Introduction

2D materials have developed rapidly since the discovery of graphene [1,2,3,4,5]. Nevertheless, 2D materials hardly have intrinsic ferromagnetism [6] according to the Mermin–Wagner theory [7]. Fortunately, large magnetic anisotropic energy (MAE) may offset the thermal fluctuations and maintain long-range magnetic ordering. In addition, the achievement and maintaining of FM order at room temperature and generation of high spin current are crucial for spintronic devices [8]. Particularly, intrinsic 2D FM semiconductors and half-metals with high \(T_{{\text{C}}}\), large MAE hold promising application prospect in spintronics and the related applications [8,9,10,11,12,13].

Recently, FM order and semiconducting feature were observed in monolayers CrX3 (X = Cl, Br, I) and bilayer Cr2Ge2Te6, but they all exhibited rather low \(T_{{\text{C}}}\) (17 K for CrCl3 [14, 15]; 34 K for CrBr3 [16]; 45 K for CrI3 [17]; and 30 K for Cr2Ge2Te6 [18]). Meanwhile, computational studies have predicted various FM semiconducting monolayers, such as Cr2I3X3 (X = Cl, Br) (TC = 26 and 33 K [19]); MnX2 (X = S, Se) (TC = 225 and 250 K [20]); CrXTe3 (X = Si, Ge) (TC = 35.7 and 57.2 K [21]); CrOX (X = F, Cl, Br) (TC = 150, 160 and 129 K [22, 23]). Due to the incompatibility between ferromagnetism and semiconducting [10], the TC of 2D FM semiconductors is generally below room temperature except for the recent prediction on 1 T’-CrS2 (TC = 1000 K [24]). FM half-metals may possess higher \(T_{{\text{C}}}\) than FM semiconductors due to the carrier-driven strong exchange interaction [25, 26]. FM half-metallic monolayers, such as Co2Se3 (\(T_{{\text{C}}}\) = 600 K [27]) and MnX (X = P, As) (TC = 495 and 711 K [28]) with high TC, have been predicted. It is worth noting that half-metals can provide 100% spin-polarized current and are desirable candidates as pure spin injection [29] and spin transport materials [30].

The intensive investigations on 2D ferromagnets have been focused on transition metal compounds. Nevertheless, lanthanide elements having strong spin–orbit coupling and highly localized 4f electrons [31], offer more potential to generate larger magnetic moments and larger MAE. It is inspiring that 2D FM GdSi2 and EuSi2 were successfully exfoliated from their AFM layered bulks [32]; layered EuGe2 and GdGe2 were synthesized and evolve from typical 3D AFM to 2D FM as their thickness is reduced to few atomic layers [31]. Meanwhile, it has been predicted that monolayer GdI2 has a large magnetic moment (8 \({\upmu }_{{\text{B}}} /{\text{f}}.{\text{u}}.\)) [33]; monolayer Gd2B2 is a FM metal with a large magnetic moment (7.30 \({\upmu }_{{\text{B}}} /{\text{Gd}}\)) and high \(T_{C}\) up to 550 K [34]. These findings inject new vitality into the applications of lanthanide compounds.

Among the lanthanide elements, Gd is the most noteworthy one. It is the only lanthanide metal with room temperature ferromagnetism [33]. The TC of Gd metal and its compound Gd5Si4 is 293 K [35] and 336 K [36], respectively. The layered Gd2C, 2D ferromagnet with TC of 350 K, was experimentally prepared [37]. Additionally, the electronic property of few-layer Gd2C was recently studied via first-principles calculations as electrode [38, 39]. The emergence of Weyl semimetal (WSM) phase was discovered in the layered FM Gd2C. The WSM phase of time-reversal symmetry breaking is driven by the combined effects of crystal field, ferromagnetism and spin-orbital coupling (SOC) [39]. Remarkably, 2D layered materials can be stacked in different orders owing to the weak interlayer interaction [40], leading to various properties [41]. The influence of stacking order is particularly outstanding in 2D magnetic materials [12, 17, 18, 42]. Not only varying magnetic orders and electronic properties [17, 18, 43], even new physical phenomena may be obtained via adjusting the stacking order [12, 42].

