Valence of Ti cations and its effect on magnetic properties of spinel ferrites Ti_xM_1−xFe₂O₄ (M = Co, Mn)

Abstract

Powder samples of Ti_xCo_1−xFe₂O₄ (0.0 ≤ x ≤ 0.4) and Ti_xMn_1−xFe₂O₄ (0.0 ≤ x ≤ 0.3) were synthesized using a conventional method for preparing ceramics. X-ray diffraction analysis confirmed that the samples consisted of a single phase with a cubic (A)[B]₂O₄ spinel structure. The average molecular magnetic moment (μ_exp) measured at 10 K decreased monotonically with increasing x for two series of samples. According to previous investigations, Ti²⁺ and Ti³⁺ cations are present in these ferrites, but there are no Ti⁴⁺ cations; the magnetic moments of the Ti²⁺, Ti³⁺, and Mn³⁺ cations are assumed to couple antiferromagnetically with those of the Mn²⁺, Co²⁺, Co³⁺, Fe²⁺, and Fe³⁺ cations whenever they are at the (A) or [B] sublattice. The dependence of μ_exp of the two series of samples on the doping level x was fitted using a quantum-mechanical potential barrier, and the cation distributions in the two series of samples were obtained.

---

1 Introduction

Spinel ferrites have received much attention in recent years because of their application in spintronics and multiferroics.^1–7 In a (A)[B]₂O₄ spinel ferrite, each unit cell contains eight formula units, in which the 32 larger oxygen anions form a close-packed face-centered-cubic structure with the 24 smaller metal cations occupying two types of interstitial position: the tetrahedral (8a) or (A) sites and the octahedral (16d) or [B] sites,^8–11 which form the (A) and [B] sublattices.

Many studies were carried out on the magnetic moment of and cation distribution in Ti-doped spinel ferrites.^12–16 In these investigations, all of the Ti cations were assumed to be tetravalent, but there have been disputes regarding the cation distribution. Dwivedi et al. prepared a series of samples, Co(Fe_1−xTi_x)₂O₄ (x = 0, 0.05, or 0.1), by conventional solid-phase reactions; using X-ray photoelectron spectroscopy (XPS) they discovered that all Ti cations went into the octahedral sites.¹² Srinivasa Rao et al. prepared samples of the CoTi_xFe_2−xO₄ (0.0 ≤ x ≤ 0.3) series; they thought that the Ti⁴⁺ ions had the tendency to go to the [B] site, which affected the cation distribution in the samples.¹³ Schmidbauer prepared samples of the Fe_1+xCr_2−2xTi_xO₄ (0 ≤ x ≤ 1) series and concluded that there were Fe²⁺ ions at the (A) and [B] sites, and all Cr and Ti cations occupied the B-sites.¹⁴ Schmidbauer also prepared samples of two spinel ferrite series, Fe_2.4−tCr_0.6Ti_tO₄ (0 ≤ t ≤ 0.7) and Fe_2.1−tCr_0.9Ti_tO₄ (0 ≤ t ≤ 0.55), and assumed that all of the Ti⁴⁺ ions entered the [B] sites.¹⁵ However, when Kale et al. prepared Ti_xNi_1+xFe_2−2xO₄ (0.0 ≤ x ≤ 0.7), they estimated the cation distribution at the (A) and [B] sites using X-ray diffraction and came to the conclusion that the fraction of Ti⁴⁺ cations entering the (A) sites increased with increasing x, and it reached 0.5 when x = 0.7.¹⁶

In order to resolve these discrepancies regarding cation distributions in spinel ferrites, Xu et al. investigated the valence, distribution of cations and the magnetic structure of Ti-doped spinel ferrites^17–19 by using an O 2p itinerant-electron model.^20–22 They found an additional antiferromagnetic phase when Ti cations replaced a portion of the Ni or Fe cations in the spinel ferrites Ni_0.68Fe_2.32O₄ (ref. 17 and 18) and NiFe₂O₄,¹⁹ and they offered the following explanation for the phenomenon: most of the Ti cations were Ti²⁺ cations that occupied the [B] sites; the remaining Ti cations were Ti³⁺ cations and there were no Ti⁴⁺ cations; the magnetic moments of the Ti cations coupled antiferromagnetically with those of Fe and Ni cations whenever they were at the (A) or [B] sites.

The absence of Ti⁴⁺ in an oxide has been confirmed by theoretical and experimental investigations. Cohen²³ and Cohen and Krakauer²⁴ used density functional theory to calculate the densities of states for valence electrons in the perovskite oxide BaTiO₃. Their results indicated that the average valence of Ba is +2, which is the same as the traditionally accepted value, but the average valences of Ti and O are +2.89 and −1.63, respectively, which are different from the conventional results of +4 and −2, respectively. This calculation result was confirmed by the X-ray photoelectron spectra obtained by Wu et al.,²⁵ who found that the average valence of O anions, V_alO, is −1.55, which is close to the value (−1.63) calculated by Cohen. In addition, using XPS analysis, Dupin et al. found that the average valence of O anions is −1.15 for TiO₂,²⁶ which indicates that there are Ti²⁺ and Ti³⁺ cations, but no Ti⁴⁺ cations, in TiO₂. Ji et al. proposed a method to estimate the valences of cations and anions in (A)[B]₂O₄ spinel ferrites; they obtained estimated values between −1.6 and −1.8 for V_alO of spinel ferrites, and they also defined the ionicity of an oxide as f_i = |V_alO|/2, accompanied by calculated values of the iconicity of several cations in spinel ferrites.²⁷

