Impact of strain, pressure, and electron correlation on magnetism and crystal structure of Mn₂GaC from first-principles

Abstract

The atomically laminated Mn₂GaC has previously been synthesized as a heteroepitaxial thin film and found to be magnetic with structural changes linked to the magnetic anisotropy. Related theoretical studies only considered bulk conditions and thus neglected the influence from possible strain linked to the choice of substrate. Here we employ first principles calculations considering different exchange–correlation functionals (PBE, PW91, PBEsol, AM05, LDA) and effect from use of + U methods (or not) combined with a magnetic ground-state search using Heisenberg Monte Carlo simulations, to study influence from biaxial in-plane strain and external pressure on the magnetic and crystal structure of Mn₂GaC. We find that PBE and PBE + U, with U_eff ≤ 0.25 eV, gives both structural and magnetic properties in quantitative agreement with available experimental data. Our results also indicate that strain related to choice of substrate or applied pressure is a route for accessing different spin configurations, including a ferromagnetic state. Moreover, the easy axis is parallel to the atomic planes and the magnetocrystalline anisotropy energy can be increased through strain engineering by expanding the in-plane lattice parameter a. Altogether, we show that a quantitative description of the structural and magnetic properties of Mn₂GaC is possible using PBE, which opens the way for further computational studies of these and related materials.

Introduction

In the 1960s, Nowotny and coworkers^1,2 discovered a family of inherently atomically laminated materials, which decades later were coined MAX phases. It is a ternary phase with the general formula M_n+1AX_n (n = 1–3) where M is a transition metal (e.g. M = Ti, Cr, Mo, Zr), A is typically a group 13 to 16 element (e.g. A = Al, Si, Ga, Ge), and X is carbon or nitrogen. The material family, however, did not receive much attention until the mid-1990s and early 2000s when Ti₃SiC₂³ and Ti₄AlN₃^4,5, respectively, was demonstrated to possess a unique combination of metallic and ceramic characteristics. Since then, MAX phases have been synthesized both as bulk material and in thin film form, and have been shown to exhibit extraordinary physical, chemical, electrical and mechanical properties⁶. Due to this, the MAX phases are being considered for protective coatings, electrical contacts, sensors, and high-temperature structural applications. They are also used as precursors for MXenes, their two-dimensional (2D) counterparts exhibiting extraordinary properties and are considered for a host of different applications^7,8,9.

The first experimental evidence of magnetic MAX phases were quaternary Mn-doped Cr₂GeC^10,11,12, Cr₂GaC^{13,14,15,16,17} and Cr₂AlC^14,18,19 followed by (Mo,Mn)₂GaC^20,21 and (V,Mn)₃GaC₂²². Examples of ternary MAX phases with demonstrated magnetic properties are Cr₂AlC^23,24, Cr₂GeC^11,24, Cr₂GaC^13,25, Cr₂GaN²⁵, and Mn₂GaC^26,27,28,29. The Cr-based phases have all been made in bulk form and as thin films, whereas Mn₂GaC have been synthesized in thin film form only. Potential applications of magnetic MAX phases range from spintronics to refrigeration, even though the research efforts have so far been focused solely on the discovery of new magnetic phases and compositions, and fundamentals of magnetic properties as summarized in Ref.³⁰.

It has previously been shown that Mn₂GaC simultaneously undergoes a magnetic and structural transition between 230 and 250 K²⁷. With decreased temperature, the out-of-plane c axis contracts by 0.2% with an asymmetric change of the crystal structure as demonstrated from both X-ray diffraction²⁷ and neutron diffraction²⁸. It has also been shown that at 214 K Mn₂GaC undergoes a first order magnetic phase transition from antiferromagnetic (AFM) at higher temperature to a non-collinear AFM spin structure with a large uniaxial c-axis magnetostriction of 450 ppm²⁹. Moreover, Mn₂GaC exhibits neutron-diffraction peaks consistent with long-range AFM order with a periodicity of two structural unit cells²⁸. Also, a local magnetic moment of ~ 1.7 µ_B per Mn atom at 3 K and 5 T have been measured²⁷. Based on supercell calculations, the magnetic critical order–disorder temperature T_c has been predicted to be 660 ± 133K³¹. Subsequent measurement found a Néel temperature of 507 K, at which Mn₂GaC changes from a (suggested) collinear AFM state to the paramagnetic (PM) state²⁹.

