Introduction

The coexistence of ferromagnetism and conventional superconductivity in a single material is extremely rare because the strong ferromagnetic exchange field tends to align the spins of singlet Cooper pairs and destroy them1. In the few cases where it has previously been observed, e.g., in rare earth-based rhodium borides2 and ternary molybdenum chalcogenide Chevrel phases3, coexistence only occurs over a very narrow range (ΔT < 0.5 K) of rather low temperatures (T < 1.5 K) and consists of a spatially modulated magnetic state with a very short period rather than a true ferromagnetic one4,5. However, the recent discovery of several europium-containing iron pnictide superconductors has completely transformed this field6,7,8,9. In particular, it has been shown that isovalent P-doping in EuFe2(As1−xPx)2 leads to the emergence of a dome (Fig. 1a) of high-temperature superconductivity (\({T}_{{{{\rm{c}}}}}(\max )\simeq 25\) K at x ≃ 0.2) associated with the Fe-3d electrons whose critical temperature can significantly exceed the ferromagnetic ordering temperature of the Eu2+ spins, TFM ≃ 19 K10,11. Phosphorus doping also causes the Eu2+ magnetic moments to cant out of their initial antiferromagnetic alignment in the ab plane at x = 0, tilting them very close to the crystalline c-axis at x ≃ 0.2, and resulting in a large net out-of-plane ferromagnetic moment12,13,14,15,16. Remarkably, due to the spatial separation of the superconducting electrons in the FeAs layers and the Eu2+ magnetic sub-lattice, as well as an unusually weak exchange interaction, these superconducting and ferromagnetic phases can coexist over a very broad temperature range (ΔT ≤ 19 K)15,17,18. In samples close to optimal doping, this offers a unique opportunity to study the influence of uniaxial ferromagnetic order on the superconducting state as it emerges below TFM ≃ 19 K.

Fig. 1: Magnetic characterisation of EuFe2(As1−xPx)2 samples.
figure 1

a Schematic phase diagram of EuFe2(As1−xPx)2 with approximate positions of samples S1, S2, and SD indicated, after refs. 17,21. b Crystal structure of EuFe2(As1−xPx)2 with direction of Eu moments indicated by green arrows for x ≈ 0.2. c Zero-field-cooled (ZFC) and field-cooled (FC) measurements of magnetisation for S1 (solid blue and dash-dot green) and SD (dashed orange), in applied magnetic fields of 10 Oe and 5 Oe respectively, oriented parallel to the c-axis. Arrows indicate the superconducting critical temperature Tc and ferromagnetic ordering temperature TFM. d ZFC (solid red) and FC (dash-dot purple) measurements of S2 measured under the same conditions as S1 in (c). e Temperature dependence of the susceptibility \(\chi =\frac{dM}{dH}{| }_{M = 0}\) as determined from MHLs for S1 (blue circles) and S2 (green diamonds). The vertical, dotted grey line indicates TFM. Inset: example MHLs from S1 at various fixed temperatures. f Coercive field Hc (M(Hc) = 0) for S1 (blue circles) and S2 (green diamonds), as determined from MHLs. Inset shows expanded view of data in the range 15–23 K to highlight peak in Hc(T) for S1 near TFM.

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In a seminal, low-temperature magnetic force microscopy (MFM) imaging study on EuFe2(As0.79P0.21)2, Stolyarov et al.19 revealed the striking, cooperative nature of superconductivity and ferromagnetism in this material. As the temperature was lowered below TFM, their MFM images resolved a ferromagnetic stripe domain structure emerging in the Domain Meissner State (DMS). In this state, the natural domain width was strongly reduced due to the presence of Meissner screening currents flowing near domain walls. At lower temperatures, a first-order transition to the Domain Vortex State (DVS) was identified, in which dense arrays of vortices and antivortices were observed to spontaneously nucleate within the ferromagnetic domains. Moreover, the concurrent suppression of Meissner screening currents in this state also led to the abrupt growth of domain widths. The presence of the DMS and DVS as bulk phases in EuFe2(As0.8P0.2)2 was later confirmed by small-angle neutron scattering measurements that also revealed the suppression of the two phases at high magnetic fields20. In contrast, a follow-up MFM study on a sample with composition x = 0.25 and Tc ≈ 18.4 K  < TFM revealed a substantially different local magnetic structure that was attributed to the domination of ferromagnetism over superconductivity for this composition21.

Previous MFM works have so far focused on elucidating the subtle ways in which the two electronically ordered phases interact in the absence of an applied magnetic field19,21. Thus, the influence of the emerging ferromagnetic order on the dynamics of superconducting vortices in an applied magnetic field remains completely unexplored. A comprehensive understanding of this could underpin important applications in high-performance superconducting tapes and/or wires for operation at very high magnetic fields. Here we combine systematic temperature-dependent magnetisation and magnetic relaxation measurements with nanowire MFM imaging experiments. We reveal the vortex dynamics in EuFe2(As1−xPx)2 crystals in two different doping regimes; the first with x ≈ 0.21 close to optimal doping with Tc > TFM and the second with x ≈ 0.28 in the overdoped regime with TFM > Tc. Remarkably, we find that strong magnetic irreversibility only appears in our samples once both ordering phenomena are present, i.e., T < Tc and T < TFM, clearly highlighting the cooperative nature of the interaction between them.

