Spontaneous Running Waves and Self-Oscillatory Transport in Dirac Fluids

TL;DR

The work addresses whether a DC current can drive self-oscillatory transport in a current-carrying Dirac fluid. It develops a hydrodynamic framework with density-dependent dissipation near charge neutrality, predicting a modulational (Turing-type) instability that selects a finite wavelength $\lambda=2\pi/k_*$ and triggers downstream running waves above a critical drift velocity $u_c = u_0/|R|$, where $u_0=\sqrt{Un/m}$ and $R=\frac{n}{\gamma}\left|\frac{d\gamma}{d n}\right|$. Observable signatures include a second-order-like onset in the time-averaged current and narrow-band EM emission at the washboard frequency $f = u/\lambda$, with $f$ tunable by current and carrier density. Owing to the small Dirac mass, the resulting frequencies lie in tens to hundreds of GHz, highlighting Dirac materials as a platform for high-frequency intrinsic electron-hydrodynamic self-oscillations; the mechanism is intrinsic and disorder-free, analogous to Kapitsa roll waves in viscous systems.

Abstract

We predict hydrodynamic Turing instability of current-carrying Dirac electron fluids that drives spontaneous self-oscillatory transport. The instability arises near charge neutrality, where carrier kinetics make current dissipation strongly density dependent. Above a critical drift velocity, a uniform electronic flow becomes unstable and undergoes a dynamical transition to a state with coupled spatial modulation and temporal oscillations--an electronic analogue of Kapitsa roll waves in viscous films. The transition exhibits two clear signatures: a nonanalytic, second-order-like onset in the time-averaged current and narrow-band electromagnetic emission at a tunable washboard frequency $f=u/λ$. Although reminiscent of sliding charge-density waves, the mechanism is intrinsic and disorder independent. Owing to the small effective mass of Dirac carriers, hydrodynamic time scales translate into emission frequencies in the tens to hundreds of gigahertz range, establishing Dirac materials as a platform for high-frequency self-oscillatory electron hydrodynamics.

Figures (5)

• Figure 1: (a) Current dissipation in graphene bilayer (BLG) band vs. electron density. Dissipation is high in the Dirac plasma near charge neutrality point (CNP) but drops rapidly upon doping away from CNP. Density-dependent dissipation drives the Turing–Kapitsa instability. Insets illustrate carrier population in BLG band for densities at CNP and away from it, respectively. (b) Instability growth rate ${\rm Im}\,\omega_\pm(k)$ for the modes $\omega_+(k)$ and $\omega_-(k)$, Eq. \ref{['eq:modes1']}, representing density and velocity perturbations. The density mode satisfies $\omega_+(0)=0$ by particle-number conservation, becoming unstable above a critical flow velocity (Eq.\ref{['eq:u_critical']}); the growth rate ${\rm Im}\,\omega_+(k)>0$ peaks at the wave numbers $\pm k_*$ marked by arrows. The velocity mode remains damped, ${\rm Im}\,\omega_-(k)<0$. Small initial perturbations grow into a periodic modulation with wavelength $\lambda\sim 2\pi/k_*$ propagating downstream, as shown in Fig. \ref{['fig3']}.
• Figure 2: Wavelength selection at instability. Shown is the long-time modulation (orange) evolved from different initial perturbations (blue): $\delta n(x)=\sum_m a_m\cos k_m x+b_m\sin k_m x$, $\delta p(x)=\sum_m a'_m\cos k_m x+b'_m\sin k_m x$, with $k_m=\frac{2\pi}{L}\times 3,5,8,13,21,55$. Irrespective of initial conditions, the system converges to the same spatial pattern, with wavelength set by the wavenumber $k$ of maximal growth (marked by arrows in Fig. \ref{['fig2']}), consistent with Turing’s maximum-growth rule.
• Figure 3: Signatures and observables for current-driven electronic Turing–Kapitsa instability. Plotted are the running waves modulation amplitude and time-averaged current near the instability onset, obtained numerically for a graphene bilayer model (see Eqs. \ref{['eq:transport_eqs_p']}, \ref{['eq:transport_eqs_n']} and accompanying text). The red line shows the theoretical square-root behavior $\Delta n\propto (u_d-u_d^*)^{1/2}$ expected on the general symmetry grounds LL Fluid Mechanics. The sharp slope change in the time-averaged (DC) current–field dependence marks the instability threshold and the added current from the running waves. Insets (a) and (b) illustrate a typical modulation and the current–field relation.
• Figure 4: Modulation wavelength $\lambda = 2\pi/k_*$, where $\pm k_*$ (blue arrows) maximizes the instability growth rate $\mathrm{Im}\,\omega_+(k)$, is governed by the behavior of $\mathrm{Im}\,\omega_+(k)$ at small and large $k$. For small $k$, the growth rate (red parabola) behaves as $\alpha k^2$, with $\alpha>0$ for $u>u_c$ and $\alpha<0$ for $u<u_c$. For large $k$, it saturates to a $k$-independent value (horizontal red line), again positive for $u>u_c$ and negative for $u<u_c$. Matching these small- and large-$k$ asymptotics yields Eq. \ref{['eq:zeta_large-k']}, and hence Eq. \ref{['eq:k*_result']}. The instability growth rate is shown for $u=0.55>u_c$ and $u=0.2<u_c$ (blue and pink curves); solid and dashed lines correspond to $D_{n,p} = 0.1$ and $D_{n,p}=0$.
• Figure 5: Critical velocity and modulation wavelength vs. electron density at the Turing-Kapitsa instability for BLG band, Eq.\ref{['eq:BLG_spectrum']}. The results shown were obtained using the current relaxation rate $\Gamma_J$ given in Eq.\ref{['eq:Gamma_full']}.