Introduction

Half-metallic ferromagnets (HMFs), coexisting metallic nature for electrons in one spin channel and insulating nature in the other, can generate completely spin-polarized electrical current, and are considered as key candidates for many spintronic applications from magnetic memories to spin-polarized tunneling devices.1,2,3 In order to develop high-performance spintronic devices, three criteria are highly required for a HMF: (1) a high Curie temperature (Tc) for room-temperature (RT) applications, (2) a wide half-metallic gap (ΔHM) to efficiently prevent the spin-flip transition of carriers due to thermal excitation, and (3) a large bulk magnetocrystalline anisotropy energy (MAE) to overcome the thermal-fluctuation-induced random and uncontrollable switching of spins.4 Although some HMFs have been reported theoretically and experimentally,2,3,5,6,7,8 to our knowledge, most of them cannot fulfill all these criteria.

Inorganic metal halide perovskites, which are advantaged in their exceptional defect tolerance, low-cost solution processing and tunable emission across the visible spectrum, have multiple optoelectronic applications, e.g., solar cells, photodetectors, light-emitting diodes, transparent conductors, transistors, and lasing applications.9,10,11,12,13,14,15,16,17,18,19,20,21,22 Very recently, a new type of 〈111〉-oriented layered halide double perovskite (LHDP), e.g., Cs4CuSb2Cl12 and Cs4MnSb2Cl12, have been synthesized successfully in experiments with good tolerance towards humidity, heat and light.23,24,25,26 Interestingly, it is found that Cs4MnSb2Cl12 exhibits weak antiferromagnetic (AFM) ordering between the first nearest-neighboring (NN) Mn ions. The observation of AFM ground-state in Cs4MnSb2Cl12 opens up new possibilities in discovering ferromagnetic (FM) ordering, even half-metallicity, in LHDPs.

In this article, we have explored the possibility of the existence of half-metallicity in LHDPs using first-principles calculations based computational material screening in a large number of LHDPs with the stoichiometries of Cs4MB2X12 (\({\mathrm{Cs}}_{4}{{\mathrm{M}}^{2 + }}{{\mathrm{B}}^{3 + }}{}_2{{\mathrm{X}}^{\mathrm{VII}}}{}_{12}\) and \({\mathrm{Cs}}_{4}{{\mathrm{M}}^{4 + }}{{\mathrm{B}}^{2 + }}{}_2{{\mathrm{X}}^{\mathrm{VII}}}{}_{12}\)). Among several compounds that have good dynamic and thermal stabilities, interestingly, we identify that the Cs4FePb2Cl12 can exhibit a half-metallic ground-state with a calculated Tc above RT (Tc ~ 450 K). Meanwhile, Cs4FePb2Cl12 has a wide ΔHM ~ 0.55 eV and a large MAE ~ 380 μeV/Fe, which ranks Cs4FePb2Cl12 as one of the best half-metallic materials for spintronics. Remarkably, when the thickness of layered Cs4FePb2Cl12 is reduced to monolayer, it can still sustain the half-metallicity with a high Tc ~ 370 K and a strong MAE ~ 318 μeV/Fe. The external strain can be applied to efficiently manipulate the magnetic coupling and MAE of Cs4FePb2Cl12.

Results and discussion

The family of Cs4MB2X12 (\({\mathrm{Cs}}_4{\mathrm{M}}^{2 + }{\mathrm{B}^{3 + }}_{\!2}{\mathrm{X}^{{\mathrm{VII}}}}_{\!12}\) or \({\mathrm{Cs}}_4{\mathrm{M}}^{4 + }{\mathrm{B}^{2 + }}_{\!2}{\mathrm{X}^{{\mathrm{VII}}}}_{\!12}\)) is a unique mixed metal 〈111〉-oriented layered perovskite with each B-M-B layer consisting of three sublayers (one M octahedral layer and two B octahedral layers) (see Fig. 1a). The inclusion of Cs is very critical to stabilize the perovskite structure.27 Since the distances between two NN M ions are quite large (>7 Å) in Cs4MB2X12 family LHDPs, the direct exchange coupling between them is extremely weak.28,29,30 Instead, from the orbital-projected density of states (DOS), we can find that a strong coupling between M d and X p orbitals generally exists in Cs4MB2X12 (Supplementary Fig. 1). The two NN M ions could interact with each other through the bridging BX6 octahedron via super-superexchange mechanism, as shown in Fig. 1b.

