Abstract
The technology disclosed herein is based on a novel multilayer material having a plurality of internal charge dipoles and in-plane conductivity.
Claims
1.-31. (canceled)
32. A material in a form of a multilayer of two or more layers of a doped 2D material exhibiting in-plane conductivity, wherein each layer having a polarization pointing in a direction normal to the multilayer plane and opposite to the direction of polarization of any adjacent layer.
33. The material according to claim 32, wherein each of the material layers is formed of a 2D semiconductor material.
34. The material according to claim 33, wherein the 2D material is a diatomic hexagonal material.
35. The material according to claim 34, wherein the diatomic hexagonal material is selected from hexagonal-boron-nitride (h-BN), transition-metal-dichalcogenides (TMD), hexagonal-aluminum-nitride (h-AlN), hexagonal-zinc-oxide (h-ZnO), and hexagonal-gallium-nitride (h-GaN).
36. The material according to claim 34, wherein the 2D material is a transition-metal-dichalcogenides (TMD) selected from MoS.sub.2, WS.sub.2, MoSe.sub.2 and WSe.sub.2.
37. A conductive stacked multilayer diatomic hexagonal material or a conductive stacked multilayer structure formed by orienting any two stacked layers of a doped diatomic hexagonal material into a stacked (substantially) parallel lattice orientation to induce internal interfacial electric field normal to the layers plane at an interface between the two stacked material layers and in-plane conductivity, wherein the multilayer is n-doped or p-doped.
38. The material according to claim 32, wherein the doped multilayer is formed by chemical doping of the 2D materials prior to forming the multilayer structure.
39. The material according to claim 32, wherein the doped multilayer is formed by electrostatic doping of a preformed multilayer.
40. The material according to claim 32, comprising two or more stacked layers of a 2D material exhibiting out-of-plane switchable polarization and comprising free charge carriers of a density that is at least 10.sup.10 cm.sup.2 evenly distributed in the multilayer.
41. The material according to claim 40, comprising two or more layers of a TMD material stacked in a substantially parallel lattice orientation and exhibiting out-of-plane switchable polarization and comprising free electrons or holes of a density that is at least 10.sup.10 cm.sup.2 evenly distributed in the multilayer.
42. A device implementing a material according to claim 32.
43. A multi-switch polarization device having in-plane conductivity, the device comprising a plurality of out-of-plane switchable polarization states, the multilayer material comprising two or more stacked layers of a 2D material, wherein at least one of the layers formed of the 2D material is doped with charge carriers or holes (electrons or holes) that are (substantially) evenly distributed in the material layer(s).
44. The device according to claim 43, comprising a pair of electrodes, each of said electrodes being positioned at an edge of the layers of the multilayer material.
45. The device according to claim 43, comprising a top electrode and a bottom electrode.
46. The device according to claim 43, being selected from non-volatile memory devices, MEMS, photovoltaic cells, field effect transistors, memristors, and polar diodes.
47. The device according to claim 43, comprising two or three or more layers of one or more transition metal dichalcogenide (TMD), wherein the layers are artificially stacked in a parallel lattice orientation and encapsulated by thin flakes of a non-polar hexagonal boron nitride (h-BN), placed atop a graphite or gold metallic electrode.
48. The device according to claim 43, being a photovoltaic cell, an electro-mechanical generator, a dense information manipulation and storage device, a motion detection device, an opto-mechanical modulator, or an electronic device combining in-plane conductivity and internal out-of-plane polarization.
49. The device according to claim 43, being a photovoltaic device.
50. A photovoltaic device comprising a multi-switch polarization arrangement having in-plane conductivity, the arrangement being a multilayer stack of two or more layers of at least one 2D material, wherein at least one of the layers formed of the 2D material is doped with charge carriers (electrons or holes) that are (substantially) evenly distributed in the material layer(s), the device comprising a pair of electrodes positioned at the stacked layers edges and optionally a top electrode and a bottom electrode.