The need for using DFT + U methods for studying magnetic MAX phases have been debated, see Ref. ³⁰ for further details. The first studies motivate the use of + U to get a better correlation between measured and calculated bulk modulus for Cr₂AC (A = Al, Ga, Ge)^{32,33,34,35,36,37}. We have later shown that a better match can be achieved without any + U but through extended unit cells to describe non-trivial magnetic configurations^38,39. Most investigations using + U for studying magnetic MAX phases use U from 0 to 2 eV, while, e.g., the hypothetical 2D Mn₂C MXene was studied using U = 4 eV and J = 1 eV, motivated by studies of Mn-based 3D compounds such as La(Mn,Zn)AsO, (Ga,Mn)N⁴⁰.

For the comparatively unexplored (theoretically and experimentally) Mn₂GaC phase, it is therefore motivated to investigate if the use of the ordinary generalized gradient approximation (GGA) is enough to describe the electron correlation, or if other functionals or DFT + U methods are necessary to describe the material. To address these questions, we here chose to investigate the structure and magnetic characteristics of Mn₂GaC under thin film constraints from choice of substrate, by applying tensile and compressive strain, both in- and out-of-plane, while considering different exchange–correlation functionals as well as use of DFT + U methods. In addition, we explore the easy axis and elaborate on its correlation to applied strain.

Result and discussion

In previous work of Mn₂GaC, we presented theoretical results valid for bulk synthesis conditions and thermodynamic equilibrium, by energies and volumes for various spin configurations obtained through complete relaxation of the unit cell volume, lattice parameters and atomic positions using the PBE functional^26,27,28,31. Thin films, on the other hand, are known to realize also metastable states, sometimes far from thermodynamic equilibrium, with the lattice parameter of the film material influenced by the choice of substrate. Since Mn₂GaC has been synthesized as epitaxial thin films only, we here use the approach of performing structural relaxation under constraints that mimic thin film conditions. Depending on the substrate used, different lattice parameters may be achieved due to strain, which in turn may influence the magnetic properties.

As a first step, we include collinear spin configurations defined in Fig. 8 in Computational Details and in Table S1, using the PBE functional. We relax the structure (i) as function of volume and (ii) under the constraint of keeping the in-plane lattice parameter a fixed (at compressed (a = 2.87), equilibrium (a₀ = 2.90 Å), and tensile (a = 2.93 Å) strains) while changing the out-of-plane lattice parameter c. Figure 1 show the rather large spread in energy for the various spin configurations. For both volume relaxation and with Mn₂GaC constrained in-plane we find qualitatively similar results, with FM and ${\text{AFM}}{\left[0001\right]}_{\alpha }^{A}$ being lowest in energy. Optimized lattice parameters for considered collinear spin configurations are found in Table 1.

Figure 1

Total energy as a function of (a) volume and out-of-plane lattice parameter c for in-plane lattice parameter a being (b) 2.87 Å, (c) 2.90 Å, and (d) 2.93 Å assuming different spin configurations. All data presented are based on first-principles calculations employing the PBE exchange–correlation functional. The experimentally measured volume and lattice parameter c at room temperature (RT) and 150 K is represented by the vertical dashed line^26,27.

Table 1 Calculated equilibrium volume V, lattice parameters a and c, and absolute magnetic moment per Mn atom for considered magnetic spin configurations of Mn₂GaC using the PBE exchange–correlation functional.

Based on Fig. 1 the focus will from this point forward be on magnetic spin configurations in the low-energy region and investigate how biaxial in-plane strain, i.e., effect from using different substrates, and pressure applied perpendicular to the film surface, i.e., along the surface normal (the c-axis), may alter the crystal structure and magnetic properties of Mn₂GaC. In addition, we also include a magnetic ground-state search at different biaxial strains and look at the magnetic anisotropy energies. Moreover, we herein also evaluate the effect from choice of different exchange correlation functionals.

Biaxial in-plane strain

We first investigate the impact from biaxial in-plane strain on the magnetic and the crystal structure of Mn₂GaC using PBE, PW91, PBEsol, AM05, and LDA functionals. We consider seven representative spin configurations (FM, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{1}$, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{X}$, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{X}$, and in-AFM2) along with the non-magnetic (NM) solution, at various strains. Independent of strain or functional used we find that the overall low-energy spin configurations are FM, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$, and $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$, as shown in Fig. S1. These three spin configurations are therefore chosen for further evaluation.

