Programmable Extreme Pseudomagnetic Fields in Graphene by a Uniaxial Stretch

Many of the properties of graphene are tied to its lattice structure, allowing for tuning of charge carrier dynamics through mechanical strain. The graphene electromechanical coupling yields very large pseudomagnetic fields for small strain fields, up to hundreds of Tesla, which offer new scientific opportunities unattainable with ordinary laboratory magnets. Significant challenges exist in investigation of pseudomagnetic fields, limited by the nonplanar graphene geometries in existing demonstrations and the lack of a viable approach to controlling the distribution and intensity of the pseudomagnetic field. Here we reveal a facile and effective mechanism to achieve programmable extreme pseudomagnetic fields with uniform distributions in a planar graphene sheet over a large area by a simple uniaxial stretch. We achieve this by patterning the planar graphene geometry and graphene-based heterostructures with a shape function to engineer a desired strain gradient. Our method is geometrical, opening up new fertile opportunities of strain engineering of electronic properties of 2D materials in general.

Figure 1

Producing uniform pseudomagnetic fields in a planar shaped graphene strip under a uniaxial stretch. (a) Schematic showing a graphene nanoribbon of varying width under a uniaxial stretch producing a pseudomagnetic field, $B_{\mathrm{ps}}$. The red circle denotes cyclotron orbits in the field giving rise to pseudo-Landau levels in (h). (b)–(d) Contour plots of the resulting strain components in the graphene, ${\epsilon }_{xx}$, ${\epsilon }_{yy}$, and ${\epsilon }_{xy}$, respectively, under a 5% uniaxial stretch. (e) Resulting pseudomagnetic fields in the graphene nanoribbon shown in (a) under a uniaxial stretch of 5%, 10%, and 15%, respectively. (f) Intensity of the pseudomagnetic field as the function of location along the centerline of the graphene ribbon for various applied uniaxial stretches. (g) Intensity of the pseudomagnetic field is shown to be linearly proportional to the applied uniaxial stretch and inversely proportional to the length of the graphene ribbon $L$. (h) Local density of states of unstrained graphene and graphene with a constant strain gradient determined by density functional theory calculations. $N=0$ and $N=\pm 1$, $\pm 2$, $\pm 3$ Landau levels, corresponding to cyclotron motion in a magnetic field are seen to emerge in the strained graphene, demonstrating a uniform pseudomagnetic field. The wiggles in the results for the unstrained case result from finite size effects in the calculations. See Supplemental Material for further discussion [37].

Figure 2

Producing uniform pseudomagnetic fields in planar graphene-based heterostructures under a uniaxial stretch. (a) Schematic showing a 2D heterostructure consisting of graphene and graphane (or h-BN) bonded to a center piece of graphene under a uniaxial stretch. (b) Left: Intensity of the resulting pseudomagnetic field in the graphene domain of a graphene-graphane and a graphene–h-BN heterostructure, respectively, under a 15% uniaxial stretch. Here the top-bottom width ratio of the graphene domain is $f_r=0.5$. Right: Contour plot of the resulting pseudomagnetic field in the graphene-graphane heterostructure. (c) Left: Intensity of the resulting pseudomagnetic field in the graphene domain of a graphene-graphane and a graphene–h-BN heterostructure, respectively, under a 15% uniaxial stretch. Here the top-bottom width ratio of the graphene domain is $f_r=0$. Right: Contour plot of the resulting pseudomagnetic field in the graphene-graphane heterostructure.