Van der waals heterostructure transport device for gate-tunable quantum anomalous hall state

Abstract

A multiple gate system for realizing the quantum anomalous Hall effect, including a top channel electrode which controls carrier density and electric field in a conduction channel, a first dielectric layer for the top channel gate, a patterned contact gate electrode which controls earner density on the contact and screens the top channel electrode's electric field on the contact, and a second dielectric layer for the contact gate, an electrically tunable material comprising semiconducting molybdenum ditelluride in the form of a first atomic sheet and a second atomic sheet, wherein first atomic sheet is rotated planarly from the second atomic sheet by about 2.5 to about 5 degrees, to form a geometrically stacked homobilayer moiré superlattice configured to host ferromagnetic states that spontaneously break time-reversal symmetry, where a perpendicular electric field is applied to the electrically tunable material, tuning the quantum anomalous Hall effect.

Description

VAN DER WAALS HETEROSTRUCTURE TRANSPORT DEVICE FOR GATE- TUNABLE QUANTUM ANOMALOUS HALL STATE CROSS-REFERENCES TO RELATED APPLICATIONS

[0001] This application claims the benefit of and priority to U.S. Provisional Patent Applications No. 63 / 509913 filed June 23, 2023, and No. 63 / 585926 filed September 20, 2023, the contents of which are hereby incorporated by reference in their entirety. STATEMENT OF GOVERNMENT LICENSE RIGHTS

[0002] This invention was made with government support under Grant Nos. FA9550-19-1-0390 and FA9550-21-1-0177, awarded by the Air Force Office of Scientific Research and Grant No. DGE-2140004, awarded by the National Science Foundation and Grant Nos. DE-SC0018171 and DE-SC0019443, awarded by the U.S. Department of Energy. The government has certain rights in the invention. BACKGROUND

[0003] Topological materials are quantum phases of matter characterized by a bulk electronic band structure that exhibits nontrivial topology. An elegant example is the integer quantum Hall (QH) effect realized in a two-dimensional electron gas. When subjected to a large magnetic field, Landau levels form, resulting in the first known topologically nontrivial condensed matter system. The topology is characterized by an invariant, known as the Chern number C, which determines the number of topologically protected chiral channels along the edge of a sample. When strong electron interaction effects are introduced into the QH system, fractional QH (FQH) states can emerge. These states can host collective charge excitations obeying fractional statistics. Both integer and fractional QH states require time reversal symmetry breaking, achieved under a strong external magnetic field.

[0004] In 1988, Duncan Haldane proposed the existence of integer QH states at zero magnetic field known as quantum anomalous Hall (QAH) states. Unlike the QH effect, these states do not rely on Landau level formation. Experimental realization of the integer QAH effect was first realized in magnetically doped topological insulator thin films. More recently, the QAH effect has also been realized in topological insulators with intrinsic magnetism and van der Waals assembled moiré materials. Motivated by the path from QHto FQH effects, a natural question arises as to whether a fractional QAH (FQAH) state can be realized when strong interactions are introduced into a QAH insulator. These FQAH states would be the lattice analog of FQH states, emerging at zero magnetic field without Landau level formation.

[0005] Moiré superlattice engineering has proven an incredibly effective approach to create new phases of matter with exotic physical properties. Transition metal dichalcogenide (TMD) moiré superlattices have attracted particular attention in recent years, as their realization of highly tunable effective Hubbard models enables study of a range of strongly correlated electronic phases. These are evidenced by observation of correlated insulators at integer and fractional moiré fillings, gate-controllable spontaneous spin ordering, and topologically nontrivial phases of matter – a striking example being the recent discovery of zero-field fractional Chern insulators and the fractional quantum anomalous Hall effect in homobilayer MoTe2. A major advantage of TMD moirés lies in the properties of their constituent TMD layers, which have a spin-valley locked degree of freedom arising from spin-orbit coupling, a strong optical response, and valley-dependent optical selection rules. This has allowed for a variety of optical techniques to be employed in investigating TMD moiré physics, in addition to transport and local probe measurements. The range of tuning knobs and experimental techniques possible with TMD superlattice devices have established them as versatile and powerful platforms to engineer new physical properties.

[0006] It has been conjectured that electrical current could be used to control the magnetic order and light emission helicity of a TMD moiré. Current control of emission helicity is highly sought after, as it involves coupling the relevant degrees of freedom in electronics, spintronics, and photonics. This behavior has recently been achieved in complex multilayer architectures. For these devices, charge current in a layer with strong spin-orbit coupling generates a spin current, causing spin-orbit torque flipping of an adjacent ferromagnetic layer. This allows for injection of spin-polarized carriers into a separate optically active layer, which emits with a finite degree of circular polarization. A substantial advantage of TMD moiré devices, in contrast, is that the necessary ingredients – spin-orbit torque control of magnetic order and spin polarization dependent helical emission – can in principle be found in a single system. The first of these, spin / valley Hall torque switching of magnetic order, has been observed in a MoTe2 / WSe2 heterobilayermoiré. The second has been recently demonstrated in homobilayer MoTe2, where the combination of moiré ferromagnetism and valley-dependent optical selection rules leads to nearly 100% circularly polarized emission at zero magnetic field.

[0007] Accordingly, multiple gate systems for quantum anomalous Hall effect are needed. SUMMARY

[0008] This summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. This summary is not intended to identify key features of the claimed subject matter, nor is it intended to be used as an aid in determining the scope of the claimed subject matter.

[0009] In one aspect, disclosed herein is a multiple gate system for realization of the integer and fractional quantum anomalous Hall effect, including a top channel electrode which controls carrier density and electric field in a conduction channel, a first dielectric layer for the top channel gate, and a contact region, including a patterned contact gate electrode which controls carrier density on the contact and screens the top channel electrode’s electric field on the contact, and a second dielectric layer for the contact gate, an electrically tunable material comprising semiconducting molybdenum ditelluride in the form of a first atomic sheet and a second atomic sheet, wherein first atomic sheet is rotated planarly from the second atomic sheet by about 2.5 to about 5 degrees, to form a geometrically stacked homobilayer moiré superlattice configured to host ferromagnetic states that spontaneously break time-reversal symmetry, wherein the material of atomic sheets define an xy plane, and a set of Hall bar contacts for electrical contact to a sample in a Hall bar geometry where the carrier density can be tuned by the patterned contact gate electrode, where the Hall bar contacts are arranged adjacent to each other along an x direction in an xy plane and opposite to each other in a y direction in the xy plane, a third dielectric layer for the bottom channel gate, a bottom channel electrode, where the top and bottom channel electrodes form a dual gate geometry, where an input voltage Vtg is placed on the top channel electrode, an input voltage Vbg is placed on the bottom channel electrode, an input voltage Vcg is placed on the contact gate electrode, a bias voltage Vbias is placed on the source Hall bar contact of the set of Hall bar contacts, where Vxx is longitudinal potential between two Hall bar contacts of the set of Hall bar contacts and Vxy is the Hall potential between two Hall bar contacts of the set of Hall bar contacts, wherethe patterned contact electrode spaced from the source Hall bar contact of the set of Hall bar contacts is grounded as a drain Hall bar contact, and where a perpendicular electric field is applied to the electrically tunable material, tuning the quantum anomalous Hall effect.

[0010] In some embodiments, the multiple gate system is a split gate system. In some embodiments, the top electrode and the bottom electrode comprise graphite. In some embodiments, the patterned gate electrode comprises platinum, titanium, chromium, gold, or a combination thereof.

[0011] In some embodiments, the first dielectric layer, the second dielectric layer, the third dielectric layer, or a combination thereof comprise hexagonal boron nitride (hBN). In some embodiments, the stacked homobilayer is rhombohedral. In some embodiments, the moiré superlattice is honeycomb.

[0012] In some embodiments, the contact region is hole doped to about - 1×1013cm-2. In some embodiments, the carrier density and the electric field on the sample are converted from a top gate voltage and a bottom gate voltage using a parallel plate capacitor model: n = (VtgCtg + VbgCbg) / e-noffsetand D / ɛ0 = (VtgCtg − VbgCbg) / 2ɛ0-Doffset, where Ctg and Cbg are a top gate capacitance and a bottom gate capacitance, e is an electron charge, and ɛ0is a vacuum permittivity.

[0013] In some embodiments, the first atomic sheet and the second atomic sheet each have from one to three atomic layers of semiconducting molybdenum ditelluride. In some embodiments, the first atomic sheet and the second atomic sheet are rotated planarly from each other by about 3.52 degrees. In some embodiments, the homobilayer has electrically tunable topological phase transitions. In some embodiments, the homobilayer has fractional quantum anomalous Hall states. In some embodiments, the fractional quantum anomalous Hall states are at -1 / 2 and -2 / 3 filling of the moiré superlattice. In some embodiments, the homobilayer is photoluminescent with a linewidth about 3 meV. In some embodiments, the homobilayer is photoluminescent between 1.13 and 1.09 eV.