Motived by the promising potential of lanthanide compounds as spintronic materials, we explored the electronic structure and magnetic coupling of both monolayer and bilayer Gd2C. Via performing the theoretical investigations based on the first-principles methods, we demonstrate the robust ferromagnetism of monolayer Gd2C. It is worth noting that monolayer Gd2C is an FM half-metal with a large energy gap in the semiconducting spin-channel and a high TC. Under 5% biaxial strain, its TC increases to 392 K. In addition, we also studied the possible stacking configurations and interlayer magnetic coupling in bilayer Gd2C. It turns out that bilayer Gd2C has two stable stacking configurations, AB- and AA-stacking; both of them are FM metals with high TC.

Computational methods

All calculations were performed based on the spin-polarized density function theory (DFT) implemented in the Vienna ab-initio calculation package (VASP) [44]. The Perdew–Burke–Ernzerhof (PBE) functional of generalized gradient approximation (GGA) [45] was chosen to describe the exchange correlation interaction. A vacuum space of 15 Å was used to eliminate the interaction between adjacent layers in periodic model [46]. The plane wave cutoff energy was set as 500 eV. All the structures were fully optimized; the convergence criteria for energy and force were set to \(10^{ - 6}\) eV and 0.01 eV/Å, respectively. The gamma-centered 16 × 16 × 1 and 8 × 8 × 1 k-point grids based on the Monkhorst–Pack method [46] were adopt for unit cell and 2 × 2 × 1 supercell of monolayer and bilayer Gd2C, respectively. The damped van der Waals correction (DFT-D2) method was used to correct the weak interlayer interaction for bilayer Gd2C. Considering the strong correlation interaction of the Gd-4f orbital, we adopted the PBE + U method and set the parameters as U = 9.2 eV and J = 1.2 eV, which has been proved to be reasonable in the previous works [47, 48]. The phonon dispersion calculation was performed on 2 × 2 × 1 supercell using PHONOPY code based on finite-displacement method [49]. For the ab-initio molecular dynamics (AIMD) simulation, the NVT ensemble with the Nosé-Hoover thermal bath method was employed to control the temperature [50].

Results and discussion

Structure and stability of monolayer Gd2C

Monolayer Gd2C consists of triple atomic layers forming the sandwich structure where magnetic Gd atoms form two atomic layers of honeycomb lattice. The atomic structure of monolayer Gd2C that belongs to the \({\text{ D}}_{{3{\text{d}}}}\) point group is depicted in Fig. 1a and e. From Table 1, we can see that the in-plane lattice constant of monolayer Gd2C is a = b = 3.623 Å, agreeing well with the experimental result of the layered bulk Gd2C (3.639 Å) [37, 51]. To discern the magnetic ground state of monolayer Gd2C, we calculated the FM configuration and three typical AFM configurations of the hexagonal lattice by using a 2 × 2 × 1 supercell, AFM-zigzag, AFM-stripe and AFM-Néel [52]. The energy differences between FM and AFM configurations shown in Figure S1 indicate that monolayer Gd2C has a FM ground state. Its magnetic moment is ~ 8 μB per Gd atom, implying large information storage density. The spin-resolved charge density shows that its magnetic moment mainly attributes to the Gd atoms; the contribution of C atoms is negligible.

Figure 1
figure 1

Top and side views for the atomic structures of monolayer and bilayer Gd2C. (a) and (e) monolayer Gd2C, (b) and (f) AB-, (c) and (g) AA-, (d) and (h) AD-stacking bilayer Gd2C. The solid parallelogram indicates the unit cell of monolayer Gd2C. The light purple (dark purple) and light green (dark green) balls represent the Gd and C atoms in the top (bottom) layer. The high-symmetrical directions [100] and [1 \(\overline{1}\) 0] are marked with blue and red translucent arrows, respectively. In the AB-stacking bilayer Gd2C, the bottom Gd atom in the top Gd2C layer is directly above the C atom in the bottom Gd2C layer. The numbers in the square brackets represent the shift of bottom Gd atom in the top layer along the x and y directions relative to the C atom in the bottom Gd2C layer

