Phase diagrams, critical temperatures, and cuprate superconductors

One of the most iconic plots in condensed-matter physics, if not in all of physics, is the phase diagram of the high-Tc superconductors. As in the example below, the diagram depicts the various electronic phases of a high-Tc superconducting material, such as bismuth strontium calcium copper oxide (BSCCO), as a function of temperature and doping—that is, the addition of atoms that contribute charge carriers (oxygen in the case of BSCCO).

I’d seen the phase diagram so many times that I never questioned it—until last week, when I encountered a paper in Science by Xucun Ma of Chinese Academy of Sciences’ Institute of Physics, Xi Chen of Tsinghua University, and their collaborators.

The Ma–Chen collaboration reported the results of applying scanning tunneling microscopy and spectroscopy to iron selenide. The superconductor is interesting not because of its critical temperature—8 K at ambient pressure; 37 K at 8.9 gigapascals—but because it’s the simplest member of the newest family of unconventional superconductors based on iron.

The discovery of iron-based superconductors three years ago touched off an explosion of activity among condensed-matter physicists. What excited them—and continues to excite them—is the prospect that playing with an additional family of unconventional superconductors will lead to a successful microscopic theory of high-Tc materials and to room-temperature superconductivity.

Before getting to what Ma, Chen, and their collaborators found—and to why it got me thinking about the phase diagram—I should explain that one of the key features of a superconductor is the shape of its so-called superconducting gap.

You can think of the superconducting state as a low-lying, slippery surface that electrons slide along without resistance. Above the superconducting state lies a normal state, a rough surface that electrons require help from an applied voltage to cross. The wider the gap between the two states, the harder it is for electrons in the superconducting state to gain enough energy from their surroundings to jump up and lose their superconductivity.

The superconducting gap isn’t the same height in all directions with respect to the crystal’s symmetry axes. In fact, in BSCCO and other superconducting copper oxides, the gap vanishes at nodal points. Whether FeSe and the other iron-based superconductors also have nodes remains an open question.

For its STM study, the Ma–Chen collaboration grew very pure FeSe on a silicon substrate. When the researchers used their microscope to probe the material’s superconducting gap, they discovered that it has nodes. They also confirmed that bulk FeSe, which usually contains a small admixture of tellurium, doesn’t have nodes.

Ma and Chen suggest an explanation for why the presence of Te might remove the nodes. The archetypal iron-based superconductor, LaFeAsO, has a Tc of 26 K and lacks nodes. Its near relative, LaFePO, has a Tc of 7 K and had nodes. In those and other iron-based superconductors the conduction electrons occupy occupy a complicated band structure derived from a mix of Fe orbitals and those of its pnictide (P, As, Sb) or chalcogen (Se) partner.

Structural comparisons among the iron-based superconductors suggest that the separation between the Fe plane and the pnictide or chalcogen plane is correlated with both Tc and the presence of nodes. The P and Fe planes are closer in LaFePO than the As and Fe planes are in LaFeAsO. And the Se and Fe planes are closer in FeSe than they are in Fe(Se,Te).

But it isn’t the interplane separation per se that governs the value of Tc and the presence of nodes. Rather, the separation determines the relative propensity of electrons prefer to pair up with electrons on nearest-neighbor atoms or on next-nearest-neighbor atoms. Nearest-neighbor pairing leads to nodal gaps. Next-nearest-neighbor pairing leads to nodeless gaps.

Is it really a phase diagram?

I haven’t seen phase diagram that shows how the Tc of Fe(Se,Te) varies with the proportion of Te, but now I wonder to what extent such plots really are phase diagrams. Each ordinate corresponds to a single crystalline material whose electronic state changes continuously with temperature. That sounds like proper phase behavior to me.

But each abscissa corresponds to a series of different crystalline materials. In the case of Fe(Se,Te), a low-temperature abcissa could include nodeless and nodal superconductors. Although you can think of a continuous transformation between the two—which is what the phase diagram implies—you can’t realize that transformation in practice.

Does that distinction matter? Maybe not. But it might affect the quest to devise a complete microscopic theory of high-Tc superconductivity. If you regard the abscissae in the phase diagram as corresponding to different materials, then a theory should aim to account for the ordinates. It should predict the temperature dependence of a material’s normal and superconducting states, including, most important of all, the value of Tc.

For conventional, BCS conductors such a predictive formula exists. Published in 1968 by William McMillan, it remains one of the great triumphs of theoretical condensed-matter physics.