Fig. 1
figure 1

Magnetic mechanism. a Crystal structure of Cs4MB2X12 LHDP. Voids between the octahedrons are filled with Cs atoms, which are omitted for clarity. b Schematic diagram of the super-superexchange interaction in Cs4MB2X12 LHDPs. c Schematic diagram of AFM coupling in experimental synthesized Cs4MnSb2Cl12. Red and blue bars represent spin-up and spin-down states, respectively.

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Origin of AFM ground-state in Cs4MnSb2Cl12

To explore how to achieve FM in LHDPs, first, we bring the origin of the AFM behavior in Cs4Mn2+Sb3+2Cl12 LHDP, as observed in recent experiment,24 to light. Based on the crystal field theory, the five d orbitals of Mn will split into a singlet Alg \((d_{z^2})\), doublet Eg (dxz + dyz), and doublet \(E_{\mathrm{g}}^\prime\) \((d_{xy} + d_{x^2 - y^2})\) orbitals under a D3d crystal symmetry. The order of d-orbital energies (Alg, Eg, \(E_{\mathrm{g}}^\prime\)) is determined by the interaction strength of X p orbital and M d orbital.31 Moreover, it is found that the Mn2+ in Cs4MnSb2Cl12 favors a high spin configuration of d5↑d0↓, resulting in the magnetic moments of 5 μB/Mn. As shown in Fig. 1c, the orbital coupling between the two NN Mn ions, i.e., Mna2+ and Mnb2+, can gain energy only if they have antiparallel spin alignment (AFM coupling). For example, the interaction between the occupied \({\mathrm{Mn}}_{\mathrm{a}}^{2 + } - E_{\mathrm{g}}^\prime \uparrow\) and empty \({\mathrm{Mn}}_{\mathrm{b}}^{2 + } - E_{\mathrm{g}}^\prime \uparrow\) would lead to a doublet low-lying bonding states and another doublet high-lying antibonding states. A similar orbital coupling occurs between the occupied \({\mathrm{Mn}}_{\mathrm{b}}^{2 + } - E_{\mathrm{g}}^\prime \downarrow\) and empty \({\mathrm{Mn}}_{\mathrm{a}}^{2 + } - E_{\mathrm{g}}^\prime \downarrow\). Finally, the five electrons would occupy five low-lying bonding states to maximize the energy gain of the system. As demonstrated in Cs4MnSb2Cl12, the magnetic ground-states in Cs4MB2X12 are mainly determined by the orbital occupations of M ions via super-superexchange interactions. Therefore, it is interesting to further explore the possibility of the existence of long-range FM ordering in a large number of unknown Cs4MB2X12 LHDPs.

Materials screening for stable Cs4MB2X12 compounds

We have considered the combinations of (Cs, M2+, B3+, XVII) with M2+ = Ti2+/V2+/Cr2+/Mn2+/Fe2+/Co2+/Ni2+/Cu2+, B3+ = Sb3+/In3+/Bi3+, and XVII = Cl−/Br−/I− in Cs4M2+B3+2XVII12 LHDPs and (Cs, M4+, B2+, XVII) with M2+ = Ti2+/V2+/Cr2+/Mn2+/Fe2+/Co2+/Ni2+, B2+ = Pb2+/Ge2+, and XVII = Cl−/Br−/I− in Cs4M4+B2+2XVII12 LHDPs (see Fig. 2a). In order to discover possible promising magnets for spintronics in unknown Cs4MB2X12 compounds, material stability is an important issue to be clarified (Fig. 2b). Firstly, to evaluate the crystallographic stability of materials in the perovskite structure, Goldschmidt’s empirical criterion is adopted with two empirical quantities: the Goldschmidt tolerance factor (t) and the octahedral factor (μ).27,32,33,34 It is found that the formation of stable perovskite structures requires 0.81 < t < 1.11 and 0.41 < μ < 0.90.27,32 Among all the 114 compounds, we found that 57 of them fall into the empirical stable areas of perovskite structures (Supplementary Note 1 and Supplementary Fig. 2).

