Multiple charge density waves and superconductivity nucleation at antiphase domain walls in the nematic pnictide Ba_{1-x}Sr_{x}Ni_{2}As_{2}
Sangjun Lee, John Collini, Stella X.-L. Sun, Matteo Mitrano, Xuefei Guo, Chris Eckberg, Johnpierre Paglione, Eduardo Fradkin, Peter Abbamonte
MMultiple charge density waves and superconductivity nucleation at antiphase domainwalls in the nematic pnictide Ba − x Sr x Ni As Sangjun Lee, John Collini, Stella X.-L. Sun, Matteo Mitrano, Xuefei Guo, ChrisEckberg, Johnpierre Paglione,
2, 3
Eduardo Fradkin,
1, 4 and Peter Abbamonte ∗ Department of Physics and Materials Research Laboratory,University of Illinois, Urbana, Illinois 61801, USA Maryland Quantum Materials Center, Department of Physics,University of Maryland, College Park, Maryland 20742, USA Canadian Institute for Advanced Research, Toronto, ON M5G 1Z8, Canada Institute of Condensed Matter Theory, University of Illinois, Urbana, Illinois 61801, USA
How superconductivity interacts with charge or nematic order is one of the great unresolvedissues at the center of research in quantum materials. Ba − x Sr x Ni As (BSNA) is a charge orderedpnictide superconductor recently shown to exhibit a six-fold enhancement of superconductivity dueto nematic fluctuations near a quantum phase transition (at x c = 0 .
7) [1]. The superconductivity is,however, anomalous, with the resistive transition for 0 . < x < x c occurring at a higher temperaturethan the specific heat anomaly. Using x-ray scattering, we discovered a new charge density wave(CDW) in BSNA in this composition range. The CDW is commensurate with a period of two latticeparameters, and is distinct from the two CDWs previously reported in this material [1, 2]. We arguethat the anomalous transport behavior arises from heterogeneous superconductivity nucleating atantiphase domain walls in this CDW. We also present new data on the incommensurate CDW,previously identified as being unidirectional [2], showing that is a rotationally symmetric, “4 Q ”state with C symmetry. Our study establishes BSNA as a rare material containing three distinctCDWs, and an exciting testbed for studying coupling between CDW, nematic, and SC orders. One of the perennial questions in quantum materials isto what extent superconductivity (SC) may be enhancedby coupling to other Fermi surface instabilities, such ascharge density wave (CDW), spin density wave (SDW)or nematic orders. This issue has been investigated mostwidely in cuprate and iron-based superconductors [3–6].While spin fluctuations are considered a primary ingre-dient in Cooper pairing in both materials, the CDW incuprates [7–9] and nematic fluctuations in iron-based ma-terials [10, 11] are pervasive and suggest a close interrela-tion between SC and these other electronic instabilities.The importance of such orderings for SC is still unclear sothere is a great need for new materials that can providefresh insight.Ba − x Sr x Ni As is a nickel-based pnictide supercon-ductor that is an ideal system to investigate the rela-tionship between charge instabilities and SC. The parentcompound, BaNi As , is a structural homologue of theprototypical iron-based superconductor BaFe As withall Fe atoms replaced by Ni, that undergoes a first-orderstructural phase transition at T s = 136 K from tetragonalto triclinic symmetry [12, 13]. We recently showed thatBaNi As is an intriguing material in which two distinctCDW instabilities arise sequentially on lowering temper-ature [2]. In the tetragonal phase, an incommensurateCDW (IC-CDW), previously identified as unidirectional,forms along the in-plane H axis with wave vector q = 0 . T s , the IC-CDW is replaced by the second CDW,which we denote C-CDW-1, that is commensurate with q = 1 /
3. This CDW arises via a lockin transition from anincipient, slightly incommensurate CDW with q ∼ . As is substituted with Co or Sr, thetriclinic phase that hosts C-CDW-1 is suppressed anda SC dome emerges [1, 2, 13], suggesting C-CDW-1plays a similar role to antiferromagnetism in chemically-substituted BaFe As . Further, full suppression of theCDW by Sr substitution, which occurs at the criticalcomposition x c = 0 .
