Electrical resistivity across a nematic quantum critical point

Abstract

Correlated electron systems are highly susceptible to various forms of electronic order. By tuning the transition temperature towards absolute zero, striking deviations from conventional metallic (Fermi-liquid) behaviour can be realized. Evidence for electronic nematicity, a correlated electronic state with broken rotational symmetry, has been reported in a host of metallic systems^1,2,3,4,5 that exhibit this so-called quantum critical behaviour. In all cases, however, the nematicity is found to be intertwined with other forms of order, such as antiferromagnetism^5,6,7 or charge-density-wave order⁸, that might themselves be responsible for the observed behaviour. The iron chalcogenide FeSe_1−xS_x is unique in this respect because its nematic order appears to exist in isolation^9,10,11, although until now, the impact of nematicity on the electronic ground state has been obscured by superconductivity. Here we use high magnetic fields to destroy the superconducting state in FeSe_1−xS_x and follow the evolution of the electrical resistivity across the nematic quantum critical point. Classic signatures of quantum criticality are revealed: an enhancement in the coefficient of the T² resistivity (due to electron–electron scattering) on approaching the critical point and, at the critical point itself, a strictly T-linear resistivity that extends over a decade in temperature T. In addition to revealing the phenomenon of nematic quantum criticality, the observation of T-linear resistivity at a nematic critical point also raises the question of whether strong nematic fluctuations play a part in the transport properties of other ‘strange metals’, in which T-linear resistivity is observed over an extended regime in their respective phase diagrams.

Main

The crystal structure of pristine FeSe is shown in Fig. 1a. At the structural phase-transition temperature T_s ≈ 90 K, FeSe undergoes a tetragonal-to-orthorhombic transition that is accompanied by a lowering of the symmetry of the electronic state, manifest as a deformation of the Fermi surface from fourfold to twofold symmetry¹¹. At T = T_s, the nematic susceptibility (extracted from elasto-resistance measurements) is found to be strongly peaked and follows a Curie–Weiss law⁹. Crucially, there is no evidence for magnetic order or enhanced spin fluctuations around T_s (ref. ¹⁰). With S substitution, T_s is suppressed and ultimately, driven to zero temperature at a ‘critical’ S concentration x_c ≈ 0.16 (Fig. 1b), at which the Fermi surface distortion vanishes^12,13, the nematic susceptibility is maximized and the effective Curie–Weiss temperature is zero⁹.

Fig. 1: Crystal structure and phase diagram of FeSe_1−xS_x.

a, The crystal structure of FeSe. b, The composition versus temperature phase diagram of FeSe_1−xS_x extracted from our transport measurements. The orange region indicates the nematic phase, the blue area depicts the superconducting phase and the region below the purple dashed line represents the range of quasi-T-linear resistivity above T_c. The closed squares show the actual T_s values for our crystals, obtained from the position of the asymmetric kink in the temperature derivative curves displayed in Extended Data Fig. 8. For x ≥ 0.16, the asymmetric kink vanishes and is replaced by a symmetric minimum that gradually softens with increasing x, but extends out to x = 0.25. The dashed red line is a guide to the eye. c, Zero-field resistivity curves for a series of FeSe_1−xS_x single crystals in the temperature range 4.2–150 K. The curves are shifted vertically with respect to each other for clarity. The curves for x = 0.18 and x = 0.2 are divided by a factor of 1.2 for convenience.

Source Data

The nematic phase (orange region in Fig. 1b) is identified by a kink in the derivative of the in-plane resistivity ρ_ab(T, x), the raw data for which are displayed in Fig. 1c (for the derivatives themselves, see Extended Data Fig. 8). In contrast to what is found, say, in the Fe-pnictides¹⁴, the superconducting phase (blue area in Fig. 1b) shows little variation with x; its transition temperature T_c ≈ 8–10 K for 0 ≤ x ≤ 0.25, the range of x studied here. Between T_s and T_c, ρ_ab(T) of FeSe_1−xS_x exhibits an anomalous quasi-linear T-dependence over a wide range of x (ref. ⁹). This is reminiscent of the ‘strange metal’ behaviour found in the systems highlighted above and raises the question: does the limiting low-T behaviour of ρ_ab(T) in FeSe_1−xS_x also exhibit the hallmarks of quantum criticality? To address this question, it is necessary to suppress the superconductivity in FeSe_1−xS_x, for example, using high magnetic fields, and to follow the evolution of ρ_ab(T, x) down to low temperatures.

