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[Saizheng Cao](https://orcid.org/0000-0002-1201-6961), Xin Ma, [Dongsheng Yuan](https://orcid.org/0000-0001-9650-2272), Zhen Tao, Xiang Chen, Yu He, Patrick N. Valdivia, [Shan Wu](https://orcid.org/0000-0001-9935-0657), Hang Su, Wei Tian, [Adam A. Aczel](https://orcid.org/0000-0003-1964-1943), [Yaohua Liu](https://orcid.org/0000-0002-5867-5065), [Xiaoping Wang](https://orcid.org/0000-0001-7143-8112), Zhijun Xu, Huiqiu Yuan, [Edith Bourret-Courchesne](https://orcid.org/0000-0002-8487-0112), Chao Cao, Xingye Lu, Robert Birgeneau, [Yu Song](https://orcid.org/0000-0002-8835-9071)

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[Superstructures and magnetic order in heavily Cu-substituted (Fe1−𝑥⁢Cu𝑥)1+𝑦⁢Te](https://mdr.nims.go.jp/datasets/9c245379-e103-45ad-9436-c856dd101039)

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Superstructures and magnetic orders in heavily Cu-substituted (Fe1−xCux)1+yTeSaizheng Cao,1 Xin Ma,1 Dongsheng Yuan,2 Zhen Tao,3 Xiang Chen,4, 2 Yu He,5, 4, 2 Patrick N. Valdivia,4, 2Shan Wu,4, 2 Hang Su,1 Wei Tian,6 Adam A. Aczel,6 Yaohua Liu,6, 7 Xiaoping Wang,6 Zhijun Xu,8, 9 HuiqiuYuan,1 Edith Bourret-Courchesne,2 Chao Cao,1 Xingye Lu,3 Robert Birgeneau,4, 2 and Yu Song1, 4, 2, ∗1Center for Correlated Matter and School of Physics, Zhejiang University, Hangzhou 310058, China2Materials Science Division, Lawrence Berkeley National Lab, Berkeley, California 94720, USA3Center for Advanced Quantum Studies, Applied Optics Beijing Area Major Laboratory,and Department of Physics, Beijing Normal University, Beijing, China4Physics Department, University of California, Berkeley, California 94720, USA5Department of Applied Physics, Yale University, New Haven, Connecticut 06511, USA6Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA7Second Target Station, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA8NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA9Department of Materials Science and Engineering,University of Maryland, College Park, Maryland 20742, USAMost iron-based superconductors exhibit stripe-type magnetism, characterized by the orderingvector Q = ( 12, 12). In contrast, Fe1+yTe, the parent compound of the Fe1+yTe1−xSex superconduc-tors, exhibits double-stripe magnetic order associated with the ordering vector Q = ( 12, 0). Here, weuse elastic neutron scattering to investigate heavily Cu-substituted (Fe1−xCux)1+yTe compoundsand reveal that for x & 0.4, short-range magnetic order emerges around the stripe-type vector, atQ = ( 12± δ, 12± δ, 12) with δ ≈ 0.05. We identify the short-range magnetic order to be associatedwith a phase that exhibits superstructure modulations at Q = ( 13, 13, 12), with the correlation lengthfor the magnetic order shorter than that for the superstructure. For x & 0.55, we observe an ad-ditional inter-grown phase with higher Cu-content, characterized by a superstructure modulationvector Q = ( 13, 13, 0) and magnetic peaks at Q = ( 23, 13, 12)/( 13, 23, 12). The positions of the superstruc-ture peaks suggest that relative to the tetragonal unit cell of Fe1+yTe, heavy Cu-substitution leadsto an expansion of the unit cell by√2 × 3√2 times in the ab-plane. First-principles calculationssuggest the formation of spin chains and spin ladders from Fe-Cu ordering. Our findings show thatstripe-type magnetism is common in magnetically diluted iron pnictides and chalcogenides, despitethe varying associated atomic orderings.I. INTRODUCTIONMagnetic fluctuations may facilitate unconventionalsuperconductivity [1], thus the character and origin ofmagnetism are pivotal issues in understanding unconven-tional superconductors. For most iron-based supercon-ductors (FeSCs), superconductivity appears in the vicin-ity of stripe-type antiferromagnetic (AFM) order, char-acterized by the ordering vector Q = (12, 12) in the two-Fetetragonal unit cell [Figs. 1(a) and (b)] [2, 3]. In contrast,for the Fe1+yTe1−xSex superconductors [4, 5], their par-ent compound Fe1+yTe exhibits bicollinear magnetic or-der with the ordering vector Q = (12, 0) [Fig. 1(b)] [6, 7].The stripe-type magnetism in the FeSCs may arise fromthe nesting of Fermi surfaces [8, 9] or local-moments thatexhibit frustrated magnetic interactions [10], whereas aninterplay between local-moments and itinerant carriersis suggested to be important for understanding the mag-netism in Fe1+yTe [11].The magnetism of FeSC parent compounds such asBaFe2As2 and NaFeAs can be effectively tuned via dop-ing on the Fe site. For example, Co- and Ni-substitutionleads to electron-doping and results in the suppressionof AFM order and the emergence of superconductivity[12, 13]. The effect of Cu-substitution is more complex.At low substitution levels, in addition to electron-dopingand suppressing AFM order, the Cu atoms also lead tostrong impurity potentials [14, 15] and weakened super-conductivity [16, 17]. With increasing Cu substitutionlevel, the electron-doping contributed per Cu decreases[15, 18], and the Cu dopants eventually become nonmag-netic and no longer contribute to the electronic densityof states at the Fermi level [19].In heavily Cu-substituted NaFe1−xCuxAs, a correlatedinsulator appears for x & 0.2 [20–23], accompanied by Fe-Cu ordering into chains and magnetic order atQ = (12, 12)[Fig. 1(c)] [22, 24]. In Ba(Fe1−xCux)2As2, ω/T scalingis found near a putative AFM quantum critical pointat x = 0.043 [25], and short-range AFM order at thestripe-type vector reemerges for x & 0.15, accompaniedby insulating-like electrical transport, without detectableFe-Cu ordering [26]. In the case of (Fe1−xCux)1+yTe(FCT), AFM order at the double-stripe vector (12, 0) be-comes suppressed for x ≈ 0.1 [27] while a spin glass statewith insulating-like electrical transport appears for largerCu content (up to x ≈ 0.5) [28, 29]. Whether the spinglass state in FCT is associated with short-range mag-netic order (mSRO), and whether heavy Cu-substitutionin FCT leads to Fe-Cu ordering as in NaFe1−xCuxAs, areyet unclear.In this work, we examine these issues using elas-tic neutron diffraction and first-principles calculations.2We find that FCT with x & 0.4 exhibits mSRO atQ = (12± δ, 12± δ, L), with δ ≈ 0.05 and half-integer-L. The mSRO is associated with a superstructure mod-ulation at half-integer-L (13, 13) positions [Fig. 1(d)]. Forx & 0.55, we discover an inter-grown phase with super-structure peaks at integer-L (13, 13) positions, and mag-netic peaks at half-integer-L (13, 