# Talk:Fermi level

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It has seemed a good idea to replace the former re-direct page with a page that explained some of the basic facts about how the concept of Fermi level is used in electric engineering and semiconductor physics. There is a lot of confusion, both in textbooks and on Wikepedia, about the distinction between "Fermi level" and "Fermi energy" (RGForbes (talk) 03:09, 23 March 2009 (UTC))

## Merge to electrochemical potential

This is a page about the electrochemical potential of electrons. It should be made into a section of that page. --Steve (talk) 17:06, 27 March 2009 (UTC) Never mind, that's a silly idea, there's plenty to be said about electrons in particular, to warrent a dedicated article. :-) --Steve (talk) 06:58, 29 March 2009 (UTC)

## Neutral point of View on the definition of Fermi energy and Fermi level

The page says that it's "incorrect" to say that Fermi level and Fermi energy have the same definition. But lots of semiconductor-physics textbooks and reliable sources specifically state and use that definition. A definition can't be right or wrong, it can only be conventional or unconventional. There's no basis for saying that the definition used by one large group of well-regarded physicists is "correct" and the definition used by a different large group of well-regarded physicists (i.e. semiconductor physicists) is "incorrect". We should replace the term "correct" with "most common" and the term "incorrect" with "less common" or "specific to semiconductor physics" or something like that. --Steve (talk) 17:06, 27 March 2009 (UTC)

No difficulty with this in principle. However, the very widely used and influential textbook "Physics of Semiconductor Devices" by Sze uses the term "Fermi level" rather than "Fermi energy" and I have the subjective impression (though I could be wrong) that "Fermi level" is the majority usage in semiconductor physics. (RGForbes (talk) 17:11, 9 April 2009 (UTC))(Richard)

## Don't forget about chemical potential

Article claims:

"Difference in voltage between conductor A and Earth = − (Difference in Fermi level between conductor A and Earth) / e"

This is false, because it neglects possible chemical potential differences. For example, solder an aluminum wire to a copper wire and put them on a table. The Fermi level will equilibrate between the two. But the voltage will not be the same, because of the volta potential at the junction arising from their different work functions. The word "voltage" has to be replaced with "electrochemical potential". --Steve (talk) 02:17, 29 March 2009 (UTC)

Hi Steve ! I am not disagreeing with your physics - certainly there are (what are often called) "contact potential differences" between materials of different work-function. [The term "Volta potential" is not normally used in the research literature of my subject area, so I don't use it.] These contact potential differences give rise to electrostatic fields at the junction of the two materials of different local work function. Also, when current is flowing, then there may be contact resistance - let's ignore this complication.] I do, however, disagree with your use of the term "voltage". In my view, it almost universal throughout electrical and electronic engineering, experimental physics, and circuit analysis as part of physics, to use the term "voltage difference" to describe a thermodynamic potential difference that is (-1/e) times the difference in "total thermodynamic potential" that applies to electrons (sometimes called "chemical potential", sometimes called "electrochemical potential"). The quantity that you call "voltage difference" is probably what I would call "electrostatic potential difference". I agree that there is some tendency to call an electrostatic potential difference a "voltage difference" (presumably because it is measured in volts), but in my experience this is very much a minority use of the term. It generates less confusion if a difference in electrostatic potential is simply called a "difference in electrostatic potential". (RGForbes (talk) 15:42, 9 April 2009 (UTC)) (Richard)

## Definition.

I didn't read the definitions very attentively, but it should be mentionned explicitly that in solid-state physics, the Fermi energy is defined as chemical potential as T → 0. The article should reflect this usage over others (which should be at chemical potential). Headbomb {ταλκκοντριβς – WP Physics} 07:12, 29 March 2009 (UTC)

Yes, that's the most common definition of Fermi energy in solid-state physics. This, however, is an article about the Fermi level, which is universally (I believe) in both solid-state physics and semiconductor physics a synonym for the chemical potential of electrons. Do you have any reference that uses "Fermi level" to refer specifically to a zero-temperature value? Anyway, there's already an article Fermi energy.
The chemical potential article is already quite crowded, I think it makes plenty of sense to have a dedicated article on the chemical potential of electrons (as opposed to the chemical potential of water molecules or whatever else). This is that article. --Steve (talk) 08:42, 3 April 2009 (UTC)

Yes I agree, however to introduce the Fermi level, you need to talk about the chemical potential first. As for the ref, I do have it (Ashcroft/Mermin Solid State Physics 1976, and possibly C. Kittel 6th edition), but I don't have it at hand. I'll pick it up tomorrow and I'll quote the relevant passages. But to ease your mind, the Fermi level represents the highest level occupied by an electron when in the system is in the lowest energy possible. You'll see that in metals both coincide when T--> 0K. In intrinsic semiconductors/insulators, the chemical potential is right smack in the middle of the gap when T -->0, but the Fermi level is at top of the conduction band. In extrinsic semiconductors, I forget where μ is (probably middle of the band, shifted towards donors/acceptors) the Fermi level will coindice with the donors level. It does not move, ever.

What is of use is the chemical potential, which is often called Fermi level because it is ridiculously close to the chemical potention in metals, which were historically "easier" to tackle. The fermi level concept first made its apparition in the drude model and sommerfeld model, well before the Bloch's band theory ever got around, where distinguishing between the chem pot and fermi energy introduces an error which is a fraction of kT (0.026eV at room temp [might have one more zero], which is quite negligeable). Then, when semiconductors/insulators got around, it became a serious error to refer to the chemical potential as the fermi level (at low temperatures), but not so much at room temp, where the FD distribution regains its Maxwell-Boltzmann-like characteristics. So people used both terms interchangeably anyway, as most people worked at RT, and people understood what was meant when they moved to low temps. And so began the huge mess of solid state physics.

