Drude conductivity and the plasma edge

Let $n$ be the carrier number density, $m$ an effective inertial mass and $-e$ the carrier charge. In a relaxation-time description, scattering removes average momentum at rate $\gamma=1/\tau$, where $\tau$ is a positive time. This phenomenological term represents an ensemble average, not a statement that every collision occurs at equal intervals. Under a weak homogeneous field, the mean velocity equation is linear and the current is charge density times mean velocity.

For $e^{-i\omega t}$, the velocity denominator is $\gamma-i\omega$. Multiplying by the negative charge again gives a positive conductivity numerator. At low frequency the current is nearly in phase with the field and its real part dissipates energy. At high frequency the carriers cannot equilibrate within one cycle; the current becomes mainly reactive. The model omits interband transitions, detailed collision energy dependence and nonlocal transport, so it should be used only over a justified measurement range.

One can keep the carrier current explicitly on the right side of Ampère's law or absorb it into displacement current. The two descriptions are equivalent if the carrier term appears exactly once. Calling it both a free conductivity and part of a fitted complex permittivity in the same equation counts the same response twice. Let a real, nearly constant background relative permittivity describe bound processes outside the frequency band being modeled.

Equating the current to the time derivative of a carrier polarization gives a plus sign before $i\sigma/(\epsilon_0\omega)$ in relative permittivity. The resulting Drude term has negative real part and positive imaginary part. In particular, negative real permittivity is not a declaration of gain: passivity is controlled by absorption, which remains nonnegative for positive damping. The formal divergence of carrier polarization at zero frequency is compatible with a finite dc current and should not be interpreted as a static bound dipole.

Separating real and imaginary parts shows that the real permittivity crosses zero only if the squared plasma frequency exceeds the background times the squared damping rate. In the weak-damping limit the crossing lies near the plasma frequency divided by the square root of the background permittivity. This is screening by the bound-charge response. It distinguishes the unscreened carrier parameter from an observed optical feature, even before reflection at a surface is considered.

A real reflectivity spectrum also depends on damping, background dispersion, the external medium, sample thickness and surface quality. Therefore the algebraic zero of real permittivity is a useful material diagnostic rather than an exact universal boundary between perfect reflection and perfect transmission. In the collisionless local model with positive permeability, a negative real permittivity gives an evanescent bulk wave instead of ordinary propagating light. Finite damping makes propagation and attenuation complex rather than dividing frequencies into two perfectly sharp categories.