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Drude conductivity, passive energy and plasma cutoff

Physical scope

Conditional harmonic response and transverse bulk dispersion for the phenomenological Drude current closure in a fixed homogeneous material rest frame. The energy statement is conditional on the proved Drude branch of the augmented-energy theorem and the smooth finite-energy Maxwell model. The cutoff applies to nonzero real-wavevector transverse bulk modes in the collisionless branch only; it asserts no interface modes, longitudinal plasma oscillations, experimental validation or microscopic carrier theory.

Statement

Assume Countable Choice, the hypothesis carried by the finite-energy Maxwell supplier. Work in a fixed homogeneous material rest frame in SI and standard right-handed Cartesian coordinates with orthonormal basis $\hat{x}, \hat{y}, \hat{z}$ . Let $ε_{\infty} > 0$ be in F m $^{- 1}$ , $μ > 0$ in H m $^{- 1}$ , $a > 0$ in F m $^{- 1}$ s $^{- 2}$ , and $γ \geq 0$ in s $^{- 1}$ as in the adopted Drude current closure. For a spatially uniform harmonic electric field $E (t) = Re (e e^{- i ω t})$ , with $ω > 0$ in s $^{- 1}$ and $e$ in V m $^{- 1}$ , consider the harmonic particular solution of the Drude equations, written $V (t) = Re (v e^{- i ω t})$ and $P (t) = Re (p e^{- i ω t})$ , where $v$ is in A m $^{- 2}$ and $p$ is in C m $^{- 2}$ . The frequency-domain response below refers to this harmonic particular solution; arbitrary initial data may add homogeneous solutions, which are not included in these amplitudes. The conductivity $σ (ω)$ is the complex scalar mapping $e$ to $v$ , in S m $^{- 1}$ , and the absolute permittivity $ε (ω)$ is the scalar mapping $e$ to the displacement amplitude in $D = ε_{\infty} E + P$ , in F m $^{- 1}$ .

For every $ω > 0$ ,

$σ (ω) = \frac{a}{γ - i ω}, Re σ (ω) = \frac{a γ}{γ^{2} + ω^{2}} \geq 0, ε (ω) = ε_{\infty} - \frac{a}{ω (ω + i γ)} .$

The carrier energy density and the loss term in the smooth Maxwell energy balance are $∣ V ∣^{2} / (2 a)$ and $γ ∣ V ∣^{2} / a$ , respectively; the energy density has units J m $^{- 3}$ and the loss density has units W m $^{- 3}$ .

In the collisionless branch $γ = 0$ , set $ω_{p} : = \sqrt{a / ε_{\infty}} > 0$ (so $ω_{p}$ is in s $^{- 1}$ ). For a source-free nonzero transverse bulk plane wave with real $k \in R^{3} ∖ {0}$ in m $^{- 1}$ , frequency $ω > 0$ in s $^{- 1}$ , and nonzero complex electric amplitude $e$ satisfying $k \cdot e = 0$ , the Maxwell amplitude equations imply

$∣ k ∣^{2} = μ ε_{\infty} (ω^{2} - ω_{p}^{2}) .$

Thus $ω > ω_{p}$ admits source-free transverse modes with nonzero real wavevectors, while $0 < ω < ω_{p}$ admits none with real wavevector. Extending the same harmonic ansatz to a complex wave number $κ$ in m $^{- 1}$ along a coordinate $z$ in m, the positive-imaginary branch below cutoff gives a transverse mode that decays exponentially as $z$ increases; this is the evanescent meaning used here. At $ω = ω_{p}$ the symbol gives $k = 0$ , outside the stated nonzero-wavevector plane-wave class; no claim about the separate spatially uniform endpoint or longitudinal plasma oscillations is made.

