Intrinsic ferromagnetism with high temperature, strong anisotropy and controllable magnetization in the CrX (X = P, As) monolayer

An-Ning Ma , Pei-Ji Wang and Chang-Wen Zhang *
School of Physics and Technology, University of Jinan, Jinan, Shandong 250022, People's Republic of China. E-mail:

Received 5th December 2019 , Accepted 6th February 2020

First published on 6th February 2020


2D ferromagnetic (FM) materials with high temperature, large magnetocrystalline anisotropic energy (MAE), and controllable magnetization are highly desirable for novel nanoscale spintronic applications. Herein by using DFT and Monte Carlo simulations, we demonstrate the possibility of realizing intrinsic ferromagnetism in 2D monolayer CrX (X = P, As), which are stable and can be exfoliated from their bulk phase with a van der Waals layered structure. Following the Goodenough–Kanamori–Anderson (GKA) rule, the long-range ferromagnetism of CrX is caused via a 90° superexchange interaction along Cr–P(As)–Cr bonds. The Curie temperature of CrP is predicted to be 232 K based on a Heisenberg Hamiltonian model, while the Berezinskii–Kosterlitz–Thouless transition temperature of CrAs is as high as 855 K. In contrast to other 2D magnetic materials, the CrP monolayer exhibits a significant uniaxial MAE of 217 μeV per Cr atom originating from spin–orbit coupling. Analysis of MAE reveals that CrP favors easy out-of-plane magnetization, while CrAs prefers easy in-plane magnetization. Remarkably, hole and electron doping can switch the magnetization axis in between the in-plane and out-of-plane direction, allowing for the effective control of spin injection/detection in 2D structures. Our results offer an ideal platform for realizing 2D magnetoelectric devices such as spin-FETs in spintronics.

I. Introduction

Spin-based electronic devices that use the spin of electrons are of great significance in both fundamental physics and information storage.1–3 To build spintronic devices, the selection of ferromagnetic (FM) materials and control of magnetism are crucial yet challenging.4–6 In this respect, intrinsic two-dimensional (2D) FM semiconductors with a high Curie temperature (Tc), large magnetocrystalline anisotropic energy (MAE), and high carrier mobility have great potential for spintronics and magneto-optoelectronics.7–15 However, the coexistence of long-range ferromagnetism and semiconductor characteristics in single 2D films is nontrivial, leading to a significant obstacle in the design of future magnetic storage devices.16–28

According to the GKA rule, 2D ferromagnetism at finite temperature is generally prohibited in systems with continuous spin symmetries due to thermal fluctuations from gapless spin waves.29–32 To lift the restriction of spin symmetry, finite magnetic anisotropy such as exchange anisotropy and single-ion anisotropy is necessary, as observed experimentally in van der Waals crystals CrI3[thin space (1/6-em)]9 and Cr2Ge2Te6.12 However, their Curie temperature decreases significantly with a decrease in the number of atomic layers; the corresponding Tc in single-layer and few layers is only 45 K for CrI3and 25 K for Cr2Ge2Te6. Therefore, it is crucial to design new 2D films with high Tc and controllable magnetism for nanoscale spintronic devices.

In the present work, we demonstrate the possibility of realizing intrinsic 2D ferromagnetism in the semiconductor CrX (X = P and As), which possess large spin polarization, large MAE, and a high Tc. Following the GKA rules, the long-range FM ordering of 2D CrX can be explained via a 90° superexchange interaction along Cr–P(As)–Cr bonds. In particular, the predicted Curie temperature Tc of CrP reached up to 232 K, while the Berezinskii–Kosterlitz–Thouless transition temperature of CrAs is as high as 855 K. Further MAE calculations reveal that CrP favors easy out-of-plane magnetization, while CrAs favors easy in-plane magnetization. Remarkably, hole and electron doping can switch the magnetization easy axis of CrP in between the in-plane and out-of-plane direction, providing a means to control the efficiency of spin injection/detection in 2D magnetic semiconductors. The observed high Tc and electrically controllable magnetism pave a way for realizing spin field-effect-transistors (spin-FETs) from 2D materials.33