Magnetic relaxation measurements in the DMS phase reveal a pronounced peak in the vortex creep activation energy, more than a factor of two larger than the background value at lower temperatures. We attribute this observation to the formation of a vortex polaron, when the widths of up and down domains are locally perturbed by the presence of a nearby superconducting vortex. MFM images provide further evidence for an attractive vortex-vortex interaction due to the distortion of the domain structure by vortex polarons, and we also show how penetrating vortices and antivortices lead to shearing and radical restructuring of the underlying ferromagnetic stripe domains. Note that the vortex field in magnetic superconductors induces a polarisation of the localised magnetic moments, resulting in some shrinkage of the vortex diameter22. When in motion, such vortices polarise the surrounding moments non-uniformly and re-polarise them; these vortices are termed “polaron-like” vortices23,24. In our case, the vortex polaron is somewhat different, manifesting as a localised distortion of the domain structure. Additionally, the interaction between the vortex and domain magnetic fields leads to a highly unusual short-range attractive vortex-vortex potential and can even stabilise multi-quantum vortices that would not normally exist. Vortex-vortex attraction has been predicted in hybrid superconductor-ferromagnet superlattices25, particularly when the magnetic system exhibits strong spatial dispersion. In some sense, our short-period domain structure acts in a similar way, with the scale of magnetic non-locality corresponding to the domain width. Other exotic cooperative phenomena are also predicted to appear in superconductor-chiral magnet heterostructures, where the mutual interaction of the two states can give rise to a rich variety of mesoscopic states including clusters and stripes26.

As the temperature is lowered into the DVS phase, we see a rapid increase in the magnetic remanence and coercivity, which we link to a temperature-dependent critical current density governed by giant flux creep over a thermal activation barrier of  ~240 K. This observation is reminiscent of earlier ac susceptibility studies of vortex-antivortex dynamics in EuFe2(As1−xPx)2, where several thermally activated vortex/antivortex hopping mechanisms were identified as being important27,28. However, remagnetisation of the stripe domain structure in the DVS phase explicitly requires vortex-antivortex annihilation at domain walls, and we associate the observed thermally activated behaviour with the existence of a Bean-Livingston barrier for this process29.

Our results have important implications for the development of high-current superconducting tapes and wires, which are pivotal in applications such as magnetic resonance imaging, maglev, and fusion reactors. Although iron-based superconductors generally exhibit lower critical temperatures when compared to the cuprate family of superconductors, their lower anisotropy and better chemical stability present attractive properties that are well suited to industrial-scale fabrication of high-current superconducting tapes and wires30. A key engineering challenge is the realisation of materials that can sustain high critical current densities while subject to very high magnetic fields31, an attribute that is strongly dependent on the material’s vortex pinning properties. The high-current performance of a superconductor can typically be enhanced through a wide variety of extrinsic modifications, e.g., the introduction of non-magnetic32 or magnetic33 pinning centres, through high-energy particle irradiation34,35 or via the magnetic textures in superconductor-ferromagnet multilayers36. Our findings indicate that by careful control of the magnetic domain structure in ferromagnetic superconductors, it should be possible to exploit the intrinsic phenomena we observe to significantly enhance vortex pinning over a wide range of temperatures and achieve far superior high magnetic field performance.

Results

Magnetic characterisation

Magnetisation data for three single crystals of EuFe2(As1−xPx)2 are shown in Fig. 1c, d. Samples S1 and SD both have a doping level close to x ≈ 0.21 and exhibit identical superconducting onset and ferromagnetic ordering temperatures of Tc ≈ 24.5 K and TFM ≈ 19.3 K, respectively, as shown in the zero-field-cooled (ZFC) curves. In contrast, sample S2 with a doping level of x ≈ 0.28 exhibits the same magnetic ordering temperature of TFM ≈ 19.3 K, but has a much lower superconducting Tc ≈ 12.5 K. The field-cooled (FC) curves of S1 and S2 are, however, very similar, exhibiting a crossover from paramagnetic to ferromagnetic behaviour at TFM. This is clearer in measurements with larger applied fields (Supplementary Fig. 1). Given the identified values of Tc and TFM, the approximate locations of these samples are indicated on the phase diagram shown in Fig. 1a, where S1 and SD correspond to ferromagnetic superconductors (Tc > TFM) and S2 represents a superconducting ferromagnet (Tc < TFM).

To characterise the magnetic properties of our samples, families of magnetic hysteresis loops (MHLs) were measured at various fixed temperatures for S1 (Tc > TFM) and S2 (Tc < TFM), examples of which are shown for S1 in the inset of Fig. 1e. At the 5 K base temperature, the MHLs of the two samples exhibit features of both superconductivity and ferromagnetism: superconductivity leads to the opening of the hysteresis loop (magnetic irreversibility) and an initial increase in the magnitude of the magnetisation upon reversal of the sweep direction at the maximum field excursions, while ferromagnetism is reflected in the steep, linear M(H) behaviour in a window of applied field centred around H = 0, the width of which increases as the temperature is reduced below TFM. Above Tc and TFM the MHLs of all samples become fully reversible and exhibit a weak, paramagnetic response.

Figure 1e illustrates the behaviour of the ferromagnetic contribution to the MHLs as a function of temperature, where \(\chi =\frac{dM}{dH}{| }_{M = 0}\) is the local slope where the curves pass through M = 0. S1 and S2 both display very similar behaviours, showing a rapid increase in χ as the temperature is reduced from 25 K, which saturates in a cusp at the magnetic ordering temperature TFM ≈ 19.3 K, and exhibits only very weak changes at lower temperatures. The temperature at these cusps is very close to that of the features associated with the onset of magnetic order in the ZFC magnetisation curves shown in Fig. 1c, d, and in Supplementary Fig. 1.

