The superconductivity of materials always emerges from their normal state, and is closely related to the resistivity of that normal state.1,2 However, measuring the resistivity of a material under high pressure has long presented a significant technical challenge, owing to the difficulty of measuring the pressure-induced changes in the crystallographic directions, especially for samples with anisotropic layered structures, such as high-T_c superconductors and other intriguing quantum materials. Here, we propose a novel and effective method for determining high-pressure resistivity. The validity of the method has been confirmed through our investigations of pressurized copper-oxide superconductors.3 This successful application of the proposed method provides encouraging evidence that it can provide an effective approach for quantitatively investigating the connections between normal-state and ground-state properties in various other quantum materials under high pressure, which has been becoming an increasingly fruitful research direction in condensed matter physics and materials science.

Apart from chemical doping, pressure is the most important way to tune the transport properties of materials. Applying pressure to superconductors or other quantum materials not only reveals numerous novel physical phenomena,4–17 thereby providing vital insights for the exploration of new superconductors,18,19 but also can effectively assist in uncovering the underlying physics.20,21

Electrical resistivity provides a fundamental characterization of superconducting and numerous other quantum materials. It is well established that the resistivity ρ can be described by the following equation:$$\rho =\frac{1}{\sigma }=R\times \frac{S}{L},$$(1)where σ is the conductivity, R is the resistance, L is the distance between the electrical leads used for the voltage measurements, and S is the cross-sectional area of the sample. In practical terms, determining the resistivity of a compressed material in a diamond anvil cell (DAC) is challenging, owing to the high-pressure experimental setup, which, as mentioned in Sec. I, makes direct measurement of the pressure-induced size changes of the sample in the three crystallographic directions difficult (see Fig. 1), especially when anisotropic layered crystal structures are present. Consequently, it has been necessary to use resistance, an extensive quantity, to describe the conducting property of compressed materials, which hinders the identification of universal trends through comparison with earlier experimental results obtained from ambient-pressure chemical doping. An important example that underscores the significance of our proposed method is the investigation of correlations among the superconducting transition temperature T_c, the superfluid density ρ_sc, and the normal-state conductivity σ at T_c [where, as indicated in Eq. (1), σ is the reciprocal of the resistivity ρ], which is a crucial aspect of the understanding of “why T_c is high” in high-T_c materials, through the application of Homes’s law, which establishes a scaling relation between T_c and the density ρ_sc and resistivity ρ, namely, ρ_sc = AT_c/ρ.1,2 Therefore, the development of a practical method to determine high-pressure resistivity is crucial to facilitate high-pressure quantitative studies of the wide variety of enigmatic quantum materials that are currently emerging.

First, let us consider the ambient-pressure case [see Fig. 1 and Eq. (1)], where the distance between the electrical leads for voltage measurements L, the cross-sectional area S, and the resistance R₀ of an ambient-pressure sample can be directly measured, allowing determination of the ambient-pressure resistivity ρ₀. Given that S = na₀ × mc₀ and L = lb₀, ρ₀ can be represented as follows:$${\rho }_0=R_0\times \frac{n{a}_0\times m{c}_0}{l{b}_0}=R_0(nm/l)(a_0{c}_0/b_0),$$(2)where a₀, b₀, and c₀ are the ambient-pressure lattice parameters of the sample’s unit cell, while n, m, and l are the numbers of the unit cell of the sample in the three crystallographic directions. The value of nm/l can be calculated from the initial size of the loaded sample. As the pressure is applied, a₀, b₀, and c₀ change, but nm/l remains constant in the absence of a crystal structure phase transition. Therefore, the relationship between the resistivity ρ_i and the measured resistance R_i at fixed pressure P_i can be expressed as$${\rho }_i=R_i(nm/l)({a_i{c}_i/b}_i),$$(3)where a_i, b_i, and c_i are the lattice parameters at P_i, which can be obtained from high-pressure X-ray diffraction (XRD) measurements. By employing Eq. (3), we can derive ρ_i at different pressures.

It is evident that there are two sources of error in high-pressure resistivity measurements. One is associated with the lattice parameters a_i, b_i, and c_i, leading to errors typically around 5%.17 The other is associated the numbers of the unit cell of the measured sample in the three crystallographic directions (n, m, and l). The error attributed to these is about 5%–10%. Hence, in the most extreme scenario, the errors are around 10%.

It is important to note that when employing the proposed method, the sample must meet the following criteria:1.The measured sample should be a bulk single crystal, and its ambient-pressure dimensions S and L, lattice parameters a₀, b₀, and c₀, and resistivity, as well as its high-pressure lattice parameters a_i, b_i, and c_i, and the resistance R_i, should be known.2.Samples with cubic, tetragonal, and orthorhombic structures are suitable for obtaining more accurate ρ_i(P_i).3.No pressure-induced structural phase transition should occur within the pressure range investigated.4.When studying the low-temperature properties of materials, the use of low-temperature XRD to measure the lattice parameters at both ambient pressure and high pressure should yield more precise results than room-temperature XRD. This is because the values of a₀, b₀, and c₀, as well as those of a_i, b_i, and c_i, collected at low temperature may differ to some extent from those measured at room temperature.

The proposed method provides a viable way to overcome the technical challenges associated with determining the resistivities of compressed materials. It not only introduces researchers to a new means of obtaining the resistivities of pressurized samples, but also contributes to understand the underlying superconducting mechanism by generating a substantial amount of high-pressure data regarding the correlation of the superconducting transition temperature with superfluid density and resistivity. We have demonstrated the effectiveness of this approach by examining the pressure-induced coevolution of superconductivity with resistivity and superfluid density in bismuth-based cuprate superconductors.3 It is anticipated that this method can also be utilized in high-pressure studies on other quantum materials, thereby providing a basis for further advances in condensed matter physics and materials science.

Acknowledgment. This work was supported by the National Key Research and Development Program of China (Grant Nos. 2022YFA1403900 and 2021YFA1401800) and the NSF of China (Grant Nos. U2032214 and 12104487).