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{{short description|Model of electrical conduction}}
[[File:Electrona in crystallo fluentia.svg|thumb|300 px|right|Drude model electrons (shown here in blue) constantly bounce between heavier, stationary crystal ions (shown in red).{{Citation needed|date=July 2023}}]]

The '''Drude model''' of [[electrical conduction]] was proposed in 1900<ref>{{cite journal |last= Drude |first= Paul|year= 1900 |title= Zur Elektronentheorie der Metalle |journal= [[Annalen der Physik]] | volume= 306 | issue=3 | pages=566–613 |doi= 10.1002/andp.19003060312|bibcode = 1900AnP...306..566D |doi-access= free }}{{dead link|date=February 2019|bot=medic}}{{cbignore|bot=medic}}</ref><ref>{{cite journal |last= Drude |first= Paul |year= 1900 |title= Zur Elektronentheorie der Metalle; II. Teil. Galvanomagnetische und thermomagnetische Effecte |url= http://www3.interscience.wiley.com/cgi-bin/fulltext/112485893/PDFSTART |journal= [[Annalen der Physik]] |volume= 308 |issue=11 |pages=369–402 | doi= 10.1002/andp.19003081102| bibcode = 1900AnP...308..369D }}{{dead link|date=February 2019|bot=medic}}{{cbignore|bot=medic}}</ref> by [[Paul Karl Ludwig Drude|Paul Drude]] to explain the transport properties of [[electron]]s in materials (especially metals). Basically, [[Ohm's law]] was well established and stated that the current ''J'' and voltage ''V'' driving the current are related to the resistance ''R'' of the material. The inverse of the resistance is known as the conductance. When we consider a metal of unit length and unit cross sectional area, the conductance is known as the conductivity, which is the inverse of [[Electrical resistivity and conductivity|resistivity]]. The Drude model attempts to explain the resistivity of a conductor in terms of the scattering of electrons (the carriers of electricity) by the relatively immobile ions in the metal that act like obstructions to the flow of electrons.

The model, which is an application of [[kinetic theory of gases|kinetic theory]], assumes that the microscopic behaviour of electrons in a solid may be treated classically and behaves much like a [[pinball]] machine, with a sea of constantly jittering electrons bouncing and re-bouncing off heavier, relatively immobile positive ions.

In modern terms this is reflected in the [[valence electron]] model where the sea of electrons is composed of the valence electrons only,<ref>{{Cite book |title=Free electrons in solid |year=2009 |isbn=978-3-540-93803-3 |editor=springer |pages=135–158 |chapter="Free" Electrons in Solids |doi=10.1007/978-3-540-93804-0_6}}</ref> and not the full set of electrons available in the solid, and the scattering centers are the inner shells of tightly bound electrons to the nucleus. The scattering centers had a positive charge equivalent to the [[Valence (chemistry)|valence number]] of the atoms.<ref name=":9" group="note">{{Harvnb|Ashcroft|Mermin|1976|pp=3 page note 4 and fig. 1.1}}</ref>
This similarity added to some computation errors in the Drude paper, ended up providing a reasonable qualitative theory of solids capable of making good predictions in certain cases and giving completely wrong results in others. 
Whenever people tried to give more substance and detail to the nature of the scattering centers, and the mechanics of scattering, and the meaning of the length of scattering, all these attempts ended in failures.<ref name=":10" group="note">{{Harvnb|Ashcroft|Mermin|1976|pp=3 page note 7 and fig. 1.2}}</ref>

The scattering lengths computed in the Drude model, are of the order of 10 to 100 inter-atomic distances, and also these could not be given proper microscopic explanations.

Drude scattering is not electron-electron scattering which is only a secondary phenomenon in the modern theory, neither nuclear scattering given electrons can be at most be absorbed by nuclei. The model remains a bit mute on the microscopic mechanisms, in modern terms this is what is now called the "primary scattering mechanism" where the underlying phenomenon can be different case per case.<ref name=":11" group="note">{{Harvnb|Ashcroft|Mermin|1976|pp=3 page note 6}}</ref>

The model gives better predictions for metals, especially in regards to conductivity,<ref name=":12" group="note">{{Harvnb|Ashcroft|Mermin|1976|pp=8 table 1.2}}</ref> and sometimes is called Drude theory of metals. This is because metals have essentially a better approximation to the [[free electron model]], i.e. metals do not have complex [[band structures]], electrons behave essentially as [[free particle]]s and where, in the case of metals, the [[Effective field theory|effective number]] of de-localized electrons is essentially the same as the valence number.<ref name=":13" group="note">{{Harvnb|Ashcroft|Mermin|1976|pp=5 table 1.1}}</ref>

The two most significant results of the Drude model are an electronic equation of motion,
<math display="block">\frac{d}{dt}\langle\mathbf{p}(t)\rangle = q\left(\mathbf{E}+\frac{\langle\mathbf{p}(t)\rangle}{m} \times\mathbf{B} \right) - \frac{\langle\mathbf{p}(t)\rangle}{\tau},</math>
and a linear relationship between [[current density]] {{math|'''J'''}} and electric field {{math|'''E'''}},
<math display="block">\mathbf{J} = \frac{n q^2 \tau}{m} \, \mathbf{E}.</math>

