Palmer Craig's 1926 claim, covered in Anomalous Low-Field Hall Coefficient in Bismuth, was that bismuth's Hall coefficient does something unexpected when the magnetic field becomes very weak. C. W. Heaps disputed it the following year. Nearly a century later, a group at the Institute of Physics of the Polish Academy of Sciences in Warsaw, the University of Warsaw and the Gleb Wattaghin Physics Institute in Campinas, Brazil, also reported an unexpected low-field Hall behaviour in bismuth. The paper is "Anomalous Hall Effect in Bismuth" by Bruno Cury Camargo, Piotr Gierlowski, Andrei Alaferdov, Iraida N. Demchenko, Maciej Sawicki, Katarzyna Gas and Yakov Kopelevich. Version 2 of the arXiv preprint (2002.00656) is dated 2 March 2021 and was submitted to the Journal of Magnetism and Magnetic Materials.
Readers should not treat it as a vindication of Craig. It studies a different kind of sample at a different temperature and over a different field range, and it describes a different physical effect. It does show that later physicists still found low fields in bismuth worth examining.
What the paper reports
The authors say they saw an anomalous Hall effect (AHE) in bismuth crystals at fields below 30 mT. They call it "ferromagnetic-like" because the transverse resistance traced a hysteresis loop, the kind of history-dependent curve normally produced by magnetic materials such as iron. The paper points out that bismuth is not expected to show this. It describes bismuth as one of the most strongly diamagnetic materials known that is not a superconductor. Diamagnetic materials are weakly repelled by magnetic fields, not magnetised by them.
The paper also reports a second effect. In weak fields, and only when the sample had been cooled without a field applied, the ordinary resistance fell sharply below about 4 K, by up to 25%. Fields of about 20 mT suppressed this drop. Samples cooled with a field already applied showed no drop below 5 K.
How the measurements were made
The samples were cut from large bismuth crystals grown by the Bridgman method and described as 5N (99.999%) pure. They were bars about 5 mm by 3 mm and 0.5 to 1 mm thick. Their surfaces had many grain boundaries, the interfaces between differently oriented crystallites. The team measured seven samples, labelled Bi1 to Bi7, between 2 K and 10 K and in fields from −30 mT to +30 mT. The resistance measurements were run in two separate cryostats, which the paper says gave similar results. Magnetisation was measured in a SQUID magnetometer.
To isolate the anomalous part of the Hall signal, the authors subtracted a straight-line background with a slope of about −43 mΩ/T. A loop resembling ferromagnetic hysteresis remained below 4 K. They fitted the temperature dependence of the coercive field, the field needed to reverse the loop, with a model of non-interacting, randomly oriented single-domain magnetic particles. The fit gave a characteristic temperature of about 4.0 K. The paper also reports that the same instruments had not produced this behaviour in other high-magnetoresistance materials such as graphite, and argues that this rules out an instrumental origin.
What the evidence does and does not show
The paper's main difficulty is that the magnetisation measurements did not confirm the transport results. The SQUID data showed diamagnetism with only a weak paramagnetic signal at the lowest temperatures, and no ferromagnetic-like behaviour. Elemental analysis found no magnetic impurities. From the detection limit, the authors estimate that any iron contamination could be at most 45 parts per million by mass. From the weak paramagnetic background they estimate an atomic impurity level of roughly 0.2 ppm. They argue that so little impurity cannot explain a resistance change of about 20% or the Hall hysteresis loops. Their conclusion is that the magnetism is confined to a tiny fraction of the sample.
They suggest that fraction lies at surfaces or grain boundaries. Heating the samples in vacuum (10⁻³ Torr at 370 K for 4 hours) changed nothing, which they say argues against adsorbed gases and points to grain boundaries. The rest of the paper's interpretation is openly speculative. It proposes that the electronic structure is reconstructed along those boundaries into "flat bands", and that magnetism and superconductivity there may compete. The authors call their supporting evidence "circumstantial". Their own estimate of the magnetic exchange parameter comes out at about 10 eV when they use bulk values, which they say is unrealistically high. They bring it down to about 10 meV only by assuming that the density of electronic states is a thousand times higher at grain boundaries. They also say they cannot tune their samples at will, so the proposed mechanism remains untested.
The paper is a preprint with reported results from one group. The document itself does not establish that other laboratories have reproduced the signal. It also reports that samples grown the same way behaved "similar, but non-identical".
How it bears on Craig's claim
The 2021 paper does not mention Craig or Heaps. The earlier precedent it cites is G. Conn and B. Donovan's 1948 Nature report of anomalous negative magnetoresistance in bismuth below 40 mT. Conn and Donovan attributed that to an anomalous Hall contribution but did not attempt hysteresis measurements.
The paper differs from Craig's work in several concrete ways:
- Field range. Since 1 mT equals 10 gauss, the anomaly the paper reports appears below about 300 gauss. Craig's reported rise in −R happened between 0.07 and 0.3 gauss, three to four orders of magnitude lower. The paper does not report resolving features at fields that small.
- Temperature. Every measurement in the paper was made between 2 K and 10 K, and the hysteresis appeared only below about 4 K.
- Samples. The paper used millimetre-thick crystal bars. Craig used thin films prepared as described in Thin Bismuth Film Preparation.
- Kind of anomaly. The paper reports a hysteretic extra Hall signal on top of a linear background. Craig reported a coefficient that rose steeply as the field fell, with no hysteresis described.
The paper therefore leaves the Craig–Heaps dispute unresolved. It shows that low-field Hall measurements on bismuth can turn up features that need explaining, and it attributes those features to structural disorder at surfaces and grain boundaries, a sample-dependent factor. That doesn't confirm Craig's numbers, and nothing in the paper relates to his proposed rectifier.
References
- B. C. Camargo, P. Gierlowski, A. Alaferdov, I. N. Demchenko, M. Sawicki, K. Gas, Y. Kopelevich, "Anomalous Hall Effect in Bismuth," arXiv:2002.00656v2 [cond-mat.str-el], 2 March 2021. https://arxiv.org/pdf/2002.00656
- G. Conn, B. Donovan, "Anomalous Magneto-Resistance Effects in Bismuth," Nature 162 (1948) 336, as cited in the paper.
- See also Palmer H. Craig, C. W. Heaps, Hall Effect.
Source notes & attribution
- https://arxiv.org/pdf/2002.00656