S. Patent and Trademark Office patent file or records, but otherwise reserves any and all copyright rights.

TECHNICAL FIELD

[0004] This patent relates to electron waveguides, and more particularly to guiding electrons in graphene with carbon nanotubes.

BACKGROUND

[0005] Individual photons can be used as carriers of information. For example, individual photons can be instantiated in a particular quantum state and transmitted over long distances through optical fibers. The quantum state of the photon can be read out to discern the carried information. [0006] Like photons, the quantum state of individual electrons can be used as carriers of information. However, there is a limited number of tools to control electrons. In addition, guiding the electrons in a solid, like an optical fiber for light, is technically difficult. While one-dimensional materials such as semiconducting nanowires can provide guidance for electrons, such materials can only transmit electrons over short distances before losing information. Other techniques such as transmitting an electron through the edge channel of a two-dimensional electron gas in the quantum Hall regime requires a large magnetic field for the channel to carry only a single mode, which is important for ensuring that the carried information is not to be distorted during transmission.

SUMMARY

[0007] In some embodiments a system includes an upper insulating layer having an upper surface and a lower surface opposite the upper surface; a lower insulating layer; at least one nanotube on the upper surface of the upper insulating layer, the nanotube having a nanotube axis extending in a first direction; a conductive layer between the upper insulating layer and the lower insulating layer and on the lower surface of the upper insulating layer, the conductive layer conductively connected to at least one conductive layer electrode; and at least two nanotube electrodes conductively connected to the nanotube to produce a voltage across the nanotube, wherein the voltage across the nanotube produces a potential well in the conductive layer such that at least some electrons within the conductive layer are confined in a direction perpendicular to the first direction and parallel to the lower surface of the upper insulating layer.

[0008] In some embodiments the conductive layer has a thickness of 1-10 nm.

[0009] In some embodiments one or both of the upper and lower insulating layers have a thickness of 1-10 nm. [0010] In some embodiments one or both of the upper and lower insulating layers comprises hexagonal boron nitride (h-BN).

[0011] In some embodiments the conductive layer comprises graphene.

[0012] In some embodiments the potential well has a depth of 0-0.3 eV.

[0013] In some embodiments the potential well has a width of 1-10 nm.

[0014] In some embodiments the width of the potential well is less than or equal to 10 nm.

[0015] In some embodiments the system further comprises an electron propagating in a single mode in the potential well. In some embodiments the single mode propagating electron is distinguishable from the continuum states outside of the potential well. In some embodiments the single mode propagating electron propagates in a single mode at room temperature.

[0016] In some embodiments the nanotube has a diameter between 1-3 nm.

[0017] In some embodiments, the nanotube includes a carbon nanotube.

[0018] A method of operating the system comprising: applying a first electric potential to the conductive layer with the at least one conductive layer electrode; applying a second electric potential to the nanotube with the at least two nanotube electrodes, wherein the second electric potential is different from the first electric potential; and forming a potential well in the conductive layer by tuning the difference between the first and second electric potentials.

[0019] In some embodiments, a method includes disposing a conductive layer between a lower surface of an upper insulating layer and an upper surface of a lower insulating layer, the conductive layer conductively connected to at least one conductive layer electrode; disposing at least one nanotube on an upper surface of the upper insulating layer opposite the lower surface of the upper insulating layer, the nanotube having a nanotube axis extending in a first direction; and electrically coupling the nanotube to at least two nanotube electrodes configured to produce a voltage across the nanotube, wherein the voltage across the CNT produces a potential well in the conductive layer; such that at least one electron within the potential well is confined in the conductive layer in a direction perpendicular to the first direction and parallel to the lower surface of the upper insulating layer.

BRIEF DESCRIPTION OF THE FIGURES [0020] Various objectives, features, and advantages of the disclosed subject matter can be more fully appreciated with reference to the following detailed description of the disclosed subject matter when considered in connection with the following drawings, in which like reference numerals identify like elements.

[0021] FIG. 1 A is a graphical representation of a simplified waveguide, according to some embodiments.

[0022] FIG. IB are schematics of the band structure of a conductive layer and the density of states, according to some embodiments.

[0023] FIG. 1C is a diagram showing exemplary conditions of the diameter d and fluctuation energies in meV for a waveguide to host a Dirac single mode and non-relativistic single mode, according to some embodiments.

[0024] FIG. ID is an optical image of an electron waveguide, according to some embodiments.

[0025] FIG. IE is a schematic of an electron waveguide, according to some embodiments.

[0026] FIG. IF is a cross section view of an electron waveguide, according to some embodiments.

[0027] FIG. 2A is a schematic showing a measurement scheme, according to some embodiments. [0028] FIG. 2B is a schematic showing an operational principle of a carbon nanotube sensor, according to some embodiments.

[0029] FIG. 2C is a graph showing the conductance of an exemplary carbon nanotube, according to some embodiments.

[0030] FIGS. 2D-2E are graphs that show local quantum capacitance measured as a function of global charge carrier density, according to some embodiments.

[0031] FIG. 2F is a graph showing electronic energy levels of a carbon nanotube, according to some embodiments.

[0032] FIGS. 3 A- 3B are graphs showing the evolution of the graphene local density of states, according to some embodiments.

[0033] FIG. 3C shows graphs plotting computed dispersion relation, according to some embodiments.

[0034] FIG. 3D is a graph showing the positions of a local Dirac point and a first guided mode, according to some embodiments.

[0035] FIGS. 4A-4D show a method of making an electron waveguide, according to some embodiments.

[0036] FIG. 5 shows a method of making an electron waveguide, according to some embodiments.

[0037] FIGS. 6A-6B are a schematic and a graph showing characterization of carbon nanotubes, according to some embodiments.

[0038] FIGS. 7A-7C are illustrations showing a method mechanically transferring carbon nanotubes to an electron waveguide, according to some embodiments

[0039] FIGS. 8A-8B show representations of an electron waveguide circuit, according to some embodiments. [0040] FIGS. 9A-9D are graphs characterizing electron waveguides having wide potential wells, according to some embodiments.

DETAILED DESCRIPTION

[0041] Aspects of the present disclosure relate to electron waveguides, such as waveguides in graphene using carbon nanotubes. In some embodiments, a carbon nanotube can be used to generate a guiding potential for electrons in a nearby conductive layer, such as a 2D graphene sheet, to create a single mode electronic waveguide. The nanotube and graphene can be separated by a few nanometers and can each be controlled and measured independently. Charging the nanotube can cause formation of a single guided mode in graphene through which electrons can be guided. The same nanotube can be used as a probe to detect the propagation and properties of the guided electrons. This single electronic guided mode in graphene can be sufficiently isolated from other electronic states of the linear Dirac spectrum continuum, allowing the transmission of information with minimal distortion.