[0014] In some embodiments, the input voltage is about -10 V to about 10 V. In some embodiments, a critical electric field DC / ɛ0 is less than or equal to about 150 mV / nm.

[0015] In another aspect, disclosed herein is a transistor comprising a multiple- gate system for quantum anomalous Hall effect as shown and described herein.

[0016] In some embodiments, the transistor operates by electric-field induced phase transition to regulate current flow between a source electrode and a drain electrode of the transistor. DESCRIPTION OF THE DRAWINGS

[0017] The foregoing aspects and many of the attendant advantages of this invention will become more readily appreciated as the same become better understood by reference to the following detailed description, when taken in conjunction with the accompanying drawings, wherein:

[0018] FIGURE 1A is an example device schematic having a multiple gate design, in accordance with the present technology;

[0019] FIGURES 1B-1C show longitudinal (Rxx) and Hall (Rxy) resistance measurements as a function of doping n and perpendicular electric field D / ε0 at 1.6 K, in accordance with the present technology;

[0020] FIGURES 1D-1E show Rxx and Rxy, respectively, as a function of magnetic field µ0H at v=-1 and D=0, in accordance with the present technology;

[0021] FIGURE 1F shows MX and XM high symmetry sites on which two energy degenerate moiré orbitals are localized, forming a honeycomb lattice, in accordance with the present technology;

[0022] FIGURE 1G is a Hartree-Fock calculation of topological bands at v=-1 and D = 0 with |C| = 1, in accordance with the present technology;

[0023] FIGURE 1H is another example device having a multiple gate design, in accordance with the present technology;

[0024] FIGURE 2A shows magnetic field dependent Rxxand Rxyat selected filling factors around v=-1 with electric field D = 0, in accordance with the present technology;

[0025] FIGURE 2B shows magnetic field dependent Rxxand Rxyat selected electric field D / ε0 at v=-1, in accordance with the present technology;

[0026] FIGURE 2C shows an antisymmetrized Rxyand symmetrized Rxxat |µ0H|=100mT as a function of v at D = 0. Rxy evolves from quantized value h / e2near v=- 0.9, then reduces and eventually drops below 1 kΩ as hole density increases to v=-1.5, in accordance with the present technology;

[0027] FIGURE 2D shows an antisymmetrized |Rxy| and symmetrized Rxxat |µ0H|=100mT as a function of D / ε0 at v=-1, in accordance with the present technology;

[0028] FIGURE 3A shows reflective magnetic circular dichroism (RMCD) measurements versus D / ε0 and v at a finite magnetic field of 100 mT, in accordance with the present technology;

[0029] FIGURE 3B shows Hartree-Fock calculations of the out-of-plane spin (Sz) and energy gap normalized to the hopping t1 as a function of electric field, in accordance with the present technology;

[0030] FIGURE 4A shows Rxxand Rxyat D=0 versus magnetic field sweeps at selected temperatures, in accordance with the present technology;

[0031] FIGURE 4B shows Rxxversus temperature at selected electric fields D / ε0, in accordance with the present technology;

[0032] FIGURE 4C is an illustration of gap extraction for three different displacement fields D / ε0 = -70, -50, and 0 mV / nm, in accordance with the present technology;

[0033] FIGURE 4D is an energy gap as a function of electric field extracted from FIG.4B, in accordance with the present technology.

[0034] FIGURE 5A-5B show symmetrized longitudinal (Rxx) and,antisymmetrized Hall (Rxy) resistance, respectively, at magnetic fields |µ0H|=200mT as a function of electric field (D / ε0) and carrier density (n), in accordance with the present technology;

[0035] FIGURE 5C-5D show Rxxand Rxy, respectively, as a function of magnetic field µ0H at v=-1 / 2 and D=0, in accordance with the present technology;

[0036] FIGURE 6A is an atomic force microscope (AFM) image of a twisted MoTe2 device equipped with Pt contacts for current injection, in accordance with the present technology;

[0037] FIGURE 6B is an RMCD signal as a function of μ0H swept down and up, in accordance with the present technology;

[0038] FIGURE 6C is an RMCD spatial map in the ferromagnetic (FM) metal regime (ν=-0.77) at μ0H = 0T, taken after initialization at μ0H > 0T, in accordance with the present technology;

[0039] FIGURE 6D is an RMCD spatial map measured under the same conditions as FIG.6C but with a current of I = -35nA flowing between the contacts on the upper left of the device, in accordance with the present technology;

[0040] FIGURE 6E is an RMCD spatial map measured under the same conditions as FIG. 6D but with the current direction reversed, in accordance with the present technology;

[0041] FIGURE 6F is a schematic of the transverse spin-valley current in moiré MoTe2, in accordance with the present technology;

[0042] FIGURE 7A is an RMCD signal vs ν and D at μ0H = 0T and for I = -20nA, taken with the beam spot on the sign reversed domain in FIG. 6D, in accordance with the present technology;

[0043] FIGURE 7B is an RMCD signal measured under the same conditions as FIG.7A, but with I = 20nA, in accordance with the present technology;

[0044] FIGURE 7C is an RMCD signal vs I swept down, as a function of ν, with μ0H = 0T and D / ε0 = 0.17V / nm, in accordance with the present technology;

[0045] FIGURE 7D is an RMCD signal measured under the same conditions as FIG.7C, but with I swept up, in accordance with the present technology;

[0046] FIGURE 7E is hysteretic component of RMCD vs I, ΔRMCD, as a function of ν, in accordance with the present technology;

[0047] FIGURE 7F is RMCD vs I swept down and up at ν=-0.976, in accordance with the present technology;

[0048] FIGURE 8A is a spatial map of PL degree of circular polarization ρ with current I = 35nA and at μ0H = 0T, taken after initialization at μ0H > 0T, in accordance with the present technology;

[0049] FIGURE 8B is a spatial map of PL measured under the same conditions as FIG.8A, but with I = -35nA, in accordance with the present technology;

[0050] FIGURE 8C shows circular polarization resolved PL spectra at I = 35nA and μ0H = 0T, taken on the black dot in FIG.8A, in accordance with the preset technology;

[0051] FIGURE 8D shows circular polarization resolved PL spectra measured under the same conditions as FIG. 8C but with I = -35nA, in accordance with the present technology;

[0052] FIGURE 8E shows ρ vs μ0H swept down and up, with I = 0nA, in accordance with the present technology;

[0053] FIGURES 8F-8G show ρ vs μ0H swept down and up, with I = 35nA and I = -35nA, respectively, in accordance with the present technology;

[0054] FIGURE 9A shows ρ vs I swept down, as a function of μ0H, starting from positive μ0H, in accordance with the present technology; and

[0055] FIGURE 9B shows ρ as a continuous function of I, at μ0H = 0T, initialized with μ0H > 0T, and with insets showing circular polarization resolved PL spectra for large positive and negative current, in accordance with the present technology. DETAILED DESCRIPTION

[0056] Twisted MoTe2bilayers, which form a synthetic honeycomb lattice, have recently arisen as a highly promising material platform for investigating integer and fractional quantum anomalous Hall (QAH) effects. Gate tunable correlated magnetic insulating states at both integer and fractional filling (v) of the moiré unit cell have been observed in a twist angle range of about 3o-4o. The spontaneous ferromagnetic order provides the time reversal symmetry breaking necessary for the realization of QAH and fractional quantum anomalous Hall (FQAH) effects. Utilizing a trion sensing technique, a fan diagram of three ferromagnetic insulating states (v=-1, -2 / 3, and -3 / 5) was obtained. Remarkably, these three states were found to evolve with slopes corresponding to Chern numbers C = −1, −2 / 3 and −3 / 5, respectively, consistent with the characteristics of integer QAH and FQAH states according to the Streda formula C = Φ0∂n / ∂B (where Φ0 is the magnetic flux quantum, n is the carrier density of the gapped state, and B ismagnetic field). The C=-1 and -2 / 3 states have also been confirmed through exciton sensing measurements, indicating robustness of both the phenomena and the material platform. While they provide valuable insights, optical probes are not a direct probe of the quantized anomalous Hall effect. Electrical transport measurements are crucial for establishing the topological nature of the underlying bands and the presence of dissipation-free chiral edge states. However, forming good electrical contacts to a 2D semiconductor under zero applied electric field, a prerequisite to reach the honeycomb lattice regime in twisted TMDs where topological phases arise, is a known challenge in the community.

[0057] Disclosed herein are electrical transport measurements of rhombohedral- stacked twisted MoTe2 bilayers, focusing near the moiré filling factor of v=-1. Provided herein is direct evidence of a correlated QAH insulator, that establishes a topological phase diagram controlled by doping and electric field.

