Large Landau-level splitting in a tunable one-dimensional graphene superlattice probed by magnetocapacitance measurements

The unique zero-energy Landau level of graphene has a particle-hole symmetry in the bulk, which is lifted at the boundary leading to a splitting into two chiral edge modes. It has long been theoretically predicted that the splitting of the zero-energy Landau level inside the bulk can lead to many interesting physics, such as quantum spin Hall effect, Dirac-type singular points of the chiral edge modes, and others. However, so far the obtained splitting with high magnetic field even on a $h$-BN substrate is not amenable to experimental detection, and functionality. Guided by theoretical calculations, here we produce a large-gap zero-energy Landau-level splitting ($\sim 150\mathrm{meV}$) with the usage of a one-dimensional (1D) superlattice potential. We have created tunable 1D superlattice in a $h$-BN encapsulated graphene device using an array of metal gates with a period of $\sim 100\mathrm{nm}$ and carried out magnetocapacitance spectroscopy as a function of superlattice potential. At zero magnetic field we observe the modification of the density of states in our capacitance measurement which is consistent with the existing literature. At finite perpendicular magnetic field, we monitor the splitting of the zeroth Landau level as a function of superlattice potential. The observed splitting energy is an order higher in magnitude compared to the previous studies of splitting due to the symmetry breaking in pristine graphene. The origin of such large Landau-level splitting in 1D potential is explained with a degenerate perturbation theory. We find that owing to the periodic potential, the Landau level becomes dispersive, and acquires sharp peaks at the tunable band edges. Our study will pave the way to create the tunable 1D periodic structure for multifunctionalization and device application like graphene electronic circuits from appropriately engineered periodic patterns in near future.

Figure 1

(a) Schematic of one-dimensional periodic potential. The periodic potential is applied along the $x$ direction with a potential strength of $U_0$. $W_b$ and $W_w$ correspond to the regions of potential barrier and no barrier, respectively. (b) Band dispersion of a graphene superlattice with barrier width $W_b=0.6$ and $W_w=0.4$ and $U=130\mathrm{meV}$. (c) Schematic of the device architecture. Bottom inset shows the scanning electron microscope image of the grids. The scale bar is $200\mathrm{nm}$. Left inset shows equivalent capacitance model between the top gate and graphene. (d) DOS for $W_b=0.5$ and $W_w=0.4$ adopted from Ref. [17] (blue). The DOS without superlattice potential is shown in dashed red line.

Figure 2

(a) Color plot of two-probe resistance as a function of top-gate voltage ($V_{\mathrm{tg}}$) and grid voltage ($V_{\mathrm{grid}}$) at $T=2\mathrm{K}$. (b) Resistance cut lines for $V_{\mathrm{grid}}=0\mathrm{V},\pm 1.5\mathrm{V},\pm 2.5\mathrm{V}$. (c) Derivative of the total measured capacitance as a function of top-gate voltage ($V_{\mathrm{tg}}$) and grid voltage ($V_{\mathrm{grid}}$). (d) Total capacitance as a function of top-gate voltage ($V_{\mathrm{tg}}$) for grid voltage $V_{\mathrm{grid}}=0\mathrm{V},\pm 1.5\mathrm{V},\pm 2.5\mathrm{V}$. The dashed horizontal line is the geometrical capacitance between the top gate and graphene.

Figure 3

(a) Blue solid curve shows the extracted quantum capacitance as a function of Fermi energy $E_F$ for $V_{\mathrm{grid}}=0\mathrm{V}$. The red dashed line shows the single-particle quantum capacitance versus $E_F$ of monolayer graphene with the experimental value of $C_g=55fF$. The olive solid line shows the resistance plot with $E_F$. (b) Blue solid line shows the quantum capacitance as a function of $E_F$ for $V_{\mathrm{grid}}=-2\mathrm{V}$. Red line shows the theoretical DOS as a function $E_F$, calculated for asymmetric potential with $W_b=0.6$ and $W_w=0.4$ as described by Ref. [17], with added Fermi energy broadening of $\sim 40\mathrm{meV}$.

Figure 4

(a) Measured total capacitance ($C_t$) as a function of top-gate voltage ($V_{\mathrm{tg}}$) at $B=9.8\mathrm{T}$ (blue line). The red line shows the corresponding change in chemical potential as a function of $V_{\mathrm{tg}}$ using the charge conservation relation. (b) Fan diagram for $V_{\mathrm{grid}}=0\mathrm{V}$, where we plot $C_t$ as a function of $V_{\mathrm{tg}}$ for different $B$.

Figure 5

(a) The Landau-level spectrum (measured quantum capacitance) for $V_{\mathrm{grid}}=0\mathrm{V}$ as a function of magnetic field. The solid line corresponds to the theoretically predicted one. The right panel shows the LLs energies for the hole side as a function of magnetic field. There is a mismatch between the theory (solid line) and the experiment as discussed in the main text. (b) $C_q$ vs $E_F$ as function of magnetic field for $V_{\mathrm{grid}}=-1.0\mathrm{V}$. (c) $C_q$ vs $E_F$ as function of magnetic field for $V_{\mathrm{grid}}=-1.5\mathrm{V}$. (d) $C_q$ vs $E_F$ as function of magnetic field for $V_{\mathrm{grid}}=-2.0\mathrm{V}$.

Figure 6

(a) Calculated DOS as a function of $E_F$ for $U=0\mathrm{meV}$. (b) Calculated DOS as a function of $E_F$ for $U=50\mathrm{meV}$. (c) Energy separation of the $N=0$ LL as a function of superlattice potential (blue). The red curve shows the theoretically calculated energy separation.