Introduction

A common characteristic of unconventional superconductors is their proximity to another ground state of broken symmetry, fluctuations of which can both mediate superconductivity and drive non-Fermi-liquid (nFL) behavior in the vicinity of its associated quantum critical (QC) point. Nematicity—a lowering of rotational symmetry without breaking translational symmetry—is one form of order that has been observed in a variety of systems, including iron-based1,2,3, cuprate4, heavy fermion5, and Moiré6 superconductors. The extent to which nematic order and its fluctuations are responsible for pairing and QC phenomena has proved a challenging question, however, largely due to the fact that nematicity often occurs in the vicinity of another, possible primary, instability. In iron-pnictides, for example, nematicity is claimed to be a spin-driven effect7 while QC phenomena observed in Sr3Ru2O7—initially attributed to a nematic quantum critical point (NQCP)8—were later found to arise in the presence of a field-tuned spin-density wave9.

FeSe is unusual in that nematic order stabilizes in the absence of static magnetism3. Below a tetragonal-to-orthorhombic distortion at Ts = 90 K, both its normal10 and superconducting (SC)11 state properties exhibit marked two-fold anisotropy. Although widely believed to be electronic in origin12, it remains unclear whether the nematic transition is driven by charge13, orbital14, or magnetic15 correlations. Nevertheless, its discovery offers a unique opportunity to test theoretical predictions for nFL or “strange metallic” behavior arising solely from critical nematic fluctuations16,17,18,19,20,21,22. To this end, a large effort has been made to elucidate the respective roles of nematic and magnetic fluctuations in shaping the normal and SC properties of FeSe23,24,25,26.

High-pressure studies on FeSe have proved to be highly instructive in this pursuit. As pressure increases, Ts is suppressed (to Ts = 0 K at p = pc) but the SC transition temperature Tc is not enhanced at pc27. Beyond the nematic state (p > pc), however, there is a marked (four-fold) increase in Tc27,28,29 that has been naturally linked to strengthening magnetic interactions30. The role of nematicity in driving nFL/QC phenomena has proved more controversial. At p = pc, the critical nematic fluctuations in FeSe are quenched13, presumably due to the emergence of long-range magnetic order before the nematic phase terminates31. In FeSe1-xSx, nematicity is also suppressed with increasing sulfur substitution, vanishing at a critical S concentration xc = 0.1732 where the nematic susceptibility also diverges12 and quantum critical transport is observed33. Since no magnetic order develops at any point across the substitution series (at ambient pressure), this divergence suggests that a genuine NQCP exists in FeSe1-xSx.

The question remains, however, whether the emergent critical nematic fluctuations are responsible for the strange metal transport seen at ambient pressure in FeSe1-xSx33,34,35,36,37,38. Although static magnetism is not stabilized at ambient pressure, low-energy spin-fluctuations, for example, are known to persist to p = 0 at low T and low x39. Moreover, quantum oscillation studies indicating a lack of divergence in the effective mass m* on approaching the NQCP40 have led to the suggestion that the critical nematic fluctuations may also be quenched at ambient pressure—in this case, due to nemato-elastic coupling or local strain effects41,42—the nFL transport then being attributed to scattering of the residual spin fluctuations. To date, however, the full evolution of m*(x) from x = 0 to x > xc is only known for a single oscillation frequency40 leaving open the question of whether or not mass enhancement occurs at other locations on the Fermi surface.

With increasing x, pc falls while pm, the onset pressure for magnetic order, increases43, leading ultimately to a separation of the nematic and magnetic phases in the (p, T) plane at higher x. Previous NMR measurements appeared to confirm such a separation at x = 0.12 (<xc)44. Detailed transport studies45 on pressurized FeSe1-xSx with x = 0.11 then revealed the absence of nFL transport or m* enhancement across pc, supporting the picture of quenched nematic criticality due to strong nemato-elastic coupling45. A more recent µSR study, however, found that magnetism at x = 0.11 is stabilized before nematicity is destroyed (the discrepancy between µSR and NMR likely reflects the different timescales of the two probes)46. Hence, it is unclear whether the suppression of nematic criticality near x = 0.12 under pressure is due to coupling to the lattice or to slowly fluctuating moments. In order to determine whether critical nematic fluctuations alone can drive nFL transport in FeSe1-xSx, pressure studies on samples with higher x values, where the nematic and magnetic phases are fully separated, are required.

