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For a weakly absorbing doped semiconductor with one dominant carrier species and approximately constant background dielectric function ${ϵ}_b>1$, the Drude model gives a normal-incidence reflectivity minimum near ${ϵ}_r=1$. Thus ${\omega }_{pc}^2=({ϵ}_b-1){\omega }_{\min }^2$, with ${\omega }_{pc}^2=n{e}^2/({ϵ}_0{m}^{\ast })$ and ${\omega }_{\min }=2\pi c/{\lambda }_{\min }$. Eliminating the plasma frequency proves the displayed mass estimate. The real-dielectric zero instead has ${\omega }_0\simeq {\omega }_{pc}/\sqrt{{ϵ}_b}$ and is the nearby screened plasma edge, so its wavelength requires a different denominator. With appreciable loss, fit the full complex Fresnel equations rather than identifying the minimum with an exactly real dielectric value.

Charge carrier mobility is the magnitude of steady drift velocity per unit weak electric field in a scalar transport model. In the Drude model, ${\mathbf{v}}_d=\mathrm{sgn}(q)\mu \mathbf{E}$ and $\sigma =n|q|\mu$. The positive charge carrier mobility separates ease of motion from the sign of electric charge; its SI unit is ${\mathrm{m}}^2/(\mathrm{V}\mathrm{s})$.

A motivating laboratory realization is The Hall Effect and Conductivity in Germanium by Rex Tayloe. The handout specifies a germanium sample holder with temperature and magnetic field control, a low-voltage supply, digital meters, and a current-reversing switch. Measure the sample dimensions, map the contacts, and establish the directions of electric current, magnetic field, and transverse voltage. At fixed temperature and zero magnetic field, sweep positive and negative electric currents and record longitudinal electric potential differences. A straight voltage-versus-electric current relation tests Ohm's law; sample geometry converts its slope to electrical resistivity. For the Hall effect, minimize the zero-field transverse offset and measure transverse voltage at all four combinations of $\pm I$ and $\pm B$, repeating for several electric currents. The readouts test longitudinal proportionality, transverse polarity, and transverse proportionality to electric current. These are protocol instructions, not reported measurement outcomes.^[ref]

For interpreting such measurements, an ideal uniform bar of wire cross-sectional area $A$ with longitudinal sensing contacts separated by $L$ gives $\sigma =LI/(A{V}_L)$. Four-terminal sensing suppresses voltage drops in current contacts when the voltage meter draws negligible electric current; it does not fix uncertain dimensions or a nonuniform current density. Reverse electric current and use $[V_L(+I)-V_L(-I)]/2$ to reject a constant voltage offset. Keep temperature stable, allow the current density to settle, and reduce electric current to test for Joule heating. Those are controls for interpreting the material coefficient, rather than reasons to assume every measured curve is linear.

For the transverse measurement define $\mathbf{J}=J_x\hat{\mathbf{x}}$, $\mathbf{B}=B_z\hat{\mathbf{z}}$, width $w$ along $y$, and thickness $d$ along $z$. Define $V_H=V(y=-w/2)-V(y=+w/2)=E_{y}w$, so $J_x=I/(wd)$ givesExtract the part odd in both electric current and magnetic field usingThis combination rejects a constant offset and longitudinal pickup even in magnetic field. It cannot reject another signal with the same two reversal symmetries. Check field polarity and magnitude independently; a sign error in the contact map changes the inferred Hall coefficient.

For one Drude model species in transverse open circuit, the steady Lorentz force balance is $0=q(\mathbf{E}+{\mathbf{v}}_d\times \mathbf{B})-m{\mathbf{v}}_d/\tau$ with $v_{d,y}=0$. Its $y$ component gives $E_y=v_{d,x}{B}_z$. ConsequentlyLongitudinal electrical conductivity and the Hall coefficient therefore separate number density from charge carrier mobility in this single-species model. Inferring a momentum relaxation time additionally requires an appropriate mass. A constant-field longitudinal measurement alone cannot supply all three.

