Spin-canting driven Weyl physics in EuCd 2 As 2
K.M. Taddei, L.Y. Lin, L.D. Sanjeewa, J. Xing, C. dela Cruz, A.S. Sefat, D. Parker
SSpin-canting driven Weyl physics in EuCd As K.M. Taddei, ∗ L.Y. Lin, ∗ L.D. Sanjeewa, J. Xing, C. dela Cruz, A.S. Sefat, and D. Parker Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 † Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 (Dated: December 7, 2020)Though rare, magnetic Weyl semimetals stand as the best platform to study elusive Weyl physicsas they can host the minimal allowable number of Weyl points. Here we present neutron diffractionand density functional theory work elucidating the magnetic structure realized in the magnetic Weylsemimetal EuCd As . Our work shows an unanticipated magnetic structure (magnetic space group C (cid:48) /m (cid:48) ) with an in-plane [210] moment direction and a slight out-of-plane canting. This cantedstructure indicates that subtle tuning (rather than a phase transition) may be able to stabilize thesought c-polarized state. Our density functional theory work shows that though Weyl physics shouldexist for a purely in-plane [210] structure, even a slight canting drastically alters the relevant bandsleading to well defined Weyl points. Furthermore, we find that relative to the c -polarized state the[210] order with a small canting brings the Weyl points closer to the Fermi level and thus may leadto clearer signatures of the Weyl physics. PACS numbers: 74.25.Dw, 74.62.Dh, 74.70.Xa, 61.05.fm
I. INTRODUCTION
Despite implying (along with long-range entangledstates) the presence of classifications beyond Landau the-ory, symmetry protected topological states (SPT) are de-lightfully enumerable in suitable materials and toy mod-els via familiar and intuitive symmetry considerations.For instance, starting from a Dirac semimetal (DSM)with a single Dirac point (DP), one may break inversion-symmetry (IS), time-reversal symmetry (TRS), both oreven combinations of either with crystal symmetries tobreak the DP into varying numbers of Weyl points (WP).Alternatively, one could instead shift the DP off a crystalsymmetry operation to render the DSM instead a topo-logical insulator [1–3].Enticingly, each of these situations leads to differentmaterials’ properties and, for some of the cases, differentemergent physics heralding in a ‘zoo’ of exotic topologi-cal quasi-particles [1, 4]. Of these, the chiral Weyl quasi-particle is especially interesting both as a unique real-ization of a solution to relativistic wave-equations andas a tool for enabling new or enhancing existing tech-nologies such as photovaltics, valleytronics and quantumcomputers [5–10]. Yet the numerous routes to and re-alizations of Weyl semimetals (WSM) are not all equal,with TRS breaking providing the only route to a minimalnumber of WP and thus providing their clearest signa-tures. However, magnetic WSM have proven somewhatrare with several of the candidate materials encumberedby less than ideal symmetries leading to numerous WP[11–15]Recently, an analysis of magnetic space groups (MSG)led to the prediction that EuCd As (centrosymmetric ∗ These authors contributed equally † corresponding author [email protected] space group P m
