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Magnetic hysteresis and superantiferromagnetism in ferritin nanoparticles
Journal of Magnetism and Magnetic Materials 241 (2002) 430–440
Magnetic hysteresis and superantiferromagnetism in ferritin nanoparticles C. Gillesa, P. Bonvillea,*, H. Rakotob, J.M. Brotob, K.K.W. Wongc, S. Mannc b
a ! 91191 Gif-sur-Yvette, France CEA, C.E. Saclay, Service de Physique de l’Etat Condense, INSA, LPMC-SNCMP, Complexe Scientifique de Rangueil, 31077 Toulouse Cedex, France c School of Chemistry, University of Bristol, England, UK
Received 27 September 2000; received in revised form 25 June 2001
Abstract The magnetic behaviour of nanoparticles of antiferromagnetic ferritin has been investigated by means of magnetic measurements. First, using a combination of low-field susceptibility and magnetisation measurements (up to 5:5 104 Oe) in the superparamagnetic regime (30–250 K), we propose a method to estimate the N!eel temperature of ferritin, by which we obtained TN C500 K. The hysteresis loop at 2.5 K has been measured, and interpreted with a simple model, where the irreversibilities are due to the switching of the uncompensated moments. The magnetisation curve at 2.5 K up to 30 104 Oe has also been measured, and we show that it can be interpreted in terms of superantiferromagnetism, which is a finite size effect predicted by N!eel in 1961, but which has not been previously experimentally evidenced. Superantiferromagnetism occurs in small antiferromagnetic particles with an even number of ferromagnetic reticular planes, and results in an enhancement of the antiferromagnetic susceptibility and in a non-linear field dependence of the magnetisation at higher fields. r 2002 Elsevier Science B.V. All rights reserved. PACS: 75.50.Ee; 75.50.Tt; 75.60.Ej Keywords: Ferritin; Antiferromagnets; Uncompensated magnetisation; Hysteresis; Superantiferromagnetism
1. Introduction Ferritin, the iron-storage protein of mammals, is made of a hollow protein shell, with a diameter about 12 nm, in which a natural single crystal of a ferric oxihydroxide is embedded [1]. The single crystal, or ferritin core, has a diameter varying between 2–3 nm and 7 nm. The Fe3þ spin arrange*Corresponding author. Tel.: +33-169-087517; fax: +33169-088786. E-mail address: [email protected] (P. Bonville).
ment in the core is antiferromagnetic (AF), with a Ne! el temperature which has been estimated to be . 240 K by a Mossbauer study [2] and 460 K by a recent magnetisation study [3]. In zero field, the direction of the moments (easy antiferromagnetic axis) is determined by the magnetocrystalline anisotropy. An assembly of ferritin proteins, either natural as extracted for instance from horse spleen, or artificially reconstituted from apoferritin (the empty protein shell) via the injection of Fe2þ ions and in situ bio-oxidisation [4], constitutes an ensemble of antiferromagnetic nanoparticles,
0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 4 6 1 - 9
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distributed in size, whose magnetic properties have been investigated by a number of authors in the last decade [3,5–9]. The ferritin cores are superparamagnetic [10], i.e. the two AF sublattice magnetisations, which are strongly coupled by the exchange interaction, undergo thermally activated fluctuations across the anisotropy barrier. In the presence of a magnetic field much smaller than the exchange field, the magnetisation of an ensemble of randomly oriented ferritin particles (or more generally antiferromagnetic nanoparticles) is found to be the sum of two contributions [11]. The first arises from the slight canting of the two AF sublattices and gives rise to a term linear in field: wAF H; where wAF is the powder AF susceptibility. The second arises from the uncompensated magnetic moments occurring within the body of the particle or at the surface, which confers a non-zero net moment to the particle. Owing to the strong exchange coupling, this excess moment can be assumed to follow the dynamics of the AF sublattices, at least for temperatures well below TN [11]. As our measurements are performed for To0:5TN ; we will consider that the uncompensated net moment is rigidly coupled to the staggered AF magnetisation, with probably a small misalignment due to surface moments. In a previous work [8], we showed that, at thermal equilibrium with respect to the characteristic time scale of the magnetic measurements (superparamagnetic regime), the uncompensated magnetisation in ferritin does not follow a Langevin law as is usually assumed. Instead, we proposed a new function which is better adapted for antiferromagnetic nanoparticles because it takes into account the predominance of anisotropy over the Zeeman effect, characteristic of weakly uncompensated antiferromagnets. In the present work, we will first show how one can determine the Ne! el temperature in ferritin, by combining low-field and high-field magnetisation data, in the temperature range 30–250 K, and using the new field and temperature dependence of the uncompensated magnetisation derived in Ref. [8]. Then, we will focus on the low temperature (2.5 K) properties, namely the hysteresis loop and the very high-field (up to 30 104 Oe) magnetisation curve. The hysteresis loop in ferritin has been
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previously measured [3,5,12], but no quantitative interpretation has been put forward. We present here a simple model, based on the description of an antiferromagnetic particle given by Mrup [13], which ascribes the irreversibilities (i.e. the hysteresis) to the switching of the uncompensated moments. Our model reasonably reproduces the main features of the loop, i.e. the irreversibility field and the remnant moment. Finally, we present the magnetisation curve at 2.5 K with a magnetic field up to 30 104 Oe, and show that it can be accounted for by the phenomenon of superantiferromagnetism, predicted by Ne! el 40 years ago [11,14,15]. To our knowledge, this is the first experimental evidence of superantiferromagnetism up to now.
2. Experimental The susceptibility at H=80 Oe and the magnetisation curves for fields up to 5:5 104 Oe, as well as the hysteresis loop, were measured in a commercial SQUID magnetometer (2.5–300 K). The magnetisation curve at 2.5 K up to 30 104 Oe was measured at the Service National des Champs Magne! tiques Pulse! s (SNCMP, INSA) at the University of Toulouse (France). A solution of commercial ferritin (from SIGMA corporation) was used for the susceptibility and 5:5 104 Oe magnetisation curves, with a protein concentration of 100 mg/ml. The 30 104 Oe curve was measured in the same commercial ferritin, but with a concentration of 300 mg/ml to ensure a larger signal, and it was calibrated by scaling with the signal of an identical sample measured up to 5:5 104 Oe in the SQUID magnetometer. The hysteresis loop was measured in a solution of artificial ferritin, made at the University of Bristol, in which the core contained an average of 2570 Fe atoms [8], with a protein concentration of 7.5 mg/ml. Interaction effects between ferritin cores play a very minor role, as far as their magnetic properties are concerned, as demonstrated in Ref. [5] for commercial ferritin with the standard 100 mg/ml concentration. In our samples, with different concentrations and thus different interparticle distances, the dipolar fields
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are lower than 1 Oe and therefore they will be neglected. Size histograms were obtained in the artificial ferritin sample with 2570 Fe atoms per core from Transmission Electron Microscope pictures. The cores have a mean diameter d0 =5.7 nm, with a standard log-normal diameter deviation sd =0.15– 0.20. As for horse spleen ferritin, its FC-ZFC susceptibility and magnetisation curves are almost identical to that of our artificial ferritin sample with 2570 Fe atoms per core. This indicates that the size distribution and the mean Fe loading in both samples are similar, and so we will use the above mean diameter and standard deviation for commercial ferritin as well. In our experiments, the samples were cooled in zero field, implying that, in the zero field configuration, the easy antiferromagnetic axes are oriented at random in space.