In anisotropic or quasi-two-dimensional superconductors, the magnetic field H is typically applied perpendicular to the conducting planes, where the upper critical field H_c2 is smallest^15,16,17. In this orientation, however, there is invariably a sizeable transverse orbital magnetoresistance that in some cases follows a non-analytical form, thus making it difficult to extract the intrinsic (zero-field) normal state resistivity. The anisotropy in H_c2 in FeSe_1−xS_x, however, is only a factor of two, making it possible to access the normal state using static (direct current) magnetic fields while at the same time minimizing the magnetoresistance by studying the evolution of ρ_ab(T, x) in the longitudinal configuration. To illustrate this point, we show in Fig. 2a a comparison of the normal-state magnetoresistance (for x = 0.16) with B//c and B//ab up to 15 T. While the transverse magnetoresistance is already around 20%, the longitudinal magnetoresistance is negligible (and remains so, in fact, up to 35 T), implying that the resistivity measured at maximal field is the same as the resistivity that would have been measured in zero field in the absence of superconductivity. For all x > x_c, the longitudinal magnetoresistance is negligible, as exemplified in Extended Data Fig. 2a for x = 0.2. In samples with lower S concentration, the longitudinal magnetoresistance is finite, though even here, the magnetoresistance merely shifts the resistivity vertically, without adversely affecting its overall temperature dependence. A representative set of longitudinal magnetoresistance curves is displayed in Fig. 2b for x = 0.13 (see Extended Data Fig. 3 and discussion in Methods for more details). The only exception to this behaviour is pure FeSe, where the longitudinal magnetoresistance is largest (about 25%) in the normal state above T_c but becomes negligible at temperatures below 4 K (Extended Data Fig. 2b).

Fig. 2: Longitudinal magnetoresistance in FeSe_1−xS_x.

a, Magnetoresistance curves for FeSe_1−xS_x (x = 0.16) up to 15 T at T = 20 K with the magnetic field B//c (thin orange line) and B//ab (thick black line). The inset shows the B//ab curve up to 35 T. b, Set of longitudinal magnetoresistance curves up to 35 T for x = 0.13 at selected temperatures. c–f, Zero-field (thin blue lines) and high-field (thick black lines) resistivity curves below T = 25 K for x = 0, 0.13, 0.16 and 0.25, respectively (the magnetic field orientation is always B//ab). The purple solid circles in c and d are obtained from individual field sweeps taken below 1.5 K in a He-3 cryostat. The red dashed line in e is a linear fit whereas the green dashed line in f is a quadratic fit.

Source Data

Figure 2c–f displays resistivity curves measured in both zero field and in a 35 T field (B//ab) for four representative x values. (The data for all eight crystals are shown in Extended Data Fig. 4). For samples with x < x_c (Fig. 2c and d), ρ_ab(T) at 35 T (thick black curve) decreases linearly with temperature down to approximately 10 K, at which point it passes through an inflection point. For x_c = 0.16 (Fig. 2e), ρ_ab(T) is strictly T-linear from 13.5 K down to at least 1.5 K, below which superconducting fluctuations are still discernible (as shown in Extended Data Fig. 10, H_c2 exhibits a slight peak at x ≈ x_c). For x > x_c, ρ_ab(T) ≈ T² over an extended temperature range (Fig. 2f).

To determine the precise form of ρ_ab(T), we examine its temperature derivative dρ_ab/dT. This is plotted in Fig. 3a for FeSe (at 35 T). (Again, other derivatives are shown in Extended Data Fig. 5). Below a temperature marked T₂ = 4.5 ± 0.5 K, dρ_ab/dT is T-linear with a zero intercept, implying that ρ_ab(T) is strictly quadratic below T₂, as confirmed by plotting ρ_ab(T) as a function of T² (inset of Fig. 3a). Note that the magnitude of the T² component of ρ_ab(T) far exceeds the residual resistivity. Above a second temperature scale T₁ = 10.5 ± 1.5 K, dρ_ab/dT is constant, implying that ρ_ab(T) is strictly T-linear. As mentioned above, the longitudinal magnetoresistance for x = 0 is negligible below 4 K. Thus, the crossover from T-linear to T²-resistivity with decreasing T is a robust feature of the data—only the inflection point in ρ_ab(T) becomes less pronounced with decreasing field strength.

