Multi-Q mesoscale magnetism in CeAuSb_2
Guy G. Marcus, Dae-Jeong Kim, Jacob A. Tutmaher, Jose A. Rodriguez-Rivera, Jonas Okkels Birk, Christof Niedermeyer, Hannoh Lee, Zachary Fisk, Collin L. Broholm
MMulti-Q mesoscale magnetism in CeAuSb Guy G. Marcus, ∗ Dae-Jeong Kim, Jacob A. Tutmaher, Jose A. Rodriguez-Rivera,
3, 4
JonasOkkels Birk,
5, 6
Christof Niedermeyer, Hannoh Lee, Zachary Fisk, and Collin L. Broholm
1, 4 Institute for Quantum Matter and Department of Physics and Astronomy,The Johns Hopkins University, Baltimore, MD 21218, USA Department of Physics and Astronomy, University of California at Irvine, Irvine, California 92697, USA Department of Materials Sciences, University of Maryland, College Park, Maryland 20742, USA NIST Center for Neutron Research, Gaithersburg, MD 20899, USA Laboratory for Neutron Scattering and Imaging,Paul Scherrer Institut, CH 5232 Villigen-PSI, Switzerland Department of Physics, Technical University of Denmark (DTU), DK-2800 Kgs. Lyngby, Denmark (Dated: August 27, 2018)We report the discovery of a field driven transition from a striped to woven Spin Density Wave(SDW) in the tetragonal heavy fermion compound CeAuSb . Polarized along c , the sinusoidalSDW amplitude is 1.8(2) µ B /Ce for T (cid:28) T N =6.25(10) K with wavevector q = ( η, η, / ) ( η =0 . H (cid:107) c , harmonics appearing at 2 q evidence a striped magnetic texture below µ ◦ H c = 2 . H c , these are replaced by woven harmonics at q + q = (2 η, ,
0) + c ∗ until µ ◦ H c = 5 . µ B /Ce for µ ◦ H ≈ From micelles and vesicles in surfactant solutions [1, 2]to mixed phase type-II superconductors [3, 4], the spon-taneous formation of large scale structure in condensedmatter is a subject of great beauty, complexity, and prac-tical importance. The phenomenon is often associatedwith competing interactions on distinct length scales andsensitivity to external fields that shift a delicate balance.Heavy fermion systems epitomize this scenario in met-als, which place f -electrons with strong spin orbit inter-actions near the transition point between localized anditinerant [5, 6]. Whether described in terms of oscilla-tory Ruderman-Kittel-Kasuya-Yosida (RKKY) exchangeinteractions or Fermi-surface nesting, these strongly in-teracting Fermi liquids are prone to the development oflong wave length modulation of spin, charge, and elec-tronic character with strong sensitivity to applied mag-netic fields.Here we examine the magnetism of the heavy fermionsystem CeAuSb , which was previously shown to havetwo distinct phases versus field ( H ) and temperature ( T ).By establishing the corresponding magnetic structures,we gain new insight into the interactions and mechanismsthat control the phase diagram and give rise to electronictransport anomalies. Specifically, we show that the ap-plication of a magnetic field along the tetragonal axis ofCeAuSb induces a transition from a striped to a wovenmodulation of magnetization, both c -polarized and mod-ulated on a length scale exceeding the lattice spacing byan order of magnitude.CeAuSb is part of the ReTX series(Re=La,Ce,Pr,Nd,Sm; T=Cu,Ag,Au; X=Sb,Bi) [7–10], which crystallizes in spacegroup P4/nmm (see ∗ Figure 1(b)). Metamagnetic transitions with transportanomalies are common in these compounds, so ourfindings may have broader relevance. CeAuSb isIsing-like with an (001) easy axis and lattice parameters a = 4 .
395 ˚A and c = 10 .