51. A process for constructing a multilayer structure according to claim 32, the process comprising: orienting any two stacked layers of one or more 2D materials into a stacked (substantially) parallel lattice orientation, wherein one or more of the layers of the structure are formed of a doped 2D material; or orienting any two stacked layers of one or more 2D materials into a stacked (substantially) parallel lattice orientation, to obtain the multilayer structure and exposing said structure to electrostatic doping to induce in-plane conductivity.
Description
BRIEF DESCRIPTION OF THE DRAWINGS
[0107] In order to better understand the subject matter that is disclosed herein and to exemplify how it may be carried out in practice, embodiments will now be described, by way of non-limiting example only, with reference to the accompanying drawings, in which:
[0108] FIGS. 1A-F: Excess charge distribution calculations for polar bilayers of MoS.sub.2 (left panels) and Wse.sub.2 (right panels). (A,B) Laterally averaged excess carrier density profiles of the undoped bilayers, .sub.0.sup.ex(z) (dashed black line), defined as the density of the bilayer stack after subtracting the superposition charge density of two isolated layers. Doping-induced variations in the excess carrier density, .sup.ex(z), for different hole densities are represented by the solid lines (C,D). The results for MoS.sub.2 are divided by 10 for clarity of the representation. (C,D) Doping induced antisymmetric part of the charge carrier density variation profiles defined as .sup.ex(z).sup.ex(z), where z=0 is set at the bilayer center for several doping densities. (E,F) Band structures of the undoped bilayer systems, colored according to the projection of the corresponding crystal momentum state on the top (light grey) and bottom (black) layers (see text). K momentum states split into layer-specific bands (prominent grey\black) that resemble the dashed black bands of a monolayer, only separated by ![custom-character]()
A larger splitting due to strong interlayer-delocalization (designated by dark grey) appears for valence and conduction Q states.
[0109] FIGS. 2A-B: (A) DFT calculations as in FIG. 1, for relaxed multilayers of 3R-MoS2, 3R-WSe2 and 3R-BN. The pristine BN result suggests potential accumulations as high as 4V, significantly larger than the TMD crystals. The dashed lines represent the band gaps of the periodic structures. (B) Measured average surface potential as a function of the number of layers N for 2 representatives 3R MoS.sub.2 flakes, each containing regions of various thicknesses (triangles). DFT calculations of the total potential difference across separate 3R crystals containing 2-20 layers with co-polarized interfaces (empty squares). A Schrodinger-Poisson equation solution of the surface potential for the same structures while considering a finite doping level of 13.5e10 cm.sup.2 (circles).
[0110] FIGS. 3A-E: Multiple polarization states in Wse.sub.2 multi-layers. A. Schematic illustration of the Kelvin probe force microscopy setup including a multi-layered TMD structure (lightest grey), encapsulated between h-BN layers (dark grey) that are placed on a graphitic or gold (black) gate (V.sub.g) electrode lying atop a silicon oxide substrate (bottom dark grey). B. Electric potential map at the Wse.sub.2 tri-layer surface. The dashed white lines mark borders between regions consisting of single and two active interfaces. Arrows denote the out-of-plane polarization orientation in five domains with different stacking configurations. C. Typical line cuts of the lateral potential drop across domain walls separating single (solid) and two (dashed) active interface regions (see Fig. S1f and panel b). Line cuts crossing domain walls that separate smaller domains are illustrated by the solid and dashed lines in panel A. The stacking configurations at the corresponding interfaces (shown schematically for the bilayer) are marked aside each potential step, with the corresponding interface polarizations marked by black arrows. D. Calculated laterally-averaged vertical potential profile along an ABC stacked trilayer with co-oriented (black arrows) interfacial polarization. E. Calculated potential difference across a multi-layer Wse.sub.2 structure compared to the experiment. The potential accumulates linearly with each extra layer for ABC stacking (full dots) and remains unaltered for anti-parallel AA stacking (empty squares).
