Anomalous conductance quantization of a one-dimensional channel in monolayer WSe2

Among quantum devices based on 2D materials, gate-defined quantum confined 1D channels are much less explored, especially in the high mobility regime where many body interactions play an important role. We present results of measurements and theory of conductance quantization in a gate-defined one dimensional channel in a single layer of transition metal dichalcogenide material WSe 2 . In the quasi-ballistic regime of our high mobility sample, we report conductance quantization steps in units of e 2 /h for a wide range of carrier concentrations. Magnetic field measurements show that as the field is raised higher conductance plateaus move to accurate quantized values and then shift to lower conductance values while the e 2 /h plateau remains locked. Based on microscopic atomistic tight-binding theory, we show that in this material, valley and spin degeneracies result in 2 e 2 /h conductance steps for non-interacting holes, suggesting that symmetry breaking mechanisms such as valley polarization dominate the transport properties of such quantum structures.


INTRODUCTION
Realization of quantum devices based on two-dimensional (2D) materials has attracted significant interest in recent years 1,2 ; in particular, advances in fabrication techniques for devices based on transition metal dichalcogenides (TMDs) enabled the realization of building blocks of quantum circuits such as gate-controlled quantum dots in monolayer and few-layer MoS 2 3-6 and WSe 2 7,8 as well as one-dimensional (1D) channels based on split gate technology [9][10][11][12][13] . 1D channels are of great interest in quantum information science because they have been established as valuable tools for noninvasive readout of semiconducting charge and spin qubits in GaAs 14 , SiGe 15 , graphene [16][17][18][19] , bilayer graphene 20,21 , and WSe 2 22 . In 1D channels, the Landauer-Buttiker formalism explains the quantized conductance in units of n e 2 /h, where n is the number of available transport channels, which depends on the degeneracies of the system; for example, twofold spin degeneracy for GaAs 23 and fourfold spin and valley degeneracy for graphene 24 . Beyond this picture, much less is known about the mechanisms through which interaction effects can add complexity and play a role in transport anomalies 23 . Therefore, 1D channels based on high-mobility 2D materials offer the possibility to access interaction regimes.
Here, we present results of investigation of a 1D channel based on high-mobility monolayer WSe 2 , and find that the conductance is quantized in units of e 2 /h. This is surprising since in monolayer TMDs, due to spin-valley locking, we expect conductance quantization for noninteracting holes in units of 2 e 2 /h. Our results are in agreement with reports using few-layer MoS 2 9,11 , trilayer WSe 2 13 and monolayer MoS 2 10 , but the origin of the e 2 /h quantization remains unexplained. We attribute this broken valley and spin degeneracy to the formation of valley and spin-polarized states of holes, which have been predicted for WS 2 25,26 and in laterally gated MoS 2 quantum dots 27,28 .
Although there is already a substantial body of work explaining non-universal conductance quantization, a full understanding of the so-called "0.7 anomaly" in quantum point contacts remains elusive and is assumed to be due to electron-electron interactions [29][30][31][32][33][34][35][36][37][38] . While the full theory of conductance in the presence of hole-hole interactions is in progress, here we show a complete single-particle model based on atomistic tight-binding theory for holes in a WSe 2 monolayer, confined in an electrostatically defined 1D channel. It demonstrates that a channel potential does not break valley degeneracy and without hole-hole interactions conductance is expected to be quantized in units of 2 e 2 /h. Hence the observed anomalous quantization in units of e 2 /h can likely be explained in terms of broken symmetry valley-polarized ground state induced by interactions 25,26,39 .

