Introduction

In spintronics, it is still a challenge in experiments to realize room temperature ferromagnetic (FM) semiconductors. The Curie temperature TC of intrinsic two- and three-dimensional FM semiconductors are still far below the room temperature1,2,3,4,5,6,7,8,9, which largely limit their applications.

Doping is an effective approach to control the physical properties of materials. By doping a small amount of magnetic impurities into non-magnetic semiconductors, the magnetic properties of the materials can be dramatically improved, these materials are called dilute magnetic semiconductors (DMS)10,11,12,13,14,15,16,17. For the classic DMS (Ga, Mn)As its highest TC can reach 200 K18. High TC DMSs have been reported in recent experiments, such as TC = 230 K in (Ba, K)(Zn, Mn)2As2 with 15% doping of Mn19,20, TC = 340 K in (Ga, Fe)Sb with 25% doping of Fe21, TC = 385 K in (In, Fe)Sb with 35% doping of Fe22, TC = 280 K in (Si0.25Ge0.75, Mn) with 5% doping of Mn23, etc.

In contrast to DMS, there are also some studies on the magnetic impurities doped antiferromagnetic (AFM) insulators and semiconductors in experiments. Some AFM insulators and semiconductors with high Neel temperature TN have been obtained experimentally, as shown in Table 124,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39. Being a high TN AFM insulator, LaFeO3 has attracted a lot of attention due to its interesting properties. LaFeO3 has a perovskite structure with chemical formula of ABO324,25,26. A high TN = 740 K has been observed in LaFeO326, where the magnetic ground state is G-AFM with intralayer and interlayer AFM order. LaFeO3 has a large optical band gap of 2.05–2.51 eV in experiments27,40. Room temperature ferroelectricity of LaFeO3 has also been observed41. In addition, the doped LaFeO3 has also been studied, such as (La, X)FeO3 with x = Sr42, Al43, Bi44,45, Ca46, Ba46, and La(Fe, D)O3 with D = Mo47, Ni48, Cr49,50,51,52, Ti40,53,54, Zn27,55, Cu56, Mn57, Mg58, Co59, etc. It shows a high tolerance to impurities, the doping concentration at both La and Fe sites could reach to about 50%. Some magnetic impurities doped AFM insulators and semiconductors with high TN are shown in Table 2. The experimental studies of La(Fe1−xDx)O327,40,47,48,49,50,51,52,53,54,55,56,57, Bi(Fe1−xDx)O360,61,62,63,64,65, and (Ni1−xDx)O66,67,68,69 have shown very tiny net magnetic moments, although the high concentrations of magnetic impurities can be realized.

Table 1 Some antiferromagnetic (AFM) insulators and semiconductors with high Neel temperatrue TN in experiments
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Table 2 Some magnetic impurities doped AFM insulators and semiconductors with high TN
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As shown in Table 2, there is an increase of net magnetic moment in AFM materials after doping, which was explained as the formation of clusters49,50,52,55, enhancement of interface effects40,52,53, change of magnetic coupling50,51,52,56, etc. However, their net magnetic moment is still negligible, which can be understood from the symmetry of spin up and down sublattices of AFM host materials. As shown in Fig. 1, magnetic impurities were equally doped into the spin up and down sublattices of the AFM materials, resulting in zero net magnetic moment. On the other hand, as shown in Table 2, only a few theoretical studies focus on the magnetic impurities doped AFM insulators and semiconductors, and nearly have not discussed the theoretical results of magnetic properties, such as TN70,71,72,73,74,75. Is there a way to break the symmetry of spin up and down sublattices of AFM host materials?

Fig. 1
figure 1

Schematic diagram of spin-independent doping (left) with zero net magnetic moment and spin-dependent doping (right) with non-zero net magnetic moment, for the antiferromagnetic (AFM) materials doped with magnetic impurities.

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In this paper, we propose that the effective magnetic field from the FM substrate can break the symmetry of spin up and down sublattices and make spin-dependent doping possible in AFM materials, as schematically shown in Fig. 1. To demonstrate our proposal, we study the Mn-doped AFM insulator LaFeO3 with FM substrate of Fe metal by the density functional theory (DFT) calculations. The calculation results for the supercell La(Fe, Mn)O3/bcc-Fe show that the doped magnetic Mn impurities prefer to occupy one sublattice of AFM insulator, and introduce large magnetic moments in La(Fe, Mn)O3. By this way, some ferrimagnetic (FIM) semiconductors with Curie temperature TC above room temperature are predicted for La(Fe1−xDx)O3 with D = 3d, 4d transition metal impurities and x = 0.125 and 0.25. In addition, La(Fe0.75D0.25)O3 shows large magneto-optical Kerr effect. The variation of TC in the FIM La(Fe1−xDx)O3 as a function of elements D can be well understood by a formula of mean-field theory. Our results propose a way to obtain high-temperature FIM semiconductors by spin-dependent doping in high-temperature AFM insulators and semiconductors.

