Mode mixing induced by disorder in a graphene $pnp$ junction in a magnetic field

We study the electron transport through the graphene $pnp$ junction under a magnetic field and show that mode mixing plays an essential role. By using the nonequilibrium Green's function method, the space distribution of the scattering state for a specific incident mode as well the elements of the transmission and reflection coefficient matrices are investigated. All elements of the transmission (reflection) coefficient matrices are very different for a perfect $pnp$ junction, but they are the same at a disordered junction due to the mode mixing. The space distribution of the scattering state for the different incident modes also exhibits similar behaviors, i.e., that they distinctly differ from each other in the perfect junction but are almost the same in the disordered junction. For a unipolar junction, when the mode number in the center region is less than that in the left and right regions, the fluctuations of the total transmission and reflection coefficients are zero, although each element has a large fluctuation. These results clearly indicate the occurrence of perfect mode mixing and play an essential role in a graphene $pnp$ junction transport.

Figure 1

(a) Schematic of a multiterminal scattering system. The center scattering region is connected to an injecting lead (lead $\beta$) and several outgoing leads (lead $\alpha$). (b) Wires with primitive cell consisted of a simple layer (upper) and multiple layers (lower). Suppose the Hamiltonian between each of the two adjacent layers is invertible. A primitive cell in the upper wire contains only one layer, so that the Hamiltonian $H_1$ between two adjacent primitive cells is invertible. However, the Hamiltonian $H_1$ is noninvertible in the lower wire whose primitive cell contains multiple layers.

Figure 2

This is a map of onsite energy for a random disorder configuration. The graphene $pnp$ junction consists of the left and right $p$ regions and the center $n$ region. The center scattering region includes the center $n$ region and a part of the left and right $p$ regions. The disorder only exists in the center scattering region near the boundary of nanoribbon and $pn$ interface.

Figure 3

(a) The conductance versus onsite energy $V_L$ with different disorder strengths, while the theoretical conductance is given as black dashed line. (b) The conductance versus disorder strength with different onsite energy $V_L$ (i.e., filling factors ${\nu }_L$). The onsite energy $V_L$ is 1.0, 0.3, $-0.3$, 2.2, and 2.8 eV, from upper to lower in the legend. The onsite energy $V_G$ is fixed on 0.15 eV with the filling factor ${\nu }_G=3$, and the magnetic field $\varphi =0.1$ for a hexagonal lattice.

Figure 4

Numerical simulation for (3,5,3) armchair graphene $pnp$ junction. (a), (b) Show the nine elements of the transmission coefficient matrix $T_{jl}$ and the nine elements of the reflection coefficient matrix $R_{jl}$ versus the disorder strength $W$. In panel (a), the three curves with the value being about 1 at $W=0$ are $T_{11},T_{22}$, and $T_{33}$. While at $W\sim 0.6\mathrm{eV}$, nine curves in (a) and (b) converge together and they almost overlap in $W>0.6\mathrm{eV}$. $V_L=-0.3\text{eV}$ and the other parameters are same as in Fig. 3.

Figure 5

Numerical simulation for (3,5,3) armchair graphene $pnp$ junction. Each panel shows the nine elements of the transmission coefficient matrix $T_{jl}$ versus the disorder strength $W$ under different conditions. Panels (a) and (b) are under short-range disorder like in Fig. 4, while the magnetic flux $\varphi =0.05$ and 0.18, respectively. When the magnetic flux $\varphi$ varies, the intervals between the Landau levels change also. In order to let the junction keep in the (3,5,3) case, the onsite energies are set $V_G=-0.6\mathrm{eV}$ and $V_L=-0.3\mathrm{eV}$ while $\varphi =0.05$, and $V_G=0.3\mathrm{eV}$ and $V_L=0.9\mathrm{eV}$ while $\varphi =0.18$. Panels (c) and (d) are under the common magnetic flux $\varphi =0.1$, while the disorder is long range with $\eta =2$ and 5, respectively. The curve which converges slowest (dark red) in (c) and (d) stands for $T_{11}$. The other parameters are same as in Fig. 3.

Figure 6

The space distribution of the wave functions $|{\psi }^{\left(lL\right)}{|}^2$ [(a)–(c) and (g)–(i)] and the corresponding current density [(d)–(f) and (j)–(l)] of scattering states in (3,5,3) armchair graphene $pnp$ junction. The color indicates the intensity of wave function and current, and the arrows in current density pictures indicate the orientation. Panels (a)–(f) are for the perfect $pnp$ junction without disorder ($W=0\mathrm{eV}$), where all the three scattering states are perfect conducting channel. The disorder strength at (g)–(l) is $W=0.5\mathrm{eV}$. The columns from left to right are for the first, second, and third incident modes, respectively. $V_L=-0.3$ eV and the other parameters are same as in Fig. 3.

Figure 7

Numerical simulation for (3,1,3) armchair graphene $pnp$ junction. (a), (b) Present the nine elements of transmission and reflection coefficient matrices vs disorder strength $W$, respectively. Here, the total transmission and reflection coefficients, the sum of all nine elements, are shown also (see the black line). (c), (d) Show the fluctuation of each element and total transmission and reflection coefficients vs disorder strength $W$. Notice that the fluctuation of the total transmission (reflection) coefficient $T$ ($R$) is not equal to the sum of the fluctuation of the nine elements $T_{jl}$ ($R_{jl}$), although $T={\sum }_{jl}{T}_{jl}$ ($R={\sum }_{jl}{R}_{jl}$). $V_L=1.0$ eV and the other parameters are same as in Fig. 3.

Figure 8

The space distribution of the wave functions $|{\psi }^{\left(lL\right)}{|}^2$ of scattering states in (3,1,3) armchair graphene $pnp$ junction. The color indicates the intensity of wave function. Panels (a)–(c) are for the perfect $pnp$ junction without disorder ($W=0\mathrm{eV}$), while the disorder strength at (d)–(f) is $W=1\mathrm{eV}$. The columns from left to right are for the first, second, and third incident modes, respectively. $V_L=1.0$ eV and the other parameters are same as in Fig. 3.

Figure 9

(a), (b) Present the nine elements of transmission and reflection coefficient matrices vs disorder strength $W$ in (3,$-1,3$) armchair graphene $pnp$ junction, respectively. $V_L=2.2$ eV and the other parameters are same as in Fig. 3.

Figure 10

Numerical simulation for (3,1,3) Stone-Wales edge-reconstructed zigzag graphene $pnp$ junction. (a), (b) Show the nine elements of transmission and reflection coefficient matrices as well the total transmission and reflection coefficients vs disorder strength $W$. (c), (d) Show the fluctuation of each element and total transmission and reflection coefficients vs $W$. The parameters are same as in Fig. 7.