7, results in a quantum phase tran-sition (QPT) characterized by nematic fluctuations thatdrive a sixfold enhancement of the superconducting T c [1, 14]. Substituted BaNi As is therefore an excitingnew system in which the interaction between SC, CDW,and nematic order can be studied in detail.Recent studies revealed several peculiar properties ofBa − x Sr x Ni As that are not fully understood [1, 2].First, for the composition range of 0 . < x < x c , theSC transition in transport measurements is broadenedand occurs at a higher temperature than the specific heatanomaly [1]. Why the resistive and thermodynamic tran-sitions should be split in this manner is not known. Sec-ond, the IC-CDW was identified in Ref. [1] as a lattice-driven instability, without electronic character, since itshows no precursor response in the nematic susceptibility.However, it was also observed that the elastoresistancebecomes hysteretic whenever the IC-CDW is present,suggesting the two phases are coupled. These two obser-vations are seemingly contradictory, since a purely struc-tural phase transition should not influence the nematicresponse in this way. The nature of the interaction be-tween the IC-CDW and the nematic phase remains un-clear.Here, using x-ray scattering, we present the discoveryof a third CDW in Ba − x Sr x Ni As , which we denote C-CDW-2, in the composition range 0 . < x < x c , where a r X i v : . [ c ond - m a t . s t r- e l ] F e b x c = 0 .
7. This CDW is commensurate with a periodof two lattice parameters (period-2). A CDW with ageneric wave vector has a complex order parameter [15].The period-2 CDW state is special in that its orderingwave vector lies at the edge of the Brillouin zone (BZ)and, hence, its order parameter is real. Thus, the topo-logical defects of a period-2 CDW are domain walls wherethe order parameter changes sign and therefore it mustvanish at the location of the domain wall. We will showbelow that competing (weaker) SC state can be nucle-ated on these domain walls. The implication is that thephase at 0 . < x < x c may be a heterogeneous state inwhich SC resides at the topological defects of the CDWorder.Further, we report a wider x-ray momentum surveyshowing that the IC-CDW phase is in fact bidirectionalwith 90 ◦ rotational symmetry. We observed additionalsatellite reflections demonstrating a coherent “4 Q ” statewith C symmetry that does not break the rotationalsymmetry of the underlying tetragonal lattice. This ob-servation explains the absence of a precursor nematic re-sponse in elastoresistivity measurements [1], and suggeststhis CDW could be electronic and strongly coupled to thenematic order [1].Single crystals of Ba − x Sr x Ni As with x = 0, 0.27,0.42, 0.47, 0.65, and 0.73 were measured in this study.Three-dimensional x-ray surveys of momentum spacewere obtained using a Mo K α (17.4 keV) microspot x-raysource and a Mar345 image plate detector by sweepingcrystals through an angular range of 20 ◦ . All tempera-ture evolution measurements in this study are conductedwhile warming the samples starting from low tempera-ture (see Supplemental Material for experimental details[16]).The parent compound, BaNi As , has tetragonal I /mmm symmetry at room temperature and undergoesa phase transition to triclinic P ¯1 at T s = 136 K [2, 12].This transition is observed in x-ray measurements as asplitting of the tetragonal reflections into four peaks dueto the formation of triclinic twin domains [2, 12, 16].Measurements in the triclinic phase were indexed by se-lecting a single subset of these four reflections, empha-sizing scattering from a single domain. Figures 1(a) and(b) show the conventional unit cells and the BZ bound-aries in the tetragonal and triclinic phases. Note that the H - K planes of these two phases are not parallel, but aretilted by about 20 ◦ from one another. Throughout thisletter, we use ( H, K, L ) tet and ( H, K, L ) tri to denote mo-mentum space locations indexed with the tetragonal andtriclinic unit cells, respectively (see Supplemental Mate-rial for more details on the structure and indexing [16]).Here, we find that when BaNi As is substituted withSr, the triclinic transition as seen with x-rays initially in-creases in temperature, reaching a maximum T s = 141 Kat x = 0 .
