Disorder-induced universality and scaling in hole-doped iron-based superconductors

Abstract

Iron-based superconductors exhibit various magnetic and electronic phases that are highly sensitive to structural and chemical modifications. Elucidating the origins of these phases remains a central challenge. Here, using neutron and x-ray diffraction, we uncover a universal phase diagram that identifies disorder as a hidden tuning parameter governing these phase transitions. By analyzing nine hole-doped phase diagrams, we observe the emergence of a double-Q tetragonal magnetic phase in proximity to ideal FeAs₄ tetrahedral configurations, thereby demonstrating a strong link between bond-angle stabilization and magnetic transitions. Beyond stabilizing the double-Q phase, atomic disorder also influences charge doping and magnetic anisotropy. We further observe similar scaling behavior of the transition temperatures of the double-Q and the more prevalent orthorhombic single-Q magnetic phases, evidencing a unified origin of structural and magnetic properties linked to itinerant nesting instability. Our findings establish a comprehensive basis for understanding how chemical disorder, charge doping, and structural features collectively shape the magnetic and superconducting properties of iron-based superconductors.

Introduction

Following the 2006 discovery of superconductivity in iron-based pnictides¹, these materials garnered considerable attention due to their rich array of physical phenomena^{2,3,4,5,6,7,8}, including the realization of unconventional superconductivity^9,10, spin-density waves (SDW or ${C}_{2M}^{a}$ where the subscript and superscript denote the two-fold symmetry of the magnetic phase and the direction of the magnetic moments, respectively)^2,11,12,13, charge-spin density waves (CSDW or ${C}_{4M}^{c}$)^14,15, spin vortex crystal (SVC or ${C}_{4M}^{{ab}}$)^15,16,17 and quantum criticality (QC)^18,19,20,21. The discovery of a structurally re-entrant tetragonal magnetic phase, ${C}_{4M}^{c}$^5,22, and the determination of its double-Q magnetic structure²³, have driven extensive theoretical^{24,25,26,27,28,29,30,31,32,33} and experimental investigations. This breakthrough resolved a long-standing debate over whether magnetic fluctuations, rather than orbital or nematic fluctuations, are responsible for simultaneous first-order structural and magnetic transitions, highlighting the universal significance of these tetragonal states.

The ${C}_{4M}^{c}$ phase has been observed in several hole-doped AFe₂As₂ systems (“122” where A = Ba_1−xNa_x, Ba_1−xK_x, Sr_1−xNa_x, or Ca_1−xNa_x), where it arises from a first-order in-plane to out-of-plane spin reorientation^22,34, occurring within a narrow range of Na substitution. This phase resides within a limited dome in the orthorhombic ${C}_{2M}^{a}$ regime³⁵, where magnetic and superconducting states coexist microscopically^6,36,37. Prior studies on the isovalently-substituted LaFeAs_1−xP_xO (“1111” system)³⁸ revealed the sequential occurrence of ${C}_{4M}^{{ab}}$ and ${C}_{4M}^{c}$ phases outside the ${C}_{2M}^{a}$ magnetic phase space, underscoring the potential shared mechanisms between these tetragonal states across different pnictide systems.

Further studies have demonstrated^5,35,39 that in the “122” systems, the ${C}_{4M}^{c}$ phase occurs over a range of Na content (equivalent to hole doping), with the re-entrant maximal transition temperature, T_{r max}, rising from ≈48 K to 68 K before declining back to ≈52 K for the Ba_1−xNa_xFe₂As₂ (BNFA), Sr_1−xNa_xFe₂As₂ (SNFA), and Ca_1−xNa_xFe₂As₂ (CNFA) solid solution systems, respectively. This trend mirrors the antiferromagnetic transition temperature, T_N, of the primary ${C}_{2M}^{a}$ phase, illustrating how A-site ionic size variance influence the electronic properties of the materials. The flexible extent of the ${C}_{2M}^{a}$ phase space and the concurrent shifts of ${C}_{4M}^{c}$ and superconducting domes reinforce the idea of a shared underlying mechanism between these phases. However, key structural factors governing magnetic anisotropy and long-range magnetic order remain unresolved, despite theoretical studies⁴⁰ linking doping and T_N to changes in magnetic order parameters.