Statement references

Drude current closure and collisionless limit

The Axiom of Countable Choice ( ${A C}_{ω}$ )

Real plane-wave fields encoded by complex amplitudes

Framework-local SI units, the vacuum constants and the exact/measured status of their values

Augmented energy and passive absorption

The macroscopic Maxwell equations in terms of D, H and the free sources

Diagram

This schematic projects a right-handed orthonormal triad for the real-wavevector transverse mode: $k$ is along $+ \hat{z}$ , the electric amplitude $e$ is along $\hat{x}$ , and the magnetic-induction amplitude $b$ is along $+ \hat{y}$ . The axes are drawn in a two-dimensional projection; their right-handed three-dimensional convention is stated above. The mode construction and cross-product signs are proved in steps 3.1–4.1. The diagram does not depict the below-cutoff complex wave number as a real vector.

\begin{tikzpicture}[>=stealth,scale=1.0]
  \draw[->,gray] (0,0) -- (2.4,0) node[below] {$\hat x$};
  \draw[->,gray] (0,0) -- (0.6,1.1) node[above right] {$\hat y$};
  \draw[->,gray] (0,0) -- (0,2.3) node[left] {$\hat z$};
  \draw[->,thick,blue] (0,0) -- (1.8,0) node[below=4pt] {$e$};
  \draw[->,thick,green!60!black] (0,0) -- (0.45,0.82) node[above right] {$b$};
  \draw[->,thick,red] (0,0) -- (0,1.8) node[left=4pt] {$k$};
\end{tikzpicture}

Facts & Assumptions

Given: Countable Choice; the real Drude state equations $P_{t} = V$ , $V_{t} + γ V = a E$ , $D = ε_{\infty} E + P$ , $B = μ H$ in the material rest frame; the smooth macroscopic Maxwell equations in that framework; the SI constants and units; $ω > 0$ ; for the frequency-domain response, the harmonic particular solution rather than a general solution with arbitrary initial data; and, for the plane-wave claim, a source-free nonzero transverse harmonic mode with real $k \neq 0$ .

[A1]

The free-carrier current and polarization obey $P_{t} = V$ , $V_{t} + γ V = a E$ with $a > 0$ , $γ \geq 0$ ; $D = ε_{\infty} E + P$ and $B = μ H$ with constant positive scalars; the initial polarization is separately specified (Drude current closure and collisionless limit). The frequency-domain calculation uses the harmonic particular solution, not a general initial-state solution.

[A2]

In the smooth macroscopic model, $curl E = - \partial_{t} B$ and $curl H = J_{f} + \partial_{t} D$ , with the free sources and the $D, H$ definitions of the framework (The macroscopic Maxwell equations in terms of D, H and the free sources, clause (b)).

[F1]

The real plane-wave encoding uses $E (x, t) = Re (e e^{i (k \cdot x - ω t)})$ and $B (x, t) = Re (b e^{i (k \cdot x - ω t)})$ , with real $k \neq 0$ , $ω > 0$ , and complex amplitudes; the componentwise exponential encoding is real-field notation (Real plane-wave fields encoded by complex amplitudes).

[F2]

For real vectors, $k \times (k \times e) = k (k \cdot e) - ∣ k ∣^{2} e$ ; extending the cross product complex-bilinearly in the amplitude preserves this componentwise identity (The vector triple product identity $u \times (v \times w) = ⟨ u, w ⟩ v - ⟨ u, v ⟩ w$ ).

[F3]

Countable Choice is the set-theoretic assumption used by the finite-energy Maxwell energy supplier and is carried through the augmented-energy result (The Axiom of Countable Choice ( ${A C}_{ω}$ ), Augmented energy and passive absorption).

[F4]

Every positive real number has a unique positive square root (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ).

[L1]

The current augmented-energy carrier's Drude branch gives carrier storage $∣ V ∣^{2} / (2 a)$ and local loss $γ ∣ V ∣^{2} / a$ when $P_{t} = V$ and $V_{t} + γ V = a E$ . Its complete current proof has been inspected; this use retains the exact weak-Poynting and Countable Choice hypotheses declared there (Augmented energy and passive absorption, Drude branch of the Statement and Proof).

