Investigation of magnetic ordering and cation distribution in the spinel ferrites CrxFe3−xO4 (0.0≤x≤1.0)

Physica B 438 (2014) 91–96

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Investigation of magnetic ordering and cation distribution in the spinel ferrites CrxFe3
xO4 (0.0 rx r1.0) G.D. Tang, Q.J. Han, J. Xu, D.H. Ji, W.H. Qi, Z.Z. Li n, Z.F. Shang, X.Y. Zhang Hebei Advanced Thin Films Laboratory, Department of Physics, Hebei Normal University, Shijiazhuang City 050024, People0 s Republic of China

art ic l e i nf o

a b s t r a c t

Article history: Received 29 August 2013 Received in revised form 7 January 2014 Accepted 8 January 2014 Available online 19 January 2014

Ferrite powder samples of CrxFe3
xO4 (0.0r x r1.0) were prepared by chemical co-precipitation, and calcined in a tube furnace with argon-flow at 1723 K for 2 h. X-ray diffraction patterns indicated that all the samples had an (A)[B]2O4 single phase cubic spinel structure with a Fd3m space group. Magnetic measurements indicated that the magnetization of the samples decreased with the Cr doping level. A new model for the magnetic ordering in these samples was employed to explain the dependence of the magnetization and cation distribution on the Cr doping level; namely, taking into consideration constraints arising from Hund0 s rules and from the spin direction of the itinerant 3d electrons, the directions of the Cr2 þ and Cr3 þ cation magnetic moments were taken to lie antiparallel to the moments of the Fe cations within the same sub-lattice (A or B sub-lattice). & 2014 Elsevier B.V. All rights reserved.

Keywords: Spinel ferrite Magnetic ordering Crystal structure Ionicity

1. Introduction Over the last few decades, many researchers have investigated the magnetic and dielectric properties of cubic spinel ferrites [1–3]. In recent years, cubic spinel ferrites have attracted a renewed interest because they are a kind of multiferroic material [4,5]. (A) [B]2O4 spinel ferrites have a unit cell structure containing eight formula units, where large oxygen anions form a close-packed facecentered-cubic structure with the smaller metal cations occupying the interstitial positions. There are two types of interstitial sites that are occupied by metal atoms: the tetrahedral (8a) or the (A) sites, and the octahedral (16d) or [B] sites [6]. Traditionally, if MFe2O4 ferrites have eight divalent cations occupying the (A) sites and 16 trivalent cations occupying the [B] sites they are described as having a normal spinel structure, but if the eight divalent cations occupy the [B] sites and the 16 ferric ions are evenly divided to occupy the (A) and the [B] sites, the system is classified as having an inverse spinel structure. In ferrites with an inverse spinel structure, all cation magnetic moments in the [B] sites are parallel to each other at low temperatures due to itinerant electrons hopping between neighboring cations mediated by O2p electrons. In addition, the cation magnetic moments at the (A) sites are antiparallel to those of the [B] sites due to the superexchange interaction between cations at (A) sites and cations at [B] sites being mediated by oxygen anions [2,7].

n

Corresponding author. Tel.: þ 86 311 8078 7330. E-mail address: [email protected] (Z.Z. Li).

0921-4526/$ - see front matter & 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2014.01.010

