Editing Free electron model

{{short description|Simple model for the behaviour of valence electrons in a crystal structure of a metallic solid}}
{{about|the solid-state model for metals|the model of a free electron gas|Fermi gas}}In [[solid-state physics]], the '''free electron model''' is a [[Quantum mechanics|quantum mechanical]] model for the behaviour of [[charge carrier]]s in a [[metal]]lic solid. It was developed in 1927,<ref>{{Cite journal|last=Sommerfeld|first=Arnold|author-link=Arnold Sommerfeld|date=1928-01-01|title=Zur Elektronentheorie der Metalle auf Grund der Fermischen Statistik|journal=Zeitschrift für Physik|language=de|volume=47|issue=1–2|pages=1–32|doi=10.1007/bf01391052|issn=0044-3328|bibcode=1928ZPhy...47....1S}}</ref> principally by [[Arnold Sommerfeld]], who combined the [[Classical physics|classical]] [[Drude model]] with quantum mechanical [[Fermi–Dirac statistics]] and hence it is also known as the '''Drude–Sommerfeld model'''.<ref name=":1" group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|loc=Ch. 2 & Ch. 3}}</ref><ref name=":1" group="Kittel">{{Harvnb|Kittel|1972|loc=Ch. 6}}</ref>

Given its simplicity, it is surprisingly successful in explaining many experimental phenomena, especially
* the [[Wiedemann–Franz law]] which relates [[electrical conductivity]] and [[thermal conductivity]];
* the temperature dependence of the [[electron heat capacity]];
* the shape of the electronic [[density of states]];
* the range of binding energy values;
* electrical conductivities;
* the [[Seebeck coefficient]] of the [[thermoelectric effect]];
* [[thermionic emission|thermal electron emission]] and [[field electron emission]] from bulk metals.

The free electron model solved many of the inconsistencies related to the Drude model and gave insight into several other properties of metals. The free electron model considers that metals are composed of a quantum electron gas where [[ion]]s play almost no role. The model can be very predictive when applied to [[alkali metal|alkali]] and [[noble metal]]s.

==Ideas and assumptions==

In the free electron model four main assumptions are taken into account:<ref name=":5" group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=60}}</ref>
*Free electron approximation: The interaction between the ions and the valence electrons is mostly neglected, except in boundary conditions. The ions only keep the charge neutrality in the metal. Unlike in the Drude model, the ions are not necessarily the source of collisions.
*[[Independent electron approximation]]: The interactions between electrons are ignored. The electrostatic fields in metals are weak because of the [[screening effect]].
*Relaxation-time approximation: There is some unknown scattering mechanism such that the electron probability of collision is inversely proportional to the relaxation time <math>\tau</math>, which represents the average time between collisions. The collisions do not depend on the electronic configuration.
*[[Pauli exclusion principle]]: Each quantum state of the system can only be occupied by a single electron. This restriction of available electron states is taken into account by [[Fermi–Dirac statistics]] (see also [[Fermi gas]]). Main predictions of the free-electron model are derived by the [[Sommerfeld expansion]] of the Fermi–Dirac occupancy for energies around the [[Fermi level]].

The name of the model comes from the first two assumptions, as each electron can be treated as [[free particle]] with a respective quadratic relation between energy and momentum.

The crystal lattice is not explicitly taken into account in the free electron model, but a quantum-mechanical justification was given a year later (1928) by [[Bloch's theorem]]:<!-- Is this Bloch theorem? We must check the validity of this paragraph --> an unbound electron moves in a periodic potential as a free electron in vacuum, except for the [[electron mass]] ''m<sub>e</sub>'' becoming an [[effective mass (solid-state physics)|effective mass]] ''m*'' which may deviate considerably from ''m<sub>e</sub>'' (one can even use negative effective mass to describe conduction by [[electron hole]]s). Effective masses can be derived from [[band structure]] computations that were not originally taken into account in the free electron model.{{Cn|date=April 2024}}

== From the Drude model ==
{{main|Drude model}}
Many physical properties follow directly from the [[Drude model]], as some equations do not depend on the statistical distribution of the particles. Taking the [[Maxwell–Boltzmann distribution#Distribution for the velocity vector|classical velocity distribution]] of an ideal gas or the velocity distribution of a [[Fermi gas]] only changes the results related to the speed of the electrons.<ref name=":0" group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=49-51}}</ref>

