Abstract
Fe1+yTe1−xSex is characterized by its complex magnetic phase diagram and highly orbital-dependent band renormalization. Despite this, the behavior of nematicity and nematic fluctuations, especially for high tellurium concentrations, remains largely unknown. Here we present a study of both B1g and B2g nematic fluctuations in Fe1+yTe1−xSex (0 ≤ x ≤ 0.53) using the technique of elastoresistivity measurement. We discovered that the nematic fluctuations in two symmetry channels are closely linked to the corresponding spin fluctuations, confirming the intertwined nature of these two degrees of freedom. We also revealed an unusual temperature dependence of the nematic susceptibility, which we attributed to a loss of coherence of the dxy orbital. Our results highlight the importance of orbital differentiation on the nematic properties of iron-based materials.
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Introduction
The intricate interplay between magnetism and nematicity in different families of iron-based superconductors has attracted great interest in the past few years1,2,3,4. In iron pnictides, magnetism, and nematicity are tightly coupled; the antiferromagnetic (AFM) transition is always coincidental with, or closely preceded by, a tetragonal-to-orthorhombic structural transition. The proximity of the two transitions can be naturally explained within the spin-nematic scenario, where the structural transition is driven by a vestigial nematic order arising from fluctuations associated with the antiferromagnetic stripe transition (see Fig. 1a)5,6,7. In iron chalcogenides, the coupling between magnetism and nematicity is less obvious. FeSe undergoes a nematic phase transition without any long-range magnetic order8,9, which has been interpreted as evidence that the nematic order in FeSe is of orbital origin10. Nevertheless, spin stripe fluctuations do develop below the nematic transition10,11, and static stripe order can be induced by hydrostatic pressure12,13.
While there are ongoing debates on the mechanism by which nematicity forms without static magnetism in FeSe14,15,16,17,18, Fe1+yTe1−xSex provides another platform to approach this problem. As selenium is replaced by tellurium (i.e., x is changed from 1 to 0), the nematic phase transition is suppressed and the superconducting critical temperature reaches optimal near the putative nematic quantum critical point (x ~ 0.5)19,20. In the high tellurium concentration regime (x < 0.5), inelastic neutron scattering experiments revealed a complex evolution of the spin correlations associated with different magnetic patterns21,22,23,24. Close to optimal doping, the wave-vector of spin fluctuations at low temperatures is (π, π) [in the crystallographic Brillouin zone], identical to the AFM order in the iron pnictides. As the tellurium concentration increases, both superconductivity and the (π, π) spin fluctuations disappear. The latter are replaced by short-range magnetic correlations with checkerboard character near (0, π), and eventually at low temperatures the competing double-stripe phase forms in Fe1+yTe (Fig. 1b) by virtue of the ferro-orbital ordering that leads to the formation of ferromagnetic Zigzag chain25,26,27. Previous elastoresistivity measurements revealed a diverging B2g nematic susceptibility in optimally doped Fe1+yTe1−xSex20,28. While the diverging nematic susceptibility is naturally expected as a consequence of the nematic quantum critical point, it is also consistent with the existence of (π, π) spin fluctuations found in the same doping. This finding suggests that nematic and magnetic fluctuations remain strongly intertwined even in the absence of static nematic and magnetic orders, consistent with previous inelastic neutron scattering study23. Nevertheless, in contrast to the magnetic sector, the behavior of nematic fluctuations for high tellurium concentrations (i.e., x < 0.5) is still poorly characterized. The compositional dependence of the nematic susceptibility in Fe1+yTe1−xSex would therefore constitute an important step in the effort to elucidate the relationship between nematicity and magnetism.
Another motivation to study Fe1+yTe1−xSex is to understand the influence of orbital selectivity on nematic instability. Orbital selectivity (or orbital differentiation) refers to the fact that different orbitals are renormalized differently by electronic correlations, a characteristic property of Hund’s metals that appears to be much more prominent in the iron chalcogenides in comparison with the pnictides29,30,31. Experimentally, recent scanning tunneling microscopy (STM) measurements revealed the impact of orbital differentiation on the superconducting state32. Theoretically, it has been suggested that orbitally selective spin fluctuations may be the origin of nematicity without magnetism in FeSe33. The relation between orbital hybridization, spin fluctuations, and nematicity, was also suggested by an earlier inelastic neutron scattering experiment23. Nematic order was also proposed to enhance orbital selectivity by breaking the orbital degeneracy, leading to asymmetric effective masses in different d-orbitals34. The effect of orbital differentiation becomes even more extreme as selenium is replaced by tellurium. In Fe1+yTe1−xSex, angle-resolved photoemission spectroscopy (ARPES) revealed a strong loss of spectral weight of the dxy orbital at high temperatures, which was interpreted in terms of proximity to an orbital-selective Mott transition35. Similar drastic changes were also observed as a function of doping36,37, mimicking the evolution of spin fluctuations. Nevertheless, the impact of orbital incoherence on nematicity remains little explored38.
