Hydrodynamics of plastic deformations in electronic crystals

We construct a hydrodynamic framework describing plastic deformations in electronic crystals. The framework accounts for pinning, phase, and momentum relaxation effects due to translational disorder, diffusion due to the presence of interstitials and vacancies, and strain relaxation due to plasticity and dislocations. We obtain the hydrodynamic mode spectrum and correlation functions in various regimes in order to identify the signatures of plasticity in electronic crystal phases. In particular, we show that proliferation of dislocations depins the spatially resolved conductivity until the crystal melts, after which point a new phase of a pinned electronic liquid emerges. In addition, the mode spectrum exhibits a competition between pinning and plasticity effects, with the damping rate of some modes being controlled by pinning-induced phase relaxation and some by plasticity-induced strain relaxation. We find that the recently discovered damping-attenuation relation continues to hold for pinned-induced phase relaxation even in the presence of plasticity and dislocations. We also comment on various experimental setups that could probe the effects of plasticity. The framework developed here is applicable to a large class of physical systems including electronic Wigner crystals, multicomponent charge density waves, and ordinary crystals.

Figure 1

Real part of the transverse optical conductivity at nonzero wavevector for increasing rate of plasticity-induced relaxation ${\mathrm{\Omega }}_G$ in the absence of pinning (left) and in the presence of pinning (right). The peak in the optical conductivity widens and shortens until the solid to liquid phase transition point, sharpening and rising back up again at a lower frequency after the transition. The black curves represent the phase transition point.

Figure 2

Circuit representation of the rheology equations for a plastic crystal in the shear sector. The circuit describes a Jeffrey material. In the limit $\eta \to 0$, the dashpot in the lower arm disappears and we get a Maxwell material, whereas in the limit ${\mathrm{\Omega }}_G\to 0$, the dashpot in the upper arm becomes infinitely rigid and we get a Kelvin-Voigt material. The bulk sector of the rheology equations behaves similarly, with $G, {\mathrm{\Omega }}_G, \eta$ replaced with $B, {\mathrm{\Omega }}_B, \zeta$, respectively.

Figure 3

Climb motion of a dislocation perpendicular to its Burgers vector (given in black) takes the lattice configuration (a) to (b), creating a new lattice site in the process. On the other hand, glide motion of a dislocation parallel to its Burgers vector takes the lattice configuration (a) to (c), without the need to create or remove a lattice site.

Figure 4

Circuit representation of the rheology equations for a pinned plastic crystal in the shear sector. Pinning introduces a weak spring component parallel to the Jeffrey circuit, causing the material to return to its original state at very late time scales similar to Zener materials. The bulk sector behaves similarly, with $G, {\mathrm{\Omega }}_G, \eta$ replaced with $B, {\mathrm{\Omega }}_B, \zeta$, respectively.

Figure 5

Real parts of the longitudinal optical conductivity and charge/particle susceptibility at nonzero wavevector for increasing rate of plasticity-induced relaxation ${\mathrm{\Omega }}_G$. The peaks in both the plots widen and shorten until the solid to liquid phase transition point, sharpening and rising back up again at a lower frequency after the transition. The black curves represent the phase transition point. The imaginary parts of the longitudinal optical conductivity and susceptibility are proportional to the imaginary parts of susceptibility and longitudinal optical conductivity respectively.

Figure 6

Real part of frequency-dependent shear viscosity $\eta \left(\omega \right)$ for nonzero rate of plasticity-induced relaxation ${\mathrm{\Omega }}_G$. The viscosity asymptotes to its “solid value” $\eta$ for $\omega \gg {\mathrm{\Omega }}_G$ and to its “liquid value” ${\eta }_l$ for $\omega \ll {\mathrm{\Omega }}_G$. The bulk viscosity $\zeta \left(\omega \right)$ also shows a similar qualitative behavior with the bulk sector relaxation rate ${\mathrm{\Omega }}_B$.

Figure 7

Circuit representation of the rheology equations from the perspective of total matter displacement.