Taking into account that there are O¹⁻ ions in addition to O²⁻ ions, our group uses the O 2p itinerant-electron model²⁰ and the quantum mechanical potential barrier method^21,22 to investigate the cation distribution in several series of spinel ferrites.^28–35 In the study reported here, we prepared spinel ferrite samples of Ti_xCo_1−xFe₂O₄ (0.0 ≤ x ≤ 0.4) and Ti_xMn_1−xFe₂O₄ (0.0 ≤ x ≤ 0.3) and measured the magnetic moment, μ_exp, of the samples at 10 K. The cation distribution in the samples was estimated by fitting the measured values of μ_exp.

2 Experimental

2.1 Sample preparation

Spinel ferrites Ti_xCo_1−xFe₂O₄ (0.0 ≤ x ≤ 0.4; hereafter referred to as the Co-series) and Ti_xMn_1−xFe₂O₄ (0.0 ≤ x ≤ 0.3; hereafter referred to as the Mn-series) were prepared using the method of solid-phase reaction.¹⁷ The analytical reagent (AR)-grade chemicals CoO, MnO₂, Fe₂O₃, and TiO₂ were used as the starting materials. First, stoichiometric amounts of each chemical were mixed together, ground for 8 h in an agate mortar, and then calcined at 1173 K for 5 h. The calcined materials were then ground again for 1 h. The ground powder was calcined at 1473 K for an additional 5 h, and then further ground for 1 h. Next, the twice calcined and thrice ground powder was pressed into pellets at a pressure of 10⁴ kg cm⁻² and then sintered at 1673 K for 10 h in a tube furnace under an argon flow. The sintered pellets were then ground for 30 min in an agate mortar, and the resulting powder was used for the measurements.

2.2 Sample characterization

The crystal structure of the samples was determined by analyzing their X-ray diffraction (XRD) patterns, which were measured with an X-ray diffractometer (X'pert Pro, PANalytical, The Netherlands) with Cu K_α (λ = 1.5406 Å) radiation at room temperature. The data were collected in the 2θ range of 15–120° with a step size of 0.0167°. The working current and voltage were 40 mA and 40 kV, respectively. The magnetic hysteresis loops of the samples were measured using a physical properties measurement system (PPMS, Quantum Design Corporation, USA) at 10 and 300 K.

3 Experimental results

3.1 Analysis of X-ray diffraction patterns

Fig. 1(a) and (b) show the XRD patterns of the Ti_xCo_1−xFe₂O₄ (0.0 ≤ x ≤ 0.4) and Ti_xMn_1−xFe₂O₄ (0.0 ≤ x ≤ 0.3) samples, which indicate that they consisted of a single-phase with a cubic spinel structure of space group Fd3̄m. The XRD data were fitted using the X'Pert HighScore Plus software (PANalytical, The Netherlands) and the Rietveld powder-diffraction profile-fitting technique.³⁶ The ions O (32e), A (8b) and B (16c) were located at the positions (u, u, u), (0.375, 0.375, 0.375), and (0, 0, 0), respectively. We obtained the crystal structure data, including the crystal lattice constant, a, the oxygen position parameters, u, the distances from the O anions to the cations at the (A) and [B] sites, d_AO and d_BO; and the distance between the cations at the (A) sites and those at the [B] sites, d_AB; the data are summarized in Table 1. For the cubic spinel structure, the ideal values (assuming u = 0.25) of d_AO, d_BO, and d_AB are , a/4, and , respectively; however, the observed values of d_AO and d_BO (Table 1) are 1.0400 and 0.9805 (or 1.0918 and 0.9565) times, respectively, of the ideal values for the Co-series (or Mn-series) samples. On the other hand, the observed values of d_AB are equal to the ideal values for the two series. The volume-averaged crystallite sizes of all samples were calculated using the X'Pert HighScore Plus software, and they were found to be greater than 100 nm. Therefore, surface effects of the crystallites are expected to be very weak in all samples.

Fig. 1 X-ray diffraction patterns of various samples: (a) Ti_xCo_1−xFe₂O₄ (0.0 ≤ x ≤ 0.4); (b) Ti_xMn_1−xFe₂O₄ (0.0 ≤ x ≤ 0.3).

Table 1 Rietveld fitting results of XRD patterns of the two series of samples, obtained using the X'Pert HighScore Plus software. a is the lattice parameter; d_AO and d_BO are the distances from the O anion to the cations at the (A) and [B] sites, respectively; and d_AB is the distance from the cations at the (A) sites to those at the [B] sites

x	a (Å)	dAO (Å)	dBO (Å)	dAB (Å)	u (Å)
TixCo1−xFe2O4
0.0	8.3871	1.888	2.056	3.477	0.24503
0.1	8.3987	1.891	2.059	3.482	0.24501
0.2	8.4089	1.893	2.061	3.486	0.24500
0.3	8.4203	1.896	2.064	3.491	0.24499
0.4	8.4343	1.898	2.066	3.497	0.24498
TixMn1−xFe2O4
0.0	8.5197	2.014	2.037	3.532	0.23856
0.1	8.5190	2.016	2.038	3.531	0.23857
0.2	8.5172	2.012	2.036	3.530	0.23858
0.3	8.5118	2.011	2.035	3.529	0.23859

---

Fig. 2 shows the dependence of the lattice parameter a on the Ti-doping level, x, in the two series of samples. It can be seen that with increasing x, a increased for the Co-series and decreased for the Mn-series. The different trends in the lattice constant were related to the cation radii, magnetic ordering, and cohesive energies of the samples.