Mainly, the free electron model and the Drude model predict the same DC electrical conductivity ''σ'' for [[Ohm's law]], that is<ref group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=|p=7}}</ref>
:<math>\mathbf{J} = \sigma \mathbf{E}\quad</math> with <math>\quad\sigma = \frac{ne^2\tau}{m_e},</math>

where <math>\mathbf{J}</math> is the [[current density]], <math>\mathbf{E}</math> is the external electric field, <math>n</math> is the [[electronic density]] (number of electrons/volume), <math>\tau</math> is the [[mean free time]] and <math>e</math> is the [[elementary charge|electron electric charge]].<!-- , and  <math>m_e</math> is the [[electron rest mass|electron mass]]. To include if paragraph above about Bloch's theorem disappears -->

Other quantities that remain the same under the free electron model as under Drude's are the AC susceptibility, the [[plasma oscillation|plasma frequency]],  the [[magnetoresistance]], and the Hall coefficient related to the [[Hall effect]].<ref name=":0" group="Ashcroft & Mermin" />

== Properties of an electron gas ==
{{main|Fermi gas}}
Many properties of the free electron model follow directly from equations related to the Fermi gas, as the independent electron approximation leads to an ensemble of non-interacting electrons. For a three-dimensional electron gas we can define the [[Fermi energy]] as<ref group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=32-37}}</ref>
:<math>E_{\rm F} = \frac{\hbar^2}{2m_e}\left(3\pi^2n\right)^\frac{2}{3},</math>

where <math>\hbar</math> is the reduced [[Planck constant]]. The [[Fermi energy]] defines the energy of the highest energy electron at zero temperature. For metals the Fermi energy is in the order of units of [[electronvolt]]s above the free electron band minimum energy.<ref>{{Cite web|url=http://hyperphysics.phy-astr.gsu.edu/hbase/Tables/fermi.html|title=Fermi Energies, Fermi Temperatures, and Fermi Velocities|last=Nave|first=Rod|publisher=[[HyperPhysics]]|access-date=2018-03-21}}</ref>

[[File:Free-electron DOS.svg|thumb|In three dimensions, the [[density of states]] of a gas of [[fermion]]s is proportional to the square root of the kinetic energy of the particles.]]

=== 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">{{Harvnb|Ashcroft|Mermin|1976|pp=|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>

where <math display="inline">E \geq 0</math> is the energy of a given electron. This formula takes into account the spin degeneracy but does not consider a possible energy shift due to the bottom of the [[Valence and conduction bands|conduction band]]. For 2D the density of states is constant and for 1D is inversely proportional to the square root of the electron energy.

=== Fermi level ===
The [[chemical potential]] <math>\mu</math> of electrons in a solid is also known as the [[Fermi level]] and, like the related [[Fermi energy]], often denoted <math>E_{\rm F}</math>. The [[Sommerfeld expansion]] can be used to calculate the Fermi level (<math>T>0</math>) at higher temperatures as:<ref group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=45-48}}</ref>
:<math>E_{\rm F}(T) = E_{\rm F}(T=0) \left[1 - \frac{\pi ^2}{12} \left(\frac{T}{T_{\rm F}}\right) ^2 - \frac{\pi^4}{80} \left(\frac{T}{T_{\rm F}}\right)^4 + \cdots \right], </math>

where <math>T</math> is the temperature and we define <math display="inline">T_{\rm F} = E_{\rm F}/k_{\rm B}</math> as the [[Fermi temperature]] (<math>k_{\rm B}</math> is [[Boltzmann constant]]). The perturbative approach is justified as the Fermi temperature is usually of about 10<sup>5</sup> K for a metal, hence at room temperature or lower the Fermi energy <math>E_{\rm F}(T=0)</math> and the chemical potential <math>E_{\rm F}(T>0)</math> are practically equivalent.

=== Compressibility of metals and degeneracy pressure ===
The total energy per unit volume (at <math display="inline">T = 0</math>) can also be calculated by integrating over the [[phase space]] of the system, we obtain<ref name=":3" group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=38-39|p=}}</ref>
:<math>u(0) = \frac{3}{5}nE_{\rm F},</math>

which does not depend on temperature. Compare with the energy per electron of an ideal gas: <math display="inline">\frac{3}{2}k_{\rm B}T</math>, which is null at zero temperature. For an ideal gas to have the same energy as the electron gas, the temperatures would need to be of the order of the Fermi temperature. Thermodynamically, this energy of the electron gas corresponds to a zero-temperature pressure given by<ref name=":3" group="Ashcroft & Mermin" />
: <math>P = -\left(\frac{\partial U}{\partial V}\right)_{T,\mu} = \frac{2}{3}u(0),</math>

where <math display="inline">V</math> is the volume and <math display="inline">U(T) = u(T) V</math> is the total energy, the derivative performed at temperature and chemical potential constant. This pressure is called the [[electron degeneracy pressure]] and does not come from repulsion or motion of the electrons but from the restriction that no more than two electrons (due to the two values of spin) can occupy the same energy level. This pressure defines the compressibility or [[bulk modulus]] of the metal<ref name=":3" group="Ashcroft & Mermin" />
:<math>B = -V\left(\frac{\partial P}{\partial V}\right)_{T,\mu} = \frac{5}{3}P = \frac{2}{3}nE_{\rm F}.</math>

This expression gives the right order of magnitude for the bulk modulus for alkali metals and noble metals, which show that this pressure is as important as other effects inside the metal. For other metals the crystalline structure has to be taken into account.