27. Further substitution decreases T s until, at x = 0 .
73, the structure transition is no longer observed,indicating a quantum phase transition at x c ∼ .
7, con-
FIG. 1. Tetragonal and triclinic structure ofBa − x Sr x Ni As . (a) The crystal structure and the unit cellsof tetragonal (black dashed lines) and triclinic phases (reddashed lines). (b) The BZ boundaries of tetragonal (blacklines) and triclinic (red lines) phases. The planes colored inblack and red represent H - K planes in the tetragonal and tri-clinic phases, respectively. (c), (d) The H - K planes of tetrag-onal and triclinic phases, respectively, showing the location ofthree CDWs in momentum space in each phase. sistent with conclusions from transport experiments [1].In Fig. 5 we compare our results for the triclinic transi-tion to the transport phase diagram of Ba − x Sr x Ni As .The triclinic transition determined by x-ray scatteringwhile warming (purple line) is slightly higher in temper-ature than the cooling transition determined by transport(black dashed line) [1], demonstrating the first order na-ture of the transition. Nevertheless, we see no evidencefor coexistence of tetragonal and triclinic phases, unlikeCo-substituted compounds, Ba(Ni − x Co x ) As , in whichan extended region of coexistence occurs [2].The primary result of this study is the discovery inthe composition range 0 . < x < x c of a third CDW,which we denote C-CDW-2. This CDW is distinct fromthe IC-CDW and C-CDW-1 phases reported previously[1, 2]. Figure 2 shows x-ray measurements of the samplewith x = 0 .
42, in which all three CDWs are observedsequentially upon cooling. The IC-CDW forms at thehighest temperature, at T IC = 132 . ± . q = (0 , . , tet , which is the same reported in the par-ent compound [2]. No corresponding reflection was ob-served at (0 . , , tet in this zone, which previously ledus to the conclusion that the IC-CDW is unidirectional[2]. Below, we present data revising this conclusion.Upon further cooling, the IC-CDW is replaced by C-CDW-1 at T C1 = 117 . ± . q = (0 , / , tri , which is commensurate with a period FIG. 2. Three distinct CDWs in Ba . Sr . Ni As . (a), (b), (c) H - K maps of IC-CDW, C-CDW-1, and C-CDW-2,respectively, at a selection of temperatures. (d), (e), (f) Line momentum scans of the CDW reflections shown in (a), (b), and(c), respectively, along the corresponding modulation direction. of three lattice parameters. As in the case of IC-CDW,no (1 / , , tri peak was observed. This CDW is alsoobserved in Co-substituted compounds, where it exhibitsa lock-in effect [2]. However, we observe no lock-in effectin Sr-substituted materials.A third, previously unobserved CDW, which we callC-CDW-2, appears at lower temperature, T C2 = 95 ± q = (0 , / , tri is commensurate with a period of two lattice param-eters. Again, no corresponding peak was observed at q = (1 / , , tri . At T C2 , the intensity of C-CDW-1 dras-tically drops, and the intensity C-CDW-2 continuouslyincreases down to our base temperature. We summarizethe momentum space locations of all three CDWs in Fig.1(b)-(d). We emphasize that these three CDWs reside invery different locations in momentum space; the differ-ence between the three CDW wave vectors is not merelydue to a change of indexing coordinates.We now discuss the rotational symmetry of the threeCDWs. The satellite reflections in all three phases ap-pear in only one direction in a given BZ, which wouldnormally be interpreted as evidence that all three CDWsare unidirectional. This is unsurprising for C-CDW-1and C-CDW-2 since they reside in the triclinic phase inwhich C (rotational) symmetry is broken