The stability of the ${C}_{4M}^{c}$ phase has been tied to optimal nesting conditions driven by spin-orbit coupling and quantum fluctuations near magnetic quantum criticality^41,42. The experimentally confirmed presence of a quantum critical point (QCP) in several “122” systems, where SDW ${C}_{2M}^{a}$ magnetic order is suppressed and superconductivity emerges [see references^{16,18,19,20,21,43,44,45}, for example], suggests a strong coupling between the ${C}_{4M}^{c}$ phase, superconductivity, and quantum criticality within the magnetic-superconducting coexistence region. These findings deepen our understanding of precursor states leading to superconductivity, but important questions remain about the universality, stability, and competitive nature of the ${C}_{2M}^{a}$, ${C}_{4M}^{{ab}}$ and ${C}_{4M}^{c}$ phases.

In this study, we address these unresolved questions by exploiting the structural flexibility of the pnictides and investigating the behavior of the ${C}_{2M}^{a}$ and ${C}_{4M}^{c}$ magnetic states in engineered hole-doped “122” compounds through Na-substituted solid solutions. These materials introduce controlled structural disorder, charge doping, and chemical pressure. A scaling relationship between magnetic and nuclear structures was established in samples with optimal conditions for maximum ${C}_{4M}^{c}$ transition temperatures, leading to the derivation of an empirical equation that describe the experimental behavior of T_N and T_r. Our findings highlight structural disorder as a hidden scaling parameter, culminating in a universal phase diagram that captures the intricate details of “122”-hole-doped materials.

Results

In addition to the previously studied Ba_1−xNa_xFe₂As₂ (BNFA), Sr_1−xNa_xFe₂As₂ (SNFA) and Ca_1−xNa_xFe₂As₂ (CNFA) series^5,35,36,39, six previously-unreported partial phase diagrams with the general formula [(Ba_1−ySr_y)_1−xNa_x]Fe₂As₂ (BSNFA) and [(Sr_1−yCa_y)_1−xNa_x]Fe₂As₂ (SCNFA) were synthesized. Numerous samples, covering a relatively broad range of Na substitution levels (x = 0.25–0.45), were prepared to explore the potential occurrence of the ${C}_{4M}^{c}$ phase, as previously observed in the parent systems. The ratios of isovalent Ba_1−ySr_y and Sr_1−yCa_y mixtures were fixed at 3:1, 1:1 and 1:3 in BSNFA and SCNFA to manipulate the A-site ionic radius and introduce disorder without altering the charge doping, which was solely controlled by sodium substitution. These constraints are reflected in the ternary phase diagram in Fig. 1, where the term ternary refers to the triply occupied A-site. The synthesis procedures followed the methods described in earlier studies^5,35,36,39.

Fig. 1: Ternary phase diagrams.

Ternary phase diagrams illustrating the doubly-substituted (Ba_1−ySr_y)_1−xNa_x and (Sr_1−yCa_y)_1−xNa_x samples selected for this study (indicated by crosses). Blue and red symbols represent the ${C}_{2M}^{{a}}$ and ${C}_{4M}^{c}$ phases, respectively. ${C}_{4M}^{{c}\max }$ samples are depicted by the white-highlighted dark-red spheres. Partially filled red and blue symbols denote samples that exhibit partial ${C}_{4M}^{c}$ phase characteristics. The color map highlights the approximate regions of the ${C}_{2M}^{{a}}$ (dark yellow), ${C}_{4M}^{c}$ (green), and superconducting (pink) phases. The narrow, dark pink bands flanking the green region indicate overlapping ${C}_{2M}^{{a}}$ and superconducting phases. The sodium content of the ${C}_{4M}^{{c}\max }$ samples ranges from 0.27 for the Ba_1−xNa_x series to 0.44 in the Ca_1−xNa_x series. Error bars are smaller than the symbol size.

