Magnetism of Mg1−xMnxO4 spinels by SQUID magnetometryand muon spin rotation spectroscopy

Paolo Ghigna,1 R. De Renzi,2 Maria C. Mozzati,3 Alessandro Lascialfari,3,4 G. Allodi,2 Marco Bimbi,2 Claudio Mazzoli,2,*Lorenzo Malavasi,1 and Carlo B. Azzoni3

1Dipartimento di Chimica Fisica “M. Rolla,” Università di Pavia, Viale Taramelli 16, I-27100, Pavia, Italy2Dipartimento di Fisica e Unità CNISM, Università di Parma, Viale delle Scienze 7a, I-43100, Parma, Italy3Dipartimento di Fisica “A. Volta” e Unità CNISM, Università di Pavia, Via Bassi 6, I-27100, Pavia, Italy

4Istituto di Fisiologia Generale e Chimica Biologica e Unità CNISM, Università di Milano, Via Trentacoste 2, 20134, Milano, Italy�Received 24 August 2005; revised manuscript received 13 March 2006; published 1 May 2006�

The magnetic properties of the substituted spinel Mg1−xMnxMn2O4, with 0�x�1, were investigated bymeans of superconducting quantum interference device �SQUID� magnetometry and �SR spectroscopy. Azero-field muon spin precession is detected in the two end members and the appearance of static internal fieldsis evidenced in all samples below a weakly x-dependent ordering temperature TN, different from that detectedby macroscopic magnetization, TM. For x close to 1 a noncollinear spin structure as in Mn3O4 is compatiblewith our low temperature results. At the opposite end �x close to zero� SQUID and �SR data suggest anantiferromagnetic arrangement of the Mn ions on the B spinel site. A nontrivial finding is that the uniform�q=0� component of the magnetic moment scales linearly with x. This evidence is discussed on the basis of theknown magnetic structure of Mn3O4.

DOI: 10.1103/PhysRevB.73.184402 PACS number�s�: 75.50.Ee, 75.50.Gg, 75.50.Lk, 76.75.�i

I. INTRODUCTION

Transition metal spinels have been studied for a longtime, but renewed interest was recently attracted to old prob-lems, such as the true nature of the Verwey transition1,2 in thebest known example of magnetic spinels, Fe3O4 �magnetite�.Another reason to revisit this family of materials is that theirstructure can give rise to alternative half metals of potentialimpact in magneto-electronic applications,3 starting againfrom magnetite, but possibly searching across the many pureand mixed compositions available. In this framework the roleof competing interactions in the two crystallographically dis-tinct magnetic sites of the spinel becomes crucial. A fullunderstanding of this aspect is far from being reached, de-spite the very large number of different magnetic spinelswhich have been investigated in the past. In the present paperwe focus on this aspect for one specific case, that of themixed Mg–Mn spinel. The joint use of macroscopic magne-tization and of the microscopic muon probe allows us toidentify the evolution of the spin structure with nonmagneticsubstitution in this direct spinel, as well as the role of all theexchange interactions.

Transition metal spinels of formula AB2O4 are often fer-rimagnets, but the presence of Jahn-Teller �JT� active ions isknown to give rise to more complex magnetic structures, dueto the action of the Goodenough-Anderson-Kanamori ruleson the anisotropic 3d orbitals. This is noticeably the case ofMn3O4, whose magnetic structure was investigated in detailby neutron scattering.4

The structure of a direct spinel is usually described by acubic close-packed arrangement of oxygen ions, with the di-valent A cations occupying the tetrahedral sites, and thetrivalent B cations occupying the octahedral sites. The ferri-magnetic behavior of the non JT active oxides is mainly dueto antiferromagnetic interactions between the A-B

sublattices.5 Although non negligible, the A-A interactionsare usually smaller, because tetrahedra are not directly con-nected. B-B interactions are usually also weaker than theA-B one’s, but they control the B sublattice magnetic con-figuration. Furthermore, there is an inherent frustration of theA site in an antiferromagnetic B structure, where each A siteis connected by equivalent superexchange paths to two Bsites on each sublattice, giving rise to the same type of de-generacy of the prototypal geometrically frustrated lattices�triangular and pyrochlore�.6

