Spatial inhomogeneity of magnetic moments in the cobalt oxide spinel ...

PHYSICAL REVIEW B 75, 054424 共2007兲

Spatial inhomogeneity of magnetic moments in the cobalt oxide spinel Co3O4 Y. Ikedo,1 J. Sugiyama,1 H. Nozaki,1 H. Itahara,1 J. H. Brewer,2 E. J. Ansaldo,3 G. D. Morris,3 D. Andreica,4 and A. Amato4 1Toyota

Central Research and Development Laboratories, Inc., Nagakute, Aich, 480-1192 Japan TRIUMF, CIAR, and Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, V6T 1Z1 Canada 3TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, V6T 2A3 Canada 4Laboratory for Muon-Spin Spectroscopy, Paul Scherrer Institut, Villigen PSI, Switzerland 共Received 31 May 2006; revised manuscript received 8 December 2006; published 28 February 2007兲

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The antiferromagnetic 共AF兲 nature of the normal spinel Co3O4 with Néel temperature 共TN兲 = 30 K was investigated by means of positive muon spin rotation and relaxation 共␮+SR兲 techniques using a polycrystalline sample. Clear muon spin precession signals due to a quasistatic long-range AF order were found in the zero-field ␮+SR spectra below TN. The spectra consist of two oscillating signals with frequencies at T → 0 K of 80 and 60 MHz, respectively, indicating an incommensurate 共IC兲 AF order in Co3O4. A possible reason for the appearance of the IC-AF order in Co3O4 would be local structural transitions due to a charge and/or a spin state change of Co ions. DOI: 10.1103/PhysRevB.75.054424

PACS number共s兲: 76.75.⫹i, 75.25.⫹z, 75.50.Ee

I. INTRODUCTION

The magnetic properties of the cobalt oxide spinel 共Co3O4兲 have been extensively studied previously by 59Co nuclear magnetic resonance 共 59Co-NMR兲,1,2 magnetic susceptibility 共␹兲,3 neutron diffraction,4 Mössbauer effect,5 and heat capacity6 共Cp兲 measurements, because of its unique magnetic structure and relatively strong superexchange interaction, which arises in spite of the large distance 共3.89 Å兲 between magnetic Co ions.4 In the last decade, both superparamagnetism and ferrimagnetism in Co3O4 nanoparticles have also attracted much attention not only for their fundamental interest but also for their potential applications in the realm of nanotechnology.7–9 The crystal structure of Co3O4 is a cubic normal spinel ¯ m, lattice constant a = 8.084 Å, with a space group O7h-Fd3 and oxygen parameter u = 0.392 at ambient temperature.10 The charge distribution for Co3O4 is represented by 2− 关Co2+兴8a关Co3+ 2 兴16d关O4 兴32e, in which 8a denotes the tetrahedral site and 16d the octahedral site surrounded by O2− ions at 32e sites 共see Fig. 1兲. Past work showed that Co3+ ions at 6 关low-spin 共LS兲, S = 0兴 the 16d site are in a diamagnetic t2g state due to a strong octahedral cubic field and consequent large crystal-field splitting between t2g and eg levels in the 3d orbitals, while Co2+ ions at the 8a site are in a high-spin 3 state with S = 3 / 2.4 It is also well known that 共HS兲 e4gt2g Co3O4 undergoes a magnetic transition from a high-T paramagnetic state to a low-T long-range-ordered antiferromagnetic 共AF兲 state at TN = 30 K. In the Co3O4 spinel lattice, Co2+ ions at the 8a site form a diamond lattice, which consists of two displaced facecentered-cubic 共fcc兲 sublattices with origins at 共0,0,0兲 and at 共 41 , 41 , 41 兲. The Co2+ ion at 共0,0,0兲 is thus tetrahedrally surrounded by four nearest-neighboring 共NN兲 Co2+ ions. Since the four NN Co2+ ions form a fcc sublattice and the four Co2+ spins align antiparallel to the Co2+ spin at 共0,0,0兲, which belongs to the other fcc sublattice, the spin arrangement in each sublattice is ferromagnetic 共FM兲 but it is AF between adjacent sublattices.4 As a result, the magnetic space 1098-0121/2007/75共5兲/054424共8兲

