AHE-Driven SIT in CsV₃Sb₅ Nanoflakes

• The paper demonstrates that protonic gating in CsV₃Sb₅ nanoflakes enables tunable transitions between robust superconductivity and a bosonic superconductor–insulator transition driven by anomalous Hall effect modulation.
• It reveals that extrinsic skew scattering and intrinsic Berry curvature govern the anomalous Hall effect, with gate voltage tuning causing sign reversal and amplitude changes in conductivity.
• The study highlights that enhanced disorder in thin flakes leads to a direct bosonic SIT characterized by finite-size scaling and a critical sheet resistance near R₍Q₎, indicative of Cooper pair localization.

The anomalous Hall effect (AHE)–driven superconductor–insulator transition (SIT) refers to the interplay between giant AHE and disorder-induced SIT in kagome metals, particularly in CsV₃Sb₅ nanoflakes. Electrical modulation of carrier density and local disorder via a protonic gate enables the systematic investigation of unconventional superconductivity (SC), chiral charge density wave (CDW) order, and topological band effects in these materials. The direct, bosonic SIT is observed in strongly disordered thin flakes and correlates with a collapse of the gate-tunable AHE, revealing the intertwined nature of quantum orders in the AV₃Sb₅ family (Zheng et al., 2021).

1. Experimental Framework: Protonic Gating and Device Architecture

CsV₃Sb₅ nanoflakes (thickness 20–100 nm) are patterned into Hall-bar geometries and deposited onto a solid proton conductor, functioning as a solid-proton field effect transistor (SP-FET) with a Pt back gate. Application of a gate voltage $V_{g}$ induces intercalation of protons into the flakes, simultaneously injecting mobile charge and generating local disorder by distorting the V–Sb lattice, creating interstitials, vacancies, and paramagnetic impurities. The modulation of hole density $p$ spans $\sim10^{22}$ cm⁻³, crossing the CDW gap and altering the Fermi surface topology from hole- to electron-like.

The protonic gate also enhances disorder, especially in thin ($\sim20$ nm) flakes, by increasing spatial inhomogeneity and scattering centers—factors crucial to the eventual SIT. Transport measurements use four-terminal configurations for longitudinal resistivity $\rho_{xx}$ and perpendicular magnetic field sweeps for Hall resistivity $\rho_{xy}$, allowing decomposition of ordinary and anomalous Hall contributions. The anomalous term, $\rho_{xy}^{AHE}$, is isolated by background subtraction and converted to conductivity:

$$\sigma_{xy}^{AHE} = \frac{\rho_{xy}^{AHE}}{\rho_{xx}^2 + (\rho_{xy}^{AHE})^2}$$

2. Characterization of the Anomalous Hall Effect (AHE)

AHE in CsV₃Sb₅ nanoflakes displays a pronounced gate voltage dependence and nontrivial sign reversal. In thick flakes (80 nm), $\sigma_{xy}^{AHE}$ attains a maximum ($\sim 1.25 \times 10^3~\Omega^{-1}\text{cm}^{-1}$) at $V_{g} = +4.5$ V (hole density $p \approx 2.5 \times 10^{22}$ cm⁻³), with positive sign. As $V_{g}$ is swept negative across the CDW gap at the van Hove singularity (VHS), the sign reverses ($V_{g} \approx -4.6$ V), generating a negative AHE in the emergent electron pocket with amplitude $\sim 5 \times 10^2~\Omega^{-1}\text{cm}^{-1}$. This evolution mirrors crossings of highly asymmetric CDW sub-bands near the $M$-point in momentum space.

AHE in this context arises from two mechanisms:

• Extrinsic Skew Scattering: At high conductivity ($$\sigma_{xx} > 5 \times 10^3~\Omega^{-1}\text{cm}^{-1}$$, $V_{g}=4.5$ V), $\sigma_{xy}^{AHE}$ scales linearly with $\sigma_{xx}$: $\sigma_{xy}^{AHE} \propto \sigma_{xx}$, implying the dominance of skew scattering by multiple impurities within nearly flat, Berry-curved bands near VHS. The scaling:

$$\sigma_{xy}^{skew} \sim n_i V^3 [D(E_F)]^2 \Omega_k$$

where $n_i$ is impurity density, $V$ the scattering potential, $D(E_F)$ the Fermi-level density of states, and $\Omega_k$ the Berry curvature.