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Table 1 The lattice constant, the bond length between the nearest neighboring (NN) Gd atoms (\({\mathrm{d}}_{\mathrm{Gd}-\mathrm{Gd}}\)), the interlayer distance and the magnetic ground state of bulk, monolayer and bilayer Gd2C
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To explore the structural stability of monolayer Gd2C, we calculated the cohesive energy as \(E_{{{\text{coh}}}} = (E_{{{\text{Gd}}_{2} {\text{C}}}} - 2E_{{{\text{Gd}}}} - E_{{\text{C}}} )/3\), where \(E_{{{\text{Gd}}_{{2}} {\text{C}}}}\) is the total energy of Gd2C; \(E_{{{\text{Gd}}}}\) and \(E_{{\text{C}}}\) are the energy of isolated Gd and C atoms, respectively. The calculated cohesive energy of monolayer Gd2C is − 5.94 eV/atom, suggesting that the Gd–C bonding is quite strong. In addition, the elastic stiffness tensors are calculated to be C11 = 76.46 N/m (C11 = C22), C12 = 23.87 N/m and C66 = 26.29 N/m, satisfying the Born-Huang criterion (C11 > 0, C11C22–C12 > 0 and C66 > 0) [53] and indicating mechanical stability. The Young’s modulus and Poisson’s ratio were calculated to be 69.01 N/m and 0.31, via \(Y = \left[ {C_{11} C_{22} - C_{12}^{2} } \right]/C_{11}\) and \(v = C_{12} /C_{11}\), respectively. These values are comparable to that of silicene (61 N/m and 0.33) [54] and much lower than that of graphene (342 N/m) [55], suggesting that monolayer Gd2C is soft and can sustain large strain. As shown in Fig. 2a, there are no imaginary phonon modes, indicating that monolayer Gd2C is dynamical stable. Furthermore, the thermal stability of the monolayer Gd2C was assessed by conducting ab-initio molecular dynamics (AIMD) [56] simulations at 300 K; the integrity of original configuration and small energy fluctuations confirm its good thermal stability (Fig. 2b).

Figure 2
figure 2

a The phonon dispersion of monolayer Gd2C. b Evolution of total energy of monolayer Gd2C during ab-initio molecular dynamics (AIMD) simulation at 300 K for 5000 fs. The insert diagrams represent the atomic structures of monolayer Gd2C at the beginning and end of AIMD simulation

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Electronic and magnetic properties of monolayer Gd2C

Since PBE-level calculations generally underestimate the band gaps, as shown in Fig. 3a and b, we calculated the electronic structures of the FM monolayer Gd2C by using the more accurate Heyd–Scuseria–Ernzerhof (HSE06) method [57]. Compared with the results obtained via the PBE + U calculations [38, 39], HSE06 produces similar band structures but with a wider band gap. Notably, the state of the spin-up channel passes through the Fermi level, while the spin-down channel shows semiconducting feature with a band gap of 1.68 eV. Our result indicates that monolayer Gd2C is an intrinsic FM half-metal. It demonstrates 100% spin polarization at Fermi level, which is highly desirable for pure spin current generation and injection.

Figure 3
figure 3

The band structures and density of states in a spin-up and b spin-down channels of monolayer Gd2C calculated by the HSE06 method. The Fermi level (EF) is set to 0 eV. c The first Brillouin zone of monolayer Gd2C. Illustration of d direct-exchange and e super-exchange interactions between Gd atoms

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The orbital-projected density of states around Femi energy is also plotted in Fig. 3a and b. The states around the Fermi level mainly come from Gd-5d orbitals. The conduction band minimum (CBM) and valence band maximum (VBM) in the semiconducting spin-channel are mainly contributed by Gd atoms. Additionally, there is strong hybridization between Gd-5d and C-2p states at the VBM, implying large exchange interaction between Gd-5d and C-2p electrons. The orbital-projected density of states in the wider energy range (Figure S2) shows that the magnetic moment of monolayer Gd2C is mainly contributed by the Gd-4f electrons, agreeing well with the results of layered Gd2C [38]. They are highly localized, showing narrow and high peaks at position far away from the Fermi level.