So I'm sorta structuring the article into "Rigourous definition of chem pot/fermi energy" "Why it got confused (which becomes clear when you know the rigourous definitions)" and then the "meat" can be tackled. The important quantity is the chem pot, which is often, but wrongly, called the fermi energy. So basically, when people speak about the Fermi level/energy above T = 0, make the switch EF --> μ and then you have what people actually mean. I've spent two of my three master's seminaries dealing very directly with the fermi level [and I dealt with it in my third one as well] and related concepts so I'm quite familiar with it.Headbomb {ταλκκοντριβς – WP Physics} 09:55, 3 April 2009 (UTC)

I don't think you understand what I'm saying: "Fermi level" and "chemical potential" are synonyms. There's no approximation in saying that they're equal, because they are two terms for the exact same thing. Neither has anything to do with absolute zero. "Fermi energy" is different, it does have to do with absolute zero. You'll find that Ashcroft and Mermin agree with exactly this terminology. I'm not sure about Kittel. Semiconductor books agree with the definitions above for Fermi level and chemical potential, but would also say that "Fermi energy" means the same thing too. Electrochemistry books would agree with everyone else as to what Fermi level means, but have a different definition of chemical potential. In fact, seems to me that "Fermi level" is the only term that's unambiguous, which is why the article should use it.
And by the way can we please not pass judgment on different definitions? If two fields use different definitions of the same term, we can't say one definition is the correct definition and the other definition is an incorrect definition. How could a widely-used definition be "incorrect", just because people in a different field use a different definition? Are economists incorrect to use the word "derivative" in a way that's inconsistent with the calculus definition? --Steve (talk) 15:30, 3 April 2009 (UTC)

That's the thing. The Fermi level and the chem potential are NOT synonymous. EF is the energy of the most energetic electron when the system is its lowest energy configuration.

Take the simple case of a free electron gas [neglect states due to spin] contained in a cubic box of side L. You have

${\displaystyle E(k)={\frac {\hbar ^{2}k^{2}}{2m}}}$

where

${\displaystyle k_{x}={\frac {2\pi n_{x}}{L}}\,}$ ${\displaystyle k_{y}={\frac {2\pi n_{y}}{L}}\,}$ ${\displaystyle k_{z}={\frac {2\pi n_{z}}{L}}\ .}$

If you have N electrons, the lower energy configuration allowed is a sphere of radius

${\displaystyle k_{F}^{3}=6\pi ^{2}{\frac {N}{V}}}$

This ${\displaystyle k_{F}}$ defines the Fermi energy

${\displaystyle E_{F}={\frac {\hbar ^{2}k_{F}^{2}}{2m}}}$

and the equation

${\displaystyle E(k)={\frac {\hbar ^{2}k_{F}^{2}}{2m}}}$

defines the Fermi surface [of a free electron gas]. It also defines the Fermi temperature

${\displaystyle E_{F}=k_{B}T_{F}}$.

Now the chem potential is defined as the value that adjust the FD distribution so you have the following condition on f(E)

${\displaystyle \int _{0}^{\infty }P(E)dE=\int _{0}^{\infty }g(E)f(E)dE=N}$.

In our case (free electron model), the density of levels is

${\displaystyle g(E)={\frac {m}{\hbar ^{2}\pi ^{2}}}{\sqrt {\frac {2mE}{\hbar ^{2}}}}}$ if ${\displaystyle E>0}$.
${\displaystyle g(E)=0}$ if ${\displaystyle E<0}$.

At T = 0, the electrons are all in a sphere of radius ${\displaystyle k_{F}}$, i.e

${\displaystyle P(E)=g(E)f(E)=1}$ if ${\displaystyle E
${\displaystyle P(E)=g(E)f(E)=0}$ if ${\displaystyle E.

But ${\displaystyle \lim _{T\to 0}f(E)=1}$ if ${\displaystyle E<\mu }$ and 0 otherwise. So in this case ${\displaystyle \mu (T=0K)=E_{F}}$.

However, f(E) changes as T increases. The and the condition\

${\displaystyle \int _{0}^{\infty }P(E)dE=\int _{0}^{\infty }g(E)f(E)dE=N}$

does not yield the same result for ${\displaystyle \mu }$, ${\displaystyle \mu }$ is shifted to slightly lower energies. In the free electron model, the chemical potential is given by

${\displaystyle \mu =E_{F}\left[1-{\frac {1}{3}}\right({\frac {\pi k_{B}T}{2E_{F}}}\left)^{2}\right]+O(T^{4})}$

In this case, at RT, the difference between the two is about 0.01%. Which is utterly negligible unless you go to temperature comparable to the Fermi temperature (~105K). This is why Fermi energy/chem potential have been used interchangeably. This behaviour (where EF and μ are comparable) is not applicable to materials in general, especially in the case of extrinsic semiconductors at low temperatures. The concept of the Fermi energy as "the energy of the most energetic electron when the system is in its lowest energy level" is ill-adapted for non-metals, as the useful quantity (chemical potential) always lies somewhere in the bandgap, and it therefore cannot correspond to the energy of any electron (even in the T=0 case). So instead, the Fermi energy is defined as the limit of the chemical potential when T = 0. This reproduces the usual definition of the "energy of the most energetic electron when the system is in its lowest energy configuration" when working with metals, but also yields a useful quantity when working with non-metals. See Ashcroft/Mermin Solid State Physics p.32-49 for a probably clearer explanation of all this.