¯ m to low-T group of Co3O4 changes from high-T O7h-Fd3 2 ¯ Td-F43m at TN, although no structural phase transition was found down to the lowest T measured.4 In this paper, we use ¯ m even in the AF state for simplicity. site notations for Fd3 One of our motivations for investigating the magnetism of Co3O4 is the fact that Co3O4 is the parent compound for layered cobalt oxides such as NaxCoO2 RS 共Ref. 11兲, 关Ca2CoO3兴0.62 关CoO2兴 共Ref. 12兲, and RS 关Ca2Co4/3Cu2/3O4兴0.62 关CoO2兴 共Ref. 13兲 共RS denotes a rocksalt-type sublattice兲, which exhibit metallic conductivity and an extraordinarily large thermopower. These features make them promising candidates as the basic material for thermoelectric power generation systems. In order to understand the origin of good thermoelectric performance in these layered cobalt oxides, we previously carried out positive muon spin rotation and relaxation 共␮+SR兲 experiments on these materials,14–22 and found that a short-range incommensurate spin-density-wave 共IC-SDW兲 order appears below on 兲 and a long-range order forms below 30 K 100 K 共=TSDW end RS on 共=TSDW兲 for 关Ca2CoO3兴0.62 关CoO2兴,18 and TSDW = 200 K and end RS TSDW = 140 K for 关Ca2Co4/3Cu2/3O4兴0.62共CoO2兲.20 Since a long-range commensurate AF order is also observed for

FIG. 1. 共Color online兲 The crystal structure of Co3O4. 8a 共tetrahedron兲 and 16d 共octahedron兲 sites are occupied by Co2+ and Co3+ ions, respectively. 8b and 48f sites are empty tetrahedra and the 16c site is empty octahedron.

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Na0.75CoO2,15 we proposed that the IC-SDW lies not in the rocksalt-type 关Ca2CoO3兴 and/or 关Ca2Co4/3Cu2/3O4兴 sublattice but in the CoO2 planes. More recent ␮+SR and neutron scattering results on improved single-crystal samples, however, indicate that the magnetic ordering is more complex than postulated above, because of the following. 共i兲 The magnetic order for NaxCoO2 with x = 0.75 and 0.82 was found to be A-type AF—i.e., FM in the CoO2 plane but AF between adjacent CoO2 planes, by recent neutron experiments.23,24 共ii兲 The internal magnetic field at T → 0 K 共Hint,0 K兲 corresponds to a muon spin precession frequency of 54 MHz for RS 关CoO2兴 共Ref. 18兲 and 60 MHz for 关Ca2CoO3兴0.62 RS 关Ca2Co4/3Cu2/3O4兴0.62 关CoO2兴,20 whereas it is 8 MHz for 22 LiCoO2, 艋3 MHz for NaxCoO2,15 and ⬍1.5 MHz for K0.49CoO2.21 共iii兲 The layered cobalt oxides 关Sr2−xPbxBi2O4兴0.5关CoO2兴, which lack Co ions in the rocksalt-type sublattice, do not exhibit IC-SDW order but only a static Kubo-Toyabe muon spin relaxation below ⬃3 K, while the resistivity-T curves show a broad minimum around 60 K, which was thought to indicate the appearance of the IC-SDW order in that system.25 共iv兲 According to other past ␮+SR work, ␯共Hint兲 ⬃ 55 MHz for rocksalt CoO 共TN = 290 K兲.26 Since implanted muons in oxides bind preferentially with O2− ions to form a O-␮+ bond 共similar to the O-H+ hydrogen bond兲, the magnitude of Hint detected by ␮+SR is determined by both the distance 共bond length兲 between Co and O ions 共dCo-O兲 and the magnitude of the Co spin—i.e., moment. In RS 关Ca2CoO3兴0.62 关CoO2兴, dCo-O was reported to range from 1.8 to 2.0 Å in the 关CoO2兴 sublattice and from 1.6 to 2.7 Å in the 关Ca2CoO3兴 sublattice.27 The large variation in dCo-O in 关Ca2CoO3兴 is caused by a misfit between the two sublattices. Although the charge distribution of Co ions in RS RS 关CoO2兴 and/or 关Ca2Co4/3Cu2/3O4兴0.62 关CoO2兴 is 关Ca2CoO3兴0.62 still unknown, the longer dCo-O in 关Ca2CoO3兴 than in 关CoO2兴 and common transport properties of the layered cobalt oxides suggest that both Co3+ and Co4+ coexist in 关CoO2兴, whereas the average valence is almost 3+ for Co ions in the RS sublattice. In the Co3O4 spinel, on the other hand, dCo3+-O is ⬃1.92 Å and dCo2+-O ⬃ 1.93 Å. The spinel Co3O4 is considered to be a cation-deficient rocksalt; that is, O2− ions at the 32e site and Co3+ ions at the 16d site form an RS lattice together with the vacant 16c site, while Co2+ ions locate at the interstitial 8a site. By comparison with the layered cobalt oxides, Co3O4 possesses a simple structure with a well-known charge distribution in the lattice. We therefore initiated the present study of the magnetic properties of Co3O4 in order to obtain more insights into the nature of the layered cobalt oxides. Furthermore, ␮+SR, as it is very sensitive to the local magnetic environment,26,28,29 provides unique information on spatially inhomogeneous long-range ordering, such as an ICspin-density-wave 共SDW兲 state.30,31 This is because the local magnetic field at a crystallographic ␮+ site varies depending on the spatial inhomogeneity, naturally generating a wide field distribution and as a result a rapidly relaxing oscillation.

