First-Principles Study of Ti-Doping Effects on Hard Magnetic Properties of RFe11Ti Magnets

Abstract

Due to the rare earth supply shortage, ThMn₁₂-type RFe₁₂-based (R is the rare earth element) magnets with lean rare earth content are gaining more concern. Most ThMn₁₂-type RFe₁₂ structures are thermodynamically metastable and require doping of the stabilizing element Ti. However, the Ti-doping effects on the hard magnetic properties of RFe₁₁Ti have not been thoroughly investigated. Herein, based on density functional theory calculations, we report the Ti-doping effects on the phase stability, intrinsic hard magnetic properties and electronic structures of RFe₁₁Ti (R = La, Ce, Pr, Nd, Sm, Y, Zr). Our results indicate that Ti-doping not only increases their phase stability, but also enhances the magnetic hardness of ground-state RFe₁₂ phases. Particularly, it leads to the transition of CeFe₁₁Ti and PrFe₁₁Ti from easy-plane to easy-axis anisotropy. Charge density distributions demonstrate that Ti-doping breaks the original symmetry of the R-site crystal field, which alters the magnetic anisotropy of RFe₁₁Ti. Projected densities of states reveal that the addition of Ti results in the shift of occupied and unoccupied f-electron energy levels of rare earth elements, affecting their magnetic exchange. This study provides an insight into regulating the hard magnetic properties of RFe₁₂-based magnets by Ti-doping.

1. Introduction

ThMn₁₂-type RFe₁₂-based magnets have received widespread attention since the 1990s due to their low cost and lean rare earth content [1,2,3,4,5,6]. As Nd-Fe-B rapidly developed into a high-performance permanent magnet because of its more effective coercivity, researchers suspended the development and industrialization of the ThMn₁₂-type rare earth permanent magnet [7,8,9,10,11]. However, influenced by rare earth supply shortages such as Pr, Nd, Dy and Tb, ThMn₁₂-type R(Fe, M)₁₂ magnets have become a research hotspot again in recent years due to their leaner rare earth content and heavy rare earth free nature [12,13]. Especially, the heavy rare earth free SmFe₁₂-based permanent magnet has high intrinsic magnetic properties and high magnetic thermal stability [14,15]. Although the saturation magnetization of SmFe₁₂-based permanent magnets is lower than that of Nd-Fe-B at room temperature, it reaches higher saturation magnetization than that of Nd-Fe-B when temperatures are above 350 K [16,17]. Therefore, it is expected that the RFe₁₂-based permanent magnet with sufficiently high coercivity can replace Nd-Fe-B-based magnets in high temperature service environments.

One issue is that most ThMn₁₂-type RFe₁₂ structures are thermodynamically metastable. The binary RFe₁₂ phase can only be prepared as thin film samples using magnetron sputtering, while the rapid quenching process is unable to fabricate stable single-phase bulk RFe₁₂ magnets [14,18]. In order to prepare stable ThMn₁₂-type RFe₁₂ bulks, researchers chose the addition of stabilized elements (Ti, Co, Cr, V, Mn, Mo, Si, Al, etc.) or R-site substitution (Y, Zr, Ce etc.); among them, Ti-doping improves the stability most significantly [12,19,20,21,22,23,24,25]. Due to the doping of Ti, the stabilized RFe₁₁Ti compounds with ThMn₁₂-type structure can be prepared by arc-melting or strip casting [3,26,27]. However, with the increasing concentration of Ti-doping, the anisotropy field saturation magnetization and Curie temperature of RFe₁₁Ti compounds will decrease, such as with Sm and Nd compounds [3,28,29,30,31]. Generally, former experiments and theorical calculations focus more on the Ti-doping effect on the improvement of the stability of ThMn₁₂-type RFe₁₂-based magnets and the corresponding preparation processes [20,21]. However, the Ti-doping effects on the hard magnetic properties of RFe₁₁Ti compounds from the aspect of magnetic and electronic structure have not been systematically investigated.

