Strain-induced spin-nematic state and nematic susceptibility arising from 2×2 Fe clusters in KFe 0.8 Ag 1.2 Te 2
Yu Song, Dongsheng Yuan, Xingye Lu, Zhijun Xu, Edith Bourret-Courchesne, Robert J. Birgeneau
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov Strain-induced spin-nematic state and nematic susceptibility arising from × Feclusters in KFe . Ag . Te Yu Song,
1, 2, ∗ Dongsheng Yuan, Xingye Lu, Zhijun Xu,
4, 5
Edith Bourret-Courchesne, and Robert J. Birgeneau
1, 2, 6 Department of Physics, University of California, Berkeley, California 94720, USA Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA Center for Advanced Quantum Studies and Department of Physics,Beijing Normal University, Beijing 100875, China NIST Center for Neutron Research,National Institute of Standards and Technology, Gaithersburg MD 20899, USA Department of Materials Science and Engineering,University of Maryland, College Park, MD 20742, USA Department of Materials Science and Engineering,University of California, Berkeley, California 94720, USA
Spin nematics break spin-rotational symmetry while maintaining time-reversal symmetry, analo-gous to liquid crystal nematics that break spatial rotational symmetry while maintaining transla-tional symmetry. Although several candidate spin nematics have been proposed, the identificationand characterization of such a state remain challenging because the spin-nematic order parame-ter does not couple directly to experimental probes. KFe . Ag . Te (K Fe Ag Te , KFAT) is alocal-moment magnet consisting of well-separated 2 × × PACS numbers: 74.25.Ha, 74.70.-b, 78.70.Nx
Quantum materials often adopt electronic groundstates that break rotational-symmetry of their under-lying crystal structures, analogous to liquid crystal ne-matics [1]. Such electronic nematic states are immenselyinteresting in their own right, and have additionally gar-nered recent attention due to their ubiquity near un-conventional superconducting states [2–6]. One formof such a state is the spin nematic, which maintainstime-reversal symmetry but breaks spin-rotational sym-metry through an even-order spin order parameter suchas h ( S x ) i−h ( S y ) i 6 = 0 [7–9], in contrast to conventionalmagnetic order such as h S x i − h S y i 6 = 0 that breaks bothsymmetries. Examples of the spin-nematic state havebeen proposed in frustrated magnets [9, 10], rare-earthmagnets [11, 12] and iron-based superconductors (IBS)[13–15]. However, because the order parameters for pro-posed spin-nematic states do not couple directly to ex-perimental probes, a direct detection of the spin-nematicstate remains elusive.The parent compounds of iron-based superconductors(PC-IBS) are bad metals made up of stacked Fe squarelattices forming a tetragonal structure, and exhibit an or-thorhombic distortion below T S as well as a stripe-typemagnetic ordering below T N ( T N ≤ T S ) [16, 17]. In theparamagnetic tetragonal state above T S , N , a sizable resis-tivity anisotropy characteristic of an electronic nematicstate ( η ) can be induced by a small strain ( ǫ ) [4, 18], and divergence of the nematic susceptibility ( dη/dǫ ) demon-strates the phase transitions below T S , N are driven byan electronic nematic order parameter that couples tothe lattice [19]. The electronic nematic order parametermay be associated with ferro-orbital ordering [20, 21] or aspin-nematic state [13, 14], and in the latter anisotropy inspin correlations between the two Fe-Fe directions breaksspin-rotational symmetry while retaining time-reversalsymmetry [13, 14, 22–24]. Experimental evidence forspin-nematicity in these materials include scaling of theshear modulus and spin-lattice relaxation rate [25] andnematic spin correlations observed in neutron scattering[26, 27], however the metallic nature of these materialsmeans the shape [28, 29] and orbital content [30] of theFermi surface may play major roles accounting for theseobservations. In addition, while anisotropy of the uni-form magnetic susceptibility is anticipated to scale withthe resistivity anisotropy in the spin-nematic scenario[22, 23], experimentally such anisotropies have not beendetected [31, 32].KFAT is structurally similar to PC-IBS, but ratherthan a metal it is a semiconductor with localized mag-netism [33, 34], and instead of planes of Fe square latticesit consists of 2 × T S , N ≈
