Elastoresistivity of Heavily Hole-Doped 122 Iron Pnictide Superconductors

Nematicity in heavily hole-doped iron pnictide superconductors remains controversial. Sizeable nematic fluctuations and even nematic orders far from magnetic instability were declared in RbFe₂As₂ and its sister compounds. Here, we report a systematic elastoresistance study of a series of isovalent- and electron-doped KFe₂As₂ crystals. We found divergent elastoresistance on cooling for all the crystals along their [110] direction. The amplitude of elastoresistivity diverges if K is substituted with larger ions or if the system is driven toward a Lifshitz transition. However, we conclude that none of them necessarily indicates an independent nematic critical point. Instead, the increased nematicity can be associated with another electronic criticality. In particular, we propose a mechanism for how elastoresistivity is enhanced at a Lifshitz transition.

1 Introduction

The “122” family, an abbreviation coined for BaFe₂As₂ and its substituted sister compounds, played a central role in the study of iron-based superconductors [1]. Those tetragonal ThCr₂Si₂-type structured compounds have the advantage that sizeable single crystals with continuous tunable doping can be prepared in a wide range, which is a crucial merit for the systematic investigation of various ordered states. Within the extended phase diagram of 122 compounds, the heavily hole-doped region, including the end-members K/Rb/CsFe₂As_2,, is of particular interest. The superconducting transition temperature T_c of Ba_1−xK_xFe₂As₂ peaks at optimal doping x = 0.4 and continuously decreases toward the overdoped (larger x) region. T_c remains finite in the end-member x = 1, while a change of the Fermi surface topology (Lifshitz transition) exists around x = 0.6 ∼ 0.8 [2]. Although the T_c vs. x trend seems to be smooth across the Lifshitz transition, there are quite a lot of things happening here. Vanishing electron pockets for x > 0.8 destroy the basis of the interpocket scattering induced-S± pairing symmetry which is generally believed as a feature of most iron-based superconductors. As a result, a change in the superconducting gap structure across the Lifshitz transition was observed experimentally [2–4]. Comparable pairing strength at the transition can foster a complex pairing state that breaks the time-reversal symmetry. Such an exotic state was also demonstrated to exist around the Lifshitz transition [5, 6]. Very recently, a so-called “Z₂ metal state” above T_c at the Lifshitz transition has been unveiled, with an astonishing feature of spontaneous Nernst effect [7].

Electronic nematicity, a strongly correlated electronic state of electrons breaking the underlying rotational symmetry of their lattice but preserving translation symmetry, has been a wave of research in unconventional superconductors, particularly in iron-based superconductors [8, 9]. Consistent experimental efforts have identified nematicity in all the different iron-based superconductor families [10–15], accompanied by theoretical proposals of the intimate relationship between nematicity and superconducting pairing [16–19]. However, according to the previous background, we should not simply extend what is known in the under- and optimal-doped 122s to the over-doped region. Whether nematicity exists and how it develops in this region needs independent censoring.

Indeed, nematicity in the heavily hole-doped 122 turns out to be more elusive. Heavily hole-doped 122s stand out as a featured series because of their peculiar Fermi surface topology, isostructural phase transition, and possible novel pairing symmetries [20–23]. Nematically ordered states were suggested by nuclear magnetic resonance spectroscopy and scanning tunneling microscopy on CsFe₂As₂ and RbFe₂As₂, and they were found to develop in different wave vectors other than the underdoped 122s [24, 25]. Such a nematic state far away from magnetic ordering challenges the prevailing idea that nematicity is some kind of vestigial order of magnetism [26]. An elastoresistance study further claims that a tantalizing isotropic (or XY-) nematicity is realized in the crossover region from dominating [100] nematicity in RbFe₂As₂ to [110] nematicity at the optimal doping [27]. However, many works pointed out that elastoresistance in K/Rb/CsFe₂As₂ is actually contributed by the symmetric A_1g channel, having little to do with the B_1g or B_2g channels which are related to nematicity [28, 29]. Overall, the debate is still on for this topic.