In model AF cases, on the other hand, a long-lived oscillating signal is normally observed; i.e., the relaxation rate is very small, because the local magnetic field is in principle identical for each ␮+ site. In this paper, we report on both weak 共relative to the spontaneous internal fields in the ordered state兲 transverse-field 共wTF-兲␮+SR and zero-field 共ZF-兲␮+SR for a polycrystalline Co3O4 sample at temperatures between 1.8 and 60 K, with the aim of elucidating the nature of the AF phase of Co3O4 with TN = 30 K. II. EXPERIMENT

A polycrystalline sample of Co3O4 was synthesized by a tape-casting method using ␤-Co共OH兲2 crystal platelets as a starting material. The sample was sintered at 1200 K for 2 h in an O2 atmosphere followed by annealing in an O2 gas flow. The spinel structure of the sample was confirmed by x-ray diffraction measurements. The diffraction pattern showed only peaks due to Co3O4 spinel structure. Magnetic susceptibility was measured using a superconducting quantum interference device 共SQUID兲 magnetometer 共MPMS, Quantum Design兲 at temperatures between 400 and 5 K. Heat capacity was measured using relaxation technique 共PPMS, Quantum Design兲 in the temperature range between 300 and 1.9 K. The ␮+SR experiments were performed on the surface muon beamline of M20 at TRIUMF and the ␲E1 beamline at PSI using the instrument Dolly. Experimental setup and techniques are described elsewhere.32 III. RESULTS A. Magnetic susceptibility and specific heat

The magnetic susceptibility ␹ of the polycrystalline Co3O4 sample was measured in field cooling mode with 10 kOe at temperatures from 400 to 5 K. Figure 2 shows the temperature dependence of 共a兲 ␹ and 共b兲 ␹−1. The inset of Fig. 2共a兲 shows the slope of ␹ 共d␹ / dT兲 as a function of T. The ␹共T兲 curve exhibits a broad maximum at around 40 K, whereas the d␹ / dT shows a sharp peak at 29 K. Above 40 K, ␹ decreases monotonically with increasing T, while ␹−1 increases almost linearly with increasing T, indicating a Curie-Weiss paramagnetic behavior. Indeed, the T dependence of ␹ above 100 K is well represented by

␹共T兲 = ␹0 + C/共T − ⌰p兲,

共1兲

where ␹0 is the temperature-independent susceptibility, C is the Curie-Weiss constant, and ⌰p is the paramagnetic Curie temperature. The fit for the ␹共T兲 curve using Eq. 共1兲 over the T range above 100 K yields ⌰p = 95± 2 K, ␹0 = 共1.86± 0.07兲 and C = 2.86± 0.04 K emu/ ⫻ 10−3 emu/ 共mol Co2+兲, 共mol Co2+兲. The effective magnetic moment per mol Co 2 共␮eff兲 is calculated by using the formula C = N␮eff / 3kB, 2+ where N is the number density of Co ions per mol and kB is the Boltzmann constant. We obtain ␮eff = 共4.79± 0.02兲␮B, where ␮B is the Bohr magneton. The current results are very consistent with the data estimated from the ␹共T兲 curve below 300 K: that is, ⌰p

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FIG. 2. 共Color online兲 Temperature dependences of 共a兲 magnetic susceptibility ␹ and 共b兲 inverse magnetic susceptibility ␹−1 for Co3O4. Solid lines represent the fit results using Eq. 共1兲. The inset of 共a兲 shows slope of ␹ 共d␹ / dT兲 as a function of T.