Overall there are five exchange coupling constants com-peting. Substitution of a nonmagnetic cation at the A sitemust greatly simplify this complex interaction, canceling twoof the constants and partial substitution may shed additionallight on the hierarchy of the other three. Therefore, we haveprepared a series of spinels that are isostructural with Mn3O4by substituting the Mn�II� ions with nonmagnetic cations�namely Mg�II�, that have been used as its ionic radiusmatches that of Mn�II� quite well�. We then have investi-gated the magnetic properties of the Mg1−xMnxMn2O4spinels with x=0, 0.25, 0.50, 0.75 and 1 with either a localprobe �i.e., muon spin rotation, �SR, at different tempera-tures between 1.7 and 300 K�, and using superconductingquantum interference device �SQUID� magnetometry in thesame temperature range to probe the total magnetization ofeach sample. In this way we are able to give a rationale forthe spin state evolution of the compounds with x.

The paper is organized as follows: Sec. II provides somerelevant experimental details; we review the spinel structurein Sec. III; Sec. IV presents and then discusses our results,starting from the macroscopic magnetic behavior of thesamples as it appears from the SQUID measurements �Sub-section IV A�, then describing the findings obtained with thelocal muon probe �Subsection IV B�, which shed a differentlight on magnetic order �Subsection IV C�. We finally draw

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our conclusions on the complex magnetic behavior ofMg1−xMnxMn2O4 in Sec. V.

II. EXPERIMENT

Mg1−xMnxMn2O4 samples were synthesized by solid statereaction starting from stoichiometric amounts of Mn2O3 �Al-drich, 99.999%�, MnO �Aldrich, 99.9%� and MgO �Aldrich,99.9%�. Pellets were prepared from the thoroughly mixedpowders and allowed to react at 1200 °C for a total time ofat least six days during which they were re-ground and re-pelletized at least twice. X-ray powder diffraction �XRPD�and electron microprobe analysis �EMPA� were performed7,8

to check the phase purity of the obtained materials. XRPDpatterns were acquired on a Philips 1710 diffractometerequipped with a Cu anticathode, adjustable divergence slit,graphite monochromator on the diffracted beam and propor-tional detector. EMPA measurements were carried out usingan ARL SEMQ scanning electron microscope, performing atleast 10 measurements in different regions of each sample. Inaddition, to confirm the homogeneity of the chemical com-position, backscattered scanning electron microscopy imageshave been acquired for all the samples. According to all themeasurements mentioned above, each sample was foundsingle phase and homogeneous in the chemical composition,which was met in agreement with the nominal one. In addi-tion, all the Mg1−xMnxMn2O4 samples were found to be di-rect spinels for x�0, with Mg in the tetrahedral A site only.7

A considerable exception is MgMn2O4 �x=0� for which amaximum inversion degree equal to 0.2 was found.7,8 How-ever, since the above figure means that only 10% of Mgenters the octahedral B site, this small inversion degree canbe regarded as negligible for what concerns the magneticinteraction of the B sublattice.

The zero-field �SR �ZF-�SR� experiments were carriedout at the ISIS pulsed muon source of the Rutherford Apple-ton Lab, Chilton, UK, on the MUSR instrument and on theGPS instrument of the continuous muon source, at the Labo-ratory for Muon Spectroscopy of the Paul Scherrer Institut�Villigen, Switzerland�.

