Introduction

Intertwined spin and charge orders have been widely observed in cuprate and iron-based superconductors and their parent compounds1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24. As superconductivity and these orders are often governed by the same interactions, they could be intimately related. For example, in the static form, spin and charge orders are sometimes considered as competing orders to superconductivity, while in their dynamic form, the spin/charge/orbital fluctuations of some of these orders are suggested to mediate the superconducting pairing1,2,3,4,5,25, so the ordered phases are sometimes considered as the parent phases.

Antiferromagnetic and nematic order are the main form of intertwined orders in iron-based superconductors with both electron and hole Fermi surfaces, such as iron pnictides and bulk Fe(Se, Te)13,14,15,16,17,18,19,20,21,22,23,24. The nematic spin and orbital fluctuations are related to the scattering between electron and hole Fermi surfaces, which facilitate superconducting pairing in various theories26,27,28,29. However, such correspondence among intertwined orders, fluctuation and Fermi surface topology fails in heavily electron-doped FeSe superconductors, such as K1−xFe2−ySe2, (Li0.84Fe0.16OH)Fe1−xSe, and monolayer FeSe/SrTiO3, where only electron Fermi surfaces are present. Although a \(2\times 2\) charge order with respect to the Fe lattice was found in KxFe2−ySe230,31, the ordering wave vector does not fit their electron-only Fermi surface topology, and its relation to superconductivity is uncertain. Moreover, a \(\sqrt{5}\times \sqrt{5}\) Fe-vacancy order was observed in the insulating regions of K1−xFe2−ySe2, and its relation to superconductivity is remote5. On the other hand, inelastic neutron scattering (INS) experiments found that the low energy spin fluctuations are quite concentrated around (π ± δπ, π) and (π, π ± δπ) in RbxFe2-ySe2 and (Li0.84Fe0.16OD)Fe1-xSe with δ ~ 0.5 and 0.38 (refs. 32,33,34,35), respectively, which seems to be unrelated to any known order in these systems. Spin fluctuation is widely accepted as the clue of electron pairing in cuprate and iron pnictide superconductors, but it is still under strong debate in heavily electron-doped FeSe compounds36,37,38,39,40,41,42.

Recently, it was shown that the superconducting transition temperature (Tc) of (Li0.84Fe0.16OH)Fe1-xSe can be gradually suppressed by increasing the Fe-site defect concentration (x) in FeSe layers43,44,45,46,47,48. It thus provides a venue to search for the possible intertwined orders in heavily electron-doped iron selenides, especially the possible magnetic order in the parent compound. Here, we systematically studied (Li0.84Fe0.16OH)Fe1−xSe films with various Tc values by using low-temperature scanning tunneling microscopy (STM). With an increased x, we find that the superconducting gap feature is gradually suppressed, and a static checkerboard charge pattern emerges firstly as a defect-pinned state and subsequently extends to the whole defect-free regions, illustrating a competing behavior with superconductivity. Combining with our simulations, the charge order is likely driven by multiple-Q spin density waves originating from the spin fluctuations observed by INS experiments. Our results demonstrate the potential of charge order as a tool to detect spin fluctuations, and suggest that spin fluctuations may also play an important role in heavily electron-doped FeSe-based superconductors.

Results

Influence of Fe-site defect on superconductivity

Figure 1a–d shows the typical topographic images of FeSe-terminated surfaces for three (Li0.84Fe0.16OH)Fe1–xSe films with Tc of 42 K, 28 K, and 8 K, and another non-superconducting one, respectively (Supplementary Fig. 1). There is no obvious difference except for different amounts of dumbbell-shaped defects located at Fe-sites. These Fe-site defects can be roughly classified into two types according to their impurity potentials, Fe vacancies (type-I) or substitutional defects (type-II), and they both suppress superconductivity40,49,50,51,52 (see Supplementary Fig. 2 for more details). The statistical Fe-site defect concentration, x, increases from 0.54% in Fig. 1a to 6.35% in Fig. 1d. Figure 1e shows the typical dI/dV spectra within ±100 meV measured on defect-free regions of these Fe1-xSe surfaces. These spectra display similar line shape, with a relatively low tunneling conductance near EF but which increases rapidly above +70 meV and below −50 meV. As reported previously, these dI/dV upturns are from the onsets of an unoccupied electron band and an occupied hole band at the Brillouin zone (BZ) center, respectively; and the states near EF are contributed by electron bands at M points of BZ53,54. The almost identical dI/dV spectra observed for different defect concentrations suggest that Fe-site defects have little influence on carrier doping.