= 108 K and ␮eff = 4.74␮B.33 It should be, however, noted that Cosse reported ⌰p = 53 K and ␮eff = 4.14␮B from a ␹ measurement up to 1000 K.3 The discrepancy between the data is probably due to the spin-state transition around 600 K—i.e., the spin state of Co3+ ions at the 16d site changes from the low-T LS state 共S = 0兲 to the high-T HS state 共S = 2兲.34 In order to explain the nature of Co3O4 at low T, which we are primarily interested in, the data below 600 K are therefore considered to be suitable. In addition, ␮eff of the spinel CoAl2O4 共4.65␮B兲,35 in which Co2+ ions locate only at the 8a site, is in good agreement with the present result, supporting the present determination of ⌰p and ␮eff for Co3O4. Assuming that Co2+ ions are responsible for the Curie-Weiss paramagnetic behavior, the estimated ␮eff = 4.79␮B is greater than the spin-only value for free Co2+ ions 共3.88␮B兲, indicating a contribution of the spin-orbit coupling of Co2+, as explained by Roth.4 Figure 3共a兲 shows the T dependence of Cp in zero field in the T range between 400 and 1.9 K. The inset of Fig. 3共a兲 is a magnification of the Cp共T兲 curve around 30 K. A clear ␭-type peak with the maximum at 29.8± 0.3 K in the Cp共T兲 curve yields the correct value of TN for the present sample. For Co3O4, Cp has both phonon Cph and magnetic Cm contributions. In order to estimate Cm, we employ the combined Einstein-Debye model for Cph as6,35 Cph =

−3 9RxD

冕

xD

0

2 xE2 exEi x 4e x i dx + R 兺 ai x . 共ex − 1兲2 共eE − 1兲2 i=1

FIG. 3. 共Color online兲 Temperature dependences of 共a兲 Cp and 共b兲 Cm. The solid line in 共a兲 indicates the fit result using Eq. 共2兲 and in 共b兲 the magnetic entropy calculated using Eq. 共3兲. R ln共4兲 is the expected value of the magnetic entropy for S = 3 / 2 system. The inset of 共a兲 shows the magnification of the Cp共T兲 curve in the vicinity of TN.

and the Einstein temperature 共⌰Ei兲, respectively. R = 8.31 J / 共mol K兲 is the molar gas constant, and ai is the number of degree of freedom for each Einstein mode; for the cubic spinel, a1 = 3 and a2 = 15.6 Above 50 K, the Cp共T兲 curve is well fitted by Eq. 共2兲, and as a result we obtain ⌰D = 234± 3 K, ⌰E1 = 311± 4 K, and ⌰E2 = 707± 1 K. Then the Cm共T兲 curve is calculated by subtracting Cph共T兲 from the measured Cp共T兲 curve 关see Fig. 3共b兲兴. The Cm共T兲 curve naturally exhibits a sharp maximum at TN = 29.8± 0.3 K. The entropy change from a magnetically ordered state to a disordered state 共⌬Sm兲 is calculated by the formula ⌬Sm共T兲 =

The first term represents the Debye-type phonon contribution 共acoustic mode兲 and the second term the Einstein-type contribution 共two optical modes兲. Both xD = ⌰D / T and xEi = ⌰Ei / T are the reduced T−1 for the Debye temperature 共⌰D兲

T

0

Cm共T兲 dT. T

共3兲

Figure 3共b兲 also shows the ⌬Sm共T兲 curve estimated using Eq. 共3兲 and the magnetic entropy change expected for the S = 3 / 2 system—i.e., R ln共2S + 1兲 = 11.5 J K−1 mol−1, while the measured ⌬Sm is estimated as 8.1 J K−1 mol−1 even at 60 K. The difference is probably due to grain boundary effects in the polycrystalline sample and/or the difficulty to fit Cph, as reported in Ref. 6.

共2兲

i

冕

B. Weak transverse field ␮+SR

Figure 4 shows wTF-␮+SR spectra for Co3O4 obtained at 60 K 共squares兲, 30 K 共triangles兲, and 8 K 共circles兲 in a magជ is perpendicular to the netic field of H = 50 Oe, where H ជ initial muon spin direction 共S␮兲. A clear muon oscillation signal due to the external field is observed at 60 K. As T

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FIG. 4. 共Color online兲 Weak transverse field 共wTF-兲␮+SR time spectra for Co3O4 at 60 K 共squares兲, 30 K 共triangles兲, and 8 K 共circles兲. The magnitude of the applied wTF is 50 Oe.

