Ferrimagnetic antiperovskite Mn $ 4$N has received growing interest due to room-temperature observation of large perpendicular magnetic anisotropy, low saturation magnetization, and ultrafast response to external magnetic fields. Comprehensive understanding of the underlying magnetic structure is instrumental in design and fabrication of computer memory and logic devices. Magneto-optical spectroscopy provides deeper insight into the magnetic and electronic structure than magnetometry. Simulations of a magneto-optical Kerr effect in biaxially strained Mn $ 4$N are performed using density functional theory and linear response theory. We consider three ferrimagnetic phases, two collinear and one noncollinear, which have been investigated separately by earlier studies. The simulated spectra are compared to measured magneto-optical data available in recent literature. One of the collinear ferrimagnetic phases is found to be consistent with the measured spectra. We show that an admixture of the noncollinear phase, which is the ground state of unstrained Mn $ 4$N, further improves the agreement with measured spectra, and at the same time, it could explain the lower than predicted net moment and magnetic anisotropy observed in thin films on various substrates.

## I. INTRODUCTION

Traditionally, ferrimagnets have been applied in magneto-optical recording,^{1–3} where a laser beam is used to increase the temperature close to the Curie point so that the magnetization can be reversed by a small applied magnetic field. However, more recently, ferrimagnetic materials have attracted much attention for high-density magnetic random access memories (MRAMs) and logic devices, which benefit from high perpendicular magnetic anisotropy (PMA), saturation magnetization much lower than in typical ferromagnets, and Curie temperature well above room temperature.^{4–7}

Ferrimagnets usually consist of two magnetic sublattices with antiferromagnetic coupling between them, which allows for fast spin dynamics motivating extensive research in antiferromagnetic spintronics.^{8–10} The absence of net magnetic moment in antiferromagnetic materials allows for closely packed memory arrays but precludes control of the magnetic domain structure by an external magnetic field. The small but finite net magnetization of ferrimagnetic materials alleviates this problem.

We avoid the reliance on common ferrimagnets containing rare earth elements^{11} by turning to metallic antiperovskite nitrides, some of which have a collinear or noncollinear ferrimagnetic structure.^{12–14} Mn-based antiperovskite nitrides are a broad family of materials hosting a range of phenomena, including magneto-transport,^{15–17} magneto-caloric,^{18–20} or magneto-optical properties, tunable by chemical composition or lattice strain.^{21–23} Mn $ 3$GaN, a widely studied member of this family with a triangular antiferromagnetic ground state, can develop a collinear ferrimagnetic (FIM) phase under compressive biaxial strain at room temperature.^{24} Subsequently, magneto-optical Kerr spectroscopy has revealed the magnetic structure of Mn $ 3$NiN, a closely related antiperovskite. The measured data are consistent with the presence of a collinear FIM phase at room temperature.^{25}

The magnetic structure of antiperovskite Mn $ 4$N has even further complexity than Mn $ 3$NiN as Ni on the $1a$ site is replaced by another Mn with a large magnetic moment as shown in Figs. 1(a)–1(c). Large PMA reaching 0.1 MJ/m $ 3$ and an ultrafast response to an external field has been observed in Mn $ 4$N recently.^{26–36} In addition, low saturation magnetization, $ \mu 0 M s\u22480.1$ T, has been demonstrated by numerous studies of Mn $ 4$N thin films listed in Table I. Therefore, in devices controlled by spin-transfer torque (STT), such as magnetic tunnel junctions (MTJs), the critical STT-current density, $ J c\u221d\alpha M st H k$ (where $\alpha $ is the damping constant, $t$ is the magnetic layer thickness, and $ H k$ is the anisotropy field proportional to the PMA energy density, $ K u$), is expected to be small.^{29,37–39} More favorable material properties of Mn-based antiperovskite nitrides have been reviewed by Isogami and Takahashi^{40} recently.

Substrate . | c/a
. | t (nm)
. | Method . | K_{u} (MJ/m^{3})
. | M_{s} (kA/m)
. |
---|---|---|---|---|---|

MgO^{27} | 0.987 | 35 | PLD | 0.16 | 157 |

MgO^{26} | 0.99 | 26 | MBE | 0.22 | 145 |

MgO^{42} | 0.99 | 100 | Sputtering | 0.88 | 110 |

STO^{28} | 0.99 | 25 | MBE | 0.1 | 80 |

MgO^{43} | 0.983 | 9 | MBE | 0.18 | 127 |

MgO^{44} | 0.991 | 30 | MBE | 0.075 | 100 |

MgO^{45} | … | 10 | MBE | 0.11 | 118 |

STO^{45} | … | 10 | MBE | 0.11 | 105 |

STO^{29} | … | 10 | MBE | 0.11 | 71 |

MgO^{30} | 0.993 | 18 | MBE | 0.06 | 63 |

STO^{30} | 0.989 | 17 | MBE | 0.126 | 73 |

LAO^{30} | 0.998 | 19 | MBE | 0.045 | 53 |

MgO^{37} | 0.99 | 30 | Sputtering | 0.1 | 80 |

MgO^{46} | 0.99 | 28 | Sputtering | 0.17 | 156 |

Glass^{34} | 0.993 | 45 | Sputtering | 0.022 | 36 |

Glass^{47} | 0.988 | 48 | Sputtering | 0.073 | 99 |

MgO/VN^{36} | 0.987 | 28 | Sputtering | 0.043 | 85 |

Bulk^{41} | 1.0 | … | Homog. powder | … | 178 |

FIM_{A}^{37} | 0.99 | … | DFT simul. | 0.5 | 6.3 |

FIM_{B}^{37} | 0.99 | … | DFT simul. | 3.8 | 143 |

ncFIM^{48} | 1.0 | … | DFT simul. | … | 194 |

FIM_{B}a | 0.99 | … | DFT+U simul. | 3.78 | 191 |

ncFIMa | 0.99 | … | DFT+U simul. | 1.81 | 114 |

Substrate . | c/a
. | t (nm)
. | Method . | K_{u} (MJ/m^{3})
. | M_{s} (kA/m)
. |
---|---|---|---|---|---|