Here, we study the low-T resistivity ρ(T) of FeSe1-xSx with x = 0.18 and 0.20 (>xc) under applied pressures up to 15 kbar (<pm). Whilst the form of ρ(T) cannot differentiate easily between nematic and magnetic fluctuations, tracking its evolution with p may reveal an approach to or a retreat from a QCP associated with either order parameter. In this way, their respective influences can be disentangled. For both samples studied here, we find two distinct T2 components in ρ(T) (due to quasiparticle-quasiparticle scattering) which extend over different T ranges and whose coefficients show contrasting p-dependencies. The term that grows with increasing p is attributed to the dressing of quasiparticles by critical magnetic fluctuations that strengthen upon approach to the magnetic QCP43,47. Its coefficient at ambient pressure, however, is found to be negligible. This implies that the source of the large and strongly x-dependent T2 coefficient observed at ambient pressure is the scattering of quasiparticles that are dressed purely by the orbital nematic fluctuations. Finally, this coexistence of two distinct components to ρ(T) also suggests that, in contrast to what is observed in the iron-pnictides, the critical nematic and magnetic fluctuations in FeSe1-xSx are completely decoupled.

Results

Nematic quantum critical resistivity

Figures 1a, b show, respectively, the zero-field ρ(0, T) (pale) and high-field ρ(35 T, T) curves for samples with nominal x values of 0.18 and 0.20 oriented H//I//ab at various pressures 0 ≤ p ≤ 14.4 kbar. The suppression of superconductivity by the magnetic field is apparent in all data sets. For T > Tc, there is almost complete overlap between ρ(0, T) and ρ(35 T, T), confirming that the magnetoresistance in this field orientation is negligible beyond xc33,36, in marked contrast to the large magnetoresistance seen for H//c35,36. The broadening and structure of the superconducting transitions in ρ(0, T) is highly reproducible between subsequent cooldowns at different pressures, between samples of similar dopings48, and between measurements performed by different groups47,49 indicating that non-hydrostaticity is unlikely to be playing a role here. We also note that the transitions sharpen again at higher pressures (~3 GPa)43 suggesting that this behavior is in fact intrinsic.

Fig. 1: Pressure dependence of the high-field in-plane resistivities of FeSe1-xSx beyond the nematic quantum critical point.
figure 1

a, b Zero-field resistivity ρ(0, T) (pale curves) and high-field resistivity ρ(35 T, T) (dark curves) measured at indicated pressures p between 0 and 14.4 kbar for x = 0.18 (x = 0.20) and up to a temperature T of 40 K with magnetic field H and current I both oriented in-plane and parallel to each other (H//I//ab). The curves have been offset by 10 µΩ cm for each subsequent pressure for clarity. c, d Corresponding first derivatives of the high-field resistivity for doping x = 0.18 (x = 0.20) at the same pressures as those in a, b. For clarity, the curves have been offset by 0.5 µΩ cm K−1 at each pressure. The dashed lines indicate the ρ = 0 or dρ/dT = 0 position of each curve. In panels c, d, the black lines are straight-line fits to the lowest temperature data chosen to both avoid the crossover to T + T2 behavior at ~10 K and the onset of superconductivity at low temperatures and to the higher temperature data above 20 K from which the resistivity coefficients At (A), B, and A′ have been deduced. The enhancement of superconductivity prevents At from being determined at the highest applied pressures. The increasingly broad superconducting transitions manifest themselves as shallow peaks in the derivatives that are most visible in dρ/dT(x = 0.18) above 9.8 kbar (panel c) but may influence the data at lower pressures as well. The fits at low-T have been forced through the origin. The small finite intercepts due to superconductivity at intermediate pressures are accounted for in the errors.