In this brief report, we will not touch upon the nature of the possible nematicity of K/Rb/CsFe₂As₂. Instead, we confirm phenomenologically the existence of elastoresistance (χ^er) in K/Rb/CsFe₂As₂ and find that its amplitude diverges exponentially with growing substituted ion size. Moreover, we present χ^er data on a series of Ba_1−xK_xFe₂As₂ crystals crossing the Lifshitz transition. We observe, unexpectedly, a clear enhancement of χ^er from both sides of the Lifshitz point. Although a presumptive nematic quantum critical point (QCP) might be of relevance, here we propose a rather more conventional explanation based on a small Fermi pocket effect. Our results add a novel phenomenon to the Lifshitz transition of the Ba_1−xK_xFe₂As₂ system and highlight another contributing factor of elastoresistance which has been almost ignored so far.

2 Experimental Details

Single crystals of heavily hole-doped Ba_1−xK_xFe₂As₂ were grown by the self-flux method [30–32]. The actual doping level x was determined by considering their structural parameters and T_c values. Elastoresistance measurements were performed as described in Ref. s [10–12]. Thin stripe-shaped samples were glued on the surface of piezo actuators. The strain gauge were glued on the other side of the piezo actuators to monitor the real strain generated. In most cases, the samples were mounted to let the electric current flow along the polar direction of the piezo actuators (R_xx), along which direction the strain was measured by the gauge. For one sample (x = 0.68), an additional crystal was mounted at 90° rotated according to the polar direction (R_yy). More details are described in Section 3.3. The sample resistance was collected with a combination of a high-precision current source and a nanovoltage meter. Because of the very large RRR (R_300K/R₀) values of the samples, special care was taken to avoid a temperature drift effect, and the electric current was set in an alternating positive/negative manner to avoid artifact.

We point out that noisy and irreproducible elastoresistance results can be acquired if DuPontⓇ 4922N silver paint is used for making contacts to the samples. On the other hand, samples contacted with EPO-TEKⓇ H20E epoxy or directly tin-soldering gave perfectly overlapping results. Given that DuPontⓇ 4922N silver paint is widely used for transport measurements and is indeed suitable for elastoresistance experiments of other materials (for example) the LaFe_1−xCo_xAsO series [12], we have no idea why it does not work for heavily hole-doped Ba_1−xK_xFe₂As₂ crystals. In this work, the presented data were collected by using the H20E epoxy. To avoid sample degradation, the epoxy was cured inside an Ar-glove box. A similar silver paint contact problem of K/Rb/CsFe₂As₂ crystals was also noticed by another group [29].

3 Results and Discussions

3.1 Elastoresistance Measurement

The elastoresistance measured along the [110] direction of the KFe₂As₂ single crystal is shown in Figure 1. The sample resistance closely followed the strain change of the piezo actuator when the voltage across the piezo actuator is tuned. As presented in Figure 1B, the relationship between resistance change (ΔR/R) and strain (ΔL/L) is linear. This fact ensures that our experiments were performed in the small strain limit. In such a case, the elastoresistance χ^er, defined as the ratio between ΔR/R and the strain, acts as a measurement of the nematic susceptibility [10]. It is worthwhile to note that χ^er in KFe₂As₂ is positive (sample under tension yields higher resistance), consistent with the previous reports [27, 28] and opposite to that of BaFe₂As₂ [10]. It is to be noted that sign reversal of the elastoresistance was reported to occur in the underdoped region [33].

FIGURE 1

FIGURE 1. Representative example of elastoresistance under strain for KFe₂As₂. (A) Resistance and strain change according to the voltage applied across the piezo actuator at a fixed temperature T = 50 K. The strain was applied along the [110] direction. (B) Change of resistance ΔR/R as a function of strain ΔL/L at several temperatures.