23, L)/(23, 13, L) positions[Fig. 1(e)]. Possible structural models consistent withthe observed superstructure peaks are studied using first-principles calculations, suggesting candidate phases withFe-Cu ordering into chains or ladders. These resultssuggest three phases in the phase diagram of FCT, onewithout Fe-Cu ordering (FCT I), one with a superstruc-ture modulated by half-integer-L (13, 13) (FCT II), andone with a superstructure modulated by integer-L (13, 13)(FCT III) [Fig. 1(f)]. Our findings suggest that stripe-type magnetism is common in magnetically-diluted ironpnictides and chalcogenides, despite the varied atomic or-derings (or lack thereof) in different systems. Moreover,the tendency towards atomic ordering in these magneti-cally diluted materials provides a pathway for the studyof correlated electrons in quasi-one-dimensional and clus-ter systems.II. METHODSA. Experimental detailsSingle crystals of (Fe1−xCux)1+yTe were synthesizedusing the modified Bridgeman method [29, 30]. A se-ries of samples were synthesized at Beijing Normal Uni-versity with the nominal compositions ynom = 0.1 andxnom = 0.2, 0.3, 0.4, 0.5, and 0.6 (Growth A). Sampleswith the nominal composition xnom = 0.5 and ynom = 0.1were synthesized at the Lawrence Berkeley National Lab-oratory (Growth B). Two samples from Refs. [29, 30](Growth C) were also studied in this work. The elec-trical resistivity ρ(T ) of samples from Growth A weremeasured using the standard four-probe method.The samples’ compositions were determined usingenergy-dispersive spectroscopy (EDS), by assuming a fulloccupancy of the Te site. The resulting values of x andy, as determined from EDS, are summarized in Table. I,together with a summary of superstructure and magneticpeaks. Multiple points on each sample were probed, andthe standard deviation between different points are usedto estimate the uncertainties of x and y. Compared tosamples from Growth B and C, those from Growth Ahave a higher occupation of the interstitial site (largervalues of y). In this work, we reference the values of xand y determined from EDS measurements.Elastic neutron scattering measurements on samplesfrom Growth A were carried out using the Fixed-IncidentEnergy Triple-Axis Spectrometer (HB-1A) at the High-Flux Isotope Reactor (HFIR), Oak Ridge National Labo-ratory (ORNL). Consistent experimental setups are usedin these measurements, and the measured intensities areFe/CuHKBaFe2As2/NaFeAs Fe1+yTe0.5-0.50.5-0.5-111-1K0.5-0.50.5-0.5-111-1NaFe0.5Cu0.5AsH2 Fe/Cu unit cellmagnetic magnetic+superstructuresuperstructureK0.5-0.50.5-0.5-111-1(Fe1-xCux)1+yTe IIH(a) (b)(c) (d)magnetic peak positions0.5-0.50.5-0.5-111-1(Fe1-xCux)1+yTe III(e)KH(f)FCT I FCT IIFCT II+FCT III0 0.1 0.2 0.3 0.4 0.5 0.6x in (Fe1-xCux)1+yTe FIG. 1. (a) 2-Fe unit cell of the FeSCs. With Cu-substitution,the Fe site may be occupied by either Fe or Cu. (b) Schematiccomparison of magnetic peak positions in BaFe2As2/NaFeAsand Fe1+yTe. Schematic diagrams for the positions ofmagnetic and superstructure peaks in (c) NaFe0.5Cu0.5As,(d) (Fe1−xCux)1+yTe II, and (e) (Fe1−xCux)1+yTe III. (f)The schematic phase diagram for (Fe1−xCux)1+yTe (FCT).FCT I does not exhibit superstructure peaks, FCT II exhibits( 13, 13, 12) superstructure peaks, and FCT III exhibits ( 13, 13, 0)superstructure peaks. For x & 0.55, FCT II and III are inter-grown. The red circles represent samples studied in this work.The y-axis does not correspond to a physical quantity.reported in arbitrary units after normalization by thesample mass. Samples from Growth B were studied onthe Elastic Diffuse Scattering Spectrometer (CORELLI)and the Single-Crystal Diffractometer (TOPAZ) at theSpallation Neutron Source (SNS), ORNL. The CORELLIand TOPAZ data were transformed into the 3D recip-rocal space via the standard reduction algorithm avail-able at the beamlines. Samples from Growth C werestudied using the Double-Focusing Triple-Axis Spectrom-eter (BT-7) at the NIST Center for Neutron Research(NCNR) and HB-1A at the HFIR. The measured inten-sities of samples from Growths B and C are reported inarbitrary units.3Growth xnom ynom x y( 13, 13, 12)-typesuperstructure( 13, 13, 0)-typesuperstructure( 12± δ, 12± δ, 12)-typemSRO( 13, 23, 12)-typemagnetic orderA 0.2 0.1 0.25(2) 0.20(3) × × × —A 0.3 0.1 0.42(1) 0.24(4) X × X —A 0.4 0.1 0.48(3) 0.30(5) X × X —A 0.5 0.1 0.58(2) 0.37(3) X X X —A 0.6 0.1 0.61(1) 0.32(3) X X X —B 0.5 0.1 0.55(1) 0.20(3) X X X XC 0.46 0.08 0.49(5) 0.22(3) X — X —C 0.42 0.18 0.47(3) 0.22(3) X — X —TABLE I. The nominal and measured sample compositions for FCT samples studied in this work. The compositions forsamples from Growth C are from Ref. [29]. The superstructure and magnetic peaks are also summarized for comparison, withthe symbols X and × respectively corresponding to their presence and absence. The — symbol corresponds to peaks that werenot examined in this study.B. First-principles calculationsTo study the possible superstructures of FCT, we per-formed first-principles density functional theory (DFT)simulations as implemented in the VASP package [31, 32].The ion-electron interactions were approximated with theprojected augmented wave (PAW) method [32, 33], andthe Perdew, Burke and Enzerhoff (PBE) flavor of general-ized gradient approximation to the exchange-correlationfunctional were chosen [34]. The energy cutoff of plane-wave basis was set to 590 eV, and Γ centered k-mesheswith spacing less than 0.02 Å−1 were used to perform theintegration of the Brillouin zone. All lattice constantsand internal atomic coordinates were fully relaxed untilforces on each atom is smaller than 0.001 eV/Å. Since weare interested in the formation of superstructures duringsample crystallization, which takes place at temperaturesmuch higher than TN, magnetism and spin-orbit couplingwere not considered in our calculations.III. RESULTSA. Electrical transport of FCTElectrical resistivity for FCT samples with x & 0.25(Growth A) are shown in Fig. 2(a), with semiconductingbehaviors observed in all cases. FCT becomes more in-sulating as x increases from 0.25 to 0.42 and 0.48, butthen becomes less insulating as x is further increasedto 0.58 and 0.61. The evolution of electrical trans-port in Fig. 2(a) is broadly consistent with results inRef. [29], which finds that FCT becomes more insulat-ing with increasing Cu-content up to x ≈ 0.47. Thisbroad consistency is also seen in the similar behaviorsof ρ(T )/ρ(300 K) for FCT samples (with x between 0.42and 0.48) from Growths A and C [Fig. 2(b)].x=0.43x=0.47Valdivia et al.00.10.20.30.4ρ(Ω×cm)50 100Temperature(K)0Sample Growth Ax=0.42x=0.48x=0.58x=0.25x=0.6100.0050.010.0150.02ρ(Ω×cm)50 1000Temperature(K)Sample Growth Cx=0.48x=0.420 150 30012345ρ(T)/ρ(300K)Temperature(K)Sample Growth A(a) (b)FIG. 2. (a) In-plane electrical resistivity ρ(T ) for FCT sam-ples. The inset zooms in around low temperatures. (b) Com-parison of ρ(T )/ρ(300 K) for FCT samples from Growths Aand C [29].B. mSRO in FCTFigs. 