Compositions exhibiting the ${C}_{4M}^{c}$ phase were identified within the ${C}_{2M}^{a}$ region of each of the nine investigated series, with transition temperatures (T_r) ranging from 38 K to 68 K, below the Néel temperature (T_N). These compositions are situated near the onset of superconductivity and optimal nesting conditions^41,42, suggesting that samples with the highest T_r values, ${C}_{4M}^{{c}\max }$, exhibit similar free energies regardless of their Na content. This observation led to the formation of a subset of 13 compositions, referred to as the ${C}_{4M}^{{c}\max }$ series, with comparable electronic properties, which follows Vegard’s law, as shown in the insets of Fig. 2 (data shown at 10 K).

Fig. 2: Lattice parameters.

Lattice parameters (a, b) and unit cell volume (c), at 10 K, of all samples as a function of average A-site ionic radius ${r}_{ < A > }$. Inset: Lattice parameters, unit cell volume, and select bond lengths for samples exhibiting maximum transition temperatures to the ${C}_{4M}^{c}$ magnetic state. Orthorhombic a and b lattice parameters were transformed to tetragonal-equivalent lattice parameter a_tet using the relationship: ${a}_{{tet}}=\scriptstyle\sqrt{\left({a}^{2}+{b}^{2}\right)}/2$. Error bars are smaller than the symbol size.

Despite the occurrence of the ${C}_{4M}^{{c}\max }$ at different Na values, this subset of samples exhibits a linear dependence of key structural variables, including unit cell volume, lattice parameters, As-As and A-As bond-lengths, on the average ionic radius ${r}_{ < A > }$ as displayed in Figs. 2 and 3. However, the Fe-As bond-length, d_Fe−As, remains initially unchanged for ${r}_{ < A > }$ > 1.22 Å, consistent with the stiff structural properties reported for similar Fe-based pnictides^{36,46,47,48,49}. For compositions with ${r}_{ < A > }$ values below 1.22 Å (Fig. 3a), this bond length decreases by ≈0.8%, signaling a transition in structural rigidity at this threshold.

Fig. 3: Structural and magnetic parameters.

Bond-length (a), bond-angles (b), and refined magnetic moment (d) for all the ${C}_{4M}^{{c}\max }$ samples at 10 K. Linear scaling between $\frac{{T}_{{{{\rm{N}}}}}}{\sqrt{M}}$ and $\frac{{T}_{{{{\rm{r}}}}}}{\sqrt{M}}$ and ${r}_{ < A > }$ (c) and between the bond-angles and bond-length and ${r}_{ < A > }$ (f). e Magnetic and superconducting transition temperatures. Observed (closed symbols) values of T_N and T_r are compared with those calculated by the scaling function (Eq. 1) (open symbols). Error bars are smaller than the symbol size, if not clearly visible.

This change in the Fe-As bond length is further reflected in the behavior of other structural parameters. The linear contraction of the out-of-plane As-As bond length with decreasing ${r}_{ < A > }$ results in a much larger compression of the c-axis (|Δc|≈ 1.15 Ǻ or ≈8.9%), compared to a smaller contraction of the a-axis (|Δa| ≈ 0.05 Ǻ or ≈1.3%), as shown in the inset of Fig. 2a, b. The tetrahedral As-Fe-As bond angles (α and β in Fig. 3b) are also significantly affected by these structural changes, particularly the displacement of As ions and the shortening of the As-As bond length by up to 0.6 Å. Initially, α and β vary in opposite directions with decreasing ${r}_{ < A > }$, achieving a perfect tetrahedral angle of 109.47° at ${r}_{ < A > }$ ≈ 1.32 Å. However, below ${r}_{ < A > }\,$≈ 1.22 Å, the two angles diverge again and stabilize, remaining statistically unchanged down to the lowest ${r}_{ < A > }$ value.