[0111] FIGS. 4A-F: Multi-polarization states in naturally-grown 3R MoS.sub.2. A-D. Topography (A,C) and surface potential (B,D) maps of two typical flakes composed of 2-7, and 7-13, respectively. The potential is measured relative to the value above an ABA stacked tri-layer region. E. Line cuts, as marked in A-B, showing the flake thickness (empty squares, left axis) and surface potential (full circles, right axis). The horizontal grids show evenly-spaced steps. F. Surface potential values and excess number of active interfaces (N.sub.N.sub.) above different positions (as marked in a-d) versus the number of layers at each point. The dashed black line connects points of the fully co-aligned polar interfaces, where the symbols correspond to those appearing in A, B and E. Other points (with symbols corresponding to those appearing in panels C and D) show fixed, evenly spaces values corresponding to multi-polarization configurations. For example, the four values measured above 7 layers with 6 active interfaces correspond to the 6.sub., 5.sub.1.sub., 3.sub.3.sub., and 2.sub.4.sub. combinations. Calculated maximal polarization values are indicated by the full squares.
[0112] FIGS. 5A-D: Effect of gate bias on the polarization. A-C. Surface potential maps obtained for bilayer MoS.sub.2 under different gate biases. The center of the scale bar in each map is set to the corresponding average potential, V.sub.avg, as indicated in the respective panel. The color scale is centered at the average potential of the two polarizations, V.sub.avg. D. Potential drop V.sub.KP across domains of opposite polarization (see FIG. 4C), as a function of the external displacement field D and the corresponding 2D carrier density (lower/upper horizontal axis). Data from one MoS.sub.2 sample (stars) and two different WSe.sub.2 samples (triangles and circles) are compared to the calculated 2 values (solid, dashed lines, respectively).
[0113] FIGS. 6A-D: Doping and depolarization measurements. (A-C) Examples of surface potential maps obtained from a parallel WSe.sub.2 bilayer under several gate biases V.sub.g (Device 3). Note the domain wall motion at high doping density (C) that extends bright areas over dark domains. (D) The average KP potential of the TMD bilayers, V.sub.avg, as a function of the applied gate potential, V.sub.g.
[0114] FIGS. 7A-F: Further device characterization. (A-B) Optical microscope image of a typical device. Two circuit configurations of gating and KPFM measurement are illustrated. (C) The electric current along the MoS.sub.2 bilayers is measured versus the source-drain bias (V.sub.sd) and plotted for several fixed gate voltages (V.sub.g) at room temperature. (D-E) Simultaneous measurement of topography and KPFM maps for a tri-layer WSe.sub.2 structure (also shown in FIG. 4 in the main text). The dark regions in the topography map indicate a crack in one of the layers, separating regions of trilayers with one and two active interfaces. (E-F) Examples of ferroelectric-like coupling between the two active interfaces in large-area domains. The (ABA, neutral color, 0.2V) domains cover a smaller area than the (CBA, dark, 0.02V) domains (circumscribed in white). The same is found for (ABC, bright, 0.3V) regions marked in black. The solid grey and dashed lines in (F) mark the line cut position of the data shown in the main-text, FIG. 5C.
[0115] FIG. 8: Potential and charge density profiles of bilayer WSe.sub.2. Difference between bilayer and isolated monolayer plane-averaged potential (solid) and charge density (dashed) for an AB stacked WSe.sub.2. The dashed lines represent the vertical location of the ions. The origin of the horizontal axis is set to the midpoint between the layers.
[0116] FIGS. 9A-C: Convergence tests. Convergence tests of the binding energy (black curve, left vertical axis) and electrostatic potential difference (grey curve, right vertical axis) of a WSe.sub.2 bilayer (same structure as in FIG. 8) as a function of: (A) vacuum size; (B) energy cutoff, and (C) number of k-points.