[0042] FIG. 1 A is a simplified graphical representation of an electron waveguide, according to some embodiments. As shown in FIG. 1 A, a nanowire, such as a carbon nanotube (“CNT”) 110 can be charged at a desired potential near a conductive structure, such as a 2D conductive layer 120 (e.g., a graphene sheet). The charge on the CNT 110 can cause a potential well in the conductive layer 120 in the x direction for each location in the conductive layer 120 along the length of the CNT 110 in the y direction, which is graphically represented by potential 111. In such an embodiment, an electron 190 traveling within the conductive layer 120 is confined within the direction Z by the 2D nature of the conductive layer 120. In addition, if the potential well 111 has a sufficient depth Uo and width d, the electron 190 in the conductive layer 120 can also be confined in the x direction. Accordingly, as described in more detail throughout the present disclosure, such a configuration can confine the electron 190 to travel only in the y-direction along the CNT 110 (e.g., in one dimension), thereby creating an electron waveguide similar to optical waveguides for photons. Furthermore, as described in more detail throughout the present disclosure, the proximity of the CNT 110 to the conductive layer 120, the charge on the CNT, and/or other characteristics can optionally be controlled so as to ensure that the electron 190 propagates in a single waveguide mode, thereby decreasing distortion and permitting the electron to travel over long distances through the electron waveguide without changing state, and thus without losing information.

[0043] FIG. IE is a schematic of an electron waveguide 100, according to some embodiments. The electron waveguide apparatus can include a CNT 110, two CNT electrodes 112 in contact with the CNT 110, insulating layers 140 and 142, a conductive layer 120, and an electrode 130 in contact with the conductive layer. In some embodiments, only one electrode is in contact with the conductive layer to apply an electric potential to the conductive layer. In some embodiments, more than one electrode (additional electrodes not shown) can be in contact with the conductive layer so that electrical transport properties of the conductive layer can also be characterized, for example based on electronic measurements (current, voltage, resistance, power, conductance, etc.) at one or more of the electrodes. The electrodes can be placed at any location, for example, on opposite sides of the conductive layer 120. As shown in FIG. IE, the conductive layer 120, which can be, for example, a 2D graphene sheet, can be disposed between two insulating layers 140 and 142, which can be, for example, hexagonal boron nitride (“h-BN”), according to some embodiments. Other insulating materials for either or both of layers 140 and 142 and for conductive layer 120 are contemplated. A CNT 110 can be located on top of the insulating layer 140 opposite the conductive layer 120. The CNT can be held at a potential, for example through conductive contact with two CNT electrodes 112. The conductive layer 120 can be connected to at least one electrode 130, which electrically connects the conductive layer 120 to a voltage source 160 which can be at a different potential from that of the CNT 110. A person of ordinary skill in the art would understand, based on the present disclosure, that other techniques could be used to control the potential of the conductive layer 120. As described in more detail throughout the present disclosure, the potential difference between the CNT 110 and the conductive sheet 120 forms a potential well in the conductive sheet 120 which can be used as an electron waveguide

[0044] FIG. IF shows a cross-sectional view of the electron waveguide 100, according to some embodiments. As shown in FIG. IF, the conductive layer 120 is encapsulated above and below by insulating layers 140 and 142, respectively, and is electrically connected to at least one electrode 130. The CNT 110 is shown above the insulating layer 140 and is connected to one of the CNT electrodes 112.

[0045] As shown in FIGS. IE and IF, during operation, the CNT 110 can be held at a potential across CNT electrodes 112, which can form a corresponding potential well in the conductive layer 120 as discussed throughout the present disclosure, for example, with reference to FIG. 1 A. The depth of the potential well can be adjusted by the voltage difference applied between the CNT 110 and the conductive layer 120. This potential well can form a guided channel in the conductive layer 120. Without being bound by theory, the width d of the guided channel (see FIG. 1 A) can be roughly equal to the radius of the CNT 110, (for example, lnm), plus the thickness of insulating layer 140 separating the CNT 110 and the conductive layer 120. This potential well, in combination with the thin conductive layer 120, can confine electrons to travel in direction 119, thereby forming an electron waveguide in the conductive layer 120. As described in more detail throughout the present disclosure, the characteristics of the electron waveguide 100, such as but not limited to the potential on the CNT 110, the thickness of the insulating layer 140, and the materials used can be selected to ensure that electrons traveling through the electron waveguide are limited to only a few modes, such as a single mode. Such configurations can ensure that information carried by the quantum state of an electron propagating within the electron waveguide is preserved over long distances.

[0046] In some embodiments a conductive layer 120 made up of a 2D graphene sheet can be separated from a CNT 110 by an insulating layer 140 made up of an insulating material, such as h-BN. One or more of these features can allow for large potentials on the CNT 110 without a breakdown or leakage to conductive layer 120, even when the insulating layer 140 is very thin (e.g., between 4-10 nm). In some embodiments, the electric potential across the insulating layer is in the range of 0.01-5 V, in the range of 0.1-4 V, or in the range of 0.5-2 V. Furthermore, this large potential, which spans across the thickness of the insulating layer 140, can be tunable by controlling the potential on the CNT 110 (and/or the potential on the conductive layer 120). Such a large potential on the CNT 100 in close proximity to the conductive layer 120 can create a potential well that is sufficiently deep and thin so as to confine electrons traveling through the electron waveguide to a single mode. Such a configuration can preserve the wave nature (or “coherence”) of an electron traveling through the electron waveguide such that information is preserved over long distances even at room temperature. In some embodiments, a CNT 110 with a small diameter, such as a single- walled CNT, can be used to further reduce the width d of the potential well. A person of skill in the art would understand, based on the present disclosure, that other materials and configurations can be used to form potential wells with sufficient characteristics to form monomode electron waveguides.

[0047] In some embodiments, implementing insulating layers 140 and 142 using h-BN can help ensure that the conductive layer 120, such as a 2D graphene sheet, is sufficiently flat, thin, and clean so as to allow for propagation of electrons in 2 directions. Furthermore, h-BN can allow for a high potential on CNT 110 without allowing current to leak to the conductive layer 120 and without altering the mobility of electrons through the conductive layer 120. In some embodiments, other conductive layers can be used, such as but not limited to gold. Such conductive layers can have a thickness of approximately 1-10 nm so as to confine the electrons to a single mode.