[0058] FIG.1A is an example device schematic having a multiple gate design, in accordance with the present technology. Graphite (Gr) top and bottom gates form a dualgate geometry, while patterned local gates for metal contacts are used to improve the electrical contacts to the twisted MoTe2 bilayer. The main challenge encountered in reported twisted 2D semiconductor transport studies to date has been achieving low contact resistance at zero electric field. As shown in FIG.1A, the device utilizes a split gate scheme to significantly improve the contact resistance by heavily hole doping the contact region to a level of about 1×1013cm-2. In some embodiments, alongside a global bottom gate, a separate top gate is fabricated to control carrier density and electric field in the conduction channel without affecting the doping level in the metal contact area. This design enables sufficient contact quality at zero electric field to measure the QAH effect through electrical transport. In some embodiments, the device has a twist angle of ~3.52°, as shown in FIG. 1A.

[0059] FIGS. 1B-1C show longitudinal (Rxx) and Hall (Rxy) resistance measurements as a function of doping n and perpendicular electric field D / ε0at 1.6 K, in accordance with the present technology. The data are symmetrized (Rxx) and antisymmetrized (Rxy) at a small out-of-plane magnetic field µ0H of ±100 mT. The applied magnetic field is merely for avoiding magnetic instability when sweeping the gates. The black shaded area marks insulating behavior that is inaccessible at this temperature (R>1MΩ). The resistance in black shaded regions cannot be reliably probed due to their highly insulating nature. Filling factor v is also indicated in the plots. The dashed region in FIG. 1C denotes the phase space within which Rxyis > 95 % of the von Klitzing constant (h / e2). Centered around v=-1 and electric field D = 0, there is a region of phase space where Rxxnearly vanishes and |Rxy| reaches the quantized value of h / e2~ 25.8kΩ, i.e., the von Klitzing constant, with h the Planck constant and e the elementary charge. This implies a QAH state with Chern number |C| = 1 in the density-electric field phase diagram. The QAH insulator region resembles that of the ferromagnetic region established by magnetic circular dichroism measurements on twisted MoTe2 at a similar twist angle, as will be discussed later.

[0060] FIGS 1D-1E show Rxxand Rxy, respectively, as a function of magnetic field µ0H at v=-1 and D=0, in accordance with the present technology. To confirm the QAH state, the magnetic field dependence of Rxx (FIG. 1D) and Rxy (FIG. 1E) were measured at v = -1 and D = 0. The quantized Rxywith Rxx<200 Ω at zero magnetic field demonstrates a quantum anomalous Hall insulator. The data were taken at T = 500 mK. Rxyshows a hysteresis loop with coercive field μ0HC = ~10 mT and a quantized value of resistance ±h / e2at zero magnetic field (FIG. 1E). Correspondingly, Rxx nearly vanishes at zero magnetic field, and shows finite resistance jumps (~0.5 h / e2) near where magnetization orientation flips at μ0HC (FIG. 1D). This observation demonstrates the realization of the QAH effect in twisted MoTe2, corroborating the evidence of a |C| = 1 QAH insulator obtained fromdiagram measurements.

[0061] FIG. 1F shows the MX and XM high symmetry sites on which two degenerate moiré orbitals are localized, forming a honeycomb lattice, in accordance with the present technology. FIG.1G is a Hartree-Fock calculation of topological bands at v=- 1 and D = 0 with |C| = 1, in accordance with the present technology. In some embodiments, R-stacked MoTe2hosts two degenerate moiré orbitals localized in opposite layers, which form an effective honeycomb lattice (FIG.1F). FIG.1G shows spin / valley-resolved moiré valence bands for a hole filling of v=-1. The associated Chern numbers of the bands have opposite signs for the two valleys. At this filling, the system is a correlated charge insulator with long range ferromagnetic order. The associated spontaneous time reversal symmetry breaking lifts the spin / valley degeneracy. As the Fermi level is tuned to reside between top two moiré flat bands, the Chern number of the insulating ground state is |C| = 1. Consequently, a QAH insulator emerges, akin to a Haldane QAH insulator.

[0062] FIG.1H is another example device 100 having a multiple gate design, in technology. In some embodiments, the device 100 includesfour electrodes 105A, 105B, 105C, 105D, a first dielectric layer 110A, a second dielectric layer 110B, a third dielectric layer 110C, a first atomic sheet 115A, and a second atomic sheet 115B. In some embodiments, the four electrodes 105A, 105B, 105C, 105D include a top channel electrode 105A, a patterned contact gate electrode 105B, a Hall bar contact (also referred to as a Hall bar electrode) 105C, and a bottom channel electrode 105D. In some embodiments, the first atomic sheet 115A and the second atomic sheet 115B are molybdenum ditelluride. In some embodiments, the first atomic sheet 115A is rotated planarly from the second atomic sheet 115B by about 2.5 to about 5 degrees to form a geometrically stacked homobilayer moiré superlattice. In some embodiments, the device includes a set of Hall bar contacts 105C for electrical contact to a sample. In some embodiments, the top channel electrode 105A and the bottom channel electrode 105D form a dual gate geometry. In some embodiments, the first atomic sheet and the second atomicsheet each have from one to three atomic layers of semiconducting molybdenum ditelluride.

[0063] In some embodiments, one or more of the dielectric layers (first dielectric layer 110A, second dielectric layer 110B, and third dielectric layer 110C) is transparent.In some the dielectric layers (first dielectric layer 110A, second dielectric layer 110B, and third dielectric layer 110C) are independently selected from silicon dioxide, hafnium dioxide, aluminum oxide, zirconium dioxide, titanium dioxide, nitrogen- doped silicon dioxide, high-k dielectric materials such as barium strontium titanate and hafnium bases high-k dielectrics, graphene oxide, hexagonal boron nitride molybdenum oxide, tungsten oxide, titanium nitride, germanium oxide, aluminum nitride, gallium oxide, zinc oxide, indium oxide, magnesium oxide, and combinations thereof.

[0064] In some embodiments, one or more of the electrodes (top channel electrode 105A, patterned contact gate electrode 105B, Hall bar contacts 105C, and bottom channel electrode 105D) are transparent. In some embodiments, the electrodes (top channel electrode 105A, patterned contact gate electrode 105B, Hall bar contacts 105C, and bottom channel electrode 105D) are independently elected from gold, platinum, graphene, graphite, indium tin oxide, aluminum zinc oxide, lithium doped nickel oxide, indium doped cadmium oxide, magnesium nitrogen doped chromium trioxide, indium or aluminum doped magnesium zinc oxide, magnesium doped copper chromium oxide, zinc indium tin oxide, zinc oxide, polystyrene sulfonic acid and salts thereof, poly(3,4- ethylenedioxythiophene) (PEDOT), Nafion, polyaniline, polyacetylene, polypyrrole, and combinations thereof.

[0065] In some embodiments, an input voltage Vtgis placed on the top channel electrode 105A, an input voltage Vbgis placed on the bottom channel electrode 105D, an input voltage Vcgis placed on the contact gate electrode 105B, and a bias voltage Vbiasis placed on the source Hall bar contact of the setof Hall bar contacts 105C. In some embodiments, Vxxis the longitudinal potential between two Hall bar contacts of the set of Hall bar contacts 105C and Vxyis the Hall potential between two Hall bar contacts of the set of Hall bar contacts 105C.

[0066] FIG.2A shows magnetic field dependent Rxx and Rxy at selected filling factors around v=-1 with electric field D = 0, in accordance with the present technology. FIG.2B is a magnetic field dependent Rxx and Rxy at selected electric field D / ε0 at v=-1, inaccordance with the present technology. FIG.2C shows an antisymmetrized Rxy and symmetrized Rxx at |µ0H|=100mT as a function of v at D = 0. Rxy evolves from a quantized value h / e2near v = -0.9, then reduces and eventually drops below 1 kΩ as hole density increases to v=-1.5, in accordance with the present technology. FIG.2D shows an antisymmetrized |Rxy| and symmetrized Rxx at |µ0H|=100mT as a function of D / ε0 at v=-1, in accordance with the present technology. As D / ε0 increases above about -50 mV / nm, Rxy reduces from the quantized value while Rxxincreases rapidly to above 100 kΩ. This demonstrates an electric field induced phase transition from a QAH insulator to a topologically trivial correlated insulator. Inset: Symmetrized Rxxat |µ0H|=100mT versus electric field D / ε0 swept back and forth near the phase transition. The absence of hysteresis implies a second order topological phase transition. The data in the inset are taken at 1.6 K to minimize electrical noise from the contacts.