In powder samples the time-dependent asymmetry of themuon decay has a different appearance in paramagnetic andin magnetically ordered states. Above the ordering transitiontemperature, Tc, the total experimental asymmetry of the ini-tial 100% spin polarized implanted muons decays slowly intime according to Ap�t�=A0 exp�−t /T1�, whereas below Tc

two distinct components appear in a 2:1 amplitude ratio �dueto the powder average of their weights�

Am�T� =Am

3�2e−�t/T2m�2/2 cos �Bt + e−t/Tlm� . �1�

corresponding, respectively, to the muon spin precession ofthe transverse spin component around the local muon fieldBm, with its spin-spin relaxation rate 1 /T2m, and to the spin-lattice relaxation of the longitudinal component. This simplepicture may be replicated for magnetically inequivalentmuons sites, each characterized by a contribution Am to thefull experimental muon asymmetry A0. It is therefore rela-

tively easy to: �i� detect the magnetic order transition, irre-spective of its structure; �ii� measure the temperature depen-dence of the local field Bm; �iii� detect changes in the spindynamics from the temperature dependence of the relaxationrates.

In this respect the MUSR instrument has a cutoff in theobservable frequencies and rates somewhat below �=10 MHz, due to the pulsed nature of the muon beam, withthe trade-off of a higher resolution for low relaxation rates,thanks to the extreme reduction of background counts in themeasuring time interval between muon pulses. The GPS in-strument with its much higher frequency passband is bestsuited for measuring the precessing components of the muonasymmetry and their 1 /T2 rates.

Magnetization �M� measurements were performed in the1.8–300 K temperature range, after a zero-field cooling�ZFC� process and during a field cooling �FC� process, bymeans of a MPMS-XL7 Quantum Design SQUID dc mag-netometer in a static magnetic field �H� produced by a 0–7 Tsuperconducting magnet. Hysteresis cycles were also col-lected at 2 K for all the investigated samples. Relaxationmeasurements of the macroscopic magnetization at 2 K wereperformed in zero field, after a ZFC process and a subse-quent application of H=5 T for 5 min: after the switchingoff of the applied field, the magnetization was recorded as afunction of time.

III. MAGNETIC STRUCTURE AND INTERACTIONS

At low temperature and room pressure, Mn3O4 �Haus-mannite� has a tetragonally distorted spinel structure �spacegroup I41/amd�, with two formula units per unit cell. Twodistinct manganese atoms are present in the cell: Mn�A�,located at �0,0,0� which is a 4-a tetrahedral site and Mn�B�,sitting at �0,1 /4 ,5 /8� in an 8-d octahedral site. The ionicstructure is that of a direct spinel, with Mn�II� in the A tet-rahedral sites, and Mn�III� in the B octahedral sites. All theoxygen atoms are at �0,y ,z�, which is a 16-h position, andare equivalent. The MnO6 octahedron is distorted, with twoapical distances equal to 2.282 Å and four equatorial dis-tances equal to 1.930 Å. All the O-Mn�B�-O angles deviatefrom 90° by about 5°. Also the distorted MnO4 tetrahedronpresents two different O-Mn�A�-O angles �103.4 and 112.6°�and four equivalent Mn�A�-O distances, all equal to 2.040 Å.Each oxygen is fourfold coordinated to three Mn�B� and oneMn�A�. Furthermore, each octahedron shares edges with sixdifferent surrounding octahedra. A salient feature of thestructure is that chains of octahedra run parallel to the a andb lattice vectors. The a and b chains are orthogonal, andconnected by shared edges.

Figure 1 displays the local coordination of oxygen and A�tetrahedral� and B �octahedral� manganese atoms. Atoms 2,3, 4, and 5 lie in the equatorial plane �short Mn-O distance�;oxygens 1 and 6 are apical �long distance�. Oxygen 3, rep-resentative of all others, is fourfold coordinated through twoshort �b� and one long �b�� octahedral bonds, and one �a�tetrahedral bond. Four superexchange paths are representedwith cyan color in the figure: b-3-b, b-3-b�, b-3-a, b�-3-a.The corresponding superexchange coupling constants will be

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indicated as JBB, JBB� JAB and JAB� . Two A sites are not directlyconnected �the path could be a-3-b-4-a�; nevertheless, thecorresponding JAA coupling constant is not negligible. Theexperimental values are JAA=−4.9 K, JAB=−6.8 K, JBB=−19.9 K,9 confirming the general expectation �JAB�JBB�, al-though this experimental determination neglects the distinc-tion between b and b� bonds.