In Fig. 2 the energy difference ΔE relative to a FM state, with a₀ = 2.90 Å, and local magnetic moments as function of biaxial in-plane strain is shown for FM, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$, and $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$. Looking at ΔE in the top panels in Fig. 2, four functionals, PW91, LDA, PBEsol, and AM05, show the same spin configuration order with FM lowest in energy, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$ highest in energy, and $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$ in-between. For PBE, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$ is lowest in energy at compressive strains and up to + 1% tensile strain. Above + 1.3%, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$ is found to be of lowest energy. Only two functionals, PBE and PW91, results in an energy minimum at or close the reported experimental value of a = 2.90 Å. LDA, PBEsol, and AM05 all overbind with a minimum ΔE at compressive strains below − 1.5%.

Figure 2

Energy relative to the FM state with a₀ = 2.90 Å (upper panels), lattice parameter c (mid panels), and local magnetic moment (bottom panels) as function of biaxial in-plane strain for FM, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$, and $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$ spin configurations using (a) PBE, (b) PW91, (c) PBEsol, (d) AM05, and (e) LDA exchange–correlation functionals. The experimentally measured lattice parameter a of 2.90 Å is represented by the vertical dashed line and c of 12.55 Å is represented by the horizontal dashed line in the mid panels^{26, 27}.

From a structural point of view, we find a decrease in lattice parameter c with increasing a, see the mid panels in Fig. 2. For all functionals we find c to be smallest for FM and largest for $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$. The reported value of c (12.55 Å) for synthesized Mn₂GaC, with a = 2.90 Å, is indicated by the dashed horizontal line. Both PBE and PW91 give c values close to reported, while overbinding (resulting in too small c) is large for PBEsol, AM05, and especially LDA.

Calculated local moments, bottom panels of Fig. 2, are in general highest for $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$ and lowest for FM. For $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$ we observe two different values which can be related to its spin structure that on a local scale resembles both FM and $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$. This can be seen by comparing the magnetic structures in Fig. 7d with Fig. 6a, c in Computational Details, where the lower value corresponds to a FM surrounding, with Mn atoms of parallel spins across the Ga layer, while those of higher value resemble $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$, with Mn atoms of antiparallel spins across the Ga layer.

The internal crystal structure of Mn₂GaC can be described by two different Mn–Mn interlayer distances illustrated in Fig. 8q in Computational Details; (i) d_X which represents the distance between two Mn layers interleaved with carbon, or X in general, and (ii) d_A which represents the distance between two Mn layers interleaved with Ga, or A in general. Fig. S2 show interlayer distances d_X and d_A as function of biaxial in-plane strain. For $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$ there are two different d_A values which can be related to the difference in the spin configuration close to the Mn layer. Similar to the two different magnetic moments in Fig. 2. The lower d_A value is similar to those found for FM, as are their local moments, while the larger d_A value is close to those obtained for $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$. For d_X, only one set of values is found for each of the three spin configurations, and these configurations are, on a local scale, equivalent.

Based on the large tendencies for overbinding when using PBEsol, AM05, and LDA functionals, we chose to investigate impact from biaxial strain upon addition of + U, using the approach by Dudarev⁴¹. It should be noted that in a theoretical study of the hypothetical 2D Mn₂C MXene⁴⁰, which is the 2D counterpart of Mn₂GaC after removal of Ga, U = 4 eV and J = 1 eV was used. Such value of U can not directly be extrapolated to also be valid for the 3D Mn₂GaC. We have therefore used several values for U_eff, though for the results presented herein, U_eff is chosen based on an identified minimum ΔE close to a₀ = 2.90 Å and values for c close to the reported value of 12.55 Å. A moderate value of U_eff = 1 eV is chosen for PBE and PW91 as their numbers are already close to or slightly larger than reported experimental values^26,27. Both PBEsol and AM05 require U_eff = 2.5 eV, while for LDA a value of U_eff = 4 eV is needed. To ensure that no other spin configurations than FM, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$, and $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$ may be a relevant low-energy solution, we also consider $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{1}$, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{X}$, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{X}$, in-AFM2, and NM. As shown in Fig. S3, the low-energy spin configurations are those with parallel spins within each Mn-C-Mn trilayer, i.e., FM, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$, and $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$. Interesting to note is that for all six functionals we find the order of the spin configurations to be the same, with $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$ lowest in energy followed by $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$ and FM as seen in Fig. 3. Moreover, for $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$ in Fig. 3 and Fig. S4 there is no longer two distinct values of neither the local magnetic moment or d_A, as compared to Fig. 2 and Fig. S4, when + U is not considered. Instead, these values are similar to values found for both FM and $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$.

