TheDrude model ofelectrical conduction was proposed in 1900^[1][2] byPaul Drude to explain the transport properties ofelectrons in materials (especially metals). Basically,Ohm's law was well established and stated that the currentJ and voltageV driving the current are related to the resistanceR 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 ofresistivity. 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 ofkinetic theory, assumes that when electrons in a solid are exposed to the electric field, they behave much like apinball machine. The sea of constantly jittering electrons bouncing and re-bouncing off heavier, relatively immobile positive ions produce a net collective motion in the direction opposite to the applied electric field. This classical microscopic behaviour forms within severalfemtoseconds^[1] and affects optical properties of solids such asrefractive index or absorption spectrum.

In modern terms this is reflected in thevalence electron model where the sea of electrons is composed of the valence electrons only,^[3] 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 thevalence number of the atoms.^{[Ashcroft & Mermin 1]}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.^{[Ashcroft & Mermin 2]}

The scattering lengths computed in the Drude model, are of the order of 10 to 100 interatomic 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.^{[Ashcroft & Mermin 3]}

The model gives better predictions for metals, especially in regards to conductivity,^{[Ashcroft & Mermin 4]} and sometimes is called Drude theory of metals. This is because metals have essentially a better approximation to thefree electron model, i.e. metals do not have complexband structures, electrons behave essentially asfree particles and where, in the case of metals, theeffective number of de-localized electrons is essentially the same as the valence number.^{[Ashcroft & Mermin 5]}

The two most significant results of the Drude model are an electronic equation of motion,$${\displaystyle \frac{d}{dt}⟨\mathbf{p}(t)⟩=q\left(\mathbf{E}+\frac{⟨\mathbf{p}(t)⟩}{m}\times \mathbf{B}\right)-\frac{⟨\mathbf{p}(t)⟩}{\tau },}$$and a linear relationship betweencurrent densityJ and electric fieldE,$${\displaystyle \mathbf{J}=\frac{n{q}^2\tau }{m}\mathbf{E}.}$$

Heret is the time, ⟨p⟩ is the average momentum per electron andq, n, m, andτ are respectively the electron charge, number density, mass, andmean free time between ionic collisions. The latter expression is particularly important because it explains in semi-quantitative terms whyOhm's law, one of the most ubiquitous relationships in all of electromagnetism, should hold.^{[Ashcroft & Mermin 6][4][5]}

Steps towards a more modern theory of solids were given by the following:

• TheEinstein solid model and theDebye model, suggesting that the quantum behaviour of exchanging energy in integral units orquanta was an essential component in the full theory especially with regard tospecific 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.^{[Ashcroft & Mermin 7]}

Drude usedMaxwell–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 correctFermi Dirac statistics,Sommerfeld significantly improved the predictions of the model, although still having asemi-classical theory that could not predict all results of the modern quantum theory of solids.^{[Ashcroft & Mermin 8]}

German physicistPaul 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.^[6] In his original paper, Drude made an error, estimating the Lorenz number ofWiedemann–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.^{[Ashcroft & Mermin 9]}

The firstdirect proof of atoms through the computation of theAvogadro number from a microscopic model is due toAlbert Einstein, thefirst modern model of atom structure dates to 1904 and theRutherford model to 1909. Drude starts from the discovery of electrons in 1897 byJ.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.^{[Ashcroft & Mermin 10]} The model was extended in 1905 byHendrik Antoon Lorentz (and hence is also known as theDrude–Lorentz model)^[7] to give the relation between thethermal conductivity and theelectric conductivity of metals (seeLorenz number), and is aclassical model. Later it was supplemented with the results of quantum theory in 1933 byArnold Sommerfeld andHans Bethe, leading to theDrude–Sommerfeld model.

Nowadays the Drude andSommerfeld models are still significant to understanding the qualitative behaviour of solids and to get a first qualitative understanding of a specific experimental setup.^{[Ashcroft & Mermin 11]} This is a generic method insolid 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-blownquantum 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).^[8]

Drude used thekinetic theory of gases applied to the gas of electrons moving on a fixed background of "ions"; this is in contrast with the usual way of applying the theory of gases as a neutral diluted gas with no background. Thenumber density of the electron gas was assumed to be$${\displaystyle n=\frac{N_{\text{A}}Z{\rho }_{\text{m}}}{A},}$$whereZ is the effective number of de-localized electrons per ion, for which Drude used the valence number,A is the atomic mass per mole,^{[Ashcroft & Mermin 10]}${\displaystyle {\rho }_{\text{m}}}$ is the mass density (mass per unit volume)^{[Ashcroft & Mermin 10]} of the "ions", andN_A is theAvogadro constant.Considering the average volume available per electron as a sphere:$${\displaystyle \frac{V}{N}=\frac{1}{n}=\frac{4}{3}\pi r_{\mathrm{s}}^3.}$$The quantity${\displaystyle r_{\text{s}}}$ is a parameter that describes the electron density and is often of the order of 2 or 3 times theBohr radius, foralkali 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.^{[Ashcroft & Mermin 12]}