Device structure
Using standard dry transfer methods 40,41 , a van der Waals heterostructure consisting of a monolayer WSe 2 flake encapsulated between two hexagonal boron nitride (hBN) flakes was assembled on a p-doped silicon substrate with 285 nm of thermally grown silicon dioxide (SiO 2 ). The silicon substrate was utilized as a back gate to introduce carriers in the WSe 2 layer. Electron beam lithography was used to define electrical contacts [Cr (2 nm)/Pt (8 nm)] 42 , which contact the WSe 2 from the bottom, and to pattern a 4-component top gate. Two of the top gates, labeled as V CG in Fig. 1a, cover the area where there is overlap between the WSe 2 flake and the electrical contacts, and are used to activate the electrical contacts. The two other top gates, labeled 1 as V SG in Fig. 1a, were used to define the 1D channel in the WSe 2 with a lithographic width of 200 nm and length of 600 nm as confirmed by a scanning electron micrograph (Fig. 1b, inset). Many important cleaning techniques were employed throughout the fabrication procedure and are detailed in "Methods". Figure 1b shows an optical micrograph of the completed device.
Electrical transport in a WSe 2 1D channel We first demonstrate that we can create a conduction region with a sufficiently low resistance to study quantized conductance at a temperature of 4 K. Figure 2a is a 4-point measurement of the device resistance R SD as a function of the split gate voltage V SG at a constant source-drain current of 10 nA and constant back gate voltage V BG = −33 V. At low V SG , when the voltages on the split gate are not sufficiently high to deplete the underlying WSe 2 regions from carriers, we measure a resistance of 480 Ω from which we are able to extract a contact resistance of 9.76 kΩ (see Supplementary Fig. 1), a value that is comparable to others reported using more complex fabrication techniques 13,43,44 . As the voltage is increased on the split gates, we observe an increase in resistance, indicating that a constriction has been formed and that holes are responsible for transport in the device.
The change in the conductance of the device in units of e 2 /h as a function of the split gate voltage V SG is shown in Fig. 2b, from which we obtain a gate depletion value of V SG = 2.7 V. At this point, the 1D channel is formed, and the total measured resistance is 5030 Ω. When indicated, this back gate-dependent series  c Color map of conductance as a function of split gate voltage V SG and back gate voltage V BG . A lead resistance (resistance at depletion) is subtracted. d Line cuts from (c) plotted in a "waterfall" style indicating a first step at e 2 /h and a second step around 2 e 2 /h. The leftmost trace is taken at V BG = − 29 V and the right most trace is taken at V BG = − 35 V. The green and magenta lines are highlighted by similarly colored dashed lines in (c) and correspond to back gate voltages of −30.5 V and −33 V, respectively. These two curves feature the regimes where we observe the 1 e 2 /h plateau (green), and the 1 e 2 /h and 2 e 2 /h plateaus (magenta).
resistance (see Supplementary Fig. 3) is subtracted from the raw data to obtain the conductance related only to the 1D channel 10 . In Fig. 2b, quantized conductance steps close to e 2 /h and 2 e 2 /h can be observed for a fixed hole concentration of 1.82 × 10 12 cm −2 (see Supplementary Fig. 2). Tuning the concentration allows us to reach two different regimes, as demonstrated in Fig. 2c. At higher hole concentrations, below V BG = −30 V, we see the appearance of the first conductance step at e 2 /h. As the carrier concentration increases (V BG decreases), the difference between the depletion and pinch-off values also increases and we observe the appearance of a second conductance step at a value of 2 e 2 /h. These conductance steps are more clearly observed in a waterfall plot as shown in Fig. 2d. The observation of quantized conductance through the 600 nm long channel along with the high measured field-effect carrier mobility of μ FE ≈ 8000-8500 cm 2 V −1 s −1 (see refs. 13,42,44,45 ) (see Supplementary Fig. 2), hints toward quasi-ballistic transport and demonstrates that a high-quality monolayer WSe 2 sample has been achieved. At lower carrier concentrations, we observe the onset of a distinct transport regime that remains to be elucidated by further investigations (see Supplementary Fig. 7).
Magnetic field dependence of the 1D channel We further investigate the quantized conductance features by applying a magnetic field perpendicular to the plane of the twodimensional material. Due to the large hole spin and valley g-factor which has been reported to be up to 12 in monolayer WSe 2 46 , we explore the low magnetic field regime between −100 and 100 mT (Fig. 3a) to eliminate any possible spin or valley polarization occurring due to a small magnetic field offset at 0 T. The quantization remains constant at low field even at a high resolution of 4 mT. We therefore conclude that the measured lifting of spin and valley degeneracies at low fields is inherent to this particular system leading to a ground state where only one conducting channel is available. At higher fields, as depicted in Fig. 3b, we observe that the first plateau remains constant between −5 and 8 T with a very small correction around 0 T which we attribute to magnetoresistance effects in the leads. This magnetoresistance correction is more significant at higher conductance, but after 1.5 T we observe the second and third quantization plateaus aligning with the values of 2 e 2 /h and 3 e 2 /h. This data set demonstrates that each conductance channel up to the third one allows transport for only one quantum of conductance. We cannot determine, however, what is the ground state in terms of spin or valley. In addition, at higher magnetic fields, we observe an evolution of the second and third plateaus in which they lower in conductance and seem to merge into the first plateau. No such change in conductance is observed for the first plateau. Measurements performed at 10 mK in a dilution refrigerator show the same behavior (see Supplementary Fig. 4).