Results

Spin-dependent doping

LaFeO3 has a G-AFM ground state and shows very weak ferromagnetism due to the spin canting caused by the Dzyaloshinskii-Moriya (DM) interaction76. The net magnetic moment per Fe atom in LaFeO3 is about 10−4 μB. Experiments found that doping at Fe sites will increase the net magnetic moment to 10−4~10−2 μB per Fe atom, while it’s still in the G-AFM state, as shown in Table 2. The thin films of LaFeO3 maintain the AFM properties in experiments77,78,79,80.

To break the symmetry of spin up and down sublattices in LaFeO3, we study the AFM insulator LaFeO3 with FM substrate of Fe metal and consider a LaFeO3/bcc-Fe heterojunction, as shown in Fig. 2. The lattice constant is a = 2.87 Å for bcc-Fe, and a = 5.60 Å, b = 5.66 Å for LaFeO3. The lattice of 2 × 2 × 1 bcc-Fe and LaFeO3 fit well with a small lattice mismatch about 1%. The optimized lattice constants of LaFeO3/bcc-Fe heterojunction are a = b = 5.56 Å, where three layers of LaFeO3, one layer of bcc-Fe along (001) direction, and a vacuum layer of 20 Å are considered. For simplicity, we fix the spin of the bcc-Fe substrate as spin up.

Fig. 2
figure 2

Crystal structure of the supercell La(Fe, Mn)O3/bcc-Fe, where the Mn impurity are doped at Fe site of layer L3.

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It is both structural and electronic for the preferential doping sites. As shown in Fig. 2, there are six Fe layers in the 1 × 1 × 3 LaFeO3 supercell, labeled as L1~L6. For each layer, there are both spin up and down sites. As an example, the supercell of La(Fe, Mn)O3/bcc-Fe with Mn at Layer 3 and spin down sublattice is shown in Fig. 2. The differences of total energy of the supercells La(Fe, D)O3/bcc-Fe with dopants D = 3d and 4d transition metals at layer α and spin up and down sublattices are calculated as shown in Table 3. There are significant energy differences with dopants D at layer α and spin up and down sublattices, indicating the stability of the spin-dependent doping.

Table 3 Total energy difference and formation energy Eformation of the supercells La(Fe, D)O3/bcc-Fe with dopants D at layer α and sublattice spin σ
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The energy difference of the supercells La(Fe, Mn)O3/bcc-Fe with Mn at spin up and down sublattices is still significant when impurities Mn are doped at layer 4, i.e., 1.6 nm to the interface. Since the LaFeO3 nanosheets could be as thin as 5 nm81,82, the influence of Fe substrate is effective. The spin-dependent doping will lead to spin polarization of dopants and induce AFM-FIM transition. Experiment found that the magnetic field will significantly increase the net magnetic moment of ZnO doped with 2% Cr83.

To study the formation energy of La(Fe, Mn)O3 with Fe substrate, as shown in Fig. 2, the supercell La12Fe19MnO36 is used. The formation energy is calculated by \({E}_{{\rm{formation}}}=({E}_{{\rm{La}}({\rm{Fe}},{\rm{Mn}}){{\rm{O}}}_{3}/{\rm{bcc}}-{\rm{Fe}}}-12{E}_{{\rm{La}}}-19{E}_{{\rm{Fe}}}\,-\,{E}_{{\rm{Mn}}}-36{E}_{{\rm{O}}})/68\), where \({E}_{{\rm{La}}({\rm{Fe}},{\rm{Mn}}){{\rm{O}}}_{3}/{\rm{bcc}}-{\rm{Fe}}}\) is energy of supercell La(Fe, Mn)O3/bcc-Fe with one dopant Mn at layer α and sublattice spin σ. ELa, EFe, and EMn are energies per atom for bulks of La, Fe, and Mn with symmetries of P63/mmc, Im\(\overline{3}\) m and I\(\overline{4}\) 3m, respectively. EO is energy per atom for O2 gas with symmetry C2/m. The formation energies for La(Fe, D)O3/bcc-Fe with dopants D = 3d and 4d transition metals are calculated in the same way, and the results are shown in Table 3. For dopants D = 3d and 4d transition metals, the obtained formation energies are negative, and lower than the formation energy of −2.435 eV atom−1 for host material LaFeO3/bcc-Fe, indicating the stability of doping. In addition, the difference of total energy of the supercells La(Fe, D)O3/bcc-Fe with dopants D at layer α and spin up and down sublattices is also shown in Table 3. There are significant energy differences with dopants D at spin up and down sublattices, indicating the stability of spin-dependent doping.