Full size image

The corresponding derivatives dρ/dT(35 T) of the high-field curves, shown in panels c and d of Fig. 1, reveal a systematic evolution of ρ(T) under applied pressure. To better orientate our discussion, we focus initially on the form of dρ/dT at ambient pressure. For T < 10 K, ρ(35 T, T) = ρ0 + AtT2 with At coefficients that are determined by fitting the dρ/dT traces at the lowest temperatures to a straight line through the origin (black lines in Figs. 1c, d). The temperature range fitted to was chosen in order to avoid being influenced by the clear crossover in behavior that occurs at ~10 K and the pressure-induced enhancement of superconductivity at the lowest temperatures. We argue below that At reflects the total quasiparticle-quasiparticle scattering cross-section enhanced by both magnetic and nematic critical fluctuations. Above the T2 regime, dρ/dT is essentially flat, implying that ρ(T) becomes T-linear (with coefficient B). Such a T2 to T-linear crossover is characteristic of a metallic system in the vicinity of a QCP1,50,51,52,53.

Pressure-induced growth of a purely T 2 component

A notable change in the derivative plots with increasing p is the emergence of a finite linear slope in dρ/dT at higher temperatures, indicative of a second T2 component that (i) coexists with the T-linear term, (ii) has a coefficient A′ that is around one order of magnitude smaller than At, and (iii) extends over a much broader temperature range. A′ and B are determined by fitting the dρ/dT data between 20 and 40 K to another straight line (high-T black lines in Fig. 1c, d). The fitting range was increased to 25 to 40 K for x = 0.18 at the highest pressures to again avoid being influenced by the onset of superconductivity. Whilst this second T2 component is most evident in the derivative data at high T, the expectation is, as for a correlated Fermi liquid, that it extends down to the lowest temperatures. In this way, At is most naturally interpreted as the sum of two T2 components, i.e., At = A + A′; the first component persisting up to ~10 K, the second component up to the highest temperature measured in our study (~40 K).

The p-dependence of coefficients A (At), B, and ρ0 (the latter obtained by extrapolating fits of the low-T ρ(T) curves at 35 to 0 K) is shown in Fig. 2a–c, respectively. It is immediately apparent that the relative slopes of all three quantities are the same, indicating that their p-dependencies share a common origin. The p-dependence of A′ and Tc is shown in Fig. 2d, e respectively. The strong anticorrelation of A′(p) with A(p) and B(p) indicates that its origin is distinct. It, therefore, appears that there are two distinct components: one that crosses from T2 (with coefficient A) to T-linear (with coefficient B) and a second that remains purely T2 up to at least 40 K (with coefficient A′).

Fig. 2: Pressure dependence of the resistivity coefficients and superconductivity.
figure 2

a Pressure p-dependence of the low-temperature T2 coefficient At (circles) obtained from linear fits of the derivative of the resistivity dρ/dT below 10 K (black lines in Fig. 1c, d). Also shown are the coefficients A = At – A′ (diamonds), the component of At attributed to electron–electron scattering dressed by critical nematic fluctuations. Dashed lines are linear fits to the data. The strengthening superconductivity prevents At (and A) from being determined at the highest applied pressures. b Pressure dependence of the T-linear coefficient B was obtained by fitting dρ/dT measured between 20 and 40 K to a straight line. Dashed lines are linear fits to the data. c Pressure dependence of the residual resistivity ρ0 was obtained by extrapolating the low-T ρ(T) curves at 35 T to 0 K. Values were only obtained up to the pressures at which superconducting fluctuations do not influence ρ(T). The dashed lines are extrapolations of straight-line fits to the data points. d Pressure dependence of the high-T T2 coefficient A′ as obtained from straight-line fits to dρ/dT at 35 T and above 20 K. e Pressure dependence of the superconducting transition temperature Tc defined as the temperature at which the zero-field resistivity reaches 90% of its value at 35 T. Tc in both samples exhibits an enhancement by a factor of around two. The error bars in panels a–d are reflective of the variation of the obtained coefficients to details of the fitting procedure (principally the precise choice of temperature range being fitted to). We estimate there to be an additional 30–50% systematic error due to uncertainty in sample and contact geometry. The error in the obtained values in panel e are within the size of the data points.