In this report, based on density functional theory (DFT) calculations, we conducted research on the effects of Ti-doping on the phase stability, intrinsic hard magnetic properties and electronic structures of RFe₁₁Ti (R = La, Ce, Pr, Nd, Sm, Y, Zr). Our results demonstrate that Ti-doping not only enhances the stability of the RFe₁₂ ground state, but also increases the magnetic hardness of the Sm, Pr, Ce and Zr compounds. Particularly, Ti-doping leads to the transition of Ce and Pr from an easy-plane to easy-axis anisotropy. Charge density distributions demonstrate that Ti-doping breaks the original symmetry of the R-site crystal field, which weakens the in-plane interactions between Fe’s 3d and R-site 4f electrons, thereby changing the magnetic anisotropy of RFe₁₂. Projected densities of states reveal that the addition of Ti results in the shift of occupied and unoccupied f-electron energy levels of rare earth elements towards Fermi level, except for Sm, making the exchange coupling strength of the La, Ce, Y, and Zr compounds increase. This study provides a theoretical basis for the subsequent experiment on ThMn₁₂ magnets.

2. Materials and Methods

All DFT calculations were performed by using the open source package for Material eXplorer (OpenMX3.9), based on the norm-conserving pseudopotentials and pseudoatomic-orbital basis sets [32,33,34,35]. The Perdew Burke Enzerhof (PBE) form in generalized gradient approximation (GGA) is used to describe the exchange correlation of electrons [36]. A 7 × 7 × 13 K-point grid with a cut-off energy of 400 Ry was adopted by testing the cut-off energy, and the force on each atom was less than 3 × 10⁻⁴ Hartree/Bohr as the convergence criterion for structural optimization. The energy convergence threshold used for self-consistent calculations is 10⁻⁷ Hartree. The pseudo-potential of the open core approximation was used for the rare earth elements, and the 4f electrons of rare earth elements were regarded as core electrons. The selection of local basis set was as follows: La8.0-s2p2d2f1, Ce8.0-s2p2d2f1, Pr8.0-s2p2d2f1, Nd8.0_OC-s2p2d2f1, Sm8.0_OC-s2p2d2f1, Y10.0-s3p2d2, Zr7.0-s3p2d2, Fe6.0S-s2p2d1, Ti7.0-s3p3d3f1. The number behind the element is the cutoff radius of the element pseudopotential, followed by the basis set. OC means the open core approximation, which treats f-orbital electrons as the core states in pseudopotentials. The stability of the lattice structure was evaluated by calculating the formation energy of the compound, where R = La, Ce, Pr, Nd, Sm, Y, Zr; M = Fe, Ti; the forming energy $∆E\left[R{Fe}_{12}\right]$ and $∆E\left[R{Fe}_{11}\mathrm{T}i\right]$ is defined as follows:

$$\begin{gathered} ∆E\left[R{Fe}_{12}\right]=E_{tot}\left[R{Fe}_{12}\right]-E_{tot}\left[R\right]-12E_{tot}\left[Fe\right] \\ ∆E\left[R{Fe}_{11}Ti\right]=E_{tot}\left[R{Fe}_{11}Ti\right]-E_{tot}\left[R\right]-11E_{tot}\left[Fe\right]-E_{tot}\left[Ti\right] \end{gathered}$$

(1)

In Formula (1), $E_{tot}\left[R{Fe}_{12}\right]$ is the total energy of $R{Fe}_{12}$, $E_{tot}\left[R{Fe}_{11}Ti\right]$ is the total energy of the $R{Fe}_{11}Ti$ compound, $E_{tot}\left[R\right]$ is the ground-state energy of the rare element R bulk, $E_{tot}\left[Fe\right]$ is the energy of bcc structure of the Fe bulk and $E_{tot}\left[Ti\right]$ is the energy of the hcp structure of Ti bulk. These energies are optimized based on the local basis set selected above for different elements.

For the calculation of magnetic properties, a denser K-point grid of 11 × 11 × 21 was used. The selected basis set regards the 4f electrons of rare earth elements as valence electrons, and the local basis set was selected as follows: La8.0-s3p3d2f1, Ce8.0-s3p3d2f1, Pr8.0-s3p3d2f1, Nd8.0-s3p3d2f1, Sm8.0-s3p3d2f1, Y10.0-s3p3d2f1, Zr7.0-s3p2d2f1, Fe6.0H-s3p2d2f1, Ti7.0-s3p2d2f1. The magnetic anisotropy energy (MAE) of a magnet bulk was evaluated in a self-consistent manner based on the total energy. In addition, the densities of states and magnetic exchange coupling constants were calculated with the DFT+U method [37]; the U value was 6 eV which is consistent with previous studies [38] and the J value was 0 eV for rare earth elements. MAE K_u is defined as:

$$\begin{array}{c}{K}_u=E_a-E_c\end{array}$$

(2)

