Quantum Hall states in graphene from strain-induced nonuniform magnetic fields

We examine strain-induced quantized Landau levels in graphene. Specifically, arc-bend strains are found to cause nonuniform pseudomagnetic fields. Using an effective Dirac model which describes the low-energy physics around the nodal points, we show that several of the key qualitative properties of graphene in a strain-induced pseudomagnetic field are different compared to the case of an externally applied physical magnetic field. We discuss how using different strain strengths allows us to spatially separate the two components of the pseudospinor on the different sublattices of graphene. These results are checked against a tight-binding calculation on the graphene honeycomb lattice, which is found to exhibit all the features described. Furthermore, we find that introducing a Hubbard repulsion on the mean-field level induces a measurable polarization difference between the $A$ and the $B$ sublattices, which provides an independent experimental test of the theory presented here.

Figure 1

Semiclassical trajectories of the propagating states in a two-dimensional electron gas in a gradient magnetic field. Besides the semicircular edge states, also present in the case of a constant magnetic field, there is a bulk propagating state.

Figure 2

(a) Energy spectrum around the $K$ and $K^{\prime }$ points for $H=1$ and $\tilde{L}=10$ [defined in Eq. (7)] in the presence of a linearly varying external magnetic field. The solid (dashed) red curves correspond to the lowest Landau level ($n=0$) around the $K$ ($K^{\prime }$) point, and the solid (dashed) blue curves correspond to the $n=1$ level around the $K$ ($K^{\prime }$) point. (b) Magnitudes of the wave functions of the $n=0$ edge states and $n=1$ bulk state. The blue curves around $\tilde{y}=0$ are the LLLs evaluated at ${\tilde{k}}_x=-6.6297$ and the red curves around $\tilde{y}=\tilde{L}$ are the LLLs evaluated at ${\tilde{k}}_x=-48.9882$ corresponding to a dimensionless energy [Eq. (8)] of ${\tilde{\varepsilon }}^2\approx 0.22$. The curves around $\tilde{y}=4$ are $n=1$ bulk states evaluated at ${\tilde{k}}_x=-30$ around the $K$ point [solid (blue) curves] and $K^{\prime }$ point [dashed (red) curves] corresponding to a dimensionless energy [Eq. (8)] of ${\tilde{\varepsilon }}^2\approx 12.61$.

Figure 3

(a) Energy spectrum around the K and $K^{\prime }$ points for $H=1$, $\tilde{L}=10$ [defined in Eq. (6)] in the presence of a strain-induced pseudomagnetic field. The solid (dashed) red curves correspond to the lowest Landau level ($n=0$) around the $K$ ($K^{\prime }$) point, and the solid (dashed) blue curves correspond to the $n=1$ level around the $K$ ($K^{\prime }$) point. (b) Magnitude of the wave functions of the LLL edge states and $n=1$ bulk state. The curves around $\tilde{y}=0$ are the LLLs evaluated at ${\tilde{k}}_x=48.98707$ (blue curves) and ${\tilde{k}}_x=-48.98707$ (red curves) corresponding to a dimensionless energy of ${\tilde{\varepsilon }}^2\approx 0.22$. The curves around $\tilde{y}=4$ are $n=1$ bulk states evaluated at ${\tilde{k}}_x=-30$ around the $K$ point [solid (blue) curves] and $K^{\prime }$ point [dashed (red) curves] corresponding to a dimensionless energy [Eq. (8)] of ${\tilde{\varepsilon }}^2\approx 12.61$.

Figure 4

Contributions to the local density of states from the $K$ and $K^{\prime }$ point states with ${\tilde{\varepsilon }}^2\in (0,0.25]$ for four different gradient pseudomagnetic fields. (a) LDOS on the $A$ sublattice. (b) LDOS on the $B$ sublattice.

Figure 5

Illustration of an inhomogeneous arc-bend strain, leading to an approximately constant gradient in the induced pseudomagnetic field. An originally rectangular graphene ribbon of length $L$ and width $W$ is distorted into a fan-shaped segment of a circular shell with inner radius $R$ and outer radius $R+L$.

Figure 6

The two lowest-lying energy levels calculated using the tight-binding model ($U=0$). (a) Energy spectrum of the tight-binding model on the graphene lattice without a strain. (b) Energy spectrum of the tight-binding model on the graphene lattice in the presence of an arc-bend strain with $R=5W$.

Figure 7

Local density of states (LDOS) as a function of energy in a graphene sheet with arc-bend strain, calculated using the tight-binding model ($U=0$). (a, b) Darker shades correspond to a higher intensity LDOS. (c, d) LDOS near the edges in the tight-binding mode of a graphene ribbon. The blue curve is for an arc-bend strain of $R=5.1W$, the orange curve corresponds to $R=3.6W$, and the red curve corresponds to $R=3W$. (a) Arc-bend strain with $R=5W$. (b) Arc-bend strain with $R=3W$. (c) LDOS near the smaller inner edge integrated over a small energy window around $\tilde{\varepsilon }=0$. (d) LDOS near the stretched outer edge integrated over a small energy window around $\tilde{\varepsilon }=0$.

Figure 8

Local magnetization in a graphene sheet calculated using the Hubbard model. Red symbols denote negative $z$ polarization, whereas blue symbols denote positive $z$ polarization. (a) Unstrained case. (b) Strained case.