Magnetic structure of GdBiPt: A candidate antiferromagnetic topological insulator
R. A. Müller, N. R. Lee-Hone, L. Lapointe, D. H. Ryan, T. Pereg-Barnea, A. D. Bianchi, Y. Mozharivskyj, R. Flacau
MMagnetic structure of GdBiPt: A candidate antiferromagnetic topological insulator
R. A. M¨uller, N. R. Lee-Hone, L. Lapointe, D. H. Ryan, T. Pereg-Barnea, A. D. Bianchi, Y. Mozharivskyj, and R. Flacau D´epartement de physique, Universit´e de Montr´eal, Montr´eal, QC, Canada ∗ Department of Physics, McGill University, 3600 University St., Montr´eal, QC, Canada ∗ D´epartement de physique,Universit´e de Montr´eal, Montr´eal, QC, Canada ∗ Department of Chemistry and Chemical Biology McMaster University, Hamilton, ON, Canada Canadian Neutron Beam Centre, Chalk River Laboratories, ON, Canada (Dated: June 4, 2018)A topological insulator is a state of matter which does not break any symmetry and is characterizedby topological invariants, the integer expectation values of non-local operators. Antiferromagnetismon the other hand is a broken symmetry state in which the translation symmetry is reduced and timereversal symmetry is broken. Can these two phenomena coexist in the same material? A proposalby Mong et al. asserts that the answer is yes. Moreover, it is theoretically possible that the onset ofantiferromagnetism enables the non-trivial topology since it may create spin-orbit coupling effectswhich are absent in the non-magnetic phase. The current work examines a real system, half-HeuslerGdBiPt, as a candidate for topological antiferromagnetism. We find that the magnetic moments ofthe gadolinium atoms form ferromagnetic sheets which are stacked antiferromagnetically along thebody diagonal. This magnetic structure may induce spin orbit coupling on band electrons as theyhop perpendicular to the ferromagnetic sheets. PACS numbers: 75.25.-j, 75.50.Ee, 73.20.-r
The discovery of the quantum Hall effect (QHE) led toa new way of classifying matter - a phase transition doesnot have to be bound to spontaneous symmetry break-ing. Two years after von Klitzing’s discovery, Thouless,Kohmoto, Nightingale and den Nijs (TKNN) developedthe concept of topological invariants in their descriptionof the same effect. The TKNN number represents thetopology of the system in the form of an integral ofthe Bloch wave functions over the Brillouin zone. Thisnon-local operation results in an integer number whichalso corresponds to the number of dissipation-less edgemodes. The edge modes are guaranteed by the topol-ogy and are also protected by it. The TKNN number,however, explicitly breaks time reversal symmetry and istherefore zero in a time reversal invariant system.In 2005, Kane and Mele proposed a new state of mat-ter: A topological system which does not break timereversal (TR) symmetry. In their example, the topo-logical number is defined modulo 2 and the edge modesgives rise to the quantum spin Hall effect. Many excit-ing developments have been presented since, both exper-imentally and theoretically but many questions still re-main . One such question is whether the topological or-der can coexist with a broken symmetry state. Moreover,is it possible for a local order parameter which breaksone or more symmetries to give rise to topological order?The answer to this question, theoretically, is a tentative‘yes’ .In 2010 Mong et al. came forward with the conceptof an antiferromagnetic topological insulator (AFTI). Incontrast to an ordinary topological insulator, in an AFTIthe presence of magnetic order breaks TR symmetry Θas well as primitive-lattice translational symmetry T / ,yet their product S = Θ T / is preserved. This allows FIG. 1: The Gd atoms are shown in black (blue), the Bi asgray (gray), and the Pt as white (yellow). The spins on the Gdatoms are oriented in ferromagnetic planes which are stackedantiferromagnetically along the magnetic propagation vector(
12 12 12 ). the definition of a topological invariant which preservesthe S symmetry. In three dimensions the result is a topo-logical state with antiferromagnetic order. Depending onwhether the surface breaks the S symmetry or not, metal-lic surface states may arise within the band gap and ahalf-integer quantum Hall effect is expected . Moreover,in certain systems, the presence of the topological phase a r X i v : . [ c ond - m a t . s t r- e l ] J un is bound to the antiferromagnetic phase and so vanishesabove the N´eel temperature. This makes the AFTI par-ticularly interesting, as the topological state appears onlyafter the system undergoes a classical phase transition.Therefore, changing the temperature allows one to turnthe topological state on and off resulting in a quantumphase transition at T N . Mong et al. propose in their“model B” that the spin-orbit interaction may result fromthe N´eel