A relevant quote from the book would be

on page 43. [Yesterday's post sometimes conflated the Fermi energy definitions in terms of highest electron energy when the system is in its lowest energy config with μ when T=0 definition, apologies]. Headbomb {ταλκκοντριβς – WP Physics} 04:44, 4 April 2009 (UTC)

For yet a third time, I'm claiming that Fermi energy and Fermi level are not synonymous in the solid-state physics literature, Ashcroft and Mermin in particular. You gave a quote about Fermi energy, not Fermi level.
I'm well aware of the fact that the electron chemical potential changes with energy. I've done some work in thermopower, in which this is an extremely important effect, even at and below room temperature.
You say "The Fermi level and the chem potential are NOT synonymous. EF is the energy of the most energetic electron when the system is its lowest energy configuration." If the second sentence is true, then I agree that the first sentence is true. But any semiconductor physicist would emphatically disagree with the second sentence. This is what I'm saying: The term "Fermi energy" means different things to different people. Just like the term "derivative" means something different to an economist and a mathematician. Here's a book with a plot of "Fermi energy EF versus temperature", for example. --Steve (talk) 06:44, 4 April 2009 (UTC)
I'm getting confused by all this discussion. Let's edit the article, and we'll see if we're clashing or saying the same thing in different words.Headbomb {ταλκκοντριβς – WP Physics} 07:08, 4 April 2009 (UTC)
OK, then I'm going to undo all your edits, because I think the definition you're using is the wrong one for this article. If you want to describe the zero-temperature quantity, go to the article Fermi energy and edit there. :-) --Steve (talk) 20:01, 5 April 2009 (UTC)

Undo for now. However, the Fermi level is defined relative to EF (as the limit of μ→0) and not as μ. μ is the chemical potential, the the Fermi level, although both are sometimes used interchangeably. EF is the Fermi level/energy. Headbomb {ταλκκοντριβς – WP Physics} 05:49, 6 April 2009 (UTC)

Right now (2010-09-09), there is no definition of EF in the section "Conduction band referencing and the parameter ζ" or the section "Earth-based referencing and the parameter µ". EF doesn't get "defined" until the section ""Fermi level" in semiconductor physics," and there it is not really defined. It seems to me that for a section entitled "Fermi Level" the variable that corresponds to it should be defined early and clearly. Since I don't fully follow all of these discussions here about definitions of Fermi levels, would one of you all please make this correction? Orange1111 (talk) 06:43, 9 September 2010 (UTC)

## Electrostatic potential

Whether we like it or not, the term "voltage difference" is very very commonly used to denote purely the electrostatic potential difference between two points. Remember, V (the quantity whose gradient is E) is usually called "voltage", so a difference between two values of V is called "voltage difference". I think it's unhelpful to say this ubiquitous terminology is "wrong", better to address directly what it is and what it means.

The article addresses internal vs. external chemical potentials, but says "There is, however, no method of measuring these components separately." This isn't true. It's not easy to measure, but it's sure possible, and people do it all the time (through IV curve modeling, photoemission, kelvin probes, etc.) Let's remember that the electrostatic potential difference is a perfectly well-defined quantity: There's a physical E-field everywhere, and you can line integral to get an electrostatic potential difference. In principle, there's no conceptual problem at all: All three quantities (internal chemical potential, external chemical potential, total chemical potential) are well-defined and uniquely defined (other than what you call "zero"). It's just a matter of coming up with an appropriate method to tease them out. Agree? --Steve (talk) 08:52, 3 April 2009 (UTC)

(1) I would agree that adequate conceptual distinctions can be made between your three quantities. Whether we know how to accurately calculate the values of them from first principles in all circumstances is another matter. (2) Electrostatic potential is very easy to define in free space. It is less easy to define in a real conductor, but in a homogeneous conductor you can (say) take the conduction band-edge as defining electrostatic potential. You then do have a certain practical problem matching up the definitions inside and outside the conductor in such a fashion that the difference can be easily measured, but let's leave that aside. Maybe saying "there is no method of measuring these components separately" is too strong a phrase, but it is certainly true (in my view) that usually there is no simple method of measuring these components inside a conductor. I know of no general method. For example, so-called IV measurements are nearly always a measurement of current as a function of difference in the (electro)chemical potential of electrons at different points in electrical instruments.
This is not to say, for example, that you cannot make (at least partly) verifiable predictions about the variation of band-edge energy with position: but (in my view) whether this constitutes a measurement is another matter. "Coming up with an appropriate method" is not easy, in my view. There really are fundamental difficulties in doing so - essentially because what normal electrical instruments measure is a difference in the relevant total thermodynamic potential (call it chemical potential or electrochemical potential, as you prefer). Possibly, it might be better to write "Usually, there is no simple method of measuring these components separately", or something like this. (RGForbes (talk) 16:57, 9 April 2009 (UTC))(Richard)

