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We typically analyze transistor's work based on a work of a simple pn-junction diode, with some modifications if needed, as it is basically a particular configuration of pn-junctions. We assume the depletion regions form at the junctions and may use things like Shockley equation to calculate current.

However, in order to obtain all these properties, one should start from the basics of statistical quantum mechanics, which assumes that we have A LOT of particles in order to make some of the statements valid.

Furthermore, we even make many assumptions besides statistical mechanics like the fact that the charge can be treated as continiuum and also that the depletion region width is much smaller than the diode crossection in order to assume that the $E$-field in the depletion region is directed solely along the diode (an analogy to a parallel-plate capacitor).

How can a modern field-effect transistor used in microprocessors with the width of only a few atoms behave like a large one, if we have made many assumptions regarding large number of particles and large scale?

Do the results of these assumptions even hold there? Why?

Since the models I described obviously don't work, which models are used to describe such systems?

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    $\begingroup$ Another problem is that the usual textbook model is based on infinite lattices. $\endgroup$
    – John Doty
    May 10 at 18:06
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    $\begingroup$ An FET and a BJT operate quite differently. When you hear about very thing gates, those are in a FET. But the fact that students are taught certain material does not mean there are not other ways to model device behavior that do not rely on those simple assumptions. $\endgroup$
    – Jon Custer
    May 10 at 18:17
  • $\begingroup$ @JonCuster Could you plaese link me to some models you're talking about. Are they theoretical or purely experimental? I added a question in the end of my post about that to make clear what I need $\endgroup$
    – Sgg8
    May 10 at 18:33
  • $\begingroup$ @JonCuster also, both BJT's and FET's have semiconductor junctions in them. That's all I state $\endgroup$
    – Sgg8
    May 10 at 18:35
  • $\begingroup$ You can look into BSIM models. Although a lot of the simplifying assumptions used to treat the ideal long-channel MOSFET may not apply to modern transistors (which causes engineers much headache), the characteristics of a well-designed logic transistor will typically not look very different. There can still be several prominent non-ideal behaviors like various leakage mechanisms, velocity saturation, drain-induced barrier lowering and other short-channel effects, and these are usually treated in textbooks with varying levels of detail. $\endgroup$
    – Puk
    May 10 at 19:05

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How can a modern field-effect transistor used in microprocessors with the width of only a few atoms behave like a large one, if we have made many assumptions regarding large number of particles and large scale?

A few points.

  1. Just because you have derived some rules (like the behavior of a transistor) under one set of assumptions, it doesn't follow that it won't work in any other situation. E.g. you give the example of treating charge as a continuum. That assumption is just there to make the math easier --- not because the charge actually has to be a continuum! So you have to differentiate between assumptions that are really necessary for the situation and assumptions which exist to make the math easier.
  2. Many of the equations for transistors are kind of phenomenological. E.g., the Shockley diode equation you linked to has an "ideality factor", which is basically a fudge factor that allows you to tweak the equation to better represent actual diodes. Once you take phenomenological factors into account, that equation is basically just "an exponential minus 1", which can describe a lot of things! For example, the equation is often derived for p-n diodes, which don't include quantum tunneling. But quantum tunneling also generally has an exponential dependence on voltage, which is why the equation works for both p-n diodes (which don't involve quantum tunneling) and Schottky diodes (which rely on quantum tunneling). The equation is simply quite general and thus can be fit to very different effects. Many of the equations used to describe transistors have similar fudge factors and general forms and can thus handle a wide range of situations.
  3. You can always add more terms to existing equations so that they capture behavior that the equations previously couldn't. You can even treat quantum tunneling with the same framework that you treat recombination and generation in old-school semiconductor devices. It's a hack, but it works.
  4. Despite all this, modern transistors are now far enough away from the transistors of old that they don't always behave like old-school transistors. Semiconductor companies don't talk much publicly about this because how they design their transistors are highly guarded secrets. But there's a reason that Silvaco, a company that makes commonly used commercial products for designing semiconductor devices, licenses quantum transport software [1,2] developed by an academic group at Purdue [3]. It turns out that cutting-edge semiconductor devices sometimes need simulation tools beyond traditional transistor models.
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  • $\begingroup$ "Just because you have derived some rules (like the behavior of a transistor) under one set of assumptions, it doesn't follow that it won't work in any other situation" - by no means, that's definitely not what I meant) $\endgroup$
    – Sgg8
    May 15 at 13:31
  • $\begingroup$ By "how can it behave like that?" I meant "what is the reason behind this?" $\endgroup$
    – Sgg8
    May 15 at 13:32
  • $\begingroup$ A large FET and a tiny FET both work on the same basic principle: putting electric charge on a "gate" can impede or enhance current thru a nearby channel by changing the electric potential in the channel. Electric potentials work basically the same way for both macroscopic and microscopic transistors, so I would expect both large and small FETs to have similar behaviors even tho the details are significantly different. E.g. tiny FETs may have quasi-1D channels. But gates still work on quasi-1D channels basically the same way they work on giant channels. $\endgroup$
    – lnmaurer
    May 16 at 3:25
  • $\begingroup$ I'll add one more thought: one of the reasons transistors behave the way they do is that they're designed to behave that way. There are physical limitations to what transistors can do, but engineers have a fair amount of leeway to adjust their behavior. If they wanted small transistors to behave differently than large transistors, they totally could. But given that modern computers (made with tiny transistors) still implement the same logic gates as old computers (made with larger transistors), it's probably simpler to try to keep transistor behavior fairly consistent. $\endgroup$
    – lnmaurer
    May 16 at 15:41
  • $\begingroup$ Just to give an example, you can make tiny "reconfigurable" FETs that can quickly switch between behaving like p-channel MOSFETs and n-channel MOSFETs (e.g. doi.org/10.1021/nl203094h). These behave differently than normal MOSFETs, and to my knowledge, there's not a way to make good "large" reconfigurable transistors. However, you don't see reconfigurable transistors in commercial use because normal MOSFETs work fine for commercial uses, so people keep using them. So, nanoscale transistors can work differently than large transistors, but engineers choose not to do it that way. $\endgroup$
    – lnmaurer
    May 16 at 15:54
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It is not clear what question refers to. For example,

  • there is much work on quantum devices, which typically have dimensions of dozens of atoms in solid state (see Quantum nanoscience)
  • there are also devices where current is passed through a single molecule or a chain of atoms, which may act as transistors, if there is a third gate controlling the current flowing through a device (see Molecular scale electronics)

Obviously, neither type of these devices can be described by the theory used for conventional transistors developed nearly a century ago (but which are still the basis of modern electronics.) Yet, they are real transistors, as much as we limit our definition to (source):

A transistor is a (semiconductor) device used to amplify or switch electrical signals and power.

(I put "semiconductor" in parentheses, since this is obviously not the case for molecular devices or atomic chains.)

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  • $\begingroup$ I edited my question to specify that I'm asking about modern field-effect transistors used in microprocessors $\endgroup$
    – Sgg8
    May 11 at 10:00
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    $\begingroup$ However, I found your answer very informative regardless, thanks $\endgroup$
    – Sgg8
    May 11 at 10:00
  • $\begingroup$ @Sgg8 could you add a reference regarding the size of transistors? $\endgroup$
    – Roger V.
    May 11 at 10:00
  • $\begingroup$ I added a hyperlink $\endgroup$
    – Sgg8
    May 11 at 10:15

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