Fig. 3: T²-resistivity in FeSe_1−xS_x below the superconducting dome.

a, The temperature derivative of the resistivity for FeSe measured in a field of 35 T (B//ab). The solid line is obtained from the temperature sweep at 35 T. The solid circles are obtained from individual field sweeps taken below 1.5 K in a He-3 cryostat. The blue dashed line indicates that the form of ρ_ab(T) is strictly quadratic up to a temperature T₂ indicated by the arrow. The slope is equivalent to 2A, where A is the coefficient of the T² term. The second arrow identifies the temperature T₁ above which ρ_ab(T) becomes T-linear, as highlighted by the black dashed horizontal line. The inset shows the same ρ_ab(T) data plotted as a function of T². The green dashed line is a linear fit from which the A coefficient can also be determined. b, Evolution of A (solid black triangles) as a function of x, showing a marked reduction beyond x_c = 0.16. (For details on the procedure used to extract the A coefficients, see Extended Data Figs. 5–7 and accompanying section in Methods). For x = 0.16 we show only an estimate for the lower bound of the error bar, extracted from the temperature derivative of the longitudinal resistivity assuming that ρ_ab(T) crosses over to a T² behaviour (see Extended Data Fig. 11). For x < x_c, A shows a modest increase with increasing x, although once the nematic order is suppressed, A experiences a fivefold reduction. Vertical error bars represent a compound of the difference in slope maxima and minima associated with the scatter in the data and of uncertainties in geometrical factors. The open green triangles show, for comparison, the magnitude of the superconducting gap 2Δ (on the hole pocket) inferred from scanning tunnelling microscopy²¹.

Source Data

A quadratic T-dependence of the limiting low-T resistivity is a hallmark of a (correlated) Fermi liquid with dominant electron–electron scattering. The coefficient of the T² term in FeSe is A = 260 ± 55 nΩ cm K⁻². This coefficient contains important information about the enhancement of the quasiparticle effective mass m^*, as encapsulated in the empirical Kadowaki–Woods ratio that links the coefficient of the T²-resistivity to the square of the electronic specific heat coefficient¹⁸. As described in Methods, the magnitude of A in FeSe is found to be in good agreement with the known Fermi surface parameters for the two cylindrical pockets revealed by quantum oscillation experiments.

A T² resistivity was found for all x ≠ x_c at low T. The resulting variation of A with x, plotted (as solid black triangles) in Fig. 3b, follows closely that of the superconducting energy gap Δ (open triangles in Fig. 3b) on the hole band¹⁹. The marked asymmetry in both A and Δ across x_c implies that the coupling of quasiparticles to nematic fluctuations is different on either side of x_c and that this in turn may influence the pairing strength. This is not the whole story, however. The evolution of the low-T form of the resistivity in FeSe_1−xS_x with increasing x, from Fermi liquid to non-Fermi liquid and back to Fermi-liquid-like, together with the variation in T₂ with doping and its depression at x = x_c (see Fig. 4b), are classic signatures of quantum criticality in metals. Typically, the reduction in T₂ is accompanied by a divergence in A. Although the raw data do not show a divergence in A, the analysis outlined below suggests that this third key characteristic of critical behaviour is also realized in FeSe_1−xS_x.

Fig. 4: Quantum critical behaviour in the charge transport of FeSe_1−xS_x.

a, Modified coefficient A^* versus x. The black symbols are extracted from the dρ_ab/dT curves, including the lower limit for x = 0.16 (see Supplementary Fig. 11). The dashed line is a fit to a phenomenological model for the divergence of the effective mass near a nematic quantum critical point²¹. Vertical error bars are the same as in Fig. 3b, but rescaled by a factor A^*/A. b, Evolution of the temperature scales T₁, defining the onset of T-linear resistivity (orange hexagons), and T₂, marking the extent of the T² resistivity (blue circles). The dashed lines are guides to the eye. (For more insight on the determination of T₂ and T₁ for x = 0 16, see section ‘A coefficients and the temperature scales T₁ and T₂ for FeSe_1−xS_x’ in Methods). Vertical error bars originate from the standard deviation of the experimental data. c, d(ħ/τ)/dT versus x. The calculated values fall at or slightly above k_B for all the samples studied. Vertical error bars represent a compound of the standard deviation in the derivatives and of uncertainties in geometrical factors. d, Schematic phase diagram of FeSe_1−xS_x, highlighting the critical behaviour of A^*, T₂ and T₁ around x_c, together with the T-dependence of the resistivity across the phase diagram.

Source Data

While the proportionality of A with (m^*)² is well established, the magnitude of A also depends on the carrier concentration n (ref. ¹⁸). In FeSe_1−xS_x, it has been shown that both the electron and hole pockets expand with increasing S content^12,20. Were m^* to remain constant across the series, A would fall as both x and the effective carrier density increases. By contrast, A is found to grow with increasing x, at least up to x_c = 0.16 (see Fig. 3b). To compensate for the increase in carrier density, m^* must increase markedly over the same range of x. To be more quantitative, we first take into account the influence of n on A. Using experimental data for the Fermi surface areas in FeSe_1−xS_x (ref. ²⁰), we extract a new parameter A^* that more faithfully tracks the evolution of m^*2 across the phase diagram (see Extended Data Fig. 9 and accompanying explanation in Methods for more details). The result is shown in Fig. 4a. The variation in A^* at low x exhibits a clear enhancement on approaching x_c from either side, indicating a tendency for m^* to diverge at x_c. This enhancement in A^* is matched by a corresponding decrease in T₂ (and T₁), which both approach zero at x ≈ x_c (Fig. 4b). All these observations are consistent with those found in other quantum critical metals and suggest a strong coupling of itinerant fermions to quantum fluctuations of the relevant order parameter.