FIG. 6. 共Color online兲 Early times part of ZF-␮+SR spectra for Co3O4 at above and below TN = 30 K. The full spectra extend up to 10 ␮s. Solid lines represent fits using Eq. 共5兲

decreases below 30 K, the amplitude of this signal decreases drastically and at 8 K there remains only a nonoscillatory, slowly relaxing signal. Actually the wTF-␮+SR spectra were well fitted in the time domain using a combination of an exponentially relaxing cosine oscillation, due to the external field, and an exponentially relaxing nonoscillatory signal caused by the appearance of random internal fields in the ordered state:

independent of T down to 40 K, then slightly increases, leaps by 0.1 ␮s−1 at 30 K, and then disappears with further lowering T below 30 K. ␭TF above 40 K is mainly due to the nuclear magnetic moments of 59Co nuclei in the paramagnetic state, while the slight increase below 40 K would imply the formation of short-range AF order of Co ionic moments. On the other hand, ␭slow decreases monotonically with decreasing T below the transition, showing typical critical behavior for a transition at TN. These indicate a sharp transition to a long-range AF order which completes below 30 K—that is, TN = 30 K, which is very consistent with the result of the Cp measurement. Moreover, Aslow共5 K兲 corresponds to 1 / 3 of the maximum value of ATF, as expected for a powder sample since 1 / 3 of the muons experience the internal field component parallel to Sជ␮ 共part of the signal known as the “1 / 3 tail”兲. The remaining 2 / 3 muon spins oscillate very rapidly due to the components of the internal field perpendicular to Sជ␮, but such oscillating signals are averaged out within the time resolution in Fig. 4 and do not contribute to the signal at longer times. Since ATF is proportional to the volume fraction of paramagnetic phases in the sample, the wTF results show that the whole sample volume enters into the AF phase.

A0 P共t兲 = ATF exp共− ␭TFt兲cos共␻␮t + ␾兲 + Aslow exp共− ␭slowt兲, 共4兲 where A0 is the total initial asymmetry, P共t兲 is the muon spin polarization function, ␻␮ is the muon Larmor frequency, ␾ is the initial phase of the muon precession, and An and ␭n 共n = TF and slow兲 are asymmetries and relaxation rates of the two signals. Figures 5共a兲 and 5共b兲 show the T dependence of An and ␭n for Co3O4 below 60 K. Above 30 K, ATF is almost constant at its maximum value 共⬃0.24兲, showing that the whole sample is paramagnetic. As T decreases from 30 K, ATF suddenly decreases and vanishes below 29 K, while Aslow appears at 30 K and seems to level off to the constant value 共⬃0.075兲 below 20 K. As T decreases from 60 K, ␭TF is also

C. Zero-field ␮+RS

FIG. 5. Temperature dependences of 共a兲 asymmetries 共ATF and Aslow兲 and 共b兲 relaxation rates 共␭TF and ␭slow兲 for Co3O4. The data were obtained by fitting the wTF-␮+SR spectra using Eq. 共4兲.

ជ 兲 of In order to investigate the internal magnetic field 共H int + the AF phase for Co3O4 in detail, ZF-␮ SR experiments were performed below 31 K. Figure 6 shows ZF-␮+SR time spectra at 30.5 K 共top兲, 29.7 K 共middle兲, and 8.1 K 共bottom兲. A clear spontaneous muon spin oscillatory signal due to a ជ quasistatic H int is observed below 29.7 K 共“quasistatic” means static at least within the time scale of ␮+SR—i.e. up to ⬃10 ␮s兲, whereas there is no oscillation signal at 30.5 K, further indicating the appearance of a long-range AF-ordered state below 29.7 K. The Fourier transforms of ZF-␮+SR time spectra clearly show two distinct frequency components in each spectrum 共see Fig. 7兲. The spectrum at 8 K consists of a main peak at ␯2 = 51.6 MHz and a second broad weaker maximum at ␯1 = 72.7 MHz, corresponding to Hint,2 = 3.8 kOe and Hint,1 = 5.4 kOe, respectively. Both ␯1 and ␯2 monotonically de-

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␯1 ⬎ ␯2 ,

FIG. 7. 共Color online兲 Fourier amplitude of ZF-␮+SR spectra at several temperatures below TN.

