Evolution of the nematic susceptibility in LaFe$_{1-x}$Co$_x$AsO

TL;DR

This study investigates how nematic susceptibility evolves with cobalt doping in LaFeAsO, revealing a suppression and sign change near optimal doping, suggesting a nematic quantum critical point beneath the superconducting phase.

Contribution

It provides a systematic elastoresistivity analysis of LaFe$_{1-x}$Co$_x$AsO, highlighting the doping-dependent evolution of nematic susceptibility and its complex features.

Findings

Nematic susceptibility is Curie-Weiss-like in all samples.

Nematic temperature decreases with cobalt doping and changes sign near optimal doping.

A double-peak feature in nematic susceptibility amplitude suggests multiple contributing effects.

Abstract

The identification of electronic nematicity across series of iron-based superconductors raises the question of its relationship with superconductivity and other ordered states. Here, we report a systematic elastoresistivity study on LaFe$_{1-x}$Co$_x$AsO single crystals, which have well separated structural and magnetic transition lines. All crystals show Curie-Weiss-like nematic susceptibility in the tetragonal phase. The extracted nematic temperature is monotonically suppressed upon cobalt doping, and changes sign around the optimal doping level, indicating a possible nematic quantum critical point beneath the superconducting dome. The amplitude of nematic susceptibility shows a peculiar double-peak feature. This could be explained by a combined effect of different contributions to the nematic susceptibility, which are amplified at separated doping levels of LaFe$_{1-x}$Co$_x$AsO.

Equations

$$\tilde{n}={\tilde{n}}_0+\frac{C}{T-T^{nem}},$$

Full text

Unconventional superconductivity is intimately related to other electronic symmetry breaking states. A common feature among unconventional superconductors is that by varying a tuning parameter $x$, such as pressure or chemical substitution, the superconductivity exists under a dome-like region in the $T-x$ phase diagram, and the extrapolation of some transition line hits zero temperature inside or close to that dome Pfleiderer ; Keimer ; Stewart ; DaiPC . The formation of a spin density wave is the most recognized order in the normal state of unconventional superconductors, and its ubiquitousness stimulated theories to explain the origin of unconventional superconductivity mediated by fluctuations of the magnetic order Scalapino ; Louis ; Lohneysen ; Pines ; Schmalian ; Sachdev . Recently, the report of newly recognized electronic orders beside magnetism in unconventional superconductors is infectious Keimer ; Fradkin , including nematicity, an electronic ordered state that spontaneously breaks the rotational symmetry of its host crystal Kivelson ; Fradkin ; Lilly . Nematic fluctuation by themselves are considered as possible mediator of electronic pairing Lederer ; Metlitski ; Maier ; Labat ; Maslov . Indeed, a nickel pnictide Ba1-xSrxNi2As2 was just discovered to be a nematicity-boosted superconductor JP .

In the case of iron-based superconductors, the stoichiometric parent compounds are always antiferromagnetically (AFM) ordered below $T_{N}$, following a structural transition at $T_{S}$ Stewart ; DaiPC . The paramagnetic orthorhombic phase between $T_{S}$ and $T_{N}$ acquires highly anisotropic electronic properties regardless of the small lattice anisotropy FisherRev . In this electronic nematic phase, all ordered states are highly intertwined, making it hard to discern the leading instability among them. One can find clues by studying their pertinent fluctuations at higher temperatures Cano ; FernendesSUST ; NematicRev . For nematicity, its fluctuations have been probed by various approaches, including the elastoresistivity measurements Chu2012 ; RSI . The divergent nematic susceptibility has been found in different families of iron-based compounds Kuo2016 ; Hosoi ; LiSL . These facts clearly eliminated the structural instability as the driving force for other transitions Chu2012 ; NematicRev .

