Hot-Carrier Distribution Spectroscopy by Transconductance in Two-Dimensional Field-Effect Transistors

The transconductance $g_m = dI_D/dV_G$ of a field-effect transistor (FET) is conventionally read as a proxy for carrier density. We show that it is instead a spectroscopic probe of the carrier distribution: because $g_m$ weights the spectral current $j(E)$ by the gate-voltage derivative $\partial f(E)/\partial V_G$ and integrates over energy, it is sensitive to the \emph{shape} of $f(E)$, not merely its integrated weight $n$. We develop an energy-resolved transport framework for two-dimensional (2D) FETs and, within a gate-independent spectral-kernel approximation, derive the decomposition $g_m = g_m^{(n)} + g_m^{(\alpha)}$ into the conventional density-modulation term $g_m^{(n)}$ and a distribution-shape-driven term $g_m^{(\alpha)}$. The latter, obtained as the residual after subtracting the smooth density-modulation background from the measured $g_m$, exhibits a characteristic anomalous peak at a gate voltage $V_G^{\rm pk}$. This peak has no counterpart in equilibrium transport and \emph{cannot be explained by carrier density modulation alone}. With the spectral kernel calibrated, the peak position and height -- extracted from standard DC/lock-in $g_m$ sweeps -- constrain the hot-carrier energy $E_0$, spectral width $\sigma$, and generation threshold $n_c$, realizing a steady-state, all-electrical spectroscopy of the carrier distribution. An optional time-resolved extension further recovers the carrier relaxation time $\tau$ from the transient response following a pump excitation, establishing the 2D FET as a distribution-function spectrometer that requires no optical readout.