We considered the intra-atomic and the inter-atomic exchange interactions including the direct-exchange and super-exchange interactions to study the magnetic coupling in monolayer Gd2C. The intra-atomic interaction between Gd-4f and Gd-5d electrons is very strong according to Hund's rule. Consequently, the 5d electrons are spin-polarized near the Fermi level (Fig. 3), and the 4f electrons generate an inherent large magnetic moment through the surrounding polarized 5d electrons [58]. As shown in Fig. 3d and e, the inter-atomic super-exchange interactions between the Gd atoms are mediated through the C atom. According to the Goodenough–Kanamori–Anderson (GKA) rules [59,60,61], the super-exchange interaction prefers FM coupling because the bond angle of Gd–C–Gd (86.8°) is close to 90°. The considerable hybridization between Gd-5d and C-2p orbitals leads to the strong FM super-exchange interaction. In contrast, the AFM direct-exchange interaction is relatively weak due to the large distance (3.42 Å) between the nearest neighboring (NN) Gd atoms.

Based on Heitler–London model [62], the magnetic exchange parameter J can be represented as \(J \approx 2k + 4\beta S\), where \(k\) presents the potential exchange; \(\beta\) and \(S\) are the hopping integral and overlap integral between atomic orbitals. Because the Gd–C–Gd angle is close to 90°, C-2p orbitals are almost orthogonal to Gd-5d orbitals, resulting in S approaching zero. \(J\) can be reduced to \({ }2k\), and \(k\) is a positive value according to the Hund’s rule. Therefore, monolayer Gd2C prefers FM state and the magnetic coupling is dominated by FM super-exchange interaction.

MAE is defined as the energy difference between the magnetization in the in-plane and out-of-plane directions. Firstly, total energies of monolayer Gd2C with magnetization directions along the [100], [010] and [001] were calculated to estimate the MAE. Table S1 shows that the MAE along the [100] and [010] directions is 703.5 μeV and 703 μeV per Gd atom, respectively. Our result indicates that the easy magnetization axis of monolayer Gd2C is along the out-of-plane [001] direction, and the MAE is isotropic along the in-plane directions. Secondly, due to the uniaxial tetragonal symmetry of monolayer Gd2C, the MAE can be described as function of azimuth angle [63]:

$${\text{MAE}}\left( \theta \right) = K_{1} \sin^{2} \theta + K_{2} \sin^{4} \theta$$

where \(K_{1}\) and \(K_{2}\) are the anisotropy constants, and θ is the azimuth angle in the xz and yz planes. If \(K_{1}\) is positive and predominates over \(K_{2}\), the magnetization direction is biased to the z-axis [001]. Otherwise, the magnetization direction is perpendicular to the z-axis. Figure 4a demonstrates the angular dependence of the MAE, which has a good fit with the above equation.

Figure 4
figure 4

a The magnetic anisotropy energy (MAE) of monolayer Gd2C, AB- and AA-stacking bilayer Gd2C as a function of the azimuth angle θ. The out-of-plane [001] magnetization energies were taken as a reference. b Schematic diagram showing MAE of monolayer Gd2C, AB- and AA-stacking bilayer Gd2C as function of polar angle and azimuthal angle. The φ and θ are the polar angle in the xy plane, and the azimuthal angle in the xz and yz planes. The color saturation and radical length represent the energy required to rotate the magnetic axis into a certain direction

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It is sensitive to azimuthal angles, but isotropic in the xy plane. Both \(K_{1}\) and \(K_{2}\) are positive, as listed in table S1. This is similar to monolayers Fe3GeTe2 [43] and CrI3 [17]. Thirdly, the MAE in the whole space shows strong dependence on the azimuthal angles and isotropic in xy plane. The maximum value of MAE is 703 μeV/Gd atom, which is larger compared with the most known 2D magnetic materials [64, 65]. Such a large perpendicular magnetic anisotropy (PMA) is enough to resist thermal fluctuations and beneficial for higher-density nonvolatile magnetic storage [66, 67].

In order to estimate the \(T_{{\text{C}}}\) of monolayer Gd2C, we performed Monte Carlo (MC) [68] simulation based on the Heisenberg model [69]. The spin Hamiltonian is expressed as:

$$H = - \mathop \sum \limits_{ij} J_{1} S_{i} S_{j} - \mathop \sum \limits_{ik} J_{2} S_{i} S_{k} - \mathop \sum \limits_{ih} J_{3} S_{i} S_{h} - AS_{i}^{Z} S_{i}^{Z}$$