Figure 3

Energy relative ferromagnetic solution with a₀ = 2.90 Å, ΔE (top panels), and local magnetic moment (bottom panels) as function of strain for FM, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$, and $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$ spin configurations using (a) PBE + U, (b) PW91 + U, (c) PBEsol + U, (d) AM05 + U, and (e) LDA + U exchange–correlation functionals. Considered U_eff values are given at the top of each panel. The experimentally measured lattice parameter a of 2.90 Å is represented by the vertical dashed line and c of 12.55 Å is represented by the horizontal dashed line in the mid panels^26,27.

The results presented for Mn₂GaC at biaxial strain shows that PBEsol, AM05, and LDA all overbind and with FM as the lowest energy spin configuration. PBE and PW91, on the other hand, results in an energy minimum at or close the reported experimental value of a = 2.90 Å and with magnetic moments slightly overestimated than experimentally reported²⁷. The use of + U, with U_eff values chosen in such a way to mimic experimentally reported lattice parameters, results in local magnetic moments much larger than those reported experimentally²⁷. We also note, independent of functional or if using + U or not, that the corresponding low-energy spin configurations always corresponds to those with FM ordering within Mn-C-Mn trilayer, i.e., FM and ${\text{AFM}}{\left[0001\right]}_{\alpha }^{A}$ (α = 2 and 4). Such FM ordering within Mn-C-Mn trilayer has previously been experimentally demonstrated for quaternary MAX phases (Cr_0.5Mn_0.5)₂GaC^15,16 and (Mo_0.5Mn_0.5)₂GaC^20,21 using ferromagnetic resonance (FMR) measurements. We thus conclude that using PBE, without + U or a with a small U_eff, gives crystallographic structures, including the asymmetric structural changes during the magnetic transition around 250 K and below²⁷, and magnetic structures comparable to those found from vibrating sampling magnetometry, X-ray and neutron diffraction ^26,27,28. Note that the use of no U or a small value of U_eff for 3D Mn₂GaC, as compared to U = 4 eV and J = 1 eV used for 2D Mn₂C⁴⁰, could be related to the interference by the Ga layer, imposing periodic screening of the Mn-C-Mn trilayers.

Magnetic ground-state search at biaxial in-plane strain

To this point, only collinear spin configurations have been considered. From analysis of magnetic measurements of Mn₂GaC there are indications that the magnetic structure is, or at least bears strong resemblance to, AFM ${\left[0001\right]}_{4}^{A}$. One approach to identify or get inspiration for other possible and more complex noncollinear spin configurations is to perform a representative magnetic ground-state search. In Ref.²⁷ such a search was performed and several magnetic spin configuration close to degenerate in energy was found as a function of volume.

Motivated by the thin films of Mn₂GaC synthesized to date, a representative magnetic ground-state search needs to keep a static to mimic thin film conditions for different substrates. We used the same approach as in Ref.²⁷ and performed a magnetic ground-state search using a coarse-grained super-moment representation, depicted in Fig. 9 in Computational Details, aiming to find additional magnetic spin configurations. To preserve control of the crystal structure, and the corresponding energies, we made the following approximations;

• (i) a and interlayer distance d_X were kept fixed while varying the interlayer distance d_A.

• (ii) we only considered low-energy candidates with parallel spins within the Mn-C-Mn trilayer, i.e., FM, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{6}^{A}$ and $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{8}^{A}$.

• (iii) we used the PBE exchange–correlation functional with no + U.

Based on these constraints we calculated the total energies, shown in Fig. S5, and used these along with the Connolly-Williams structure inversion method^42,43 to derive the exchange interactions J_ij’s for the first four super-moment interlayer coordination shells, see Fig. S6. In order to find possible long-range magnetic spin configurations, we considered supermoment chains with up to 40 beads. This range also ensures that we minimize effects related to size and boundary conditions. The energy dependence of the number of beads included in the chain at various values of d_A is shown in Fig. S8. From this point forward, we only present the low-energy solutions for each set of a, d_X, and d_A, independent on the number of beads used.

