Scaling laws for nonlinear electromagnetic responses of Dirac fermion

We theoretically propose that the Dirac fermion in two dimensions shows the giant nonlinear responses to electromagnetic fields in the terahertz region. A scaling form is obtained for the current and magnetization as functions of the normalized electromagnetic fields $E/E_{\omega }$ and $B/B_{\omega }$, where the characteristic electric (magnetic) field $E_{\omega }\left(B_{\omega }\right)$ depends on the frequency $\omega$ as $\hslash {\omega }^2/e{v}_F(\hslash {\omega }^2/e{v}_F^2)$, and is typically of the order of 80 V/cm (8 mT) in the terahertz region. Applications of the present theory to graphene and surface state of a topological insulator are discussed.

Figure 1

Schematic picture of nonlinear responses of a Dirac fermion.

Figure 2

Two regimes of nonlinear responses. Strong nonlinearity appears when $E>E_{\omega }$ and $E>E_c$. Weak nonlinearity appears otherwise. A similar phase diagram applies for the $\omega -B$ plane.

Figure 3

Diagrams for the nonlinear response ${\chi }_{1,2}({\omega }^{\prime },\omega )$. Panels (a), (b), and (c) correspond to $A_1,A_2$, and $A_3$ in Eq. (B14), respectively.