Here {{mvar|t}} is the time, ⟨'''p'''⟩ is the average momentum per electron and {{mvar|q, n, m}}, and {{mvar|τ}} are respectively the electron charge, number density, mass, and [[mean free time]] between ionic collisions. The latter expression is particularly important because it explains in semi-quantitative terms why [[Ohm's law]], one of the most ubiquitous relationships in all of electromagnetism, should hold.<ref group="note" name=":0">{{Harvnb|Ashcroft|Mermin|1976|pp=6–7}}</ref><ref>{{cite book | author = Edward M. Purcell | year = 1965 | title = Electricity and Magnetism | url = https://archive.org/details/electricitymagne00purc | url-access = registration | publisher = McGraw-Hill | pages = [https://archive.org/details/electricitymagne00purc/page/117 117–122] | isbn = 978-0-07-004908-6}}</ref><ref>{{cite book | author = David J. Griffiths | year = 1999 | title = Introduction to Electrodynamics | publisher = Prentice-Hall | pages = [https://archive.org/details/introductiontoel00grif_0/page/289 289] | isbn = 978-0-13-805326-0 | url = https://archive.org/details/introductiontoel00grif_0/page/289 }}</ref>

Steps towards a more modern theory of solids were given by the following:
* The [[Einstein solid]] model and the [[Debye model]], suggesting that the quantum behaviour of exchanging energy in integral units or [[Quantum|quanta]] was an essential component in the full theory especially with regard to [[specific heats]], where the Drude theory failed.
* In some cases, namely in the Hall effect, the theory was making correct predictions if instead of using a negative charge for the electrons a positive one was used. This is now interpreted as holes (i.e. quasi-particles that behave as positive charge carriers) but at the time of Drude it was rather obscure why this was the case.<ref name=":14" group="note">{{Harvnb|Ashcroft|Mermin|1976|pp=15 table 1.4}}</ref>

Drude used [[Maxwell–Boltzmann statistics]] for the gas of electrons and for deriving the model, which was the only one available at that time. By replacing the statistics with the correct [[Fermi Dirac statistics]], [[Sommerfeld]] significantly improved the predictions of the model, although still having a [[Semiclassical physics|semi-classical]] theory that could not predict all results of the modern quantum theory of solids.<ref name=":15" group="note">{{Harvnb|Ashcroft|Mermin|1976|pp=4}}</ref>

== History ==
German physicist [[Paul Drude]] proposed his model in 1900 when it was not clear whether atoms existed, and it was not clear what atoms were on a microscopic scale.<ref>{{Cite web | title=Niels bohr Nobel Lecture| url=https://www.nobelprize.org/uploads/2018/06/bohr-lecture.pdf}}</ref> In his original paper, Drude made an error, estimating the Lorenz number of [[Wiedemann–Franz law]] to be twice what it classically should have been, thus making it seem in agreement with the experimental value of the specific heat. This number is about 100 times smaller than the classical prediction but this factor cancels out with the mean electronic speed that is about 100 times bigger than Drude's calculation.<ref group="note">{{Harvnb|Ashcroft|Mermin|1976|pp=|p=23}}</ref>

The first [[Brownian motion#Einstein's theory|direct proof of atoms]] through the computation of the [[Avogadro number]] from a microscopic model is due to [[Albert Einstein]], the [[Plum pudding model|first modern model]] of atom structure dates to 1904 and the [[Rutherford model]] to 1909. Drude starts from the discovery of electrons in 1897 by [[J.J. Thomson]] and assumes as a simplistic model of solids that the bulk of the solid is composed of positively charged scattering centers, and a sea of electrons submerge those scattering centers to make the total solid neutral from a charge perspective.<ref group="note" name=":8">{{Harvnb|Ashcroft|Mermin|1976|pp=2–3}}</ref> The model was extended in 1905 by [[Hendrik Antoon Lorentz]] (and hence is also known as the '''Drude–Lorentz model''')<ref>{{Cite journal |last=Lorentz |first=Hendrik |date=1905 |title=The motion of electrons in metallic bodies I |url=https://dwc.knaw.nl/DL/publications/PU00013989.pdf |journal=KNAW, Proceedings |volume=7 |pages=438–453 |via=KNAW}}</ref> to give the relation between the [[thermal conductivity]] and the [[electric conductivity]] of metals (see [[Lorenz number]]), and is a [[Classical physics|classical]] model. Later it was supplemented with the results of quantum theory in 1933 by [[Arnold Sommerfeld]] and [[Hans Bethe]], leading to the [[Free electron model|Drude–Sommerfeld model]].