=== Magnetic response ===
According to the [[Bohr–Van Leeuwen theorem]], a classical system at thermodynamic equilibrium cannot have a magnetic response. The magnetic properties of matter in terms of a microscopic theory are purely quantum mechanical. For an electron gas, the total magnetic response is [[paramagnetism|paramagnetic]] and its [[magnetic susceptibility]] given by{{Cn|date=April 2024}} 
:<math>\chi=\frac{2}{3}\mu_0\mu_\mathrm{B}^2g(E_\mathrm{F}),</math>
where <math display="inline">\mu_0</math> is the [[vacuum permittivity]] and the <math display="inline">\mu_{\rm B}</math> is the [[Bohr magneton]]. This value results from the competition of two contributions: a [[Diamagnetism|diamagnetic]] contribution (known as [[Diamagnetism#Theory|Landau's diamagnetism]]) coming from the orbital motion of the electrons in the presence of a magnetic field, and a paramagnetic contribution (Pauli's paramagnetism). The latter contribution is three times larger in absolute value than the diamagnetic contribution and comes from the electron [[Spin (physics)|spin]], an intrinsic quantum degree of freedom that can take two discrete values and it is associated to the [[electron magnetic moment]].

== Corrections to Drude's model ==

=== Heat capacity ===
{{Further|Electronic specific heat}}
One open problem in solid-state physics before the arrival of quantum mechanics was to understand the [[heat capacity]] of metals. While most solids had a constant [[volumetric heat capacity]] given by [[Dulong–Petit law]] of about <math>3nk_{\rm B}</math> at large temperatures, it did correctly predict its behavior at low temperatures. In the case of metals that are good conductors, it was expected that the electrons contributed also the heat capacity.

The classical calculation using Drude's model, based on an ideal gas, provides a volumetric heat capacity given by
:<math>c^\text{Drude}_V = \frac{3}{2}nk_{\rm B}</math>.

If this was the case, the heat capacity of a metals should be 1.5 of that obtained by the Dulong–Petit law.

Nevertheless, such a large additional contribution to the heat capacity of metals was never measured, raising suspicions about the argument above. By using Sommerfeld's expansion one can obtain corrections of the energy density at finite temperature and obtain the volumetric heat capacity of an electron gas, given by:<ref group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=47}} (Eq. 2.81)</ref>
:<math>c_V=\left(\frac{\partial u}{\partial T}\right)_{n}=\frac{\pi^2}{2}\frac{T}{T_{\rm F}} nk_{\rm B}</math>,

where the prefactor to <math>nk_B</math> is considerably smaller than the 3/2 found in <math display="inline">c^{\text{Drude}}_V</math>, about 100 times smaller at room temperature and much smaller at lower <math display="inline">T</math>.

Evidently, the electronic contribution alone does not predict the [[Dulong–Petit law]], i.e. the observation that the heat capacity of a metal is still constant at high temperatures. The free electron model can be improved in this sense by adding the contribution of the vibrations of the crystal lattice. Two famous quantum corrections include the [[Einstein solid]] model and the more refined [[Debye model]]. With the addition of the latter, the volumetric heat capacity of a metal at low temperatures can be more precisely written in the form,<ref group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=|p=49}}</ref> 
:<math>c_V\approx\gamma T + AT^3</math>, 
where <math>\gamma</math> and <math>A</math> are constants related to the material. The linear term comes from the electronic contribution while the cubic term comes from Debye model. At high temperature this expression is no longer correct, the electronic heat capacity can be neglected, and the total heat capacity of the metal tends to a constant given by the Dulong–petit law.