An analysis of the As-Fe-As bond angles at 10 K across all samples reveals a universal trend with the samples that exhibit the ${C}_{4M}^{{c}}$ phase clustering within a confined region, marked by a tetrahedral angular separation of less than one degree (see the shaded rectangular box in Fig. 4a). This finding suggests a strong correlation between nearly perfect tetrahedral angles and the occurrence of the ${C}_{4M}^{{c}}$ state, particularly in regions of optimal nesting. This correlation may explain the absence of the ${C}_{4M}^{{c}}$ magnetic phase in the electron-doped “122” analogs, where the angles deviate significantly from the ideal tetrahedral value.

Fig. 4: Phase diagram universality.

a As-Fe-As bond angles (α and β) highlighting the region where samples exhibiting the ${C}_{4M}^{c}$ phase cluster (shaded rectangular box), with angles approaching those of a perfect tetrahedron. Symbols without edges represent α angles, while β angles are shown with thick, black-edged symbols of the same color. Upward triangles (magenta, green and olive green) denote the BNFA, SNFA and CNFA series, respectively. Rightward triangles (light green, dark yellow, dark cyan) represent the BNSFA 3:1, 1:1, and 1:3 series, and leftward triangles (blue, cyan, and orange) represent the SCNFA 3:1, 1:1 and 1:3 series. Bold red and blue spheres correspond to the α and β angles of samples exhibiting ${C}_{4M}^{{c}\max }$ properties. b A universal phase diagram with normalized x and y axes (see text and methods for details), combining data from all nine BNFA, SNFA, CNFA, BSNFA, and SCNFA series. Dark blue, olive green (and white with olive green edges), and red symbols represent normalized T_N, T_r and T_C values, respectively. The yellow, pink, dark yellow and green regions correspond to the non-magnetic tetragonal I4/mmm (C₄), superconducting tetragonal I4/mmm (S-C₄), magnetic orthorhombic Fmmm ${C}_{2M}^{a}$, and magnetic tetragonal I4/mmm ${C}_{4M}^{c}$ phases, respectively. Unscaled phase diagram shown in the inset of (b). Error bars are smaller than the symbol size.

Building on these structural correlations, a general phase diagram was constructed by normalizing the sodium content of each sample to that of the ${C}_{4M}^{{c}\max }$ sample of the corresponding series (more details in Methods). This normalization allowed the magnetic and superconducting properties of all samples to collapse into a universal phase diagram, shown in Fig. 4b. The transition temperatures (T_N, T_r and T_C) are scaled relative to T_N of the x_(Na) = 0 sample in each series. Here, T_N (x_(Na) = 0), extracted from ref. ⁵⁰, represents the antiferromagnetic transition temperature of the undoped parent material.

Additionally, the refined magnetic moments of the ${C}_{4M}^{{c}\max }$ samples, along with their magnetic and superconducting transition temperatures, are illustrated in Fig. 3d, e, respectively. Both T_N and T_r peak around ${r}_{ < A > }$ ≈ 1.19-1.20 Å, demonstrating a remarkable tracking behavior between the two. The absence of a ${T}_{{{{\rm{N}}}}}^{\max }$ plateau, as seen in Kirshenbaum’s⁵⁰ isovalently-substituted Ba_1−xSr_xFe₂As₂ and Sr_1−xCa_xFe₂As₂ systems, together with the significant suppression of T_N for ${r}_{ < A > }$ > 1.20 Å, can be attributed to modifications in the Fermi surface due to charge doping and disorder^{51,52,53,54,55,56,57,58,59} induced by the widely contrasting ionic radii in our triply substituted A-site samples.

Exploring further the key parameters controlling magnetic properties, we found a linear relationship between ${T}_{{{{\rm{N}}}}}/\sqrt{M}$ (where M is the magnitude of the magnetic moment) and ${r}_{ < A > }$, as shown in Fig. 3c, underscoring the substantial influence of the average A-site ionic radius in tuning magnetic interactions. Moreover, we discovered a correlation between ${r}_{ < A > }$, As-Fe-As bond-angle, and Fe-As bond-length, namely ${r}_{ < A > }$ ~ $\left(\frac{\sin (\alpha )\sin (\beta )}{{({{{\rm{d}}}}_{{{{\rm{Fe}}}}-{{{\rm{As}}}}})}^{1/2}}\right)$ (see Fig. 3f), which resembles the well-known electronic bandwidth equation^60,61,62 $W\propto \frac{\cos \left(\omega \right)}{{({\rm{d}}_{{{TM}}-O})}^{3.5}}$, where TM represents common transition metal elements (e.g., Mn or Ni), and $\omega =\frac{1}{2}\left(\pi -\left\langle \beta \right\rangle \right)$, with $\left\langle \beta \right\rangle$ and d_TM−O incorporating the average TM-O bond-angle and bond-length, respectively. This equation successfully describes the effects of chemical substitution and pressure on the electronic and magnetic properties of distorted rare-earth perovskite nickelates⁶³ and manganites^64,65,66.

