ARTICLE IN PRESS

Journal of Magnetism and Magnetic Materials 300 (2006) e294–e297
www.elsevier.com/locate/jmmm

SQUID-measurements of relaxation time of Fe3O4 superparamagnetic
nanoparticle ensembles
I. Volkova,, S. Gudoshnikovb,c, N. Usovb,c, A. Volkovd, M. Moskvinad, A. Maresova,
O. Snigireva, S. Tanakae
aPhysics Department of Moscow State University, Moscow 119992, Russia
Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation RAS (IZMIRAN), Troitsk, Moscow region 142190, Russia
cMagnetic and Cryoelectronic Systems Ltd., 4/1, Bozhenko Street, Moscow 121351, Russia
dChemical Department of Moscow State University, Moscow 119992, Russia
eToyohashi University of Technology, Toyohashi, Japanb

Available online 15 November 2005

Abstract
High-TC scanning SQUID magnetometer is applied for measuring the magnetization relaxation of the ensembles of spherical colloidal
Fe3O4 nanoparticles dispersed in the rigid polymer matrix preventing nanoparticles from agglomeration. According to the transmission
electron microscopy data, the size distribution of nanoparticles is well described by the Gaussian proﬁle with the mean size and thedispersion of about 7.5 and 2.3 nm, respectively. High sensitivity of SQUID magnetometer allows us to study samples at low contents ofthe magnetic component (0.1–1.0 vol%) magnetized in low magnetic ﬁeld (10–4 T) produced by a low-inductance coil with shortswitching time (12 ms). A low content of the magnetic component provides the absence of interparticle dipolar interactions, thus
simplifying signiﬁcantly the theoretical description of the magnetization relaxation process. The measured magnetization relaxation
curves are found to be in reasonable agreement with the numerical simulation data.
r 2005 Elsevier B.V. All rights reserved.
PACS: 75.20.g; 85.25.Dq; 75.50.Tt; 75.40.Mg
Keywords: Magnetic nanoparticles; Superparamagnetic relaxation; SQUID magnetometry; Numerical simulation

The magnetic nanomaterials come into ever increasing
use in various areas of science and technology. Magnetic
storage of information [1,2] and immunoassays utilizing
magnetic labeling technique [3,4] are one of the mostadvanced applications of magnetic nanomaterials. In orderto predict and analyze the behavior of single-domain
nanoparticles, one must take into account the superparamagnetic phenomena occurred due to thermally
activated switching of their magnetic moments. In thiscontext, SQUID magnetometry has been successfully
applied to measure the switching times of individual
nanoparticles [5,6] and the spontaneous magnetic noise
arising in ﬁlms of self-assembled Co nanoparticles [7].

Corresponding author. Tel.: +7 095 939 3343; fax: +7 095 939 3000.

E-mail address: volkov256@yandex.ru (I. Volkov).
0304-8853/$ - see front matter r 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.jmmm.2005.10.104

In the present paper, the high-TC scanning SQUID
magnetometer [8] is used to measure the magnetization
relaxation curves of a dilute ensemble of spherical Fe3O4nanoparticles. A series of samples with the mean size of
nanoparticles of about 7 nm was studied. The nanoparticles
were dispersed in the rigid polymer matrix (polyvinylalcohol+chitosan) preventing them from agglomeration.
The low content of the magnetic component (0.2 vol%) in
the samples enabled us to neglect the interparticle dipolarinteractions in the theoretical analysis. The transmission
electron microscopy (TEM) was used to determine the size
distribution of magnetic nanoparticles and to control thehomogeneity of their spatial distribution in the matrix.
According to the TEM image presented in Fig. 1, theensemble consists of well-separated particles with nearspherical shape. Note that the areas of irregular shape inthis image are caused by the overlap of projections of

ARTICLE IN PRESS
I. Volkov et al. / Journal of Magnetism and Magnetic Materials 300 (2006) e294–e297

F(Di)

0.4
F(Di) and S(D), nm-1

S(D)
0.3

0.2

0.1

0.023456789Particle size D, nm

10111213Fig. 2. Discrete F ðDi Þ and continuous SðDÞ functions of the size
distribution of the volume fraction of magnetic component.