The key differences between the two samples become evident in the intermediate temperature regime between 10 K and 20 K. S1 starts to exhibit strong irreversibility below TFM ≈ 19.3 K while S2 remains almost completely reversible until T < Tc = 12.5 K. Evidently the requirement for strongly irreversible behaviour is that both forms of electronic ordering be present. This is illustrated in the plots of temperature-dependent coercive field shown in Fig. 1f. Note that the extremely small coercivity of S2 in the regime Tc < T < TFM indicates that the material is an exceptionally soft ferromagnet with very weak domain wall pinning. Similarly, soft ferromagnetism has also been observed in the end-member of the series, EuFe2P237. A more detailed comparison of the reversible component of MHLs for S1 and S2 in this intermediate regime (Supplementary Note 2), suggests that S1 is also a soft ferromagnet with very similar properties to S2, and we therefore deduce that any irreversibility Mirr must be due to the superconductivity. Moreover, the inset to Fig. 1f shows an expanded view of the coercivity for S1 and S2 in the region around TFM. We see that S1 exhibits a very pronounced coercive field peak in the DMS close to TFM, something that we attribute to the formation of vortex polarons.

Magnetic relaxation measurements

To further explore the influence of the underlying ferromagnetism on the superconducting state, particularly in the region of the DMS, we performed magnetic relaxation measurements38 on sample S1 for T < Tc and for various final measurement fields, Hf, after magnetic saturation at H = +10 kOe. The time dependence of the irreversible magnetisation, Mirr(T) was observed to decay logarithmically (Supplementary Note 4), from which the normalised relaxation rate, \(S(T)=-d\ln {M}_{{{{\rm{irr}}}}}/d\ln t=d\ln J/d\ln t\), was extracted. In the context of the Anderson-Kim model of flux creep39, where the creep activation energy, U0, is linearly reduced by the presence of a bulk current density, the critical current density is expressed by

$${J}_{c}(T)={J}_{c0}[1-(T/{U}_{0})\ln (t/{t}_{{{{\rm{eff}}}}})],$$
(1)

where Jc0 is the temperature-dependent critical current density in the absence of flux creep and teff is the effective hopping attempt time. Correspondingly, the normalised relaxation rate, achieved by the logarithmic derivative of equation (1), is

$$S(T)=-T/[{U}_{0}-T\ln (t/{t}_{{{{\rm{eff}}}}})\left.\right)].$$
(2)

At low temperatures, the activation energy is well approximated by U0 ≈ T/∣S∣, and it is useful to determine an effective activation energy38 Ueff = T/∣S(T, H)∣ to understand the qualitative evolution of the creep activation energy with both temperature and magnetic field.

This is shown for S1 in Fig. 2a and exhibits two distinct regimes. For 17 K < T < TFM, the activation energy shows a very pronounced peak centred on 19.5 K ~TFM) with a magnitude more than twice as large as the extrapolated low-temperature background at the lowest measurement field. Moreover, this peak rapidly reduces in height as the magnitude of Hf is increased until eventually collapsing towards zero for  −Hf ≥ 2 kOe. In stark contrast, for T < 17 K, Ueff(T) shows a very weak temperature dependence with almost no field dependence up to  −Hf = 2 kOe, reducing from approximately 600 K to 300 K as the temperature is lowered. The very different behaviour in these two regimes is further emphasised in the plot of Ueff(H) in Fig. 2b at two characteristic temperatures, and we note that the crossover between the two at T ≈ 17 K is close to the expected transition between the DMS and DVS phases19.

Fig. 2: Magnetic relaxation and critical current density.
figure 2

a Effective vortex creep activation energy Ueff(T) at various final measurement fields Hf for S1. For Hf close to zero, Ueff exhibits a pronounced peak centred on 19.5 K, which decreases rapidly as the magnitude of Hf increases, before eventually collapsing at ∣Hf∣ ≥ 2 kOe. Error bars indicate the uncertainty in the linear fit to the relaxation data, i.e., the slope of \(\ln {M}_{{{{\rm{irr}}}}}(T)\) vs. \(\ln t\). Inset shows normalised relaxation rate S(T) for Hf = −76 Oe. Data for T ≤ 17 K (purple downward triangles) are fitted to equations (5) simultaneously with Jc(T) (solid red line) over the same range of T. Data above 17 K are not fitted (solid grey downward triangles). b Effective vortex creep activation energy for S1 as a function of Hf from (a), at 5.0 and 19.5 K. Ueff shows a rapid suppression with Hf at 19.5 K, while the dependence is only very weak at the lower temperature. c Critical current density Jc(T) for S1 in the limit of zero applied magnetic field as determined from MHLs. The solid red line is a fit to equation (4), simultaneously with S(T), for data with T ≤ 17 K (blue circles), while data above are excluded from the fit (solid grey circles). From the fit, we derive a value of Jc(0) ≈ 173 kA/cm2. The dashed black line is the temperature dependence of Jc in the absence of flux creep.

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Phenomenological analysis of Abrikosov vortices in ferromagnetic stripe domains

The presence of a short-period domain structure significantly modifies the properties and mutual interactions of Abrikosov vortices. The origin of the DMS itself lies in the competition between electromagnetic energy, driven by Meissner screening, and the energy associated with magnetic domain walls29,40. Since the energy of superconducting vortices is also governed by Meissner screening, a strong interaction between vortices and the magnetic domain structure can be anticipated. A vortex located within one of the domains will, within a characteristic radius ∣r∣ ≤ λ, expand neighbouring domains aligned with the orientation of its magnetic moment, and contract those with the opposite orientation, resulting in a deformation of the local domain structure. This interaction leads to the formation of a state we refer to as a vortex polaron.

