ELECTROMECHANICAL RESPONSE OF SLIDING CDWs: VOLTAGE-INDUCED TORSIONAL STRAIN IN TaS 3 Hao Zhang, John Nichols, C. Sandumali Weerasooriya, and Joseph Brill University of Kentucky Thanks to Vadim Pokrovskii and Damir Dominko for helpful discussions and Robert Thorne for samples.

Outline Review of previous electromechanical effects in orthorhombic TaS 3 * Our experiments on voltage-induced torsional strain (VITS) in TaS 3 Our model Problem: Time constants Large effects Crystal sizes (typically (1mm-1cm) x 10  m x 1  m) make them “easy” to deform (bend, twist) crystal. * Orthorhombic TaS 3 : the “fruit fly” of electromechanical effects in CDW materials:

Shear compliance, J = 1/G Zhan and Brill (1997) In 1984, it was discovered (using vibrating reed measurements) that the Young’s modulus decreased by ~ 2% with CDW depinning. In 1987, it was discovered (torsional oscillator measurements) that the shear modulus (G) decreases by ~ 20%. In both cases, the “internal friction” = tan  [damping, e.g. tan  = Im(J)/Re(J)] also increased when the CDW was depinned.

Mozurkewich (1990): Voltage dependence of elastic moduli is due to strain dependence of k F changing the configuration of CDW phase domains, which change with relaxation time  (V). Relaxation strengths Young’s: A Y = (Y CDW /Y 0 ) (1+ ∂lnk F/ ∂ε // ) 2 Shear: A G = (Y CDW /G 0 ) (∂lnk F /∂ε  ) 2 e.g. Simple (Debye) Relaxation: (M( ,V)/M 0 ) -1 = 1 + A/(1 + i  ) For V<V T,  = ∞, so the CDW spring is locked in place. Zhan and Brill (1997)

1991: Hoen, et al, found that the length of TaS 3 crystal changes as you depin the CDW, with  L/L ~ Changes are hysteretic, occur near threshold, and then saturate for large voltages (unlike changes in modulus). Model: Electric field near threshold strains the CDW, which couples to strain of the lattice.

The CDW compresses at the positive contact and stretches at the negative contact. Most of this change is hysteretic, due to pinning of the CDW phase for voltages below threshold. (V T – V on is the “phase-slip voltage”.) The changes in q put longitudinal stress on the lattice, with longitudinal “trans-modulus”  // ~ 6Y CDW ~ 40 GPa. [Golovnya et al (2002)]. (The total length changes are small because the changes in q at the two ends are only slightly different.) Itkis, et al (1986) phase

By applying square waves to the sample, we measured how quickly the CDW q can cross the hysteresis loop. The change in q is relaxational: near V on,  0 < 10 ms and is determined by diffusion of the CDW phase. [  0 (Von) = L 2 /  2 D, where D is proportional to the square of the phason velocity -- (ECRYS-2008 proceedings).] At higher voltages, relaxation is enhanced (  0 falls) due to CDW phase-slip (i.e. current). Rai and Brill, (2004)

Pokrovskii, et al, (2007) found that crystals of TaS 3 also twist under applied dc voltage. The observed twist: occurs in the CDW state has onset for V ~ V T and doesn’t go away if V → 0, but reverses if V ~ -V T (i.e. has hysteresis similar to length and CDW deformations is “sluggish”: if the CDW is depinned with ac current, torsion disappears at high frequencies. (There is also a smaller, faster and non-hysteretic voltage induced strain.) Since TaS 3 is orthorhombic and CDW is “linear”, not clear what could give rise to “voltage-induced torsional strain” (VITS). (How does it know which way to turn?)

Our Measurements Gold film on half the sample so E=0, pinning the CDW on this side. Magnetic Wire (Transducer) sample Triangle-wave or square-wave voltages applied to sample Sample inside RF (430 MHz, Q=300) cavity; when sample twists, it changes resonant frequency. Measure cavity frequency shifts (  twist angle) as functions of time or frequency of applied voltage. Application of dc B allows us to apply external torque. Application of ac B allows us measure V- dependence of shear compliance (J) and V T and to approximately calibrate the cavity response. B ~ 2 mm

Applying triangle wave (period T) to sample. Applying dc voltage to sample and small sinusoidal ac to magnet. Hysteresis loops very sluggish (take > 10 sec to develop) (Torsional resonant frequency ~ 530 Hz). Hysteresis loops are not symmetric functions of V (both shape and dynamics different for two polarities). V on (↑↓), where twist starts changing, is typically below V T (where compliance starts changing). -V on Sample C, T = 78 K

The onset voltage for the VITS is more easily seen by applying symmetric square waves (frequency  ) to the sample, driving the sample across the hysteresis loop. [The voltage and frequency dependent VITS response (in-phase and in quadrature with the applied square wave) is called  .] Sample C, T = 78 K

Frequency Dependence of Square-Wave Reponse (   ) Sample D, T = 78 K Curves show fits to modified relaxation expression:     / [1+ (-i  0 )  ]. [  < 1 corresponds to a distribution of relaxation times.] Average relaxation time (  0 ) is ~ 1 sec at V T but falls with voltage.