## Questionable formula

${\displaystyle \zeta =\zeta _{0}\left[1-{\frac {\pi ^{2}}{12}}\left({\frac {k_{\mathrm {B} }T}{\zeta _{0}}}\right)^{2}-{\frac {\pi ^{4}}{80}}\left({\frac {k_{\mathrm {B} }T}{\zeta _{0}}}\right)^{4}+\cdots \right]}$

This formula is only true if you make certain assumptions about what the density of states is. It's a correct series expansion if you have a free electron gas, but can be way off in a metal with funny density of states, as plenty of metals do. I think it would be best to just leave it out, but if the assumptions were clearly stated that would also be OK. Agreed? :-) --Steve (talk) 08:58, 3 April 2009 (UTC)

I'll pick up ashcroft and will explicit these tomorrow. The formula is used and of a certain historical importance if I recall, so it should IMO be included. But yes, there are assumptions made.Headbomb {ταλκκοντριβς – WP Physics} 09:58, 3 April 2009 (UTC)
I've checked things out and this is the free electron model formula. Ashcroft mentions that this relies on the fact that g(E) [density of levels] is differentiable at E = μ. In the case of the free electron model, which has a singularity near E = 0, the expansion will neglect terms of the order of ${\displaystyle exp[-\mu /k_{b}T]}$, which are typically of order e−100 ≈ 10−43. It obviously also relies on the assumptions that electrons can move freely. The specific condition is that g(E) must vanish as E → −∞, and that it doesn't diverge faster than some power of E as E → +∞. What you're doing is essentially taking a taylor expansion of g(E) about E=μ and working with that. As long as the taylor expansion is valid, so is your approximation.Headbomb {ταλκκοντριβς – WP Physics} 05:30, 4 April 2009 (UTC)
I'd say, more importantly, it relies on the assumption that ${\displaystyle g(E)\propto E^{1/2}}$. Is this even approximately true for any real material? For example, here's iron's DOS, hardly a smooth parabola. --Steve (talk) 07:03, 4 April 2009 (UTC)
Well sqrt(E) is "some power of E". The key point in this case is that g(E) is infinitely differentiable near E = μ, and behaves nicely at ±∞, so the expansion is well defined. The formula above is specific to ${\displaystyle g(k)={\frac {m}{\hbar ^{2}\pi ^{2}}}{\sqrt {\frac {2mE}{\hbar ^{2}}}}}$
As for the free electron model works fairly well for metals, although that is sort of an accident. It underestimates the mean speed by a factor of roughly 100, but overestimate electron densities by a factor of roughly 100. These to cancel out, and gives correct results within a factor of of π2/6 in the case of electronic specific heat. There's also a factor of π2/6 involved in electrical and thermal conductivities. Quite small when you think about how wrong the free electron model is.
The assumptions made for the expansion are generally valid in real materials as you either work with insulators or semiconductors, for which g(E) ≈ 0 near μ (as μ lies in a forbidden band), or in metals, where g(E) is by far and large well-behaved.I may be thinking of bang structure, so may metals are evil when g(E) is concerned. I'll check. The only case when I'd see a potential problem is in extrinsic semiconductors at low temperatures, where the chemical potential will lie dangerously close to either donor levels or acceptor levels, but all it would mean is that you would need a very large number of terms to correctly describe the chemical potential. In all of these cases, the expansion will be different from the one above, depending on the exact form of g(E), and the leading term to will be the first derivative of g(E) [an extremly important quantity].Headbomb {ταλκκοντριβς – WP Physics} 07:56, 4 April 2009 (UTC)
I think the true expansion assumes g(E) is analytic, and is:
${\displaystyle \zeta =\zeta _{0}\left[1+{\frac {\pi ^{2}\zeta _{0}^{3}}{3}}g''(\zeta _{0})+\cdots \right]}$
I may be missing some factor, but it's something along those lines. When ${\displaystyle g(E)\propto E^{1/2}}$, you get the formula above. When the chemical potential is in the gap of a semiconductor (for example), there's no such series expansion that works. Agree? --Steve (talk) 18:53, 4 April 2009 (UTC)

(unindent) g(E) doesn't have to be analytic everywhere, only about E = μ because you're taking the Taylor expansion of g(E) about E = μ. Additional requirement (because this does not only involve the Taylor expansion) is that g(E) vanishes as → −∞ and does not diverge faster than some power of E at E → +∞. Using a g(E) ~ E12, gives the above formula, yes. Headbomb {ταλκκοντριβς – WP Physics} 19:42, 4 April 2009 (UTC)

Here's why it has to be analytic not just at ${\displaystyle \zeta _{0}}$: Imagine that it's analytic within ${\displaystyle \pm k_{B}T/1000}$ of ${\displaystyle \zeta _{0}}$. The information in ${\displaystyle g(\zeta _{0}),g'(\zeta _{0}),\ldots }$ contains no information whatsoever about what ${\displaystyle g(\zeta )}$ is outside of the ${\displaystyle \pm k_{B}T/1000}$ range of analyticity. However, those values will obviously impact the value of ${\displaystyle \zeta }$. --Steve (talk) 19:58, 5 April 2009 (UTC)
Ah yes, well that's the Taylor conditions. These obviously need to apply for the Taylor expansion to be valid. I thought you were talking about the extra considerations introduced by Sommerfeld.Headbomb {ταλκκοντριβς – WP Physics} 07:06, 8 April 2009 (UTC)

## Importance of energy differences

A funny, seemingly-incorrect claim in the article: "In semiconductor physics it is conventional to work mainly with unreferenced energy symbols. This is possible because the relevant formulae of semiconductor physics mostly contain differences in energy levels, for example (EC-EF). Thus, for developing the basic theory of semiconductors there is little merit in introducing an absolute energy reference zero."

