Theory of Landau level mixing in heavily graded graphene $p\text{−}n$ junctions

We demonstrate the use of a quantum transport model to study heavily graded graphene $p\text{−}n$ junctions in the quantum Hall regime. A combination of $p\text{−}n$ interface roughness and delta function disorder potential allows us to compare experimental results on different devices from the literature. We find that wide $p\text{−}n$ junctions suppress mixing of $n\ne 0$ Landau levels. Our simulations spatially resolve carrier transport in the device, revealing separation of higher order Landau levels in strongly graded junctions, which suppresses mixing.

Figure 1

A schematic of a graphene device with buried split gates. The device consists of a silicon dioxide substrate into which two metal gates are formed. A gate oxide on top of the silicon and metal gates electrically isolates the graphene from the gates. Contacts on each side of the graphene device allow for measurements of carrier transport. The split gates capacitively couple to the graphene sheet and modulate the local Fermi level. By varying the voltage of the two gates separately, it is possible to form different types of junctions, including a $p\text{−}n$ junction. Applying a strong external magnetic field, perpendicular to the graphene sheet, causes the carriers to be constric-ted into Landau levels, which travel around the edges of the graphene. The arrows depict the flow of carriers in the lowest Landau level.

Figure 2

We benchmark our model by simulating conductance between source and drain contacts, which is equivalent to Hall conductance measured in typical experiments. We compare our simulated conductance with experiments which show several plateaus in ambipolar conduction [3] and those which do not [9]. All curves are for $L=320$ nm, $W=200$ nm, and $B=4$ T. Each point in every curve is the ensemble average of the conductance for 400 different disorder realizations. (a) The diagonal slice of the conductance map typically measured. We set the on-site energy of the left side of the junction, $E_1$, to be the negative of the right side, i.e., $E_1=-E_2$. This produces a symmetric $n\text{−}p$ junction. (b) The on-site energy of the left side of the junction is fixed to $E_1=-0.05$ eV, and the right side on-site energy is varied. (c) The junction is configured as in (b), but now $E_1=-0.09$ eV. In each plot, when the junction width is small $(D_W=30$ nm), all the Landau levels are mixed. Plateaus occur when the filling factors jump from 2 to 6 to 10 (the transitions are indicated by vertical dashed lines). When the junction widths are longer, mixing is suppressed and only the lowest Landau level fully mixes. This is consistent with experimental results in the literature. (c) Inset: Two terminal conductance simulated in the unipolar regime for $D_W=30\mathrm{and}125$ nm with $E_1=E_2$ (solid line), $E_1=0.09$ eV (dotted line), and $E_1=0.05$ eV (dashed line). The $x$ axis is the same as the parent plot. The line colors are the same as given by the legend in (a). Due to the weak dependence of the unipolar conductance on $D_W$, the lines for $E_1=0.05$ eV run on top of each other.

Figure 3

Hall conductance as a function of junction width $D_W$. The device is configured in the same way as Fig. 2, but instead of varying the on-site energies in the device, we fix the filling factors. In this case, the device is configured as $E_1=-E_2=0.0875$ eV, or ${\nu }_n=-{\nu }_p=6$. The effect is demonstrated for two different magnetic fields: $B=4$ and 14 Tesla. When the junction fully mixes all of the Landau levels, for junction widths shorter than about 40 nm, the conductance is approximately 3.$0e^2/h$. For junction widths over 100 nm, the device only mixes the lowest Landau level, yielding a conductance of approximately 1.$0e^2/h$. Our simulations indicate that stronger magnetic fields cause junction width to have a more pronounced effect in restricting mixing of higher order Landau levels.

Figure 4

Maps of the nonequilibrium carrier density injected from the left contact for two different junction widths, (a) 30 nm and (b) 125 nm. The full model described in the text is included; there is a finite junction width, interface roughness, and delta function disorder in the transition between the $n$ region (left side) and the $p$ region (right side). The edges of the junction are indicated by white dashed lines. The electron density enters from the bottom left of the junction, travels along the edge, turns up at the junction interface, and then turns either left or right at the top of the junction. In this case we configure the junction the same as in Fig. 3, with $E_1=-E_2=0.0875$ eV and $B=4$ T. In (a), the two Landau levels are essentially located on top of each other. This is an example of a junction which would mix Landau levels and show plateaus in the ambipolar conductance. In (b), the first Landau level turns up at the junction interface, while the $0\mathrm{th}$ Landau level continues until it turns at the middle of the junction. At a junction width of 125 nm, the Landau levels are spatially separated and the device will only show a single plateau from the mixing of the $0\mathrm{th}$ Landau level.

Figure 5

Spatial maps of the electron probability density injected from the left contact. In these simulations we fix the on-site energy variation (solid black lines in the lower panels) and vary the Fermi energy (dotted lines in the lower panels). The junction width is 100 nm and the applied magnetic field is $B=10$ T. The position where each Landau level turns at the junction is closely predicted by (2). In (a), ${\nu }_n=2$, only the $0\mathrm{th}$ Landau level transports. The current turns at the charge neutrality point. In (b), ${\nu }_n=6$, so two Landau levels transport. The $0\mathrm{th}$ Landau level current turns at the charge neutrality point and the first Landau level current turns close to the edge of the left junction interface. (c) Similar to (b) but for ${\nu }_n=10$. The Landau level spacing in these plots is fixed by the slope of the on-site energy and the Landau level energies given by (1). (d-f) Energy band diagrams corresponding to (a-c), respectively, with scatter points depicting the turning point of current in each Landau level.

Figure 6

(a) Spatial map of the electron probability density injected from the left contact. The applied magnetic field is $B=4$ T and $E_1=-E_2=0.125$ eV, yielding a filling factor of ${\nu }_n=-{\nu }_p=14$. We see four Landau levels transporting, each turning up at the junction at positions determined by (2). No roughness or delta function disorder is used, for clarity. Interestingly, the higher order Landau levels do not take on the form of Gaussians, instead having a more complex structure. In (b) we plot a slice of the simulation in (a) taken at $y=100$ nm (denoted by a white dashed line). In addition, we plot $P\left(x\right)=M_n{\mathrm{\Psi }}_{n,k_y}{\left(x\right)}^{†}{\mathrm{\Psi }}_{n,k_y}\left(x\right)$, using the analytical form of the wave function (14) which was calculated in Ref. [25].