Fig. 1: Typical topographic images and dI/dV spectra for Fe1−xSe regions with varying x values.
figure 1

a–d Atomically resolved topographic images of four Fe1−xSe regions. Measurement conditions: a Vb = 0.5 V, It = 60 pA; b Vb = −0.55 V, It = 20 pA; c Vb = 0.1 V, It = 10 pA; d Vb = 0.1 V, It = 50 pA. e Typical dI/dV spectra within ±100 meV collected in defect-free FeSe regions (Vb = 100 mV, It = 80 pA, ΔV = 2 mV). Black arrows indicate the onsets of energy bands. f Averaged dI/dV spectra within ±40 meV collected in defect-free FeSe regions. The diamonds mark the positions of coherent peaks. Measurement conditions: Vb = 30 mV, It = 120 pA, ΔV = 0.1 mV for x = 1.34%; Vb = 40 mV, It = 120 pA, ΔV = 0.1 mV for other x values. g Evolution of dI/dV spectra across a type-I Fe-site defect in the Fe1−xSe region with x = 0.54%, taken along the green arrow in the inset. The distance from the spatial locations where the spectra are collected to the defect center is indicated and color-coded with the spectra. Inset: topographic image of a type-I Fe-site defect.

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The dI/dV spectra within ±40 meV measured in defect-free FeSe regions are shown in Fig. 1f (spatially averaged spectra, and see Supplementary Fig. 3 for point spectra). For x = 0.54%, a full superconducting gap with two pairs of coherence peaks at ±9 meV and ±15 meV is observed. With increasing x, the coherence peaks are weakened quickly, and the gap feature becomes broadened. Such overall suppression of superconductivity by Fe-site defects is similar to that observed in superconducting KFe2Se2 films55, confirming that Fe-site defect concentration is the key parameter controlling the superconductivity of (Li0.84Fe0.16OH)Fe1−xSe45. Meanwhile, the spectral weight within the whole energy range of ±40 meV is also strongly suppressed with increased x, and a broad V-shaped spectrum with almost zero density of states (DOS) around EF shows up for the non-superconducting x = 6.35% sample, which evidence a spectral gap opening. Such a partial gap is likely related to the formation of a charge order discussed in the next section. As shown in Fig. 1g, the superconducting gap is almost completely suppressed and strong in-gap states are observed on the type-I Fe-site defects, while the superconducting gap recovers beyond 1.5 nm away from the defect center.

Observation of the checkerboard charge patterns

dI/dV maps were acquired on different Fe1-xSe regions to reveal the local density of state (LDOS) distribution. Figure 2a, b shows the typical dI/dV maps collected in the Fe1-xSe (x = 0.54%) region in Fig. 1a (see Supplementary Fig. 4 for additional datasets under other energies). One can see clear quasiparticle interference (QPI) patterns around type-I Fe-site defects, which contribute to nine ring-like dispersive features around the BZ center and Bragg spots (qFe and qSe) in the fast Fourier transforms (FFT) image (Fig. 2c). These QPI patterns arise from intra- and interpocket scatterings of the electron pockets at M points of BZ, which have been widely observed in heavily electron-doped FeSe materials40,41,42,56,57. Intriguingly, besides the common QPI patterns, we can resolve additional FFT features around \(\frac{1}{2}\)qFe (labeled as q2Fe) as indicated by the yellow arrows in Fig. 2c. This feature is nondispersive over the measured energy range of ±45 meV, as indicated by the red dashed line in Fig. 2d. It is not clear near the Fermi energy though, which is likely due to the strong in-gap state induced by the impurity.

