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

Abstract

The topological magnetoelectric effect (TME) in three-dimensional topological insulators (TIs), described by $ΔP = \frac{e^2}{2h} N_{\rm Ch}^{(2)} ΔB$, 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.

Figures (4)

• Figure 1: Principles of active capacitive compensation and the charge accumulation measurement in QAH systems.a, Transport conductivities of the QAH sample as a function of the DC magnetic field ($\mu_0 H_{\text{DC}}$). The system exhibits a robust QAH state, characterized by a Hall conductivity ($\sigma_{xy}$) strictly quantized to $\pm e^2/h$ and a vanishing longitudinal conductivity ($\sigma_{xx}$) outside the coercive field region. b, Field-induced surface charge accumulation ($\eta_\text{a} / B_{\text{AC}}$) versus $\mu_0 H_{\text{DC}}$ measured without compensation in an ultra-low $\sigma_{xx}$ regime at $f = 277.777$ Hz, measured on a simple disk-shaped sample. In-phase and quadrature components are plotted separately. Data from different field sweeping directions are distinguished by colors. c, Schematic of the field-induced charge accumulation driven by an out-of-plane AC magnetic field ($B_{\text{AC}}$). The red disks represent the sample region with a uniformly distributed charge accumulation ($Q_\text{a} = A_\text{S} \eta_\text{a}$, where $\eta_\text{a} = \sigma_{xy} B_\text{AC}$ is the charge density and $A_\text{S}$ is the sample area). The top yellow disks represent the top gate. Top: conventional uncompensated measurement ($U_{\text{TG}} = 0$), where a finite sample potential drives dissipation current toward the contacts (yellow rings) via the longitudinal conductivity ($\sigma_{xx}$). Bottom: active capacitive compensation applies a feedback voltage ($U_{\text{TG}} = -Q_\text{a}/C_1$) to suppress the internal potential gradient and reduce dissipation; the feedback can be viewed as a negative capacitance ($-C_1$) in series with the gate circuit. d, Circuit diagram detailing the experimental realization of the feedback voltage. The circuit establishes an effective negative capacitance governed by components: $C_1 = C_x R_x / R_0$. The output voltage of the operational amplifier ($U_\text{out}$) is used to calculate $Q_\text{a}$ as $Q_\text{a} = -U_\text{out}/(i \omega R_0 + 1/C_1)$.
• Figure 2: Quantitative verification of the active capacitive compensation method in the QAH charge accumulation measurements. Measurements were performed on the simple disk-shaped sample at $\mu_0 H_{\text{DC}} = 0.5$ T. $\sigma_{xx} = 2.35 \times 10^{-8}$ S is kept constant across the frequency sweep. Theoretical prediction curves are calculated from the quantitative dissipation model described in Supplementary Materials, assuming an intrinsically quantized charge value. a, b, Frequency dependence of the in-phase (a) and quadrature (b) components of the surface charge accumulation signal ($\eta_\text{a} / B_{\text{AC}}$). The experimental data (symbols) are plotted for several compensation ratios ($\alpha = C_{\text{TG}} / C_1$). Solid lines represent theoretical predictions. c, The measured charge accumulation signal as a function of the compensation ratio $\alpha$, measured at $f = 83.7777$ Hz. Blue squares and red circles denote the experimental in-phase and quadrature components, respectively, while solid lines show the theoretical predictions. Horizontal error bars represent the estimated uncertainty in $\alpha$ arising from systematic errors in circuit elements.
• Figure 3: Quantized QAH charge accumulation realized by the capacitive compensation. The data of the field-induced surface charge accumulation ($\eta_\text{a} / B_{\text{AC}}$) as a function of $\mu_0 H_{\text{DC}}$ are compared between the uncompensated case (circuit in Fig. \ref{['fig:fig1']}c top) and the $\alpha = 0.9$ compensated case (circuit in Fig. \ref{['fig:fig1']}c bottom), measured on the simple disk-shaped sample. All measurements were performed at $f = 277.777$ Hz and a constant temperature ($\sigma_{xx} =2.35 \times 10^{-8}$ S at $\mu_0 H_{\text{DC}} = 0.5$ T). In both panels, the in-phase and quadrature components are plotted separately. a, Uncompensated measurement ($\alpha = 0$). The measured