II. Computational details and methods

We performed first-principles calculations on monolayer and bulk CrX (X = P, As) by the projector augmented wave (PAW) method,34,35 as implemented in the Vienna ab initio simulation package (VASP)36,37 within the local density approximation (LDA).38 For bulk CrX, we also performed several comparison calculations utilizing the Perdew–Burke–Ernzerhof (PBE)39 and vdW-DF-optB8840,41 exchange–correlation functionals. To approximately describe the strongly correlated interactions of the transition metal Cr, structural optimizations were performed by using the spin-dependent GGA plus Hubbard U (GGA+U), where the Hubbard U parameter of the Cr atom was set to 3.0 eV. The screened hybrid Heyd–Scuseria–Ernzerhof (HSE06) functional42 without Hubbard U correction was used to compare the results with a higher level of theory. A plane wave basis set with a cutoff energy of 500 eV is used. The first Brillouin-zone integration is carried out by using an 18 × 18 × 1 Γ-centered Monkhorst–Pack grid for both systems. For all calculations, a vacuum spacing of 20 Å sufficiently reduces the interlayer interactions due to the periodic boundary conditions. The atomic positions are fully optimized until the Hellman–Feynman forces on each atom are smaller than 0.01 eV Å−1. To calculate the MAE, we include the spin–orbit coupling (SOC) in the computation with a full k-point grid, i.e., a total of 324 k points. Charge doping was simulated by adding electrons/holes into the system, together with a compensating uniform positive/negative background to maintain electrical neutrality. The phonon calculations are carried out by using DFT perturbation theory as implemented in the PHONOPY code.43

III. Results and discussion

Monolayers CrX (X = P, As) are isostructural to the FeSe sheet exfoliated from the layered FeSe bulk.44,45 The top and side views of CrX are given in Fig. 1(b), belonging to the P-nmm space group with 2D networks of a rectangular sublattice in the xy plane. Structural optimization reveals that the lattice constant of CrP is 4.21 Å with a Cr–Cr distance of 2.98 Å. The structural stability can be checked by calculating the formation energy as Ecoh = ECrXECrEX, where ECrX is the total energy of CrX, and ECr and EX are the energies of isolated Cr and X atoms, respectively. The obtained negative values, −5.21 eV and −4.60 eV per atom, are comparable to those of graphene (−7.85 eV per atom). This also indicates that the bonding is quite strong. To test the kinetic stability of CrP, we perform phonon spectrum calculations. As shown in Fig. 1(e), no appreciable imaginary phonon modes in the whole Brillouin zone are observed, indicating that the CrP lattice is kinetically stable. Additionally, the thermal stability of the CrP lattice is assessed by performing ab initio molecular dynamics (MD) simulations. Snapshots of CrP at 0 fs and 10[thin space (1/6-em)]000 fs are plotted in Fig. 1(d). It is obvious that 2D planar networks are well maintained and no phase transition is observed within 10 ps, suggesting that CrX is thermally stable at 300 K. Similar results for the CrAs monolayer are also presented in Fig. S1 and S2 in the ESI. This is further confirmed by the time-dependent evolution of total energies, which shows a very small fluctuation.
image file: c9nr10322h-f1.tif
Fig. 1 (a) Crystal structure of layered ternary compounds ACr2X2 (A = Ba, X = P and As). (b) Top and side views of CrX (X = P and As). (c) Cleavage energy of CrP and CrAs. (d) Evolution of total energy and snapshots of CrP from AIMD simulations at 0 and 10 ps. (e) Phonon dispersion of monolayer CrP.

