In the vortex core, quasiparticles(QPs) are confined, and form quantized levels, called CdGM modes[1]. The confined QPs form quantized energy levels, which are expressed as En = ℏω0(n+1/2), ℏω0 ≡Δ²/EF , where n is an integer, and Δ and EF are the superconducting gap and the Fermi energy, respectively. These quantized levels have a finite width, δE≡ℏ/τ where τ is the scattering time of the QPs in the vortex core. The nature of the core is expressed in terms of ω0τ , which is the ratio of the quantized energy level spacing ΔE≡ℏω0 to the level width δE. For example, high-Tc cuprate superconductors have the large superconducting gap and the small Fermi energy. Thus, large ω0τ is expected, and new phenomena related to this quantum nature may appear.

The parameter ω0τ is closely related to the steady motion of the vortex caused by an external driving current, the flux flow [2]. In the conventional understanding, the direction of the motion can be expressed in terms the Hall angle θ as tanθ=|v_x|/|v_y|=αH/η=ω0τ. Thus, if we measure the flux flow in the longitudinal direction and transverse direction, flux-flow Hall effect, we will find ω0τ. Alternatively, an effective viscous drag coefficient, ηeff =η+α²H/η=πℏnω0τ, also provides information on ω0τ. From an experimental point of view, the latter is much easier. Thus, there have been many studies of the flux flow along this direction [3–9]. Here it should be noted that in flux-flow measurement it is necessary to eliminate the effect of pinning [10]. One way is to measure flux flow by using high frequencies (typically microwaves) and analyze it using a model that includes the pinning effect. Previous studies of the effective viscous drag coefficienthave been performed on various superconductors, such as high-Tc cuprate superconductors and iron-based superconductors by means of the cylindrical cavity perturbation method. Surprisingly, those results show that the inside of the core is uniformly moderately clean (ω₀τ∼1) in various superconductors (for cuprate superconductors; ω₀τ∼0.1−0.3 and for iron-based superconductors; ω₀τ∼1) [3–9], which is contrary to the expectation that the vortex core is clean (ω₀τ≫1). This has not been satisfactorily explained by existing theories. Therefore, it is important to experimentally investigate ω₀τ in a different way, measuring flux-flow Hall effect. Due to technical difficulties, however, no measurements had been made so far. Based on the above background, we recently developed a new method to measure the microwave Hall effect for materials with high conductivity by using cross shaped bimodal cavity [11]. It enables us to measure the flux-flow Hall effect and investigate the nature of the quasiparticle state in the vortex core from a new perspective.

To investigate of the nature of the quasiparticles state in the vortex core, we performed the flux-flow Hall effect measurement for the cuprate superconductors (BSCCO and YBCO) and the iron-based superconductor (FeSe) by using the cross-shaped bimodal cavity. As a result, for the cuprate superconductors, we found that the different magnetic field dependence of the flux-flow Hall angle due to the difference the influence of the pinning, which originated from the difference in the vortex state (liquid vs. solid) [12]. In addition, and more importantly, we obtained a large tangent of the Hall angle (ω₀τ∼3) at low temperatures, which was larger by an order of magnitude than those obtained in the effective viscous drag coefficient measurements. We discussed the origin of the discrepancy between two measurements methods both in terms of the nonlinearity of the viscous drag coefficient and additional dissipation mechanisms, which is not represented in the simple equation of motion but does contribute to the vortex motion. The latter might have something to do with the dissipation caused by the distribution of the electromagnetic field outside the core [13], a new contribution from elastic scattering [14], or other unknown dissipation mechanisms. But it is still an open question and deserves further investigation. On the other hand, as for the iron-based superconductor, we fabricated high quality FeSe single crystal by the vapor transport method [15], which is another candidate material of super-clean core. FeSe shows the similar value as those obtained in the effective viscous drag coefficient measurements [9]. Considering the fact that the magnetic field dependence of the Hall angle is nonlinear, we consider that it is because of the compensation of electron and holes.

References
[1] C. Caroli, et al., Physics Letters 9, 307 (1964).
[2] G. Blatter, et al., Reviews of Modern Physics 66, 1125 (1994).
[3] Y. Tsuchiya, et al., Physical Review B 63, 184517 (2001).
[4] A. Shibata, et al., Physical Review B 68, 060501 (2003).
[5] T. Hanaguri, et al., Physical Review Letters 82, 1273(1999).
[6] A. Maeda, et al., Physica C: Superconductivity 460-462, 1202 (2007).
[7] A. Maeda, et al., Journal of the Physical Society of Japan 76, 094708 (2007).
[8] T. Okada, et al., Physical Review B 86, 064516 (2012).
[9] T. Okada, et al., Journal of the Physical Society of Japan 90, 094704 (2021).
[10] J. I. Gittleman et al., Journal of Applied Physics 39, 2617 (1968).
[11] R. Ogawa, et al., Journal of Applied Physics 129, 015102 (2021).
[12] R. Ogawa, et al., Physical Review B 104, L020503 (2021).
[13] V. G. Kogan and N. Nakagawa, Physical Review B 103, 134511 (2021).
[14] M. Smith, et al., Physical Review B 102, 180507 (2020).
[15] A. E. Böhmer, et al., Physical Review B 94, 1 (2016).