Strain-derivative of thermoelectric properties: a sensitive probe for nematicity

The nematic instability of iron-based superconductors is an undebatable ingredient of the physics of iron-based superconductors. Yet, its origin remains enigmatic as it involves a fermiology with an intricate interplay of lattice-, orbital- and spin- degrees of freedom. It is well known that thermoelectric transport is an excellent probe for revealing even subtle signatures of instabilities and pertinent fluctuations. In this paper, we report a strong response of the thermoelectric transport properties of two underdoped 1111 iron-based superconductors to a vanishingly small strain. By introducing the strain-derivative of the Seebeck and the Nernst coefficients, we provide a novel description of the nematic order parameter, proving the existence of an anisotropic Peltier-tensor beside an anisotropic conductivity-tensor. Our measurements reveal that the transport nematic phenomenology is the result of the combined effect of both an anisotropic scattering time and Fermi surface distortions, pointing out that in a realistic description, abreast of the spin-fluctuations also the orbital character is a fundamental ingredient. In addition, we show that nematic fluctuations universally relax in a Curie-Weiss fashion above T_S in all the elasto-transport measurements and we provide evidences that nematicity must be band-selective.

ν in the limit of vanishingly small strain . Experimentally, we take advantage of an innovative setup, which combines a standard thermoelectric measurement configuration, with the highly controlled uniaxial strain offered by a piezoelectric device. By measuring directly the slope of S and ν vs , we have access to the respective strain derivatives (Figure 1, see Materials and Methods for details on the experimental setup), representing the nematic susceptibility of the system. In particular, in this work, we investigate the transport nematic phenomenology of two single crystals of LaFe 1−x Co x AsO with x=0 and 0.035 17,18 , respectively. The former is the parent compound and the latter an electron underdoped sample of the La-1111 family, so far almost unexplored due to the lack of sizeable single crystals.
We will show that the thermoelectric transport properties react extraordinarily to a tiny strain well above the nematic transition, revealing an extended zone of nematic fluctuations with a diverging behavior at the structural transition. Moreover, by analyzing the contribution of the different transport coefficients we demonstrate that a band-selective response of the Peltier tensorᾱ is indispensable to quantitatively explain the behavior of the thermoelectric coefficient under strain, pointing out the fundamental role of the Fermi surface distortions caused by electron-nematic order 19 . In addition, we experimentally demonstrate that, within the validity of a single-band approximation, the elasto-Nernst coefficient can be predicted as a combination of the strain-derivatives of the Peltierᾱ and the conductivitȳ σ tensors. This lets us paint a self-consistent scenario in which all the elasto-transport coefficients are bound to each other.

Elasto-Seebeck
Figures 2a and 2b show the temperature-dependence of Seebeck coefficient S of the x=0 and the x=0.035 sample, respectively. Their amplitude and trend are consistent with previous reports on polycrystalline compounds 9 . In the x=0 sample a sign change of S occurs at around 170 K, caused by the multi-band nature of this material. Since hole-like and electron-like pockets contribute to the Seebeck coefficient with opposite sign, they tend to compensate their respective effect, possibly generating a change in the sign of S, as in our case. In the x=0.035 compound, the electron doping, obtained by Co-substitution, pushes it closer to the condition of single-carrier transport and the Seebeck coefficient remains always negative, as expected for a system dominated by electrons. This has been already shown in La(Fe,Co)AsO and Sm(Fe,Co)AsO series of polycrystalline samples, where a departure from the carrier compensation in favour of an electron-like transport due to Co-doping has been demonstrated 20,21 .
Figures 2c and 2d present the temperature dependence of the strain ( ) derivative of the Seebeck coefficient δ(∆S/T )/δ , with ∆S=S( )-S( =0), for the x=0 and x=0.035 compound, respectively. The normalization to T is introduced to get rid of the entropy contribution. Interestingly, in both the compounds a change of regime appears in correspondence of the structural transition, whose onset is around T S =155 K in the parent compound and T S =80 K for the underdoped one 18 . While δ(∆S/T )/δ of the x=0 compound exhibits a sharp cusp-like transition, in the x=0.035 compound the change of regime at T S looks like a crossover. However, for T > T S , where the crystalline cell is tetragonal and nematic fluctuation are expected, δ(∆S/T )/δ is finite and large in both the compounds, evidencing a strong response of the Seebeck coefficient to a vanishingly small strain. Moreover, δ(∆S/T )/δ exhibits a diverging trend by approaching T S . This behavior is typically detected in the context of elasto-resistivity measurements, where a Curie-Weiss fashion of the elasto-resistivity is interpreted as the fingerprint of large nematic fluctuations with an electronic origin 5,6,24 .
In addition, with our experimental approach we could also take advantage of the fact that S is sensitive to the sign of the charge carriers. Indeed, the most striking result of our measurement concerns the sign of δ(∆S/T )/δ : In the parent compound, though S changes sign, δ(∆S/T )/δ is always positive and it remains finite also when S crosses the zero. This points out that not all the different Fermi pockets are responsible for the electronic nematic phenomenology but only some of them contribute. Interestingly, the strain derivative persists to be positive also in the x=0.035 sample, in which the transport properties are dominated by the electron-like carriers. This suggests that if the nematicity is band-selective, the electron-pockets play the major role. ν undergoes an abrupt increase at T S , consistent with previous reports 9, 11 . This has been sometimes attributed to the band reorganization caused by the development of the long range magnetic ordering and the consequent appearance of Dirac-cone-like bands 9 .