Fig. 2: Spatial LDOS modulations in the Fe1−xSe region with x = 0.54%.
figure 2

a, b Typical dI/dV maps (Vb = 45 mV, It = 50 pA, ΔV = 2 mV). Type-II Fe-site defects are enclosed by the yellow dashed circles, while the others are the type-I defects. c FFT image of the dI/dV map shown in (a). Additional FFT features around q2Fe are marked by yellow arrows. d FFT line cuts extracted along the yellow dashed line in (c), taken at various energies and shown in false color. The red dashed line indicates the nondispersive nature of q2Fe pattern. e, f Enlarged images of areas A and B marked by yellow boxes in (a) and (b), with grids of a 2aFe period superimposed. g iFFT image with features around q2Fe and their second-order components around qFe considered. Insert: enlarged image of the area marked by yellow box in (g), with a grid of a 2aFe period superimposed.

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The LDOS distribution around type-II Fe-site defects exhibits a subtle structure, as enclosed in the yellow dashed circles in Fig. 2a, b. As more clearly illustrated in Fig. 2e, f, it is a checkerboard pattern orientated similarly along the two Fe-Fe lattice directions with a period approximately twice that of the Fe lattice (2aFe), which naturally leads to FFT intensities around q2Fe. We conducted inverse FFT (iFFT) of the features around q2Fe and their second-order components around qFe to remove the interference of QPI signals, which gives Fig. 2g and Supplementary Fig. 4. The checkerboard patterns now became more distinguishable and show up around both type-I and type-II Fe-site defects. As moving away from a defect, the checkerboard pattern weakens quickly and disappears beyond ~1.5–2.0 nm, which is comparable to the length scale of superconducting gap suppression around Fe-site defects (Fig. 1g). One may wonder if the pinned checkerboard pattern is due to the Yu-Shiba-Rusinov (YSR) states induced by Fe-site defects. We have excluded this possibility by carefully examining the spatial distribution of the YSR states, which turns out to behave very differently (See Supplementary Fig. 5). Moreover, as shown later for the higher x cases (Figs. 3 and 4), the checkerboard pattern still exists and extends into defect-free regions (even in the non-superconducting samples), implying that the checkerboard pattern is a static order instead of YSR states. Consistently, such a checkerboard pattern is absent near Se vacancies, where superconductivity is intact (Supplementary Fig. 6). As STM measures tunneling-matrix-element weighted LDOS, the checkerboard pattern could reflect either a charge density modulation or an orbital reconstruction.

Fig. 3: Spatial LDOS modulations in an Fe1−xSe region with x = 2.2%.
figure 3

a Atomically resolved topographic image (Vb = 40 mV, It = 20 pA). b Typical dI/dV map at E = 40 meV (setpoint: Vb = 40 mV, It = 50 pA, ΔV = 2 mV). c Enlarged image of the area marked by yellow box in (b). The black box shows the unit cell of the checkerboard pattern with a 2aFe period. d Spatial LDOS profiles taken along cuts #1 and #2 in (c). e FFT image of the dI/dV map in (b). Additional FFT features around q2Fe are marked by yellow arrows. f FFT line cuts extracted along cut #3 in (e) under various energies. A fitted Lorentz-shaped background was subtracted for each curve, and the curves were shifted vertically for clarity. The two blue dashed lines indicate the positions of qFe and q2Fe. g Comparison of FFT line cuts at E = 40 meV extracted along cuts #3 and #4 in (e). h Schematic illustration of the fine structure of the checkerboard pattern atop the Fe and Se lattices. Unit cell of this structure is indicated by black box. The magenta squares represent higher LDOS observed by STM, and the dark and light colors show difference in LDOS intensity.

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Fig. 4: Spatial LDOS modulations in Fe1−xSe regions with x = 3.2% and 6.35%.
figure 4

a, b Atomically resolved topographic image and corresponding dI/dV map at E = 40 meV for an Fe1-xSe region with x = 3.2%. Measurement conditions for a: Vb = 40 mV, It = 50 pA and b: Vb = 40 mV, It = 50 pA, ΔV = 2 mV. c The enlarged image of the area marked by the yellow box in (b). The black box shows the unit cell of the charge order with a 2aFe period. d Spatial LDOS profiles taken along cut #5 in (c). e, f Atomically resolved topographic image and corresponding dI/dV map at E = 20 meV for an Fe1−xSe region with x = 6.35%. Measurement conditions for e: Vb = 100 mV, It = 50 pA and f: Vb = 100 mV, It = 50 pA, ΔV = 10 mV. g The enlarged image of the area marked by the yellow box in (f). The black box shows the unit cell of the charge order with a 2aFe period. h Spatial LDOS profiles taken along cut #6 in (g).