[0067] The QAH phase diagram and its transition to topologically trivial states with the application of doping (FIGS 2A, 2C) and electric field (FIGS 2B, 2D) was investigated. FIG. 2A presents the magnetic field dependence of Rxy and Rxx at D = 0 for selected values of v. The data reveal that as doping deviates from v = -1, both Rxyand coercive field μ0HC decrease, accompanied by an increase in Rxx. The antisymmetrized Rxy and symmetrized Rxxat |µ0H|=100 mT, were extracted and plotted in FIG.2C as a function of filling. The data quantitatively show that the QAH insulator survives throughout the doping range between v=-0.9 and -1.1, validated by the quantized Rxyand nearly vanishing Rxx. As hole doping further increases beyond v=-1.1, Rxx continues to increase to ~20-30 kΩ until approximately v=-1.5. Concurrently, Rxyreduces but remains appreciable, which suggests that the system transitions into an anomalous Hall metal. Similar phenomena can be also reproduced in another contact geometry, in which the magnetic hysteresis has a different coercive field due to differences in channel width.

[0068] FIG. 2B shows magnetic field dependence of Rxy and Rxx at v=-1 for selected D / ɛ0. The extracted antisymmetrized Rxy and symmetrized Rxx at |µ0H|=100 mT are plotted in FIG. 2D as a function of D / ɛ0. Upon increasing the electric field from zero to a critical electric field DC / ɛ0 of about -50 mV / nm, Rxy remains quantized, concomitant with nearly vanishing Rxx. This observation demonstrates that the QAH insulator survives within this range. Above DC / ɛ0~ -50mV / nm, however, Rxybegins fluctuating before quickly falling to near zero. Concurrently, Rxx increases rapidly to greater than 100kΩ at 15 mV / nmbeyond DC / ɛ0. These observations demonstrate an electric field induced topological phase transition from a QAH insulator to a trivial correlated insulator. To investigate the nature of this phase transition, Rxx was measured while sweeping the electric field up and down near DC / ɛ0. The data are taken at T = 1.6K to minimize the effects of contact resistance during the sweep. As shown in the inset of FIG. 2D, the absence of hysteresis during the sweep suggests a continuous electric-field-driven topological phase transition from a QAH insulator to a trivial correlated insulator.

[0069] FIG.3A shows reflective magnetic circular dichroism (RMCD) measurements versus D / ε0and v at a finite magnetic field of 100 mT, in accordance with the present technology. The comparison with FIG.1C shows that the ferromagnetic phase space is larger than that of the QAH insulator (the area circled by the dashed line from FIG. 1C). FIG.3B shows Hartree-Fock calculations of the out-of-plane spin (Sz) and energy gap normalized to the hopping t1as a function of electric field, in accordance with the present technology. The measured value of DC / ɛ0, ~ -50 mV / nm, which destroys the QAH state is much smaller than the recently reported critical field of the ferromagnetic state, DFM / ɛ0, which is near 120 mV / nm. To explore this discrepancy, reflective magnetic circular dichroism (RMCD) measurements of the same sample under the same experimental conditions, i.e., at T = 1.6K and µoH = 100 mT were taken. Unlike Rxy,which vanishes in a topologically trivial magnetic insulator, RMCD probes the out-of-plane magnetization of a two-dimensional magnet. FIG. 3A shows the RMCD signal intensity plot as a function of doping and electric field. The phase space of the ferromagnetic state corresponds to the region with non-vanishing RMCD signal. This magnetic phase diagram is consistent with previous reports, and resembles the experimental QAH phase diagram in FIGS 1B and 1C, albeit with a larger ferromagnetic phase space in v and D. The difference between the QAH and ferromagnetic phase spaces implies that between the two critical electric fields DC / ɛ0 and DFM / ɛ0, there is a topologically trivial ferromagnetic insulator phase. This understanding is corroborated by the Hartree-Fock calculations of the topological and magnetic phases as a function of electric field at v=-1 (FIG. 3B) which is also consistent with recent theory papers. The system is in topologically trivial ferromagnetic insulator state between Dc and DFM and becomes non valley polarized above DFM. The calculated Chern number below the critical field Dcis C=-1 (left region), while above Dcthe Chern number becomes C = 0 (right region).

[0070] FIG.4A shows Rxxand Rxyat D=0 versus magnetic field sweeps at selected temperatures, in accordance with the present technology. The gap size of the QAH insulator was estimated by performing temperature dependent electrical transport measurements. FIG. 4A shows the magnetic field dependence of Rxy and Rxx at selected temperatures. For v=-1, at low temperature, Rxy of ~h / e2persists down to zero applied magnetic field, shows hysteresis, and Rxx is small. As temperature increases, the Rxy hysteresis loop shrinks and disappears. Though at zero magnetic field Rxydeviates from h / e2, it remains quantized at small magnetic field (>20 mT) up to 3K. Above this temperature, Rxyis no longer quantized under a finite magnetic field.

[0071] FIG.4B shows Rxxversus temperature at selected electric fields D / ε0, in accordance with the present technology. A small magnetic field (100 mT) is applied to suppress noise due to magnetic domain fluctuations. FIG. 4B plots Rxx as a function of temperature for selected electric fields at µ0H = 100 mT. Below DC / ɛ0( ~ -50 mV / nm), Rxxdecreases with temperature as the system is in the QAH DC / ɛ0, Rxx increasesas temperature is lowered, consistent with the topologically trivial ferromagnetic insulator discussed in FIGS 2A-2D. Near DC, Rxxshows little temperature dependence and is in the vicinity of h / e2. A universal longitudinal resistivity near h / e2around the critical point has been observed in phase transitions from a quantum Hall liquid to a Hall insulator and from a QAH to an Anderson insulator. Our observation at the phase transition between a QAH and a correlated ferromagnetic insulator is in good agreement with previous reports.

[0072] FIG.4C is an illustration of gap extraction for three different displacement fields D / ε0= -70, -50, and 0 mV / nm, in accordance with the present technology. The Arrhenius equation and extracted gap is represented next to each plot, except the critical field Dc / ε0= -50 mV / nm where the longitudinal resistance is almost constant with temperature. To extract the gap, Rxx was fitted with ^^^^∆ / ^^^^(^∆ / ^^^^) below (above) DC / ɛ0, where Ro is a constant, kB is the Boltzmann constant, and Δ is the insulating gap. Representative examples of the fitting are shown in FIG.4C.

[0073] FIG.4D is the energy gap as a function of electric field extracted from FIG. 4B, in accordance with the present technology. Error bars are obtained from fitting variance and enlarged by five times to enhance visibility. Grey dashed lines are guides to the eye. The closing and reopening of the gap versus D / ε0demonstrate the continuous topological quantum phase transition between a QAH insulator and a ferromagneticinsulator. FIG. 4D plots the extracted Δ versus electric field D / ɛ0. Two distinct phases, those of the QAH and topologically trivial insulators, are clearly resolved below and above DC / ɛ0. In the QAH phase, Δ initially decreases gradually as D / ɛ0 is increased from zero, before shrinking rapidly as D / ɛ0 approaches DC / ɛ0. At Dc / ɛ0, the gap nearly vanishes. The exact value of the gap near DC / ɛ0 is difficult to extract, as Rxx is nearly independent of temperature in this region. The gap reopens as D / ɛ0 is increased beyond DC / ɛ0, corresponding to the formation of the topologically trivial correlated insulator. This observed closing and reopening of the gap as a function of D / ɛ0 is a hallmark of a continuous topological quantum phase transition. These experimental results are consistent with the calculated Chern number, phase transition, and evolution of the gap versus D (FIG. 3B).

[0074] In summary, disclosed herein is unambiguous evidence that R-stacked twisted MoTe2bilayer near a doping of one hole per moiré unit cell is a QAH insulator. These results concur with the conclusion drawn from optical fan diagram measurements and pave the way for electrical transport measurements of FQAH states. Beyond the great promise of FQAH states in twisted MoTe2, the measured QAH state is unique among all known QAH insulators. Compared to magnetically doped QAH insulators, the topological gap in twisted MoTe2is much larger due to interaction induced ferromagnetism, and the QAH insulator phase can be electrically tuned to other correlated states. Currently, MnBi2Te4is the only known QAH insulator with intrinsic magnetism. However, repeatability and crystal quality remain an open challenge. Repeatability is also an issue for twisted graphene QAH insulators, presumably due to their sensitive dependence on twist angle and strain. In comparison, twisted MoTe2 is less sensitive to twist angle. Therefore, with this system, routine study of QAH physics may become feasible. In addition, the demonstration of tuning magnetic and topological orders with displacement field opens a ground for exploring topological electronics and spintronics devices. For the MoTe2 / WSe2 system, very large electric fields (~700 mV / nm) are required to invert the gap. This poses a challenge in device fabrication. For instance, very thin hBN needs be used to achieve these large electric fields. The short gate-sample distance can also introduce screening effects, which may fundamentally limit investigation of the QAH insulator in the strong correlation limit. With the expected improvement of the sample quality and metalcontacts, the family of twisted MoTe2 structures has the potential to be an active platform to investigate the interplay between correlation, topology, and magnetism.

