Someone tells it like it is, or at least pretends to… the 100%-real data is nice (read the whole article, it’s not all in the quote below)Kovar/Hall;
Electrons in germanium are confined to well-defined energy bands that are separated by “forbidden regions” of zero charge-carrier density. You can read about it yourself if you want to, although I don’t recommend it. You’ll have to wade through an obtuse, convoluted discussion about considering an arbitrary number of non-coupled harmonic-oscillator potentials and taking limits and so on. The upshot is that if you heat up a sample of germanium, electrons will jump from a non-conductive energy band to a conductive one, thereby creating a measurable change in resistivity. This relation between temperature and resistivity can be shown to be exponential in certain temperature regimes by waving your hands and chanting “to first order”.