Journal of Magnetism and Magnetic Materials 262 (2003) 445–451

Magnetization reversal in weak ferrimagnets and canted
antiferromagnets
H. Kageyamaa, D.I. Khomskiib, R.Z. Levitinc, M.M. Markinac, T. Okuyamad,
T. Uchimotod, A.N. Vasil’evc,*aMaterials Design and Characterization Laboratory, Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa,Chiba 277-8581, Japanb
Solid State Physics Laboratory, Materials Science Center, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
cPhysics Faculty, Moscow State University, Moscow 119899, RussiadInstitute of Fluid Science, Tohoku University, Katahira 2-1-1, Aoba-ku, Sendai 980-8577, Japan

Abstract
In some ferrimagnets the total magnetization vanishes at a certain compensation temperature T*. In weak magnetic
ﬁelds, the magnetization can change sign at T* (the magnetization reversal). Much rarer is observation of ferrimagneticlike response in canted antiferromagnets, where the weak ferromagnetic moment is due to the tilting of the sublatticemagnetizations. The latter phenomenon was observed in nickel (II) formate dihydrate Ni(HCOO)2  2H2O. Theobserved weak magnetic moment increases initially below TN ¼ 15:5 K, equals zero at T ¼ 8:5 K and increases again
at lowering temperature. The sign of the low-ﬁeld magnetization at any given temperature is determined by the sample’s
magnetic prehistory and the signs are opposite to each other at ToT and T  oToTN :r 2003 Elsevier Science B.V. All rights reserved.
PACS: 75.50.Gg; 75.60.Jk
Keywords: Weak ferrimagnetism; Compensation point; Magnetization reversal; Ni(HCOO)2  2H2OThe phenomena of ferrimagnetism are associated with a partial cancellation of antiferromagneticallyaligned magnetic sublattices with different values of magnetic moments and/or different temperaturedependencies of magnetization. It is well studied in ferrites [1] and recently in various inorganic [2,3] andmolecular transition metal complexes [4,5]. Ferrimagnetic behaviour is observed frequently in compounds
containing different magnetic ions. It can be seen also in materials containing only one type of magnetic
ions, which are in different valence states or crystallographic positions [6,7]. In the latter case the origin offerrimagnetism lies in the difference of molecular ﬁelds acting on non-equivalent magnetic sites.
In some ferrimagnets (in Ne! el’s classiﬁcation N-type ferrimagnets [8]) the total magnetization of a
substance vanishes at a certain compensation temperature T  : Both above and below this temperature themagnetization of different sublattices prevails. In this case magnetization reversal can be observed. In a
weak magnetic ﬁeld (less than the ﬁeld of coercitivity) the magnetization changes sign at the compensation*Corresponding author.
E-mail address: vasil@mig.phys.msu.ru (A.N. Vasil’ev).
0304-8853/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved.
doi:10.1016/S0304-8853(03)00076-3

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H. Kageyama et al. / Journal of Magnetism and Magnetic Materials 262 (2003) 445–451

temperature. The metastable ‘‘diamagnetic’’ state can be ﬁxed by the magnetocrystalline anisotropy in acertain temperature range.
Much rarer is the observation of ferrimagnetic-like response in canted antiferromagnets, where a weakferromagnetic moment is due to the tilting of sublattice magnetizations. The single-ion magnetic anisotropy
itself is able to ﬁx the directions of sublattice magnetizations tilted from antiparallel alignment [9]. On theother hand, the canting of antiferromagnetic sublattices can be due to the antisymmetric Dzyaloshinsky–Moriya interaction [10–12], which favours perpendicular orientations of the sublattice magnetizations. Veryfew canted antiferromagnets show a compensation point and magnetization reversal [13–15].
We report the observation of weak ferrimagnetism, compensation point and magnetization reversal innickel (II) formate dihydrate Ni(HCOO)2  2H2O. This compound crystallises in the monoclinic P21/c spacegroup, and includes four formula units in the unit cell with a ¼ 8:60 A, b ¼ 7:06 A, c ¼ 9:21 A, b ¼ 96:5
[16]. The structure contains two kinds of Ni2+ ions: the Ni1 with six oxygen of different formate ions asnearest neighbors, the Ni2 with four water molecules and two formate oxygen coordinated. In both Ni1 andNi2 subsystems the octahedrons coordinating the Ni ions are tilted with respect to each other. The fourfold
axes of NiO6 octahedrons are oriented differently with respect to the crystal lattice axes. The crystal
structure of Ni(HCOO)2  2H2O in polyhedral NiO6 representation is shown in Fig. 1. In Ni1 subsystem thefourfold axes lie mainly in a2b crystal plane and are tilted over c1 =2 ¼22.5 in opposite directions from thea-axis. In the Ni2 subsystem the fourfold axes also lie mainly in the a2b plane and are tilted overc2 =2 ¼57.5 from the a-axis.
Magnetic properties of commercially available (Waco Co.) powder samples of Ni(HCOO)2  2H2O were
measured in a ‘‘Quantum Design’’ SQUID magnetometer in 2–300 K temperature range, in ﬁelds up to 7 T.
X-rays analysis conﬁrmed the single phase of the title compound in the measured sample.
The temperature dependencies of the magnetization, expressed as M=H; of a Ni(HCOO)2  2H2O powder
sample taken in zero-ﬁeld-cooling (ZFC) and ﬁeld-cooling (FC) regimes at H ¼ 0:01 T are shown in Fig. 2.
At heating in the ZFC regime, the sample shows ﬁrstly a large ‘‘paramagnetic’’ response at lowtemperatures. Then, the magnetic moment gradually decreases with increasing temperature, changes to‘‘diamagnetic’’ at T ¼ 8:5 K, and once again becomes paramagnetic above 15.5 K. At subsequent coolingin the FC regime, the magnetic response of the sample at low temperatures shows mirror-like behaviour
with respect to the magnetization sign as compared with ZFC measurements. The temperature dependence