crease with increasing T and disappear simultaneously at 30 K. The full widths at half maximum 共FWHM兲 of the two peaks at ␯1 and ␯2 increase with increasing T, and the former is broader than the latter in the whole T range measured. In the layered cobalt oxides, such as NaxCoO2 and RS 关CoO2兴, coexisting Co3+ and Co4+ ions in the 关Ca2CoO3兴0.62 CoO2 planes naturally induce the wide distribution of Hint. On the contrary, in the Co3O4 spinel, Co2+ ions locate only at the 8a site. As a result, the FWHM 共T = 8 K兲 at ␯1 is 6.0 MHz and at ␯2 1.6 MHz, respectively, while the FWHM RS 关CoO2兴 is of the main peak 共␯ = 55 MHz兲 for 关Ca2CoO3兴0.62 about 20 MHz at 2.5 K and for Na0.75CoO2 1 MHz 共␯ = 3.3 MHz兲 at 2.5 K. This shows that the distribution of Hint in Co3O4 is rather small compared with those in the layered cobalt oxides, as expected. There are two possible reasons for the appearance of the two oscillatory components in the ZF time spectra: namely, the presence of two magnetically inequivalent ␮+ sites and an incommensurate 共IC兲 magnetic order in the AF phase of Co3O4. In the former case, a phenomenological relaxation function for the oscillatory components in the ZF time spectrum is a combination of two exponentially damped cosine signals, 兺An exp共−␭nt兲cos共2␲␯nt + ␾n兲 共n = 1 and 2, ␯1 ⬎ ␯2兲. Actually, the ZF time spectra for Co3O4 below TN are well fitted by this phenomenological function except that the fit results exhibit a large negative phase shift for both oscillatory signals; i.e., the delay of the initial phases are ␾1 = −70° ± 5° and ␾2 = −48° ± 4° at 8 K. This indicates the appearance of the IC magnetic order below TN for Co3O4, because ␾ should be zero for commensurate 共C-兲AF order.36 Furthermore, as T decreases from 30 K, A1 共A2兲 increases 共decreases兲 monotonically with T from 0.06 共0.10兲 at 30 K to 0.11 共0.06兲 at 8 K. In order to explain the A1共T兲 and A2共T兲 curves, one would have to assume the existence of two muon sites and a situation whereby the population of ␮+ at each site changes monotonically with T. Such behavior is very unlikely to occur at low T. The ZF time spectra were therefore fitted with the following formula:37 A0 P共t兲 = A1J0共2␲␯1t兲exp共− ␭1t兲 + A2 exp共− ␭2t兲 ⫻cos共2␲␯2t + ␾兲 + Aslow 1 exp共− ␭slow 1t兲 + Aslow 2 exp共− ␭slow 2t兲,

共5兲

where A0 is the total initial asymmetry, P共t兲 is the muon polarization function, An, ␭n, and ␯n 共n = 1 and 2兲 are the asymmetries, relaxation rates, and oscillation frequencies of the two oscillating signals, respectively, ␾ is the initial phase of the exponentially damped term, and Aslow,n and ␭slow,n 共n = 1 and 2兲 are the asymmetries and relaxation rates of the non-oscillating signals, respectively. J0共2␲␯1t兲 is the zerothorder Bessel function of the first kind. This is because according to the lattice sum calculation, a generalized IC field distribution at the muon site H is given by30,37,38 P共H兲 =