MgO^{27} | 0.987 | 35 | PLD | 0.16 | 157 |

MgO^{26} | 0.99 | 26 | MBE | 0.22 | 145 |

MgO^{42} | 0.99 | 100 | Sputtering | 0.88 | 110 |

STO^{28} | 0.99 | 25 | MBE | 0.1 | 80 |

MgO^{43} | 0.983 | 9 | MBE | 0.18 | 127 |

MgO^{44} | 0.991 | 30 | MBE | 0.075 | 100 |

MgO^{45} | … | 10 | MBE | 0.11 | 118 |

STO^{45} | … | 10 | MBE | 0.11 | 105 |

STO^{29} | … | 10 | MBE | 0.11 | 71 |

MgO^{30} | 0.993 | 18 | MBE | 0.06 | 63 |

STO^{30} | 0.989 | 17 | MBE | 0.126 | 73 |

LAO^{30} | 0.998 | 19 | MBE | 0.045 | 53 |

MgO^{37} | 0.99 | 30 | Sputtering | 0.1 | 80 |

MgO^{46} | 0.99 | 28 | Sputtering | 0.17 | 156 |

Glass^{34} | 0.993 | 45 | Sputtering | 0.022 | 36 |

Glass^{47} | 0.988 | 48 | Sputtering | 0.073 | 99 |

MgO/VN^{36} | 0.987 | 28 | Sputtering | 0.043 | 85 |

Bulk^{41} | 1.0 | … | Homog. powder | … | 178 |

FIM_{A}^{37} | 0.99 | … | DFT simul. | 0.5 | 6.3 |

FIM_{B}^{37} | 0.99 | … | DFT simul. | 3.8 | 143 |

ncFIM^{48} | 1.0 | … | DFT simul. | … | 194 |

FIM_{B}a | 0.99 | … | DFT+U simul. | 3.78 | 191 |

ncFIMa | 0.99 | … | DFT+U simul. | 1.81 | 114 |

^{a}

Last two rows show data calculated in this work.

In order to utilize the potential of Mn $ 4$N for spintronic applications, such as MTJs or skyrmionic devices,^{31} it is crucial to attain a thorough understanding of the microscopic origin of these advantageous properties and to be able to grow thin films with a well defined magnetic structure on substrates compatible with CMOS technology.

In experimental studies listed in Table I, Mn $ 4$N has been deposited since 2014 on a range of substrates, e.g., MgO, SrTiO $ 3$ (STO), and LaAlO $ 3$ (LAO) with a mismatch of approximately $\u22126%$, $\u22120.1%$, and $+2%$,^{30} respectively, assuming (001) surfaces and a lattice constant of cubic Mn $ 4$N equal to 0.3865 nm.^{41} It has been observed using x-ray diffraction that the films have $c/a\u22480.98\u22120.99$, where $a$ and $c$ are the in-plane and out-of-plane lattice constant, respectively, despite the different magnitude and sign of a lattice mismatch.

Therefore, experimental studies generally conclude that the origin of PMA in Mn $ 4$N films is the tetragonal distortion.^{26,27,30,42} $Abinitio$ studies of Ref. 28 and 37 investigate two collinear ferrimagnetic phases with PMA, the so called type A (FIM $ A$) and type B (FIM $ B$) shown in Fig. 1(b) and 1(c), respectively. Both studies deduce the potential presence of FIM $ A$ and FIM $ B$ magnetic phases (both with magnetic space group 123.345, P4/mm $ \u2032$m $ \u2032$) from neutron diffraction experiments.^{41} However, Zhang *et al.*^{49} and Gercsi *et al.*^{50} point out that there is no experimental evidence for FIM $ B$ in bulk powder samples of Ref. 41, which sees two (as in FIM $ A$) rather than three (as in FIM $ B$) inequivalent Mn sites, in agreement with a later nuclear magnetic resonance (NMR) study.^{51} A subsequent diffraction study with polarized neutrons identified a noncollinear variant of the FIM $ A$ phase (two inequivalent Mn sites) and a triangular ferrimagnetic phase (ncFIM).^{14} Magnetic moments of Mn atoms in this “umbrella-like” structure with magnetic space group 166.101, R $ 3 \xaf$m $ \u2032$ (cubic) or 12.62, C2’/m $ \u2032$ (biaxially strained) shown in Fig. 1(a) do not mutually compensate as in cubic Mn $ 3$NiN or Mn $ 3$GaN due to the extra local magnetic moment on the $1a$ site, $ m 1 a$. The moments in face-center $3c$ positions, $ m 3 c$, are tilted out of the (111) plane (where the Mn atoms form a kagome lattice) by approximately 20 $ \xb0$ to have a component along the [111] axis, antiparallel to $ m 1 a$. The ncFIM structure was confirmed computationally by Uhl *et al.*^{52} and more recently by Zhang *et al.*,^{49} including the net moment along the [111] axis, $ m n e t=1.14 \mu B$.^{41} The same team also proposed related noncollinear ferrimagnetic phases in the Mn $ 4$N film on MgO^{46} based on ncFIM and the coplanar triangular antiferromagnetic structure of Mn-based antiperovskite nitrides.