Fig. 1. Crystal structure of Ni(HCOO)2  2H2O in polyhedral NiO6 representation. Light-shaded octahedrons belong to the Ni1subsystem, and dark-shaded octahedrons belong to the Ni2 subsystem. The C ions are shown by black circles and the pathways ofexchange interaction Ni–O–C–O–Ni are shown by solid lines.

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447

1.0ZFCFC

M/H (emu/mol)

0.5

0.0

-0.5

-1.0
05

1015

202530T (K)Fig. 2. Temperature dependence of the magnetization, expressed as M=H; of Ni(COOH)2  2H2O at H ¼ 0:01 T taken in ZFC (J) andFC () regimes.

1251/χ (mol/emu)

1007550250-250255075

100

125150T (K)Fig. 3. Inverse magnetic susceptibility of Ni(HCOO)2  2H2O at T > TN : The solid line represents the linear extrapolation from thehigh temperature region. The dotted line represents the numerical ﬁt by Eq. (5).

of the inverse susceptibility w1 of Ni(HCOO)2  2H2O at T > TN is shown in Fig. 3. At high temperatures w
follows the Curie–Weiss law with paramagnetic Curie temperature Y ¼ 15:5 K and effective magnetic
moment meff ¼ 3:14 mB : This value of meff is a typical one for a magnetic moment of Ni2+ ions. The negativevalue of Y implies that the main exchange interaction in Ni(HCOO)2  2H2O is antiferromagnetic.
The magnetization curves taken in the ZFC regime at ToTN are shown in Fig. 4. These curves areferromagnetic-like, i.e. they show spontaneous magnetic moments, are weakly non-linear at low ﬁelds andtend to linear dependencies at high magnetic ﬁelds. Note that the absolute values of the magnetization areby two orders of magnitude smaller than those corresponding to a parallel alignment of the Ni2+ magnetic

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H. Kageyama et al. / Journal of Magnetism and Magnetic Materials 262 (2003) 445–451
1800

2K

1600

4K

1400

M (emu/mol)

1200

6K
1000

8K
10 K
12 K
14 K

800
600
400
200
0
0.0

0.2

0.4

0.6

0.8

1.0

H (T)

Fig. 4. Magnetization curves of Ni(COOH)2  2H2O taken in the ZFC regime at ToTN .

0.16

0.14

χ (emu/mol)

0.12

0.10

0.08

0.060.04
0246810121416T (K)

Fig. 5. Temperature dependence of the high-ﬁeld magnetic susceptibility of Ni(HCOO)2  2H2O at ToTN :moments. The temperature dependence of the high-ﬁeld magnetic susceptibility w in the magnetic orderedstate, obtained from the slope of the magnetization curves at high ﬁelds, is shown in Fig. 5.
Therefore, the experimental data presented suggest that below TN ¼ 15:5 K a weakly ferrimagnetic stateis established in Ni(HCOO)2  2H2O, and that T ¼ 8:5 K is a compensation temperature.
According to crystal structure considerations the superexchange pathways between Ni ions can berealized through Ni–O–C–O–Ni links. As noted earlier, both the Ni1 and the Ni2 subsystem contain twokinds of Ni ions with different orientations of octahedral environments. Consequently, it is natural torepresent each Ni subsystem as consisting of two sublattices, namely Ni10 , Ni100 and Ni20 , Ni200 . The
superexchange Ni–O–C–O–Ni links exist between Ni10 and Ni100 and also between Ni10 and Ni20 , and Ni100and Ni200 . Therefore, each Ni10 ion interacts with four nearest neighboring Ni100 ions and two nearest