Finally, by combining these independent scaling relations, an empirical equation was derived to describe the magnetic transition temperature in terms of the magnetic moment, As-Fe-As bond angle, and bond length:

$${T}_{{{{\rm{N}}}}}({T}_{{{{\rm{r}}}}})=\left[a+b\left(\frac{\sin (\alpha )\sin (\beta )}{{({\rm{d}}_{{{Fe}}-{As}})}^{1/2}}\right)\right]\sqrt{M}$$

(1)

where a and b are constants. For T_N, a = −5611(47) K μ_B^-0.5 and b = 9966(80) K Å^0.5 μ_B^-0.5, while for T_r, a = −950(54) K μ_B^−0.5 and b = 1784(94) K Å^0.5 μ_B^-0.5. This equation agrees remarkably well with the observed T_N and T_r values, as shown in Fig. 3e.

It is worth emphasizing that the functional form of Eq. (1) was determined empirically by fitting experimental results and examining their structural dependencies, as demonstrated in Fig. 3c, f. This formulation captures well-established correlations between transition temperatures and Fe-As bond lengths and angles, as supported by both experimental and theoretical studies^{67,68,69,70,71,72,73}. Although a full theoretical derivation remains an open question, multiple first-principles calculations and experimental findings demonstrate that Fe-As bond lengths, bond angles, and magnetic ordering temperatures are strongly correlated in the hole-doped iron pnictides.

The trigonometric terms sin($\alpha$) and sin($\beta$) in Eq. (1) arise from the geometric dependence of exchange interactions on bond angles, reflecting how deviations from the ideal tetrahedral angle influence magnetic interactions. Such angular distortions modulate Fe 3d-As 4p orbital hybridization and exchange pathways, an effect noted in both experimental and theoretical studies. Yin et al.⁶⁷, for example, showed that structural distortions affect orbital differentiation, with the Fe d_xy orbital being particularly sensitive to bond-angle changes, supporting the use of trigonometric functions to model these effects. Similarly, tight-binding and hybridization-based descriptions suggest that orbital overlaps have a trigonometric dependence, justifying the use of sin(α) and sin(β) rather than simple linear terms such as α and $\beta$. Alternative functional forms were tested, but the demonstrated success of this formulation to fit the T_N and T_r experimental values, as shown in Fig. 3e, supports its validity.

The As-Fe-As bond angle is a key structural parameter in iron pnictide superconductors and has been shown to influence the strength of orbital fluctuations⁶⁸. These fluctuations are strongest when the FeAs₄ tetrahedron approaches a regular configuration, corresponding to an optimal As-Fe-As bond angle. This result explains the experimentally observed correlation between the optimal bond angle and the superconducting transition temperature (T_C) (Lee et al.⁷⁴). Since maximum superconductivity is associated with the suppression of magnetic order, the As-Fe-As bond angle plays a significant role in tuning the strength of magnetic correlations, as also noted in studies on spin fluctuations⁷⁰.

The influence of Fe-As bond lengths on magnetic ordering stems from their direct role in exchange interactions. Shorter Fe-As bonds enhance covalent bonding between Fe 3 d and As 4p orbitals, increasing bonding-antibonding splitting and reducing local magnetic moments, which promotes greater itinerancy of the electrons⁷¹. This behavior drives long-range super-exchange interactions that contribute dominantly to antiferromagnetic interactions between nearest neighbors. Using DFT + DMFT calculations, Yin et al.⁶⁷ demonstrated that Fe-As bond lengths and angles are critical in controlling electronic correlations and ordered magnetic moments in iron pnictides and chalcogenides, providing theoretical justification for the structural dependencies incorporated in Eq. (1).

