Neutron diffraction in Fe(Al) reveals incommensurate spin density waves (SDWs) in alloys known to be spin glasses. The wave vectors for crystals of Fe(34Al), Fe(40Al), and Fe(43Al) show n varying from 11 to 6 for →q=2π(h±1/n,k±1/n,l±1/n)/a0, where (h,k,l) and a0 characterize the parent bcc lattice of the CsCl structure. The magnetic reflections are present far above the spin-glass freezing temperatures. These SDWs keep the spins on nearest-neighbor Fe atoms close to parallel, in contrast with SDWs in Cr, which keep nearest-neighbor spins close to antiparallel

Arrott, A. S

Belk, M. G

Deevi, S. C

Noakes, D. R

English

Dartmouth Digital Commons (Dartmouth College)

Dartmouth CollegeDartmouth Digital CommonsOpen Dartmouth: Faculty Open Access Articles11-21-2003Incommensurate Spin Density Waves in IronAluminidesD. R. NoakesA. S. ArrottM. G. BelkS. C. DeeviFollow this and additional works at: https://digitalcommons.dartmouth.edu/facoaPart of the Physical Sciences and Mathematics CommonsThis Article is brought to you for free and open access by Dartmouth Digital Commons. It has been accepted for inclusion in Open Dartmouth: FacultyOpen Access Articles by an authorized administrator of Dartmouth Digital Commons. For more information, please contactdartmouthdigitalcommons@groups.dartmouth.edu.Recommended CitationNoakes, D. R.; Arrott, A. S.; Belk, M. G.; and Deevi, S. C., "Incommensurate Spin Density Waves in Iron Aluminides" (2003). OpenDartmouth: Faculty Open Access Articles. 2858.https://digitalcommons.dartmouth.edu/facoa/2858Incommensurate Spin Density Waves in Iron AluminidesD. R. Noakes,1 A. S. Arrott,1,2 M. G. Belk,1 S. C. Deevi,3 Q. Z. Huang,4 J.W. Lynn,4 R. D. Shull,2 and D. Wu51Center for Interactive Micromagnetics, Virginia State University, Petersburg, Virginia 23806, USA2MSEL, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA3Research, Development and Engineering Center, Philip Morris USA, Richmond, Virginia 23261, USA4NCNR, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA5Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire 03755, USA(Received 5 August 2003; published 18 November 2003)Neutron diffraction in Fe(Al) reveals incommensurate spin density waves (SDWs) in alloys known tobe spin glasses. The wave vectors for crystals of Fe(34Al), Fe(40Al), and Fe(43Al) show n varying from11 to 6 for ~q  2h 1=n; k 1=n; l 1=n=a0, where h; k; l and a0 characterize the parent bcclattice of the CsCl structure. The magnetic reflections are present far above the spin-glass freezingtemperatures. These SDWs keep the spins on nearest-neighbor Fe atoms close to parallel, in contrastwith SDWs in Cr, which keep nearest-neighbor spins close to antiparallel.DOI: 10.1103/PhysRevLett.91.217201 PACS numbers: 75.25.+z, 75.30.Fv, 75.50.BbWe report neutron diffraction results showing antifer-romagnetic order in iron aluminides. The structure couldbe generated by a set of spin density waves (SDWs) withwave vectors ~q  2 1n ; 1n ; 1n=a0, where a0 is thelattice parameter of the parent bcc lattice underlying theB2(CsCl) structure of the atomically ordered iron alumi-nides, Fe(22Al) to Fe(55Al). The denominator n takesvalues from 11 to 6 as the concentration of Al is increasedfrom Fe(34Al) to Fe(43Al). The continuous variation of nwith concentration and temperature shows the wave vec-tors are incommensurate with the lattice.Iron aluminides have been the focus of recent discus-sions of the limits to density functional theory in thelocal spin density approximation [1,2]. We present argu-ments that affirm the statement ‘‘SDWs and antiferromag-netism are complex subjects with pervasive ramificationsin condensed-matter physics’’ [3].In 1958, Sato and Arrott predicted an antiferromageticstate for these