=== Mean free path ===
Notice that without the relaxation time approximation, there is no reason for the electrons to deflect their motion, as there are no interactions, thus the [[mean free path]] should be infinite. The Drude model considered the mean free path of electrons to be close to the distance between ions in the material, implying the earlier conclusion that the [[Diffusion|diffusive motion]] of the electrons was due to collisions with the ions. The mean free paths in the free electron model are instead given by <math display="inline">\lambda=v_{\rm F}\tau</math> (where <math display="inline">v_{\rm F}=\sqrt{2E_{\rm F}/m_e}</math> is the Fermi speed) and are in the order of hundreds of [[ångström]]s, at least one order of magnitude larger than any possible classical calculation.<ref name=":6" group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=52}}</ref> The mean free path is then not a result of electron–ion collisions but instead is related to imperfections in the material, either due to [[Crystallographic defect|defects]] and impurities in the metal, or due to thermal fluctuations.<ref>{{Cite web|url=https://unlcms.unl.edu/cas/physics/tsymbal/teaching/SSP-927/Section%2008_Electron_Transport.pdf|title=Electronic Transport|last=Tsymbal|first=Evgeny|date=2008|website=University of Nebraska-Lincoln|access-date=2018-04-21}}</ref>

=== Thermal conductivity and thermopower ===
While Drude's model predicts a similar value for the electric conductivity as the free electron model, the models predict slightly different thermal conductivities.

The thermal conductivity is given by <math>\kappa=c_V \tau\langle v^2\rangle/3 </math> for free particles, which is proportional to the heat capacity and the mean free path which depend on the model (<math>\langle v^2\rangle^{1/2} </math> is the mean (square) speed of the electrons or the Fermi speed in the case of the free electron model).<ref name=":6" group="Ashcroft & Mermin" />  This implies that the ratio between thermal and electric conductivity is given by the [[Wiedemann–Franz law]],

:<math>\frac \kappa \sigma = \frac{m_{\rm e}c_V \langle v^2 \rangle  }{3n e^2} = L T</math>

where <math>L </math> is the Lorenz number, given by<ref name=":10" group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=|p=23 and 52|ps=(Eq. 1.53 and 2.93)}}</ref>

:<math>L=\left\{\begin{matrix}\displaystyle \frac{3}{2}\left(\frac{k_{\rm B}}{e}\right)^2\;, & \text{Drude}\\
\displaystyle\frac{\pi^2}{3}\left(\frac{k_{\rm B}}{e}\right)^2\;,&\text{free electron model.}
\end{matrix}\right.</math>

The free electron model is closer to the measured value of <math>L=2.44\times10^{-8} </math> V<sup>2</sup>/K<sup>2</sup> 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 |last1=Xu |first1=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_{\mathrm{H}} = -1/|ne|</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.

Other inadequacies are present in the Wiedemann–Franz law at intermediate temperatures and the frequency-dependence of metals in the optical spectrum.<ref name=":4" group="Ashcroft & Mermin" />

More exact values for the electrical conductivity and Wiedemann–Franz law can be obtained by softening the relaxation-time approximation by appealing to the [[Boltzmann equation|Boltzmann transport equations]].<ref name=":4" group="Ashcroft & Mermin" />

The [[exchange interaction]] is totally excluded from this model and its inclusion can lead to other magnetic responses like [[ferromagnetism]].{{Cn|date=April 2024}}

An immediate continuation to the free electron model can be obtained by assuming the [[empty lattice approximation]], which forms the basis of the band structure model known as the [[nearly free electron model]].<ref name=":4" group="Ashcroft & Mermin" />

Adding repulsive interactions between electrons does not change very much the picture presented here. [[Lev Landau]] showed that a Fermi gas under repulsive interactions, can be seen as a gas of equivalent quasiparticles that slightly modify the properties of the metal. Landau's model is now known as the [[Fermi liquid theory]]. More exotic phenomena like [[superconductivity]], where interactions can be attractive, require a more refined theory.{{Cn|date=April 2024}}

==See also==
* [[Bloch's theorem]]
* [[Electronic entropy]]
* [[Tight binding]]
* [[Two-dimensional electron gas]]
* [[Bose–Einstein statistics]]
* [[Fermi surface]]
* [[White dwarf]]
* [[Jellium]]

==References==
;Citations
<references group="Ashcroft & Mermin" />
<references group="Kittel" />

;References
{{reflist}}

;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 | last1 = Ashcroft | first1 = Neil | author-link1 = 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 | last1 = Sommerfeld | first1 = Arnold | title = Elektronentheorie der Metalle | publisher = Springer Verlag | location = Berlin Heidelberg | year = 1933 | isbn = 978-3642950025 |first2=Hans|last2=Bethe|author-link1=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}}

{{Condensed matter physics topics}}

{{DEFAULTSORT:Free Electron Model}}
[[Category:Quantum models]]
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[[Category:Electronic band structures]]
[[Category:Electron]]
[[Category:Arnold Sommerfeld]]