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Justin Boddison-Chouinard, Alex Bogan, Pedro Barrios, Jean Lapointe, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Jarosław Pawłowski, Daniel Miravet, Maciej Bieniek, Pawel Hawrylak, Adina Luican-Mayer, Louis Gaudreau

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[Anomalous conductance quantization of a one-dimensional channel in monolayer WSe2](https://mdr.nims.go.jp/datasets/6c4cd9f5-0207-4824-8588-31306be8bb4d)

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Anomalous conductance quantization of a one-dimensional channel in monolayer WSe2ARTICLE OPENAnomalous conductance quantization of a one-dimensionalchannel in monolayer WSe2Justin Boddison-Chouinard1,2, Alex Bogan2, Pedro Barrios3, Jean Lapointe3, Kenji Watanabe 4, Takashi Taniguchi 5,Jarosław Pawłowski 6, Daniel Miravet 1, Maciej Bieniek 6,7, Pawel Hawrylak1, Adina Luican-Mayer 1✉ and Louis Gaudreau2✉Among quantum devices based on 2D materials, gate-defined quantum confined 1D channels are much less explored, especially inthe high-mobility regime where many-body interactions play an important role. We present the results of measurements andtheory of conductance quantization in a gate-defined one-dimensional channel in a single layer of transition metal dichalcogenidematerial WSe2. In the quasi-ballistic regime of our high-mobility sample, we report conductance quantization steps in units of e2/hfor a wide range of carrier concentrations. Magnetic field measurements show that as the field is raised, higher conductanceplateaus move to accurate quantized values and then shift to lower conductance values while the e2/h plateau remains locked.Based on microscopic atomistic tight-binding theory, we show that in this material, valley and spin degeneracies result in 2 e2/hconductance steps for noninteracting holes, suggesting that symmetry-breaking mechanisms such as valley polarization dominatethe transport properties of such quantum structures.npj 2D Materials and Applications            (2023) 7:50 ; https://doi.org/10.1038/s41699-023-00407-yINTRODUCTIONRealization of quantum devices based on two-dimensional (2D)materials has attracted significant interest in recent years1,2; inparticular, advances in fabrication techniques for devices based ontransition metal dichalcogenides (TMDs) enabled the realization ofbuilding blocks of quantum circuits such as gate-controlledquantum dots in monolayer and few-layer MoS23–6 and WSe27,8as well as one-dimensional (1D) channels based on split gatetechnology9–13. 1D channels are of great interest in quantuminformation science because they have been established asvaluable tools for noninvasive readout of semiconducting chargeand spin qubits in GaAs14, SiGe15, graphene16–19, bilayergraphene20,21, and WSe222. In 1D channels, the Landauer–Buttikerformalism explains the quantized conductance in units of n e2/h,where n is the number of available transport channels, whichdepends on the degeneracies of the system; for example, twofoldspin degeneracy for GaAs23 and fourfold spin and valleydegeneracy for graphene24. Beyond this picture, much less isknown about the mechanisms through which interaction effectscan add complexity and play a role in transport anomalies23.Therefore, 1D channels based on high-mobility 2D materials offerthe possibility to access interaction regimes.Here, we present results of investigation of a 1D channel basedon high-mobility monolayer WSe2, and find that the conductanceis quantized in units of e2/h. This is surprising since in monolayerTMDs, due to spin-valley locking, we expect conductancequantization for noninteracting holes in units of 2 e2/h. Ourresults are in agreement with reports using few-layer MoS29,11,trilayer WSe213 and monolayer MoS210, but the origin of the e2/hquantization remains unexplained. We attribute this broken valleyand spin degeneracy to the formation of valley and spin-polarizedstates of holes, which have been predicted for WS225,26 and inlaterally gated MoS2 quantum dots27,28.Although there is already a substantial body