In Formula (2), $E_a$ is the total energy when magnetization is in the a-b plane, and $E_c$ is the total energy when magnetization is along the c-axis. Only Nd compounds present easy-cone magnetic anisotropy, for which its lowest energy at the magnetized angle of 70 is chosen to replace $E_c$ in the calculation of K_u. The positive and negative values of K_u represent uniaxial anisotropy and planar anisotropy, respectively. The total magnetic moment includes the spin magnetic moment and the orbital magnetic moment. Since the orbital magnetic moment is in the form of a spin-orbit coupling (SOC), the total magnetic moment is calculated along the c-axis using non-collinear DFT calculations [39]. In addition, the charge densities are visualized by VESTA3.5.7 [40]. The magnetic exchange couplings are calculated by OpenMX based on Green’s functional representation of the Liechtenstein formula [41,42]. The Powder X-ray diffractograms are obtained based on the optimized unit cell of RFe₁₁Ti by using VESTA [40].

3. Results and Discussions

3.1. ThMn₁₂-Type Phase Stability

The RFe₁₂-based permanent magnet alloy has a ThMn₁₂ crystal structure, belonging to the tetragonal crystal system, and the space group is I4/mmm. As shown in Figure 1a, the ThMn₁₂ structure contains two formula units, as two R atoms are located at the 2a site of the Wyckoff position, and 24 iron atoms occupy three unequal positions of the unit cell, represented as Fef, Fei, Fej, and 8f, 8i, and 8j sites of the Wyckoff position, respectively. Since the binary RFe₁₂ compound is a metastable phase, the doping element Ti with a larger atomic radius is used as the stabilized element that occupies the 8i site (with the largest interatomic distance) of the RFe₁₁Ti structure, shown in Figure 1b [3]. The X-ray diffraction was analyzed based on the optimized structure of RFe₁₁Ti, as shown in Figure S1 (See in Supplementary Materials). The results show that the La, Ce, Nd and Sm compounds have similar main peaks, with a difference in the second strongest peak at 30°. In addition, there is a significant decrease in the intensity of the main peak at 44° for the Pr compound, and a slight decrease in the intensity of the main peak at 41° for the Y, Zr compound. Doped elements with a large atomic radius can reduce the distance between doped elements and Fe atoms, and facilitate the hybridization of electrons between them, thus making the structure stable. In addition, doping Ti in the RFe₁₂ compounds have three different substituted sites of 8f, 8i and 8j. As shown in Figure S2, among all of the RFe₁₁Ti compounds, the formation energy of the 8i site is the lowest among the three sites, indicating that the 8i site is the most favorable doping site for Ti thermodynamically.

To investigate the Ti-doping effect on structural stability, we calculate the formation energies of RFe₁₂ and RFe₁₁Ti. As shown in Figure 1c,d, the results indicate that, in the absence of Ti, only ZrFe₁₂ shows a negative formation energy, while the formation energy of other RFe₁₂ compounds is positive. This suggests that, except for Zr, none of the R elements can form thermodynamically stable 1:12 phase binary compounds with Fe. When Ti is doped in the 8i site of RFe₁₂, the formation energies of RFe₁₁Ti, except for Nd, become negative, which implies that the doping of Ti enables R elements to form stable ThMn₁₂-type ternary compounds with Fe. Furthermore, the substitution of Ti atoms will increase the c/a value of the ThMn₁₂-type structure and reduce the local mismatch of the distance between Fe atoms in the structure, thus improving the stability. The nearest distance of Fe-Fe/Ti in the compound is listed in Table S1, which indicate that the distance is significantly reduced after the addition of Ti [43]. The lattice parameters after the optimization of the above structures are also listed in Table S1.

3.2. Hard Magnetic Properties

In order to investigate the Ti-doping effect on the hard magnetic properties of RFe₁₁Ti compounds, the orbital magnetic moments M_orb, spin magnetic moments M_spin, and total magnetic moments M_tot of RFe₁₂ and RFe₁₁Ti were calculated, as shown in Figure 2a,b and listed in

Table 1. Total magnetic moment M_tot, spin magnetic moment M_spin, orbital magnetic moment M_orb, magnetic anisotropy K_u, saturation magnetization μ₀M_s and magnetic hardness κ of RFe₁₂ and RFe₁₁Ti.