3.2 Elastoresistance of K/Rb/CsFe₂As₂

We start by showing our χ^er(T) data measured along the [110] direction (${\chi }_{[110]}^{er}$) for a set of (K/Rb/Cs)Fe₂As₂ crystals. As shown in Figure 2, all the ${\chi }_{[110]}^{er}$(T) curves follow a divergent behavior over the whole temperature range. A Curie–Weiss (CW) fit

${\chi }^{er}={\chi }_0+\frac{\lambda /a_0}{T-T^{nem}}(1)$

can record the data. A slight deviation can be discriminated at low temperature, which is typical for elastoresistance data and is understood as a disorder effect [13]. It is to be noted that the amplitude of the elastoresistance grows substantially from KFe₂As₂ to CsFe₂As₂, nearly 5-fold at 30 K. The extracted parameters from the CW fit are shown in Figures 2E,F. While the amplitude term shows a diverging trend, the T^nem of all four samples is of a very small negative value, which practically remains unchanged if experimental and fit uncertainties are taken into account, which is at odds with a possible nematic criticality in this isovalent-doping direction. The enhanced ${\chi }_{[110]}^{er}$ might be a result of a presumptive QCP of an unknown kind or a coherence–incoherence crossover [34–36]. These cannot be discriminated by our technique and thus are beyond the scope of this report.

FIGURE 2

FIGURE 2. Temperature dependence of elastoresistance ${\chi }_{[110]}^{er}$ for (A) KFe₂As₂, (B) RbFe₂As₂, (C) K_0.046Cs_0.095Fe₂As₂, and (D) CsFe₂As₂. Solid lines are CW-fit of the data (see text). The fit parameters T^nem and λ/a₀ are summarized in (E,F), respectively. The dotted line in (F) is a guide to the eye.

3.3 Elastoresistance of Overdoped Ba_1−xK_xFe₂As₂

Next, we present a set of χ^er(T) data of five overdoped Ba_1−xK_xFe₂As₂ (0.55 ≤ x ≤ 1) across the Lifshitz point. The elastoresistance, measured only for the R_xx direction, as has been performed regularly in many reports [10, 12, 14], has been argued to be inconclusive for the end members (K/Rb/Cs)Fe₂As₂, as a result of the dominating A_1g contribution, instead of a B_2g (or B_1g) component which is related to nematicity [28, 29]. However, such complexities are ruled out by taking R_yy into account for calculating χ^er(T) for one representative example x = 0.68 (Figure 3B). The χ^er(T) curves calculated by the two different methods match well.

FIGURE 3

FIGURE 3. Doping evolution of the elastoresistance in overdoped Ba_1−xK_xFe₂As₂. χ^er measured along the [110] direction is presented in the upper panels for Ba_1−xK_xFe₂As₂ single crystals with (A) x = 0.55, (B) x = 0.68, (C) x = 0.78, (D) x = 0.86, and (E) x = 0.97. The red dashed lines are CW-fit to the data. χ^er was also measured along the [100] direction for three of the samples. The data are presented in the lower panels of (A) and (C,D). In panel (B), χ^er extracted by using both (ΔR/R)_xx and (ΔR/R)_yy (filled light blue circles) and (ΔR/R)_xx (open blue squares) shows indistinguishable results. A doping dependence of the fit parameters is displayed in (F) λ/a₀ and (G) T^nem of the [110] ${\chi }_{[110]}^{er}$ data. The thick lines are a guide to the eye. A Lifshitz transition around x = 0.8 is highlighted by the green bar. Hole pockets around the M point of the Brillouin zone transform into electron lobes across the Lifshitz point [2]. T^nem at x = 0.4 is extracted from Ref. [13].

After checking the potential A_1g contribution to χ^er for a doping level close to the Lifshitz transition, we turn now to the data. As shown in Figure 3, the ${\chi }_{[110]}^{er}$(T) curves of Ba_1−xK_xFe₂As₂ also follow a CW-like feature. One can see a clear dip at around 50 K in Figure 3A for the x = 0.55 sample. In some reports [27], such a feature was taken as a signal for a nematic order. Since no other ordering transition (structural, magnetic, and so on) has been ever reported in this doping range, we refrain from claiming an incipient nematic order solely based on such a feature. This might be equally well-explained by different origins. However, we also cannot exclude its possibility.