Mostly? My contention is that in all of condensed-matter physics, there is no situation where you need to discuss an absolute energy: Energy differences are the only things that will ever enter any formula. Can anyone come up with a counterexample? :-) --Steve (talk) 09:08, 3 April 2009 (UTC)

Yes, I agree that the things that go into semiconductor formulae are energy differences. So we can leave out the "mostly", or change it to "always". What I originally had in mind was the situation where the semiconductor is in electrical contact with Ground, and the Fermi level is referenced relative to Ground, but I agree that there is still an energy difference involved. (RGForbes (talk) 12:56, 10 April 2009 (UTC)) (Richard)

## Proposed change in pedagogical presentation

How would people feel about changing the presentation of the sections:

• 1.2 Conduction band referencing and the parameter ζ
• 1.3 Earth-based referencing and the parameter µ
• 1.4 The distinction between ζ and µ
• 1.5 "Fermi level" in semiconductor physics

1. Define internal chemical potential, external chemical potential, and total chemical potential
2. Present ζ-referencing as a way to define internal chemical potential
3. Present Earth-referencing as a way to define total chemical potential
4. Present the semiconductor physics approach as another way to define total chemical potential.

Would other people agree that this is a better way to lay out the information? :-) --Steve (talk) 09:26, 3 April 2009 (UTC)

My understanding was that the Fermi level referred to a non-interacting quasi-particle picture (it is tied fundamentally to the Fermi-Dirac distribution and its theoretical statistical origins e.g. free electrons; the derivation does not allow the energy levels of the system to change with the number of particles present), while the chemical potential has no such restriction and includes all sources of interaction. Thus the Fermi level is a much more restricted notion. If that is so, the solution to this article is to deal with the simple Fermi level concept here, and relegate all the wonderful confusions of the electrochemical potential to a different article. Right now, the article treats the two concepts as equivalent. Maybe we have here another of those annoying multiple-use debates where vague terminology relies on context for meaning? The same difficulty spills over into Work function, which I take as a many body concept, while the article on work function doesn't seem too clear about this. Brews ohare (talk) 15:56, 4 April 2009 (UTC)
Chemical/electrochemical potential is easy to explain, even with correlation effects...you just talk about diffusive equilibrium and maybe cite the appropriate article for details. Certainly most of the article should be in the non-interacting approximation, as it's simpler and more common. But I don't think much is gained by a priori restricting the scope...we can always throw in a section at the end about how the description changes in a strongly correlated system. At the end of the day, strongly-correlated systems do have a Fermi level, after all. What are the "wonderful confusions" you're referring to?
Multiple-use issues will inevitably come up in this article, which is fine with me, as multiple uses exist and should be explained and not denied or ignored. It will be very good for people to be able to learn on wikipedia what fields use which terms for which meanings. I view explanations of that sort as a unique strength of Wikipedia, being a "universal" encyclopedia, filling a role that subfield-specific textbooks usually don't bother to do. --Steve (talk) 20:28, 5 April 2009 (UTC)

I'd agree that the place to start is the non-interacting system, where most of the ideas you need to develop can be presented in mathematical rigor. It would be well to make the intro clearly separate this discussion from the interacting case. Then the quasi-particle formulation could be dealt with. Then the complexities of real systems could be brought up, and ramifications for the electrochemical potential and work function. That is a lot to do, and I'd guess the last part is where most of the present article is situated, without adequate attention to the independent particle foundations, where much of the present discussion would be unnecessary. Brews ohare (talk) 12:46, 6 April 2009 (UTC)

## Definition of Fermi Level

I don't recall ever seeing the definition of Fermi level as the "state" that is given in the first sentence of the article, "In thermodynamics and solid-state physics, the Fermi level is the most energetic state occupied by an electron when the system is in the lowest energy configuration (i.e. at absolute zero)." I only recall seeing Fermi level defined as the energy of the top most filled state (Intro to Sol St Phys, Kittel) or as the chemical potential (Prin of Mod Phys, Leighton; Semicon Stat, Blakemore). Although these definitions of Kittel, Leighton, and Blakemore are inconsistent, they are consistent in defining Fermi level as an energy, not a state.

Can anyone give a source that says that the Fermi level is the "state", rather than the energy of the state? Otherwise we should change it to energy, which would make it the same Wikipedia definition as Fermi energy.

Also, it seems that some work is needed to coordinate or combine the articles Chemical potential, Electrochemical potential, Fermi level, and Fermi energy, at least to make them consistent in their definitions. --Bob K31416 (talk) 15:09, 6 April 2009 (UTC)