1) when coupled with an out-of-planeA-type antiferromagnetism (AFM) should exhibit a sin-gle pair of WP [16, 17]. However, experimental work hasgenerated conflicting stories, with early reports of the de-sired magnetic structure giving way to suggestions of aless ideal in-plane AFM arrangement or even DSM ratherthan WSM physics [18–20]. This conflict is in-part dueto the highly neutron absorbing Eu and Cd which pre-vent the use of neutron scattering to provide a definitivemagnetic structure solution leaving less comprehensivetechniques as the only available probes [18, 19]. As a fur-ther complication, several recent papers have shown thatdifferent synthesis procedures can lead to different mag-netic orders with new seemingly purely ferromagnetic(FM) EuCd As samples appearing [21, 22]. Nonethe-less, throughout its study EuCd As has shown promis-ing hints at topological physics such as WP in ARPESmeasurements, the quantum anomalous Hall Effect, ef-fective TRS breaking in the paramagnetic phase, anda large negative transverse magnetoresistance indicat-ing the need for more work settling the actual magneticstructure[17, 22–24].In this communication, using isotopic Eu and
Cdwe report the zero field and 2 T magnetic structures ofFM EuCd As determined by neutron diffraction and an-alyze the topology of the resulting band structures. Ourdata reveal a k = (0 , ,
0) type FM order for both thezero field and 2 T structures. Qualitative analysis of themagnetic intensities suggests that for the zero field struc-ture the Eu moments point along the in-plane [210] di-rection with a slight out-of-plane canting (MSG C (cid:48) /m (cid:48) )while the 2 T structure exhibits the expected c -polarizedstate. Using this canted magnetic structure to performDFT calculations we find that the band structure is ex-traordinarily sensitive to the moment canting with evensmall canting angles enhancing the Weyl physics. Fora 10 ◦ canting we find a well-defined WP with a closeproximity to the Fermi level. As the canting is further a r X i v : . [ c ond - m a t . s t r- e l ] D ec - 0 . 5 0 . 0- 101 ( K , 2 K , 0 ) ( r . l . u . ) (H,0,0) (r.l.u.) I n t . ( a . u . ) - 0 . 5 0 . 0- 101 ( K , 2 K 0 ) ( r . l . u . ) (H,0,0) (r.l.u.)
I n t . ( a . u . ) - 0 . 5 0 . 0- 101 ( K , 2 K , 0 ) ( r . l . u . ) (H,0,0) (r.l.u.)
I n t . ( a . u . ) - 2 - 1 0- 202 ( K , 2 K , 0 ) ( r . l . u . ) (H,0,0) (r.l.u.)
I n t . ( a . u . ) ( b ) ( c ) ( d )( a )
FIG. 1. Neutron diffraction patterns of the HK0 plane for an isotopic EuCd As single crystal over a broad range of H andK for data collected at (a) 50 K, 0 T. Zoomed in regions of the HK0 plane focusing on the { } series of reflections for datacollected at (b) 50 K,0 T; (c) 5 K, 0 T; and (d) 5 K, 2 T. increased, the WP is pushed higher in energy, indicatingthe best signature of the Weyl physics might exist notin the c -polarized structure but for an in-plane modelwith slight canting. These results suggest that the Weylphysics might be optimized in EuCd As by tuning thecanting angle - a gambit allowed by the symmetry of theMSG.Single crystals of EuCd As were grown following theprocedure reported in Ref. 22. To mitigate the neutronabsorption of naturally occurring Eu and Cd (absorp-tion cross-sections of 2520 b and 4530 b respectively,)isotopic Cd and
Eu (of purity ∼
99% ) were used inthe reactions (with neutron absorption cross-sections of0.075 b and 312 b respectively) [25]. Doing so improvesthe transmitted signal for a 2 mm diameter crystal, and1.48 ˚A neutrons from 0.0003% to 66% . To further miti-gate neutron absorption (and the need for the associatedcorrections) a nominally isotropically shaped crystal ofmass < ∼ < . diffractometer of OakRidge National Laboratory’s High Flux Isotope reactorwith incident wavelength 1.48 ˚A [26, 27]. Data were col-lected by rotating the sample in the crystallographic ab plane allowing for peaks in the (HK0) plane to be ob-served. Temperature and field dependent data sets