An anisotropic van der Waals dielectric for symmetry engineering in functionalized heterointerfaces

Van der Waals dielectrics are fundamental materials for condensed matter physics and advanced electronic applications. Most dielectrics host isotropic structures in crystalline or amorphous forms, and only a few studies have considered the role of anisotropic crystal symmetry in dielectrics as a delicate way to tune electronic properties of channel materials. Here, we demonstrate a layered anisotropic dielectric, SiP2, with non-symmorphic twofold-rotational C2 symmetry as a gate medium which can break the original threefold-rotational C3 symmetry of MoS2 to achieve unexpected linearly-polarized photoluminescence and anisotropic second harmonic generation at SiP2/MoS2 interfaces. In contrast to the isotropic behavior of pristine MoS2, a large conductance anisotropy with an anisotropy index up to 1000 can be achieved and modulated in SiP2-gated MoS2 transistors. Theoretical calculations reveal that the anisotropic moiré potential at such interfaces is responsible for the giant anisotropic conductance and optical response. Our results provide a strategy for generating exotic functionalities at dielectric/semiconductor interfaces via symmetry engineering.


Symmetry and its roles in electrical conductance in crystals
When an electric field is applied to a material, an electrical current density will be generated, and the motion of free charges can be characterized by the electrical conductance tensor through Ohm's law 1 : The electrical conductance tensor can be regarded as a scalar if the material is isotropic.
Based on the spatial symmetry of crystals, the direction and magnitude of the electrical current density for certain crystals can be determined by the direction of the applied external electric field: where the subindices and denote the in-plane components of the Cartesian coordinate in two dimensions. Note that the Einstein summation convention is applied here, and the summation is understood over all three components of Cartesian coordinates for those subindices appearing twice within a single term.
Mathematically, each element of the electrical conductance tensor can be qualitatively known to be zero or nonzero, and should be invariant under the symmetry operation of the crystal. Importantly, for the system holding the time reversal symmetry, Onsage's reciprocal law indicates that the electrical conductance tensor must be symmetric, that is = .
Therefore, we only consider the spatial symmetry of the crystal below, such as the rotation and the reflection operations.
For the rotation symmetry along the z-axis (labeled as ), the rotation operation for the angle takes the form: For the vertical mirror symmetry perpendicular to the y-axis , the reflection operation for the mirror takes the form: Therefore, should be invariant under the rotation operation = −1 ( ) ( ) (5) and the reflection operation = −1 (6) Note that the electrical conductance tensor is symmetric and positive semidefined, and there always exist two orthogonal main axes that diagonalize the . On such a basis, one can clearly see that, only the materials with rotation operations of one and two folds are anisotropic, and the material with rotation operations of higher folds must be isotropic.
Specifically, in our case, the isotropic conductance in 1L-MoS2 is protected by its C3v symmetry, while once the MoS2 is placed on the SiP2, the symmetry of the heterostructure reduces to C1 symmetry, which is expected to result in the anisotropic transport in 1L-MoS2/SiP2. Such phenomenological descriptions cannot reveal detailed information on the magnitude of elements of the electrical conductance tensor and their dependence on other tuning parameters, such as temperature and gate voltage. The magnitude of is the main target of the theory of the microscopic mechanism for electrical conductance. (2 ) -= ( 11 12 12 22 ) Anisotropic

Transfer characteristics of SiP2-gated MoS2 transistors
To evaluate the gating tunability of the SiP2 dielectric, we performed electrical transport measurements under dual-gate configurations in MoS2/SiP2 field-effect transistor (FET) devices on a SiO2/Si substrate ( Supplementary Fig. 1a). Supplementary Figure 1b Fig. 2c), which is comparable to the high mobility reported in HfO2-gated 1L-MoS2 devices 2 (174 cm 2 V -1 s -1 at 4 K). Such high mobility in few-layer MoS2 indicates that the SiP2 dielectric, as a typical vdW material, might form an atomically-flat interface and reduce charge scattering 3,4 in MoS2, preserving the intrinsic properties of 2D materials and allowing the fabrication of high-performance devices. Supplementary Table 2 summarizes typical parameters of the MoS2 FETs based on different dielectrics, including the on/off ratio, field-effect mobility, operation voltage, and breakdown field. We find that our SiP2-gated MoS2 FETs host a high on/off ratio, high mobility, low operation voltage, and large breakdown field, 7 which are comparable to those of widely-used high-κ dielectric gated MoS2 FETs 2,5-8 .

Gate-tuned activation energy in 1L-MoS2/SiP2 device
To further understand the insulator-to-metal transition in the SiP2-gated 1L-MoS2 device, we replot the temperature-dependent resistance at varying tg−SiP 2 in an Arrhenius plot 11 . As shown in Supplementary Fig. 3a, the resistance follows a thermal activation dependence. We extract the activation energy for charge transport in Supplementary Fig. 3b. The activation energy decreases monotonically from ~ 7 meV to near 0 meV when increasing tg−SiP 2 from 2.5 V to 3.8 V. Such an activation energy decrease directly indicates the insulator-to-metal transition in 1L-MoS2/SiP2 device.

