## 2 Experimental Section  
We deposited graphene on pure copper foil (99.999%, Alfa Aesar) using atmospheric-pressure chemical vapor deposition (AP-CVD). Two gases, methane and ethylene, were used as precursors. The synthesis took place in a cylindrical quartz reactor with a diameter of 14 mm. We used a custom-made CVD setup (Figure S1e). The sample size was equal to 35 \(\times\) 45 mm\({}^{2}\). The temperature and duration of the catalyst annealing were equal to 1 h at 1050 \({}^{\circ}\)C, respectively. Two gases, nitrogen N\({}_{2}\) and hydrogen H\({}_{2}\), were involved during annealing. Their flow rates were equal to 100 and 150 cm\({}^{3}\) min\({}^{-1}\), respectively. Single-layer (SLG)/few-layer (FLG) and multilayer graphene (MLG) (>15 layers) samples were obtained from the reactor at 1054 \({}^{\circ}\)C within 20 min. Subsequent cooling of the reactor in the presence of gaseous nitrogen flow was conducted at a rate of 50 \({}^{\circ}\)C min\({}^{-1}\). For SLG, in order to preserve the integrity of the graphene during wet-chemical transfer, we used a polymer support, which was subsequently removed using special techniques. In the case of multilayer graphene, a wet-chemical transfer was carried out without polymer support. After the process of graphene synthesis, the copper catalyst was completely etched in an aqueous solution of FeCl\({}_{3}\). After removal of the catalyst, the graphene films were carefully transferred to the surface of 3-inch silicon wafers with 500 nm-thick thermaloxide (SiO\({}_{2}\)/Si). The methods of scanning electron microscopy and scanning probe microscopy were used to study the morphology and surface relief of the experimental samples. For all AFM measurements, a Solver Open microscope (NT-MDT) was applied in the semi-contact scanning mode with a NSG01-brand cantilever. The typical tip curvature radius for this cantilever is 10 nm. The analysis of the crystal structure and crystalline perfection was carried out by the electron backscatter diffraction (EBSD) and high-resolution X-ray diffraction (HRXRD) methods. An Ultima IV (Rigaku) diffractometer provided XRD patterns of the considered samples. We used the wavelength \(\lambda\) corresponding to CuK\(\alpha\) (0.15406 nm). Deformations, doping, defectiveness, thickness, and quality of graphene were estimated via Raman spectroscopy. We used a confocal micro-Raman spectrometer ConfotecNR500. It provides an excitation wavelength, spectral resolution and spectral accuracy of about 532 nm, 1 cm\({}^{-1}\), and \(\sim\)1 cm\({}^{-1}\), respectively. The number of graphene layers was also estimated using the classical light transmission method. Here, optical quartz plates were used as a substrate. Spectral measurements were performed using a UV-2600 Shimadzu spectrometer. In this work the plasma-assisted molecular beam epitaxy system Veeco GEN 930 was used to carry out the process of III-nitrides graphene-assisted growth. After low-temperature annealing (200 \({}^{\circ}\)C) in the load-lock chamber, the wafer with the graphene buffer was attached to a substrate holder in the growth module. This was followed by high-temperature annealing at growth temperature (in the range of 700-720 \({}^{\circ}\)C) under ultrahigh vacuum conditions. First, a 35 nm-thick AlN wetting layer was deposited. The temperature of the Al effusion cell was 1060 \({}^{\circ}\)C providing nitrogen enriched conditions. Both shutters (for Al and N cells) were opened simultaneously and remained open throughout the entire time of the AlN nucleation growth (480 s). After that, a 500 nm-thick GaN layer was grown. The temperature of the Ga cell was 970 \({}^{\circ}\)C providing metal-rich conditions. A Metal Modulated Epitaxy (MME) technique was used which consists in the pulsed gallium supply to prevent its accumulation on the surface in the form of liquid metal droplets, while the flow of active nitrogen remained constant throughout the entire GaN growth process.