of work explainingnon-universal conductance quantization, a full understanding ofthe so-called “0.7 anomaly” in quantum point contacts remainselusive and is assumed to be due to electron–electron interac-tions29–38. While the full theory of conductance in the presence ofhole–hole interactions is in progress, here we show a completesingle-particle model based on atomistic tight-binding theory forholes in a WSe2 monolayer, confined in an electrostatically defined1D channel. It demonstrates that a channel potential does notbreak valley degeneracy and without hole–hole interactionsconductance is expected to be quantized in units of 2 e2/h.Hence the observed anomalous quantization in units of e2/h canlikely be explained in terms of broken symmetry valley-polarizedground state induced by interactions25,26,39.RESULTSDevice structureUsing standard dry transfer methods40,41, a van der Waalsheterostructure consisting of a monolayer WSe2 flake encapsu-lated between two hexagonal boron nitride (hBN) flakes wasassembled on a p-doped silicon substrate with 285 nm ofthermally grown silicon dioxide (SiO2). The silicon substrate wasutilized as a back gate to introduce carriers in the WSe2 layer.Electron beam lithography was used to define electrical contacts[Cr (2 nm)/Pt (8 nm)]42, which contact the WSe2 from the bottom,and to pattern a 4-component top gate. Two of the top gates,labeled as VCG in Fig. 1a, cover the area where there is overlapbetween the WSe2 flake and the electrical contacts, and are usedto activate the electrical contacts. The two other top gates, labeled1Department of Physics, University of Ottawa, Ottawa, ON K1N 9A7, Canada. 2Emerging Technologies Division, National Research Council of Canada, Ottawa, ON K1A 0R6,Canada. 3Advanced Electronics and Photonics, National Research Council of Canada, Ottawa, ON K1A 0R6, Canada. 4Research Center for Functional Materials, National Institutefor Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 5International Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba305-0044, Japan. 6Institute of Theoretical Physics, Wrocław University of Science and Technology, Wrocław, Poland. 7Institute of Theoretical Physics, WürzburgUniversity, Würzburg, Germany. ✉email: luican-mayer@uottawa.ca; louis.gaudreau@nrc-cnrc.gc.cawww.nature.com/npj2dmaterialsPublished in partnership with FCT NOVA with the support of E-MRS1234567890():,;http://crossmark.crossref.org/dialog/?doi=10.1038/s41699-023-00407-y&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41699-023-00407-y&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41699-023-00407-y&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41699-023-00407-y&domain=pdfhttp://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0003-3638-3966http://orcid.org/0000-0003-3638-3966http://orcid.org/0000-0003-3638-3966http://orcid.org/0000-0003-3638-3966http://orcid.org/0000-0003-3638-3966http://orcid.org/0000-0002-2908-4645http://orcid.org/0000-0002-2908-4645http://orcid.org/0000-0002-2908-4645http://orcid.org/0000-0002-2908-4645http://orcid.org/0000-0002-2908-4645http://orcid.org/0000-0003-4505-1998http://orcid.org/0000-0003-4505-1998http://orcid.org/0000-0003-4505-1998http://orcid.org/0000-0003-4505-1998http://orcid.org/0000-0003-4505-1998http://orcid.org/0000-0001-9537-4600http://orcid.org/0000-0001-9537-4600http://orcid.org/0000-0001-9537-4600http://orcid.org/0000-0001-9537-4600http://orcid.org/0000-0001-9537-4600https://doi.org/10.1038/s41699-023-00407-ymailto:luican-mayer@uottawa.camailto:louis.gaudreau@nrc-cnrc.gc.cawww.nature.com/npj2dmaterialsas VSG in Fig. 1a, were used to define the 1D channel in the WSe2with a lithographic width of 200 nm and length of 600 nm asconfirmed by a scanning electron micrograph (Fig. 1b, inset).Many important cleaning techniques were employed throughoutthe fabrication procedure and are detailed in “Methods”. Figure 1bshows an optical micrograph of the completed device.Electrical transport in a WSe2 1D channelWe first demonstrate that we can create a conduction region witha sufficiently low resistance to study quantized