35 K, similar to the PC-IBS such as BaFe As [Fig. 1(a), (b)] [35, 36]. This suggests that similar toIBS [19], an electronic nematic order parameter coupledto the lattice may be present in KFAT; and given itssemiconducting nature, such an electronic nematic orderparameter should arise from localized spin and orbitaldegrees of freedom, allowing the interplay between dif-ferent orders in IBS to be probed in the strong-couplinglimit [37–39] within 2 × × T S , N , indicating that spin-nematicitydrives the ordered ground state. The strong couplingbetween lattice and spin-nematicity results from an un-derlying spin-orbital entangled state, allowing for a novelform of elastomagnetic effect, in which strain inducesanisotropic uniform magnetic susceptibility in a param-agnetic state.Single crystals of KFAT are grown from a stoichio-metric mixture of elemental materials using a modi-fied Bridgeman method [33]. Magnetization measure-ments were carried out using a Quantum Design MPMS3SQUID under a field of 5 T. The magnetization in KFATis linear with field up to at least 5 T [33]. Neutronscattering measurements were carried out using the BT-7 triple-axis spectrometer at the NIST Center for Neu-tron Research (NCNR), aligned in the [ H, K,
0] plane.Strain is applied by gluing ∼ √ × √ × I mmm unit cell appropriate fortetragonal BaFe As to describe our results [Fig. 1(c)],in this notation the Fe-Fe bonds are along (110)/(1¯10)directions.Below T S , N , a collinear stripe-type spin configurationforms inside each 2 × c -axis [Fig. 1(b)] [35]. When projectedinto the ab -plane, the magnetic structure within eachcluster resembles the PC-IBS [Fig. 1(a), (c)], with therelative easy-axis along the AF Fe-Fe direction. Magneticsusceptibilities of freestanding KFAT along three high- BaFe As KFe Ag Te Fe Ag χ χ χ χ ( - e m u g - O e - ) Temperature (K) (a)(b)(c) (d)
FIG. 1: (Color online) (a) The magnetic structure ofBaFe As , a prototypical PC-IBS. (2) The crystal and mag-netic structures of KFAT. The shaded circles represent themagnetic easy plane. (c) The magnetic structure of KFATprojected into the ab -plane. The solid box is the I mmm unit cell used in this work. (d) The magnetic susceptibilityof freestanding KFAT measured along three high-symmetrydirections. χ and χ almost overlap. symmetry directions are shown in Fig. 1(d), in goodagreement with previous report [33]. The Curie-Weissbehavior of the magnetic susceptibilities above T S , N withan effective moment µ eff ≈ . µ B /Fe ( S ≈
1) [40] pointsto local-moment magnetism, consistent with its semicon-ducting transport [33]. The larger drop of the magneticsusceptibility along the c -axis below T S , N is consistentwith it spanning the easy-plane; however due to the for-mation of twin domains below T S , N , the magnetic sus-ceptibility appear isotropic in the ab -plane for the free-standing sample.To probe the intrinsic magnetic susceptibility of KFAT,it is necessary to detwin the sample. The structural dis-tortion accompanying the magnetic order in KFAT dif-ferentiates the lattice spacings along (110)/(1¯10), withthe AF (FM) Fe-Fe direction elongated (contracted) be-low T S , N ; it is therefore possible to detwin the sample byapplying strain, similar to the PC-IBS [41–43]. By glu-ing the sample onto a GFRP substrate, strain is applieddue to anisotropic thermal contraction of the substrate[31]. Upon cooling, the direction parallel (perpendicular)to the unidirectional fibers will be the relatively longer M agne t i z a t i on ( e m u ) M agne t i z a t i on ( e m u ) χ ( - e m u g - O e - ) Temperature (K)