H. Kageyama et al. / Journal of Magnetism and Magnetic Materials 262 (2003) 445–451

bZ2''

449

Z2'

Z1'

Z1''

ψ2/2

M1'

ϕ1'

ϕ1''

αα

M2''

ββ
ϕ2' 'ϕ2'

M1''

ψ1/2

aM2'Fig. 6. The model for the weak ferrimagnetic structure of Ni(HCOO)2  2H2O. The magnetic moments M10 ; M100 ; M20 ; M200 of different
Ni ions are assumed to lie in the a2b plane. The dotted lines show the orientations of the principal axes (easy directions ofmagnetization) Z10 ; Z100 , Z20 , Z200 of different NiO6 octahedrons. The angles between these easy directions and the a axis are c10 ¼
c100 ¼ c1 and c20 ¼ c200 ¼ c2 : The deviations from antiparallel alignment of magnetic moments within the Ni1 and Ni2 subsystems aredesignated as a and b; respectively. The angles j10 ¼ j100 ¼ j1 and j20 ¼ j200 ¼ j2 give the deviations of the corresponding magneticmoments from the easy directions of magnetization.

neighboring Ni20 ions. The same is true for Ni100 ions. The pathway parameters, i.e. the interatomic
distances and angles, for all the superexchange interactions mentioned are close to each other. There are no
Ni–O–C–O–Ni superexchange pathways within the Ni2 subsystem. It means that in the absence of Ni1–Ni2exchange interactions the Ni2 subsystem can be considered paramagnetic down to the lowest temperatures.
The superexchange pathways are shown in Fig. 1 by solid lines. The crystal ﬁeld ﬁxes the orientations ofdifferent Ni magnetic moments in different directions with respect to the crystallographic axes and results inthe tilting of these moments from antiparallel alignment.
The properties of Ni(HCOO)2  2H2O in the framework of the model suggested can be described using thefollowing expression for a free energy:
E ¼ 4IM 10 M 100 þ 2IðM 10 M 20 þ M 100 M 200 Þ þ KNi1 sin2 j01 þ KNi1 sin2 j001
þ KNi2 sin2 j02 þ KNi2 sin2 j002  MðM 10 þ M 100 þ M 20 þ M 200 Þ:

ð1Þ

Here, I is the exchange interaction coefﬁcient, =M 10 = ¼ =M 100 = ¼ M1 and =M 20 = ¼ =M 200 = ¼ M2 are themagnetic moments of the Ni1 and Ni2 sublattices. The ﬁrst term in Eq. (1) describes the exchangeinteractions within the Ni1 subsystem. The second term describes the exchange interactions between the Ni1and Ni2 subsystems and these interactions are assumed to be the same on the basis of crystal latticeconsiderations. The next four terms describe of the magnetic anisotropy. The angles j01 ¼ j001 ¼ j1 andj02 ¼ j002 ¼ j2 are the angles between the easy axes of magnetization and the magnetic moments of thecorresponding Ni ions. It is natural to assume that the easy axes of magnetization coincide with theprincipal axes of the NiO6 octahedrons. The last term in Eq. (1) is a Zeeman energy.
The minimization of Eq. (1) with respect to j1 and j2 gives the equilibrium orientations of the Ni
magnetic moments. It was found that these moments lie in the a2b plane and are oriented as shown inFig. 6. According to this arrangement each Ni subsystem is weakly ferromagnetic and these weakly
ferromagnetic moments are oriented antiparallel to each other. Therefore, the resulting magnetic structureof Ni(HCOO)2  2H2O is a weakly ferrimagnetic one. In the assumption that the crystal ﬁeld energy for Ni

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450

ions is much smaller than the exchange interaction, the deviations from the antiparallel alignment will besmall. In the Ni1 and Ni2 subsystems these deviations will be denoted a and b; respectively.
The temperature dependencies of the sublattice magnetizations M1 and M2 can be calculated within theexchange approximation through the corresponding Brillouin functions BS :M1 ¼ gSmB BS ½IgSmB ð4M1 þ 2M2 Þ=kB T ;ð2Þ