Fig. 2
figure 2

Materials design strategy and materials screening. a Schematic diagram of the formation of Cs4MB2X12 (Cs4M2+B3+2XVII12 and Cs4M4+B2+2XVII12) LHDPs with different components. b Predicting stable Cs4M2+B3+2XVII12 and Cs4M4+B2+2XVII12 compounds. The “red check” indicates the compounds that have been synthesized in experiments, whereas the “green plus” and “gray cross” mean these compounds that are unreported (UR) in literatures.

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Secondly, to evaluate the thermodynamic stabilities of the above 57 Cs4MB2X12 compounds, we calculate their decomposition enthalpies (ΔH) with respect to possible decomposition pathways involving all the existing binary and ternary secondary phases from the Inorganic Crystal Structure Database (ICSD)35 and Automatic Flow (AFLOW).36 Besides of the existing Cs4CuSb2Cl12, Cs4MnSb2Cl12, we found that there are another eight compounds (Cs4MnBi2Cl12, Cs4CuBi2Cl12, Cs4CuIn2Cl12, Cs4MnPb2Cl12, Cs4FePb2Cl12, Cs4NiPb2Cl12, Cs4MnPb2Br12, and Cs4NiPb2Br12) exhibiting good thermodynamic stabilities with positive ΔH (Supplementary Note 2 and Supplementary Fig. 3). Finally, we have confirmed their dynamical and thermal stabilities. It is found that except for Cs4CuIn2Cl12, Cs4NiPb2Cl12, and Cs4NiPb2Br12, the other five compounds exhibit good phonon stability evidenced by no imaginary modes in the whole Brillouin zone (Supplementary Fig. 4) and thermally dynamical stability at RT evidenced by the ab initio molecular dynamic (AIMD) simulations (Supplementary Fig. 5).

Origin of FM ground-state in Cs4FePb2Cl12

For the stable Cs4MB2X12 compounds, we have explored all the possible magnetic configurations, including one FM configuration, three collinear antiferromagnetic order (AFM-I, AFM-II, and AFM-III) and non-collinear antiferromagnetic order (AFM-N), as shown in Fig. 3a. In AFM-I configuration, each M-B-M layer is ferromagnetic, but antiferromagnetic with respect to its neighboring M-B-M layers. For AFM-II configuration, each [MX6] along the [010] direction is ferromagnetic but with antiferromagnetic coupling to its neighboring [MX6]. For AFM-III configuration, each [MX6] along the [010] direction is antiferromagnetic.37 In AFM-N configuration, the spin directions on the three NN M atoms lie 120 degrees apart in space, and each M atom has the same magnetic moment, resulting in zero net magnetization in the system. After the total energy calculations, we find that the magnetic ground-states of all the stable Cs4MB2X12 are AFM, except for Cs4FePb2Cl12 (Fig. 3b and Supplementary Note 3).

Fig. 3
figure 3

Magnetic ground-states of stable compounds. a A Cs4MB2X12 with different AFM configurations. Voids between the octahedrons are filled with Cs atoms, which are omitted for clarity. Arrows denote the spin directions. b Magnetic ground-states for the seven stable Cs4MB2X12 compounds. c Schematic diagram of electron occupation of M2+ in Cs4M2+Bi2Cl12 (M2+ = Mn2+/Fe2+/Cu2+) and M4+ in Cs4M4+Pb2Cl12 (M2+ = Mn2+/Fe2+). d Schematic diagram of FM coupling in Cs4FePb2Cl12.

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As shown in Fig. 3c, in Cs4MnBi2Cl12, Cs4CuBi2Cl12, Cs4MnPb2Cl12, Cs4MnPb2Br12, and Cs4FePb2Cl12, the M atoms favor high spin configurations of d5↑d0↓, d5↑d4↓, d3↑d0↓, d3↑d0↓, and d4↑d0↓, respectively. Interestingly, it is found that the local magnetic moments in Cs4MnBi2Cl12, Cs4CuBi2Cl12, Cs4MnPb2Cl12, and Cs4MnPb2Br12 are contributed by fully-filled M d orbitals. Similar to the case of Cs4MnSb2Cl12, the existence of AFM ground-states in these four Cs4MB2X12 can be well understood. While in Cs4FePb2Cl12 with a local magnetic moment contributed by a half-filled Fe d orbital, when the NN Fea4+ and Feb4+ have the same spin alignment (FM coupling), the interaction between the half-filled Fea4+ and the half-filled Feb4+ would result in an energy lowering (Fig. 3d). Therefore, the magnetic ground-state of Cs4FePb2Cl12 could be a FM phase.