Considering that both CrP and CrAs monolayers exhibit an excellent stability, practical synthetic methods, such as mechanical cleavage, liquid exfoliation, and selective chemical etching, are expected to be attractive to experiments.46,47 In fact, the ternary layered compounds ACr2X2 (A = Ca, Sr, Ba; X = P, As) have been synthesized,48,49 in which the CrX layer and the A atomic layer are alternatively stacked in the c-axis. Because the combination of the A and CrX layers is relatively weak, monolayers CrP and CrAs can be obtained by selective chemical etching of the A atomic layer. Here, we predict the exfoliation energies of CrX by modeling the exfoliation process from ACr2X2, as shown in Fig. 1(a) and (c). The exfoliation energies of CrP and CrAs are 0.085 and 0.094 eV Å−2, respectively, which are lower than those of MXenes (0.086–0.205 eV Å−2).50–54 Therefore, it is expected that monolayer CrX can be obtained by exfoliation from bulk ACr2X2 and survive at room temperature.

After having established that monolayers CrP and CrAs are structurally stable, we focus on their electronic and magnetic properties. Fig. 2(a) shows the spin-dependent band structures with GGA + U and the corresponding density of states (DOS) of monolayer CrP. One can see that it shows a FM semiconductor with a spin-up band gap of 0.12 eV (0.26 eV for HSE) and a spin-down band gap of 2.29 eV (3.47 eV for HSE). Interestingly, both its conduction band minimum (CBM) and valence band maximum (VBM) come from the majority spins, which are predominantly contributed by the Cr-3d orbitals. This is also confirmed by the spin-up charge density in Fig. 2(b). Additionally, we find a stronger dispersion along the Γ–S direction, revealing a large anisotropy along the x and y directions. This is supported by a small effective electron mass of only 0.16m0 and an associated large electron mobility of 11457cm2 V−1 s−1, calculated by using a phonon-limited scattering approach.55 Regarding the spin-down channels, the highest occupied bands come mainly from the Cr-3d and P-2p orbitals, in line with the distributions of the spin-down charge density in Fig. 2(c). Additionally, most of the spin-polarized electrons locate around the Cr ions, leading to a large magnetic moment of 3.0μB per Cr atom. Furthermore, to check the effect of the adsorption atom on electronic properties, we performed calculations on the OH-functionalized CrP monolayer, as shown in Fig. S3(a) in the ESI. We found that the lattice parameter a is 4.223 Å, larger by 0.1 Å than that of pristine CrP. Spin-polarized calculations reveal that it still possesses magnetic properties with a local magnetic moment of 7.22μB. Another prominent property is that it shows a metallic feature, instead of the half-metallic feature of prinstine CrP, as shown in Fig. S3(b). The strong magnetization of Cr ions could be well understood by the localized spin-wave functions (see Fig. 2(e)). As the Fermi level lies almost in the middle of the spin gap, 100% spin-filter efficiency can be maintained in a wide positive or negative bias range, which makes CrX (X = P, As) an attractive candidate for spin-injection.

image file: c9nr10322h-f2.tif
Fig. 2 (a) Electronic band structures of CrP at the HSE level with spin-up and spin-down states, as well as spin-resolved projected DOS. The Fermi level is denoted by a dashed line at 0 eV. (b and c) The spin-dependent charge densities. (d) Spin configuration for evaluating the exchange-interaction constants. J1 and J2 are NN and NNN interaction parameters, respectively. (e) The spin wave functions of CrP.

