Scalable Tight-Binding Model for Graphene

Artificial graphene consisting of honeycomb lattices other than the atomic layer of carbon has been shown to exhibit electronic properties similar to real graphene. Here, we reverse the argument to show that transport properties of real graphene can be captured by simulations using “theoretical artificial graphene.” To prove this, we first derive a simple condition, along with its restrictions, to achieve band structure invariance for a scalable graphene lattice. We then present transport measurements for an ultraclean suspended single-layer graphene $pn$ junction device, where ballistic transport features from complex Fabry-Pérot interference (at zero magnetic field) to the quantum Hall effect (at unusually low field) are observed and are well reproduced by transport simulations based on properly scaled single-particle tight-binding models. Our findings indicate that transport simulations for graphene can be efficiently performed with a strongly reduced number of atomic sites, allowing for reliable predictions for electric properties of complex graphene devices. We demonstrate the capability of the model by applying it to predict so-far unexplored gate-defined conductance quantization in single-layer graphene.

Figure 1

Schematic of a sheet of (a) real graphene and (b) scaled graphene and their conical low-energy band structures. In (a), the lattice spacing $a_0\approx 0.142\mathrm{nm}$, the hopping energy $t_0\approx 2.8\mathrm{eV}$, and the Fermi velocity $v_F^0\approx 10^8\mathrm{cm}{\mathrm{s}}^{-1}$.

Figure 2

Band structure consistency check using an armchair graphene ribbon with width 200 nm (a) in the absence of magnetic field and (b) in the presence of a uniform magnetic field $B_z=5\mathrm{T}$. The comparison is done for both (a) and (b) between the genuine graphene with $s_f=1$ and scaled graphene with $s_f=4$, which correspond to chain numbers $N_a=800$ and $N_a=200$, respectively.

Figure 3

(a) Sketch of the suspended graphene $pn$ junction device with suspension height $h=600\mathrm{nm}$, contact spacing $L=1680\mathrm{nm}$, and average flake width $W=2125\mathrm{nm}$. (b) Mean carrier density as a function of $V_g=V_L=V_R$, based on a 3D electrostatic simulation. Experimental and theoretical data of the two-terminal conductance at (c),(d) $B_z=0$ and (e),(f) $B_z=0.2\mathrm{T}$. Both (c) and (e) were measured at temperature $T=1.4\mathrm{K}$, while the simulations were done at zero temperature using (d) $s_f=100$ and (f) $s_f=50$ scaled graphene.

Figure 4

(a) Two-terminal transmission $T$ as a function of backgate carrier density $n_{\text{bg}}$ at fixed bottom gate carrier density $n_{\text{bog}}$ for a hexagonal-boron-nitride-sandwiched ballistic graphene device (left inset), assuming a flake size of $2\times 2\mu {\mathrm{m}}^2$ subject to the carrier density profile sketched in the right inset. (b) 2D map of the transmission function $T(n_{\text{bg}},n_{\text{bog}})$; the dashed white line indicates the line trace of (a). (c)–(f) Band structures computed by a unit cell cut from the right part of the same model flake, as marked by the dashed stripe shown in the right inset of (a). The carrier density configurations of (c)–(f) are indicated by the symbols ($▵,▿,\diamond$ and $\square$) marked at the middle corresponding to those marked in (a) and (b). Both $T$ and $E_k$ are computed based on clean armchair graphene scaled by $s_f=50$.