Out-of-plane corrugations in graphene based van der Waals heterostructures

Two-dimensional (2D) materials are usually envisioned as flat, truly 2D layers. However out-of-plane corrugations are inevitably present in these materials. In this paper, we show that graphene flakes encapsulated between insulating crystals (hexagonal boron nitride, ${\mathrm{WSe}}_2$), although having large mobilities, surprisingly contain out-of-plane corrugations. The height fluctuations of these corrugations are revealed using weak-localization measurements in the presence of a static in-plane magnetic field. Due to the random out-of-plane corrugations, the in-plane magnetic field results in a random out-of-plane component to the local graphene plane, which leads to a substantial decrease of the phase coherence time. Atomic force microscope measurements also confirm a long-range height modulation present in these crystals. Our results suggest that phase coherent transport experiments relying on purely in-plane magnetic fields in van der Waals heterostructures have to be taken with serious care.

Figure 1

(a) Out-of-plane corrugations in a graphene lattice (exaggerated) with lateral correlation length $R$ and rms height $Z$. A uniform in-plane magnetic field $B_{\parallel }$ (black) will lead to a random surface normal ($\delta B_{\perp }$, blue) and a parallel ($\delta B_{\parallel }$, red) component. A homogeneous out-of-plane magnetic field $B_z$ is used to study phase coherent transport. (b) Optical image of device 1 (hBN/Gr/hBN). (c) Two-terminal conductivity of device 1 as a function of charge carrier density. (d) Ensemble-averaged weak-localization correction at different temperatures at a hole doping of $-2.2\times {10}^{12}<n<-1.8\times {10}^{12}{\mathrm{cm}}^{-2}$ and zero in-plane magnetic field.

Figure 2

(a) WL of device 1 for different values of in-plane field $B_{\parallel }$ at a temperature of 1.8 K. The fitted dephasing rate ${\tau }_{\varphi }^{-1}$ as a function of $B_{\parallel }^2$ is shown in (b).

Figure 3

(a) Fitted dephasing rate of device 2 as a function of $B_{\parallel }^2$. A ripple volume of $1.6{\mathrm{nm}}^3$ is extracted. (b) AFM image of the graphene flake on top of the bottom hBN crystal. The graphene flake outline is marked by the red arrow. The location of the Hall bar is indicated by the black dashed line, and the area used for the AFM analysis is highlighted by a red dashed rectangle. (c) Height-height correlation of device 2 extracted from the AFM measurement in (b). A correlation length of 210 nm and an average height fluctuation of $94.4\pm 1.2\mathrm{pm}$ are found.

Figure 4

(a) WAL of device 3 for different values of in-plane magnetic field $B_{\parallel }$ at a temperature of 1.8 K. (b) The fitted dephasing rate ${\tau }_{\varphi }^{-1}$ as a function of $B_{\parallel }^2$.

Figure 5

(a) Optical image and (b) AFM image of the full vdW heterostructure of device 1 with the device outline overlaid.

Figure 6

(a) Optical image and (b) AFM image of the full vdW heterostructure of device 3 that show that the device area is essentially bubble free. Data presented in the main text were taken from the Hall bar outlined in (b).

Figure 7

Two-terminal conductivity as a function of charge carrier density of device 2. A two-parameter model is fit to extract charge carrier mobility and series resistance ($\sim 190\mathrm{\Omega }$).

Figure 8

Magnetoconductivity for various in-plane magnetic fields for electron doping of device 2. The black curves are fits with a parabola to extract the phase coherence time using Eq. (B1).

Figure 9

The height distribution (red crosses) evaluated in the area marked by the red dashed rectangle in Fig. 3. A Gaussian fit (black solid line) yields $Z=94.4\pm 1.2\mathrm{pm}$.