where \(J_{1}\), \(J_{2}\) and \(J_{3}\) are the magnetic exchange parameters between NN, second NN (2NN) and third (NN) (3NN) Gd atoms (Fig. 5a), respectively. \(S_{i}\) is the spin vector of magnetic atom, \(S_{i}^{Z}\) is the \(Z\) component of the spin vector, and \(A\) represents the anisotropy parameter, which was calculated via the MAE. A supercell of 32 × 32 × 1 was used. We considered FM and three AFM configurations, and their total energies are descried as

$$E\left( {{\text{FM}}} \right) = E_{0} - \left( {12J_{1} + 24J_{2} + 12J_{3} } \right)\left| S \right|^{2}$$
$$E\left( {\text{AFM-zizag}} \right) = E_{0} - \left( {4J_{1} - 8J_{2} - 12J_{3} } \right)\left| S \right|^{2}$$
$$E\left( {\text{AFM-stripe}} \right) = E_{0} - \left( { - 4J_{1} - 8J_{2} + 12J_{3} } \right)\left| S \right|^{2}$$
$$E\left( {\text{AFM-N}\acute{\text{e}}\text{el}} \right) = E_{0} - \left( { - 12J_{1} + 24J_{2} - 12J_{3} } \right)\left| S \right|^{2}$$
Figure 5
figure 5

The variation of magnetic moment of Gd atoms and the specific heat as a function of temperature for a monolayer Gd2C, b AB- and c AA-stacking bilayer Gd2C based on the Heisenberg model. The inset diagram shows the magnetic exchange interaction between the nearest neighboring (NN) (J1), second NN (2NN) (J2) and third NN (3NN) (J3) Gd atoms

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Through the above four equations, the magnetic exchange parameters \(J_{1}\), \(J_{2}\) and \(J_{3}\) are calculated to be 1.07, 0.73 and 0.53 meV, respectively. The positive values indicate the FM coupling between the NN, 2NN and 3NN Gd atoms.

Figure 5a shows the evolutions of magnetic moment of Gd atom and heat capacity (\(C_{V}\)) as temperature. The heat capacity is defined as \(C_{V} = \frac{{\left( {\left\langle {E^{2} } \right\rangle - \left\langle E \right\rangle^{2} } \right)}}{{k_{B} T^{2} }}\). The \(T_{{\text{C}}}\) was estimated from the position where the magnetic moment drops sharply and the peak positions of \(C_{V}\). Our calculation shows that the \(T_{{\text{C}}}\) of the monolayer Gd2C is 322 K, higher than that of Gd metal (293 K) [35], slightly lower than those of compound Gd5Si4 (336 K) [36] and the layered Gd2C (350 K) [37]. The above room temperature of \(T_{{\text{C}}}\) and half-metallic behavior of monolayer Gd2C make it a promising candidate material for spintronic devices.

The magnetic property of monolayer Gd2C under biaxial strain

2D materials are usually synthesized on suitable substrates. The lattice mismatch between 2D material and substrate usually causes strain on 2D material and changes its properties [70]. Here we studied monolayer Gd2C under − 5% ~ 5% biaxial strain. Figure S3 shows that under strain of − 5% ~ 5%, the CBM of spin-down channel is slightly across the Fermi level around Γ point. Our calculation shows that monolayer Gd2C under 5% strain still can generate 100% spin-polarized current by applying a small electric field to low down the Fermi level, as shown in Figure S4.

Figure 6a shows the energy difference (\(\Delta E = E_{{{\text{AFM}}}} - E_{{{\text{FM}}}}\)) between the FM and AFM configurations of monolayer Gd2C under biaxial strain. The ΔE is positive and increases monotonically under the biaxial strain from − 5% to 5%. This indicates that the FM coupling between Gd atoms is weakened and strengthened under the increased compressive and tensile strain, respectively. Figure 6b shows the distance (d1) between the NN Gd atoms is almost constant under − 5% ~ 5% biaxial strain. The distances between 2NN (d2) and 3NN (d3) Gd atoms decrease monotonically with the increasing compressive strain, making the direct-exchange interactions stronger. The bond angles θ of Gd–C–Gd increase monotonically approaching 90°, leading to enhanced super-exchange interactions. However, the direct- and super-exchange interactions weaken, while d2 and d3 increase and the bond angle θ of Gd–C–Gd decreases away from 90° as the tensile strain is increased. The delicately competing nature of direct- and super-exchange interactions leads to the robust FM coupling of monolayer Gd2C under − 5% ~ 5% biaxial strain. We also calculated the MAE. Its value decreases as the biaxial strain is changing from − 5% to 4%. Under 5% tensile strain, the value of MAE changes to be negative, which indicates that the easy magnetization axis converts from the out-of-plane direction to the in-plane direction. Our results suggest that strain can be used to tune the magnetization orientation of monolayer Gd2C.

