Quantum Geometry and the Electric Magnetochiral Anisotropy in Noncentrosymmetric Polar Media

The electric magnetochiral anisotropy is a nonreciprocal phenomenon accessible via second harmonic transport in noncentrosymmetric, time-reversal invariant materials, in which the rectification of current, I, can be controlled by an external magnetic field, B. Quantum geometry, which characterizes the topology of Bloch electrons in a Hilbert space, provides a powerful description of the nonlinear dynamics in topological materials. Here, we demonstrate that the electric magnetochiral anisotropy in noncentrosymmetric polar media owes its existence to the quantum metric, arising from the spin-orbit coupling, and to large Born effective charges. In this context, the reciprocal magnetoresistance βB² is modified to $R(I,P,B)={{0ex}{0ex}}{R}₀[1+{β}{B}²+{{γ}}^{±{}}I·{}(P×{}B)]$, where the chirality dependent ${{γ}}^{±{}}$ is determined by the quantum metric dipole and $P$ is the polarization. In 2D, we predict a universal scaling ${{γ}}^{±{}}(V){~}{V}^{{-}5/2}$, which we compare to available phase sensitive, second harmonic transport measurements on hydrothermally grown tellurium films under applied gate voltage, $V$. The control of rectification by varying $I$, $P$, $B$, and $V$, demonstrated in this work, opens up new avenues for the building of ultrascaled complementary metal-oxide-semiconductor circuits.