100 200 300-0.10-0.050.000.05 η Temperature (K) χ para χ perp χ para −χ perp M para M perp (a) (b)(c) (d)(e) (f) perppara M GFRP (100)/(100) strain(110)/(110) strain M para M perp M GFRP η = χ para +χ perp χ para −χ perp χ ( - e m u g - O e - ) FIG. 2: (Color online) (a) Schematic of our experimentalsetup with the sample glued onto a GFRP substrate. Uponcooling, the direction parallel (perpendicular) to the glassfibers become relatively longer (shorter), applying strain tothe sample (arrows). This setup is used for magnetizationmeasurements parallel and perpendicular to the fibers, for (b)(100)/(010) strain and (c) (110)/(1¯10) strain. (d) Magneticsusceptibilities parallel ( χ para ) and perpendicular ( χ perp ) tothe fibers. (e) χ para − χ perp . (f) η = ( χ para − χ perp ) / ( χ para + χ perp ). (shorter) axis [Fig. 2(a)], and since the AF (FM) alignedFe-Fe direction elongates (contracts) below T S , N , the ma-jority domain has the AF (FM) Fe-Fe direction parallel(perpendicular) to the fibers.Using this setup, we measured the magnetizations par-allel ( M para ) and perpendicular ( M perp ) to the fibers withstrain along (110)/(1¯10), revealing a clear anisotropy[Fig. 2(c)]. The magnetic susceptibilities χ para and χ perp are obtained by subtracting the magnetization due tothe substrate ( M GFRP ) and dividing by the applied field[Fig. 2(d)]. Upon cooling below T S , N , χ para exhibits amore prominent drop compared to χ perp , suggesting thatfor the two in-plane directions, spins order along theAF Fe-Fe direction (which dominates χ para ). Combinedwith χ in Fig. 1(d), these measurements demonstratethat below T S , N , KFAT exhibits a magnetic easy-planespanned by the AF Fe-Fe direction and the c -axis, con-sistent with previous diffraction results [35].Unexpectedly, we also observed sizable magnetic sus-ceptibility anisotropy over an extended temperaturerange for T & T S , N [Fig. 2(d)], in contrast to similarexperiments on BaFe As above T S , N and FeSe above T S [31, 32]. To quantify the observed anisotropy, the dif-ference χ para − χ perp and the dimensionless anisotropy η = ( χ para − χ perp ) / ( χ para + χ perp ) are respectively shown in Fig. 2(e) and 2(f). As can be seen, the sign of theanisotropy changes from χ para < χ perp for T . T S , N to χ para > χ perp for T & T S , N . As discussed above, χ para < χ perp below T S , N is because χ para probes themagnetic easy-axis (AF Fe-Fe direction), which has asmall magnetic susceptibility in the ordered state; thereversed anisotropy for T & T S , N with χ para > χ perp in-stead indicates a paramagnetic state, in which a largermagnetic susceptibility is expected along the easy-axis.Our results therefore show that a small strain induces asizable spin-rotational symmetry-breaking without con-ventional magnetic order, realizing a strain-induced spin-nematic state.Similar measurements were carried out for (100)/(010)strain [Fig. 2(b)]; we find M para and M perp to be es-sentially identical for all temperatures (difference lessthan 0.05% of their values for T & T S , N ), in contrastto (110)/(1¯10) strain [Fig. 2(c)]. This demonstratesthat strain along Fe-Te directions does not detwin thesample below T S , N , and more importantly, the strain-induced spin anistropy in the paramagnetic state exhibitsa prominent Ising character (at least ≈
100 times largerfor (110)/(1¯10) strain). Such a prominent Ising-type re-sponse is similar to resistivity anisotropies in Sr Ru O (field-induced) [44] and BaFe As (strain-induced) [45],but contrasts with CeRhIn (field-induced) [6] and heav-ily electron- or hole-doped IBS (strain-induced) [45–49],in which the Ising character of the response is weaker andeven XY-like.Having shown that strain induces a sizable spinanisotropy in KFAT, we elucidate its origin with neutronscattering using the same strain setup. For a samplewith (110)/(1¯10) strain, we monitored its (220) (paral-lel to fibers) and (2¯20) (perpendicular to fibers) Braggpeaks as a function of temperature, with results shownin Figs. 3(a) and (b) and compared for selected tem-peratures in Figs. 3(c)-(e). The scattering angles forthe two Bragg peaks