Modulation of quantum geometry and its coupling to pseudo-electric field by dynamic strain

Abstract

Two-dimensional materials are a fertile ground for exploring quantum geometric phenomena, with Berry curvature and its first moment, the Berry curvature dipole, playing a central role in their electronic response. These geometric properties influence electronic transport and result in the anomalous and nonlinear Hall effects, and are typically controlled using static electric fields or strain. However, the possibility of modulating quantum geometric quantities in real-time remains unexplored. Here, we demonstrate the dynamic modulation of Berry curvature and its moments, as well as the generation of a pseudo-electric field using time-dependent strain. By placing heterostructures on a membrane, we introduce oscillatory strain together with an in-plane AC electric field and measure Hall signals that are modulated at linear combinations of the frequencies of strain and electric field. Our measurements reveal modulation of Berry curvature and its first moment. Notably, we provide direct experimental evidence of pseudo-electric field that results in an unusual dynamic strain-induced Hall response. This approach opens up a new pathway for controlling quantum geometry on demand, moving beyond conventional static perturbations. The pseudo-electric field provides a framework for external electric field-free anomalous Hall response and opens new avenues for probing the topological properties.

Figures (10)

• Figure 1: Schematic illustration of time-dependent strain-induced modulation of quantum geometry and pseudo-electric field generation.a, Parabolic bands with Berry curvature ($\Omega_z$) distribution indicated by the red–blue colormap. b, Time-dependent strain tilts the parabolic bands periodically, leading to a modulation of the Berry curvature dipole (BCD, $\Lambda(t)$) over time. The right panels show the instantaneous Berry curvature distributions at different times within one strain cycle, with black arrows indicating the magnitude of the BCD. c, Strain also modulates the momentum-space separation b(k,t) between the two valleys over time. This time-dependent valley displacement produces a pseudo-electric field.
• Figure 1: a, Diagram illustrating the different physical processes induced by time-modulation of strain. b, Schematic representation of different mixed frequency signals and their origin.
• Figure 2: Tailoring of band structure and transport properties – Berry curvature, BCD in TDBG induced by controlled static strain.a, Schematic of the device on a silicon nitride membrane, where static strain $u_0$ and oscillatory strain $\delta u_m$ are independently applied via two piezoelectric stacks on a strain cell. Hall voltages ($V_{xy}^\omega$) are measured at characteristic mixed frequencies. b, Illustration of a honeycomb lattice under uniaxial strain, which breaks the $C_3$ symmetry. Gray indicates the unstrained lattice, while brown and green represent lattice under tensile and compressive strain, respectively. The right panel shows the tilted band structure under strain with a colormap denoting Berry curvature and a dashed line marking the Fermi level. c, Theoretically calculated Berry curvature at a $k$ point near the valence band maxima in the $K$ valley of TDBG under uniaxial strain. d, Strain dependence of the $x$ and $y$ components of the Berry curvature dipole ($\Lambda_x$, $\Lambda_y$) at the valence band maxima in the $K$ valley. e, Measured longitudinal resistance $R_{xx}$ is plotted as a function of carrier density $n$ (top $x$-axis), filling factor $\nu$ (bottom $x$-axis) and perpendicular electric field $D/\epsilon_0$ ($y$-axis), at $T = 1.5~\text{K}$. The two high-resistance peaks correspond to full filling ($\nu = \pm 4$) of the flat bands. The schematic insets on the right depict the corresponding density of states (DOS) configurations, where colored markers indicate the Fermi level positions at selected points in the phase diagram. Red and blue colors represent Berry curvature of opposite sign in the flat bands, which reverse upon flipping the the perpendicular electric field. f, Variation in $R_{xx}$ with DC piezo voltage ($V_{\text{piezo}}^{dc}$) at $n = 3.46\times10^{12}\mathrm{cm^{-2}}$ ($\nu = 4.356$), $D/\epsilon_0 = 0~\text{V/nm}$, and $T = 1.5~\text{K}$ confirms strain application. Negative and positive $V_{\text{piezo}}^{dc}$ correspond to compressive and tensile strain, respectively.