Fig. 2 Curves of lattice constant, a, versus the Ti-doping level, x, for the two series of samples.

3.2 Analysis of magnetic properties of the samples

Fig. 3 and 4 show the magnetic hysteresis loops of the two series of samples measured at 10 and 300 K. From these figures, we obtained the specific saturation magnetization (σ_S) measured at 10 and 300 K and the magnetic moment (μ_exp) per formula unit of each sample at 10 K, as listed in Table 2. It can be seen that the values of σ_S for the two series of samples gradually decreased with increasing x at both 10 and 300 K.

Fig. 3 Magnetic hysteresis loops measured at (a) 10 K and (b) 300 K for samples of Ti_xCo_1−xFe₂O₄.

Fig. 4 Magnetic hysteresis loops measured at (a) 10 K and (b) 300 K for samples of Ti_xMn_1−xFe₂O₄.

Table 2 Specific saturation magnetization measured at 10 K (σ_{S-10 K}) and 300 K (σ_{S-300 K}) for the two series of samples; μ_exp is the experimental magnetic moment per formula unit of a sample, which was calculated using σ_S-10K

x	σS-10 K (A m^2 kg^−1)	σS-300 K (A m^2 kg^−1)	μexp (μB per formula)
TixCo1−xFe2O4
0.0	77.73	77.39	3.266
0.1	74.19	76.32	3.102
0.2	66.30	67.34	2.759
0.3	59.97	58.03	2.484
0.4	51.02	50.12	2.103
TixMn1−xFe2O4
0.0	105.23	74.41	4.346
0.1	92.28	68.47	3.799
0.2	81.06	64.31	3.327
0.3	72.91	58.18	2.983

---

4 Estimation of cation distributions by fitting the samples' magnetic moments at 10 K

Following the procedure reported by Xu et al.,^18–20 we used the O 2p itinerant-electron model²⁰ and the quantum mechanical potential barrier method^21,22 to fit the magnetic moments measured at 10 K as a function of x and estimate the cation distribution in all samples. During the fitting process, the following factors were taken into account:

Factor 1: since there were O¹⁻ ions in addition to O²⁻ ions, the ionicity of the cations in the samples was distinctly lower than 1.0, as shown in Table 3; the values listed in Table 3 were calculated using the method reported by Ji et al.²⁷ In (A)[B]₂O₄ spinel ferrites, the total valence and the total number of trivalent cations per formula unit (N₃) are both less than the traditional values of 8 and 2, respectively.

Table 3 Cation parameters used in the magnetic-moment fitting process, including the second and third ionization energies, V(M²⁺) and V(M³⁺); effective radii, r, of the divalent cations with coordination number 6; ionicity, f_i;²⁷ and the magnetic moments of the divalent and trivalent cations, m₂ and m₃

Element, M	V(M^2+) (eV)	V(M^3+) (eV)	r^37 (nm)	fi^27	m2 (μB)	m3 (μB)
Ti	13.58	27.49	0.0860	0.9716	−1	−2
Mn	15.64	33.67	0.0830	0.8293	5	−4
Fe	16.18	30.65	0.0780	0.8790	4	5
Co	17.06	33.50	0.0745	0.8314	3	4

---

Factor 2: the O 2p itinerant-electron model is characterized by certain features:²⁰ (i) in a given sublattice, an O 2p electron with constant spin direction can hop from an O²⁻ anion to the O 2p hole of an adjacent O¹⁻ anion, with a cation acting as an intermediary. (ii) The two O 2p electrons in the outer orbit of an O²⁻ anion, which have opposite spin directions, become itinerant electrons in the two different sublattices (the (A) or [B] sublattice). (iii) In a given sublattice that is constrained by Hund's rules and by the fact that an itinerant electron has constant spin direction, the direction of the magnetic moments of cations with the 3d electron number of n_d ≤ 4 will couple antiferromagnetically to those of the cations with n_d ≥ 5 at either the (A) sites or the [B] sites of a spinel ferrite. Therefore, the directions of the magnetic moments of Ti³⁺(3d¹), Ti²⁺(3d²), and Mn³⁺(3d⁴) were antiparallel to those of the magnetic moments of Mn²⁺, Co²⁺, Co³⁺, Fe³⁺, and Fe²⁺ in the same sample at either the (A) sublattice or the [B] sublattice. Therefore, in the following calculations, we set the moment of the cations to the values shown in Table 3.

