Strain-engineering of twist-angle in graphene/hBN superlattice devices

The observation of novel physical phenomena such as Hofstadter's butterfly, topological currents and unconventional superconductivity in graphene have been enabled by the replacement of SiO$_2$ with hexagonal Boron Nitride (hBN) as a substrate and by the ability to form superlattices in graphene/hBN heterostructures. These devices are commonly made by etching the graphene into a Hall-bar shape with metal contacts. The deposition of metal electrodes, the design and specific configuration of contacts can have profound effects on the electronic properties of the devices possibly even affecting the alignment of graphene/hBN superlattices. In this work we probe the strain configuration of graphene on hBN contacted with two types of metal contacts, two-dimensional (2D) top-contacts and one-dimensional (1D) edge-contacts. We show that top-contacts induce strain in the graphene layer along two opposing leads, leading to a complex strain pattern across the device channel. Edge-contacts, on the contrary, do not show such strain pattern. A finite-elements modelling simulation is used to confirm that the observed strain pattern is generated by the mechanical action of the metal contacts clamped to the graphene. Thermal annealing is shown to reduce the overall doping whilst increasing the overall strain, indicating and increased interaction between graphene and hBN. Surprisingly, we find that the two contacts configurations lead to different twist-angles in graphene/hBN superlattices, which converge to the same value after thermal annealing. This observation confirms the self-locking mechanism of graphene/hBN superlattices also in the presence of strain gradients. Our experiments may have profound implications in the development of future electronic devices based on heterostructures and provide a new mechanism to induce complex strain patterns in 2D materials.

The high charge carrier mobility attained at room temperature in graphene encapsulated in hexagonal Boron Nitride (hBN) 1 has enabled the observation of ballistic transport over macroscopic distances [2][3][4] holding the promise for the development of room-temperature electrical equivalents of optical circuits. As opposed to suspended graphene structures 5 in which low-energy flexural phonons impose severe limitations on the maximum value of charge carrier mobility observable at room temperature, 6,7 in supported structures the optical phonons of the substrate play a central role. Compared to SiO 2 , the optical phonons in hBN have higher energy and this results in an increase of the charge carrier mobility in graphene by an order of magnitude. 1 Phonon scattering is not the only limiting factor to carrier mobility on SiO 2 , scattering from adsorbates such as water and substrate roughness also dominate its value. Being free from dangling bonds and lattice matched to graphene within δ ∼ 1.7%, hBN also allows for an atomically clean interface to be formed. Crucially, the van der Waals attraction between these two-dimensional (2D) materials is strong enough to push contamination outside of the overlap region, resulting in an atomic-scale self cleaning mechanism which was shown to work for mechanically exfoliated flakes as well as large area graphene grown by chemical vapour deposition. 3,4,8 Another major breakthrough made possible by the encapsulation of graphene in hBN has been the realization of high-quality one-dimensional (1D) electrical contacts to graphene. In these edge-contact geometries, low temperature ballistic transport was reported over 15 µm together with substrate-phonon limited roomtemperature charge carrier mobility. 2 Moiré interference patterns are observed for graphene on hBN owing to the small lattice mismatch between the two crystals. The rotation of graphene with respect to the underlying hBN produces patterns each with a different Moiré wavelength, 9,10 suggesting that effective periodic potentials are formed. For massless Dirac fermions this results in the formation of new Dirac points in the electronic band structure whose energy is determined by the Moiré wavelength. 11 Superlattice structures have led to the observation of several physical phenomena including Hofstadter's butterfly, [12][13][14] topological currents, 15 correlated insulator behaviour 16 and unconventional superconductivity. 17 Critical to these observations is the formation of a commensurate state in which graphene is locally stretched in domains separated by sharp domain walls. Previous works have reported a commensurate-incommensurate transition at twist-angles θ (i.e. the angle formed between the lattice vectors of graphene and hBN) of the order of the lattice mismatch (∼ 1 • ). 18 For θ < δ graphene forms these domains of strong van der Waals interaction with hBN, whilst in the opposing case (θ > δ) local strain is not observed. Thermal annealing has been shown to induce an incommensuratecommensurate transition over micrometer scales providing the initial twist-angle is small (θ ≤ 2 • ). For flakes which do not align, a 1D network of wrinkles emerges due to the difference in thermal expansion coefficients between hBN and graphene. 19 The quantum transport characteristics of twist-angle structures are commonly probed in transistor-like geometries. These generally consist of a graphene flake etched into a multiterminal Hall-bar shape with metal contacts. However, the deposition of metal films onto graphene is known to induce structural defects, doping and strain. [20][21][22] Metal contacts are possibly responsible for the failure of the devices upon thermal annealing, cooling at cryogenic temperatures or further processing such as encapsulation in ionic gates. 23,24 Ascertaining the role of the contacts on the properties of graphene/hBN superlattice structures is of pivotal importance and presently the focus of growing interest. For example, although phase-and growth-engineered 1D contacts have been explored in several atomically-thin materials, [25][26][27] the potential of 1D metallic edge-contacts for other encapsulated heterostructures has not yet been fully explored.
In this work we study the effect of strain induced by metal contacts in graphene/hBN superlattice devices. Semi-encapsulated graphene/hBN Hall-bars have been fabricated from a single flake with two different types of contact geometry: (1) two-dimensional (2D) topcontacts and (2) one-dimensional (1D) edge-contacts. Raman spectroscopy mapping was used to determine the strain and doping levels of the semi-encapsulated graphene. The absence of a top hBN layer allows to compare the two contact types in the same device. Top-contacts were found to induce strain levels up to 0.12%, localised between two opposing leads, whilst edge-contacts do not induce any measurable strain. Surprisingly electrical transport measurements in devices encompassing the two types of contact geometry showed different twist-angles depending on the contact type, despite being fabricated on the same flake. Postprocessing thermal annealing is shown to reduce the overall doping across the device, whilst increasing the overall strain. At the same time, the twist-angle is shown to change with thermal annealing resulting in its convergence to the same value for both contacts type.
Our results are supported by finite elements method (FEM) simulations which correctly predict the observed strain pattern induced in graphene/hBN by top contacts geometries.
Our studies unveil the interplay between strain and contact geometry in semi-encapsulated graphene on hBN. This knowledge could be instrumental in experimentally accessing the rich physics arising from the introduction of a gauge potential in the effective Hamiltonian of two-dimensional materials, such as the realisation of a purely strain-based valley filter 28,29 or a charge-funnelling device for energy applications. 30  an in-plane force applied normal to each contact region. Figure 2a shows the result of this analysis, where the difference between the simulated trace of the strain tensor and the initial compressive strain is plotted in a colour map (∆tr(ε) = (ε xx + ε yy ) − |ε 0 |). Relaxation of the initial compressive strain is observed at the contacts and, more interestingly, a bowtie feature is observed between opposing contacts. Both these features are present in the experimentally measured strain map, figure 2b. Figure