Therefore, in Mn3O4, intra- and intersublattice interac-tions are comparable, and the collinear magnetic structure inthe B chains is destabilized.10,11 As a matter of fact, the over-all magnetic structure changes as the temperature is loweredbelow TN=42 K.4 For T�42 K, Mn3O4 presents a paramag-netic Curie-Weiss-like behavior. Between Tc and T*=39 K,the magnetic order is apparently of the ferrimagnetic Néeltype; between T* and Tr=33 K, the Mn�III� moments de-velop an incommensurate helimagnetic structure, with thepropagation vector aligned with the �010� axis; below 33 K,the magnetic structure consists of a canted spinarrangement,12 coinciding with the commensurate locking ofthe helix period: according to neutron diffraction,11 the spinsof Mn�II� are directed along the �010� axis, while the spins ofthe Mn�III� ions subdivide into two sets, each lying in the�100� plane, and making an angle of �69° with respect to the

�0 1̄ 0� direction: the magnetic structure is noncollinear butthe net moment of the B sublattice is antiparallel with respectto that of the A sublattice. In addition, ab initio Hartee-Fockinvestigations on the magnetic properties of Mn3O4

13 predicta ferrimagnetic ground state, but show that a number of dis-tinct structures, including the ferromagnetic one, lie veryclose to the ground state in energy. The calculation almostentirely accounts for these small energy differences by asmall spin polarization of the bridging anions.

IV. RESULTS AND DISCUSSION

A. Magnetic susceptibility

The molar magnetic susceptibility ��� at H=10 mT of allthe samples investigated in this work is shown in Figs. 2�a�and 2�b�. The magnetic transition to the ferrimagnetic statearound 42 K in Mn3O4 is clearly visible, both in the ZFC andin the FC measurements. With increasing Mg content, themagnetization transition, TM, is shifted to lower tempera-tures, and for the Mg0.75Mn0.25Mn2O4 and MgMn2O4samples a residual transition is detectable only on a muchenlarged scale �see Fig. 2�b��. All samples show irreversibil-ity, whereby ZFC and FC susceptibilities are different belowa given temperature. This temperature is field dependent, asit is shown in Fig. 2�c� for the Mg0.50Mn0.50Mn2O4 com-pound: in a weak field �i.e., 2 mT�, the difference betweenFC and ZFC is evident, while larger fields �50 mT� reducethe difference.

Figure 3 shows the hysteresis curves at T=2 K for all thesamples investigated in this work. Two regimes are easilydistinguished in the figure: a nonlinear low field region, anda linear high field region. This behavior is typical of thecompetition between exchange and Zeeman energy in cantedspin structures, as is known to be the case for x=1. The lowfield nonlinear magnetization is connected with the domainsof the ferromagnetic spin component, while the high fieldlinear magnetization is due to a progressive reduction of thecanting angle. Therefore, a canted picture seems to applythroughout the phase diagram and the uniform component ofthe magnetic moment, obtained by extrapolating the highfield behavior to H=0, decreases almost linearly with theMnB content, x.

FIG. 1. �Color� Sketch of the local structure of the A and Bsublattices in the Mn3O4 spinel structure. In cyan color four super-exchange paths are represented. FIG. 2. Molar magnetic susceptibility � �open symbols: FC

data, full symbols: ZFC data� of: �a� x=1,0.75,0.5 samples at H=10 mT; �b� x=0.25,0 samples at H=10 mT; �c� x=0.5 sample atthree different fields.

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The second most relevant feature in Fig. 3 is the dramaticdecrease in the width of the hysteresis loops with x. The x=0 and 0.25 compounds, in particular, present no detectablehysteresis. Their susceptibility at H=10 mT �Fig. 2�b�� dis-plays a change of shape with respect to that of higher xsamples, showing a maximum both for FC and for ZFC.

B. Zero-field �SR

The �SR zero-field asymmetries for the two end membersMn3O4 and MgMn2O4 are shown in Fig. 4 at low tempera-tures �T=4.5 and 1.7 K, respectively� together with their bestfit.

