DOI:
10.1039/C9NR10322H
(Paper)
Nanoscale, 2020,
12, 54645470
Intrinsic ferromagnetism with high temperature, strong anisotropy and controllable magnetization in the CrX (X = P, As) monolayer†
Received
5th December 2019
, Accepted 6th February 2020
First published on 6th February 2020
Abstract
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 longrange 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 outofplane magnetization, while CrAs prefers easy inplane magnetization. Remarkably, hole and electron doping can switch the magnetization axis in between the inplane and outofplane 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 spinFETs in spintronics.
I. Introduction
Spinbased 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 twodimensional (2D) FM semiconductors with a high Curie temperature (T_{c}), large magnetocrystalline anisotropic energy (MAE), and high carrier mobility have great potential for spintronics and magnetooptoelectronics.^{7–15} However, the coexistence of longrange 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 singleion anisotropy is necessary, as observed experimentally in van der Waals crystals CrI_{3}^{9} and Cr_{2}Ge_{2}Te_{6}.^{12} However, their Curie temperature decreases significantly with a decrease in the number of atomic layers; the corresponding T_{c} in singlelayer and few layers is only 45 K for CrI_{3}and 25 K for Cr_{2}Ge_{2}Te_{6}. Therefore, it is crucial to design new 2D films with high T_{c} 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 T_{c}. Following the GKA rules, the longrange 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 T_{c} 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 outofplane magnetization, while CrAs favors easy inplane magnetization. Remarkably, hole and electron doping can switch the magnetization easy axis of CrP in between the inplane and outofplane direction, providing a means to control the efficiency of spin injection/detection in 2D magnetic semiconductors. The observed high T_{c} and electrically controllable magnetism pave a way for realizing spin fieldeffecttransistors (spinFETs) from 2D materials.^{33}
II. Computational details and methods
We performed firstprinciples 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 vdWDFoptB88^{40,41} exchange–correlation functionals. To approximately describe the strongly correlated interactions of the transition metal Cr, structural optimizations were performed by using the spindependent 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) functional^{42} 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 Brillouinzone integration is carried out by using an 18 × 18 × 1 Γcentered 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 kpoint 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 Pnmm space group with 2D networks of a rectangular sublattice in the x–y 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 E_{coh} = E_{CrX} − E_{Cr} – E_{X}, where E_{CrX} is the total energy of CrX, and E_{Cr} and E_{X} 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 10000 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 timedependent evolution of total energies, which shows a very small fluctuation.

 Fig. 1 (a) Crystal structure of layered ternary compounds ACr_{2}X_{2} (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 ACr_{2}X_{2} (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 caxis. 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 ACr_{2}X_{2}, 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 ACr_{2}X_{2} 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 spindependent 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 spinup band gap of 0.12 eV (0.26 eV for HSE) and a spindown 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 Cr3d orbitals. This is also confirmed by the spinup 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 0.16m_{0} and an associated large electron mobility of 11457cm^{2} V^{−1} s^{−1}, calculated by using a phononlimited scattering approach.^{55} Regarding the spindown channels, the highest occupied bands come mainly from the Cr3d and P2p orbitals, in line with the distributions of the spindown charge density in Fig. 2(c). Additionally, most of the spinpolarized electrons locate around the Cr ions, leading to a large magnetic moment of 3.0μ_{B} per Cr atom. Furthermore, to check the effect of the adsorption atom on electronic properties, we performed calculations on the OHfunctionalized 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. Spinpolarized calculations reveal that it still possesses magnetic properties with a local magnetic moment of 7.22μ_{B}. Another prominent property is that it shows a metallic feature, instead of the halfmetallic feature of prinstine CrP, as shown in Fig. S3(b).† The strong magnetization of Cr ions could be well understood by the localized spinwave functions (see Fig. 2(e)). As the Fermi level lies almost in the middle of the spin gap, 100% spinfilter efficiency can be maintained in a wide positive or negative bias range, which makes CrX (X = P, As) an attractive candidate for spininjection.

 Fig. 2 (a) Electronic band structures of CrP at the HSE level with spinup and spindown states, as well as spinresolved projected DOS. The Fermi level is denoted by a dashed line at 0 eV. (b and c) The spindependent charge densities. (d) Spin configuration for evaluating the exchangeinteraction constants. J_{1} and J_{2} 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 Cr^{3+}. 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 (Cr_{1}–Cr_{2}) and superexchange (Cr_{1}–P–Cr_{3}) interactions mediated through P ions, as shown in Fig. 2(d). The nearest neighboring (NN) exchange interaction, J_{1}, and the next nearestneighboring (NNN) exchange interaction, J_{2}, are both positive, indicating that CrP prefers the FM state. The NN interaction J_{1} comes from the direct Cr_{1}–Cr_{3} exchange interactions and the NNN interaction J_{2} is introduced from the superexchange interaction between Cr_{1} and Cr_{2} ions linked by the neighboring P ions. Mermin and Wagner^{56} 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 bondangle (92°) of Cr_{1}–P–Cr_{2}, as shown in Fig. 2(d), is close to 90°, which means that the Crd 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 J_{2} in 2D systems should be expressed as J_{2} ≈ 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 J_{2} = 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 longrange 2D ferromagnetism, in which singleion anisotropy and exchange anisotropy are two important factors. Singleion 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 outofplane 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 (μ_{B}) per Cr atom, anisotropy constants K (μeV), and Curie temperatures T_{c} (K) for monolayers CrP and CrAs
System 
E(100)–E(001) 
E(010)–E(001) 
E(111)–E(001) 
T
_{c}

