Tunneling microscope maps the melting transition of a two-dimensional vortex lattice

At absolute zero, a two-dimensional array of identical interacting particles crystallizes into a hexagonal close-packed lattice. But if the particles are warmed even a little, long-wavelength fluctuations will destroy the lattice’s perfect translational symmetry. Orientational order persists, however. Although the particles may no longer be equally spaced, they’ll still line up in crisscrossing diagonal rows.

What happens to a 2D crystal as the temperature increases further has interested physicists for decades. Eventually, of course, the crystal melts. But in between, its more or less uniform orientational order begins to crumble and a topologically distinct state, a hexatic state, appears: Pairs of point defects called dislocations sprout and spread, fragmenting the crystal into patches of local orientational order demarcated by defects called disclinations.

That 2D melting proceeds via a hexatic state was predicted in 1978 by Bertrand Halperin and David Nelson ¹ and observed 20 years later in a system of colloidal particles by Georg Maret, Ralf Lenke, and Klaus Zahn. ² Now, Isabel Guillamón of the Autonomous University of Madrid, Spain, and her collaborators have imaged the whole melting process—from a crystal, through a hexatic state, to a liquid—in a quite different 2D system, a lattice of superconducting vortices that formed in a thin film of tungsten subjected to a perpendicular magnetic field. ³

So-called type I superconductors expel magnetic fields. But tungsten, high-T _c cuprates, and other type II superconductors can accommodate magnetic fields by wrapping the flux lines in vortices of supercurrent. If the type II superconductor is thin, the vortices, like other particles in a 2D system, will crystallize and, at higher temperatures, become mobile: The crystal will melt.

In principle, vortex melting can be observed with a scanning tunneling microscope. Vortex cores are in the normal, non-superconducting state. As the STM scans across a core, its tip can pull out electrons even at low voltage. But when the tip scans across the surrounding swirl of supercurrent, the electrons, trapped by the superconducting energy gap, can’t tunnel across. To map the tip’s conductivity over the sample is to map the vortices.

Imaging a single, 10-nm vortex is not especially challenging. But melting is a many-body phenomenon. To follow its progress, Guillamón needed to scan an atomically flat, defect-free surface that encompassed tens of vortices at least. That requirement is so stringent that the idea for the melting experiment came to her only after high-quality samples became available from the University of Zaragoza’s nanoscience group.

The Zaragoza group uses an ion beam focused down to a nanoscale patch to make samples that are perfectly flat over areas of order 100 × 100 nm². The technique yields thin films that are amorphous, not crystalline. That property is fortuitous: Amorphous metal films have higher critical temperatures and more isotropic superconducting gaps than crystalline metal films do. For Guillamón’s study, amorphous samples have another advantage: no crystalline axes to bias the orientational order.

The 220-nm panels show one of the melting transitions that Guillamón observed. At the lowest temperature (0.1 K; not shown), slight surface imperfections pin and distort the vortex lattice. But at 1.2 K, well below the film’s 4-K critical temperature, the lattice gains just enough thermal energy to become unpinned and undistorted. The hexatic state starts to appear at 1.9 K. Two of the vortices in the corresponding image have lost their sixfold coordination and acquired lattice-disrupting fivefold and sevenfold coordination. As the temperature increases, the lattice becomes more distorted and the vortices begin to move. By 2.1 K, some of the vortices move faster than the STM tip can scan; they appear as stripes. Interestingly, the stripes are bent, demonstrating a kind of restricted symmetry that’s analogous to the smectic order in liquid crystals. The stripes could also represent another hallmark of the hexatic phase: movement along disclinations. By 2.8 K, the stripes have almost vanished; by 3.0 K, no order remains. The last image’s gray, undifferentiated blur corresponds to the average of the white superconducting state and the black normal state.

The Madrid experiment not only confirms the essential features of 2D melting theory, but it also appears consistent with attempts to model experiments that followed electrical resistance through the vortex melting state. Next on Guillamón’s to-do list is to measure the resistance while observing the vortices.