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Melting of the vortex lattice and collapse of its shear modulus

When a magnetic field is applied to a type II superconductor, the flux enters as a dense array of vortices (quantized flux lines).  In a low-Tc superconductor such as Nb, the vortices are strongly attracted to each other to form a hexagonal 'Abrikosov' lattice.  In this vortex 'solid' phase, the motion of an individual vortex is greatly restricted since it involves displacement of a large number of neighbors, if not the whole lattice (vortex motion produces undesirable Joule heating).  The resistance of the vortex lattice to shearing forces is measured by the shear modulus c66.  The large value of c66 in Nb justifies classifying the vortex lattice as a solid.
 


 

In the cuprates, however, the superconductiviting properties are highly anisotropic (the phase rigidity is very stiff within each copper layer (light blue sheets in figure), but much less so between adjacent layers).  In each layer, a small supercurrent loop (dark blue circle) surrounds the magnetic flux.  This prevents the flux from spreading out in the layer (this constitutes the 2D pancake vortex).  Between layers, the flux bulges out because supercurrents are much weaker.

At sufficiently high temperature T (and flux density B), the anisotropy leads to a melting of the vortex solid into a vortex-liquid phase in which the shear modulus vanishes.   Several key experiments have shown that this melting transition occurs in all the cuprates (and also in low-Tc superconductors, notably 2H-NbSe2).  An anomaly was detected in the heat capacity, consistent with a first-order transition.  In a second experiment, a microHall probe was used to demonstrate that the vortex 'matter' expands on freezing, much like ice.   In single crystals (especially of YBaCuO), the melting coincides with an abrupt increase in the voltage across the sample produced by the rapid diffusion of vortex lines in the liquid state (the finite dissipation is called flux-flow resistivity).

To demonstrate that an actual solid-to-liquid transition occurs, however, it is desirable to show that the shear modulus in fact vanishes at Hm.   Our approach is to impose an oscillatory force on the lattice, and to measure its ac velocity response v.   Since the velocity translates into an ac electric field E = B x v, this is equivalent to measuring the complex impedance of the crystal in an applied field B.

 

An ac current J (at frequency w) exerts a Lorentz force F = J x B (red arrow in figure) on each vortex core (this force is the analog of the Magnus force that acts on a spinning tennis ball).  For a high-purity crystal (without twin boundaries), we expect that only a small fraction of the vortices are pinned to impurities.  The remaining vortices respond to the driving Lorentz force as if they are unpinned, except they interact with their nearest neighbors (this is absent if the shear modulus c66 vanishes).  Finally, all the vortices are damped by a friction force that is proportional to w (green arrow).  The experiment is very much like shaking a piece of Jello that is nailed to the table at only a few places, and measuring the average velocity response.  The ac velocity amplitude and phase averaged over all vortices gives the voltage observed.  It depends on the parameters characterizing the 3 competing forces.  By varying the frequency w of J, one goes from the low-w regime where the friction force is negligible to the high-w regime where it is dominant.  Hence by measuring the spectrum of the complex impedance, one may derive detailed information on how the various force parameters vary with increasing flux density B.

A striking result is observed in the out-of-phase component of the voltage response (the inductance L).   Initially, L rises linearly with B (the vortex density).  The rate of increase slows as B approaches Hm.  When H exceeds Hm, L abruptly collapses to zero, as the vortex solid becomes a liquid.  We developed a classical model using the arguments discussed above.   To solve the interacting vortex model, we adopt the mean-field assumption that the vortices (only) at pinning sites all share the same velocity response.  Solving the equations numerically, we obtain spectra for the complex impedance that fit remarkably well to the experimental curves.   This enables us to extract the shear modulus c66, and to show that it in fact collapses over 3 decades to zero at Hm.

The experiment is challenging because we have to measure a complex impedance of 10 to 100 mOhms at w up to 20 MHz with a resolution of about 10 microOhm.  Moreover, this has to be done in a superconducting magnet with cables of length 3-4 meters in length.  We developed a scattering method in which the sample shunts the incident cable to ground.  We phase-detect both the transmitted and reflected waves phase using a high-w lock-in amplifier   The initial experiment by Hui Wu involved a 2-probe method that allows the experiment to be performed at a fixed temperature while H is varied.  However, for measurements in which T is varied with H fixed, this method is inadequate (background changes are too severe).  Peter Matl overcame this problems by generalizing the scattering technique to handle a true 4-probe sample, using rf transformers to float the sample ground.

See
Hui Wu, N.P. Ong and Y.Q. Li, “Frequency dependence of the vortex-state resistivity in YBa2Cu3O7-d.”, Phys. Rev. Lett. 71, 2642  (1993).
Hui Wu, N. P. Ong, R. Gagnon and L. Taillefer, “The Complex Resistivity Spectra and the Shear Modulus of the Vortex Solid in Untwinned YBa2Cu3O7.”, Phys. Rev. Lett. 78, 334 (1997).
N.P. Ong and Hui Wu, “Relating the ac complex resistivity of the pinned vortex lattice to its shear modulus.”, Phys. Rev. B 56, 458 (1997).
Hui Wu, P. Matl, N. P. Ong, R. Gagnon and L. Taillefer, “Collapse of the Vortex Lattice Inductance in Untwinned YBCO Observed by Complex Resistivity Technique.”, Proceedings of M2S-HTSC V, Beijing 1997, Physica C 282-287, 2183 (1997).
Hui Wu, Ph.D thesis, Princeton, 1996.
Peter Matl, Ph.D thesis, Princeton, 1998.