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Figure 3 examines the spatial LDOS modulation of an Fe1-xSe region with x = 2.2%. The topography of this region is shown in Fig. 3a, and the corresponding dI/dV maps are displayed in Fig. 3b and Supplementary Fig. 7. A checkerboard pattern is ubiquitously present in the defect-free areas over the measured energy range of ±40 meV; it propagates along the two Fe-Fe lattice directions and exhibits a 2aFe period, as illustrated by the LDOS profiles in Fig. 3c, d. Accordingly, the FFT intensities at q2Fe are greatly enhanced (Fig. 3e), corresponding to the growing checkerboard patterns with increasing x. The checkerboard pattern is nondispersive as indicated by the local maxima at q2Fe of the FFT line cuts under varying energies in Fig. 3f; it is C4-symmetric as can be directly seen from the FFT images and is further confirmed by the similar spectral weight of FFT features around q2Fe along the two Fe-Fe lattice directions at E = 40 meV (Fig. 3g). Moreover, the checkerboard pattern is more clearly resolved in the iFFT images based on the features around q2Fe and their second-order components around qFe (see Supplementary Fig. 7 and Supplementary Fig. 8). Such an extended checkerboard pattern was also observed in another Fe1-xSe region with a similar x ~ 1.8% (Supplementary Fig. 9). By comparing the atomically resolved Se lattice in Fig. 3a and the LDOS distribution of the checkerboard pattern in Fig. 3c, we infer a possible fine structure of the checkerboard order atop the Fe and Se lattice as sketched in Fig. 3h—neighboring FeSe4 tetrahedrons differ in LDOS intensity, exhibiting a period of 2aFe along the two Fe-Fe directions.

Figure 4 shows the results of Fe1−xSe regions with higher x values. With further increased defect density (Fig. 4a, e), the checkerboard charge pattern persists in the defect-free FeSe regions and dominates even in the non-superconducting samples with x = 6.35%, as shown in Fig. 4b, f and Supplementary Figs. 10–13. Figure 4c, g shows the enlarged images of the areas indicated by yellow boxes in Fig. 4b, f, exhibiting clear charge modulations with 2aFe-period for both cases, which are further confirmed by the LDOS profiles shown in Fig. 4d, h. Overall, the static checkerboard charge pattern is a general feature of heavily electron-doped FeSe, regardless of the defect density and superconducting transition temperature.

Sometimes we can see unidirectional charge stripes in the vicinity of Fe-site defects, as shown in Supplementary Fig. 14. It seems likely that in a small sample region, a 2aFe-period modulation is preserved along one of the Fe-Fe lattice directions, whereas it is smeared in the perpendicular direction. The appearance of unidirectional charge stripes seems strongly depend on the local environment. Its origin is not clear now, and we speculate that local strains induced by gathered Fe-site defects may be responsible.

Here we summarize how the Fe-site defect concentration dominates the behavior of superconductivity and charge order in (Li0.84Fe0.16OH)Fe1−xSe. At first, when x is low, Tc can be as high as 42 K. Superconductivity is only suppressed at the defect sites40,50,51,52, and a checkerboard pattern is pinned there (Fig. 2). With increased x, the overall superconductivity is suppressed and Tc drops, the checkerboard pattern extends to the whole defect-free areas (Fig. 3 and Fig. 4b) and persists even after the superconductivity disappears when x > 5% (Fig. 4f). Such an evolution clearly shows that the checkerboard order competes with superconductivity, which flourishes when superconductivity is partly or fully suppressed by Fe-site defects. The formation of such charge orders could explain the suppression of spectral weight around EF for higher x (Fig. 1f). Moreover, we find the checkerboard charge order is short-ranged and the estimated in-plane correlation length is approximately 4aFe ~5aFe, as discussed in detail in Supplementary Fig. 15. Besides, the period of the checkerboard charge order is doping independent and maintains a value of ~2aFe with varying x, which is consistent with the unaffected carrier doping level and the doping independent band structures.