However, it is much more difficult to explain the dependence of the magnetization on the basis of the cation moment arrangement in Cr doped ferrites. Lee et al. [7] observed that the Curie temperatures of MgFe2
xCrxO4 powder samples decrease linearly with increasing Cr doping concentration. They explained this phenomenon in terms of the superexchange interaction in the Fe (A)–O–Fe[B] chain being stronger than that in the Fe(A)–O–Cr[B] chain. Kim et al. [8] prepared CrxFe3
xO4 (x r0.95) film samples with thicknesses of 700–800 nm. They found decreasing saturation magnetization (MS) with increasing x, and explained this observation as a decrease in the net spin magnetic moment due to the substitution of Cr3 þ (3μB) for octahedral Fe3 þ (5μB). Using Mössbauer spectroscopy, Krieble et al. [9] investigated the local environments of Fe ions in five CoCrxFe2
xO4 (x¼0.0–0.8) powder samples. Their study indicated that Cr has an even stronger [B] site preference than does Mn, and displaces more of the Co from the [B] to the (A) sites. Fu et al. [10] investigated the magnetic properties of Li0.5Fe2.5
xCrxO4 (x ¼0.0–1.0) samples. They found that the saturation magnetization decreases linearly with increasing chromium concentration. They claimed that this occurred because the non-magnetic ions weakened the inter-site exchange interaction. Magalha~es et al. [11] prepared a series of Crcontaining magnetites, Fe3
xCrxO4 (x ¼0.00, 0.07, 0.26, 0.42 and 0.51), with a conventional co-precipitation method. Using Mössbauer spectroscopy and powder X-ray diffraction measurements, they determined that the samples had spinel crystalline phases, in which only Fe3octþ ([B] sites) ions were substituted by Cr3 þ when there was a low Cr concentration, and then both Fe2octþ ([B] sites) and Fe3tetþ ((A) sites) ions were substituted by chromium ions at higher Cr concentrations.

92

G.D. Tang et al. / Physica B 438 (2014) 91–96

In this paper, we propose a new model on the basis of Hund0 s rule to explain the dependence of the saturation magnetization of CrxFe3
xO4 (0.0 rx r1.0) on the Cr doping level. On the basis of this model, we believe that the magnetic ordering of all spinel and other ferrites can be more easily understood than by using traditional theory.

2. Experiment and results for CrxFe3
xO4 (0.0rx r1.0) Ferrite powder samples with the composition CrxFe3
xO4 (0.0r xr1.0) were prepared by chemical co-precipitation. The process used was similar to that of Refs. [12,13]. The resulting powders were calcined in a tube furnace with argon-flow at 1723 K for 2 h, and then cooled to room temperature while in the furnace. The crystal structure of the samples was identified using an X0 pert Pro X-ray diffractometer with Cu Kα (λ ¼1.5406 Ǻ) radiation at room temperature. The data were collected in the 2θ range 10– 1201 with a step size of 0.01671. X-ray diffraction patterns for the samples are shown in Fig. 1, which indicates the presence of a single phase, cubic spinel structure with space group Fd3m. The X-ray diffraction pattern (XRD) data were fitted using the X0 Pert HighScore Plus software, and we obtained the crystal lattice constant a, the distances, dAO and dBO, between the O anion and the cations at the (A) and [B] sites, and the distance between the cations at the (A) and [B] sites, dAB (see Table 1). It can be seen that the lattice constant is the same for all the samples, a ¼0.8390 (70.0006) nm. For the ideal values of dAO, pffiffiffi the spinel structure, pffiffiffiffiffiffi dBO and dAB are 3a=8, a=4 and 11a=8, respectively. However, the observed values of dAO and dBO in Table 1 are 1.036 and 0.982 times the ideal values, respectively, while the value of dAB is equal to its ideal value.

Fig. 1. X-ray diffraction patterns for samples of CrxFe3
xO4 (0.0 rx r 1.0).

The magnetic properties of the samples were measured using a Physical Properties Measurement System (Quantum Design Corporation, USA). Fig. 2 shows the temperature dependence of the magnetization of the samples under an applied magnetic field of 0.05 T. For the sample with x ¼0, it can be seen that there is a Verwey transition [6] at 116 K. Fig. 3 shows the magnetic hysteresis loops of the samples when measured at 10 K, except for the sample x¼ 0 which was measured at a temperature of 116 K, below which the Verwey transition occurs. The behavior of the magnetic moment μ (μB) per formula vs. the Cr doping level x is shown in Table 1. It can be seen that μ decreases with increasing x, which is similar to the results reported by Kim et al. [8].