3. Determination of the N!eel temperature of ferritin The superparamagnetic regime, where the sublattice magnetisations of all the ferritin cores fluctuate rapidly with respect to the characteristic time of the measurement technique, is reached, in the ferritin samples under consideration, at 30 K for the magnetic measurements (tw C100 s) and at . 90 K for the 57 Fe Mossbauer spectroscopy measurements (tL C5 109 s) [3,8]. This makes the determination of the Ne! el temperature by usual methods impossible. We present here susceptibility (H=80 Oe) and magnetisation measurements in the temperature range 30–250 K and show how these data allow the Ne! el temperature to be determined. The magnetisation curves above 30 K are shown in Fig. 1. As stated in the introduction, they consist in the sum of an antiferromagnetic canting term, linear in the field, and of a contribution due to uncompensated moments. This latter contribution is analysed according to Refs. [8,9], i.e. by replacing the traditional Langevin function by a new function GðxÞ which can be written as Z p GðxÞ ¼ 12 dy sin y cos y tanhðx cos yÞ: ð1Þ 0
Fig. 1. Isothermal magnetisation curves in commercial horse spleen ferritin. The solid lines are fits to expression 2.
The reason why the GðxÞ function is a better approximation than the Langevin function in AF particles, and for fields lower than the spin-flop field, is that due to the smallness of the net uncompensated moment, the anisotropy energy is greater than the Zeeman energy. Then, the sublattice moments, and hence the uncompensated magnetisation, are not free to fluctuate randomly, but they fluctuate along the easy magnetic axes (oriented at random in the sample). Then, taking into account the size distribution of the particles, the field and temperature dependence of the magnetisation can be written as: Z Vmax 1 mðH; TÞ ¼ wAF ðTÞ H þ /VS Vmin m H dVf ðVÞ mnc ðV; TÞ G nc ; ð2Þ kB T where mnc ðV; TÞ is the uncompensated moment of a particle with volume V and f ðVÞ is the log-normal function describing the volume distribution. For the volume dependence of the uncompensated pffiffiffiffimoment mnc ; we chose: mnc ðV; TÞ ¼ uðTÞm0 N ; according to an argument given by Ne! el [16] when the uncompensated moments arise from a random occupation of the two sublattices by the magnetic ions. In the above
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expression, m0 ¼ 5 mB is the Fe3þ ionic moment, N is the number of Fe atoms in the particle proportional to the volume through the relation: N ¼ ðNa =/VSÞV; where Na is the mean number of Fe atoms in the core, and uðTÞ is a proportionality parameter. The fits of the magnetisation curves to expression (2) are very satisfactory and yield the two quantities wAF ðTÞ and uðTÞ; where both decrease as the temperature is increased. The thermal decrease of wAF in AF particles, opposite to the bulk behaviour, has been attributed by Ne! el [15] to a finite size effect occurring in those particles that show T ¼ 0 superantiferromagnetism, to be described in Section 5. The thermal variation of the susceptibility with a field of 80 Oe is shown in Fig. 2, together with that of the AF susceptibility wAF : The difference between the two quantities is the contribution of the uncompensated moments, as is easily seen from the low field limit of expression (2) wðTÞ wAF ðTÞ ¼
1 /mnc ðTÞS2 ; /VS 3kB T
ð3Þ
where the mean uncompensated pffiffiffiffiffiffi moment is defined as: /mnc ðTÞS ¼ uðTÞm0 Na : From both magnetisation and susceptibility data, a thermal varia-
Fig. 2. Thermal variation of the 80 Oe susceptibility w and of the antiferromagnetic susceptibility wAF extracted from the isothermal magnetisation data. The dashed pffiffiffiffi line represents the empirical law: wAF ðTÞ ¼ 1:68 103 = T ; which correctly reproduces the thermal variation of the AF susceptibility between 30 and 250 K.
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tion of the mean uncompensated moment /mnc ðTÞS can be extracted. The two sets of points are shown in Fig. 3. They do not coincide, but they are close to each other, and present a thermal decrease which reflects that of the sublattice magnetisation. In the temperature range of the measurements, the thermal variation of the susceptibility derived set can be reproduced by a T 2 law, reminiscent of the magnetisation decrease due to antiferromagnetic magnons. In order to obtain an estimation of the Ne! el temperature, we plotted a T 2 law through both sets of the data points up to 300 K (solid lines on Fig. 3), and we extrapolated these laws to higher temperature (dashed lines in Fig. 3). The intercept of these curves with the Taxis yields an estimation of TN ; probably slightly overestimated. Thus we take the value TN C500 K as a correct magnitude for the Ne! el temperature in ferritin. This value is close to the estimation made in Ref. [3] (460 K), although the analysis of the magnetisation curves with a modified Langevin law in this latter work yields a linear thermal decrease for the parameter identified as the uncompensated magnetisation. The mean saturated uncompensated moment we obtain (/mnc ð0ÞSC150 mB ) is by contrast much smaller than that derived in Ref. [3] (345 mB ).
Fig. 3. Thermal variations of the mean uncompensated moment in ferritin derived from magnetisation and susceptibility measurements. The solid lines are fits to a law: /mnc ðTÞS ¼ /mnc ð0ÞSð1 aT 2 Þ between 0 and 300 K, and the dashed lines are extrapolations of these laws.
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From this TN value, we can obtain the first neighbour exchange field HE (assuming that the second neighbour exchange is negligible) from the molecular field expression: kB TN ¼ 13m0 HE : We get HE ¼ 320 104 Oe. The 0 K perpendicular AF susceptibility in the molecular field approximation is given by
4. The hysteresis loop at 2.5 K
measured in ferritin cores [3,5,12] and in NiO small particles [18]. The AF hysteresis loop is quite different from that in ferro- or ferrimagnets: it remains open up to much higher fields (a few 104 Oe), and it is superposed onto the field linear AF contribution, whereas the ferromagnetic magnetisation (almost) saturates quite rapidly for fields of a few kOe. As perfectly compensated antiferromagnets do not show hysteresis, it seems reasonable to assign its presence in AF single domain nanoparticles, in a way similar to ferromagnets, mainly to the history-dependent switching of the uncompensated moments. The hysteresis loop at 2.5 K in the artificial ferritin sample with 2570 Fe atoms per core is shown in Fig. 4, in a field ranging from 5:5 up to 5:5 104 Oe. The irreversibility field, where the ‘‘up’’ and ‘‘down’’ branches coalesce, is: Hirr C3 104 Oe; the remanent moment mr ðC4 emu=cm3 ) amounts to about 0:25/mnc S: The interpretation of hysteresis in uncompensated antiferromagnets is not well documented in the literature, so here we will develop our model at some length, and present the characteristic features of the irreversibilities in these systems.
In the single domain particles of nanometric size, hysteresis in the magnetisation curve is known to arise from the competition between the particle anisotropy energy and the Zeeman energy, after the early work of Stoner and Wohlfarth [17]. Irreversibility is due to history-dependent magnetisation reversal, and complications may arise from surface effects in very small particles, where the surface moments can behave differently from those in the core. Hysteresis loops in ferro- or ferrimagnetic particles have been intensively studied, in view of potential applications for information storage, and their characteristics are well known: the coercive (or reversal) field is of the order of K=Ms (K is the anisotropy energy density and Ms the magnetisation, supposed to be uniform throughout the particle volume), i.e. of a few kOe, and the T ¼ 0 remanent magnetisation (after the field has been decreased) is Ms /2 in the case of uniaxial anisotropy. In the single domain antiferromagnetic particles, the low temperature hysteresis loops of the magnetisation have been
Fig. 4. Hysteresis loop at 2.5 K in the artificial ferritin sample with a mean Fe loading of 2570 atoms per core. The solid line is the fit using the model of reversal of uncompensated moments, with a degree of uncompensation a ¼ 0:018 and an exchange field HE ¼ 320 104 Oe.
w> ¼
Na m0 M0 ¼ 2/VSHE HE
ð4Þ
where M0 is the magnetisation of one AF sublattice, and the 0 K powder AF susceptibility is: wAF ¼ 23w> : With the values: Na ¼ 2570 and HE ¼ 320 104 Oe, we obtain a theoretical estimate for the powder AF susceptibility in ferritin: wAF C15 105 emu/cm3 : By extrapolating the measured wAF to 0 K, we get the experimental value: wAF C40 105 emu/cm3 ; which is 2.5 times larger than expected from the molecular field expression. We will come back to this problem in Section 5, which deals with very high field magnetisation curve and superantiferromagnetism.
