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

The quantum Griffiths singularity (QGS) is a phenomenon driven by quenched disorders that break conventional scaling invariance and result in a divergent dynamical 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-Tc 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-Tc superconductor CaFe1-xNixAsF (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 field. Our findings reveal the universality of QGS in 3D superconducting systems and unconventional high-Tc SCs, thereby substantially expanding the range of applicability of QGS.


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][2][3][4][5][6][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 .
The 1111-type FeAs-based superconductors are the earliest discovered iron-based unconventional high-Tc superconductors with the highest Tc 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 .
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 3 / 17 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 CaFe1-xNixAsF (x << 1).However, single crystal of CaFe1-xNixAsF has been challenging to grow, impeding further exploration of this material.
In this study, superconducting single crystals of CaFe1-xNixAsF (x < 5%) is 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" versus 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 CaFe1-xNixAsF, respectively, consistent with theoretical predictions and numerical simulations [31][32][33][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-Tc superconductor, which serves to catalyze new research efforts on substantially extended physical grounds of the QGS effect.

Magnetic-field-driven SMTs with multiple QCPs in CaFe1-xNixAsF single crystals
High quality CaFe1-xNixAsF single crystals were synthesized and characterized as shown in Methods and Supplementary Information S1&S2.For clarity, CaFe1-xNixAsF samples with x = 3.1%, 3.5%, 3.7% and 4.9% are denoted as sample A, B, C and D, respectively.Figs.1a-d -d, lower panels), it is evident that B// larger than 14T may be required to fully suppress superconductivity and to achieve SMT.The brown arrows in Figs.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 .

Doping tunable quasi-2D to 3D anisotropic crossover in CaFe1-xNixAsF single crystals
Electronic dimension is an important character that governs material properties and detectable via transport techniques.Here we show that Ni doping can induce an electronic dimensional crossover in the CaFe1-xNixAsF single crystals.Among the four samples investigated in this study, sample A has 3D anisotropic electronic structure while samples B, C and D have quasi-2D electronic structures, in both the normal states and the superconducting states.We shall focus on samples A and C in the main text; data from samples B and D can be found in Supplementary Information S5-8.Figs.2a-b show the normal state magnetotransport data MR vs. Bcosθ for samples A and C, respectively.Here, the magnetic field ranges from 0 T 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 > Tc.As shown in Fig. 2a, the MR curves of sample A do not scale into a single curve when plotted with Bcosθ, 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 Bcosθ (Fig. 2b), indicating the dominance of two-dimensional Fermi surface in sample C above Tc [35,36].For the superconducting state, Figs.2c-d show the angular dependence of the upper critical field (Hc2) at T = 1.58K < Tc for sample A and C, respectively.Here, Hc2 is defined as the magnetic field where the resistance becomes 50% of the normal state resistance, which is extracted from Supplementary Information S13 and S9 for sample A and C, respectively.In Fig. 2c, Hc2(θ) of sample A can be well fitted by the 3D anisotropic mass model 37 , where 2 () = 2 // /( 2 + 2 2 ) 1/2 with = 2 // / 2 ⊥ , but not the 2D Tinkham model 38 , where sample A is still three-dimensional.Here 2 // and 2 ⊥ represent the in-plane and out-of-plane upper critical field, respectively.In contrast, Fig. 2d shows the Hc2(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 (Supplementary Information S10), it is evident that sample C has a quasi-2D superconducting electronic state.
Similar analysis is performed for sample B and D as shown in Supplementary Information S5a and S7a, respectively, also showing quasi-2D electronic structure for the two samples.