Factor 3: we assumed that there is a square potential barrier between a pair of anion and cation.²¹ The height and the width of the potential barrier are related to the cation ionization energy and the distance between the cation–anion pair. The content ratio (R) of the different cations is therefore related to the probability of the last ionized electrons transmitted through the potential barriers, and the following equation can be obtained:

where nanometers (nm) and electron-volts (eV) are used as the units of length and energy; P_C (or P_D) stands for the probability of the last ionized electron of the C (or D) cation jumping to the anions through the potential barrier with the height V_C (or V_D) and the width r_C (or r_D). V_C and V_D are the ionization energies of the last ionized electron of the cations C and D, and r_C and r_D are the distances from the cations C and D to the anions. 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. When V_C = V_D and r_C = r_D, it is obvious that c = 1.0.

Factor 4: we considered the Pauli repulsion energy of the electron cloud between adjacent cations and anions. This can be taken into account using the effective ionic radius:³⁷ smaller ions tend to enter the sites with smaller available space in the lattice. It is worth noting that the volumes of the (A) sites are smaller than those of the [B] sites in spinel ferrites.

Factor 5: during the thermal treatment of the samples, the tendency to balance the electrical charge density forced some of the divalent cations (with large effective ionic radii) to enter the (A) sites (with smaller available space) from the [B] sites (with large available space), jumping over an equivalent potential barrier, V_BA, because cations at the (A) sites have four adjacent oxygen ions while cations at the [B] sites have six adjacent oxygen ions. V_BA is related to the ionization energy, ionic radius, and the thermal-treatment temperature. We assumed V_BA of the ferrite samples can be expressed by the following equations:³⁴

where M = Co or Mn; V(M³⁺), V(Ti³⁺), and V(Fe³⁺) are the third ionization energies of Co, Mn, Ti and Fe, respectively; and r(M²⁺), r(Ti²⁺), and r(Fe²⁺) are the effective radii of the divalent cations with coordination number 6, as shown in Table 3.

The chemical formulae of the ferrite samples Ti_xCo_1−xFe₂O₄ and Ti_xMn_1−xFe₂O₄, are rewritten here as Ti_x1M_x2Fe_3−x1−x2O₄ (M = Co, Mn) so that the cation distributions can be described by the equation

(Ti_y1³⁺M_y2³⁺Fe_y3³⁺Ti_y4²⁺M_y5²⁺Fe_y6²⁺)[Ti_{x1−y1−y4−z1}²⁺M_{x2−y2−y5−z2}²⁺Fe_{3−x1−x2−y3−y6−z3}²⁺Ti_z1³⁺M_z2³⁺Fe_z3³⁺]O₄. (4)

It can be seen from eqn (4) that

y₁ + y₂ + y₃ + y₄ + y₅ + y₆ = 1, (5)

y₁ + y₂ + y₃ + z₁ + z₂ + z₃ = N₃, (6)

where N₃ is the number of trivalent cations per formula unit. The parameters f_Ti, f_Fe, and f_M = f_Co (or f_Mn) represent the ionicities of the Ti, Fe, and Co (or Mn) ions,²⁷ whose values are shown in Table 3. Eqn (7) suggests that when the ionicity of all cations are 1.00, the sum of the valence of all cations is 8.00, while N₃ = 2.00. In fact, the ionicity of each cation is lower than 1.00 (see Table 3), resulting in N₃ < 2.00. From eqn (4), we havewhere R_A1, R_A2, R_A4, R_A5, and R_A6 represent the probability ratios of the Ti³⁺, Co³⁺(Mn³⁺), Ti²⁺, Co²⁺(Mn²⁺), and Fe²⁺ ions, respectively, with respect to the Fe³⁺ ions at the (A) sites, while R_B1 and R_B2 represent the probability ratios of the Ti³⁺ and Co³⁺(Mn³⁺) ions with respect to the Fe³⁺ ions at the [B] sites. From eqn (5) and (8), we can obtain

From eqn (6) and (9), we have

According to the above-mentioned quantum mechanical potential barrier method for estimating the cation distributions in spinel ferrites,^21,22 which is similar to eqn (1), the content ratios R_A1, R_A2, R_A4, R_A5, and R_A6 at the (A) sites and R_B1 and R_B2 at the [B] sites can be rewritten as

where M = Co or Mn; and V(M²⁺), V(M³⁺), V(Ti²⁺), V(Ti³⁺), V(Fe²⁺), and V(Fe³⁺) are the second and third ionization energies of Co, Mn, Ti, and Fe, respectively, as shown in Table 3. The parameter c_v is a barrier shape-correcting constant related to the potential barrier of Ti²⁺ and Ti³⁺ cations; we assume that c_v = 1.0 for other cations because the second and third ionization energies of Ti cations are distinctly lower than those of other cations. V_BA(M²⁺), V_BA(Ti²⁺), and V_BA(Fe²⁺) are the heights of the equivalent potential barriers (all have a width of d_AB), which must be transmitted through by the M²⁺, Ti²⁺, and Fe²⁺ ions as they move from the [B] sites to the (A) sites during thermal treatment. The values of, d_AO, d_BO, and d_AB are the observed values in the XRD patterns, as listed in Table 1.