To provide an insight into the energetics of this scenario, we employ a phenomenological analysis of the free energy of a single Abrikosov vortex sitting within a stripe domain structure with width l and magnetisation oriented along the z direction (see Supplementary Note 5 for full details of the calculation). We find that the energy of a vortex polaron is lower than a standard Abrikosov vortex by an amount

$$\Delta E=-\frac{{\Phi }_{0}^{2}}{64\pi \lambda l}$$
(3)

where Φ0 is the magnetic flux quantum. Thus, there is a substantial lowering in energy if the domain width is smaller than the size of the vortex, as characterised by the penetration depth, λ.

The motion of a vortex polaron involves moment reversal near domain walls, resulting in an effective vortex pinning potential and strikingly modified vortex dynamics. Furthermore, the interaction between two vortex polarons can be dramatically modified, and the usual repulsive inter-vortex interaction can give way to vortex attraction at short distances smaller than λ (but larger than the domain width), favouring vortex clustering.

Giant flux creep in the domain vortex state

The inset to Fig. 2a shows S(T) at Hf = −76 Oe, which is much larger than previously observed in BaFe2(As0.68P0.32)2 single crystals41 but similar in magnitude to other electron-42 and hole-doped35 iron-based superconductors, as well as the giant flux creep regime of high-Tc cuprate superconductors43. The quasi-exponential shape of the critical current density, Jc(T, H = 0), shown in Fig. 2c, is also reminiscent of that seen in the cuprates43,44 and iron-based superconductors34,35, suggesting that giant or collective flux creep is important in this material. To describe the behaviour of both S(T) and Jc(T), we base our analysis on a phenomenological model used by Thompson et al.43 to describe thermally activated flux motion in cuprates, which has also been utilised effectively for similar analysis in iron-based superconductors34. The authors give the following expressions for Jc and S:

$${J}_{c}(T)={J}_{c0}/{[1+(\mu T/{U}_{0})\ln (t/{t}_{{{{\rm{eff}}}}})]}^{1/\mu },$$
(4)
$$S(T)=-T/[{U}_{0}+\mu T\ln (t/{t}_{{{{\rm{eff}}}}})],$$
(5)

where μ is a characteristic, glassy exponent that expresses how U0 depends on the current density. The temperature dependence of Jc0 and U0 are assumed to take the forms

$${J}_{c0}(T)={J}_{c00}{[1-{(T/{T}_{{{{\rm{c}}}}})}^{2}]}^{{n}_{1}}$$
(6)

and

$${U}_{0}(T)={U}_{00}{[1-{(T/{T}_{{{{\rm{c}}}}})}^{2}]}^{{n}_{2}},$$
(7)

with Jc00 = Jc0(0) and U00 = U0(0). Following Thompson et al.43, the exponent n1 is set to be 3/2, such that Jc0(T) ~ Jdepairing(T). However, we allow n2 to be a free fit parameter that reflects the unusual magnetic nature of the creep potential barrier in our samples.

We simultaneously fit Jc(T, H = 0) and \(S\left.\right(T,{H}_{f}=-76\) Oe) for T ≤ 17 K to equations (4) and (5) respectively and the results are shown by the solid red lines in the inset of Fig. 2a, c For the exponent describing the temperature evolution of the activation energy (equation (7)), we determine a value n2 ≈ 3, revealing that U0(T) is much more rapidly suppressed at high temperatures than when n = 3/2 as assumed by Thompson et al.43. This may indicate that the relevant temperature scale of the flux creep mechanism is not Tc, but the lower temperature of TFM. We also determine U00 ≈ 235 K, in very good agreement with the low temperature value of Ueff, and μ ≈ 1.3, which is suggestive of vortex-glass45 or collective-pinning46 scenarios.

Magnetic imaging

To directly visualise how the ferromagnetic state influences the magnetic irreversibility in EuFe2(As1−xPx)2, we have undertaken a magnetic force microscopy (MFM) imaging study at a range of different temperatures and magnetic field histories. These measurements were performed using sample SD, which has nominally the same phosphorus composition as S1 and identical values of Tc and TFM (Fig. 1c). All images were captured in a plane parallel to the a-b surface of the platelet-shaped sample with the field applied along the c-axis direction. Figure 3 shows a series of these MFM images in very low applied magnetic field at several fixed temperatures, where the magnetic contrast is manifest as a shift in the resonant frequency of a nanowire with a ferromagnetic tip (see “Methods”).

Fig. 3: Magnetic texture of sample SD in very low and zero applied field.
figure 3

a 2 × 2 μm2 μm scan at T = 21.1 K, after cooling from above Tc in applied magnetic field H ≈ 44 Oe, showing Abrikosov vortices nucleated on a predominantly flat magnetic background. b Line scans of a vortex in two orthogonal directions, as indicated by the lines (i) [green squares] and (ii) [pink circles] in (a). The solid blue line is a fit to a modified variational Clem model. c Series of 2 × 2 μm2 scans at decreasing temperatures in approximately zero applied field. d Average domain period as a function of temperature as determined from MFM measurements in (c). Error bars indicate the range (minimum to maximum) of domain widths observed.

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In Fig. 3a (T = 21.1 K), the sample is in the purely superconducting state (Tc > T > TFM), reached by cooling the sample from above Tc in a small residual applied field (H ≈ 44 Oe). The residual field causes nucleation of Abrikosov flux vortices on a predominantly flat magnetic background. The lines (i) and (ii) indicate line scans across a typical vortex, the profiles of which are shown in Fig. 3b. The MFM system is configured such that the magnetic contrast (Δf) is proportional to (∂Bx/∂x + ∂By/∂y) in the x-y plane at some height z above the sample. As a result, the measured profiles of the vortices, which are proportional to ∂Bz/∂z, appear significantly narrower than if one measured the magnetic field directly. In order to model this shape, we employ the variational Clem model47 while taking into account the specifics of the contrast mechanism (see Supplementary Note 6 for further details). The solid blue line in Fig. 4b is a fit of this model to both profiles (i) and (ii), resulting in a very good description of the vortex shape and yielding a scan height z = 83 nm, consistent with known measurement parameters.