Figure 4 summarizes the result from our Heisenberg Monte Carlo simulations. Figure 4a shows a schematic illustration of the two individual angles defining the spin-orientation between nearest and next nearest neighbour supermoment spin vectors, θ₁ and θ₂. Details of θ₁ and θ₂ extracted from low-energy solutions when varying a, d_X, and d_A is shown in Fig. S8. A selection of representative low-energy solutions is illustrated in Fig. 4b along with related information such as interlayer distance d_A, corresponding lattice parameter c, and average and/or individual angle between nearest and next nearest neighbour supermoment spin vectors, ${\stackrel{-}{\uptheta }}_{1}$ and ${\stackrel{-}{\uptheta }}_{2}$. Note that for d_A = 4.058 Å (c = 12.30 Å), two different solutions were found with equal energy, III and IV, each with two distinct values of ${\uptheta }_{1}$, as given within parenthesis, though with an average value of $90^\circ$. Similar results are found when Mn₂GaC is under compressive (-1%, a = 2.87 Å) and tensile (+ 1%, a = 2.93 Å) strain, shown in Fig. S8, although shifted to larger and smaller d_A, respectively. When smaller(larger) values of d_X are considered, the results are shifted to larger(smaller) value of d_A, shown in Fig. S8.

Figure 4

(a) Schematic illustration individual angles defining the spin-orientation between nearest and next nearest neighbour supermoment spin vectors, θ₁ and θ₂. (b) Selected low-energy spin configurations extracted from the Heisenberg Monte Carlo simulations at various d_A where a = 2.90 Å and d_X = 2.09 Å.

Inspired by the obtained noncollinear spin spiral configurations depicted in Fig. 4b, we considered a small set of representative configurations for evaluation using first-principles calculations. For a₀ = 2.90 Å, we use PBE + U with U_eff from 0 to 2 eV and compare energies, structural parameters and magnetic moments, see Fig. 5. The calculated energies relative to the FM state is shown in in Fig. 5a. With ordinary PBE (U_eff = 0 eV), the considered noncollinear spin configurations are all in between FM and the lowest energy $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$. For PBE + U, with increasing U_eff up to 0.5 eV, all AFM configurations, collinear and noncollinear, decreases in energy relative to FM.

Figure 5

(a) Energy relative to the FM state, (b) lattice parameter c, (c) local magnetic moment, and (d) interlayer distances d_A for Mn₂GaC, with a₀ = 2.90, Å as function U_eff for collinear FM, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$, and $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$ and noncollinerar $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A,60}$, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A,90}$, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A,120}$, and $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A,135}$ spin configurations using the PBE + U exchange–correlation functional.

Measured lattice parameter c at room temperature (RT) and at 150 K, indicated by horizontal lines in Fig. 5b, clearly shows that using ordinary PBE gives too small c while U_eff ≥ 0.5 eV gives too large values of c. Moreover, not only does c, the interlayer distance d_A and the local magnetic moment increase with increasing U_eff, but we also find that they merge towards similar values independent of considered spin configuration. This is most clear for U_eff ≥ 1 eV. This is indicative of a parameter forcing the magnetic moment to be localized, raising a question of the proper value of U_eff to describe Mn₂GaC in particular and magnetic MAX phases in general. These results indicate the PBE or PBE + U (with U_eff ≤ 0.25 eV) is the most appropriate functionals to use until proven otherwise for describing Mn₂GaC when compared to experimental results^26,27,28,29.

Mn₂GaC under pressure

Here we investigate how the magnetic properties and crystal structure of Mn₂GaC can be altered when pressure is applied perpendicular to the film surface, i.e., along the surface normal (the c-axis). We only consider low energy spin configurations FM, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$, and $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$, and choose to not include noncollinear spin configurations since the energies for these are all found in between those of FM, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$ and $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}.$ Starting with the thermodynamic stability of Mn₂GaC, the possibility of phase transition between considered spin configurations is investigated. The enthalpies are calculated according to

$$H\left[p\right]=E\left[p\right]+pV\left[p\right],$$

(1)

where E[p] and V[p] is the equilibrium energy and volume, respectively, at given pressure p.

Figure 6a depicts the pressure vs U_eff where the color represents the spin configuration of lowest enthalpy for a₀ = 2.90 Å. For ordinary PBE (U_eff = 0) we find that at 3.8 GPa there is a transition from $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$ to FM. When U_eff increases $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$ becomes the favoured configuration at small applied pressures. FM is still accessible for U_eff > 0 but an increased pressure is required.