[0048] Without being bound by theory, the configurations shown in FIGS. IE and IF can take advantage of massless quasiparticles in graphene. The quasi-relativistic linear energy dispersion in graphene allows the wavefunction of the Dirac fermions to travel with minimal distortion. Furthermore, high mobility allows electrons to be transmitted ballistically over several microns even at room temperature.

[0049] In some embodiments, it is possible to use the configurations shown in FIGS. IE and IF to create a single mode electronic guide with a deep potential well having a width much smaller than the wavelength of electrons in order to suppress scattering in the core of the waveguide. In some embodiments, the wavelength can reach around one hundred nanometers at certain electron densities, which makes it helpful to be able to place extremely narrow gates close to the electron gas. In some embodiments, the electron density in the conductive layer can be tuned by adjusting a back gate potential (e.g., Vb_g shown in FIG. 2A, not shown in FIG. IE). The electronic modes generated by such a 1 -dimensional (ID) potential well are manifested in the band structure of the graphene as branches similar to optical modes, which are separated from the continuum up to the energy that roughly corresponds to the depth of the potential well Uo.

[0050] The left panel of FIG. IB is a schematic of the band structure of an exemplary graphene conductive layer 120 as a function of momentum k_y, according to some embodiments. The schematic shows the allowed momentums k_y for a given energy (vertical axis). As shown in FIG. IB, the continuum bulk states (shown in solid lines) available outside the potential well are separated from the guided modes (shown in dotted lines). Furthermore, unlike the continuum of states in the bulk population, the guided modes in the potential well are isolated from one another by large gaps. Due to such isolation between bulk states and between individual guided modes, the guided modes are unlikely to mix with one another and with the continuum states, and therefore constitute reliable carriers of information. The right panel of FIG. IB shows the global density of states (“DOS”) in bulk graphene (dotted line) and local density of states (“LDOS”) within the potential well (solid line), according to some embodiments. The guided modes form locally at the center of the potential well such that they do not affect the overall graphene density of states (DOS) but appear as resonances in the local density of states (LDOS) close to the LDOS minimum which indicates the position of the local Dirac point.

[0051] In some embodiments, it is desirable to obtain a single guided mode within the potential well. Under such conditions, electrons traveling through the waveguide do not easily mix with the surrounding bulk electrons and cannot change state to another mode. Such a single guided mode is well suited for carrying information over long distances. Without being bound by theory, the number of modes in the potential well can be approximately given by the ratio Uod/hvF, where Uo is the potential at the bottom of the potential well, d is the width of the potential well, and v_F is the Fermi velocity. In some embodiments, this quantity is approximately one for a single mode waveguide. In principle, this condition can be fulfilled for very wide and shallow potentials, but for the mode to be well-defined it is helpful to form a potential depth Uo which is much greater than the fluctuations of chemical potential caused by disorder. This explains why it can difficult to guide electrons in disordered materials, such as disordered graphene. For graphene encapsulated in h-BN, these fluctuations are on the order of a few meV, making it easier to implement a well-defined single mode. [0052] Without being bound by theory, FIG. 1C is a diagram showing exemplary conditions of the diameter d and fluctuation energies in meV for a waveguide to host a relativistic single model (i.e., Dirac single mode) and non-relativistic single mode, according to some embodiments. In some embodiments, the Dirac single mode carries Dirac fermions (e.g., massless electrons) whose velocity equals the Fermi velocity and does not depend on its energy. In some embodiments, the non-relativistic single mode carries non-relativistic electrons (e.g., massive electrons) whose velocity is energy dependent. In some embodiments, Dirac single mode can propagate over a much larger distance (e.g., 100 microns) than non-relativistic single mode (e.g., less than a micron). In order to obtain a single mode waveguide that is immune to disorder, the depth of the potential well, therefore, can be approximately a few tens of meV, which corresponds to a width on the order of 10 nm or less. Such conditions can be satisfied, for example using the gate made with a single- walled CNT in close proximity to a conductor, such as graphene, as described above with reference to FIGS. IE and IF. In some embodiments, one may utilize a linear dispersion between the energy and momentum for Dirac fermion to create a single mode for electrons. Without being bound by theory, for non-relativistic electrons in semiconductors, a single mode can be produced under the condition that Uod² « h²/m, where m is the effective mass of the electron, leading to a much shallower potential well (for example, on the order of 1-10 meV) even for smaller width d< 10 nm.

[0053] In some embodiments, in addition to generating a potential well, the same CNT 110 can also be used as a local probe to measure the graphene LDOS utilizing the capacitive coupling between CNT and the guided modes in the graphene. For example, the CNT can be operated as a single electron transistor (“SET”), which can perform like a charge sensor. In some embodiments, the CNT is sensitive to its surroundings. For example, when the electrical environment around the carbon nanotube changes, the carbon nanotube will react to the change. In some embodiments, the CNT reacts by having its electrons move. By monitoring how the electrons in the carbon nanotube behave, one can deduce an electrical environment around the carbon nanotube. In some embodiments, the electrical environment consists of the lower graphene layer 120, which means that one can deduce what’s going on in the graphene layer 120 by monitoring the carbon nanotube electrons. It is therefore possible to sense the presence or absence of a single guided mode in graphene by monitoring the carbon nanotube. In some embodiments, to detect the guided mode, the electron density of state of the graphene can be filled (or emptied) using a voltage applied on the back gate. For example, if a large negative voltage is used first and increased towards positive values, the graphene can be filled with electrons and the CNT can be measured to sense how many electrons are added for a given voltage increment. In some embodiments this increment depends on the slope of the dispersion relation (see FIG. IB). In some embodiments, it is not easy to tell whether continuum states or a guided mode is filled because their dispersion relations are both linear with similar slopes. In some embodiments, at the beginning of the guided mode (see FIG. IB), the dispersion relation is curved and the slope is much weaker, which means that for a given voltage increment it is possible add many more electrons into the guided mode than the continuum states in graphene. Therefore, the CNT can sense the presence of a guided mode.

[0054] FIG. 2A shows a schematic of the measurement scheme 200 where the electrostatic potential of CNT can be controlled by both graphene gate voltage ( VG ) (for example, the voltage applied by electrode 130) and the global back gate voltage (V_bg). In some embodiments, the graphene gate voltage (VG) changes the electric potential of the graphene with respect to ground (and thus, the CNT as well). In some embodiments, the global back gate voltage (Vbg) changes the carrier density of the graphene. The two voltages can be controlled by a user input, such as knobs on a voltage application device, that serve to vary the state of the CNT and graphene based on user input.