Simulations of spin and charge orders

The ordering wave vector of the checkerboard order does not match the nesting vectors between Fermi surface pockets in (Li0.84Fe0.16OH)Fe1−xSe. Therefore, one needs to examine how Fe-site defects may induce the observed charge orders. Although Fe-site defects in Fe1−xSe layer do not alter the itinerant carriers effectively, they have been shown to expand the in-plane lattice and reduce the out-of-plane lattice parameter45,47, so that the electronic structure will be altered. Moreover, the Fe oxidation state changes from slightly below +2 to higher values45,58; thus, the Fe orbital configuration is driven toward the half-filled 3d5 state. Consequently, both the electron correlation and the local magnetic moment of Fe ions in the Fe1-xSe layer would be enhanced59,60. We notice that weak magnetic behavior was observed in superconducting (Li0.84Fe0.16OH)Fe1−xSe43, and an enhanced local Fe moment was deduced from a Curie-Weiss-like magnetic susceptibility in insulating (Li0.84Fe0.16OH)Fe0.7Se58.

The enhanced electron correlation and local moment would promote magnetism; thus, the magnetic fluctuations may be stabilized into static order. A \(2\times 2\) charge order in the FeSe layer was reported in K1-xFe2-ySe2 before30,31, which was considered to be related to a block-antiferromagnetic order suggested in the parent compound of K1-xFe2-ySe2 (ref. 61), corresponding to a Bragg spot located at (π/2, π/2). For (Li0.84Fe0.16OH)Fe1-xSe, there are no low-energy spin fluctuations at (π/2, 0), (0, π/2), and (π/2, π/2), so the observed charge orders are unlikely to be driven by stabilizing spin fluctuations at these momenta. Instead, INS measurements on optimally doped (Li0.84Fe0.16OD)Fe1-xSe have found low energy magnetic excitations centered around four momenta (π ± δπ, π) and (π, π ± δπ) (δ ~ 0.38) (ref. 32) with similar spectral weights, as reproduced in Fig. 5a. We note that such a momentum distribution is broad, covering commensurate values such as δ = 0.5, which corresponds to the scatterings between certain sectors of the Fermi surfaces with distinct orbital characters (Fig. 5b). The horizontal \({{{{{{\bf{Q}}}}}}}_{3},{{{{{{\bf{Q}}}}}}}_{4},{{{{{{\bf{Q}}}}}}}_{3}^{{\prime} },{{{{{{\bf{Q}}}}}}}_{4}^{{\prime} }\) and vertical \({{{{{{\bf{Q}}}}}}}_{1},{{{{{{\bf{Q}}}}}}}_{2},{{{{{{\bf{Q}}}}}}}_{1}^{{\prime} },{{{{{{\bf{Q}}}}}}}_{2}^{{\prime} }\) patterns correspond to the scatterings between the Fermi surface sectors of dxy orbitals with that of dxz or dyz orbitals, respectively. In heavily electron-doped FeSe, the lattice is C4-symmetric and the dxz or dyz orbitals are degenerate, thus the scatterings between dxy orbitals and dxz or dyz orbitals are degenerate as well, leading to the same spectral weights of different magnetic excitations. Therefore, we will consider the superposition of multiple magnetic excitations with equal weightage.