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We use a formulation introduced by Mrup [13] to describe an AF nanoparticle, with the assumption that the uncompensated moments are rigidly linked to the AF sublattices moments by a strong exchange interaction. The degree of uncompensation is defined by the parameter a ¼ mnc =M0 V: The anisotropy is assumed to be of uniaxial type, and an anisotropy field is defined as: HA ¼ K=M0 : At T ¼ 0; in the presence of a magnetic field H, the equilibrium sublattice moments lie in the (H,e) plane, where e is the easy magnetic axis. Defining as yb the initial orientation of the easy axis in zero field and y the orientation of the sublattice moments with respect to H, the energy of the particle can be written as:
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this symmetry is broken and the energy profiles become asymmetrical. For yb ¼ 1001; the energy profile, shown in the upper panel of Fig. 5, has two well-defined potential wells, in the intervals ½p; 0 and ½0; p; for all applied field values. Therefore, when the field is increased or decreased, the system remains in the initial potential well, contained in the interval ½0; p: There is no reversal of the
Eðy; yb ; aÞ ¼ M0 HE ½ð1 þ aÞ þ 12b2 sin2 y þ ab cos y þ kð1 þ aÞ cos2 ðy yb Þ; ð5Þ where b ¼ H=HE and k ¼ HA =HE : This expression is valid for b51; k51 and when the ‘‘canting angle’’ 2e of the two sublattices has its equilibrium (small) value: 2e ¼ b sin y: In order to study the irreversibilities occurring when the field is increased and decreased, one must investigate the energy profiles as a function of the field and of the angle yb : In small ferromagnetic particles, all the moments lying in the lower halfplane (yb > p2) will reverse when the field reaches a critical value Hc ðyb Þ; i.e. the energy profiles present a switching of the potential well from the interval [0,p] to the interval [p;0] at the critical value. The situation is not exactly the same in small AF particles. To illustrate this, we consider two particular values of the easy axis initial orientation: yb ¼ 1001 and yb ¼ 1501: The energy profiles, given by Eq. (5), were calculated for a ¼ 0:02; HE ¼ 320 104 Oe and HA ¼ 0:1 104 Oe, which are typical values in ferritin (the anisotropy constant in artificial ferritin with 2570 Fe atoms per core [8] is: K ¼ 3:5 105 ergs/cm3 ). For a fully compensated AF particle, the energy profiles present two equally deep potential wells corresponding to the invariance of the system by inversion of the two sublattice moments. When the particle possesses an uncompensated moment,
Fig. 5. Energy profiles for an uncompensated antiferromagnet for different values of the applied field (represented by the numbers near the curves, in units 104 Oe), for a degree of uncompensation a ¼ 0:02 and an exchange field HE ¼ 320 104 Oe. Upper panel: yb ¼ 1001–the two wells present a minimum for all field values. Lower panel: yb ¼ 1501–for H=5 and 10 104 Oe, the well on the righthand side has no minimum, implying irreversibility of the magnetisation reversal.
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moment and thus no irreversibility. The energy profiles corresponding to yb ¼ 1501; shown in the lower panel of Fig. 5, are different. The two potential wells are still clearly present for fields below 2 104 Oe, but at higher fields (5210 104 Oe in the figure), the less shallow well in [0; p] is smeared out. This implies that the uncompensated moment ‘‘reverses’’ into the second well in the interval [p; 0] for a critical field value, and that it remains in this well when the field is further increased and then decreased. This gives rise to an irreversibility, and for this yb value there is hysteresis. So, in small uncompensated AF particles, and contrary to ferromagnetic particles, the reversal of magnetic moments does not occur for all values of yb > 901: There is a threshold angle ylim for reversal which depends on a (or rather on a=K). Its variation with a is shown in Fig. 6. For a ¼ 0; ylim ¼ 1801; as in fully compensated AF particles, the energy profile presents symmetrical wells, and there is no hysteresis; ylim decreases as a increases, and rapidly saturates at the value 901; corresponding to the ferromagnetic case. As can be seen in Fig. 5, there is a limited field range where only one potential well is present. Its boundaries can be approximately set by saying that the Zeeman energy associated with the uncompensated moment mnc H must be larger than
Fig. 6. Variation of the threshold angle for moment reversal ylim in uncompensated AF nanoparticles, as a function of the degree of non-compensation a; for an exchange field HE ¼ 320 104 Oe.
the anisotropy energy KV but lower than the Zeeman energy w> H 2 V associated with the canting of the two antiferromagnetic sublattices. Using w> ¼ M0 =HE ; one gets the field range HA oHoaHE : a
ð6Þ
The regions in the (H,yb ) plane with one or two potential wells are shown in Fig. 7 for a ¼ 0:02: The lower curve in Fig. 7 represents the variation with yb of the reversal field, i.e. of the irreversibility field. It is similar to the well known angular variation of the coercive field in ferromagnetic materials, with the difference that there is no moment reversal for p=2pyb pylim in AF nanoparticles. The irreversibility field in AF particles shows little angular variation and is approximately equal to HA =a; which is 3 104 Oe in ferritin. Finally, in Fig. 8 we represent the variation with a of the ratio of the remanent moment mr to the uncompensated moment. In the case of ferromagnetic particles in the axial anisotropy case, the remanent moment is equal to the half of the saturated moment. This is not so in AF particles, where mr omnc =2: As can be seen in Fig. 8, the curve mr =mnc ðaÞ saturates rather rapidly, as a increases, to the limiting value mr =mnc ¼ 12; which corresponds to the ferromagnetic case.
Fig. 7. Boundaries of the regions in the (H,yb ) plane where one or two potential wells are present, for a degree of uncompensation a ¼ 0:02; and for Na ¼ 2570 and HE ¼ 320 104 Oe.
C. Gilles et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 430–440
Fig. 8. Variation of the ratio mr =mnc with a; for uncompensated AF particles.
Using the above-described model, we calculated the different branches of the hysteresis curve of the magnetisation and applied it to ferritin (see Fig. 4). For a given yb value, the magnetisation is obtained by qE ¼ w> sin2 ymin H qH þ aM0 cos ymin ;
mðyb ; HÞ ¼
ð7Þ
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wAF value, an anisotropy constant K ¼ 3:5 105 ergs/cm3 (yielding /HA S ¼ 0:06 104 Oe), and a mean degree of uncompensation /aS ¼ 0:018: The irreversibility field HA =aC3 104 Oe and the remanent moment are well reproduced by our calculation, whereas the ‘‘pinching’’ of the loop (the change in concavity of a given branch as the H-axis is crossed) is not. This can be due to the overall simplicity of the model which, for instance, does not allow for surface moments to behave differently from those in the bulk. Recently, a more complex micromagnetic-like calculation has been developed for computing the hysteresis loop in antiferromagnetic NiO [18], which reproduces rather well the experimental data in this compound.
