Theory of nonlinear response for charge and spin currents

The nonlinear Hall effect, which is the second-order harmonic charge Hall effect from the Berry curvature dipole in momentum space, has received much attention recently. As the responses to higher harmonics of the driving ac electric field are prominent and measurable, we develop a general nonlinear theory by taking the charge and spin currents as well as the longitudinal and transverse effects into account. We introduce the expansion order of the electric field and Berry curvature multipole moment, where the Berry curvature dipole is a particular one manifesting itself at the second harmonic order and the second expansion order of the electric field. There are four cases with conserving or breaking the time-reversal symmetry (TRS) and inversion symmetry (IS). We find a specific “selection rule” that only longitudinal odd harmonic order charge currents exist for conserving both the TRS and IS, and with breaking both symmetries, all harmonic order charge and spin currents are nonzero. With conserving TRS and breaking IS, the charge Hall current exists at even harmonic order, and the longitudinal charge current occurs at odd harmonic order. Only the longitudinal spin current survives at even harmonic order. With breaking TRS and conserving IS, only odd harmonic order charge and spin currents can appear. Moreover, we observe that every harmonic order current contains a series of infinite-order expansion of the electric field. We further show that the Berry curvature dipole and quadrupole can be determined by measuring the second and fourth harmonic order currents in experiments. This may open a way to explore the higher responses of an ac driving system.

Figure 1

(a) The energy structure of the tilted 2D massive Dirac model in Eq. (13) at $k_y=0$. (b), (d), and (f) The distribution of the Berry curvature $\mathrm{\Omega }$, Berry dipole density $d_{xz}^{+}={\partial }_x{\mathrm{\Omega }}_z^{+}$, and Berry quadrupole density $q_{xxxz}^{+}={\partial }_x^3{\mathrm{\Omega }}_z^{+}$ in momentum space. The Berry curvature dipole and quadrupole vary with the Fermi energy $E_f$ in (c) and (e), respectively. All these are only for the conduction band. The other parameters are $v=1$ eV Å, $t/v=0$, 0.2, 0.4, 0.6, and $\mathrm{\Delta }=0.1$ eV.

Figure 2

(a) The charge current and (b) spin current as a function of Fermi energy $E_f$. Here, the driving electric field is in the $x$ direction and the magnitude is ${\mathcal{E}}_x=4\times {10}^5$ V/m, and $\tau =1$ ps. The other parameters are $v=1$ eV Å, $t/v=0.2$, and $\mathrm{\Delta }=0.1$ eV.