Nowadays the Drude and [[Free electron model|Sommerfeld]] models are still significant to understanding the qualitative behaviour of solids and to get a first qualitative understanding of a specific experimental setup.<ref group="note" name=":7">{{Harvnb|Ashcroft|Mermin|1976|pp=2}}</ref> This is a generic method in [[solid state physics]], where it is typical to incrementally increase the complexity of the models to give more and more accurate predictions. It is less common to use a full-blown [[quantum field theory]] from first principles, given the complexities due to the huge numbers of particles and interactions and the little added value of the extra mathematics involved (considering the incremental gain in numerical precision of the predictions).<ref>{{Cite web  | url=https://www.youtube.com/watch?v=VF79QNugSEk&list=PLd9hKAUC3AZuo7is-aN45pmfDwJHOqKAj&index=3 | title=Solid State Physics, Lec ture 3: Drude Theory and Sommerfeld Free Electron| website=[[YouTube]]}}</ref>

== Assumptions ==
Drude used the [[kinetic theory of gases]] applied to the gas of electrons moving on a fixed background of "[[ion]]s"; this is in contrast with the usual way of applying the theory of gases as a neutral diluted gas with no background. The [[number density]] of the electron gas was assumed to be 
<math display="block"> n  = \frac{N_\text{A} Z \rho_\text{m}}{A},</math>
where ''Z'' is the effective number of de-localized electrons per ion, for which Drude used the valence number, ''A'' is the atomic mass per mole,<ref name=":8" group="note" /> <math>\rho_\text{m}</math> is the mass density (mass per unit volume)<ref name=":8" group="note" /> of the "ions", and ''N''{{sub|A}} is the [[Avogadro constant]].
Considering the average volume available per electron as a sphere:
<math display="block">\frac{V}{N} = \frac{1}{n} = \frac{4}{3} \pi r_{\rm s}^3 .</math>
The quantity <math>r_\text{s}</math> is a parameter that describes the electron density and is often of the order of 2 or 3 times the [[Bohr radius]], for [[alkali metals]] it ranges from 3 to 6 and some metal compounds it can go up to 10.
The densities are of the order of 1000 times of a typical classical gas.<ref group="note" name=":1">{{Harvnb|Ashcroft|Mermin|1976|pp=2–6}}</ref>

The core assumptions made in the Drude model are the following:
* Drude applied the kinetic theory of a dilute gas, despite the high densities, therefore ignoring electron–electron and electron–ion interactions aside from collisions.<ref group="note" name=":5">{{Harvnb|Ashcroft|Mermin|1976|pp=4}}</ref>
* The Drude model considers the metal to be formed of a collection of positively charged ions from which a number of "free electrons" were detached. These may be thought to be the [[Electron shell|valence electrons]] of the atoms that have become delocalized due to the electric field of the other atoms.<ref group="note" name=":1">{{Harvnb|Ashcroft|Mermin|1976|pp=2–6}}</ref>
* The Drude model neglects long-range interaction between the electron and the ions or between the electrons; this is called the independent electron approximation.<ref name=":1" group="note" />
* The electrons move in straight lines between one collision and another; this is called free electron approximation.<ref name=":1" group="note" />
* The only interaction of a free electron with its environment was treated as being collisions with the impenetrable ions core.<ref name=":1" group="note" />
* The average time between subsequent collisions of such an electron is {{mvar|τ}}, with a [[Memorylessness|memoryless]] [[Poisson distribution]]. The nature of the collision partner of the electron does not matter for the calculations and conclusions of the Drude model.<ref name=":1" group="note" />
* After a collision event, the distribution of the velocity and direction of an electron is determined by only the local temperature and is independent of the velocity of the electron before the collision event.<ref name=":1" group="note" /> The electron is considered to be immediately at equilibrium with the local temperature after a collision.

Removing or improving upon each of these assumptions gives more refined models, that can more accurately describe different solids:
* Improving the hypothesis of the [[Maxwell–Boltzmann statistics]] with the [[Fermi–Dirac statistics]] leads to the [[Drude–Sommerfeld model]].
* Improving the hypothesis of the Maxwell–Boltzmann statistics with the [[Bose–Einstein statistics]] leads to considerations about the specific heat of integer spin atoms<ref name="einstein24">{{cite journal|title=Quantum Theory of the Monatomic Ideal Gas|author=Einstein|editor=Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse| year=1924 |pages=261–267}}</ref> and to the [[Bose–Einstein condensate]].
* A valence band electron in a semiconductor is still essentially a free electron in a delimited energy range (i.e. only a "rare" high energy collision that implies a change of band would behave differently); the independent electron approximation is essentially still valid (i.e. no electron–electron scattering), where instead the hypothesis about the localization of the scattering events is dropped (in layman terms the electron is and scatters all over the place).<ref>{{Cite web | url=https://www.youtube.com/watch?v=e8BsQyafDh4&list=PLd9hKAUC3AZuo7is-aN45pmfDwJHOqKAj&index=17 | title=Solid State Physics, Lecture17: Dynamics of Electrons in Bands| website=[[YouTube]]}}</ref>

== Mathematical treatment ==

=== DC field ===
The simplest analysis of the Drude model assumes that electric field {{math|'''E'''}} is both uniform and constant, and that the thermal velocity of electrons is sufficiently high such that they accumulate only an infinitesimal amount of momentum {{math|''d'''''p'''}} between collisions, which occur on average every {{mvar|τ}} seconds.<ref name=":0" group="note" />

Then an electron isolated at time {{mvar|t}} will on average have been travelling for time {{mvar|τ}} since its last collision, and consequently will have accumulated momentum
<math display="block">\Delta\langle\mathbf{p}\rangle= q \mathbf{E} \tau.</math>

During its last collision, this electron will have been just as likely to have bounced forward as backward, so all prior contributions to the electron's momentum may be ignored, resulting in the expression
<math display="block">\langle\mathbf{p}\rangle = q \mathbf{E} \tau.</math>

Substituting the relations
<math display="block">\begin{align}
\langle\mathbf{p}\rangle &= m \langle\mathbf{v}\rangle, \\
\mathbf{J} &= n q \langle\mathbf{v}\rangle,
\end{align}</math>
results in the formulation of Ohm's law mentioned above:
<math display="block">\mathbf{J} = \left( \frac{n q^2 \tau}{m} \right) \mathbf{E}.</math>

=== Time-varying analysis ===
[[File:DrudeResponse.gif|thumb|300 px|right|Drude response of current density to an AC electric field.]]