Estimation of the dielectric constant of SiP2 via Hall measurements
To experimentally evaluate the dielectric constant r of vdW SiP2, we measured the 2D carrier density ( 2D ) as a function of tg−SiP 2 based on the Hall effect measurements, and finally deduced the dielectric constant via the relation 2D = 0 r tg−SiP 2 /( SiP 2 ) (details in Methods), where is the electron charge, 0 is the vacuum permittivity, SiP 2 = 20 nm is the thickness of SiP2. As shown in Supplementary Fig. 4a, the 2D values remain nearly unchanged before the threshold voltage of 1.7 V and start to increase linearly with the top gate afterward.
This value of threshold voltage tg−SiP 2 = 1.7 V is close to its pinch-off voltage estimated from the transfer curve ( Fig. 1c in main text). One can see that 2D can be continually modulated from 5×10 9 cm -2 to 8×10 12 cm -2 by applying tg−SiP 2 to 5 V ( Supplementary Fig. 4a). We can obtain the effective dielectric constant r = 8.1 for SiP2 by fitting the linear part of the 2Dtg−SiP 2 data using 2D = 0 r tg−SiP 2 /( SiP 2 ) . More interestingly, the 2D values can be further modulated by dual-gate control with tg−SiP 2 and bg−SiO 2 , and a maximum 2D close to 10 13 cm -2 can be achieved. An insulator-metal transition (phase diagram shown in Supplementary Fig. 4b) of MoS2 can be obtained in such a dual-gate configuration, where the critical 2D value for the transition is estimated to be 6.7×10 12 cm -2 (black dashed line) according to previous work 12 .

SiP2 and other dielectric materials
To quantitatively highlight the difference between the SiP2 dielectric with non-symmorphic S2

Berry curvature dipole and circular photo-galvanic effect in symmetry-mismatched heterointerface
The Berry curvature dipole is a crucial topological quantity for characterizing the electron wavefunction in solids and plays a significant role in realizing exotic nonlinear phenomena of quantum materials. In particular, in such symmetry-mismatched van der Waals heterointerfaces, the band structure with interfacial hybridization of atomic orbitals is more sensitive to Berry curvature dipole generation due to symmetry breaking 31

Linear dichroic PL spectra generated in 1L-MoS2 and 1L-WS2 via symmetry engineering
To understand the symmetry engineering at the interface, we performed polarized PL measurements on 1L-MoS2/SiP2 at 77 K. The PL spectra of 1L-MoS2, SiP2 (50 nm), and 1L-MoS2 stacked on SiP2 are all fitted by the multiple Voigt functions 32 to clarify the peak energies and integrated intensities ( Supplementary Fig. 6). For 1L-MoS2, there are two PL emission peaks located at 1.91 eV and 1.82 eV, of which the higher energy peak (peak 1) corresponds to the exciton of MoS2 while the lower energy peak (peak 2) corresponds to the defect states 33 .
For SiP2, there are three PL emission peaks, of which the higher energy peak located at 2.06 eV (peak 3) corresponds to the intrinsic exciton of SiP2, while the other two (peaks 4 and 5) correspond to the defect states 14 in SiP2. When 1L-MoS2 is placed on SiP2, we find a strong lower energy peak on the side of the MoS2 exciton peak (peak 1) in the PL spectra of the 1L-MoS2/SiP2. This exciton state might be related to the defect states 33 in intrinsic 1L-MoS2 (peak 2), which might be enhanced at the heterointerface.

SHG signals and rotational symmetry of 1L-WS2/SiP2 and 1L-MoS2/SiP2
To confirm symmetry breaking at the interface of monolayer WS2 with C3 symmetry and SiP2 with non-symmorphic S2 symmetry, we performed polarization-resolved SHG measurements on 1L-WS2, SiP2, and 1L-WS2/SiP2 heterostructure as shown in Supplementary Fig. 10. The detection polarization is set to be parallel to the excitation polarization. Similar to the 1L-MoS2 case, the SHG signal of 1L-WS2 shows a typical sixfold pattern (blue plot), with maxima appearing when the polarizations lie along the armchair directions ( Supplementary Fig. 10b), revealing the C3 symmetry of WS2 (ref. 35). In contrast, the shape of the six petals becomes asymmetric for the SHG signal of 1L-WS2/SiP2 heterostructure ( Supplementary Fig. 10a Fig. 10c).
To investigate the influence of SiP2 thickness on the anisotropic SHG response on the heterostructure, we performed polarized SHG measurements on 1L-MoS2/SiP2 by changing the bottom SiP2 thickness. As shown in Supplementary Fig. 11