Figure 6
figure 6

a The energy difference (ΔE) between ferromagnetic (FM) and three antiferromagnetic (AFM) configurations, b the distance between the nearest neighboring (NN) (d1), second NN (2NN) (d2) and third NN (3NN) (d3) Gd atoms, and Gd–C–Gd bond angle θ, c the magnetic anisotropy energy (MAE) and d the exchange parameters of monolayer Gd2C under biaxial strain of − 5% ~ 5%. MAE greater/less than zero represents the out-of-plane/in-plane magnetization direction

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The calculated \(J_{1}\), \(J_{2}\) and \(J_{3}\) of monolayer Gd2C under − 5% ~ 5% biaxial strain is shown in Fig. 6d. It indicates that \(J_{1}\), \(J_{2}\) and \(J_{3}\) increase monotonically under the biaxial strain from − 5% to 5%. Correspondingly, Figure S5 shows that the \(T_{{\text{C}}}\) of the monolayer Gd2C drops to 182 K at the compressive biaxial strain of − 5%, but rises to 392 K at the tensile biaxial strain of 5%. This is consistent with the monotonic enhancement of \(T_{{\text{C}}}\) for most systems under tension [20, 21, 71, 72].

Configurations of bilayer Gd2C

There are three high-symmetrical sites on the surface of monolayer Gd2C: above the C atoms (\(M_{{\text{C}}}\)), above bottom Gd atoms (\(B_{{{\text{Gd}}}}\)), above the top Gd atoms (\(T_{{{\text{Gd}}}}\)). Correspondingly, three configurations of bilayer Gd2C with the bottom Gd atom in the top Gd2C layer locating on the three sites are constructed. As shown in Fig. 1, the three configurations of bilayer Gd2C are named as AB-, AA- and AD-stacking. Notably, AB-stacking is indeed the stacking configuration of the layered bulk Gd2C. Our calculations show that the interlayer distances of the AB-, AA- and AD-stackings bilayer Gd2C are 3.04, 3.01 and 3.51 Å, respectively. AB-stacking is energetically most favorable; the energies of AA- and AD-stackings are 1.6 and 147.9 meV/Gd atom higher than that of AB-stacking, respectively. The results show that the interlayer distance and the interlayer stacking energy of bilayer Gd2C are sensitive to the stacking configuration.

Various stacking configurations may be obtained via physical, structural and chemical approaches [73], which provides a possible way to adjust the electronic and magnetic properties of bilayer Gd2C. In this context, by rigidly and laterally sliding the top Gd2C layer along the high symmetry directions [100] and [1 \(\overline{1}\) 0], we explored the other possible stacking configurations of bilayer Gd2C. The interlayer stacking energy and the interlayer exchange energy were calculated for each stacking configuration, and the results are shown in Fig. 7a and b. AB-stacking has the lowest energy; the energy of the AA-stacking is slightly higher than that of the AB-stacking; AD-stacking has the highest energy. Except for the AA- and AB-stacking, no other energetically stable stacking configuration was identified.

Figure 7
figure 7

The evolution of the interlayer stacking energy and exchange energy of bilayer Gd2C with respect to the rigidly and laterally sliding of one Gd2C layer along the high-symmetrical [100] (solid blue line) and [1 \(\overline{1}\) 0] (solid red line) directions (a) and (b), and the xy plane (c) and (d). In the AB-stacking bilayer Gd2C, the bottom Gd atom in the top Gd2C layer is directly above the C atom in the bottom Gd2C layer. In the AC-, AA-, AD- and AE-stacking bilayer Gd2C, the bottom Gd atom in the top Gd2C layer is displaced by [1/6, − 1/6], [1/3, 2/3], [2/3, 1/3] and [1/2, 0] relative to the C atom in the bottom Gd2C layer