3(a) and (b) show elastic scans along (H,H, 32)for FCT samples with x = 0.48 and 0.61 (Growth A), atboth 5 K and T > 200 K. As can be seen, a pair of broadpeaks centered around (12, 12, 32) are present at 5 K, whichvanish when T > 200 K. The 5 K data, with the corre-sponding T > 200 K data subtracted as background, areshown in Fig. 3(c) for x = 0.42, 0.48, 0.58 and 0.61, andreveal the presence of similar mSRO in all these samples.A x = 0.25 sample was also studied, but no detectablemagnetic peaks were found. Given Cu becomes nonmag-netic in heavily Cu-substituted Sr(Fe1−xCux)2As2 [19]and NaFe1−xCuxAs [22], we assume that Cu is nonmag-netic in FCT samples with x & 0.4, and collective mag-netism such as the mSRO arise from Fe.The data in Fig. 3(c) are well described by two4Lorentzian peaks centered around (12, 12, 32):I(Q) =A1[H − (12− δ1)]2 + κ2+A2[H − (12+ δ2)]2 + κ2,(1)where Ai are the intensity scale factors for the two peaks,δi are the shifts from (12, 12) along the in-plane longitu-dinal direction, and κ is the half-width-at-half-maximum(HWHM) of the Lorentzian peaks. κ is related to thecorrelation length through ξ110 = a/(√2πκ), where thefactor of√2 rather than 2 is due to the scan being alongthe (110) direction. The fit values δ1 and δ2 are similar,with an average value δ = (δ1+δ2)/2 ≈ 0.05. Scans alongL centered on the right peak in Fig. 3(c) are presentedin Fig. 3(d), revealing a single peak centered at L = 32for each sample. These L-scans are well-described by asingle Lorentzian peak, and the correlation length can beextracted via ξc = c/(2πκ). We note instrumental reso-lution was not considered in our analysis, so the fit κ areupper limits, and the corresponding correlation lengthsare lower limits.The temperature dependence of the magnetic inten-sity at (12+ δ, 12+ δ, 32) are shown in Fig. 3(e), revealinga gradual onset of magnetic intensity below T ≈ 170 K.Fig. 3(f) shows that the scaled temperature dependenceoverlap for different samples. This contrasts with be-haviors found in NaFe1−xCuxAs, where the temperaturedependence of the magnetic intensity varies significantlywith Cu substitution level [22].The mSRO is also studied for a x = 0.47 FCT samplefrom Growth C, with the 4.5 K data in multiple Brillouinzones shown in Figs. 4(a)-(d), after subtracting the 200 Kdata. In all cases, a pair of diffuse peaks are observed atpositions equivalent to (12± δ, 12± δ, 12). In Figs. 4(a)and (c), the intensities of the two peaks are highly asym-metric, and is not captured by the magnetic form factoralone. It is interesting that the maximum intensity oc-curs at L = 32rather than L = 12, further confirmed inthe scan along (0.55, 0.55, L) [Fig. 4(f)]. We find that themomentum dependence of magnetic intensity in Fig. 4(f)is reasonably described by a lattice sum of Lorentzianfunctions polarized along the in-plane longitudinal (110)direction (red dashed line):I(Q) = f(Q)2(1−Q2110/Q2)∞∑i=−∞A(L− (i+ 12))2 + κ2,(2)where A is an intensity scale factor, f(Q) is the Fe2+magnetic form factor, Q110 is the component of the mo-mentum transfer along the (110) direction, (1−Q2110/Q2)is the polarization factor, and κ is the HWHM of theLorentzian peaks. The scans along (110) in Figs. 4(a)-(d) are also reasonably described by a similar model (red0.35 0.4 0.45 0.5 0.55 0.6 0.65(H,H,1.5) (r.l.u.)00.020.040.060.080.10.120.14Normalized Counts0.35 0.4 0.45 0.5 0.55 0.6 0.65(H,H,1.5) (r.l.u.)00.010.020.030.040.050.060.07Normalized Countsx = 0.48 x = 0.610.35 0.4 0.45 0.5 0.55 0.6 0.65(H,H,1.5) (r.l.u.)00.020.040.060.08Normalized Counts1 1.25 1.5 1.75 2(0.5+δ,0.5+δ,L) (r.l.u.)00.020.040.060.08Normalized Counts0 50 100 150 200Temperature (K)-0.0100.010.020.030.040.050.060.07Normalized Counts0 50 100 150 200Temperature (K)-0.200.20.40.60.811.2Scaled Intensityx = 0.42x = 0.48x = 0.58x = 0.61x = 0.42, δ = 0.05 x = 0.48, δ = 0.05x = 0.58, δ = 0.05x = 0.61, δ = 0.066(a) (b)(c) (d)(e) (f)Sample Growth A, HB-1A5 K220 K5 K241 K5 K - high T 5 K - high Tx = 0.42, δ = 0.05 x = 0.48, δ = 0.05x = 0.58, δ = 0.05x = 0.61, δ = 0.06x = 0.42, δ = 0.05 x = 0.48, δ = 0.05x = 0.58, δ = 0.05x = 0.61, δ = 0.06(0.5+δ,0.5+δ,1.5)(0.5+δ,0.5+δ,1.5)FIG. 3. Scans along (H,H, 32) for FCT with (a) x = 0.48 and(b) x = 0.61 at 5 K and T > 200 K. (c) (H,H, 32) and (d) ( 12+δ, 12+ δ, L) scans for FCT, subtracted by the correspondinghigh temperature data (T = 220 K for x = 0.42 and 0.48, T =241 K for x = 0.58 and 0.61). (e) Temperature dependence ofthe elastic intensity at ( 12+δ, 12+δ, 32) for FCT, the average ofdata at T ≥ 170 K have been subtracted as background. (f)The data in panel (e) are scaled together by further dividingthe average of data with T ≤ 30 K.dashed lines):I(Q) = f(Q)2(1−Q2110/Q2)×∞∑i=−∞A(H − (i + 12− δ))2 + κ2+A(H − (i+ 12+ δ))2 + κ2.(3)The observation that the elastic magnetic signal inFCT can be reasonably modeled by Eqs. 2 and 3 sug-gest that the spins are dominantly polarized along thein-plane longitudinal direction, similar to the stripe-typemagnetic order in the FeSCs [2].Figs. 4(a)-(d) unequivocally demonstrate that themSRO in FCT is split along the in-plane longitudi-50.35 0.4 0.45 0.5 0.55 0.6 0.65-200204060801000.35 0.4 0.45 0.5 0.55 0.6 0.65-0.2 -0.1 0 0.1 0.2(0.55+h,0.55-h,1.5) (r.l.u.)-20020406080100120Counts (arb. units)Counts (arb. units)Counts (arb. units)Counts (arb. units)(a) (b)(c) (d)(e) (f)Sample Growth C, x = 0.47, HB-1A4.5 K - 200 K-20020406080100Counts (arb. units)(H,H,0.5) (r.l.u.)0.35 0.4 0.45 0.5 0.55 0.6 0.65-200204060801001.35 1.4 1.45 1.5 1.55 1.6 1.65(H,H,0.5)-20020406080100(H,H,1.5) (r.l.u.)(H,H,2.5) (r.l.u.)