by the underly-ing lattice. However, it is puzzling that IC-CDW shouldalso be unidirectional, since it resides in the tetragonalphase where C symmetry is preserved. Our claim thatIC-CDW is unidirectional [2] led the authors of Ref. [1] to conclude that it is a purely structural phase transition,uncoupled to the valence electron system, since they ob-served no precursor response in the nematic susceptibil-ity expected of an electronic phase with broken rotationalsymmetry.We reexamined this issue by performing a wide, three-dimensional x-ray survey of a 20 ◦ wedge of momentumspace, analyzing where CDW satellites reside in multi-ple BZs. We found that the C-CDW-1 and C-CDW-2show the same unidirectionality in all zones observed (seeSupplemental Material [16]), affirming that these triclinicCDWs are unidirectional as claimed [2]. However, theIC-CDW exhibits the more complicated pattern shownin Fig. 3. While any given BZ has only two satellites,their orientation is different in different BZs. The over-all pattern is invariant under 90 ◦ rotations around theorigin, and therefore exhibits the same C symmetry asthe underlying tetragonal lattice. We conclude that theIC-CDW is not unidirectional, but is a coherent “4 Q ”state in which two reflections in each BZ are extinguishedby some additional symmetry. This result implies thatIC-CDW may be electronic in origin after all, and maycouple strongly to the nematic order.Figure 4 summarizes the behavior of the three CDWsover the full range of Sr composition investigated. TheIC-CDW is present from x = 0 to 0 .
47 and is strongestwithin a ∼
20 K range above the tetragonal-triclinic phaseboundary. The C-CDW-2 phase is first observed at x =0 .
42, where it replaces C-CDW-1 in the triclinic phase,
FIG. 3. Bidirectionality of the IC-CDW phase. (a) H - K map around ( − , − , tet showing the IC-CDW satellites at( ± . , , tet . The satellites are absent at (0 , ± . , tet .(b) H - K map around (1 , , tet showing the satellites at(0 , ± . , tet . The satellites are absent at ( ± . , , tet .(c) Illustration of the symmetry pattern of IC-CDW in H - K plane at odd-numbered L . Black dots represent Bragg peaklocations, and red dots represent IC-CDW peak locations.FIG. 4. Warming temperature dependence of the inte-grated intensities of the IC-CDW (blue circles), C-CDW-1(purple squares), and C-CDW-2 (green diamonds) reflectionsin Ba − x Sr x Ni As . The Sr composition, x , is shown in eachpanel. The shaded region represents the temperature rangeof the triclinic phase. Each curve is scaled to the maximumCDW intensity at low temperature. The solid lines are guidesto the eye. FIG. 5. Phase diagram of Ba − x Sr x Ni As comparing theIC-CDW (blue circles), C-CDW-1 (purple squares), C-CDW-2 (green diamonds) to the transport measurements of Ref.[1]. The hollow purple square at x = 0 represents the on-set of the incipient CDW that undergoes a lock-in transitionand evolves to C-CDW-1. The triclinic phase boundary onwarming coincides with the boundary between C-CDW-1 andC-CDW-2 and the tetragonal phase. The cooling transition, T s , cooling (dashed black line), determined by transport mea-surements shows the hysteresis region (shaded). The overlaidcolor scale represents the nematic susceptibility, χ nem , and theonset temperature of the strain-hysteresis of the nematic re-sponse, T nem,hyst , (black hollow squares) are plotted together.The superconducting transition temperatures, T c (red hollowsquares), determined by transport measurements are shownfor comparison. χ nem , T nem,hyst , T s , cooling , and T c are fromRef. [1]. and persists up to x = 0 .
65 [17]. At x = 0 .