werecollected using a cryomagnet. To monitor for absorp-tion artifacts, each data set was collected over > ◦ of rotation. Symmetry analysis was carried out usingthe Bilbao Crystallographic Server, SARAh and ISODIS-TORT [28–33]. Simulated diffraction intensities were calculated using the FullProf software suite and crys-tal structure visualization was performed using VESTA[34, 35]. First-principles calculations were performed us-ing density functional theory (DFT) with spin-orbit cou-pling as implemented in the Vienna Ab initio Simula-tion Package (VASP) [36, 37]. Projector augmented wavepseudo-potentials were applied with the Perdew-Burke-Ernzerhof exchange correlation functional and an energycutoff of 318 eV [21, 38–40]. The Brillouin zone wassampled with a Γ-centered 11 × × k point mesh. Toaccount for the strong localization of the Eu 4 f orbitals,a Hubbard U of 5.0 eV was applied [17, 40]. In all cal-culations, convergence criteria 10 − eV and 0.001 eV/˚Awere used for the energy and atomic forces, respectively.To start our analysis we consider data collected above T c (i.e. >
30 K) to identify the nuclear peaks and checkfor any absorption effects. In Fig. 1(a) we show the ob-tained (HK0) slice collected at 50 K over a broad rangeof H and K. We note that the small size and high qualityof our sample led to resolution limited peaks which havevery slight extent in reciprocal space complicating theirvisualization. Nonetheless peaks are seen which are con-sistent with the P m Q constant rings (e.g. the { } set of reflec-tions) we see no significant modulation over the plotted180 ◦ of rotation indicating no significant anisotropic ab-sorption effect [41].Shown in Fig. 1(b) is a region centering on the { } series of reflections which are accidentally forbidden inthe nuclear structure. As the sample is cooled below T c we observe an increase in intensity on this series as P m ' M = ( M z ) C m , M = ( M y ,0) C m ' , M = ( M y , M y , M z ) P M = ( M x , M y , M z ) FIG. 2. Visuallizations of the magnetic structures allowed by k = (0 , ,
0) and P m ◦ canting angle is displayed. In all panels the originalnuclear unit cell is shown. shown in the data collected at 5 K (Fig. 1(c)), no addi-tional new reflections are seen at fractional coordinates(including along the L direction which the out-of-planedetector coverage allows us to check to L= 0 ± . T c is indicative of a magnetic transition with orderingvector k = (0 , , a Wyckoff position of the Eu sitein the P m k = (0 , ,
0) order is purelyFM. This is consistent with the descriptions of the mag-netic structure arising from magnetic susceptibility mea-surements but inconsistent with the previous results fromresonant elastic x-ray scattering experiments which sug-gested AFM order with k = (0 , , ) [19, 21, 22]. How-ever, as discussed in Refs. 21 and 22 this disagreementis likely due to specifics of the sample growth which canproduce either AFM or FM samples.To identify the magnetic structure we performed sym-metry analysis and considered all allowed subgroups of P m k = (0 , ,
0) which produced unique magneticstructures (Fig. 2.) Of the four allowed MSG P m (cid:48) c -polarized state, C /m is a purely b -polarizedstate and the remaining two allow mixing between in-plane and out-of-plane components with C (cid:48) /m (cid:48) lockingthe in-plane moment component to the crystallographic[210] direction and the final, and lowest symmetry, MSG P { } series. In Fig. 3(a) P 3 m ' 1 C 2 ' / m ' ( c a n t e d ) C 2 / m P 1 C 2 ' / m '
Calculated Int. (a.u.)
I n d e x ( H K L )
Integrated Int. (a.u.)