The inclusion of the magnetic moment term $\sqrt{M}$ accounts for its influence on exchange interactions and transition temperatures. First-principles calculations⁷² have shown that the Fe-spin state is strongly coupled to the Fe-As bonding environment, where a reduction in the Fe magnetic moment weakens Fe-As bonding and enhances As-As hybridization. This structural modification directly impacts the stability of magnetic ordering. Furthermore, studies on Stoner factors in doped “122” iron pnictides⁷³ have shown a direct correlation between Fe-As bond length and magnetic instability, providing additional theoretical justification for incorporating this dependence into Eq. (1). Further, Yin et al.⁶⁷ presented strong evidence that the magnetic moment of Fe is closely linked to the Fe-As bonding environment and electronic correlations, rather than being a mere consequence of itinerancy or localization. Similarly, theoretical studies by Usui and Kuroki⁷⁰ highlight how changes in Fe-As bond length modulate the density of states at the Fermi level, affecting both magnetic interactions and superconductivity.

Discussion

To better understand the scaling behavior and universality of our phase diagram, we first examined the structural impact of substitution, particularly focusing on the average A-site ionic radius, ${r}_{ < A > }$. Initially, the dominant influence arises from changes in the inversely evolving tetrahedral bond angles, which drive modifications in the conduction band at the Fermi surface. These changes lead to an increase in the Néel temperature (T_N), in agreement with theoretical studies that highlight the sensitivity of the Fermi surface and the material’s resistive and magnetic properties to small topological variations in the electron and hole pockets⁷⁵, as exemplified in BaFe₂(As_1−xP_x)₂^{4,76,77,78,79}.

However, as ${r}_{ < A > }$ decreases to approach 1.22 Å, the bond angles stabilize and remain largely unchanged. This stabilization presents a challenge in explaining the observed downturns of T_N and T_r. Upon further analysis (Fig. 3a), we found that in this region, the Fe-As bond length contracts continuously by ≈0.8% as the lattice compresses further, despite the stabilization of the bond angles. This contraction emerges as the principal structural parameter influencing the material’s magnetic properties. Simultaneously, the refined Fe magnetic moment decreases by ~0.4μ_B, as shown in Fig. 3d, consistent with earlier studies and full potential DFT calculations^39,70,76 that report similar behavior for the magnetic moment in response to compression of the Fe-As bond length.

Disorder induced by the difference in the ionic radii of the A-site cations plays a critical role in shaping the magnetic properties of hole-doped pnictides. This disorder suppresses the spin density wave (SDW) ${C}_{2M}^{a}$ ordering, favoring the formation of the ${C}_{4M}^{{c}}$ phase, and emerges as a key scaling parameter when normalizing the nine phase diagrams investigated in this study, Fig. 4b. Supporting this idea, experiments on isovalently-substituted LaFeAs_1−xP_xO³⁸ have demonstrated that disorder induced by differences in As/P ionic sizes stabilizes the double-Q magnetic phases. However, it remains unclear whether perfectly random local disorder or some form of local ordering produces distinct effects on electronic properties. For example, in the 1111-LaFeAs_1−xP_xO system, variations in synthesis methods result in vastly different electronic phase diagrams, where the suppression or preservation of superconducting and magnetic orders can be controlled despite identical long-range structures⁸⁰. Similarly, disorder has been shown to have contrasting effects on the superconducting and magnetic properties of pnictide superconductors⁸¹ depending on whether the material is underdoped or overdoped.