compositions [4]. By 1980, probably every-one, including them, became convinced that this systemis best described by the term spin glass, for, indeed, thesealloys have all the properties associated with that term;see Shapiro [5]. Peaks in the temperature dependence ofthe magnetic susceptibility that signal the onset of spin-glass freezing have been observed [6–8].A dramatic transition from paramagnetism to ferro-magnetism back to paramagnetism with decreasing tem-perature occurs for Fe(30.5Al). The inverse Curietemperature is 180 K. A ‘‘spin-glass state’’ appears below80 K. Sato and Arrott attributed this behavior to compe-tition between nearest neighbors with ferromagnetic(FM) coupling and third neighbors with antiferromag-netic (AFM) coupling, where the Fe(I) atoms on oppositecube corners interact by a 180 superexchange when thecenter site of the bcc cube is occupied by an Al atom.Their predicted long-range antiferromagetism was notseen in neutron diffraction studies of their Fe(Al) alloys[9] nor in detailed studies of Fe(30.5Al) [10,11].The CsCl structure naturally divides into two simplecubic lattice complexes, one filled by Fe(I) atoms whilethe predominantly Al sublattice has the remaining Fe(II)atoms occupying sites at random, leading to frustration ofthe spin system in trying to resolve the competitionbetween the two interactions.In Fe3Al the Al sublattice itself becomes ordered,breaking into two fcc lattice complexes; one is filled byFe(II) and the other by Al. The symmetry is DO3. Thematerial is ferromagnetic, but, as noted by Grest [12], ifthe 180 superexchange interaction were stronger, Fe3Alwould have the AFM state shown in Fig. 1(a), correspond-ing to wave vectors ~q  2 14 ; 14 ; 14=a0. Figure 1(b)applies to hypothetical Fe9Al7 [2], where the nodes of theSDW pattern are at the positions of the Al atoms and theantinodes center on the clusters of nine Fe atoms.The change in structural unit cell from fcc in Fig. 1(a)to primitive cubic in Fig. 1(b) frees the cube of Fe(I)atoms to adjust its positions with respect to the Al atoms,changing the volumes of Fe atoms and affecting theirFIG. 1 (color). Magnetization patterns with cubic symmetryfor (a) Fe3Al and (b) Fe9Al7. The green and red atoms are Fe(I)sites with opposite spin directions. The gold atoms are Al. Thegrey atoms sit in positions of magnetic frustration. In (a) theseare Fe(II) and in (b) they are Al. The first three layers of the128-atom cubic unit cells are shown in perspective.P H Y S I C A L R E V I E W L E T T E R S week ending21 NOVEMBER 2003VOLUME 91, NUMBER 21217201-1 0031-9007=03=91(21)=217201(4)$20.00  2003 The American Physical Society 217201-1magnetic moments. This applies also for the B2 structureaway from stoichiometry.The appearance of a SDW pattern with wave vectors~q  2 14 ; 14 ; 14=a0 would not have been surprisingif Fe(43Al) had the cubic Fe9Al7 structure, but it does not.[The diffraction pattern shows only weak diffuse inten-sity at the positions, e.g., 12 ; 12 ; 12, where the Fe3Al order-ing would give Bragg peaks.] If n were 8 for the Fe9Al7structure, the magnetization pattern would be that ofFig. 2, where the antinodes of the SDW pattern are atthe centers of eight clusters of nine Fe atoms each. Thiscomplex pattern has a unit cell with 1024 atoms. If themoment of each Fe atom depends on the strength of themolecular field forcing the SDWs, then the atoms nearthe center of the eight clusters of nine Fe atoms wouldhave the highest moment and those at the edges of thesupercluster would have the lowest moment.The randomness of occupation by Fe(II) of the Alsublattice precludes the long-range AFM structure ofFig. 