conductance at atemperature of 4 K. Figure 2a is a 4-point measurement of thedevice resistance RSD as a function of the split gate voltage VSG at aconstant source-drain current of 10 nA and constant back gatevoltage VBG=−33 V. At low VSG, when the voltages on the splitgate are not sufficiently high to deplete the underlying WSe2regions from carriers, we measure a resistance of 480Ω fromwhich we are able to extract a contact resistance of 9.76 kΩ (seeSupplementary Fig. 1), a value that is comparable to othersreported using more complex fabrication techniques13,43,44. As thevoltage is increased on the split gates, we observe an increase inresistance, indicating that a constriction has been formed and thatholes are responsible for transport in the device.The change in the conductance of the device in units of e2/h asa function of the split gate voltage VSG is shown in Fig. 2b, fromwhich we obtain a gate depletion value of VSG= 2.7 V. At thispoint, the 1D channel is formed, and the total measured resistanceis 5030Ω. When indicated, this back gate-dependent seriesFig. 1 Monolayer WSe2 device structure. a Schematic of the monolayer WSe2 device structure. b Optical micrograph of the device used inthis study. The WSe2 monolayer flake is outlined in red for clarity. Inset: Scanning electron micrograph of the top gate structure (scalebar is400 nm). The left and right gates are used to activate the contact region (VCG). The top and bottom gates (VSG) are used to form the 1Dchannel. The 1D channel is lithographically defined to have a width of 200 nm and a length of 600 nm.Fig. 2 Formation of a 1D channel and its dependence on the hole concentration. a 4-point measurement of resistance vs. split gate voltage(VSG) at VBG=− 33 V. The dashed line indicates the resistance where the channel is formed (at depletion). b Conductance in units of e2/h vs.split gate voltage (VSG) at VBG=− 33 V. The dashed line represents the conductance where the channel is formed (at depletion). c Color mapof conductance as a function of split gate voltage VSG and back gate voltage VBG. A lead resistance (resistance at depletion) is subtracted.d Line cuts from (c) plotted in a “waterfall” style indicating a first step at e2/h and a second step around 2 e2/h. The leftmost trace is taken atVBG=− 29 V and the right most trace is taken at VBG=− 35 V. The green and magenta lines are highlighted by similarly colored dashed linesin (c) and correspond to back gate voltages of −30.5 V and −33 V, respectively. These two curves feature the regimes where we observe the1 e2/h plateau (green), and the 1 e2/h and 2 e2/h plateaus (magenta).J. Boddison-Chouinard et al.2npj 2D Materials and Applications (2023)    50 Published in partnership with FCT NOVA with the support of E-MRS1234567890():,;resistance (see Supplementary Fig. 3) is subtracted from the rawdata to obtain the conductance related only to the 1D channel10.In Fig. 2b, quantized conductance steps close to e2/h and 2 e2/hcan be observed for a fixed hole concentration of 1.82 × 1012 cm−2(see Supplementary Fig. 2). Tuning the concentration allows us toreach two different regimes, as demonstrated in Fig. 2c. At higherhole concentrations, below VBG=−30 V, we see the appearanceof the first conductance step at e2/h. As the carrier concentrationincreases (VBG decreases), the difference between the depletionand pinch-off values also increases and we observe theappearance of a second conductance step at a value of 2 e2/h.These conductance steps are more clearly observed in a waterfallplot as shown in Fig. 2d. The observation of quantizedconductance through the 600 nm long channel along with thehigh measured field-effect carrier mobility ofμFE ≈ 8000–8500 cm2 V−1 s−1(see refs. 13,42,44,45) (see Supplemen-tary Fig. 2), hints toward quasi-ballistic transport and demon-strates that a high-quality monolayer WSe2 sample has beenachieved. At lower carrier concentrations, we observe the onset ofa distinct transport regime that remains to be elucidated byfurther investigations (see Supplementary Fig. 7).Magnetic field dependence of