Nonuniform Hydrodynamic Plasma-Wave Instability in a Gated Electron Channel

1. Introduction

• Different source and drain electrical boundary conditions can produce net round-trip wave gain, which competes with momentum relaxation and electron-viscosity losses.These processes jointly shape the instability threshold, frequency, and growth rate.Source: §1 · p. 2
• Drainward voltage reduction lowers carrier density while increasing drift velocity, and local charge compressibility also makes plasma velocity vary along the channel.Earlier work identified the inviscid accelerating branch's termination at the local sonic condition.Source: §1 · p. 2
• Near the sonic region, drain-local approximations can mislead because growth depends on spatial wave-amplitude evolution, making the complete linearized problem a needed reference.A phase integral may describe the frequency shift despite appreciable nonuniformity, but growth requires more than endpoint values or local attenuation added to reflection.Source: §1 · p. 2

2. Hydrodynamic Model

• The one-dimensional transport model uses continuity and Navier–Stokes equations, with momentum relaxation and a constant kinematic viscosity in each calculation.The active model fixes temperature and includes neither an electron energy equation nor a separate Poisson equation.Source: §2 · p. 3
• The unified charge-control model relates carrier density to local gate-to-channel voltage and supplies the local plasma velocity.The full expression is used in quantitative calculations; the strong-inversion gradual-channel approximation is used only for stated analytical limits.Source: §2 · p. 3
• Current-driven operation without an external AC signal fixes source density and drain particle current as the electrical boundary conditions.The equations and boundary conditions are discretized and solved numerically using a one-dimensional non-uniform mesh.Source: §2 · p. 3

3. Current-driven nonuniform steady state

• On the accelerating subsonic branch, drift velocity rises toward the drain while carrier density and plasma velocity decrease, with nonuniformity pronounced near sonic conditions.The inviscid steady-state profiles in Fig. 1 illustrate this current-induced redistribution.Source: §3 · p. 5, Figure 1 · p. 6
• For the retained parameter convention, solving Eq. (13) with the stated condition gives a steady-state sonic Mach number Ms,sonic ≃ 0.51198, distinct from the dynamical-instability condition ω2 = 0.Source: §3 · p. 6

3.1 Strong-inversion limit and the nonuniformity factor

• In the strong-inversion limit, the factor 1 − M^2 cancels the large local phase density in the frequency integral, so a narrow near-sonic region does not force total cavity frequency to vanish.Quantitative profiles and the updated comparison instead use the complete unified charge-control model.Source: §3.1 · p. 7

4.1 Local viscous dispersion and the cubic correction

• The local viscous dispersion yields quadratic wave-number roots, while retaining its cubic term adds a third spatial root and a perturbative correction.The correction is not uniformly valid in viscosity at every drift velocity.Source: §4.1 · pp. 7–8

4.2 Global phase condition

• The leading WKB phase condition accumulates propagation through the nonuniform channel, but neglects separate spatial transport of wave amplitudes.The resulting frequency expressions retain nonuniform coefficients and are implicit in the real trial frequency.Source: §4.2 · pp. 8–9

4.3 Principal reduced forms of the oscillation frequency

• The reduced frequency forms retain the friction-induced nonuniform DC profile while dropping direct dissipative AC-phase corrections; a particular next correction raises frequency on the regular subsonic branch.The hierarchy’s levels isolate physical contributions and should not be read as a uniform sequence of powers of viscosity near the sonic point.Source: §4.3 · pp. 9–11

4.4 Comparison of the frequency approximations

• The nonuniform phase-integral curves capture the pronounced current-induced frequency reduction and remain substantially closer to simulation than the source-matched uniform-channel reference.The comparison uses the exact-UCCM inviscid DC profile for the frequency hierarchy; the largest improvement over a drain-local estimate is associated with the nonuniform phase integral.Source: §4.4 · p. 11, Figure 2 · p. 12

5.1 Complete finite-viscosity eigenvalue problem

• The complete finite-viscosity eigenvalue condition retains nonuniform amplitude evolution and requires neither weak growth nor elimination of a viscous root.It is an implicit spectral expression, not an elementary closed formula.Source: §5.1 · pp. 13–14

5.2 Inviscid transfer-matrix limit

• The inviscid growth rate peaks near Ms=0.4398 at approximately 2.207 × 10^12 s−1, then rolls over while remaining positive toward the inviscid sonic endpoint.This is exact for the stated inviscid linear problem, not for the viscous device.Source: §5.2 · p. 15

5.3 Weak-growth reduction without a WKB amplitude approximation

• The residual-based weak-growth reduction keeps finite-viscosity nonuniform propagation without replacing it by local WKB amplitudes, and requires a small correction with nonzero residual derivative.The plotted finite-viscosity curves use the complete complex root instead.Source: §5.3 · p. 15

5.4 Nonuniform two-wave growth expression and its reduction

• The nonuniform two-wave expression combines reflection gain, momentum-relaxation loss, and distributed viscous loss, all weighted by the DC profile.It is a reduction of the quadratic two-root dispersion, not the complete first-order viscous expansion because a cubic term is omitted.Source: §5.4 · p. 16