particles located at different heights. Fig. 2 shows the size
distribution histogram F ðDi Þ approximated by a normalized Gaussian proﬁle SðDÞ with the mean diameter D0 ¼
7:47 nm and the dispersion s ¼ 2:32 nm.
The SQUID sensors were fabricated from high-Tc
superconductor ﬁlms (YBa2Cu3OX) deposited on SrTiO3
bicrystal substrates with misorientation angles of 241 and301. The fabrication process included photo- and e-beamlithographies followed by wet and ion beam etchings[9–11]. The SQUIDs had a geometry of square washer withthe following parameters: the hole dimension 40 mm andthe outer dimension 120 mm.
The magnetization relaxation measurements were performed on the samples with dimensions of 0.6  0.6 
0.06 mm3. The SQUID sensor was located close to thesample surface at a height of about 300 mm. The SQUID
plane was roughly parallel both to the sample surface andto the applied ﬁeld direction. The measurements werecarried out at a temperature of T ¼ 77 K inside a m-metalshield with the remanent magnetic ﬁeld as high as 107 T.
In order to magnetize the sample, a weak external magneticﬁeld (68 A/m) generated by a coil was applied parallel tothe sample surface. After several minutes allowing thesample magnetization to attain its equilibrium, the SQUID
sensor scanned the z-component of the stray magnetic ﬁeld
Bz ðx; yÞ produced by the sample and the obtained data
were saved in the ﬁle. Finally, the SQUID sensor waslocated at the maximum of Bz-component (close to thesample edge) and the external magnetic ﬁeld was switched
off. Note that the initial sample magnetization wasquantitatively retrieved from the Bz ðx; yÞ data using
mathematical procedure mentioned in Ref. [12].
In order to switch off the magnetizing ﬁeld, a special
electronic circuit was designed. It enabled us to switch off
the bias current ﬂowing through the coil to within 300 ns.
Since the residual current and residual magnetic ﬁeld
cannot become zero immediately after the bias current isswitched off, the following condition must be fulﬁlled: themagnetizing ﬁeld must relax much faster than the sample
magnetization. In our experiment, we have achieved arelaxation time of the external ﬁeld (99% decay of themagnitude) of about 12 ms. After a 12-ms time gap, westarted registration of useful relaxing signal produced by
the sample. Within this time gap, the amplitude of theuseful signal decreased from 5% to 20% of its initial value.
Nevertheless, the measured portion of the relaxation curvescontained very useful information on the distribution ofthe relaxation times in the ensemble under investigation.
Curve 1 in Fig. 3 demonstrates the measured magnetization
relaxation of the Fe3O4 ensemble (0.2 vol%) presented in
the logarithmic time scale.
In order to understand the experimental results obtained, we have calculated the relaxation curves of Fe3O4
nanoparticles with diameters D in the range from 5 to
11 nm corresponding to the size distribution histogram
presented in Fig. 2. At T ¼ 77 K, below the Verwey’stransition temperature T V ¼ 125 K, the particles aresupposed to be roughly uniaxial with the magnetocrystalline anisotropy constant K u ¼ 4:5  104 J=m3 and the

1.6Sample magnetization, A/mFig. 1. TEM image of Fe3O4 nanoparticles dispersed in the rigid polymermatrix at a content of 0.5 vol%.

e295

T = 77 K
H|| = 68 A/m
CV = 0.2 vol %

1.2

0.8
10.4
2
0.0-6-4-2lg (t/t0)

02Fig. 3. Magnetization relaxation in the dilute ensemble of Fe3O4
nanoparticles presented in logarithmic time scale: (1) experimental data;(2) numerical simulation result.

ARTICLE IN PRESSI. Volkov et al. / Journal of Magnetism and Magnetic Materials 300 (2006) e294–e297

e296

Table 1
Reduced energy barriers, relaxation times and average magnetization of Fe3O4 nanoparticles at T ¼ 77 K
D, nm