the 1D channelWe further investigate the quantized conductance features byapplying a magnetic field perpendicular to the plane of the two-dimensional material. Due to the large hole spin and valleyg-factor which has been reported to be up to 12 in monolayerWSe246, we explore the low magnetic field regime between −100and 100mT (Fig. 3a) to eliminate any possible spin or valleypolarization occurring due to a small magnetic field offset at 0 T.The quantization remains constant at low field even at a highresolution of 4 mT. We therefore conclude that the measuredlifting of spin and valley degeneracies at low fields is inherent tothis particular system leading to a ground state where only oneconducting channel is available. At higher fields, as depicted inFig. 3b, we observe that the first plateau remains constantbetween −5 and 8 T with a very small correction around 0 T whichwe attribute to magnetoresistance effects in the leads. Thismagnetoresistance correction is more significant at higherconductance, but after 1.5 T we observe the second and thirdquantization plateaus aligning with the values of 2 e2/h and 3 e2/h.This data set demonstrates that each conductance channel up tothe third one allows transport for only one quantum ofconductance. We cannot determine, however, what is the groundstate in terms of spin or valley. In addition, at higher magneticfields, we observe an evolution of the second and third plateaus inwhich they lower in conductance and seem to merge into the firstplateau. No such change in conductance is observed for the firstplateau. Measurements performed at 10 mK in a dilutionrefrigerator show the same behavior (see Supplementary Fig. 4).TheoryIn order to determine whether the anomalous quantizationbehavior originates from the spin-valley locking mechanism inTMDs, we developed a single-particle model of a 1D channel. Tocalculate single-hole states, we use a tight-binding model forWSe2 monolayer27,47–49 in a basis of three d-orbitals localized ontungsten atoms and three p-like orbitals describing Se2 dimers (6even orbitals in total). A single-particle channel wavefunction for ahole state s satisfies the Schrödinger equation48,49:Hbulk þ jejϕð Þ Ψsj i ¼ Es Ψsj i; (1)where the 1D electronic confinement defined within the WSe2monolayer lattice is determined by an applied gate-definedpotential ϕ.To obtain the channel potential landscape, we performed self-consistent Poisson–Schrödinger calculations50,51 using parameterscorresponding to the device used. The model includes the sameFig. 3 Out-of-plane magnetic field B⊥ dependence of the 1D channel. a Dependence of the quantized conductance plateaus on aperpendicular low magnetic field ranging from −100 to 100mT and a resolution of 4mT. b Dependence of the quantized conductanceplateaus on a perpendicular magnetic field ranging from −5 to 8 T. In both panels, a horizontal offset is introduced between adjacent linetraces for clarity. VBG=− 33.2 V for all runs with the corresponding lead resistance of RL= 5029Ω subtracted.J. Boddison-Chouinard et al.3Published in partnership with FCT NOVA with the support of E-MRS npj 2D Materials and Applications (2023)    50 arrangement of layers represented by different dielectric con-stants and the same gate layout with the corresponding voltagesapplied. Further details can be found in the SupplementaryInformation. The obtained potential landscape results in a channeloriented along the x axis, and is presented together with anelongated rhombohedral tight-binding computational box thatcovers the channel area in Fig. 4a. A hole is free to move in thechannel direction, while movement in the perpendicular directionis constrained by a Gaussian-like potential along the y axis. Tosimplify further calculations, we fit the channel profile atx= 500 nm with a Gaussian function (Uch ¼ U0ð1� expð� y22σ2ÞÞ)resulting in a potential depth U0= 1600 mV and width σ= 65 nm.In the following calculations, we will use Uch as a realisticapproximation of the ϕ potential.We define the finite tight-binding computational box in a formof a rhomboid of the WSe2 