[0055] In some embodiments, when a CNT is connected to metallic electrodes and at sufficiently low temperature, the CNT can enter the Coulomb blockade regime and become sensitive to external charges (as shown in FIG. 2F). In some embodiments, the temperature at which the CNT can be cooled down to enter the Coulomb blockade regime depends on the CNT diameter: the smaller the diameter, the higher the temperature. In some embodiments, the temperature at which the CNT can be cooled down to enter the Coulomb blockade regime depends on the distance between the two electrodes: the closer the electrodes, the higher the temperature. In some embodiments, the temperature at which the CNT can be cooled down to enter the Coulomb blockade regime is in the range of 1 K to 100 K, or in the range of 10 K to 50 K. In some embodiments, in the Coulomb blockade regime, the CNT acts as a charge sensor that can detect a change in a surrounding electrical environment. For example, if a charge is brought close to the CNT, the electrons in the CNT can ‘feel’ it and redistribute themselves in response to the introduction of the external charge. By monitoring the configuration of these electrons, it is possible to deduce what's going on with the electrical environment around the CNT.

[0056] FIG. 2F is a graph showing electronic energy levels of a carbon nanotube, according to some embodiments. By measuring the conductance GNT of the CNT 110 as a function of the gate voltage V_bg or the potential applied to the graphene sheet VG , it is possible to observe a series of peaks corresponding to the different electronic energy levels of the CNT 110. An exemplary spectrum of peaks corresponding to different electronic energy levels of the CNT 110 is shown in FIG. 2F, according to some embodiments. Each of these exemplary energy levels can contain one electron. These energy levels can be used as individual local probes sensitive to the electrostatic environment and therefore to the local charge density of graphene located below the CNT.

[0057] An exemplary operational principle of these probes, informed by direct measurements of Fermi energy performed in graphene and bilayer graphene, is illustrated in FIG. 2B. In some embodiments, the Fermi energy EF 210 corresponds to the energy of the electrons responsible for the electrical transport. In some embodiments, the back gate voltage Vbg can be adjusted to empty or fill the graphene conductive layer 210 with electrons (positive voltage will attract electrons from outside and fill the graphene while negative voltage will repulse electrons and empty the graphene). As shown in step 212, when increasing the back gate potential Vbg, the graphene band structure can be filled by increasing the number of carriers by Snc with the corresponding change of Fermi energy δE_F .

[0058] If the total electrochemical potential of graphene m 218 (electrostatic potential added to the Fermi energy EF) exceeds the energy of one of the electronic states of the CNT, then the electronic state of the CNT is filled. Subsequently in step 214, the graphene electrostatic potential can be lowered with VG and therefore the energy of all the electrons in the graphene can be reduced by an amount SEF. If the electrochemical potential of graphene m , adjusted by a change of the graphene bias SVG, becomes lower than the energy of the same CNT electronic level, it consequently empties and goes back to its original state 216. By measuring the charge state of the CNT between each step, it is then possible to deduce the energy change SEF = eSVc corresponding a charge variation Snc, where e the charge of an electron. Without being bound by theory, this procedure yields the local quantum capacitance of graphene:

[0059] Note that e²C_q at finite temperature is the compressibility of a mesoscopic system dnc/ m , which can be associated with the many body DOS. Since capacitive coupling between graphene and CNT can be strongly localized in the vicinity of the CNT, the measured C_q can be proportional to the LDOS of graphene underneath of the CNT. This technique can provide an absolute measurement of quantum capacitance without any scaling parameters or adjustment of the origin of energies.

[0060] FIG. 2C is a graph showing the conductance of an exemplary CNT conductance GNT as a function of Vbg and VG , according to some embodiments. For this exemplary device characterized in FIG. 2C, the h-BN spacer between CNT and graphene is 4 nm thick and measurements were performed at 1.6 K. The measured peaks in the GNT exhibit trajectories in the Vb_g - VG plane that yield the evolution of the Fermi energy as described above. The slope of these trajectories can yield directly the local quantum capacitance C_q.

[0061] The line 222 on the left in FIG. 2C shows an exemplary trajectory obtained following the steps in FIG. 2B. In some embodiments, the trajectory shows how the Fermi energy evolves as the graphene is filled with electrons. Starting from step 212, one can first increase Vbg which controls the number of extra electrons in graphene no. The conductance of the CNT may increase as a result of the extra electrons in graphene. Therefore, a kink may be observed. In step 214, one can lower VG which controls the Fermi energy in graphene. If the electrochemical potential of graphene m , adjusted by a change of the graphene bias 6VG, becomes lower than the energy of the same CNT electronic level, the CNT electronic level may consequently empty and go back to its original state 216. As a result, anther kink in the CNT conductance may be observed.

[0062] In some embodiments, for bare-graphene, a simple S-shape trajectory can be observed with a single kink like the white dashed line 224 in the center. In some embodiments, if there is a guided mode, there can be an extra kink as in the white dashed line 226 on the right or in the inset of FIG. 2C. If there is a guided mode, there is a point at which it is desirable to add more electrons to increase the Fermi energy of the same quantity. This can lead to the extra kink 228, which is thus a signature of the presence of a guided mode. When VG ~ 0, the potential difference between the CNT and the graphene is small and, consequently, the potential well generated by the presence of the CNT is shallow. The inset shows dGNT/dVG over a small region in order to highlight a double kink corresponding to the Dirac point followed by a guided mode resonance (blue arrow).

[0063] FIGS. 2D and 2E are graphs that show local quantum capacitance measured as a function of global charge carrier density n<, for two different voltage differences VG = 1.45 and -0.2 V, respectively, between the CNT and graphene, according to some embodiments. The peak 230 in FIG. 2D corresponds to the extra kink as shown in FIG. 2C. When the graphene is exposed to a large enough potential from the CNT (e.g., V_g = -1.45 V), a single guided mode is created and it’s possible to observe a peak in the local quantum capacitance. As shown in FIG. 2D, when the potential is low (e.g., V_g = —0.2 V), no single guided mode is created and no peak is observed. The LDOS measured is the one of bare graphene with a minimum at zero energy, following |«G| ^1/2 on the hole and electron sides. Note that n<, denotes the global charge density of graphene since Vb_g controls the charge density over the entire graphene sheet. With the minimum of LDOS being very close to no = 0, the doping underneath the CNT can be assumed to be low, suggesting low impurity levels in the example measurement.