Full size image

Carrier density inferred from the residual resistivity

The drop in A (At), B, and ρ0 with increasing pressure could signify either a reduction in scattering or an increase in the plasma frequency ωp2 (i.e., n/m*), or some combination thereof. In the first scenario, the fall in A (At), B, and ρ0 with increasing p (depicted in Fig. 2) would be attributed directly to a reduction in the dressing of quasiparticles by the relevant critical fluctuations. While this interpretation can support a typical quantum critical scenario in which A(p) (and perhaps ρ0) drops as the system is tuned away from the NQCP, the scattering rate associated with the linear-in-T coefficient is not expected to decrease too. Indeed, the T-linear resistivity inside of the quantum critical fan in FeSe1-xSx at ambient pressure has been shown to be governed by a doping-independent scattering rate 1/τ that is tied to the Planckian limit, i.e., ħ/τ = akBT with 1 ≤ a ≤ 233.

In the second scenario, the change in all three coefficients can be ascribed wholly to an increase in n/m*. A sizeable increase in n with pressure has been deduced in both FeSe53 and FeSe0.89S0.1145 from quantum oscillation studies. Indeed, for x = 0.11, six of the eight observed oscillation frequencies (corresponding to the largest Fermi pockets) increase appreciably (50–75%) between 0 and 17 kbar45. To account for this, a rescaling factor ρ0(0)/ρ0(p) (dashed lines in Fig. 2c) can be found that assumes the decrease in ρ0 reflects a change in carrier density (and not a reduction in enhancement from the NQCP). Figure 3a (Fig. 3b) shows the p-dependence of A*, B* (A′*), the coefficients A, B, and A′ rescaled by multiplying each quantity by ρ0(0)/ρ0(p). As can be seen, the resultant A* and B* coefficients are either p-independent (for x = 0.18) or fall slightly (for x = 0.20) (note, however, the large error bars for the data at highest pressures). The near-constancy and magnitude of B*(p) is then consistent with the notion that the effective scattering rate remains at the Planckian bound with increasing pressure, in agreement with what had been found at ambient pressures33. Within a QC scenario, the near-constancy of A*(p) is also consistent with the fact that the extent of the (low-T) T2 regime in both samples does not vary with p. This is consistent with pressure tuning parallel to the nematic phase boundary in the p-T plane as indicated in Fig. 3c. By contrast, at ambient pressure At exhibits a marked decrease with increasing x beyond the NQCP (see Fig. 3d) while the temperature of the T2 to T-linear crossover increases as the system is tuned away from the NQCP by chemical substitution33,35.

Fig. 3: Decoupled nematic and magnetic criticality in FeSe1-xSx.
figure 3

a Renormalized A* coefficient (upper) and similarly renormalized B* coefficient (lower) as a function of pressure having rescaled A and B respectively by the pressure-induced change in ρ0. b Similarly renormalized A′* coefficient as a function of pressure. c Schematic T = 0 phase diagram in the p-x plane showing the nematic and magnetic phase boundaries. Vertical arrows indicate the pressure-induced approach of the two studied samples to the magnetic quantum critical point and the pressure ranges over which A and A′* can be determined. The horizontal arrow represents tuning away from the nematic quantum critical point (NQCP) with increasing x (relevant to panel d). Near x = xc, the nematic phase boundary is shown as a dashed line to reflect its putative weak first-order nature. d Variation of At*, the total low-temperature T2 coefficient at ambient pressure rescaled by the relative growth in the carrier density, and x beyond the nematic quantum critical point near xc  ~ 0.17 (red dotted line) with data from this work and literature sources33,36,37. The error bars represent the estimated uncertainty in both x and the reported coefficients. We have estimated the error in the values reported in this work to be 50% due to the constraints on sample size in a pressure cell and 30% elsewhere. The uncertainty in x has been assumed to be ±0.015 and representative of the typical variation in x within an individual batch of samples12. See Supplementary Note 1 for details of the rescaling procedure. The dashed line is a guide to the eye.

