Unusual $H\text{−}T$ phase diagram of $\mathrm{Ce}{\mathrm{Rh}}_2{\mathrm{As}}_2$: The role of staggered noncentrosymmetricity

Superconductivity in a crystalline lattice without inversion is subject to complex spin-orbit-coupling effects, which can lead to mixed-parity pairing and an unusual magnetic response. In this study, the properties of a layered superconductor with alternating Rashba spin-orbit coupling in the stacking of layers, hence (globally) possessing a center of inversion, is analyzed in an applied magnetic field, using a generalized Ginzburg-Landau model. The superconducting order parameter consists of an even- and an odd-parity pairing component which exchange their roles as dominant pairing channel upon increasing the magnetic field. This leads to an unusual kink feature in the upper critical field and a first-order phase transition within the mixed phase. We investigate various signatures of this internal phase transition. The physics we discuss here could explain the recently found $H\text{−}T$ phase diagram of the heavy-fermion superconductor $\mathrm{Ce}{\mathrm{Rh}}_2{\mathrm{As}}_2$.

Figure 1

Superconducting multilayer system motivated by $\mathrm{Ce}{\mathrm{Rh}}_2{\mathrm{As}}_2$ [22]. The lack of local inversion symmetry, illustrated by the alternating orientation of the triangles, gives rise to Rashba effects causing the formation of Cooper pairs of mixed parity within each layer. The minimization of the interlayer coupling causes the dominant order-parameter component to maintain its sign, whereas the other is forced to adopt the alternating sign of the spin-orbit coupling. The A and B order-parameter structures shown are the solutions permitted by symmetry, where the latter phase is favored at high magnetic fields.

Figure 2

$B\text{−}T$ phase diagram obtained from the linearized coupled GL equations. On the vertical axis, $B$ is scaled with $B_{c2}$ and the temperature axis is measured in units of the bare critical temperature of the even component $T_{c,e}$. The outer phase boundaries drawn in solid gray are in excellent agreement with the critical field found for the full numerical treatment in Fig. 5. The dashed lines mark the continuations of the crossing phase boundary lines with the A (blue) and B (green) phase and do not represent physical phase transitions. The red line corresponds to the critical field for the internal phase transition between A and B phases as derived in Sec. 2c.

Figure 3

$B\text{−}T$ phase diagram for the free energy including paramagnetic depairing as given in Eq. (28). The field is measured in units of the zero-temperature paramagnetic critical field for a monolayer $B_p={(1+1/4\pi {\chi }_n)}^{-1/2}$ [24]. The A and B phases are shaded in blue and green, respectively. Note the dashed segment of the critical field of the A phase does not coincide with the critical field of the A-B transition (red line) described by Eq. (31). For this figure, we chose $T_{c,o}=0$ and $Q{\chi }_n=2\times {10}^{-3}$ to guarantee real solutions in the whole temperature range and have an effective Maki parameter ${\alpha }_M\approx 20$.

Figure 4

Numerical minimization of the variational parameters ${\eta }_{e,\infty }$, ${\eta }_{o,\infty }$ (left) as well as the vortex-core size ${\xi }_c$ (right). The magnetic induction on the horizontal axis is normalized by the orbital upper orbital critical field $B_{c2}={\kappa }_0$ and the vortex core size by ${\xi }_{B_{c2}}=1/{\kappa }_0$, its value at $B_{c2}$. The parameters ${\eta }_{e,\infty }$ and ${\eta }_{o,\infty }$ are shown for a Maki parameter ${\alpha }_M=14$, while the core size ${\xi }_c$ is additionally shown for ${\alpha }_M=20$. The initial increase in ${\eta }_{e,\infty }$ is the result of a numerical artifact. In both plots the internal phase transition is clearly visible.

Figure 5

$B\text{−}T$ phase diagram with the solid black line showing the second-order phase transition. The color map displays the amplitude of the superconducting order parameter $\mathrm{\Delta }=\sqrt{{\eta }_{e,\infty }^2+{\eta }_{o,\infty }^2}$, and the dashed black line marks the A-B transition. The inset shows the comparison of the numerically determined critical field for the A-B transition (red points) with the analytical approximation in Eq. (31).

Figure 6

Minimal Maki parameter ${\alpha }_M$ required for phase B to form at $T=0$ as a function of the spin-orbit-coupling strength $\epsilon$ for different interlayer coupling strengths $J$. A strong spin-orbit coupling is beneficial for phase B to form, whereas the interlayer coupling stabilizes phase A. Here we used $a_{0,o}=0.6$ and $\mathcal{Q}=0.15\times 4\pi a_e/b$.

Figure 7

(a) Typical magnetization and (b) susceptibility curves. Both curves show a jump at the internal first-order phase transition. The relative jump in magnetization is on the order of $1\%$ for typical parameter values. (c) Absolute and (d) relative magnetization jump $\mathrm{\Delta }\mathcal{M}$ as a function of Josephson coupling $J$, which in this plot is set equal to the spin-orbit-coupling strength. The blue lines correspond to the approximation in Eq. (52), whereas the red points are results obtained from the numerical solution of the full model. The magnetization jump increases with growing $J=\epsilon$. For the plots (a) and (b) we used $a_{o,0}=0.8$ and $T_{c,o}/T_{c,e}=0.7$, whereas we used the usual parameter values for plots (c) and (d).

Figure 8

Entropy as a function of temperature at $B/B_{c2}=0.187$. The entropy shows a discontinuity in accordance with the first-order nature of the internal transition. Inset: order parameters ${\eta }_{e,\infty }$ and ${\eta }_{o,\infty }$ as a function of temperature at constant field showcasing the swapping of order parameters at $T_c$.