Ingredient	Mspin (μB/f.u.)	Morb (μB/f.u.)	Mtot (μB/f.u.)	Ku (MJ m−3)	μ0Ms (T)	κ (KJ m−3)
LaFe12	27.58	0.61	28.19	−0.62	1.85
CeFe12	26.78	0.88	27.66	−0.04	1.85
PrFe12	25.46	1.71	27.17	−0.28	1.82
NdFe12	24.45	2.05	26.50	−2.27	1.76
SmFe12	22.11	2.62	24.73	8.07	1.68	1.90
YFe12	27.06	0.57	27.63	−0.26	1.88
ZrFe12	26.73	0.65	27.38	0.32	1.91	0.33
LaFe11Ti	23.13	0.60	23.72	−0.32	1.53
CeFe11Ti	22.34	0.88	23.22	0.33	1.53	0.42
PrFe11Ti	20.94	1.65	22.58	0.04	1.49	0.15
NdFe11Ti	19.86	2.15	22.01	−0.83	1.44
SmFe11Ti	17.86	2.60	20.46	7.11	1.35	2.21
YFe11Ti	22.79	0.56	23.34	−0.11	1.56
ZrFe11Ti	22.37	0.63	23.00	0.56	1.58	0.53

. The results indicate that the addition of the non-magnetic stabilizing element Ti reduces the total magnetic moment of RFe₁₂. Compared to the spin magnetic moments, the reduction in orbital magnetic moments is negligible. The total magnetic moments of RFe₁₂ and RFe₁₁Ti are listed in Table 1, in which the total magnetic moments decreased by 15.8% (La), 16.1% (Ce), 16.9% (Pr), 18.0% (Nd), 17.3% (Sm), 15.5% (Y) and 15.0% (Zr). This phenomenon is attributed to the magnetic moment decrease of Fe atoms surrounding the 8i site after Ti substitution, which is revealed by the local magnetic moment, as shown in Tables S5 and S6.

As shown in Figure 2c,d, in contrast to the magnetic moment, Ti-doping increases the magnetic anisotropy for all RFe₁₁Ti compounds except Sm. The K_u of the RFe₁₁Ti compounds is enhanced by 0.31 MJ/m³ (La), 0.36 MJ/m³ (Ce), 0.32 MJ/m³ (Pr), 1.44 MJ/m³ (Nd), 0.15 MJ/m³ (Y), 0.24 MJ/m³ (Zr), and the Sm compound is weakened by 0.96 MJ/m³. Particularly, Ti-doping causes the transition of the Ce and Pr compounds from the easy-plane to easy-axis. In order to further analyze the influence of Ti-doping on magnetic anisotropy, we consider the magnetic anisotropic energy E_ani($\mathsf{\theta }$) as the function of magnetization direction, which are calculated and fitted according to Formula (3) [44].

$$\begin{array}{c}K\left(\theta \right)=K_1{\sin \theta }^2+K_2{\sin \theta }^4+K_3{\sin \theta }^6\end{array}$$

(3)

Formula (3) is the calculation of single-ion anisotropy, $K\left(\theta \right)$ is the magnetic anisotropy energy when magnetization is along the θ direction, K₁, K₂, K₃ are magnetic-crystalline anisotropy constants, and θ is the azimuthal angle of the magnetization direction, which are listed in Table S3. K₁ and K₂ dominate the magnetic anisotropy while K₃ is negligible. The results shown in Figure 2e and Figure S3 indicate that only the E_ani of NdFe₁₂ has a minimum value at 80°, exhibiting the easy conical anisotropy, while the other compounds present easy-axis or easy-plane. The Ti-doping significantly enhances the K₁ value of the Nd compound by 79%. This improvement results in K₂ < K₁/2, leading to a transition of the Nd compound from easy-cone to easy-plane. In addition, both the K₁ and K₂ of Ce and Pr compounds increase after Ti-doping, thus causing the magnetic anisotropy to change from easy-plane to easy-axis. Furthermore, the K_u of the Sm compound decreased due to a 17% decrease in K₁, despite a 10% increase in K₂.