Effect of annealing on strain patterns
Thermal annealing is commonly used to enhance the electrical properties of graphene fieldeffect transistors by improving the metal-graphene interface and reducing contamination (e.g from polymer residues). However such procedure can lead to contact failure. In view of the observed strain pattern in our device, we study the effect of thermal annealing in this structure. Figure 3a shows the ω G /ω 2D distribution of the data from figure 1. In the asfabricated device, the data are distributed along the iso-strain axis with a vertical shift away from the pristine case. This is thought to be due to Fermi velocity reduction, previously reported for graphene on hBN, which arises from van der Waals interlayer interaction. 36,37 Upon thermal annealing for 2 hours in forming gas (H 2 /Ar, 10%/90%) at 200 • C the data- Previous reports have shown that graphene on hBN can undergo a rotation upon thermal annealing, 19 increasing the crystallographic alignment, which occurs as the system tries to minimise the interlayer van der Waals energy. This suggests that there is a competition between flake rotation and mechanical clamping from the metal electrodes. With increased clamping from top contacts a smaller change in strain occurs in these regions. This observation highlights the potential negative role played by this phenomenon on the failure of Table 1: Comparison of strain and doping values before and after annealing extrapolated from the Gaussian fits in figure 3b,c. Top (contact) refers to the region between two opposing metal contacts. Top (channel) refers to the graphene region between two adjacent metal contacts (see figure 1).
ε exp (%) n h (10 12 cm −2 ) Contact Type As-fabricated  In contrast to the strain statistics, a single Gaussian can be fitted to the data indicating uniform doping across both contact regions with n ∼ 5.6 · 10 12 cm −2 . As expected, following annealing this reduces to n ∼ 1.5 · 10 12 cm −2 , validating the usefulness of this common processing step in enhancing the electrical properties of graphene devices. This indicates that the charge carrier mobility is limited by Coulomb impurities and not by strain-induced modifications to the deformation potential acoustic phonon scattering.