Theory
In order to determine whether the anomalous quantization behavior originates from the spin-valley locking mechanism in TMDs, we developed a single-particle model of a 1D channel. To calculate single-hole states, we use a tight-binding model for WSe 2 monolayer 27,47-49 in a basis of three d-orbitals localized on tungsten atoms and three p-like orbitals describing Se 2 dimers (6 even orbitals in total). A single-particle channel wavefunction for a hole state s satisfies the Schrödinger equation 48,49 : where the 1D electronic confinement defined within the WSe 2 monolayer lattice is determined by an applied gate-defined potential ϕ.
To obtain the channel potential landscape, we performed selfconsistent Poisson-Schrödinger calculations 50,51 using parameters corresponding to the device used. The model includes the same Fig. 3 Out-of-plane magnetic field B ⊥ dependence of the 1D channel. a Dependence of the quantized conductance plateaus on a perpendicular low magnetic field ranging from −100 to 100 mT and a resolution of 4 mT. b Dependence of the quantized conductance plateaus on a perpendicular magnetic field ranging from −5 to 8 T. In both panels, a horizontal offset is introduced between adjacent line traces for clarity. V BG = − 33.2 V for all runs with the corresponding lead resistance of R L = 5029 Ω subtracted. arrangement of layers represented by different dielectric constants and the same gate layout with the corresponding voltages applied. Further details can be found in the Supplementary Information. The obtained potential landscape results in a channel oriented along the x axis, and is presented together with an elongated rhombohedral tight-binding computational box that covers the channel area in Fig. 4a. A hole is free to move in the channel direction, while movement in the perpendicular direction is constrained by a Gaussian-like potential along the y axis. To simplify further calculations, we fit the channel profile at x = 500 nm with a Gaussian function (U ch ¼ U 0 ð1 À expðÀ y 2 2σ 2 ÞÞ) resulting in a potential depth U 0 = 1600 mV and width σ = 65 nm. In the following calculations, we will use U ch as a realistic approximation of the ϕ potential.
We define the finite tight-binding computational box in a form of a rhomboid of the WSe 2 lattice. We wrap the computational box on a torus, apply the periodic (Born-von Karman) boundary conditions, and obtain a set of allowed, discretized k-vectors over which we diagonalize the bulk Hamiltonian H bulk . The valence band (VB) wavefunction for each allowed k-vector is a linear combination of Bloch functions on the W and Se 2 dimer sublattices l(l = 1, …, 6): where χ σ j i is the spinor part of the wavefunction, and are Bloch functions built with atomic orbitals φ l . N is the number of unit cells, while R l defines the position of atomic orbitals in the computational rhomboid. By diagonalising the 6 by 6 Hamiltonian H bulk at each allowed value of k, we obtain the bulk energy bands E VB kσ and wavefunctions A VB kσl . The topmost valence band energy surface is shown in Fig. 4b, where two non-equivalent maxima, K and K 0 valleys, are clearly visible.
In the next step, we expand the channel wavefunction in terms of the lowest energy valence band states given by Eq. (2): Finally, we solve the Schrödinger Eq. (1) with the potential term U ch by converting it to an integral equation for the coefficients B VB;s kσ , and obtain the single-hole eigenstates confined within the channel, where U ch l ðq; kÞ ¼ 1 N P N R l ¼1 U ch ðR l Þe iðkÀqÞÁR l is the Fourier transform of the channel confinement on each sublattice l. The structural and the tight-binding Slater-Koster parameters in the calculations 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 the tight-binding model for WSe 2 monolayer rhombus with applied 1D gate-defined channel potential. The hole eigenstates, characterized by their energy, wave vector k x , and spin, form characteristic parabolic subbands located in (c) K 0 valley with spin-up (red dots), or (d) K valley with spin-down (blue dots). Subsequent subbands are used to calculate the channel conductance (e) via the Landauer formula. Resulting conductance curve has a characteristic stepped-like shape with subsequent plateaus at multiples of 2 e 2 /h. Those states (numbered by s quantum number) are occupied by hole states with spatial densities (f) resembling harmonic oscillator modes. are directly listed in the "Methods" section under tight-binding parameters.
The channel states of energy E s σ are characterized by subband index s, by the spin σ, and also by the valley K or K 0 determined by the expectation value of the wave vector 〈k x 〉 ≡ k x in the channel direction. They belong to two different valleys: K 0 with spin-up- Fig. 4c, and K with spin-down-Fig. 4d, creating degenerate spinvalley locked states. Characteristic parabolic subbands are formed due to the lateral quantization within the channel represented by different values of s quantum number.
We note that the VB maximum (for the free hole Hamiltonian H bulk ) is shifted to zero and the highest energy state (ground state for holes in the channel) is located below, at −12 meV. The second spin-valley pair (not presented) is split-off by 0.5 eV below on the energy scale by the strong intrinsic spin-orbit coupling present in this material.
After calculating the single-particle eigenstates of the Hamiltonian H, we are ready to numerically estimate the conductance through the 1D channel. For each subband s, we calculate the density of states within the given subband dNs dE , and then estimate the particle velocity as v s ðEÞ ¼ 1 _ ∂Es ∂kx . The conductance, which combines contributions from states from different occupied subbands s filled up to the Fermi energy, is calculated using the 2-terminal Landauer formula 52 : The factor 1/2 is due to the fact that we take only states with k x > K x (or k x > K 0 x for the second valley), i.e., active under the given source-drain bias. In the above formula, we assume that no backscattering occurs. The calculated conductance is presented in Fig. 4e. It has the conductance plateaus at multiples of 2 e 2 /h. The corresponding charge densities for each subband are plotted in Fig. 4f. The final result shows that a microscopic theory of a 1D channel predicts conductance steps in units of 2 e 2 /h. Hence, we see that the measured e 2 /h conductance quantization cannot be explained in the single-particle picture.