3. A model for magnetic ordering in spinel ferrites MxFe3
xO4 (0.0rx r1.0) In order to understand better the magnetic ordering mechanism of the ferrites MxFe3
xO4 (0.0rx r1.0), three factors need to be considered simultaneously. Firstly, the O2p electrons serve as intermediaries for the itinerant 3d electrons between the cations [14]; secondly, magnetic ordering in spinal ferrites is dependent on the number of 3d electrons, including local and the itinerant 3d electrons in the cations, subject to constraints arising from Hund0 s rules; thirdly, the spin directions of the itinerant electrons remain constant during the hopping process between the cations. This can be understood as follows: (i) There is a certain probability of finding 2p electrons with opposite spin directions in the outer orbits of the oxygen anions (taking into consideration the ionicity), which serve as intermediaries for the itinerant 3d electrons between the

Fig. 2. Temperature dependence of the magnetization of the samples CrxFe3
xO4 (0.0r x r1.0) under a magnetic field of 0.05 T.

Table 1 The parameters of the CrxFe3
xO4 samples obtained by XRD and magnetic measurements: a is the lattice parameter; dAO and dBO are the distances between the O anion and the cations at the (A) and [B] sites; dAB is the distance between the cations at the (A) and [B] sites; and μexp is the magnetic moment of the samples measured at 10 K. x

a (Å)

dAO (Å)

dBO (Å)

dAB (Å)

μexp (μB)

1.0 0.8 0.6 0.4 0.2 0.0

8.394 8.387 8.385 8.386 8.389 8.396

1.883 1.881 1.881 1.881 1.882 1.883

2.061 2.060 2.059 2.059 2.060 2.062

3.480 3.477 3.476 3.476 3.478 3.481

1.883 2.528 2.870 3.200 3.685 3.927a

a

Measured at 116 K for the sample x ¼0, below which the Verwey transition occurs.

Fig. 3. Magnetic hysteresis loops for samples measured at 10 K. For the sample with x ¼ 0, the hysteresis loop was measured at 116 K, below which the Verwey transition occurs.

G.D. Tang et al. / Physica B 438 (2014) 91–96

cations. Therefore, the metal cations around an oxygen anion can be divided into two groups with opposite spin direction itinerant electrons. Those cations with the same bond lengths and bond angles between the oxygen anion and the metal cations will compose a single group, in order to minimize the energy change of the system when the itinerant electron, with constant spin direction, hops between the cations. For example, the spin direction of itinerant electrons in the (A) sublattice is opposite to that in the [B] sublattice in spinel ferrites, because the pffiffiffi distance between cation and anion in the (A) sublattice is ð 3=8Þa, while the distance between the cation and anion in the [B] sublattice is a=4, where a is the lattice constant of the spinel unit cell. (ii) In a group of metal cations, the spin direction of the itinerant electron remains constant when it moves to any cation, and the spin directions of the 3d electrons (including itinerant and local 3d electrons) are constrained by Hund0 s rules. Hence the directions of cation magnetic moments are determined. By Hund0 s rules, the spins of the electrons in a subshell of a free atom tend to align in one direction until the maximum multiplicity is attained. After that, the spins of the electrons will align in the opposite direction. Thus in the 3d subshell of free transition metal atoms, a maximum of five electrons can have their spins aligned in one direction [15]. Therefore, when a itinerant electron hops to a cation with local 3d electron number, nd r4, the spin direction of this itinerant electron will be parallel to that of the local 3d electrons (major spin). However, when a itinerant electron hops to a cation with nd Z5, the spin direction of this itinerant electron will be antiparallel to that of the local 3d electrons (major spin). Thus, for example, the magnetic moment directions of Cr3 þ and Cr2 þ with nd r4, will be antiparallel to those of Fe3 þ and Fe2 þ with nd Z5.

4. Cation distribution and moment dependence on the Cr doping level x in CrxFe3
xO4 (0.0rx r1.0) According to the quantum mechanical potential barrier model proposed earlier by our group [16,17] for estimating the cation distribution in oxides, there are three important factors that affect the cation distribution in spinel ferrites doped with magnetic cations. First, there is a square potential barrier between a cation–anion pair that determines the probability of an electron moving between the cation and anion. The height and the width of the potential barrier are related to the ionization energy of the last ionized electron and the distance between neighboring cations and anions, respectively. Therefore, the content ratio (R) of the different cations is related to the probability of their last ionized electrons transmitting through the potential barrier, and takes the form R¼

TC VD 1=2 1=2 ¼ exp½10:24ðr D V D
cr C V C Þ; TD VC

where nanometers (nm) and electron-volts (eV) are used as the units of length and energy, respectively; TC (TD) represents the probability of the last ionized electron of the C (D) cations jumping to the anions through the potential barrier of height VC (VD) and width rC (rD); VC and VD are the ionization energies of the last ionized electron of the cations C and D, respectively. Finally, rC and rD are the distances from the cations C and D to the anions, respectively. The parameter c is a barrier shape correcting constant related to the different extents to which the shapes of the two potential barriers deviate from a square barrier. It is obvious that parameter c¼1 when VC ¼VD and rC ¼ rD.

