Drude model

The simplest analysis of the Drude model assumes that electric field 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 dp between collisions, which occur on average every τ seconds.[note 1]

Then an electron isolated at time t will on average have been travelling for time τ since its last collision, and consequently will have accumulated momentum

${\displaystyle \Delta \langle \mathbf {p} \rangle =q\mathbf {E} \tau .}$

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

${\displaystyle \langle \mathbf {p} \rangle =q\mathbf {E} \tau .}$

Substituting the relations

${\displaystyle \langle \mathbf {p} \rangle =m\langle \mathbf {v} \rangle ,}$

${\displaystyle \mathbf {J} =nq\langle \mathbf {v} \rangle ,}$

results in the formulation of Ohm's law mentioned above:

${\displaystyle \mathbf {J} =\left({\frac {nq^{2}\tau }{m}}\right)\mathbf {E} .}$

Drude response of current density to an AC electric field.

The dynamics may also be described by introducing an effective drag force. At time t = t₀ + dt the average electron's momentum will be

${\displaystyle \langle \mathbf {p} (t_{0}+dt)\rangle =\left(1-{\frac {dt}{\tau }}\right)\left(\langle \mathbf {p} (t_{0})\rangle +q\mathbf {E} \,dt\right),}$

because, on average, a fraction of 1 − dt/τ of the electrons will not have experienced another collision, and the ones that have will contribute to the total momentum to only a negligible order.[note 3]

With a bit of algebra and dropping terms of order dt², this results in the differential equation

${\displaystyle {\frac {d}{dt}}\langle \mathbf {p} (t)\rangle =q\mathbf {E} -{\frac {\langle \mathbf {p} (t)\rangle }{\tau }},}$

where ⟨p⟩ denotes average momentum and q the charge of the electrons. This, which is an inhomogeneous differential equation, may be solved to obtain the general solution of

${\displaystyle \langle \mathbf {p} (t)\rangle =q\tau \mathbf {E} (1-e^{-t/\tau })+\langle \mathbf {p(0)} \rangle e^{-t/\tau }}$

for p(t). The steady state solution, d ⟨p⟩/dt = 0, is then

${\displaystyle \langle \mathbf {p} \rangle =q\tau \mathbf {E} .}$

As above, average momentum may be related to average velocity and this in turn may be related to current density,

${\displaystyle \langle \mathbf {p} \rangle =m\langle \mathbf {v} \rangle ,}$

${\displaystyle \mathbf {J} =nq\langle \mathbf {v} \rangle ,}$

and the material can be shown to satisfy Ohm's law ${\displaystyle \mathbf {J} =\sigma _{0}\mathbf {E} }$ with a DC-conductivity σ₀:

${\displaystyle \sigma _{0}={\frac {nq^{2}\tau }{m}}}$

Complex conductivity for different frequencies assuming that τ = 10⁻⁵ and that σ₀ = 1.

The Drude model can also predict the current as a response to a time-dependent electric field with an angular frequency ω. The complex conductivity is

${\displaystyle \sigma (\omega )={\frac {\sigma _{0}}{1-i\omega \tau }}={\frac {\sigma _{0}}{1+\omega ^{2}\tau ^{2}}}+i\omega \tau {\frac {\sigma _{0}}{1+\omega ^{2}\tau ^{2}}}.}$

Here it is assumed that:

${\displaystyle E(t)=\Re \left(E_{0}e^{i\omega t}\right);}$

${\displaystyle J(t)=\Re \left(\sigma (\omega )E_{0}e^{i\omega t}\right).}$

In engineering, i is generally replaced by −i (or −j ) in all equations, which reflects the phase difference with respect to origin, rather than delay at the observation point traveling in time. The imaginary part indicates that the current lags behind the electrical field. This happens because the electrons need roughly a time 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 σ(ω) are shown in the graph.

If a sinusoidally varying electric field with frequency ${\displaystyle \omega }$ 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

${\displaystyle \varepsilon ={\frac {D}{\varepsilon _{0}E}}=1+{\frac {P}{\varepsilon _{0}E}}}$

where ${\displaystyle D}$ is the electric displacement and ${\displaystyle P}$ is the polarization density.

The polarization density is written as

${\displaystyle P(t)=\Re \left(P_{0}e^{i\omega t}\right)}$

and the polarization density with n electron density is

${\displaystyle P=-nex}$

After a little algebra the relation between polarization density and electric field can be expressed as

${\displaystyle P=-{\frac {ne^{2}}{m\omega ^{2}}}E}$

The frequency dependent dielectric function of the solid is

${\displaystyle \varepsilon (\omega )=1-{\frac {ne^{2}}{\varepsilon _{0}m\omega ^{2}}}}$

At a resonance frequency ${\displaystyle \omega _{\rm {p}}}$, called the plasma frequency, the dielectric function changes sign from negative to positive and real part of the dielectric function drops to zero.

${\displaystyle \omega _{\rm {p}}={\sqrt {\frac {ne^{2}}{\varepsilon _{0}m}}}}$

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.[5] 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.

The characteristic behavior of a Drude metal in the time or frequency domain, i.e. exponential relaxation with time constant τ or the frequency dependence for 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 τ⁻¹ is in the infrared frequency range, where other features that are not considered in the Drude model (such as band structure) play an important role.[6] But for certain other materials with metallic properties, frequency-dependent conductivity was found that closely follows the simple Drude prediction for σ(ω). These are materials where the relaxation rate τ⁻¹ is at much lower frequencies.[6] This is the case for certain doped semiconductor single crystals,[7] high-mobility two-dimensional electron gases,[8] and heavy-fermion metals.[9]

• Ashcroft, Neil; Mermin, N. David (1976). Solid State Physics. New York: Holt, Rinehart and Winston. ISBN 978-0-03-083993-1.CS1 maint: ref=harv (link)