Twist-angle in strained G/hBN superlattices
Two peaks in resistivity are observed in figure 4a. The CNP appears at V g ∼ −20 V (ρ xx = 4.5 kΩ/sq) whilst a second, satellite peak, appears at V g ∼ −60 V. This second peak arises due to the emergence of additional Dirac points in the band structure of graphene on hBN, as previously observed in low-temperature transport experiments [11][12][13][14]38 and more recently at room temperature 39,40 (see also figure S8, Supporting Information, for measurements on a second device).
The interaction between hBN and graphene, which modifies the band structure of the latter, is tuned by the crystallographic angle between the two materials. This is reflected in a change of the separation of these two peaks in transport measurement which can be correlated to this angle. 40 This separation is examined more closely in figure 5a,b. In the as-fabricated where δ ∼ 0.017 is the lattice mismatch between graphene and hBN and a = 0.246 nm is the lattice constant of graphene. Due to the spin and valley degeneracies in graphene, full-filling occurs at a density of four electrons per superlattice cell (n = 4n 0 ), with the unit cell area 1/n 0 = √ 3λ 2 /2. 12 Since the carrier density is n = C g (V g −V CNP )/e where C g is the geometric gate capacitance, V CNP is the position of the charge-neutrality point and e is the electron charge, we find: Using equation (1) and equation (2) it is therefore possible to extrapolate the twist-angle between graphene and hBN from the data in figure 5a,b. Figure 5c is a schematic illustration of the Moiré superlattice structure formed by rotating the graphene with respect to the hBN.

Summary and discussion
To summarise, we have analysed the strain induced in single-layer graphene deposited on hBN by 2D top-contacts and 1D edge-contacts. Using Raman spectroscopy mapping, we have shown that top-contacts induce strain in the graphene flake, pulling in opposite directions.
On the contrary, edge-contacts do not induce such strain. Our observation is supported by FEM simulations. We associate this strain to the shrinking of Au and graphene which, due to their different thermal expansion coefficients, lead to a net pull by the metal contacts after thermal evaporation of the metal. Thermal annealing on such devices reduces the overall doping, as expected from the removal of contaminants, but it also increases the overall compressive strain on the graphene. Such increase can lead to contacts failure given the observed strain pattern on the flake. Furthermore, on aligned samples, where the graphene and hBN lattice vectors are rotated by a small angle (θ < 1 • ), we have shown that the angle θ is different for the two types of contacts, although it converges to the same angle after thermal treatment. The convergence of the twist-angle confirms the self-locking mechanism observed in aligned graphene/hBN heterostructures. 19 The interplay between rotation and contacts-induced strain can also lead to contacts failure. Full encapsulation in hBN with edge-contacts may affect the way the angle θ changes upon annealing, however the negligible levels of strain observed in the edge-contacts suggest that this would not play a major role in this kind of devices. plasma. This processing step increases both the yield and lateral size of exfoliated flakes.
Graphene was exfoliated onto a polymer bilayer (PMMA/PVA) and placed on the hBN by dry transfer. 41 The lithography steps performed to create both top-and edge-contacts on the same flake are described in figure S1 and associated Supporting text. Electron beam lithography (Nano Beam NB5) was performed using 500 nm-thick PMMA (MicroChem 950K A6), developed using a 3:1 solution of isopropanol (IPA) and 4-Methyl-2-pentanone (MIBK).
Contacts were deposited using Cr/Au (15/60 nm) thermal evaporation at pressure < 5 · Electrical transport measurements AC lock-in measurement techniques (Ametek Signal Recovery 7270) were employed to accurately probe changes in resistivity with small excitation voltages (V ac ∼ 1 mV) minimising Joule heating in the device in a custom-built measurement chamber. 42 The excitation voltage was modulated at a frequency of 72.148 Hz. Two-(V 2T ) and four-terminal (V 4T ) voltages allowed the simultaneous measurement of channel resistivity, ρ xx , field-effect mobility, µ, and contact resistance, (R 2T − R 4T ) /2. The graphene channel was capacitively coupled to the Si + backgate through a 280 nm-thick SiO 2 layer, allowing the modulation of carrier density n by applying a DC voltage. All measurements were performed in vacuum (P < 10 −6 mBar) at room temperature.

Raman spectroscopy
Raman spectra were acquired using a custom-built set-up with a 514 nm excitation CW solid-state laser as source, focussed thorough a ×50 lens (NA= 0.9, Olympus MPLFLN). 43 Back-scattered light was collected by the same lens and, after filtration of the excitation line, dispersed by a 1800 g/mm grating mounted in a Princeton Instruments Acton SP2500 spectrometer and the spectra measured by a Princeton Instruments PIXIS400 CCD camera. The laser spot-size is 484 nm, compatible with the lateral step-size of the motorised stage (500 nm). 43 All measurements were performed in vacuum (P < 10 −6 mBar) at room temperature.
We have tested our system to perform the analysis outlined in the text in order to confirm the ability to resolve strain and doping levels, as detailed in figure S2. As shown in figure   S3, for our test we employed a graphene flake deposited half on hBN and half on SiO 2 .
Our analysis correctly showed a difference in both strain and doping in these two regions and highlighted an average compressive strain for graphene on hBN of ε 0 ∼ −0.12 %, in agreement with literature. 33

Graphical TOC Entry
Strain G hBN Cr/Au