Breakdown of the critical state in the ferromagnetic superconductor EuFe$_2$(As$_{1-x}$P$_x$)$_2$

TL;DR

This paper addresses how ferromagnetic domain textures in EuFe2(As1−xPx)2 influence irreversible vortex dynamics and the superconducting-critical state. Using a nanoscale Hall sensor array, the authors spatially resolve flux penetration and hysteresis across coexisting ferromagnetic and superconducting states, revealing a domain Meissner state with chaotic vortex polaron dynamics just below $T_{FM}$ and a domain vortex state with smoother flux fronts at lower temperatures. The key findings show that the underlying ferromagnetic domain width and structure critically alter flux front propagation and irreversibility, causing departures from the Brandt–Indenbom critical-state model and highlighting a strong material-parameter dependence. The work underscores the potential to control vortex–domain interactions via domain engineering (e.g., strain or in-plane fields) to access new magnetic behaviors in ferromagnetic superconductors with robust $T_c$.

Abstract

There are very few materials in which ferromagnetism coexists with superconductivity due to the destructive effect of the magnetic exchange field on singlet Cooper pairs. The iron-based superconductor EuFe$_2$(As$_{1-x}$P$_x$)$_2$ is therefore unique in exhibiting robust superconductivity with a maximum critical temperature of 25 K and long-range ferromagnetism below $T_\mathrm{FM}\approx19$ K. Here we report a spatially-resolved study of the irreversible magnetisation in this system that reveals a variety of novel behaviours that are strongly linked with underlying ferromagnetic domain structures. In the superconducting-only state, hysteretic magnetisation due to irreversible vortex motion is consistent with typical weak vortex-pinning behaviour. Just below $T_\mathrm{FM}$, very narrowly-spaced stripe domains give rise to highly erratic and irreproducible fluctuations in the irreversible magnetisation that we attribute to the dynamics of multi-vortex clusters stabilised by the formation of vortex polarons. In contrast, at lower temperatures, ferromagnetic domains become wider and saturated with spontaneously nucleated vortices and antivortices, leading to a smoother but unconventional evolution of the irreversible state. This observation suggests that the penetrating flux front is roughened by the presence of the magnetic domains in this regime, presenting a clear departure from standard critical state models. Our findings indicate that the mechanism governing irreversibility is strongly influenced by the precise nature of the underlying ferromagnetic domains, being very sensitive to the specific material parameters of EuFe$_2$(As$_{1-x}$P$_x$)$_2$. We consider the possible microscopic origins of these effects, and suggest further ways to explore novel vortex-domain magnetic behaviours.

Figures (6)

• Figure 1: (a) Schematic phase diagram of EuFe$_2$(As$_{1-x}$P$_x$)$_2$ with approximate positions of samples #1 and #2 indicated. (b) Tetragonal crystal structure of EuFe$_2$(As$_{1-x}$P$_x$)$_2$. The pink arrows indicate the direction of the Eu$^{2+}$ moments in the ferromagnetic state. (c) Schematic of experimental setup with a sample placed on top of a patterned HSA device. The inset shows an expanded, top-down view of the sample on the device, and the nominal positions of the centre (C), intermediate (I) and edge (E) sensors are indicated by the red crosses.
• Figure 2: Hysteresis loops of local magnetisation $\mu_0M_l$ for samples (a) #1 and (b) #2. In each, the row corresponds to a particular sensor position - centre (C), intermediate (I) and edge (E) - beneath the sample, and the column corresponds to the given temperature. The primary and subsequent loops are indicated by the orange and blue curves in (a), and the green and purple curves in (b). The $M_l$ curve at 21.0 K for the edge sensor (E) in panel (a) has been scaled up by a factor of 2 for clarity.
• Figure 3: Penetration field $H_p(T)$ as determined at the edge sensor for sample #1 (orange circles) and intermediate sensor for sample #2 (blue squares). The inset shows two example measurements at 17.0 K of $B_l(H_a)$ in sample #1. The sample is zero-field cooled to the target temperature and the field is increased to a positive (red) or negative (green) maximum value. The penetration field $H_p$ is indicated as the field at which $B_l$ deviates from the Meissner-like response, and the error bars indicate the uncertainty in identifying $H_p$.
• Figure 4: Local magnetic induction $B_l$ measured at different temperatures for (a) edge sensor, sample #1, and (b) intermediate sensor, sample #2. The hollow circles are the measured data and the dashed black lines are fits to the modified Brandt-Indenbom model (equation \ref{['eq:Brandt']}). The data in (a) at 18.5 and 20.5 K, and (b) at 7.0 and 9.0 K, have been offset by $+5$ and $+10$ mT for clarity. The solid green line in (a) is the expected response from the model with $z_s=0$. (c) Critical sheet current density $j_c(T)$ determined from the fits for sample #1.
• Figure 5: Local magnetisation hysteresis loops measured in sample #1 at 18.5 K using the centre sensor. (a)-(c): After ZFC from above $T_c$, the orange (a), blue (b) and pink (c) curves are the first, second and third loops measured in immediate succession. The previous loops to (b) and (c) are shown with increasing transparency. (d), (e): The light blue and red curves are independent repeats with the sample ZFC from above $T_c$ between each loop. The curve from (a) is shown as transparent in (d), and similarly (a) and (d) are shown as transparent in (e). (f): The difference between the response in (a) and in (d), i.e. $\Delta M_l = M_l^{(a)} - M_l^{(d)}$. The data have been expanded by a factor of 2 for clarity. The green and blue arrows indicate the sequence of successive measurements.