Free electron model: Difference between revisions

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=== Density of states ===

The 3D [[density of states]] (number of energy states, per energy per volume) of a non-interacting electron gas is given by:<ref group="Ashcroft & Mermin">{{CnHarvnb|dateAshcroft|Mermin|1976|pp=April 2024|p=44}}</ref>

:<math>g(E) = \frac{m_e}{\pi^2\hbar^3}\sqrt{2m_eE} = \frac{3}{2}\frac{n}{E_{\rm F}}\sqrt{\frac{E}{E_{\rm F}}},</math>

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The free electron model is closer to the measured value of <math>L=2.44\times10^{-8} </math> V²/K² while the Drude prediction is off by about half the value, which is not a large difference. The close prediction to the Lorenz number in the Drude model was a result of the classical kinetic energy of electron being about 100 smaller than the quantum version, compensating the large value of the classical heat capacity.

However, Drude's mode predicts the wrong order of magnitude for the [[Seebeck coefficient]] (thermopower), which relates the generation of a potential difference by applying a temperature gradient across a sample <math>\nabla V =-S \nabla T</math>. This coefficient can be showed to be <math>S=-{c_{\rm V}}/{|ne|}</math>, which is just proportional to the heat capacity, so the Drude model predicts a constant that is hundred times larger than the value of the free electron model.<ref name=":7" group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=|p=23|ps=}}</ref> While the latter get as coefficient that is linear in temperature and provides much more accurate absolute values in the order of a few tens of μV/K at room temperature.<ref name=":6" group="Ashcroft & Mermin" /><ref name=":7" group="Ashcroft & Mermin" /> However this models fails to predict the sign change<ref name=":4" group="Ashcroft & Mermin" /> of the thermopower in [[lithium]] and noble metals like gold and silver.<ref>{{Cite journal |lastlast1=Xu |firstfirst1=Bin |last2=Verstraete |first2=Matthieu J. |date=2014-05-14 |title=First Principles Explanation of the Positive Seebeck Coefficient of Lithium |url=https://link.aps.org/doi/10.1103/PhysRevLett.112.196603 |journal=Physical Review Letters |volume=112 |issue=19 |pages=196603 |doi=10.1103/PhysRevLett.112.196603|pmid=24877957 |arxiv=1311.6805 |bibcode=2014PhRvL.112s6603X }}</ref>

==Inaccuracies and extensions==

The free electron model presents several inadequacies that are contradicted by experimental observation. We list some inaccuracies below:<ref name=":4" group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=58-59}}</ref>

; Temperature dependence: The free electron model presents several physical quantities that have the wrong temperature dependence, or no dependence at all like the electrical conductivity. The thermal conductivity and specific heat are well predicted for alkali metals at low temperatures, but fails to predict high temperature behaviour coming from ion motion and [[phonon]] scattering.

; Hall effect and magnetoresistance: The Hall coefficient has a constant value {{ <math|''R''_H R_{\mathrm{=H}} –1= -1/&#124;''|ne''&#124;}}|</math> in Drude's model and in the free electron model. This value is independent of temperature and the strength of the magnetic field. The Hall coefficient is actually dependent on the [[band structure]] and the difference with the model can be quite dramatic when studying elements like [[magnesium]] and [[aluminium]] that have a strong magnetic field dependence. The free electron model also predicts that the traverse magnetoresistance, the resistance in the direction of the current, does not depend on the strength of the field. In almost all the cases it does.

; Directional: The conductivity of some metals can depend of the orientation of the sample with respect to the electric field. Sometimes even the electrical current is not parallel to the field. This possibility is not described because the model does not integrate the crystallinity of metals, i.e. the existence of a periodic lattice of ions.

; Diversity in the conductivity: Not all materials are [[electrical conductor]]s, some do not conduct electricity very well ([[Insulator (electricity)|insulators]]), some can conduct when impurities are added like [[semiconductor]]s. [[Semimetal]]s, with narrow conduction bands also exist. This diversity is not predicted by the model and can only by explained by analysing the [[valence and conduction bands]]. Additionally, electrons are not the only charge carriers in a metal, electron vacancies or [[Electron hole|holes]] can be seen as [[quasiparticle]]s carrying positive electric charge. Conduction of holes leads to an opposite sign for the Hall and Seebeck coefficients predicted by the model.

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;General

*{{cite book | last = Kittel | first = Charles | author-link=Charles Kittel | title = [[Introduction to Solid State Physics]] | ___location = University of Michigan | year = 1972|publisher=Wiley & Sons|isbn=978-0-471-49024-1}}

*{{cite book | lastlast1 = Ashcroft | firstfirst1 = Neil | author-linklink1 = Neil Ashcroft | last2 = Mermin | first2 = N. David | author-link2 = N. David Mermin | title = Solid State Physics | publisher = Holt, Rinehart and Winston | ___location = New York | year = 1976 | isbn = 978-0-03-083993-1 | url-access = registration | url = https://archive.org/details/solidstatephysic00ashc }}

*{{cite book | lastlast1 = Sommerfeld | firstfirst1 = Arnold | title = Elektronentheorie der Metalle | publisher = Springer Verlag | ___location = Berlin Heidelberg | year = 1933 | isbn = 978-3642950025 |first2=Hans|last2=Bethe|author-linklink1=Arnold Sommerfeld|author-link2=Hans Bethe}}

* {{cite book|last1=Ziman|first1=J.M.|title=Principles of the theory of solids|edition=2nd|publisher=Cambridge university press|year=1972|isbn=0-521-29733-8}}