Magnetic hardness is an important parameter to describe the hard magnetic properties of permanent magnets. For a magnetic material with uniaxial anisotropy, κ > 1 means that the material can become a permanent magnet resisting self-demagnetization. A value of 0.1 < κ < 1 means that the material is semihard and can be made into shape-limited oriented magnets [45], and κ < 0.1 is usually used as soft magnetic. The expression of κ is shown as following:

$$\begin{array}{c}\mathsf{\kappa }=\sqrt{(K_u/{\mu }_0{M_s}^2)}\end{array}$$

(4)

K_u is the magnetic anisotropy value, μ₀ is the vacuum permeability, and ${\mu }_0$M_s is the saturation magnetization. The calculated results are shown in Figure 2e and listed in Table 1.

$$\begin{array}{c}{\mu }_0{M}_s=\frac{{4\pi {\mu }_{B}M}_{tot}}{{\mathrm{V}}_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}}\end{array}$$

(5)

Formula (5) calculates the saturation magnetization ${\mu }_0{M}_s$, where $M_{tot}$ is the total magnetic moment which is the sum of the spin magnetic moment M_spin and the orbital magnetic moment M_orb, ${\mu }_B$ is the Bohr magneton, and ${\mathrm{V}}_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}$ is the volume of the unit cell of the compound. Ti-doping increases the cell volume, leading to a decrease in M_s. This indicates that the doping of Ti enhances the magnetic hardness of the Pr and Ce-contained RFe₁₂ compounds, which have κ values greater than 0.1. Furthermore, the κ values of CeFe₁₁Ti are very close to 0.5 and ZrFe₁₁Ti breaks through the κ value limit of 0.5, which can serve as oriented magnets. For the Sm compound, although its K_u is reduced, M_s is reduced more, which ultimately leads to an increase in magnetic hardness and increase in κ value by 14.0%, which is consistent with previous work [46]. For Zr compounds, its K_u increases and M_s decreases, so the magnetic hardness increases more and the κ value increases by 37.7%. In summary, the addition of Ti enhances the magnetic hardness and makes the three compounds of Pr, Ce and Zr become semihard permanent magnets.

3.3. Electronic Structures

Since the crystal field of RFe₁₂ influences the MAE of ThMn₁₂-type magnets, the total charge density distributions of RFe₁₂ and RFe₁₁Ti were investigated. Magnetic anisotropy is mainly determined by the ThMn₁₂-type crystal field parameters $A_0^2$ and the Stevens coefficient of the rare earth element ϑ₂. The Stevens coefficient describes the shape of the ellipsoidal 4f electron real-space wavefunction of rare earth elements and depends only on the number of 4f electrons [47]. The cross-section of the structure diagram in Figure 3 is an Fe-elongated six-ring structure, corresponding to the crystal field around the rare earth atom. Figure 3 and Figure S4 present the total charge density distributions as magnetization along the c-axis direction. They depict that with the increase of the rare earth f electrons, the charge density and electron cloud anisotropy of the rare earth atom increase. Furthermore, Ti-doping breaks the symmetry of the original crystal field in the Fe-elongated six-ring structure, thus regulating the single-ion anisotropy of the rare earth atom. It is worth noting that the charge density distribution in the Nd compounds turns 45° after the doping of Ti, due to the fact that the 4f electrons of Nd are inclined to interact with Fe 3d electrons. This mechanism results in a significant increase in the K_u value in NdFe₁₂ after the addition of Ti. In contrast, the magnetic anisotropy value of the Sm compound, due to its elliptical 4f electron cloud distribution, makes the interaction between the electron cloud of the 4f electron of Sm and the electron cloud of the 3d electron of Ti less strong than that between Sm and Fe, so the value of in-plane anisotropy E_a decreases, finally resulting in a decrease in K_u [48].

