L. Ladino, J.W. Brill, University of Kentucky Supporting measurements: M. Uddin, M. Freamat,

1. Electro-Optic Search for Threshold Divergence of the CDW Diffusion Constant in Blue Bronze (K0.3MoO3) L. Ladino, J.W. Brill, University of Kentucky Supporting measurements: M. Uddin, M. Freamat, D. Dominko, E. Bittle Samples from R.E. Thorne Helpful discussions with O. Narayan Support from U.S. National Science Foundation

2. The infrared transmission (q) and reflectance (R) (for hn < 2D) of blue bronze vary with position and voltage in the CDW state. T = 80 K, n = 820 cm-1 VT: threshold for non-Ohmic dc current, depinning at contacts, phase-slip Von: onset of CDW polarization (elastic strain), depinning in bulk: x=0 (adjacent to current contact) Phase-slip voltage Vps ~ VT - Von

3. CDW: n = n0 + n1cos(Qx+φ); polarization (strain) ≡ 1/Q ∂φ/∂x +V -V T = 80 K, n = 820 cm-1 Changes in IR due to: a) changes in density of quasiparticles screening CDW strain and b) changes in phonons in strained CDW: Dq/q DR/R∂φ/∂x Extra strains near current contacts when CDW slides Strain varies linearly with position if no sliding

4. Solitons & Phase Dislocations The strain in the CDW phase may occur through soliton-like 2p-changes. These line up on neighboring chains to form extended defects (e.g.phase dislocation lines/loops). Phase-slip occurs by the growth of dislocation loops by climb across the crystal cross-section. ? Does changing polarization require 1) glide of solitons/dislocations along the conducting chains or 2) smooth, gradual changes in phase?

5. Use the electro-optic response to measure the frequency, voltage, and spatial dependence of CDW “repolarization” IR Microscope DQ/Q = V(w) / V(W) Spot size = 50 mm Electro-Reflectance: DR = R(V+) – R(V-) Electro-Transmittance: DQ = Q(V+) – Q(V-) w

6. T = 80 K, n = 890 cm-1 Fits to: Dq/q = Dq/q)0 / { 1- (w/w0)2 + (-iwt0)g} t0 = average relaxation time, decreases with increasing voltage 1/w0 = delay time (→ 0 near contact) g<1: distribution of relaxation times

7. Dq/q = Dq/q)0 / {1 - (w/w0)2 + (-iwt0)g} • Relaxation time t0 ~ V-p (1<p<2), and increases slowly with distance from contact. • Delay time 1/w0 decreases more slowly with voltage, but increases rapidly with distance from contacts: strain flows from contacts. • g often decreases with decreasing current: distribution of time constants (current?). T = 80 K #1 #2 #3 820 890 820 cm-1 X = 0 X = 100 mm X = 200 mm Can we understand these effects from “simple” model of CDW phase dynamics?

8. Adelman et al successfully used a 1D model to describe changes in phase after a current reversal in NbSe3: Pinning field x ∂f/∂t = [Q/enc(r0+rc)] [r0Jtotal – Eon + (Q/enc)K∂2f/∂x2] - ∫rps(f(x’,t)dx’ { -∞ phase slip Electric Damping Elasticity Phase slip rate: Normal and (high-field) CDW resistivities Strain: Note: this is the (elastic) diffusion equation for phase changes, with the extra, strongly nonlinear phase-slip term: ∂f/∂t = b(Jtotal – Eon/r0) + D ∂2f/∂x2 - ∫rpsdx’, with diffusion constant D = (Q/enc)2K / [r0 + rc] We solved for the periodic strain response to a square wave current as a function of frequency, and found the first Fourier component, ew, to simulate the lock-in response.