Figure 6

Spin configuration of lowest enthalpy for (a) a pressure vs U_eff grid, and for a pressure vs biaxial-in-plane strain grid for U_eff equal to (b) 0 eV, (c) 0.25 eV, and (d) 0.5 eV, using the PBE + U exchange–correlation functional.

To explore potential pathways for attaining a FM state, we also probe for lowest enthalpy spin configuration on a pressure vs biaxial-in-plane strain grid for a range of lattice parameter a, from compressive (a = 2.86 Å, − 2.07%) to tensile (a = 2.94 Å, + 2.07%) strain. The various values of a thus represent substrates of different size. Figure 6b depicts pressure vs bi-axial in-plane strain for ordinary PBE (U_eff = 0 eV) where the color represents the spin configuration of lowest enthalpy. At zero pressure and from compressive up to 1.25% tensile biaxial strain, we find $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$, while further increase in tensile strain results in a transition to $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$. When pressure is applied, the enthalpy difference between $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{\alpha }^{A}$ and FM decreases. For a₀ = 2.90 Å (0%) we find that at 3.8 GPa there is a transition from $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$ to FM. This transition is associated with a corresponding decrease in c, magnetic moment, and interlayer distances d_A and d_X, see details in Fig. S9, and is found at both compressive (down to -2%) and tensile (up to + 1.25%) biaxial strain for an applied pressure up to 4 GPa. Between tensile strain from + 0.75 to 1.25% there is a transition from $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$ to $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$ with increasing pressure before accessing FM. Furthermore, for a tensile biaxial strain above + 1.25% and at zero or low pressure we find $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$. When pressure is applied there is a transition from $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$ to FM, though shifted to higher pressure for an increased tensile strain.

For U_eff ≥ 0.25 eV, see Fig. 6c, d, we find similar magnetic transitions as for ordinary PBE, but shifted towards more compressive strain and higher pressure. $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$ is the favoured spin configuration at low pressure. Focusing on U_eff = 0.25 eV at a₀ = 2.90 Å (0%), we find that at 9.1 GPa there is a transition from $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$ to FM associated with a significant change in both magnetic moment and crystal structure, shown in Fig. 5b, c, as compared to a $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$ to FM transition.

These results thus indicate that through manipulation of the strain within the thin film plane, e.g., through the choice of substrate, it is possible to strain-engineer accessible spin configurations. Moreover, through exertion of an external force, e.g., by applying a rather moderate pressure perpendicular to the film plane, one would enable a magnetic transition from AFM to FM. The significant change both in terms of crystal and magnetic structure during this transition should allow experimental verification.

Magnetocrystalline anisotropy energy

There are so far no studies reported on the magnetocrystalline anisotropic energy (MAE) for Mn₂GaC, neither experimental nor theoretical. Measurements on a related phase where 50% Mn in Mn₂GaC is substituted for Mo in (Mo_0.5Mn_0.5)₂GaC shows that the easy axis is in-plane, despite having a small MAE of 4.5 ± 0.3 kJ/m³ at 100 K favoring an out-of-plane orientation ²¹. Vibrating sampling magnetometry (VSM) measurements on Mn₂GaC demonstrates that spins are aligned within the Mn planes^26,27. We chose to investigate the MAE for Mn₂GaC by including the spin–orbit coupling and by using the PBE + U exchange–correlation functional for 0 ≤ U_eff ≤ 1 eV. Based on results presented in previous sections of this work and in related experimental results^26,27,28, we only consider the three low-energy collinear spin configurations, i.e., FM, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$, and $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$.

The MAE in Fig. 7 was calculated as the energy difference between the configurations with spins aligned in-plane, i.e. parallel to the Mn layer, and normal to the Mn layers, MAE = E( →) − E(↑). To ensure convergence for calculated MAE we considered various k-point densities, see Fig. S11. From this we estimated an error of ± 0.1 MJ/m³. The structures used are for a₀ = 2.90 Å and for each U_eff we used an optimized structure with c given Fig. 5b.

Figure 7

Magnetic anisotropy energy (MAE) as a function U_eff when considering FM, $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$, and $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$ spin configurations and using the PBE + U exchange–correlation functional. The estimated error bar from k-point convergence is 0.1 MJ/m³. The schematic shows the preferential spin alignment dependent on sign of the MAE.