• Intrinsic Berry Curvature: As $V_{g}$ approaches $-4.6$ V, reduced $D(E_F)$ causes conductivity to drop and $\sigma_{xy}^{AHE}$ to decouple from $\sigma_{xx}$, indicative of intrinsic Berry curvature effects from chiral CDW bands becoming dominant.

3. Superconductor–Insulator Transition (SIT): Bosonic Scaling and Disorder Effects

In thin flakes ($\sim$21 nm), increasing disorder from enhanced proton intercalation drives a direct SIT. At gate voltages $V_{g} < -11$ V, the sheet resistance $R_\Box(T)$ displays a low temperature upturn, flattening near the critical value $R_Q = h/4e^2 \approx 6.45$ k$\Omega$, associated with Cooper-pair quantum resistance. The phase boundary is characterized by finite-size scaling of over 20 $R_\Box(T, V_g)$ curves:

$$R_\Box(T, V_g) = R_c\,F\left(\frac{|V_g - V_g^c|}{T^{1/(z\nu)}}\right)$$

with exponent product $z\nu \approx 1.85 \pm 0.14$, distinct from classical or fermionic SIT scenarios. The absence of a single-particle gap and $R_c \approx R_Q$ indicate a bosonic SIT, consistent with localized but intact Cooper pairs undergoing phase decoherence due to disorder.

4. Intertwined Evolution of AHE and SIT

There is a direct correlation between the electronic evolution of the AHE and the emergence of the SIT. The region of maximal positive AHE (hole band, $V_{g} \approx +4.5$ V) coincides with a robust superconducting dome ($T_c \approx 3$ K) and strong CDW order. As $V_{g}$ becomes negative and the Fermi level is tuned downwards, both CDW and $T_c$ are progressively suppressed.

In the parameter space where the bosonic SIT is observed ($V_{g} < -11$ V), $\sigma_{xy}^{AHE}$ collapses towards zero, reflecting the concurrent loss of coherent Berry-curvature structures and large density of states at VHS. This simultaneous suppression signifies the fundamental interplay between disorder, phase coherence, and topological aspects of the underlying band structure.

Disorder from protonic gating not only localizes Cooper pairs, destroying superconducting phase coherence, but also broadens the nearly flat kagome bands, thereby smearing out Berry-curvature "hot spots" crucial to both extrinsic skew scattering and intrinsic AHE. Conversely, the loss of superconducting screening alters impurity scattering dynamics and can further modify $\sigma_{xy}^{skew}$.

5. Global Picture and Implications for Quantum Correlated Materials

Protonic gating in CsV₃Sb₅ nanoflakes enables simultaneous and continuous control over both carrier density and disorder, providing a tunable platform for exploring the coupling of unconventional superconductivity, CDW order, and topological transport phenomena. The following interrelated regimes can be delineated:

Flake Type	Dominant AHE Mechanism	SIT Behavior
Thick	Skew scattering (extrinsic)	No SIT; robust SC and CDW
Thin	Intrinsic Berry curvature	Direct bosonic SIT; suppressed AHE

In thick flakes, gate tuning drives transitions between skew-scattering–dominated and Berry-curvature–dominated AHE through the CDW mini-gap. In thin, highly disordered flakes, phase-fluctuation–driven localization of Cooper pairs induces a bosonic SIT, while the disorder collapses both the density of states and Berry curvature responsible for the giant AHE.

A plausible implication is that the AV₃Sb₅ kagome system is a prototype for investigating the interconnectedness of band topology, many-body quantum order, and disorder-driven localization, providing a fertile platform for future studies on the fundamental interplay between topological transport and correlated electron phenomena (Zheng et al., 2021).