Editing Drude model (section)

== 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="Ashcroft & Mermin" />

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>\textstyle 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>\textstyle \frac{dt}{\tau}\mathbf{f}(t)dt</math> which is of second order.<ref group="Ashcroft & Mermin" 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''|''dt''}}⟨'''p'''⟩ {{=}} 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 {{math|−''i''}} (or {{math|−''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="Ashcroft & Mermin" 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 {{math|''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_r = \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 {{math|''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_r(\omega) = 1 - \frac {n e^2}{\varepsilon_0m \omega^2}</math>

{{math proof|title=Proof using Maxwell's equations<ref group="Ashcroft & Mermin" 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.
The following are Maxwell's equations without sources (which are treated separately in the scope of [[plasma oscillation]]s), in [[Gaussian units]]:
<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>\varepsilon(\omega)</math> in the Helmholtz form 
<math display="block"> - \nabla^2 \mathbf{E} = \frac{\omega^2}{c^2} \varepsilon(\omega) \mathbf{E}</math>
where the refractive index is <math display="inline">n(\omega) = \sqrt{\varepsilon(\omega)}</math> and the phase velocity is <math> v_\text{p} = \frac{c}{n(\omega)}</math>  
therefore the complex dielectric constant is 
<math display="block">\varepsilon(\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">\varepsilon(\omega) = \left( 1 - \frac{\omega_{\rm p}^2}{\omega^2} \right); \omega_{\rm p}^2 = \frac {4\pi n e^2}{m}  \text{(Gaussian units)}.</math> In [[International_System_of_Units | SI units]] the <math>4 \pi</math> in the numerator is replaced by <math>\varepsilon_0</math> in the denominator and the dielectric constant is written as <math>\varepsilon_r</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="Ashcroft & Mermin" 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} \, \mathrm{W{\cdot}\Omega/K^2}</math>
Experimental values are typically in the range of <math>2-3 \times 10^{-8} ~ \mathrm{W{\cdot}\Omega/K^2}</math> for metals at temperatures between 0 and 100 degrees Celsius.<ref group="Ashcroft & Mermin" name=":19">{{harvnb|Ashcroft|Mermin|1976|pp=18 table 1.6}}</ref>
{{math proof|title=Derivation and Drude's errors<ref group="Ashcroft & Mermin" 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>\varepsilon[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 {{math|''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 {{math|''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 {{math|''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 {{math|''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 {{math|''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="Ashcroft & Mermin" 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 {{math|''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} \mathrm{~V/K} </math>
where the typical thermopowers at room temperature are 100 times smaller, of the order of microvolts.<ref group="Ashcroft & Mermin" name=":22">{{harvnb|Ashcroft|Mermin|1976|pp=25}}</ref>
{{math proof|title=Proof together with the Drude errors<ref group="Ashcroft & Mermin" 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}~\mathrm{V/K} </math>
where the typical thermopowers at room temperature are 100 times smaller of the order of microvolts.<ref group="Ashcroft & Mermin" name=":22">{{harvnb|Ashcroft|Mermin|1976|pp=25}}</ref>
}}