Here, we focus on experimental data available for thin films. We are not aware of neutron diffraction studies of thin Mn $ 4$N films; therefore, we work with FIM $ A$ and FIM $ B$ studied earlier.^{28,37} Both Ito *et al.*^{28} and Isogami *et al.*^{37} found that FIM $ B$ has significantly lower total energy than FIM $ A$ in the range $c/a\u2208(0.96\u22121.1)$, in agreement with our simulations shown in Fig. 1(d). Both studies suggest that this intrinsic tetragonal phase, FIM $ B$, explains why $c/a\u2208(0.98\u22120.99)$ has been reported in Mn $ 4$N films epitaxially grown on different substrates regardless of the film thickness and the lattice mismatch. However, negligible dependence of PMA on $c/a$ was predicted by Isogami *et al.*^{37} in disagreement with PMA measured by Hirose *et al.*^{30} in Mn $ 4$N on MgO, STO, and LAO. Moreover, very recent x-ray magnetic circular dichroism (XMCD) measurements of a magnetic structure of thin Mn $ 4$N films on STO^{53} reveal the FIM $ B$ phase for film thickness $t>5$ nm and a noncollinear ferrimagnetic structure for $t=4$ nm. Ito *et al.*^{28} and Isogami *et al.*^{37} did not consider the ncFIM phase identified by Fruchart *et al.*^{14} as cubic anisotropy was expected instead of PMA seen in Mn $ 4$N films. This represents a gap in the available simulations, which hinders interpretation of new experimental findings.

Therefore, in this study, we compare the collinear (FIM $ A$, FIM $ B$) and noncollinear FIM (ncFIM) phases of strained Mn $ 4$N based on total energy, magnetocrystalline anisotropies, magnetic moments, and magneto-optical Kerr effect (MOKE). The immediate motivation is a recent study of MOKE in 23 nm thick Mn $ 4$N films sputtered on an MgO substrate, which proposed that a superposition of different ferrimagnetic phases could exist.^{54} Moreover, an in-plane component of magnetization has been detected in an Mn $ 4$N film deposited on MgO by MBE,^{55} and the authors explained it in the presence of a FIM phase with magnetization along the [111] axis.

## II. COMPUTATIONAL METHODS

We employ noncollinear spin polarized density functional theory (DFT) combined with linear response theory following the approach of Ref. 56 and our earlier work.^{25} We use the projector augmented wave method as implemented in the VASP code^{57} with generalized gradient approximation (GGA) parameterized by Perdew–Burke–Ernzerhof.^{58} Our results were obtained using a 500 eV energy cutoff and a $23\xd723\xd723$ k-mesh (for a unit cell with 5 atoms) to ensure convergence (in agreement with numerical settings of Ref. 37). The valence configurations of manganese and nitrogen are 3 $ d 6$4 $ s 1$ and 2 $ s 2$2 $ p 3$, respectively.

As has been done for MOKE spectra of Mn $ 3$NiN^{25} and earlier MOKE studies of some collinear antiferromagnets, such as CuMnAs,^{59} we modify the intra-atomic Coulomb interaction within GGA through the rotationally invariant approach to GGA+U as proposed by Dudarev *et al.*^{60} We explore values of U from 0.2 to 2.2 eV on the Mn-3 $d$ orbital. (Most data shown here were simulated using $U=0.7$ eV refined in Ref. 25.) This repulsion lifts the unoccupied manganese $3d$-states further away from the Fermi level, resulting in a blueshift in the optical and magneto-optical responses, which improves the agreement with the available measured data.^{54}

MOKE spectra offer a valuable insight into the magnetic and electronic structure of thin films and reflect the symmetry of the magnetic structure as detailed in Appendix D. We evaluate the Kerr rotation, $\theta (\omega )$, and ellipticity, $\eta (\omega )$, from the complex permittivity tensor, $ \epsilon i j(\omega )$, computed using VASP.^{25,56} In the polar configuration, they are related as follows: $\theta +i\eta =\u2212 \epsilon x y/[( \epsilon 0\u22121) \epsilon 0]$, where $ \epsilon 0=( \epsilon x x+ \epsilon y y)/2$. The films are conductive, 23 nm thick;^{54} therefore, the light is mostly absorbed within the layer justifying the use of semi-infinite medium approximation.

## III. RESULTS AND DISCUSSION

The unit cells and corresponding total energies as functions of the $c/a$ ratio are shown in Fig. 1. Available experimental data listed in Table I provide a range of values of Poisson’s ratio, $\nu $. Therefore, for each $c/a$ ratio, we calculate the total energy assuming $\nu =0.33$^{46} [labeled by $\nu $ in Fig. 1(d)] as well as constant $a= a 0$ [labeled by $ a 0$ in Fig. 1(d)], where $ a 0=0.389$ nm is the equilibrium lattice parameter of ncFIM (the ground state at $c/a=1$) from our DFT simulations. This is close to the experimental value $ a 0=0.3865$ nm and to an earlier DFT calculation, $ a 0=0.382$ nm.^{49} We note that our conclusion is independent of this choice: The energy minimum for the FIM $ A$ phase obtained at $c/a>1$ is more than 0.2 eV/f.u. higher than the energy minimum of FIM $ B$ at $c/a=0.98$, in agreement with earlier DFT studies,^{28,37} which suggest that the Mn–Mn direct AFM interaction might be stabilizing the FIM $ B$ structure. However, the ground state of the ncFIM phase ( $c/a=1$) is another 30 meV/f.u. lower than the energy minimum of FIM $ B$.