For U_eff ≤ 0.25 eV, we find negative values for MAE which is indicative of preferential spin alignment within the Mn planes. Note that this is consistent with experimental reports^26,27. At larger U_eff, MAE change sign and for U_eff ≥ 0.75 eV all spin configurations show preferential for out-of-plane spin alignment. Here we again notice how sensitive the results are with respect to value of U_eff and that results for U_eff > 0.25 are inconsistent with experiments. In addition, for ordinary PBE, we find a negative MAE for various a and c values, see Fig. S12, showing that FM and $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$ having spins aligned in-plane become energetically more favorable upon tensile in-plane and compressed out-of-plane strain. $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$, on the other hand, show a MAE value being almost constant for here considered lattice parameters.

Our results of having preferential spin alignment within the Mn planes, for U_eff ≤ 0.25 eV, are consistent with VSM measurements on Mn₂GaC showing that spins are aligned within the Mn planes^26,27. For $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$ and $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$ the calculated MAE for U_eff ≤ 0.25 eV is in the range − 100 to − 300 kJ/m³. We suggest that the MAE for a Mn₂GaC thin film could be increased by expanding the in-plane lattice parameter a and thus reducing the value of c to minimize the stress in the c direction, as exemplified in Fig. S12(c).

Conclusion

We here present a first principles study of the structural and magnetic properties of the atomically laminated Mn₂GaC, with emphasis on effect of biaxial in-plane strain, pressure, and choice of different exchange–correlation functionals, aiming towards a relevant description of the structural and magnetic properties attainable for Mn₂GaC in thin film form. Here we note that previous theoretical results were primarily relevant for bulk synthesis conditions, i.e. for an equilibrium volume and energy. It is found that use of the PBE functional or PBE + U (U_eff ≤ 0.25 eV) gives both structural and magnetic properties in good quantitative agreement with available experimental data. PW91, PBEsol, AM05, and LDA functionals and/or use of an + U approach result in structures and magnetic characteristics not compatible with the same experimental reports. Through a magnetic ground-state search we find several noncollinear magnetic configurations which bears structural and magnetic resemblance to low-energy collinear candidates. Notable is that through strain-engineering by choice of substrate or applied pressure, different spin configurations may be accessible, which in turn suggest a tuning potential of the magnetic properties, including attaining a FM spin state. Furthermore, we also suggest that the easy axis in Mn₂GaC is parallel to the atomic planes and that the MAE can be increased through strain engineering by expanding the in-plane lattice parameter a, thus reducing the value of the out-of-plane lattice parameter c, which strengthen the spin orientation parallel to the atomic planes. Finally, the results presented here suggest that a quantitative description of the structural and magnetic properties of Mn₂GaC is possible using PBE, which opens the way for further computational studies of these and related materials.

Computational details

All first-principles density functional theory (DFT) calculations are performed using the projector augmented wave method^44,45 as implemented within the Vienna ab initio simulation package (VASP)^46,47,48, with a plane-wave energy cutoff of 400 eV. For sampling of the Brillouin zone we used the Monkhorst–Pack scheme⁴⁹. Within this work we use different exchange–correlation functionals to study their dependence on structure and magnetic properties. If not stated otherwise, the generalized gradient approximation (GGA) as parameterized by Perdew-Burke-Ernzerhof (PBE)^50,51 has been used. In parts of the work we also used the PBE revised for solids (PBEsol)⁵², the Perdew–Wang 91 (PW91)⁵³, and AM05^54,55,56 GGAs, and the local density approximation (LDA)⁵⁷ functionals. In addition, we also used the rotationally invariant approach as proposed by Dudarev⁴¹. Note that within this formalism the onsite Coulomb parameter U and the exchange parameter J are spherically averaged into a single effective interaction parameter U_eff = U − J which does not depend on their individual values. The equilibrium structures are obtained by minimization of the total energy for an a and c lattice parameter grid with full relaxation of atomic positions until forces are converged below 10^–4 eV Å⁻¹. Calculations including spin–orbit coupling (SOC) have been performed in the mode implemented in VASP by Hobbs et al.⁵⁸ and Marsman and Hafner⁵⁹, in a two-step procedure. First, we performed a scalar-relativistic calculation to obtain the correct geometry, followed by calculations including spin–orbit coupling with spins aligned along the c-axis, i.e. parallel/antiparallel to [0001] axis, and perpendicular to the c-axis, i.e. parallel/antiparallel to both [1–100] and [1–210] axis.