73, no CDWtransition is observed. No two CDWs are ever optimizedat the same composition or temperature, indicating thatthe three phases likely compete with one another in theLandau sense.A summary phase diagram of Ba − x Sr x Ni As is pre-sented in Fig. 5. The anomalous superconducting phaseat 0 . < x < x c , in which transport and thermodynamicsignatures occur at different temperatures [1], resides en-tirely within the C-CDW-2 phase. By contrast, at x > x c when no CDW is present, the superconducting signaturesare normal. This implies that the peculiar superconduct-ing state reported in Ref. [1] is connected to the presenceof C-CDW-2.A period-2 CDW competing with superconductivityis prone to developing a heterogeneous mixed state (seeSupplemental Material Sec. V [16]). A CDW is highlysensitive to disorder [18], trace amounts of which canlead to the formation of domain walls with a π phaseshift, across which the order parameter changes its signand, thus, vanishes. Since the CDW is suppressed at thedomain walls, these locations are favorable for compet-ing superconductivity to emerge, resulting in a hetero-geneous or filamentary superconducting state. In turn,Josephson coupling between these superconducting re-gions then leads to a globally coherent state at lowertemperatures. Thus, the onset of the resistive transitionis where the CDW domain walls become superconduct-ing, with macroscopic superconductivity being achievedat lower temperatures, driven by the Josephson couplingbetween the domain walls. A similar mechanism waspreviously proposed in TiSe [19], which also exhibits aperiod-2 CDW (see also Ref. [20]), as well as in cupratesuperconductors in high magnetic fields [21]. This phe-nomenon bears a close analogy with the CDW order seenin the SC halos of vortices of high T c superconductors[22, 23], and iron-based superconductors with structuraltwin domains [24] or antiferromagnetic domains [25]. Wetherefore identify this unusual SC state as a heteroge-neous state in which SC is locally formed at domain wallsof C-CDW-2 that are consequence of any disorder presentin the system.Another feature of the phase diagram (Fig. 5) is thatthere is a close correspondence between the presence ofthe IC-CDW and irreversible behavior in the nematicsusceptibility measured with elastoresistance techniques[1]. When the IC-CDW is present, the strain responseis hysteretic. When absent, the behavior is reversible.As discussed above, it is possible that the IC-CDW cou-ples strongly to the nematic order parameter, and there-fore pins the nematic fluctuations. The hysteretic re-sponse then can be understood as the training of thesestatic nematic domains by the applied elastic strain field.When the IC-CDW is absent, the nematic domains be-come dynamic, the response becomes reversible, and thefluctuations enhance superconductivity near x c = 0 .
7, asclaimed in Ref. [1].In summary, we showed that Ba − x Sr x Ni As exhibitsthree distinct charge density waves, IC-CDW, C-CDW-1and C-CDW-2. The order parameter of C-CDW-2, whichis period-2, is always zero at antiphase domain walls,allowing the competing superconductivity to emerge lo-cally. This results in a heterogeneous superconductingstate for 0 . ≤ x ≤ x c = 0 .
