  Electromagnetic (EM) instabilities and turbulence driven by the
electron-temperature gradient are considered in a local slab model of a
tokamak-like plasma. The model describes perturbations at scales both larger
and smaller than the flux-freezing scale $d_e$, and so captures both
electrostatic and EM regimes of turbulence. The well-known electrostatic
instabilities -- slab and curvature-mediated ETG -- are recovered, and a new
instability is found in the EM regime, called the Thermo-Alfv\'enic instability
(TAI). It exists in both a slab version (sTAI, destabilising kinetic Alfv\'en
waves) and a curvature-mediated version (cTAI), which is a cousin of the
(electron-scale) kinetic ballooning mode (KBM). The cTAI turns out to be
dominant at the largest scales covered by the model (greater than $d_e$ but
smaller than $\rho_i$), its physical mechanism hinging on the fast equalisation
of the total temperature along perturbed magnetic field lines (in contrast to
KBM, which is pressure balanced). A turbulent cascade theory is then
constructed, with two energy-injection scales: $d_e$, where the drivers are
slab ETG and sTAI, and a larger (parallel system size dependent) scale, where
the driver is cTAI. The latter dominates the turbulent transport if the
temperature gradient is greater than a certain critical value, which scales
inversely with the electron beta. The resulting heat flux scales more steeply
with the temperature gradient than that due to electrostatic ETG turbulence,
giving rise to stiffer transport. This can be viewed as a physical argument in
favour of near-marginal steady-state in electron-transport-controlled plasmas
(e.g., the pedestal). While the model is simplistic, the new physics that is
revealed by it should be of interest to those attempting to model the effect of
EM turbulence in tokamak-relevant configurations with high beta and large
electron temperature gradients.