9. We used Adelman’s 90 K NbSe3 parameters: (blue bronze parameters not determined) Eon = 170 mV/cm We considered two different boundary conditions for the phase:

10. Response is delayed relaxation, with delay time increasing away from contacts. • Poorer fits at x=0, because w0→∞. • Allowing phase slip beyond the contacts (“free contacts”) slows the response at all points; i.e. phase slip beyond the contacts “holds the strain back”. delay relaxation Solid symbols: Pinned contacts Open symbols: Free contacts Fits to ew = e0 / {1-w2/w02 + (-iwt0)g}

11. ew = e0 / {1-w2/w02 + (-iwt0)g} • Relaxation time and delay time (1/w0) decrease with increasing current (t0 has much stronger dependence on j). • Both time constants increase with distance from contact; 1/w0→ 0 at x=0. • g ~ independent of current: single relaxation time. (Smaller value at x=0 is artifact of fit.) For free contacts,, the phase-slip beyond the contact causes peak in t0 at low current. For pinned contacts, t0 saturates at the diffusion value:t0 = L2/p2D Solid symbols: Pinned contacts Open symbols: Free contacts

12. In 1983, Fisher proposed that CDW depinning was a dynamic critical transition. His model assumes that near depinning onset, CDW deformations are elastic, i.e. phase-slip is negligible. In 3d, renormalization group calculation of critical properties, Narayan and Fisher found that the CDW velocity, coherence length, and diffusion constant vary as: v  (V/Von-1)5/6, ξ  (V/Von-1)-1/2, D  (V/Von-1)-1/6, Previous experimental searches for critical behavior have concentrated on the CDW velocity, for which it is difficult to escape effects of contacts and phase-slip, and have had mixed results. We instead attempted to use electro-optic response to look for divergence of D (i.e. decrease of t0 (V/Von-1)1/6) near Von.

13. Expected behavior of t0 This required a sample with a large region (Von < V < VT) with ~ elastic deformations (i.e. “no” phase-slip) to observe saturation of t0.

14. x=0, T = 78 K, n = 850 cm-1 T = 78 K, n = 850 cm-1, w/2p = 25 Hz, x=0 Worked adjacent to contact so delay time small (i.e. w0 ~ ∞) and relaxation peak most clearly defined. Fits to DR/R = A0 / [1 + (-iwt0)g]

15. DR/R = A0 / [1 + (-iwt0)g] We observe expected saturation of t0, but no decrease for V/Von≥ 1.06; →(V/Von-1)crit < 0.06. t0(sat) ~ 20 ms → D = L2/p2t0 = 0.02 cm2/s

16. D = (Qen)2K/(r0+rCDW) →the CDW elastic constant K = 0.1 eV/Å, (one order of magnitude greater than (semimetallic) NbSe3). 2 bands condense into CDW Taking K = 4(Q/2p)mFvf2, with Fröhlich mass mF = 250 me → phason velocity vf = 14 km/s consistent with neutron scattering results of Hennion et al [PRL 68, 2374 (1992)]. → If CDW polarization is due to motion (glide) of dislocation lines, they diffuse with same constant as the CDW has for small phase perturbations. OR CDW polarization involves smooth, continuous changes in phase (like phasons).

17. Summary • Electro-optic measurements used to measure the dynamics of changes in CDW phase. Response to alternating voltages: “delayed relaxation”. • Relaxation time decreases rapidly with voltage and increases slowly with distance from current; delay time decreases slowly with voltage and increases rapidly with distance: consistent with model in which phase changes are governed by diffusion + phase-slip. • The diffusion constant is consistent with the measured phason velocity →polarization controlled by same “intrinsic” elastic constant as phasons. • No decrease in relaxation time (i.e. divergence of the diffusion constant) is observed near depinning onset, implying that the critical region: (V/Von-1)crit < 0.06 !! Thank you, Natasha, Pierre, and Serguei !!

18. TaS3, T = 80 K Want to look for correlations between current induced torsional strain and gradient of CDW phase (electro-optic response): Need wide (> 20 mm) crystals of TaS3 !!!

19. Time Dependence of Voltage for Square-Wave Currents “overshoot”