why are good conductors of heat good conductors of electricity as well?
what factors govern the thermal drift of electrons(if any)
In physics and electrical engineering, a conductor is a material which contains movable electric charges. In metallic conductors, such as copper or aluminum, the movable charged particles are electrons (see electrical conduction). Positive charges may also be mobile in the form of atoms in a lattice that are missing electrons (known as holes), or in the form of ions, such as in the electrolyte of a battery. Insulators are non-conducting materials with fewer mobile charges, which resist the flow of electric current.
Metals (e.g. copper, platinum, gold,etc.) are usually the best conductors of thermal energy. This is due to the way that metals are chemically bonded: metallic bonds (as opposed to covalent or ionic bonds) have free-moving electrons which are able to transfer thermal energy rapidly through the metal. The "electron fluid" of a conductive metallic solid conducts nearly all of the heat flux through the solid. Phonon flux is still present, but carries less than 1% of the energy.
Are you familiar with concept of Fermi-surface? Electrons are fermions, which cannot occupy the same state. That means that all of the states with low kinetic energies are taken, forcing most electrons to occupy energy states with sizable kinetic energies. If temperature is exactly 0, the cutoff is very sharp, and the boundary is called the Fermi-surface.
What's interesting is that if temperature is non-zero, the cutoff is smooth. The higher the temperature, the smoother the cutoff. So if temperature in different parts of material is different, this distorts the Fermi-surface. The distortion is similar to one that happens when you apply voltage, so an electric potential gradient (electric field) and thermal potential gradient are both affecting the movement of electrons in similar ways, relating electric and heat conduction properties.
Of course, a metal also conducts heat via phonons in the metallic lattice. That conduction mode can contribute nearly as much or sometimes more than electrons, and this mode of heat conduction is unrelated to electric properties. So while being a good electric conductor guarantees good heat conductivity, reverse is not always true.