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.^{[Ashcroft & Mermin 13]}
• 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 thevalence electrons of the atoms that have become delocalized due to the electric field of the other atoms.^{[Ashcroft & Mermin 12]}
• The Drude model neglects long-range interaction between the electron and the ions or between the electrons; this is called the independent electron approximation.^{[Ashcroft & Mermin 12]}
• The electrons move in straight lines between one collision and another; this is called free electron approximation.^{[Ashcroft & Mermin 12]}
• The only interaction of a free electron with its environment was treated as being collisions with the impenetrable ions core.^{[Ashcroft & Mermin 12]}
• The average time between subsequent collisions of such an electron isτ, with amemorylessPoisson distribution. The nature of the collision partner of the electron does not matter for the calculations and conclusions of the Drude model.^{[Ashcroft & Mermin 12]}
• 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.^{[Ashcroft & Mermin 12]} 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 theMaxwell–Boltzmann statistics with theFermi–Dirac statistics leads to theDrude–Sommerfeld model.
• Improving the hypothesis of the Maxwell–Boltzmann statistics with theBose–Einstein statistics leads to considerations about the specific heat of integer spin atoms^[9] and to theBose–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).^[10]

The simplest analysis of the Drude model assumes that electric fieldE is both uniform and constant, and that the thermal velocity of electrons is sufficiently high such that they accumulate only an infinitesimal amount of momentumdp between collisions, which occur on average everyτ seconds.^{[Ashcroft & Mermin 6]}

Then an electron isolated at timet will on average have been travelling for timeτ since its last collision, and consequently will have accumulated momentum$${\displaystyle \mathrm{\Delta }⟨\mathbf{p}⟩=q\mathbf{E}\tau .}$$

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$${\displaystyle ⟨\mathbf{p}⟩=q\mathbf{E}\tau .}$$

Substituting the relationsresults in the formulation of Ohm's law mentioned above:$${\displaystyle \mathbf{J}=\left(\frac{n{q}^2\tau }{m}\right)\mathbf{E}.}$$

The dynamics may also be described by introducing an effective drag force. At timet =t₀ +dt the electron's momentum will be:$${\displaystyle \mathbf{p}(t_0+dt)=\left(1-\frac{dt}{\tau }\right)\left[\mathbf{p}(t_0)+\mathbf{f}(t)dt+O(d{t}^2)\right]+\frac{dt}{\tau }\left(\mathbf{g}(t_0)+\mathbf{f}(t)dt+O(d{t}^2)\right)}$$where${\displaystyle \mathbf{f}(t)}$ can be interpreted as generic force (e.g.Lorentz force) on the carrier or more specifically on the electron.${\displaystyle \mathbf{g}(t_0)}$ is the momentum of the carrier with random direction after the collision (i.e. with a momentum${\displaystyle ⟨\mathbf{g}(t_0)⟩=0}$) and with absolute kinetic energy$${\displaystyle \frac{⟨|\mathbf{g}(t_0)|{⟩}^2}{2m}=\frac{3}{2}KT.}$$

On average, a fraction of${\displaystyle 1-\frac{dt}{\tau }}$ 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${\displaystyle \frac{dt}{\tau }\mathbf{f}(t)dt}$ which is of second order.^{[Ashcroft & Mermin 14]}

With a bit of algebra and dropping terms of order${\displaystyle d{t}^2}$, this results in the generic differential equation$${\displaystyle \frac{d}{dt}\mathbf{p}(t)=\mathbf{f}(t)-\frac{\mathbf{p}(t)}{\tau }}$$

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

At timet =t₀ +dt the average electron's momentum will be$${\displaystyle ⟨\mathbf{p}(t_0+dt)⟩=\left(1-\frac{dt}{\tau }\right)\left(⟨\mathbf{p}(t_0)⟩+q\mathbf{E}dt\right),}$$and then$${\displaystyle \frac{d}{dt}⟨\mathbf{p}(t)⟩=q\mathbf{E}-\frac{⟨\mathbf{p}(t)⟩}{\tau },}$$where⟨p⟩ denotes average momentum andq the charge of the electrons. This, which is an inhomogeneous differential equation, may be solved to obtain the general solution of$${\displaystyle ⟨\mathbf{p}(t)⟩=q\tau \mathbf{E}(1-e^{-t/\tau })+⟨\mathbf{p}(0)⟩e^{-t/\tau }}$$forp(t). Thesteady state solution,⁠d/dt⁠⟨p⟩ = 0, is then$${\displaystyle ⟨\mathbf{p}⟩=q\tau \mathbf{E}.}$$

As above, average momentum may be related to average velocity and this in turn may be related to current density,and the material can be shown to satisfy Ohm's law${\displaystyle \mathbf{J}={\sigma }_0\mathbf{E}}$ with aDC-conductivityσ₀:$${\displaystyle {\sigma }_0=\frac{n{q}^2\tau }{m}}$$

The Drude model can also predict the current as a response to a time-dependent electric field with an angular frequencyω. 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: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 timeτ 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 forσ(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 distancex 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.