[0075] FIGS 5A-5B show symmetrized longitudinal (Rxx) and, antisymmetrized Hall (Rxy) resistance, respectively, at magnetic fields |µ0H|=200mT as a function of electric field (D / ε0) and carrier density (n), in accordance with the present technology. FIGS 5C- 5D show Rxx and Rxy, respectively, as a function of magnetic field µoH at v=-1 / 2 and D=0, in accordance with the present technology. Quantized Rxy~ 2h / e2at zero magnetic field demonstrates a correlated fractional quantum anomalous Hall insulator. Data are taken at a temperature of 100 mK.

[0076] Further disclosed herein is spin Hall torque driven magnetization switching and current controlled helical light emission in near-AA stacked homobilayer moiré MoTe2.

[0077] FIG.6A is an atomic force microscope (AFM) image of a twisted MoTe2device equipped with Pt contacts for current injection, in accordance with the present technology. A dual gated device architecture equipped with electrical contacts, as shown in FIG. 6A, enables current injection into the moiré MoTe2system tuned to the ferromagnetic regime. Spatially resolved reflective magnetic circular dichroism (RMCD), a technique sensitive to spin / valley polarization, establishes a transverse spin Hall current and spin Hall torque switching of magnetic order. Polarization-resolved trion photoluminescence (PL) measurements demonstrate electrical current control of emission helicity, which can be tuned between nearly complete left (σ-) and right (σ+) circular polarization at zero applied magnetic field.

[0078] The ferromagnetism in moiré MoTe2using RMCD was investigated. Valley-dependent optical selection rules enable this optical probe to read out spin / valley polarization of our system, and thus observe ferromagnetic order. A ferromagnetic phase arises over a large range of filling ν upon hole doping into the first moiré valence band, for small out-of-plane electric field (or displacement field) D / ε0.

[0079] FIG.6B is an RMCD signal as a function of μ0H swept down and up, in accordance with the present technology. As shown in FIG. 6B, RMCD was measured at filling ν=-0.77 and displacement field D / ε0=0, where the system is in the FM metal regime. All measurements are taken at 1.6K unless otherwise indicated. As the applied magneticfield μ0H is swept down and up, hysteresis and sharp transitions in the RMCD signal were observed, which remains finite at μ0H=0 – clear hallmarks of ferromagnetism.

[0080] FIG.6C is an RMCD spatial map in the ferromagnetic (FM) metal regime (ν=-0.77) at μ0H = 0T, taken after initialization at μ0H > 0T, in accordance with the present technology. To determine the effect of injecting current via the contacts on the spin / valley polarization, a spatial map of the RMCD signal over the entire sample was created. FIG. 6C shows this map for the same ν and D / ε0as in FIG. 6B, measured at μ0H=0 after initializing with positive μ0H. Finite, positive RMCD signal is visible across the entire sample area, with contacts visible as regions of decreased RMCD (outlined with grey dashed lines). This is the expected behavior for a homogeneous sample tuned to the FM regime. In stark contrast, when an RMCD spatial map is taken under the same conditions as FIG.6C, but with current flowing between the two contacts on the top left, clear domains with an opposite sign of RMCD are observed (FIGS 6D-6E). This sign flipped domain appears on one side of the current channel, and changes to the opposite side when the direction of current flow is reversed. In FIG.6C, the RMCD across the sample is positive, with a maximum value of approximately 15%, with darker regions having a higher value of RMCD.

[0081] Grey dashed lines indicate Pt contact area. FIG.6D is the RMCD spatial map taken under the same conditions as FIG.6C, but with a current of I = -35nA flowing between the contacts on the upper left of the device, in accordance with the present technology. In FIG. 6D, the domain on the upper left side of the current channel flowing between the contacts has a value of RMCD of about 0 to +15%, and the domain on the lower right side of the current channel has a value of RMCD of about 0 to -15%. The darkness of each region indicates a larger absolute value of RMCD signal. The RMCD sign change indicates the formation of a domain with the opposite spin / valley polarization. FIG.6E is the RMCD spatial map measured under the same conditions as FIG.6D but with the current direction reversed, in accordance with the present technology. FIG.6F is a schematic of the transverse spin-valley current in moiré MoTe2, in accordance with the present technology. In FIG. 6E, the domain on the upper left side of the current channel flowing between the contacts has a value of RMCD of about 0 to -15%, and the domain on the lower right side of the current channel has a value of RMCD of about 0 to +15%. That is, the sign of the domains on either side of the current channel have switched with reversalof current flow. As with FIGS.6C-6D, the darkness of each region indicates larger absolute RMCD, as explained herein.

[0082] Current flow between contacts produces a spin / valley polarized transverse current, resulting in the formation of a domain with opposite spin / valley polarization. This behavior, consistent with that previously observed in heterobilayer TMD moirés, is a signature of a spin / valley current transverse to the injected charge current – that is, a spin Hall effect. This behavior can be attributed to the large Berry curvature which arises in the moiré valence bands. The Berry curvature produces a transverse valley current, which is also a spin current due to spin / valley locking. As shown schematically in FIG.6F, the spin Hall effect results in a flow of spin / valley polarized carriers, producing domains with opposite polarization on either side of the current channel, as observed in the RMCD spatial maps.

[0083] Accordingly, current control of spin / valley polarization in the FM metal regime has been disclosed herein. To explore the effect of the injected current in other regions of parameter space, the beam spot on one of the domains (i.e., the sign reversed domain in FIG.6D) is parked and RMCD vs ν and D / ε0for selected current I is measured.

[0084] FIG.7A is an RMCD signal vs ν and D at μ0H = 0T and for I = -20nA, taken with the beam spot on the sign reversed domain in FIG. 6D, in accordance with the present technology. A clear sign reversal is visible both in the partially filled first moiré band and in the FM tails at hole fillings above ν=-1. FIG.7A shows this measurement with I=-20nA and at μ0H=0, after initializing with positive μ0H. When compared to the behavior for I=0nA, the general outline of finite RMCD is the same, but the sign is reversed both for hole doping less than ν =-1 and in the FM ‘tails’ beyond ν =-1 and at finite D / ε0. This behavior implies that the spin / valley Hall current induced domains occur not only in the FM metal regime explored in FIGS 6A-6F, but more generically throughout the ferromagnetic phase space. Worth noting is the lack of sign reversal in the region around ν =-1. It is possible that in this regime, the system is strongly gapped, preventing current from flowing through the bulk. This region corresponds to a Chern insulator in the quantum anomalous Hall regime. Here, the current travels via edge modes, and therefore cannot produce the bulk domain flipping seen in other regions of phase space. Though similar behavior would be expected for the gapped fractional Chern insulator at ν =-2 / 3, it is possible that the substantially smaller gap prevents this from being observed.

[0085] FIG.7B is the RMCD signal measured under the same conditions as FIG. 7A but with I = 20nA, in accordance with the present technology. Sign switching is visible near ν=-1 and around D / ε0 ≈ 0.17V / nm. This value is close to the critical field DC where the ν=-1 gap closes. FIG. 7B shows positive RMCD in the partially filled moiré valence band and in the FM tails. Looking more closely at FIG. 2B, it is demonstrated the sign of RMCD changes over a small region of phase space near ν =-1 and DC – i.e., the critical displacement field where the phase transition from the topological to trivial phase occurs, and ferromagnetism disappears.

[0086] FIG.7C is an RMCD signal vs I swept down, as a function of ν, with μ0H = 0T and D / ε0 = 0.17V / nm, in accordance with the present technology. A sharp sign reversal is visible at positive I for a range of fillings, indicating a current driven switching of the spin-valley polarization. FIG.7D is the RMCD signal measured under the same conditions as FIG.7C but with I swept up, in accordance with the present technology. Sharp transitions occur at finite current for fillings slightly below ν =-1.

[0087] FIG.7E is hysteretic component of RMCD vs I, ΔRMCD, as a function of ν, in accordance with the present technology. Hysteresis occurs over a range of hole fillings slightly below ν=-1. FIG.7F is an RMCD vs I swept down and up at ν=-0.976, in accordance with the present technology. Clear hysteresis is visible, indicating current- driven magnetization switching. As the RMCD sign change corresponds to a flipping of the spin / valley polarization, these data indicate that injecting charge current produces a spin Hall torque switching of the magnetization at μ0H=0.

