Velocity and attenuation of stress waves in GdBa2Cu3O7 near the superconducting transition

Abstract We have measured the Young's modulus (E) and quality factor (Q) using a resonant bar method in the ceramic superconductor GdBa 2 Cu 3 O 7 as a function of temperature and magnetic field. In contrast to previous pulse-echo and resonant ultrasound results for YBa 2 Cu 3 O 7−δ , we find no dramatic stiffening below T c . The associated Q increases smoothly by approximately an order of magnitude from ambient temperature to 100 K. Below T c , Q is a strong function of magnetic field, suggesting significant attenuation from flux motion in the sample.

Thermodynamic probes have proven to be of value in the study of the superconducting phase transition because the free energy of superconductors can be determined independently of a microscopic formalism.
The bulk thermodynamic properties derived from the free energy then constrain a microscopic description and provide a good test for its validity.
The sound velocities, or more precisely, the elastic moduli, are particularly valuable because extremely small changes can be detected, on the order of O.1 parts per million (ppm). This is sufficiently sensitive to enable observation of changes related to variations of the free energy near the superconducting critical temperature TC. I A more easily observed feature of conventional superconductivity is the rapid decrease in ultrasonic attenuation below Tc .2 This is related directly to the quasi-particle density, because the principal loss mechanism for sound waves in metals at low temperatures is dissipation from electronic currents induced by the displacement of the ions by the sound field.
The recent interest in oxide superconductors has motivated several ultrasound studies of the compounds (La-Sr)2Cu04 ~ and Heasurements of the bulk modulus B of VBa2C~307 show a substantial increase in IdB/dT] below T c, indicating that there are physical processes associated with the superconductivity that require a more complicated treatment than simple thermodynamic arguments can provide (the standard assumption being that only the difference in free energy of the electrons varies near T c while the lattice is an unchanging background).
In this Conm~mication, we report on acoustical measurements in sintered GdBa2Cu3Ov_ ~ using resonant ultrasound at frequencies near 320 kHz.
This low frequency circumvents some of the problems, described below, associated with the granular nature of the samples.
We achieved quality factors (the Q of the resonance) in excess of 103 at 100 K, enabling us to obtain a precision of slightly better than 1 ppm. In contrast, the higher frequencies used in pulseecho measurements are attenuated heavily, most likely from dissipation and scattering at grain boundaries.
Because good sensitivity to velocity changes depends on a large number of echoes, at best 1 part in 104 could be detected in our 30 MHz pulse-echo system.
The samples were prepared from the oxide powders Gd203 and CuO, and barium carbonate (BaC03).
The starting materials were ground and fired twice at 950C, and then pressed into a right circular cylinder (6.25 cm x 6.00 cm) at a pressure of 3 kbar.
We were able to achieve 84% density by this method.
To ensure parallel faces for the acoustic measurements, the ends were machined with a special fixture on a milling machine.
Sample characterization was performed with a Quantum Design 6 susceptometer, from which we found Tc = 94.5 K ± 0.5 K. The Meissner signal was 35~ of ~ = -1/4v. Upon cooling to 10 K in zero field, complete shielding of the field in the interior of the sample was observed.
The transdvcers were LINbO 3 from Valpey-Fisher, v 1/4" in diameter having coaxial electrodes (1/8" active area), with crystal axes oriented to excite the longitudinal modes of the sample.
A transducer on one end was used to drive the sample and a second transducer on the other end was used to detect.
To prevent epoxy from diffusing in, 2000 ~ Cu was evaporated on each end of the sample before the transducers were attached.
In the resonant method, we excite the fundamental compressional mode of the bar, and therefore the sound velocity v is approximately related to the resonant frequency fo by where X = 2L (L is the sample length), E is Young's modulus, and p is the mass density. This value of v is approximately 20% lower than for a sound beam in an infinite medium, s Moderate drive levels (>200 mV) caused some backbendin § in the tuning curves of the resonance, so we always measured with less than 100 mV applied to the driving transducer. Although many resonances could be driven, we concentrated on the fundamental longitudinal mode which we were able to identify with the help of a pulse-echo experiment at ambient temperature.
We obtained v = 4.4 i 0.2 km/s from pulse-echo, and v = 4.3 i 0.1 km/s from the resonant experiments.
Our transducers could also weakly excite shear resonant modes.
The shear velocity we obtained was 2.6 km/s i 0.1 from the resonant method (no number was obtained from pulse-echo).