[0117] FIGS. 10A-B: Potential and charge density profiles for WSe.sub.2 and MoS.sub.2 bilayer. Difference between bilayer and isolated monolayer plane-averaged potential (solid) and charge density (dashed) for AB stacked (A) WSe.sub.2 and (B) MoS.sub.2 bilayers. The vertical dashed lines represent the vertical location of the ions. The origin of the horizontal axis is set to the midpoint between the layers.
[0118] FIGS. 11A-F: Convergence tests. Convergence tests of the binding energy (black curve, left vertical axis) and electrostatic potential difference (grey curve, right vertical axis) of WSe.sub.2 (top panels) and MoS.sub.2 (bottom panels) bilayers (same structures as in FIG. 10) as a function of (A, D) vacuum size; (B, E) energy cutoff, and (C, F) number of k-points.
[0119] FIG. 12: Effect of doping on the interlayer potential drop. The potential drop as a function of electron (n, filled squares) and hole (p, empty circles) doping density for AB stacked bilayer WSe.sub.2 (grey) and MoS.sub.2 (black). The doping is introduced via the metal pseudo nuclei.
[0120] FIGS. 13A-F: Band structure and Fermi level variations with doping charge density. (A,D) The band structures of undoped (black), n-doped (dark grey), and p-doped (light grey) WSe.sub.2 (A) and MoS.sub.2 (D). For WSe.sub.2 (MoS.sub.2) the n-doped and p-doped band-structures are plotted for a charge density of n.sub.2D=1.110.sup.13 cm.sup.2 (9.310.sup.13 cm.sup.2), respectively. The origins of the vertical axes are set to the topmost -point valence band energy (E.sub.VBM). (B, E) The variation of the difference between the topmost K and valence band energies as a function of doping density for (B) WSe.sub.2 and (E) MoS.sub.2. Results are presented with respect to the energy difference obtained for the undoped system: E=[E.sub.KVBME.sub.VBM].sub.doped[E.sub.KVBME.sub.VBM].sub.undoped. (C, F) The Fermi level position of (C) WSe.sub.2 and (F) MoS.sub.2 as a function of n-doping (grey) and p-doping (black) charge densities. The origins of vertical axes are set to the conduction band minimum energy for n-doping and the valence band maximum energy for p-doping. The doping is introduced via the metal pseudo nuclei.
[0121] FIGS. 14A-B: Comparison of different doping schemes. The polarization as a function of n-doping (grey) and p-doping (black) charge density for (A) WSe.sub.2 and (B) MoS.sub.2. Different doping schemes are applied including: only metal ions (filled squares), only chalcogen ions (open circles), or all ion doping (open triangle). For WSe.sub.2, the W ion doping charge excess values were =10.sup.4, 210.sup.4, 510.sup.4, 810.sup.4, 10.sup.3, 210.sup.3, 510.sup.3, 810.sup.3, 0.01; the Se ion doping charge values were =510.sup.5, 10.sup.4, 2.510.sup.4, 410.sup.4, 510.sup.4, 10.sup.3, 2.510.sup.3, 410.sup.3, 0.005; and the all ion doping charge excess values used were =3.310.sup.5, 6.610.sup.5, 1.610.sup.4, 2.610.sup.4, 3.310.sup.4, 6.610.sup.4, 1.610.sup.3, 2.610.sup.3, 0.0033. The electron configuration of Se was given by [Ar]3d.sup.104s.sup.24p.sup.4, and the corresponding nuclear charge was Z=34. For MoS.sub.2, the Mo ion doping charge excess values were =10.sup.4, 10.sup.3, 210.sup.3, 510.sup.3, 810.sup.3, 0.01, 0.02, 0.03, 0.04; the S ion doping charge excess values were =510.sup.5, 510.sup.4, 10.sup.3, 2.510.sup.3, 410.sup.4, 0.005, 0.01, 0.015, 0.02; and the all ion doping charge excess values used were =3.310.sup.5, 3.310.sup.4, 6.610.sup.4, 1.610.sup.3, 2.610.sup.3, 3.310.sup.3, 6.610.sup.3, 0.01, 0.013. The electron configuration of S was given by [Ne]3s.sup.23p.sup.4, and the corresponding nuclear charge was Z=16.