A Hall coefficient is not universally the reciprocal of signed number density. For example, adding independent electron and electron hole contributions with densities $n,p$ and positive charge carrier mobilities ${\mu }_e,{\mu }_h$ in the weak-magnetic field limit givesThe Hall coefficient can vanish even while electrical conductivity is positive, and its sign weights charge carrier mobility as well as number density. The germanium handout explicitly asks students to track possible transverse sign changes with temperature; it does not establish a universal single-carrier interpretation. Real electronic band structure and unequal scattering also limit that interpretation.^[ref]

Switch on a constant electric field ${\mathbf{E}}_0$ at $t=0$. For prescribed initial current density $\mathbf{J}(0)$, the exact Drude model solution is the displayed formula. For electrons, the corresponding drift velocity isStarting from zero drift velocity, the current density reaches $1-e^{-1}\approx 0.632$ of its limiting value after one momentum relaxation time. Switching the electric field off at $T$ instead gives $\mathbf{J}(t)=\mathbf{J}(T)e^{-(t-T)/\tau }$ for $t\ge T$. A finite step of electric field changes the acceleration immediately but leaves drift velocity and current density continuous.

More generally, multiplying the current density equation by an integrating factor givesThis is the causal response of a first-order relaxation equation: only earlier electric fields enter the integral. It also shows why instantaneous Ohm's law is an approximation when the electric field changes appreciably during a momentum relaxation time.

Use physical electric fields $\mathbf{E}(t)=\mathrm{Re}[\hat{\mathbf{E}}{e}^{-i\omega t}]$ and current densities $\mathbf{J}(t)=\mathrm{Re}[\hat{\mathbf{J}}{e}^{-i\omega t}]$, with peak complex amplitudes. After the initial transient has decayed, substitution into the Drude model equation gives the complex conductivityHere the mobile species is electrons. With the alternative $e^{+i\omega t}$ convention, the denominator is $1+i\omega \tau$ and the imaginary part changes sign; the physical response is identical.^[ref]

For $E_x(t)=E_0\cos \omega t$ with $\omega >0$, the real-time current density isThus a positive argument of complex conductivity in the $e^{-i\omega t}$ convention means that current density lags the electric field. At $\omega \tau =1$, its real part and imaginary part are both ${\sigma }_0/2$, its modulus is ${\sigma }_0/\sqrt{2}$, and the lag is $\pi /4$.

For $\omega \tau \ll 1$, complex conductivity tends to the positive constant ${\sigma }_0$: current density nearly follows the electric field. For $\omega \tau \gg 1$ within this model,The current density amplitude decreases and its response becomes predominantly inertial. Average power absorbed per volume is $\overline{p}=\frac{1}{2}\mathrm{Re}\sigma |\hat{\mathbf{E}}{|}^2\ge 0$. These limits describe the material response to a specified local electric field, not the complete electromagnetic response of a finite specimen with contacts. The Drude response has its pole at $\omega =-i/\tau$, consistent with the decaying causal response of a first-order relaxation equation.

The simple Drude model does not predict the microscopic momentum relaxation time. It neglects electronic band structure and Fermi-Dirac statistics; a useful semiclassical extension replaces $m$ by an appropriate band effective mass, but neither a single scalar mass nor one constant momentum relaxation time describes every material. Several charge carriers, energy-dependent scattering, and transitions between electronic bands can spoil the simple Drude frequency response. A successful low-angular frequency fit does not justify extrapolation to arbitrarily large angular frequencies.

The Drude model explains finite electrical conductivity by balancing charge carrier acceleration in an electric field against loss of directed momentum through collisions. Its useful surprise is that continuous microscopic motion can produce a constant macroscopic current density: the applied electric field continually restores the directed momentum lost to the material. The simplest version has one mobile species, constant number density $n$, electric charge $q$, mass $m>0$, and momentum relaxation time $\tau >0$, in a stationary, homogeneous, isotropic material. At zero magnetic field, its mean drift velocity and current density obeyFor electrons, $q=-e$ with positive elementary charge $e$, so ${\sigma }_0=n{e}^2\tau /m$. The electron drift velocity points against a constant electric field, whereas conventional current points along it. Random microscopic velocities can have zero mean without being zero individually; drift velocity is this population mean, rather than each particle's speed.^[ref]

The same first-order linear differential equation predicts both Drude current relaxation after switching an electric field and the Drude frequency response to a sinusoidal electric field. Together they let a measured electrical conductivity be interpreted in terms of a density, a mass, and a time scale; a single constant-field measurement determines only their combination $n{q}^2\tau /m$.