Full size image

Discussion

Irrespective of which scenario is the most appropriate, the marked increase in A′ (or in A′*) with pressure, in both samples, is a robust observation. The order of magnitude change in A′*, in particular, is even greater than that seen in At* upon approach to the NQCP at ambient pressure (Fig. 3d) and comparable to that observed in other quantum critical systems with well-established magnetic QCPs50,54,55. Moreover, the fact that A′ is anti-correlated with A and B implies that the former has a distinct origin. The marked rise in A′ is consistent with an enhancement in the quasiparticle-quasiparticle scattering cross-section upon approach to a second, distinct QCP. The absolute magnitude of A′ over our experimental pressure range (~5 nΩ cm K−2), however, is much smaller than the value that At reaches (>200 nΩ cm K−2)33,35 upon approaching the ambient pressure NQCP, as shown in Fig. 3d. This, coupled with the more extended temperature range over which this T2 term persists, suggests that the second QCP is likely to be situated at a critical pressure far beyond those accessible here. As illustrated in Fig. 2e, the approach to the second QCP also coincides with a marked (factor of 2) growth in Tc for both samples, the growth in A′ and Tc being largest for x = 0.18. As mentioned in the introduction, a marked increase in Tc with pressure at lower sulfur concentrations has been linked previously to strengthening magnetic interaction30. Indeed, it has been suggested that Tc is maximized at the magnetic QCP43 and it is known that magnetism is stabilized at higher pressures43,53. Although there have been no reports to date confirming the presence of magnetic order with increasing pressure beyond xc = 0.17, resistivity data presented in Matsuura et al.43. show that the magnetic ordering temperature at 5 GPa remains doping independent up to xc. Thus, it seems reasonable to expect magnetic order to be stabilized under pressure beyond xc and we associate this second QCP with the pressure-induced antiferromagnetic phase, and ascribe the p-dependence of the second T2 component in ρ(T) to quasiparticle-quasiparticle dressing by critical spin fluctuations in the quantum disordered regime.

Of course, there are other scattering mechanisms that are capable of generating T2 resistivity with a variable coefficient, such as non-critical electron–electron scattering near a Mott metal-insulator transition56, electron–phonon scattering in disordered systems57 or short-range spin fluctuation scattering58. However, the order-of-magnitude increase in A′* over a relatively narrow pressure range is difficult to reconcile with any of these mechanisms. One would need to invoke a pressure-induced suppression of disorder by one order of magnitude for electron–phonon scattering to be sufficient57, there is no evidence for Mottness and while spin fluctuations are found to be pressure independent in pure FeSe39, they become suppressed with pressure up to 2 GPa (the pressure range of this study) at x = 0.1259. Clearly, further studies will be required to definitively rule out these alternative explanations. However, given the known emergence of a magnetic phase boundary in FeSe1-xSx at higher pressures as well as a precedent for magnetic quantum criticality in other Fe-based55 or heavy fermion51,54 systems manifesting in a divergent T2 coefficient of the low-T resistivity, a magnetic QCP seems the most plausible.

These contrasting x- and p-dependencies (At(x) and A*(p)) may be reconciled by considering the proposed T = 0 phase diagram shown schematically in Fig. 3c. The vertical solid- and open-headed arrows represent, respectively, the pressure tuning of the x = 0.18 and 0.20 samples, while the horizontal arrow represents tuning away from the NQCP with increasing x at ambient pressure. The near-constancy of A* (within the second scenario above) may indicate that pc(x)—the phase boundary for nematic order in the (p, x) plane—is very steep near x = xc. This seems plausible given the steepness of Ts(x) near xc—see Fig. 1a in M. Čulo et al.38, for example. Consequently, with increasing p, samples with x > xc track effectively parallel to the nematic phase boundary, rather than away from it. At the same time, the application of pressure tunes each sample towards pm(x)—the magnetic phase boundary—resulting in a marked increase in A′. In this way, the contrasting variation in A(p) and A′(p) can be understood. The steepness of the pc(x) boundary might also indicate a crossover in the nematic phase transition from second-order to weakly first-order near x = xc. Such a crossover, intimated in Fig. 3c by the dashed nematic phase boundary, would lead to a cutoff in the nematic fluctuations, thereby providing an alternative explanation for the p-independence of A* and B*. It is noted that in pure FeSe, Ts(p) terminates at a first-order structural and magnetic phase transition at ~2 GPa (a divergence of 1/T1T at low T is lost)60. The T = 0 endpoint of the magnetic transition, however, appears to remain second-order60. Thus one anticipates that the magnetic phase boundary at the higher dopings measured in this study is also second-order and capable of hosting a QCP.