order. Their model contains itinerant electronsand fixed spins. When the electrons hop between lat-tice sites they may do so through intermediate magneticsites. For certain paths of the conduction electrons themagnetic moments serve to create an Aharonov-Bohm-like flux which in turn acts as Rashba spin-orbit cou-pling, responsible for the topological order. The theoret-ical model is inspired by systems like GdBiPt which havebeen proposed to be topological based on first principlescalculations . In order for the S symmetry to be pre-served together with a significant spin orbit coupling themodel requires a specific magnetic structure. The mo-ments should be aligned ferromagnetically in layers whichare stacked antiferromagnetically. For the system to begapped, the hopping between layers should be larger thanthe hopping within the layer. For the half-Heusler struc-ture, this spin-orbit term is maximal if the moments arealigned ferromagnetically in the (111) plane and stackedantiferromagnetically along the [111] space diagonal asshown in Figure 1 . The Heusler and the derivative half-Heusler structures favour half-metallic band structureswith just one band crossing at the Fermi level, while leav-ing all the other bands well separated and have been alsoproposed as candidate materials for conventional topo-logical insulators . The purpose of the current workis to test whether the desired magnetic structure doesindeed occur in GdBiPt. We report on powder neutronscattering measurements of GdBiPt which shows a mag-netic structure very similar to the one proposed in , withthe magnetic moments arranged in ferromagnetic sheets,perpendicular to the [111] space diagonal. This makesGdBiPt a strong candidate for this new state of matter.GdBiPt crystallizes in the cubic half-Heusler crystalstructure with the space group F ¯43 m . Members of the RE BiPt family show many interesting properties suchas superconductivity, antiferromagnetic order and super-heavy-fermion behaviour. Band structure calculationsand ARPES experiments on Lu, Nd, and GdBiPt in-dicate the presence of metallic surface states that dif-fer strongly from the band structure in the bulk. How-ever, the authors found that within their resolution aneven number of bands cross the Fermi level at the sur-face, making these states sensitive to disorder unlike instrong topological insulators where an odd number ofcrossings is expected, protecting surface states from be-ing backscattered by a non-magnetic impurity. An X-rayresonant magnetic scattering (XRMS) study on GdBiPtindicated a doubling of the unit cell along its [111] spacediagonal, however the authors were unable to establishthe exact direction of the magnetic moments , infor- Temperature (K) R e s i s t i v i t y ( m Ω c m ) R H ( c m / C ) R H ( c m / C ) FIG. 2: The solid points show the resistivity ρ ( T ) of GdBiPtat zero magnetic field for a temperature range of 10 K to300 K. The open circles show the temperature evolution ofthe Hall coefficient from 1.8 K to 300K, revealing a kink wellabove the 9 K N´eel temperature (shown in more detail in theinset). mation that is essential in determining whether GdBiPtcould be an AFTI.The half-Heusler structure consists of four interpene-trating fcc lattices shifted by [ , , ], three of them oc-cupied by a different element while the fourth forms anordered vacancy. We carried out combined refinement ofour X-ray and neutron scattering data, which yields thelowest χ , if the atoms in GdBiPt take the same posi-tions as reported for YbBiPt and CeBiPt - platinumlocated on the [0 , ,
0] site (4 a ), Gd on the [ , , ] (4 c ),and Bi on the [ , , ] position (4 d ) (See Table I of ).These atomic positions are in agreement with the onesthat have been previously reported by Kreyssig et al. .In addition, we also carried out a single crystal X-raydiffraction experiment. Due to the non-centrosymmetricnature of the F m space group, we also tested an in-verted structure (racemic twin) with Pt on the 4 a , Bi onthe 4 c and Gd on the 4 d site in order to see if such astructure could account for the observed intensities. Ina non-centrosymmetric structure, anomalous X-ray scat-tering leads to different intensities for so-called Friedel pairs, such as ( hkl ) and (¯ h ¯ k ¯ l ). The refinement confirmedthe original structure, resulting in R1 = 0.0241, where R1is the difference between the experimental observationsand the ideal calcluated values, and a Flack parameter,which is the absolute structure factor, of -0.13(2) for thecurrent structure in contrast to R1 = 0.0806 and Flackparameter of 1.2(1) for the inverted structure (please notethat a Flack parameter is 0 for the correct structure and1 for the inverted structure).GdBiPt has a low carrier density ( ∼ · cm − / C).Figure 2 shows that there is a gradual increase in theHall coefficient as the temperature is reduced, with aclear kink near 25 K. The Hall coefficient was measured
