Tunneling Spectroscopy of Quantum Hall States in Bilayer Graphene $p\text{−}n$ Junctions

We report tunneling transport in spatially controlled networks of quantum Hall (QH) edge states in bilayer graphene. By manipulating the separation, location, and spatial span of QH edge states via gate-defined electrostatics, we observe resonant tunneling between copropagating QH states across incompressible strips. Employing spectroscopic tunneling measurements and an analytical model, we characterize the energy gap, width, density of states, and compressibility of the QH edge states with high precision and sensitivity within the same device. The capability to engineer the QH edge network also provides an opportunity to build future quantum electronic devices with electrostatic manipulation of QH edge states, supported by rich underlying physics.

Figure 1

Gate-defined $p\text{−}n\text{−}p$ networks in bilayer graphene. (a) Microscope image of the device, consisting of $h$-BN-encapsulated bilayer graphene with three horizontal local back gates and one top gate. (inset) (b),(c) Top gate $V_T=8\mathrm{V}$ and side back gates $V_E=-10\mathrm{V}$ are set to define the insulating region $J2$ and $L2$. Only the center gate $V_B$ is swept in the experiment, and we define $\mathrm{\Delta }{V}_B=0$ at the charge neutrality point of center region $K2$ ($V_B=-10\mathrm{V}$). Simulated carrier density and edge state configuration at (b) $\mathrm{\Delta }{V}_B=-5$ and (c) $\mathrm{\Delta }{V}_B=5\mathrm{V}$. The dashed lines depict the spatial distribution of edge state current under magnetic field. When $\mathrm{\Delta }{V}_B<0$ (b), current can flow through edge states whose number equals the filling factor of the center region of the device, and conventional quantum Hall transport behavior with quantized conductance plateaus is expected. However, when $\mathrm{\Delta }{V}_B>0$ (c), the $N$-type center region ($K2$) is surrounded by a finite-width insulating region with a well-defined $\nu =0$ Landau gap and displacement-field-induced band gap. Magnetotransport across the sample is dominated by quantum tunneling and the device enters the nonequilibrium regime. (d) As a result, while quantized plateaus are clearly visible in the $p\text{−}p\text{−}p$ regime ($\mathrm{\Delta }{V}_B<0$), a steep change in measured magnetoresistance occurs when transitioning to the $p\text{−}n\text{−}p$ regime ($\mathrm{\Delta }{V}_B>0$).

Figure 2

Quantum Hall phase transition in $p\text{−}n$ configuration. (a) Measured (top panel) resistance and (bottom panel) conductance across the device as a function of magnetic field $B$ and central bottom gate voltage $\mathrm{\Delta }{V}_B$ [and equivalently in the electron density of the center region $K2$ ($n_N$)]. While the fan diagram is visible in negative $\mathrm{\Delta }{V}_B$, the device shows an unusually sharp magneto resistance rise around $B_C=4.5\mathrm{T}$ at positive $\mathrm{\Delta }{V}_B$ (dashed red box). At lower magnetic field, resonant tunneling between the Landau levels on the $p$ and $n$ side of the $p\text{−}n$ junction occurs. A zoom in higher resolution scan with optimized color scale can be found in Fig. 3. (b) Measured resistance ($R$), expected resistance with full equilibrium ($R_e$), fitting of the low field data using our analytical quantum tunneling model in semiclassical regime ($R_S$), and their ratio ($R/R_e$) as a function of magnet field sweep at $\mathrm{\Delta }{V}_B=5\mathrm{V}$ [equal carrier density for $p$ and $n$ region, along the double dashed line in (a)]. (c) Magnetic length $l_B$ and calculated incompressible strip width $d_{\mathrm{IS}}$ as a function of field. (d) $B<B_C$, $l_B>d_{\mathrm{IS}}$. The system is in the semiclassical regime, where tunneling occurs only across the gate-defined $\nu =0$ tunnel barrier, agreeing with $R_S$. (e) $B>B_C$, $l_B<d_{\mathrm{IS}}$. Incompressible strips become well-defined due to reduced coupling between them. Cascade tunneling occurs across all ISs, resulting in a steep double exponential increase in $R$ and $R/R_e$, a deviation from semiclassical description ($R_S$).

Figure 3

Resonant Landau tunneling in the semiclassical regime ($B<B_C$). (a) When sweeping $B$, Landau level alignment changes from resonance (down triangle) to off resonance (up triangle), resulting in oscillations of the measured resistance $R$ periodically with inverse magnetic field $1/B$. When sweeping $\mathrm{\Delta }{V}_B$, while Landau levels are in resonance, the Fermi level can be temporarily unpinned and jump across the Landau gap (diamond), resulting into modulations of oscillation amplitude. (b) $R$ as a periodic function of $1/B$ at different electron densities in the center region $K2$ ($n_n$) and hole density in region $K1$ and $K3$ ($n_p$) (c) 2D scan of the resonant tunneling features verifying that the Landau level alignments are only sensitive to the $p\text{−}n$ junction height instead of individual carrier densities in each respective region. (Inset) Differentiated data in the range of the dashed region, a modulation in oscillation amplitude has been observed with a periodicity of the electron filling factor ${\nu }_n=4$. (d) By tracing the peak positions as a function of field and gate voltage, we extract the spectroscopy band gap of bilayer graphene as a function of displacement field $D$.