Therefore, it is conceivable that the ncFIM phase, energetically favorable for $c/a\u2208(0.99\u22121.0)$, coexists with the FIM $ B$ phase in films where the lattice mismatch with the substrate does not induce large tensile biaxial strain, $c/a<0.99$. This failure to induce strain can be due to (a) dislocations in the lattice structure at the interface,^{27,28} (b) nitrogen deficiency at the interface,^{37} or (c) relaxation due to greater film thickness. Areas of the film with less dislocations (better epitaxy) are then more likely to stabilize the tetragonal FIM $ B$ phase by the smaller $c/a$ ratio. The films relevant to our MOKE study show $c/a=0.989$;^{54} therefore, the ncFIM could coexist with the FIM $ B$ phase.

### A. MOKE spectra

Figure 2 presents the main result of the work. It shows Kerr rotation and Kerr ellipticity as a function of energy, $\omega \u2208(1\u22127)$ eV. (Our model does not include the intraband contribution, which dominates below 1 eV.) We calculated the spectra for several ratios $c/a\u2208(0.97\u22121.03)$, but the dependence on tetragonal distortion appears to be much smaller than the differences between the three FIM phases; therefore, we plot only spectra for $c/a=0.99$ and 1.01. The spectra in this figure include fine features as we use small Gaussian smearing, $\sigma =0.01$ eV, to treat the partial occupancies in k-space integration. This approach would correspond to experimental data measured in a high quality monocrystalline (epitaxial) film with low density of structural defects at low temperatures.

In order to interpret our spectra based on features of the band structure, we plot the projected DOS (PDOS) for all three phases in Fig. 3. We resolve PDOS only for Mn-3 $d$ orbitals as the other contributions are small and too far from the Fermi energy to play an important role in the visible magneto-optical response. Mn $ 1$, Mn $ 2$, and Mn $ 3$ occupy the $3c$ sites with Cartesian coordinates (0.0, 0.5, 0.5), (0.5, 0.0, 0.5), and (0.5, 0.5, 0.0), respectively, whereas Mn $ 4$ occupies the $1a$ site with coordinates (0, 0, 0) as shown in Fig. 1. We note that our PDOS for FIM $ B$ phase is in agreement with Fig. 7(a) of Ref. 37.

Figure 3(a) for FIM $ A$ shows PDOS with one dominant transition indicated by a gray arrow between a peak in occupied states of Mn $ 1 \u2212 3$ and a peak in excited states of Mn $ 4$. This transition described by energy difference, dE $\u22482$ eV, corresponds to the sharp peak in the magneto-optical response at $\omega \u22482$ eV. Figure 3(b) for FIM $ B$ shows PDOS with two dominant transitions indicated again by arrows described by dE $\u22482$ and 3 eV, which correspond to a dip and a peak in Kerr rotation, respectively.

Figure 3(c) for ncFIM shows PDOS lacking prominent peaks around 2 eV below the Fermi level, which are present in the case of FIM $ B$. The peak in PDOS of FIM $ A$ around 0.5 eV above Fermi energy is shifted to 1 eV in ncFIM. Such a band structure results in the absence of prominent peaks in the magneto-optical response at photon energies below 4 eV. We conclude that the predicted spectral features can be interpreted qualitatively based on transitions from Mn-3 $d$ orbitals on $3c$ sites to Mn-3 $d$ orbitals on site $1a$.

In order to compare our data to MOKE spectra measured by Sakaguchi *et al.,*^{54} we interpolate our curves plotted in Fig. 2 using the UnivariateSpline function from Python SciPy package with a smoothing parameter set to 0.003 on a linear grid with 200 points. In addition, we include analogous MOKE spectra for $U=2.2$ eV. (Please see Appendix B for this spectra without smoothing.)

The smeared spectra for $c/a=0.99$ are shown in Fig. 4. Our Kerr rotation and ellipticity can be compared directly to Figs. 5(a) and 5(b) of Ref. 54, respectively, where the boron content in Mn $ 4$N is zero.

First, we note that the amplitude of the measured spectra is approximately 0.1 $ \xb0$, an order of magnitude larger than in the related noncollinear Mn-based systems, such as Mn $ 3$NiN^{25} and Mn $ 3$Sn.^{61} The amplitude of the simulated spectra is only a factor of two larger, indicating high quality of the Mn $ 4$N film.