One prevailing understanding of the nematicity in iron-based compounds is to treat it as the result of fluctuating AFM order CFang ; FeSe ; Hackl ; Dai . This scenario received strong support from the discovery of a good scaling between the lattice softening and the magnetic fluctuating amplitude in Ba(Fe1-xCox)2As2 over a wide doping range scaling . As illustrated in Fig. 1(a), the $T_{S}$ and $T_{N}$ transition lines are very close to each other in Ba(Fe1-xCox)2As2. The extension of these lines penetrate into a sign-reversed $s$-wave superconducting dome Stewart , which is the fingerprint of AFM fluctuation driven superconductivity Inosov ; LiS . In stark contrast, as Fig. 1(b) shows, there is no static magnetic order over the whole phase diagram of FeSe1-xSx. Theories proposed a different kind of superconductivity of sign-preserved pairing, based on unequal orbital occupancy Brink ; Phillips ; KuW ; Devereaux ; Onari . The nuclear magnetic resonance measurements saw no slowing down of spin fluctuations above $T_{S}$ in FeSe and much smaller changes of spin-lattice relaxation rates across $T_{S}$ than in other iron-based superconductors 3pb ; Baek , lent support to the orbital scenarios and alternative origin of nematicity. However, subsequent neutron scattering studies can identify spin fluctuations above $T_{S}$ QiSi . Moreover, these spin fluctuations are found changing across $T_{S}$ and being anisotropic in the nematic phase QiSi2 ; Chen . Thus, the importance of magnetic fluctuations is restored in FeSe. The leading instability and therefore the primary pairing mediator of the seemingly simple FeSe system remain elusive.

In this letter, we report the study of nematicity of a series of LaFe1-xCoxAsO single crystals. As Fig. 1(c) indicates, the $T_{S}$ and $T_{N}$ of LaFe1-xCoxAsO are well separated parent ; Grafe ; WangLR . Superconductivity emerges after the total suppression of the AFM order Grafe ; Francesco , making LaFe1-xCoxAsO interesting to search for separated magnetic and nematic fluctuation regions. By utilizing elastoresistivity measurements, the persistence of a Curie-Weiss-like nematic susceptibility deep in the tetragonal phase of LaFe1-xCoxAsO was found. The Curie-Weiss temperature changes sign around the optimal doping level $x$ = 0.06, with an enhanced nematic susceptibility on its top. This indicates the existence of a nematic quantum critical point (nQCP) under the superconducting dome, and supports the proposals of nematicity as the driver for superconductivity Lederer ; Metlitski ; Maier ; Labat ; Maslov . Besides, another enhancement of the nematic susceptibility is resolved around $x$ = 0.04, close to the end point of the AFM order.

LaFe1-xCoxAsO single crystals were prepared by the solid state single crystal growth method parent . The oriented samples were cut into a typical size of 1.2 mm $\times$ 0.5 mm in plane and cleaved to around 20 $\mu$m thickness to ensure an efficient strain transmission to the sample Chu2012 . The experimental setup is illustrated in the insert of Fig. 2(c). The sample was glued to a commercial piezoelectric actuator (Piezomechanik PSt 150 / 5$\times$5$\times$7) by using an adhesive epoxy (Devcon, No. 14250), after electrically contacting voltage and current leads directly on the fresh surface with sliver paint. A strain gauge was glued on the other side of the actuator and was measured with a built-in Wheatstone bridge circuit. To eliminate a possible temperature change caused by driving the piezoelectric actuator, data were taken after waiting several seconds at each strain step.

Elastoresistivity measured along the [110] direction of the LaFeAsO single crystal is shown in Fig. 2. By changing the voltage across a piezo actuator, one can precisely tune the strain applied to the sample glued on the actuator’s edge. In the nematic ordered state, the resistivity anisotropy of the in-plane axes is very large Chu2010 ; Tanatar . This anisotropy represents the electronic nematicity. Due to the electron-lattice coupling, strain can induce a resistivity anisotropy above the nematic ordered state if nematic fluctuations exist Chu2012 . As illustrated in Fig. 2 (b), the fractional change of sample resistivity ($\eta=\Delta\rho/\rho$) shows a perfect linear relationship to the strain ($\varepsilon_{x}=\Delta L_{x}/L_{x}$) applied along the current direction. As demonstrated in the references Chu2012 ; RSI ; Kuo2016 ; Hosoi , the slope of $\eta(\varepsilon_{x})$ curve probes the nematic susceptibility of the sample. The negative slope means that the in-plane resistivity of LaFeAsO is higher along the shorter orthorhombic direction. For simplicity, we use $-(\delta\eta/\delta\varepsilon_{x})$ in the small strain limit as the definition of $\tilde{n}$, which is a gauge of the nematic susceptibility. The temperature dependence of $\tilde{n}$ in LaFeAsO is shown in Fig. 2(c). A clear kink is resolved at the structural transition $T_{S}$, above which the $\tilde{n}(T)$ curve can be well fitted to the Curie-Weiss type temperature dependence

[TABLE]

in which $\tilde{n}_{0}$ is the intrinsic piezoresistivity effect unrelated to electronic nematicity, $T^{nem}$ is the nematic transition temperature in the mean field theory, and $C$ is the Curie constant which indicates the magnitude of the nematic susceptibility.

