Uniformity of the pseudomagnetic field in strained graphene

We present a numerical study on the uniformity of the pseudomagnetic field in graphene as a function of the relative orientation between the graphene lattice and straining directions. We calculated the deformation of a regular micron-sized graphene hexagon by symmetrically displacing three of its edges. We found that the pseudomagnetic field is strongest if the strain is applied perpendicular to the armchair direction of graphene. For a hexagon with a side length of $1\mu \text{m}$, the pseudomagnetic field has a maximum of 1.2 T for an applied strain of 3.5% and it is uniform (variance $<1$%) within a circle with a diameter of $\sim 520$ nm. This diameter is on the order of the typical diameter of the laser spot in a state-of-the-art confocal Raman spectroscopy setup, which suggests that observing the pseudomagnetic field in measurements of shifted magnetophonon resonance is feasible.

Figure 1

(a) The lattice structure of graphene, in which $\theta$ indicates the rotation angle between the armchair direction and the $x$ axis. The red and blue atoms indicate the two different sublattices, and the ${\mathbf{d}}_{\mathbf{n}}$ indicates the nearest neighbor vectors. (b) The geometry used in our calculations is a regular hexagon with side length $L$. The force $F$ is applied to the indicated edges.

Figure 2

(a) The strain in the geometry for zero rotation angle $\theta$. There are three edges with zero strain and three with $15\%$ of strain. The corners of the geometry exhibit the largest amount of strain (up to $50\%)$, whereas the strain reduces to 0% in the center. (b) The pseudomagnetic field as a function of the rotation angle $\theta$. At $\theta ={30}^{\circ }$ the pseudomagnetic field disappears in the center of the geometry, as the atomic displacement is parallel to the nearest neighbor vectors. (c) The pseudomagnetic field in the center of the geometry is in excellent agreement with that from Eqs. (12) and (13). The pseudomagnetic field is normalized with respect to the field at $\theta =0^{\circ }$.

Figure 3

(a) The average pseudomagnetic field and its corresponding normalized standard deviation are calculated within a circle of diameter $d$ that is centered in the geometry. (b) The upper panel shows the pseudomagnetic field as a function of the diameter $d$ for different rotation angles. The bottom panel shows that all these curves fall almost on top of each other if one normalizes each curve with respect to the value of the pseudomagnetic field in the center. (c) The standard deviation of the pseudomagnetic field as a function of the diameter $d$ for the rotation angles shown in (b). The generated pseudomagnetic field is uniform within $d=34$ nm for $\theta =0^{\circ }$. The inset shows a zoom for small $d$.

Figure 4

(a) The computed strain field in a graphene sheet that can be reached with a state-of-the-art MEMS device shows values $(\sim 20\%)$ close to the maximum strain that graphene can withstand. The force $F$ is applied to the indicated edges. (b) The corresponding pseudomagnetic field for various rotation angles $\theta$. (c) The pseudomagnetic field as a function of $d$ and (d) the corresponding standard deviation. In this case, the pseudomagnetic field is uniform within $d=520$ nm.

Figure 5

The solution for the radius $r$ for $c=1$ $\mu \text{m}$ as a function of the angle $\theta$ (a) and its corresponding geometry for $\left|r\right|$ (b). In contrast to what was thought, the resulting pseudomagnetic field is not uniform (c). (d) The linearly varying forces (left panel) on two opposing boundaries of a rectangle that generate a uniform pseudomagnetic field. The right panel shows the strain in a grayscale as well as the resulting shape of the geometry.