Fig. 4: Evolution of the domain Meissner state with applied field.
figure 4

a, b Sample SD, series of 2 × 2 μm2 MFM images captured at T ≈ 19.8 K. a Starting from the ZFC state and increasing the field up to 10 kOe (i.e., the initial branch), and b decreasing from 10 kOe, through zero field, and reversing the field to −10 kOe (i.e., the upper branch). In (b), H = 50 Oe, three line scans (i) [brown], (ii) [orange] and (iii) [dark blue] are indicated (shown in d and e). c Average domain period as a function of applied magnetic field H from (a and b) (arrows indicate direction of change of H in the two branches). The shaded areas indicate the approximate magnetic field regions of the various magnetic domain structures observed at this temperature. Error bars indicate the range of domain widths observed. d Data from line scans (i) [brown squares] and (iii) [dark blue circles] indicate a closely spaced vortex pair and a single vortex (i.e., Δf > 0); the former (i) clearly has a larger amplitude and width compared to the latter (iii). e Data from line scans (i) and (ii) [orange circles] showing similarly sized vortex objects (amplitude and dimension) but with opposite sense (up vs. down). The solid blue and green lines are fits to (i) and (ii), respectively, using the bound vortex pair profile, with the dashed red and pink lines indicating the profiles of each vortex in the pair for profile (i).

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In Fig. 3c, the sample is cooled again from above Tc now in nominally zero applied field, minimising vortex nucleation. As the sample is cooled below TFM, a fine stripe domain structure emerges (T = 20.0 K), characteristic of the DMS phase. Light and dark domains have opposite directions of the magnetisation, \(\overrightarrow{M}\), oriented approximately out of (up) and into (down) the sample surface. Cooling further to T = 19.0 K, but remaining above the transition to the DVS, a few spontaneous vortex-antivortex pairs can be seen nucleating around Y-shaped defects in the magnetic domain structure. Vortices(antivortices) appear as much brighter(darker) regions and sit within the up(down) domains. Further cooling to T = 18.0 K sees the partial appearance of the DVS, characterised by domains which are much wider (Fig. 3b) and exhibit much stronger magnetic contrast, owing to their high density of spontaneously nucleated vortices and antivortices, and the suppressed Meissner screening currents. The sample, however, does not undergo a uniform transition from the DMS to the DVS as the temperature is reduced due to the first-order nature of the transition19,29, with the DVS component continuing to grow in both domain width and fractional occupation as the temperature is lowered to T = 5.0 K. The evolution of the domain size with temperature is shown in Fig. 4d, where the error bars indicate the range in measured domain period of both the DMS and DVS. The observed zero-field evolution of the DMS and DVS is in good qualitative agreement with previous reports19,21.

Field evolution of the domain Meissner state

Figure 4a, b show a series of MFM images captured in the DMS phase at T = 19.8 K for a sequence of magnetic fields chosen to recreate the field history of the MHLs of S1 and S2. The sample was first cooled at H = 0 to the target temperature, after which the field was increased up to a maximum of H = 10 kOe (Fig. 4a) building the initial branch of the MHL. After ferromagnetic saturation, the field was decreased to zero and then reversed to negative saturation at H = −10 kOe, creating the upper branch of the MHL (Fig. 4b).

The initial branch begins in the pure DMS state, but only a very modest increase of field to H = 70 Oe leads to a radical change; a line of up vortices penetrating from the sample edge has buckled the stripe domain structure leading to a pronounced cusp-like distortion associated with a line of Y-shaped domain defects. As the field is increased further (H = 100 and 250 Oe) this process leads to a complete rearrangement of the domain structure until above 750 Oe the stripes start to align close to the vertical direction, driven by a small in-plane component of the applied field due to an unintentional tilt of the surface normal with respect to the field direction. At the same time, the width of the up domains increases with H while the width of the down domains decreases, leading to an overall increase in period, which is well understood in the context of stripe domain structures in a ferromagnet with uniaxial magnetic anisotropy48. At high fields, the sample has become penetrated by so much light up flux that it is no longer possible to resolve individual vortices, and dark down stripes begin to break up into shorter segments and, ultimately, isolated bubbles. Eventually, at H > 4 kOe, the ferromagnet becomes saturated and the domain structure is no longer visible. Any residual contrast in the saturated image is believed to be linked to the surface topography of our samples, with additional contrast arising from the stray field at steps and edges on the sample surface.

Upon decreasing the field, following the upper MHL branch (Fig. 4b), the dark down domains reappear via the penetration of magnetic bubbles, presumably containing integer numbers of flux quanta. Further reduction of the field sees these bubbles join up into chains and then fuse into continuous dark stripes. Around zero applied field (H = 50, −50 and −100 Oe), the very short period DMS state is restored, decorated by small numbers of uncorrelated vortices and antivortices. Up vortices are confined to up domains and vice versa, and all flux structures have slightly elliptical shapes, predominantly due to the superposition of the domain magnetic fields on the vortex fields.