[0122] FIGS. 15A-B: Effect of polarization on the band structure. The band structures of (A) WSe.sub.2 and (B) MoS.sub.2 bilayers at their anti-parallel AA (solid black curve) and parallel AB (dashed grey curve) stacking modes.
DETAILED DESCRIPTION OF EMBODIMENTS
[0123] As disclosed herein, the inventors now demonstrate stacked 2D layers that support robust interfacial polarization which features three unique characteristics: (i) it supports polarization as high as 0.5 pC/m per interface; (ii) it exhibits distinct and switchable multiple polarization configurations; and (iii) it sustains charge carrier densities up to 10.sup.14 cm.sup.2, a computational prediction that is confirmed by the experiments disclosed herein for charge carrier densities as high as 10.sup.13 cm.sup.2. The coexistence of polarization and conductivity is attributed to the interfacial localization of the polarization and the excess charge carriers delocalization on both layers, which inhibits strong depolarization fields. Notably, the measured polarization and density values are nearly ten times larger than those found for non-hexagonal TMDs to date and may support rich correlated electronic phases. The cumulative distinct multi-polar ladder of states reported here thus paves the way to bottom-up construction of 3D multi-ferroic structures out of well-defined 2D building blocks in a controllable, position-and orientation-specific manner.
Materials and Methods
Device Fabrication
[0124] h-BN flakes of various thicknesses were exfoliated onto a Si/SiO.sub.2 substrate. MoS.sub.2 and WSe.sub.2, obtained from HQ Graphene, were exfoliated onto polydimethylsiloxane (PDMS). Large single-layer flakes (20 m or more) of transition metal dichalcogenides (TMDs) were identified using optical contrast. h-BN flakes were picked up from the substrate and placed on few-layered graphene or predesigned gold electrodes (FIG. 7A). Subsequently, parallel bilayers of TMDs were prepared on the h-BN surface using the tear and stack technique. In this process, a fragment of a chosen TMD flake is first stamped on h-BN, followed by successive stacking of the remaining flake on it. The entire stack is then encapsulated with another h-BN flake. The bottom graphene or gold substrate acts as a reference electrode for Kelvin probe force microscopy (KPFM) measurements and as a gate electrode. In the tri-layer measurements (without doping), the stack was placed directly on the conducting electrode without the bottom h-BN.
TABLE-US-00001 TABLE 1 Details of devices according to some embodiments of the invention. h-BN spacer Device Name Electrode thickness (nm) Dev 1: MoS.sub.2 Au 6.3 Dev 2: WSe.sub.2 Few layer graphene 12 Dev 3: WSe.sub.2 Au 5.1
AFM Measurements
[0125] Topography and KPFM measurements were acquired simultaneously, using Park System NX10 AFM in non-contact scanning mode. The electrostatic signal was measured at a side-band frequency using a built-in lock-in amplifier. A PointProbe Plus Electrostatic Force Microscopy (PPP-EFM) n-doped tips with a conductive coating was used. The mechanical resonance frequency of the tips was 75 kHz and the force constant was 3 N/m. The cantilever oscillated mechanically with an amplitude ranging from 20 to 5 nm. In several experiments, the average height above the surface, h, was controlled via a two-pass measurement. The first pass records the topography, whereas in the second pass the tip follows the same scan line with a predefined lift (typically 4-5 nm) and measures the KPFM signal. The cantilever was excited with an AC voltage to perform KPFM measurements, with an amplitude of 1.5-4 V and a frequency of 2-4 kHz. In the closed-loop measurements, the DC voltage was controlled by a bias servo to obtain the surface potential. Images were acquired using the Park SmartScan software and the data analyzed with Gwyddion program.