The average magnetic moment of Fe atoms in bulk bcc-Fe is 2.95 μB. For the supercell La(Fe, Mn)O3/bcc-Fe with Mn at L3 and spin down sublattice, as shown in Fig. 2, there are two Fe layers in bcc-Fe. The average magnetic moment is 3.37 μB for Fe at the interface and 3.12 μB for Fe at the layer next to the interface. Thus, the magnetic moment of Fe substrate near the interface has been slightly enhanced.

T C in LaFeO3-based FIM semiconductors

The band structure of LaFeO3 is shown in Fig. 3a, with a band gap of 2.38 eV, consistent with the experimental value of 2.05~2.51 eV27,40. Since LaFeO3 is AFM with zero net magnetic moment, we determine its TN through energy and specific heat by Monte Carlo simulations. The results are shown in Fig. 3c, with a sharp peak of specific heat at TN = 650 K, close to the experimental value of 740 K26.

Fig. 3: Band structures and Monte Calor results of LaFeO3 and La(Fe0.75Mn0.25)O3.
figure 3

DFT results of band structure for a LaFeO3 with a band gap of 2.38 eV and b La(Fe0.75Mn0.25)O3 with a band gap of 0.56 eV. Monte Carlo results of energy and specific heat as a function of temperature for c LaFeO3 with Neel temperature TN = 650 K and d La(Fe0.75Mn0.25)O3 with Curie temperature TC = 603 K. Results here are obtained without Fe substrate.

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For the La(Fe0.75Mn0.25)O3 where one of the four Fe atoms is replaced by a Mn atom in a LaFeO3 unit cell. DFT results show that its magnetic ground state is FIM. Mn has a magnetic moment of 3.73 μB, smaller than Fe (4.18 μB), induces a net magnetic moment near 0.12 μB per LaFeO3 unit cell. In addition, La(Fe0.75Mn0.25)O3 is a FIM semiconductor with a band gap of 0.56 eV, and a high Curie temperature TC = 603 K is estimated by the Monte Carlo simulation, as shown in Fig. 3d.

To study the formation energy of La(Fe0.75Mn0.25)O3 without Fe substrate, the unit cell La4Fe3MnO12 is used. The formation energies are calculated by \({E}_{{\rm{formation}}}=({E}_{{\rm{La}}({\rm{Fe}},{\rm{Mn}}){{\rm{O}}}_{3}}-4{E}_{{\rm{La}}}-3{E}_{{\rm{Fe}}}-{E}_{{\rm{Mn}}}-12{E}_{{\rm{O}}})/20\), where \({{{E}}}_{{\rm{La}}({\rm{Fe}},{\rm{Mn}}){{\rm{O}}}_{3}}\) is the energy of La4Fe3MnO12 with one dopant Mn at a LaFeO3 unit cell without substrate. The formation energies for La(Fe, D)O3 with dopants D = 3d and 4d transition metals are calculated in the same way, and the results are shown in Table 4. For dopants D = Sc, Ti, V, Cr, Y, Zr, Nb, Mo, Tc, the obtained formation energies are negative, and lower than the formation energy of −2.71 eV atom−1 for host material LaFeO3, indicating the stability of doping. It is noted that without Fe substrate, the formation energies for D at spin up and down sublattices are the same.

Table 4 The calculated results of the average magnetic moment per magnetic atom 〈M〉, band gap, formation energy Eformation, and Curie temperature TC for La(Fe0.75D0.25)O3 and La(Fe0.875D0.125)O3
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In addition, the formation energies of Bi(Fe, D)O3, Sr(Tc, D)O3, and Ca(Tc, D)O3 are calculated in the same way, and the results are shown in Supplementary Tables 4–6 in Supplementary Material, respectively. All the obtained formation energies are negative, and some are lower than the formation energy of the host materials, indicates their stability of doping.

