Dynamics of the electro-optic response of blue bronze ${\mathrm{K}}_{0.3}\mathrm{Mo}{\mathrm{O}}_3$

We have studied the charge density wave (CDW) repolarization dynamics in blue bronze by applying square-wave voltages of different frequencies to the sample and measuring the changes in infrared transmittance, proportional to CDW strain. The frequency dependence of the electrotransmittance was fit to a modified harmonic oscillator response and the evolution of the parameters as functions of voltage, position, and temperature are discussed. Resonant frequencies decrease with distance from the current contacts, indicating that the resulting delays are intrinsic to the CDW with the strain effectively flowing from the contact. For a fixed position, the average relaxation time has a voltage dependence given by ${\tau }_0\sim V^{-p}$, with $1<p<2$. The temperature dependence of the fitting parameters shows that the dynamics are governed by both the force on the CDW and the CDW current: for a given force and position, both the relaxation and delay times are inversely proportional to the CDW current as temperature is varied. The long relaxation and delay times $(\sim 1\mathrm{ms})$ suggest that the strain response involves the motion of macroscopic objects, presumably CDW phase dislocation lines.

Figure 1

Dependence of the dc resistance and relative change in transmittance $(\nu =820{\mathrm{cm}}^{-1})$ for sample 3 at $T\sim 80\mathrm{K}$ on voltage across the sample. The transmission is measured at a point adjacent to a current contact $(x=0)$ in-phase with a bipolar square-wave at $25\mathrm{Hz}$, for which the quadrature changes are negligible. The low-field “ohmic” resistance associated with quasiparticle current $\left(R_0\right)$, threshold voltage for nonlinear current $\left(V_{\mathrm{T}}\right)$ and onset voltage for the electro-optic response $\left(V_{\text{on}}\right)$ are indicated.

Figure 2

Comparison of the (a) spatial, (b) voltage, and (c) frequency dependences of the electrotransmittance (solid symbols, $\nu =820{\mathrm{cm}}^{-1}$) and electroreflectance (open symbols, $\nu =850{\mathrm{cm}}^{-1}$) for sample 1 at $T\sim 80\mathrm{K}$ at the frequencies, voltages, and positions indicated. Both the response in-phase and in quadrature with the driving bipolar square-waves are shown.

Figure 3

The spatial dependence of the electrotransmittance $(\nu =820{\mathrm{cm}}^{-1})$ of sample 1 at $T\sim 80\mathrm{K}$ in-phase with bipolar square waves at several voltages at $25\mathrm{Hz}$, for which the quadrature response is negligible. The sample was $550\mu \mathrm{m}$ long and the light spot was $50\mu \mathrm{m}$ wide. Each data set is vertically offset by 0.2%; the dashed zero line for each data set is shown, with the voltage given on the right. The solid lines through the data points are for reference only. The open symbols show the response for a positive unipolar square wave (multiplied by 10).

Figure 4

(a),(b) The frequency dependence of the electrotransmittance $(\nu =820{\mathrm{cm}}^{-1})$ of sample 1 at a few bipolar square-wave voltages at a position (a) adjacent to a current contact and (b) $200\mu \mathrm{m}$ from the contact; both the response in-phase and in quadrature with the square wave are shown. The arrows indicate the high-frequency inverted in-phase response associated with delay and the low-frequency inverted quadrature response associated with long-time decay of the electro-optical signal. The curves are fits to Eq. (1). (c) Electrotransmittance vs time for $V=3.6V_{\text{on}}$, $\omega ∕2\pi =3.2\mathrm{kHz}$ bipolar square wave at two positions; the applied square wave is shown for reference.

Figure 5

The frequency dependence of the electrotransmittance $(\nu =890{\mathrm{cm}}^{-1})$ of sample 2 at a few bipolar square-wave voltages at a position (a) adjacent to a current contact and (b) $200\mu \mathrm{m}$ from the contact; both the in-phase and quadratures responses are shown. The arrow indicates the high-frequency inverted in-phase response associated with delay for $x=200\mu \mathrm{m}$. The curves are fits to Eq. (1).

Figure 6

The frequency dependence of the electrotransmittance $(\nu =820{\mathrm{cm}}^{-1})$ of sample 3 at a few bipolar square-wave voltages at a position (a) adjacent to a current contact and (b) $200\mu \mathrm{m}$ from the contact; both the in-phase and in quadrature responses are shown. The arrow indicates the high-frequency inverted in-phase response associated with delay for $x=200\mu \mathrm{m}$. The curves are fits to Eq. (1).

Figure 7

Voltage dependence of fitting parameters for Eq. (1) for sample 1 (open symbols), sample 2 (black symbols), and sample 3 (gray symbols) at $T\sim 80\mathrm{K}$: (a) relaxation times, (b) resonant frequencies, (c) amplitudes, (d) exponents. The triangles are for $x=0$, diamonds for $x=100\mu \mathrm{m}$, and squares for $x=200\mu \mathrm{m}$. Reference lines showing $1∕V$ and $1∕V^2$ behavior are shown in (a) and $V^{1∕2}$ behavior in (b). The vertical arrows in (a) indicate the nonlinear current threshold voltages. In (c), the curves are guides to the eye, with extrapolated arrows showing the onset voltages.

Figure 8

Temperature dependence of frequency-dependence fitting parameters for Eq. (1) for sample 3 $(\nu =820{\mathrm{cm}}^{-1})$ for two positions and voltages, as indicated: (a) relaxation times, (b) resonant frequencies (only determined for $x=200\mu \mathrm{m}$), (c) amplitudes, (d) exponents. The curves in (a,b,c) show the temperature dependence of the low-field resistance and conductance.

Figure 9

Relaxation times and resonant frequencies for sample 3 $(\nu =820{\mathrm{cm}}^{-1})$ plotted as functions of the temperature dependent CDW current at $V=V_{\text{on}}+10\mathrm{mV}$ and $V_{\text{on}}+20\mathrm{mV}$. A line with $\text{slope}=1$ is shown for reference. Inset: CDW current vs voltage at a few temperatures.

Figure 10

Temperature dependence of frequency-dependence fitting parameters for Eq. (1) for sample 3 $(\nu =820{\mathrm{cm}}^{-1})$ for $I_{\mathrm{CDW}}=100\mu \mathrm{A}$ and two positions: (a) relaxation times, (b) resonant frequencies (only determined for $x=200\mu \mathrm{m}$), (c) amplitudes, (d) exponents.

Figure 11

Relaxation times for sample 4 $(\nu =775{\mathrm{cm}}^{-1})$ plotted as functions of the temperature dependent CDW current at $V=V_{\text{on}}+10\mathrm{mV}$ and $V_{\text{on}}+20\mathrm{mV}$. A line with $\text{slope}=1$ is shown for reference. Inset: Temperature dependence of the relaxation times.

Figure 12

The frequency dependence of the electrotransmittance $(\nu =820{\mathrm{cm}}^{-1})$ of sample 3 at $V=2V_{\mathrm{T}}=3.1V_{\text{on}}$ and $T=101\mathrm{K}$. Shown are the measured in-phase and quadrature responses for bipolar square-waves at $x=0$ and $x=200\mu \mathrm{m}$ and positive and negative unipolar square-waves at $x=0$. Also shown is the difference between the $x=0$ bipolar and (difference of the) unipolar responses. The curves are fits to Eq. (1). Inset: The spatial dependence of the (in-phase) bipolar $(V=1.5V_{\mathrm{T}})$ and negative unipolar $(V=2V_{\mathrm{T}})$ responses at $25\mathrm{Hz}$, for which the quadrature responses are negligible.