are obtained through Gaussian fits[Fig. 3(f)]. Upon cooling, the scattering angles for bothpeaks increase monotonically due to thermal contractiondown to T ≈ T S , N , below which (220) moves to lowerscattering angles and (2¯20) moves to higher scatteringangles. The disparate responses are due to detwinningof the sample, in contrast to a twinned sample in whichsplittings (or broadenings when resolution is insufficient)of both (220) and (2¯20) are observed [35]. From thescattering angles we obtained the corresponding latticespacings d and d , and extracted the temperaturedependence of the strain ǫ = ( d − d ) / ( d + d )[Fig. 3(g)], which is a dimensionless measure of the latticeanisotropy. Compared to a freestanding sample, the sim-ilar values of ǫ (orthorhombicity for T < T S , N ) indicatesthat our strain setup mainly acts to detwin the samplebelow T S , N , while for T & T S , N the structural transitionis smeared out and a non-zero strain is induced up toroom temperature. C oun t s / s scattering angle 2 θ (degrees)98 100 102 98 100 102 1040100200300400 C oun t s / s Temperature (K) sc a tt e r i ng ang l e θ ( deg r ee s ) θ ( deg r ee s ) ε = d − d d + d Temperature (K) (220) perp(220) para strainedfreestanding5 K 50 K 300 K (a)(b)(c) (d) (e) (220)(220) (f) (g) (220)(220) ε C oun t s / s C oun t s / s FIG. 3: (Color online) Pseudo-color plots of longitudinalscans for (a) (220) and (b) (2¯20) Bragg peaks as a functionof temperature. Representative scans are compared for (c) T = 5 K, (d) T = 50 K and (e) T = 300 K. (f) Scatteringangles 2 θ for longitudinal scans in (a) and (b), obtained fromfits to Gaussian peaks. (g) The lattice anisotropy ǫ (strain),for a sample under strain and a freestanding sample. Thesolid lines in (c), (d) and (e) are the results of fits to Gaussianpeaks. To clarify the effect of strain on the magnetic order inKFAT, we studied two magnetic Bragg peaks related bya 90 ◦ rotation, with domains associated with Q ( Q )having the longer AF (shorter FM) Fe-Fe bonds par-allel to the fibers [Fig. 4(a)]. The temperature depen-dence of these two magnetic Bragg peaks are shown inFig. 4(b), from their ratio a detwinning ratio of ≈ Q Bragg peak in a strained sample and a freestand-ing sample [Fig. 4(c)], we find that strain enhances theonset temperature of the magnetic transition by severalKelvins, similar to the PC-IBS [41–43]. The increase ofonset temperature of the magnetic transition under straincan also be seen from our magnetic susceptibility mea-surements, with results in Figs. 1(d) and 2(d) magnified freestanding strained N o r m a li z ed I n t en s i t y Temperature (K) 0 20 40 60Temperature (K) 02468 C oun t s / s χ ( a r b . un i t s ) −1 −0.5 0 0.5 1−1−0.500.51 H (r. l. u.) K (r . l . u . ) Q Q Q Q Q Q Q χ para χ perp χ free (a) (b)(c) (d) FIG. 4: (Color online) (a) Allowed structural and magneticpeaks in the [
H, K,
0] scattering plane of KFAT. Superstruc-ture peaks occur due to the √ × √ Q and Q are two magnetic Braggpeaks related by a 90 ◦ rotation, corresponding to magneticdomains with AF spins along (110) and (1¯10), respectively.Only peaks associated with one of two √ × √ Q and Q magnetic Bragg peaks measured with (110) parallel tofibers. (c) Temperature dependence of the Q magnetic peakin a freestanding and a strained sample compared, after nor-malizing by the intensity at low temperatures. (d) Uniformmagnetic susceptibilities of a freestanding and a strained sam-ple compared near T S , N . and compared in Fig. 4(d). We note that while magneticorder onsets below ≈