Electronic and magnetic properties of Cs4FePb2Cl12

We now look into the electronic properties of Cs4FePb2Cl12. As shown in Fig. 4 and Supplementary Fig. 7, interestingly, we find that Cs4FePb2Cl12 is a HMF with 100% spin-polarized conduction electrons near the Fermi level irrespective of the spin-orbit coupling (SOC). We also confirm that Cs4FePb2Cl12 remains half-metallic by using the PBE+U method.38 The ΔHM, which is defined as the minimum of the bottom energy of minority spin conduction bands with respect to the Fermi level and the absolute values of the top energy of minority spin valence bands,39,40 is estimated to be 0.55 eV in Cs4FePb2Cl12. The value of ΔHM in Cs4FePb2Cl12 is among one of the best values reported in other intrinsic HMFs, e.g., quaternary Heusler alloys CoFeCrAl and CoFeCrSi (0.16 and 0.28 eV),41,42 double perovskites Sr2FeMoO6 (~0.5 eV),6 and zincblende phases of the transition metal chalcogenides CrTe, CrSe, and VTe (<0.60 eV).7,40 A wide ΔHM can efficiently prevent the spin-flip transition of carriers due to thermal excitation and thus enhance the stability of its half-metallicity.

Fig. 4
figure 4

Electronic properties of Cs4FePb2Cl12. a Calculated band structure and b orbital-projected DOS for Cs4FePb2Cl12. δ is the half-metallic gap. Fermi level (Ef) is set to zero. c Spin density of Cs4FePb2Cl12. Voids between the octahedrons are filled with Cs atoms, which are omitted for clarity. d Temperature, magnetization, and e ΔHM as functions of time-steps in AIMD simulation at targeted 300 K for Cs4FePb2Cl12. The AIMD simulation is performed by using a supercell containing 8 unit cells. The red dashed line in (e) shows the ΔHM value at 0 K.

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From the spin densities (Fig. 4c), it is seen that only Fe d orbitals contribute to the magnetism of Cs4FePb2Cl12. To further confirm the electronic stability of Cs4FePb2Cl12 at finite temperature, an AIMD simulation on Cs4FePb2Cl12 is performed at 300 K. As shown in Fig. 4d, under thermal fluctuations, the FM ground-state can be well maintained at 300 K after 1 ps. The oscillation of ΔHM values at 300 K is in the range of 0.46–0.61 eV (Fig. 4e), which is large enough for RT applications.

To develop practical spintronic applications, the magnetic properties of the compounds at finite temperatures should be explored. We adopt the following classical Heisenberg Hamiltonian:

$$H = - \mathop {\sum}\limits_{ < i,j > } {J_1} S_i \cdot S_j - \mathop {\sum}\limits_{ < k,l > } {J_2} S_k \cdot S_l - \mathop {\sum}\limits_{ < m,n > } {J_3} S_m \cdot S_n - \mathop {\sum}\limits_{ < i > } D (S_{iz})^2$$

where J1, J2, and J3 are the first, second and third nearest-neighbor magnetic exchange parameters, respectively. Si (j,k,l,m,n) is the spin magnetic moment on the magnetic atom at site i (j, k, l, m, n). D is the single-site magnetic anisotropy parameter, and Siz represent components of S along z (out-of-plane) orientations. The positive and negative J (D) values represent FM coupling (out-of-plane easy axis) and AFM couplings (in-plane easy axis), respectively. The calculated J1, J2, and J3 of Cs4FePb2Cl12 are J1 = 5.1 meV, J2 = 1.3 meV, and J3 = 0.6 meV. Interestingly, the interlayer exchange interaction (J3) is much weaker than the intralayer interaction (J1 and J2), due to the large vdW interlayer distance.

Based on Monte Carlo (MC) simulations,43 the Tc of Cs4FePb2Cl12 is estimated to be 442 K (Fig. 5d), which is well above RT and among the highest Tc of intrinsic HMFs reported in the literatures.8,44,45,46 As a benchmark, the Neel temperature of Cs4MnSb2Cl12 (AFM ground-state) is calculated to be ~1 K, which is consistent with the experimental measurements24 and thus testify the accuracy of our calculations.