The original 1900 paper Zur Elektronentheorie der Metalle by Paul Drude sought a common account of electrical transport and heat transfer using mobile charged particles and ideas from the kinetic theory of gases. Its introduction, especially printed pp. 568–569, explains that comparing different transport phenomena could constrain the populations of free and bound charged particles. The modern momentum relaxation time formulation above is a useful presentation of the model, not a verbatim translation of the original notation or its historical assumptions about mass.^[ref]

The positive electron mobility relates the drift velocity of conduction electrons to an electric field: ${\mathbf{v}}_d=-{\mu }_e\mathbf{E}$. In the isotropic Drude model, ${\mu }_e=e\tau /m^{\ast }$, where $\tau$ is the momentum relaxation time and $m^{\ast }$ the band effective mass. The minus sign describes negative charge; the electron mobility itself is positive.

For a spin-degenerate parabolic two-dimensional band, the Drude model gives $\tau =m^{\ast }/(n{e}^2{\rho }_{\square })$ and $v_F=\hslash k_F/m^{\ast }$. Their product is the displayed mass-independent mean free path. A Fermi-surface transport interpretation requires a sufficiently degenerate electron gas; a quantum lifetime need not equal the transport relaxation time.

In a one-carrier Drude model, the Hall coefficient is $R_H=1/(nq)$. Combining its inferred electron density with atomic density ${\rho }_m{N}_A/M$ gives the displayed number of mobile electrons per atom. The molar mass and mass density must be supplied independently. Multiple bands, anisotropic carrier motion and Hall factors limit this simple interpretation; field reversal helps distinguish the transverse response from voltage offsets.

In an isotropic single-carrier Drude model, let the electric current density point along $x$, with zero transverse current and magnetic field along $z$. The Hall coefficient is $R_H=1/(nq)$ and the longitudinal electrical conductivity is $\sigma =n|q|\mu$, where the charge carrier mobility $\mu$ is positive. Consequently $E_y/E_x=R_H\sigma B=\mathrm{sgn}(q)\mu B$. This signed Hall angle convention describes the angle of the electric field relative to the current; a convention measuring current relative to field has the opposite sign.

In a one-carrier isotropic Drude model, $R_H=1/(nq)$ and an inferred carrier density is $n_H=1/(e|R_H|)$. Dividing by the atomic number density gives an effective Hall count per atom. A noninteger result need not mean a literal fractional count of valence electrons: multiple energy bands, anisotropic Fermi surfaces and different charge carrier mobilities can change the Hall coefficient. In a two-carrier isotropic approximation,Its sign identifies the dominant Hall response rather than necessarily the most numerous charge carrier.

The Hall coefficient relates a transverse electric field to longitudinal current density and transverse magnetic field, with oriented axes as in the displayed formula. Its SI unit is ${\mathrm{m}}^3/C$. For one isotropic Drude model species it is $1/(nq)$; multiple charge carriers need not follow that formula.

The Drude model relative permittivity is $1-{\omega }_p^2/[\omega (\omega +i/\tau )]$. At frequencies large compared with the collision rate but below ${\omega }_p$, its leading real part is negative and the physical spatial wave-number branch has positive imaginary part. The field therefore decays into the conductor; close to the threshold the full complex response must be retained.

Collisional absorption of electromagnetic wave energy by charged particles scattering from other charges. In a weakly collisional unmagnetized Drude model, with $X={\omega }_p^2/{\omega }^2<1$, the intensity absorption coefficient is ${\alpha }_I\simeq ({\nu }_{ei}/c)X/\sqrt{1-X}$. The corresponding electric-field amplitude decays at half that rate.

The momentum relaxation time is the exponential relaxation time of mean directed momentum after the driving electric field is removed. In the simplest Drude model, independent collisions occur with probability $dt/\tau$ during a short time interval $dt$ and reset the mean velocity of the collided population to zero. The remaining fraction preserves its previous mean momentum. To first order in $dt$, averaging gives $d⟨\mathbf{p}⟩/dt=q\mathbf{E}-⟨\mathbf{p}⟩/\tau$. This is a population equation: it does not assert that every particle stops at equally spaced instants.^[ref]

For real materials, a collision that changes a particle's direction only slightly need not erase its directed momentum, so the momentum relaxation time need not equal the time between all scattering events. This transport time is also distinct from the bulk charge relaxation time $ϵ/\sigma$: one describes loss of directed motion, the other redistribution of charge density.

