Engineering high quality graphene superlattices via ion milled ultra-thin etching masks

Nanofabrication research pursues the miniaturization of patterned feature size. In the current state of the art, micron scale areas can be patterned with features down to ~30 nm pitch using electron beam lithography. Here, we demonstrate a nanofabrication technique which allows patterning periodic structures with a pitch down to 16 nm. It is based on focused ion beam milling of suspended membranes, with minimal proximity effects typical to standard electron beam lithography. The membranes are then transferred and used as hard etching masks. We benchmark our technique by electrostatically inducing a superlattice potential in graphene and observe bandstructure modification in electronic transport. Our technique opens the path towards the realization of very short period superlattices in 2D materials, but with the ability to control lattice symmetries and strength. This can pave the way for a versatile solid-state quantum simulator platform and the study of correlated electron phases.


Supplementary Figure 2. Fabrication of a 30 nm period hexagonal lattice on a FLG. a.
A 15x15 um 2 and 10 nm thin Si membrane is patterned with a He FIB. The scale bar is 10 µm. b. The membrane is picked up with a PPC/PDMS stamp and c. dropped on the target FLG flake (around 3 nm thick). d. An O2/Ar RIE process is followed (20W, O2/Ar, 40/40 sccm, 10 Pa, 1'40") and e. finally the silicon membrane is removed by acetone sonication for 2 minutes, followed by 15 minutes of acetone cleaning plus 3 hours vacuum annealing at 600 C. f. AFM topography image of the resulting 30 nm hexagonal lattice. The scale bar is 250 nm.

Supplementary Note 2. Limitations of the technique
There are some limitations that compromise the resolution limit when patterning highly dense lattices  One of the main limitations is the mechanical stability of the suspended mask. Due to its nanometric thickness, milling a highly dense lattice where a big fraction of the material is removed results in holes merging or big areas breaking due to stress relaxation. In particular, producing patterns with very fine periods and large overall areas is further complicated by any minimal amount of drift or deformation in the long range order of the milled pattern, which cause neighboring holes (or hole rows) to overlap. These issues can largely be alleviated by using alternate custom-made membrane windows with thinner membrane thickness or by using mechanically stronger materials (e.g. non stochiometric SiN). Thinning the membrane contributes to reducing the milling dose, thereby reducing secondary ion collision damage and allowing the usage of higher resolution (lower current) HIM beam conditions.  Related to the mechanical stability of the membrane is the aspect ratio of the patterned membrane -the ratio of the hole size (lattice period) to the membrane thickness. While He FIB milling is known to achieve high aspect ratios, we have observed that increasing the membrane thickness decreases the achievable resolution. This is most likely due to variations in the hole size between the top and bottom of the membrane (in addition to the increased aforementioned secondary ion collision damage).
 It is therefore expected that using dedicated Si membranes with thinner Si windows will enable further reduction in the minimal lattice period. Similarly, alternate membrane window materials can be used, which are mechanically stronger than polycrystalline Si (e.g. ultrathin nonstoichiometric SiN). In this regard, it should be noted that while the membranes used are nominally 5 nm thick, the actual thickness measured in AFM (after transfer to substrate) is larger, on the order of 10 nm. Importantly, the membrane thickness (and the milling dose) does not change appreciably between different membranes (including those produced in different fabrication batches).
 Another part of the process which suffers from limited mechanical stability is the transfer process to the target substrate, where the membrane is picked up (broken) from the window frame and attached to a polymer. Due to the adherence properties of PPC, we change the temperature between the different steps in the transfer process. Even though the mask survives the transfer, we sometime observe long range disorder appearing in highly dense lattices due to stretching/compression of the mask. We attribute this to the thermal expansion of PPC. We expect that further optimization of the transfer process may reduce the number of such disordered lattices.
 As explained in the previous section, a buffer layer on the target substrate is needed to allow for a reliable lift off after etching. This results in a higher aspect ratio for the etching process, limiting the minimum feature size. It is conceptually possible to avoid using the buffer PMMA layer and remove the membrane by etching. We made experiments using SF6 plasma for etching, but found the process less reliable then the PMMA based liftoff technique, due to the very sensitive optimization needed to remove the Si mask completely without fluorinating and partially etching the graphite flake.

Supplementary Note 3. Heterostructures and device's micrographs
We provide optical micrographs before patterning the hall bars and after, for Dev 1 (Supplementary

Supplementary Note 4. Electrostatic potential profile calculation
We estimated the electronic-density profile induced in graphene by the patterned gate by solving the equation of electrostatic using the Finite Element Method. For the sake of simplicity, we performed 2D simulations in a 1D-periodic geometry with the same lateral dimension of the real devices. We do not expect the solution of the full 3D equation to change the results qualitatively.
We considered the non-linearity due to the quantum capacitance of graphene, and we used the following values of the dielectric constants of SiO2 ( 3.9) and hBN ( 3.56, 6.7). We By adjusting the patterned hole radius + and the smoothing parameter ,, we are able to reproduce the potential profile calculated in the previous section.
In Supplementary Figure 6 we plot the calculated band structure for both devices discussed in the main paper, and the inverse Drude weight, which is proportional to the band conductivity.
Importantly, the effective potential modulation induced in the graphene is also influenced by in-plane screening, to a degree that depends on the lattice period. Furthermore, the typical dependence of screening in monolayer graphene on the carrier density means that the experienced modulation will be smaller for the higher order Dirac peaks, which in general can have a bearing on interpreting experimental results.
To account for the in-plane screening, we consider the random phase approximation (RPA), similar to its application in 3 . The effective magnitude of the potential in the graphene layer, $ QRR is reduced by a factor of 1/S, where S 1 T < ? U V(G ; X 0),  Figure 7 shows the screening effect in our devices, where the potential reduction is slightly stronger for the shorter period device, Dev 2, than for Dev 1. The effect of the screening depends on the SL filling fraction, i.e. carrier density. However, the variation is relatively slow. For example, the induced potential does not change dramatically (in the RPA) between filling fractions 4 and 8.
Supplementary Figure 6. Calculated band structure, density of states and inverse of Drude weight for Dev 1 and 2. We consider the potential modulation for the calculation obtained from our electrostatics model. In the top panels, we plot the bands along the high symmetry point in the BZ of a square lattice, together with the density of states for the case with the modulation (black line) and the zero-modulation case (green dashed line). Bottom panels display the inverse of the Drude weight as a function of carrier density and filling fraction of the SL / .

Supplementary Note 7. Extra data for Dev 2
In Fig. 3e we show hall effect measurements for Dev 1. In Supplementary Figure 9 we show a similar measurement (hall effect and longitudinal resistivity) for Dev 2 for two values of the PBG.
In Supplementary Figure 10, we display as well the relative change in resistivity respect to the n op< 0 V case. This figure shows with more clarity the effect of the patterned gate on Dev 2.
In generating the plot in Fig. 3f of the main text, as well the plot above, it should be noted that we have manually removed parts of the plot where a measurement artifact appears, which obscures the measured signal. This artifact is specific to the measurement configuration and not to the patterned region of our device. In a TG-PBG voltage resistance map of our device (Supplementary Figure 11), this artifact appears as a diagonal line with different slope to our main Dirac peak line. The different slope means a different capacitance, indicating that this line is a measurement artifact not related to our patterned region.

Supplementary Note 8. AFM image filtering
Since AFM topography images suffer from substrate (SiO2) height variations at the sub nanometer scale, we apply a high pass filter with cut off frequency & 1/3 • & N vw by performing a 2D FFT. In Supplementary Figure 12 we show the AFM image from Fig. 2 before and after processing.