To verify the magnetic ground state, we evaluate the relative stability of FM and antiferromagnetic (AFM) states for both systems using a 4 × 4 supercell, and find that the FM coupling is energetically more stable than AFM coupling. The spin density in Fig. 2(e) reveals that the ferromagnetism mainly comes from the Cr ion, consistent with the high spin state of Cr3+. In contrast, P ions carry a small opposite spin moment, i.e., they are hardly magnetized. The origin of FM ordering can be understood by the competition of direct exchange (Cr1–Cr2) and superexchange (Cr1–P–Cr3) interactions mediated through P ions, as shown in Fig. 2(d). The nearest neighboring (NN) exchange interaction, J1, and the next nearest-neighboring (NNN) exchange interaction, J2, are both positive, indicating that CrP prefers the FM state. The NN interaction J1 comes from the direct Cr1–Cr3 exchange interactions and the NNN interaction J2 is introduced from the superexchange interaction between Cr1 and Cr2 ions linked by the neighboring P ions. Mermin and Wagner56 pointed out that the AFM state is energetically more stable than the FM state due to the conventional 180° superexchange interaction. However, this doesn't hold true for all cases, especially for 2D magnetic structures. According to the GKA rule,19–22 FM coupling is favored for the 90° superexchange interaction between two magnetic ions, while AFM coupling is preferred for the 180° superexchange interaction. The bond-angle (92°) of Cr1–P–Cr2, as shown in Fig. 2(d), is close to 90°, which means that the Cr-d orbitals are nearly orthogonal to the p orbital of P(As) ions, leading to a negligible overlap integral S. According to the suggestions from Launay et al.,57 the exchange integral J2 in 2D systems should be expressed as J2 ≈ 2k + 4βS, where k is the potential exchange, and β and S are the hopping and overlap integrals, respectively. Because the overlap integral S is close to zero, we can infer that J2 = 2k, where k is positive due to Hund's rule. As a result, CrP adopts a FM ground state.

Magnetic anisotropy is crucial for establishing long-range 2D ferromagnetism, in which single-ion anisotropy and exchange anisotropy are two important factors. Single-ion anisotropy, known as MAE, which determines the easy/hard magnetization axis, can be evaluated by total energies as a function of magnetization direction with SOC. Table 1 shows the angular dependent MAE in CrP, which possesses a high magnetic anisotropy with an easy axis along c, distinct from CrAs with an out-of-plane easy axis. The observed large MAE indicates that CrX has the potential for application in magnetic storage devices.

Table 1 Magnetic anisotropy energies (μeV) per Cr atom of different directions against the (001) direction, magnetic moment (μB) per Cr atom, anisotropy constants K (μeV), and Curie temperatures Tc (K) for monolayers CrP and CrAs
System E(100)–E(001) E(010)–E(001) E(111)–E(001) T c K1 K 2 M
CrP 217 199 139 232 208.45 0.93 3
CrAs −389 −404 −40 855 379.92 −4.23 3

Based on the uniaxial tetragonal symmetry of 2D systems, the angular dependence of MAE58 can be described by:

MAE(θ) = K1[thin space (1/6-em)]sin2[thin space (1/6-em)]θ + K2[thin space (1/6-em)]sin4[thin space (1/6-em)]θ(1)
where K1 and K2 are system-dependent anisotropy constants and θ is the azimuthal angle of rotation. Positive values of K1 and K2 indicate strong Ising ferromagnetism with an out-of plane easy axis, whereas K1 < 0 means that the magnetic direction is perpendicular to the z-axis. As listed in Table 1, we find that both K1 and K2 are positive for CrP, and MAE reaches a maximum value of 217 μeV per Cr pair at θxz = θyz = π/2, belonging to the family of 2D Ising magnets. It is larger than those of Co monolayer/Pt(111) (100 meV)47 and monolayer FeCl2 (60 meV).62 The evolution of MAE as the spin axis rotates through the whole space is illustrated in Fig. 3(a). The MAE exhibits a strong dependence on the azimuthal angle θ and a much weaker dependence on the polar angle ϕ, which confirms again the strong magnetic anisotropy. Thus, we can infer that the large MAE will be sufficient to stabilize ferromagnetism against heat fluctuations at certain temperature.

image file: c9nr10322h-f3.tif
Fig. 3 Angular dependence of the MAE for CrP and CrAs with the direction of magnetization lying on three different planes (a and b) and the whole space (c and d). The inset illustrates that the spin vector S on the xy, yz, and xz plane is rotated at an angle θ around the x, y, and z axes, respectively.