5. The high field magnetisation curve and superantiferromagnetism The magnetisation curve in ferritin at 2.5 K for fields up to 30 104 Oe is shown in Fig. 9; also shown is the expected molecular field response (dashed line) for a classical (compensated) powder antiferromagnet with an exchange field HE ¼ 320 104 Oe, corresponding to a Ne! el
where ymin is the position of the potential well corresponding to the path under consideration. The powder magnetisation is obtained by integration over the easy axis orientations yb Z p mðHÞ ¼ 12 Mðyb Þcos ymin sin yb dyb : ð8Þ 0
In Eq. (7), w> ¼ M0 =HE is the classical AF transverse susceptibility and gives rise to the overall slope of the loop. When trying to reproduce the experimental hysteresis curve, it had to be considered as a fitting parameter, because the classical value M0 =HE yields a much too low slope wAF ¼ 23w> with respect to the experimental data. The fitted value is: wAF C45 105 emu/cm3 ; which is about 3 times larger than the expected value. This enhanced AF susceptibility, already pointed to in the preceding section, is due to superantiferromagnetism (see Section 5). The solid line in Fig. 4 is the theoretical hysteresis loop obtained with the above quoted
Fig. 9. Magnetisation curve in horse spleen ferritin at 2.5 K. The dashed line is the calculated powder magnetisation for a compensated antiferromagnet with an exchange field HE ¼ 320 104 Oe and Na ¼ 2570:
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temperature TN ¼ 500 K such as determined for ferritin in Section 3, and a number of magnetic ions Na ¼ 2570: For a classical antiferromagnet, the powder magnetisation curve at high field is linear with a slope w> : At low field, it is linear with a slope wAF ¼ 23w> in the fully compensated case, and, in the presence of a small uncompensated moment, a small extra contribution arises which rapidly vanishes as the field is increased towards pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the spin-flop field Hsf ¼ 2HA HE : It is clear that the standard molecular field model cannot account for the magnetisation at 2.5 K in ferritin: the experimental mðHÞ curve is not linear with the field, and its magnitude is somehow larger than the molecular field curve. Thus, even in the hypothesis of a large overestimation of the exchange field, which would yield a larger w> ; the experimental data could not be reproduced. In 1961, Ne! el described what he called ‘‘superantiferromagnetism’’ [11,14,15], which consists of an enhancement of the AF susceptibility in small AF particles having an even number 2N of ‘‘active’’ (ferromagnetic) reticular planes. This phenomenon is a boundary effect which actually occurs for any particle size, but Ne! el showed that it is only observable below a limiting field
where l1 and l2 are the exchange constants with the first and second neighbours, the induced magnetisation for an AF particle with an even number of active reticular planes is written as Mp k msAF ðHÞ ¼ Mp cos j0 þ tan y Z p=2 sin2 j dj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð11Þ j0 1 k2 sin2 j where Mp is the magnetic moment of a reticular plane, and j0 and k are quantities depending on the reduced field h ¼ H=Hl and on the parameter y; which are found by solving the two equations 8 Z p=2 h tan y dj > > < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; ¼ k j0 ð12Þ 1 k2 sin2 j > > : k sin j0 ¼ cos y: At low field, Ne! el shows that the susceptibility is given by: l1 wsAF ¼ 2w> 1 þ : ð13Þ l1 l2
i.e. it cannot be observed in bulk materials where 2N is very large. In ferritin, where HE C320 104 Oe and 2NC10 20; Hl amounts to a few 10 104 Oe, and superantiferromagnetism should be observable in the high field magnetisation experiments. Essentially, superantiferromagnetism consists in a continuous rotation of the direction of the staggered magnetisation within the particle, when a magnetic field is applied perpendicular to the AF axis. This yields an enhancement of the AF susceptibility by a factor which is exactly 2 when one takes into account only first neighbour exchange, and which can be larger if second neighbour exchange is considered. Introducing the parameter y defined by
The magnetisation curves calculated from Eqs. (11) and (12) for different values of the number 2N of active reticular planes and for a constant atomic density within the particles, are represented in Fig. 10. The susceptibility at low fields is twice as great as that of the bulk, in agreement with expression (13), and the magnetisation at high field is not linear for the small values of 2N: In order to apply the superantiferromagnetic calculation to our ferritin sample, we first have to perform a powder average; for this purpose, we usedthe equilibrium orientation ye of the AF sublattices as obtained for a perfect antiferromagnet in Ref. [19]. The induced powder magnetisation is then obtained, for each initial orientation yb of the AF axis, by considering the component of the applied field perpendicular to it and by integrating over the random directions of the AF axes: Z p mðHÞ ¼ 12 msAF ðH sin ye Þsin ye sin yb dyb : ð14Þ
l1 tan y ¼ ; l1 4l2
We then assume that, statistically, there is an equal number of particles with an even and odd
Hl ¼
HE ; N
ð9Þ
0
2
ð10Þ
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Fig. 10. Magnetisation curves for antiferromagnetic particles with increasing even number 2N of active reticular planes (2N=10, 20, 100, solid lines) and for a bulk antiferromagnet (dashed line), illustrating the phenomenon of superantiferromagnetism. The curves were calculated with l2 =0 and with parameters close to that of ferritin: an exchange field HE ¼ 320 104 Oe, an atomic density: r ¼ 2:5 1022 cm3 ; and an interreticular spacing e ¼ 0:5 nm.
number of active planes. Those with even number of active planes show a superantiferromagnetic response, and those with odd number of active planes show a classical antiferromagnetic response. The best fit to the experimental data, shown as a solid line in Fig. 11, is obtained with 2N ¼ 8; which is compatible with the mean size of the particles of ferritin, and with a ratio l2 =l1 ¼ 0:14: For fields below 7 T, the calculated curve lies below the experimental data; this corresponds to the contribution of the uncompensated moments, which are not taken into account in the model. At high field, this contribution vanishes because the spin configuration is mainly of spin-flop type, i.e. the uncompensated moments are perpendicular to the applied field. So we believe that, although approximations have been made in the calculation of the high field magnetisation in our ferritin sample, its behaviour shows the phenomenon of superantiferromagnetism, which is thus experimentally evidenced for the first time. Superantiferromagnetism should also be observable at low fields, but it is usually masked by the contribution
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Fig. 11. The Magnetisation curve in ferritin at 2.5 K and the theoretical curve (solid line) assuming 50% of superantiferromagnetic particles (with even number of active planes) and 50% of classical AF particles (with odd number of active planes).
of the uncompensated moments, in the frozen regime as well as in the superparamagnetic regime.
6. Conclusion Magnetisation and susceptibility measurements have been performed in antiferromagnetic nanoparticles of ferritin, both natural and artificial, in the temperature range 2.5–250 K, and with a magnetic field up to 30 104 Oe. We measured the thermal variation of the mean uncompensated moment of the particles, and estimated the Ne! el temperature of ferritin: TN C500 K. The hysteresis loop at 2.5 K was interpreted by means of a simple model where the irreversibilities are attributed to the switching of the uncompensated moments; we show that the irreversibility field for an antiferromagnetic particle is: Hirr CHA =a; where HA is the anisotropy field and a the degree of uncompensation, i.e. the ratio of the uncompensated moment to the sublattice moment. A 30 104 Oe magnetisation curve was measured at 2.5 K and we show that it evidences the phenomenon of superantiferromagnetism, predicted by Ne! el in 1961, which consists in an enhancement of the antiferromag-
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netic susceptibility in small particles with an even number of reticular active planes.
References [1] T.G. St. Pierre, J. Webb, S. Mann, in: S. Mann, J. Webb, R.J.P. Williams (Eds.), Biomineralization: Chemical and Biochemical Perspectives, Chapter 10, VCH, New York, 1989. [2] E.R. Bauminger, I. Nowik, Hyperfine interactions 50 (1989) 489. [3] S.A. Makhlouf, F.T. Parker, A.E. Berkowitz, Phys. Rev. B 55 (1997) R14 717. [4] F.C. Meldrum, V.J. Wade, D.L. Nimmo, B.R. Heywood, S. Mann, Nature 349 (1991) 684. [5] S.H. Kilcoyne, R. Cywinsky, J. Magn. Magn. Mater. 140 (1995) 1466. [6] R.A. Brooks, J. Vymazal, R.B. Goldfarb, J.W.M. Bulte, P. Aisen, Magnetic resonance in medicine 40 (1998) 227.
