Visual images can have a powerful impact on the viewer, and this is certainly the case when the images are atomic-scale pictures of the surface of a high-temperature copper-oxide superconductor. The unusual properties of these materials have vexed condensed-matter theorists for the last 15 years. Now a new flurry of theoretical papers has been stimulated by the latest images of a cuprate superconductor, which reveal that an applied magnetic field can induce “stripes” of charge. The experimental results provide intriguing and important clues to the nature of these ever-surprising materials – although the theorists have still to agree on their meaning.

Many different families of copper-oxide compounds exhibit superconductivity. All of them are layered materials containing planes of copper and oxygen atoms. The copper forms a square lattice with an oxygen atom bridging each pair of copper atoms. By varying the electronic charge density within these layers, one can convert from a superconducting phase, in which electrical current can flow without resistance, to an electrical insulator. This is typically achieved by modifying the chemical composition of the compound, and the resulting insulating state is accompanied by antiferromagnetic ordering in which the magnetic moments of neighbouring copper atoms point in opposite directions.

It is this apparent competition between two states with such opposing properties that has generated much of the lasting interest and debate regarding the nature of superconductivity in the cuprates.

In recent years, there has been a great deal of discussion about other possible types of order – beyond superconductivity and antiferromagnetism – that might intervene between these phases and compete with superconductivity. Some of the proposals include exotic electronic states that have not yet been observed in any system. And these new experimental results have added considerable fuel to the debate.

The images that are proving so provocative were obtained by Seamus Davis of the University of California at Berkeley and co-workers using a scanning tunnelling microscope (J E Hoffman et al. 2002 Science 295 466). In a scanning tunnelling microscope (STM), a sharp metallic tip is brought within a nanometre or two of the surface of the sample. Images are formed by scanning the tip across the surface and measuring the electrical current that tunnels quantum mechanically between it and the surface as a function of applied bias voltage. The nature of the image depends on how the measurement is obtained. One can choose to measure the positions of atoms or – as Davis and co-workers did – to map the electronic density of states at the surface.

The team investigated bismuth strontium calcium copper oxide (Bi₂Sr₂CaCu₂O_{8 + δ} or BSCCO for short) – the same material used in nearly all STM studies of the cuprates. The beauty of BSCCO for surface studies is that it can be cleaved very easily, making it practical to prepare a fresh, uncontaminated surface with little effort. This cleavage occurs between the weakly bonded bismuth oxide layers. A copper-oxide layer can then be imaged through the non-metallic bismuth-oxide layer at the surface.

Davis and co-workers performed the measurements with a magnetic field applied to the sample. One of the hallmarks of the superconducting state is the Meissner effect, in which supercurrents are spontaneously generated such that they cancel out the applied field inside the superconductor. However, cuprates fall into the class of so-called type-II superconductors in which quantized units of magnetic flux thread through the sample above a threshold field. A supercurrent flows about each quantum of flux, cancelling the field within the bulk of the sample in a manner reminiscent of the Meissner effect. The amplitude of the superconducting wavefunction falls to zero at the centre of this vortex, and it gradually recovers to full strength outside a circular region called the vortex core, which has a radius equal to the coherence length of the superconducting state.

In BSCCO, the diameter of the vortex core is roughly five times the lattice spacing between copper atoms. The magnetic field is strongest at the vortex core and it only becomes fully screened some 400 lattice spacings away. The strength of the magnetic field used by Davis and co-workers produces a vortex spacing that is smaller than the screening distance, so that the magnetic field is nearly uniform outside the vortex core.

Within the vortex core, where the superconductivity is suppressed, the electronic state is usually assumed to correspond to the “normal” or non-superconducting state. However, early STM studies of the vortex core by Øystein Fischer’s group at the University of Geneva in Switzerland demonstrated anomalous behaviour. And much recent work has been stimulated, in part, by the theoretical prediction of Shou-Cheng Zhang of Stanford University and co-workers that the core should exhibit antiferromagnetic order.

Davis and colleagues found that the local density of states around each vortex is modulated in a checkerboard pattern (see figure). Moreover, the modulation has a period of four lattice spacings along the direction parallel to the copper-oxygen bonds. Several periods of the modulation are apparent and they extend well outside the vortex-core region, indicating that the modulation overlaps and coexists with the superconducting state.

The implications of this image are striking. One expects new features induced by the magnetic flux to be associated with suppression of the superconducting state. However, a spatially modulated local density of electronic states – like the one observed by the Berkeley group – does not occur in the normal state of conventional metallic superconductors. The modulation indicates a competing type of order that can also coexist with superconductivity. Much of the theoretical outpouring that has been stimulated by this work is aimed at explaining the nature of the competing order.

One empirical hint about the nature of the order comes from recent neutron-scattering experiments with a different cuprate compound: lanthanum strontium copper oxide (LSCO). An international team led by Bella Lake of the Oak Ridge National Laboratory recently made a particularly relevant discovery (2002 Nature 415 299). When Lake and co-workers applied a magnetic field to a sample of LSCO, they managed to induce local antiferromagnetic order that was modulated with a period of eight lattice spacings – exactly twice the period of the charge-related modulation in BSCCO.

Although the neutron measurements average over the entire sample, the magnetic scattering increases with the field strength as one would expect for local order associated with the vortices. Regions of antiferromagnetic and charge order – commonly referred to as stripes – with the periods of eight and four lattice spacings, respectively, have also been observed in closely related modifications of LSCO. Stripe order can coexist with superconductivity, but it tends to be associated with a drop in the superconducting transition temperature.

Not all theoretical models require the simultaneous presence of magnetic and charge-density modulations, but it would obviously be interesting to see STM and neutron-scattering measurements done on the same materials. So why don’t the two experimental groups just swap samples and repeat their measurements? This is easier said than done. LSCO crystals are not easily cleaved, which is a requirement for STM studies. On the other hand, it is challenging to obtain sufficiently large crystals of BSCCO for successful neutron-scattering measurements. Of course, overcoming these challenges is all part of the fun of doing science.

The images of field-induced modulations of charge and spin states obtained in the STM and neutron-scattering experiments are stimulating. Theorists are eagerly evaluating models and making new predictions, while experimentalists are busy trying to repeat the studies on different materials and to extend them in new directions. One can expect interesting developments in the near future, and the new results are likely to change the image of cuprate superconductors.