Nanoscale decoupling of electronic nematicity and structural anisotropy in FeSe thin films

In a material prone to a nematic instability, anisotropic strain in principle provides a preferred symmetry-breaking direction for the electronic nematic state to follow. This is consistent with experimental observations, where electronic nematicity and structural anisotropy typically appear hand-in-hand. In this work, we discover that electronic nematicity can be locally decoupled from the underlying structural anisotropy in strain-engineered iron-selenide (FeSe) thin films. We use heteroepitaxial molecular beam epitaxy to grow FeSe with a nanoscale network of modulations that give rise to spatially varying strain. We map local anisotropic strain by analyzing scanning tunneling microscopy topographs, and visualize electronic nematic domains from concomitant spectroscopic maps. While the domains form so that the energy of nemato-elastic coupling is minimized, we observe distinct regions where electronic nematic ordering fails to flip direction, even though the underlying structural anisotropy is locally reversed. The findings point towards a nanometer-scale stiffness of the nematic order parameter.

The modulation lines observed in multilayer FeSe/SrTiO 3 (001) are roughly along 110 orientation (the nearest-neighbor Fe-Fe direction). Assuming the film surface is a free surface with a bulk lattice constant a f that is smaller than the substrate lattice constant a s , a straightforward geometrical condition dictates that n times the substrate lattice constant a s can accommodate n + 1 times a f . Therefore and the dislocation lines spacing is Note that the dislocation lines are along 110 orientation. In our case, a f and a s in this direction are √ 2 of the in-plane lattice constants of FeSe and SrTiO 3 , which are 0.377 nm and 0.39 nm, respectively. Using the formula above, l is calculated to be ∼16 nm, which is within the range of experimental dislocation network spacings observed. We note that the system exhibits small structural buckling along the c-axis, but as this does not break any in-plane symmetries, we do not expect it to play a role in determining if an electronic nematic domain would be oriented along a-or b-axis.

Supplementary Note 2 Non-dispersing charge stripes in conductance maps
In contrast to the C 2 -symmetric dispersing features pinned around the impurities, the charge-ordered stripes do not show evidence of dispersion in a range of sample biases measured (Supplementary Figure 1). Supplementary Note 3 Determination of the strain tensor maps from STM topographs An STM topograph is a quasi-periodic function T (r) defined in a 2-dimentional realspace. A perfectly periodic atomic lattice can be expressed using a discrete Fourier series: where g represents reciprocal lattice vectors. A small spatially-varying distortion from this ideal lattice can be accounted for by changing H g into H g = A g (r)e iP (r) . The phase P (r) is equivalent to the displacement field u(r), defined as: −g · u(r) = P (r). Combing these, we have: With u(r) determined by the algorithm, one can apply a transformation: and the topograph expressed in terms of new coordinates r : will be a perfectly periodic lattice. We determine u(r) from atomically resolved STM topographs using the Lawler-Fujita algorithm described in Ref. 1. The coarsening length scale L = 1/Λ u , as defined in Ref. 1, is indicated in the caption of each strain analysis figure. The strain tensor u ij (r) is defined as the gradient of the displacement field u(r): Empirically, certain precautions need to be taken when calculating the strain tensor from the displacement field. The real strain caused by lattice mismatch constitutes a portion of u(r), while another significant contribution comes from piezoelectric nonlinearity, thermal drift, and hysteretic effects of the STM scanner. These effects can be represented by a slowly varying "background" in u(r) 2,3 , and here we apply 2 nd -degree polynomial fit to remove it.

Supplementary Note 4 Bias-independence of the strain maps and data reproducibility
We show that quasi-periodic structural modulations and strain variations are intrinsically of a structural origin by showing the bias-independence of the strain maps ( Supplementary Figures 2 and 4).
We also demonstrate data reproducibility on multiple samples (sample #1 in the main text and Supplementary Figure 2; sample #2 in Supplementary Figures 4 and 5). While the sample shown in the main text was capped with ∼50 nm thick amorphous Se, transferred to the STM in air and then de-capped in UHV, the sample shown in Supplementary Figures 4 and 5 was grown and transferred in a vacuum suitcase to STM, entirely in UHV conditions. Structural modulations and electronic nematicity were observed on thicker layers in both samples, indicating they are independent of the de-capping procedure.     Supplementary Figure 2(b). The resolution is ∼7 pixels per atom along Se-Se directions. We determine the √ 13 × √ 13 R33.7 • SrTiO 3 surface reconstruction by a RHEED image of SrTiO 3 and STM topograph of 1 ML FeSe (Supplementary Figure 6(a,b)). Similar RHEED pattern has been reported in Ref. 4, where the surface reconstruction was confirmed by low energy electron diffraction (LEED) as well. On the 1 ML FeSe, we observed a stripe-like modulation. The angle between the stripes and the Se-Se lattice vector is 34.8 • and the width of the stripes is ∼ √ 13a Se-Se . As the direction of Se-Se is along the same direction as Ti-Ti in SrTiO 3 , we ascribe the stripe-like reconstruction to a √ 13 × √ 13 R33.7 • SrTiO 3 surface reconstruction.