I n d e x ( H K L ) ( a ) ( b )
FIG. 3. (a) Integrated intensities of the { } series ofreflections for data collected at 5K,0T (green) and 5K,2T(black). (b) simulated intensities for the various possible mag-netic structures. Calculations for the C (cid:48) /m (cid:48) structure wereperformed with zero canting (blue triangles) and with 30%canting (green upside down triangle). we plot the integrated intensity of the { } reflectionsand in Fig. 2(b) we do the same but for simulated scat-tering from the various possible magnetic structures witha magnetic moment of 6 . µ B per Eu [21, 22]. In panel(a) we see that there is an intensity modulation wherethe (100) and (100) reflections are weaker than the (010)and (110) reflections. This observation alone rules outthe purely c -polarized state which has equivalent scat-tering intensities for the { } series (Fig. 3(b).)Next, we compare the relative peak intensities of the { } series to those predicted by the various magneticstructures. For the b -polarized C /m structure we findthe six reflections bifurcate into a doublet with four re-flections ((010) , (110) , (110) , (010)) having a smaller in-tensity than the remaining two ((100) , (100)) (Fig. 3(b)).While our data do not contain all six reflections, we cansee from the four we do observe that this pattern doesnot hold - our peaks appear split in the opposite man-ner with four reflections having larger intensity than the(100) and (100) reflections. This observation is consis-tent with the C (cid:48) /m (cid:48) MSG, whose [210] moment direc-tion weakens (100) and (100) peaks. The final MSG ( P C (cid:48) /m (cid:48) .Looking closer at the purely in-plane C (cid:48) /m (cid:48) model,we see a discrepancy between the simulated and observedintensities - while the model predicts a (010) / (100) inten-sity ratio of ∼
10, our measured intensities have a ratioof ∼
2. Considering the magnetic model, we can iden-tify two possible causes for this disagreement. The firstis that the moment is not purely in-plane but canted to-wards the c -axis. This would increase the intensity onthe (100) and (100) reflections by increasing the momentcomponent perpendicular to their scattering vector. Thesecond possibility, is that this increased intensity is dueto multiple magnetic domains (associated with the nu-clear C symmetry) with unequal populations.This second possibility merits deeper discussion. The ( a )( b ) ( c ) ( d ) ( e )( f ) ( g ) FIG. 4. The band structures of ferromagnetic EuCd As for magnetization along the (a) [001] and (b) [210] directions.Expanded view of the green dashed region shown in (a) and (b) for (c) the in-plane [210] structure with (d) 10 ◦ , (e) 20 ◦ and(f) 30 ◦ canting (with respect to the ab -plane) as well as (e) for the fully c -polarized [001] structure. symmetry allowed magnetic domains transform the in-plane moment constraint of M domain = (2 M y , M y ) to M domain = ( M x , − M x ) and M domain = ( M x , M x ).This permutes which set of reflections has the lower inten-sity from (100) , (100) to (110) , (110) and (010) , (010). Itis therefore, possible to achieve the observed (010) / (100)intensity ratio by carefully tuning the domain popula-tions with 63% of the sample in domain 1 and an equalsplit of the remaining sample into 18.5% of domain 2 andof domain 3. While such a situation is not impossible, itis unlikely that the sample would tune so carefully thedomain populations - domain 2 and 3 must balance nearperfectly to give the observed intensity split. Further-more, while this is able to produce the intensity ratio,in our modeling it was unable to adequately capture si-multaneously both the ratios between the magnetic re-flections and the magnetic and nuclear reflections. Wetherefore, favor the simpler explanation of a spin-canting.This discussion might bring up the question of why weapparently see only a single magnetic domain, or at leastone largely dominant domain. Indeed in Ref. 21 multi-ple domains were optically imaged in the magneticallyordered state of a FM sample. To explain this we pointout the previously reported sensitivity of these samples tofield cooling protocols. In Refs. 19 and 22, magnetizationmeasurements on EuCd As revealed a significant split-ting between field-cooled and zero-field cooled protocolswhere even a small cooling field of tens of Oe producedlarge 50% value changes in the base temperature signalostensibly due to domain alignment effects. In our exper-iment, a cryomagnet was used which is known to exhibitremnant fields on the order of dozens of Oe providing aplausible explanation for this observation.In Fig. 3(b) we show the simulated intensities of the C (cid:48) /m (cid:48) model with a moment canting of 30 ◦ (Fig. 2(c).)As discussed, this increases the intensities on the (100) and (100) reflections and gives a (010) / (100) intensity ra-tio close to that observed. We