Fig. 5
figure 5

Magnetic properties of bulk and monolayer Cs4FePb2Cl12. a Top and side views of Cs4FePb2Cl12 monolayer. b Calculated exfoliation energy (Eef) for Cs4FePb2Cl12 and several layered compounds for comparison. c Orbital-projected DOS for Cs4FePb2Cl12 monolayer. Ef is set to zero. d Magnetic moment per Fe site as a function of temperature for bulk Cs4FePb2Cl12 (orange line) and Cs4FePb2Cl12 monolayer (blue line) from MC simulations. The vertical dashed lines indicate the Tc.

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As for a layered structure, it is interesting to further understand the electronic and magnetic properties of monolayer Cs4FePb2Cl12 (Fig. 5a). As shown in Fig. 5b, the exfoliation energy (Exf) of Cs4FePb2Cl12, i.e., the energy required to separate an individual layer from the bulk compound to infinity, is calculated to be 0.16 J/m2, which is comparable to that for a large number of layered compounds (around 0.32 J/m2).47,48 It indicates that Cs4FePb2Cl12 monolayer might be exfoliated from bulk Cs4FePb2Cl12, once bulk Cs4FePb2Cl12 can be synthesized in the experiments.

As shown in Fig. 5c, Cs4FePb2Cl12 monolayer remains half-metallicity with ΔHM ~ 0.56 eV. Most importantly, the Tc of Cs4FePb2Cl12 monolayer can sustain around RT (370 K), as shown in Fig. 5d, which is a novel advantage over many other magnetic 2D materials. For example, the Tc of Fe3GeTe2 with a FM ground-state is dramatically reduced from 180 K (in the 3D bulk) to 20 K (in a monolayer).49 The reduction in the Tc as the samples are thinned down can be explained by the thermal fluctuations of spin waves, whose energy distribution is intimately connected to the dimensionality.50,51 However, for Cs4FePb2Cl12, the interlayer exchange interactions (J3) is already weak in the bulk due to the large vdW interlayer distance, and the intralayer exchange interactions (J1 and J2) in the monolayer remain strong enough as that in the bulk. Therefore, the Tc only slightly decreases from 442 K in the bulk to 370 K in the monolayer.

Besides of ΔHM and Tc, we also investigate another important property for spintronic materials, which is related to MAE. From the spatial dependence of the MAE for Cs4FePb2Cl12 (Fig. 6a), it is found that Cs4FePb2Cl12 can exhibit an in-plane easy axis and the hardest axis of magnetization (largest MAE value) aligns along the out-of-plane direction. Within the framework of second order perturbation theory, the value of MAE scales with the square of the SOC scaling parameter.52,53 Thus, the compounds containing the heavy elements which have significant SOC effect are expected to have large MAE. The calculated MAE is 380 μeV/Fe for Cs4FePb2Cl12, which is comparable to or even larger than that in Fe (−1.4 μeV per atom), Co (−65 μeV per atom), Ni (2.7 μeV per atom) bulks,54 and monolayer of Fe, Co deposited on substrates (80–370 μeV per metal atom).55,56 The large MAE in bulk Cs4FePb2Cl12 may be due to the significant SOC effect of Pb element and large structural anisotropy. The large bulk MAE makes Cs4FePb2Cl12 promising for magneto-electronic applications, and is expected to stabilize the long-range FM order in its 2D systems, which is consistent with our above discussions.

Fig. 6
figure 6

MAE and strain effects. Spatial (angular) dependence of MAE per unit cell (i.e., per Fe) for (a) bulk Cs4FePb2Cl12 and (b) Cs4FePb2Cl12 monolayer. Point A indicates the easy axis of magnetization. c Exchange energy (ΔE = EAFM − EFM) and d MAE for Cs4FePb2Cl12 monolayer as a function of in-plane (ab plane) biaxial strain (ε). Calculated Tc for Cs4FePb2Cl12 monolayer under −4% compressive strain and under 4% tensile strain from MC simulations are also shown in (c).