[0064] A deeper potential well can be formed by increasing VG, which can cause the LDOS to develop a more pronounced characteristic resonance, corresponding to a single guided mode as manifested by the peak 230 shown in FIG. 2D. Compared to the measurement performed at VG = 0, the minimum of quantum capacitance has shifted from the global charge neutrality point and towards the electron side ( nc > 0), as expected for a positive voltage applied on graphene while the CNT is maintained at ground potential. The resonance lies between this minimum and the global charge neutrality point of graphene (nc = 0), a region where the doping caused by the potential well can lead to an NPN junction configuration. The appearance of this resonance can be understood in the following manner: as a guided mode detaches from the Dirac cone, it generates a peak in the LDOS due to the ID van Hove singularity appearing at the extrema of the single mode energy dispersion E(k_y) where k_y is the wave vector along the CNT (see Fig.3C). Our measurements are in excellent agreement with numerical tight- binding simulations where the only fitting parameters are the depth and width of the potential well (see supplementary information). Theory predicts the appearance of multiple successive modes that could give rise to additional resonances. However, due to presumably disorder induced broadening, unambiguously identifying multiple resonances is challenging within our experimental noise limit.

[0065] A continuous evolution from bare graphene to a single mode waveguide can be observed as the potential depth Uo is tuned, according to some embodiments. For example, as shown in the measurements of FIG. 3 A, the graphene LDOS appears to be affected by tuning Uo, by changing the potential difference VG between the CNT and graphene becomes non-zero. At low Uo, the minimum corresponding to the Dirac point (between n_<, = -0.5 to ~0) is less pronounced, and that an asymmetry is formed between the electron and hole. As shown in FIG. 3 A, the measured quantum capacitance appears to be symmetric about the center point where nc= 0, when VG is low. However, as VG increases, the asymmetry in the measured quantum capacitance becomes more and more clear. Without being bound by theory, this evolution, also predicted by numerical simulations shown in FIG. 3B, can be explained in some embodiments by the formation of closely packed guided modes whose branches are too close to the continuum, preventing the development of sharp resonances in the LDOS.

[0066] FIG. 3C shows computed dispersion relation as a function of k_y momentum along the CNT direction. A branch corresponding to the ID guided mode gradually and continuously separates from the Dirac cone as Uo increases from the left pane at Uo = 0, to the right pane at Uo = 180 meV. In some embodiments, it is possible to access a single branch without inadvertently accessing the other branch. For example, for the exemplary device measured herein, it is possible to choose Uo = 90 meV.

[0067] In some embodiments, for larger VG , a resonance as marked by arrow 306 gradually increasing in amplitude and shifting from the charge neutrality point appears. This can reflect the formation of a branch in the dispersion relation of graphene, which can become increasingly more detached from the continuum, represented as 302 and 304 shown in FIGS. 3A and 3B. The curvature of this branch at its beginning becomes flat until it acquires a minimum located around k_y ~ 1/d, giving rise to a sharp resonance in LDOS. In embodiments of the relativistic Dirac fermionic system, the ID guide mode can exhibit a potential strength threshold for the appearance of the first guided mode.

[0068] FIG. 3D shows the positions of the local Dirac point and the first guided mode as a function of VG for a simulated system (lines 316 and 318) and an exemplary experimental systems (lines 312 and 314), according to some embodiments. The vertical axis E corresponds to an energy difference that separates the guided mode from the continuum states. In some embodiments, it is possible to choose a large E for better access to the guided mode. FIG. 3D suggests that the appearance of a guided mode 312 starts at finite Uo. In some embodiments, a single guided mode can appear in systems with an insulating layer 140 of h-BN having a thickness of 6 nm or thinner.

[0069] Guided modes with larger energy separations have many technological applications. For example, such a system can be used to transmit information over large distances using the quantum state of an electron traveling through the waveguide. Larger separations between modes and from the continuum allow for electrons to transmit information transmission robustly along the guide while avoiding processes that scatter electrons, leading to loss of information. For applications operating at room temperature, the energy separation between the guided modes and the energy separation between the guided mode and the continuum states can be above the thermal energy, which in some embodiments is approximately 25 meV. This separation is directly given by the energy position of the resonance with respect to the global Dirac point of graphene. In some embodiments, this energy separation can be larger than the thermal fluctuation room temperature (around 300 K, i.e., approximately 25 meV) to permit operation at room temperature. In some embodiments, an energy separation can be used that is as large as possible in order to access a single guided mode and reduce the likelihood of accessing other states. As shown in the exemplary embodiment of FIG. 3D, it is possible to control this energy continuously up to approximately 0.1 eV, well above thermal fluctuations at room temperature. Thus, like existing optical waveguides which use photons as carriers of information, the guided mode waveguides disused throughout the present disclosure can be used in electronic devices at room temperature to carry information using electrons in the waveguide. In addition, electron waveguides like those discussed in the present disclosure can be used in plasmonics applications and as test-beds for relativistic simulation. In some embodiments, electron waveguides like those discussed in the present disclosure can be used in transmission of information with electrons in a quantum processor, single electron logic, analog of optical devices dedicated to electrons (e.g., beam splitter, directional coupler, etc.).

Exemplary Waveguide and Fabrication Techniques

[0070] FIG. ID is an optical image of an electron waveguide, according to some embodiments. As shown in FIG. ID, graphene is encapsulated between two layers of h-BN where the upper one is only a few nm thick and on which a CNT is deposited. Since the CNT diameter is between approximately 1 and 3 nm, and the thickness of the top h-BN layer is only a few nanometers, the characteristic width d of the well is less than 10 nm. This exemplary configuration can be and has been used to drive the device into a single guided mode in the graphene beneath the CNT. The graphene and CNT are both connected to separate electrodes, which allows them to be independently controlled and measured as discussed above. Without being bound by theory, the length of the waveguide can be expressed as the distance between electrodes connecting the CNT, i.e. 500 nm. Exemplary details of fabrication are discussed below.