According to the O 2p itinerant-electron model, the magnetic moments of the Mn³⁺, Ti²⁺, and Ti³⁺ cations are antiparallel to those of Co²⁺, Co³⁺, Mn²⁺, Fe³⁺, and Fe²⁺ cations in the same sublattice of a spinel ferrite (see Table 3). Therefore, we can calculate the average magnetic moment per formula unit of a sample from eqn (4):

where μ_C is the calculated magnetic moment of a sample; μ_AT and μ_BT are the magnetic moments of the (A) and [B] sublattices; and μ_B1, μ_B2, and μ_B3 are magnetic moments contributed by the Ti, Co (or Mn), and Fe ions, respectively, at the [B] sublattice.

For each sample, there are 22 parameters: y₁–y₆; z₁–z₃; N₃; R_A1, R_A2, R_A4, R_A5, and R_A6; R_B1 and R_B2; V_BA(Ti²⁺), V_BA(M²⁺), and V_BA(Fe²⁺); μ_C; and c_v. Altogether, there are 20 independent equations, including eqn (2), (3), (5)–(9), and (12)–(19), where eqn (8) contains five equations and eqn (9) contains two equations. Therefore, we needed to obtain the values of at least two independent parameters, such as c_v and V_BA(Ti²⁺), in order to fit the observed values of μ_exp of a sample at 10 K.

Using the above parameters and equations, we fitted the dependence of μ_exp on x for the two series of samples. The points and curves in Fig. 5 represent the observed and calculated magnetic moments, μ_exp and μ_C, of the samples. It can be seen that the fitted curves are very close to the experimental results. In the fitting process, we obtained the cation distribution and other data, as listed in Tables 4 and 5. The cation distribution is shown as a function of x for the two series of samples in Fig. 6 and 7.

Fig. 5 Fitted magnetic moments, μ_C (line), and observed values, μ_exp (points), as functions of x for the two series of samples.

Table 4 Cation distributions obtained by fitting the dependence of the magnetic moments in the samples Ti_xCo_1−xFe₂O₄ on x. The parameters V_BA(Ti²⁺), V_BA(Co²⁺), and V_BA(Fe²⁺) are the heights of the potential barriers that must be transmitted though by the Ti²⁺, Co²⁺, and Fe²⁺ ions when moving from the [B] sites to the (A) sites during thermal treatment of the samples; N₃ is the total average number of trivalent cations per formula unit; μ_AT and μ_BT are the magnetic moments per formula unit of the (A) and [B] sublattices, respectively; and μ_C = μ_BT − μ_AT is the calculated magnetic moment per formula unit

x	0.0	0.1	0.2	0.3	0.4
N3	0.9070	0.9496	0.9906	1.0297	1.0675
VBA(Ti^2+) (eV)	1.0933	1.2900	1.4867	1.6833	1.8800
VBA(Co^2+) (eV)	1.1542	1.3618	1.5694	1.7770	1.9847
VBA(Fe^2+) (eV)	1.1056	1.3045	1.5034	1.7022	1.9011
A sites
Ti^3+	0.0000	0.0168	0.0372	0.0602	0.0849
Co^3+	0.1164	0.1193	0.1172	0.1106	0.1002
Fe^3+	0.4159	0.4822	0.5392	0.5864	0.6244
Ti^2+	0.0000	0.0178	0.0283	0.0332	0.0343
Co^2+	0.1286	0.0919	0.0637	0.0429	0.0280
Fe^2+	0.3392	0.2706	0.2121	0.1639	0.1252
B sites
Ti^2+	0.0000	0.0578	0.1202	0.1859	0.2536
Co^2+	0.6674	0.6173	0.5610	0.4992	0.4331
Fe^2+	0.9578	0.9935	1.0219	1.0424	1.0553
Ti^3+	0.0000	0.0091	0.0165	0.0228	0.0289
Co^3+	0.0931	0.0739	0.0586	0.0468	0.0377
Fe^3+	0.2816	0.2483	0.2219	0.2029	0.1914
μBT (μB per formula)	7.6142	7.2385	6.8574	6.4745	6.0925
μAT (μB per formula)	4.2909	4.2040	4.1225	4.0427	3.9621
μC (μB per formula)	3.3233	3.0345	2.7349	2.4318	2.1303

---

Table 5 Cation distributions obtained by fitting the dependence of the magnetic moments in the samples Ti_xMn_1−xFe₂O₄ on x. The parameters V_BA(Ti²⁺), V_BA(Mn²⁺), and V_BA(Fe²⁺) are the heights of the potential barriers that must be transmitted though by the Ti²⁺, Mn²⁺, and Fe²⁺ ions when moving from the [B] sites to the (A) sites during thermal treatment of the samples; N₃ is the total average number of trivalent cations per formula unit; μ_AT and μ_BT are the magnetic moments per formula unit of the (A) and [B] sublattices, respectively; and μ_C = μ_BT − μ_AT is the calculated magnetic moment per formula unit