-200204060801000 1 2 3 4(0.55,0.55,L) (r.l.u.)Counts (arb. units)(f)FIG. 4. Scans for a x = 0.47 FCT sample along [110]centered at (a) ( 12, 12, 12), (b) ( 12, 12, 32), (c) ( 12, 12, 52), and (d)( 32, 32, 12). (e) Scan along [11̄0] centered at (0.55, 0.55, 32) forthe x = 0.47 FCT sample. (f) (0.55, 0.55, L) scan for thex = 0.47 FCT sample over a large L-range. All scans aremeasured at 4.5 K, and the corresponding 200 K scan havebeen subtracted as background. The solid lines are fits toEq. 1 or a single Lorentzian peak, the dashed line in (f) is afit to Eq. 2, and the dashed lines in (a)-(d) are fits to Eq. 3.nal direction, forming two peaks at (12± δ, 12± δ) withδ ≈ 0.05. To examine whether a similar splitting of mag-netic peaks occurs along the in-plane transverse direc-tion, we mounted the x = 0.47 sample in the [113]× [11̄0]scattering plane, which in combination with a small tiltof the sample table allows the (0.55 + h, 0.55 − h, 1.5)scan to be carried out. As can be seen in Fig. 4(e), incontrast to the in-plane longitudinal direction, a singlepeak is observed along the in-plane transverse direction.Based on data in Figs. 3 and 4, we find that the mSROpeaks in FCT are split by δ ≈ 0.05 along the in-plane lon-gitudinal direction, but are centered at half-integer posi-tions along the in-plane transverse direction and along L,0.2 0.3 0.4 0.5 0.6 0.7 0.8(H,H,1.5) (r.l.u.)02004006008001000120014001600Coutns (arb. units)180 K150 K120 K90 K60 K3 K0 50 100 150 200048121620Correlation Length (Å)Temperature (K)(a) (c)Sample Growth C, x = 0.49, BT-7230 K  datasubtractedFIG. 5. (a) Scans along [110] centered at ( 12, 12, 32) for ax = 0.49 sample at various temperatures, with the 230 Kdata subtracted as background. The data have been shiftedvertically for clarity. (b) The temperature evolution of thecorrelation length, obtained by fitting the data in panel (a)to Eq. 1.as schematically depicted in Fig. 1(d). We note mSROswith peaks transversely split around (12, 12) are observedin Co- and Ni-doped BaFe2As2 [35, 36].Scans along (H,H, 32) at various temperatures are car-ried out for a x = 0.49 sample (Growth C), with re-sults shown in Fig. 5(a). As can be seen, the peak shapehardly changes with temperature. By fitting the data inFig. 5(a) to Eq. 1, we find an essentially temperature-independent correlation length of ≈ 10 Å [Fig. 5(b)].C. Superstructure in FCTMotivated by the observation of superstructure peaksat half-integer-L (12, 12) positions in NaFe1−xCuxAs[Fig. 1(c)], we searched for superstructure peaks in FCTalong high symmetry directions, with findings summa-rized in Figs. 6 and 7. In Fig. 6, scans along [110] cen-tered around (13, 13, L) and (23, 23, L), with L = 0 and 12,are shown for FCT samples with x = 0.42, 0.48, 0.58,and 0.61 (Growth A). For x = 0.42 and 0.48, clear su-perstructure peaks are observed at (13, 13, 12) and (23, 23, 12)positions, whereas little or no intensity is seen at L = 0positions [Figs. 6(a)-(d)]. While a small peak is seenat (13, 13, 0) in the x = 0.42 sample, a scan along L re-veals that it is in fact associated with a scattering rod[Fig. 7(a)]. Therefore, we conclude that in the x = 0.42and 0.48 samples, the superstructure peaks are observedat half-integer-L (13, 13) and (23, 23) positions.For the x = 0.58 and 0.61 samples, in addition tothe half-integer-L superstructure peaks, integer-L su-perstructure peaks are also observed, although with in-plane momenta slightly smaller than (13, 13) and (23, 23)[Figs. 6(e)-(h)]. We rule out this being an incommensu-rate structural modulation, since in such a case the peaksshould be at (13− ǫ, 13− ǫ) and (23+ ǫ, 23+ ǫ), inconsistentwith the experimental observation. As will be discussedbelow, these are in fact integer-L (13, 13) and (23, 23) su-perstructure peaks associated with an inter-grown phase.60.10.20.30.4Normalized CountsL = 0.50.10.20.30.4Normalized Counts0.28 0.3 0.32 0.34 0.36(H,H) (r.l.u.)00.10.20.3Normalized Counts0.050.10.150.20.250.3Normalized Counts0.62 0.64 0.66 0.68 0.7(H,H) (r.l.u.)00.20.40.60.8Normalized Counts0.10.20.30.40.5Normalized Counts0.010.020.030.040.050.06Normalized Counts0.010.020.03Normalized CountsL = 0x = 0.42x = 0.48x = 0.58x = 0.61(a) (b)(c) (d)(e) (f)(g) (h)Sample Growth A, T = 5 K, HB-1AFIG. 6. Scans at 5 K along [110] around Q = ( 13, 13, L), forFCT samples with (a) x = 0.42, (c) x = 0.48, (e) x = 0.58,and (g) x = 0.61. Similar scans around Q = ( 23, 23, L) areshown in panels (b), (d), (f) and (g). The solids lines are fitsto Gaussian peaks on linear backgrounds.With the increase in x, these integer-L (13, 13) superstruc-ture peak gains intensity, suggesting an increase in thevolume fraction of the inter-grown phase.Our observation of superstructure peaks is corrobo-rated by L-scans shown in Fig. 7. In the x = 0.42 sam-ple, clear peaks are observed at (13, 13, 12) and (23, 23, 12), al-though the latter is much weaker [Fig. 7(a)]. These peaksare also observed in the x = 0.48 sample [Fig. 7(b)]. Onthe other hand, the L-scan across (13, 13, 0) in the x = 0.42FCT sample is completely flat [Fig. 7(a)], despite thepresence of a peak in the scan along [110] [Fig. 6(a)].This suggests the integer-L peak in the scan along [110]in Fig. 6(a) results from a scattering rod, evidencing thepresence of stacking faults associated with the (13, 13, 12)0.10.20.30.40.5Normalized Counts0.10.20.30.4Normalized Counts0.10.20.3Normalized Counts0 0.2 0.4 0.6 0.8 1L (r.l.u.)00.20.40.60.8Normalized Counts(0.330,0.330,L)(0.335,0.335,L)(0.665,0.665,L)(a)(b)(c)x = 0.42(0.340,0.340,L)x = 0.48(0.666,0.666,L)(d)(0.335,0.335,L)x = 0.58(0.325,0.325,L)(0.335,0.335,L)(0.325,0.325,L)(0.650,0.650,L)x = 0.61Sample Growth A, T = 5 K, HB-1AFIG. 7. L-scans of superstructure peaks at 5 K in FCT sam-ples with (a) x = 0.42, (b) x = 0.48, (c) x = 0.58, and (d)x = 0.61. The solids lines are fits to Lorentzian peaks onlinear backgrounds.superstructure. In the x = 0.58 and 0.61 samples, super-structure peaks are observed in L-scans at both integer-Land half-integer-L positions [Figs. 7(c) and (d)], consis-tent with observations in Fig. 6.The superstructure peaks in Figs. 6 and 7 are scaledto unit height and overplotted in Fig. 8, centered at thefit peak positions. As can be seen, upon increasing xfrom 0.48 to 0.58, the half-integer-L (13, 13) peak broadensalong the [110] direction, while a sharpening along L isobserved [Figs. 8(a) and (b)]. In comparison, the peakwidths along the two directions changes very little when xis increased from 0.42 to 0.48, and from 0.58 to 0.61. Forthe integer-L (13, 13) peak, which is only observed in thex = 0.58 and 0.61 samples, the peak is found to sharpenalong both [110] and L directions with the increase of x[Figs. 8(c) and (d)].7-0.1 -0.05 0 0.05(h,h) (r.l.u.)