26
$\begingroup$

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?

$\endgroup$
7
  • 13
    $\begingroup$ I see two separate questions here, and it's not clear (to me) which one you're really asking. Are you wearing your I-want-to-know-just-enough-to-build-a-useful-device hat? Or are you wearing your I-want-to-deeply-understand-how-nature-works hat? $\endgroup$ Jun 28 '21 at 22:05
  • 2
    $\begingroup$ I think the question is, beyond providing theoretical motivation was QM really that useful in developing semiconductor devices? Can you remove basically 99% of theory built up around QM and still have gotten a working NP diode? $\endgroup$ Jun 28 '21 at 22:20
  • 5
    $\begingroup$ @FourierFlux When you say "developing" do you mean "pioneering" or do you mean "optimizing"? $\endgroup$
    – DKNguyen
    Jun 28 '21 at 22:40
  • 1
    $\begingroup$ What book are you talking about? It would be helpful to find some context. It's hard to imagine device physics without quantum mechanics. $\endgroup$
    – garyp
    Jun 29 '21 at 11:16
  • $\begingroup$ Friendly reminder: if you’d like to post a brief answer to the question, please don’t use the comments. $\endgroup$
    – rob
    Jun 29 '21 at 11:34
48
$\begingroup$

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).

$\endgroup$
6
  • 6
    $\begingroup$ There are aspects of device behaviour even in simple devices that require quantum mechanics to understand. A standard MOSFET exhibits Fowler-Nordheim tunnelling into and through the gate oxide in charge-dependent dielectric breakdown, a failure mode used as a reliability metric by manufacturers. $\endgroup$
    – Hearth
    Jun 29 '21 at 15:39
  • 2
    $\begingroup$ True, but a lot of this can be swept under the table. In fact, you can sweep the tunneling aspect of TFETs under the table: just replace the tunneling with some magic generation and recombination rates, and your TFET model will work --- even tho it totally hides what's really going on. $\endgroup$
    – lnmaurer
    Jun 29 '21 at 19:18
  • 10
    $\begingroup$ Many people might consider "quantum mechanics" and "magic" as synonyms, so replacing "quantum tunneling" by "magic" doesn't change anything :) $\endgroup$
    – alephzero
    Jun 30 '21 at 13:25
  • 2
    $\begingroup$ @alephzero: Insert reference to Arthur C. Clarke here $\endgroup$
    – Joe Bloggs
    Jun 30 '21 at 15:30
  • 1
    $\begingroup$ "Why the heck are they using NEGF to model a fricking MOSFET" Well I don't know what the talk was about, however some (actually, many) aspects concerning modern ultra-scaled MOSFET require the use of SE to understand e.g. the quantized inversion layer. $\endgroup$
    – edmz
    Jul 1 '21 at 9:14
15
$\begingroup$

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.

$\endgroup$
7
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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.

$\endgroup$
3
  • 1
    $\begingroup$ Infinite barriers are still trapping even in QM. But infinite barriers are an idealization anyway. $\endgroup$
    – Ian
    Jun 29 '21 at 15:23
  • 1
    $\begingroup$ Whoops, yes I didn't mean infinite in extent but rather a barrier higher than existing potential. These do allow for tunnelling. True, infinite barriers are an idealisation. However an insulating layer is close to infinite. As such while there might be some current through it, it would be negligable and probably wouldn't be seen for thermal effects. The appreciable current that can be achieved is consistent with QM tunnelling. $\endgroup$
    – Tman
    Jun 30 '21 at 10:22
  • 2
    $\begingroup$ "semiconductor devices, particularly FET, are based on the principal of quantum tunneling" There certainly are FETs that rely on quantum tunneling (i.e. TFETs), but your standard MOSFET does not rely on quantum tunneling. In fact, tunneling is usually seen as undesirable in a MOSFET. Remember that in a system at non-zero temperature, tunneling is not the only way to get past a barrier. Thermal energy can get electrons over the barrier the old fashioned way. $\endgroup$
    – lnmaurer
    Jun 30 '21 at 20:12

Not the answer you're looking for? Browse other questions tagged or ask your own question.