therefore continue with theassumption that EuCd As ’s magnetic structure includesa canting of the moments out of the ab plane. Admittedlywe are unable to unequivocally rule out some contribu-tion from secondary domains and so refrain reporting afirm canting angle, instead giving a range of θ = 5 − ◦ based on the simulated relative magnetic reflection inten-sities and the relative magnetic/nuclear Bragg intensities.It is important to emphasize that even in the absenceof this observation, since a c -direction component is al-lowed by the MSG then by symmetry arguments somefinite (albeit possibly small) moment along c should beexpected.If we turn to the data collected under a 2 T appliedfield, we see a different story (Fig. 1(d).) While we simi-larly see intensity on the { } series, we do not observeany statistically significant intensity modulation aroundthe series (this also functions as evidence that our sampleexhibits no significant anisotropic absorption (Fig. 3(a).)This indicates a metamagnetic transition which must beto the c -polarized P m (cid:48) C (cid:48) /m dueto suggestion of a phase transition in previous studies[21, 22].To predict the topological properties we performedDFT calculations of the electronic band structure for thepurely in-plane [210] direction, the [001] direction as wellas for several canting angles in between (Fig. 4.) Startingwith the [001] structure (Fig. 4(a) and (g)), our resultsare consistent with previous calculations showing a WP ∼ ab -plane, we find thatthis flat band quickly drops away and begins to resemblethe [001] band structure. For even a minor 10 ◦ canting(Fig. 4(d)) the dispersion in the immediate vicinity ofthe WP closely resembles the [001] structure. As themoment is further canted, we find this trend continues.Looking at the position of the WP relative to the Fermilevel, we find that the [210] structure is closest at ∼ ◦ the WP moves up in energy to ∼ ∼ As our results suggest otherwise. As shown, onelikely does not need to stabilize the fully c -polarized stateto realize a single set of WP - this appears to be achievedwith an in-plane [210] type order. Furthermore, whilethe band structure for the purely in-plane order is lessthan ideal, our neutron data suggest that the momentis canted by 5 − ◦ . This canting creates a situationsimilar to the [001] band structure while also moving theWP closer to the Fermi level. This seemingly indicatesthat the ideal configuration is closer to the realized zero-field ground state magnetic structure without any needfor significant tuning. In summary, we report that EuCd As orders in theFM C (cid:48) /m (cid:48) MSG with Eu moments pointing along thecrystallographic [210] direction with a slight out-of-planecanting. DFT analysis of this structure shows that thewell defined WPs previously reported for the purely c -polarized structure are achieved by the slight out-of-planecanting of the moment. Furthermore, we find that theWP lies closest to the Fermi level for the in-plane struc-ture and so the minimal canting possible which leads toa well-defined WP is the ideal magnetic structure for op-timizing the Weyl physics. Notably, the identified MSGallows canting by symmetry and so continuous tuning ofthe canting angle should be possible via small perturba-tions which tune the magnetic interactions. This resultincreases the intrigue of our recent report on Ba sub-stitution (i.e. Eu − x Ba x Cd As ) which suggested smallBa substitutions led to moment canting and enhancedWeyl physics as identified by transport measurements[22]. This indicates that routes such as Ba substitu-tion may be valid approaches to tune the Weyl physicsin EuCd As . Our results confirm the interest in thismaterial as a magnetic WSM with a single pair of WPand present a route forward for optimizing the desiredphysics. ACKNOWLEDGMENTS
The research is partly supported by the US DOE, BES,Materials Science and Engineering Division. The partof the research conducted at ORNL’s High Flux IsotopeReactor was sponsored by the Scientific User FacilitiesDivision, Office of Basic Energy Sciences (BES), US De-partment of Energy (DOE). [1] N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyland Dirac semimetals in three-dimensional solids, Rev.Mod. Phys. , 015001 (2018).[2] A. Burkov, Topological semimetals, Nature materials ,1145 (2016).[3] B. Yan and C. Felser, Topological materials: Weylsemimetals, Annual Review of Condensed Matter Physics , 337 (2017).[4] X.-G. Wen, Colloquium: Zoo of quantum-topologicalphases of matter, Rev. Mod. Phys. , 041004 (2017).[5] G. B. Osterhoudt, L. K. Diebel, M. J. Gray, X. Yang,J. Stanco, X. Huang, B. Shen, N. Ni, P. J. Moll, Y. Ran, et al. , Colossal mid-infrared bulk photovoltaic effect in atype-I Weyl semimetal, Nature materials , 471 (2019).[6] C.-K. Chan, P. A. Lee, K. S. Burch, J. H. 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