T (K) M agne t i c I n t en s i t y ( c t s / h r) C m ag ( J / m o l K ) d ( 𝜒 ・ T ) / d T T (K ) C / T ( J / m o l K ) S = R ln(8) FIG. 3: Inset: The specific heat is shown as
C/T vs. T .The solid line is a fit to determine the phonon contribution C ph = βT and the electronic specific heat C el = γT . Mainfigure: The open circles show the magnetic specific heat C m = C − C ph − C el , solid diamonds show the temperature derivativeof the magnetic susceptibility ddT ( χT ). Solid green circlesshow the intensity of the first magnetic peak (
12 12 12 ) plottedas a function of temperature. The solid line is a fit to thesquare of the magnetic moment, obtained from numericallysolving a Weiss model for a J of . using a Quantum Design PPMS, which was also usedfor the specific heat measurements. CeBiPt also showssuch a kink followed by a stronger increase of R H . InCeBiPt this kink appears at the transition temperature T N and was ascribed to the development of a superzonegap in the ordered state and consequently a reductionof the number of charge carries . In GdBiPt a similarkink seems to be present, however it occurs around 25 Kwhich is above T N ∼ χ of Gd shows a Curie-Weiss behaviourwith a Curie-Weiss temperature θ W of − . µ eff of 7 . µ B consis-tent with the 7 . µ B expected for Gd . The data weretaken in an applied field of 0.05 T using a Quantum De-sign VSM squid magnetometer. The magnetic entropy S mag shown as the dashed line reaches 0 . R ln(8) at T N indicative of the absence of frustration in contrast to thepredictions of . Here S mag was calculated by integratingthe magnetic specific heat C − C ph − C el after subtractingthe phonon C ph and electronic contributions C el , respec-tively. Fig.3 also shows that ddT ( χT ) exhibits a peakat 8.5 K which confirms the antiferromagnetic orderingwith a N´eel temperature T N of 8.5 K. In fact, all threemeasurements: Specific heat C p ( T ), electrical resistivity ddT ρ ( T ) (not shown), as well as the magnetic suscepti-bility ddT ( χT ), show discontinuities at the same criticaltemperature T N , giving evidence to the high quality ofour samples .At fit to a straight line of C/T as a function of T for temperatures above 15 K yields a C ph = βT with a β of 2 . × J /mol K . This value of β corresponds
10 20 30 40 5002 . . Θ (deg) C oun t s . . . . C oun t s . . . . C oun t s
20 K3.6 KDifference ( , , ) ( , , )( / , / , / ) ( / , / , / ) ( / , / , / ) FIG. 4: Neutron powder diffraction patterns for GdBiPttaken above (20 K, top panel) and below (3.6 K, middle panel)the N´eel temperature. The bottom panel emphasises the formof the magnetic scattering by showing the difference betweenthe 20 K and 3.6 K patterns. The solid line through the datais a fit (described in the text) while the solid line below eachpattern shows the residuals. In the 20 K pattern (top), theupper set of Bragg markers are for the nuclear contributionfrom GdBiPt. The second row indicates the position of Biflux. In the 3.6 K pattern (middle), the first row of Braggmarkers is the nuclear contribution, and the bottom row isthe magnetic contribution. As the difference pattern (bot-tom) only has magnetic peaks, the Bragg markers are for themagnetic pattern. to a Debye temperature θ D of 188(5) K. The same fitresults in Sommerfeld coefficient γ of only 2 mJ/mol K ,which is low for a metallic compound containing heavyelements such as Gd and Bi. In contrast, the heavyFermion YbBiPt shows a γ of 8 J/mol K , which was as-signed to low lying crystal field levels . Since in GdBiPtthe angular momentum L of the 4 f configuration iszero, crystal fields are not expected to play a signifi-cant role. Consequently, we should observe the full mag-netic moment of the Gd ion. This is supported by the0.9 R ln(8) entropy release observed in the phase transi-tion.Our GdBiPt crystals were grown from non-enrichedGd containing the natural abundance of the differentGd-isotopes which lead to an extreme absorption crosssection of GdBiPt . In order to be still able to carryout our neutron diffraction experiment, we used a thinlydispersed sample on a large flat Si sample plate witha very low background (for details see ). The neu-tron diffraction pattern in the top panel of Figure 4 wastaken at 20 K, well above the N´eel temperature. It there-fore shows only nuclear reflections which can be indexedwith the MgAsAg-type fcc structure. On cooling below T N to 3.6 K the gadolinium moments order and sev-eral magnetic reflections appear in the middle panel ofFigure 4. All of the magnetic peaks can be indexed as( n −
12 2 n −