Second, the measured $\theta (\omega )$ is positive below 1.5 eV, has a dip around 2.2 eV, and returns to positive values above 3 eV, where the studied interval ends. Such a trend is observed only in data simulated for the FIM $ B$ phase with a corresponding dip at 2.1 eV (2.3 eV) for $U=0.7$ eV ( $U=2.2$ eV). The crossing points, $\theta (\omega )=0$, are at 1.4 and 2.6 eV (1.5 and 3 eV) for $U=0.7$ eV ( $U=2.2$ eV). We note that there is a shift to lower energies for $U=0.7$ eV and a perfect agreement with experiment for $U=2.2$ eV. A shift of optical and magneto-optical responses to higher energies is commonly achieved in DFT studied by increasing the Coulomb repulsion on sites with more localized states (here the Mn-3 $d$ states). Therefore, we simulated the spectra for a range of Hubbard parameters, $U$, from $U=0.2$ to $2.2$ eV. The upper limit is large in comparison with Mn $ 3$NiN^{25} and CuMnAs^{59} where $U=0.7$ and $U=1.7$ eV were used, respectively. $U=2.2$ eV does not significantly affect the local magnetic moments and MAE of FIM $ B$ as shown in Appendix A, but the properties of ncFIM become unrealistic, including the prediction of a ferrimagnetic compensation (zero saturation magnetization), which has never been observed experimentally. We believe that it is unlikely that the localization of Mn-3 $d$ electrons should be described by very different values of $U$ in different magnetic phases.

Alternatively, the crossing point at 2.6 eV for FIM $ B$ could be shifted to higher energy by assuming an admixture of ncFIM, which is negative throughout the energy interval. Such a superposition of spectra would also shift the crossing point at 1 eV to lower energies in contrast to the experiment. However, our predictions are less reliable below 1.5 eV as our model does not include the intraband contributions (the Drude peak).

We check the agreement in the case of Kerr ellipticity, $\eta (\omega )$, which is Kramers–Kronig-related to $\theta (\omega )$. Figure 5(b) of Ref. 54 shows a monotonous decrease of $\eta (\omega )$ to zero around 3 eV. As expected, FIM $ B$ is the only phase that shows such a trend in the simulated spectrum, but the crossing point is shifted to slightly lower energy. To conclude the comparison with Fig. 4, we caution that the smoothing parameter reflects the unknown density of structural defects and thermal excitations in the sample—we have to use a large value to obtain the smooth spectra observed experimentally, which turns the exact crossing-point values less reliable.

In conclusion, the comparison of our MOKE spectra to Ref. 54 strongly suggests the presence of FIM $ B$ in agreement with a recent NMR study,^{53} but it cannot confirm or rule out the presence of ncFIM. In order to attain a deeper understanding of the nontrivial magnetic structure giving rise to the MOKE spectra presented above, we proceed by comparing the simulated magneto-crystalline anisotropy and the saturation magnetization with experimental values. The film is measured in a magnetic field applied parallel and antiparallel to the [001] axis (out-of-plane), and the two spectra are subtracted to eliminate nonlinear magneto-optical effects.^{54} The film has to undergo a rotation of magnetic moments between the two measurements, i.e., overcoming a certain energy barrier. Sakaguchi *et al.*^{54} measured the anisotropy field, $ H k=1.5$ T, which is a typical value in Mn $ 4$N on MgO, but we do not know the saturation magnetization. We have to assume that the anisotropy follows the general trend shown in Table I (K $ u=0.1$ MJ/m $ 3$) so that the sample could be fully aligned with the applied field during MOKE measurement.

### B. Magnetocrystalline anisotropy

We calculate the total energies of magnetic structures shown in Figs. 1(a) and 1(c) coherently rotated to align the net magnetic moment, $ m n e t$, with the [001] axis (perp-to-plane), the [110] axis (in-plane), and the [00 $ 1 \xaf$] axis. In the case of FIM $ B$, the anisotropy within the (001) plane is weak (e.g., the difference between energies for $ m n e t\Vert [110]$ and $ m n e t\Vert [100]$ is below 0.03 meV/f.u. $\u22482%$ of the predicted PMA); therefore, we choose the rotation to the [110] axis, which goes through the easy axis of the ncFIM phase, $ m n e t\Vert [111]$.

In Fig. 5(a), the total energy is plotted as a function of angle $\varphi $ between the net magnetization and the [001] axis (perpendicular to the film). The insets show the orientation of the local moments of the FIM $ B$ phase for net magnetization pointing along [001], [110], and [00 $ 1 \xaf$]. This choice is relevant for the polar-MOKE experiment, where the sample is measured in a magnetic field applied parallel and antiparallel to the [001] axis.^{54} The film has to undergo a rotation of magnetic moments driven by the reversal of the perpendicular applied field between the two measurements. Therefore, it has to overcome the energy barrier of the in-plane orientation, $\varphi =\pi /2$, which is K $ u=1.4$ meV per formula unit or 3.78 MJ/m $ 3$ for the FIM $ B$ at $c/a=0.99$. This PMA is in perfect agreement with earlier calculations^{37} as shown in Table I. However, the value is an order of magnitude larger than experimental PMA $\u22480.1$ MJ/m $ 3$.

Isogami *et al.*^{37} speculate that this discrepancy is due to imperfect chemical ordering of nitrogen, which can lead to nitrogen deficient impurity phases, such as nano-crystals of pure Mn, or the formation of dislocations in the initial growth layer. More generally, the initial growth layer of the Mn $ 4$N film on the substrate with a large lattice mismatch is prone to dislocations, which degrade the PMA due to their weak crystallinity. On the other hand, the anisotropy, K $ u$, of FIM $ B$ is not predicted to depend strongly on biaxial strain;^{37} therefore, we cannot expect lower PMA for substrates of a low lattice mismatch with Mn $ 4$N. At the same time, the shape anisotropy competes with PMA, but it is negligible due to the low saturation magnetization. We note that experimental PMA is determined using the crossing point of saturated out-of-plane (easy axis) and in-plane (hard axis) hysteresis curves; therefore, magnetization switching facilitated by domain-wall propagation cannot explain the smaller experimental values either.

