Graphene Dirac fermions in one-dimensional inhomogeneous field profiles: Transforming magnetic to electric field

We show that the low-energy electronic structure of graphene under a one-dimensional inhomogeneous magnetic field can be mapped into that of graphene under an electric field or vice versa. As a direct application of this transformation, we find that the carrier velocity in graphene is isotropically reduced under magnetic fields periodic along one direction with zero average flux. This counterintuitive renormalization has its origin in the pseudospin nature of graphene electronic states and is robust against disorder. In magnetic graphene superlattices with a finite average flux, the Landau level bandwidth at high fields exhibits an unconventional behavior of decreasing with increasing strength of the average magnetic field due to the linear energy dispersion of graphene. As another application of our transformation relation, we show that the transmission probabilities of an electron through a magnetic barrier in graphene can directly be obtained from those through an electrostatic barrier or vice versa.

Figure 1

(Color online) (a) Structure of pristine graphene. (b) Band dispersion of pristine graphene near the $K$ point. (c) Structure of a 1D MGS, with the darker regions denoting a magnetic field pointing along the $-z$ direction and lighter regions denoting a magnetic field pointing along the $+z$ direction. This structure repeats itself in both the $x$ and $y$ directions. (d) Isotropically renormalized band structure of a 1D MGS of the kind shown in (c).

Figure 2

(Color online) Ratio of the Fermi velocity $v_g$ in the presence of a periodic magnetic field to the Fermi velocity $v_0$ of pristine graphene near the $K$ point is plotted as a function of the vector potential magnitude $A_p$ for both a $\delta$-function magnetic field and a piecewise constant magnetic field Kronig-Penney superlattices. The analytical (lines) and numerical (symbols) results are in agreement. $l_0$ is the superlattice period and $B_l=\hslash c/\left(e{l}_0^2\right)$ is the characteristic magnetic field strength associated with $l_0$. The group velocity is identical in all directions. For $l_0=100\text{mm}$ and a magnetic field of 1.8 T, $v_0$ is renormalized by a factor of 1/2.

Figure 3

(Color online) (a) Energy bands for a piecewise constant magnetic field pattern with magnetic field strength $B_p/B_l=1.4$, immersed in a uniform background field of $B_0/B_l=0.6$, where $B_l=\hslash c/\left(e{l}_0^2\right)$ is the characteristic magnetic field strength associated with the superlattice periodicity. The first 10 bands are plotted. (b) Bandwidths $\Delta E$ of the first three bands, as a function of $B_0$, of a piecewise constant magnetic field with magnetic field strength $B_p/B_l=2$, immersed in a uniform background field of strength $B_0$.

Figure 4

(Color online) (a) Comparison of the period of oscillation of the Landau bands in $k_y$ as obtained from the numerical calculations with the analytic prediction that this period is equal to $l_0/l_B^2$, where $l_0$ is the size of the unit cell and $l_B=\sqrt{\hslash c/\left(e⟨B⟩\right)}$ is the magnetic length associated with the average background magnetic field strength. (b) Three representative wave functions for the system with $B\ne 0$. The same parameters are used as in Fig. 3a: a piecewise constant magnetic field pattern with magnetic field strength $B_p/B_l=1.4$, immersed in a uniform background field of $B_0/B_l=0.6$, where $B_l=\hslash c/\left(e{l}_0^2\right)$ is the characteristic magnetic field strength associated with the superlattice periodicity. The index n in this figure refers to the $n\text{th}$ Landau level as defined in Fig. 3a.

Figure 5

(Color online) (a) Vector potential and (b) electrostatic barriers in graphene and the transmission probabilities through the (c) vector potential and (d) electrostatic barriers, as a function of incident angle.

Figure 6

(Color online) Ensemble-averaged density of states of a random magnetic superlattice, with randomness parameters $r=0.05$ and $r=0.1$ (see text), compared to a perfectly periodic superlattice $\left(r=0\right)$. The density of states of pristine graphene is also shown ($r=0$, B field off). The velocity reduction factor that corresponds to this change in density of states is $v/v_0=0.58$. The energy range in this plot is 1/2 the bandwidth of an empty-lattice graphene superlattice.