With different 3d and 4d dopants, the magnetic ground states of La(Fe0.75D0.25)O3 maintain FIM. Because the magnetic moments of Fe are almost constant compared with different dopants, the net magnetic moment are from the broken of the symmetry of the AFM spin sublattices, which can be calculated as Mtot = ∣Mdopant − MFe∣, the detailed magnetic moments see Supplementary Table 1 in Supplementary Material. The average magnetic moment per lattice 〈M〉 of La(Fe0.75D0.25)O3 is defined as 〈M〉 = Mtot/N, the magnetic lattice number N = 4 for the LaFeO3 unit cell, and the results are shown in Fig. 4a. The Curie temperature TC of La(Fe0.75D0.25)O3 which was estimated by the Monte Carlo simulations, as shown in Fig. 4b. It is noted that most of TC with 3d and 4d dopants are above room temperature.

Fig. 4: Average magnetic moment per magnetic atom 〈M〉 and Curie temperature TC for La(Fe0.75D0.25)O3 and La(Fe0.875D0.125)O3.
figure 4

a Average magnetic moment per magnetic atom 〈M〉 and b Curie temperature TC for La(Fe0.75D0.25)O3. c 〈M〉 and d TC for La(Fe0.875D0.125)O3. The impurity D is taken as 3d and 4d transition metal elements. For comparison, the TN = 650 K of host LaFeO3 is also included in (b) and (d). Results here are obtained without Fe substrate.

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The band structure of bulk LaFeO3 show spin splitting in k paths Γ − R2 and Γ − U2. The spin splitting without spin-orbit coupling (SOC) that happens in antiferromagnetic (AFM) materials requires broken θIT and UT symmetry, where θ, I, T, U are the time inverse, space inverse, translation, and spin inverse operations, respectively84,85,86. The crystal space group and magnetic space group of G-AFM LaFeO3 are pnma and P21/c, respectively, allow the spin splitting without soc in part of Brillouin zone85. The spin splitting in AFM materials could be k-dependent, according to the symmetry of k space84. Similar band structures with k-dependent band splitting in LaFeO3 is obtained and discussed in the previous study87.

To discuss the effect of concentrations, the material La(Fe0.875D0.125)O3 is studied. A 2 × 1 × 1 supercell is considered, where one of eight Fe atoms is replaced by the D (3d or 4d) atom. DFT results show that its magnetic ground state maintain FIM with different dopants. The 〈M〉 of La(Fe0.875D0.125)O3 is about half to that of La(Fe0.75D0.25)O3, as shown in Fig. 4c. It is expected since the concentration of dopants decreases from 1/4 to 1/8. It is interesting to note that the TC of La(Fe0.875D0.125)O3 are higher than that of La(Fe0.75D0.25)O3, as shown in Fig. 4d.

The calculated values of average magnetic moment per lattice 〈M〉, Curie temperature TC, and band gap of La(Fe0.75D0.25)O3 and La(Fe0.875D0.125)O3 are summarized in Table 4.

MOKE in LaFeO3-based FIM semiconductors

We investigated the magneto-optical Kerr effect for La(Fe0.75D0.25)O3. The Kerr rotation angle is given by:

$${\theta }_{{\rm{K}}}(\omega )=Re\frac{{\varepsilon }_{xy}}{(1-{\varepsilon }_{xx})\sqrt{{\varepsilon }_{xx}}},$$
(1)

where εxx and εxy are the diagonal and off-diagonal components of the dielectric tensor ε, ω is the frequency of incident light. The dielectric tensor ε can be obtained by the optical conductivity tensor σ as \(\varepsilon (\omega )=\frac{4\pi i}{\omega }\sigma (\omega )+I\), where I is the unit tensor. The calculated ε(ω) as a function of photon energy for LaFeO3, and La(Fe0.75D0.25)O3 with D = Ni, Cu, Zn, Mo, and Pd is shown in Fig. 5. The experimental result for Fe88 and our DFT result for Fe bulk are also included for comparison. There are a big Kerr angle for La(Fe0.75D0.25)O3 with ω < 2 eV, about 10 times bigger than bcc-Fe. It is worth noting that LaFeO3 shows small but non-zero Kerr angle, despite its collinear AFM order, this may be related to the room temperature ferroelectricity of LaFeO341. Detailed results of the Kerr angle are given in Supplementary Fig. 9 in Supplementary Material.

