The meaning of the temperature in the Shockley Equation In the Shockley equation, which is the model of the $I$-$V$ curve of a p-n junction, what does the Temperature refer to, carrier temperature or lattice temperature? When a p-n junction subjected on a forward current, is the carrier temperature higher than lattice temperature as it does in the laser exciting case?
 A: The Shockley diode equation doesn't distinguish between carrier ($T_{eh}$) and lattice temperature $T$; it assumes that they are in equilibrium, $T_{eh} = T$.
Just a word of caution. We can't really say we have a single hot-carrier because temperature is a property of a large number of particles. You can say that you have a hot electron gas which has a range of velocities given by the Fermi-Dirac distribution with $T_{eh} > T$.
Regarding hot-carrier effects at forward bias. It very much depends on the device structure you are considering (more on this later). But in general the best way to think about this is as an energy balance. A forward bias accelerates carriers, so they are gaining energy from the field at rate $R_{field}$, however, carriers can lose energy by emitting phonons at rate $R_{phonons}$. Usually, $R_{field} \ll R_{phonons}$ so the carrier temperature remains in equilibrium with the lattice. At extreme forward bias conditions it's possible $R_{field} > R_{phonon}$ allowing $T_{eh} > T$. Until voltage saturation occurs.
To generate hot-carriers at modest forward voltages you need to design your semiconductor heterostructure with a potential cliff. For example, see the diagram below. Here electrons are injected into the low bandgap region, where they join a hot-distribution. You have to go to quite extreme lengths to generate hot-carriers in semiconductors because they lose energy by emitting phonons very quickly. For example, in GaAs the LO-phonon energy is 36meV and $R_{phonon}=1/1ps$. Therefore, if an electron has 1eV of excess energy (above the band edges) it can cool to the band edges within around 30ps (after emitting around 30 phonons)!
If you are interested in hot-carrier effects in semiconductors you should read about hydrodynamic transport equations.