In order to understand the temperature effect on magnetism before implementing a CrP monolayer into practical spintronic devices, we perform Monte Carlo simulations on the basis of the 2D Heisenberg Hamiltonian model to examine the transition temperature from FM to PM states. Here, the spin Hamiltonian is expressed as:

image file: c9nr10322h-t1.tif(2)
where J1 and J2 are the nearest and next-nearest magnetic exchange interaction parameters, respectively, Si is the spin vector of each atom, A is the anisotropy energy parameter, and Siz is the Z component of the spin vector. A supercell of 100 × 100 × 1 with a periodic boundary condition is used here. Fig. 4(a) shows the temperature-dependent magnetic moment per unit cell. The magnetic moment begins to drop dramatically at 232 K, implying the formation of the PM state. To better understand the FM–PM transition, we further calculate the heat capacity (Cv) using the equation
image file: c9nr10322h-t2.tif(3)
where E is the corresponding energy of each magnetic moment. As can be seen from the inset of Fig. 4(a), the FM–PM phase transition occurs at 232 K, which is much higher than the recently observed 2D CrI3[thin space (1/6-em)]9 (45 K) and Cr2Ge2Te6 (30 K).12 To further confirm the robustness of the magnetic stability, we calculate the variation of the total energies of the FM and AFM configurations with the biaxial compressive strain from 0 to −6%. The results show that the ground state of CrP remains FM. The exchange energy increases with the increase of strain (when ε = −2%, J1 = 82.8 meV and J2 = 74.8 meV; when ε = −4%, J1 = 85.4 meV and J2 = 76.6 meV; when ε = −6%, J1 = 87.6 meV and J2 = 78.3 meV), and the Curie temperature is estimated to be 285 K under a moderate strain (−6%). The dependency of the Curie temperature on strain is similar to 2D FeCl2[thin space (1/6-em)]59 and NbSe2.60 Therefore, the predicted high Tc indicates that CrP may be a promising spintronic material at room temperature.

image file: c9nr10322h-f4.tif
Fig. 4 (a) Specific heat Cv and the corresponding magnetism with respect to temperature. (b) Curie temperature of CrP in comparison with CrI3,9 Cr2Ge2Te6,12 GaMnAs,64 CrOCl,65 and CrWI6.66 The transition temperatures Tc are denoted by dashed lines.

Motivated by the prediction that Ising ferromagnetism at finite temperature is stabilized by anisotropic SOC in CrP, we next turned our attention to substituting the P ion with As to tune the noncollinear spin behavior. In sharp contrast to CrP, we find that CrAs exhibits an easy magnetization plane, such that there is no energetic barrier to the rotation of spins within the xy plane of CrAs. Fig. 3(b) shows the angular dependence of the MAE, which is zero in-plane (θ = 90°) and reaches a maximum of 404 μeV per Cr pair perpendicular to the plane, but it is zero for all azimuthal angle θ and is only weakly dependent on ϕ in the plane orientations. It is because of the continuous O(2) spin symmetry in the plane that there is no FM ordering at finite temperature, which prohibits spontaneous symmetry breakage in systems with continuous symmetry with dimensions ≤2. Thus, a Berezinskii–Kosterlitz–Thouless (BKT) transition to a quasi-long-range phase occurs at low temperature. In this case, the critical temperature of the BKT transition can be obtained according to the XY model,61,62 which gives:

image file: c9nr10322h-t3.tif(5)
where kB is Boltzmann's constant, and EFM and EAFM are the energies of the collinear FM and AFM states, respectively. The energy difference is EAFMEFM = 889.3 meV per Cr, which leads to a BKT transition temperature of 855 K. A similar BKT transition has been reported in the nitride MXene Mn2NO2 monolayer.63