I'd add work function to the list. Brews ohare (talk) 17:47, 6 April 2009 (UTC)
I concur that some coordination is definitely needed - but it's probably best to try to reach a consensus on terminlogy and notation first.
On the question of "state" - this "unreferenced" usage certainly occurs in experimental practice, though maybe it hasn't been described in the best way. Example 1: "The oscilloscope trace on our energy spectrometer shows that the difference between the emitter Fermi level and Earth is -1.9 eV." Example 2: "The voltage (difference) across the resistor is 2 V - so the Fermi level is 2 eV higher at the right-hand side". Obviously, if you define an energy reference zero then all "level"s can be converted to "energies", but the reality is that in some contexts people do not define an energy reference zero. I would also argue that, in practice, many people use the term "Fermi level" as a label for the unreferenced energy level of the local electron state(s) that have occupancy 0.5. This is an observation about the working verbal practice of scientists and engineers.
One of the central problems for this article is that - taking physical science and engineering as a whole - the label "Fermi level" is in practice being applied/used in at least four logically different ways [corresponding in the original article to the usages of the symbols EF, ζ, ζ0, and μ]. Part of the role of this article - which certainly should be called "Fermi level" in my view - is to explain these slightly different uses, and to also explain briefly the related terms, including Fermi energy, quasi-Fermi level, imref, chemical potential, electrochemical potential and voltage. My view would be to keep the article and the mathematics to a reasonable minimum, and put complications (such as many-body effects - which is a valid point) into a separate article. The terminology issue of "chemical potential" vs "electrochemical potential" needs sorting, but maybe the best answer to this is to rework the relevant articles. (RGForbes (talk) 19:35, 9 April 2009 (UTC))(Richard)
Re "state", perhaps the best way to see that we are on the same page, is to consider the following change of adding the underlined part to the first sentence of the article,
...the Fermi level is the energy of the most energetic state occupied by an electron when the system is in the lowest energy configuration...
Do you agree? --Bob K31416 (talk) 20:22, 9 April 2009 (UTC)

Concept What chemists call it What solid-state physicists call it What semiconductor physicists call it Current locale Steve's ideal locale
Total chemical potential of electrons Electrochemical potential of electrons Chemical potential of electrons Fermi level or Fermi energy Chemical potential, Fermi level Fermi level (plus a short description and "main article" link in chemical potential)
Internal chemical potential of electrons Chemical potential of electrons Electrochemical potential of electrons No name ("Fermi level relative to vacuum / conduction-band-minimum / etc.") Little bits of Fermi level and Fermi energy, A section of Chemical potential Maybe Work function, although they're only approximately the same...
Internal chemical potential of electrons at 0K N/A Fermi energy (common), Fermi level (rare) "Fermi level at zero kelvin" or something like that Fermi energy and Fermi level, (redundantly) Just Fermi energy
Total chemical potential of non-electrons Electrochemical potential Chemical potential N/A Chemical potential and Electrochemical potential Just Chemical potential
Internal chemical potential of non-electrons Chemical potential Electrochemical potential N/A A section of chemical potential A section of chemical potential

I vote for reducing the electrochemical potential article down to a short discussion of terminology and a link to the chemical potential article, which already covers the exact same concept. Headbomb, since you're opposed to using "Fermi level" as the article-title for the concept in first row (total chemical potential of electrons), what do you propose instead? Chemical potential is already taken. Total chemical potential of electrons is awkward. Chemical potential of electrons is ambiguous. Any better ideas? --Steve (talk) 02:49, 7 April 2009 (UTC)