339 ˚A at T = 2 K. The Ne´eltemperature is T N =6.25(10) K and the lower (upper)critical field is µ ◦ H c = 2 . µ ◦ H c = 5 . via the self-flux method and used 8.5(1) mgand 114.5(1) mg crystals for diffraction in the ( hh(cid:96) )and ( hk
0) reciprocal lattice planes, respectively. Todetermine the magnetic structure, we mapped neutrondiffraction intensity in the ( hh(cid:96) ) and ( hk
0) planes usingthe MACS instrument at NIST [12]. The sample wasrotated by 180 degrees about the vertical axis andthe intensity data mapped to one quadrant. Fielddependence with H (cid:107) c was studied in the ( hk
0) planeon MACS with a vertical field magnet and in the ( hh(cid:96) )plane using RITA-II at PSI [13] using a horizontal fieldmagnet.The difference between diffraction data acquired be-low (2 K) and above (8 K) T N is shown in Figure 3(a).Three out of a quartet of satellite peaks are apparentaround (111) and a single satellite is visible near theorigin. These peaks are indexed by q = ( ηη / ) with η = 0 . c andis modulated in the basal plane with a wave length λ m = ( a/ √ η ) = 23 ˚A, as depicted in Figure 2(a). Theabsence of satellite peaks of the form ( η, η, / + n ), forinteger n ≥ c -axis. To check this hy-pothesis and establish the size of the ordered moment, weextracted the integrated intensity of the magnetic Braggpeaks by integrating over the relevant areas of the two- a r X i v : . [ c ond - m a t . s t r- e l ] J u l (b) Sb'SbAuCe ab c
FIG. 1. (a) Phase diagram for CeAuSb with boundariesdetermined from magnetization (squares), resistivity (dia-monds) and neutron diffraction (circles). The symbol fill in-dicates the scan direction within the H − T phase diagram asshown in the legend. (b, inset) Crystallographic unit cell oftetragonal CeAuSb . Magnetic moments shown on Ce sitesare associated with the Γ irreducible representation. dimensional intensity maps. The corresponding magneticdiffraction cross sections at µ ◦ H =0 T are compared to astriped model with spins oriented along c in Figure 3(c),which provides an excellent account of the data with aspin density wave amplitude of m q = 1 . µ B . Herenormalization was achieved through comparison to thenuclear diffraction data acquired in the same experimentand compared to expectations for the accepted chemicalstructure [7].Figure 4(a) reports the ordered moment versus T asextracted from the wave vector integrated magnetic neu-tron diffraction intensity at (1 − η, − η, / ) and for µ ◦ H =20 mT. Near T N these data can be described as m q ( T ) ∝ (1 − T /T N ) β where β =0.32(5), consistent withthe β =0.326 for the 3D Ising model [14], but also with β = 0 . G q , associatedwith q . A description of the diffraction data in Fig-ure 3(a) by either Γ ( ↑↑ ) or Γ ( ↑↓ ) is consistent withthis tenet. Application of the P4/nmm symmetry oper-ations generates a second distinct wavevector (arm) ofthe star { q i } namely q = ( η ¯ η / ). The observation of q satellite peaks leaves open whether distinct, single- q domains or a multi- q modulation describes the zero-fieldmagnetic structure. As we shall now show, this is re-solved by analysis of magnetic diffraction data in a field H (cid:107) c .We enter the striped phase by zero-field cooling (ZFC)to 100 mK. Initial application of a magnetic field along c has little effect on the ordered moment m q until an FIG. 2. Panels (a-c) show the low field striped magnetic struc-ture and (d-f) show the high field woven structure. Through-out (a-f), the color scale indicates the component of magne-tization along c for a single square lattice layer of Ce atoms.False color images in (a) and (d) show the magnetic structurewithin the basal plane while the lower frames (b-c) and (e-f)show the modulation of magnetization along particular linesthrough the basal plane indicated in frames (a) and (d). Panel(g) depicts the maximum and minimum values of local Ce magnetization at 100 mK within this model. The maximumvalues expected from an alternate model (see SupplementaryInformation) is overlaid for comparison along with the themeasured uniform magnetization. abrupt reduction of 0.65(5) µ B at µ ◦ H c =2.78(1) T (Fig-ure 4(b)). Continuing this isothermal field-sweep (IFS)to higher fields, the staggered magnetization m q is con-tinually suppressed as in the approach to a second orderphase transition before eventually falling below the de-tection limit above µ ◦ H c =5.42(5) T.While no hysteresis was detected in the field depen-dence of the ordered moment, we do find hysteresis in thefield dependence of the characteristic wavevector. Fig-ure 4(d) shows that η locks into two distinct plateaus forincreasing IFS each terminated by regimes where η , towithin resolution, decreases continuously with increas- (c) -1 10 (scaled x 2) (b)(a) (d) FIG. 3. Constant field maps of symmetrized, magnetic dif-ferential scattering cross-section are shown above for 0 T (a),and 4 T (b). The quality of nuclear and magnetic refinementof these data is demonstrated by a plot of the experimen-tal integrated intensities in absolute units of cross-section ( (cid:101) σ ,see