Editing Free electron model (section)

== 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>