DISCUSSION
A possible explanation for the e 2 /h conductance quantization can be obtained in terms of a broken symmetry valley-polarized ground state 25,26,39 . A valley-polarized state is expected to be the ground state of a 2D hole gas in a single layer of WSe 2 for sufficiently strong interaction strength and hole density (in analogy to WS 2 26 ). The breaking of valley degeneracy naturally lifts the degeneracy of bands in the mean-field picture. Therefore, conductance plateaus in conductance quantization are expected to be half of the degenerate system, giving e 2 /h instead of 2 e 2 /h steps as a function of Fermi energy. This might explain the e 2 /h plateau for a large range of hole densities, as shown in Fig. 2d.
In summary, we have fabricated high-quality monolayer WSe 2 devices where an electrostatically confined 1D hole transport channel is formed. At a temperature of 4 K, hole quasi-ballistic transport is observed, revealing an unexpected conductance quantization in steps of e 2 /h instead of 2 e 2 /h taking into account the electronic effects from the band structure of TMDs such as spin-valley locking. We have compared the experimental results to a single-particle atomistic tight-binding model for holes in WSe 2 with realistic channel profile and demonstrated that a 1D confining potential does not reproduce the measured conductance quantization in units of e 2 /h. Recent experiments 25 and theory 27 suggest the existence of a valley-polarized state of the hole gas in TMDs. Such a state would explain the measured plateau at e 2 /h. A theoretical model which includes hole-hole correlations is therefore necessary to explain quantum transport in 2D TMDs. These results show that the electronic properties of monolayer TMDs are still insufficiently understood and require further research to elucidate.

Device fabrication
The device was assembled on a p-doped silicon substrate with 285 nm of thermally grown silicon dioxide (SiO 2 ). Using standard dry transfer methods 40,41 , a flake of hexagonal boron nitride (hBN) (26 nm) was picked-up with a polypropylene carbonate (PPC) coated polydimethylsiloxane (PDMS) stamp and transferred onto lithographically patterned local gates. Electrical contacts [Cr (2 nm)/Pt (8 nm)] 42 were subsequently patterned on top of the hBN. To remove any contaminants on the surface of the hBN and the electrical contacts, the sample was thermally annealed in a vacuum furnace (10 −7 Torr) at 300°C for 30 min and further cleaned mechanically using an atomic force microscope tip (AFM) in contact mode 8,22,53,54 . A second polymer stamp was used to subsequently pick-up an hBN flake (34 nm) then a monolayer WSe 2 flake. To ensure proper contact to the electrical contacts, the stack of flakes and stamp were placed in an AFM such that the surface of the WSe 2 flake was exposed where it was then mechanically cleaned using an AFM tip. The hBN/WSe 2 stack was then dropped off onto the patterned contacts. At this stage, the device was thermally annealed in a vacuum furnace following the same recipe as before. A final lithographic step was performed to pattern the top gates which include the two contact top gates (V CG ) and the split gates (V SG ).

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.