= Drude frequency response
{c}
{title2=$\sigma(\omega)=\sigma_0/(1-i\omega\tau)$}
Use physical <electric fields> $\mathbf E(t)=\operatorname{Re}[\widehat{\mathbf E}e^{-i\omega t}]$ and <current densities> $\mathbf J(t)=\operatorname{Re}[\widehat{\mathbf J}e^{-i\omega t}]$, with peak <complex amplitudes>. After the initial transient has decayed, substitution into the <Drude model> equation gives the <complex conductivity>
$$
\sigma(\omega)=\frac{ne^2\tau/m}{1-i\omega\tau},\qquad
\operatorname{Re}\sigma=\frac{\sigma_0}{1+\omega^2\tau^2},\qquad
\operatorname{Im}\sigma=\frac{\sigma_0\omega\tau}{1+\omega^2\tau^2}.
$$
Here 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.<Lecture notes on the electrodynamics of metals and superconductors>[p. 2, Eqs. (1)–(11), and pp. 5–6, Eqs. (23)–(27)]{ref}
For $E_x(t)=E_0\cos\omega t$ with $\omega>0$, the real-time <current density> is
$$
J_x(t)=\frac{\sigma_0E_0}{\sqrt{1+\omega^2\tau^2}}\cos(\omega t-\phi),\qquad
\phi=\arctan(\omega\tau).
$$
Thus 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 $\sigma_0/\sqrt2$, and the lag is $\pi/4$.
\Image[/drude-complex-conductivity-frequency-response.png]
{title=<Drude frequency response>: dissipative component, amplitude, and <phase> lag}
{description=Exact dimensionless <Drude model> curves with $e^{-i\omega t}$ time dependence. The <real part> controls average electrical <power> absorption; the positive <imaginary part> describes delayed <current density>. The dotted line marks $\omega\tau=1$.}
{source=drude-complex-conductivity-frequency-response.py}
For $\omega\tau\ll1$, <complex conductivity> tends to the positive constant $\sigma_0$: <current density> nearly follows the <electric field>. For $\omega\tau\gg1$ within this model,
$$
\sigma(\omega)\sim\frac{i ne^2}{m\omega},\qquad
\operatorname{Re}\sigma\sim\frac{ne^2}{m\omega^2\tau},\qquad
\phi\longrightarrow\frac{\pi}{2}.
$$
The <current density> amplitude decreases and its response becomes predominantly inertial. Average <power> absorbed per volume is $\overline p=\tfrac12\operatorname{Re}\sigma\,|\widehat{\mathbf E}|^2\geq0$. 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>.
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