Understanding magnetic focusing in graphene $p\text{−}n$ junctions through quantum modeling

We present a quantum model which provides enhanced understanding of recent transverse magnetic focusing experiments on graphene $p\text{−}n$ junctions. Spatially resolved flow maps of local particle current density show quantum interference and $p\text{−}n$ junction filtering effects, which are crucial to explaining the device operation. The Landauer-Büttiker formula is used alongside dephasing edge contacts to give exceptional agreement between simulated nonlocal resistance and the recent experiment by Chen et al. [Science 353, 1522 (2016)]. The origin of positive and negative focusing resonances and off-resonance characteristics are explained in terms of quantum transmission functions. Our model also captures subtle features from experiment, such as the $p\text{−}{p}^{-}$ to $p\text{−}{p}^{+}$ transition and the second $p\text{−}n$ focusing resonance.

Figure 1

(a) Model of the graphene device depicting the first TMF resonance for $p\text{−}{p}^{\prime }$ and $p\text{−}n$ junctions. The graphene devices we compare with in this paper were encapsulated by hexagonal boron nitride; however, since we do not consider that crystal in our model, we only depict the graphene lattice. (b) Schematic of device with four-terminal measurement configuration. The simulated device has dimensions $D_C=W=200$ nm, $D_W=50$ nm, and $L_C=60$ nm. The red rectangles indicate dephasing contacts used in the simulation. (c) Real-space energy band diagram of the device.

Figure 2

Maps of local particle current density (5) for a $p\text{−}{p}^{\prime }$ $\left[p\text{−}n\right]$ junction in (a) [(b)] the first TMF resonance and (c) [(d)] the second TMF resonance. Darker colors indicate higher magnitude of local particle current density. The $p\text{−}{p}^{\prime }$ junctions in (a) and (c) are configured as $E_1=50$ and $E_2=75$ meV. For the first $p\text{−}n$ TMF resonance in (b), the junction is configured as $E_1=-E_2=50$ meV. The second $p\text{−}n$ TMF resonance in (d) is configured as $E_1=50$ and $E_2=-100$ meV. The scale bars of all figures in this paper are 60 nm.

Figure 3

Table of mode-resolved particle current density for each panel of Fig. 2. The modes of each column are summed to give the final result in Fig. 2. By looking at each mode individually, the interplay between the semiclassical and quantum mechanical nature of the system is visible.

Figure 4

Comparison of particle current density for the device configured as in Fig. 2, both (a) with and (b) without dephasing edge contacts. When the dephasing edge contacts are removed, in (b), carrier density which is not focused into contact four will skip around the edge of the device until it exits out one of the small contacts. The carrier density, which is not dephased, will interfere with the incoming waves and destroy the resonance condition. This results in the extremely chaotic pattern seen in (b), with no observable focusing resonances.

Figure 5

Nonlocal resistance as a function of magnetic field in an (a) $p\text{−}{p}^{\prime }$ and (b) $p\text{−}n$ junction. The junctions are configured as in Fig. 2. To compare to the carrier densities shown in Fig. 7, the left side of (a) is set to $-0.18\times {10}^{12}{\mathrm{cm}}^{-2}$ and the right side is set to $-0.41\times {10}^{12}{\mathrm{cm}}^{-2}$. The left side of (b) is configured the same as (a), but the right side is now set to $+0.18\times {10}^{12}{\mathrm{cm}}^{-2}$. Important transmission (3) functions are plotted for each configuration, as explained in the text.

Figure 6

Off-resonance condition particle current density for the symmetric $p\text{−}n$ junction studied in Fig. 2. The junction is configured as $E_1=-E_2=50$ meV. (a) $B=0.11$ T; (b) $B=0.16$ T. When the magnetic field is not strong enough to focus the carriers into contact four, the transmitted wave collides with the top of the device, to the right of contact four, and skips into contact three. When the magnetic field is too strong, as in (b), the transmitted wave is focused to the left of contact four and again skips into contact three.

Figure 7

(a) Nonlocal resistance map for a fixed $p$-type doping as a function of the carrier density of the right side of the junction and magnetic field. Scatter points mark configurations where we have demonstrated spatially resolved particle current density in Fig. 2. Our results show exceptional agreement with (b) experimental measurements of Chen et al. [8], reproduced with copyright permission.