Finite Ferroelectric-Like Coupling of Two Active Interfaces
[0126] While the internal electric fields are mostly confined to the interfacial volume, as discussed in the main text, we find indications of finite coupling between adjacent regions of two active interfaces structures. This is achieved by comparing the average area coverage of co-aligned (ABC/CBA) and anti-aligned (ABA) domains (FIGS. 7E-F). The higher adhesion phase naturally expands on the expense of other stacking configurations. Indeed, we find that regions in the map of large area domains show a clear preference for ABC or CBA stackings with or (bright or dark) polarization, respectively, at the expense of the anti-aligned ABA and BAB domains (with neutral-color) of or polarization, respectively. A close look at two active interface regions with smaller domains (outside the marks) also shows a reduced area of the neutral domains even away from the physical edge of the layers (although the area difference here is minor). Recently, we reported a similar behavior in a single active interface system of parallel h-BN bilayers, where domain wall sliding in response to an externally applied electric field promoted larger domains that align with the external field at the expense of the anti-aligned configuration (see also FIGS. 4A-C). The dynamics of this phenomenon is governed by the loss of adhesion energy in the domain wall network and the pinning from the disorder at the interface. The internal out-of-plane coupling reported here (with no external field) reveals a more stable ferroelectric coupling (ABCABC . . . ) in comparison to the antiferroelectric order in the Bernal (ABAB . . . ) configuration.
Doping and De-Polarization Measurements
[0127] A precise measurement of the out-of-plane polarization at the high doping limit is challenging due to the KPFM signal sensitivity to long-range coulomb forces. The latter interacts with the tip's cantilever and cone rather than its local apex only. While the side-band measurement mode overcomes this challenge to provide quantitative information at zero gate bias, its reliability drops as the external potential on the gate electrode and correspondingly the doping charge density on the TMD increase. Crucially, this measurement limitation can only underestimate (by averaging out) the local potential drop, V.sub.KP, between domains and its corresponding polarization magnitude. To minimize this underestimation, we used two complementary gating schemes, where either the sample potential is grounded, and the gold electrode is biased or vice versa (FIGS. 7B and 7A a respectively). Data shown in the main text and in the configuration of FIG. 7A. We also focused on domains located next to the electrode's edges (while placing the cantilever outside them) and controlled the potential on the global silicon substrate independently. Additional limitations arise from the motion of domain walls at high charge doping and displacement fields, surface chemical adsorption, and surface degradation (see FIGS. 6A-C). The latter hinders our quantitative analysis even with a very thin gate dielectric (down to 5 nm thick), where the maximum doping level (at the h-BN breakdown electric field) is reached at moderate gate potentials.
[0128] Lastly, localized defect states at the host crystals may reduce the occupation of delocalized states by the gate bias. This may result in some overestimate of the precise doping, if extracted from the geometric capacitance only. To eliminate this overestimation, we extracted the doping density in FIG. 5B from the change in the average potential (V.sub.avg), measured on oppositely polarized domains (as marked in FIG. 3C), rather than directly from the applied V.sub.g. V.sub.avg grows with V.sub.g beyond some threshold value, and in one direction only for each particular sample (FIG. 6D). We attribute this behavior to unpinning of the Fermi-level from gap states associated with native dopants in each sample. The latter seems to prevent achieving electron and hole doping in the same sample. Importantly, V.sub.avg is only sensitive to the mobile charge density that accumulates to screen the bottom electrode, regardless of internal properties of the electrodes such as localized defect states, Schottky barriers, or quantum capacitance. The deviation from the ideal V.sub.avg=V.sub.g (dashed black) slope at high |V.sub.avg| is attributed to the underestimation of the local KPFM signal in case of large doping levels and spatially alternating potentials at the edges of the electrodes, as discussed above. Altogether, the measurements in FIG. 5D provide an underestimation of the polarization magnitude and the mobile charge density.
