Emerging symmetric strain response and weakening nematic fluctuations in strongly hole-doped iron-based superconductors

Electronic nematicity is often found in unconventional superconductors, suggesting its relevance for electronic pairing. In the strongly hole-doped iron-based superconductors, the symmetry channel and strength of the nematic fluctuations, as well as the possible presence of long-range nematic order, remain controversial. Here, we address these questions using transport measurements under elastic strain. By decomposing the strain response into the appropriate symmetry channels, we demonstrate the emergence of a giant in-plane symmetric contribution, associated with the growth of both strong electronic correlations and the sensitivity of these correlations to strain. We find weakened remnants of the nematic fluctuations that are present at optimal doping, but no change in the symmetry channel of nematic fluctuations with hole doping. Furthermore, we find no indication of a nematic-ordered state in the AFe2As2 (A = K, Rb, Cs) superconductors. These results revise the current understanding of nematicity in hole-doped iron-based superconductors.

Supplementary Fig. 3. No evidence for a nematic phase transition in RbFe2As2. a The temperature derivative of the resistance of RbFe2As2 (normalized at 300 K), showing no sharp anomalies associated with a possible phase transition [3] at the temperature of maximummA 1g around 40-50 K. The inset shows the same data, with a linear background subtracted. b The uniaxial in-plane thermal expansion coefficient αa of RbFe2As2, showing no sharp anomalies associated with a possible phase transition around 40-50 K. The inset shows the same data, with a linear background subtracted. Note that the α is expected to be a sensitive probe for pressure sensitive phase transitions.

SUPPLEMENTARY NOTE 2: QUANTITATIVE COMPARISON BETWEEN ELASTORESISTANCE AND THERMODYNAMIC QUANTITIES
The resistance of KFe 2 As 2 can be fit to R = R 0 + AT 2 at low temperature. In this Fermi-liquid formula, A is a measure of the effective mass according to A = c KW γ 2 , where c KW = 2 × 10 −6 µΩ cm(K mol/mJ) 2 is the Kadowaki-Woods constant appropriate for this system [4]. Using this constant, we can compare the strain dependence of the A coefficient ( Fig. 3b of the main text) with the strain dependence of γ inferred from the thermal expansion.
We consider here the case of KFe 2 As 2 with strain applied along [100] (Fig. 3 ), we obtain ∆γ = 2.67 mJ/molK 2 for the expected change of γ in our experiment. Note that this calculation assumes that γ is unaffected by the B 1g component of strain.
To calculate ∆γ from resistance measurements, we use ∆γ = ∆A/2c KW γ. Defining A A1g ≡ (A xx + A yy )/2 we find that the change in A A1g under strain xx is given by dA yy /d xx = 0.219 µΩ cm/K 2 is the slope of Fig. 3b of the main text. dA xx /d xx = 0.088 µΩ cm/K 2 is found similarly from the longitudinal elastoresistance data (not shown). To obtain these numbers, we use ρ(300 K) = 300 µΩ cm [5]. Using γ = 100 mJ/molK 2 for freestanding KFe 2 As 2 [6,7], we then predict ∆γ = ∆A A1g /2c KW γ = 1.61 mJ/molK 2 . If we now correct for the fact that the true strain differs from the nominal strain in our strain cell according true = 0.6 disp (Fig. 1), we obtain ∆γ = 1.61/0.6 mJ/molK 2 = 2.68 mJ/molK 2 from the elastoresistance, in agreement with the value ∆γ = 2.67 mJ/molK 2 expected from the thermodynamic data calculated above. A similar comparison for RbFe 2 As 2 and CsFe 2 As 2 is not possible because the low-temperature resistance is well fit by R = R 0 + AT n with n = 2, due to non-Fermi liquid behavior. The simple interpretation of the A coefficient as effective mass does not hold in this case.
We can also compare the strain dependence of T c with values inferred from thermodynamic measurements via Ehrenfest relations [4]. For the case of uniaxial stress, we have that

S4
For the case of KFe 2 As 2 , we have ∂T c /∂ a = 162.12 K and ∂T c /∂ c = 53.24 K from thermodynamic data in Ref. [4].  (Fig. 1), we obtain ∆T c /∆ a = 62.3±7.0 K in reasonable agreement with the thermodynamic value extracted from Ref. [4] (∆T c /∆ a = 82.18 K).  The coherence-incoherence crossover in these materials is clearly seen as a maximum in the thermal expansion coefficient divided by temperature α/T [6,7], which is a measure of the pressure dependence of the entropy. Since the resistance is known to be sensitive to the electronic entropy in these materials [8,9], the strain derivative of resistance is expected to be related to α/T . In particular, α a /T measures the change in entropy in response to in-plane symmetric stress, and therefore should probe similar physics as the A 1g elastoresistance coefficient m A1g . To compare the temperature dependence of m A1g with the temperature dependence of α a /T , we start from the Fisher-Langer relation for the resivitivy ρ in quantum critical magnets [10][11][12][13] ∂ρ

SUPPLEMENTARY NOTE 3: QUALITATIVE COMPARISON BETWEEN ELASTORESISTANCE AND THERMAL EXPANSION
where C mag is the magnetic specific heat capacity. Using resistance R in place of ρ, it follows that Taking the pressure derivative of both sides while treating R and S mag as multivariable functions of T and in-plane pressure p, we find that The thermal expansion coefficient α is defined as α = (−1/V m )∂S/∂p, where S is the molar entropy and V m is the molar volume. Here, we make the assumption that ∂S/∂p is dominated by the magnetic contribution in quantum critical magnets. Therefore, we have One can integrate this equation to obtain Here, the integration constant is chosen that α → 0 as T → 0, in accordance with the third law of thermodynamics.
To make contact with elastoresistance, we use (∂R/∂ ) A1g = −c A1g (∂R/∂p) A1g , where the minus sign arises because pressure is defined as positive for compression, while strain is defined as positive for tension. We then obtain where c A1g is the elastic constant for symmetric in-plane stress, which we take to be temperature independent for simplicity. The elastoresistance coefficient is m A1g = (1/R)(∂R/∂ ) A1g , so that m A1g R = (∂R/∂ ) A1g . In Fig. 4, we make this comparison and find qualitative agreement between thermal expansion and elastoresistance. However, for RbFe 2 As 2 and KFe 2 As 2 the coherence-incoherence crossover is not as clear in elastoresistance. This may relate to the fact the Fisher-Langer relation applies only in a quantum critical region. CsFe 2 As 2 is known to be the most critical of these materials [4,14]. Discrepancies may also be due to a temperature dependence of the elastic constant c A1g .