[0071] FIGS. 4A-4D are diagrams showing a method of fabricating the electron waveguide of FIGS. 1D-1F, according to some embodiments. In the example of FIGS. 4A- 4D, an electron waveguide is fabricated using a graphene conductive layer 420 encapsulated between two insulating layers of h-BN. For example, as shown in FIG. 4A, an h- BN/graphene/h-BN sandwich of layers 442, 420, and 440 is formed, for example, on an n- doped silicon wafer with 285 nm Si02. In some embodiments, the graphene layer 420 can be a layer of pre-fabricated graphene flake, which is placed onto the insulating layer 442. In some embodiments, the graphene layer is transferred through mechanical methods, such as, but not limited to a polymer transfer process. In some embodiments the thickness of the top h-BN layer 440 can be chosen between 4 and 100 nm and the bottom h-BN layer 442 can be chosen to be around 20 nm.

[0072] As shown in FIG. 4B, electrodes 430 can be disposed in contact with the conductive graphene layer 420, according to some embodiments. In some embodiments, only one electrode is in contact with the conductive graphene layer to apply an electric potential to the conductive graphene layer. In some embodiments, more than one electrode can be in contact with the conductive graphene layer so that electrical transport properties of the conductive graphene layer can be characterized. E-beam lithography can be used to design the electrodes 130 contacting the graphene flake. For example, the edges of the graphene flake can first be exposed by reactive ion etching through a resist mask and subsequently evaporation of a metallic trilayer Cr(5nm)/Pd(15nm)/Au(5nm) through the same mask. A second step of lithography can then be performed to design electrodes 412 (for example, using the same or a different metallic trilayer) on top of the top h-BN layer 440, as shown in FIG. 4C. These electrodes can be used to contact the carbon nanotube during the transfer step described in more detail throughout the present disclosure. As shown in FIG.

4D, the sample can be covered with a lOOnm thick layer of resist 470 (for example, PMMA A4 495K) except for areas where the electrodes are to be connected to the CNT. The resist can help increase the efficiency of the transfer of the carbon nanotube described throughout the present disclosure.

[0073] In some embodiments, CNTs can be grown and characterized according to any known or yet-to-be developed technique. In some examples, CNTs can be grown on 5 x 5mm² silicon chip with a slit in the center, as shown in FIG. 5 using standard technique of chemical vapor deposition. For example, a substrate 510 can be prepared with an upper surface 512 having a slit 514, as shown in step 501 of FIG. 5. A catalyst 516 can be deposited on one side of the slit, as shown in step 502 of FIG. 5 such that CNTs 518 grow suspended over the slit 514 as shown in step 503 of FIG. 5. One of example of these CNTs 518, suspended over a slit 514 that is 65 im wide and 1cm long, is shown in the optical picture 505 of FIG. 5. The CNT 518 of the optical picture 505 of FIG. 5 is covered with 30 nm of Au, so it can be seen optically.

[0074] After growth, CNTs 518 can be characterized using Rayleigh scattering to identify whether CNTs are metallic or semiconducting. For example, as shown in FIG. 6 A, broadband laser light 618 can be shone on the CNTs 518. The scattered light 682 can be collected with a detector to identify a resulting spectrum. FIG. 6B shows an exemplary spectrum showing how strongly the CNT is scattering incident light with different energy, which can be used to identify the CNT chirality. In some embodiments, positions of the peaks in the spectrum are used to determine the CNT chirality. In the example of FIG. 6B, the CNT was metallic with a (16, 4) chirality. Moreover, the scattered light 682 can also be used to measure the position of the carbon nanotube along the slit such that it can be aligned with the circuit for subsequent transfer. In some embodiments, the position of the CNT is measured by sweeping the beam of the broadband laser along the slit (using a micromanipulator that moves the chip) starting from the edge of the slit and detecting when the light is scattered by a carbon nanotube.

[0075] FIGS. 7A-7C are illustrations showing a method for providing CNTs to form an electron waveguide 700, according to some embodiments. As shown in FIG. 7A, a surface 785 with a slit, such as the surface 512 with slit 514 shown in FIG. 5, and with a suspended CNT 710 can be placed above the electron waveguide 700 in order to align the CNT 710 with the area of interest with electrodes. The surface 785 with a slit can be pressed on the sample as shown in FIG. 7B. After obtaining sufficient mechanical contact, electron waveguide 700 can be warmed up, for example to 180°C for 5 minutes, in order to melt the resist to help the CNT 710 transfer from the surface 785 with a slit to the target electron waveguide 700. The surface 785 can then be separated after cooling to room temperature, which leaves the CNT 710 behind on the electrodes of the electron waveguide 700. It should be noted that similar techniques can be used to encapsulated graphene with sheets of h-BN.

Quantum Capacitance Measurements

[0076] Without being bound by theory, FIG. 8A is an equivalent circuit of the electron waveguide discussed above with reference to FIGS. 1D-1F, according to some embodiments. As shown in FIG. 8 A, a CNT 810 is held at a potential relative to the conductive layer 820 at VG and Vb_g of the back gate 860. Without being bound by theory, FIG. 8B shows the same hybrid nanotube-graphene device as a network of capacitances including geometric and quantum capacitances. This schematic is equivalent to the following set of equations

where, as discussed above, Vbg is the voltage applied on the back gate, VG is the voltage applied on the graphene flake, no is the number of carriers in the graphene flake, nNT is the number of carriers in the CNT, C_sio2 is the capacitance between the graphene flake and back gate, C_BN is the capacitance between the CNT and back gate, is the Fermi energy of

graphene, and is the Fermi energy of the CNT. In some embodiments, these equations can be obtained from an electrostatic description of the circuit where the total energy of the circuit Etot is given by:

Equation 2: E_tot

where m_g is the extra amount of charge accumulated on the back gate, and DG(NT) (E) is the density of states of graphene (resp. nanotube) as a function of energy E. The total energy Etot can contain the following terms: The first two can be the electrostatic energies of the two geometric capacitors formed, for the first one, by the back gate and the ensemble graphene- nanotube (e.g., the graphene and CNT as one entity) and, for the second one, by the graphene flake and the CNT. The next two terms can be the energies due to the fillings of electronic levels in the nanotube and in the band structure of graphene. The last two terms can be the energy provided by the two voltage sources applying respectively a potential VG and Vbg on the graphene and back gate. This energy can be at a minimum when dEtot/dnm = 0 and which, combined with the condition that TINT + TIG

= -m_g since the circuit is a closed system, can lead to the system of equations 1 described above. Note that in some embodiments, this follows from

[0077] In some examples, without being bound by theory, equations 1 can simplify using the approximations that are often valid. These

conditions can be valid in some embodiments since tens of volts are applied on the back gate while only hundreds of mV are applied on the graphene flake, since the nanotube contains in some examples only tens of electrons while the graphene flake contains tens of thousands of electrons, and since the Fermi energy of graphene rarely exceeds a few hundreds of meV. By introducing the nanotube quantum capacitance , the equations 1 simplify to:

with Without being bound by theory, from equations 3, it is possible

to see that, at a fixed number of charges in nanotube (TINT constant), the number of charges in graphene no and its Fermi energy can be identified for given values of Vb_g and VG.