93

The second factor is the Pauli repulsion energy of the electron cloud between neighboring cations and anions. It can be taken into account using the effective ionic radius [18]: the smaller ions should be located at the sites with smaller available space in the lattice. For instance, the volumes of the (A) sites are smaller than those of the [B] sites in the spinel ferrites. Third, a tendency toward charge density balance forces a part of the M2 þ ions (with large effective ionic radius) to enter (A) sites from the [B] sites (with large available space) by jumping a potential barrier VBA, which arises from the Pauli repulsion energy of the electron cloud and from the magnetic ordering energy. Taking into consideration that there are both ionic and covalent bonds in oxides, the ionicity has been defined as the fraction of ionic bonds [19] and a method for estimating the ionicity for spinel ferrites has been proposed [20]. In Ref. [20], the ionicity of Fe3O4 was calculated to be 0.8790 and that of CrFe2O4 was found to be 0.8769. Therefore, for Fe3O4 the total valence is 7.032 (¼8  0.8790) and that of CrFe2O4 is 7.015 ( ¼8  0.8769) and the total numbers (N3) of trivalent cations per formula are 1.032 and 1.015, respectively, rather than 2.000. Subject to the constraints arising from the above factors, the cation distributions in CrxMFe2
xO4 (M¼ Fe, 0.0 rxr 1.0) can be represented by the formula 2þ ðFe3y1þ Cr3y2þ M3y3þ Fe2y4þ Cr2y5þ M2y6þ Þ½Fe22 þ
x
y1
y4
z1 Crx
y2
y5
z2 3þ 3þ 3þ M21 þ
y3
y6
z3 Fez1 Crz2 Mz3 O4

ð1Þ

It can be seen from Eq. (1) that y1 þ y2 þ y3 þ y4 þ y5 þ y6 ¼ 1;

ð2Þ

y1 þ y2 þ y3 þ z1 þ z2 þ z3 ¼ N 3

ð3Þ

where N3 is the number of trivalent cations per formula. 8 N3 ¼ ½f M þ f Cr x þð2
xÞf Fe 
6; 3

ð4Þ

where fFe (fM ¼fFe) and fCr, representing the ionicities of the Fe and Cr ions, are 0.8790 and 0.8769, respectively [20]. From Eq. (1), we have RA1 ð2
xÞ ¼

RB1

y1 y y y y ; RA2 x ¼ 2 ; RA4 ð2
xÞ ¼ 4 ; RA5 x ¼ 5 ; RA6 ¼ 6 ; y3 y3 y3 y3 y3

2
x
y1
y4 z1 x
y2
y5 z2 ¼ ; RB2 ¼ ; 1
y3
y6 z3 1
y3
y6 z3

ð5Þ

ð6Þ

where RA1, RA2, RA4, RA5 and RA6 represent the probability ratios of the Fe3 þ , Cr3 þ , Fe2 þ , Cr2 þ and M2 þ ions taken with respect to the M3 þ ions at the (A) sites, and RB1 and RB2 represent the probability ratios of the Fe3 þ and Cr3 þ ions taken with respect to the M3 þ ions at the [B] sites, respectively. From Eqs. (2) and (5), we have y3 ¼ 1=½RA1 ð2
xÞ þ RA2 x þ 1 þ RA4 ð2
xÞ þ RA5 x þ RA6 

ð7Þ

and from Eqs. (3) and (6), we have Z3 ¼

N3
y3 ½RA1 ð2
xÞ þ RA2 x þ 1 x
y1
y4 x
y2
y5 1 þ RB1 2
1
y
y þ RB2 1
y
y 3

6

3

ð8Þ

6

By using the quantum mechanical potential barrier model to estimate the cation distribution in ferrites as proposed by our group [17] (as an approximation, let barrier shape correction constant c¼ 1), the content ratios RA1, RA2, RA4, RA5 and RA6, and the ratios RB1 and RB2, can be derived to be: RA1 ¼