Thedielectric constant of the sample is expressed as$${\displaystyle {\epsilon }_r=\frac{D}{{\epsilon }_{0}E}=1+\frac{P}{{\epsilon }_{0}E}}$$where${\displaystyle D}$ is theelectric displacement and${\displaystyle P}$ is thepolarization density.

The polarization density is written as$${\displaystyle P(t)=\mathrm{\Re }\left(P_0{e}^{i\omega t}\right)}$$and the polarization density withn electron density is$${\displaystyle P=-nex}$$After a little algebra the relation between polarization density and electric field can be expressed as$${\displaystyle P=-\frac{n{e}^2}{m{\omega }^2}E}$$The frequency dependent dielectric function of the solid is$${\displaystyle {\epsilon }_r(\omega )=1-\frac{n{e}^2}{{\epsilon }_{0}m{\omega }^2}}$$

At a resonance frequency${\displaystyle {\omega }_{\mathrm{p}}}$, called theplasma frequency, the dielectric function changes sign from negative to positive and real part of the dielectric function drops to zero.$${\displaystyle {\omega }_{\mathrm{p}}=\sqrt{\frac{n{e}^2}{{\epsilon }_{0}m}}}$$The plasma frequency represents aplasma oscillation resonance orplasmon. 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.^[11] 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 ofultraviolet radiation.^{[Ashcroft & Mermin 17]}

One great success of the Drude model is the explanation of theWiedemann-Franz law. This was due to a fortuitous cancellation of errors in Drude's original calculation. Drude predicted the value of the Lorenz number:$${\displaystyle \frac{\kappa }{\sigma T}=\frac{3}{2}{\left(\frac{k_{\mathrm{B}}}{e}\right)}^2=1.11\times {10}^{-8}\mathrm{W}\cdot \mathrm{\Omega }/{\mathrm{K}}^2}$$Experimental values are typically in the range of${\displaystyle 2-3\times {10}^{-8}\mathrm{W}\cdot \mathrm{\Omega }/{\mathrm{K}}^2}$ for metals at temperatures between 0 and 100 degrees Celsius.^{[Ashcroft & Mermin 18]}

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:$${\displaystyle \mathbf{E}=Q\mathrm{\nabla }T}$$andQ is called thermopower. The estimates by Drude are a factor of 100 low given the direct dependency with the specific heat.$${\displaystyle Q=-\frac{c_v}{3ne}=-\frac{k_{\mathrm{B}}}{2e}=0.43\times {10}^{-4}\mathrm{V}/\mathrm{K}}$$where the typical thermopowers at room temperature are 100 times smaller, of the order of microvolts.^{[Ashcroft & Mermin 20]}

The Drude model provides a very good explanation of DC and AC conductivity in metals, theHall effect, and themagnetoresistance^{[Ashcroft & Mermin 14]} in metals near room temperature. The model also explains partly theWiedemann–Franz law of 1853.

Drude formula is derived in a limited way, namely by assuming that the charge carriers form aclassicalideal gas. When quantum theory is considered, the Drude model can be extended to thefree electron model, where the carriers followFermi–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.^[12][13]

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

The characteristic behavior of a Drude metal in the time or frequency domain, i.e. exponential relaxation with time constantτ or the frequency dependence forσ(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 frequencyτ⁻¹ is in the infrared frequency range, where other features that are not considered in the Drude model (such asband structure) play an important role.^[12] But for certain other materials with metallic properties, frequency-dependent conductivity was found that closely follows the simple Drude prediction forσ(ω). These are materials where the relaxation rateτ⁻¹ is at much lower frequencies.^[12] This is the case for certaindoped semiconductor single crystals,^[14] high-mobilitytwo-dimensional electron gases,^[15] andheavy-fermion metals.^[16]

• Free electron model
• Arnold Sommerfeld
• Electrical conductivity

• Ashcroft, Neil;Mermin, N. David (1976).Solid State Physics. New York: Holt, Rinehart and Winston.ISBN 978-0-03-083993-1.
• Kittel, Charles (2005).Introduction to Solid State Physics (8th ed.). John Wiley & Sons Inc.ISBN 0-471-41526-X.
• Ziman, J.M. (1972).Principles of the theory of solids (2nd ed.). Cambridge university press.ISBN 0-521-29733-8.

• Heaney, Michael B (2003)."Electrical Conductivity and Resistivity". In Webster, John G. (ed.).Electrical Measurement, Signal Processing, and Displays. CRC Press.ISBN 9780203009406.

• Electric current
• Material dispersion models
• Plasmonics

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