Steve, I appreciate the work you've done with the table and I'll be studying it. Regarding the suggestion you made re the electrochemical potential article, perhaps it should be discussed at Talk:Electrochemical potential. --Bob K31416 (talk) 19:36, 7 April 2009 (UTC)
First the Fermi level as a state is my mistake. Someone (Steve?) said they were reverting my edits, so I assumed they did, but apparently not. I don't oppose building this article on the topic that shall remain nameless for now. What I would suggest is building the article in steps. The first thing that needs to be talked about IMO, is the absolute zero scenario. This gives context. AKA what are the links and distinctions between EF and μ. Talk about EF as the energy of the most energetic electron when the system is in its lowest energy config (absolute zero) [which is how the concept of the fermi energy originally was conceived], then to its generalization as the limit of μ as T→0, etc... Here the free electron model should be mentionned for an example of the energy of the most ... blablahblah ... and how the T→0 limit of μ is equivalent, and how band gaps make the old definition inadequate, and the new one preferable.
Then it needs to be made clear that it is μ that is the important quantity, which can in many, but not all, cases be approximated to EF. When this is done, then the concept at hand can be introduced. One dimensional case (Fermi level) then higher dimension cases (Fermi surface). I've always seen those defined by the equation E(k)=EF [and thus E(k)=μ(T=0K)] and never as E(k)=μ(T), although I've seen them used as E(k)=μ(T), as the distinction is often not important, or implicit.
Then the relationship between the Fermi level, band diagram and band bending (that last one being an entirely loathsome concept if you ask me) can be talked about. Then comes discussion of some classic cases such as pn junctions, Schottky diode, etc... After the Fermi level is done with, moving on to the Fermi surface and its relation to things like Landau levels, the de Haas–van Alphen effect, Cyclotron resonance and so on.Headbomb {ταλκκοντριβς – WP Physics} 06:54, 8 April 2009 (UTC)
Headbomb, what do you mean by "band gaps make the old definition inadequate"? The statement (found in some textbooks) that ${\displaystyle \mu (0)}$ lies in the middle of the gap is a wrong one, obtained when incorrectly using the thermodynamical limit.[1][2][3]--Evgeny (talk) 09:18, 8 April 2009 (UTC)
Assuming a large N so the FD distribution is valid. Very interesting to see that the detailed calculations leads to a near consistency with the "old definition", (aka energy is not that of the most energetic electron when the system is its lowest energy config, but rather the energy of the next one you would add). I never saw this result before, at least when under this form, as I've never considered the small N case. I now realize that this reproduce the "thermodynamical" definition of the chemical potential (in terms of entropy and whatnot). Now I have some notes to correct. Headbomb {ταλκκοντριβς – WP Physics} 09:26, 8 April 2009 (UTC)
It's also worth noting that the total ${\displaystyle N}$ can be large; small is the number of occupied states in the conduction band. Indeed, even for a sample of the size of the Universe (!) ${\displaystyle \mu (T)}$ varies significantly when approaching the absolute zero on a very sizable range of few kelvins. Of course, this is due to a huge ${\displaystyle e^{E_{g}/kT}}$ factor. In the end, it all boils down to the strongly non-analytical form (i.e., discontinuity) of the density of states in the (ideal) semiconductor.--Evgeny (talk) 11:46, 8 April 2009 (UTC)
Headbomb, I'm not opposed to starting at absolute zero and distinguishing EF and μ. But I think a discussion of the Fermi surface belongs in the Fermi surface article. In an article about μ, it's important to describe how lower-energy states are more likely to be occupied, but I don't see how it helps to discuss in detail the particular electronic states with energies near μ. When do people use the approximation ${\displaystyle \mu \approx E_{F}}$? Is this common? I've never seen it in semiconductor physics, maybe in metals or something...? I definitely agree with talking about band diagrams and band bending, maybe with one example (the p-n junction?), but the details can be put into improved pages at band diagram and band bending. --Steve (talk) 19:54, 8 April 2009 (UTC)
Well the Fermi surface could be a simple paragraph, with a {{main}} pointing to Fermi surface. The other point of contention is whether this is about the use of the Fermi level as E(k)=μ, or the use of the Fermi level as E(k)=EF. As for its use in semiconductor physics, from what I can tell, what EF and μ means is rarely well defined. Many use both interchangeably. For example, this page [1] uses EF to mean μ.
In the Drude model (free electron, classical distributions), the approximation is valid, as the difference between the two is very small. This is also of relevance in the Sommerfeld model (free electron, fermi distribution). For example, the electrothermic conductivity tensor has elements involving terms of the form ${\displaystyle \int dE\left(-{\frac {\partial {f}}{\partial {E}}}\right)\left(E-\mu \right)^{a}\mathbf {\sigma } \left(E\right)}$ (a is 0, 1 or 2 depending on the element). Since the derivative of f is only importance in the neighbourhood of μ ≈ EF, conductivites are often expressed in terms of EF, rather than in terms of μ, yielding results within ~(kBT/EF)^2. Ashcroft/Mermin says this is usually valid in metals (neglecting crystallographic defects, impurities and similar things), but not in semiconductors. I've μ and EF be equated very often whenever the context is metal, but the distinction is always clear. In the context of semiconductors, I've seem people use EF when μ is meant, and talk about things like the temperature dependance of EF.Headbomb {ταλκκοντριβς – WP Physics} 00:13, 9 April 2009 (UTC)
Yes...What I've been saying over and over is that in semiconductor physics EF is the universal term for what you're calling μ, and never has anything to do with absolute zero. The semiconductor physics textbooks that I've read make this perfectly clear and well-defined, e.g. "EF is defined as the energy at which a state would have a 50% chance of being occupied", etc. See the chart above. You agree? Or do you have counterexamples? --Steve (talk) 17:53, 9 April 2009 (UTC)

The table above is nonsense without being properly sourced. It is also nonsense, period, as its sources, even if listed, are highly suspect. The care in definition depends on the quality of the source. Fermi Energy/Fermi Level is not a concept from electrochemistry, it is a concept from physics. Electrochemists have a long historical tendency for perverting physics terminology, and that is the main problem here.

Kittel is purely an introductory text. You can find the more rigorous definition, and more thorough historical context, in Ashcroft and Mermin, or other references that actually delineate the history. This article concerns the Fermi level, not the chemical potential and all its interpretations (which go well beyond practical electrochemistry). Therefore, the article should stick to the mission at hand - rigourous/reliable definition of the "Fermi Level." — Preceding unsigned comment added by Wikibearwithme (talkcontribs) 20:22, 28 May 2016 (UTC)

## On the need to define "chemical potential" (and can this definition be made to work for ζ)

In chemistry, the symbol μ is very widely used for partial molar energy (measured in J/mol), and is being used in the present context for the related quantity measured in eV; μ is usually called "chemical potential". If we are going to use the term "chemical potential" as the basis for discussion, then we should define it clearly, and ideally we should give a definition that is capable of being carried out in the real world to give a specific number. The definition should probably take a form something like: μ is the work needed to place an electron at the Fermi level, starting from a state in which the electron is ******(in some specified initial state)****.

I have a problem in trying to formulate a definition of this kind for the quantity ζ0, because (in a free-electron model) the starting position has to be at the bottom of a Sommerfeld box (preferably not the one it is going to end up in). Also I cannot imagine any real-world process that could perform the transfer required. This is why I have doubts as to whether ζ0 really does correspond well to the quantity that the chemists call (electro-)chemical potential, (and is also why I do not use the symbol μ0 for Fermi energy). Sommerfeld, in his textbook on Thermodynamics and Statistical Mechanics, calls ζ the free enthalpy of an electron. The point is that, if we wish to start from definitions of the various contributions to chemical potential, then we will need to be careful (and probably lengthy) in our choice of words. The present article should probably be an introductory article about the usages of the term Fermi level, so it may be easier to start from the Fermi-Dirac distribution function, and put the considerations about components of chemical potential into a separate restructured article covering "chemical potential" and "electrochemical potential". (RGForbes (talk) 20:30, 9 April 2009 (UTC)) (Richard)