TðFe3 þ Þ TðM3 þ Þ

¼

VðM3 þ Þ V ðFe3 þ Þ

n o exp 10:24½dAO VðM3 þ Þ1=2
dAO VðFe3 þ Þ1=2  ;

ð9Þ

94

RA2 ¼

G.D. Tang et al. / Physica B 438 (2014) 91–96

TðCr3 þ Þ TðM

3þ

Þ

¼

VðM3 þ Þ V ðCr

3þ

Þ

n o exp 10:24½dAO VðM3 þ Þ1=2
dAO V ðCr3 þ Þ1=2  ;

where μAT and μBT are the magnetic moments of the (A) and [B] sublattices. μB1, μB2 and μB3 are the magnetic moments of the Fe, Cr and M ions at the [B] sites, respectively, and μM2 and μM3 are the magnetic moments of M2 þ and M3 þ ions, where M ¼Fe. The magnetic moments of the Cr2 þ , Cr3 þ , Fe2 þ and Fe3 þ ions are 4μB, 3μB, 4μB and 5μB, respectively. Because it is difficult for divalent cations with high ionization energies and large effective radii to enter the (A) sites, we assume for CrMFeO4 (x ¼1.0) that

ð10Þ RA4 ¼

TðFe2 þ Þ TðM

3þ

Þ

¼

VðM3 þ Þ V ðFe

2þ

Þ

n exp 10:24½dAO VðM3 þ Þ1=2
dAO VðFe2 þ Þ1=2


dAB V BA ðFe2 þ Þ1=2 g; RA5 ¼

TðCr2 þ Þ TðM

3þ

Þ

¼

VðM3 þ Þ V ðCr

2þ

Þ

2 þ 1=2


dAB V BA ðCr RA6 ¼

TðM2 þ Þ TðM

3þ

Þ

¼

VðM3 þ Þ 2þ

VðM


dAB V BA ðM RB1 ¼

TðFe3 þ Þ TðM

3þ

Þ

¼

Þ

Þ

2 þ 1=2

Þ

ð11Þ

n exp 10:24½dAO VðM3 þ Þ1=2
dAO V ðCr2 þ Þ1=2

g;

V BA ðFe2 þ Þ ¼ V BA ðCr2 þ Þ

TðCr3 þ Þ 3þ

TðM

Þ

¼

V BA ðM 2 þ Þ ¼ V BA ðCr2 þ Þ

n exp 10:24½dAO VðM3 þ Þ1=2
dAO VðM2 þ Þ1=2 g;

n o exp 10:24½dBO VðM3 þ Þ1=2
dBO VðFe3 þ Þ1=2  ; VðFe3 þ Þ

n h io exp 10:24 dBO VðM3 þ Þ1=2
dBO VðCr3 þ Þ1=2 ; Þ

VðM3 þ Þ VðCr

ð15Þ 2þ

2þ

3þ

where V(Fe )¼16.18 eV, V(Cr ) ¼15.5 eV, V(Fe ) ¼30.65 eV and V(Cr3 þ ) ¼30.96 eV are the second and third ionization energies of Fe and Cr. The distances between the cation and anion in the (A) sites and the [B] sites, dAO and dBO, and the distance between the cations in the (A) sites and [B] sites, dAB, are the observed values obtained from the XRD patterns and given in Table 1. Finally, VBA(Fe2 þ ), VBA(Cr2 þ ) and VBA(M2 þ ) are the heights of the equivalent potential barriers (all have width dAB) which must be jumped by the Fe2 þ , Cr2 þ and M2 þ ions from the [B] to the (A) sites. Because the numbers of local 3d electrons in Cr2 þ , Cr3 þ , Fe2 þ and Fe3 þ ions are 4, 3, 6 and 5, respectively, using the model of Section 3, the magnetic moment directions for Cr2 þ (nd ¼4) and Cr3 þ (nd ¼3) are found to be antiparallel to those of Fe2 þ (nd ¼6) and Fe3 þ (nd ¼5) in the same sublattice. Therefore, we can calculate the average magnetic moment per formula in samples of CrxMFe2
x O4 (0.0rx r1.0) from Eq. (1) to be 9 μC ¼ μBT
μAT ; > > > > > μAT ¼ 5y1
3y2 þ μM3 y3 þ 4y4
4y5 þμM2 y6 ; > > > > μB1 ¼ 4ð2
x
y1
y4
z1 Þ þ 5z1 ¼ 4ð2
x
y1
y4 Þ þz1 ; = ð16Þ μB2 ¼ 4ðx
y2
y5
z2 Þ þ 3z2 ¼ 4ðx
y2
y5 Þ
z2 ; > > > > > > μB3 ¼ μM2 ð1
y3
y6
z3 Þ þ μM3 z3 ; > > > ; μ ¼ μ
μ þμ ; BT