The dynamics may also be described by introducing an effective drag force. At time {{math|''t'' {{=}} ''t''<sub>0</sub> + ''dt''}} the electron's momentum will be:
<math display="block">\mathbf{p}(t_0+dt) = \left( 1 - \frac{dt}{\tau} \right) \left[\mathbf{p}(t_0) + \mathbf{f}(t) dt + O(dt^2)\right] + \frac{dt}{\tau} \left(\mathbf{g}(t_0) + \mathbf{f}(t) dt + O(dt^2)\right)</math>
where <math>\mathbf{f}(t)</math> can be interpreted as generic force (e.g. [[Lorentz Force]]) on the carrier or more specifically on the electron. <math>\mathbf{g}(t_0)</math> is the momentum of the carrier with random direction after the collision (i.e. with a momentum <math>\langle\mathbf{g}(t_0)\rangle = 0</math>) and with absolute kinetic energy 
<math display="block">\frac{\langle|\mathbf{g}(t_0)|\rangle^2}{2m} = \frac{3}{2} KT.</math>

On average, a fraction of <math>1-\frac{dt}{\tau}</math> of the electrons will not have experienced another collision, the other fraction that had the collision on average will come out in a random direction and will contribute to the total momentum to only a factor <math>\frac{dt}{\tau}\mathbf{f}(t)dt</math> which is of second order.<ref group="note" name=":2">{{Harvnb|Ashcroft|Mermin|1976|p=11}}</ref>

With a bit of algebra and dropping terms of order <math>dt^2</math>, this results in the generic differential equation
<math display="block">\frac{d}{dt}\mathbf{p}(t) = \mathbf{f}(t) - \frac{\mathbf{p}(t)}{\tau}</math>

The second term is actually an extra drag force or damping term due to the Drude effects.

=== Constant electric field ===
At time {{math|''t'' {{=}} ''t''<sub>0</sub> + ''dt''}} the average electron's momentum will be
<math display="block">\langle\mathbf{p}(t_0+dt)\rangle=\left( 1 - \frac{dt}{\tau} \right) \left(\langle\mathbf{p}(t_0)\rangle + q\mathbf{E} \, dt\right),</math>
and then
<math display="block">\frac{d}{dt}\langle\mathbf{p}(t)\rangle = q\mathbf{E} - \frac{\langle\mathbf{p}(t)\rangle}{\tau},</math>
where {{math|⟨'''p'''⟩}} denotes average momentum and {{mvar|q}} the charge of the electrons. This, which is an inhomogeneous differential equation, may be solved to obtain the general solution of
<math display="block">\langle\mathbf{p}(t)\rangle = q \tau \mathbf{E}(1-e^{-t/\tau}) + \langle\mathbf{p}(0)\rangle e^{-t/\tau}</math>
for {{math|''p''(''t'')}}. The [[steady state]] solution, {{math|{{sfrac|''d'' ⟨'''p'''⟩|''dt''}} {{=}} 0}}, is then
<math display="block">\langle\mathbf{p}\rangle = q \tau \mathbf{E}.</math>

As above, average momentum may be related to average velocity and this in turn may be related to current density,
<math display="block">\begin{align}
\langle\mathbf{p}\rangle &= m \langle\mathbf{v}\rangle, \\
\mathbf{J} &= n q \langle\mathbf{v}\rangle,
\end{align}</math>
and the material can be shown to satisfy Ohm's law <math>\mathbf{J} = \sigma_0 \mathbf{E}</math> with a [[Direct current|DC]]-conductivity {{math|''σ''<sub>0</sub>}}:
<math display="block">\sigma_0 = \frac{n q^2 \tau}{m}</math>

=== AC field ===
[[File:DrudeModelComplexConductivity.png|thumb|300 px|right|Complex conductivity for different frequencies assuming that {{math|''τ'' {{=}} 10<sup>−5</sup>}} and that {{math|''σ''<sub>0</sub> {{=}} 1}}.]]
The Drude model can also predict the current as a response to a time-dependent electric field with an angular frequency {{mvar|ω}}. The complex conductivity is
<math display="block">\sigma(\omega) = \frac{\sigma_0}{1 - i\omega\tau}= \frac{\sigma_0}{1 + \omega^2\tau^2}+ i\omega\tau\frac{\sigma_0}{1 + \omega^2\tau^2}.</math>

Here it is assumed that:
<math display="block">\begin{align}
E(t) &= \Re{\left(E_0 e^{-i\omega t}\right)}; \\
J(t) &= \Re\left(\sigma(\omega) E_0 e^{-i\omega t}\right).
\end{align}</math>
In engineering, {{mvar|i}} is generally replaced by {{mvar|−i}} (or {{mvar|−j}}) in all equations, which reflects the phase difference with respect to origin, rather than delay at the observation point traveling in time.