charge accumulation deviates from quantization with both in-phase and quadrature components, reaching an amplitude significantly lower than the ideally quantized value (about 50%) due to dissipation. The field-dependent tilt of the data plateaus stems from the variation of $\sigma_{xx}$. b, Compensated measurement utilizing a compensation ratio of $\alpha = 0.9$. The compensation successfully recovers a robustly quantized signal. The in-phase component of the data exhibits flat plateaus with the amplitude reaching above 95% of the intrinsically quantized value. The quadrature component of the data, stemming from the dissipation, is successfully suppressed within the plateau.
• Figure 4: Theoretical framework of the dual-gate capacitive compensation for TME polarization measurements.a,b, Schematics of the dual-gate capacitive-compensation measurement circuit. $C_\text{TG}$ and $C_\text{BG}$ are the top gate and bottom gate dielectric capacitances, respectively. $C_\text{S}$ is the geometric capacitance of the sample between top and bottom sample surfaces. The intrinsic opposite surface charges ($\pm Q_\text{4D-QHE} = \pm B_\text{AC} A_\text{S} \cdot e^2/(2h)$) of the TME induce gate signals $q_1$ and $q_2$, which are measured on the respective gate lines with tailored feedback negative capacitances ($-C_1$ and $-C_2$). $q_1$ and $q_2$ can be calculated from the effective circuit (b; see Supplementary Materials for details), where $\pm Q_\text{4D-QHE}$ are treated as current sources. $q_1$ and $q_2$ are transformed to the average charge density $\eta_\text{TME}^\text{TG}$ and $\eta_\text{TME}^\text{BG}$ for quantization analysis in c and d. The dashed line connecting the sample edge denotes the configurable option to either ground or float the sample. c, Simulated TME polarization signals on the top gate (average charge density per unit field, $\eta_\text{TME}^\text{TG}/B_\text{AC} = q_1/(B_\text{AC}A_\text{S})$) and dissipation time constants ($\tau_{\text{decay}}$) as a function of $(1-\alpha)$ (the gap from the full compensation), where $\alpha = (1/C_1 + 1/C_2)/(1/C_{\text{TG}} + 1/C_{\text{BG}})$ denoting the compensation ratio. The black curve represents simulated $\eta_\text{TME}^\text{TG}/B_\text{AC}$ ignoring the dissipation, purely governed by the geometric-capacitance-induced attenuation factor, $C_{\text{total}}^{\text{eff}}/C_{\text{S}}$. The blue and red curves plot $\tau_{\text{decay}}$ for the TME polarization [$\tau_\text{TME} = (C_\text{TG}^\text{eff}/\sigma_{xx})(C_{\text{S}}/C_{\text{total}}^{\text{eff}})$] and the net charge accumulation ($\tau_\text{QAH} = C_\text{TG}^\text{eff}/\sigma_{xx}$), respectively. The colors of background shadings denote practical measurement sensitivity boundaries: $(1-\alpha) \sim 10^{-3}$ is the capacitance calibration limit, while $\eta_\text{TME}^\text{TG}/B_\text{AC} \sim 5 \times 10^{-3}\, e^2/h$ is the signal amplitude limit. The circled region designates the optimal compensation target that balances both sensitivities. Simulations employ $\sigma_{xx} = 10^{-7}$ S, $C_{\text{S}} = 100$ nF, and $C_{\text{TG}} = C_{\text{BG}} = 1$ nF. d, Simulated magnetic field dependence of the compensated TME signal measured on the top gate ($\eta_\text{TME}^\text{TG}/B_\text{AC}$), utilizing the optimal compensation parameters from the circled region in c ($C_\text{TG}^\text{eff} = C_{\text{BG}}^{\text{eff}} = C_{\text{S}} =100$ nF). The model assumes a sandwich-magnetic-doped topological insulator where the top and bottom surfaces possess different coercive fields. Small black arrows indicate the magnetization configurations of the surfaces, while blue and red arrows denote the field sweep directions. The analysis contrasts the contact-grounded (top panel) and contact-floating (bottom panel) configurations. Gray shaded field regions correspond to the opposite surface magnetization case (axion insulator state of the sample), where the TME exhibits. Both configurations give a quantized TME polarization signal of $\eta_\text{TME}^\text{TG}/B_\text{AC} = \pm C_{\text{total}}^{\text{eff}}/C_{\text{S}} \cdot e^2/(2h)$ in the axion insulator regions. Only the grounded configuration permits a QAH charge accumulation signal.