-0.200.20.40.60.811.2Scaled Intensity -0.05 0 0.05(h,h) (r.l.u.)x = 0.42x = 0.48x = 0.58x = 0.61Q = (1/3,1/3,1/2)x = 0.58x = 0.61Q = (1/3,1/3,0)(a) (c)-0.2 -0.1 0 0.1 0.2l (r.l.u.)x = 0.42x = 0.48x = 0.58x = 0.61Q = (1/3,1/3,1/2)(b)-0.2 -0.1 0 0.1 0.2l (r.l.u.)x = 0.58x = 0.61Q = (1/3,1/3,1)(d)-0.200.20.40.60.811.2Scaled Intensity Sample Growth A, T = 5 K, HB-1AFIG. 8. A comparison of the superstructure peak widths at5 K in FCT samples, for scans along (a) [110] and (b) L forthe half-integer-L superstructure peaks, and scans along (c)[110] and (d) L for the integer-L superstructure peaks. Allpeaks are fit to a single Lorentzian peak on a linear back-ground, shown as the solid lines. In these plots, the x-axis issubtracted by the fit peak center, the y-axis is subtracted bythe fit background, and the maximum intensity is scaled tounity.The appearance of integer-L peaks near (13, 13) and(23, 23) in the x = 0.58 and 0.61 samples, and the factthese peaks are positioned at slightly smaller in-planemomenta compared to the half-integer-L superstructurepeaks, suggest that they may be associated with aninter-grown phase (FCT III) that emerges with increas-ing x. To investigate this possibility, a scan along [110]centered at (2, 2, 0) is presented in Fig. 9(a) for thex = 0.61 sample, revealing a smaller side peak cen-tered around (1.94, 1.94, 0). A rocking scan centered at(1.945, 1.945, 0) [inset of Fig. 9(a)] excludes this peakbeing associated with aluminum sample holders or amisaligned grain, and instead evidences an inter-grownphase (FCT III) with similar lattice parameters and thesame orientation as the main phase (FCT II). Since the(2, 2, 0) peak for the inter-grown phase appears at asmaller in-plane momentum compared to the main phase,it has a larger in-plane d-spacing (by ∼ 3%). When theinteger-L superstructure peaks in Fig. 6 are indexed usingthe larger in-plane lattice parameter of the inter-grownphase, they are identified as (13, 13) and (23, 23) peaks.To examine whether the inter-grown phase (FCT III)1.9 1.95 2 2.05(H,H,0) (r.l.u.)051015202530354045Normalized Counts(a)44 46 48 50 52ω (degrees)0123456Q = (1.945,1.945,0)2.85 2.9 2.95 3 3.05 3.1 3.15L (r.l.u.)-0.200.20.40.60.811.21.4Scaled Intensity(0.65,0.65,L)(0,0,L)(b)Sample Growth A, x = 0.61, HB-1AT = 5 KT = 5 KFIG. 9. (a) Longitudinal scan around the (2, 2, 0) peak for ax = 0.61 FCT sample at 5 K. The solid line is a fit to thesum of two Gaussian peaks on a linear background. The in-set shows a rocking scan centered at (1.945, 1.945, 0), and thesolid line is a fit to a Gaussian peak on a linear background.(b) Comparison of L-scans at (0, 0, 3) and (0.65, 0.65, 3) forthe x = 0.61 FCT sample. The solid lines are fits to Gaussianpeaks on linear backgrounds, the fit backgrounds are sub-tracted and the maximum intensities are scaled to unity.also exhibits a different c-axis lattice parameter com-pared to the main phase (FCT II), a longitudinal scancentered at (0, 0, 3) is carried out in the x = 0.61 sample[blue square symbols in Fig. 9(b)], revealing an absenceof detectable side peaks. On the other hand, a L-scancentered on an integer-L superstructure peak associatedwith FCT III [red circle symbols in Fig. 9(b)] reveals thatit is centered at a slightly larger L-value, compared to the(0, 0, 3) peak. Assuming that the (0, 0, 3) peak is domi-nated by the main phase (FCT II), we estimate that theout-of-plane d-spacing for FCT III is ∼ 0.4% smaller thanthat of FCT II. A larger in-plane d-spacing and a smallerout-of-plane d-spacing suggest that FCT III has a largerCu-content compared to FCT II, based on the evolutionof lattice parameters with Cu-content in NaFe1−xCuxAs[22] and Sr(Fe1−xCux)2As2 [19].These results suggest the presence of three differentphases in FCT, one that does not exhibit any super-structures (FCT I), one with the half-integer-L (13, 13)superstructure (FCT II), and one with the integer-L(13, 13) superstructure (FCT III). FCT I persists up to atleast x = 0.25 without detectable superstructure peaks,FCT II appears for x & 0.4, and FCT III is inter-grownwith FCT II for x & 0.55. These findings are summa-rized in Table. I and are shown schematically in Fig. 1(f).Comparing Fig. 1(f) with the evolution of ρ(T ) in Fig. 2suggests while all FCT phases with x & 0.25 are semi-conducting, FCT II is more insulating than FCT I andFCT III.D. Reciprocal space maps for a x = 0.55 FCTsampleTo further probe the mSRO and superstructure peaksin FCT, a x = 0.55 sample (Growth B) was studied using87K  L = 0250K  L = 07K  L = 0.5250K  L = 0.57K  L = 10 1 2 3-1-2-3 0 1 2 3-1-2-3 0 1 2 3-1-2-3 0 1 2 3-1-2-3(H,0,0) (r.l.u.) (H,0,0) (r.l.u.) (H,0,0) (r.l.u.) (H,0,0) (r.l.u.)-3-2-10123-3-2-10123250K  L = 1.57K  L = 1.5250K  L = 1(K,0,0) (r.l.u.)(K,0,0) (r.l.u.)(H,-H,0) (r.l.u.)0 1 2-1-2250K7K-3-2-10123(L,0,0) (r.l.u.)-44-3-2-10123-44(L,0,0) (r.l.u.)min maxSample Growth B, x = 0.55, CORELLI(a)(b)(c)(d)(e)(f)(g)(h)(i)(j))(d) = 0.5 = 0.500((FIG. 10. Diffraction data of a x = 0.55 FCT sample measured using CORELLI, with the 7 K (H,K)-maps at L = 0, 12, 1,and 32respectively shown in panels (a), (c), (e), and (g). The corresponding (H,K)-maps at 250 K are respectively shown inpanels (b), (d), (f) and (h). (i) and (j) show the (H,−H,L)-maps at 7 K and 250K, respectively. Panels (a)-(h) are obtained bybinning data L± 0.1 within the central L-value, and panels (i) and (j) are obtained by binning (H,H) within ±(0.025, 0.025).Peaks that fall on the dashed gray bands in (c) and (d) do not change significantly with temperature, whereas those on thesolid gray bands do.CORELLI, with results summarized in Fig. 10. Com-paring the integer-L (H,K)-maps at 7 K and 250 K[Figs. 10(a), (b), (e), (f)], it can be seen that the integer-L(13, 13) peaks (associated with FCT III) are present acrossmultiple Brillouin zones, do not change significantly inintensity upon warming, and do not weaken at large mo-mentum transfers. These observations confirm that theyare superstructure (rather than magnetic) peaks.Furthermore, the integer-L (H,K)-maps clearly revealthat the (13, 13) and (13,− 13) peaks have distinct shapes,with the (13, 13) peaks more elongated along the [110]direction, and the (13,− 13) peaks more elongated alongthe [11̄0] direction. This difference in peak shape, whichis consistently observed across multiple Brillouin zones,suggest that the superstructure modulation does not ex-hibit fourfold rotational symmetry. Instead, the apparentfour-fold rotational symmetry of the diffraction data re-sults from the twinning of superstructures that exhibittwo-fold rotational symmetry, similar to the twinning ofAFM/orthorhombic domains