We report on a quantitative analysis of the magnetic field generated by a continuous current running in metallic microwires fabricated on an electrically insulating diamond substrate. A layer of nitrogen-vacancy (NV) centers engineered near the diamond surface is employed to obtain spatial maps of the vector magnetic field, by measuring Zeeman shifts through optically detected magnetic resonance spectroscopy. The in-plane magnetic field (i.e., parallel to the diamond surface) is found to be significantly weaker than predicted, while the out-of-plane field also exhibits an unexpected modulation. We show that the measured magnetic field is incompatible with Ampère's circuital law or Gauss's law for magnetism when we assume that the current is confined to the metal, independent of the details of the current density. This result was reproduced in several diamond samples, with a measured deviation from Ampère's law by as much as 94(6)% (i.e., a $15\sigma$ violation). To resolve this apparent magnetic anomaly, we introduce a generalized description whereby the current is allowed to flow both above the NV sensing layer (including in the metallic wire) and below the NV layer (i.e., in the diamond). Inversion of the Biot-Savart law within this two-channel description leads to a unique solution for the two current densities that completely explains the data, is consistent with the laws of classical electrodynamics, and indicates a total NV-measured current that closely matches the electrically measured current. However, this description also leads to the surprising conclusion that in certain circumstances the majority of the current appears to flow in the diamond substrate rather than in the metallic wire, and to spread laterally in the diamond by several micrometers away from the wire. No electrical conduction was observed between nearby test wires, ruling out a conventional conductivity effect. Moreover, the apparent delocalization of the current into the diamond persists when an insulating layer is inserted between the metallic wire and the diamond or when the metallic wire is replaced by a graphene ribbon. The possibilities of a measurement error, a problem in the data analysis, or a current-induced magnetization effect are discussed, but do not seem to offer a more plausible explanation for the effect. Understanding and mitigating this apparent anomaly will be crucial for future applications of NV magnetometry to charge transport studies.

• Received 2 October 2018
• Revised 18 December 2018

DOI:https://doi.org/10.1103/PhysRevB.99.014436