2, but an approximation to such a pattern could bemaintained by shifting the phase of the pattern withrespect to the lattice over the distances of coherencefound in the diffraction results. The coherence length isseen in the breadth of the 17 ; 17 ; 17 peak in the diffractionpattern of polycrystalline Fe(40Al) shown in Fig. 3(a).There is a decrease in the small-angle neutron scatteringfrom 120 to 1.7 K. The difference between the patterns atthese temperatures shows that most of the decrease inscattering intensity reappears as a broadened peak with aperiodicity of 1.17 nm and a coherence length of 5 nm.The peak position yields the magnitude of the scatteringvector. The single crystals provide the direction; seeFig. 3(b). There is negligible difference in results betweenthe polycrystalline material, a well-characterized homo-geneous alloy of remarkable ductility [13], and the singlecrystal of Fe(40Al). The crystals were grown from themelt using the Bridgeman method [14].The temperature dependence for the scattering ampli-tude of the 1n ; 1n ; 1n peaks is shown in Fig. 4. There is noobvious sign of the freezing temperatures, known fromthe ac susceptibility work of Shull et al. [7] and Takahashiet al. [8]. These data resemble neutron scattering from aweak ferromagnetic material in a large applied field. Itmatches the behavior of the dynamically fluctuating SDWclusters seen in Cu(Mn), Pd(Mn), and Pd(Cr) spin-glassalloys [15–17], for which greatly refining the energyselection in triple axis neutron spectrometry leads tosome resolution of the freezing process.The solid lines in Fig. 4 describe the response of a lo-calized moment to an applied field plus a cooperative fieldproportional to the resulting moment. The neutron scat-tering intensity I  q=q02Iq0  IBG  IBG, whereFIG. 2 (color). Magnetization patterns with cubic symmetryfor Fe9Al7 with the magnetic unit cell doubled in each directionto form a structure that would produce magnetic scatteringrelated to that observed in Fe(43Al) . The first three layers ofthe 1024-atom unit cell are shown with the coding of Fig. 1.FIG. 3. Neutron scattering intensity for (a) polycrystallineFe(40Al) at 120 and 1.7 K with their difference shown in theinset and (b) single crystal Fe(40Al) at 12 K shown as contoursabout the 17 ; 17 ; 17 peak.P H Y S I C A L R E V I E W L E T T E R S week ending21 NOVEMBER 2003VOLUME 91, NUMBER 21217201-2 217201-2q, the moment at wave vector ~q, is given implicitly bymq  q=q0  BJq0Hq qq=kBT, where BJis the Brillouin function. The relation between T andmq is simplified by using the approximation B3x 	1=9=4x2  1p to giveT  TH=mq  T1m2qq: (1)The fitting yields estimates of T  q2q0=kB andTH  q0Hq=kB. T is a measure of the cooperativeinteraction of the Fe moments. TH is a measure of thestrength of Hq. These are compared in Table I with thefreezing temperatures Tf from Takahashi et al. [8].The field Hq is assumed here to be temperature indepen-dent. If Hq is interpreted as coming from the spin densitywaves, this implies a high value for the Ne´el temperatureof the SDWcompared to T. The SDW field Hq appears topersist far above T.The breadth of the SDW peaks, e.g., see Fig. 3, can beinterpreted as shifts in the phase of the SDWs with respectto the lattice with the phase changing at the rate of0:1 rad=nm. Conceptually there is a difference betweena picture where the SDWs have abrupt shifts of phase atdomain boundaries or gradual shifts as the SDWs adjust totake into account the positions of the clusters with anFe(II) atom surrounded by eight Fe(I) atoms. While neu-tron diffraction cannot distinguish between these, micro-magnetic measurements might.Fe(34Al), Fe(40Al), and Fe(43Al) have wave vectorsgiven by n 11, 7, and 6, respectively, with n in-creasing with temperature by less than 1 over the range 4to 120 K. The width