Elasto-Nernst
Figures 3b and 3e present the temperature dependence of the strain derivative of the Nernst coefficient δ(∆ν/T )/δ as a function of T , where ∆ν=ν( )-ν( =0). In analogy to the strain-derivative of the Seebeck effect, also δ(∆ν/T )/δ presents a diverging behavior in the tetragonal phase and a change of regime at around T S . Differently from the Seebeck coefficient, the ambipolar nature of the Nernst effect prevents ν to cross the zero value in our compounds. This allows to normalize δ(∆ν/T )/δ to the value of the unstrained ν without incurring the risk of an unphysical divergency and define a Nernst nematic susceptibility χ ν =δ(∆ν/ν)/δ . χ ν can be directly compared to the susceptibility calculated from the elasto-resistivity χ ρ =−δ(∆ρ/ρ)/δ 24 , whose value is generally assumed as the response of the electronic nematic order parameter to the applied strain 5,6,24 . The result is presented in Let's now focus on the absolute value of χ ν and χ ρ . Interestingly, χ ν tends to decrease from x=0 to x=0.035, reaching the respective maximal value of around 230 and 120 close to T S . On the contrary χ ρ increases with doping. The increase of χ ρ due to Co-doping in the Ba(Fe,Co) 2 As 2 series was attributed to an increase of the nematic fluctuation intensity towards the optimally-doped composition, which maximizes the superconducting critical temperature 5,6 . By assuming that both χ ν and χ ρ should reflect the response of the nematic order parameter to the applied strain, the reason for their mismatch deserves some consideration. The Nernst coefficient is a complex quantity which results from a non-trivial combination of of the resistivity tensorρ and the Peltier thermoelectric tensorᾱ, reading ν=α xy ρ yy − α xx ρ xy 25 . In this notation x is the direction along which strain and heat gradient (or electric current) are applied, while y is the transverse direction. In complex materials, such as the iron based superconductors, the experimental prediction of ν is usually unsuccessful due to the complications of the multi-band nature. However, in the next section, we show that the behavior of the elasto-Nernst of the x=0.035 compound (closer to the single band condition thanks to the doping) can be reasonably obtained from the other transport coefficients, in a self-consistent scenario.
Analysis of the transport coefficients of the x=0.035 compound First of all, in a single-band approximation, it is possible to evaluate also a Seebeck susceptibility χ S =δ(∆S/S)/δ , shown in Figure 4a as a function of T . For T > T S , we interpolated the χ S curve with a Curie-Weiss function. The fitting parameter T * =25±8 is in very good agreement with the values obtained by fitting χ ν and χ ρ . The Seebeck coefficient is explicited in terms of transport coefficients as S=α xx ρ xx , where α xx and ρ xx are the diagonal terms of the Peltier and the resistivity tensors, respectively. It is immediate to verify that χ S =χ ρ +χ α , where χ α =δ(∆α xx /α xx )/δ . From these relations, one can evaluate the temperature dependence of α xx and χ α , presented in Figure 4b. It must be noticed that χ αxx is of the same order of magnitude as χ ρ . However, they exhibit an opposite sign, which is understandable, considering that α xx ∼ dσ xx /dE = −ρ −1 xx dρ xx /dE, where σ xx is the electrical conductivity and E is the energy. At this point, one can consider χ ν and verify wether it is experimentally obtainable by a combination of the other transport coefficients.
In the limit of small strain, in which we operate, one can safely state that ρ xx ρ yy and