Twist-angle dependent anisotropic SHG and PL responses in 1L-MoS2/SiP2
To investigate the effect of twist angle on the anisotropic properties of the heterostructure, we performed polarization-resolved PL and SHG measurements on various 1L-MoS2/SiP2 heterostructures with different twist angles. Specifically, in those 1L-MoS2/SiP2 heterostructures with different twist angles (for example 0° and 50° shown in Supplementary   Fig. 12), the polarized SHG signals displays the strong anisotropic SHG behavior, which is remarkably different from the six-fold symmetric SHG signals in pristine 1L-MoS2. Such

Anisotropic transport properties of 1L-MoS2 device gated with SiP2 dielectric
To investigate the anisotropic transfer characteristics of MoS2 FETs using SiP2 dielectric, we measured the four-terminal conductance along the x ( ) and y ( ) directions of the 1L-MoS2/SiP2 heterostructure ( Supplementary Fig. 14a,b). Note that the SiP2/MoS2 channel used in such measurements is intentionally designed as the square-shape to make the current flow symmetric between x and y directions. We also designed two electrode pads side by side as the source (or drain) electrode, assuming these two big electrode pads can serve as one large electrode for injecting uniform current ( Supplementary Fig. 15a,c). Note that such two special designs can make the current uniformly flow through the sample channel and ensure that measurement geometry between x and y directions is equivalent. Specifically, based on the measurement geometry with current flowing along x direction of a typical MoS2/SiP2 FET ( Supplementary Fig. 15c), we shorted the E1 and E2 as the drain electrode (E3 and E4 as source electrode) to inject a uniform current across the sample, and we measured the conductance between the local electrodes E5 and E6 (or between E7 and E8). Supplementary Fig. 15d shows the conductance between E5 and E6 (or between E7 and E8) as a function of top-gate voltage via SiP2 dielectric. One can see that the two different channels exhibit almost the same conductance, indicating that the current flow through the sample is relatively uniform. We thus believe such a symmetric measurement geometry can minimize the difference between the measured anisotropic conductance and the anisotropic conductivity. Supplementary Fig. 14c Schematic illustration of our four-wire measurement geometry with current flow along y direction. We short the E1 and E2 as the drain electrode while E3 and E4 as the source electrode to inject a relatively uniform current across the sample, and we measure the conductance with the local electrodes E5 and E6 (E7 and E8). b, Conductance of channel 1 (E5 and E6) and channel 2 (E7 and E8) as a function of SiP2 top gate voltage under the measurement geometry in (a). c, Schematic illustration of our four-wire measurement geometry with current flow along x direction. We short the E5 and E6 as the drain electrode while E7 and E8 as the source electrode to inject a relatively uniform current across the sample, and we measure the conductance with the local electrodes E1 and E2 (E3 and E4). d, Conductance of channel 1 (E1 and E2) and channel 2 (E3 and E4) as a function of SiP2 top gate voltage under the measurement geometry in (c).

Thickness-dependent anisotropic conductance in SiP2-gated MoS2 devices
To understand the origin of the anisotropic conductance in MoS2/SiP2, we fabricated MoS2/SiP2 heterostructures with different thicknesses of MoS2 and compared their anisotropic conductance ratios. Supplementary Figures 16 and 17

The detailed illustration of the moiré patterns of case-I and case-II
To show the moiré pattern on the heterostructures, we performed first-principles calculations to quantitatively obtain the atomic lattice structures therein.

Band alignment of the unstrained 1L-MoS2/SiP2 heterostructure
In this section, we use DFT calculations to get the band alignment for the unstrained 1L-MoS2 and 1L-SiP2, the detailed lattice constant parameters can be found in the Methods section. The work function ( ) is defined as the energy difference between the valence band maximum (VBM) and the vacuum level ( vac ) . As listed in Supplementary Table 5,  Since the 1L-MoS2 is stacked on the SiP2 thin flakes instead of the SiP2 monolayer in the experiments, we first investigate the thickness dependence of the work function of SiP2 with the PBE functional. As shown in Supplementary Fig. 22a, the work function of SiP2 decreases with increasing thickness, and saturates when the thickness increases to six layers. Therefore, we treat the work function of the SiP2 thin flakes as that with the thickness of 6-layer.
Although the values of the work function of SiP2 multilayers will be different between the results under the PBE functional and HSE06 functional, the thickness dependence of the work function of SiP2 should be the same. The work function difference between SiP2 monolayer and one with the thickness of six-layer is defined as  (labeled in Supplementary Fig. 22a).
Thus, approximately the work function of the SiP2 thin flake is estimated by subtracting the absolute value of  from the work function of 1L-SiP2 under the HSE06 functional. Moreover, the bandgap of 2.14 eV of the bulk SiP2 sample 14 is approximately treated as the bandgap of the SiP2 thin flake. The corrected band alignment between the 1L-MoS2 and SiP2 thin film is shown in Supplementary Fig. 22b. One can see that the 1L-MoS2/SiP2 heterostructure belongs to type II heterostructure, the conduction band edge is mainly contributed by 1L-MoS2, which is consistence with results from optical and transport measurements.
Importantly, as shown in Supplementary Fig. 22c,