[7] J.G.E. Harris, J.E. Grimaldi, D.D. Awschalom, A. Chiolero, D. Loss, Phys. Rev. B 60 (1999) 3453. [8] C. Gilles, P. Bonville, K.K.W. Wong, S. Mann, Eur.Phys. J. B 17 (2000) 417. [9] C. Gilles, Ph.D. thesis, Universit!e de Paris 6 (2000), unpublished. [10] C.P. Bean, J.D. Livingston, J. Appl. Phys. 30 (1959) 120S. [11] L. N!eel, J. Phys. Soc. Jpn. 17 (Supp. B 1) (1962) 676. [12] S. Gider, D.D. Awschalom, T. Douglas, K.K.W. Wong, S. Mann, G. Cain, J. Appl. Phys. 79 (1996) 5324. [13] S. Mrup, Surf. Sci. 156 (1985) 888. [14] L. N!eel, C.R. Academy of Science, Paris 253 (1961) 203. [15] L. N!eel, C.R. Academy of Science, Paris 253 (1961) 1286. [16] L. N!eel, C.R. Academy of Science, Paris 252 (1961) 4075. [17] E.C. Stoner, E.P. Wohlfarth, Philos. Trans. R. Soc. London Ser. A 240 (1948) 599. [18] R.H. Kodama, A.E. Berkowitz, Phys. Rev. B. 59 (1999) 6321. [19] V. Beckmann, W. Bruckner, W. Fuchs, G. Ritter, H. Wegener, Phys. Stat. Sol. 29 (1968) 781.
Magnetic hysteresis and superantiferromagnetism in ferritin nanoparticles C. Gillesa, P. Bonvillea,*, H. Rakotob, J.M. Brotob, K.K.W. Wongc, S. Mannc b
a ! 91191 Gif-sur-Yvette, France CEA, C.E. Saclay, Service de Physique de l’Etat Condense, INSA, LPMC-SNCMP, Complexe Scientifique de Rangueil, 31077 Toulouse Cedex, France c School of Chemistry, University of Bristol, England, UK
Received 27 September 2000; received in revised form 25 June 2001
Abstract The magnetic behaviour of nanoparticles of antiferromagnetic ferritin has been investigated by means of magnetic measurements. First, using a combination of low-field susceptibility and magnetisation measurements (up to 5:5 104 Oe) in the superparamagnetic regime (30–250 K), we propose a method to estimate the N!eel temperature of ferritin, by which we obtained TN C500 K. The hysteresis loop at 2.5 K has been measured, and interpreted with a simple model, where the irreversibilities are due to the switching of the uncompensated moments. The magnetisation curve at 2.5 K up to 30 104 Oe has also been measured, and we show that it can be interpreted in terms of superantiferromagnetism, which is a finite size effect predicted by N!eel in 1961, but which has not been previously experimentally evidenced. Superantiferromagnetism occurs in small antiferromagnetic particles with an even number of ferromagnetic reticular planes, and results in an enhancement of the antiferromagnetic susceptibility and in a non-linear field dependence of the magnetisation at higher fields. r 2002 Elsevier Science B.V. All rights reserved. PACS: 75.50.Ee; 75.50.Tt; 75.60.Ej Keywords: Ferritin; Antiferromagnets; Uncompensated magnetisation; Hysteresis; Superantiferromagnetism
1. Introduction Ferritin, the iron-storage protein of mammals, is made of a hollow protein shell, with a diameter about 12 nm, in which a natural single crystal of a ferric oxihydroxide is embedded [1]. The single crystal, or ferritin core, has a diameter varying between 2–3 nm and 7 nm. The Fe3þ spin arrange*Corresponding author. Tel.: +33-169-087517; fax: +33169-088786. E-mail address: [email protected] (P. Bonville).
ment in the core is antiferromagnetic (AF), with a Ne! el temperature which has been estimated to be . 240 K by a Mossbauer study [2] and 460 K by a recent magnetisation study [3]. In zero field, the direction of the moments (easy antiferromagnetic axis) is determined by the magnetocrystalline anisotropy. An assembly of ferritin proteins, either natural as extracted for instance from horse spleen, or artificially reconstituted from apoferritin (the empty protein shell) via the injection of Fe2þ ions and in situ bio-oxidisation [4], constitutes an ensemble of antiferromagnetic nanoparticles,
0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 4 6 1 - 9
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distributed in size, whose magnetic properties have been investigated by a number of authors in the last decade [3,5–9]. The ferritin cores are superparamagnetic [10], i.e. the two AF sublattice magnetisations, which are strongly coupled by the exchange interaction, undergo thermally activated fluctuations across the anisotropy barrier. In the presence of a magnetic field much smaller than the exchange field, the magnetisation of an ensemble of randomly oriented ferritin particles (or more generally antiferromagnetic nanoparticles) is found to be the sum of two contributions [11]. The first arises from the slight canting of the two AF sublattices and gives rise to a term linear in field: wAF H; where wAF is the powder AF susceptibility. The second arises from the uncompensated magnetic moments occurring within the body of the particle or at the surface, which confers a non-zero net moment to the particle. Owing to the strong exchange coupling, this excess moment can be assumed to follow the dynamics of the AF sublattices, at least for temperatures well below TN [11]. As our measurements are performed for To0:5TN ; we will consider that the uncompensated net moment is rigidly coupled to the staggered AF magnetisation, with probably a small misalignment due to surface moments. In a previous work [8], we showed that, at thermal equilibrium with respect to the characteristic time scale of the magnetic measurements (superparamagnetic regime), the uncompensated magnetisation in ferritin does not follow a Langevin law as is usually assumed. Instead, we proposed a new function which is better adapted for antiferromagnetic nanoparticles because it takes into account the predominance of anisotropy over the Zeeman effect, characteristic of weakly uncompensated antiferromagnets. In the present work, we will first show how one can determine the Ne! el temperature in ferritin, by combining low-field and high-field magnetisation data, in the temperature range 30–250 K, and using the new field and temperature dependence of the uncompensated magnetisation derived in Ref. [8]. Then, we will focus on the low temperature (2.5 K) properties, namely the hysteresis loop and the very high-field (up to 30 104 Oe) magnetisation curve. The hysteresis loop in ferritin has been
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previously measured [3,5,12], but no quantitative interpretation has been put forward. We present here a simple model, based on the description of an antiferromagnetic particle given by Mrup [13], which ascribes the irreversibilities (i.e. the hysteresis) to the switching of the uncompensated moments. Our model reasonably reproduces the main features of the loop, i.e. the irreversibility field and the remnant moment. Finally, we present the magnetisation curve at 2.5 K with a magnetic field up to 30 104 Oe, and show that it can be accounted for by the phenomenon of superantiferromagnetism, predicted by Ne! el 40 years ago [11,14,15]. To our knowledge, this is the first experimental evidence of superantiferromagnetism up to now.
2. Experimental The susceptibility at H=80 Oe and the magnetisation curves for fields up to 5:5 104 Oe, as well as the hysteresis loop, were measured in a commercial SQUID magnetometer (2.5–300 K). The magnetisation curve at 2.5 K up to 30 104 Oe was measured at the Service National des Champs Magne! tiques Pulse! s (SNCMP, INSA) at the University of Toulouse (France). A solution of commercial ferritin (from SIGMA corporation) was used for the susceptibility and 5:5 104 Oe magnetisation curves, with a protein concentration of 100 mg/ml. The 30 104 Oe curve was measured in the same commercial ferritin, but with a concentration of 300 mg/ml to ensure a larger signal, and it was calibrated by scaling with the signal of an identical sample measured up to 5:5 104 Oe in the SQUID magnetometer. The hysteresis loop was measured in a solution of artificial ferritin, made at the University of Bristol, in which the core contained an average of 2570 Fe atoms [8], with a protein concentration of 7.5 mg/ml. Interaction effects between ferritin cores play a very minor role, as far as their magnetic properties are concerned, as demonstrated in Ref. [5] for commercial ferritin with the standard 100 mg/ml concentration. In our samples, with different concentrations and thus different interparticle distances, the dipolar fields
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are lower than 1 Oe and therefore they will be neglected. Size histograms were obtained in the artificial ferritin sample with 2570 Fe atoms per core from Transmission Electron Microscope pictures. The cores have a mean diameter d0 =5.7 nm, with a standard log-normal diameter deviation sd =0.15– 0.20. As for horse spleen ferritin, its FC-ZFC susceptibility and magnetisation curves are almost identical to that of our artificial ferritin sample with 2570 Fe atoms per core. This indicates that the size distribution and the mean Fe loading in both samples are similar, and so we will use the above mean diameter and standard deviation for commercial ferritin as well. In our experiments, the samples were cooled in zero field, implying that, in the zero field configuration, the easy antiferromagnetic axes are oriented at random in space.