2 H , 2 2 冑 ␲ 共H1 − H 兲共H2 − H22兲 H2 ⬍ H ⬍ H1 ,

共6兲

where 2␲␯1 = ␥␮H1 and 2␲␯2 = ␥␮H2 共␥␮ is the muon gyromagnetic ratio兲. Since Eq. 共6兲 represents the field distribution in a frequency domain, Eq. 共6兲 is unavailable to fit ZF-␮+SR time spectra. On the other hand, the sum of J0共2␲␯t兲 and the cosine function, which corresponds to the first two terms in Eq. 共5兲, is known to well represent the fine spectrum of P共H兲. We thus used Eq. 共5兲 to fit the ZF-␮+SR spectra of Co3O4. The two nonoscillating components in Eq. 共5兲 correspond to the “1 / 3 tail,” as described in the previous section on wTF-␮+SR results. Figure 8 shows the T dependences of 共a兲 ␭n, 共b兲 A1 + A2 and Aslow1 + Aslow2, and 共c兲 ␯n and ⌬␯ = ␯1 − ␯2 of the two oscillating signals 共n = 1 and 2兲 obtained by the fitting results using Eq. 共5兲. As T decreases from 30 K, both ␭1 and ␭2 rapidly decrease with decreasing slope and reach their minima below ⬃15 K. This is reasonably explained by the critical phenomenon in the magnetically ordered system at the vicinity of TN. A1 + A2 is almost T independent below TN and levels off to its maximum value 共⬃0.15兲 for a magnetic powder sample. Aslow1 + Aslow2 is also T independent below TN and levels off to half the value of A1 + A2 共⬃0.08兲. This indicates that Aslow1 + Aslow2 represent the “1 / 3 tail,” which is consistent with the result of wTF-␮SR. The result is further evidence supporting the conjecture that the whole sample enters into the IC-AF order phase below TN. The two ␯n共T兲 curves exhibit a similar T dependence 关see Fig. 8共c兲兴. As T decreases from 31 K, both ␯1 and ␯2 suddenly appear at 30 K and increase with decreasing T, with decreasing slope. At the vicinity of TN, ⌬␯ rapidly increases with decreasing T and seems to level off to ⬃20 MHz below 20 K. Since ⌬␯ measures the field distribution of the IC order, this implies that the IC AF structure of Co3O4 is completed at 20 K. This is consistent with the result of the T dependences of the ␭n as described above. The other parameters Aslow1, Aslow2, ␭slow1, and ␭slow2 lack T dependence below TN, indicating that the implanted muon does not change the site at least in the time scale of the muon lifetime. The ␾共T兲 curve is independent of T with ␾ ⬃ 10° at

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FIG. 8. 共Color online兲 Temperature dependences of 共a兲 ␭n, 共b兲 A1 + A2 and Aslow1 + Aslow2, and 共c兲 ␯n 共n = 1 and 2兲 and ⌬␯ = ␯1 − ␯2. These results were obtained by fitting the ZF-␮+SR spectra with Eq. 共5兲. Solid curves are guide to the eye. 䉱, 䉮, 〫 and data obtained at TRIUMF and 䊏, 䊊, and ⽧ and at PSI.

whole temperatures measured. This phase shift is due to the rotation of the initial muon spin direction by 10° to avoid direct positrons from the muon source. IV. DISCUSSION A. Origin of incommensurate order

Here, we discuss the origin of the IC-AF order in Co3O4. To begin with, the magnetic interaction between Co2+ ions at the 8a site is known to proceed via multiple exchange paths involving several Co2+-O-Co3+-O-Co2+ bonds 共see Fig. 1兲.4 Additionally, among related materials, CoAl2O4 lacks a magnetically ordered state even at 4 K,39 while CoRh2O4, in which Rh3+ is in the 4d6 configuration of LS state 共S = 0兲, exhibits a long-range AF order below TN = 27 K.40 This suggests that the vacant eg orbitals at the 16d site play an important role in the AF order in Co3O4. We therefore need to take into account not only the nature of the magnetic Co2+ ions at the 8a site but also the charge and/or spin fluctuations of Co3+ at the 16d site for explaining the origin of the IC-AF order. Since the exchange path through three intervening ions produces a relatively weak superexchange interaction, which

is responsible for the AF ordering of Co3O4, either a slight local structural distortion around Co ions or the change in charge and spin of Co3+ would alter the overlap between the orbitals of Co3+ and O2−, resulting in a modulation of the local internal magnetic field of Co3O4. According to a recent neutron diffraction study for a cubic 2− 41 spinel MnSc2S4 = 关Mn2+兴8a关Sc3+ 2 兴16d关S4 兴32e with TN = 2.3 K an IC-SDW order with a spiral magnetic structure was found below TN, although there are no structural phase transitions down to 1.5 K. The origin of the IC-SDW order in MnSc2S4 was proposed as a local structural transition due to spindriven Jahn-Teller 共JT兲 effects below TN. Such a transition was, however, considered to be undetectable, because the distortion is probably far beyond the resolution of the current neutron diffraction technique. A similar scenario would be applicable for Co3O4, which also keeps a cubic symmetry down to 4.2 K,4 if Co ions are JT active. The possible reactions in the AF phase of Co3O4 are therefore as follows. 共i兲 Spin-state transition of Co3+ ions at the 16d site: Co3+共LS, S = 0兲 → Co3+ 共HS, S = 1 or S = 2兲. 共ii兲 Charge disproportionation of Co3+ ions at the 16d site into Co2+ and Co4+ ions: Co3+ 共LS兲 → Co2+ 共HS, S = 3 / 2兲 + Co4+ 共S = 1 / 2兲. 共iii兲 Exchange of Co2+ ion at the 8a site with Co3+ ion at the 16d site: Co2+ 共8a , S = 3 / 2兲 + Co3+ 共16d , S = 0兲 → Co3+ 共8a , S = 2兲 + Co2+ 共16d , S = 3 / 2兲. All the reactions listed above are assigned to an electron transfer process between Co ions with activation energies ranging from 0.3 to 0.7 eV.34 The above reactions are thus likely to occur locally even at low T and, as a result, induce a local JT distortion in the CoO6 octahedra. Such distortion naturally leads to a fluctuation of the distance between Co2+ ions and ␮+. We should here note that, although there is no evidence for IC-SDW order in the past neutron powder diffraction measurements, it is not impossible but very difficult to detect the IC-SDW order by neutron. This is because the IC modulation reduces the intensity of magnetic Bragg peaks. In order to further understanding of the origin of the inhomogeneity, additional ␮+SR experiments are required using single-crystal samples of Co3O4 to determine the ␮+ sites more precisely and to know the relationship between Hint and the easy axis of magnetization. B. Comparison with layered cobalt oxides

