Transconductance as a Probe of Valley Thermodynamics in Multilayer WSe$_2$

Transconductance is a central figure of merit in field-effect transistors, typically governed by charge accumulation and carrier mobility. In multilayer WSe$_2$ transistors, however, it is shown to carry a nonlinear transport signature of inter-valley carrier redistribution between the $K$ and $\Gamma$ valleys. This valley-crossover contribution suppresses transconductance in bilayer WSe$_2$ and reverses sign in trilayer, while remaining absent in single-valley systems. Unlike extrinsic mechanisms such as trap-state filling or contact resistance, the anomaly leaves the subthreshold swing unchanged and cannot be reproduced within conventional single-valley transport models. Introducing the valley susceptibility $\chi_v \equiv \partial f_\Gamma/\partial V_{\rm GS}$, bounded by an intrinsic thermodynamic limit $(4k_BT)^{-1}$, we quantify this response and show that it reaches ${\sim}0.20\,\mathrm{V}^{-1}$ in bilayer WSe$_2$ near threshold at room temperature. The sign, magnitude, and temperature dependence of the anomaly provide directly measurable fingerprints of valley thermodynamics, establishing transconductance as an electrical probe of internal electronic degrees of freedom and revealing a previously hidden nonlinear response in standard transistor measurements.