• Figure 2: External electric field–free anomalous Hall effect induced by pseudo-electric field in TDBG.a, Schematic representation of the external field free Hall voltage at strain modulation frequency $\omega_m$. Strain modulation induces pseudo-electric field $\vec{E}_{\omega_m}^{K}(\vec{E}_{\omega_m}^{K'})$ in $K(K')$ valley. In the two valleys the pseudo-electric field as well as the Berry curvature has opposite sign. As a result anomalous velocity due to coupling of Berry curvature ($\vec{\Omega}$) and pseudo-electric field $-e(\vec{\Omega}\times \vec{E}_{\omega_m})$ will deflect charge carriers in both the valleys in same direction, leading to an external electric field free anomalous Hall voltage. b, Hall voltage $V_{xy}^{\omega_m}$ measured as a function of carrier density ($n$) and perpendicular electric field ($D/\epsilon_0$) at $T = 1.5$ K, in the absence of an external current ($I_{xx}^{\omega_c} = 0$). A peak-to-peak piezoelectric modulation voltage of 10 V at frequency $\omega_m = 177$ Hz is applied to generate oscillatory strain. The emergence of a finite Hall voltage at the strain modulation frequency, despite the absence of applied current, demonstrates the coupling of pseudo-electric field with Berry curvature.
• Figure 3: Nonlinear Hall signals encode the modulation of BCD and pseudo-electric field due to the application of dynamic strain.a, b, Nonlinear Hall voltages at $\omega_m + 2\omega_c$ ($V_{xy}^{\omega_m + 2\omega_c}$) (a) and $\omega_m$ ($V_{xy}^{\omega_m}$) (b) as functions of carrier density ($n$) and perpendicular electric field ($D/\epsilon_0$) at $T = 1.5 \, \text{K}$. Measurements were performed using a current of 98 nA at $\omega_c = 17$ Hz and a piezoelectric excitation at $\omega_m = 177$ Hz with a peak-to-peak voltage of 10 V. Both signals exhibit antisymmetric behavior under $D$-field reversal, consistent with the sign-change of the BCD in ABAB TDBG. c, d, Linear scaling of the Hall voltages at $\omega_m + 2\omega_c$ (c) and $\omega_m$ (d) with the amplitude of strain modulation, represented by the peak-to-peak piezoelectric excitation voltage ($V_{\text{piezo}}^{\omega_m}$), confirming that the BCD modulation is directly proportional to strain modulation amplitude. e, f, Frequency dependence of the nonlinear Hall signals shows linear scaling with $\omega_m$, indicating that the signals are directly proportional to the rate of change of BCD. g, h, Linear scaling of the Hall voltages at $\omega_m + 2\omega_c$ (g) and $\omega_m$ (h) with the square of the current amplitude, $(I_{xx}^{\omega_c})^2$, indicates a second-order nonlinear Hall response with respect to the in-plane electric field. Gray dashed lines represent linear fits to the experimental data, confirming the expected quadratic dependence on $I_{xx}^{\omega_c}$. Notably, $V_{xy}^{\omega_m}$ remains finite even at zero current, an indication to the presence of pseudo-electric field. The residual finite offset at $I_{xx}^{\omega_c}=0$ is arising from a pseudo-electric field generated by time-dependent strain. i, The extracted BCD-driven component of the nonlinear Hall signal, defined as $\Delta V_{xy}^{\omega_m} = V_{xy}^{\omega_m} - V_{xy}^{\omega_m}(I_{xx}^{\omega_c} = 0)$ from panel (h), is plotted as $- \frac{1}{2} \Delta V_{xy}^{\omega_m}$ (red pentagons) against $(I_{xx}^{\omega_c})^2$. It closely overlaps with the independently measured $V_{xy}^{\omega_m + 2\omega_c}$ signal (blue triangles), confirming the strain-modulated BCD origin of the response.