For one spin-degenerate parabolic two-dimensional band, $k_F=\sqrt{2\pi n}=1.3729\times 10^8$ m${}^{-1}$. The free-electron model with $m=m_e$ givesThe given $10^3\Omega$ is a sheet resistivity, not a bulk resistivity in ohm-metres. The two-dimensional Drude sheet conductance is $n{e}^2\tau /m_e$, soThus $v_F=\hslash k_F/m_e=1.589\times 10^4$ m/s andThe sample temperature is not given. This is specifically the Fermi-surface mean free path, appropriate in the degenerate regime $T\ll 8.3$ K for the bare-mass model. At higher temperatures the characteristic thermal speed should replace $v_F$ for a thermal transport distance. If a real two-dimensional band has mass $m^{\ast }$, the stated results scale as $E_F,T_F\propto 1/m^{\ast }$, $\tau \propto m^{\ast }$ and $v_F\propto 1/m^{\ast }$; the inferred ${\ell }_F=\hslash k_F/(n{e}^2{\rho }_{\square })$ is mass independent under these same Drude assumptions. No semiconductor effective mass or valley degeneracy is silently introduced.

A conventional face-centered cubic cell contains four atoms. With three conduction electrons per atom,Using the bare electron mass,The Drude transport relation is $\rho =m_e/(n{e}^2\tau )$, givingFor a degenerate electron gas, transport samples electrons near the Fermi surface, so use $v_F=\hslash k_F/m_e=2.025\times 10^6$ m/s:This is a transport relaxation time and associated transport mean free path inferred from the specified resistivity. It need not equal a single-particle quantum lifetime when scattering is strongly angle dependent. The free-electron valence count also neglects the detailed band structure of aluminium, as the stipulated model requires. A classical thermal speed is not the appropriate velocity scale for this strongly degenerate metal.

At zero magnetic field the two circular conductivities coincide, so an isotropic material has no magnetically induced circular birefringence. In the collisionless Drude model range the dielectric function is $ϵ={ϵ}_{\infty }-{\omega }_p^2/{\omega }^2$. A high carrier density gives a large plasma frequency, often placing the screened plasma edge ${\omega }_p/\sqrt{{ϵ}_{\infty }}$ in the infrared or visible range.

Below that edge the real dielectric function is negative and a transverse wave is evanescent in the bulk. The material is strongly reflecting, with penetration through a skin depth and losses set by finite $1/\tau$. Above the plasma edge propagation becomes possible, subject to absorption and interband transitions. Carrier density alone does not specify a complete optical spectrum: the statements assume the probe frequency relative to the plasma scale and the validity of the single-band Drude model.

Use time dependence $e^{-i\omega t}$ and assume the dopants supply $N$ mobile electrons per unit volume. The Drude model equation isThe free-carrier electric polarization is ${\mathbf{P}}_c=-Ne\mathbf{x}$, so, after adding the bound-electron background,ThereforeThe symbol ${\omega }_p$ here is the background-screened plasma frequency; the unscreened carrier frequency is $\sqrt{{ϵ}_{\infty }}{\omega }_p$. For positive $\gamma$, the imaginary part of ${ϵ}_r$ is positive in this time convention, consistent with absorption. Incomplete dopant ionization would replace the nominal doping density by the actual mobile carrier density.

The Drude model treats carriers as classical independent particles, accelerated between collisions by the electric force. Collisions occur at constant rate ${\tau }^{-1}$ and randomize the directed velocity to zero mean; the ions are an effectively stationary background. Over a short time $dt$, directed velocity gains $(e/m)\mathbf{E}dt$ while the fraction $dt/\tau$ that collides loses its mean directed motion. ThereforeHere $e$ is the signed carrier charge. For the stated complex harmonic convention,The conductivity is positive in its dissipative real part regardless of the sign of $e$.

For zero charge density, Gauss's law makes the electric amplitude transverse. Faraday's and Ampère-Maxwell's laws, with displacement current retained, giveSubstitution of $e^{i\mathbf{k}\cdot \mathbf{x}-i\omega t}$ yieldsFor propagation in a fixed direction, $k$ is the possibly complex scalar wave number; equivalently the spatial equation uses $\mathbf{k}\cdot \mathbf{k}$, without complex conjugation. The printed real-vector notation $|\mathbf{k}|$ must be analytically continued in an absorbing or evanescent medium.

In the high-frequency Drude plasma cutoff regime, $\omega \tau \gg 1$ gives ${ϵ}_r\simeq 1-{\omega }_p^2/{\omega }^2$. Below the plasma frequency, the physical decaying branch isFinite collisions add a small phase component while maintaining positive attenuation. Very near the threshold one should retain the full complex permittivity rather than the purely imaginary approximation.