{{math proof|title=Proof using the equation of motion<ref group="note" name=":16">{{Harvnb|Ashcroft|Mermin|1976|pp=16}}</ref>|proof=

Given
<math display="block">\begin{align}
\mathbf{p}(t) &= \Re{\left(\mathbf{p}(\omega) e^{-i\omega t}\right)} \\
\mathbf{E}(t) &= \Re{\left(\mathbf{E}(\omega) e^{-i\omega t}\right)}
\end{align}</math>
And the equation of motion above
<math display="block">\frac{d}{dt}\mathbf{p}(t) = -e\mathbf{E} - \frac{\mathbf{p}(t)}{\tau}</math>
substituting
<math display="block">-i\omega\mathbf{p}(\omega) = -e\mathbf{E}(\omega) - \frac{\mathbf{p}(\omega)}{\tau}</math>
Given
<math display="block">\begin{align}
\mathbf{j} &= - n e \frac{\mathbf{p}}{m} \\
\mathbf{j}(t) &= \Re{\left(\mathbf{j}(\omega) e^{-i\omega t}\right)} \\
\mathbf{j}(\omega) &= - n e \frac{\mathbf{p}(\omega)}{m}=\frac{(n e^2/m)\mathbf{E}(\omega)}{1/\tau -i \omega}
\end{align}</math>
defining the complex conductivity from:
<math display="block">\mathbf{j}(\omega) = \sigma(\omega)\mathbf{E}(\omega)</math>
We have:
<math display="block">\sigma(\omega) = \frac{\sigma_0}{1-i\omega\tau};\sigma_0=\frac{ne^2\tau}{m}</math>
}}

The imaginary part indicates that the current lags behind the electrical field. This happens because the electrons need roughly a time {{mvar|τ}} to accelerate in response to a change in the electrical field. Here the Drude model is applied to electrons; it can be applied both to electrons and holes; i.e., positive charge carriers in semiconductors. The curves for {{math|''σ''(''ω'')}} are shown in the graph.

If a sinusoidally varying electric field with frequency <math>\omega</math> is applied to the solid, the negatively charged electrons behave as a plasma that tends to move a distance ''x'' apart from the positively charged background. As a result, the sample is polarized and there will be an excess charge at the opposite surfaces of the sample.

The [[dielectric constant]] of the sample  is expressed as
<math display="block">\varepsilon = \frac {D}{\varepsilon_0 E} = 1 + \frac {P}{\varepsilon_0 E} </math>
where <math>D</math> is the [[Electric displacement field|electric displacement]] and <math>P</math> is the [[polarization density]].

The polarization density is written as
<math display="block">P(t) = \Re{\left(P_0e^{i\omega t}\right)} </math>
and the polarization density with ''n'' electron density is
<math display="block">P = - n e x</math>
After a little algebra the relation between polarization density and electric field can be expressed as
<math display="block">P = - \frac{ne^2}{m\omega^2} E</math>
The frequency dependent dielectric function of the solid is
<math display="block">\varepsilon(\omega) = 1 - \frac {n e^2}{\varepsilon_0m \omega^2}</math>

{{math proof|title=Proof using Maxwell's equations<ref group="note" name=":17">{{Harvnb|Ashcroft|Mermin|1976|pp=17}}</ref>|proof=