Fig. 5: Simulation of spin and charge patterns in real space.
figure 5

a Momentum distribution of spin fluctuations in (Li0.84Fe0.16OD)Fe1−xSe measured by INS in momentum space. The red dots mark the commensurate spin fluctuation momenta, as shown in (b). b Cartoon of the corresponding scatterings between Fermi surface sections in the unfolded BZ for the momenta shown in (a). c The simulated SDW pattern based on four wave vectors: \({{{{{{\bf{Q}}}}}}}_{1}=\left(0.5,\,0.75\right),{{{{{{\bf{Q}}}}}}}_{2}=\left(0.5,\,0.25\right),\,{{{{{{\bf{Q}}}}}}}_{1}^{{\prime} }=\left(-0.5,\,0.75\right),{{{{{{\bf{Q}}}}}}}_{2}^{{\prime} }=\left(-0.5,\,0.25\right)\) (in reciprocal lattice unit (r. l. u.)), with its unit cell indicated by the black box. d–f The simulated charge order patterns by considering d:\({{{{{{\bf{Q}}}}}}}_{1},{{{{{{\bf{Q}}}}}}}_{2},{{{{{{\bf{Q}}}}}}}_{1}^{{\prime} },{{{{{{\bf{Q}}}}}}}_{2}^{{\prime} }\), e:\({{{{{{\bf{Q}}}}}}}_{3}=\left(0.75,\,0.5\right),{{{{{{\bf{Q}}}}}}}_{4}=\left(0.25,\,0.5\right),\,{{{{{{\bf{Q}}}}}}}_{3}^{{\prime} }=\left(-0.75,\,0.5\right)\,{{{{{\rm{and}}}}}}\,{{{{{{\bf{Q}}}}}}}_{4}^{{\prime} }=\left(-0.25,\,0.5\right)\), and f: eight wave vectors of \({{{{{{\bf{Q}}}}}}}_{i}\,{{{{{\rm{and}}}}}}\,{{{{{{\bf{Q}}}}}}}_{i}^{{\prime} }\,\left(i=1-4\right)\). g The charge order pattern obtained by an overlay of the patterns shown in (d) and (e). The unit cells of these patterns are indicated by the yellow boxes. h Comparison between typical LDOS profile of the charge stripe (upper panel) and the charge distributions from simulations based on quadruple-Q (\({{{{{{\bf{Q}}}}}}}_{3},{{{{{{\bf{Q}}}}}}}_{4},{{{{{{\bf{Q}}}}}}}_{3}^{{\prime} },{{{{{{\bf{Q}}}}}}}_{4}^{{\prime} }\)) SDW (middle panel, taken along cut #1 in e) and double-Q (\({{{{{{\bf{Q}}}}}}}_{4}\) and \({{{{{{\bf{Q}}}}}}}_{4}^{{\prime} }\)) SDW (lower panel, taken along cut #2 in Supplementary Fig. 16h3), respectively. The orange and blue curves show the simulated charge profiles with and without convolution with spatial resolution.

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Assuming that the spin fluctuation distributions are similar in our samples and that the δ = 0.5 spin fluctuations could be stabilized into commensurate static orders by Fe-site defects, we could qualitatively simulate the resulting spin density as \({I}_{s}={{\cos }}({{{{{{\bf{Q}}}}}}}_{i}\cdot {{{{{\bf{r}}}}}})\) (note that \({{{{{{\bf{Q}}}}}}}_{i}\) and −\({{{{{{\bf{Q}}}}}}}_{i}\) are equivalent). Moreover, it has been shown that for a collinear ferromagnet or antiferromagnet, the charge redistribution induced by spin-orbital coupling depends on the spin axis instead of its direction62; as a result, the period of the charge modulation is half of that of the spin modulation. Such a relation between spin and charge orders has been observed in cuprates7,8,9,10,11, MnP63, Cr crystals64, and also predicted for iron-based superconductors65. Therefore, the charge order driven by such a spin density wave (SDW) can be qualitatively simulated using \({I}_{c}=|{I}_{s} |=|{{\cos }}\left({{{{{{\bf{Q}}}}}}}_{i}\cdot {{{{{\bf{r}}}}}}\right)|\).