[0078] As a consequence, along a trajectory made by an electronic level of the nanotube (i.e. when the charge is fixed at half an integer) in the {VG, Vbg} plane, it is possible to assume the following:

[0079] Since in some examples, measurements have shown that C_{si02 ≈} 12n / cm², the quantum capacitance of graphene can be directly determined from the slope of these trajectories as described above. Density of States Calculation Using a Discretization of the Massless Dirac Equation

[0080] Without being bound by theory, this section explains the theoretical underpinnings behind calculations of the density of states, according to some embodiments. In some embodiments, the calculations can confirm the creation of guided modes observed in the measurements.

[0081] Dirac Hamiltonian: Without being bound by theory, in some embodiments, in order to describe the graphene waveguides described above, the following two-dimensional massless Dirac Hamiltonian can be used:

where VF is the Fermi velocity, s_c and o_y the Pauli matrices. Due to the presence of the charged nanotube, the electrostatic potential landscape U (x) can take the shape of a potential well that is invariant along the axis of the nanotube (y-axis). For simplicity, and without being bound by theory, a Lorentzian potential can be chosen, though the precise shape of the potential well can depend on how the electrons of graphene screen the electric field generated by the nanotube. The Lorentzian potential can be written as:

where x = 0 corresponds to the position of the nanotube along the x-axis, Uo is the strength of the potential and d is the width of the potential. The width d can depend on the radius of the nanotube as well as the distance between nanotube and graphene which can be set by the thickness of the h-BN between them, as discussed above. A logarithmic potential

can also be used, which corresponds to the potential generated by a one-dimensional wire in a parallel plane. The selection of this potential has not been seen to produce a qualitative difference in the resulting density of states. [0082] Discretized Dirac Equations: Without being bound by theory, in order to calculate the density of states in the graphene flake, the eigenstates of the Hamiltonian ψ (x, y) can be calculated, which obey the massless Dirac equation:

where E are the eigenenergies. Since the present non-limiting analytical description can be invariant along they direction, the operator p_y can be replaced by a classical variable p_y = hk_y where k_y is the projection of the wavevector along this axis. As a consequence, the solutions can be written as two-components spinors , which

are plane waves along the y-axis.

[0083] In some embodiments, without being bound by theory, ψ and ψ can be calculated numerically using a discretization of the Hamiltonian over a lattice whose points are separated by a step Δ. However, a naive replacement of the derivative by its discreet equivalent might not preserve the hermiticity of the Hamiltonian and cause a fermion doubling problem. To circumvent this problem, Susskind discretization can be used such that:

and

where and m is a relative integer such that m ∈

for a flake of width W = 2NΔ. In some embodiments, this means that ψ can be evaluated over the points of the lattice but

at the midpoints. The discrete version of the Dirac equation is then written as

and

and the boundary conditions can be chosen such that .

Such boundary conditions can result in the formation of states on the edge of the graphene flake, but such states will not analytically affect the local density of states below the nanotube.

[0084] Without being bound by theory, by solving equations 3 and 4, it is possible to obtain branches of eigenenergies E_n {k_y) with corresponding eigenstates whose spinor components are Iⁿ practice, this calculation is performed by first

writing equations 3 and 4 in the following form:

and then diagonalizing numerically the matrix H, which corresponds to the Hamiltonian. Here, a 4N-components vector

[0085] In these expressions 2N x 2N matrices are introduced where © is a matrix full of zeros, I is the identity matrix, B_+i is a matrix in which all the coefficients are zero except on the first upper diagonal where all the coefficients are equal to 1, B-i is a matrix in which all the coefficients are zero except on the first lower diagonal where all the coefficients are equal to 1, and is a diagonal matrix which refers to the position along the x-

axis.

[0086] By diagonalizing the

it is possible to obtain the eigenvalues E_n {k_y) and the corresponding eigenstates

[0087] Global and local density of states calculations: Without being bound by theory, in some embodiments the global density of states in graphene DOS (£) can be given by

where the factor 2 ca account for the spin degree of freedom and where a phenomenological broadening Υ for each electronic level of energy E_n {k_y) is introduced in order to smooth the density of states. In some embodiments, the value of g can be chosen such that g =

0.01 X hV_f/Δ , which roughly corresponds to the distance between two energy levels in some embodiments. The total density of states can be obtained, in some embodiments, by summing over all the eigenenergies of

(4 N in total) for a given k_y and then by summing over k_y. Here a graphene flake width W= 2NA and length L can be considered such that k_y can be considered to be quantized in steps of Ή/L in the interval In some non-

limiting exemplary calculations in the present disclosure, the following values can be selected: Δ = 1 nm, N= 200 and L = 50πΔ.

[0088] The local density of states LDOS (E) below the nanotube can be obtained in a similar fashion but taking into account the spatial distribution of the wavefunctions

where the matrix M can account for the small region below the nanotube over which the LDOS is measured. This region can be chosen to have the same width d as the electrostatic quantum well created by the same nanotube such that M can be written as:

where it is assumed that the sensitivity of the nanotube decreases with distance following a Lorentzian decay. Note that

can correspond to the surface over which the nanotube measures the local density of states, which means that LDOS ( E) can be a local density of states per unit area.

[0089] Fitting parameters Uo and d. Without being bound by theory, in some exemplary simulations described in the present disclosure, only two fitting parameters are used to describe quantitatively the measured density of states. The first one is the depth of the potential Uo that can be controlled, as discussed above, by setting the voltage difference applied between the CNTs and the conductive layer, such as a graphene layer, VG. The second exemplary parameter is the width d of the potential, which can be roughly given by the radius of the CNT plus the thickness of the h-BN spacer between CNT and conductive layer. However, in some embodiments this yields an approximate value as the shape of the potential well generated by the nanotube can be affected by the screening of the graphene electrons. For example, a voltage on the nanotube generates an electric field that is the cause of the guiding potential in graphene. This electric field can attract (or repulse) electrons in graphene. These electrons also generate an electric field that can compensate the electric field generated by the nanotube. In some embodiments, good agreement between theory and experiments can be obtained using a width of 10 nm, (e.g., d= 10Δ) in the simulations shown in FIG. 3B discussed above.