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As shown in Fig. 7b, all stacking configurations generated via top Gd2C layer sliding along the two symmetrical routes [100] and [1 \(\overline{1}\) 0] maintain the FM interlayer coupling. More stackings via sliding the top Gd2C layer along the entire xy plane were studied. As shown in Fig. 7c and d, AB-stacking still has the lowest energy; all the investigated stacking configurations energetically prefer the FM interlayer coupling. In addition, the stacking-constraint relaxations in the z-direction were performed as stacking affects the interlayer distance. As shown in Figure S6 (a) and (b), both the interlayer stacking energy and the interlayer exchange energy change with maintaining the overall trends. The AB-stacking bilayer Gd2C is still the global minimum, and the energy of AA-stacking is 1.6 meV/Gd atom higher than that of AB-stacking. The stacking order does not change the ferromagnetism of bilayer Gd2C.

Adopting the CI-NEB method [74], we further calculated the minimum energy path for the stacking transition reaction of bilayer Gd2C from AB- to AA-stacking. As shown in Figure S7, there is an energy barrier of about 200 meV/unit cell, which will prevent the transition from AB- to AA-stacking at room temperature [75]. Our calculations predict two stable stacking configurations for bilayer Gd2C which may co-exist at room temperature.

Electronic and magnetic properties of AB- and AA-stacking bilayer Gd2C

Using the HSE06 function the spin-resolved band structures of AB- and AA-stacking bilayer Gd2C were calculated and are shown in Fig. 8a and b. Unlike the half-metallic monolayer Gd2C, both AB- and AA-stacking bilayer Gd2C exhibit metallic behavior. As shown in the schematic diagram, both the spin-up and the spin-down channels pass through the Fermi level. As shown in Figure S8 (a) and (b) for the orbital-projected density of states, Gd-4f electrons of the AB- and AA-stacking bilayer Gd2C are highly localized and spin-polarized. They are the major contributor to the magnetic moment but do not participate in the intralayer or interlayer exchange interaction.

Figure 8
figure 8

The spin-resolved band structures calculated by the HSE06 method and the schematic diagram showing the spin-resolved density of states for a AB- and b AA-stacking bilayer Gd2C. The Fermi level (EF) is set to 0 eV

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As listed in Table S1, the energies of the AB- and AA-stacking bilayer Gd2C with magnetization direction along the [001] are lower than that along the [100] and [010] directions, and the energies along the [100] and [010] directions are almost same. Our results indicate that the easy magnetization axis of AB- and AA-stacking bilayer Gd2C is along the out-of-plane [001] direction. The angle dependence of MAE is shown in Fig. 4a and b. Consistent with monolayer Gd2C, the MAE of the two stacking configurations of bilayer Gd2C is all sensitive to the azimuthal angle on xz and yz planes but is independent of the polar angle on xy plane. The maximum MAE is 427 and 338 μeV/Gd atom for AB- and AA-stacking bilayer Gd2C, respectively.

The magnetic coupling in AB- and AA-stacking bilayer Gd2C was investigated by using the equation \(H = - \mathop \sum \limits_{ij} J_{{{\text{intra}}}} S_{i} S_{j} - \mathop \sum \limits_{ij} J_{{{\text{inter}}}} S_{i} S_{k}\). \(J_{{{\text{intra}}}}\) represents the intralayer exchange interaction, including the exchange interactions between intralayer NN (\(J_{11}\)), 2NN (\(J_{22}\)) and 3NN (\(J_{33}\)) Gd atoms. \(J_{{{\text{inter}}}}\) represents the interlayer exchange interaction of NN Gd atoms. The possible magnetic configuration of bilayer Gd2C is shown in Figure S9. The energy of AB-stacking bilayer Gd2C can be expressed as:

