btw the sn7400 is already a fairly advanced ic; something like an uln2003 is closer to 'the simplest real world integrated circuit'
probably the best place to start for this, in a top-down sequence, is the art of electronics by horowitz and hill. it explains how transistors and diodes act in §1.6, §2, and §3, initially explaining transistors with the simplified 'current amplifier' model (which already goes beyond the 'transistor switch' model you're thinking of), then quantitatively with the ebers–moll model; they're focused on how to use this information to design working circuits from discrete components you can put together on a circuit board. camenzind's designing analog chips (available free online) goes into how to use this information to design actual chips (not only does nand2tetris get into things like metastability and noise margin, the authors seem to be confused about what a chip even is, thinking you can make a chip by assembling other chips)
but the ebers–moll model is still not solid-state physics knowledge. so far the best overview i've found of that is madou's 'fundamentals of microfabrication and nanotechnology' which has a couple of chapters about solid-state physics, going into the historical development of quantum mechanics. but it's not really a quantum mechanics textbook; it's just an overview that shows where quantum-mechanical knowledge fits into understanding solid-state physics
'the feynman lectures on physics' is the best quantum mechanics textbook i've found so far, but because my knowledge of quantum mechanics is even more minimal, please don't trust my recommendation on this
hope this helps. good luck in your learning voyage!