Supplementary Information), versus the calculated cross-section ( σ ) in (c). Panel (d) shows scans through (110)-2 q atvarious fields illustrating the appearance of a harmonic peakfor intermediate H . ing H . For decreasing IFS, η follows a different, non-intersecting trajectory without plateaus. This hysteresisin q , and in the higher harmonics discussed below, per-sists to the lowest fields and for T s up to at least 2 K(see Supplementary Information), while no hysteresis isobserved in the field dependence of the staggered mag-netization nor in the uniform magnetization, m ◦ (Fig-ure 4(e)). m ◦ increases linearly with applied field at a rate of m (cid:48)◦ = 0 . µ B T − / Ce until an abrupt increase of∆ m ◦ = 0 . µ B /Ce at H c . Above this transition, m ◦ continues to increase linearly at a similar rate until H c , where the incommensurate magnetic peaks vanish.Interestingly, m ◦ continues to increase for H > H c untilsaturating at 1 . µ B /Ce, which is indistinguishablefrom the zero field staggered magnetization (gray bandin Figure 4(e)).Figure 4(f) shows the longitudinal magnetoresistivityversus H at 2 K where the ρ ( T ) is dominated by theresidual component. For H = H c there is an abruptincrease in resistivity that is subsequently reversed for H > H c . This decrease in ρ is approximately twice aslarge as the increase in ρ at H c . One interpretationis that parts of the Fermi surface develop a gap in theordered regimes. . . . . (a) (b)(c) (d)(e)(f) MX Γ R Z A Z R (g)
012 012
FIG. 4. Irrep constrained ordered moment (a,b) and order-ing wavevector (c,d) is shown here spanning the H-T phasediagram. Field dependence of the uniform magnetization isshown in (e). Longitudinal magnetoresistance is shown in (f),with dashed lines highlighting a factor two increase of ∆ ρ across H c . The reduced Fermi surface (g) is extracted fromDFT calculations and overlaid with potential nesting condi-tions for q (black), 2 q (red), and q + q (blue). Figure 3(d) shows representative line cuts of elasticneutron scattering along ( hh
0) for
H < H c . We find aweak, field-induced peak at (2 η, η,
0) = 2 q − c ∗ , whichindicates the spatial modulation of magnetization ceasesto follow a simple sinusoidal form in a field. The newFourier component is supported by a single q domainand is not accompanied by harmonics of the form q ± q .This constitutes evidence that the H < H c SDW stateis striped and consists of distinct q and q domains.Furthermore, the presence of a magnetic satellite peakat momentum transfer Q = (2 η, η,
0) implies the SDWmoment associated with the two Ce spins within a unitcell are in phase corresponding to the Γ ( ↑↑ ) IR.The field dependence of the amplitude of the 2 q har-monic, m q ( H ), is shown in Figure 4(b). As directlyapparent from Figure 3, there is no evidence for this har-monic in zero field with a quantitative limit of | m q | < . µ B . Upon comparison to the 0.60(7) µ B third orderharmonic expected from a square-wave structure, this in-dicates a sinusoidal modulation for H = 0. A linear in H fit to m q ( H ) yields m (cid:48) q = 0 . µ B T − / Ce, which isindistinguishable from m (cid:48)◦ so that | m q ( H ) | ≈ | m ( H ) | throughout the striped phase (Figure 4(b,e)). Combin-ing the three Fourier components we obtain m j ( r ) = m + ν j m q cos( q · r ) + m q cos(2 q · r ) on sublattice j . Here ν = 1 (IR: Γ ) and ν = − ) cannot bedistinguished in the present data.Without loss of generality we pick m > m q >
0. To ensure | m j ( r ) | does not exceed the saturationmagnetization at any r requires m q <
0. The corre-sponding m j ( r ) = ν j m q cos( q · r ) + m ◦ (1 − cos(2 q · r ))for H immediately below H c is shown in Figure 1(c-e). Here we have used our experimental finding that m ( H ) ≈ − m q ( H ). Qualitatively, we find stripeswhere m ( r ) > m j ( r ) <
0. Given only the fundamental and thefirst harmonics and assuming m q = − m , a globalmaximum in m j ( r ) that exceeds m q occurs when m exceeds m q /
4. The similarity of m ( H c ) = 0 . µ B to m q ( H c ) / . µ B indicates the phase transitionat H c is associated with reaching the maximum magne-tization possible for a striped phase dominated by justthree Fourier components m , m q , and m q .Figure 4(b) shows m q abruptly vanishes for H >H c . The false color map of the ( hk
0) plane at µ H = 4 Tin Figure 3(b) shows the 2 q harmonic is replaced bysatellites spanned by q ± q that surround (110), (1¯10),and (000). These indicate the simultaneous presence atthe atomic scale of both q and q and a field inducedharmonic that transforms as Γ . Figure 4 shows m q ± q ,abruptly jumps to and then holds an essentially con-stant value of 0.7(1) µ B /Ce for H c < H < H c . Thesimilarity to the plateau-like dependence of the residualmagneto-resistivity is consistent with both phenomenaarising from the opening of an additional gap on theFermi surface: Two nesting wavevectors, rather than one,gap out twice as much of the Fermi surface, thereby dou-bling the residual resistivity as observed (Figure 4(f)).For H c < H < H c the c -oriented staggered magneti-zation can be described as m j ( r ) = m + ν j m q (cos( q · r ) + cos( q · r )) (1)+ m q ± q (cos(( q + q ) · r ) + δ cos(( q − q ) · r )) . Here m and m q = m q can again be chosen positive without loss of generality. There are two qualitativelydifferent structures δ = ± δ = 1, m ( r ) describes a checkered patternwith four fold symmetry and m q ± q < site falls considerablybelow saturation immediately above H c (Figure 2(g)).While this might be possible if gradient terms dominateover quartic terms in a Landau free energy, the choiceof δ = −