50 K under strain, the uniform mag-netic susceptibility anisotropy χ para > χ perp extends tomuch higher temperatures.An important question to address in KFAT is whetherits structural transition results from a lattice instabilityor an underlying electronic nematic order parameter. Toresolve this issue, consider the free energy F = a φ + b φ + c ǫ + d ǫ − λφǫ, where φ is an electronic nematic order parameter and ǫ is the lattice distortion, with φ and ǫ coupled through λ [19]. As previously shown, the strain-nematic susceptibil-ity dφ/dǫ will exhibit a divergence only if the structuralphase transition is driven by an instability in φ [19]. Inthe case of KFAT, we have obtained the dimensionlessspin [ η , Fig. 2(f)] and lattice [ ǫ , Fig. 3(g)] anisotropiesunder strain. The measured spin anisotropy η ∝ φ , andwithin linear response dφ/dǫ ∝ dη/dǫ = η/ǫ . From ourmeasurements, η/ǫ for KFAT [Fig. 5(a)] exhibits a cleardivergence-like increase upon cooling towards T S , N , indi-cating the coupled phase transitions at T S , N are drivenby an instability in φ [40]. The nematic susceptibilityfor T ≥
50 K can be described by a Curie-Weiss form η/ǫ = a/ ( T − T ∗ ) + b with T ∗ = 29(3) K. Since both η and ǫ are dimensionless anisotropies, η/ǫ provides a η / ε Temperature (K) (a) (b) + +− −
110 para110 perp majority1 23 4 + + − − minority1 23 4
FIG. 5: (Color online) (a) η/ǫ . The red line is a Curie-Weissfit for T ≥
50 K, with the Weiss temperature T ∗ = 29(3) K.We note T ∗ also has a dependence on the temperature rangeused in fitting. (b) Two stripe-type configurations of 2 × quantitative comparison between the two. Cooling to-wards T S , N , η/ǫ ≈
30 at T ≈
50 K, the much large spinanisotropy indicates an inherent tendency towards break-ing spin-rotational symmetry; although in the absenceof strain, time-reversal symmetry simultaneously breaksand results in conventional magnetic order.Compared to our findings in KFAT, similar magneticsusceptibility anisotropies were not detected in strainedBaFe As and FeSe [31, 32], likely due to intrinsicallysmaller magnetic susceptibilities in IBS [50, 51], and de-velopment of the spin-nematic order parameter at thestripe-type ordering vector is accompanied by an in-crease in the magnetic correlation length [52] that re-duces effects of the former in the uniform limit. In con-trast, the local-moment nature KFAT results in a largermagnetic susceptibility, and the 2 × T S , N , a natural candidate to account for the spin-nematicity is an Ising degree of freedom emerging fromthe 2 × h S · S i + h S · S i−h S · S i−h S · S i [Fig. 5(b)], a vestigial order of magnetic order[53] similar to the spin-nematic state proposed for theIBS [13]. Our observation of strain having a strong ef-fect on the spin anisotropy immediately points to a strongcoupling between the spin and orbital degrees of freedom.One form of such coupling is of the Kugel-Khomskii type[39, 54, 55], in which strain-induced lattice anisotropyresults in both ordering between d xz / d yz orbitals andΦ = 0, and the shape of d xz / d yz orbitals locks the AF(FM) Fe-Fe direction with the elongated (contracted)crystal axis due to bond-dependent exchange couplings.Another essential ingredient for our observation is the presence of a strong local spin anisotropy [easy-axis alonggrey slabs in Fig. 5(b)], present in the magnetically or-dered ground state and persisting well above T S , N whenthe clusters are fluctuating independently between twodynamic configurations with Φ > <
0. Wenote while the magnetic order in the PC-IBS is collinearwith spins along the AF Fe-Fe direction, the c -axis po-larized spin waves are lowest in energy [56, 57]. Thisindicates a common hierarchy of spin anisotropy energies∆ AF ≤ ∆ c < ∆ FM in PC-IBS and KFAT. These twoeffects in combination give rise to an spin-orbital entan-gled state, resulting in a unusually large coupling betweenspin and the lattice, allowing the spin-nematic state todirectly manifest through uniform magnetic susceptibil-ity measurements under strain.In conclusion, we have demonstrated that the 2 × ∗ Electronic address: [email protected][1] Eduardo Fradkin, Steven A. Kivelson, Michael J. Lawler,James P. Eisenstein, and Andrew P. Mackenzie, Annu.Rev. Condens. Matter Phys. , 153-178 (2010).[2] Yoichi Ando, Kouji Segawa, Seiki Komiya, and A. N.Lavrov, Phys. Rev. Lett. , 137005 (2002).[3] Sourin Mukhopadhyay, Rahul Sharma, Chung Koo Kim,Stephen D. Edkins, Mohammad H. 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