On the other hand, we measured χ^er along the [100] direction $({\chi }_{[100]}^{er})$ for several samples. They are shown in the shaded panels of Figure 3. All of them are negative and small in amplitude, and none of them shows a CW-like feature. Such an observation is incompatible with the existence of B_1g nematicity in the heavily doped Ba_1−xK_xFe₂As₂ series, which is in sharp contrast to what is reported for the closely related Ba_1−xRb_xFe₂As₂ series [27, 37, 38]. As a result, the so-called XY-nematicity is clearly ruled out in the Ba_1−xK_xFe₂As₂ series.

One remarkable feature, however, that can be safely concluded is that the amplitude of ${\chi }_{[110]}^{er}$ has a clear tendency to peak around x = 0.8, close to the Lifshitz transition. This becomes clearer in Figure 3F, where the CW fit parameters are plotted against the doping level. The question is why ${\chi }_{[110]}^{er}$ is increased at the Lifshitz transition? A nematic QCP is a potential explanation. However, as Figure 3G shows, T^nem drops from $\sim 45$ K of the x = 0.4 (optimal-doped) to $\sim 0$ K at the Lifshitz transition. Further doping does not drive T^nem to the more negative side within our experimental resolution. This is not typical QCP behavior. Moreover, since T_c across the Lifshitz transition is quite smooth, it seems not to be boosted by pertinent potential nematic fluctuations. Furthermore, three-point bending experiments did not show any anomaly in this doping range [39]. All these facts seem to be incompatible with the more understood nematic QCP in the electron-doped side [40]. Hence, if it is a nematic QCP, novel mechanisms need to be invoked. This motivated us to seek for alternative explanations for enhanced χ^er in heavily doped Ba_1−xK_xFe₂As₂. In the following section, we propose a conventional argument based on the small pocket effect, exempting from invoking a QCP to exist at the Lifshitz transition.

3.4 Theory for Enhanced Elastoresistance at the Lifshitz Transition

To study the effect of a Lifshitz transition to elastoresistance, we have calculated this quantity based on a minimal model of iron-based superconductors [41] with a very small Fermi surface. The corresponding dispersion which was used is shown in Figure 4A for the normal state along a cut (π, k_y). We considered the two orbital models in Ref. [41] with the same hopping matrix elements but having set the nematic interaction equal to a very small value. Thus, the nematic interaction accounts here only for the temperature dependence of the susceptibility according to the Curie–Weiss law. Moreover, we introduced a very small lattice distortion in the x direction which is coupled with the electron system. Using the first-order perturbation theory with respect to this coupling (linear response), we then calculated the elastoresistivity. We have considered two different cases of the coupling between distortion and electrons (strength g): (i) The conventional coupling with the local electron density (electron–phonon coupling), where we denote the corresponding response with χ_ph. (ii) A direct coupling of the distortion with the hopping matrix element t_x in the x direction. The corresponding response is denoted by χ_t.

$\begin{array}{cc}\hfill {\chi }_t& \propto g\underset{\mathrm{\Delta }{t}_x\to 0}{\lim }\frac{{\sigma }_{xx}^{-1}\left(t_x+\mathrm{\Delta }{t}_x\right)-{\sigma }_{yy}^{-1}\left(t_x+\mathrm{\Delta }{t}_x\right)}{\mathrm{\Delta }{t}_x}\hfill \\ \hfill {\chi }_{ph}& \propto \frac{1}{N}\sum _{\mathbf{k}}{\left(\frac{g}{\omega +{\epsilon }_{\mathbf{k}}-{\epsilon }_{\mathbf{k}+\mathbf{q}}}\right)}^2\frac{f_{\mathbf{k}}-f_{\mathbf{k}+\mathbf{q}}}{{\epsilon }_{\mathbf{k}+\mathbf{q}}-{\epsilon }_{\mathbf{k}}}.\hfill \end{array}(2)$

Here, t_x is the hopping matrix element in the x direction and σ_xx, σ_yy are the conductivities in the x and y directions [41]. The phonon energy ω in ${\chi }_{ph}^{er}$ is in general renormalized by the coupling to the electrons and becomes soft for a particular mode if the system is near a structural phase transition[42, 43]. The electron dispersion ɛ_k considered here is shown in Figure 4A. The function f_k is the Fermi distribution with respect to ɛ_k. Thus, for most systems investigated, this term dominates χ^er. It is to be noted how magnetic fluctuations impact the phonons and the nematicity has been investigated [9, 16].