B1

B2

VðM2 þ ÞrðM2 þ Þ

; VðCr2 þ ÞrðCr2 þ Þ

ð18Þ

where r(Cr2 þ ) and r(M2 þ )¼r(Fe2 þ ) are the effective radii of the Cr2 þ and Fe2 þ cations, and V(Cr2 þ ) and V(M2 þ )¼V(Fe2 þ ) are their second ionization energies. There are 20 independent equations in Eqs. (2)–(6) and Eqs. (9)–(18) for each value of the Cr doping level x. However, there are 21 parameters altogether: y1–y6, z1–z3, N3, RA1, RA2, RA4, RA5 and RA6, RB1 and RB2, VBA(Fe2 þ ), VBA(Cr2 þ ) and VBA(M2 þ ), and μC. Therefore, we need to obtain a value for at least one parameter in order to solve or fit the system of equations. The most direct approach is to fit experimental data for the magnetic moments. In this way, for Fe3O4 (x ¼0.0), we obtained VBA(M2 þ )¼VBA(Fe2 þ ) ¼ 0.815 eV by fitting the known experimental moment (4.2μB) of Fe3O4; for CrFe2O4 (x ¼1.0), we obtained VBA(Cr2 þ )¼ 0.892 eV by fitting the experimental moment (1.88μB) of CrFe2O4. We also obtained VBA(Fe2 þ )¼0.908 eV, for the case x¼ 1.0, using Eq. (17). We can then calculate the dependence of the magnetic moments on the Cr doping level x, by assuming that both VBA(Cr2 þ ) and VBA(Fe2 þ ) increase linearly with increasing x (see Table 2). Fig. 4 shows how the fitted (line) and the observed data (points) for the magnetic moments are dependent on the Cr doping level x. It can be seen that the fitted results are very close to the observed data, which indicates that the proposed model is reasonable. Table 2 lists the calculated moments (μC), the number of trivalent cations per formula (N3), the parameters VBA(Fe2 þ ) and VBA(Cr2 þ ) for the samples, and the concentrations of various cations at the (A) and [B] sites. In the table, F3, F2, C3 and C2 are the concentrations of Fe3 þ , Fe2 þ , Cr3 þ and Cr2 þ cations, respectively. Fig. 5(a)–(c) shows the dependence of the cation distributions on the Cr doping level x. From these data, it can be seen that the numbers of Cr3 þ and Cr2 þ cations at the (A) and [B] sites increase linearly with increasing x. It can also be seen that C2[B] (the concentration of Cr2 þ cations at the [B] sites) is larger than C3[B], C2(A) and C3(A), which is similar to the experimental results from Mössbauer spectroscopy reported by Krieble [9] and Magalha~ es [11].