A crucial question then emerges: what is the relevant order parameter? The dashed line in Fig. 4a shows the expected enhancement in A near a nematic quantum critical point²¹. Although this appears consistent to within experimental uncertainty, as discussed below, there is no known theory for a T-linear resistivity down to T = 0 at a nematic quantum critical point in a clean system. Whereas FeSe exhibits only nematic order below T_s, a spin-density-wave state is stabilized under relatively modest pressures²², and the spin-lattice relaxation rate (at ambient pressure) shows evidence for enhanced spin fluctuations below T_s (ref. ²³), though not at T_s itself. Moreover, critical behaviour can be seen in the muon spin depolarization rate for T_s > T > T^* (about 10 K)²⁴, the same range over which ρ_ab(T) is quasi-T-linear, suggesting a possible link between T-linear resistivity and antiferromagnetic, rather than nematic, fluctuations.

Although there exists a region in the temperature–pressure phase diagram of pure FeSe, in which nematic and spin-density-wave states overlap or coexist, with increasing x, these two phases become decoupled²⁵. The range of nematic order shrinks, eventually vanishing at x_c while the dome of spin-density-wave orders shifts to higher pressures²⁵. Thus, at x = x_c, the spin-density-wave phase is located far from the ambient pressure axis at which our experiments are conducted. Moreover, nuclear magnetic resonance (NMR) experiments have shown that spin fluctuations, although present in FeSe_1−xS_x at low x values, are strongly suppressed by S substitution²³. These combined results imply that the critical behaviour we observe at x = x_c cannot be associated with proximity to a magnetic phase.

The T² resistivity appearing below the quantum critical fan suggests a predominance of umklapp-type processes involving large-Q momentum transfer. Such processes ought to be negligible in low-carrier-density metals near a Q = 0 Pomeranchuk instability where the effective interaction is long-range²⁶, particularly metals with convex and simply connected Fermi surfaces²¹. Thus, naively, one would not expect signatures of nematic quantum criticality in ρ(T). Such arguments, however, neglect the role of impurity scattering, which is an important source of momentum relaxation, particularly in multi-band systems like FeSe that contain bands of different mass²⁷. Whenever impurity scattering is more efficient than umklapp processes at relaxing momentum, both quantum critical scaling of the T² resistivity and a non-Fermi-liquid form of the resistivity at the quantum critical point^28,29 can emerge. It is unlikely, however, that such conditions are satisfied in pristine FeSe where the T² resistivity is by far the dominant contribution to ρ(T), yet its low-T resistivity is identical to that of more disordered FeSe_1−xS_x.

These arguments suggest therefore that we may need to look beyond the long-wavelength critical fluctuations associated with the Pomeranchuk instability. Recently it has been argued that FeSe possesses Q = (π, 0) antiferroquadrupolar order, which could also lead to a breaking of C₄ symmetry³⁰. According to Yu and Si³⁰, coupling of this antiferroquadrupolar order to the itinerant fermions will suppress any tendency towards antiferromagnetism and thus offers an explanation of why the structural transition at T_s is not accompanied by static antiferromagnetism. More importantly, this coupling could provide a channel for momentum relaxation and clearly motivates a need to calculate the electrical resistivity within this model. Antiferroquadrupolar order may also exist in the pnictides, though nematicity is not necessarily the dominant order parameter there. Nevertheless, the evolution of the normal state in-plane resistivity reported here and summarized in the phase diagram of Fig. 4d raises the prospect that nematic fluctuations may have a much stronger influence on the transport properties of other so-called ‘strange metals’ than previously thought, though disentangling the effects of nematicity from other competing or co-existing order parameters will remain a challenge.

