Developing a theory of high-temperature superconductivity (HTS) is one of the outstanding problems in physics. It is a challenge that has defeated theoretical physicists for more than ten years, ever since researchers discovered that certain materials containing copper-oxide layers remain superconducting at temperatures up to 125 K. Indeed, it is often remarked that there are as many theories of HTS as there are theoretical physicists working on the problem. Arguments between the different theories and theorists are common, and the debate can sometimes reach temperatures that experimentalists can only dream of.
However,it is the last paragraph that makes the paper unique: “The tragedy of beautiful theories, ” write Chakraverty et al., “is that they are often destroyed by ugly facts. One perhaps can add that the comedy of not so beautiful theories is that they cannot even be destroyed; like figures in a cartoon they continue to enjoy the most charming existence until the celluloid runs out.” Provocative and unusual stuff for the world’s leading physics journal.
The paper provoked an immediate response from Sasha Alexandrov of Loughborough University in the UK, one of the champions of the bipolaron theory. On the same day that the “ugly facts” paper appeared in PRL, Alexandrov submitted a “comment” on the paper to PRL and to the Los Alamos e-print server, in which he argued that the objections in the paper were “the result of an incorrect approximation… and the misuse of the bipolaron theory”. He concluded by stating: “What is clear, however, is that any theory, beautiful or not, cannot be destroyed by ‘ugly’ artefacts as those in Chakraverty et al.”
So what is a bipolaron? And why has it polarized the HTS community so strongly?
The term polaron was coined by the late Lev Landau to describe the lattice distortion or polarization caused by the charge on the electron, and which follows the electron as it moves through the solid. One side effect is that the electron’s effective mass is increased. Bipolarons are bound pairs of polarons, mutually attracted by the lattice distortion. In the 1970s various physicists, including Chakraverty, used bipolarons to explain certain properties of different solids.
In 1981, before HTS was discovered, Ranninger and Alexandrov, both then working in Grenoble, published a paper in which they suggested that “bipolarons might be superconducting”. Ranninger later abandoned this theory, but Alexandrov continued to champion the bipolaronic approach in collaboration with the late Nevill Mott at Cambridge.
Fast forward to 1998. Physicists now have lots of theories of HTS but they are not sure which, if any, of them is correct. They know that HTS involves pairs of charge carriers, but the precise nature of these carriers and the pairing mechanism holding them together remain unclear. Chakraverty, Ranninger and Feinberg, however, are sure that the bipolaron theory is wrong.
According to Ranninger, their paper points out two main flaws in the bipolaron approach: the superconducting transition temperature, T c, is inversely proportional to the effective mass of the bipolarons, but the effective mass of the bipolarons in HTS materials is so high – at least 210 electron masses – that T c cannot be higher than 10 K, which is clearly too low to explain HTS. The paper also points out: “The existence of a Fermi surface in the high-T c materials has now been established experimentally beyond doubt. Bipolarons being bosons do not have a Fermi surface.”
Alexandrov replies that Chakraverty et al. have “misused” the theory by using onsite bipolarons, in which both electrons are near the same lattice site, rather than intersite bipolarons, in which the electrons are near neighbouring sites. He says that the effective mass of the bipolarons is about 12 electron masses and that this is consistent with experiments in HTS materials. “I am pretty sure we are correct, ” says Alexandrov. “There are no experimental facts that destroy our theory at present.”
He also disputes that experiments have shown that HTS materials have Fermi surfaces, pointing out that experiments have shown that the Fermi surface is destroyed in underdoped superconductors. (Doping refers to the addition of oxygen or some other element to provide charge carriers for the superconducting material. Optimal doping is the amount of doping which gives the highest T c.) And he claims that other properties of bipolarons – in particular the fact that they are mobile – can explain some of the other experimental results cited by Chakraverty et al . However, Alexandrov admits that not all theorists agree that bipolarons are mobile.
Chakraverty et al. have submitted a response to Alexandrov’s “comment”, and were reluctant to discuss this before PRL had reached a decision on publication. However, Ranninger said that their paper had already addressed the situation of intersite bipolarons and found that Tc was still only 5 K.
But why did they conclude with such a provocative paragraph? Ranninger says the last paragraph was written specifically to “calm the situation” and does not think that it was provocative. “We could have had a devastating statement at the end and that would have been a lot worse.” But Alexandrov, for one, says that he found the final paragraph “unhealthy and not motivated by any reason”.
Reaction to the paper in the HTS community has been mixed. In a letter to Ranninger, Alexei Abrikosov of the Argonne National Laboratory wrote: “I would like to express my pleasure upon reading your paper about bipolaronic superconductivity. I completely agree with it, and I appreciated the last two sentences.”
Philip Anderson of Princeton University, a long-term critic of the bipolaron theory, also welcomed the paper. “It had worried me that two theorists as competent as Ranninger and Chakraverty kept on with some support for the bipolaron theory of high-T c [when] the rest of the serious many-body community had long since rejected it.” The tone of the paper also went down well with Anderson: “I rather like physicists to express these kinds of sociological and methodological ideas, when the editors let them get away with it, and when it is done as eloquently as this.”
Gene Wells, managing editor of Physical Review Letters, agrees that the last paragraph of the paper was unusual. “I am not surprised that the letter caused a few raised eyebrows, ” says Wells, “but in context, I do not find it misleading or wrong. I doubt that an author who had never positively contributed to bipolaron theories would have been ‘allowed’ such a conclusion. Ranninger is criticized nearly as much as Alexandrov in the letter.”
However, the tone was “unhelpfully polemic” according to Alan Bishop of the Los Alamos National Laboratory. “I might comment in the same vein [that] ‘beauty is in the eye of the beholder’. In this case there are several beholders!”
The reaction among HTS experimentalists has also been mixed. Besides the question of the Fermi surface, there are “many other severe disagreements” between the bipolaron theory and experiment according to Juan-Carlos Campuzano of the University of Illinois at Chicago. For instance, he says, experiments find an effective mass of 2-3 free electron masses, compared with 12 electron masses predicted by the bipolaron theory. Campuzano adds that he found the PRL paper “almost charming” and that the original authors of the polaron theory had shown “a wonderful sense of humour” in pointing out that their own theory was wrong. “Would science not be a much more pleasant enterprise if more of us were willing to admit our faux pas ?” he asks.
Guo-meng Zhao and Hugo Keller of the University of Zurich put the lack of experimental support for the bipolaron theory down to the fact that most experiments have been carried out on optimally doped cuprates. Experiments have shown, they say, that two types of charge carriers – Fermi-liquid-like carriers and polarons or bipolarons – co-exist in optimally doped cuprates. Since the bipolaron theory is based on only one type of carrier, it will not be consistent with these experiments.
However, the situation is different for deeply underdoped cuprates. “To our knowledge, ” say Zhao and Keller, “no other theory can explain the physical properties of deeply underdoped cuprates in a more consistent way than the bipolaron theory.”
And Alan Bishop, for one, does not agree that Chakraverty et al. have demolished the bipolaron theory. But he does not think that Alexandrov and co-workers have proven it to be “the” theory either. “The situation remains experimentally and theoretically complicated, ” he says. “In my opinion, there is no such thing as ‘the theory of HTS’ presently available.”
