Classical and quantum statistical mechanics
The classical theory of electrons in a metal was off by almost two orders of magnitude, and it took twenty-seven years to learn why: Pauli forbids almost all of them from moving. The Fermi level that comes out of that is the central idea of the unit that closes this subject.
01Why a statistical physics is needed
A cubic centimeter of copper has on the order of \( 10^{23} \) free electrons. Neither Newtonian mechanics nor the Schrödinger equation is designed to keep track of each one: what is needed is a description that says, instead, what fraction of those electrons occupies each available energy level. That is statistical mechanics, and the previous unit left open the question that makes it different for electrons than for anything else: the Pauli exclusion principle forbids two electrons from sharing the same state.
02The classical distribution and why it fails for electrons
Before Pauli, Drude (1900) treated the electrons of a metal as a classical gas: distinguishable particles, with no limit on how many share a state, distributed according to Maxwell-Boltzmann.
The prediction fails by a factor close to one hundred: the electrons of a real metal contribute much less heat capacity than the classical model demands. The error was not one of detail, it was conceptual, and it took twenty-seven years to resolve.
03Fermi-Dirac: what Pauli imposes
Sommerfeld solved the problem in 1927 by applying Pauli to the letter. If two electrons cannot share a state, most of them are “trapped” far below the top of the energy range: they cannot absorb a little thermal energy because the neighboring state, just slightly higher, is already occupied. Only the electrons near the top have empty states available nearby to jump into.
You choose the metal —it changes \( E_F \)— and the temperature. The Fermi-Dirac curve looks almost vertical because \( kT \) is tiny compared with \( E_F \): that is, in a single image, the reason for the error of the classical model.
04Bose-Einstein: when sharing a state is not forbidden
Photons —and in general integer-spin particles, the bosons— do not obey Pauli: any number of them can occupy exactly the same state, and in fact they tend to do so. That is Bose-Einstein statistics, and it has already appeared twice in this subject without being named: Planck's blackbody law is the Bose-Einstein distribution applied to the photons of a cavity, and a Bose-Einstein condensate —the ultracold helium atom from the de Broglie topic— is the extreme case, with a macroscopic fraction of atoms in the same quantum state.
Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein share the same exponential form in the high-energy limit —far from \( E_F \), Fermi-Dirac reduces to the classical one— and differ only in how they treat the possibility of sharing a state: forbidden for fermions, free for bosons, and the question does not even arise if the particles are distinguishable. Which one to use depends exclusively on the spin of the particle being counted.
05Where it appears in electronics
Everything a metal or a semiconductor does —conduct, or insulate— depends on where \( E_F \) falls relative to the allowed bands, and on how many electrons the Fermi-Dirac tail has above that band. The next and last unit of this subject rests entirely on this.
The same reasoning, applied to a semiconductor instead of a metal, is the next unit: how many electrons cross the band gap into the conduction band depends on where \( E_F \) is and on the Fermi-Dirac tail at the working temperature.
06In the lab
Calculate the Fermi temperature \( T_F = E_F/k \) of copper and compare it with room temperature. Explain why that comparison justifies using Fermi-Dirac and not Maxwell-Boltzmann.
With the lab on this page, measure at what distance from \( E_F \) the occupation drops from 90 % to 10 %, at 300 K and at 1000 K. Verify that the result is proportional to \( kT \).
With the Richardson-Dushman equation, calculate the current density of a tungsten cathode (\( \varphi = 4.5 \) eV) at 1000 K, 2000 K and 2700 K —a typical operating temperature— and compare the three results.
07Common mistakes
- Applying Maxwell-Boltzmann to the electrons of a metal. It is the correct statistics for a dilute classical gas, not for degenerate fermions.
- Believing that \( E_F \) is the maximum energy an electron can have. At \( T>0 \) there is always a fraction, a small one, above \( E_F \).
- Thinking that \( f_{FD}(E_F) \) depends on temperature. It is always 1/2, by definition of the Fermi level.
- Confusing the Fermi level with an actual occupied energy level. It can fall right in a band gap, where there is no state at all —which is exactly what happens in an intrinsic semiconductor.
- Using Bose-Einstein statistics for electrons, or Fermi-Dirac for photons. The spin of the particle decides which one applies, without exceptions.
- Forgetting that Fermi-Dirac and Maxwell-Boltzmann coincide far from \( E_F \). Classical statistics is not “wrong”: it is the limit of the quantum one when the occupation is low.
08Self-assessment
Why does Drude's classical theory predict the heat capacity of the electrons in a metal poorly?
Because it assumes that any electron can absorb thermal energy freely, when Pauli prevents almost all of them from doing so: only those near \( E_F \) have empty states available.
What does the Fermi level \( E_F \) represent?
The energy where the Fermi-Dirac occupation probability is exactly 1/2, at any temperature.
What does \( f_{FD}(E) \) look like at \( T=0 \)?
Like a perfect step: occupation 1 for all \( E
What distinguishes Bose-Einstein from Fermi-Dirac?
That bosons do not obey the exclusion principle: any number can occupy the same state, whereas fermions are forbidden to.
What becomes of the Planck blackbody distribution, seen from this unit?
It is Bose-Einstein statistics applied to the photons of a thermal cavity.
What part of the Fermi-Dirac distribution explains thermionic emission?
The high-energy tail, far above \( E_F \), where \( f_{FD} \) already resembles the classical exponential: the few electrons there have enough energy to escape from the metal.
09Further reading
- Charles Kittel. Introduction to Solid State Physics. 8th ed., Wiley, 2004. The table of Fermi energies of the metals and the complete derivation of the Sommerfeld model.
- Neil W. Ashcroft and N. David Mermin. Solid State Physics. Cengage, 1976. Chapter 2 develops the Sommerfeld gas and compares it point by point with the classical Drude theory.
- Robert Eisberg and Robert Resnick. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles. 2nd ed., John Wiley & Sons, 1985 (Spanish edition, “Física cuántica: átomos, moléculas, sólidos, núcleos y partículas”, Limusa). The three statistics —Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein— presented side by side, with the same approach as the whole subject.
- Simon M. Sze and Kwok K. Ng. Physics of Semiconductor Devices. 3rd ed., Wiley, 2007. Chapter 1 applies Fermi-Dirac to semiconductors, the direct bridge to the last topic of this subject.