Finally, we discuss the character of the T-linear resistivity above T₁. Combining the Fermi surface parameters for FeSe_1−xS_x with our values for dρ_ab/dT, we extract an estimate for the transport scattering rate ħ/τ = αk_BT within the quantum critical fan (see Methods for more details) where ħ is the reduced Planck constant and k_B is the Boltzmann constant. Intriguingly, α ≈ 1 for all x (Fig. 4c). Recently, this inequality was found to be a generic property of a host of strongly correlated electron systems exhibiting a T-linear resistivity³¹. Comparison with the metamagnetic metal Sr₃Ru₂O₇ is quite informative in this respect as both FeSe_1−xS_x and Sr₃Ru₂O₇ show T-linear resistivity at the critical magnitude despite the fact that thermodynamically, not all sections of the Fermi surface show enhanced masses on approach to the quantum critical point^20,31. As discussed in ref. ³¹, this strongly suggests that the quantum criticality in both FeSe_1−xS_x and Sr₃Ru₂O₇ is associated with the onset of efficient scattering, with strength proportional to T, which affects all of the quasiparticles.

Methods

Growth and characterization of samples

Single crystals of FeSe_1−xS_x were grown by the chemical vapour transport technique. A mixture of Fe, Se and S powders together with KCl and AlCl₃ powders was sealed in an evacuated SiO₂ tube and heated to 390–450 °C on one end, while the other end was kept at 140–200 °C. The actual sulphur composition x was determined by energy dispersive X-ray spectroscopy and found to be about 80% of the nominal x. All x values quoted are the nominal values.

Measurements

The crystals were cut into rectangular-shaped samples and six electrical contacts applied to each sample in a Hall bar geometry. The contacts were made manually by indium-soldering 25-μm-diameter Au wires onto the samples with a sharp hot tip. To ensure a uniform current flow, a layer of Dupont 4929 Ag paint was added on the two edges around the indium point contacts employed for the current flow. The resistances of both sides of the samples were always measured simultaneously and showed consistent behaviour, confirming the homogeneous flow of current through each sample. The data were acquired using a standard four-probe technique, applying a constant current of 1 mA for temperatures above 4.2 K, or 500 μA below, with lock-in amplifiers and transformers used to measure the response of the samples.

All magnetotransport measurements were carried out at the High Field Magnet Laboratory (HFML) in Nijmegen. All samples were preliminarily characterized in a superconducting magnet with a maximum magnetic field of 16 T. The high-field data were acquired in a resistive Bitter magnet with a maximum field of 35 T using a combination of He-4 and He-3 cryostats. All high-field measurements were performed employing the same transport probe, equipped with a rotating sample holder, a Cernox CX-SD temperature sensor, an RS Pro foil strain gauge heater and an Arepoc s.r.o. LHP-NUs Hall probe. The orientation of the samples with respect to the applied magnetic field was determined by first using the Hall probe signal to orient the rotating platform, and then the magnetoresistance of the sample itself to locate the longitudinal field orientation more precisely. See Extended Data Fig. 1 and Methods section ‘Thermometer calibration in field’ for details about the temperature determination in the high-field environment.

Thermometer calibration in field

The Cernox resistance thermometer used for the high-field experiments was calibrated in zero magnetic field in the range 0.3–300 K. To obtain a reliable temperature reading while varying the temperature in a constant magnetic field of 35 T, the magnetoresistance of the Cernox had to be taken into account. Field sweeps at constant temperature were used to record the Cernox output at zero field (T(0 T)) and at 35 T (T(35 T)) at numerous temperatures. To stabilize temperature in the range 0.3 K < T < 1.4 K, the He-3 vapour pressure was varied without applying any heat into the sample space. For 1.4 K < T < 4.2 K, He-4 vapour pressure was similarly used. The 4.2 K temperature point was recorded by putting the sample in good thermal contact with a liquid He-4 bath at ambient pressure. For temperatures above 4.2 K, a constant heater power was applied through a strain gauge, controlled by a Lakeshore temperature controller. Once the temperature had been stabilized at zero field (typically within a precision of a few millikelvin for 10 min), the field was then ramped. In this way, the plot in Extended Data Fig. 1 was obtained. The data points are divided into four intervals (identified by different colours) in which the temperature dependence of the magnetoresistance has a slightly different form. Third-order polynomial fits, with different fitting parameters, were then calculated for the first three ranges, while a linear fit was calculated for the highest temperature range. The fits are represented by the solid lines in Extended Data Fig. 1. The inset displays a blow-up at the lowest temperatures. After acquiring the 35 T temperature ramps shown in the main paper, the temperature was then corrected by means of the fitting functions plotted in Extended Data Fig. 1.

Longitudinal magnetoresistance of FeSe_1−xS_x

To probe the normal state of FeSe_1−xS_x in a high magnetic field, it is crucial to ensure as far as possible that the effect of the field is only to suppress superconductivity, without introducing any extrinsic contribution to the normal state resistivity. In our case, this was achieved by measuring the longitudinal magnetoresistance (that is, with B//I//ab) of all samples. For all x, the magnetoresistance was either vanishingly small or merely introduced a (small) offset with a negligible temperature dependence.