7. We also showed that theIC-CDW can strongly couple to the nematic order pa-rameter, and promote static nematic domains by pinningthe fluctuations. Our study establishes BSNA as a rareexample of a material containing three distinct CDWs,and an exciting testbed for studying coupling betweenCDW, nematic, and SC orders.We thank P. Ghaemi for an early collaboration, andS. A. Kivelson for discussions. X-ray experiments weresupported by the U.S. Department of Energy, Office ofBasic Energy Sciences grant no. DE-FG02-06ER46285(PA). Theory work was supported in part by the U.S.National Science Foundation grant DMR 1725401 (EF).Materials synthesis was supported by the National Sci-ence Foundation Grant no. DMR1905891 (JP). P. A. and J. P. acknowledge the Gordon and Betty Moore Founda-tion’s EPiQS Initiative through grant nos. GBMF9452and GBMF9071, respectively. ∗ [email protected][1] C. Eckberg, D. J. Campbell, T. Metz, J. Collini,H. Hodovanets, T. Drye, P. Zavalij, M. H. Christensen,R. M. Fernandes, S. Lee, P. Abbamonte, J. W. Lynn,and J. Paglione, Nature Physics , 346 (2020).[2] S. Lee, G. de la Pe˜na, S. X.-L. Sun, M. Mitrano, Y. Fang,H. Jang, J.-S. Lee, C. Eckberg, D. Campbell, J. Collini,J. Paglione, F. M. F. de Groot, and P. Abbamonte, Phys.Rev. Lett. , 147601 (2019).[3] E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Rev.Mod. Phys. , 457 (2015).[4] J. M. Tranquada, Physica B: Condensed Matter , 136(2015).[5] J. Paglione and R. L. Greene, Nat. Phys. , 645 (2010).[6] P. Dai, Rev. Mod. Phys. , 855 (2015).[7] P. Abbamonte, A. Rusydi, S. Smadici, G. D. Gu, G. A.Sawatzky, and D. L. Feng, Nature Physics , 155 (2005).[8] G. Ghiringhelli, M. Le Tacon, M. Minola, S. Blanco-Canosa, C. Mazzoli, N. B. Brookes, G. M. De Luca,A. Frano, D. G. Hawthorn, F. He, T. Loew, M. M. Sala,D. C. Peets, M. Salluzzo, E. Schierle, R. Sutarto, G. A.Sawatzky, E. Weschke, B. Keimer, and L. Braicovich,Science , 821 (2012).[9] R. Comin and A. Damascelli, Annual Review of Con-densed Matter Physics , 369 (2016).[10] J.-H. Chu, H.-H. Kuo, J. G. Analytis, and I. R. Fisher,Science , 710 (2012).[11] H.-H. Kuo, J.-H. Chu, J. C. Palmstrom, S. A. Kivelson,and I. R. Fisher, Science , 958 (2016).[12] A. S. Sefat, M. A. McGuire, R. Jin, B. C. Sales, D. Man-drus, F. Ronning, E. D. Bauer, and Y. Mozharivskyj,Phys. Rev. B , 094508 (2009).[13] C. Eckberg, L. Wang, H. Hodovanets, H. Kim, D. J.Campbell, P. Zavalij, P. Piccoli, and J. Paglione, Phys.Rev. B , 224505 (2018).[14] S. Lederer, E. Berg, and E.-A. Kim, Phys. Rev. Res. ,023122 (2020).[15] W. L. McMillan, Phys. Rev. B , 1187 (1975).[16] See Supplemental Material [URL will be inserted by pub-lisher] for the experimental details, the lattice parametersof samples and structural transitions, the symmetry pat-terns of CDWs, and for details of the theory.[17] J. Collini, et al., to be published.[18] Y. Imry and S.-K. Ma, Phys. Rev. Lett. , 1399 (1975).[19] Y. I. Joe, X. M. Chen, P. Ghaemi, K. D. Finkelstein,G. A. de la Pe˜na, Y. Gan, J. C. T. Lee, S. Yuan, J. Geck,G. J. MacDougall, T. C. Chiang, S. L. Cooper, E. Frad-kin, and P. Abbamonte, Nature Physics , 421 (2014).[20] C. Chen, L. Su, A. H. Castro Neto, and V. M. Pereira,Phys. Rev. B , 121108 (2019).[21] Y. Yu and S. A. Kivelson, Phys. Rev. B , 144513(2019).[22] J. E. Hoffman, E. W. Hudson, K. M. Lang, V. Madhavan,H. Eisaki, S. Uchida, and J. C. Davis, Science , 466(2002).[23] S. A. Kivelson, D.-H. Lee, E. Fradkin, andV. Oganesyan, Phys. Rev. B , 144516 (2002). [24] B. Kalisky, J. R. Kirtley, J. G. Analytis, J.-H. Chu,A. Vailionis, I. R. Fisher, and K. A. Moler, Phys. Rev.B , 184513 (2010).[25] H. Xiao, T. Hu, A. P. Dioguardi, N. apRoberts Warren, A. C. Shockley, J. Crocker, D. M. Nisson,Z. Viskadourakis, X. Tee, I. Radulov, C. C. Almasan,N. J. Curro, and C. Panagopoulos, Phys. Rev. B85