In the best fit for the pure Mn3O4 spinel we identified twocomponents like that of Eq. �1�, corresponding to two non-equivalent muon sites. We did not force the amplitude rela-tion implicit in Eq. �1�, fitting instead up to four independentpartial asymmetries, up to two precessing transverse compo-nents ATk�t� and up to two purely relaxing longitudinal com-ponents ALk, k=1,2. The two latter where distinguishedwhenever their relaxation rates 1 /T1 were sufficiently differ-ent. Similarly a distinction of the transverse components isnot possible when the relaxation rate 1 /T2 becomes toolarge, as temperature increases.

Figure 5 shows the temperature dependence in Mn3O4 ofthe best fit parameters, asymmetries, relaxation rates, andinternal fields. Note that the fit displays the two transversecomponents only below Tr=33.6 K and two longitudinalcomponents only in an intermediate range of temperatures�22�T�36 K�.

The present fit is an improvement over that described in apreliminary account of the present results,14 where only oneprecessing component was identified and the missing com-ponent was erroneously attributed to a Muonium-like radical.The improvement in the analysis is made possible by a care-ful calibration of the time t=0 in the acquired time-differential counts, which yields the same correct initial

phase, within errors, for both precessions, resulting in a re-duction of the number of fit parameters, hence of parametercorrelation. The sum of all the fitted partial asymmetries,shown in Fig. 5�b� by the red symbols, is always equal,within errors, to the full asymmetry A0 characteristic of theGPS spectrometer �measured far above Tc�, i.e., all the im-planted muon polarization is also detected in the orderedphase. Furthermore, the total longitudinal asymmetry AL�blue circles in Fig. 5�b�� is always very close to A0 /3, asexpected for the average over powder samples.

Three features are apparent in the temperature evolutionsof Fig. 5: �i� the onset of magnetic order at TN�42 K, whichcorresponds to the appearance of a net local muon field B, tothe corresponding reduction of the nonprecessing asymmetryto A0 /3, and to a peak in both relaxation rates, due to criticalslowing down of spin fluctuations;14 �ii� a clear peak at Tr=33.6 K in the spin-lattice 1/T1 rate of the fast relaxingcomponent �as measured more easily from the low noiseISIS data� is accompanied by the distinction, below Tr, oftwo inequivalent muon sites, as shown by their internalfields; �iii� a kink in the temperature dependence of thesingle internal field, at Tl=36 K corresponds to a clear peakin 1/T2 and a smaller anomaly in 1/T1. The two lower tem-peratures Tl, Tr correspond to transitions already identifiedby neutron diffraction4 and attributed, respectively, to theonset of an incommensurate spin density wave order on sub-lattice B�, and to its commensurate locking. All this is fullycoherent with the known magnetic properties of Hausman-nite.

Muons detect magnetic order also in the opposite end-member MgMn2O4, as is clear from the top panel of Fig. 4.The temperature dependent internal field and asymmetryshown in Figs. 6�a� and 6�b�, respectively, reveal the onset oforder around Tc=49�2� K. Also shown �blue triangles, Fig.6�a�� is the rather large width of the local field distribution,extracted as B=1/2�T2 from the Gaussian time decay ofthe precession, Eq. �1�. The width may be attributed to staticdisorder, but it is definitely smaller than average local field,indicating the presence of long range �LR� order below TN=43�3� K. Figures 6�c� and 6�d� analogously show that atransition to magnetic order is evident also in the muon datafrom Mg0.75Mn2.25O4, marked by the appearance of an inter-nal field and by the reduction of the asymmetry �although thelatter indicates that the transition is less sharp than in bothother end members�. On the x=0.25 sample �not on the x=0� we have collected ISIS data, best suited for investigatingthe spin-lattice relaxation rates, and they additionally reveala peak in 1/T1, around TN. Although, strictly speaking, theLR nature of order may only be directly demonstrated byq-sensitive probes, the appearance of internal fields, theasymmetry drop and the 1/T1 peak detected at the same TNfor the x=0 sample, as well as the two similar features de-tected for the x=0 and 0.25 samples, are conclusive15 for LRorder.