Extended Data Fig. 2a shows a set of field sweep curves for x = 0.2. As can be seen, the variation in ρ_ab(H) is essentially constant (within the field-accessed normal state) over the entire temperature range studied, implying that the values of the resistivity measured at 35 T and at 0 T have the same temperature dependence. For 0.05 ≤ x ≤ 0.13, the magnetoresistance is finite and positive over the entire field range, but only results in a constant shift in ρ_ab(T). This is illustrated in Extended Data Fig. 3, where we plot the slope of the magnetoresistance sweeps for x = 0.13 between 34 T and 35 T (extracted from linear fits) at different temperatures. We find a broad temperature region (1.5 K ≤ T ≤ 30 K) in which the slope is constant to within our experimental uncertainty. Thus, the only effect of the field is to shift ρ_ab to higher values, without any influence on its variation with temperature. Below 1.5 K, the maximum field is no longer strong enough to fully suppress superconductivity, resulting in a sharp increase in the slope. A similar behaviour is found for Δρ_ab = ρ_ab(35 T) – ρ_extrap(0 T), where ρ_extrap(0 T) is the extrapolation at 0 T of the linear fits calculated between 34 T and 35 T. Only for pure FeSe does the magnetoresistance vary with temperature, but even this vanishes at lower T, as shown in Extended Data Fig. 2b.

A coefficients and the temperature scales T ₁ and T ₂ for FeSe_1−xS_x

The A coefficients and the temperature scales T₁ and T₂ for each x value were determined primarily from the temperature derivative dρ_ab/dT of the 35 T resistivity curves. To calculate these derivatives, the raw ρ_ab(T) curves of Extended Data Fig. 4 were first binned and smoothed. As shown in Extended Data Fig. 5, for every x ≠ x_c, dρ_ab/dT exhibits three distinct regimes. At the lowest temperatures, dρ_ab/dT has a linear slope with a zero intercept. This corresponds to a quadratic temperature dependence for ρ_ab(T). The temperature at which dρ_ab/dT first deviates from linearity is then defined as T₂ and the slope itself is equal to 2A. The uncertainty in A comes from both the scatter in the derivative data and the geometric uncertainty in the sample dimensions. Once T₂ is obtained, a second determination of A can be obtained from a plot of ρ_ab versus T²—see Extended Data Fig. 6 (raw data) and Extended Data Fig. 7 (smoothed data)—and the linear fit drawn between the lowest measured temperature point and (T₂)².

At higher temperatures, the derivative attains a constant value, implying that the resistivity has become T-linear. The onset temperature of this T-linear resistivity is labelled T₁. The constant value of dρ_ab/dT can then be compared with the one obtained when fitting the curves in Extended Data Fig. 4 with straight lines down to T = T₁ to obtain two estimates of the T-linear slope. For T₂ < T <T₁, ρ_ab(T) exhibits a gradual crossover from the T² resistivity at low T to the T-linear resistivity at higher temperatures. This crossover is rather smooth for x > x_c but for x < x_c, it presents a peak, presumably due to the finite magnetoresistance in the latter case.

For x_c = 0.16, we estimate T₁ to be lower than 1.5 K (the lowest temperature that we could access with our current magnetic field range). Looking at the behaviour of samples with x < x_c, we estimate the maximum value of T₂ to be at least half of T₁, but with an error bar extending between 0 K and 0.75 K (see Fig. 4b). For an estimate of the lower bound of A for x_c = 0.16, see Extended Data Fig. 11 and the ‘Estimation of the A coefficient and the extent of the T-linear regime for x = 0.16’ section in Methods.

Calculation of A for FeSe from existing Fermi surface parameters

The A coefficient of the T² resistivity contains important information about the enhancement of the quasiparticle effective mass m^*, as encapsulated in the empirical Kadowaki–Woods ratio that links the coefficient of the T²-resistivity to the square of the electronic specific heat coefficient and is expressed for a multiband quasi-two-dimensional metal as¹⁸:

$$A=\frac{8{{\rm{\pi }}}^{3}\mathop{a}\limits^{ \sim }c{k}_{{\rm{B}}}^{2}}{N{e}^{2}{\hbar }^{3}}\frac{1}{\sum _{i}({k}_{\mathrm{Fi}}^{3}/{m}_{i}^{\ast 2})}$$

(1)

where c is the c-axis lattice parameter, N is the number of formula units per unit cell, k_Fi is the Fermi wave vector for the individual Fermi pockets and $\mathop{a}\limits^{ \sim }$ is the effective minimum mean-free-path ${\ell }$ (the Mott–Ioffe–Regel limit), that can be set, depending on the appropriate definition, to be equal to a, the a-axis lattice parameter (${\ell }$ = a) or 1/k_F (k_F${\ell }$ = 1).