Fig. 5: Results of magneto-optical Kerr rotation angle.
figure 5

a DFT results of Kerr angle for Fe, LaFeO3 and La(Fe0.75D0.25)O3 with D = Ni, Cu, and Zn. b DFT results of Kerr rotation angle for La(Fe0.75D0.25)O3 with D = Mo, Pd, and Cd. Experimental Kerr rotation angle of Fe88 is also included for comparison. Results here are obtained without Fe substrate.

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Other high T C FIM semiconductors

In addition to LaFeO3, we also study the doping of other high TN AFM insulators and semiconductors, including BiFeO3, SrTcO3, CaTcO3. The calculation results are shown in Table 5. When 25% of the 3d transition metal element of the host are replaced by other 3d or 4d impurities, many room temperature FIM semiconductors are obtained in LaFeO3, BiFeO3, SrTcO3, and CaTcO3. All of these host materials are perovskite with TN above 550 K and band gap bigger than 1.5 eV. Detailed results are given in Supplementary Figs. 4–8 and Supplementary Tables 4–6 in Supplemental Material. For the same impurity and concentration, TC and band gap obtained after doping are positively related to TN and band gap of AFM material. According to the calculation results, room temperature FIM semiconductors could be obtained by doping in AFM semiconductors, and a high TN and a large band gap are needed.

Table 5 The calculated band gap and TN for some high TN AFM insulators and semiconductors with chemical formula ABO3, and the calculated band gap, TC and 〈M〉 for their doped materials A(B0.75D0.25)O3
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Mean-field theory of the effect of doping on T C

To study the influence of different impurities on TC, as shown in Fig. 4, we use the Weiss molecular field approximate89. By the simple AFM Heisenberg model and the mean-field approximation (MFA), we get TN of G-AFM LaFeO3 as

$${T}_{{\rm{N}}}=2\frac{{J}_{0}{S}_{0}({S}_{0}+1)}{{k}_{{\rm{B}}}},$$
(2)

where J0 represents the nearest-neighbor coupling constant of Fe–Fe in LaFeO3, S0 is the magnetic moment of Fe in LaFeO3, and kB is the Boltzmann constant. By the help of DFT calculation, J0 = 2.25 meV, S0 = 4.15 μB. By Eq. (2), it has TN = 1115 K. It is noted that the TN = 1115 K by mean-field theory of Eq. (2) is much higher than the TN = 650 K by the Monte Carlo simulation with the same J0 and the TN = 740 K of LaFeO3 in experiment26.

By the similar mean-field theory, we can obtain the expression of TC for FIM semiconductors La(Fe, D)O3. For simplicity, we only discuss the case of one impurity per unit cell without disorder, and only the nearest-neighbor coupling are considered.

The ratio of TC and TN is expressed as:

$$\begin{array}{ll}\frac{\;\;\;{T}_{{\rm{C}}}}{{T}_{{\rm{N}}}}\,=\,{t}_{0}\sqrt{\frac{a+\sqrt{{a}^{2}-b}}{8}},&\\ \quad\;\;{a}\,=\,\frac{1}{9}\left[6(6-{z}_{{\rm{AB}}}){t}_{{\rm{D}}}+{z}_{{\rm{AB}}}{z}_{{\rm{BA}}}+6(6-{z}_{{\rm{BA}}})\right],\\ \quad\;\;{b}\,=\,\frac{16}{9}{t}_{{\rm{D}}}(6-{z}_{{\rm{AB}}})(6-{z}_{{\rm{BA}}}),\\ \quad\,{t}_{0}\,=\,\frac{{J}_{1}}{{J}_{0}}\frac{S(S+1)}{{S}_{0}({S}_{0}+1)},{t}_{{\rm{D}}}={\left(\frac{{J}_{2}}{{J}_{1}}\right)}^{2}\frac{{S}_{{\rm{D}}}({S}_{{\rm{D}}}+1)}{S(S+1)},\end{array}$$
(3)

where J0, J1 are the nearest-neighbor coupling constants of Fe–Fe in LaFeO3 and La(Fe, D)O3, respectively, J2 is the nearest-neighbor coupling constants between Fe and D in La(Fe, D)O3. S0, S are the magnetic moments of Fe in LaFeO3 and La(Fe, D)O3, respectively, and SD is the magnetic moment of D in La(Fe, D)O3, zij is the coordination number of the site j near the site i. Supposing dopants at spin down sites, sublattice A mean Fe atoms spin up with nearest-neighbor impurities, sublattice B mean Fe atoms spin down without nearest-neighbor impurities, respectively. Here t0 describes the ratio of Fe–Fe couplings in La(Fe, D)O3 and LaFeO3, tD describes the ratio of Fe–D coupling and Fe–Fe coupling in La(Fe, D)O3. See detailed information in Supplementary Sections 12 and 13 in Supplemental Material.