For the two other intermediate compositions �SR datawitness broader distributions of internal fields at the muonsite�s�, so that the transverse components of the muon asym-metries do not provide useful information, whereas the tem-perature dependence of the longitudinal asymmetry and thepeak in 1/T1, shown in Fig. 7, yield an evaluation of the

FIG. 3. Magnetic hysteresis curves at T=2 K for all the samplesinvestigated in this work. Looking at highest field, from top tobottom: Mn3O4, Mg0.25Mn2.75O4, Mg0.50Mn2.50O4, Mg0.75Mn2.25O4

and MgMn2O4. Inset: relaxation of the magnetization of MgMn2O4,performed at 2 K in zero field, after a ZFC process and a subse-quent application of H=5 T for 5 minutes.

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transition temperature for the onset of magnetic order. In thiscase the muon evidence of the LR nature for T�TN is maybenot stringent, but the large, slightly increasing value of thetransition temperatures excludes short range order. These

findings are summarized in the phase diagram of Fig. 8�a�,where TN, taken as the 1/T1 peak position, is reported vs x,together with the SQUID onset temperature TM.

C. Discussion of the magnetic order

The �SR results for Hausmannite provide a picture fullycoherent with the known magnetic properties of this endcompound. The internal fields B�T� shown in the top panelsof Figs. 5�a�, 6�a�, and 6�c� for the x=1, 0 and 0.25 com-pounds, respectively, reach an upper value B=0.75 T forT→0 for x=0, and slightly lower values in the mixedcompounds, quite typical for Mn compounds �e.g.,manganites16�.

Surprisingly, the onset temperature TN detected by muonsin the x�1 compounds is higher than the transition tempera-ture TM detected by SQUID magnetometry �Fig. 8�a��. Theycoincide for x=1 and nearly coincide for x=0.75, but theyare increasingly different for x�0.75. This indicates that inthe latter cases the main magnetic transition TN is to a col-linear order with no net macroscopic moment, presumablythe simple antiferromagnetic arrangement of the sole B sitesublattice, which is expected for the ideal x=0 compound.

The magnetization data suggest a rough distinction be-tween the compounds with x�0.5 and those with x�0.5. Asit was noted in Sec. IV A the temperature dependence of ��Fig. 2�a�� for samples x=0.75 and 0.5 closely resembles thatof Mn3O4, and these samples display a hysteresis in M�H� at

FIG. 4. �Color� ZF-�SR asymmetry vs time for Mn3O4 �bottom�and MgMn2O4 �top�; the transverse precession at early times isevidenced on the left, whereas the later times, rebinned on the right,show the decay of the longitudinal components; the solid curves arethe best fits.

FIG. 5. �Color� Temperature dependence of the fit parameters inMn3O4: �a� internal fields; �b� total asymmetry, separated, belowTN, in total longitudinal component �blue circles� and total longitu-dinal plus total transverse components �red squares�; �c� spin-latticerelaxation rate, with two distinct rates in the range 20–35 K; �d�spin-spin relaxation rate of the majority component.

FIG. 6. �Color� Temperature dependence of the fit parameters,for MgMn2O4: �a� internal field B �red circles� and its mean squareroot deviation B �blue triangles�, roughly coinciding above T=36 K; �b�: asymmetries, total �red triangles�, total longitudinal�blue circles� and the two longitudinal components below TN �leftand right pointing triangles�, distinguished by their relaxation rates.For Mg0.75Mn2.25O4: internal field �red triangles, left scale� andspin-lattice relaxation rate �circles, right scale�; inset: relaxation rateon an expanded scale, including the GPS data below T=36 K; �d�asymmetry.