Nearly all of the light up vortices in the H = 50 Oe image all have the same peak amplitudes and sizes, and are almost certainly single flux quantum vortices. However, one light object in the top-left corner and two dark objects near the centre of the frame have significantly higher amplitudes and are considerably longer. Additionally, the stripe domain structure appears to become distorted in the vicinity of all flux objects, particularly so for the larger objects. Line profiles of the larger up object, one larger down object and one smaller up object are indicated by the dotted lines (i), (ii) and (iii) respectively, with the profiles shown in Fig. 4d, e. The line scans are taken parallel to the direction of the domains, where the contribution of the domain to the profile is approximately constant with position. The profiles of the two up objects (Fig. 4d) display clearly distinct characteristics, with (i) being both longer and larger in amplitude than (iii). In contrast, the profiles (i) and (ii), shown in Fig. 4e, display very similar amplitudes and sizes but with opposite polarity, i.e., up vs. down.

The profiles clearly suggest that the objects in (i) and (ii) are not single flux quantum vortices, as profile (iii) is, but they are pairs of vortices (antivortices) in very close proximity, arranged along the direction of the domain. Using the same model as described for Fig. 3b, but adapted for pairs of vortices (see Supplementary Note 6), we fit profiles (i) and (ii) with the results shown as the solid blue and green lines in Fig. 4e. Curves have been generated assuming the same fitted scan height of z = 49 nm in all cases, and a vortex separation w = 98 nm for (i) and w = 96 nm for (ii), with the profiles of the individual vortices comprising profile (i) shown as the dashed red and pink lines. Critically, the separation between the vortices is much less than the estimated penetration depth λ(T = 19.8 K) = 462 nm at this temperature. This pairing of vortices in such close proximity is a key prediction of our vortex polaron theory, with the distortion of the domain structure in the vicinity of the vortex objects being another important aspect.

Following further reduction of the field (Fig. 4e, H = −50 and −100 Oe), we first observe a single chain of discrete dark antivortices, occupying the same down domain, which then fuses into a structureless stripe with a very large peak amplitude. Again, we believe that this stripe is composed of very closely spaced antivortices held together by the short-range attraction due to the vortex polaron effect. As the field is decreased further towards negative saturation, the behaviour mirrors that close to positive saturation, except now the light up regions become minority domains, shrinking in size and breaking into bubbles.

Figure 4c shows the average domain period determined from this sequence of MFM images, and the approximate regions of the different domain structures are indicated by the shaded areas. In contrast to the magnetisation of sample S1 (e.g., Hc(T), Fig. 1f), the domain period shows almost perfect reversibility between the initial and upper branches, indicating the weakness of any domain wall pinning. This further emphasises the fact that the global magnetic irreversibility is driven by superconducting flux dynamics within the ferromagnetic domain structure and not by any intrinsic irreversibility on the part of the ferromagnetism.

Field evolution of the domain vortex state

Figure 5 shows a similar series of MFM images to Fig. 4, but now captured deep in the DVS phase at T ≈ 4.3 K. While the evolution of the domain structure with field is qualitatively similar, there are several important differences. The domain width l in the initial ZFC state is now much larger and closer to the intrinsic width of the ferromagnetic domain structure due to the suppression of Meissner screening29. Furthermore, these domains are now saturated with a very high density of spontaneously nucleated vortices and antivortices such that we can no longer resolve discrete vortices in our measurements. Additionally, any field-induced vortices penetrating the sample will experience a magnetic landscape that is markedly different from that in the DMS phase. We also expect the effects of vortex polaron formation to be substantially weakened in the DVS. This is in part due to the increase of the ratio l/λ, a key measure of the strength of the effect, but also due to a drastic reduction in the Meissner screening currents in the DVS, which are central to the ability of the vortex polaron to reduce the electromagnetic energy of the domain structure.

Fig. 5: Evolution of the domain vortex state with applied field.
figure 5

a, b Sample SD, series of 3 × 3 μm2 MFM images captured at T ≈ 4.3 K, following the same ZFC protocol as in Fig. 4a, b. a After ZFC and increasing the field to 10 kOe, b decreasing the field from 10 kOe to zero and reversing the field to −10 kOe. c Average domain period as a function of applied magnetic field H from (a and b), with arrows indicating the direction of changing H for the two branches. Error bars indicate the range of domain widths observed.

Full size image

When the field is increased from the ZFC state (Fig. 5a), the up and down domains initially widen and shrink as observed in higher temperature measurements, but the domain structure now survives up to a much higher temperature-dependent saturation field of about ∣H∣ = 8 kOe when the last few magnetic bubbles disappear. Unique to this series of images is the observation of composite domain states of stripes containing chains of bubbles (cf., at H = 4.0 kOe in Fig. 5a and at H = 6 kOe in Fig. 5b). While such structures are generally metastable, they are often observed in ferromagnets with strong uniaxial anisotropy subject to specific magnetic histories48. In addition, we see a pronounced disordering of the domain structure as the applied field is reduced close to zero, with the proliferation of loops linked to Y-shaped defects. There is also noticeable hysteresis in the data; the domain period never recovers its initial small ZFC value after the first magnetisation leg (Fig. 5c), and magnetic bubbles survive to much higher fields when the applied field magnitude is increasing compared to when it is decreasing. Finally, we note that the mechanism by which the sample becomes remagnetised now explicitly involves the penetration of one sign of flux from the sample edges combined with vortex-antivortex annihilation at domain walls. This latter process involves thermal activation over a Bean-Livingston barrier that we attribute as being responsible for the flux creep behaviour observed in magnetisation and magnetic relaxation measurements at lower temperatures.