For case of 1/4 doping, the coordination number is zAB = 4, zBA = 6. For the case of 1/8 doping, the coordination number is zAB = 4, zBA = 4. Take these parameters and coupling constant and magnetic moment from DFT into Eq. (3), we obtain the ratio of TC/TN for La(Fe0.75D0.25)O3 and La(Fe0.875D0.125)O3, as shown in Fig. 6a, b, respectively. The ratio of TC/TN obtained by Eq. (3) with the mean-field approximation (MFA) and numerical calculations (DFT + MC) shown in Fig. 4 are in a good agreement. Thus, we note that it is possible to understand the effect of doping on TC in FIM semiconductors La(Fe, D)O3 by Eq. (3) of the conventional mean-field theory.

Fig. 6: The ratio of TC for La(Fe1−xDx)O3 and TN of LaFeO3.
figure 6

For TN of LaFeO3 and TC of La(Fe1−xDx)O3, the ratio of TC/TN for a x = 0.25 and b x = 0.125. The impurity D is taken as 3d and 4d transition metal elements. The numerical results (DFT+MC) are taken from Fig. 4b, d. The mean-field approximation results are obtained by Eq. (3).

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Discussion

Based on the DFT calculations, we show an approach to obtain room temperature FIM semiconductors by spin-dependent doping in high TN insulators and semiconductors with large band gap. To demonstrate spin-dependent doping, the Mn-doped AFM insulator LaFeO3 with FM sublattices bcc-Fe is studied by the DFT calculation. It is shown that the doped Mn impurities prefer to occupy one sublattice of LaFeO3 due to the effective magnetic field of substrate bcc-Fe, and obtain the FIM semiconductor La(Fe, Mn)O3 with large magnetic moment. By this method, we predict a series of room temperature FIM semiconductors in La(Fe, D)O3, where D denoted the dopant of 3d and 4d transition metals. Large magneto-optical Kerr effect were found in La(Fe0.75D0.25)O3. By the equation of mean-field approximation, the ration of TC in La(Fe, D)O3 and TN of LaFeO3 are obtained, in a good agreement with the numerical results of DFT + MC. In the same way, the FIM semiconductors with high TC are also predicted in some other high TN AFM insulators and semiconductors, such as BiFeO3, SrTcO3, CaTcO3, etc. Our results suggest that spin-dependent doping is a promising way to produce high TC FIM semiconductors from high TN AFM insulators and semiconductors.

Methods

Density functional theory calculations

Our calculations were based on the DFT as implemented in the Vienna ab initio simulation package (VASP)90. The exchange-correlation potential is described by the Perdew-Burke-Ernzerhof (PBE) form with the generalized gradient approximation (GGA)91. The electron-ion potential is described by the projector-augmented wave (PAW) method92. We carried out the calculation of GGA + U with U = 4 or 2 eV for 3d or 4d elements, respectively. The plane-wave cutoff energy is set to be 500 eV. The 4 × 4 × 1, 4 × 4 × 3, and 2 × 4 × 3 Γ center k-point meshed were used for the Brillouin zone (BZ) sampling for supercells of La(Fe0.75D0.25)O3/bcc-Fe, La(Fe0.75D0.25)O3 and La(Fe0.875D0.125)O3, respectively. The structures of all materials were fully relaxed, where the convergence precision of energy and force were 10−6 eV and 10−2 eV Å−1, respectively. The van der Waals effect is include with DFT-D3 method93. The Wannier90 code was used to construct a tight-binding Hamiltonian to calculate the Kerr rotation angle94,95.

Monte Carlo program

The Heisenberg-type Monte Carlo simulation was performed on 10 × 10 × 10 and 8 × 8 × 8 lattice with 4000 and 4096 magnetic points for La(Fe0.75D0.25)O3 and La(Fe0.875D0.125)O3, respectively. More than 8 × 104 steps were carried for each temperature, and the last one-third steps were used to calculate the temperature-dependent physical quantities.