of the diffraction peaks also increaseswith temperature, corresponding to a decrease in coher-ence length from 5 to 3.5 nm.In Fe(Al), the eight satellites about the origin appearalso around the 1; 1; 0 and 2; 0; 0 reciprocal latticevectors of the parent bcc lattice, but not around the1; 0; 0 nor the 1; 1; 1 superlattice reciprocal latticepositions; see Fig. 5. The structures of Figs. 1 and 2 alsohave eight satellite reflections about each of the reciprocallattice vectors for the parent bcc structure but not for thesuperlattice reciprocal lattice positions.The SDWs in Fe(Al) should be as important to theunderstanding of magnetism in metals as the SDWs inCr. In Cr the ~q is just short of 21; 0; 0=a0, which makesthe moments of nearest neighbors of the bcc structurealmost oppositely aligned. This ~q couples states onparallel parts of the Fermi surface to form the SDW.In the iron aluminides, the nearest-neighbor momentsare almost aligned parallel. If the principal wave vectorsare the 48 combinations of ~q  2 1 1n;1 1n;0 1n=a0, these could couple states on opposite sidesof the Fermi surface to form the SDWs. Here the wavevectors are in groups of four, each near one of the (1; 1; 0)reciprocal lattice spots of the parent bcc structure; seeFig. 5.TABLE I. Comparison with freezing temperatures fromTakahashi et al. [8].Atomic percent Al Tf (K) T (K) TH (K)34 50 50 3840 26 17 3243 15 12 10FIG. 5 (color). The reciprocal lattice spots in Fe(40Al).FIG. 4 (color). Temperature dependence of the SDW ampli-tudes of Fe(Al) single crystals [14]. The black arrows are thefreezing temperatures from Ref. [8]. Fitting with Eq. (1) yieldsthe solid curves. The red arrows are the T from the fits.P H Y S I C A L R E V I E W L E T T E R S week ending21 NOVEMBER 2003VOLUME 91, NUMBER 21217201-3 217201-3It is as if each of the 12 1; 1; 0 wave vectors of thelattice split into four wave vectors that swing away fromthe 1; 1; 0 to create 48 gaps at the Fermi surface. Thissuggests the possibility that the application of a highmagnetic field could move the wave vectors of the SDWback to the 1; 1; 0, producing field induced ferromagne-tism from a commensurate SDW.From the polycrystalline Fe(40Al) data, we can put alower bound on the moment per atom if we take as amodel the extreme case of a single ~q state with a trans-verse helical SDW. This yields a moment of 0:26B peratom if every atom (Fe and Al) were to carry the samemoment. The decrease in moment with increasing Al,seen in the single crystals, has yet to be quantified.The competition between near-neighbor FM couplingand third-neighbor AFM coupling through an Al atomhas a firm theoretical basis. The superexchange interac-tion appears in cluster calculations of Reddy et al. [18].Shukla and Wortis [19] obtained spin-glass behavior byremapping the Fe(Al) problem into one that can betreated by renormalization group methods. Grest [12]carried out Monte Carlo calculations with these twospecific interactions, obtaining results that mimickedthe data of Ref. [7]. Neither of these two studies repro-duced the reentrant behavior. The almost ferromagneticSDWs in Fe(Al) point to another interaction.The similarities between Fe(Al) and the spin-glassalloys Cu(Mn), Pd(Mn), and Pd(Cr) [15–17] should benoted in this regard. Over 40 years ago Overhauser [20,21]proved that Hartree-Fock theory, which treats exchangeas the nonlocal interaction that it is, makes a SDW statelower energy than a paramagnetic state for an electrongas, so-called jellium, when treated rigorously. There is apeak in the wave-vector dependent susceptibility  ~q forwave vectors that span the Fermi surface. Overhauserused this to predict SDWs in Cu(Mn), which were foundlater. The Cu(Mn) results compel the acceptance