[0088] Having established current control of magnetic order, a second key behavior in moiré MoTe2 – spin / valley polarization dependent emission helicity was investigated. Spin / valley polarized carriers dress excitons in the opposite valley to form singlet trions, which emit in the σ+(σ-) channel for excitons in the +K (-K) valley. In the FM phase, spontaneous spin / valley polarization of the doped holes causes emission from a single valley, leading to circularly polarized emission at μ0H=0. Thus, it follows that using current to control spin / valley polarization would also allow control of emission helicity. As shown in FIGS 8A-8B, this is indeed the case.

[0089] FIG.8A is a spatial map of PL degree of circular polarization ρ with current I = 35nA and at μ0H = 0T, taken after initialization at μ0H > 0T, in accordance with the present technology. Domains with opposite signs of ρ are visible on either side of thecurrent channel. Finite, but smaller, ρ is visible far from the current channel, on the bottom right side of the device. System is in the FM metal regime, with ν=-0.77. FIG.8B is a spatial map of PL degree of circular polarization ρ measured under the same conditions as FIG. 8A but with I = -35nA, in accordance with the present technology. Signs of the domains are switched.

[0090] FIG.8C shows circular polarization resolved PL spectra at I = 35nA and μ0H = 0T, taken on the black dot in FIG.8A, in accordance with the preset technology. PL shows clear helicity dependence of the light emission, with signal mainly in the σ- channel. FIG.8D shows circular polarization resolved PL spectra measured under the same conditions as FIG. 8C, but with I = -35nA, in accordance with the present technology. Helicity dependence is switched, with signal mainly in the σ+channel. FIG.8E shows ρ vs μ0H swept down and up, with I = 0nA, in accordance with the present technology. Hysteresis is visible, indicating that light emission helicity is controlled by the spin / valley polarization. FIGS 8F-8G show ρ vs μ0H swept down and up, with I = 35nA and I = -35nA, respectively, in accordance with the present technology. Preferred emission channel at low μ0H is switched by changing the sign of I. Spin Hall torque from the transverse spin-valley current allows for control of the light emission helicity in the moiré MoTe2system.

[0091] Here, spatial maps of the device in the FM metal regime were taken with finite I, similar to FIGS 6D-6E. However, in this case, circular polarization resolved trion photoluminescence was measured - extracting a degree of circular polarization ρ as in Equation 1. ^^^σ^^ − ^^^σ^^^ =

[0092] In these spatial maps, we also see domains with opposite signs of ρ on either side of the current channel, which switch with reversal of I. Selecting a spot on one of these domains (as shown in the black dot of FIG. 8A) the polarization resolved PL spectra was plotted (FIGS 8C-8D) for I=±35nA. There is a clear difference between σ+and σ- polarized emission, with the dependence switched for reversed I. This demonstration of the ability to switch between σ+and σ- PL at μ0H=0 by changing I establishes current control of emission helicity.

[0093] Finally, the emission helicity as a function of both applied magnetic field and current was considered. As seen in FIG.8E, ρ vs μ0H swept down and up at I =0 shows hysteretic behavior and finite ρ at μ0H=0. This makes sense, as the system is ferromagnetic, and finite ρ arises from spin / valley polarization. The same measurement with injected current (FIGS 8F-8G) shows that for small magnetic fields, a different sign of ρ is favored depending on the current direction.

[0094] FIG.9A shows ρ vs I swept down, as a function of μ0H, starting from positive μ0H, in accordance with the present technology. Emission helicity can be continuously tuned between the σ+and σ- channels by both current and magnetic field. Data are taken with the beam spot on the opposite side of the current channel than for FIGS 7A- 7F and 8A-8G. System is in the FM metal regime, with ν=-0.77. FIG.9B shows ρ as a continuous function of I, at μ0H = 0T, initialized with μ0H > 0T, in accordance with the present technology. The insets show helicity resolved PL emission for large positive and negative current. Helicity can be continuously tuned between these two limits by changing I. The tuning of the trion PL emission helicity as a function of current is highlighted in FIG. 9B; it can be continuously varied between σ+and σ- at zero magnetic field.

[0095] Using RMCD and polarization resolved PL, spin Hall torque switching of magnetism and current controlled helical light emission in moiré MoTe2has been established. The strong coupling between electronic and photonic degrees of freedom that these results imply could prove crucial in a range of device applications. One possibility is to use these properties to link magnetic information storage with optical communication within a single device. The sensitivity to the circular polarization degree of freedom within this system also holds great promise for optical sensing in addition to quantum information processing, as quantum information can be encoded in the photon polarization.

[0096] The topological phases which occur in moiré MoTe2open up additional possibilities. As the integer and fractional Chern insulators recently observed in this system have a topological index which depends on the magnetization orientation, the observed spin Hall torque implies that current injection can be used to manipulate these topological states. One interesting direction could be to use current injection to establish two magnetic domains with opposite orientations and investigate the physics arising at the edge between two zero-field FCIs with opposite topological indices. In any case, establishing current control of magnetic order in moiré MoTe2 adds an important tool to our toolbox asinvestigation of these zero-field FCIs and their anyonic excitations continue – one which may prove crucial in the eventual exploitation of the system for topologically protected quantum computation.

[0097] METHODS

[0098] As an example of sample fabrication, hBN, MoTe2, and graphite crystals were mechanically exfoliated onto p-doped Si / SiO2 substrates to obtain nm thick flakes. An optical microscope was used to find thin hBN for contact gates, thick hBN for top and bottom gates, and thin graphite for gate electrodes. The cleanliness of the flakes was confirmed by atomic force microscope (AFM) or contrast-enhanced optical microscope. The backgate heterostructure was fabricated by a standard polycarbonate-based dry transfer process.

[0099] First, an hBN flake was chosen to serve as a backgate dielectric and picked up, followed by a graphite bottom gate electrode, before putting down the structure on a silicon substrate. Standard e-beam lithography was used to define Hall bar contacts, and Ti / Pt (2 / 8 nm) electrodes were deposited by e-beam evaporation. The backgate contact structure was cleaned using contact-mode AFM and moved into a glovebox with O2and H2O levels < 0.1 ppm. In the glovebox, MoTe2 was exfoliated to find a monolayer flake that was cut using an AFM tip. A twisted MoTe2structure was formed by picking up an hBN flake, picking up part of the MoTe2 monolayer, rotating the stage by the desired angle, and then picking up the other part of MoTe2monolayer, before putting the structure down onto the prepared backgate structure. The stamp polymer was washed off with molecular sieve-cleaned anhydrous chloroform and DCM in a glovebox environment. After the finished encapsulation and wash off process, the twisted MoTe2 device was removed from the glovebox and AFM cleaned again to mechanically squeeze bubbles out from between the hBN and twisted MoTe2 layers. Local contact gates, used to improve the contact resistance, were added using standard e-beam lithography and e-beam evaporated 8 nm Pt. Another lithography and evaporation of Cr / Au (5 / 70 nm) was used to define electrodes and conductive pads to allow for wire-bonding to the sample. An additional AFM cleaning process was then performed on the entire structure to remove polymer residue for the subsequent transfer. The structure was then finished by adding a graphite / hBN top gate. Polymer from this transfer was washed off with chloroform in ambient conditions.

[0100] In some examples, transport measurements were conducted in a dilution refrigerator (Bluefors) with base phonon temperature ~20 mK (~80 mK electron temperature calibrated by other nano device with similar structure) and a 9T-1T-1T vector magnetic field (9T along the out-of-plane axis). It was found that a constant current measurement scheme can significantly increase the stability of the magnetic state. An AC current bias of ~0.2-0.5nA was generated by the lock-in amplifier in series with a large resistance of 100 MΩ. The current was monitored simultaneously with a DL1211 preamplifier. Four terminal Rxx and Rxy signals were pre-amplified by the differential mode of an SR560 with 10-50x amplification. The input resistance of SR560 is ~100 MΩ >> contact resistance. All preamplifiers were read out by SR830 lock-ins. Two terminal conductance measurement was done by applying a 200-1000 µV ac voltage bias to the source and monitoring drain current with a DL1211 preamplifier. For temperature dependence > 4K, the current on a resistance heater close to the sample was gradually ramped up. A Cernox temperature sensor close to the sample was used to read out temperature. A control experiment of a carbon film sensor on the chip carrier was performed to find the temperature difference between the sensor and sample stage temperature, which is < 50mK for T = 4-10K and thus negligible.

[0101] RMCD measurements were performed in a in a closed-loop magneto- optical exchange gas cryostat (attoDRY 2100) with an attocube xyz piezo stage, 9T out-of- plane superconducting magnet, and base temperature of 1.6K. Wavelength-tunable RMCD excitation was achieved by filtering a broadband supercontinuum source (NKT SuperK FIANIUM FIU-15) via dual-passing through a monochromator, selecting out an energy resonant with the trion feature of the hole-doped twisted MoTe2 system. RMCD signal is proportional to the difference in reflectance of right and left circularly polarized (RCP and LCP) light normalized by the total reflectance. Excitation was chopped at 1kHz and polarization was switched between RCP and LCP with a photoelastic modulator at 50kHz. Reflected signal was detected by an InGaAs avalanche photodiode and read out by two lock-in amplifiers (SR830) detecting at 50 kHz and 1 kHz. This scheme allows for continuous detection of both the reflectance difference between RCP and LCP and the total reflectance, thus allowing RMCD to be extracted. The wavelength of the incident light is chosen to be 1115nm, in resonance with the trion emission, as reported previously.