5.5 Uniform-channel and low-current limits

• The low-current scaling limits explain why increasing viscosity moves the approximate instability onset to a larger drift velocity, but they should not be extrapolated near Md ≃ 1.These reduced expressions are not the source of the reported finite-viscosity agreement.Source: §5.5 · p. 17

5.6 Comparison with the available growth-rate data

• The finite-viscosity eigenvalue curves agree with available subsonic growth-rate data, most strongly for ν=100 cm2/s, but the data are sparse and do not validate the full interval to Ms=0.5.Some rates come from fits to limited transient peaks, whose values can depend on the fitting interval.Source: Figure 3 · p. 18, Table III · p. 18, §5.6 · pp. 18–19

Finite-viscosity sonic transition

• Fractional-power scales characterize the sonic transition as a singular perturbation, rather than an ordinary expansion where no inviscid steady solution exists.The local balance yields an inner Riccati structure, but numerical steady trajectories remain necessary for contact-dependent sonic-crossing values.Source: §6 · p. 19
• The first sonic-crossing source Mach number rises from 0.51198 inviscid to 0.55086 at ν=100 cm2/s; these are steady-state thresholds, not zeros of ω2.Low- and moderate-viscosity shifts approximately follow the ν^2/3 dependence associated with Eq. (67).Source: Figure 4 · p. 20

Range of validity

• Subsonic agreement does not establish identical contact models, and negative simulation rates near or above sonic conditions cannot be assigned uniquely without matching stationary fields and DC/AC boundary closure.The correspondence remains unresolved.Source: §6 · pp. 20–21

7. Conclusion

• Finite drain current lowers carrier density and plasma-wave velocity toward the drain, making nonuniformity increasingly important near the sonic regime.The resulting redistribution strongly modifies plasma-wave propagation through the channel.Source: §7 · p. 21
• The oscillation frequency follows propagation through the entire nonuniform channel, producing a pronounced current-induced reduction that a uniform-channel approximation increasingly overestimates near the sonic regime.The relevant change is in effective round-trip propagation time, not merely the local drain conditions.Source: §7 · p. 21
• Instability reflects competition between boundary-induced amplification and losses from momentum relaxation and electron viscosity, making threshold and growth rate nonlocal properties of the channel.Both propagation and dissipation depend on the spatially varying background.Source: §7 · p. 21
• Finite viscosity smooths the inviscid singularity and reproduces subsonic growth rates well, but higher-current behavior depends increasingly on near-sonic structure and contact boundary conditions.Thus eventual instability suppression cannot yet be attributed to bulk hydrodynamics alone.Source: §7 · p. 21
• Current-induced spatial nonuniformity is an essential contributor to plasma-wave instability in gated electron channels, while the near-sonic, contact-sensitive regime remains a device-level modeling challenge.Source: §7 · p. 22

Appendix A. Quadratic discriminant and first cubic correction

• The appendix makes the connection between local dispersion and the frequency hierarchy reproducible through additional algebra.Source: Appendix A · p. 22
• The full quadratic phase includes a mixed numerator and a finite-νω_r denominator; omitting these and the imaginary-discriminant correction yields Eq. (30) only when their contributions are small.Source: Appendix A · p. 22
• The cubic correction is derived from the quadratic-root sum and product, with the polynomial derivative at the roots given as ±2D.The root separation is then adjusted using that derivative.Source: Appendix A · p. 22
• The resummed correction is distinct from the strict first correction used in Eq. (23).Source: Appendix A · p. 22

Appendix B. Closed frequency reductions and the finite-friction term

• For the strong-inversion inviscid profile, the Level II frequency integral is reduced using an antiderivative verified by differentiation.Source: Appendix B · pp. 22–23
• The finite-γ_2 correction is retained before expanding the frequency by defining two dimensional integrals, each with units of time.Source: Appendix B · p. 23
• The numerical Level III integral, its positive-root approximation, and its first explicit correction are separate reductions, not interchangeable formulas.At ν = 0 on the strong-inversion profile, I_fr = 2(M_d − M_s)/(3γ).Source: Appendix B · p. 23

Appendix C. Attenuation, cubic consistency, and the weak-growth derivative

• The attenuation formula in Eq. (59) follows from first-order expansion in γ and ν and corresponds to the earlier quadratic gain–loss derivation.Source: Appendix C · p. 23
• Equations (C1) and (C2) expand the retained cubic dispersion; their difference from Eq. (59) is of order νv_0^2 away from the sonic point.This distinction means the quadratic attenuation formula is not the complete first-order finite-viscosity result at appreciable drift.Source: Appendix C · pp. 23–24
• The diagnostic reduction in Eq. (C3) neglects exact nonuniform amplitude transport and any contact contribution of the third root, and is not extrapolated to s = v_0.Both Eqs. (60) and (C3) recover the small-current limit in Eq. (62).Source: Appendix C · p. 24
• Differentiating the linear boundary problem at fixed background provides a consistent evaluation of Eq. (54) without a new physical fitting parameter.The frequency derivative is propagated through the system and evaluated at the drain.Source: Appendix C · p. 24

• The author discloses no conflict of interest.Source: AUTHOR DECLARATIONS · p. 24