Applying Torque with Magnet Current (I B ) Sample E, T = 78 K Magnet: B/I B = 80 Gauss/A, Sample E:  /I B ~ 12 o /A. Application of torque changes the magnitude and time-constant of the square wave response, and can even change the sign! [Note that applied torque does not affect the resistance or V T. (It may have small effect on  J(V).] ▼▲▲IN-PHASE Δ Δ QUADRATURE Δ

Sample F, T = 78 K, 0.3 Hz triangle waves Similarly, application of external torque affects the size and even direction of the hysteresis loop: ~ 5 o twist reversed the direction of the loop. Effect of Torque on VITS Hysteresis Loops This result suggests that the VITS effect (even with no applied torque) may be due to residual twist in the sample, due to contacts, wire, or growth defects in the crystal. (TaS 3 is notorious for defects.) For example, in this sample, twisting with I B ~ 0.8A seems to minimize the VITS by straightening the sample.

q = q 0 ( z +  r   Before applying voltage, the CDW wavevector: After applying voltage, q 0 → q 0 +  q(z), where  q(z) is a hysteretic function of applied voltage. We know that longitudinal strains in q can cause longitudinal stress and strain in the crystal. We similarly assume that azimuthal q- strains put a (hysteretic, voltage dependent) torque (  ) on the sample:  z  ≈   /q 0  dA r (  r  q) ≈   /q 0  R 4  q(z) where A and R are the sample cross-section area and “radius” and   is the torsional “trans- modulus” relating shear CDW strain and torque on the sample. where  = d  /dz and r is distance from central axis. If the sample is twisted, the CDW q will have an azimuthal (  ) component (as viewed in the lab frame). CDW wavefronts (untwisted sample) Hysteresis loop in  q Simple Model

Consequently, the sample will twist: d  / dz ≈ d(  / dz, where and  ≈ GR 4 /z is the torsional rigidity (elastic restoring force) of a point a distance z from the fixed end of sample. ( G is the shear modulus.) If  q(z) ≈ -  q(L-z) and if  d  /dz =constant, then at the free end:  (L) ≈    L  q(L)/(G q 0 ). [e.g.: for  L ≈ 2 o /mm, G ≈ 5 x 10 9 J/m 3,  q(L  ≈ q 0 (as for NbSe 3 ) → |   | ≈ 40 GPa ≈ |  // | (Golovnya et al 2002)] [The sign changes of the hysteresis loops (observed for three samples) indicate that    q is negative (i.e. opposes  ) on the positive side of the sample. If  q is positive here (i.e. the CDW is compressed),   < 0.]

 q // is due to voltage induced changes in CDW phase (phase-slip enhanced diffusion) VITS is associated with sample strain responding to voltage-induced changes in CDW phase.  J is associated with local CDW phases rearranging in response to applied stress on sample. Expect  0 (  q // ) to be limiting time constant for VITS, but VITS stays > 100 times slower than both  q // and  J at high V. Why is VITS so slow? Expect similar  ’s ? } Comparison of Time Constants

Fits to:   =   0 / [1 + (-i  0 )  ] Temperature dependence of  0 is much (~ 20 x) stronger than temperature dependence of ohmic conductivity. However, the CDW current (at a given voltage above threshold) varies as strongly as  0 : Sample #4 Sample G (I B = 0) Temperature Dependence of VITS (square wave response)   (arb. Units)  

 0 is ~ inversely proportional to CDW current in this temperature range → VITS relaxation driven by CDW current ?? Is CDW current required to push CDW defects (e.g. domain walls) around crystal for VITS ?? Note: V T and V on are fairly constant for 90 K ≤ T ≤ 120 K

Conclusions VITS is a consequence of residual twisting of the sample (due to growth defects, mounting) giving CDW q an azimuthal component. As  q goes through hysteresis loops with applied voltage, the azimuthal CDW deformations put torque on the lattice, giving rise to the VITS:  (L) ≈    L  q(L)/(G q 0 ) VITS time constant ~ 1 second at V T and T ~ 80 K, decreasing at higher temperatures and voltages:  0 ~ (I CDW ) -1. Why is it so slow ??? Thank you Natasha, Pierre, and Serguei !!!