in the FeSCs [2].The CORELLI data further reveal the half-integer-L(13, 13) and (13,− 13) peaks (associated with FCT II) alsohave anisotropic peak shapes [Figs. 10(c), (d), (g), (h)],exactly like the integer-L superstructure peaks associatedwith FCT III. The common anisotropy in superstructurepeak shapes for FCT II and FCT III indicates that bothsuperstructures form anisotropic domains.Our results reveal remarkable similarities for FCT IIand FCT III, both phase exhibit (i) superstructure mod-ulations with the same in-plane ordering vector Q =(13, 13), (ii) two-fold rotational symmetry, and (iii) domainlengths that are shorter along the ordering vector. Forexample, the observation that (13, 13) peaks are broaderalong [110] than along [11̄0] suggests the typical size ofthe corresponding domain is shorter along [110] com-pared to along [11̄0], and may result from the more com-plex modulation along the ordering vector, which makesit more difficult to order well along this direction.The half-integer-L (H,K)-maps [Figs. 10(c), (d), (g),(h)] also reproduce the temperature-dependent mSROaround (12, 12) positions in Figs. 3 and 4, although thesplitting of peaks into pairs at (12± δ, 12± δ) is not ob-served. The CORELLI data further clearly show thatthese diffuse scattering are not detectable in higher Bril-louin zones, providing strong evidence for their magneticnature.Figs. 10(i) and (j) show (H,−H,L)-maps for the x =0.55 FCT sample at 7 K and 250 K, respectively. Con-sistent with observations described so far, the (13,− 13)peaks persist at 250 K and are clearly detectable in higherBrillouin zones, whereas the (12,− 12) peaks disappear at250 K and are limited to small momenta. In addition,it can be directly seen that the integer-L (13,− 13) peaksoccur at slightly smaller momenta compared to the half-integer-L (13,− 13) peaks. This is consistent with the HB-1A data in Fig. 6, and is a result of the integer-L peaksbeing associated with FCT III, which has a slighter largerin-plane d-spacing compared to FCT II [Fig. 9(a)]. Theseresults from the CORELLI data are also confirmed inTOPAZ measurements of a x = 0.55 sample at 100 K.90 1 2 3-1-2(0,K,0) (r.l.u.)4 5-4-3-20123(0,0,L) (r.l.u.)-140 1 2 3-1-2(0,K,0) (r.l.u.)4 5-4-3-20123(0,0,L) (r.l.u.)-14T= 250 K, CORELLIT= 7 K, CORELLI0 1 2 3L (r.l.u.)4(H,K)=(0,1), 7 K(H,K)=(0,2), 7 K(H,K)=(0,3), 7 K(H,K)=(0,1), 250 KCORELLI0 1 2 3(0,K,0) (r.l.u.)4T= 100 K, TOPAZ0123(0,0,L) (r.l.u.)4560.000.020.040.060.080.10Intensity (arb. units)(a)(b)(c)(d)Sample Growth B, x = 0.55FIG. 11. (0,K, L)-maps from the CORELLI data for ax = 0.55 FCT sample at (a) 7 K and (b) 250K. The maps areobtained by binning H within ±0.025. White arrows high-light (0, 1,± 12), where sharp magnetic peaks are observed.(c) Comparison of cuts along L obtained from the CORELLIdata. The cuts are made by binning H and K within ±0.025of the center position. (d) (0, K, L)-map of the TOPAZ datafor a x = 0.55 FCT sample at 100 K. The TOPAZ data hasbeen symmetrized by applying the H0L, HK0, and 0KL mir-ror planes, and the map is obtained by binning H within±0.005.E. Magnetic order associated with FCT IIIThe half-integer-L (H,K)-maps of the x = 0.55 samplealso reveal (1, 0)/(0, 1) peaks that are present at 7 K butnot at 250 K [Figs. 10(c), (d), (g), (h)]. The (0, 1) peaksare further confirmed in (0,K, L)-maps at 7 K and 250 K[Figs. 11(a) and (b)]. These data show that (i) the (0, 1)peaks appear systematically at half-integer-L positions,(ii) mostly disappear at 250 K, and (iii) equivalent peaksare not detected at (0, 2), (0, 3), or (0, 4). Observations(ii) and (iii) can also be seen in L-cuts of the CORELLIdata, shown in Fig. 11(c). These behaviors suggest thehalf-integer-L (1, 0)/(0, 1) peaks are magnetic. Similarbehaviors are also seen in measurements using TOPAZat 100 K [Fig. 11(d)], further indicating that while thesepeaks mostly disappear at 250 K, they persist up to atleast 100 K.In addition to the half-integer-L (1, 0)/(0, 1) peaks,Figs. 10(c), (d), (g), (h) also reveal that the half-integer-L (13, 23) and (23, 13) peaks vary significantly in intensitybetween 7 K and 250 K. This points to the presenceof magnetic peaks at these positions. Given that thehalf-integer-L (1, 0)/(0, 1) and (23, 13) peaks are sharp inmomentum and weaken or disappear upon warming to250 K, they likely have the same origin.Considering the mSRO signal at half-integer-L (12±δ, 12± δ) appear in FCT samples with x from 0.42 to0.61, in all of which the half-integer-L (13, 13) peaks due toFCT II are detected, we identify the mSRO with FCT II.This identification is consistent with the absence of bothhalf-integer-L (12± δ, 12± δ) magnetic peaks and half-integer-L (13, 13) superstructure peaks in the x = 0.25sample [Table. I]. The sharp half-integer-L (1, 0)/(0, 1),(13, 23), and (23, 13) magnetic peaks in the x = 0.55 sam-ple can then be identified with the inter-grown FCT IIIphase, and is supported by the relatively simple modeldescribed below [Fig. 12(b)], which accounts for both thesuperstructure and magnetic peaks in FCT III.In the model shown in Fig. 12(b), atoms on the in-terstitial site are ignored, and Fe-chains are separatedby Cu-ladders, leading to the composition Fe1/3Cu2/3Te.Fe-Cu ordering in this structure only expands the unitcell in the ab-plane but not along the c-axis, and ac-counts for the (13, 13) superstructure peaks that appearonly at integer-L positions. In the superstructure unitcell, which contains only a single Fe atom, the magneticmoments exhibit AFM alignment along as, bs and c (Neel-type magnetic order), which results in magnetic peaks at(12, 12)s positions in the reciprocal space expanded by a∗sand b∗s [Fig. 12(c)]. Converting from the scattering pat-tern in Fig. 12(c) to the reciprocal space associated withthe 2-Fe/Cu unit cell in Fig. 1(a), and by further con-sidering a 90 degrees rotated twin domain, leads to themagnetic scattering pattern shown in Fig. 12(d), withmagnetic peaks at (13, 23), (23, 13) and (1, 0) positions. Themagnetic peaks appear at half-integer-L positions, due toan AFM alignment of moments along the c-axis.The model in Fig. 12(b) predicts that magnetic peaksonly appear along lines with odd values of H ±K, with+ and − corresponding to two domains rotated by 90degrees [domains B and A in Fig. 12(d), respectively].The half-integer-L (H,K)-maps in Figs. 10(c), (d), (g),(h) agree remarkably well with the scattering pattern inFig. 