According to electronic structure calculations based on density functional theory, the Fermi surface of FeSe comprises five cylindrical pockets: three hole pockets in the centre of the zone and two electron pockets at the zone corners. However, two independent quantum oscillation studies carried out in high magnetic fields up to 35 T by Terashima et al.¹³ and Watson et al.¹¹ found only four fundamental frequencies that one might attribute to the extremal areas of the different cylinders. A comparison of the findings of the two studies is shown in Extended Data Table 1. As one can see, good agreement was found between the two studies. Similar frequencies were also reported (albeit with smaller masses) in an earlier pulsed-field experiment³².

The absolute value of the low-temperature electronic specific heat coefficient γ_el places a strong thermodynamic constraint on the possible assignment of these oscillation frequencies to individual pockets. For quasi-two-dimensional FeSe, γ_el can be estimated simply by multiplying the sum of the (averaged) masses of all the individual pockets by 0.7 mJ mol⁻¹ K⁻² (refs ^11,13). Moreover, in order to ensure charge neutrality in (stoichiometric) FeSe, we should ensure, within experimental uncertainty, that the (averaged) areas of the individual electron and hole pockets are equivalent. With this in mind, the best pairing of the four oscillation frequencies turns out to be α/δ and β/γ as the minimum/maximum of each pocket respectively. We note that this assignment is qualitatively consistent with the observation that the magnitude of α and β increases with increasing polar angle¹³, while that of δ and γ correspondingly decreases. This assignment also has the advantage of making the individual pockets more warped, in agreement with the relatively low electrical anisotropy or low (factor of two) anisotropy in H_c2.

With this assignment, one then obtains (averaged) effective masses m* for the two pockets of 3.35m_e and 5.55m_e from which γ_el = 6.2 mJ mol⁻¹ K⁻², that is, in good agreement with the as-measured value of 5.7–6.8 mJ mol⁻¹ K⁻² (refs ^33,34,35). Making the pairings α/γ and β/δ, as done by Terashima et al.¹³, also gives γ_el = 6.2 mJ mol⁻¹ K⁻², but does not satisfy charge neutrality. Under the assumption that there were three rather than two pockets in FeSe, Watson et al.¹¹ estimated γ_el to be 9.4 mJ mol⁻¹ K⁻², 50% higher than the measured value. In summary, the combination of quantum oscillation studies and specific heat measurements suggests that there are only two pockets in FeSe containing equal numbers of electrons and holes. We cannot completely rule out, however, the possibility of a second very small electron pocket with a light mass that has been inferred from multiband analysis of Hall and magnetotransport data^36,37. In the following, we consider a purely two-band picture.

The existence of singular cylindrical pockets of equal (averaged) cross-section simplifies our working considerably. First, our expression for A now becomes:

$$A=\frac{8{{\rm{\pi }}}^{3}\mathop{a}\limits^{ \sim }c{k}_{{\rm{B}}}^{2}}{N{e}^{2}{\hbar }^{3}{k}_{{\rm{F}}}^{3}}\frac{1}{\sum _{i}(1/{m}_{i}^{\ast 2})}$$

(2)

From the quantum oscillation frequencies reported in refs ^3,4, we find k_F = 1.07(1) nm⁻¹. Taking m* = 3.35m_e and 5.55m_e for the two pockets and noting that N = 2 and c = 5.52 Å, we obtain A = 90 nΩ cm K⁻² and 230 nΩ cm K⁻² assuming ${\ell }$ = a and k_F${\ell }$ = 1, respectively. The latter is in reasonable agreement with the measured value (A = 260 nΩ cm K⁻²) and suggests that the origin of the T² resistivity of FeSe within the nematic phase is (enhanced) electron–electron scattering.

Calculation of A* for FeSe_1−xS_x

Quantum oscillations have been measured for several members of the FeSe_1−xS_x family (0.0 ≤ x ≤ 0.19) by Coldea et al.²⁰. The same frequencies found in pristine FeSe can be tracked as a function of x and show a gradual increase in magnitude with the possible appearance of an additional frequency or frequencies at higher x. (The new frequency ε reported in ref. ²⁰ at F ≈ 400 T is assumed here to be the second harmonic of β (F ≈ 200 T), rather than arising from a third pocket which, as described above, would be inconsistent with the measured electronic specific heat coefficient). As indicated by equation (1), the A coefficient in the electrical resistivity is dependent on both (m*)² and (k_F)³. To isolate the change in A that arises from (m*)² alone, in the following we use the data of ref. ²⁰ to track the evolution of k_F and then obtain a renormalized A coefficient given by the following expression:

$${A}^{\ast }=\frac{{k}_{{\rm{F}}}^{3}(x)}{{k}_{{\rm{F}}}^{3}(0)}A$$

(3)

where k_F(0) = 1.07 nm⁻¹ is the effective Fermi wave vector for both pockets in FeSe. (Note that this expression is only strictly valid in the regime of charge neutrality where the k_F values for both the electron and hole pockets remain equivalent.)