3. Determination of the N!eel temperature of ferritin The superparamagnetic regime, where the sublattice magnetisations of all the ferritin cores fluctuate rapidly with respect to the characteristic time of the measurement technique, is reached, in the ferritin samples under consideration, at 30 K for the magnetic measurements (tw C100 s) and at . 90 K for the 57 Fe Mossbauer spectroscopy measurements (tL C5 109 s) [3,8]. This makes the determination of the Ne! el temperature by usual methods impossible. We present here susceptibility (H=80 Oe) and magnetisation measurements in the temperature range 30–250 K and show how these data allow the Ne! el temperature to be determined. The magnetisation curves above 30 K are shown in Fig. 1. As stated in the introduction, they consist in the sum of an antiferromagnetic canting term, linear in the field, and of a contribution due to uncompensated moments. This latter contribution is analysed according to Refs. [8,9], i.e. by replacing the traditional Langevin function by a new function GðxÞ which can be written as Z p GðxÞ ¼ 12 dy sin y cos y tanhðx cos yÞ: ð1Þ 0
Fig. 1. Isothermal magnetisation curves in commercial horse spleen ferritin. The solid lines are fits to expression 2.
The reason why the GðxÞ function is a better approximation than the Langevin function in AF particles, and for fields lower than the spin-flop field, is that due to the smallness of the net uncompensated moment, the anisotropy energy is greater than the Zeeman energy. Then, the sublattice moments, and hence the uncompensated magnetisation, are not free to fluctuate randomly, but they fluctuate along the easy magnetic axes (oriented at random in the sample). Then, taking into account the size distribution of the particles, the field and temperature dependence of the magnetisation can be written as: Z Vmax 1 mðH; TÞ ¼ wAF ðTÞ H þ /VS Vmin m H dVf ðVÞ mnc ðV; TÞ G nc ; ð2Þ kB T where mnc ðV; TÞ is the uncompensated moment of a particle with volume V and f ðVÞ is the log-normal function describing the volume distribution. For the volume dependence of the uncompensated pffiffiffiffimoment mnc ; we chose: mnc ðV; TÞ ¼ uðTÞm0 N ; according to an argument given by Ne! el [16] when the uncompensated moments arise from a random occupation of the two sublattices by the magnetic ions. In the above
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expression, m0 ¼ 5 mB is the Fe3þ ionic moment, N is the number of Fe atoms in the particle proportional to the volume through the relation: N ¼ ðNa =/VSÞV; where Na is the mean number of Fe atoms in the core, and uðTÞ is a proportionality parameter. The fits of the magnetisation curves to expression (2) are very satisfactory and yield the two quantities wAF ðTÞ and uðTÞ; where both decrease as the temperature is increased. The thermal decrease of wAF in AF particles, opposite to the bulk behaviour, has been attributed by Ne! el [15] to a finite size effect occurring in those particles that show T ¼ 0 superantiferromagnetism, to be described in Section 5. The thermal variation of the susceptibility with a field of 80 Oe is shown in Fig. 2, together with that of the AF susceptibility wAF : The difference between the two quantities is the contribution of the uncompensated moments, as is easily seen from the low field limit of expression (2) wðTÞ wAF ðTÞ ¼
1 /mnc ðTÞS2 ; /VS 3kB T
ð3Þ
where the mean uncompensated pffiffiffiffiffiffi moment is defined as: /mnc ðTÞS ¼ uðTÞm0 Na : From both magnetisation and susceptibility data, a thermal varia-
Fig. 2. Thermal variation of the 80 Oe susceptibility w and of the antiferromagnetic susceptibility wAF extracted from the isothermal magnetisation data. The dashed pffiffiffiffi line represents the empirical law: wAF ðTÞ ¼ 1:68 103 = T ; which correctly reproduces the thermal variation of the AF susceptibility between 30 and 250 K.
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tion of the mean uncompensated moment /mnc ðTÞS can be extracted. The two sets of points are shown in Fig. 3. They do not coincide, but they are close to each other, and present a thermal decrease which reflects that of the sublattice magnetisation. In the temperature range of the measurements, the thermal variation of the susceptibility derived set can be reproduced by a T 2 law, reminiscent of the magnetisation decrease due to antiferromagnetic magnons. In order to obtain an estimation of the Ne! el temperature, we plotted a T 2 law through both sets of the data points up to 300 K (solid lines on Fig. 3), and we extrapolated these laws to higher temperature (dashed lines in Fig. 3). The intercept of these curves with the Taxis yields an estimation of TN ; probably slightly overestimated. Thus we take the value TN C500 K as a correct magnitude for the Ne! el temperature in ferritin. This value is close to the estimation made in Ref. [3] (460 K), although the analysis of the magnetisation curves with a modified Langevin law in this latter work yields a linear thermal decrease for the parameter identified as the uncompensated magnetisation. The mean saturated uncompensated moment we obtain (/mnc ð0ÞSC150 mB ) is by contrast much smaller than that derived in Ref. [3] (345 mB ).
Fig. 3. Thermal variations of the mean uncompensated moment in ferritin derived from magnetisation and susceptibility measurements. The solid lines are fits to a law: /mnc ðTÞS ¼ /mnc ð0ÞSð1 aT 2 Þ between 0 and 300 K, and the dashed lines are extrapolations of these laws.
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From this TN value, we can obtain the first neighbour exchange field HE (assuming that the second neighbour exchange is negligible) from the molecular field expression: kB TN ¼ 13m0 HE : We get HE ¼ 320 104 Oe. The 0 K perpendicular AF susceptibility in the molecular field approximation is given by
4. The hysteresis loop at 2.5 K
measured in ferritin cores [3,5,12] and in NiO small particles [18]. The AF hysteresis loop is quite different from that in ferro- or ferrimagnets: it remains open up to much higher fields (a few 104 Oe), and it is superposed onto the field linear AF contribution, whereas the ferromagnetic magnetisation (almost) saturates quite rapidly for fields of a few kOe. As perfectly compensated antiferromagnets do not show hysteresis, it seems reasonable to assign its presence in AF single domain nanoparticles, in a way similar to ferromagnets, mainly to the history-dependent switching of the uncompensated moments. The hysteresis loop at 2.5 K in the artificial ferritin sample with 2570 Fe atoms per core is shown in Fig. 4, in a field ranging from 5:5 up to 5:5 104 Oe. The irreversibility field, where the ‘‘up’’ and ‘‘down’’ branches coalesce, is: Hirr C3 104 Oe; the remanent moment mr ðC4 emu=cm3 ) amounts to about 0:25/mnc S: The interpretation of hysteresis in uncompensated antiferromagnets is not well documented in the literature, so here we will develop our model at some length, and present the characteristic features of the irreversibilities in these systems.