However, several films show a small but finite coercive field even for the in-plane M-H loop which is not consistent with purely uniaxial anisotropy of FIM $ B$ phase^{34,43,47} and speaks in favor of our hypothesis of ncFIM and FIM $ B$ coexistence. Moreover, the admixture of ncFIM is consistent with the conclusions of Wang *et al.*^{55} and Yasuda *et al.*^{53} based on measured anomalous Hall effect (AHE) and XMCD data, respectively.

Therefore, we also study the magnetocrystalline anisotropy of ncFIM. The energy barrier during the magnetization reversal at $c/a=0.99$ is smaller, K $ u=0.67$ eV/f.u. or 1.81 MJ/m $ 3$; therefore, an admixture of ncFIM could provide another mechanism explaining the smaller experimental K $ u$. However, the spin reorientation mechanism becomes much more complicated in the case of ncFIM. There are eight variants of this phase with the net magnetization pointing parallel or antiparallel to the four body-diagonals, in perfect analogy to Mn $ 3$NiN.^{17} A large applied field can, in principle, align the net magnetization with the [001] axis, but the rotation of the moments to the opposite field orientation can go through different energy minima and energy barriers depending on the local conditions in the film. We have carried out a DFT study of the total energy landscape as a function of coherent rotations of the four local moments.

We considered all rotations that belong to the $Pm 3 \xafm$ space group of the cubic perovskite lattice: twofold and fourfold rotations about the main axes, threefold rotations about the body diagonals, and twofold rotations about the side diagonals. Details about our findings will be summarized elsewhere. Here, we show an example of a rotation between two energy minima (from [111] to [ $ 1 \xaf$1 $ 1 \xaf$]), which incurs the lowest energy barrier, K $ u=0.67$ eV/f.u. mentioned above. This rotation is a simultaneous rotation by $\pi /2$ about the [010] axis, by $\pi /2$ about the [001] axis, and by $2\pi /3$ about [ $ 1 \xaf$1 $ 1 \xaf$] axis. (An intuitive simple rotation of magnetization from [111] to [11 $ 1 \xaf$] about the [ $ 1 \xaf$10] axis would not restore the ground state magnetic structure.) The dashed–dotted line in Fig. 5 consists of (i) a simple rotation about the [ $ 1 \xaf$10] axis from a state with net moment along [001] to the ground state along [111] denoted by $ \varphi 0=\pi /2\u2212arctan\u2061(1/ (2))$ or $ \varphi 0= 55.4 \xb0$ in the case of $c/a=1$ or $c/a=0.99$, respectively, (ii) the simultaneous rotation to another ground state along [ $ 1 \xaf$1 $ 1 \xaf$] denoted by $\pi \u2212 \varphi 0$), and (iii) a simple rotation about [110] to a state with net magnetization along [00 $ 1 \xaf$]. All four significant states are depicted as insets in Fig. 5(b). Notably, the structure in the first inset is of the same type (and has the same direction of magnetization) as the noncollinear structure proposed for Mn $ 4$N in Fig. 2(d) of Ref. 46.

In summary, Fig. 5 illustrates that a film in the ncFIM phase with a cubic lattice or with slight tetragonal distortion, $c/a\u2208(0.99,1)$, can be in a multi-domain state even at saturation if the field is applied perpendicular to the film and the magnetization reversal can follow different paths for each domain. The anisotropy, K $ u$, is smaller than in FIM $ B$ but still too large to achieve saturation when the field typically applied in the experiment ( $\u22481$ T) is along a hard axis, such as the [001] axis. For simplicity, in our MOKE simulation of ncFIM presented in Fig. 2, we assume that the applied field switches between two single-domain ground states with net moment along [111] and [ $ 1 \xaf 1 \xaf 1 \xaf$] whose local moments are linked by an inversion, in analogy to our earlier work on MOKE in Mn $ 3$NiN.^{25} We have to subtract the off-diagonal elements of the dielectric tensor calculated for these two states in order to eliminate non-linear magneto-optical effects. In the case of a collinear ferrimagnet or a ferromagnet, this inversion of all moments can be achieved by a coherent rotation of all moments around an axis perpendicular to all of the moments. Such a process corresponds to experimental magnetization switching. However, in ncFIM, there is no axis perpendicular to all four magnetic moments. The coherent rotation of ncFIM shown in Fig. 5(b) may describe an experimental switching scenario, but it does not invert all the moments and it inverts only the $x$ and $z$-component of the net moment. It is beyond the scope of this work to establish if the application of the field perpendicular to the film can achieve the full inversion of magnetic moments assumed in our theory or if it can be described by coherent rotations of the noncollinear moments. We caution that this ambiguity of the magnetization switching process compromises our ability to interpret the measured MOKE spectra.

### C. Saturation magnetization

Our hypothesis of an ncFIM admixture in a dominant FIM $ B$ phase has implications also for the observed saturation magnetization, M $ s$ (net moment); therefore, we include Fig. 5(b) to show the out-of-plane magnetization component, M $ z$. Isogami *et al.*^{37} predict M $ s=180$ mT for FIM $ B$ and observe 110 mT experimentally. They are able to attribute the discrepancy to a dead layer at the interface with a substrate, nitrogen deficiency, and top surface oxidation. Here, we suggest that the lower M $ s$ could be due to the admixture of ncFIM with M $ s=0.727 \mu b$/f.u. = 143 mT as shown in Table I, which is smaller than $ M s=1.223 \mu b$/f.u. = 240 mT that we predict for FIM $ B$.