12(d). In particular, temperature-dependent peaksare seen at (0, 1) and (1, 2), but not at (0, 2) and (−1, 1);the (− 13, 23) and (− 23, 13) peaks vary strongly with tem-perature, but the (− 13, 13) peak hardly changes in inten-sity between 7 K and 250 K; peaks on the H + K = 1line [solid gray bands in Figs. 10(c) and (d)] vary signif-icantly with temperature, but peaks on the H +K = 0line [dashed gray bands in Figs. 10(c) and (d)] hardlychange in intensity between 7 K and 250 K.The structural and magnetic model in Fig. 12(b) pro-posed for Fe1/3Cu2/3Te is remarkably similar to that ofNaFe0.5Cu0.5As [Fig. 12(a)] [22], both featuring AFM Fechains separated by Cu atoms. In the proposed struc-ture of Fe1/3Cu2/3Te, the magnetic peaks at (13, 23), (23, 13)arise from superstructure domains that order at integer-L (13,− 13) positions, which have Fe chains along [110]. In10+bS*aS*NaFe0.5Cu0.5As Fe1/3Cu2/3Te(a) (b)(c) (d)aSbScdomain A domain B domains A+BH (r.l.u.)0 1 2-1-2K (r.l.u.)012-1-2++++++++++++++-------------FeCuFIG. 12. Schematic crystal and magnetic structures of (a)NaFe0.5Cu0.5As and (b) Fe1/3Cu2/3Te, with only Fe and Cuatoms shown. The ‘+’ and ‘−’ symbols represent relative spinorientations of the Fe moments. (c) The expected magneticpeak positions for Fe1/3Cu2/3Te, in the reciprocal lattice as-sociated with the superstructure unit cell spanned by as andbs in panel (b). The magnetic peaks appear at ( 12, 12)s andequivalent positions, because the magnetic structure is Neel-type. (d) The magnetic scattering pattern of Fe1/3Cu2/3Te,obtained from (c) by translating from the superstructure unitcell spanned by as and bs to the 2-Fe/Cu unit cell in Fig. 1(a),and considering the presence of a domain rotated by 90 de-grees.NaFe0.5Cu0.5As, magnetic peaks at (12, 12) are associatedwith superstructure domains that order at (12,− 12), withFe chains along [110] [22].F. Exploration of FCT phases via first-principlescalculationsTo understand the structures of FCT II and FCT III,we studied 5 candidate stoichiometric FCT superstruc-tures via first-principles calculations, without consid-ering occupancy of the interstitial site. These super-structures exhibit√2 × 3√2 expansions of the unitcell in the ab-plane relative to P4/nmm FeTe, and in-clude α-Fe2/3Cu1/3Te, β-Fe2/3Cu1/3Te, Fe1/2Cu1/2Te,α-Fe1/3Cu2/3Te, and β-Fe1/3Cu2/3Te [Figs. 13(a)-(e),SS1-SS5]. The α phases are AA-stacked with super-structure peaks at integer-L, whereas the β phases areAB-stacked with superstructure peaks at half-integer-L. Fe1/2Cu1/2Te consists of alternating layers withFe:Cu=1:2 and 2:1, with superstructure peaks at half-integer-L.We define the formation energies of these compoundsas ∆Ec(SSi) = E(SSi) − ((1 − xi)EFeTe + xiECuTe)(i = 1, · · · , 5), where xi is the Cu-content for SSi, E(SSi)is the energy per formula unit for SSi in DFT calcula-tions, and EFeTe and ECuTe are the energies of P4/nmmFeTe and CuTe. ∆Ec for the 5 superstructures are shownin Fig. 13(f), revealing SS1 and SS5 to be energeticallyunfavorable. The gray line defines the lowest energy ofFe1−xCuxTe compounds for 13≤ x ≤ 23, when the 5 su-perstructures in Figs. 13(a)-(e) are considered. As canbe seen, SS2 and SS4 fall on the gray line, whereas SS3has an energy slightly above the gray line.These first-principles results are consistent with our ex-perimental observation of two inter-grown phases in theFCT phase diagram. Particularly, SS4 is identical to thestructure for FCT III inferred from experimental data[Fig. 12(b)], which accounts for both the integer-L su-perstructure peaks and the sharp half-integer-Lmagneticpeaks in the x = 0.55 FCT sample [Figs. 10 and 11]. Onthe other hand, either SS2 [AB-stacked Fe-ladders sep-arated by Cu-chains] or SS3 [alternating Fe-chains and-ladders separated by Cu-ladders and -chains] may cor-respond to the FCT II phase with half-integer-L (13, 13)superstructure peaks and the half-integer-L (12±δ, 12±δ)mSRO peaks. This is because although the energy ofSS3 is slightly above the SS2 and SS4 mixture [gray linein Fig. 13(f)], this energy (∼ 1.5 meV) is too small torule it out in the FCT phase diagram. Although ourDFT calculations do not uniquely identify a superstruc-ture for the FCT II phase, the two candidates structuresboth contain Fe-ladders, making FCT II interesting forthe investigation of quantum magnetism and correlatedelectrons in low dimensions.IV. DISCUSSIONThe x = 0.25, 0.42, 0.48, 0.58, and 0.61 samples(Growth A) were measured on HB-1A using identicalsetups, the corresponding data in Figs. 3, 6, 7 are nor-malized by sample mass, and can be quantitatively com-pared. The integrated intensities for the L = 0 and L = 12superstructure peaks obtained from scans along [110] areshown in Fig. 14(a). The L = 12peaks are absent inthe x = 0.25 sample, become maximized in the x = 0.42and 0.48 samples, and weaken in the x = 0.58 and 0.61samples. On the other hand, the L = 0 peaks are ab-sent in the x = 0.25, 0.42, and 0.48 samples, first ap-pears in the x = 0.58 sample, and becomes more intensein the x = 0.61 sample. Such an evolution suggests avolume-wise competition between FCT II modulated byhalf-integer-L superstructure peaks, and FCT III modu-11SS5 β-Fe1⁄3Cu2⁄3TeSS2 β-Fe2⁄3Cu1⁄3TeSS3 Fe½Cu½TeSS1 α-Fe2⁄3Cu1⁄3TeSS4 α-Fe1⁄3Cu2⁄3TeFeCu(a) (b)(e) (f )(c) (d)1/3 1/2 2/3-0.4-0.38-0.36-0.34-0.32∆EC(eV)x in Fe1-xCuxTeSS1SS2SS5SS4SS3FIG. 13. (a)-(e) The 5 candidate FCT structures consid-ered in DFT calculations, with the interstitial site ignored.Only Fe and Cu atoms shown. (f) The formation energy∆Ec for the 5 candidate FCT structures, as a function ofthe Cu-content x. The circle and square symbols representsuperstructures that do and do not double the unit cell alongthe c-axis. The solid gray line represents ∆Ec of the SS2/SS4mixture, which is slightly below ∆Ec of SS3.lated by integer-L superstructure peaks.Fig. 14(b) shows the mSRO intensities as a functionof x, obtained by integrating the scans in Fig. 3(c). Themagnetic signal first appears in the x = 0.42 sample,becomes more intense in the 0.48 sample, and then de-creases with further increase of x. Such an evolutionwith x is similar to the half-integer-L (13, 13) superstruc-ture peak, and provides evidence that the mSRO aroundthe stripe-type vector is associated with FCT II.The superstructure scans at (13, 13, 0) and (13, 13, 12) arefit to Lorentzian peaks [Fig. 8], with the extracted corre-lation lengths shown in Fig. 14(c). For the half-integer-L (13, 13) peak, the in-plane and out-of-plane correlationlengths are highly anisotropic