x	0.0	0.1	0.2	0.3
N3	0.8994	0.9373	0.9753	1.0132
VBA(Ti^2+) (eV)	0.8960	1.0200	1.0800	1.1400
VBA(Mn^2+) (eV)	1.1348	1.2057	1.2766	1.3476
VBA(Fe^2+) (eV)	0.9708	1.0315	1.0921	1.1528
A sites
Ti^3+	0.0000	0.0042	0.0090	0.0145
Mn^3+	0.0872	0.0847	0.0808	0.0755
Fe^3+	0.3319	0.3579	0.3840	0.4101
Ti^2+	0.0000	0.0104	0.0200	0.0290
Mn^2+	0.1789	0.1543	0.1312	0.1097
Fe^2+	0.4019	0.3885	0.3749	0.3612
B sites
Ti^2+	0.0000	0.0795	0.1585	0.2371
Mn^2+	0.6225	0.5566	0.4911	0.4262
Fe^2+	0.8972	0.8734	0.8489	0.8235
Ti^3+	0.0000	0.0059	0.0124	0.0194
Mn^3+	0.1114	0.1044	0.0969	0.0885
Fe^3+	0.3689	0.3802	0.3922	0.4052
μBT (μB per formula)	8.1005	7.5945	7.0953	6.6036
μAT (μB per formula)	3.8129	3.7513	3.7033	3.6690
μC (μB per formula)	4.2876	3.8433	3.3921	2.9347

---

Fig. 6 Distribution of (a) Fe, (b) Ti, and (c) Co cations and (d) the total content percentages of different valence cations at the (A) and [B] sites in samples of Ti_xCo_1−xFe₂O₄ (0.0 ≤ x ≤ 0.4).

Fig. 7 Distribution of (a) Fe, (b) Ti, and (c) Mn cations and (d) the total content percentages of different valence cations at the (A) and [B] sites in samples of Ti_xMn_1−xFe₂O₄ (0.0 ≤ x ≤ 0.3).

5 Discussion

From Tables 4, 5, Fig. 6 and 7, we found that the fitting parameters and the cation distribution in the samples had certain characteristics, as discussed in the following subsections.

5.1 Fitting parameters: c_v and V_BA

During the fitting process, we determined that the potential barrier shape-correcting constant c_v was equal to 1.1 and 1.2 for the pair of ions, Ti–O, in the Co-series and Mn-series, respectively. Both values are reasonable when compared with c_v = 1.0 for other cation–anion pairs.

The Ti²⁺ ions must transmit though the equivalent potential barrier V_BA(Ti²⁺) as they moved from the [B] sites to the (A) sites during the thermal treatment. We obtained the values of V_BA(Ti²⁺) in the fitting process, and they increased from 1.093 eV (x = 0.0) to 1.880 eV (x = 0.4) for the Co-series and from 0.896 eV (x = 0.0) to 1.140 eV (x = 0.3) for the Mn-series. The values of V_BA(Fe²⁺), V_BA(Co²⁺), and V_BA(Mn²⁺) were calculated from eqn (2) and (3), and they appear reasonable in the context of a physics problem.

5.2 Valence and distribution of Ti cations

(i) The ratio of Ti²⁺ ions at the (A) and [B] sites to the Ti-doping level, x, is more than 72% in the Co-series samples (Fig. 6(b)) and about 89% in the Mn-series samples (Fig. 7(b)). This result is similar to that measured using XPS and reported by Dupin et al.; they found that the average O ionic valence is −1.15 for TiO₂, which suggests that 70% of Ti cations in TiO₂ are Ti²⁺ ions.²⁶ Therefore, the conventional view^12–16 that all Ti cations in an oxide are Ti⁴⁺ ions needs to be modified.

(ii) The ratio of Ti²⁺ cations that entered the [B] sites to x increased from 58% (x = 0.1) to 63% (x = 0.4) in the Co-series samples (Fig. 6(b)), and this ratio remained at 79% from x = 0.1 to x = 0.3 in the Mn-series samples (Fig. 7(b)). This result is similar to that reported by Xu et al., who found the ratio of Ti²⁺ cations that entered the [B] sites to x was 81% in Ti-doped ferrite Ni_0.68Fe_2.32O₄.¹⁸

(iii) The ratio of Ti cations, including Ti²⁺ and Ti³⁺, that entered the [B] sites to x increased from 67% (x = 0.1) to 72% (x = 0.4) in the Co-series samples (Fig. 6(d)), and this ratio was 85% in the Mn-series samples (Fig. 7(d)). This result appears to be a balance between the contrasting results reported by several authors:^12–16 Kale et al. concluded that 71% of Ti cations entered the (A) sites in Ti_0.7Ni_1.7Fe_0.6O₄,¹⁶ while other authors assumed that all of the Ti ions entered the [B] sites of the spinel ferrite samples.^12–15

5.3 Distribution of Co cations in Co-series

(i) The ratio of Co cations, including Co²⁺ and Co³⁺, that entered the [B] sites to the total Co cation content ranged from 76% to 78% (Fig. 6(d)). This ratio is very close to that reported by Shang et al.³⁰ for Co_1−xCr_xFe₂O₄. This result is also close to that reported by Wakabayashi et al. for a CoFe₂O₄ film with thickness of 11 nm, which was based on soft X-ray absorption spectroscopy (XAS) and X-ray magnetic circular dichroism (XMCD) combined with cluster model calculations.³⁸

(ii) The ratio of Co²⁺ cations that entered the [B] sites to the total Co cation content increased from 67% (x = 0.0) to 73% (x = 0.4). This result is similar to that reported by Shang et al., who found that the ratio of Co²⁺ cations that entered the [B] sites ranged from 64% (x = 0.0) to 59% (x = 0.8) in Co_1−xCr_xFe₂O₄.³⁰