lattice. We wrap the computational boxon a torus, apply the periodic (Born-von Karman) boundaryconditions, and obtain a set of allowed, discretized k-vectors overwhich we diagonalize the bulk Hamiltonian Hbulk. The valenceband (VB) wavefunction for each allowed k-vector is a linearcombination of Bloch functions on the W and Se2 dimersublattices l(l= 1,…, 6):ψVBkσ�� � ¼ X6l¼1AVBkσl ψk;l�� �� χσj i; (2)where χσj i is the spinor part of the wavefunction, andψk;l�� � ¼ 1ffiffiffiffiNpXNRl¼1eik�Rlφlðr� RlÞ (3)are Bloch functions built with atomic orbitals φl. N is the numberof unit cells, while Rl defines the position of atomic orbitals in thecomputational rhomboid. By diagonalising the 6 by 6 HamiltonianHbulk at each allowed value of k, we obtain the bulk energy bandsEVBkσ and wavefunctions AVBkσl . The topmost valence band energysurface is shown in Fig. 4b, where two non-equivalent maxima, Kand K 0 valleys, are clearly visible.In the next step, we expand the channel wavefunction in termsof the lowest energy valence band states given by Eq. (2):Ψsσ�� � ¼ XkBVB;skσ ψVBkσ�� �: (4)Finally, we solve the Schrödinger Eq. (1) with the potential termUch by converting it to an integral equation for the coefficientsBVB;skσ , and obtain the single-hole eigenstates confined within thechannel,EVBqσBVB;sqσ þXkBVB;skσXlAVBqσl� ��AVBkσl jejUchl ðq; kÞ ¼ EsσBVB;sqσ ; (5)where Uchl ðq; kÞ ¼ 1NPNRl¼1 UchðRlÞeiðk�qÞ�Rl is the Fourier transformof the channel confinement on each sublattice l. The structuraland the tight-binding Slater–Koster parameters in the calculationsFig. 4 Single-particle model of the 1D channel. a Potential profile from the Poisson–Schrödinger calculations. Within the WSe2 monolayer(represented by computational rhombus -- dashed line), this profile forms a 1D gate-defined channel potential applied along the x axis.b Valence band energy surface within the Brillouin zone with two maxima at K 0 and K points. c, d Single-particle states calculated within thetight-binding model for WSe2 monolayer rhombus with applied 1D gate-defined channel potential. The hole eigenstates, characterized bytheir energy, wave vector kx, and spin, form characteristic parabolic subbands located in (c) K 0 valley with spin-up (red dots), or (d) K valley withspin-down (blue dots). Subsequent subbands are used to calculate the channel conductance (e) via the Landauer formula. Resultingconductance curve has a characteristic stepped-like shape with subsequent plateaus at multiples of 2 e2/h. Those states (numbered by squantum number) are occupied by hole states with spatial densities (f) resembling harmonic oscillator modes.J. Boddison-Chouinard et al.4npj 2D Materials and Applications (2023)    50 Published in partnership with FCT NOVA with the support of E-MRSare directly listed in the “Methods” section under tight-bindingparameters.The channel states of energy Esσ are characterized by subbandindex s, by the spin σ, and also by the valley K or K 0 determined bythe expectation value of the wave vector 〈kx〉≡ kx in the channeldirection. They belong to two different valleys: K 0 with spin-up—Fig. 4c, and K with spin-down—Fig. 4d, creating degenerate spin-valley locked states. Characteristic parabolic subbands are formeddue to the lateral quantization within the channel represented bydifferent values of s quantum number.We note that the VB maximum (for the free hole HamiltonianHbulk) is shifted to zero and the highest energy state (ground statefor holes in the channel) is located below, at −12meV. The secondspin-valley pair (not presented) is split-off by 0.5 eV below on theenergy scale by the strong intrinsic spin-orbit coupling present inthis material.After calculating the single-particle eigenstates of the Hamilto-nian H, we are ready to numerically estimate the conductancethrough the 1D channel. For each subband s, we calculate thedensity of states within the given subband dNsdE , and then estimatethe particle velocity as vsðEÞ ¼ 1_∂Es∂kx. The conductance, whichcombines contributions from states from different occupiedsubbands s filled up to the Fermi energy, is calculated using the2-terminal Landauer formula52:GðEFÞ ¼ e2EFXsZ EF0dE12dNsdEvsðEÞ: (6)The factor 1/2 is due to the fact that we take only states withkx > Kx (or kx > K 0x for the second valley), i.e., active under the givensource-drain bias. In the above formula, we assume that nobackscattering occurs. The calculated conductance is presented inFig. 