Dynamical Stability Limit for the Charge Density Wave in ${\mathrm{K}}_{0.3}{\mathrm{MoO}}_3$

We study the response of the one-dimensional charge density wave in ${\mathrm{K}}_{0.3}{\mathrm{MoO}}_3$ to different types of excitation with femtosecond optical pulses. We compare direct excitation of the lattice at midinfrared frequencies with injection of quasiparticles across the low energy charge density wave gap and with charge transfer excitation in the near infrared. For all three cases, we observe a fluence threshold above which the amplitude-mode oscillation frequency is softened and the mode becomes increasingly damped. We show that all the data can be collapsed onto a universal curve in which the melting of the charge density wave occurs abruptly at a critical lattice excursion. These data highlight the existence of a universal stability limit for a charge density wave, reminiscent of the Lindemann criterion for the melting of a crystal lattice.

Figure 1

(a) Monoclinic crystal structure of ${\mathrm{K}}_{0.3}{\mathrm{MoO}}_3$. Corner-sharing ${\mathrm{MoO}}_6$ octahedra form chains along the [010] direction that are separated by sheets of potassium atoms (purple). Below $T_C=183\mathrm{K}$, incommensurate charge density wave order develops at a wave vector $q_{\mathrm{CDW}}=(1,q_b,\overline{0.5})$, with $q_b=0.748$ at 100 K. (b) Top view of the chains. The resonantly excited phonon mode comprises Mo-O stretching along the insulating [$-201$] direction, perpendicular to the chains.

Figure 2

Optical conductivity ${\sigma }_1$ of ${\mathrm{K}}_{0.3}{\mathrm{MoO}}_3$ below the transition temperature $T_C$. (a) The material is insulating along [$-201$] with a 1.3-eV charge-transfer gap, and the Mo-O stretching mode appears as a sharp peak at $1000{\mathrm{cm}}^{-1}$. (b) The CDW gap is formed along the [010] chain direction. The shaded regions denote the frequencies to which the optical excitation pulses were tuned: infrared-active phonon mode (blue), electronic excitation across the charge-transfer gap (black), and excitation of quasiparticles above the CDW gap (red). (c) Transient reflectivity changes recorded at 800 nm probe wavelength (normalized) for the three excitation schemes in a regime of weak perturbation. Zero time delay was determined from the simultaneously measured four-wave mixing signal at frequency $2{\omega }_{\text{probe}}+{\omega }_{\text{pump}}$, which yielded the pump pulse envelopes shown in the figure (shaded).

Figure 3

(a) The normalized oscillatory responses, extracted from the data plotted in Fig. 2, are shown as dots. The different optical excitations are color coded as described in Figs. 2 and 2. The extrapolation of the oscillatory response to time zero, fitted with a sum of three exponentially decaying harmonic oscillators, shows a sinelike phase for lattice excitation and a cosinelike phase for the electronic excitations. (b) Fourier transforms of the oscillatory responses, showing the CDW amplitude mode at 1.68 THz and four zone-folded phonon modes at higher frequencies. (c)–(e) Pump frequency dependent magnitudes of the coherent amplitude mode oscillations, again for the three different excitation schemes. The optical conductivity in the respective frequency ranges and along the respective directions, convolved with the excitation pulse bandwidth is plotted for comparison.

Figure 4

Fluence dependence of the amplitude, frequency, and scattering rate of the coherent oscillations of the amplitude mode for resonant phonon excitation (blue) and for electronic excitations above the charge transfer gap (black) as well as across the CDW gap (red). The corresponding energy density is given in the top axis of the upper panels. In all cases, the oscillation amplitude saturated at $\sim 2\%$ $\mathrm{\Delta }R/R_0$, which corresponds to atomic displacements of approximately 20% of the lattice distortion induced by the CDW formation at equilibrium as estimated from Refs. [6] and [7] (scale on the right side of upper panel, see text).

Figure 5

Frequency and scattering rate of the charge density wave amplitude mode, extracted from the data shown in Fig. 4 and plotted against the measured oscillation amplitude. The data fall on a universal curve for the different excitation schemes. The amplitude of the coherent oscillations is given in % ($\mathrm{\Delta }R/R_0$) at 800 nm (upper scale) and in the relative change of the lattice distortion, induced by the CDW formation (lower scale).