Given the approximations for the <math>\sigma(\omega)</math> included above
* we assumed no electromagnetic field: this is always smaller by a factor v/c given the additional Lorentz term <math> - \frac {e \mathbf{p}}{mc} \times \mathbf{B} </math> in the equation of motion
* we assumed spatially uniform field: this is true if the field does not oscillate considerably across a few mean free paths of electrons. This is typically not the case: the mean free path is of the order of Angstroms corresponding to wavelengths typical of X rays.
Given the Maxwell equations without sources (which are treated separately in the scope of [[plasma oscillation]]s)
<math display="block">\begin{align}
\nabla \cdot \mathbf{E} &= 0 ; & \nabla \cdot \mathbf{B} &= 0 \\
\nabla \times \mathbf{E} &= - \frac{1}{c}\frac{\partial \mathbf{B}}{\partial t} ; & \nabla \times \mathbf{B} &= \frac{4\pi}{c}\mathbf{j} + \frac{1}{c}\frac{\partial \mathbf{E}}{\partial t}
\end{align}</math>
then
<math display="block">\nabla \times \nabla \times \mathbf{E} = - \nabla^2 \mathbf{E} = \frac{i \omega}{c} \nabla \times \mathbf{B} = \frac{i \omega}{c} \left( \frac{4\pi \sigma}{c} \mathbf{E} - \frac{i \omega}{c} \mathbf{E} \right)</math>
or
<math display="block"> -\nabla^2 \mathbf{E} = \frac{\omega^2}{c^2}  \left( 1 + \frac {4\pi i \sigma}{\omega}\right) \mathbf{E}</math>
which is an electromagnetic wave equation for a continuous homogeneous medium with dielectric constant <math>\epsilon(\omega)</math> in the Helmoltz form 
<math display="block"> - \nabla^2 \mathbf{E} = \frac{\omega^2}{c^2} \epsilon(\omega) \mathbf{E}</math>
where the refractive index is <math display="inline">n(\omega) = \sqrt{\epsilon(\omega)}</math> and the phase velocity is <math> v_p = \frac{c}{n(\omega)}</math>  
therefore the complex dielectric constant is 
<math display="block">\epsilon(\omega) = \left( 1 + \frac {4\pi i \sigma}{\omega}\right)</math>
which in the case <math>\omega\tau \gg 1</math> can be approximated to:
<math display="block">\epsilon(\omega) = \left( 1 - \frac{\omega_{\rm p}^2}{\omega^2} \right) ;\omega_{\rm p}^2 = \frac {4\pi n e^2}{m}</math>
}}
At a resonance frequency <math>\omega_{\rm p}</math>, called the '''plasma frequency''', the dielectric function changes sign from negative to positive and real part of the dielectric function drops to zero.
<math display="block">\omega_{\rm p} = \sqrt{\frac{n e^2}{\varepsilon_0 m}} </math>
The plasma frequency represents a [[plasma oscillation]] resonance or [[plasmon]]. The plasma frequency can be employed as a direct measure of the square root of the density of valence electrons in a solid. Observed values are in reasonable agreement with this theoretical prediction for a large number of materials.<ref name="Kittel2">{{cite book|title=[[Introduction to Solid State Physics]]|author=C. Kittel|publisher=Wiley & Sons|year=1953–1976|isbn=978-0-471-49024-1}}</ref> Below the plasma frequency, the dielectric function is negative and the field cannot penetrate the sample. Light with angular frequency below the plasma frequency will be totally reflected. Above the plasma frequency the light waves can penetrate the sample, a typical example are alkaline metals that becomes transparent in the range of [[ultraviolet]] radiation.<ref group="note" name=":18">{{Harvnb|Ashcroft|Mermin|1976|pp=18 table 1.5}}</ref>

=== Thermal conductivity of metals ===
One great success of the Drude model is the explanation of the [[Wiedemann-Franz law]].  This was due to a fortuitous cancellation of errors in Drude's original calculation.  Drude predicted the value of the Lorenz number: 
<math display="block"> \frac {\kappa}{\sigma T} = \frac{3}{2}\left(\frac{k_{\rm B}}{e}\right)^2 = 1.11 \times 10^{-8} \, \text{W}\Omega/\text{K}^2</math>
Experimental values are typically in the range of <math>2-3 \times 10^{-8} \, \text{W}\Omega/\text{K}^2</math> for metals at temperatures between 0 and 100 degrees Celsius.<ref group="note" name=":19">{{Harvnb|Ashcroft|Mermin|1976|pp=18 table 1.6}}</ref>
{{math proof|title=Derivation and Drude's errors<ref group="note" name=":17">{{Harvnb|Ashcroft|Mermin|1976|pp=17}}</ref>|proof=

Solids can conduct heat through the motion of electrons, atoms, and ions. Conductors have a large density of free electrons whereas insulators do not; ions may be present in either. Given the good electrical and thermal conductivity in metals and the poor electrical and thermal conductivity in insulators, a natural starting point to estimate the thermal conductivity is to calculate the contribution of the conduction electrons.

The thermal current density is the flux per unit time of thermal energy across a unit area perpendicular to the flow.  It is proportional to the temperature gradient.
<math display="block">\mathbf{j}_q = - \kappa \nabla T </math>
where <math>\kappa</math> is the thermal conductivity.
In a one-dimensional wire, the energy of electrons depends on the local temperature <math>\epsilon[T(x)]</math>
If we imagine a temperature gradient in which the temperature decreases in the positive x direction, the average electron velocity is zero (but not the average speed). The electrons arriving at location x from the higher-energy side will arrive with energies <math>\varepsilon[T(x-v\tau)]</math>, while those from the lower-energy side will arrive with energies <math>\varepsilon[T(x+v\tau)]</math>.  Here, <math>v</math> is the average speed of electrons and <math>\tau</math> is the average time since the last collision.

The net flux of thermal energy at location x is the difference between what passes from left to right and from right to left:
<math display="block">\mathbf{j}_q  = \frac{1}{2} n v \big( \varepsilon[T(x-v\tau)] - \varepsilon[T(x+v\tau)] \big)</math>
The factor of {{sfrac|2}} accounts for the fact that electrons are equally likely to be moving in either direction.  Only half contribute to the flux at ''x''.