Extended Data Fig. 9 shows the evolution of the two sums (α + δ and β + γ) as a function of x, obtained from figure 3c of ref. ²⁰. As one can see, the charge compensation condition is found to be satisfied well for all x < 0.17. For x = 0.19, the discrepancy may indicate the appearance of a third pocket, a point we shall return to later. In the following, however, we assume that charge neutrality is preserved across the phase diagram and obtain effective k_F values for the electron and hole pocket for each x using the dashed lines shown in Extended Data Fig. 9, and finally A* using equation (3) above. The resulting plot for A*(x) is shown in Fig. 4a.

Calculation of α

It has been shown for a number of correlated metals near quantum criticality that the strength of the T-linear scattering rate ħ/τ ≈ αk_BT where α is a number close to unity³¹. In order to test this relation in FeSe_1−xS_x, we use the simple Drude expression for the electrical conductivity for a quasi-two-dimensional metal to obtain the following formula for α (again assuming perfect compensation and k_Fe = k_Fh = k_F):

$$\alpha =\frac{N{e}^{2}\hbar {k}_{\mathrm{Fi}}^{2}}{2{\rm{\pi }}c{k}_{{\rm{B}}}}\sum _{i}(1/{m}_{i}^{\ast })\frac{d\rho }{dT}$$

(4)

where dρ/dT is the slope of the T-linear resistivity and all the other labels refer to the same parameters as above. The k_F values were obtained from Extended Data Fig. 9 using the same procedure as for A* while the m* values were determined from A*(x) using the scaling relation.

$${m}^{\ast }(x)=\sqrt{\frac{{A}^{\ast }(x)}{{A}^{\ast }(0)}}{m}^{\ast }(0)$$

(5)

where A*(0) = 260 nΩ cm K⁻² (for FeSe) and m*(0) = (5.55 + 3.35)m_e/2 = 4.45m_e is the pocket-averaged effective mass for FeSe.

Determination of the upper critical field

The upper critical field H_c2(x) (for B//ab) is defined in Extended Data Fig. 10a as the field strength at which ρ_ab first deviates by 1.5σ (σ is the standard deviation) from its high-field slope. For consistency, these values were determined at the same temperature, T = 1.35 K, the lowest temperature that we could access the normal state resistivity for all x ≠ 0.16 with the maximal magnetic field available to us. For x = 0.16, we extrapolated the value of the upper critical field at 1.35 K by fitting the points at higher temperatures to the formula ${H}_{{\rm{c}}2}\left(T\right)={H}_{{\rm{c}}2}^{0}\left(1-{\left(\frac{T}{{T}_{{\rm{c}}}}\right)}^{2}\right)$. The values obtained are plotted in Extended Data Fig. 10b as a function of x. The jump in H_c2 above x = 0 is probably due to the increased disorder that modifies the effective coherence length. The peak in H_c2 at the critical doping x_c is presumed to be due to the renormalization of v_F (through m*) at the quantum critical point.

Estimation of the A coefficient and the extent of the T-linear regime for x = 0.16

As reported in Fig. 2e, the resistivity for the sample closest to the quantum critical point (x = 0.16) appears to vary linearly with temperature down to 1.5 K—the lowest temperature at which we could access the normal state. This confirms an earlier study for x = 0.16 down to 4.2 K, in which ρ_ab(T) was determined by extrapolating the positive magnetoresistance (with B//c) above H_c2 (ref. ¹⁷). To determine a lower limit for the T²-coefficient for x = 0.16, the temperature derivative of the resistivity curve acquired in 35 T was replotted in Extended Data Fig. 11. Even though it is not possible to distinguish any deviation from a T-linear behaviour down to the lowest measured temperature, an estimation of the A coefficient was extracted from the slope of a straight line passing through the origin and through the mean value of the derivative at the lowest measured temperature, thus assuming that ρ_ab would cross over to a T² behaviour below the accessible temperature range of our experiments. Extended Data Fig. 11 also defines the upper bound of the strictly T-linear regime, as indicated by the arrow.