Is semiconductor theory really based on quantum mechanics?

Question

Often semiconductors are cited as the big application of quantum mechanics (QM), but when looking back at my device physics book basically no quantum mechanics is used. The concept of a quantum well is presented and some derivations are done, but then the next chapter mostly ignores this and goes back to statistical physics along with referencing experimentally verified constants to explain things like carrier diffusion, etc.

Do we really need quantum mechanics to get to semiconductor physics? Outside of providing some qualitative motivation to inspire I don't really see a clear connection between the fields. Can you actually derive transistor behaviour from QM directly?

Answer 1

Do we really need quantum mechanics to get to semiconductor physics?

It depends what level of understanding you're interested in. For example, are you simply willing to take as gospel that somehow electrons in solids have different masses than electrons in a vacuum? And that they can have different effective masses along different direction of travel? That they follow a Fermi-Dirac distribution? That band gaps exist? Etc.

If you're willing to accept all these things (and more) as true and not worry about why they're true, then quantum mechanics isn't really needed. You can get very far in life modeling devices with semi-classical techniques.

However, if you want to understand why all that weird stuff happens in solids, then yes, you need to know quantum mechanics.

Can you actually derive transistor behavior from QM directly?

It depends on the type of transistor. If you're talking about a TFET (or other tunneling devices, like RTDs and Zener diodes), then I challenge you to derive its behavior without quantum mechanics! However, if you're talking about most common transistors (BJTs, JFETs, MOSFETs, etc.), then deriving their behavior from quantum mechanics is a lot of work because the systems are messy and electrons don't "act" very quantum because of their short coherence time in a messy environment. However, the semi-classical physics used for most semiconductor devices does absolutely have a quantum underpinning. But there's a good reason it's typically not taught from first principles.

Anecdote: One time, I was sitting next to my advisor at a conference, and there was a presentation that basically boiled down to modeling a MOSFET using non-equilibrium greens functions (which is a fairly advanced method from quantum mechanics). During the presentation, my advisor whispered to me something along the lines of: "Why the heck are they using NEGF to model a fricking MOSFET?!?" In other words, just because you can use quantum mechanics to model transistors, doesn't mean you should. There are much simpler methods that are just as accurate (if not more accurate).

Answer 2

The question in the title is quite different from the question in the text.

But, here I will refer to the Preface in Shockley's book Electrons and Holes in Semiconductors, published in 1950 by van Nostrand.

In Part I, only the simplest theoretical concepts are introduced and the main emphasis is laid upon interpretation in terms of experimental results. This material is intended to be accessible to electrical engineers or undergraduate physicists with no knowledge of quantum theory or wave mechanics.

Part III, at the other extreme, is intended to show how fundamental quantum theory leads to the abstractions of holes and electrons ... Part III also contains an introduction to statistical mechanics and other topics applicable to the theory of electronic conduction in crystals.

Clearly Shockley thought quantum mechanics was pretty important to understanding electrons and holes, the heart of understanding a junction diode.

However, silicon 'cat whisker' diodes were invented before quantum mechanics (they were patented in 1906 per Wikipedia), but were not fully understood until solid state physics (based on quantum mechanics) was developed. Final understanding of them was the basis for the operation of the original point-contact transistor.

Answer 3

Many conference talks in the last 20 years began with this point and the Moore's law diagram, claiming that we are at the point where we do need to incorporate quantum mechanics in electronics design.

Band structure and mean free path
To expand a bit more about it: band structure is the basis of semiconductor electronics, so techncially quantum mechanics is necessary, since without it we would not have band structure. However, most transport phenomena used by modern semiconductor technology can be understood in semi-classical terms: the kinetic energy is taken in the effective mass approximation and the variation in the band structure is represented as potential (potential steps, wells, etc.) This is largely due to the fast relaxation processes in semiconductors and the mean free path being smaller than the characteristic structure size. On experimental level we are beyond this limitations since a few decades, as manifested by massive research on semiconductor nanostructures: quantum wells, quantum dots, superlattices, etc. But the industry is getting there very slowly.

"Classical" is simplified quantum
Another important aspect is that many of the "classical" methods actually require rather involved quantum justifications - e.g., simple Drude model actually requires Landau Fermi liquid theory to prove its correctness (see this question).

Beyond currents
Finally, I have noted in passing that the semiclassical description works well for transport phenomena - it is not necessarily so good when we are dealing with opical phenomena, spin phenomena or even such seemingly simple things as heat capacity.

Answer 4

To understand how solid state physics works one needs two basic ingredients:

1. What are (approximate) solutions of the Schrödinger equations for a single atom of a given kind.

2. How are those solutions modified if said atom is part of a regular lattice of identical atoms (what are the mutual interactions of the nearest neighbors in the cristalline lattice on the outer electron shells of the atoms of said lattice).

The purely geometric relationship between these atoms, how they are piled up accounts for some properties of a cristalline lattice: diamond vs coal vs graphite, etc… but some other properties (e.g. the conductivity of a metal) can only be explained by how the behavior of the outer electron shells is modified by the cristalline structure.

So yes, the behavior of cristalline solids (conductivity, diffraction, reflection, etc.) requires the formalism of quantum mechanics and, a fortiori, so does the part of solid state physics dealing with semiconductors.

Answer 5

In short, semiconductor devices, particularly FET, are based on the principal of quantum tunneling. Classical behaviour would say an electron could not travel through a perfect insulator/ infinite potential barrier but it is possible in QM. Yes, you can derive behaviour from QM. The electron wave equation will give the probability of where the electron will be (at least in one interpretation of QM). From this you can find the probability of crossing the barrier and hence electrical properties. More generally electron behaviour in crystals comes from solid state physics which is itself largely dervied from QM principals.