Calculation of WSe.SUB.2 .Polarization
[0129] The electrostatic potential profile along the normal direction of AB stacked few-layered WSe.sub.2 is presented in FIG. 3. To obtain this profile, the Perdew-Burke-Ernzerhof (PBE) generalized-gradient exchange-correlation density functional approximation was used, augmented by the Grimme-D3 dispersion correction with Becke-Johnson (BJ) damping as implemented in the Vienna Ab-initio Simulation Package (VASP). The core electrons of the W and Se atoms were treated via the projector augmented wave (PAW) approach. Spin-orbit interactions were included. This level of theory was recently successfully used to calculate the polarization of transition metal dichalcogenide (TMD) bilayers.
[0130] An AB stacked WSe.sub.2 bilayer, constructed from two relaxed monolayers, was allowed to relax, yielding a lattice constant of 3.29 and an interlayer distance (defined as the normal distance between adjacent Se ions of the two layers) of 3.10 . Single-point electron density calculations were then performed on the relaxed structure with a plane wave energy cutoff of 600 eV and a k-point mesh of 12121, setting a vertical vacuum size of 10 nm to avoid interactions between adjacent bilayer images. To evaluate the vertical polarization, a dipole moment correction was employed. The potential profile along the normal direction of bilayer WSe.sub.2 is plotted in FIG. 8. The resulting difference between the electrostatic potential values obtained above the top and below the bottom surfaces is 69 meV, which defines the vertical polarization of the system.
[0131] Convergence tests of the VASP calculations (see FIG. 9) with respect to the vacuum size, energy cut-off, and number of k-points indicate that our choice of parameters leads to binding energies that are converged to within 0.006, 0.002, and 0.007 meV/atom, respectively. Correspondingly, the electrostatic potential difference converges to within 0.01, 0.05, and 0.04 meV with respect to the vacuum size, energy cut-off, and number of reciprocal space k-points, respectively.
[0132] In FIG. 3E, further explored was the thickness dependence of the system's polarization. To this end, multilayer systems were constructed including the AB stacked bilayer, atop of which a few AB or AA stacked WSe.sub.2 layers were added. Following optimization, single point potential profile calculations were performed as detailed above.
Calculation of Doping-Induced Depolarization in WSe.SUB.2 .and MoS.SUB.2
[0133] Doping calculations of bilayer WSe.sub.2 and MoS.sub.2 were performed using the fractional nuclear charge pseudoatom approach, allowing for simulating doping densities in the experimentally relevant range. To this end, we use pseudopotentials (PPs) generated for atoms with fractional nuclear charge. These calculations were performed using the open source package Quantum Espresso, instead of VASP that was used herein, allowing us to construct appropriate PPs. We first generated Rappe-Rabe-Kaxiras-Joannopoulos (RRKJ) PPs, including spin-orbit interactions, using the ld1.x program of the plane-wave pseudopotential Quantum Espresso package. The nuclear charge of the pseudoatom was set to the original charge of the neutral element plus a small fractional charge . For example, the nuclear charge of a doped pseudo W atom was set to Z=74. The valence electronic charge was changed accordingly to maintain neutrality of the unit-cell, with an electron configuration given by [Xe]4f.sup.146s.sup.26p.sup.05d.sup.4. A set of PPs were generated by setting =10.sup.9, 10.sup.8, . . . , 10.sup.2 for all W atoms in the bilayer system, corresponding to doping densities of n.sub.2D=2.110.sup.7, 2.110.sup.8, . . . , 2.110.sup.13 cm.sup.2, respectively. A similar procedure was used to generate MoS.sub.2 PPs with fractional nuclear charge and valence charge. For example, for a pseudo Mo nuclear charge of Z=42, the electron configuration was set to [Kr]5s.sup.25p.sup.04d.sup.4.
