Out-of-Plane Transport of 1T-TaS2/Graphene-Based van der Waals Heterostructures

Due to their anisotropy, layered materials are excellent candidates for studying the interplay between the in-plane and out-of-plane entanglement in strongly correlated systems. A relevant example is provided by 1T-TaS2, which exhibits a multifaceted electronic and magnetic scenario due to the existence of several charge density wave (CDW) configurations. It includes quantum hidden phases, superconductivity and exotic quantum spin liquid (QSL) states, which are highly dependent on the out-of-plane stacking of the CDW. In this system, the interlayer stacking of the CDW is crucial for interpreting the underlying electronic and magnetic phase diagram. Here, atomically thin-layers of 1T-TaS2 are integrated in vertical van der Waals heterostructures based on few-layers graphene contacts and their electrical transport properties are measured. Different activation energies in the conductance and a gap at the Fermi level are clearly observed. Our experimental findings are supported by fully self-consistent DFT+U calculations, which evidence the presence of an energy gap in the few-layer limit, not necessarily coming from the formation of out-of-plane spin-paired bilayers at low temperatures, as previously proposed for the bulk. These results highlight dimensionality as a key effect for understanding quantum materials as 1T-TaS2, enabling the possible experimental realization of low-dimensional QSLs.


Theoretical framework and discussion.
Optical contrast is a widely used technique in the field of 2D materials that allows the fast identification of flakes with different thicknesses, 1,2 being of great importance for air unstable materials, such as 1T-TaS2 and CrI3. 3 As it has been reported for graphene, 4 metal halides 1,3 and other transition metal dichalcogenides 1,5,6 -but not for 1T-TaS2-, optical contrast can be done quantitatively and its results compared with those expected theoretically. In this work we consider thin-layers of 1T-TaS2 on top of SiO2/Si substrates. The reflected light intensity (I) depends on the thickness of every layer (d), the refractive index of the different materials involved, ñ (ñ = n -ik, where n and k are the real and the imaginary component of the refractive index, respectively), and the incident wavelength (λ). An analytical expression in normal incidence can be derived from the Fresnel equations. 1,3,4 I λ r e r e r e r r r e e r r e r r e S1 I λ r r e 1 r r e S2 In the expressions above, the subindices refer to the different media involved (0, 1, 2 and 3 correspond to argon, 1T-TaS2, SiO2 and Si, being the last one considered as a semi-infinite layer), rij is the reflection Fresnel coefficient when the light travels from medium i to medium k, rij = │(ñi -ñj)/ (ñi + ñj)│, and Φi is the optical pathway length in the medium i, Φi = 2πñidi/λ.
The optical contrast can be defined as: 1,3 C λ I I I I S3 From Equation S3, the optical contrast can be estimated from both the experimental and theoretical point of view. Experimentally, it can be obtained from optical microscopy images: the Iflake value is obtained by selecting a region of interest in an RGB image and averaging its intensities (the same procedure follows for Isubstrate). From the theoretical point of view, the reflected intensities of the flake and the substrate can be estimated using the Fresnel equations as described above. Previously reported refractive indices were employed for Ar, 7 Si 8 and SiO2. 9 In the present case, we worked inside an argon glovebox; nonetheless, a similar trend in the optical contrast is expected for nitrogen or air environments due to their similarities of the refractive index in the visible range. The only available optical data for 1T-TaS2 was the dielectric constant (dispersion, ε1, and absorption, ε2). 10 The refractive index was then calculated by considering the optical relationships ε1 = n2 -k2 and ε2 = 2nk. 10 For a single layer in normal incidence (Figure S.1), optimal contrasts are found for a wavelength of 750 nm and SiO2 thickness of either 100 nm or 380 nm (C ~ -20 %). For the SiO2 thickness of 90 nm, 285 nm and 300 nm, some of the most common SiO2 thickness used in the field of 2D materials, 4,11 the optimum contrast for identifying a monolayer is obtained for 540 nm (C ~ -14 %), 565 nm (C ~ -14 %) and 590 nm (C ~ -15 %), respectively. In general, the optical contrast of single layer 1T-TaS2 is comparable to other transition metal dichalcogenides like TaSe2 (C ~ -27 % for 265 nm of SiO2 and λ = 550 nm) 1 or NbSe2 (C ~ -20 % for 90 nm of SiO2 and λ = 500 nm) 11 . Once the thickness of SiO2 is fixed, it is possible to identify thin-layers (those flakes with thicknesses below 10 nm). For this purpose, it must be identified the conditions where the variation of the optical contrast with the 1T-TaS2 thickness in the range below 10 nm is higher. As can be seen in Figure S.2, the optimal curves for the identification of thin-layers exhibit a negative contrast in the thin-layer regime that, later, changes its slope until it tends to saturate. The optimal hot spots are found for 90-100 nm of SiO2 and 285-300 nm of SiO2 and a wavelength in the range of 550 -600 nm. On the contrary, native silicon oxide (1 -3 nm) or 150 nm of SiO2 are not good conditions for the optical identification of thin-flakes at any incident wavelength.
The previous discussion is applicable for a strictly normal incidence of the light. However, it is common to determine the optical contrast by using an optical microscope where the light is focused by an objective lens and, thus, the numerical aperture (NA) of the objective has to be taken into account since the different angle of light incidence modify the optical pathway and affects the interference. The main feature is a diminishment of the magnitude as well as a blue shift of the optical contrast peak as a consequence of the destruction of the optical interference, which can be seen in Figures S. 1 and Figure S. 3. In order to consider this oblique incidence, it has to be taken into account that for s(p)-polarized light, the reflection coefficient, ñi, must be substituted by ñiꞏcosθ (ñi/ñj → ñicosθj/ ñjcosθi), where θ is the incident angle (for unpolarized light, an average is taken of both contributions), and that Φi is 2πñidicosθi/λ, where θi is the refraction angle in the layer i (θi = asin((n0/ni)ꞏsinθ), as can be derived from Snell's law). Then, for including the NA of the objective, the reflected light intensity has to be calculated by the numerical integration of the intensities over the solid angle of each NA, determined by the angle θm. Considering that θm = asin(NA) and a Gaussian distribution of the collected light (W(θ) = exp(-2sin2θ/sin2θm), equation S1 and S2 are transformed to: 12,13 The predicted contrast tendencies are validated experimentally for 285 nm of SiO2 (  -Optical contrast as a function of the incident wavelength and the 1T-TaS2 thickness for different SiO2 thicknesses for NA = 0 (normal incidence).