5.4 Distribution of Mn cations in Mn-series

The ratio of Mn²⁺ cations that entered the [B] sites to the total Mn cation content was 61%. The ratio of Mn ions, including Mn²⁺ and Mn³⁺ cations, that entered the [B] sites to the total Mn cation content was 73%. This result is similar to that reported by Xu et al.²⁰

5.5 Entry of few Co and Mn cations into the (A) sites

It can be seen from Fig. 6(c) and 7(c) that a few of the Co (Mn) cations entered the (A) sites of the Co (Mn) series samples. This is in accordance with the observed results from XRD mentioned in Section 3.1: the ratio of observed to ideal values of A–O distance for MnFe₂O₄, 1.09, is higher than that for CoFe₂O₄, 1.04, because the effective radius of Mn is greater than that of Co (see Table 3). This suggested that a few of the Mn (Co) cations entered the (A) sites of MnFe₂O₄ (CoFe₂O₄).

6 Conclusions

The single-phase spinel ferrites Ti_xCo_1−xFe₂O₄ (0.0 ≤ x ≤ 0.4) and Ti_xMn_1−xFe₂O₄ (0.0 ≤ x ≤ 0.3) were prepared using the conventional method for preparing ceramics. The samples were found to consist of a single phase with a cubic spinel structure. The lattice constant increased in the Co-series samples and decreased in the Mn-series samples with increases in the dopant level, x. The values of μ_exp of the two series of samples, measured at 10 K, decreased approximate linearly with increasing x.

The dependence of μ_exp on x for the two series of samples was fitted using a quantum-mechanical potential barrier method. The fitted magnetic moments were very close to the experimental results. In the fitting process, the cation distributions of the two series of samples were obtained.

The cation distributions and the magnetic structure obtained in this study are distinctly different from those reported by other groups: (i) there were Ti²⁺ and Ti³⁺ ions, but no Ti⁴⁺ ions, in our samples. (ii) The ratio of Ti²⁺ cations that entered the [B] sites to the Ti-doping level, x, increased from 58% (x = 0.1) to 63% (x = 0.4) in the Co-series samples, and this ratio was 79% from x = 0.1 to x = 0.3 in the Mn-series samples. (iii) The magnetic moments of Ti²⁺, Ti³⁺, and Mn³⁺ ions (with 3d electron number of n_d ≤ 4) coupled antiferromagnetically with other cations (n_d ≥ 5) whenever they were at the (A) or [B] sublattice.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (NSF-11174069), the Natural Science Foundation of Hebei Province (A2015205111), the Key item Science Foundation of Hebei Province (Grant No. 16961106D), and the Young scholar Science Foundation of the Education Department of Hebei Province (QN2016015).