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T=2 K �Fig. 3�, as expected for ferrimagnets. On the con-trary, for samples x=0 and 0.25 ��T� presents a maximumboth in FC and in ZFC �Fig. 2�b�� and no hysteresis, withinexperimental errors �Fig. 3�. Furthermore, a pronounced timerelaxation in the magnetization of the x=0 sample�MgMn2O4� on the time scale of a few hours is observed �seeinset of Fig. 3�. These three features taken together suggest aspin glass �SG� behavior of the macroscopic magnetization.Such an additional glassy behavior must clearly coexist withthe magnetic order detected by muons �cf. Fig. 4�.

All these facts, together with the phase diagram of Fig.8�a�, indicate that the long range ferrimagnetic behavior sur-vives for x close to 1, while for increasing Mg content, stillwith x�0.5, antiferromagnetic order in the B sublattice is theprecursor to a lower temperature ordered state, presumablyof the ferrimagnetic kind. The q=0 �uniform� component ofthis low temperature structure becomes eventually glassy inthe x�0.5 samples.

From the viewpoint of a local probe a spin glass transitionis characterized by slowing-down spin fluctuations for T ap-proaching a glass temperature Tg from above, and by coex-isting static and dynamic spin components, below Tg. Im-planted muons in Mg1−xMnxMn2O4 detect a ratherinhomogeneous static local field already below TN�x� and theeffect of additional static �frozen� spins for T�TM is there-fore masked. In contrast, an additional fluctuating spin com-ponent must show up in the muon 1/T1 relaxation rates.Indeed 1/T1 increases for x=0.5 �Fig. 7�a�� as T approaches30 K, roughly coinciding with the SQUID detected transi-tion, and it saturates below. For x=0.25 instead the rate in-creases down to lower temperatures �Fig. 6�c�, inset�. Finally,for x=0 there are two contributions to the longitudinal muoncomponent, with rates differing by one order of magnitude�not shown, �1 and �0.1, respectively�. Their temperature

dependence is in line with that of the other samples, butmodest, whereas there is a sizable shift in asymmetry fromthe slow to the fast relaxing component at low temperatures�magenta triangles in Fig. 6�b��, in agreement with inhomo-geneous spin freezing. Hence muons results are consistentwith the presence of an additional spin glass-like componentwhen half or more than half of the A site are occupied byMg.

This transition corresponds to a short range ordering ofthe Mn spins at the tetrahedral sites of the spinel structure,which are also marginally present in the MgMn2O4 sampledue to a small degree of inversion.7 It should be noted thatfor x�0.25 the A sublattice behavior is determined predomi-nantly by JAB, whose exchange path is not influenced bydilution, whereas JAA would vanish for x→0. The resultingSG contribution is consistent, for small x, with the notionthat the A sites are geometrically frustrated6 in a backgroundof antiferromagnetically ordered B sites.

This spin glass behavior must be reconciled with the mag-netization data of Fig. 3: we already remarked that �i� thelinear high field increase of the moment M /�B per unit for-mula, �ii� its large reduction from the average Mn value�4.3 �B�, and �iii� the roughly linear decrease of M /�B withx suggest that at low temperatures an average canted spinstructure, like that of Mn3O4, exists for all compositions. Thelast point, �iii�, further suggests that the relative spin orien-tation progressively changes with x. Note, therefore, that, asthe Mn fraction at the A site, x, is reduced to zero and as TMsimilarly decreases, it is the ferromagnetic, uniform compo-

FIG. 7. �Color online� Temperature dependence of the fit param-eters, for Mg0.5Mn2.5O4: spin-lattice relaxation rate �a� and asym-metry �b�; and for Mg0.25Mn2.75O4: spin-lattice relaxation rates �c��notice the logarithmic scale�, with two components below TN, andasymmetry, �d�.

FIG. 8. �Color online� Evolution of magnetic parameters withcomposition x: �a� phase diagram showing the magnetic order tran-sitions detected by muons TN �circles�, and the SQUID onset tem-peratures TM �triangles�; �b� macroscopic moment and low tempera-ture spin structures, left, in Mn3O4, according to Ref. 5, and right,in MgMn2O4, proposed. T is the spin vector at the A site, R3,4 andS5,6 are those at the four independent B site sublattices.