The construction of the heterostructure models used in the DFT calculations
The strain is intentionally applied to lower the lattice mismatch between the unstrained 1L-MoS2 and unstrained 1L-SiP2, which allow us to build suitable supercells and make direct DFT On the other hand, with the HSE06 functional, the band alignment of strained 1L-MoS2/1L-SiP2 heterostructure is illustrated in Supplementary Fig. 24h. Compared with the band alignment of the unstrained 1L-MoS2/1L-SiP2 heterostructure ( Supplementary Fig. 24g), one can see that the conduction band offset is not influenced by the strain and the conduction band edge is also contributed by the MoS2 layer. Although the valence band offset is reversed by applied artificial strain, the preserved conduction band offset enables us to investigate the transport behavior of the electrons correctly with the strained slab models.
In conclusion, although the inevitable influence on electronic structures is introduced when we apply the strain artificially, our model with strained 1L-MoS2 and 1L-SiP2 still can be used to 44 replace the unstrained 1L-MoS2 and 1L-SiP2 to build heterostructures that can be afforded for DFT calculations. Although the PBE functional fails to give the correct conduction band offset ( Supplementary Fig. 24i

The relaxation of the heterostructure models for the DFT calculations
Aiming to simulate the stacking configurations (AA, AB, and BA) in the moiré patterns of case-I and case-II, four structures are built and labeled as case-I-ABBA ( Supplementary Fig.   25d), case-I-AA ( Supplementary Fig. 25f), case-II-AABA ( Supplementary Fig. 25e), and case-II-AB ( Supplementary Fig. 25g). As shown in Supplementary Fig. 25a Fig. 25d-g). Such a result directly shows the significant influence of the introduced moiré potentials on the 1L-MoS2/1L-SiP2 heterointerfaces. As shown in Supplementary Fig. 25, the interlayer distance (ⅆ) is used to quantitatively describe the structural corrugations of the MoS2 layers after the relaxation.  Supplementary Fig. 25a,b.

The electronic structures of the heterostructures in the DFT calculations
To analyze the charge density distribution of the conduction band edge for the 1L-MoS2 in the heterostructures, we performed DFT calculations with the PBE functional for each strained heterostructure model. In the supercell of heterostructure, the k point of conduction band edge for the strained 1L-MoS2 is folded to the  point in the reduced BZ. Supplementary Fig. 26 shows the charge density distribution of the conduction band edge in real space. One can see

The roles of the interlayer distance and the charge density distribution on the conduction band edge of the moiré superlattices of case-I and case-II
In this section, we show the calculated structural corrugations and charge density distribution of the conduction band edge in the moiré patterns (case-I and case-II). The real space variation of the interlayer distance (ⅆ) and the distribution of are used to depict the moiré potential.
For example, for moiré pattern of case-I, we obtain the ⅆ and at the intermediate stacking configurations (I-AA, I-AB, and I-BA stacking configurations) from the DFT calculation with superlattice model (Supplementary Fig. 27a), then do the cubic spline interpolation to get the real space distribution of ⅆ and in the moiré pattern. The colormap ( Supplementary Fig. 27b) is plotted to show the structural fluctuations in the moiré superlattice of case-I. The same method is used to other moiré superlattice ( Supplementary Fig. 27c, e-f). Note that the values ( ) of the stacking configurations (AA, AB, and BA) are enlarged 1,000 times before the interpolation for convenience.
Focusing on the moiré pattern of case-I, it is evident that the stacking areas (I-BA) host the smallest interlayer distances (Supplementary Fig. 27b) and largest electron localization ( Supplementary Fig. 27c), indicating the largest moiré potential. One can clearly see that the distribution of the interlayer distance (or the distribution of charge density) shows strong anisotropy with the mirror symmetry parallel to the x direction of the heterostructure (more clearly in Fig. 4 with a wide range of moiré lattices).Thus, the anisotropic moiré potential in the 1L-MoS2/1L-SiP2 heterointerface can be described by the structural fluctuation and the real space charge density distribution of the conduction band edge. The states trapped by moiré potential mainly stay at the atomic interface in MoS2/SiP2 heterostructure. Such anisotropic moiré potential influences the hopping between the direction along and perpendicular to the mirror plane and eventually corroborates the observed anisotropic conductance. We could use the same methods to the moiré pattern of case-II ( Supplementary Fig. 27d-f) and draw the same conclusion.