of a peakin  ~q [22].When Overhauser added the corrections to Hartree-Fock theory needed for the correlation energy, he dis-covered static charge-density-wave (CDW) states with alower energy than the static SDW states in deformablejellium [23,24]. CDWs have been found recently in liquidLiNH34 [25].The SDW and CDW states are closely related [26]. Thedifference is just a shift in the phase of the static spatialoscillations for spin-up electrons with respect to those forspin-down electrons. In a sufficiently large static fieldwith wave vector ~q, the CDW should take on a SDWcomponent. The Fe clusters evident in Figs. 1 and 2 mightcause such a phase shift in the 1; 1; 0 CDWs.CDWs (also called long period superlattices or concen-tration waves) appear in related B2 alloys of Ti30V20Al50,where the wave vectors are quite similar to those ob-served here [27]. For long period superlattices the con-nection of the wave vectors to the Fermi surface wasfirmly established by Sato and Toth [28].We do not know whether there is indeed cubic symme-try from multiple ~q’s in each region or lowered symmetryfrom single- ~q states in multiple regions. Nothing has beenlearned yet about polarization axes. We have yet to carryout such tasks as finding the effects of magnetic fields,pressure, strains, and changes in alloy composition bysubstitution of other atoms. Heat capacity measurementswould show whether the moments on the Fe atoms have athermodynamic spin degree of freedom or result from amore itinerant electron model [29].This work was supported by U.S. Air Force Office ofScientific Research Grant No. F49620-00-1-0364.[1] F. Lechermann et al., Phys. Rev. B 65, 132104 (2002).[2] G. P. Das et al., Phys. Rev. B 66, 184203 (2002).[3] K. Capelle and L. N. Oliveira, Phys. Rev. B 61, 15 228(2000).[4] H. Sato and A. Arrott, Phys. Rev. 114, 1427 (1959).[5] S. M. Shapiro, in Spin Waves and Magnetic Excitations,edited by A. S. Borovik-Romanov and S. K. Sinha(Elsevier, Amsterdam, 1988), Vol. 2, p. 219.[6] A. Arrott and H. Sato, Phys. Rev. 114, 1420 (1959).[7] R. D. Shull et al., Solid State Commun. 20, 863 (1976).[8] S. Takahashi et al., J. Phys. Condens. Matter 8, 11 243(1996).[9] S. J. Pickart and R. Nathans, Phys. Rev. 123, 1163 (1961).[10] J.W. Cable et al., Phys. Rev. B 16, 1132 (1977).[11] K. Motoya et al., Phys. Rev. B 28, 6183 (1983).[12] G. S. Grest, Phys. Rev. B 21, 165 (1980).[13] S. C. Deevi, Intermetallics 8, 679 (2000).[14] D. Wu and I. Baker, Mater. Sci. Eng. A 329–331, 334(2002).[15] J.W. Cable and Y. Tsunoda, J. Appl. Phys. 73, 5454 (1993).[16] F. J. Lamelas et al., Phys. Rev. B 51, 621 (1995).[17] Y. Tsunoda et al., Phys. Rev. B 56, 11051 (1997).[18] B.V. Reddy et al., Phys. Rev. B 64, 132408 (2001).[19] P. Shukla and M. Wortis, Phys. Rev. B 21, 159 (1980).[20] A.W. Overhauser, Phys. Rev. 128, 1437 (1962).[21] A.W. Overhauser and G. Lacueva, Phys. Rev. B 66,165115 (2002).[22] O. Fajen and G. Vignale, Solid State Commun. 77, 829(1991).[23] A.W. Overhauser, Phys. Rev. 167, 691 (1968).[24] A.W. Overhauser, in Highlights of Condensed-MatterTheory, edited by F. Bassani et al. (North-Holland,Amsterdam, 1985).[25] C. A. Burns et al., Phys. Rev. Lett. 86, 2357 (2001).[26] A.W. Overhauser, Phys. Rev. B 29, 7023 (1984).[27] Gousheng Shao, Appl. Phys. Lett. 74, 2643 (1999).[28] H. Sato and R. S. Toth, Phys. Rev. 127, 469 (1962).[29] D. A. Papaconstantopoulos et al., J. Appl. Phys. 89, 6889(2001).P H Y S I C A L R E V I E W L E T T E R S week ending21 NOVEMBER 2003VOLUME 91, NUMBER 21217201-4 217201-4

Incommensurate Spin Density Waves in Iron Aluminides

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Incommensurate Spin Density Waves in Iron Aluminides

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