4e. It has the conductance plateaus at multiples of 2 e2/h. Thecorresponding charge densities for each subband are plotted inFig. 4f. The final result shows that a microscopic theory of a 1Dchannel predicts conductance steps in units of 2 e2/h. Hence, wesee that the measured e2/h conductance quantization cannot beexplained in the single-particle picture.DISCUSSIONA possible explanation for the e2/h conductance quantization canbe obtained in terms of a broken symmetry valley-polarizedground state25,26,39. A valley-polarized state is expected to be theground state of a 2D hole gas in a single layer of WSe2 forsufficiently strong interaction strength and hole density (inanalogy to WS226). The breaking of valley degeneracy naturallylifts the degeneracy of bands in the mean-field picture. Therefore,conductance plateaus in conductance quantization are expectedto be half of the degenerate system, giving e2/h instead of 2 e2/hsteps as a function of Fermi energy. This might explain the e2/hplateau for a large range of hole densities, as shown in Fig. 2d.In summary, we have fabricated high-quality monolayer WSe2devices where an electrostatically confined 1D hole transportchannel is formed. At a temperature of 4 K, hole quasi-ballistictransport is observed, revealing an unexpected conductancequantization in steps of e2/h instead of 2 e2/h taking into accountthe electronic effects from the band structure of TMDs such asspin-valley locking. We have compared the experimental results toa single-particle atomistic tight-binding model for holes in WSe2with realistic channel profile and demonstrated that a 1Dconfining potential does not reproduce the measured conduc-tance quantization in units of e2/h. Recent experiments25 andtheory27 suggest the existence of a valley-polarized state of thehole gas in TMDs. Such a state would explain the measuredplateau at e2/h. A theoretical model which includes hole–holecorrelations is therefore necessary to explain quantum transport in2D TMDs. These results show that the electronic properties ofmonolayer TMDs are still insufficiently understood and requirefurther research to elucidate.METHODSDevice fabricationThe device was assembled on a p-doped silicon substrate with285 nm of thermally grown silicon dioxide (SiO2). Using standarddry transfer methods40,41, a flake of hexagonal boron nitride (hBN)(26 nm) was picked-up with a polypropylene carbonate (PPC)coated polydimethylsiloxane (PDMS) stamp and transferred ontolithographically patterned local gates. Electrical contacts [Cr(2 nm)/Pt (8 nm)]42 were subsequently patterned on top of thehBN. To remove any contaminants on the surface of the hBN andthe electrical contacts, the sample was thermally annealed in avacuum furnace (10−7 Torr) at 300 °C for 30min and furthercleaned mechanically using an atomic force microscope tip (AFM)in contact mode8,22,53,54. A second polymer stamp was used tosubsequently pick-up an hBN flake (34 nm) then a monolayerWSe2 flake. To ensure proper contact to the electrical contacts, thestack of flakes and stamp were placed in an AFM such that thesurface of the WSe2 flake was exposed where it was thenmechanically cleaned using an AFM tip. The hBN/WSe2 stack wasthen dropped off onto the patterned contacts. At this stage, thedevice was thermally annealed in a vacuum furnace following thesame recipe as before. A final lithographic step was performed topattern the top gates which include the two contact top gates(VCG) and the split gates (VSG).Tight-binding parametersIn the tight-binding atomistic calculations, we chose the followingstructural parameters: d∥= 1.9188Å (W–Se2 dimer center dis-tance), d⊥= 1.6792Å (Se atom distance from z= 0 plane). Thetight-binding Slater–Koster parameters