First, a simple SDW and the corresponding charge order are simulated by choosing one of the \({{{{{{\bf{Q}}}}}}}_{i}\) vectors (\({{{{{{\bf{Q}}}}}}}_{1}=\left(0.5,\,0.75\right)\), \({{{{{{\bf{Q}}}}}}}_{2}=\left(0.5,\,0.25\right)\), \({{{{{{\bf{Q}}}}}}}_{3}=\left(0.75,\,0.5\right)\), \({{{{{{\bf{Q}}}}}}}_{4}=\left(0.25,\,0.5\right)\), \({{{{{{\bf{Q}}}}}}}_{1}^{{\prime} }=\left(-0.5,\,0.75\right)\), \({{{{{{\bf{Q}}}}}}}_{2}^{{\prime} }=\left(-0.5,\,0.25\right)\), \({{{{{{\bf{Q}}}}}}}_{3}^{{\prime} }=\left(-0.75,\,0.5\right)\), \({{{{{{\bf{Q}}}}}}}_{4}^{{\prime} }=\left(-0.25,\,0.5\right)\) in the reciprocal lattice unit (r.l.u.)) shown in Fig. 5a. The results are plotted in Supplementary Fig. 16a3-d3, and the tilted charge stripes differ completely from the experimental patterns. Alternatively, a spin order could result from a coherent addition of several SDWs with different wave vectors, and a double-Q SDW was proposed to account for the spin order in Sr0.63Na0.37Fe2As2 (ref. 66). A multiple-Q SDW could be simulated by a coherent addition of various SDWs: \({I}_{s}=\sum {{\cos }}({{{{{{\bf{Q}}}}}}}_{i}\cdot {{{{{\bf{r}}}}}})\). Supplementary Fig. 16e2-h2 and e3-h3 present double-Q SDWs and corresponding charge orders simulated with various pairs of \({{{{{{\bf{Q}}}}}}}_{i}\) vectors. Most of these patterns fail to reproduce the experiments in terms of the period and orientation, except the one based on \({{{{{{\bf{Q}}}}}}}_{4}\) and \({{{{{{\bf{Q}}}}}}}_{4}^{{\prime} }\) that reproduces the period and orientation of vertical stripes (see Supplementary Fig. 16h3). Furthermore, considering the complete magnetic excitation patterns in Fig. 5a, it is more natural to include the four horizontal or vertical \({{{{{{\bf{Q}}}}}}}_{i}\) vectors. The resulting quadruple-Q SDW based on \({{{{{{\bf{Q}}}}}}}_{1},{{{{{{\bf{Q}}}}}}}_{2},{{{{{{\bf{Q}}}}}}}_{1}^{{\prime} },{{{{{{\bf{Q}}}}}}}_{2}^{{\prime} }\) is plotted in Fig. 5c and has a 4aFe × 2aFe unit cell. The corresponding charge order in Fig. 5d shows the horizontal stripes with a 2aFe × aFe unit cell. Likewise, the vertical charge stripes in Fig. 5e correspond to the coherent addition of SDWs with \({{{{{{\bf{Q}}}}}}}_{3},{{{{{{\bf{Q}}}}}}}_{4},{{{{{{\bf{Q}}}}}}}_{3}^{{\prime} },{{{{{{\bf{Q}}}}}}}_{4}^{{\prime} }\). The charge distribution driven by the total eight momenta is shown in Fig. 5f, which is rotated 45° with a 2\(\surd 2\)aFe × 2\(\surd 2\)aFe unit cell, failing to reproduce the experiment. On the other hand, a direct overlap of the vertical and horizontal stripes could qualitatively reproduce the checkerboard pattern (Fig. 5g). This may also explain the observed local unidirectional charge stripes in the vicinity of Fe-site defects, when one set of stripes is stronger than the other (Supplementary Fig. 16m3). Such a broken symmetry may be caused by various reasons. For example, the local Fe-site defect distribution may favor scattering between the dxz and dxy orbitals rather between the dyz and dxy orbitals (Fig. 5b). In such a multiple-Q density wave scenario, the 2aFe × 2aFe checkerboard order and 2aFe × 1aFe charge stripe found in (Li0.84Fe0.16OH)Fe1-xSe could be naturally explained. Figure 5h illustrates that the simple simulations of the double-Q (\({{{{{{\bf{Q}}}}}}}_{4}\) and \({{{{{{\bf{Q}}}}}}}_{4}^{{\prime} }\)) and quadruple-Q (\({{{{{{\bf{Q}}}}}}}_{3},{{{{{{\bf{Q}}}}}}}_{4},{{{{{{\bf{Q}}}}}}}_{3}^{{\prime} },{{{{{\rm{and}}}}}}\,{{{{{{\bf{Q}}}}}}}_{4}^{{\prime} }\)) charge patterns could qualitatively reproduce the periodic patterns of the charge orders. Because of the complexity in tunneling matrix element, it is still premature to decide which is correct. This calls for further theoretical investigations to understand the fine structures of the checkerboard order and localized charge stripes, and to distinguish between double-Q and quadruple-Q SDW scenarios.