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We also studied the strain effect on the magnetic properties of Cs4FePb2Cl12 monolayer. As shown in Fig. 6c, it is found that the FM configuration is more stable than the AFM configuration within a large range of strain (−4% < ε < 4%). From the orbital-projected DOS (Fig. 5c), it can be seen that a pronounced hole state which is mainly formed by Fe 3d orbitals with hybridization with Cl 3p orbitals, located at 1.8 eV in the spin-down channel. This hole state, which is responsible for the FM ordering, is robust under strains. The FM coupling in the super-superexchange mechanism can be efficiently enhanced under compressive strain, and thus Tc of Cs4FePb2Cl12 monolayer can be dramatically increased to 448 K under ε = −4% (Fig. 6c and Supplementary Fig. 8). From Fig. 6b, it is found that Cs4FePb2Cl12 exhibits an in-plane easy axis (direction [010] in Fig. 5a). The hardest axis of magnetization aligns along the out-of-plane direction (direction [001] in Fig. 5a). The calculated MAE is 318 μeV/Fe for Cs4FePb2Cl12 monolayer. The value can be modulated from 215 μeV/Fe under ε = −4% to 350 μeV/Fe under ε = 4% (Fig. 6d). These MAE values are all large enough for the stability of long-range FM ordering in Cs4FePb2Cl12 monolayer.

In conclusion, through materials screening by using first-principles calculations, we have identified that Cs4FePb2Cl12 can exhibit a half-metallic ground-state with high Tc, wide ΔHM and large MAE, which make Cs4FePb2Cl12 as one of the best HMFs. Further experimental efforts are called for to synthesize this compound. Interestingly, the high Tc of layered Cs4FePb2Cl12 can be well maintained in Cs4FePb2Cl12 monolayer. The discovery of Cs4FePb2Cl12 monolayer can enrich the 2D magnets family, and this material is expected to have applications from sensing to data storage.

Methods

First-principles calculations

First-principles calculations were based on density functional theory (DFT) implemented in the Vienna Ab initio Simulation Package (VASP).57 The exchange-correlation functional was treated with the generalized gradient approximation (GGA) in the Perdew–Burke–Ernzerhof (PBE) form.58 To properly describe the strongly correlated electrons in the partially filled d subshells in Cs4M2+B3+2XVII12 (M2+ = Ti2+/V2+/Cr2+/Mn2+/Fe2+/Co2+/Ni2+/Cu2+, B3+ = Sb3+/In3+/Bi3+, XVII = Cl−/Br−/I−), and Cs4M4+B2+2XVII12 (M2+ = Ti2+/V2+/Cr2+/Mn2+/Fe2+/Co2+/Ni2+, B2+ = Pb2+/Ge2+, XVII = Cl−/Br−/I−) family compounds, in this work, we used the screened hybrid Heyd–Scuseria–Ernzerhof (HSE06) hybrid density functional59,60 during the calculations of the band structures and the magnetic exchange parameters (J1, J2, and J3) of the candidate compounds. The plane-wave cutoff energy of 400 eV was employed with the energy and force convergence criteria of 10−5 eV and 0.02 eV·Å−1, respectively. DFT-D3 method was adopted for the van der Waals correction in all our calculations.61 A 6 × 6 × 4 Γ-centered k-mesh was employed to sample the Brillouin zone of the primitive cell of Cs4M2+B3+2XVII12 and Cs4M4+B2+2XVII12 layered halide double perovskites. A sufficiently large vacuum region along the z direction of 14 Å was used in building Cs4FePb2Cl12 monolayer.

Phonon spectra and AIMD simulations

The phonon spectra were calculated by using the PHONOPY code62 with the finite displacement method.63 To verify the thermal stability of the selected materials, AIMD simulations were performed in the constant-volume and constant-temperature (NVT) ensemble at room temperature (300 K) by using the Nosé-Hoover thermostat.64 The initial configurations of Cs4MnBi2Cl12, Cs4CuBi2Cl12, Cs4MnPb2Cl12, Cs4MnPb2Br12, and Cs4FePb2Cl12 with 152 atoms (8 unit cells) were adopted. The time step was set to 1.0 fs.

MAE calculations

To determine MAE, the spins are rotated to different directions, which are represented as function of the angles (θ, ϕ) in the spherical coordinate.65 The SOC effect is taken into account. MAE is defined as the energy difference for the magnetization oriented along the easy-axis and along the hard-axis. The number of bands is set to be twice compared with the collinear calculation.