Incommensurate Systems as Model Compounds for Disorder Revealing Low-Temperature Glasslike Behavior

We show that the specific heat of incommensurately modulated crystals with broken translational periodicity presents similar features at low temperatures to those of amorphous and glass materials. Here we demonstrate that the excess to the constant $C_p(T)/T^3$ law (or Debye limit) is made up of an upturn below 1 K and of a broad bump at $T\approx 10\mathrm{K}$ that directly originates from the gapped phase and amplitude modes of the incommensurate structure. We argue that the low-energy dynamics of incommensurate systems constitute a plausible simplification of the landscape of interactions present in glasses, giving rise to their low-temperature anomalies.

Figure 1

$C_P/\beta T^3$ plots of several IC compounds, with $\beta$ the coefficient of the Debye law. The left and middle panels are charge density wave compounds [${\mathrm{K}}_{0.3}{\mathrm{MoO}}_3$ [20], $({\mathrm{TaSe}}_4{)}_2\mathrm{I}$ [22], KCP [20], and ${\mathrm{TaS}}_3$ [23] ]. The $C_P/\beta T^3$ plot of vitreous silica is displayed as a comparison [19]. Right panel: Insulating compounds, BCPS [$({\mathrm{ClC}}_6{\mathrm{D}}_4{)}_2{\mathrm{SO}}_2$] [24] and biphenyl [25]. Error bars are smaller than the size of the symbol.

Figure 2

Temperature dependence of the specific heat divided by $T^3$ in a log-log plot for ${\mathrm{ThBr}}_4$. Experimental data are displayed with full black circles. The black line represents the Debye contribution calculated from sound velocities determined in a neutron scattering experiment [36], using the Debye model of the dispersion with ${\theta }_D\approx 62\mathrm{K}$, and corresponding to $\beta =8\times {10}^{-3}\mathrm{J}/\text{mole}·{\mathrm{K}}^4$. The green line is the amplitudon contribution estimated from the measured INS dispersion. Full red circles represent the contribution remaining after subtraction of previous contributions and assigned to the phason contribution. Experimental error bars are smaller than the size of the symbol.

Figure 3

Low-energy contribution to the specific heat of ${\mathrm{ThBr}}_4$ divided by $T^3$ in a semilog plot. The red points correspond to the experimental data shown in Fig. 2 once the Debye and amplitudon contributions have been subtracted. The continuous black line is the result of the calculation from Eq. (3) with the phason gap of 46 GHz and damping of 3.8 GHz. The black dashed lines correspond to simulations using the same damping and different phason gaps of 22 and 90 GHz. The dispersion curve for the phason is drawn in the inset, with points representing the experimental data [36]. Experimental error bars are smaller than the size of the symbol.