The presence of two anti-correlated but additive T2 components in the low-T resistivity is unusual but implies the presence of two independent scattering channels of distinct origin. Given the correlation between A′ and Tc at finite pressure and the anticorrelation between A′ and A, it seems very unlikely that spin fluctuations could be responsible for both. Indeed, while measurements of the spin-lattice relaxation rate in FeSe1-xSx at ambient pressure indicate the emergence of low-lying spin fluctuations below Ts, spin fluctuations are strongly suppressed for x > xc39. Moreover, as mentioned above, there is no evidence that such fluctuations go critical at x = xc. It would appear that spin fluctuations, as parameterized by A′ (~At/10), play only a minor role in the overall low-T resistivity in FeSe1-xSx at ambient pressure.

The measurements presented here imply that the nematic fluctuations anchored at the NQCP and the magnetic fluctuations anchored at the AFM QCP act as decoupled mechanisms for the enhancement of quasiparticle-quasiparticle scattering over most of the phase diagram of FeSe1-xSx. One possible way to account for their distinct nature is to consider the particular Fermi surface topology of FeSe1-xSx. Figure 4a shows a schematic projection of the Fermi surface of FeSe1-xSx (x > xc) at kz = 0 assuming only one hole pocket centered at Γ and two-electron pockets at X and Y. Since spin fluctuations in detwinned FeSe are peaked at Q = (π, 0)15, we also assume that in the tetragonal phase, critical spin fluctuations would enhance the quasiparticle-quasiparticle scattering cross-section predominantly at four ‘hot-spots’, as shown in Fig. 4b. The precise symmetry of the nematic fluctuations in FeSe1-xSx has not yet been confirmed. Raman spectroscopy studies have indicated the presence of a d-wave Pomeranchuk instability61,62 while quasiparticle scattering interference experiments63 have revealed a highly anisotropic spectral weight (of different orbital character) on both pockets with p-wave symmetry (lightly shaded sections in Fig. 4b). For the former, critical nematic fluctuations would dress the quasiparticle states everywhere except at the AFM hot-spots (the nodes of the d-wave Pomeranchuk deformation), while for the latter, these cold-spots would reside at the “bellies” of each pocket. Such considerations might then help us to envisage how the influence of the critical nematic or magnetic fluctuations manifests themselves as two distinct components of the T2 resistivity. Intriguingly, the in-plane magnetoresistance of FeSe1-xSx (at ambient pressure) can also be decomposed into two components35; a QC component that exhibits H/T scaling and is maximal near the NQCP and a second component that remains purely H2 (up to 35 T) and shows conventional Kohler’s scaling. It is tempting to attribute these two components as arising from these distinct nematic and spin interactions, only one of which goes critical at ambient pressure.

Fig. 4: Decoupling of the nematic and magnetic interactions and the Fermi surface of FeSe1-xSx.
figure 4

a Schematic Fermi surface of FeSe1-xSx outside of the nematic phase showing the Γ-centered hole pocket (α) and X, Y-centered electron pockets (ε and δ) at kz = 0. States on different pockets can be connected via finite-Q scattering as indicated by the gray arrows. b Schematic illustrating the distinct regions of quasiparticle dressing due to critical magnetic fluctuations (grey circles) arising from the translation of the pockets through Q = (π, 0), (0, π), and nematic (Pomeranchuk) fluctuations (lighter shaded regions on the electron/hole pockets where the quasiparticle spectral weight is reduced63).