ð13Þ

VðM3 þ Þ

3þ

ð17Þ

ð12Þ

ð14Þ RB2 ¼

VðFe2 þ ÞrðFe2 þ Þ ; VðCr2 þ ÞrðCr2 þ Þ

B3

Table 2 The estimated data of the CrxFe3
xO4 samples: μC is the calculated magnetic moment; the quantities F3, F2, C3 and C2 are the concentrations of the Fe3 þ , Fe2 þ , Cr3 þ and Cr2 þ cations; N3 is the number of trivalent cations per formula as estimated in Ref. [20]; VBA(Fe2 þ ) and VBA(Cr2 þ ) are the heights of the equivalent potential barriers (all of width dAB) which must be jumped by the Fe2 þ and Cr2 þ ions from the [B] sites to the (A) sites. x

1.0 0.8 0.6 0.4 0.2 0.0

μC (μB)

1.880 2.375 2.853 3.317 3.765 4.201

(A) sites

[B] sites

F3

F2

C3

C2

F3

F2

C3

C2

0.295 0.321 0.346 0.370 0.393 0.416

0.347 0.391 0.436 0.484 0.533 0.584

0.138 0.109 0.081 0.053 0.026 0.000

0.220 0.179 0.137 0.093 0.047 0.000

0.404 0.445 0.487 0.530 0.573 0.616

0.955 1.043 1.130 1.216 1.301 1.384

0.178 0.143 0.108 0.072 0.036 0.000

0.464 0.369 0.275 0.182 0.090 0.000

N3

VBA(Fe2 þ ) (eV)

VBA(Cr2 þ ) (eV)

1.015 1.018 1.022 1.025 1.029 1.032

0.908 0.889 0.871 0.852 0.834 0.815

0.892 0.877 0.861 0.846 0.830 0.815

G.D. Tang et al. / Physica B 438 (2014) 91–96

95

5. Potential barrier VBA

Fig. 4. Fitted (line) and observed (points) magnetic moment data for CrxFe3
xO4 as a function of the Cr doping level x.

Recall that the potential barrier VBA is important in determining the number of bivalent ions that enter (A) sites from [B] sites. It should be noted that only two independent fitting parameters VBA(Fe2 þ )x ¼ 0 ¼ 0.815 eV and VBA(Cr2 þ )x ¼ 1 ¼0.892 eV were used in the above fitting process for the dependence of the magnetic moment on the Cr doping level x. The parameter VBA(Fe2 þ )x ¼ 1 ¼0.908 eV was calculated using Eq. (17). It can be seen that the value of VBA(Fe2 þ ) (0.815 eV) in Fe3O4 is close to that of VBA(Fe2 þ ) (0.908 eV) in CrFe2O4, because the ionization energies and effective radii of Cr2 þ and Cr3 þ are very close to those of the Fe cations, while the only essential difference between the Cr and Fe cations is that the magnetic moment directions of Cr (Cr2 þ and Cr3 þ ) cations are antiparallel to the Fe cations in the same sublattice. In addition, it can be seen from Fig. 5(c) that there is no obvious variation in the total amount of divalent and trivalent cations at the (A) and [B] sites as the Cr doping level increases. Therefore, our magnetic ordering model and magnetic moment fitting method are appropriate.

6. Conclusions We propose a new model for understanding magnetic ordering in spinel ferrites. Specifically, the directions of all cation magnetic moments in ferrites are determined by the number of local 3d electrons in the cations, subject to constraints arising from Hund0 s rules and the spin direction of itinerant electrons. The directions of the Cr2 þ and Cr3 þ cation magnetic moments are then found to be antiparallel to the moments of the Fe cations in the same sublattice (A or B sublattice) in the ferrites CrxFe3
xO4 (0.0rxr1.0). With this model and using calculated values for the ionicity, the dependence on the Cr doping level of the cation distribution and the saturation magnetization of the samples has been successfully modeled and a consistent physical explanation provided. Only two independent fitting parameters were used in the model for the dependence of the magnetic moment on the Cr doping level x.

Acknowledgments This work is supported by the National Natural Science Foundation of China, under Contract no. NSF-1174069, the Natural Science Foundation of Hebei Province (Grant no. E2011205083), the Key Item Science Foundation of Hebei Province (Grant no. 10965125D), and the Key Item Science Foundation of the Education Department of Hebei Province (No. ZD2010129). The authors wish to thank Dr. Norm Davison, Dr. Steven Sahyun and Prof. Ying Liu for their helpful discussions. References

Fig. 5. Dependence on the Cr doping level, x, of the cation contents per formula for Fe (a), Cr (b) and total amount of divalent and trivalent cations (c).

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