[0102] The carrier density n and electric field D on the sample can be derived from top (bottom) gate voltage Vtg (Vbg) with Equation 2: n = (VtgCtg + VbgCbg) / e-noffset and D / ɛ0 = (VtgCtg − VbgCbg) / 2ɛ0-Doffset Equation 2

[0103] where e is the electron charge, ɛ0is the vacuum permittivity, and Ctg and Cbg are the top and bottom gate capacitance obtained from the gate thickness measured by AFM. Doffset~ 0 was inferred from the RMCD 2D map shown in FIG. 3A. The filling number v = -2 can be obtained from prominent feature as the peak of Rxx in transport map (FIG 1B). The filling number v = -1 can be obtained from the prominent feature of RMCD 2D map. The gate capacitance and assignment of filling number were then independently verified by high magnetic field Landau Fan fitting at finite electric field, and the Streda slope of v = -1 at zero electric field.

[0104] At a twist angle of 3.5° in the absence of an electric field, the continuum model realizes a Kane-Mele model with the top two bands having Chern number -1 and 1. This allows construction of a tight-binding Hamiltonian on the honeycomb lattice. With the electric field, the single-particle part of the Hamiltonian reads: ^^^^^^^^^^^ = " #" (^' ' + " ^^^-. / 0 (' + " ( 6.

[0105] Here, $% is the nearest neighbor hopping between A and B sublattices, and$^ is the next nearest neighbor hopping within a single sublattice. 8^) = ±1 depends on thehopping direction. $^ = 0.2$% and = = > / 3 were chosen. The onsite Hubbard interactionis chosen as @ = 10$%. Hartree-Fock calculations were performed on this Hamiltonian atthe filling factor of 8 = −1 under different electric fields.

[0106] The disclosed multiple gate systems for realizing QAH effects can also be used as transistor structures. In this context, the device operates as a transistor when the electric field is tuned to introduce a phase transition. The transistor operates by electric- field induced phase transition, which then dictates current flow through the QAH structure, for example between two electrodes, a source and a drain. Accordingly, in one embodiment the transistor includes a multiple gate system with QAH structures as disclosed herein, and further includes source and drain electrodes in electrical contact with the system such thatcurrent flow is controlled between the source and the drain via a phase transition of the QAH structures.

[0107] The present application may reference quantities and numbers. Unless specifically stated, such quantities and numbers are not to be considered restrictive, but representative of the possible quantities or numbers associated with the present application. Also, in this regard, the present application may use the term “plurality” to reference a quantity or number. In this regard, the term “plurality” is meant to be any number that is more than one, for example, two, three, four, five, etc. The terms “about,” “approximately,” “near,” etc., mean plus or minus 5% of the stated value. For the purposes of the present disclosure, the phrase “at least one of A, B, and C,” for example, means (A), (B), (C), (A and B), (A and C), (B and C), or (A, B, and C), including all further possible permutations when greater than three elements are listed.

[0108] Embodiments disclosed herein may utilize circuitry in order to implement technologies and methodologies described herein, operatively connect two or more components, generate information, determine operation conditions, control an appliance, device, or method, and / or the like. Circuitry of any type can be used. In an embodiment, circuitry includes, among other things, one or more computing devices such as a processor (e.g., a microprocessor), a central processing unit (CPU), a digital signal processor (DSP), an application-specific integrated circuit (ASIC), a field-programmable gate array (FPGA), or the like, or any combinations thereof, and can include discrete digital or analog circuit elements or electronics, or combinations thereof.

[0109] An embodiment includes one or more data stores that, for example, store instructions or data. Non-limiting examples of one or more data stores include volatile memory (e.g., Random Access memory (RAM), Dynamic Random Access memory (DRAM), or the like), non-volatile memory (e.g., Read-Only memory (ROM), Electrically Erasable Programmable Read-Only memory (EEPROM), Compact Disc Read-Only memory (CD-ROM), or the like), persistent memory, or the like. Further non-limiting examples of one or more data stores include Erasable Programmable Read-Only memory (EPROM), flash memory, or the like. The one or more data stores can be connected to, for example, one or more computing devices by one or more instructions, data, or power buses.

[0110] In an embodiment, circuitry includes a computer-readable media drive or memory slot configured to accept signal-bearing medium (e.g., computer-readable memorymedia, computer-readable recording media, or the like). In an embodiment, a program for causing a system to execute any of the disclosed methods can be stored on, for example, a computer-readable recording medium (CRMM), a signal-bearing medium, or the like. Non- limiting examples of signal-bearing media include a recordable type medium such as any form of flash memory, magnetic tape, floppy disk, a hard disk drive, a Compact Disc (CD), a Digital Video Disk (DVD), Blu-Ray Disc, a digital tape, a computer memory, or the like, as well as transmission type medium such as a digital and / or an analog communication medium (e.g., a fiber optic cable, a waveguide, a wired communications link, a wireless communication link (e.g., transmitter, receiver, transceiver, transmission logic, reception logic, etc.). Further non-limiting examples of signal-bearing media include, but are not limited to, DVD-ROM, DVD-RAM, DVD+RW, DVD-RW, DVD-R, DVD+R, CD-ROM, Super Audio CD, CD-R, CD+R, CD+RW, CD-RW, Video Compact Discs, Super Video Discs, flash memory, magnetic tape, magneto-optic disk, MINIDISC, non-volatile memory card, EEPROM, optical disk, optical storage, RAM, ROM, system memory, web server, or the like.

[0111] The detailed description set forth above in connection with the appended drawings, where like numerals reference like elements, are intended as a description of various embodiments of the present disclosure and are not intended to represent the only embodiments. Each embodiment described in this disclosure is provided merely as an example or illustration and should not be construed as preferred or advantageous over other embodiments. The illustrative examples provided herein are not intended to be exhaustive or to limit the disclosure to the precise forms disclosed. Similarly, any steps described herein may be interchangeable with other steps, or combinations of steps, in order to achieve the same or substantially similar result. Generally, the embodiments disclosed herein are non-limiting, and the inventors contemplate that other embodiments within the scope of this disclosure may include structures and functionalities from more than one specific embodiment shown in the figures and described in the specification.

[0112] In the foregoing description, specific details are set forth to provide a thorough understanding of exemplary embodiments of the present disclosure. It will be apparent to one skilled in the art, however, that the embodiments disclosed herein may be practiced without embodying all the specific details. In some instances, well-known process steps have not been described in detail in order not to unnecessarily obscure variousaspects of the present disclosure. Further, it will be appreciated that embodiments of the present disclosure may employ any combination of features described herein.

[0113] The present application may include references to directions, such as “vertical,” “horizontal,” “front,” “rear,” “left,” “right,” “top,” and “bottom,” etc. These references, and other similar references in the present application, are intended to assist in helping describe and understand the particular embodiment (such as when the embodiment is positioned for use) and are not intended to limit the present disclosure to these directions or locations.

[0114] The principles, representative embodiments, and modes of operation of the present disclosure have been described in the foregoing description. However, aspects of the present disclosure, which are intended to be protected, are not to be construed as limited to the particular embodiments disclosed. Further, the embodiments described herein are to be regarded as illustrative rather than restrictive. It will be appreciated that variations and changes may be made by others, and equivalents employed, without departing from the spirit of the present disclosure. Accordingly, it is expressly intended that all such variations, changes, and equivalents fall within the spirit and scope of the present disclosure as claimed.

[0115] While illustrative embodiments have been illustrated and described, it will be appreciated that various changes can be made therein without departing from the spirit and scope of the invention.