M2 ¼ gSmB BS ½IgSmB ð2M1 Þ=kB T ;ð3Þ

where g is a g-factor, spin S ¼ 1 is the spin of a Ni2+ ion. These and forthcoming expressions are given fora formula unit.
Eqs. (2) and (3) allow to calculate the Ne! el temperature
2CI;ð4Þ
TN ¼ pﬃﬃﬃ
21
where C ¼ g2 SðS þ 1Þm2B =3kB :The magnetic susceptibility w at T > TN isC:
w¼T þ 4CI  4C 2 I 2 =T

ð5ÞClearly, w deviates from the Curie–Weiss law approaching the Ne! el temperature which reﬂects the presenceof two different magnetic subsystems in Ni(COOH)2  2H2O. The inverse paramagnetic susceptibility is
extrapolated down to zero at the paramagnetic Curie point
Y0 ¼ 2CIð1  O2Þ:

ð6ÞIn the exchange approximation, it is possible also to calculate the magnetic susceptibility in the magnetic
ordered state at ToT N :ð1 þ mÞ2;
ð7Þ16I
where m ¼ M2 =M1 : Note that the temperature dependence of w is not trivial. w increases at lowering
temperature due to the fact that the Ni2 subsystem itself is paramagnetic in the absence of exchangeinteractions with the Ni1 subsystem. Indeed, this behaviour was observed experimentally, as shown inFig. 5.
To calculate the weak ferromagnetic moment of Ni(HCOO)2  2H2O at ToTN ; it is necessary to step out
beyond the frames of the exchange approximation and to take the magnetic anisotropy into account. The
minimization of the free energy (1) at H ¼ 0 allows to obtain the deviations of the magnetic moments a andb from antiparallel alignment within both the Ni1 and Ni2 subsystems, and to calculate the resulting weak
ferrimagnetic moment
w¼

Mtotal ¼ M1 ða  mbÞ=2orð8Þ
M1 2q1 q2 sin c1 cos c2 þ m q1 sin c1  3q2 sin c2  2q1 q2 cos c1 sin c2  m2 q1 sin c1 þ q2 sin c2
Mtotal ¼:mð4 þ 2q1 cos c1 þ q2 cos c2 Þ þ 8q2 cos c2 þ 4q1 q2 cos c1 cos c22ð9ÞThe angles c1 and c2 are the angles between the easy axes of magnetization within the Ni1 and Ni2subsystems, q1 ¼ KNi1 =2IM12 ; q2 ¼ KNi2 =2IM12 :The experimental data available for Ni(HCOO)2  2H2O can be compared with results of analytical andnumerical calculations within the model proposed. The exchange interaction coefﬁcient I ¼ 1:77 T/mB was

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451

calculated from formula (4) for the Ne! el temperature. In this model the value of the paramagnetic Curie
temperature Y0 ¼ 2:6 K (see Eq. (6)). The magnetic susceptibility at T > TN in this model is given by Eq. (5)and experimental data could be well ﬁtted by the theoretical curve shown by the dotted line in Fig. 3. The
paramagnetic Curie temperature found through the ﬁtting procedure Y0 ¼ 3:5 K is close to that calculated
within the model approximation.
The physical scenario for the appearance of a compensation for Ni(HCOO)2  2H2O looks as follows. At
cooling below the Ne! el temperature, the weak magnetization of the Ni1 subsystem exceeds that of the Ni2subsystem. This occurs because the M1 ðTÞ dependence is much steeper than the M2 ðTÞ dependence in thevicinity of TN : It results in an initial increase of the total magnetization Mtotal below TN : At further cooling,
the magnetixation values of the Ni1 and Ni2 subsystems approach each other and compensation of theirmagnetizations occurs at T  : If the magnetic anisotropy in the Ni2 subsystem is ‘‘larger’’ than that in theNi1 subsystem, at low temperatures the weak magnetization of the Ni2 subsystem will prevail.
Concluding, the phenomenon of weak ferrimagnetism is found in nickel (II) formate dihydrate
Ni(HCOO)2  2H2O containing only one type of magnetic ions. These ions constitute two canted
antiferromagnetic subsystems whose competition results in a compensation point and magnetization
reversal. This observation widens the range of the substances and of the physical mechanisms responsible
for the appearance of a compensation point and magnetization reversal in weakly ferrimagnetic
compounds.
This work was supported by NWO Grant 047-008-012.

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