Full size image

Finally, we turn to consider the evolution of the superconductivity in FeSe1-xSx. While there is strong evidence to suggest that low-energy spin-fluctuations play a significant role in the pairing mechanism in FeSe1-xSx23,24,25,26, the increase in Tc(p) appears to be well correlated with A′(p) (panels d, e of Fig. 2), it is striking that A′ ~ At/10 at ambient pressure yet Tc remains high (~8 K). This finding may suggest some role for nematicity in the pairing in FeSe1-xSx but clearly, further work is required to confirm this. In pnictide superconductors, where nematicity and magnetism are strongly coupled, superconductivity is most likely driven by low-energy spin fluctuations, though Tc could be enhanced by a reduction in the bare intra-pocket repulsion brought about by the nematic fluctuations7. In the case of FeSe1-xSx, the decoupling of the nematic and the magnetic fluctuations means that this cooperative process is no longer viable and as a result, Tc is not enhanced at the NQCP.

Previously, pressure tuning between two distinct QCPs was reported in the heavy-fermion compounds Ge-doped CeCu2Si264,65 and YbRh2Si2 with Ir and Co doping66. To the best of our knowledge, however, FeSe1-xSx represents the first example of a correlated metal exhibiting an enhancement in the coefficient of the T2 resistivity associated with two distinct QCPs. Clearly, the task is now to determine the universality classes associated with each criticality. In order to achieve this, however, it will be necessary to study a sample with a sulfur concentration even closer to the NQCP and to extend the pressure range (e.g., using an anvil cell) until the magnetic QCP itself is crossed. At the same time, determination of the evolution of complementary resistive properties (such as the Hall effect) with pressure may help elucidate further the nature of the two components.

Methods

Samples

Single crystals were grown via a KCl/AlCl3 chemical vapor transport method. The growth was typically performed with a source temperature of 420 oC, a deposition zone temperature of 230 oC, and with a growth time of 200 h. Their nominal dopings are x = 0.18 and 0.20. The actual S content of crystals can often be lower than the nominal value12. For both of our samples, however, the zero-field \(\rho\)(T) curves (at ambient pressure) are found to agree well with previous reports on samples with similar dopings33,36,37. Specifically, there is no kink or minimum in the derivative d\(\rho /{{{{{\rm{d}}}}}}T\) that could be attributed to a finite Ts, and the T2 regime at low-T extends up to around 8-10 K with a coefficient At ~40–55 nΩ cm K−2, compared with >200 nΩ cm K−2 for x ≤ 0.1733,36. In this work, there is heightened geometrical uncertainty associated with measuring small crystals inside a pressure cell. Whilst, the as-measured At values are ~25% lower than previous reports at the same nominal doping levels, as is evident from Fig. 3d, the values obtained are in good agreement with the general trend of At(x) with data taken from multiple groups (see Supplementary Note 1 for details).

Resistivity measurement under pressure

Resistivity was measured using a standard ac lock-in technique. Electrical contact was made to the samples by first masking the samples and sputtering gold pads. Contact to the pads was made using gold wire and DuPont 4929 silver paint. Typical contact resistances were less than 1 Ω and stable over time. Both crystals were mounted together in a single piston-cylinder pressure cell and oriented such that H // I // ab. Daphne 7373, which is known to remain hydrostatic at room temperature up to 22 kbar67, was used as a pressure transmitting medium. Resistivity measurements were performed using a standard four-point ac lock-in technique in Cell 4 of the High Field Magnet Laboratory (Radboud University, Nijmegen, The Netherlands) where a maximum magnetic field of 35 T could be applied. Temperature sweeps were performed in both field orientations (positive and negative 35 T) such that the longitudinal component could be isolated from any Hall component present due to an offset in the voltage contacts (though it is noted that the Hall contribution was found to be a near-negligible part of the total signal).