These values are consistent with some recent static measurements. ~° The correction to the resonant frequency from transducer mass loading is small, reducing the resonant frequencies approximately 3%. ~ Geometry corrections were estimated to be less than 8%. 11 In the course of our measurements, we often detected small unexplained shifts in resonant frequency.
The highly anisotropic and randomlyoriented grains suggest that movement due to strain relaxation at the grain boundaries or alterations of the twin ~2 structure at some arbitrary temperature could have caused such effects.
We note that our noise floor was well below these jumps.
In Fig. I, we show velocity data for two different samples.
The sample used for the upper plot (sample I) was cycled between ambient temperature and 40 K several times.
The lower plot (sample If) is of measurements taken during the initial cooldown.
We found it typical for the samples to undergo many small "earthqua/<es" at lower temperatures (~I00 K) before stabilizing.
This effect became less prevalent with further thermal cycling.
The data shown for sample I was taken during the final run.
The 70 ppm discontinuous drop of the resonant frequency at T c was observed on four separate experimental runs.
(h~ two other runs, small jumps occured at many temperatures, masking any effects at T c. Just below the transition temperature, the resonant frequency fo increased at a rate of 30 ppm/K, with the slope decreasing in magnitude with decreasing temperature. Purely thermodynamic arguments for a second order superconducting transition give, for the discontinuity in the buIk modulus B at Tc, found AC = 3 9 d--mole--l-K -~ (one mole = one mole formula unit of GdBa2CuaOv).
Using the known unit cell parameters from x-ray measurements, '4 one obtains a specific heat jump of 3.7(10) 4 J-~-K-'.
Not forgetting the limitations of polycrystalline samples, our measured shift of 70 ppm is reasonable and con-sistent with Borges, Reeves, and thermodyr~nics In Fig. 2 we show the temperature departdance of Q, defined as For T ) 95 K, the change in Q is linear with tempera-ture, at a rate of -39 K -~.
Below 9't K, the slope increases dramatically, to --75 K--'. Because both of our samples show this effect, we associate it with the onset of superconductivity.
Although an increasing Q with decrea~;ing temperature is expected, we cammt attribute this behavior to conventioi~al BCS superconductivity.
In fact, a discontinuity {n the attenua-tion slope is expected for a PX--~ superconductor.  The high critical temperature suggests that attenuation of the sound wave is mostly from thermal phonons.
An estimation of loss indicates that the loss from phonon-electron coupling is only 10~ of the loss compared to three-phonon processes. However, the present poor understanding of these materials gives us little confidence in this estimate. Experimentally, however, we note that the estimate is reasonable, because it has been found that the thermal conductivity K increases as the temperature is lowered below Tc .~6 We conclude that either phonon scattering by electrons is significant near T c or the material is altered when the material becomes a superconductor (e.g. structurally or magnetically). 1~'17 Our original motivation for applying a magnetic field to the sample was to turn superconductivity off, thereby enabling us to subtract the normal background, facilitating the observation of small effects associated with the superconducting state.
Because the changes in sound velocity near T c were small and because the moderate fields (less that, 8 T) could not suppress the onset of superconductivity by more tl~n 4 K, we could not measure ma~mtic-field effects on sound velocity.
However. very strong effects were observed for Q. These data are shown for a series of temperatures in Fig. 3. The lowest temperatures show a large drop in Q (more than a factor of three for T = 70 K) in approximately 7 T fields and almost no temperature dependence.
The losses are far higher than a linear extrapolation made from the data above 9~ K.
In summary, we have observed a small decrease (70 ppm) of the sound velocity in the oxide superconductor GdBa2C-~3Ov_ 6 at the superconducting critical temperature, consistent with other thermodynamic measurements and considering only arguments applicable to secondorder phase transitions. This is in contrast to other work on YBa2Cu3Ov_6. 3'4 Scatter in our data arising from an intrinsic irreproducibility during the first few cooldowns indicates that the sintered samples are under a great deal of internal stress, which relaxes somewhat during each thermal cycle.
Trapped stress and twin structures are commonly found together elsewhere.
We find a change in dOJdT which begins at T e, perhaps from strong electron-phonon coupling near the transition temperature. Although the magnetic fields we applied do not destroy the superconducting state, a much lower Q is observed in the presence of magnetic fields, suggesting significant dissipation from flux motion, typical of type II superconductors.