We have considered several collinear magnetic spin configurations for Mn₂GaC, most of them have been defined previously for a general M₂AX phase^27,38,39. A schematic representation is shown in Fig. 8, and includes a ferromagnetic (FM), nine layered antiferromagnetic (AFM), and six in-plane AFM spin configurations. The notation used for layered AFM spin configurations is defined as follows; single layer AFM with spins changing sign for every M layer corresponds to AFM[0001]₁, and multilayered AFM ordering with α consecutive M layers (where α = 2, 4, 6, 8) of the same spin direction before changing sign upon crossing an A or an X layer corresponds to ${\text{AFM}}{\left[0001\right]}_{\alpha }^{A}$ and ${\text{AFM}}{\left[0001\right]}_{\alpha }^{X}$, respectively^27,38. In addition, the paramagnetic (PM) have been modelled using the disorder local moment (DLM)⁶⁰ approach where the spin-correlation functions are equal to zero for at least the first 10 M-coordination shells. The disordered magnetic moments in (Mn↑_0.5Mn↓_0.5)₂GaC is simulated by means of the special quasi-random structure (SQS) method^61,62 using a supercell with 64 Mn, 32 Ga, and 32 C atoms, i.e., 4 × 4 × 1 or 16 M₂AX unit cells. A strict definition of these collinear spin configurations is given in Table S1 where the spin correlation function Φ_i is given for the first 10 M-shells.

Figure 8

Schematic representation of collinear spin configurations for a M₂AX phase where (a) represents ${\text{FM}}$, (b) $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{1}$, (c) ${\text{AFM}}{\left[0001\right]}_{2}^{A}$, (d) ${\text{AFM}}{\left[0001\right]}_{4}^{A}$, (e) ${\text{AFM}}{\left[0001\right]}_{6}^{A}$, (f) ${\text{AFM}}{\left[0001\right]}_{8}^{A}$, (g) ${\text{AFM}}{\left[0001\right]}_{2}^{X}$, (h) ${\text{AFM}}{\left[0001\right]}_{4}^{X}$, (i) ${\text{AFM}}{\left[0001\right]}_{6}^{X}$, (j) ${\text{AFM}}{\left[0001\right]}_{8}^{X}$, (k) in-${\text{AFM}}1$, (l) in-${\text{AFM2}}$, (m) in-${\text{AFM3}}$, (n) in-${\text{AFM4}}$, (o) in-${\text{AFM5}}$, and (p) in-${\text{AFM6}}$. (q) The interlayer distances d_X and d_A is indicated. Projections are viewed along the $\left[1-210\right]$ direction.

For the magnetic ground-state search we used a coarse-grained model, shown in Fig. 9 in Computational Details, where the local moment of Mn atoms in a Mn-C-Mn trilayer plane is represented by a supermoment. This is motivated by all low-energy spin configurations of Mn₂GaC having parallel spin directions within their Mn-C-Mn trilayers. The supermoment model is thus described using only magnetic exchange interactions (MEI) across the A layer along the c axis and cannot be used to modelling critical temperature since the MEI within a Mn-C-Mn tri-layer is neglected, and hence the exact temperature from the Monte Carlo simulation becomes irrelevant as it scales with the area of each layer. The spin correlation function Φ_i for the five considered spin configurations are given in Table S2. The essence of the approach is to identify possible non-trivial spin configurations through us of Monte Carlo simulations using a Heisenberg Hamiltonian.

Figure 9

Schematic illustration of the coarse-grained super-moment model where each Mn-C-Mn trilayer is represented by a super-moment bead, red for spin up and blue for spin down, in Mn₂GaC for (a) FM, (b) $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{2}^{A}$, (c) $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{4}^{A}$, (d) $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{6}^{A}$, and (e) $\mathrm{A}\mathrm{F}\mathrm{M}{\left[0001\right]}_{8}^{A}$ spin configurations.

$$\mathcal{H}=-\sum (i\ne j){J}_{ij}{\mathbf{e}}_{{\varvec{i}}}\cdot {\mathbf{e}}_{{\varvec{j}}},$$

(2)

where ${J}_{ij}$ is the MEI between pairs of super-moments (i, j), with unit vectors e_i and e_j along the local magnetic moment at site i and j, represented by a chain of super-moments. The exchange interactions J_ij’s are derived for the first four super-moment interlayer coordination shells using the magnetic Connolly-Williams structure inversion method^42,43, shown in Table S2 in combination with energies from first-principles calculations. To avoid possible metastable solutions, we initially set the temperature to a large value, and then slow cooling towards 0 K and to allow long-range magnetic interactions we used supermoment chains including up to 40 beads.