Measurement Principle of Modulated Magnetic Susceptibility in Diamond Anvil

Superconductivity is defined by two fundamental criteria: zero electrical resistance and the Meissner effect. However, measuring the magnetic properties of samples under high pressure in diamond anvil cells—where sample sizes are limited to tens of micrometers and confined spatially—has long been a challenging task in high-pressure research. Magnetic measurements under high pressure using diamond anvil cells can generally be classified into four distinct methods. Among these, the modulated magnetic susceptibility measurement, which employs laboratory-fabricated multi-turn micro-coils and two lock-in amplifiers connected in series, has often yielded contradictory experimental results in the literature due to an insufficient understanding of its underlying measurement principles. In this work, starting from the experimental configuration and Faraday’s law of electromagnetic induction, we re-derive the expressions for the signal magnitude of the superconducting diamagnetic transition registered on the primary and secondary lock-in amplifiers. We obtain an expression for the signal amplification introduced by the modulated magnetic field, thereby clarifying the measurement principle of modulated magnetic susceptibility and identifying potential issues in previous studies.