We also investigate the R element projected densities of states (PDOS) at site 2a in RFe₁₂ and the total densities of states (TDOS) of RFe₁₂ and RFe₁₁Ti, as shown in Figure 4 and Figure S5, and the calculated results are consistent with previous studies [49]. According to the densities of states in Figure 4, RFe₁₂ and RFe₁₁Ti are metallic states with a zero-band gap. The PDOS of the R-atom occupied state is spin-down, indicating that the rare earth atoms are anti-ferromagnetic when coupling with Fe. As depicted in Figure 4b, Ti-doping causes a few spin-up electron states to shift above the Fermi level, thus resulting in the reduction of the total magnetic moment. Furthermore, the non-occupied f-electron orbitals in the Sm compounds exhibit a tendency to shift away from the Fermi level, which weaken the magnetic exchange coupling and thus reduce the Curie temperature of the magnets [50], whereas the Ce, Nd, Pr, and Zr compounds get close to the Fermi level. In general, the charge densities disclose the slight change in K_u, and the densities of states indicate the major decrease of M_s after the Ti substitution. Thus, it shows that the hard magnetism of the Ce, Pr, Sm, Zr compounds are improved after the Ti-doping.

3.4. Magnetic Exchange Coupling

Magnetic exchange couplings between R-Fe and Fe-Fe determine the temperature stability of hard magnetic properties of the permanent magnet [47]. In the RFe₁₂ and RFe₁₁Ti structures, the magnetic exchange coupling of the 3d-3d orbital between transition metal elements has the greatest weight of influence, followed by the magnetic exchange coupling of the 4f-3d orbital between the rare earth elements and the transition metal elements. As shown in Figure 5, we investigate the magnetic exchange coupling constants of the 4f-3d orbitals (which are 5d-3d orbitals when the R element does not contain 4f electrons) between the R-site elements and the Fe of three different sites in RFe₁₂ and RFe₁₁Ti. The introduction of Ti causes a change of the crystal field symmetry, which affects the magnetic exchange coupling, as shown in Figure S7. It can be deduced by comparing Figure 5a,b that the magnetic exchange coupling constants of R-Fe of La, Ce, Y, and Zr compounds increase after the Ti-doping, whereas those of other compounds decrease. The reduction of the magnetic exchange coupling in the Sm compounds is attributed to the phenomenon of non-occupied f-electron orbitals shifting away from the Fermi level. In addition, the magnetic exchange coupling constants for the RFe₁₁Ti compounds, except Nd, are significantly lower than those for R-Fe(8i). Although the magnetic exchange coupling constant of R-Fe of the Sm compound decreased after Ti-doping, its value was still much higher than that of other compounds, which means that the Curie temperature of the Sm compound is the highest among the RFe₁₂ and RFe₁₁Ti compounds. Interestingly, Nd-Fe(8f) exhibited a ferromagnetic coupling in both NdFe₁₂ and NdFe₁₁Ti; however, its magnitude decreased upon the Ti-doping. Moreover, Nd-Fe(8i) and Nd-Fe(8j) underwent a transition from an antiferromagnetic state to a ferromagnetic state after Ti-doping.

For the Sm and Zr compounds with the larger hard magnetism, we investigate the distance distribution of the magnetic exchange coupling constants between the nearest neighboring Fe atom around RFe₁₂ and RFe₁₁Ti (R = Sm, Zr), as shown in Figure 5c,d. As magnetic atoms separate further, the magnetic exchange coupling constant of Fe exhibits damped oscillation, indicating the presence of Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. The Fe-Fe distance plays a key role in the magnetic exchange coupling in RFe₁₂, which is distributed around 2.8 Å (the lattice constant of bcc-Fe). However, Ti-doping increases the Fe-Fe distance deviating from bcc-Fe, resulting in a reduction in the magnetic exchange coupling constant from 29.82 to 7.57 meV in the Sm compound and from 20.60 to 5.36 meV in the Zr compound. This results in a decrease in Curie temperature of SmFe₁₁Ti and ZrFe₁₁Ti, which weakens their hard magnetism at high temperature [3].

4. Summary

In conclusion, based on the density functional theory calculations, we report the Ti-doping effects on the phase stability, intrinsic hard magnetic properties and electronic structures of RFe₁₁Ti (R = La, Ce, Pr, Nd, Sm, Y, Zr). Our results indicate that Ti-doping enhances the phase stability and hard magnetic properties of the ground-state RFe₁₂ phase. In addition, Ti-doping changes the crystal field and electronic structure of the original structure, thereby increasing the K_u of other R compounds, except Sm. Ti-doping also improves the magnetic hardness of Sm, Pr, Ce and Zr compounds. Furthermore, the change of electronic structures also improves the magnetic exchange coupling of the La, Ce, Y, Zr compounds. Our work reveals the effects of Ti-doping on hard magnetic properties of RFe₁₁Ti magnets.