are given by (all in eV):Ed=−0.4330, Ep0=−3.8219, Ep±1=−2.3760, Vdpσ=−1.58193,Vdpπ= 1.17505, Vddσ=−0.90501, Vddπ= 1.0823, Vddδ=−0.1056,Vppσ= 0.52091, Vppπ=−0.16775, together with characterizingintrinsic spin-orbit couplings: λW ¼ 0:275; λSe2 ¼ 0:08. We notethat these parameters are chosen to reproduce ab initio electronicband structure of two spinful valence bands.DATA AVAILABILITYThe data that support the findings of this study are available from the correspondingauthor upon reasonable request.Received: 4 October 2022; Accepted: 15 June 2023;REFERENCES1. Liu, X. & Hersam, M. C. 2D materials for quantum information science. Nat. Rev.Mater. 4, 669–684 (2019).2. Alfieri, A., Anantharaman, S. B., Zhang, H. & Jariwala, D. Nanomaterials forquantum information science and engineering. Adv. Mater. 35, 2109621 (2022).3. Zhang, Z.-Z. et al. Electrotunable artificial molecules based on van der Waalsheterostructures. Sci. Adv. 3, e1701699 (2017).4. Pisoni, R. et al. Gate-tunable quantum dot in a high quality single layer MoS2 vander Waals heterostructure. Appl. Phys. Lett. 112, 123101 (2018).5. Wang, K. et al. Electrical control of charged carriers and excitons in atomically thinmaterials. Nat. Nanotechnol. 13, 128–132 (2018).6. 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M.B. acknowledges financial support from the PolishNational Agency for Academic Exchange (NAWA), Poland, grant PPI/APM/2019/1/00085/U/00001. This research was enabled in part by support provided by the DigitalResearch Alliance of Canada (alliancecan.ca). This research was supported in part byPL-Grid Infrastructure.AUTHOR CONTRIBUTIONSJ.B.-C., A.B., and L.G. designed the device architecture. J.B.-C., P.B., and J.L. fabricatedthe top gate structure, while J.B.-C. fabricated the remainder of the device with inputsfrom A.L.-M. and L.G. J.B.-C. performed the experimental measurements and the dataanalysis with consultations from A.B., A.L.-M., and L.G. Theoretical models andcalculations were completed by J.P., D.M., M.B., and P.H. K.W. and T.T. grew thehexagonal boron nitride crystals. All authors participated in the writing of themanuscript.COMPETING INTERESTSThe authors declare no competing interests.ADDITIONAL INFORMATIONSupplementary information The online version contains supplementary materialavailable at https://doi.org/10.1038/s41699-023-00407-y.Correspondence and requests for materials should be addressed to Adina Luican-Mayer or Louis Gaudreau.Reprints and permission information is available at http://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard to jurisdictional claimsin published maps and institutional affiliations.J. Boddison-Chouinard et al.6npj 2D Materials and Applications (2023)    50 Published in partnership with FCT NOVA with the support of E-MRShttps://doi.org/10.1038/s41699-023-00407-yhttp://www.nature.com/reprintshttp://www.nature.com/reprintsOpen Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in anymedium or format, as long as you giveappropriate credit to the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made. The images or other third partymaterial in this article are included in the article’s Creative Commons license, unlessindicated otherwise in a credit line to the material. If material is not included in thearticle’s Creative Commons license and your intended use is not permitted by statutoryregulation or exceeds the permitted use, you will need to obtain permission directlyfrom the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2023J. Boddison-Chouinard et al.7Published in partnership with FCT NOVA with the support of E-MRS npj 2D Materials and Applications (2023)    50 http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Anomalous conductance quantization of a one-dimensional channel in monolayer WSe2 Introduction Results Device structure Electrical transport in a WSe2 1D channel Magnetic field dependence of the 1D channel Theory Discussion Methods Device fabrication Tight-binding parameters DATA AVAILABILITY References Acknowledgements Author contributions Competing interests ADDITIONAL INFORMATION