Low dimensional materials usually present a sensitive response to external stimuli, which enables the tunability of their electronic and magnetic properties. For instance, the value of MAE and the direction of the easy magnetization axis are successfully tuned by charge doping induced orbital occupation in the Fe/graphene complex system.67 Jiang et al.68 have shown that carrier doping can be introduced into 2D CrI3 by electric gating to control magnetism. Here we find that the MAE of CrP can be tuned significantly by carrier doping, and switch the magnetization easy axis. The calculated results are shown in Fig. 5(a). One can see that the total energy changes as a function of carrier concentration (n) with different magnetization directions. With electron doping (n < 0), the energy differences between the out-of-plane and in-plane magnetization increase with an increase in electron doping. When n > −5.5 × 1014 cm−2, CrP prefers out-of-plane magnetization. In contrast, hole doping (n > 0) can make CrP become an in-plane ferromagnet with a critical hole doping above 2.817 × 1014 cm−2. Experimentally, it is feasible to achieve a carrier concentration reaching 1013–1014 cm−2 in 2D systems, thus carrier doping is an effective way to control ferromagnetism in CrP.

image file: c9nr10322h-f5.tif
Fig. 5 Electrical control of 2D ferromagnetism of CrP. (a) Energy of the FM state with different magnetization directions as a function of carrier concentration n. (b) A schematic of a 2D magnetoelectric device with a giant magnetoresistance effect that is controlled by electrostatic doping. Here the 2D FM monolayer is doubly gated, while two dielectric layers such as h-BN avoid direct tunneling.

The carrier-tuned ferromagnetism in CrP presents an ideal platform to realize 2D spin FETs. Fig. 5(b) shows the proposed model of the magnetoelectric device, where 2D CrP is double-gated by top and bottom electrodes with two dielectric layers (e.g. 2D hexagonal BN and MoS2) to prevent direct tunneling. In this model, the carrier concentration and easy axis would be controlled by the double gate, while the source–drain voltage drives spin-dependent transport. Since the easy axis of CrP switches between the in-plane and out-of-plane direction upon critical doping, the in-plane and out-of-plane FM interface appears in between the doped and undoped region. In this case, the hetero-magnetic interface exhibits a high resistance state due to strong interface scattering. In contrast, homo-magnetization below critical doping maintains a low resistance state, realizing a 2D spin-FET.

IV. Conclusions

In summary, we predict a class of 2D intrinsic FM monolayers CrP and CrAs. The predicted Tc values are 232 and 855 K for CrP and CrAs, respectively, much higher than the recently reported Tc of 2D CrI3 and Cr2Ge2Te6 lattices. The obtained monolayers CrP and CrAs are identified as intrinsically FM semiconductors with spin-down band gaps of 2.29 eV and 2.49 eV and spin-up band gaps of 0.123 eV and 0.143 eV, respectively. Both of them are dynamically and thermally stable. Their exfoliation energies are much smaller than those of graphite, suggesting that they can be fabricated by mechanical cleavage or exfoliation like graphene. In contrast to other 2D magnetic materials, CrP exhibits a significant uniaxial MAE of 217 μeV per Cr originating from SOC. Remarkably, the magnetization easy axis can be tuned from out-of-plane to in-plane by electrostatic doping. The microscopic mechanisms of both high Tc and electrical tunability provide a way for designing novel 2D FM semiconductors. These findings indicate that monolayer CrP not only provides long-desired promising alternatives to 2D magnetic materials, but also offers an avenue for 2D magneto-optoelectronic applications such as spin FETs.

Conflicts of interest

There are no conflicts to declare.


This work was supported by the Taishan Scholar Project of Shandong Province (No.: ts20190939) and the Natural Science Foundation of Shandong Province (Grant No.: ZR2018MA033).


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Electronic supplementary information (ESI) available. See DOI: 10.1039/c9nr10322h

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