In the single domain particles of nanometric size, hysteresis in the magnetisation curve is known to arise from the competition between the particle anisotropy energy and the Zeeman energy, after the early work of Stoner and Wohlfarth [17]. Irreversibility is due to history-dependent magnetisation reversal, and complications may arise from surface effects in very small particles, where the surface moments can behave differently from those in the core. Hysteresis loops in ferro- or ferrimagnetic particles have been intensively studied, in view of potential applications for information storage, and their characteristics are well known: the coercive (or reversal) field is of the order of K=Ms (K is the anisotropy energy density and Ms the magnetisation, supposed to be uniform throughout the particle volume), i.e. of a few kOe, and the T ¼ 0 remanent magnetisation (after the field has been decreased) is Ms /2 in the case of uniaxial anisotropy. In the single domain antiferromagnetic particles, the low temperature hysteresis loops of the magnetisation have been
Fig. 4. Hysteresis loop at 2.5 K in the artificial ferritin sample with a mean Fe loading of 2570 atoms per core. The solid line is the fit using the model of reversal of uncompensated moments, with a degree of uncompensation a ¼ 0:018 and an exchange field HE ¼ 320 104 Oe.
w> ¼
Na m0 M0 ¼ 2/VSHE HE
ð4Þ
where M0 is the magnetisation of one AF sublattice, and the 0 K powder AF susceptibility is: wAF ¼ 23w> : With the values: Na ¼ 2570 and HE ¼ 320 104 Oe, we obtain a theoretical estimate for the powder AF susceptibility in ferritin: wAF C15 105 emu/cm3 : By extrapolating the measured wAF to 0 K, we get the experimental value: wAF C40 105 emu/cm3 ; which is 2.5 times larger than expected from the molecular field expression. We will come back to this problem in Section 5, which deals with very high field magnetisation curve and superantiferromagnetism.
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We use a formulation introduced by Mrup [13] to describe an AF nanoparticle, with the assumption that the uncompensated moments are rigidly linked to the AF sublattices moments by a strong exchange interaction. The degree of uncompensation is defined by the parameter a ¼ mnc =M0 V: The anisotropy is assumed to be of uniaxial type, and an anisotropy field is defined as: HA ¼ K=M0 : At T ¼ 0; in the presence of a magnetic field H, the equilibrium sublattice moments lie in the (H,e) plane, where e is the easy magnetic axis. Defining as yb the initial orientation of the easy axis in zero field and y the orientation of the sublattice moments with respect to H, the energy of the particle can be written as:
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this symmetry is broken and the energy profiles become asymmetrical. For yb ¼ 1001; the energy profile, shown in the upper panel of Fig. 5, has two well-defined potential wells, in the intervals ½p; 0 and ½0; p; for all applied field values. Therefore, when the field is increased or decreased, the system remains in the initial potential well, contained in the interval ½0; p: There is no reversal of the
Eðy; yb ; aÞ ¼ M0 HE ½ð1 þ aÞ þ 12b2 sin2 y þ ab cos y þ kð1 þ aÞ cos2 ðy yb Þ; ð5Þ where b ¼ H=HE and k ¼ HA =HE : This expression is valid for b51; k51 and when the ‘‘canting angle’’ 2e of the two sublattices has its equilibrium (small) value: 2e ¼ b sin y: In order to study the irreversibilities occurring when the field is increased and decreased, one must investigate the energy profiles as a function of the field and of the angle yb : In small ferromagnetic particles, all the moments lying in the lower halfplane (yb > p2) will reverse when the field reaches a critical value Hc ðyb Þ; i.e. the energy profiles present a switching of the potential well from the interval [0,p] to the interval [p;0] at the critical value. The situation is not exactly the same in small AF particles. To illustrate this, we consider two particular values of the easy axis initial orientation: yb ¼ 1001 and yb ¼ 1501: The energy profiles, given by Eq. (5), were calculated for a ¼ 0:02; HE ¼ 320 104 Oe and HA ¼ 0:1 104 Oe, which are typical values in ferritin (the anisotropy constant in artificial ferritin with 2570 Fe atoms per core [8] is: K ¼ 3:5 105 ergs/cm3 ). For a fully compensated AF particle, the energy profiles present two equally deep potential wells corresponding to the invariance of the system by inversion of the two sublattice moments. When the particle possesses an uncompensated moment,
Fig. 5. Energy profiles for an uncompensated antiferromagnet for different values of the applied field (represented by the numbers near the curves, in units 104 Oe), for a degree of uncompensation a ¼ 0:02 and an exchange field HE ¼ 320 104 Oe. Upper panel: yb ¼ 1001–the two wells present a minimum for all field values. Lower panel: yb ¼ 1501–for H=5 and 10 104 Oe, the well on the righthand side has no minimum, implying irreversibility of the magnetisation reversal.
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moment and thus no irreversibility. The energy profiles corresponding to yb ¼ 1501; shown in the lower panel of Fig. 5, are different. The two potential wells are still clearly present for fields below 2 104 Oe, but at higher fields (5210 104 Oe in the figure), the less shallow well in [0; p] is smeared out. This implies that the uncompensated moment ‘‘reverses’’ into the second well in the interval [p; 0] for a critical field value, and that it remains in this well when the field is further increased and then decreased. This gives rise to an irreversibility, and for this yb value there is hysteresis. So, in small uncompensated AF particles, and contrary to ferromagnetic particles, the reversal of magnetic moments does not occur for all values of yb > 901: There is a threshold angle ylim for reversal which depends on a (or rather on a=K). Its variation with a is shown in Fig. 6. For a ¼ 0; ylim ¼ 1801; as in fully compensated AF particles, the energy profile presents symmetrical wells, and there is no hysteresis; ylim decreases as a increases, and rapidly saturates at the value 901; corresponding to the ferromagnetic case. As can be seen in Fig. 5, there is a limited field range where only one potential well is present. Its boundaries can be approximately set by saying that the Zeeman energy associated with the uncompensated moment mnc H must be larger than
Fig. 6. Variation of the threshold angle for moment reversal ylim in uncompensated AF nanoparticles, as a function of the degree of non-compensation a; for an exchange field HE ¼ 320 104 Oe.
the anisotropy energy KV but lower than the Zeeman energy w> H 2 V associated with the canting of the two antiferromagnetic sublattices. Using w> ¼ M0 =HE ; one gets the field range HA oHoaHE : a
ð6Þ
The regions in the (H,yb ) plane with one or two potential wells are shown in Fig. 7 for a ¼ 0:02: The lower curve in Fig. 7 represents the variation with yb of the reversal field, i.e. of the irreversibility field. It is similar to the well known angular variation of the coercive field in ferromagnetic materials, with the difference that there is no moment reversal for p=2pyb pylim in AF nanoparticles. The irreversibility field in AF particles shows little angular variation and is approximately equal to HA =a; which is 3 104 Oe in ferritin. Finally, in Fig. 8 we represent the variation with a of the ratio of the remanent moment mr to the uncompensated moment. In the case of ferromagnetic particles in the axial anisotropy case, the remanent moment is equal to the half of the saturated moment. This is not so in AF particles, where mr omnc =2: As can be seen in Fig. 8, the curve mr =mnc ðaÞ saturates rather rapidly, as a increases, to the limiting value mr =mnc ¼ 12; which corresponds to the ferromagnetic case.
Fig. 7. Boundaries of the regions in the (H,yb ) plane where one or two potential wells are present, for a degree of uncompensation a ¼ 0:02; and for Na ¼ 2570 and HE ¼ 320 104 Oe.
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Fig. 8. Variation of the ratio mr =mnc with a; for uncompensated AF particles.
Using the above-described model, we calculated the different branches of the hysteresis curve of the magnetisation and applied it to ferritin (see Fig. 4). For a given yb value, the magnetisation is obtained by qE ¼ w> sin2 ymin H qH þ aM0 cos ymin ;
mðyb ; HÞ ¼
ð7Þ
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wAF value, an anisotropy constant K ¼ 3:5 105 ergs/cm3 (yielding /HA S ¼ 0:06 104 Oe), and a mean degree of uncompensation /aS ¼ 0:018: The irreversibility field HA =aC3 104 Oe and the remanent moment are well reproduced by our calculation, whereas the ‘‘pinching’’ of the loop (the change in concavity of a given branch as the H-axis is crossed) is not. This can be due to the overall simplicity of the model which, for instance, does not allow for surface moments to behave differently from those in the bulk. Recently, a more complex micromagnetic-like calculation has been developed for computing the hysteresis loop in antiferromagnetic NiO [18], which reproduces rather well the experimental data in this compound.