FIGURE 4

FIGURE 4. Theoretical consideration of the elastoresisitivity. (A) Dispersion of the hole-like band leading to a very small hole pocket around the point (π, π) when the chemical potential μ_small (upper dotted line) is placed near the Lifshitz point. The range of momentum vectors contributing to the elastoresisitivity arising from the Fermi function is schematically shown by the dashed lines. A much larger hole pocket is indicated by a lower value of the chemical potential μ_large (lower dotted line). (B) Calculated parts of the elastoresistivity according to Eq. 3. Around the Lifshitz point μ_small the first-order part ${\chi }_t^{er}$ dominates the second-order part which is, however, the most important contribution for μ away from the Lifshitz point.

Figure 4B shows the two parts ${\chi }_t^{er}$ and ${\chi }_{ph}^{er}$ which were calculated separately as a function of the chemical potential μ to simulate different doping values. To compare with Figure 3F, we extracted the temperature behavior according to Eq. 2 and plotted the calculated value λ/a₀ in energy units of t_y. It is seen that the first-order part ${\chi }_t^{er}$ dominates the second-order part only in the narrow range of μ, where the Fermi surface around (π, π) becomes very small. Thus, only when the system has very small Fermi surfaces, as in the case of Ba_1−xK_xFe₂As₂ at the Lifshitz transition, the term ${\chi }_t^{er}$ becomes important. However, it is also seen that if the chemical potential is chosen away from the Lifshitz point corresponding to a proper doping, the second-order part ${\chi }_{ph}^{er}$ is mostly important as expected.

The enhancement of ${\chi }_t^{er}$ in the presence of a very small Fermi surface can be explained by the existence of low-energy excitations in a relatively wide range of momentum vectors. Since the conductivities are proportional to Fermi distribution functions f_k as follows

${\sigma }_{ii}\propto \sum _{\mathbf{k}}{\left(\frac{\partial {\epsilon }_{\mathbf{k}}}{\partial k_i}\right)}^2{f}_{\mathbf{k}}\left(1-f_{\mathbf{k}}\right),(3)$

we find that at low temperature, if the Fermi surface is small, the momentum range k, where f_k(1 − f_k) is non-zero, is much larger because of the tendency of the band to rapidly change the Fermi surface topology near the Lifshitz transition (compare the red dashed lines in Figure 4A) than for a usual Fermi surface.

4 Conclusion

To summarize, we reported that a CW-like χ^er(T) is observed for all kinds of heavily hole-doped 122s. There is an unexpected enhancement of the elastoresistance around the Lifshitz transition. We explained it as a small Fermi pocket effect on the nematicity. We expect that our explanation of an alternative contribution to the enhanced elastoresistance other than a nematic QCP will be considered in other systems, in particular for those with small Fermi pockets.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author Contributions

IM, SA, VG, KK, and C-HL prepared the samples. XH, FC, and MB performed the experiments. SS proposed the theoretical model. CH and BB supervised the study. XH, SS, FC, and CH analyzed the data and wrote the manuscript with input from all authors.

Funding

This work has been supported by the Deutsche Forschungsgemeinschaft (DFG) through SFB 1143 (Project No. 247310070), through the Research Projects CA 1931/1-1 (FC) and SA 523/4-1 (SA). SS acknowledges funding by the Deutsche Forschungs gemeinschaft via the Emmy Noether Program ME4844/1-1 (project id 327807255). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Program (grant agreement No. 647276-MARS-ERC-2014-CoG).

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Acknowledgments

We would like to thank Anna Böhmer, Ian Fisher, Suguru Hosoi, Rüdiger Klingeler, Christoph Meingast, Jörg Schmalian, Christoph Wuttke, Paul Wiecki, and Liran Wang for helpful discussions. We would like to thank Christian Blum and Silvia Seiro for their technical support.

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