5678

910

11

KuV/kBTt, shMi=M s

2.77
5.4  109
1.9  103

4.79
3.1  108
4.0  103

7.61
4.1  107
7.2  103

11.35
1.4  105
1.1  102

16.16
1.5  103
1.7  102

22.17
0.51
2.4  102

29.51
680
3.2  102

saturation magnetization M s ¼ 5  105 A=m [13]. For
smaller diameters, 5oDo8 nm, the calculations were madeby means of direct integration of the Landau—Lifshitz
equation augmented by a stochastic thermal ﬁeld with awhite noise spectrum [14–16]. For lager particles,
9oDo11 nm, the analytical estimate [17] for the relaxation
time has been used, because for these sizes the numericalsimulation is time consuming. Table 1 summarizes thereduced energy barriers e ¼ K 1 V =kB T (V is the particle
volume and kB is the Boltzmann constant), relaxation times
and equilibrium particle magnetizations in applied magnetic ﬁeld with the magnitude of 68 A/m as a function ofparticle size D. Note that for the given ensemble, thehighest relaxation time corresponds to particles withD ¼ 11 nm, because the percentage of particles with
D411 nm is negligibly small. The highest relaxation timeturns out to be of the order of exposition time of theensemble in applied magnetic ﬁled (texp 300 s). Therefore,
practically all particles of the ensemble attain theirequilibrium magnetizations during the exposition time.
The forth row of the Table 1 gives, in fact, themagnetization susceptibility of the particles as a functionof their diameters at T ¼ 77 K.
Fig. 3 shows the measured magnetization relaxation
curve 1 for the dilute ensemble of Fe3O4 nanoparticles incomparison with the numerical simulation result (curve 2)
presented in the logarithmic time scale with t0 ¼ 1 s. The
time interval of the measurement is given by 50 s and theinitial time gap is 12 ms in this case. The volume fraction ofthe magnetic component of the ensemble is C V ¼ 0:2 vol%.
Similar data were also obtained for the ensembles withC V ¼ 0:5 and 1.0 vol%, though the characteristic relaxation time increased with increasing content of magnetic
component due to interparticle dipolar interactions.
The theoretical relaxation curves of the ensembles understudy were obtained by convoluting the calculated relaxation curves for the particles with given D ¼ 5–11 nm withthe size distribution histogram presented in Fig. 2.
According to Fig. 1, the shapes of the particles are near
spherical. Thus, in a good approximation one can neglectthe shape anisotropy of the particles. Besides, the mean
distance between the particles is estimated to be  6 D0 for
C V ¼ 0:2 vol%, where D0  7 nm is the mean particle
diameter. Therefore, the energy of interparticle dipolarinteraction is minor as compared to the anisotropy energy.
The calculations have been carried out for small values ofthe damping parameter, k ¼ 0:0120:05, because there is anindication [18] that the low-damping regime better
corresponds to the experimental situation for an ensembleof perfect magnetic nanoparticles. It is found that thedependence of the relaxation curve 2 on this parameter isminor within this interval. On the other hand, thetheoretical relaxation curve depends considerably on thevalue of uniaxial magnetocrystalline anisotropy constant.
One can see from the Fig. 3 that for assumed value ofK u ¼ 4:5  104 J=m3 [13] there is a reasonable agreement
between the experimental and numerical simulation data.
In conclusion, high-sensitive scanning high-TC SQUIDmagnetometer is used to measure the magnetization
relaxation curves of a dilute ensembles of Fe3O4 nanoparticles at T ¼ 77 K. This technique enables us to study
samples at low contents of magnetic component(0.1–1.0 vol%) magnetized in a weak magnetic ﬁeld of theorder of 104 T. The TEM is applied to determine the size
distribution of magnetic nanoparticles and to control thehomogeneity of their spatial distribution in the matrix. The
size distribution of nanoparticles is described well by
Gaussian proﬁle with the mean size and the dispersion ofabout 7.5 and 2.3 nm, respectively. The measured relaxation curves of the dilute ensembles are shown to be inreasonable agreement with the numerical simulation results
obtained by means of direct integration of the Landau—
Lifshitz equation augmented by a stochastic thermal ﬁeld
with a white noise spectrum.
This work was supported by the ISTC project # 1991.
References[1] D. Weller, A. Moser, IEEE Trans. Magn. 35 (1999) 4423.
[2] www.research.ibm.com/resources/news/20001207_mramimages.shtml[3] R. Kötitz, W. Weitschies, L. Trahms, W. Brewer, W. Semmler,J. Magn. Magn. Mater. 194 (1999) 62.
[4] S. Katsura, T. Yasuda, K. Hirano, A. Mizuno, S. Tanaka, Supercond. Sci. Technol. 14 (2001) 1131.
[5] W. Wernsdorfer, E.B. Orozco, K. Hasselbach, A. Benoit, B. Barbara,
N. Demoncy, A. Loiseau, H. Pascard, D. Mailly, Phys. Rev. Lett. 78(1997) 1791.
[6] M. Jamet, W. Wernsdorfer, C. Thirion, D. Mailly, V. Dupuis,P. Mélinon, P. Pérez, Phys. Rev. Lett. 86 (2001) 4676.
[7] S.I. Woods, J.R. Kirtley, Sh. Sun, R.H. Koch, Phys. Rev. Lett. 87(2001) 137205.
[8] S. Gudoshnikov, L. Matveets (Eds.), High-Temperature Superconductivity 2, Springer, Germany, 2004. p. 337.
[9] I. Volkov, A. Kalabukhov, O. Snigirev, A. Zherikhin, IEEE Trans.
Appl. Supercond. 11 (2001) 292.

ARTICLE IN PRESS
I. Volkov et al. / Journal of Magnetism and Magnetic Materials 300 (2006) e294–e297[10] I. Volkov, A. Kalabukhov, O. Snigirev, A. Zherikhin, Physica C
372–376 (2002) 72.
[11] I. Volkov, M. Chukharkin, O. Snigirev, IEEE Trans. Appl. Supercond. 13 (2003) 861.
[12] I. Volkov, M. Chukharkin, O. Snigirev, A. Volkov, M. Moskvina,
S. Gudoshnikov, A. Kerimov, IEEE Trans. Appl. Supercond., 2005,in press.
[13] S. Chikazumi, Physics of Magnetism, Wiley, New York, 1964.

e297[14] J.L. Garcia-Palacios, F.J. Lazaro, Phys. Rev. B 58 (1998) 4937.
[15] W. Scholz, T. Schreﬂ, J. Fidler, J. Magn. Magn. Mater. 233 (2001)
296.
[16] D.V. Berkov, IEEE Trans. Magn. 38 (2002) 2489.
[17] W.F. Brown Jr., Phys. Rev. 130 (1963) 1677.
[18] J.L. Dormann, F. D’Orazio, F. Lucari, E. Tronc, P. Prene,J.P. Jolivet, D. Fiorani, R. Cherkaoui, M. Nogues, Phys. Rev. B 53
(1996) 14291.