QGS in quasi-2D and 3D anisotropic unconventional high-Tc 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-Tc SCs (Sample B, C and D).Fig. 3a shows the MR of sample C versus B at various temperatures ranging from 0.46 K to 7 K.Indeed, the MR isotherms exhibit a series of continuously moving crossing points (Bc, Rc) rather than a single point, indicative of QGS behavior.Fig. 3b plots the evolution of the line of "critical" points for the SMTs in terms of Bc versus T. Notably, as T decreases, Bc 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 CaFe1-xNixAsF 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, ≡ ( 0 ) −1/ with z the dynamical critical exponent, v the correlation length exponent and T0 the lowest temperature in the fitting range; Rc and Bc are the critical resistance and critical magnetic field obtained from the critical points, respectively; = − is the deviation from Bc and f(x) is an arbitrary function with f(0) = 1.For the quasi-2D sample C, the "critical" point (Bc, Rc) of a certain small critical transition region is defined as the crossing point of several R(B) curves with adjacent temperatures (details in Supplementary Information S11).The scaling results are presented in Supplementary Information S12 and Fig. 3c, which give the effective "critical" exponents zv versus T and B, respectively.Supplementary Information S12 shows that zv diverges with decreasing temperature, indicating an enhanced effect of the quenched disorder, which introduces locally ordered superconducting rare regions on the microscopic level 5,14,15 .Fig. 3c displays zv vs. B, which are found to follow the activated scaling law = * − − .Here C is a constant, = 0.6 is the 2D infinite-randomness critical exponent and * = 7.9T 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 Supplementary Information S5-8.We note that most of the Fe-based and Cu-based high-Tc superconductors exhibit only one QCP in their SMT/SIT 2,8 , with the only exception of underdoped La2-xSrxCuO4 films that exhibit two QCPs 4 .Thus, our data represents the first discovery of quasi-2D unconventional high-Tc superconductors to host QGS in their superconductor-metal quantum phase transitions.
Despite the fact that QGS is rarely found in quasi-2D unconventional high-Tc superconductors, QGS in 3D superconducting systems is even more elusive due to experimental and theoretical difficulties 41 .Fig. 3d shows MR of 3D anisotropic sample A versus B  at temperatures ranging from 0.49 K to 5.75 K; MR versus B// of sample A can be found in Supplementary Information 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//.Fig. 3e and the inset of Supplementary Information S14b show Bc versus 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 Supplementary Information S15-S16).Fig. 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 Supplementary Information S14b.The critical behavior of sample A under B  and B// is found to have only two differences: 1) Bc under B  diverges with decreasing temperature at the low-T limit, while Bc 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// (Supplementary Information 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-Tc superconductor CaFe1-xNixAsF (x = 3.1%), representing the first experimental demonstration of magnetic field driven QGS of SMT in 3D systems.

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 c onset .
curve (blue dots, extracted form 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.
As shown in Fig. 4a, the normal state in the diagram behaves as weakly localized metal above the overlapping points of c onset () and c () .The fluctuation region lies between the mean-field c2 fit .curve (red dashed line) and the c2 .curve (blue dashed line).At low T, the c onset () curve turns upwards and significantly deviates from the mean-field WHH 40 behavior below a temperature TM ~1.9 K, indicating the emerging quantum fluctuation below TM 21,42 .In particular, for temperatures below TM and for magnetic field above the mean-field WHH limit 40 , the quantum Griffiths state emerges, characterized by a diverging zv and an upturning c () at low temperatures 5 .This quantum Griffiths state can be regarded as the effect related to quenched disorder on the Abrikosov vortex lattice in the region of c2 fit < B < c * , where rare regions of large SC puddles appear when T < TM.Meanwhile, the exponentially small excitation energy causes ultraslow dynamics accompanied by diverging effective dynamical exponent around zero T, similar to 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 TM ~1.8 K in sample A to TM ~5.3 K in sample D, the later higher than any reported values in the literature (more details in Supplementary Information S7c), highlighting the peculiarity of QGS in unconventional high-Tc superconductors.
Fig. 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 c onset () or c () under B// saturates at the low-T limit, resulting in a narrow QGS region, compares to a diverging c onset () or c () under B  with decreasing T. Saturating c () 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 vortex glass state, such that the low-T divergent c () is absent.For the third case, the saturated c () 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 c () cannot be explained by weaker Josephson coupling 19 ; the anisotropic nature of the 3D electronic state in the material, on the other hand, might results in the B//-induced rare regions without the emergence of the vortex glass state 23 , leading to a saturating c ().

Summary
In summary, 3D quantum Griffiths singularities have been experimentally observed in the superconductor-metal transition of unconventional high-Tc superconductor CaFe1-xNixAsF (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.

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The robustness of QGS in FeAs-based SCs has also been confirmed.This work opens a new venue for exploring QGS physics in Fe/Cu-based high-Tc superconductors and in 3D superconducting systems.

Synthesis of CaFe1-xNixAsF single crystals
In this study, single crystals of CaFe1-xNixAsF 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℃ 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℃ and 1230℃ for 45 hours and 30 hours, respectively, to promote crystal growth.Finally, the tube was cooled down to 850℃ at a rate of 2℃/h, followed by quick cooling to room temperature.
Figs.1a-dshow similar monotonic decrease in Tc with increasing B//, but the B//-driven SMT is only observed in sample A (Fig.1a, lower panel).For samples B, C and D (Figs.1b-d, lower