[0134] Single point calculations were performed using the generated PPs to obtain the electron density and the corresponding electrostatic potential profiles. To this end, we employed the PBE generalized-gradient density functional approximation and the Grimme-D3 dispersion correction with BJ damping, as implemented in Quantum Espresso. A plane wave energy cutoff of 60 Ry (816.34 eV) was used with a k-mesh of 12121, and a vertical vacuum size of 10 nm was set to avoid interactions between adjacent bilayer images. Fermi-Dirac smearing was used to enhance the convergence of the self-consistent cycle. To obtain the electrostatic potential profiles, a dipole moment correction was used.
[0135] As in the procedure discussed herein, AB-stacked WSe.sub.2 and MoS.sub.2 bilayers were first constructed and optimized, yielding lattice constants of 3.29 and 3.16 and interlayer distances of 3.05 and 2.95 , respectively. The resulting electrostatic potential drops were 71 meV and 76 meV for the undoped WSe.sub.2 and MoS.sub.2 bilayers, respectively. Note that little difference (2 and 6 meV for WSe.sub.2 and MoS.sub.2, respectively) was found between the potential drops calculated by VASP in section S4 and those obtained using Quantum Espresso. The potential and charge density profiles along the vertical direction for the two bilayers are shown in FIG. 10.
[0136] Convergence tests for the Quantum Espresso calculations (see FIG. 11) with respect to the vacuum size, energy cut-off, and number of k-points indicate that our choice of parameters leads to WSe.sub.2 (MoS.sub.2) binding energies that are converged to within 0.0003 (0.0003), 0.004 (0.003), and 0.003 (0.0004) meV/atom, respectively. Correspondingly, the electrostatic potential difference converges to within 2.6 (0.9), 3.8 (1.1), and 3.6 (2.4) meV with respect to the vacuum size, energy cut-off, and number of reciprocal space k-points, respectively.
[0137] Doping of the WSe.sub.2 and MoS.sub.2 bilayers was performed by charging the metal nuclei. As discussed in the main text, up to a system dependent hole or electron charge density, the polarization remains mostly unaffected, following which a polarization drop is clearly seen (see FIG. 12). We note that the fractional nuclear charge pseudoatom doping approach adopted in this study remains valid as long as variations in the calculated band-structure, induced by the nuclear pseudo charging, are negligible. To confirm that our calculations satisfy this condition, we compare the bandstructures of the undoped and doped WSe.sub.2 (FIG. 13A) and MoS.sub.2 (FIG. 13D) bilayers up to the highest doping density considered. Our results clearly demonstrate merely minor deviations of the band-structures of the doped systems from those of the undoped counterparts. The energy difference between the topmost K and valence band points for the doped and undoped systems is presented in FIGS. 13B and 13E for WSe.sub.2 and MoS.sub.2, respectively. Larger energy differences at higher doping levels result from the depolarization shown in FIG. 11. As an additional validity test, the doping-induced WSe.sub.2 and MoS.sub.2 Fermi level shifts are presented in FIGS. 13C and 13F, respectively, exhibiting the expected logarithmic dependence up to doping densities of 110.sup.13 cm.sup.2.
[0138] To demonstrate that our conclusions are independent on the choice of doping only via the metal atoms, we repeated the calculations by doping only via the chalcogen nuclei or doping all nuclei (see FIG. 14). Consistent results are obtained regardless of the doping scheme.
Effect of Polarization on the Band-Structure
[0139] To evaluate the effect of the emerging polarization on the band structure, we compare in FIG. 15 the band structure of the anti-parallel AA stacked undoped bilayers with those of the parallel AB stacked counterparts, all evaluated at the same level of theory as described herein. The results clearly demonstrate band splitting of both the conduction and the valence bands at the K point. Notably, this splitting is of the order of the calculated vertical potential drops, indicating that the emerging polarization is indeed causing the splitting.