Similar measurements and analyses were conducted for LaFe1-xCoxAsO single crystals across the substitution series ($x$ up to 0.075). As shown in Fig. 3, for all the samples we have studied, their $\tilde{n}(T)$ curves diverge in a clear Curie-Weiss-like form. The kink at $T_{S}$ becomes less pronounced with doping and is unresolvable in our data set for $x>0.03$, while $T_{S}$ could be traced up to $x=0.045$ in thermal expansion measurements (indicated by green arrows) Francesco . One may attribute the smearing of the $T_{S}$ feature to the increasing impurity scattering effect. In such a case, according to Ref. Kuo2014 , $\tilde{n}$ should remain practically unaffected by the defects and impurity concentrations in the tetragonal phase. However, the blurred feature of $T_{S}$ could also be an intrinsic property of LaFe1-xCoxAsO, since the structural transition anomaly observed in the thermal expansion experiments becomes broader with increasing $x$ Francesco . Indeed, a nuclear quadrupole resonance study on LaFe1-xCoxAsO showed that competing local charge environments exist at the nanoscale in the underdoped region Grafe , similar to the sister compound LaFeAsO1-xFx Lang ; Lang2 . In that sense, it is natural that the features of $T_{S}$ from averaged properties are muddled by doping.

We analyze the $\tilde{n}(T)$ data for the finite doping levels in the same Curie-Weiss manner as for the parent compound (solid red lines). In order to choose the proper fitting range of the $\tilde{n}(T)$ curves, especially for those without (a clear) $T_{S}$, we followed the procedure described in Ref. [Kuo2016, ]. Additionally, the fitting range is allowed to alter by 20 K for each sample in order to estimate the uncertainty caused by improper fitting ranges SM , set as error bars of the fitted parameters $T^{nem}$ and $C$ in the following discussion.

The information extracted from Fig. 3 is summarized in Fig. 4, plotted together with the phase diagram of LaFe1-xCoxAsO derived from the doping dependence of $T_{S}$, $T_{N}$, and $T_{c}$ Grafe ; Francesco . First of all, one can see that the nematic transition temperature $T^{nem}$ is linearly suppressed by doping, and finally changes its sign around the optimal doping level $x\approx 0.06$. It is known that superconductivity could get boosted via fluctuations related to electronic instabilities. The most studied case is that of AFM fluctuations with certain wave vectors around a magnetic critical point. As shown in Fig. 1(c), the magnetic transition is totally suppressed before the superconductivity emerges in LaFe1-xCoxAsO Grafe ; Francesco , suggesting that the magnetic fluctuations near the critical point may not contribute to the superconducting transition temperature $T_{c}$ directly. Theoretically, a nQCP and its pertinent fluctuation can also promote the superconductivity via the attractive forces in all paring channels Lederer ; Metlitski ; Maier ; Labat , or by reducing the intra-pocket repulsion FernendesSUST ; Maslov . Our observation of the sign change of $T^{nem}$ around $x$ = 0.06 strongly corroborates those theoretical expectations.

Interestingly, upon investigating the doping dependence of the magnitude of $\tilde{n}$, one can clearly observe a significant enhancement of this quantity at the optimal doping level. This can be inferred from the data in the upper panels of Fig. 3(a-k) and also the contour plot presented in Fig. 4, revealing that the enhanced $\tilde{n}$ mimics the superconducting dome. The latter trend is also clearly visible in the evolution of the Curie constant displayed in Fig. 4, too, which represents the divergency coefficient of the $\tilde{n}(T)$ curve. This finding of an enhanced $\tilde{n}$ on top of the critical point of $T^{nem}$ and maximum $T_{c}$ strongly underpins the above notion of a putative nQCP near the optimal doping level.

