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-*T*_{c}
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 *c*_{66}.
The large value of *c*_{66} 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-*T*_{c}
superconductors, notably 2*H*-NbSe_{2}). 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 *H*_{m}.
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 *c*_{66} 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 *H*_{m}. When *H* exceeds *H*_{m},
*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 *c*_{66},
and to show that it in fact collapses over 3 decades to zero at *H*_{m}.

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,

Peter Matl, Ph.D thesis,