in the x = 0.42 and 0.48samples, with ξ110 ≈ 80 Å and ξc ≈ 20 Å. Increasing xtowards 0.58 and 0.62 leads to a reduction in ξ110 andan enhancement in ξc, so that both are around 40 Å. Forthe integer-L superstructure peak, both ξ110 and ξc areenhanced when x is increased from 0.58 to 0.61. In par-ticular, ξc for the integer-L superstructure peak in thex = 0.61 sample reaches ∼ 100 Å, and is (or close to)resolution-limited, with a similar width as a main Braggpeak [Fig. 9(b)].The in-plane magnetic correlation lengths are obtainedby fitting the data in Fig. 3(c) to Eq. 1, and the out-of-plane magnetic correlation lengths are obtained by fittingthe data in Fig. 3(d) to a single Lorentzian peak. Theextracted magnetic correlation lengths ξ110 are roughlyin the range 12−18 Å, and ξc roughly in the range 4−9 Å[Fig. 14(d)], neither changing significantly with x, con-sistent with the almost overlapping scaled temperaturedependence in Fig. 3(f). The x-dependence of the mag-netic correlation length contrasts with behaviors of thesuperstructure correlations lengths, and may result fromthe former being much shorter, such that it is insensitiveto changes in the latter.The presence of inter-grown phases and significant oc-cupation of the interstitial Fe/Cu site [Table I] in FCTare likely important in accounting for the highly variedsample properties reported in the literature [37–46]. Tomake further progress in understanding the FCT phasediagram, it is important to synthesize pure phases ofthe FCT II and FCT III with minimal interstitial Fe/Cuatoms.The possibility of FCT II realizing Fe-ladders is inter-esting, given that superconductivity has been found inpressurized Fe-ladder compounds BaFe2S3 and BaFe2Se3[47–49]. Such low-dimensional systems also offer an idealground for studying the physics of strongly correlatedelectrons, since experimental comparisons with realisticmodels are possible [50–53]. In addition, as single crys-tals of FCT can be readily made, it offers a potentialroute towards probing the spin dynamics in multi-orbitalchain or ladder systems, whereas the spin dynamics of123 Fe-ladder systems were mostly carried out on pow-der samples [54–56]. In the 123 Fe-ladder systems, Festoichiometry and sample quality is found to be impor-tant for a pressure-induced superconducting state [57]. Inthis regard, the short-range nature of superstructure andmagnetic peaks, the presence of a substantial amount in-terstitial Fe/Cu atoms, and the presence of the FCT IIIphase, may be detrimental for achieving a superconduct-ing state. A study of FCT with x = 0.5 under pressuresup to 34 GPa did not find superconductivity [45].Our discovery of superstructures in FCT underscorethe potential of magnetically diluted iron pnictides andchalcogenides as platforms for realizing low-dimensionalcorrelated electrons and quantum magnetism. Akin toFe-Cu ordering in NaFe0.5Cu0.5As which leads to Fechains [22], and Fe-Ag ordering in KFe0.8Ag1.2Te2 thatleads to 2×2 Fe blocks exhibiting nematicity [58, 59], thiswork shows that Fe-Cu ordering in FCT may lead to Fechains or ladders, although it might be an experimental1220406080100120Correlation Length (Å)102030405060Integrated Intensity (arb. units)0.4 0.45 0.5 0.55 0.6 0.65x05101520250.2 0.3 0.4 0.5 0.6x02468Integrated Intensity (arb. units)Correlation Length (Å)(a) (c)(b) (d)L = 1/2L = 0ξ110, L = 1/2ξc, L = 1/2ξ110, L = 0ξc, L = 1Q = (1/3,1/3,L)Q = (0.5±δ,0.5±δ,1.5)Q = (1/3,1/3,L)Q = (0.5±δ,0.5±δ,1.5)ξ110ξcSample Growth A, HB-1AFIG. 14. Doping dependence of (a) ( 13, 13, L) superstructureintegrated intensities, (b) ( 12± δ, 12± δ, 32) magnetic peak in-tegrated intensities, (c) ( 13, 13, L) superstructure correlationlengths, and (d) ( 12± δ, 12± δ, 32) magnetic peak correlationlengths. The superstructure correlation lengths are obtainedfrom fits in Fig. 8, and the magnetic correlation lengths areobtained from fits in Figs. 3(c) and (d). Resolution effectshave not been considered, and the correlation lengths shouldbe regarded as lower limits.challenge to achieve perfect ordering between Fe and Cu.The observation of stripe-type magnetism in FCT IIis unexpected, as Fe1+yTe exhibits double-stripe mag-netic order with the ordering vector (12, 0) [6, 7]. Incombination with previous studies, our results show thatwhile the atomic orderings between Fe atoms and thenonmagnetic ions in NaFe0.5Cu0.5As, KFe0.8Ag1.2Te2,Ba(Fe1−xCux)2As2 and FCT vary, magnetic orderingat or around the stripe-type vector appears common inmany of these systems. Given these materials are semi-conducting or semiconducting-like, the common mag-netism should arise from local moments, therefore local-ized electronic degrees of freedom may be important forunderstanding stripe-type magnetism in the FeSCs.In summary, by carrying out a systematic study ofthe (Fe1−xCux)1+yTe series using neutron scattering, wediscover integer-L and half-integer-L (13, 13, 0) superstruc-tures, associated with two distinct phases (FCT II andIII respectively). Diffuse half-integer-L (12± δ, 12± δ)(δ ≈ 0.05) magnetic peaks are detected around the stripe-type ordering vector, identified with the FCT II phase.Sharp half-integer-L (13, 23), (23, 13) and (1, 0) magneticpeaks are observed in a x = 0.55 sample, resulting fromthe FCT III phase inter-grown with FCT II. We ex-plore possible structures of FCT II and FCT III via first-principles calculations, finding that FCT III consists ofFe-chains and FCT II contains Fe-ladders. The common-ality of stripe-type magnetism in several magnetically di-luted Fe pnictide and chalcogenide compounds indicatesuch magnetism could arise from localized electronic de-grees of freedom. Furthermore, our findings highlightmagnetically diluted Fe pnictides and chalcogenides aspromising platforms for studying low-dimensional corre-lated electrons and quantum magnetism.The work at Zhejiang University was supported by thePioneer and Leading Goose R&D Program of Zhejiang(2022SDXHDX0005), the National Key R&D Programof China (No. 2022YFA1402200), the Key R&D Programof Zhejiang Province, China (2021C01002), and the Na-tional Science Foundation of China (No. 12274363 andNo. 12274364). The work at University of California,Berkeley and Lawrence Berkeley National Laboratorywas supported by the Office of Science, Office of BES,Materials Sciences and Engineering Division, of the U.S.DOE under Contract No. DE-AC02-05-CH11231 withinthe Quantum Materials Program (KC2202). 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