More details about the dependence of local magnetic moments on Mn sites and the net moment on the Hubbard parameter $U$ for all three phases are presented in Appendix A. Here, we compare our moments calculated with $U=0.7$ eV to local magnetic moments deduced by Takei *et al.*^{41} based on neutron diffraction in powder samples. They obtained a net moment, $ m n e t e x p= m 1 a+3 m 3 c=1.14 \mu B$/f.u., from magnetometry and $ m d i f f e x p= m 1 a\u2212 m 3 c=4.75 \mu B$/f.u. from the magnetic scattering with mixed indices. Solving these two equations assuming collinear order with $ m 1 a$ antiparallel to $ m 3 c$ gives $ m 1 a=3.85 \mu B$/f.u. and $ m 3 c=\u22120.9 \mu B$/f.u. in disagreement with our results for FIM $ A$. Takei *et al.* assume FIM $ A$ with a cubic lattice where we get prohibitively high total energy as shown in Fig. 1(d). Later, a polarized neutron diffraction study by Fruchart *et al.*^{14} found a noncollinear variant of FIM $ A$, possibly the ncFIM phase where we get $ m n e t=0.727 \mu b$/f.u. and $ m 1 a=3.65 \mu B$/f.u. (on the Mn $ 4$ site), which is in reasonable agreement with Takei *et al.* Our value of $ m 3 c=2.8 \mu B$/f.u. (on sites Mn $ 1$, Mn $ 2$, Mn $ 3$) is much larger than the measured 0.9 $ \mu B$/f.u. We believe that this discrepancy is due to the assumption of collinearity in Ref. 41. Increasing $U$ leads to a higher discrepancy in the case of the simulated net moment, which vanishes at $U=1.7$ eV. Such magnetic compensation has not been observed in Mn $ 4$N, to our best knowledge; therefore, we prefer to explain our MOKE spectra by a combination of FIM $ B$ and ncFIM phases rather than FIM $ B$ alone with $U=2.2$ eV.

## IV. SUMMARY

We simulated the MOKE spectrum of strained Mn $ 4$N using DFT+U assuming three ferrimagnetic phases investigated by earlier theoretical studies. We compared our results to polar-MOKE spectra measured by Sakaguchi *et al.*^{54} in Mn $ 4$N films on an MgO substrate. We found that the key features of the simulated spectra are consistent with the measured spectrum in the case of the FIM $ B$ phase. The agreement of the simulated Kerr rotation could be further improved if a fraction of the ncFIM phase was added to the dominant FIM $ B$ phase. At the same time, the admixture of ncFIM could explain the lower PMA, the lower $ M s$, and the deviation of the magnetic easy axis from the film normal, all observed experimentally. We believe that our analysis will motivate further MOKE studies where the energy interval is wider and the applied field can be inverted along a chosen path. This could shed more light on the ncFIM phase preferred at lower tensile strains and enable spintronic devices with sub-nanosecond spin dynamics at room temperature.

## ACKNOWLEDGMENTS

We acknowledge fruitful discussions with Freya Johnson, Lesley F. Cohen, Martin Veis, Jakub Železný, and Zsolt Gercsi. This work was supported by the Ministry of Education, Youth and Sports of the Czech Republic through the e-INFRA CZ (ID: 90254).

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflicts to disclose.

### Author Contributions

**J. Zemen:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX A: MAGNETIC MOMENTS

In a ferrimagnetic system, it is useful to list the predicted magnetic moments to fully describe the magnetic structure. Moreover, we use the magnetic moments to check for reasonable values of the Hubbard parameter $U$ on the Mn-3 $d$ orbitals. Figure 6 shows the dependence of local magnetic moments on Mn sites and the net moment (saturation magnetization) on $U$ for $c/a=0.99$ and all three phases. The moment of Mn $ 4$ on the corner site, $ m 1 a$, and the net moment, $ m n e t e x p$, measured by Takei *et al.*^{41} in a powder sample are included as a reference (horizontal lines) in all three panels. As discussed in the main text, there is semiquantitative agreement with predicted $ m n e t$ and $ m 1 a$. (The other predicted local moments are much larger than the experimental reference most likely due to the inappropriate assumption of collinear phase FIM $ A$ in the experiment.) In case of ncFIM, $ m n e t$ ( $ m 1 a$) decreases (increases) with $U$, and we find the best overall agreement for $U=0.7$ eV, which we also identified in an earlier MOKE study of Mn $ 3$NiN.^{25}

The best agreement with the measured moments is observed in case FIM $ B$, which is most likely a coincidence as Takei *et al.* saw only two inequivalent magnetic sites ( $1a$ and $3c$) respecting the cubic symmetry of the lattice. However, FIM $ B$ has three inequivalent magnetic sites. We note that using our methodology, FIM $ A$ and FIM $ B$ cannot be simulated with a cubic lattice—the moments relax to ncFIM during the self-consistent DFT cycle or vanish when a constraint is applied. Therefore, we compare all the moments for $c/a=0.99$, but the tetragonal distortion $c/a\u2208(0.97,1.1)$ has only an insignificant impact on the local moment sizes. As expected based on the unfavorable total energy, FIM $ A$ shows the poorest agreement with experimental net moment $ m n e t e x p$.

