Quasiclassical theory of vortex states in locally non-centrosymmetric superconductors: application to CeRh$_{2}$As$_{2}$

Abstract

CeRh$_{2}$As$_{2}$, a heavy fermion superconductor discovered in 2021, exhibits two distinct superconducting phases under a $c$-axis magnetic field. This unconventional phase diagram has been attributed to the local inversion symmetry breaking at the Ce sites. At low magnetic fields, a conventional even-parity spin-singlet superconducting state is realized, whereas at higher fields, an odd-parity spin-singlet superconducting state, in which the order parameter alternates sign between neighboring Ce layers, becomes stabilized. In this study, we employ a quasiclassical approach to investigate the vortex states of bilayer superconductors with locally broken inversion symmetry. We calculate the local density of states (LDOS) in the vortex lattice state and find that the pairing symmetry of different superconducting states is clearly manifested in the peak structure of LDOS at the vortex core. Since LDOS is experimentally observable, our work provides a pathway for experimental verification of the superconducting parity transition in CeRh$_{2}$As$_{2}$.

Figures (6)

• Figure 1: Schematic figure of the bilayer Rashba model. This model supposes the bilayer system, in which the inversion symmetry is locally broken in each layer. Interlayer hopping $t_\perp$, external magnetic field $H$, and layer-dependent ASOC $\alpha$ upon these layers are illustrated.
• Figure 2: The magnetic field-dependence of (a) order parameters and (b) free energy of superconducting states at $T/T_{\mathrm{c}0}^{e}=0.5$. In (a), the blue and orange lines show the order parameters $\{d_{j,N}\}$ for the BCS ($j=e$) and PDW ($j=o$) states, respectively. The solid, dashed, and dotted lines represents the order parameters for the Landau level indices $N=0,\,6,\,12$, respectively. In (b), the free energies of the BCS and PDW states are shown in the blue and orange line, respectively. The energies are measured in the unit of $\Omega N_0 (T_{\mathrm{c}0}^{e})^2$.
• Figure 3: The phase diagram of the bilayer Rashba model with $\alpha/t_\perp=15.0,\,\alpha_{\mathrm{M}}=4.8$ and $T_{\mathrm{c}0}^{o}/T_{\mathrm{c}0}^{e}=0.9$. The red dotted, dashed, and solid lines represent the parity transition line calculated with the Landau level cutoff $N_{\mathrm{max}}=0,\,6,\,12$. The blue and orange lines indicate the upper critical fields of the BCS and PDW states, respectively.
• Figure 4: The magnetic field-dependence of the vortex core radii for the BCS (blue) and PDW (orange) states. The length is measured in the unit of $R_0=v_{\mathrm{F}}/T_{\mathrm{c}0}^{e}$. The temperature is fixed at $T/T_{\rm c0}^{e}=0.5$ and the minimum value of the magnetic field is $\mu_{\mathrm{B}} H/T_{\mathrm{c}0}^{e}=0.5$. The endpoints of the curves on the high-field side, indicated by diamonds, correspond to the upper critical fields $H_{c2}^{j}$. The red vertical line represents the parity transition field $H^*$.
• Figure 5: The LDOS near the vortex core for the (a) BCS and (b) PDW states. The back and front sides of the figure correspond to the vortex core and the midpoint of the nearest-neighboring vortices, respectively. The distance from the vortex core $r$ is measured in the unit of $a$, which is the distance between nearest-neighboring vortices. The temperature and magnetic field are fixed at $T/T_{\mathrm{c}0}^{e}=0.74,\ \mu_{\mathrm{B}} H/T_{\mathrm{c}0}^{e}=10.25$, which is located near the multicritical point in the $H$-$T$ phase diagram (Fig. \ref{['fig:phase_diagram']}).