Quantum transport in graphene $p\text{−}n$ junctions with moiré superlattice modulation

We present simulations of quantum transport in graphene p-n junctions $\left(pn\mathrm{Js}\right)$ in which moiré superlattice potentials are incorporated to demonstrate the interplay between $pn\mathrm{Js}$ and moiré superlattice potentials. It is shown that the longitudinal and Hall resistivity maps can be strongly modulated by the $pn\mathrm{J}$ profile, junction height, and moiré potentials. Device resistance measurements are subsequently performed on graphene/hexagonal-boron-nitride heterostructure samples with accurate alignment of crystallographic orientations to complement and support the simulation results.

Figure 1

(a) Illustration of graphene device with an imposed moiré superlattice potential. The chemical potentials are labeled by ${\mu }_L$ and ${\mu }_R$ and have a randomized junction profile to more accurately represent an actual device's junction roughness. The rough $pn\mathrm{J}$ profile appears as a dashed blue and red line along the center of the device. (b) Longitudinal resistivity is plotted as a function of ${\mu }_L$ and ${\mu }_R$ for a straight $pn\mathrm{J}$ profile. (c) The same results as in panel (b) are plotted by using a rough $pn\mathrm{J}$ profile. (d) When a moiré superlattice potential is imposed, the resistivity map visibly changes in an asymmetric manner, for both the straight $pn\mathrm{J}$ profile in panel (d) and for the rough $pn\mathrm{J}$ profile in panel (e), with the addition of satellite Dirac peaks. (f) Taking the solid purple and dotted blue lines in panel (e), the behaviors of the longitudinal resistivity are compared. (g) A zoom-in of panel (e) shows that a satellite Dirac peak does exist in this region, albeit at a much smaller value of resistivity. All presented longitudinal resistivity color scales are in units of $h/2e^2$. For all maps, chemical potentials are in units of $\hslash vb$, where $v$ is the Fermi velocity and $b$ is the reciprocal superlattice constant of approximately $0.53\mathrm{n}{\mathrm{m}}^{-1}$ when the alignment between graphene and h-BN is nearly zero [14].

Figure 2

(a) Illustration of a rough $pn\mathrm{J}$ profile. The barrier height $t_j$ and the average chemical potential in the junction, μ are labeled. A moiré pattern potential was applied to the system. (b), (c) The Hall and longitudinal resistivity maps for a normal junctionless device are shown, respectively. They are plotted as a function of average chemical potential μ and the magnetic flux in units of the magnetic flux quantum ${\mathrm{\Phi }}_0$. The recursive Hofstadter's butterfly appears at the representative horizontal dashed lines at 1, 2/3, 1/2, and 1/3. This graphing style applies to the remaining resistivity maps. (d), (e) The Hall and longitudinal resistivity maps are recalculated with a 200 meV $pn\mathrm{J}$ barrier height. (f), (g) The maps are recalculated with a 400 meV $pn\mathrm{J}$ barrier height.

Figure 3

(a) Hall resistivity map plotted against chemical potentials ${\mu }_{\mathrm{L}}$ and ${\mu }_{\mathrm{R}}$ at 14 T with no moiré potential. The red and blue colors respectively indicate positive and negative fractions of $h/2e^2$. (b) Longitudinal resistivity map for the same case as in panel (a). (c) Hall resistivity map plotted after including the effects of a moiré potential. The region within the dotted green rectangle is reproduced after an order-of-magnitude rescaling of the resistivity. (d) The longitudinal resistivity map is replotted to incorporate the effects of the moiré potential.

Figure 4

(a) Illustration of a real graphene-based device showing the cross-section of the chip and the configuration of the embedded gates. (b) Optical image of an example device along with a drawing of the contact configuration for the larger device in the center of the image. The gold text in the drawing indicates the backgate contact number. (c) The data were acquired at 1.65 K with no magnetic field and are presented for the case of equal chemical potentials on the middle and rightmost regions of the device. (d) A resistance map is acquired for the voltages between −9 and 9 V for both regions of the $pn\mathrm{J}$, which is equivalent to modifying the chemical potential in Fig. 1.