34.8°√
13a Se-Se  Supplementary  Figure 7(i)). We also observe a superconducting gap of ∼9 meV magnitude in differential conductance spectra over the same region ( Supplementary Figure 7(j)) and a wide gap-like opening observed in dI /dV spectra acquired over a larger energy range, consistent with the gap at the Γ point ( Supplementary Figure 7(k)). Both the QPI peaks and the superconducting gap are consistent with previous works 5,6 . We model each modulation line as a single edge dislocation at the interface of the film and the substrate. The theoretically expected strain arising from an edge dislocation can be calculated using elasticity theory 7,8 . Since FeSe film has a smaller in-plane lattice constant compared to the SrTiO 3 (001) substrate, each edge dislocation that would form in FeSe during the growth process would consist of an extra half-plane of atoms perpendicular to the interface (Fig. 3(f)). The in-plane displacement field u(x) on the surface of FeSe, caused by a dislocation line as a function of distance x from its core is predicted to be 8 : where b is the Burgers vector and d is the thickness of the film from the origin of misfit dislocation near the interface to the surface. Taking its derivative, we can obtain the theoretically predicted strain tensor component. For example, u xx can be written as: Without loss of generality, strain tensor components can be determined along any vector directions, and in this work we choose the two Fe-Fe lattice directions (a-and b-axes). We calculate the theoretical u aa (u bb ) maps as the superposition of the strain caused by 3 dislocation lines oriented roughly along the b-axis (a-axis) ( Fig. 3(g,h)). We note that the calculated strain from elasticity theory is relative to the lattice constant of the undeformed substrate and consequently, it always takes negative values ( Fig. 3(g-i)), which should be distinguished from the experimental strains calculated from STM topographs. However, the theoretical antisymmetric strain map can be directly compared to the experimental one since it represents the difference in u aa and u bb , and they show surprising resemblance (Fig. 3(j)). We emphasize that the theoretical model does not take into account any structural anisotropy due to electronic nematic state in FeSe.
Supplementary Note 8 Discussion on the absence of superconductivity at the surface of multilayer FeSe/SrTiO 3 (001) Some of the initial MBE growth of FeSe on graphitized SiC demonstrated that both ultrathin and thick FeSe films are superconducting, with the T c ∼8 K in thick films (comparable to that in bulk single crystals) getting reduced to ∼2 K in the monolayer limit 9,10 . The reduction of T c approaching the 2D limit can be theoretically explained by an additional boundary condition in Ginzberg-Landau free energy equation 11 . However, FeSe on SrTiO 3 behaves very differently. While a monolayer FeSe on SrTiO 3 shows a dramatic increase of T c up to ∼60-100 K, the surface of the second FeSe layer on SrTiO 3 is already not superconducting as shown from STM dI /dV spectra 12,14 . Interestingly, bulk transport of thicker FeSe films (≥ 2 monolayer thickness) on SrTiO 3 still shows superconductivity, suggesting that superconductivity originates from the interface of FeSe and SrTiO 3 itself, even though the topmost layer is not superconducting 12 . There may be several reasons for the absence of superconductivity at the surface of thicker FeSe films on SrTiO 3 . First and foremost, it is possible that electronic nematicity enhanced by strain in multilayer thin films (T N in multilayer FeSe films is ∼120 K 13 compared to ∼90 K in the bulk) directly competes with superconductivity. It has also been argued that insufficient charge carrier doping away from the interface may be responsible 14 . This is consistent with experiments using K surface doping, which leads to the return of superconductivity at the surface of thicker FeSe films 15 . Lastly, it is worth noting that superconducting mutilayer FeSe films on SiC mentioned above are nearly "free-standing" (based on the fact that FeSe islands grown on SiC can be moved by an STM tip, which in turn suggests a weak coupling of FeSe to the substrate, with nearly zero effective strain) 9 . On the other hand, FeSe films grown on SrTiO 3 are more strongly bonded to the substrate -there are no reports of moving the FeSe islands on SrTiO 3 and there clearly exists large spatially varying strain. So it is likely that strain indeed plays some role in explaining why thicker films are not superconducting, possibly by enhancing electronic nematicity.