When the mean free path <math>\ell = v \tau</math> is small, the quantity
<math> \big( \varepsilon[T(x-v\tau)] - \varepsilon[T(x+v\tau)] \big) / 2 v \tau</math>
can be approximated by a derivative with respect to x.  This gives
<math display="block">\mathbf{j}_q  = n v^2 \tau \frac {d \varepsilon}{dT} \cdot \left(-\frac{dT}{dx} \right)</math>
Since the electron moves in the <math>x</math>, <math>y</math>, and <math>z</math> directions, the mean square velocity in the <math>x</math> direction is <math>\langle v_x^2 \rangle = \tfrac{1}{3} \langle v^2 \rangle</math>.  We also have <math>n \frac {d\varepsilon}{dT}=\frac{N}{V}\frac {d\varepsilon}{dT} = \frac{1}{V} \frac {dE}{dT} = c_v</math>, where <math>c_v</math> is the specific heat capacity of the material.

Putting all of this together, the thermal energy current density is
<math display="block">\mathbf{j}_q  = -\frac{1}{3} v^2 \tau c_v \nabla T</math>
This determines the thermal conductivity:
<math display="block">\kappa = \frac{1}{3} v^2 \tau c_v</math>
(This derivation ignores the temperature-dependence, and hence the position-dependence, of the speed v.  This will not introduce a significant error unless the temperature changes rapidly over a distance comparable to the mean free path.)

Dividing the thermal conductivity <math>\kappa</math> by the electrical conductivity <math>\sigma = \frac{n e^2 \tau} {m}</math> eliminates the scattering time <math>\tau</math> and gives
<math display="block">\frac{\kappa}{\sigma} = \frac{c_v m v^2}{3n e^2}</math>

At this point of the calculation, Drude made two assumptions now known to be errors.  First, he used the classical result for the specific heat capacity of the conduction electrons: <math> c_v= \tfrac{3}{2}n k_{\rm B}</math>.  This overestimates the electronic contribution to the specific heat capacity by a factor of roughly 100.  Second, Drude used the classical mean square velocity for electrons, <math>\tfrac{1}{2}mv^2=\tfrac{3}{2}k_{\rm B} T</math>.  This underestimates the energy of the electrons by a factor of roughly 100.  The cancellation of these two errors results in a good approximation to the conductivity of metals.  In addition to these two estimates, Drude also made a statistical error and overestimated the mean time between collisions by a factor of 2.  This confluence of errors gave a value for the Lorenz number that was remarkably close to experimental values.

The correct value of the Lorenz number as estimated from the Drude model is<ref group="note" name=":20">{{Harvnb|Ashcroft|Mermin|1976|pp=25 prob 1}}</ref>
<math display="block">\frac {\kappa}{\sigma T} = \frac{3}{2}\left(\frac{k_{\rm B}}{e}\right)^2 = 1.11 \times 10^{-8} \, \text{W}\Omega/\text{K}^2.</math>
}}

=== Thermopower ===
A generic temperature gradient when switched on in a thin bar will trigger a current of electrons towards the lower temperature side, given the experiments are done in an open circuit manner this current will accumulate on that side generating an electric field countering the electric current. This field is called thermoelectric field:
<math display="block">\mathbf{E} = Q \nabla T</math>
and Q is called thermopower. The estimates by Drude are a factor of 100 low given the direct dependency with the specific heat.
<math display="block">Q = - \frac{c_v}{3ne} = - \frac{k_{\rm B}}{2e} = 0.43 \times 10^{-4} \text{V}/\text{K} </math>
where the typical thermopowers at room temperature are 100 times smaller of the order of micro-volts.<ref group="note" name=":22">{{Harvnb|Ashcroft|Mermin|1976|pp=25}}</ref>
{{math proof|title=Proof together with the Drude errors<ref group="note" name=":21">{{Harvnb|Ashcroft|Mermin|1976|pp=24}}</ref>|proof=

From the simple one dimensional model
<math display="block">v_Q=\frac{1}{2}[v(x-v\tau)-v(x+v\tau)]=-v \tau \frac {dv}{dx}= - \tau \frac {d}{dx}\left(\frac{v^2}{2}\right)</math>
Expanding to 3 degrees of freedom <math>\langle v_x^2 \rangle = \frac{1}{3} \langle v^2 \rangle</math> 
<math display="block">\mathbf{v_Q}=- \frac {\tau}{6} \frac {dv^2}{dT} (\nabla T)</math>
The mean velocity due to the Electric field (given the equation of motion above at equilibrium)
<math display="block">\mathbf{v_E}=- \frac {e \mathbf{E} \tau}{m}</math>
To have a total current null <math>\mathbf{v_E} + \mathbf{v_Q} = 0</math> we have
<math display="block">Q = - \frac{1}{3e}\frac {d}{dT}\left(\frac{mv^2}{2}\right) = - \frac{c_v}{3ne}</math>
And as usual in the Drude case <math>c_v=\frac{3}{2}nk_{\rm B}</math> 
<math display="block">Q = - \frac{k_{\rm B}}{2e} = 0.43 \times 10^{-4} \text{V}/\text{K} </math>
where the typical thermopowers at room temperature are 100 times smaller of the order of micro-Volts.<ref group="note" name=":22">{{Harvnb|Ashcroft|Mermin|1976|pp=25}}</ref>
}}

== Accuracy of the model ==
The Drude model provides a very good explanation of DC and AC conductivity in metals, the [[Hall effect]], and the [[magnetoresistance]]<ref name=":2" group="note" /> in metals near room temperature. The model also explains partly the [[Wiedemann–Franz law]] of 1853.