−K_{1} 
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 MAE^{58} can be described by:

MAE(θ) = K_{1}sin^{2}θ + K_{2}sin^{4}θ  (1) 
where
K_{1} and
K_{2} are systemdependent anisotropy constants and
θ is the azimuthal angle of rotation. Positive values of
K_{1} and
K_{2} indicate strong Ising ferromagnetism with an outof plane easy axis, whereas
K_{1} < 0 means that the magnetic direction is perpendicular to the
zaxis. As listed in
Table 1, we find that both
K_{1} and
K_{2} are positive for CrP, and MAE reaches a maximum value of 217 μeV per Cr pair at
θ_{xz} =
θ_{yz} = 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 FeCl
_{2} (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.

 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 x–y, y–z, and x–z 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:

 (2) 
where
J_{1} and
J_{2} are the nearest and nextnearest magnetic exchange interaction parameters, respectively,
S_{i} is the spin vector of each atom,
A is the anisotropy energy parameter, and
S_{i}^{z} 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 temperaturedependent 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 (
C_{v}) using the equation

 (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 CrI
_{3}^{9} (45 K) and Cr
_{2}Ge
_{2}Te
_{6} (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%,
J_{1} = 82.8 meV and
J_{2} = 74.8 meV; when
ε = −4%,
J_{1} = 85.4 meV and
J_{2} = 76.6 meV; when
ε = −6%,
J_{1} = 87.6 meV and
J_{2} = 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 FeCl
_{2}^{59} and NbSe
_{2}.
^{60} Therefore, the predicted high
T_{c} indicates that CrP may be a promising spintronic material at room temperature.

 Fig. 4 (a) Specific heat C_{v} and the corresponding magnetism with respect to temperature. (b) Curie temperature of CrP in comparison with CrI_{3},^{9} Cr_{2}Ge_{2}Te_{6},^{12} GaMnAs,^{64} CrOCl,^{65} and CrWI_{6}.^{66} The transition temperatures T_{c} 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 x–y plane of CrAs. Fig. 3(b) shows the angular dependence of the MAE, which is zero inplane (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 quasilongrange 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:

 (5) 
where
k_{B} is Boltzmann's constant, and
E_{FM} and
E_{AFM} are the energies of the collinear FM and AFM states, respectively. The energy difference is
E_{AFM} −
E_{FM} = 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 Mn
_{2}NO
_{2} 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 CrI_{3} 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 outofplane and inplane magnetization increase with an increase in electron doping. When n > −5.5 × 10^{14} cm^{−2}, CrP prefers outofplane magnetization. In contrast, hole doping (n > 0) can make CrP become an inplane ferromagnet with a critical hole doping above 2.817 × 10^{14} cm^{−2}. Experimentally, it is feasible to achieve a carrier concentration reaching 10^{13}–10^{14} cm^{−2} in 2D systems, thus carrier doping is an effective way to control ferromagnetism in CrP.

 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 hBN avoid direct tunneling.  
The carriertuned 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 doublegated by top and bottom electrodes with two dielectric layers (e.g. 2D hexagonal BN and MoS_{2}) 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 spindependent transport. Since the easy axis of CrP switches between the inplane and outofplane direction upon critical doping, the inplane and outofplane FM interface appears in between the doped and undoped region. In this case, the heteromagnetic interface exhibits a high resistance state due to strong interface scattering. In contrast, homomagnetization below critical doping maintains a low resistance state, realizing a 2D spinFET.
IV. Conclusions
In summary, we predict a class of 2D intrinsic FM monolayers CrP and CrAs. The predicted T_{c} values are 232 and 855 K for CrP and CrAs, respectively, much higher than the recently reported T_{c} of 2D CrI_{3} and Cr_{2}Ge_{2}Te_{6} lattices. The obtained monolayers CrP and CrAs are identified as intrinsically FM semiconductors with spindown band gaps of 2.29 eV and 2.49 eV and spinup 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 outofplane to inplane by electrostatic doping. The microscopic mechanisms of both high T_{c} and electrical tunability provide a way for designing novel 2D FM semiconductors. These findings indicate that monolayer CrP not only provides longdesired promising alternatives to 2D magnetic materials, but also offers an avenue for 2D magnetooptoelectronic applications such as spin FETs.
Conflicts of interest
There are no conflicts to declare.
Acknowledgements
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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Footnote 
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9nr10322h 

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