In this report, we present systematic measurements of both the B1g and B2g nematic susceptibilities of Fe1+yTe1−xSex (0 ≤ x ≤ 0.53) using the elastoresistivity technique. We demonstrate that the doping dependence of the two nematic susceptibilities closely track the evolution of the corresponding spin fluctuations. In particular, a diverging B1g nematic susceptibility is observed in the parent compound Fe1+yTe, suggesting that the spin-nematic paradigm also applies to the double-stripe AFM order39,40,41. A diverging B2g nematic susceptibility is observed over a wide range of doping (0.17 ≤ x ≤ 0.53), and its magnitude is strongly enhanced by both Se doping and annealing. In addition, the temperature dependence of the B2g nematic susceptibility shows significant deviation from Curie–Weiss behavior above 50 K. This is in sharp contrast to the iron pnictides, where the Curie–Weiss temperature dependence extends all the way to 200 K. This unusual temperature dependence is captured by a theoretical calculation that includes the loss of spectral weight of the dxy orbital, revealing its importance for B2g nematic instability.
Results and discussion
Doping dependence and temperature dependence of elastoresistivity
By symmetry, the B1g and B2g nematic susceptibilities (\({\chi }_{{B}_{1g}}\) and \({\chi }_{{B}_{2g}}\)) are proportional to the elastoresistivity coefficients m11 − m12 and 2m66, respectively. We performed the elastoresistivity measurements in the Montgomery geometry, which enables simultaneous determination of the full resistivity tensor, hence the precise decomposition into different symmetry channels, as illustrated in Fig. 1c, d. Representative data of anisotropic resistivity as a function of anisotropic strain at 20 K in B2g and B1g channels are shown in Fig. 1e, f. The B1g elastoresistivity coefficient m11 − m12 and the B2g one, 2m66, can be obtained by fitting the linear slope of resistivity versus strain. Samples with high doping concentrations (x = 0.38, 0.45, 0.53) show predominantly a B2g response while the low doping ones (x = 0, 0.12) show comparable B1g and B2g responses.
Figure 2a, b shows the temperature dependence of m11 − m12 and 2m66 of annealed Fe1+yTe1−xSex for 0 ≤ x ≤ 0.53. For 0.28 ≤ x ≤ 0.53, 2m66 shows a strong temperature dependence that grows continuously as temperature decreases. For x = 0.45, 2m66 reaches a value of ~100, comparable to optimally doped pnictides. While preserving a similar diverging temperature dependence, the maximum value of 2m66 decreases rapidly as selenium concentration decreases, from 100 for x = 0.45 to 15 for x = 0.28. The magnitude of 2m66 continues to decrease but changes sign for x = 0.17. On the other hand, m11 − m12 shows a diverging response when x is below 0.17, which is in the vicinity of the double-stripe AFM order. As selenium concentration increases, m11 − m12 evolves to a temperature-independent response, with small kinks at low temperatures likely coming from contamination of 2m66 due to misalignment. Overall, our observation of the doping dependence of 2m66 and m11 − m12 is consistent with the evolution of low-temperature spin fluctuations from predominantly (π, 0) at small x to predominantly (π, π) at optimal doping x ~ 0.521,22,23,24.
To gain more insight, we fit the 2m66 and m11 − m12 to a Curie–Weiss temperature dependence:
For the parent compound Fe1+yTe, m11 − m12 can be well fitted to a Curie–Weiss behavior in the temperature range just above the double stripe AFM ordering temperature Tmag = 71.5 K (Fig. 2c). The fitted Curie–Weiss temperature T* as listed in Supplementary Table III is slightly smaller than Tmag. Despite the smaller absolute value (~10 at maximum), the behavior of m11 − m12 is reminiscent of the 2m66 in the parent phase of iron pnictides, suggesting that the spin-nematic mechanism is still at play here, in agreement with theoretical expectations39,40,41.
Figure 2d shows the Curie–Weiss fitting of 2m66 for the x = 0.45 sample. The fitting of 2m66 only works at low temperatures, as can be seen in the linear temperature dependence of \(| 2{m}_{66}-2{m}_{66}^{0}{| }^{-1}\) below 50K. It shows a significant deviation for temperature >50 K. The T* obtained from the low-temperature fitting of 2m66 is close to 0K. Intriguingly, the T* extracted from the Curie–Weiss fitting is approximately zero for all 0.17 ≤ x ≤ 0.53, while the Curie constant λ/a decreases as x decreases (Supplementary Table II). While the number of doping concentrations studied in the current work is insufficient to support a power law analysis, 2m66 at constant T = 16 K appears to be diverging as x increases from 0.17 to 0.45 (Fig. 3a). Both the doping dependence and the near zero T* are consistent with the existence of a putative nematic quantum critical point at x ~ 0.5 discovered recently20.
