Writing and detecting topological charges in exfoliated Fe$_{5-x}$GeTe$_2$

Fe$_{5-x}$GeTe$_2$ is a promising two-dimensional (2D) van der Waals (vdW) magnet for practical applications, given its magnetic properties. These include Curie temperatures above room temperature, and topological spin textures (TST or both merons and skyrmions), responsible for a pronounced anomalous Hall effect (AHE) and its topological counterpart (THE), which can be harvested for spintronics. Here, we show that both the AHE and THE can be amplified considerably by just adjusting the thickness of exfoliated Fe$_{5-x}$GeTe$_2$, with THE becoming observable even in zero magnetic field due to a field-induced unbalance in topological charges. Using a complementary suite of techniques, including electronic transport, Lorentz transmission electron microscopy, and micromagnetic simulations, we reveal the emergence of substantial coercive fields upon exfoliation, which are absent in the bulk, implying thickness-dependent magnetic interactions that affect the TST. We detected a ``magic" thickness $t \sim$30 nm where the formation of TST is maximized, inducing large magnitudes for the topological charge density ($6.45 \times 10^{20}$ cm$^{-2}$), and the concomitant anomalous ($\rho_{xy}^{\text{A,max}} \simeq 22.6$ $\mu \Omega$cm) and topological ($\rho_{xy}^{\text{u,T}} \simeq 15$ $\mu \Omega$ cm) Hall resistivities at $T$ ~ 120 K. These values for $\rho_{xy}^{\text{A,max}}$ and $\rho_{xy}^{\text{u,T}}$ are higher than those found in magnetic topological insulators and, so far, the largest reported for 2D magnets. The hitherto unobserved THE under zero magnetic field could provide a platform for the writing and electrical detection of TST aiming at energy-efficient devices based on vdW ferromagnets.