The material dependence of superconducting transition temperature ($T_c$) provides useful clues to the mechanism of high-temperature superconductivity. In cuprate high-temperature superconductors, it is known that $T_c$ depends on the number of CuO₂ planes in a unit cell, n, and is generally maximized at $n$ = 3. On the other hand, even within the $n$ = 3, trilayer cuprate family, there exist substantial $T_c$ variations, ranging from 110 K in Bi₂Sr₂Ca₂Cu₃O_10+$\delta$ (Bi2223) to 134 K in HgBa₂Ca₂Cu₃O_8+$\delta$ (Hg1223), which is also known as the highest record of $T_c$ at ambient pressure among any superconductors. In this context, comparative studies on such trilayer cuprates are desired to understand key elements governing their $T_c$s and to explore possible paths to realize superconductivity at higher temperatures.

One powerful experimental probe of electronic structure is angle-resolved photoemission spectroscopy (ARPES). ARPES is particularly effective for the study of trilayer cuprates as it is capable of observing the electronic states of two inequivalent CuO₂ planes [inner plane (IP) and outer plane (OP), see Fig. 1(a)] separately. ARPES is a momentum-resolved technique and thus requires single-crystalline samples. Furthermore, due to high surface sensitivity of the technique, crystals should be cleaved under ultra-high vacuum to expose a fresh surface. Bi2223 is easy to cleave and hence has been intensively studied by ARPES. Previous ARPES works [1] revealed an imbalance between the electronic states of IP and OP; larger superconducting gap for the IP by a factor of $\sim$ 1.5. Thus, it has been conjectured that strong superconductivity at the IP is responsible for high $T_c$ of trilayer cuprates. However, that alone does not account for the $T_c$ variation within the trilayer cuprates and significantly higher $T_c$ of Hg1223, necessitating direct ARPES investigations into Hg1223.

Although it has been challenging to synthesize high-quality single crystals of Hg1223, Mino et al. [2] solved the problem by making partial Re substitution for Hg. They succeeded in growing sizable single crystals of (Hg,Re)Ba₂Ca₂Cu₃O_8+$\delta$ [(Hg,Re)1223)] with $T_c$s exceeding 130 K. Since (Hg,Re)1223 lacks natural cleavage plane unlike Bi2223 and thus cleaved surface is expected to be highly disordered, we utilized tightly focused (10 µm $\times$ 10 µm) beam available at the Bloch beamline of MAX IV to target a flat region out of rough surfaces.

We initiated our ARPES investigation by making a spatial map of spectral intensity [3]. As can be seen in Fig. 1(b), the intensity varies drastically over space, reflecting the disordered cleaved surface. However, after careful examination, we found that it is possible to obtain ARPES spectra with decent intensity and sharpness at the optimal position [Fig. 1(c)]. Fixing the sample position, we quantitatively evaluated the magnitude of spectral gaps at various momentum positions separately for the IP and OP. The obtained gap values are plotted as a function of $d''-''$wave order parameter in Fig. 2. While the proportionality breaks down at larger $d''-''$wave parameters due to pseudogap opening, superconducting properties can be evaluated by extrapolating the linear dependence to $d''-''$wave parameter $=$ 1 and defining gap $\Delta_{0}$. Comparing the values of $\Delta_{0}$ to those of Bi2223 [1] (Fig. 2), $\Delta_{0}$ (IP) was virtually identical. On the other hand, $\Delta_{0}$ (OP) was found to be significantly larger for (Hg,Re)1223. While large $\Delta_{0}$ (IP) has been highlighted as the key characteristic of trilayer cuprates, the present results imply that $\Delta_{0}$ (OP) is a significant factor governing the $T_c$ variations within trilayer cuprates. The outcome of this work [3] may provide useful information to establish a designing principle of higher $T_c$ compounds at ambient pressure.

References

• [1] S. Ideta et al., Phys. Rev. Lett. 104, 227001 (2010).
• [2] Y. Mino et al., J. Phys. Soc. Jpn. 93, 044707 (2024).
• [3] M. Horio et al., Phys. Rev. Lett. 135, 046501 (2025).

Authors

• M. Horio, M. Miyamoto, Y. Mino^a,b, S. Ishidab, B. Thiagarajan^c, C. M. Polley^c, C. H. Lee^b, T. Nishio^a, H. Eisaki^b, and I. Matsuda
• ^aTokyo University of Science
• ^bNational Institute of Advanced Industrial Science and Technology (AIST)
• ^cMAX IV Laboratory