This deviation from Curie–Weiss at high temperatures is very unusual. In the iron pnicitides, such a deviation was only observed at low temperatures in transition-metal doped BaFe2As2 (Fig. 2e) and LaFeAsO42. This unusual temperature dependence of 2m66 is consistent with the thermal evolution of spin correlations observed by neutron scattering43 and appears to echo the coherent-incoherent crossover observed by ARPES35, where the spectral weight of the dxy orbital is strongly suppressed as the temperature increases or as the selenium concentration decreases. To further confirm this correlation, we measured the Hall coefficient RH, which has been demonstrated to be a good indicator of this incoherent-to-coherent crossover36,44,45. The recovery of the dxy spectral weight is generally correlated with a sign-change of RH45 from positive to negative. Figure 3a shows the low-temperature RH and 2m66 as a function of doping, whereas Fig. 3b–d contain the full temperature and doping dependence of RH, m11 − m12, and 2m66, respectively. These plots reveal the strong correlation between a negative RH and an enhancement of 2m66.
Effect of annealing
The properties of Fe1+yTe1−xSex also depend on the amount of excess iron, which can only be removed by annealing46. Taking x = 0.45 as an example, the resistance of the annealed sample is metallic for temperatures below 150 K (Fig. 4a). As Fig. 4b shows, at around 40 K the Hall coefficient of the annealed sample turns from positive to negative, which is a signature of incoherent to coherent crossover36,44,45,47. In contrast, the resistance of the as-grown sample shows a weakly insulating upturn at low temperatures (Fig. 4a black dashed curve), and the Hall coefficient remains positive at all temperatures (Fig. 4b black circles), indicating that the dxy orbital is still incoherent at low temperatures. Interestingly, at the same temperature where the resistance and the Hall coefficient of the as-grown and annealed samples depart from each other, the elastoresistivity coefficient 2m66 shows a pronounced enhancement for the annealed sample (Fig. 4c). Such an enhancement was observed in all annealed samples (Supplementary Note 2), providing further evidence of the correlation between the enhancement of the nematic susceptibility and the coherence of the dxy orbital.
Effect of orbital selectivity on nematic fluctuations
The doping and annealing dependences of 2m66 presented above indicate that the B2g nematic susceptibility also have an orbitally selective character. This is in agreement with a recent theoretical calculation that reveals a strongly orbitally dependent nematic susceptibility. In particular, the dxy orbital contributes most to the overall nematic instability48. To gain more insight, we calculated the nematic susceptibility with and without a reduced spectral weight in the dxy orbital to simulate the orbital correlation effect in Fe1+yTe1−xSex and BaFe2(As1−xPx)2, respectively. The calculated B2g nematic susceptibility and its inverse are plotted in Fig. 5a, b. The excellent agreement with the experimentally measured 2m66 and \(| 2{m}_{66}-2{m}_{66}^{0}{| }^{-1}\) (Fig. 5c, d) confirms the orbitally selective nature of the nematic susceptibility.
Indeed, previous theoretical works have highlighted the impact of orbital degrees of freedom on spin-driven nematicity17,33,34,49,50,51. Using a slave-spin approach, ref. 38 found a suppression of the orbital-nematic susceptibility due to orbital incoherence. To model our data, we employ the generalized random phase approximation (RPA) of ref. 48 to compute the spin-driven nematic susceptibility for the five-orbital Hubbard-Kanamori model (details in Supplementary Note 6). For fully coherent orbitals, it was found that the largest contribution to the nematic susceptibility χnem comes from the dxy orbital. Thus, one expects that χnem would be suppressed if the dxy orbital were to become less coherent.