1, labeled as the “woven” phase, has the virtuethat max[ m j ( r )] remains virtually constant through H c and all the way up to H c when the measured field de-pendent amplitudes and magnetization in Figure 4(b,e)are considered in Equation 2. While diffraction cannotprovide definite proof for this structure, illustrated inFigure 2(d-g), there is circumstantial evidence.Figure 2(d) and Eq. 2 show that the woven SDW, justas the crystal structure, is not invariant under C : Lobesof c -polarized spins extend along a ( b ) for m q ± q > m q ± q < c , the woven pat-tern shifts within the basal plane by half of its periodin the direction of the prolate axis of lobes with magne-tization antiparallel to H . As was the case for H < H c ,there are two spatially separated domains only now com-posed of c -polarized lobes of spins extended along a or b . However, the development of magnetization in the wo-ven structure is qualitatively distinct. This is apparentin Figure 4 where the fundamental amplitude m q de-creases with field at a rate of m (cid:48) q = − . µ B T − / Cewhile the harmonic m q ± q is field independent. m (cid:48)◦ =0 . µ B T − / Ce maintains the same value in the wo-ven phase as it had in the striped phase. Figure 2(g)shows this corresponds to increasing the magnetizationof negatively magnetized regions only.Throughout the magnetization process shown in Fig-ure 4(c-d), the magnetic wavelength, λ m , shows lessthan a 5% variation. This contrasts with other cerium-based Ising systems. For example, CeSb undergoes aseries of field driven phase transitions that alter the di-rection of magnetization of entire planes of spins from ↑↑↓↓ ( q = (001 / ↑↑↓↓↑↑↓ ( q = (004 / ↑↑↓↑↑↓ ( q = (002 / q = 0)[16]. These square-wave structures are modulated alongthe easy-axis and can be accounted for by the ANNNI(Anisotropic Nearest and Next Nearest Neighbor Ising)model, where 4f electrons are described as localized Isingdegrees of freedom subject to oscillatory, anisotropicRKKY interactions.A model of competing near neighbor exchange inter-actions that reproduces the critical wave vectors q and q as well as the Weiss temperature and the upper criti-cal field is possible. At a minimum, it involves antiferro-magnetic J c ¿0 between Ce sites on the c -bond and basal-plane interactions J > a -bond, J < J / a + b )-bond, and J = − J / πη on the 2 a -bond[17]. However, the absence of harmonics at zero field andlow temperatures appears inconsistent with such a frame-work and points to a Fermi surface nesting induced SDW.To examine this possibility, we calculated the Fermi sur-face using the generalized gradient approximation (seeSupplementary Information). The result is shown in Fig-ure 4(g). Near the Fermi level, the band structure isdominated by f -electrons with contributions to the lowenergy density of states from sharply dispersing p -bandsan order of magnitude smaller. While there are no idealnesting conditions, q , 2 q , and q ± q do connect ar-eas of the f -electron dominated Fermi-surface, consistentwith an SDW instability.The distinct hysteresis of the small changes in SDWwave vector versus field (Figure 4(c,d)) is indicative ofthe profound rearrangement of static magnetism at H c and H c . Upon reducing field at low T , nucleation ofthe woven state from the paramagnetic state at H c canbe expected to allow for greater adherence to constraintsimposed by impurities and defects than when nucleatingthe woven state within the striped state upon increasingfield past H c at low T . Within the SDW picture thecorresponding subtle differences in magnetic order pro-vide a natural explanation for field-hysteretic electronictransport.Our results provide a simple phenomenological descrip-tion of the magnetization process in CeAuSb that de-termines the critical magnetization at the metamagnetictransitions. Net magnetization is achieved by addingboth a uniform and a single first harmonic componentto a sinusoidal magnetization wave while maintainingthe fundamental wave length and maximum amplitude.A SDW picture would appear to be appropriate andmight allow for a unified understanding of the manymeta-magnetic transitions in the ReTX family of heavyfermion compounds.We are glad to thank Christian Batista, Martin Mouri-gal, Sid Parameswaran, Chandra Varma, Yuan Wan, andAndrew Wills for helpful discussions. This research wasfunded by the U.S. Department of Energy, Office of BasicScience, Division of Materials Sciences and Engineering,Grant No. DE-FG02-08ER46544. 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