Comparative of in-plane vs. out-of-plane devices.
Reported transport data of 1T-TaS2 thin-layers (from Yu et al. 14  out-of-plane measurements than in the reported in-plane ones. As well, the thermal variation of Rnorm presents a sharper change at low-temperatures in the out-of-plane configuration whereas in the in-plane configuration Rnorm increases for temperatures below 350 K. The change in resistance observed in the hysteresis at 350 K is larger in the in-plane configuration that in the out-of-plane one for bulk 1T-TaS2. In

Junction geometrical factors, Arrhenius and hopping analysis.
The area of the junction and the thickness of the 1T-TaS2 flake as well as the bottom and top few-layer graphene contacts measured by atomic force microscopy (AFM) for all the devices are summarized in Table S.1. As well, it is indicated the number of layers of 1T-TaS2 were STEM cross sectional images have been performed together with the theoretical thickness in nm (considering the crystal structure of 1T-TaS2, 17 the thickness of 1 layer of 1T-TaS2 is 0.59 nm). It can be observed that the measured thickness by AFM is higher than the one observed by STEM. This is ascribed to the presence of adsorbates and residues from the exfoliation and transfer process that yield to a thickness higher than the real one in AFM measurements. 18 Resistivity curves with the geometrical factors in Table S The different devices were fitted following an Arrhenius expression for the conductance: where G0 is a prefactor, Ea is the activation energy, kB is the Boltzmann constant and T is the temperature. 19 The fitting for the different devices (from  and m varies from 0.8 to 1 (we did not find significant changes by varying m from 0.8 to 1 and we show the results for m = 1). 20,21 The exponent x determines the scaling behavior. We consider the cases of x = 1/3 (2D-VRH), x = 1/4 (3D-VRH) and x = 1 (nearest-neighborhopping, NNH). We note that, formally, the analysis with x = 1 takes the same expression as