To verify this scenario, we calculated how χnem changes upon suppressing the spectral weight Zxy of the dxy orbital. For our purposes, the reduction in Zxy acts phenomenologically as a proxy of the incoherence of this orbital, similarly to ref. 32, but its microscopic origin is not important. Fig. 5a, b contrasts the nematic susceptibility for 0.7 ≤ Zxy ≤ 1. We note two main trends arising from the suppression of dxy spectral weight: first, as anticipated, the nematic susceptibility (and the underlying nematic transition temperature, which is non-zero in the model) is reduced (Fig. 5a). Second, its temperature dependence changes from a Curie–Weiss-like behavior over an extended temperature range to a behavior in which the inverse nematic susceptibility quickly saturates and strongly deviates from a linear-in-T dependence already quite close to the nematic transition (Fig. 5b). These behaviors are similar to those displayed by the elastoresistance data shown in Fig. 5c, d, with Zxy = 1 mimicking the behavior of optimally P-doped BaFe2As2 and Zxy < 1, of optimally doped Fe1+yTe1−xSex. Interestingly, the susceptibility associated with (π, π) fluctuations is also suppressed by the decrease in Zxy, in qualitative agreement with the neutron scattering experiments52 (for a more detailed discussion, see Supplementary Note 6). Of course, since Zxy in our model is an input, and not calculated microscopically, our model is useful to capture tendencies, but not to extract the experimental value of Zxy. Furthermore, note that in our calculation Zxy is temperature-independent, while in the experiment it changes with temperature.
Conclusions
In summary, our results reveal the close connection between nematic fluctuations and spin fluctuations in Fe1+yTe1−xSex for both B1g and B2g channels. Additionally, the unusual temperature dependence of the B2g nematic susceptibility can be attributed to the coherent-to-incoherent crossover experienced by the dxy orbital, providing direct evidence for the orbital selectivity of the nematic instability. Our work presents Fe1+yTe1−xSex as an ideal platform to study the physics of intertwined orders in a strongly correlated Hund’s metal.
Methods
Crystal growth
Single crystals of Fe1+yTe1−xSex were grown by the modified Bridgeman method. The electrical, magnetic, and superconducting properties of Fe1+yTe1−xSex are known to sensitively depend on y, the amount of excess iron. To study these effects, crystals were annealed in selenium vapor to reduce the amount of excess iron. Crystals are cleaved into thin slices (~1 mm), loaded in a crucible with another crucible of an appropriate amount (~the amount of excess iron in atomic weight) of selenium powder beneath it, sealed in quartz tubes, and annealed at 500 ∘C for a week.
Elastoresistivity measurements
The elastoresistivity measurements were performed in the Montgomery geometry, which enables simultaneous determination of the full resistivity tensor28. Crystals are prepared into thin square plates with edges along the Fe-Fe (B2g) and Fe-Chalcogen (B1g) bonding directions. The crystal orientation is determined by polarization-resolved Raman spectroscopy, as shown in Supplementary Note 1. Samples were glued on the sidewall of a piezoelectric stack, which generates a linear combination of anisotropic and isotropic strain. Electrical contacts made at the four corners enable the simultaneous determinations of the resistivities along two perpendicular directions of the same piece of sample. This setup allows the precise decomposition into the anisotropic resistivity change (B1g and B2g) Δρxx − Δρyy and isotropic resistivity change (A1g) Δρxx + Δρyy in response to different symmetries of strain (see details in Supplementary Note 3).
Data availability
All data needed to evaluate the conclusions are present in the paper and supplementary materials. Additional data are available from the corresponding author on reasonable request.
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Acknowledgements
We thank Ming Yi for the fruitful discussion. The work at UW was supported by NSF MRSEC at UW (DMR-1719797). The material synthesis was supported by the Northwest Institute for Materials Physics, Chemistry, and Technology (NW IMPACT) and the Gordon and Betty Moore Foundation’s EPiQS Initiative, Grant GBMF6759 to J.-H.C. J.H.C. acknowledge the support of the David and Lucile Packard Foundation, the Alford P. Sloan Foundation and the State of Washington funded Clean Energy Institute. Theory work (M.H.C. and R.M.F.) was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences, and Engineering Division, under Award No. DE-SC0020045. The Raman measurement is partially supported by the Department of Energy, Basic Energy Sciences, Materials Sciences and Engineering Division (DE-SC0012509).
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J.-H.C. proposed and designed the research. J.-H.C., Q.J., and Y.S. carried out the elastoresistivity measurements with the help of Z.-Y.L. and P.M. The elastoresistivity data were analyzed by J.-H.C. and Q.J. Single crystals of Fe1+yTe1−xSex were synthesized and annealed by Q.J., Y.S., and J.J.S. The polarized Raman spectroscopy was carried out by Q.J., B.H., Z.L., and X.X. Theoretical calculations were carried out by R.M.F. and M.H.C. Q.J. and J.-H.C. wrote the paper with input from all co-authors. J.-H.C. oversaw the project.
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Jiang, Q., Shi, Y., Christensen, M.H. et al. Nematic fluctuations in an orbital selective superconductor Fe1+yTe1−xSex. Commun Phys 6, 39 (2023). https://doi.org/10.1038/s42005-023-01154-8
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DOI: https://doi.org/10.1038/s42005-023-01154-8
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