Berry Curvature Dipole in Strained Graphene: A Fermi Surface Warping Effect

It has been recently established that optoelectronic and nonlinear transport experiments can give direct access to the dipole moment of the Berry curvature in nonmagnetic and noncentrosymmetric materials. Thus far, nonvanishing Berry curvature dipoles have been shown to exist in materials with substantial spin-orbit coupling where low-energy Dirac quasiparticles form tilted cones. Here, we prove that this topological effect does emerge in two-dimensional Dirac materials even in the complete absence of spin-orbit coupling. In these systems, it is the warping of the Fermi surface that triggers sizable Berry dipoles. We show indeed that uniaxially strained monolayer and bilayer graphene, with substrate-induced and gate-induced band gaps, respectively, are characterized by Berry curvature dipoles comparable in strength to those observed in monolayer and bilayer transition metal dichalcogenides.

Figure 1

(a) Top view of the uniformly strained Bernal-stacked bilayer graphene lattice. Red and blue carbon atoms correspond to different layers and are not equivalent due to the applied voltage that breaks inversion symmetry. Black sites indicate the overlap between atoms of the two different layers. By applying an electric field ${\mathcal{E}}_x$ parallel to the dipole $D_x$, a nonlinear Hall current $J_{nlH}$ is generated. (b) Strain deformation of the Brillouin zone. The warped low-energy dispersion is shown in the vicinity of the two unstrained high symmetry points $K$ and $K^{\prime }$.

Figure 2

Berry curvature $\mathrm{\Omega }$ and dipole density ${\partial }_{k_x}\mathrm{\Omega }$ of the conduction band corresponding to the Hamiltonian of Eq. (2) for unstrained (a),(c) and strained (b),(d) monolayer graphene. When strain is present the Fermi surface is deformed shifting the dipole moment from zero to a finite value. Momenta are measured in units of the inverse of the lattice constant $a$; the Berry curvature in units of $a^2$ while the dipole density in units of $a^3$.

Figure 3

Berry curvature $\mathrm{\Omega }$ and dipole density ${\partial }_{k_x}\mathrm{\Omega }$ of the conduction band corresponding to the low-energy Hamiltonian of Eq. (3) for different values of the strain: (a) $w=-5{ϵ}_L$, (b) $w=-1{ϵ}_L$, (c) $w=0$, (d) $w=1{ϵ}_L$, (e) $w=5{ϵ}_L$. All plots are shown for the same carrier density fixed by placing the Fermi energy $E_F$ at the Lifschitz transition in the unstrained $w=0$ and gapped case [panel (c)]. Momenta are measured in units of ${\kappa }_L$, the Berry curvature in units of $1/{\kappa }_L^2$, and the dipole density in units of $1/{\kappa }_L^3$.

Figure 4

Berry curvature dipole, measured in units of $1/{\kappa }_L$, in bilayer graphene for $\mathrm{\Delta }=10\mathrm{meV}$ and various strains $w$ as a function of the electron density $n$ measured in units of ${\kappa }_L^2$, where ${\kappa }_L\simeq 0.035{\mathrm{nm}}^{-1}$. Densities of this order of magnitude have been experimentally reported in Refs. [30, 35, 36].