Drude formula is derived in a limited way, namely by assuming that the charge carriers form a [[Classical physics|classical]] [[ideal gas]]. When quantum theory is considered, the Drude model can be extended to the [[free electron model]], where the carriers follow [[Fermi–Dirac distribution]]. The conductivity predicted is the same as in the Drude model because it does not depend on the form of the electronic speed distribution. However, Drude's model greatly overestimates the electronic heat capacity of metals. In reality, metals and insulators have roughly the same heat capacity at room temperature. Also, the Drude model does not explain the scattered trend of electrical conductivity versus frequency above roughly 2 THz.<ref name="Dressel">{{cite journal |author1=M. Dressel |author2=M. Scheffler |year=2006 |title=Verifying the Drude response |journal=[[Annalen der Physik]] |volume=15 |issue=7–8 |pages=535–544 |bibcode=2006AnP...518..535D |doi=10.1002/andp.200510198 |s2cid=14153937}}</ref><ref>{{Cite journal |last=Jeon |first=Tae-In |last2=Grischkowsky |first2=D. |last3=Mukherjee |first3=A. K. |last4=Menon |first4=Reghu |date=2000-10-16 |title=Electrical characterization of conducting polypyrrole by THz time-domain spectroscopy |url=http://dx.doi.org/10.1063/1.1319188 |journal=Applied Physics Letters |volume=77 |issue=16 |pages=2452–2454 |doi=10.1063/1.1319188 |issn=0003-6951|hdl=11244/19868 |hdl-access=free }}</ref>

The model can also be applied to positive (hole) charge carriers.

=== Drude response in real materials ===
The characteristic behavior of a Drude metal in the time or frequency domain, i.e. exponential relaxation with time constant {{mvar|τ}} or the frequency dependence for {{math|''σ''(''ω'')}} stated above, is called Drude response. In a conventional, simple, real metal (e.g. sodium, silver, or gold at room temperature) such behavior is not found experimentally, because the characteristic frequency {{math|''τ''<sup>−1</sup>}} is in the infrared frequency range, where other features that are not considered in the Drude model (such as [[Electronic band structure|band structure]]) play an important role.<ref name="Dressel" /> But for certain other materials with metallic properties, frequency-dependent conductivity was found that closely follows the simple Drude prediction for {{math|''σ''(''ω'')}}. These are materials where the relaxation rate {{math|''τ''<sup>−1</sup>}} is at much lower frequencies.<ref name="Dressel" /> This is the case for certain [[Doping (semiconductor)|doped semiconductor]] single crystals,<ref>{{cite journal |author1=M. van Exter |author2=D. Grischkowsky |year=1990 |title=Carrier dynamics of electrons and holes in moderately doped silicon |url=https://shareok.org/bitstream/11244/19898/1/okds_Grischkowsky_PRB_1990-06-15.pdf |journal=[[Physical Review B]] |volume=41 |issue=17 |pages=12140–12149 |bibcode=1990PhRvB..4112140V |doi=10.1103/PhysRevB.41.12140 |pmid=9993669 |hdl-access=free |hdl=11244/19898}}</ref> high-mobility [[2DEG|two-dimensional electron gases]],<ref>{{cite journal |author1=P. J. Burke |author2=I. B. Spielman |author3=J. P. Eisenstein |author4=L. N. Pfeiffer |author5=K. W. West |year=2000 |title=High frequency conductivity of the high-mobility two-dimensional electron gas |url=https://authors.library.caltech.edu/2073/1/BURapl00b.pdf |journal=[[Applied Physics Letters]] |volume=76 |issue=6 |pages=745–747 |bibcode=2000ApPhL..76..745B |doi=10.1063/1.125881}}</ref> and [[Heavy fermion|heavy-fermion metals]].<ref>{{cite journal |author1=M. Scheffler |author2=M. Dressel |author3=M. Jourdan |author4=H. Adrian |year=2005 |title=Extremely slow Drude relaxation of correlated electrons |journal=[[Nature (journal)|Nature]] |volume=438 |issue=7071 |pages=1135–1137 |bibcode=2005Natur.438.1135S |doi=10.1038/nature04232 |pmid=16372004 |s2cid=4391917}}</ref>

==See also==
*[[Free electron model]]
*[[Arnold Sommerfeld]]
*[[Electrical conductivity]]

== Citations ==
<references group="note" />

==References==
{{Reflist}}

=== General ===

* {{cite book|last1=Ashcroft|first1=Neil|author-link=Neil Ashcroft|last2=Mermin|first2=N. David|author-link2=N. David Mermin|title=[[Ashcroft and Mermin|Solid State Physics]]|publisher=Holt, Rinehart and Winston|location=New York|year=1976|isbn=978-0-03-083993-1}}

==External links==
*{{Cite book|title=Electrical Measurement, Signal Processing, and Displays|last=Heaney|first=Michael B|publisher=CRC Press|year=2003|isbn=9780203009406|editor-last=Webster|editor-first=John G.|chapter=Electrical Conductivity and Resistivity|chapter-url=https://www.researchgate.net/publication/309188334}}
{{Condensed matter physics topics}}

[[Category:Electric current]]
[[Category:Electric and magnetic fields in matter]]