Devices with Wider Potential Wells

[0090] FIGS. 9A-9D shows various characteristics of electron waveguides with wider potential wells, according to some embodiments. In particular, FIGS. 9A-9D show simulations and measurements for devices with wider potential wells. Without being bound by theory, the exemplary data results from a device with 30 nm thick h-BN layer separating a CNT from a graphene layer. Similar behaviors are observed to those observed in devices with h-BN that are 10 nm or thicker.

[0091] The exemplary simulations are performed for a potential that is 100 nm wide.

FIG. 9A shows the dispersion relation of the conductive graphene layer underneath the CNT when the potential is at Uo = 0.45 eV, according to some non-limiting analytical embodiments. As shown in FIG. 9A, the branches are difficult to distinguish from the continuum. Unlike a single mode which is well isolated from the continuum, these multi modes can couple with one another as well as with the continuum, which makes them poorly guided.

[0092] FIG. 9B shows a numerically calculated C_q, according to some non-limiting analytical embodiments. As shown in FIG. 9B, resonances do not develop due to the fact that branches detaching from the continuum are too close from each other and the continuum.

FIG. 9C shows a non-limiting analytical representation of a carbon nanotube conductance measurement for a device around a high gate potential (VG = 7.25 V). The trajectories of the conductance peaks form smooth S curves which represent the Dirac point, and there are no “kinks” seen in FIG. 2C. In some embodiments, this indicates the absence of resonance in the density of states as shown in FIG. 9D, which plots quantum capacitances as a function of Vbg (compare to FIG. 2D).

[0093] However, there is an asymmetry between the electron and hole sides (e.g., on either side of nG = 0 in FIG. 9B) and a smoothing of the Dirac point (i.e., the minimum being less sharp), meaning that the electric field generated by the nanotube affects the graphene density of states. The lack of resonances in this exemplary non-limiting analytical embodiment for wider potentials is in agreement with theory and an indication of the formation of several modes rather than a single guided mode. This shows that in some embodiments, significantly sharp potential wells are helpful for the realization of a single mode electron guide. In some embodiments, this also implies that although the electric field generated by the nanotube affects the graphene density of states, but because the top h-BN layer is too thick, the electric field generated by the nanotube is not enough to generate guided modes.

[0094] Control of various parameters described herein can be input by a user using existing laboratory equipment or suitable computer program controlling the same. Measurements described herein can be performed by conventional measurement technology known to those of skill in the art. Measurements and techniques described herein can be user-controlled or automated by a processor and memory storing instructions thereon to perform techniques described herein.

[0095] While the invention has been particularly shown and described with reference to specific preferred embodiments, it should be understood by those skilled in the art that various changes in form and detail may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.

Claims

1. A system comprising: an upper insulating layer having an upper surface and a lower surface opposite to the upper surface; a lower insulating layer; at least one nanotube on the upper surface of the upper insulating layer, the nanotube having a nanotube axis extending in a first direction; a conductive layer between the upper insulating layer and the lower insulating layer and on the lower surface of the upper insulating layer, the conductive layer conductively connected to at least one conductive layer electrode; and at least two nanotube electrodes conductively connected to the nanotube to produce a voltage across the nanotube, wherein the voltage across the nanotube produces a potential well in the conductive layer such that at least one electron within the conductive layer is confined in a direction perpendicular to the first direction and parallel to the lower surface of the upper insulating layer.

2. The system of claim 1, wherein the conductive layer has a thickness of 1-10 nm.

3. The system of claim 1, wherein one or both of the upper and lower insulating layers have a thickness of 1-10 nm.

4. The system of claim 1, wherein one or both of the upper and lower insulating layers comprises hexagonal boron nitride (h-BN).

5. The system of claim 1, wherein the conductive layer comprises graphene.

6. The system of claim 1, wherein the potential well has a depth of 0-0.3 eV.

7. The system of claim 1, wherein the potential well has a width of 1-10 nm.

8. The system of claim 1, wherein the width of the potential well is less than or equal to 10 nm.

9. The system of claim 1, further comprising an electron propagating in a single mode in the potential well.

10. The system of claim 9, wherein the single mode propagating electron is distinguishable from the continuum states outside of the potential well.

11. The system of claim 10, wherein the single mode propagating electron propagates in a single mode at room temperature.

12. The system of claim 1, wherein the nanotube has a diameter between 1-3 nm.

13. The system of claim 1, wherein the nanotube comprises a carbon nanotube.

14. A method comprising: disposing a conductive layer between a lower surface of an upper insulating layer and an upper surface of a lower insulating layer, the conductive layer conductively connected to at least one conductive layer electrode; disposing at least one nanotube on an upper surface of the upper insulating layer opposite the lower surface of the upper insulating layer, the nanotube having a nanotube axis extending in a first direction; and electrically coupling the nanotube to at least two nanotube electrodes configured to produce a voltage across the nanotube, wherein the voltage across the CNT produces a potential well in the conductive layer such that at least one electron within the potential well is confined in the conductive layer in a direction perpendicular to the first direction and parallel to the lower surface of the upper insulating layer.

15. The method of claim 14, wherein the conductive layer has a thickness of 1-10 nm.

16. The method of claim 14, wherein one or both of the upper and lower insulating layers have a thickness of 1-10 nm.

17. The method of claim 14, wherein one or both of the upper and lower insulating layers comprises h-BN.

18. The method of claim 14, wherein the conductive layer comprises graphene.

19. The method of claim 14, wherein the potential well has a depth of 0-0.3 eV.

20. The method of claim 14, wherein the potential well has a width of 1-10 nm.

21. The method of claim 14, wherein the width of the potential well is less than or equal to 10 nm.

22. The method of claim 14, further comprising forming a single mode for an electron to propagate in the potential well.

23. The method of claim 22, wherein the single mode propagating electron is distinguishable from the continuum states outside of the potential well.

24. The method of claim 23, wherein the single mode propagating electron propagates in a single mode at room temperature.

25. The method of claim 14, wherein the nanotube has a diameter between 1-3 nm.

26. The method of claim 14, wherein the nanotube comprises a carbon nanotube.

27. A method of operating the system of claim 1 comprising: applying a first electric potential to the conductive layer with the at least one conductive layer electrode; applying a second electric potential to the nanotube with the at least two nanotube electrodes, wherein the second electric potential is different from the first electric potential; and forming a potential well in the conductive layer by tuning the difference between the first and second electric potentials.