Discussion

The magnetometry data for samples S1 and S2 strongly indicate that irreversible vortex dynamics within the ferromagnetic domain structure is the driving force behind magnetic irreversibility in EuFe2(As1−xPx)2. First, the purely superconducting state of sample S1 (TFM < T < Tc) shows very weak magnetic irreversibility, reflecting the absence of Coulomb scattering following isovalent P-doping, as seen in other iron-based superconductors49. Second, the purely ferromagnetic state of sample S2 is highly reversible, and comparison of the reversible magnetisation Mrev of S1 and S2 establishes the very similar nature of the ferromagnetic ordering at the two different phosphorus compositions. Additionally, the very narrow domain width, apparent in MFM images of sample SD, indicates a very small domain wall energy σw, more than an order of magnitude smaller than, e.g., yttrium-iron garnet50, that gives rise to very weak domain wall pinning. This is further emphasised by the high degree of reversibility of the domain size during the MHL shown in Fig. 4. Due to the nominally identical characteristics of samples SD and S1, we argue that this must be true for sample S1 as well. Therefore, the rapid increase of magnetic irreversibility when T < Tc & TFM is clearly a cooperative effect of both superconductivity and ferromagnetism, attributable to the magnetic control of the vortex dynamics.

The two distinct regimes of the effective vortex pinning potential Ueff(T) in S1, as well as the peak in Hc(T) near TFM, clearly indicate a fundamental change in the nature of the magnetically driven vortex pinning as the sample transitions from the DMS at higher temperatures to the DVS at lower temperatures (T ≲ 17 K). In the high-temperature regime, the rapid suppression of Ueff(H) as the sample is driven to ferromagnetic saturation is an unambiguous signature of the magnetic origin of this behaviour. In the DMS, the magnetisation of the ferromagnetic domains is screened by circulating Meissner currents, causing the domain width to shrink below its intrinsic size29,40. Upon transitioning to the DVS, the screening currents collapse in favour of the spontaneous nucleation of vortices and antivortices, and the ferromagnetic domains widen back towards their intrinsic values. The behaviour of Ueff(T) in the two regimes is therefore intimately linked to the spontaneous nucleation of vortices and antivortices as well as the underlying ferromagnetic domain size.

The magnetic irreversibility and magnetic relaxation in S1 in the DMS regime can be understood as being dominated by vortex polaron dynamics. In zero applied field, the domain width is at its narrowest, and the vortex polaron energy is at its lowest when compared with a free Abrikosov vortex. The application of, e.g., a positive magnetic field will lead to the penetration of vortices along up domains with parallel magnetisation. These domains will widen as the field increases (Fig. 4), leading to a rapid reduction of the energy of vortex polarons associated with them. Therefore, as the sample is driven towards magnetic saturation, the effective pinning potential Ueff collapses due to the diverging width of the up domains and the loss of the associated vortex polaron pinning. In the DVS regime, the domain width is at least two times larger than in the DMS state, and the vortex polaron energy is hence very much lower. In addition, the fields of the penetrating free vortices are screened by the surrounding spontaneous vortices, and vortex polarons no longer play a significant role.

In contrast, the analysis of Jc(T) and magnetic relaxation S(T) for S1 in the DVS region leads to a consistent picture of giant flux creep with a characteristic activation energy of U00 ≈ 240 K. The mechanism by which the sample becomes remagnetised in the DVS phase must explicitly involve the penetration of one sign of flux from the sample edges in conjunction with vortex-antivortex annihilation at domain walls. The latter process involves thermal activation over a Bean-Livingston barrier29, which, we believe, governs the flux creep behaviour observed in magnetisation and magnetic relaxation measurements at lower temperatures. We note that thermally activated behaviours with quite similar activation energies have previously been identified in frequency-dependent measurements of the ac susceptibility in EuFe2(As0.7P0.3)228. These were attributed to several suggested intra- and inter-domain vortex hopping mechanisms, and vortex-antivortex annihilation processes were not explicitly considered.

MFM images at T = 19.8 K reveal the prolific formation of Y-shaped defects in the domain structure, with a vortex frequently located at the intersection of the domains. These Y-defects appear to be integral to the formation of domain structure grain boundaries (Fig. 4, 70 Oe) and to the observed domain buckling (Fig. 4, 100 and 250 Oe). In these cases, we speculate that the dominant vortex penetration direction from the sample perimeter has a large vector component perpendicular to the original domain walls. Since propagation through an adjacent reverse domain has a very large associated energy barrier, it is instead easier for the vortex to distort the stripe domain structure in this direction and travel along the same up domain. Recent results from Vagov et al. have demonstrated the high mobility of these Y-defects51, and hence, it should be relatively easy for them to stack together, each carrying a vortex at the domain intersection. Furthermore, the stripe domain structure of the DMS can be viewed as a form of ordered pinning substrate and therefore, with the addition of a suitable drive current, superconducting vortices may exhibit further exotic depinning behaviours and possibly distinct, dynamic phases52.

The vortex polaron potential is inversely proportional to the ferromagnetic domain width (equation (3)), a parameter that can be tuned by modifying the sample properties. The Kooy-Enz model of ferromagnetic stripe domains53 predicts that the domain period at H = 0 varies as the square root of the sample thickness, and thinner samples should show narrower domains and a significant enhancement of vortex polaron pinning across a DMS regime that spans a wider range of temperatures. Therefore, manipulation of the domain structure by control of material parameters presents a new route to the magnetic enhancement of vortex pinning strengths in ferromagnetic superconductors that can be employed in the field of high-current tapes/wires for industrial applications.

Methods

Sample growth

Single crystals of EuFe2(As1−xPx)2 were grown using a self-flux method. Stoichiometric amounts of FeAs, FeP, and Eu (99.99%) powders were mixed and loaded into alumina crucibles, which themselves were placed and sealed in stainless steel tubes under Ar atmosphere. The sealed tubes were heated under N2 atmosphere to ≥1300 °C and held for 12 h, then cooled slowly to 1050 °C at 2 °C per hour before allowing to cool naturally to room temperature. This produces platelet-shaped single crystals with the larger two dimensions corresponding to the ab-plane and the shortest dimension to the c-axis.

