Effects of nucleation at a first-order transition between two superconducting phases: Application to ${\mathrm{CeRh}}_2{\mathrm{As}}_2$

Recent experiments observed a phase transition within the superconducting regime of the heavy-fermion system ${\mathrm{CeRh}}_2{\mathrm{As}}_2$ when subjected to a $c$-axis magnetic field. This phase transition has been interpreted as a parity switching from even to odd parity as the field is increased, and is believed to be of first order. If correct, this scenario provides a unique opportunity to study the phenomenon of local nucleation around inhomogeneities in a superconducting context. Here, we study such nucleation in the form of sharp domain walls emerging on a background of spatially varying material properties and hence, critical magnetic field. To this end, we construct a spatially inhomogeneous Ginzburg-Landau functional and apply numerical minimization to demonstrate the existence of localized domain-wall solutions and study their physical properties. Furthermore, we propose ultrasound attenuation as an experimental bulk probe of domain-wall physics in the system. In particular, we predict the appearance of an absorption peak due to domain-wall percolation upon tuning the magnetic field across the first-order transition line. We argue that the temperature dependence of this peak could help identify the nature of the phase transition.

Figure 1

(a) Schematic phase diagram showing the even-parity low-field A (gray) and odd-parity high-field B (red) superconducting phases separated by a first-order internal phase-transition (dashed) line at $H=H_t$. The inset shows the configuration of the order parameters ${\eta }_{\mu }^j$ in layer $j$ with $\mu =0,\left(1\right)$ for even (odd) parity. Here, $u_A>v_A>0$ and $v_B>u_B>0$. (b) Schematic depiction of nucleation (left) and evolution of (normalized) absorption peak with temperature (right). Deep in the A or B phase the system is devoid of domain walls. At $H\lesssim H_t$ ($H\gtrsim H_t$) the landscape is dominated by the A (B) phase, with bubbles of B (A). Maximum domain-wall proliferation is expected at $H\approx H_t$. The absorption peak is expected to broaden as temperature is lowered ($T_1<T_2<T_3$), which is a distinguishing feature of this first-order phase boundary (see text for details).

Figure 2

Effective magnetic field profiles (top row) and the resulting OP configurations (bottom row) computed via numerical minimization. (a) Hyperbolic tangent inhomogeneity with width $4{\xi }_0$, and (b) a double DW configuration. All $\tilde{H}\left(x\right)$ profiles are symmetric with respect to ${\tilde{H}}_t$ with a maximum deviation of $5\%$. For the numerical calculations, we fix temperature to $T/T_{c,0}=0.5$ and use the parametrization $b_{\mu }=b$, $K_{\mu }=K$, $-a_0/b=K/\left|a_0\right|=1$, $a_1/a_0=0.105$, $J/|a_0|=1$, and $\epsilon /|a_0|=2$. Lengths are expressed in units of ${\xi }_0=\sqrt{K/|a_0|}$ and the OP as ${\mathrm{\Psi }}_{\mu }^j=\sqrt{-b/a_0}{\eta }_{\mu }^j$.

Figure 3

DW potential (measured in units of $b/a_0^2$) for various values of ${\epsilon }_{zz}$ and $\delta$, with fitted parabolas for each data set. For clarity we shifted the data sets vertically in the following way: for ${\epsilon }_{zz}=\{0,-0.02,-0.04,-0.06,-0.08,-0.1\}$ curves are shifted by $(-1)\times \{5,4,3,2,1\}$ units of free energy, respectively.

Figure 4

(a) $c\left(\tilde{H}\right)/A$ averaged over ${\epsilon }_{zz}=-k0.02$ for $k=0,1,\cdots ,5$ (red points) and least square fit of the function $p/\delta$. The dashed line represents a fit of all available data, whereas the solid line ignores the first data point at $\delta =2{\xi }_0$, where the inhomogeneity length scale becomes too short and our leading-order approximation is less accurate. (b) Linear coefficients of the parabolic DW potential from Table 1, averaged over $\delta /{\xi }_0=2,\cdots ,7$ (red points) and a linear fit with zero intercept (black line).