$$E_{{{\text{FM}}}}^{{{\text{FM}}}} = E_{0} - \left( {24J_{11} + 48J_{22} + 24J_{33} } \right)\left| S \right|^{2} - 12J_{{{\text{inter}}}} \left| S \right|^{2}$$
$$E_{{{\text{FM}}}}^{{{\text{AFM}}}} = E_{0} - \left( {24J_{11} + 48J_{22} + 24J_{33} } \right)\left| S \right|^{2} + 12J_{{{\text{inter}}}} \left| S \right|^{2}$$
$$E_{{\text{AFM - zigzag}}}^{{{\text{FM}}}} = E_{0} - \left( {8J_{11} - 16J_{22} - 24J_{33} } \right)\left| S \right|^{2} + 4J_{{{\text{inter}}}} \left| S \right|^{2}$$
$$E_{{\text{AFM - stripe}}}^{{{\text{AFM}}}} = E_{0} - \left( { - 8J_{11} - 16J_{22} + 24J_{33} } \right)\left| S \right|^{2} - 4J_{{{\text{inter}}}} \left| S \right|^{2}$$
$$E_{{\rm{AFM-Neel}}}^{{\rm{AFM}}} = {E_0} - \left( { - 24{J_{11}} + 48{J_{22}} - 24{J_{33}}} \right){\left| S \right|^2} + 12{J_{{\rm{inter}}}}{\left| S \right|^2}$$

The superscript and subscript indicate the interlayer and intralayer magnetic coupling, respectively. The calculated exchange parameters \(J_{11}\), \(J_{22}\), \(J_{33}\) and \(J_{{{\text{inter}}}}\) for the AB-/AA-stacking bilayer Gd2C are 0.64/0.45, 0.59/0.44, 0.22/0.17 and 1.22/0.82 meV, respectively. We find that both the intralayer and interlayer magnetic coupling for AB- and AA-stacking bilayer Gd2C are FM. The interlayer exchange interaction of AB- and AA-stackings bilayer Gd2C is at the same level as those of Cr2Ge2Te6 [18] and RuCl3 [76] and is dominant over the intralayer interaction. Compared with monolayer Gd2C, the intralayer exchange interactions in AB- and AA-stacking bilayer Gd2C are significantly lower.

Differential charge density (DCD) of AB- and AA-stacking bilayer Gd2C is shown in Figure S10. Comparing with CrS2 [77], the interlayer electronic hybridization of bilayer Gd2C is quite strong. Specifically, there is significant charge reduction near the bottom Gd atom of the top Gd2C layer and the top Gd atom of the bottom Gd2C layer. Charge accumulation occurs in the middle region between top and bottom Gd2C layers. The bottom Gd atom in the top Gd2C layer and the top Gd atom in the bottom Gd2C layer share considerable electrons, weakening intralayer Gd–C bonding and reducing intralayer exchange interaction.

Based on the calculated magnetic coupling parameters and the Heisenberg model, we calculated the \(T_{{\text{C}}}\) of AB- and AA-stacking bilayer Gd2C. As shown in Fig. 5b and c, the \(T_{{\text{C}}}\) of AB-stacking bilayer Gd2C is 332 K, higher than room temperature and slightly higher than that of monolayer Gd2C. The \(T_{{\text{C}}}\) of the AA-stacking bilayer Gd2C is 235 K, lower than that of the monolayer Gd2C. Consistently, the intralayer and interlayer exchange constants of AB-stacking are greater than those of AA-stacking. The \(T_{{\text{C}}}\) of the AA-stacking bilayer Gd2C is 235 K, lower than that of the monolayer Gd2C.

Conclusions

In summary, our results demonstrate that monolayer Gd2C is an FM half-metal, having good stability and large magnetic moment (~ 8 \({\upmu }_{B} /{\text{Gd}}\)) along the out-of-plane direction. Plus, monolayer Gd2C has a large MAE and its \(T_{{\text{C}}}\) is predicted to be above room temperature (322 K). Under small biaxial strain (− 5% ~ 5%), monolayer Gd2C maintains to be FM and could generate 100% spin-polarized current via applying a small electric field. Its \(T_{{\text{C}}}\) rises to 392 K, and its magnetization direction changes from the out-of-plane direction to the in-plane direction under 5% tensile biaxial strain. We found that bilayer Gd2C prefers FM interlayer coupling, and identified two stable stacking configurations. The two staking configurations may co-exist at room temperature, and their \(T_{{\text{C}}}\) is predicted to be 332 K and 235 K based on the Heisenberg model, respectively. We propose a new stable FM monolayer and bilayer lanthanide compound Gd2C with large magnetic moment and high \(T_{{\text{C}}}\). The robust FM half-metallicity of monolayer Gd2C is highly desired to the applications in the field of spintronics.