Magnetisation measurements

A Quantum Design MPMS 3 magnetometer was used to determine Tc and TFM, and to perform measurements of magnetic hysteresis loops (MHLs) and magnetic relaxation. Measurements were conducted using a quartz half-rod on which a small quartz cube was secured, creating a flat surface the normal of which is parallel to the long axis of the rod and which is located precisely halfway along the half-rod’s length. The platelet samples were mounted on this surface so that the crystal c-axis was parallel to the axis of the half-rod and thus also to the applied magnetic field. In each measurement, the total magnetic dipole moment m is measured, from which the magnetisation M is derived: M = m/V, where V is the volume of the sample.

Zero-field-cooled (ZFC) and field-cooled (FC) measurements were performed by cooling the sample in zero field to base temperature (~5 K), after which a small field was applied and the magnetisation measured upon warming the sample to above Tc. The sample is then cooled back to base temperature with the applied field maintained.

MHLs were performed by initially warming the sample above Tc before cooling to the target temperature T in zero field. The magnetisation is measured periodically as the magnetic field is increased to 10 kOe, reduced through zero to −10 kOe and then increased back to 10 kOe. The sweep rate of the magnetic field was kept identical for all MHL measurements, with the same number of measurements within each loop. In complement, a hysteresis loop with increasing field excursions was measured at 5 K (base temperature) in order to determine the minimum field for the establishment of the critical state and full flux penetration of the sample54,55 (Supplementary Note 3). This was found to be  ≈ 4 kOe, and thus sweeping the field initially to 10 kOe is more than sufficient to achieve full flux penetration. Furthermore, at the same temperature, the full reversal of the critical state was achieved in a window of ΔH ≈ 1 kOe. Within the valid critical state portion of each MHL, we calculated the the critical current density Jc using the Bean critical state model for a slab in a perpendicular field43,54, Jc(H, T) = 20ΔM/(w(1 − w/3l)) (with Jc in units of A cm−2), where ΔM = Mupper − Mlower (in units of emu cm−3) is the width of the hysteresis loop, l is the length of the sample and w is the width of the sample (both in cm), such that l > w.

Magnetic relaxation data in sample S1 were taken by warming above Tc and cooling to target T in zero field. The magnetic field is then increased from zero up to 10 kOe, at the same rate as for the MHLs, before decreasing to the final target field Hf. Once the final field is reached, the magnetisation was recorded every as a function of time (M(t)) every  ~30 s for several minutes. The time-dependent relaxation of the irreversible magnetisation exhibits a characteristic logarithmic decay, and the normalised relaxation rate S is determined from a linear fit to \(\ln {M}_{{{{\rm{irr}}}}}\) - \(\ln t\) (Supplementary Fig. 4), where Mirr is the irreversible magnetisation38. However, the measurement is of the total magnetisation M = Mrev + Mirr, where Mrev is the time-independent reversible contribution to the magnetisation which must be accounted for, using the data from the MHLs, in order to determine the irreversible component only, Mirr(t).

Magnetic force microscopy imaging

The force microscope used in this study, detailed in ref. 56,57,58, operates with a singly clamped nanowire as cantilever in the pendulum geometry. The nanowire is made from Si, has a length of 20 μm and a width of 100 nm. Its fabrication is documented in ref. 59. It is tipped with an elongated ferromagnetic Co structure, which renders its two first-order flexural modes susceptible to the magnetic field profile. When modelled with an effective magnetic charge q57,60, the shifts in the mechanical resonance frequency of the two modes, Δfx and Δfy, are proportional to the in-plane magnetic field gradients, such that \(\Delta {f}_{x}=\frac{\partial {B}_{x}}{\partial x}\) and \(\Delta {f}_{y}=\frac{\partial {B}_{y}}{\partial y}\). Under the assumption that ∇ ⋅ B = 0, the sum of these frequency shifts, Δf = Δfx + Δfy, is proportional to the out-of-plane field gradient, yielding \(\Delta f=-q\frac{\partial {B}_{z}}{\partial z}\).

During image acquisition, the nanowire’s high sensitivity to field gradients and the small mode splitting frequently led to mode crossings, preventing the reliable use of a phase-locked loop to track the frequency shifts. Thus, the MFM images were generated by recording thermal noise spectra at each measurement point, extracting the resonance frequencies, fx and fy, and calculating the frequency shifts according to Δfx = fx − fx,0 and Δfy = fy − fy,0. fx,0 = 243 kHz and fy,0 = 245 kHz are the natural resonance frequencies of the modes in the absence of any interaction with the sample.

MFM was conducted on the platelet-shaped sample SD under varying temperatures and applied magnetic fields. The sample was mounted with its c-axis nominally aligned with the field and normal to the imaging plane. Optical microscopy revealed a small tilt of the c-axis with respect to the field, leading to a small in-plane component of the applied field.

For all measurements, the tip-sample separation was between 50 and 150 nm, and was adjusted between scans in order to compensate for the sample-tip interaction strength. For the ZFC measurements, zero applied field was calibrated by minimising the vortex density to 1–2 vortices per 10 × 10 μm2 area in the purely superconducting state of the sample. The temperature was determined using a four-point probe measurement with a calibrated Cernox® sensor. The sensor is integrated within the heater, which is connected to the sample holder. A small temperature gradient between the sensor and the sample results in a sample temperature which is slightly lower than that read by the sensor, and the magnitude of this difference decreases as the temperature is reduced to the base temperature. The temperatures reported in the manuscript, corresponding to features derived from the MFM images, are therefore the nominal temperatures, i.e., the temperatures as read by the sensor.