In earlier ␮+SR work, two major precessing signals with ␯n共0 K兲 = 54 and 78 MHz, plus a minor signal at 150 MHz, were observed in a rocksalt-type CoO 共Co2+O2−兲 crystal,26 comparable to the present experimental values of ␯n共0 K兲 2− = 60 and 80 MHz for the 关Co2+兴8a关Co3+ 2 兴16d关O4 兴32e spinel. In the AF phase of CoO below TN = 290 K, the spins of Co2+ ions align parallel to each other on 共111兲 planes, but with antiparallel spin directions in alternate 共111兲 planes. It is therefore very surprising that Hint detected by ␮+ for CoO are almost the same as for Co3O4, in spite of the fact that the magnetic structure of CoO is so different from that of Co3O4. This could suggest the existence of Co2+ ions in the 16d site in Co3O4, as in the case for the well-known Fe3O4

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2+ = 关Fe3+兴8a关Fe2+Fe3+兴16d关O2− ions are 4 兴32e case. However, Co RS most unlikely to exist in 关Ca2CoO3兴0.62关CoO2兴 based on ␹ measurements,42 a bond-valence-sum calculation using the structural refinement data,43 and a photoelectron spectroscopic analysis at ambient temperature.44 Nevertheless, assuming that a charge disproportionation of Co3+ ions 共2Co3+ → Co2+ + Co4+兲 occurs in the CoO2 planes at low T as in the case for LiCoO2,22 it would be reasonable that the IC-SDW with ␯共0 K兲 = 54 MHz exists in the CoO2 planes in RS 关Ca2CoO3兴0.62 关CoO2兴. Although the current ␮+SR results on Co3O4 do not provide crucial information concerning the position of the IC-SDW in the layered cobalt oxides, indirect evidence, if Co2+ ions exist in the 16d octahedral site, still seems to support that the IC-SDW lies in the CoO2 planes.

spectra below 30 K, both with sizable inhomogeneous broadening. The precise analyses of the ZF spectra also suggested the formation of an incommensurate antiferromagnetic order below TN. Although the origin of the inhomogeneity of the magnetic moment in the AF order state is not fully understood, it could be caused by a slight structural distortion together with a spin and/or charge state transition of Co3+ ions at the 16d site. Experiments on not only a single crystal Co3O4 but also related materials, such as, CoRh2O4, ZnCo2O4 共diamagnetic ions at 8a sites兲, and CoAl2O4 共diamagnetic ions at 16d sites兲, help to elucidate the role of Co3+ ions at the 16d site on the long-range AF order.

V. SUMMARY

The magnetic properties of Co3O4 below TN were studied in wTF- and ZF-␮+SR experiments, together with magnetic susceptibility and heat capacity measurements. Spontaneous muon spin precession signals with two different frequencies 77± 4 and 56± 3 MHz 共T → 0兲 were observed in ZF-␮+SR

1

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ACKNOWLEDGMENTS

This work was performed at both TRIUMF, Vancouver, Canada and the Swiss Muon Source, Paul Sherrer Institut, Villigen, Switzerland. We thank S. R. Kreitzmann, B. Hitti, and D. J. Arseneau of TRIUMF, P. Russo of Columbia University, and LMU staff of PSI for help with the ␮+SR experiments. This work was partially 共J.H.B.兲 supported at UBC by CIAR, NSERC of Canada, and at TRIUMF by NRC of Canada.

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