The impact of disorder is further highlighted by studies on Na-substituted compounds such as Sr_1−xNa_xFe₂As₂^82,83 and Ba_1−xNa_xFe₂As₂⁸⁴, for example. In these systems, dopant-induced disorder increases with increasing Na concentrations, with local compositional fluctuations generating random fields⁸⁵ that act on the orthorhombic structural order parameter and influence phase transitions. Pair distribution function analyses^82,83 of Sr_1−xNa_xFe₂As₂ reveal that orthorhombic distortions persist locally even at high doping levels and in regions of the phase diagram where long-range orthorhombicity is absent. Similarly, single-crystal x-ray diffuse scattering studies⁸⁴ on Ba_1−xNa_xFe₂As₂ have revealed strong local distortions induced by the large ionic size difference between Ba and Na, which lead to disorder of the electronic properties on the Fe sites.

The presence of the ${C}_{4M}^{{c}}$ magnetic phase in samples near optimal nesting conditions supports existing literature, which suggests that the ${C}_{4M}^{{c}\max }$ samples, despite their varying Na content, maintain similar electronic structures across different phase diagrams. Angle-resolved photoemission spectroscopy (ARPES) measurements^86,87, for instance, have shown that electron or hole doping affects the chemical potential, shifting it continuously in agreement with a rigid band shift model predicted by renormalized first-principle band calculations⁸⁸. This shift results in an electron-hole asymmetry in the Fermi surface nesting conditions, with the Lindhard spin susceptibility function peaking on the hole-doped side of the phase diagram near optimal doping for superconductivity. Assuming a rigid chemical potential shift, simulations by Scherer and Anderson⁴² suggest that hole doping enhances c-axis polarized low-energy magnetic fluctuations in BaFe₂As₂ and LaFeAsO and modifies the magnetic anisotropy hierarchy. This enhancement is consistent with increased magnetic susceptibility along the c-axis, in agreement with the observed out-of-plane magnetic moment of the ${C}_{4M}^{{c}}$ phase and ARPES measurements⁸⁶.

The presence of the ${C}_{4M}^{{c}}$ phase in electron-doped BaFe_2−xM_xAs₂ (where M is a transition metal) remains uncertain. As mentioned earlier, experimental studies, including our own, have confirmed the presence of this phase in several hole-doped “122” families and in LaFeAs_1−xP_xO³⁸, where strong competition between superconductivity and magnetism is observed. Similarly, this phase has been observed in FeSe when subjected to ~6.8 GPa pressure⁸⁹, resembling the behavior seen in hole-doped “122” compounds.

A broader perspective on the impact of substitution strategies was provided by Dai⁹⁰, who reviewed the effects of both A-site and B-site substitutions on spin excitations and magnetic order, offering important perspectives into the role of substitution in magnetic phase transitions. Consistent with our observations, theoretical studies^91,92,93 suggest that disorder induced by A-site hole substitution favors the double-Q magnetic state over the single-Q SDW state, highlighting the role of substitution-induced disorder in stabilizing distinct magnetic phases. Models proposed by Vavilov and Chubukov⁹⁴ identify disorder as a primary scaling parameter in generic phase diagrams. Disorder affects the Fe 3 d bands through different mechanisms, directly via Fe-site substitutions or indirectly through local distortions induced by A-site and As-site substitutions. Fe-site substitutions (e.g., Co or Ni replacing Fe) increase intraband scattering in the electron bands due to additional electrons, which disrupts the electronic structure of the Fe 3 d orbitals, weakens magnetic exchange interactions, and causes fluctuations in the As-Fe-As bond angle. These effects result in a rapid suppression of the magnetic order. In contrast, A-site substitutions (e.g., K or Na replacing Ba) induce structural distortions that alter the interlayer spacing and modify the Fe-As bond lengths and bond angles without directly affecting the Fe plane. This leads to a more gradual suppression of the single-Q SDW state and the observed emergence of competing magnetic phases, reinforcing the importance of differentiating between structural and electronic disorder when analyzing phase stability. The suppression of the single-Q SDW state by A-site substitution, in favor of a double-Q magnetic state involving two itinerant Fe electronic sublattices, clearly demonstrates the significant impact of A-site substitution on the Fe 3 d bands.

Further theoretical studies suggest that variations in the As-Fe-As bond angle influence orbital-dependent electronic structure, particularly affecting the yz/xz versus xy energy splitting^28,67,95,96, which governs magnetic interactions. Deviations from the ideal tetrahedral geometry suppress the nesting conditions required for spin-density wave formation and alter the stability of competing magnetic and superconducting phases. Thus, while both Fe-site and A-site substitutions introduce disorder, their effects on magnetism and superconductivity remain distinct, requiring careful consideration of their structural and orbital consequences.

The competing single-Q and double-Q magnetic phases underline the intimate relationship between magnetic state degeneracy and itinerant electron properties. The tetragonal collinear double-Q magnetic state emerges from the superposition of two orthogonal spin-density waves that restore the four-fold symmetry of the crystal structure²³. This wave interference redistributes the spin density such that half of the iron sites are non-magnetic, whereas the magnetic moments of the remaining sites are doubled. The itinerant character of the magnetic excitations, demonstrated experimentally^{8,23,34,40,93,97,98,99} and theoretically^{91,92,100,101,102,103}, is incompatible with localized spin models and provides clear evidence for itinerant magnetism.

Finally, the observed scaling relationship, captured by Eq. (1), between the magnetic transition temperatures of both the orthorhombic and tetragonal phases and key internal structural parameters provides strong evidence for the existence of a unified origin rooted in the nesting instabilities of the itinerant electrons. Nesting conditions, influenced by Fermi surface topology^{7,51,59,70,104,105,106} and electron-hole pocket interactions, influence the stability of the competing magnetic phases.

In conclusion, the pronounced suppression of T_N upon doping, especially near the ${C}_{4M}^{{c}\max }$ phases where optimal charge doping and nesting conditions converge, reveals a delicate balance between superconductivity and magnetism. While charge doping tunes the nesting conditions, disorder induced by contrasting A-site ionic radii emerges as a hidden but critical tuning parameter. This scaling parameter produces a universal phase diagram that explains the properties of hole-doped “122” material systems and offers a more comprehensive understanding of the interplay between structural, electronic, and magnetic characteristics.

Methods

The nuclear and magnetic structures of our samples were determined as a function of temperature using high-quality neutron powder diffraction data collected at POWGEN¹⁰⁷, HB-2A¹⁰⁸, and WISH¹⁰⁹ at the Spallation Neutron Source and High Flux Isotope Reactor at Oak Ridge National Laboratory, and the ISIS Pulsed Neutron and Muon Source at Rutherford Appleton Laboratory, respectively. High-resolution x-ray powder diffraction was conducted using beamline 11-BM-B^110,111 at the Advanced Photon Source at Argonne National Laboratory. Consistent Rietveld refinement results were obtained from multiple samples measured on all four instruments, enabling the combination of refined structural parameters without corrections. The absence of impurity phases together with the linear evolution of the refined lattice parameters and unit cell volume within each of the nine investigated series, as a function of the average ionic radius of the A-site mixed ions, ${r}_{ < A > }$, demonstrate the high quality of our stoichiometric samples, as shown in Fig. 2.

Measurements of superconductivity, displayed in Fig. 3, were performed using a Quantum Design Magnetic Property Measurement System with zero field cooling and a measuring field of 200 Oe on warming. Tabulated values¹¹² of the ionic radii of eight-coordinated Ba (1.42 Å), Sr (1.26 Å), Ca (1.12 Å) and Na (1.18 Å) were used to calculate the average ionic radius of the A-site mixed ions, ${r}_{ < A > }$.

Figure 4b was constructed by normalizing the sodium content ($x$) of each sample to that of the sample displaying maximum transition temperature (T_r) denoted as ${x}_{{C}_{4M}^{{c}\max }}$. These normalized values ($x$/${x}_{{C}_{4M}^{{c}\max }}$) were used as a common x-axis for all the phase diagrams. Similarly, the transition temperatures (T_N, T_r, and T_C) were scaled to the maximum transition temperature, T_N, of the sodium-free sample (x_(Na) = 0) within each Ba_1−xSr_x and Sr_1−xCa_x series, with the Ba/Sr or Sr/Ca ratios constrained as described earlier. The T_N values for sodium-free samples were taken from Kirshenbaum et al.⁵⁰.