Topological Surface Charge Detection via Active Capacitive Compensation: A Pathway to the 4D Quantum Hall Effect

The topological magnetoelectric effect (TME) in three-dimensional topological insulators (TIs), described by $\Delta P = \frac{e^2}{2h} N_{\rm Ch}^{(2)} \Delta B$, serves as a condensed-matter realization of the four-dimensional quantum Hall effect (4D QHE). In dual-gate axion-insulator devices, the TME-induced polarization yields a current $I_{\rm TME} \propto (C_{\rm total}/C_{\rm S})\,Q_{\rm 4D\text{-}QHE}$, where the signal is suppressed by the capacitance ratio $C_{\rm total}/C_{\rm S}$. Here we propose an active compensation scheme that introduces a tunable negative capacitance $C_{\rm comp} \approx -C_{\rm gate}$ into the gate line, effectively canceling the gate dielectric capacitance and driving $C_{\rm total}/C_{\rm S} \to 1$. We validate the method using a quantum anomalous Hall (QAH) device, which shares the same surface-state physics with the axion insulator but permits direct charge measurement via a single gate, recovering over $95\%$ of the quantized charge signal from an initially half-attenuated state. This compensation method provides a robust means of resolving minute TME signals, offering a promising pathway toward direct measurements of the 4D QHE.