### APPENDIX B: DEPENDENCE OF MOKE ON *U*

We calculated MOKE spectra for the FIM $ B$ and ncFIM phase for $c/a=0.99$ with a range of Hubbard parameter, $U\u2208(0.2,2.2)$ eV. Figure 7 shows a clear trend how the onsite repulsion lifts the unoccupied manganese $3d$-states further away from the Fermi level, resulting in a blueshift in the spectra. The dominant peak around 3 eV moves to higher energies with increasing $U$. At the same time, for higher $U$, a minor peak appears in Kerr rotation around 2.5 eV in disagreement with the measured spectra, but it is not large enough to appear in the smoothed spectra that we compare to the experiment.

Figure 8 shows an analogous set of spectral curves for the ncFIM phase. A less clear blueshift is visible due to the lack of more pronounced features in the spectrum. Two spectra are included for an ncFIM $ z$ state, which has the net moment aligned with the [001] axis as shown in Fig. 5. We explained in the main text that we do not expect the field applied perpendicular to the film to force the moment of the ncFIM domain into this hard direction due to the large anisotropy. If the applied field was large enough to achieve saturation, the Kerr rotation of the ncFIM $ z$ state could not be used to push the crossing-point of the combined (FIM $ B$+ncFIM $ z$) from 2.5 to 3 eV as seen in the experiment. However, the dip at 2.5 eV is broadly in agreement with the measured spectra. Finally, we note that ncFIM spectra with $U=2.2$ eV are highly speculative as the net moment is predicted antiparallel to Mn $ 4$, which is not consistent with experimental results and other theoretical studies.^{48}

To check the consistency of the dependence of MOKE on $U$ with the underlying electronic states, we provide a comparison of the projected DOS for $U=0.7$ and $2.2$ eV in Fig. 9. The data for $U=0.7$ eV were already presented in Fig. 3. The comparison reveals the expected general trend when all the occupied Mn-3 $d$ states shift to lower energies and an unoccupied state to higher energies. The transitions between them that correspond to peaks in the MOKE spectrum span a wider energy difference, resulting in the blueshift mentioned earlier.

### APPENDIX C: BAND STRUCTURE

To complete the study of the electronic structure of Mn $ 4$, we present the band structure for FIM $ A$ and FIM $ B$ with $c/a=0.99$ and $U=0.7$ and 2.2 eV. The spin–orbit coupling that needs to be included to simulate MOKE is not used here in order to be able to resolve spin-up and spin-down states. We do not plot the band structure of ncFIM as the noncollinear magnetic structure does not allow for comparison to the collinear phases on the same footing (spin quantization axis along [001]). Figure 10 compares the band structure for the FIM $ A$. One can observe how the bands are moving away from the Fermi level with the increased $U$. Figure 11 shows the same trend for the FIM $ B$ phase. Unfortunately, we have not been able to find plots of the band structure for Mn $ 4N$ in the literature; therefore, comparison with this level of detail is not possible.

### APPENDIX D: SYMMETRY ANALYSIS

^{25}we note that MOKE is an optical counterpart of the anomalous Hall effect (AHE). Both MOKE and the intrinsic contribution to anomalous Hall conductivity (AHC), $ \sigma \alpha \beta $, originate (within linear response theory) in an non-vanishing integral of Berry curvature $ \Omega n \alpha \beta (k)$ over the Brillouin zone,

^{25}

The presence of MOKE and AHE can be determined by analyzing the transformation properties of the Berry curvature under all symmetry operations of a particular magnetic space group. In Table II, we list the space groups of the three FIM phases of Mn $ 4$N (subject to tetragonal distortion) obtained by FINDSYM software.^{62,63} The last row of Table II presents the form of the AHC tensor in a linear response regime obtained using software Symmetr^{64} considering both sets of symmetry operations and the spin–orbit coupling. We note that both collinear FIM phases share the same form of an AHC tensor with one independent nonzero element, $ \sigma x y$, inducing the polar-MOKE.

. | ncFIM . | FIM_{A}, FIM_{B}
. |
---|---|---|

Space group | 123, P4/mmm | 123, P4/mmm |

Mag. space group | 12.62, C2′/m′ | 123.345, P4/mm′m′ |

AHC, σ_{αβ} | $ ( 0 \sigma x y \sigma x z \u2212 \sigma x y 0 \sigma x z \u2212 \sigma x z \u2212 \sigma x z 0 )$ | $ ( 0 \sigma x y 0 \u2212 \sigma x y 0 0 0 0 0 )$ |

. | ncFIM . | FIM_{A}, FIM_{B}
. |
---|---|---|

Space group | 123, P4/mmm | 123, P4/mmm |

Mag. space group | 12.62, C2′/m′ | 123.345, P4/mm′m′ |

AHC, σ_{αβ} | $ ( 0 \sigma x y \sigma x z \u2212 \sigma x y 0 \sigma x z \u2212 \sigma x z \u2212 \sigma x z 0 )$ | $ ( 0 \sigma x y 0 \u2212 \sigma x y 0 0 0 0 0 )$ |

The AHC tensor of ncFIM has two independent nonzero elements, $ \sigma x y$ and $ \sigma x z= \sigma y z$ as in case of strained Mn $ 3$NiN.^{25} Cubic Mn $ 4$N would have $ \sigma x y= \sigma x z= \sigma y z$. The listed forms of the AHC tensor are determined by the symmetry of the structure rather than the net magnetization so they would not change (AHE and MOKE would not vanish) even if the net magnetic moment vanished, i.e., if full compensation of the ferrimagnet was achieved, which is desirable when seeking ultrafast spintronic devices.^{31,49}

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