12 2 n − ) with n =1, 2, ..., indicating that themagnetic unit cell is doubled along the (1 1 1) directionof the crystallographic unit cell.Plotting the intensity of the first magnetic peak againsttemperature (Figure 3) and fitting it reveals a N´eel tem-perature of 9.4(1) K, slightly higher than derived ear-lier from heat capacity and susceptibility. The k -vector k = [ , , ] of this type-II antiferromagnetic structurebelongs to a star containing three more elements k =[ − , , ], k = [ − , − , ] and k = [ , − , ],whichare equivalent due to the cubic symmetry. We thenused the BasIreps program, which is part of the Full-prof Suite ) to find the basis functions of the irreduciblerepresentations of the F m space group with k =[
12 12 12 ].This symmetry allows two sets of basis functions whosereal and imaginary components are listed in Tab. ?? .For the basis functions listed in Tab. the magneticmoment is given by: S = C · [ BasR + i BasI ] (1)The two basis functions of set 2 represents the tworacemic structures possible. Due to the fact that we usedpowder these are indistinguishable in the refinement andwe are left with a single parameter C as refinable quan-tity.The first set of basis functions places the gadoliniummoments along the body diagonal of the cubic structure.However the (
12 12 12 ) peak is forbidden for this set sincethree of the four equivalent (
12 12 12 ) peaks are systemat-ically absent due to the translational symmetry of thespace group (face centered), and the fourth is absent dueto the magnetic polarization factor for neutron scatter-ing. However, it is clear from the difference pattern infigure 4 that this is the strongest of the observed mag-netic peaks. This allows us to rule out the first set ofbasis functions.A refinement of the second set of basis functions con-tains two equivalent basis vectors, of which the first waschosen for the refinement. The 3.6 K pattern returns aGd magnetic moment of 6 . µ B which corresponds to amoment of 7 . µ B at 0 K, which is comparable to value of 7 . µ B reported for single crystal Gd . The differencepattern in Figure 4 was also refined and gave the same6 . µ B for the Gd moment at 3.6 K.Previous resonant magnetic X-ray scattering experi-ments , were unable to determine the direction of themagnetic moment of GdBiPt. Their attempts to re-fine the actual moment direction were inconclusive asthey had several sizeable magnetic domains within the ∼ . × . beam footprint that led to incompleteaveraging over directions. By working with a powderand a much larger ( ∼ . × ) beam footprint, do-main averaging is complete in our data permitting a fullanalysis of the peak intenisties and allowing us to deter-mine the magnetic structure. Complex (e.g. cycloidal)ordering was deemed to be incompatible with the XRMSdata , and since we detected no other magnetic scatter-ing down to 2 θ = 4 ◦ , ( q ∼ . − ), we can directly ruleout long-period modulations of the magnetic structurewith periods less than about 19 ˚A (about three latticespacings). Longer-period modulations would yield satel-lites around the magnetic peaks which are also absent.We conclude that GdBiPt adopts a simple collinear typeII antiferromagnetic structure. The magnetic unit cell iseight times larger than the crystallographic unit cell, asthe k =[
12 12 12 ] propagation vector doubles all three crys-tallographic axes. The magnetic moments form ferro-magnetic sheets which are stacked antiferromagneticallyalong the [111] body diagonal (Figure 1). The samepropagation vector is found for the vanadium doped half-Heusler compound CuMnSb , but not for CeBiPt whichorders as a type I AFM with a propagation vector of[100] . The evaluation of the magnetic moment direc-tion with the program BasIreps suggests a common, sin-gle k -vector structure with the moments perpendicularto the space-diagonal. Our results make GdBiPt a srongcandidate material for an AFTI.The results presented here suggest a similar structureto that proposed by Mong et al. , with an observed spinarrangement that results in strong spin-orbit interactionalong the space diagonal. This leads to a path asymme-try for inter ferromagnetic plane hopping between non-magnetic sites. In conclusion, given its spin-structure,GdBiPt is therefore a promising candidate for an antifer-romagnetic topological insulator.The research at McGill and UdeM received supportfrom the Natural Sciences and Engineering ResearchCouncil of Canada (Canada) and Fonds Qu´eb´ecois dela Recherche sur la Nature et les Technologies (Qu´ebec).ADB and YM are also supported by the Canada Re-search Chair Foundation. The neutron diffraction mea-surements were made at the Canadian Neutron BeamCentre, Chalk River, Ontario. ∗ Regroupement Qu´eb´ecois sur les Mat´eriaux de Pointe(RQMP) R. S. K. Mong, A. M. Essin, and J. E. Moore, Physical
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