Claims

CLAIMS The embodiments of the invention in which an exclusive property or privilege is claimed are defined as follows:

1. A multiple gate system hosting the quantum anomalous Hall effect, comprising: a top channel electrode which controls carrier density and electric field in a conduction channel; a first dielectric layer for the top channel gate; a contact region, comprising: a patterned contact gate electrode which controls carrier density on the contact and screens the top channel electrode’s electric field on the contact, and a second dielectric layer for the contact gate, an electrically tunable material comprising semiconducting molybdenum ditelluride in the form of a first atomic sheet and a second atomic sheet, wherein first atomic sheet is rotated planarly from the second atomic sheet by about 2.5 to about 5 degrees, to form a geometrically stacked homobilayer moiré superlattice configured to host ferromagnetic states that spontaneously break time-reversal symmetry, wherein the material of atomic sheets define an xy plane, and a set of Hall bar contacts for electrical contact to a sample in a Hall bar geometry where the carrier density can be tuned by the patterned contact gate electrode; wherein the Hall bar contacts are arranged adjacent to each other along an x direction in an xy plane and opposite to each other in a y direction in the xy plane; a third dielectric layer for the bottom channel gate; a bottom channel electrode; wherein the top and bottom channel electrodes form a dual gate geometry; wherein an input voltage Vtg is placed on the top channel electrode, an input voltage Vbg is placed on the bottom channel electrode, an input voltage Vcg is placed on the contact gate electrode, a bias voltage Vbias is placed on the source Hall bar contact of the set of Hall bar contacts; wherein Vxx is longitudinal potential between two Hall bar contacts of the set of Hall bar contacts and Vxyis the Hall potential between two Hall bar contacts of the set of Hall bar contacts;wherein the patterned contact electrode spaced from the source Hall bar contact of the set of Hall bar contacts is grounded as a drain Hall bar contact; and wherein a perpendicular electric field is applied to the electrically tunable material, tuning the quantum anomalous Hall effect.

2. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 1, wherein multiple gate system is a split gate system.

3. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 1, wherein the top electrode and the bottom electrode comprise graphite.

4. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 1, wherein the patterned gate electrode comprises platinum, titanium, chromium, gold, or a combination thereof.

5. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 1, wherein the first dielectric layer, the second dielectric layer, the third dielectric layer, or a combination thereof comprise hexagonal boron nitride (hBN).

6. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 1, wherein the stacked homobilayer is rhombohedral.

7. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 1, wherein the moiré orbitals form a honeycomb superlattice.

8. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 1, wherein the contact region is hole doped to about -1×1013cm-2.

9. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 1, wherein the carrier density and the electric field on the sample are converted from a top gate voltage and a bottom gate voltage using a parallel plate capacitor model: n = (VtgCtg + VbgCbg) / e-noffsetand D / ɛ0 = (VtgCtg − VbgCbg) / 2ɛ0-Doffset; wherein Ctg and Cbg are a top gate capacitance and a bottom gate capacitance, e is an electron charge, and ɛ0 is a vacuum permittivity.

10. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 1, wherein the first atomic sheet and the second atomic sheet each have from one to three atomic layers of semiconducting molybdenum ditelluride.

11. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 1, wherein the first atomic sheet and the second atomic sheet are rotated planarly from each other by about 3.52 degrees.

12. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 1, wherein the homobilayer has electrically tunable topological phase transitions.

13. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 1, wherein the homobilayer has fractional quantum anomalous Hall states.

14. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 13, wherein the fractional quantum anomalous Hall states are at -2 / 3, -3 / 5, and - 4 / 7 filling of the moiré superlattice.

15. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 1, wherein the homobilayer is photoluminescent with a linewidth about 3 meV.

16. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 16, wherein the homobilayer is photoluminescent between 1.13 and 1.09 eV.

17. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 1, wherein the input voltage is about -10 V to about 10 V.

18. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 1, wherein a critical electric field DC / ɛ0 is less than or equal to about 150 mV / nm.

19. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 1, wherein the multiple gate system is operational at a temperature of 0.001 to 8 Kelvin.

20. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 1, wherein the multiple gate system is configured for use as a topological quantum computer element.

21. A transistor comprising a multiple-gate system for realizing the quantum anomalous Hall effect of Claim 1.

22. The transistor of Claim 21, wherein the transistor operates by electric-field induced phase transition to regulate current flow between a source electrode and a drain electrode of the transistor.

23. A multiple gate system for realizing the quantum anomalous Hall effect, comprising: a top channel electrode which controls carrier density and electric field in a conduction channel; a first dielectric layer for the top channel gate; a contact region, comprising:a patterned contact gate electrode which controls carrier density on the contact and screens the top channel electrode’s electric field on the contact, and a second dielectric layer for the contact gate, an electrically tunable material comprising semiconducting molybdenum ditelluride in the form of a first atomic sheet and a second atomic sheet, wherein first atomic sheet is rotated planarly from the second atomic sheet by about 2.5 to about 5 degrees, to form a geometrically stacked homobilayer moiré superlattice configured to host ferromagnetic states that spontaneously break time-reversal symmetry, wherein the material of atomic sheets define an xy plane, and a set of Hall bar contacts for electrical contact to a sample in a Hall bar geometry where the carrier density can be tuned by the patterned contact gate electrode; wherein the Hall bar contacts are arranged adjacent to each other along an x direction in an xy plane and opposite to each other in a y direction in the xy plane; a third dielectric layer for the bottom channel gate; a bottom channel electrode; wherein the top and bottom channel electrodes form a dual gate geometry; wherein an input voltage Vtg is placed on the top channel electrode, an input voltage Vbgis placed on the bottom channel electrode, an input voltage Vcgis placed on the contact gate electrode, a bias voltage Vbias is placed on the source Hall bar contact of the set of Hall bar contacts; wherein Vxx is the longitudinal potential between two Hall bar contacts of the set of Hall bar contacts and Vxyis the Hall potential between two Hall bar contacts of the set of Hall bar contacts; wherein the patterned contact electrode spaced from the source Hall bar contact of the set of Hall bar contacts is grounded as a drain Hall bar contact; and wherein a perpendicular electric field is applied to the electrically tunable material, tuning the quantum anomalous Hall effect.

24. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 23, wherein multiple gate system is a split gate system.

25. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 23 or Claim 24, wherein the top electrode and the bottom electrode comprise graphite.

26. The multiple gate system for realizing the quantum anomalous Hall effect of any one of Claims 23-25, wherein the patterned gate electrode comprises platinum, titanium, chromium, gold, or a combination thereof.

27. The multiple gate system for realizing the quantum anomalous Hall effect of any one of Claims 23-26, wherein the first dielectric layer, the second dielectric layer, the third dielectric layer, or a combination thereof comprise hexagonal boron nitride (hBN).

28. The multiple gate system for realizing the quantum anomalous Hall effect of any one of Claims 23-27, wherein the stacked homobilayer is rhombohedral.

29. The multiple gate system for realizing the quantum anomalous Hall effect of any one of Claims 23-28, wherein the moiré orbitals form a honeycomb superlattice .

30. The multiple gate system for realizing the quantum anomalous Hall effect of any one of Claims 23-29, wherein the contact region is hole doped to about -1×1013cm-2.

31. The multiple gate system for realizing the quantum anomalous Hall effect of any one of Claims 23-30, wherein the carrier density and the electric field on the sample are converted from a top gate voltage and a bottom gate voltage using a parallel plate capacitor model: n = (VtgCtg + VbgCbg) / e-noffset and D / ɛ0 = (VtgCtg − VbgCbg) / 2ɛ0-Doffset; wherein Ctgand Cbgare a top gate capacitance and a bottom gate capacitance, e is an electron charge, and ɛ0 is a vacuum permittivity.

32. The multiple gate system for realizing the quantum anomalous Hall effect of any one of Claims 23-31, wherein the first atomic sheet and the second atomic sheet each have from one to three atomic layers of semiconducting molybdenum ditelluride.

33. The multiple gate system for realizing the quantum anomalous Hall effect of any one of Claims 23-32, wherein the first atomic sheet and the second atomic sheet are rotated planarly from each other by about 3.52 degrees.

34. The multiple gate system for realizing the quantum anomalous Hall effect of any one of Claims 23-33, wherein the homobilayer has electrically tunable topological phase transitions.

35. The multiple gate system for realizing the quantum anomalous Hall effect of any one of Claims 23-34, wherein the homobilayer has fractional quantum anomalous Hall states.

36. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 35, wherein the fractional quantum anomalous Hall states are at -1 / 2 and -2 / 3 filling of the moiré superlattice.

37. The multiple gate system for realizing the quantum anomalous Hall effect of any one of Claims 23-37, wherein the homobilayer is photoluminescent with a linewidth about 3 meV.

38. The multiple gate system for realizing the quantum anomalous Hall effect of Claim 37, wherein the homobilayer is photoluminescent between 1.13 and 1.09 eV.

39. The multiple gate system for realizing the quantum anomalous Hall effect of any one of Claims 23-38, wherein the input voltage is about -10 V to about 10 V.

40. The multiple gate system for realizing the quantum anomalous Hall effect of any one of Claims 23-39, wherein a critical electric field DC / ɛ0 is less than or equal to about 150 mV / nm.

41. The multiple gate system for realizing the quantum anomalous Hall effect of any one of Claims 23-40, wherein the system is operational at a temperature of 0.001 to 8 Kelvin.

42. The multiple gate system for realizing the quantum anomalous Hall effect of any one of Claims 23-41, configured for use as a topological quantum computer element.

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