Proof

technique · solve the first-order current equation in harmonic form, use the established conditional Drude energy branch, then substitute the collisionless response into the smooth Maxwell plane-wave symbol

1.1givenA1algebra

On the harmonic particular solution with $e^{- i ω t}$ time dependence, $V_{t}$ has amplitude $- i ω v$ , so $(γ - i ω) v = a e$ . The scalar $γ - i ω$ is nonzero because its imaginary part is $- ω \neq 0$ ; hence the vector equation gives $v = (a / (γ - i ω)) e$ , including when $e = 0$ . The conductivity is the scalar multiplier $σ (ω) : = a / (γ - i ω)$ of this linear map. Multiplying numerator and denominator by $γ + i ω$ gives $Re σ = a γ / (γ^{2} + ω^{2}) \geq 0$ .

1.2A1F3L1

The Drude branch in [L1] applies to precisely the state equations in [A1] under Countable Choice [F3]; it gives the material energy $∣ V ∣^{2} / (2 a)$ and loss $γ ∣ V ∣^{2} / a$ . Together with the field energy in that same balance, these are nonnegative storage and dissipation because $a > 0$ and $γ \geq 0$ . This is a deduction within the adopted model, conditional on [L1]'s Maxwell-energy prerequisite.

2.1givenA1step 1.1algebra

The equation $P_{t} = V$ gives $- i ω p = v$ . Combining with step 1.1, $p = \frac{a}{(- i ω) (γ - i ω)} e = - \frac{a}{ω (ω + i γ)} e$ . Because $D = ε_{\infty} E + P$ , its harmonic amplitude is $(ε_{\infty} - a / [ω (ω + i γ)]) e$ , which is the stated absolute permittivity.

3.1givenA1A2F1F2F4step 2.1algebra

In the collisionless case, step 2.1 gives $ε (ω) = ε_{\infty} - a / ω^{2} = ε_{\infty} (1 - ω_{p}^{2} / ω^{2})$ . For the encoded plane wave [F1], differentiating the phase gives the amplitude equations $k \times e = ω b$ and $k \times h = - ω ε (ω) e$ , where $b = μ h$ ; the real-field equality is equivalent to equality of the complex amplitudes since the phase ranges over all real values. Substitution yields $k \times (k \times e) = - μ ω^{2} ε (ω) e$ . By $k \cdot e = 0$ and [F2], the left side is $- ∣ k ∣^{2} e$ ; since $e \neq 0$ , cancellation gives $∣ k ∣^{2} = μ ω^{2} ε (ω) = μ ε_{\infty} (ω^{2} - ω_{p}^{2})$ . If $0 < ω < ω_{p}$ this right side is negative, impossible for real $k$ . If $ω > ω_{p}$ , define $k_{0} = \sqrt{μ ε_{\infty} (ω^{2} - ω_{p}^{2})} > 0$ by [F4], set $k = k_{0} \hat{z}$ , $e = \hat{x}$ , $b = (k_{0} / ω) \hat{y}$ , and $h = b / μ$ . Then $k \cdot e = k \cdot b = 0$ , $k \times e = ω b$ , and $k \times h = - k_{0}^{2} e / (μ ω) = - ω ε (ω) e$ , so both source-free curl equations and the divergence constraints hold for a transverse plane-wave mode with real nonzero $k$ . At equality the only magnitude is zero, outside [F1].

4.1step 3.1F2F4algebra∎

For $0 < ω < ω_{p}$ , let $α = \sqrt{μ ε_{\infty} (ω_{p}^{2} - ω^{2})} > 0$ by [F4] and set $κ = i α$ . Choose $k = κ \hat{z}$ , $e = \hat{x}$ , $b = (κ / ω) \hat{y}$ , and $h = b / μ$ . Then $k \cdot e = k \cdot b = 0$ , $k \times e = ω b$ , and $k \times h = - κ^{2} e / (μ ω) = α^{2} e / (μ ω) = - ω ε (ω) e$ , using $ε (ω) = - ε_{\infty} (ω_{p}^{2} - ω^{2}) / ω^{2}$ . Thus both curl equations and the divergence constraints hold for these transverse amplitudes, while $e^{i κ z} = e^{- α z}$ for $z \geq 0$ gives exponential attenuation. This complex- $k$ branch is not a real- $k$ propagating mode and does not classify interfaces, longitudinal modes, or the uniform endpoint.

Drude conductivity, passive energy and plasma cutoff

Statement

Statement references

Diagram

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources