Three-dimensional quantum Griffiths singularity in bulk iron-pnictide superconductors

ABSTRACT

The quantum Griffiths singularity (QGS) is a phenomenon driven by quenched disorders that break conventional scaling invariance and result in a divergent dynamic critical exponent during quantum phase transitions (QPT). While this phenomenon has been well-documented in low-dimensional conventional superconductors and in three-dimensional (3D) magnetic metal systems, its presence in 3D superconducting systems and in unconventional high-temperature superconductors (high-T_c SCs) remains unclear. In this study, we report the observation of robust QGS in the superconductor-metal transition (SMT) of both quasi-2D and 3D anisotropic unconventional high-T_c superconductor CaFe_1-xNi_xAsF (x <5%) bulk single crystals, where the QGS states persist to up to 5.3 K. A comprehensive quantum phase diagram is established that delineates the 3D anisotropic QGS of SMT induced by perpendicular and parallel magnetic fields. Our findings reveal the universality of QGS in 3D superconducting systems and unconventional high-T_c SCs, thereby substantially expanding the range of applicability of QGS.

Three-dimensional quantum Griffiths singularity in bulk unconventional iron-based superconductors.

INTRODUCTION

The superconductor-insulator (metal) transition (SIT/SMT), a prototypical example of quantum phase transition (QPT), has garnered significant attention due to its implications for understanding quantum states of matter as well as its potential applications in novel low-dimensional superconducting quantum computing devices [1–7]. In conventional SIT/SMT systems, a single quantum critical point (QCP) with power-law divergence of a single spatial or temporal correlation length as a function of non-thermal control parameters is presented [2,8,9]. The critical exponents of the divergence reflect the universality class of the quantum critical behavior [10,11]. However, the presence of quenched disorders can disrupt conventional scaling invariance and alter the universality class of the QPT, leading to the emergence of the quantum Griffiths phase, which is characterized by continuously varying critical exponents with respect to temperature and control parameters, fundamentally challenging the conventional understanding of QPT in homogeneous disordered systems [12–16]. In the quantum Griffiths phase, distinct superconducting islands (‘rare regions’) emerge and the fluctuations of the order parameter within these rare regions become non-negligible [12–16]. QGS have been experimentally observed in two-dimensional (2D) [5,17–23] and quasi-one-dimensional (quasi-1D) [7] conventional superconductors. However, the fate of QGS in three-dimensional (3D) bulk superconductors and unconventional high-T_c superconductors remains unknown.

The 1111-type FeAs-based superconductors are the earliest discovered iron-based unconventional high-T_c superconductors with the highest T_c in the material family [24]. CaFeAsF, a representative parent compound of the 1111-type FeAs-based superconductors, is an antiferromagnetic bad-metal consisting of alternately stacked CaF and FeAs layers along the c-axis [24]. Quantum oscillation measurements in CaFeAsF, along with band-structure calculations, revealed a pair of symmetry-related Dirac electron cylinders and a normal hole cylinder at the Fermi surface [25]. A nontrivial topological electronic structure has been predicted in CaFeAsF arising from strong electronic correlations of the Fe 3d electrons [26] and superconductivity in CaFeAsF can be achieved through chemical doping or external pressure [24,27,28]. Although undoped CaFeAsF is not superconducting at ambient pressure, a magnetic-field-induced metal-insulator QPT near the quantum limit has been reported [29], providing a plausible precursor for SIT/SMT in doped and superconducting CaFeAsF. Electron doping of CaFeAsF can be achieved by substituting a fraction of Fe with Ni, resulting in the formation of CaFe_1-xNi_xAsF (x ≪ 1). However, single crystals of CaFe_1-xNi_xAsF have been challenging to grow, impeding further exploration of this material.

In this study, superconducting single crystals of CaFe_1-xNi_xAsF (x <5%) are successfully synthesized using the flux method (details in Methods and Supplementary Information S1 and S2). In all the samples where SMTs are realized, multiple QCPs and diverging dynamic critical exponents are observed, providing direct evidence of QGS in these materials [5,12,18,30]. By fitting the experimental divergence behavior of ‘critical exponent’ vs. magnetic field to an activated scaling law [5], the value of the extracted exponents are 0.6 and 0.4 for quasi-2D and 3D anisotropic CaFe_1-xNi_xAsF, respectively, consistent with theoretical predictions and numerical simulations [31–34]. In particular, this is the first experimental observation of B_⊥- and B_//-driven QGS of SMT in a 3D anisotropic superconductor and in an unconventional high-T_c superconductor, which serves to stimulate new research efforts on substantially extended physical grounds of the QGS effect.

RESULTS

Magnetic-field-driven SMTs with multiple QCPs in CaFe_1-xNi_xAsF single crystals

High quality CaFe_1-xNi_xAsF single crystals were synthesized and characterized as shown in Methods and S1 and S2. For clarity, CaFe_1-xNi_xAsF samples with x = 3.1%, 3.5%, 3.7% and 4.9% are denoted as sample A, B, C and D, respectively. Figure 1a–d shows the resistance vs. temperature R(T) curves of samples A–D under both B_⊥ (upper panels) and B_// (lower panels) from 0 to 14 T in log-log scale. The R(T) curves in linear coordinates are shown in S4. The onset superconducting transition temperature at zero field () for samples A, B, C and D are 5.75 K, 6.3 K, 7.5 K and 11 K, respectively. The upper panels of Fig. 1a–d show monotonically decreasing superconducting transition temperature T_c with increasing B_⊥, ultimately leading to a weakly insulating R(T) without superconducting transition at arbitrarily low temperatures, i.e. leading to a B_⊥-driven SMT. The lower panels of Fig. 1a–d show a similar monotonic decrease in T_c with increasing B_//, but the B_//-driven SMT is only observed in sample A (Fig. 1a, lower panel). For samples B, C and D (Fig. 1b–d, lower panels), it is evident that B_// larger than 14 T may be required to fully suppress superconductivity and to achieve SMT. The brown arrows in Fig. 1a–d highlight the R(T) curves at specific ‘critical’ magnetic fields where no onset of superconductivity can be detected at the lowest measured temperature, representing the emergence of magnetic field-driven SMT in each sample. Below these critical magnetic fields, continuously changing crossing points in the R(B) isotherms can be extracted from the R(T, B) data (see Fig. 3), indicative of the presence of multiple QCPs and contrasts with conventional quantum phase transitions with a single QCP [2,8,9].

Figure 1.

Magnetic-field-driven SMT with multiple QCPs in CaFe_1-xNi_xAsF. (a–d) Temperature-dependent resistance under B_⊥ (upper panels) and B_// (lower panels) from 0 to 14 T for CaFe_1-xNi_xAsF samples A, B, C and D in log-log scale, respectively. Different colors represent different magnetic field values. The nickel (Ni) contents and the onset superconducting transition temperature for each sample are marked near the top of the figures. The brown arrows indicate the magnetic field where the field tunable SMTs occur.

Figure 3.

QGS of SMT in quasi-2D and 3D anisotropic CaFe_1−xNi_xAsF single crystals. (a) B_⊥ dependent resistance isotherms for sample C at T = 1.7–7.5 K in the main panel and 0.46–1.64 K in the inset, respectively. (b) The critical magnetic field B_c(T) extracted from (a), the red dashed line is the fitting curve from the WHH theory [40]. (c) The activated quantum scaling behavior of critical exponent zv vs. B_⊥ in sample C. The red solid line is the fitting curve from the activated scaling law and gives  7.9 T (vertical dashed line). The horizontal dashed red line shows zv = 1. (d–f) Similar data for sample A as in (a–c) for sample C under B_⊥.

Doping tunable quasi-2D to 3D anisotropic crossover in CaFe_1-xNi_xAsF single crystals

Electronic dimension is an important characteristic that governs material properties and is detectable via transport techniques. Here we show that Ni doping can induce an electronic dimensional crossover in the CaFe_1-xNi_xAsF single crystals. Among the four samples investigated in this study, sample A has a 3D anisotropic electronic structure while samples B, C and D have quasi-2D electronic structures, in both normal and superconducting states. We shall focus on samples A and C in the main text; data from samples B and D can be found in S5–S8. Figure 2a and b shows the normal state magnetotransport data MR vs. B cos θ for samples A and C, respectively. Here, MR = [R(B) − R(0)]/R(0), and the magnetic field B ranges from 0 to 14 T with its angle θ from 0° (along the c-axis) to 90° (along the a-axis, defined as the in-plane direction perpendicular to the current) and the temperature T = 14 K > T_c. As shown in Fig. 2a, the MR curves of sample A do not scale into a single curve when plotted with B cos θ, pointing to a three-dimensional Fermi surface in the normal state of sample A. On the contrary, the MR curves of sample C scale remarkably well with B cos θ (Fig. 2b), indicating the dominance of two-dimensional Fermi surface in sample C above T_c [35,36]. For the superconducting state, Fig. 2c and d shows the angular dependence of the upper critical field (H_c2) at T = 1.58 K < T_c for samples A and C, respectively. Here, H_c2 is defined as the magnetic field where the resistance becomes 50% of the normal state resistance, which is extracted from S13 and S9 for samples A and C, respectively. In Fig. 2c, H_c2(θ) of sample A can be well fitted by the 3D anisotropic mass model [37], where with , but not the 2D Tinkham model [38], where , indicating that the superconducting state in sample A is still three-dimensional. Here and represent the in-plane and out-of-plane upper critical field, respectively. In contrast, Fig. 2d shows the H_c2(θ) of sample C that can be fitted by the 2D Tinkham model but not the 3D anisotropic mass model; together with the Berezinskii−Kosterlitz−Thouless (BKT) transition [39] observed in sample C (S10), it is evident that sample C has a quasi-2D superconducting electronic state. Similar analysis is performed for samples B and D as shown in S5a and S7a, respectively, also showing quasi-2D electronic structure for the two samples.

Figure 2.

Doping tunable 3D anisotropic to quasi-2D crossover in CaFe_1-xNi_xAsF single crystals. (a and b) Normal state MR vs. perpendicular magnetic field B cos θ for sample A and C at 14 K, respectively. Here MR = [R(B) − R(0)]/R(0). The magnetic field angle θ, defined in the insets, have values 3°, 13°, 31°, 49°, 57°, 73°, 80° and 87° for sample A and 0°, 16°, 34°, 52°, 60° and 76° for sample C. (c and d) Angular dependence of the upper critical fields μ₀H_c2 for sample A and C, respectively. The blue and red dashed lines are the theoretical curves of H_c2(θ) from the 2D Tinkham model [38] and the 3D anisotropic mass model [37], respectively. Insets show the zoom-in plot near θ = 90°. H_c2 is defined as the magnetic field where the resistance becomes 50% of the normal state resistance.

QGS in quasi-2D and 3D anisotropic unconventional high-T_c SCs

Since QGS has so far been observed in 2D and quasi-1D conventional superconductors, we first investigate the possibility of QGS states in quasi-2D unconventional high-T_c SCs (samples B, C and D). Figure 3a shows the MR of sample C vs. B_⊥ at various temperatures ranging from 0.46 K to 7 K. Indeed, the MR isotherms exhibit a series of continuously moving crossing points (B_c, R_c) rather than a single point, indicative of QGS behavior. Figure 3b plots the evolution of the line of ‘critical’ points for the SMTs in terms of B_c vs. T. Notably, as T decreases, B_c exhibits continuous upward displacement, deviating from the mean field Werthamer–Helfand–Hohenberg (WHH) theory [40], in agreement with previous reports in 2D QGS of SMTs in conventional superconductors [17,18].

To gain a quantitative understanding of the QGS in CaFe_1-xNi_xAsF samples, we employed finite-size scaling (FSS) analysis to investigate the critical points of the SMTs. The resistance of the sample near these critical points follows a scaling form [1]: . Here, where z is the dynamical critical exponent, v is the correlation length exponent and T₀ is the lowest temperature in the fitting range; R_c and B_c are the critical resistance and critical magnetic field obtained from the critical points, respectively; is the deviation from B_c and f(x) is an arbitrary function with f(0) = 1. For the quasi-2D sample C, the ‘critical’ point (B_c, R_c) of a certain small critical transition region is defined as the crossing point of several R(B) curves with adjacent temperatures (details in S11). The scaling results are presented in S12 and Fig. 3c, which give the effective ‘critical’ exponents zv vs. T and B, respectively. S12 shows that zv diverges with decreasing temperature, indicating an enhanced effect of the quenched disorder, which introduces locally ordered superconducting rare regions at the microscopic level [5,14,15]. Figure 3c displays zv vs. B, which are found to follow the activated scaling law . Here C is a constant, is the 2D infinite-randomness critical exponent and is the divergent critical field, as obtained from the fitting. The value of in 2D systems with infinite-randomness are predicted to be ∼0.6 ( ≈ 1.2 and ≈ 0.5) [33,34], which agrees well with our experiment, providing strong evidence for the existence of QGS in quasi-2D sample C. Additional data on QGS in other quasi-2D samples B and D are presented in S5–S8. We note that most of the Fe-based and Cu-based high-T_c superconductors exhibit only one QCP in their SMT/SIT [2,8], with the only exception of underdoped La_2-xSr_xCuO₄ films that exhibit two QCPs [4]. Thus, our data represents the first discovery of quasi-2D unconventional high-T_c superconductors to host QGS in their superconductor-metal quantum phase transitions.

Despite the fact that QGS is rarely found in quasi-2D unconventional high-T_c superconductors, QGS in 3D superconducting systems is even more elusive due to experimental and theoretical difficulties [41]. Figure 3d shows MR of 3D anisotropic sample A vs. B_⊥ at temperatures ranging from 0.49 K to 5.75 K; MR vs. B_// of sample A can be found in S14. Surprisingly, the MR isotherms exhibit continuous movement of crossing points as sample temperature changes, indicative of QGS behavior of sample A under both B_⊥ and B_//. Figure 3e and the inset of S14b shows B_c vs. T for B_⊥ and B_// configurations, respectively, which show similar deviation from the WHH theory at the low T regime. FSS analysis is carried out for data collected from 3D anisotropic sample A (details in S15–S16). Figure 3f shows the resulting zv vs. B_⊥ curve for sample A, following the same activated scaling law with = 4.07 T and υψ ≈ 0.4. Numerical simulations of 3D random quantum magnets [31,32] have predicted a correlation length exponent υ ≈ 0.98 and a tunneling critical exponent ψ ≈ 0.46, resulting in υψ ≈ 0.45, in close agreement with our experiment. The analysis of zv vs. B_// for sample A can be found in S14b. The critical behavior of sample A under B_⊥ and B_// is found to have only two differences: (1) B_c under B_⊥ diverges with decreasing temperature at the low-T limit, while B_c under B_// appears to saturate at low temperatures. This phenomenon will be discussed in details in the Discussion Section 2) the for the diverging zv is 4.07 T for B_⊥ (Fig. 3f) and 12.41 T for B_// (S14b), highlighting the weakly anisotropic nature of the 3D electronic structure of sample A. Apart from these two differences, the behavior of sample A under B_⊥ and B_// is essentially the same, including the fitted critical exponents υψ ≈ 0.4, which are both in agreement with numerical simulations [31,32]. This remarkable consistency between the experimental data and the numerical simulations provides compelling evidence for the presence of QGS in the 3D unconventional high-T_c superconductor CaFe_1-xNi_xAsF (x = 3.1%), representing the first experimental demonstration of magnetic field driven QGS of SMT in 3D systems.

B-T phase diagram and discussions

Based on data from sample A, B-T phase diagrams of bulk 3D anisotropic unconventional Fe-based superconducting system with QGS under B_⊥ (Fig. 4a) and B_// (Fig. 4b) are constructed. The phase diagrams are characterized by three curves: (1) the superconducting onset curve (purple squares) which coincides with the curve (orange dots); (2) the mean field WHH [40] upper critical field curve (red dashed line); (3) the upper critical field curve (blue dots, extracted from 90% of the normal resistance). These three curves divide the phase diagram into four regions: the SC state, the fluctuation region (including ‘QF’: quantum fluctuations and ‘TF’: thermal fluctuations), the QGS state and the normal state.

Figure 4.

B–T phase diagram of QGS in a 3D anisotropic superconductor. (a, b) The phase diagrams are constructed with data from CaFe_1−xNi_xAsF sample A under B_⊥ and B_//, respectively. The QGS phase is the new discovery from this work. Purple squares and orange dots: from R(B) vs. T curves and vs. T curves, respectively. Blue dots: H_c2 vs. T defined as 90% of the normal state resistance. Red and blue dashed lines: fitted curve of to the WHH theory [40]. ‘QF’ and ‘TF’ denotes quantum fluctuations and thermal fluctuations, respectively. (c) Robust QGS in CaFe_1-xNi_xAsF single crystals. We plotted the QGS emerging temperature T_M for sample A, B, C and D together with T_M for all the other reported QGS superconductor-metal transitions in the literature [refs. 5,7,17–21,23,44–48]. It can be seen that T_M for CaFe_1-xNi_xAsF single crystals are much higher than other reported values, underscoring the robustness of the QGS in unconventional high-T_c superconductors.

As shown in Fig. 4a, the normal state in the diagram behaves as weakly localized metal above the overlapping points of and . The fluctuation region lies between the mean-field curve (red dashed line) and the curve (blue dashed line). At low T, the curve turns upwards and significantly deviates from the mean-field WHH [40] behavior below a temperature T_M ∼1.9 K, indicating the emerging quantum fluctuation below T_M [21,42]. In particular, for temperatures below T_M and for magnetic field above the mean-field WHH limit [40], the quantum Griffiths state emerges, characterized by a diverging zv and an upturning at low temperatures [5]. This quantum Griffiths state can be due to the quenched disorder on the Abrikosov vortex lattice in the region of < B < . Such quenched disorder produces rare regions of large SC puddles below T_M, and the exponentially small excitation energy gives rise to ultra-slow dynamics with diverging effective dynamical exponent around zero T. This is consistent with the behavior of large clusters in the random transverse field Ising model [43]. It is worth noting that QGS in samples A–D are conspicuously robust [18] (shown in Fig. 4c), with T_M ∼1.8 K in sample A to T_M ∼5.3 K in sample D, the latter higher than any reported values in the literature (more details in S7c), highlighting the peculiarity of QGS in unconventional high-T_c superconductors. Besides, different from the QGS mostly found in superconducting films, flakes, or interfaces (shown in Fig. 4c), we have found QGS of SMT in large-sized bulk single crystals, which may enable several experiments that are very challenging for thin films, such as heat capacity and magnetization measurements. This may bring new perspectives to the study of quantum Griffith phase in this class of materials in the future.

Figure 4b depicts the B-T phase diagram of the same sample under B_//, which exhibits behavior similar to that under B_⊥, with one key difference: the or under B_// saturates at the low-T limit, resulting in a narrow QGS region, compared to a diverging or under B_⊥ with decreasing T. Saturating has been previously reported in only three superconducting systems, the B_//-driven QGS of SMT in few-layer PdTe₂ films [23] and in β-W films [21], as well as the B_⊥-driven QGS of SIT in TiO films [19]. The former two cases [21,23] are considered to be QGS without the formation of a vortex glass state, such that the low-T divergent is absent. For the third case, the saturated is attributed to weaker Josephson coupling of the local rare regions in an insulating normal state background [19]. In our case, since the resistance of sample A is much less than previous reports of SMT/SIT in QGS [19,21,23], a saturating cannot be explained by weaker Josephson coupling [19]; the anisotropic nature of the 3D electronic state in the material, on the other hand, might result in the B_//-induced rare regions without the emergence of the vortex glass state [23], leading to a saturating .

CONCLUSION

In summary, 3D quantum Griffiths singularities have been experimentally observed in the superconductor-metal transition of unconventional high-T_c superconductor CaFe_1-xNi_xAsF (x = 3.1%) single crystals. A comprehensive quantum phase diagram for the 3D-QGS of SMT was established, which demonstrates the universality and similarity of QGS in 2D and 3D SC systems. The robustness of QGS in FeAs-based SCs has also been confirmed. This work opens a new avenue for exploring QGS physics in Fe/Cu-based high-T_c superconductors and in 3D superconducting systems.

METHODS

Synthesis of CaFe_1-xNi_xAsF single crystals

In this study, single crystals of CaFe_1-xNi_xAsF were synthesized using the CaAs self-flux method. Initially, a mixture of Ca granules and As grains in a 1 : 1 ratio was heated at 700°C for 10 hours in an evacuated quartz tube to obtain the CaAs precursor. The CaAs precursor was then subjected to a grinding and sintering process, repeated three times to ensure complete mixing and uniformity of CaAs, which is critical for obtaining high-quality single crystals. Subsequently, Fe powder, Ni powder, FeF₂ powder, and the homemade CaAs flux were mixed together in a stoichiometric ratio of 1−x: x: 1 : 15 (x = 4%, 6%, 8%, 10%), placed in an alumina crucible, and sealed in a quartz tube under vacuum. The sealed quartz tube was then heated at 950°C and 1230°C for 45 hours and 30 hours, respectively, to promote crystal growth. Finally, the tube was cooled down to 850°C at a rate of 2°C/h, followed by quick cooling to room temperature.

Characterization of CaFe_1-xNi_xAsF superconducting single crystals

The CaFe_1-xNi_xAsF crystals exhibited a tetragonal structure, consisting of alternating CaF and (Fe_1-xNi_x)As layers along the c-axis, as depicted in S1a. A typical high quality CaFe_1-xNi_xAsF single crystal (sample C, x = 3.7%) has a rectangular shape and ∼4 ×3 × 0.15 mm in size, as shown in the inset of S1b. X-ray diffraction of the CaFe_1-xNi_xAsF crystals (S1c) confirmed the high crystallinity of the as-grown crystals with (001) orientation, consistent with the parent CaFeAsF compound [49]. Further characterization using scanning electron microscopy and energy dispersive spectrometer analysis (details in S2) revealed flat as-grown (001) surfaces with definite Ni contents for different crystals. The above results prove the first successful growth of high-quality CaFe_1-xNi_xAsF single crystals.

Transport measurements

Transport measurements were conducted in the temperature range of 0.3 K to 300 K, under magnetic fields of up to 14 T, using both an Oxford Teslatron cryostat and a Quantum Design PPMS. The resistance was measured at a frequency of 17.77 Hz using conventional lock-in techniques.

FUNDING

This work was supported by the National Key R&D Program of China (2019YFA0308402), the Innovation Program for Quantum Science and Technology (2021ZD0302403) and the National Natural Science Foundation of China (11934001 and 92265106 and 11921005).

AUTHOR CONTRIBUTIONS

J.-H.C. & S.L. conceived the idea; J.-H.C. directed the experiment; S.L., Y.C. & C.C. provided high-quality crystals; S.L. fabricated most of the devices and performed the transport measurements; C.T., H.C., X.W., M.C., J.L., Y.F., J.C., Y.Z., Y.S. & T.H. aided in sample preparation and transport measurement; J.-H.C., S.L., C.T., Y.Z. & Y.S. collected and analyzed the data; S.L. & J.-H.C. wrote the manuscript; all authors commented and modified the manuscript.

Conflict of interest statement. None declared.