5. The high field magnetisation curve and superantiferromagnetism The magnetisation curve in ferritin at 2.5 K for fields up to 30 104 Oe is shown in Fig. 9; also shown is the expected molecular field response (dashed line) for a classical (compensated) powder antiferromagnet with an exchange field HE ¼ 320 104 Oe, corresponding to a Ne! el
where ymin is the position of the potential well corresponding to the path under consideration. The powder magnetisation is obtained by integration over the easy axis orientations yb Z p mðHÞ ¼ 12 Mðyb Þcos ymin sin yb dyb : ð8Þ 0
In Eq. (7), w> ¼ M0 =HE is the classical AF transverse susceptibility and gives rise to the overall slope of the loop. When trying to reproduce the experimental hysteresis curve, it had to be considered as a fitting parameter, because the classical value M0 =HE yields a much too low slope wAF ¼ 23w> with respect to the experimental data. The fitted value is: wAF C45 105 emu/cm3 ; which is about 3 times larger than the expected value. This enhanced AF susceptibility, already pointed to in the preceding section, is due to superantiferromagnetism (see Section 5). The solid line in Fig. 4 is the theoretical hysteresis loop obtained with the above quoted
Fig. 9. Magnetisation curve in horse spleen ferritin at 2.5 K. The dashed line is the calculated powder magnetisation for a compensated antiferromagnet with an exchange field HE ¼ 320 104 Oe and Na ¼ 2570:
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temperature TN ¼ 500 K such as determined for ferritin in Section 3, and a number of magnetic ions Na ¼ 2570: For a classical antiferromagnet, the powder magnetisation curve at high field is linear with a slope w> : At low field, it is linear with a slope wAF ¼ 23w> in the fully compensated case, and, in the presence of a small uncompensated moment, a small extra contribution arises which rapidly vanishes as the field is increased towards pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the spin-flop field Hsf ¼ 2HA HE : It is clear that the standard molecular field model cannot account for the magnetisation at 2.5 K in ferritin: the experimental mðHÞ curve is not linear with the field, and its magnitude is somehow larger than the molecular field curve. Thus, even in the hypothesis of a large overestimation of the exchange field, which would yield a larger w> ; the experimental data could not be reproduced. In 1961, Ne! el described what he called ‘‘superantiferromagnetism’’ [11,14,15], which consists of an enhancement of the AF susceptibility in small AF particles having an even number 2N of ‘‘active’’ (ferromagnetic) reticular planes. This phenomenon is a boundary effect which actually occurs for any particle size, but Ne! el showed that it is only observable below a limiting field
where l1 and l2 are the exchange constants with the first and second neighbours, the induced magnetisation for an AF particle with an even number of active reticular planes is written as Mp k msAF ðHÞ ¼ Mp cos j0 þ tan y Z p=2 sin2 j dj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð11Þ j0 1 k2 sin2 j where Mp is the magnetic moment of a reticular plane, and j0 and k are quantities depending on the reduced field h ¼ H=Hl and on the parameter y; which are found by solving the two equations 8 Z p=2 h tan y dj > > < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; ¼ k j0 ð12Þ 1 k2 sin2 j > > : k sin j0 ¼ cos y: At low field, Ne! el shows that the susceptibility is given by: l1 wsAF ¼ 2w> 1 þ : ð13Þ l1 l2
i.e. it cannot be observed in bulk materials where 2N is very large. In ferritin, where HE C320 104 Oe and 2NC10 20; Hl amounts to a few 10 104 Oe, and superantiferromagnetism should be observable in the high field magnetisation experiments. Essentially, superantiferromagnetism consists in a continuous rotation of the direction of the staggered magnetisation within the particle, when a magnetic field is applied perpendicular to the AF axis. This yields an enhancement of the AF susceptibility by a factor which is exactly 2 when one takes into account only first neighbour exchange, and which can be larger if second neighbour exchange is considered. Introducing the parameter y defined by
The magnetisation curves calculated from Eqs. (11) and (12) for different values of the number 2N of active reticular planes and for a constant atomic density within the particles, are represented in Fig. 10. The susceptibility at low fields is twice as great as that of the bulk, in agreement with expression (13), and the magnetisation at high field is not linear for the small values of 2N: In order to apply the superantiferromagnetic calculation to our ferritin sample, we first have to perform a powder average; for this purpose, we usedthe equilibrium orientation ye of the AF sublattices as obtained for a perfect antiferromagnet in Ref. [19]. The induced powder magnetisation is then obtained, for each initial orientation yb of the AF axis, by considering the component of the applied field perpendicular to it and by integrating over the random directions of the AF axes: Z p mðHÞ ¼ 12 msAF ðH sin ye Þsin ye sin yb dyb : ð14Þ
l1 tan y ¼ ; l1 4l2
We then assume that, statistically, there is an equal number of particles with an even and odd
Hl ¼
HE ; N
ð9Þ
0
2
ð10Þ
C. Gilles et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 430–440
Fig. 10. Magnetisation curves for antiferromagnetic particles with increasing even number 2N of active reticular planes (2N=10, 20, 100, solid lines) and for a bulk antiferromagnet (dashed line), illustrating the phenomenon of superantiferromagnetism. The curves were calculated with l2 =0 and with parameters close to that of ferritin: an exchange field HE ¼ 320 104 Oe, an atomic density: r ¼ 2:5 1022 cm3 ; and an interreticular spacing e ¼ 0:5 nm.
number of active planes. Those with even number of active planes show a superantiferromagnetic response, and those with odd number of active planes show a classical antiferromagnetic response. The best fit to the experimental data, shown as a solid line in Fig. 11, is obtained with 2N ¼ 8; which is compatible with the mean size of the particles of ferritin, and with a ratio l2 =l1 ¼ 0:14: For fields below 7 T, the calculated curve lies below the experimental data; this corresponds to the contribution of the uncompensated moments, which are not taken into account in the model. At high field, this contribution vanishes because the spin configuration is mainly of spin-flop type, i.e. the uncompensated moments are perpendicular to the applied field. So we believe that, although approximations have been made in the calculation of the high field magnetisation in our ferritin sample, its behaviour shows the phenomenon of superantiferromagnetism, which is thus experimentally evidenced for the first time. Superantiferromagnetism should also be observable at low fields, but it is usually masked by the contribution
439
Fig. 11. The Magnetisation curve in ferritin at 2.5 K and the theoretical curve (solid line) assuming 50% of superantiferromagnetic particles (with even number of active planes) and 50% of classical AF particles (with odd number of active planes).
of the uncompensated moments, in the frozen regime as well as in the superparamagnetic regime.
6. Conclusion Magnetisation and susceptibility measurements have been performed in antiferromagnetic nanoparticles of ferritin, both natural and artificial, in the temperature range 2.5–250 K, and with a magnetic field up to 30 104 Oe. We measured the thermal variation of the mean uncompensated moment of the particles, and estimated the Ne! el temperature of ferritin: TN C500 K. The hysteresis loop at 2.5 K was interpreted by means of a simple model where the irreversibilities are attributed to the switching of the uncompensated moments; we show that the irreversibility field for an antiferromagnetic particle is: Hirr CHA =a; where HA is the anisotropy field and a the degree of uncompensation, i.e. the ratio of the uncompensated moment to the sublattice moment. A 30 104 Oe magnetisation curve was measured at 2.5 K and we show that it evidences the phenomenon of superantiferromagnetism, predicted by Ne! el in 1961, which consists in an enhancement of the antiferromag-
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netic susceptibility in small particles with an even number of reticular active planes.
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