References

1. W. M. Xu, G. R. Hearne, S. Layek, D. Levy, J.-P. Itié, M. P. Pasternak, G. Kh. Rozenberg and E. Greenberg, Phys. Rev. B: Condens. Matter Mater. Phys., 2017, 96, 045108.
2. K. Ugendar, S. Samanta, S. Rayaprol, V. Siruguri, G. Markandeyulu and B. R. K. Nanda, Phys. Rev. B: Condens. Matter Mater. Phys., 2017, 96, 035138.
3. O. Kuschel, R. Buß, W. Spiess, T. Schemme, J. Wollermann, K. Balinski, A. T. N'Diaye, T. Kuschel, J. Wollschlager and K. Kuepper, Phys. Rev. B: Condens. Matter Mater. Phys., 2016, 94, 094423.
4. G. Lavorato, E. Winkler, B. Rivas-Murias and F. Rivadulla, Phys. Rev. B: Condens. Matter Mater. Phys., 2016, 94, 054405.
5. M. Abes, C. T. Koops, S. B. Hrkac, J. McCord, N. O. Urs, N. Wolff, L. Kienle, W. J. Ren, L. Bouchenoire, B. M. Murphy and O. M. Magnussen, Phys. Rev. B: Condens. Matter Mater. Phys., 2016, 93, 195427.
6. T. Watanabe, S. Takita, K. Tomiyasu and K. Kamazawa, Phys. Rev. B: Condens. Matter Mater. Phys., 2015, 92, 174420.
7. D. Fritsch and C. Ederer, Phys. Rev. B: Condens. Matter Mater. Phys., 2012, 86, 014406.
8. D. Venkateshvaran, M. Althammer, A. Nielsen, S. Geprägs, M. S. Ramachandra Rao, S. T. B. Goennenwein, M. Opel and R. Gross, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 79, 134405.
9. M. Reehuis, M. Tovar, D. M. Többens, P. Pattison, A. Hoser and B. Lake, Phys. Rev. B: Condens. Matter Mater. Phys., 2015, 91, 024407.
10. C. W. Chen, Magnetism and Metallurgy of Soft Magnetic Materials, North-Holland Publishing Company, 1977, pp. 171–417.
11. J. M. D. Coey, Magnetism and Magnetic Materials, Cambridge University Press, 2010, pp. 414–427.
12. G. D. Dwivedi, A. G. Joshi, H. Kevin, P. Shahi, A. Kumar, A. K. Ghosh, H. D. Yang and S. Chatterjee, Solid State Commun., 2012, 152, 360–363.
13. K. Srinivasa Rao, A. Mahesh Kumar a, M. Chaitanya Varma, G. S. V. R. K. Choudary and K. H. Rao, J. Alloys Compd., 2009, 488, 6–9.
14. E. Schmidbauer, Solid State Commun., 1976, 18, 301–303.
15. E. Schmidbauer, Phys. Chem. Miner., 1983, 9, 124–126.
16. C. M. Kale, P. P. Bardapurkar, S. J. Shukla and K. M. Jadhav, J. Magn. Magn. Mater., 2013, 331, 220–224.
17. J. Xu, D. H. Ji, Z. Z. Li, W. H. Qi, G. D. Tang, Z. F. Shang and X. Y. Zhang, J. Alloys Compd., 2015, 619, 228–234.
18. J. Xu, D. H. Ji, Z. Z. Li, W. H. Qi, G. D. Tang, X. Y. Zhang, Z. F. Shang and L. L. Lang, Phys. Status Solidi B, 2015, 252, 411–420.
19. J. Xu, W. H. Qi, D. H. Ji, Z. Z. Li, G. D. Tang, X. Y. Zhang, Z. F. Shang and L. L. Lang, Acta Phys. Sin., 2015, 64, 017501.
20. J. Xu, L. Ma, Z. Z. Li, L. L. Lang, W. H. Qi, G. D. Tang, L. Q. Wu, L. C. Xue and G. H. Wu, Phys. Status Solidi B, 2015, 252, 2820–2829.
21. G. D. Tang, D. H. Ji, Y. X. Yao, S. P. Liu, Z. Z. Li, W. H. Qi, Q. J. Han, X. Hou and D. L. Hou, Appl. Phys. Lett., 2011, 98, 072511.
22. S. R. Liu, D. H. Ji, J. Xu, Z. Z. Li, G. D. Tang, R. R. Bian, W. H. Qi, Z. F. Shang and X. Y. Zhang, J. Alloys Compd., 2013, 581, 616–624.
23. R. E. Cohen, Nature, 1992, 358, 136–138.
24. R. E. Cohen and H. Krakauer, Phys. Rev. B: Condens. Matter Mater. Phys., 1990, 42, 6416–6423.
25. L. Q. Wu, Y. C. Li, S. Q. Li, Z. Z. Li, G. D. Tang, W. H. Qi, L. C. Xue, X. S. Ge and L. L. Ding, AIP Adv., 2015, 5, 097210.
26. J. C. Dupin, D. Gonbeau, P. Vinatier and A. Levasseur, Phys. Chem. Chem. Phys., 2000, 2, 1319–1324.
27. D. H. Ji, G. D. Tang, Z. Z. Li, X. Hou, Q. J. Han, W. H. Qi, R. R. Bian and S. R. Liu, J. Magn. Magn. Mater., 2013, 326, 197–200.
28. G. D. Tang, Q. J. Han, J. Xu, D. H. Ji, W. H. Qi, Z. Z. Li, Z. F. Shang and X. Y. Zhang, Phys. B, 2014, 438, 91–96.
29. L. L. Lang, J. Xu, W. H. Qi, Z. Z. Li, G. D. Tang, Z. F. Shang, X. Y. Zhang, L. Q. Wu and L. C. Xue, J. Appl. Phys., 2014, 116, 123901.
30. Z. F. Shang, W. H. Qi, D. H. Ji, J. Xu, G. D. Tang, X. Y. Zhang, Z. Z. Li and L. L. Lang, Chin. Phys. B, 2014, 23, 107503.
31. X. Y. Zhang, J. Xu, Z. Z. Li, W. H. Qi, G. D. Tang, Z. F. Shang, D. H. Ji and L. L. Lang, Phys. B, 2014, 446, 92–99.
32. L. L. Lang, J. Xu, Z. Z. Li, W. H. Qi, G. D. Tang, Z. F. Shang, X. Y. Zhang, L. Q. Wu and L. C. Xue, Phys. B, 2015, 462, 47–53.
33. G. D. Tang, Z. F. Shang, X. Y. Zhang, J. Xu, Z. Z. Li, C. M. Zhen, W. H. Qi and L. L. Lang, Phys. B, 2015, 463, 26–29.
34. L. C. Xue, L. L. Lang, J. Xu, Z. Z. Li, W. H. Qi, G. D. Tang and L. Q. Wu, AIP Adv., 2015, 5, 097167.
35. L. L. Ding, L. C. Xue, Z. Z. Li, S. Q. Li, G. D. Tang, W. H. Qi, L. Q. Wu and X. S. Ge, AIP Adv., 2016, 6, 105012.
36. H. M. Rietveld, J. Appl. Crystallogr., 1969, 2, 65–71.
37. R. D. Shannon, Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr., 1976, 32, 751–767.
38. Y. K. Wakabayashi, Y. Nonaka, Y. Takeda, S. Sakamoto, K. Ikeda, Z. Chi, G. Shibata, A. Tanaka, Y. Saitoh, H. Yamagami, M. Tanaka, A. Fujimori and R. Nakane, Phys. Rev. B: Condens. Matter Mater. Phys., 2017, 96, 104410.

---