Discussion and conclusions
Once established the existence of a finite χ α beside a finite χ ρ , one can conjecture on the microscopic mechanisms that determine the transport nematic phenomenology. Generally, a pure spin-nematic scenario mainly supports an anisotropic scattering time as a source for transport anisotropy [26][27][28][29] , while the pure orbital-ordering description takes mainly into account anisotropies of the Fermi surface parameters, such as the Fermi velocity [30][31][32][33] .
In the context of cuprate superconductors, it was explicitly predicted that the Nernst effect anisotropy is a very sensitive probe of Fermi surface distortions caused by electronnematic order 19 . This is caused by a large anisotropy in α, which overcomes the anisotropy of ρ and results particularly enhanced in correspondence of a change of the Fermi surface topology 19 . This means that, if only an orbital anisotropy dominates the transport, a χ α substantially larger than χ ρ can be expected. Since in our case |χ ρ | ≥ |χ α |, it is likely that a significant contribution from an anisotropic scattering time must be present.
On the other hand, it has been reported that the NMR spin-lattice relaxation rate (T 1 T ) −1 of LaFeAsO is finite well above T S , as a signature of persistent spin fluctuations 34 . For T < T S , (T 1 T ) −1 increases a lot, before diverging in correspondence of T N , where spin fluctuations freeze 34 . Interestingly, neither χ ρ nor χ ν seem to be sensitive to the magnetic transition and they do not follow the trend of (T 1 T ) −1 . This suggests that they are not mimicking the spin susceptibility of the system. As a consequence, the existence of an anisotropic scattering time, directly linked to anisotropic spin fluctuations, is not sufficient to explain the transport anisotropy, but an orbital contribution from the distortion of Fermi surface must be included. Hence, to shed light on the cryptic nematic phase of iron-based superconductors, it is evident that a theoretical picture which includes different microscopic mechanism must be adopted. In this sense the orbital-selective spin-nematic model is a promising candidate, since it predicts that, once the orbital character of the spin-fluctuations is taken into account, both the anisotropy in scattering rate and in the Fermi surface parameters (i.e. the Fermi velocities) must play a substantial role [35][36][37] .
In summary, we measured for the first time the strain-derivative of the Seebeck and the Nernst effect of two single crystals belonging to the 1111 family of iron-based superconductors. We observed that thermoelectric properties, in proximity of a nematic instability, are strongly susceptible to a vanishingly small strain. The inspection of the Seebeck effect provided a clear signature of the band-selective character of the nematic phenomenology in case of a multi-band system, which is a fundamental information for the definition of a nematic order parameter. In addition, by defining a Nernst nematic susceptibility, we experimentally demonstrated that an anisotropy in the resistivity tensorρ is not enough to explain the behavior of the thermoelectric properties, but a finite anisotropy in the Peltier tensorᾱ must be included. This suggests that the transport nematic phenomenology is likely to be the result of the combined effect of both an anisotropic scattering time and Fermi surface distortions, pointing out that in a realistic description, beside the spin-fluctuations also the orbital character is a fundamental ingredient. We expect that these results will trigger novel theoretical insights, setting new bounds for the anisotropic transport models and giving a substantial contribution to the understanding of the nematic puzzle.

Crystal growth
The crystals were obtained using the solid-state single crystal growth method at ambient pressure using Na-As as a liquid phase promoting an abnormal grain growth due to enhanced interfacial anisotropy by introducing a liquid-solid interface. This is a different strategy from the usually used flux growth. As this growth is based on polycrystalline starting materials, a polycrystalline sample of LaFeAsO was prepared using a two-step solid-state reaction. The obtained polycrystalline pellets and Na-As powder were layered into an alumina crucible.
The molar ratio of LaFeAsO to Na-As used was 1:4, which corresponds to a ratio in volume of about 1:1. The material was heated to 1080 • C and annealed for 200 h. By using this method single crystals sized up to 2x3x0.4 mm 3 were obtained. Ref. 17 gives a detailed description of the synthesis process of all the investigated compounds. The crystals were analyzed using SEM with EDX, Laue backscattering, powder X-ray diffraction and SQUID magnetometry measurements.