Theory of the strain-induced magnetoelectric effect in planar Dirac systems

The magnetoelectric response in inversion-breaking two-dimensional Dirac systems induced by strain is analyzed. It is shown that, in the same way that the piezoelectric response in these materials is related to the valley Chern number, the magnetic field and strain-induced electric polarization is related to the nontrivial Berry curvature and the derivative of the orbital magnetic moment per valley. We also show that the elastic vector field ${\mathcal{A}}^{el}$ present in these systems drives the system out of equilibrium in presence of a magnetic field, allowing for such electrical response. In contrast, we show that the strain plays a very different role in the generation of a nonequilibrium magnetization and optical activity. In this case, it is the modification of the band structure induced by strain that allows for the magnetic response to external electric perturbations.

Figure 1

Behavior of the function $h(x,\delta )$ of the magnetoelectric tensor with the chemical potential normalized to the mass parameter $m$ for massive Dirac, $\delta =0$ (blue solid line), ${\mathrm{MoS}}_2$ (orange dashed line) with $\delta =\frac{1}{20}$, and for comparison, the case with $\delta =\frac{1}{3}$ (green solid line).

Figure 2

(a) Kerr angle as a function of the frequency $\omega$ centered at $\omega =2\mu$, in units of the Dirac mass $m$, for chemical potential $\mu =\{0.5,1.1,1.5,2,3,10\}m$ as the black, purple, blue, green, yellow, and red curves, respectively, for the parameters given in the text. In the inset we show the imaginary part of ${\sigma }_{12}$ for the same parameters, showing the direct proportionality of the Kerr angle with the imaginary part of ${\sigma }_{12}$ for small Hall conductivities. Note that we only have ${\sigma }_{12}\ne 0$ (and therefore ${\theta }_K\ne 0$) if $\mu >\left|m\right|$. (b) Kerr angle as a function of the frequency $\omega$ centered at $\omega =2\mu$, for $\mu =10m$ and scattering time $\mu =\{1,{10}^{1/2},10,{10}^{3/2},{10}^2\}{\tau }_0$ as the black, purple, blue, green, and red curves, respectively, where ${\tau }_0$ is the scattering time given in the text. For high $\tau$, the peak in ${\sigma }_{12}$ grows and narrows as described in the text, and the nonlinear dependence of ${\theta }_K$ with ${\sigma }_{\mu \nu }$ becomes apparent. In the inset we show the scaling law of $\mathbb{I}\text{m}\left[{\sigma }_{12}(x_{\tau },\tau )\right]$ [Eq. (33)].

Figure 3

(a) Three-dimensional Gaussian deformation of a 2D material. (b) In-plane deformation of the 2D material induced by the Gaussian deformation. (c) Piezoelectric-induced charge density (divided by $\frac{e}{4\pi }$) and (d) magnetic-field-induced charge density (in units of $\tau \frac{3e^2}{4\pi }{B}_3\frac{3}{8m}{v}^2$), both cases for $\mu =2m$.