## References

1. ^ Shegelski, Mark R.A. (1986-05). "New result for the chemical potential of intrinsic semiconductors: Low-temperature breakdown of the Fermi-Dirac distribution function". Solid State Communications. 58 (6): 351–354. doi:10.1016/0038-1098(86)90803-3. ISSN 0038-1098. Check date values in: |date= (help)
2. ^ Shegelski, Mark R. A. (2004-05). "The chemical potential of an ideal intrinsic semiconductor". American Journal of Physics. 72 (5): 676–678. doi:10.1119/1.1629090. Check date values in: |date= (help)
3. ^ Kaplan, T. (2006-03-01). "The Chemical Potential". Journal of Statistical Physics. 122 (6): 1237–1260. doi:10.1007/s10955-005-8067-x.

## Definition of Fermi level according to sources

We need to focus on what the sources say regarding the definition of Fermi level.

${\displaystyle f(E)={\frac {1}{1+exp({\frac {E-\phi }{kT}})}}}$
... The quantity ${\displaystyle \phi }$ is variously known as the Fermi level, Fermi surface, and the electrochemical potential.
In semiconductor theory, the electron chemical potential is always called the Fermi level.

Would anyone care to give any other sources and excerpts for the definition of Fermi level? My feeling is that we should define Fermi level as the chemical potential ${\displaystyle \phi }$ that appears in the above equation for the Fermi-Dirac distribution for a system of electrons. Thanks. --Bob K31416 (talk) 23:32, 9 April 2009 (UTC)

Hi! In relation to the last comment, my view is that there are four related, but "not strictly equivalent", widespread usages of the term "Fermi level" around. There are also 6 or more other terms that have equivalent or closely related meanings. I now see this article as a kind of "disambiguation article" that briefly sets out the underlying concepts (or some of them) and indicates which terms are attached to which concepts in which subject areas. I do not think that it is practicable or useful to try to decide on which usage of the term "Fermi level" is the definitive one, though I do think that it would be useful to discourage the usage that "Fermi level" is the name of a zero-temperature quantity. There is also a question of where you start from. I share the view that the best starting point (for a beginner trying to understand these issues) is the Fermi-Dirac distribution function, and my personal preference is to start from the semiconductor physics convention of labeling state-energy as E and the (unreferenced) (temperature-dependent) Fermi level in this function as EF. For example, this is what is done in "Sze" (Physics of Semiconductor Devices), which is a widely used textbook. (RGForbes (talk) 13:38, 10 April 2009 (UTC)) (Richard)
From your comment, it appears that Sze defines Fermi level the same way as is done in the sources above and by the definition that I am suggesting.
Also, please consider my remark at the beginning of this section that, "We need to focus on what the sources say regarding the definition of Fermi level." So far the three sources Blakemore, Kittel, and Sze, seem to be in agreement on the definition of Fermi level. Our own opinions are OK to express on this talk page but what goes into the article should be information from sources. Thanks. --Bob K31416 (talk) 13:52, 10 April 2009 (UTC)

I changed the lead sentence of the article which defines Fermi level to define it as being a type of chemical potential for electrons in semiconductors. I gave a source for the definition. I only included semiconductors in the definition so far since I didn't have a source for metals. We can change the definition to include metals or anything else if there is a source to support those inclusions. I haven't made the corresponding changes yet in the rest of the article to allow time for discussion of this matter. Thanks. --Bob K31416 (talk) 17:22, 11 April 2009 (UTC)

There is only one chemical potential for an electron in a solid. There are not various types of chemical potential. Also, there is not a 50% chance that an electron occupies one particular energy state. There is an energy level, E, that any given electron has an equal probability of being either above or below. — Preceding unsigned comment added by 76.113.185.52 (talk) 14:51, 26 December 2012 (UTC)

## In article, discussion of energy referencing

There is a considerable amount of discussion in the article regarding energy referencing that seems to be editor original research without much substance. Does anyone know of any reliable sources for that information? If so, please give on this talk page the page numbers and excerpts from the sources that were used. Thanks. --Bob K31416 (talk) 15:05, 15 April 2009 (UTC)

## Interesting distinction between energy and energy level related to Fermi level

In the lead sentence, I almost changed The Fermi level is an energy... to The Fermi level is an energy level..., but realized that would be a mistake. Energy level is an energy of a quantum state. For an intrinsic semiconductor, there are no states in the gap between the conduction and valence bands, yet there is a Fermi level there. --Bob K31416 (talk) 22:28, 17 April 2009 (UTC)

There is not "Fermi level" any more than a "Fermi energy" there; it is a convention of specifying/estimating the Fermi level as mid-gap (a probabolistic statement). Calling it one or the other does not make it any more rigorously defined or physically accurate. Wikibearwithme (talk) 20:04, 28 May 2016 (UTC)

## Pinning of Fermi level

Does anybody mind if this section is thrown away? It belongs rather to the Fermi level article. —Preceding unsigned comment added by Evgeny (talkcontribs) 17:03, 7 May 2009 (UTC)

## Rating change

Perhaps a bit of bias on my part as I've been working on it a lot, but I upgraded this article to High importance (any semiconductor physics discussion involves Fermi level), and changed from start to C quality. --Nanite (talk) 23:40, 26 May 2013 (UTC)