The further inspection of $\tilde{n}(T)$ in Figs. 3 and 4 at lower doping clearly reveals a second peak-like enhancement of it, very close to the magnetic critical point where the AFM order disappears but still at doping levels with an AFM ground state. An alternative nematic critical point as the origin of the additional peak in the underdoped region at $x$ $\approx$ 0.04 can be ruled out because the fitting of $\tilde{n}(T)$ gives unambiguously a finite $T^{nem}\approx$ 35 K at $x$ $\approx$ 0.04 SM . One may speculate that the enhanced $\tilde{n}$ around $x$ = 0.04 is caused by magnetic fluctuations, which are critically enhanced at the doping levels just below the disappearance of the AFM order (like in many other iron-based superconductors DaiPC ), and remain small up to the optimal doping level $x$ = 0.06 Grafe . This notion is supported by the established $T^{nem}\approx$ 35 K for $x\approx$ 0.04 which roughly matches the observed magnetic ordering temperature $T_{N}$ around this doping level. Thus, altogether our data seem to reveal that in the underdoped region of the phase diagram for $x\leq$ 0.05, $\tilde{n}$ follows the trend of the magnetic fluctuations which are connected to the incipient order. At higher doping, $\tilde{n}$ is enhanced and seems decoupled from the magnetic fluctuations, while on top of the presumptive nQCP, $\tilde{n}$ and $T_{c}$ domes exist despite a reduced magnetic susceptibility. It is worthwhile to point out that the seeming appearance of two separated criticalities suggests nematicity itself can exist as a primary fluctuation and may be responsible for the superconductivity. Furthermore, the comparable Curie constant $C$ around the two critical doping levels indicates a similar strength of the elasto-electronic and elasto-magnetic couplings in this material. This is an interesting point to look at in further investigations.

The dichotomic-origin of the $\tilde{n}$ in LaFe1-xCoxAsO provides fresh input for rationalizing the importance of nematic fluctuations in other members of the iron-based superconductor family. In electron-doped BaFe2As2, the peak of $\tilde{n}$ has a broad appearance with the tendency to maximize at a slightly underdoped level Chu2012 ; Kuo2016 ; Christoph . Since the magnetic and structural transition lines of those materials are close to each other, the measured $\tilde{n}R$, a superposition of the two profiles which maximize at slightly separated end points, could appear like a broadened peak in between. We note that FeSe1-xSx has recently been proven to be a remarkable system which possesses an nQCP under its superconducting region Hosoi ; Licciardello . Although no AFM order exists in its ambient pressure phase diagram, magnetic fluctuations are still interpreted as the booster of superconductivity in FeSe1-xSx Matsuura . As we have mentioned above, however, magnetic fluctuations are suppressed at the superconducting doping levels in LaFe1-xCoxAsO, suggesting it is a more suitable prototype system to search for the relationship between superconductivity and nematicity. Finally, we point out that recent reports also claimed separated QCPs masked below the wide-spread superconducting dome in cobalt- and nickel-doped NaFeAs, extracted from the nuclear magnetic resonance and neutron scattering spectra respectively Zheng ; Wang . We therefore suspect a similar double-peak $\tilde{n}$ profile would also be detected in the NaFeAs family.

To summarize, we found that the Curie-Weiss-like $\tilde{n}$ also exists in LaFe1-xCoxAsO, in line with the other iron-based superconductors. The sign-change of $T^{nem}$ around $x$ = 0.06 and the divergent amplitude of $\tilde{n}$ by approaching this doping level are consistent with a possible nQCP at the optimal doping level. An additional peak of $\tilde{n}$ in the underdoped region around $x$ $\approx$ 0.04 is present close to the end point of the AFM transition line in LaFe1-xCoxAsO, which we attribute to a coupling of the nematic fluctuation to critical magnetic fluctuations. Altogether, the double-peak feature of $\tilde{n}$ suggests another origin of nematicity in iron-based compounds, besides the well accepted vestigial magnetism explanation CFang ; FeSe . A detailed map out of the magnetic fluctuations in LaFe1-xCoxAsO is important to further understand nematicity in this material.

We would like to thank Rüdiger Klingeler, Sven Sauerland, Liran Wang, Seung-Ho Baek, Christoph Wuttke, and Steffen Sykora for helpful discussions. This work has been supported by the Deutsche Forschungsgemeinschaft (DFG) through the SFB 1143 (project-id 247310070), through the Priority Programme SPP1458 (Grant No.BU887/15-1), under grant DFG-GRK 1621, through DFG Research Grants AS 523/4-1 (S.A.) CA1931/1-1 (F.C.), and through the Emmy Noether Programme WU595/3-3 (S.W.). X.C.H. acknowledges the support of the Leibniz-DAAD Research Fellowship. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 647276-MARS-ERC-2014-CoG) and the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter $ct.qmat$ (EXC 2147, project-id 390858490).

∗ E-mail: x.c.hong@ifw-dresden.de

† E-mail: c.hess@ifw-dresden.de