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nent of the canted arrangement which acquires a spin glasscharacter. This uniform component of the magnetic moment,�q=0�x�, may be estimated by extrapolating to H=0 the highfield linear behaviors of M /�B �Fig. 3�; the results are re-ported in Fig. 8�b�.

If we start from the actual spin structure of Mn3O4,sketched on the top left of Fig. 8�b�, which already displaysa large staggered spin component at the B sites �Sz=2.85 �B and Rz=3.23 �B, respectively4� we may explainquite naturally the nearly linear variation of �q=0�x� bymeans of the model sketched in the bottom right of Fig. 8�b�.We disregard the inhomogeneity �i.e., the spin glass behav-ior� and we assume a linear reduction with x for both theMnA moment, due to dilution by Mg, and the angle � that theMnB moments ��R=3.5 �B and �S=3.1 �B� make with the caxis, from their x=0 values, respectively, �T=4.6�1� �B and�=30°.4

The uniform moment per unit formula given by thismodel is �q=0�x�= ��Tx− ��R+�S�sin x��, and it is shownas a solid line in Fig. 8�b�. The model agrees well with thedata, and =0.74, reasonably close to 1, is the only adjust-able parameter which makes room for the simplifying �q=0extrapolation criterion. This continuous variation of the lowtemperature spin structure leads naturally to a nearly collin-ear state at x=0, as required by our findings. A small system-atic departure of the x=0.5 data point from the model possi-bly marks the change from a homogeneous noncollinearferrimagnet to the phase where the uniform component of themagnetization �mostly due to the residual MnA� becomesnonhomogeneous and glass-like in nature.

Our model confirms that canted states are dominant in theentire phase diagram below TM, whereas the collinear spinstructure detected by muons between TM and TN implies athermally activated relief of the MnA spin frustration, possi-bly via their dynamic fluctuations. The open challenge for abinitio calculations is to justify theoretically these findings.

V. CONCLUSIONS

By comparing the findings of macroscopic magnetizationmeasurements with the local picture revealed by the muonprobe, we have shown that the complete suppression of themagnetic interactions with the tetrahedral A sites of theHausmannite spinel leads to an antiferromagnetic ordering ofMn�III� at the octahedral B sites. This is in agreement withboth observation11 and predictions13 of a mean antiferromag-netic JBB exchange, which is then the only residual interac-tion. When few Mn ions share the A site with a majority ofMg ions both SQUID measurements and muons support areentrance of a low temperature spin glass-type behavior. Inthe low x limit this may be simply understood as due togeometrical frustration. Certain spin glass features, e.g., ther-mal irreversibilities, as well as a large difference between TNand TM, are still visible at half substitution, although thisparticular composition does already display a finite magnetichysteresis, typical of the long–range ordered ferrimagnetslike Mg0.25Mn2.75O4 and Mn3O4. We finally propose that theantiferromagnetic ground state in MgMn2O4 may be reachedacross the phase diagram via a small angle continuous�mean� reorientation of the noncollinear octahedral site spinstructure of Mn3O4, thus justifying the observed nearly linearreduction of M /�B�x�.

ACKNOWLEDGMENTS

We would like to thank Philip King, the staff of theMUSR instrument and of the ISIS pulsed muon source of theRutherford Appleton Lab, Chilton, UK, as well as AlexAmato, the people of the Laboratory for Muon Spectroscopyand of the synchrotron at the Paul Scherrer Institut �Villigen,Switzerland�, for their invaluable help. R.D.R., M.B., andG.A. are grateful to Germano Guidi for many fruitful discus-sions. We acknowledge the financial support of FIRB project�Grant No. RBNE017XSW�.

*Present address: European Synchrotron Radiation Facility, 6 rueJules Horowitz, BP 220, 38043 Grenoble cedex 9, France

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MAGNETISM OF Mg1−xMnxO4 SPINELS BY SQUID¼ PHYSICAL REVIEW B 73, 184402 �2006�

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