Discussion

Note that spin fluctuations are usually much stronger than charge fluctuations in low-dimensional correlated systems; thus, charge/orbital orders could form at higher temperatures than those of spin ordering13,14,63. Therefore, even though we observed charge order here, the spin order might still be dynamic. Long-range magnetic order has not been found in (Li0.84Fe0.16OH)Fe1−xSe by macroscopic measurements32,33, here we indicate that when the superconductivity is effectively suppressed, the magnetic order or the enhanced magnetic fluctuations stabilized by local defects can be indirectly detected by the presence of charge orders. Future spin-resolved STM studies could further clarify the existence of local magnetic order around the Fe-site defects. If it exists, it should be viewed as the major competing order for superconductivity.

We may also examine other possible origins for the charge order, for example, if it is driven by phonon-softening effects. Ideally, inelastic x-ray scattering may be conducted to clarify this, but it is very challenging for thin film samples. However, we note that usually one could observe the charge order in topographic image in the case of charge density wave caused by electron-phonon interactions, while we did not observe it in our case. Disorder-induced local strain is another possible cause. However, it is unlikely, because the checkerboard pattern extends to the large defect-free regions for x = 1.8% and 2.2% (Fig. 3 and Supplementary Fig. 9), where local strain should be weak. Moreover, with increased x values, disorder-induced local strain will increase in principle, but the observed checkerboard pattern does not change.

To conclude, our results establish the comprehensive relations among the observed competing order, spin fluctuations, and Fermi surface topology in heavily electron-doped FeSe superconductors. The stabilized double- or quadruple-Q spin fluctuations that scatter between different parts of the Fermi surface likely drive the observed checkerboard charge order, when the superconductivity is suppressed by Fe-site defects. Our results favor spin fluctuations rather than nematic orbital fluctuations as the driving force of the charge order, although the orbital fluctuations were suggested to be as important as spin fluctuations in iron-based superconductors with both electron and hole Fermi surfaces, such as iron pnictides and bulk Fe(Se,Te). Therefore, spin fluctuations may play an important role in Cooper pair formation in (Li0.84Fe0.16OH)FeSe and other heavily electron-doped FeSe-based superconductors. Moreover, our study suggests that the charge order revealed by STM can be used as a tool to detect the spin fluctuations or magnetic orders of the parent phase, which can be extended to other systems in the future.

Methods

Synthesis of (Li0.84Fe0.16OH)Fe1−xSe films and characterizations

High-quality single-crystalline (Li0.84Fe0.16OH)Fe1−xSe films were grown on LaAlO3 substrates by a matrix-assisted hydrothermal epitaxy method46,47. Large crystals of K0.8Fe1.6Se2 (nominal 245 phase) are grown in advance and used as the matrix for facilitating the hydrothermal epitaxial growth. By adjusting the synthesis temperature in the range from 120 to 180 °C, these samples with different Tc from 42 K down to 0 K are available. The (Li0.84Fe0.16OH)Fe1−xSe films have thicknesses of 600–700 nm, determined on a Hitachi SU5000 scanning electron microscope. X-ray diffraction patterns were collected at room temperature on a 9 kW Rigaku SmartLab diffractometer using Cu Kα1 radiation (λ = 1.5405 Å), with a 2θ range of 5°–80° and 2θ scanning steps of 0.01°. All magnetic susceptibility measurements were conducted on a Quantum Design MPMS-XL1 system.

STM measurements

(Li0.84Fe0.16OH)Fe1−xSe films were mechanically cleaved at 78 K in ultrahigh vacuum with a base pressure better than 1 × 10−10 mbar and immediately transferred into a UNISOKU cryogenic STM at T = 4.2 K. Pt-Ir tips were used after being treated on a clean Au (111) substrate. The dI/dV spectra were collected by a standard lock-in technique with a modulation frequency of 973 Hz and a typical modulation amplitude ΔV of 1~2 mV at 4.2 K.