Dislocation structures, interfacing, and magnetism in the $L1$−$MnGa$ on $η$_⊥−$Mn3N2$ bilayer

The MnGa system includes a variety of magnetic phases depending on the composition and structure. A variety of polymorphs of MnGa exist. A few examples include ferrimagnetic tetragonal D0₂₂-Mn₃Ga, antiferromagnetic hexagonal D0₁₉-Mn₃Ga, and the phase of interest to this work - ferromagnetic L1₀-MnGa.^1,2 Recently, a strong interest in developing permanent magnets without the use of rare-earth elements has developed due to clean energy and national security demands.³ Mn-based alloys offer an exciting and diverse possibility to fulfill these demands through the variety of magnetic phases and high tunability of magnetic properties.^1,4 The L1₀-MnGa alloy has attracted attention as a candidate to fulfill this role as a rare-earth-free permanent magnet.⁵ 

Additionally, MnGa alloys attracted significant attention for use in nanoscale spintronic devices due to their perpendicular magnetic anisotropy (PMA) with a uniaxial anisotropy constant (⁠$Ku$⁠) as high as $20Merg/cm3$ and a high Curie point of $350°C$⁠.^4–7 A tailorable saturation magnetization (⁠$Ms$⁠) is important for creating noninteracting, but high signal-to-noise storage bits.⁴ Understanding what growth conditions and crystal quality impact the magnetic properties is integral to bringing these materials to maturity as devices or permanent magnets.

For our investigations, we focus on the ferromagnetic $L10−MnGa$⁠, a tetragonal system consisting of equal parts of Mn and Ga. When a high-quality crystal is prepared, a layered structure of alternating layers of Mn and Ga is formed along (001).⁴ 

The growth and magnetism of $L10−MnGa$ on semiconductors like GaN and GaAs are well documented with a good lattice matching for growth along the [111] direction.^4,8,9 By altering the concentration of Ga in these systems, multiphase MnGa crystals can develop with strong consequences on the magnetic properties of the film.¹⁰ Growth on insulators like MgO(001) and sapphire(111) has been studied previously with growth orientations of [100] or [110].^9,11,12 A review detailing the growth and magnetism of $L10−MnGa$ and related $D022$ on various substrates is given by Zhu et al.⁴ Despite the well-documented growth and magnetism of MnGa phases in thin films, the dislocation structures of these compounds remain unexplored with only a few studies reporting observations of dislocations as part of larger studies.

Zheng et al. investigated the impact of molecular beam epitaxy (MBE) growth parameters on the full-width at half-maximum (FWHM) in x-ray diffraction (XRD) patterns, which is affected by dislocation density of the material; although the researchers did not address this relationship directly, a buildup of edge dislocations was observed in atomic force microscopy (AFM).¹¹ Other studies uncovered grain structures in high-resolution transmission electron microscopy images of the $L10$ and $D022$ structures.^6,13 However, overall the literature lacks studies involving atomic observations of dislocations in MnGa compounds.

Magnetic bilayers utilizing exchange bias (EB) are of great technological interest.¹⁴ However, the EB phenomenon has yet to be fully understood.^15–17 A bilayer of ferromagnetic $L10−MnGa$ and antiferromagnetic (aFM) $η⊥−Mn3N2$ offers the possibility to explore unconventional EB (⁠$TC≤TN$⁠) effects above room temperature. The exploration of unconventional EB systems with the traditional paradigm of $TN≤TC$ being reversed looks at interesting temperature effects and even the persistence of EB into the paramagnetic state.^16,18 Experimentally, EB is highly dependent on the experimental growth conditions, the materials of choice, and physical conditions of the sample. Preparing novel systems for possible EB measurements first requires an understanding of the interfacial structure and the magnetism of the bilayer in question.

$Mn3N2$ is an interesting compound for potential spin device applications and has been explored in detail chemically, crystallographically, and magnetically, even down to the atomic spin structure of the terminating surfaces via spin-polarized scanning tunneling microscopy (STM) making it a good candidate for possible EB explorations.¹⁹ Zilske et al. and Meinert et al. demonstrated giant EB in MnN heterostructures, motivating further investigations of the $MnxNy$ system for EB experiments prepared under MBE conditions.^20,21

In this report, STM and diffraction experiments are combined with density functional theory (DFT) calculations to investigate the interplay between the interface and disorder on the magnetic properties of MBE grown bilayer of $L10−MnGa$ on aFM $η⊥−Mn3N2$⁠. We determine a significant impact of the dislocation density on the magnetic ordering of the solid, where a high dislocation density nearly halves the measured moment per unit cell (i.e., $Ms$⁠). The interfacing between $L10−MnGa$ and $η⊥−Mn3N2$ is explored computationally leading to the understanding of the underlying mechanism between crystallographic disorder of the $L10$ layers and the magnetic ordering of the MnGa film. We also observe edge and screw type dislocations terminating on the surface via atomically resolved STM imaging and relate these to the interfacing. These findings are important for the tailoring of magnetic properties for spintronic devices with a low $Ms$ or strong rare-earth-free magnets by uncovering some of the interplay between disorder and magnetism.

The samples are grown by MBE in an ultrahigh vacuum (UHV) system with a base pressure of $1.0×10−10Torr$⁠. Epitaxial development during crystal formation is monitored in real-time with a 20 keV electron beam from a reflection high-energy electron diffraction system provided by Staib instruments. A custom-built room temperature STM and 5 keV Auger electron Spectrometer (AES) from Staib Instruments are operated in a separate UHV chamber that is coupled to the MBE growth chamber, allowing for in situ growth and analyses under UHV conditions. The STM uses electrochemically etched W tips for scanning.²² Conductance mapping is performed using a lock-in amplifier to measure the first harmonic with a modulation bias of $∼20mV$ and a frequency of $∼1500Hz$⁠. The MgO(001) substrate is annealed at $1100±30°C$ in a $2.0×10−5Torr$ N plasma until the RHEED pattern is streaky indicating a smooth surface, as seen in Fig. 1(a). Deposition of $η⊥−Mn3N2$ follows the work of Yang et al. with a substrate temperature of $400±30°C$⁠, a $2.0×10−5Torr$ N plasma, and an Mn:N flux ratio of $0.40±0.04$⁠.²³ The $η⊥−Mn3N2$ deposition conditions are maintained until a $50±5nm$ film grows. Then, the N plasma and Mn flux are terminated simultaneously, and the substrate cools to $325±30°C$ for MnGa deposition.

Figure 1(b) shows the streaky RHEED pattern of the $50±5nm$ $η⊥−Mn3N2$⁠, indicating that the prepared film is atomically smooth. The substrate temperate is maintained at $325±30°C$ for MnGa deposition with a flux ratio of $Mn:Ga=1.0±0.1$ until the $73±5nm$ thick film forms. The Mn and Ga fluxes are then terminated, and the sample cools to room temperature prior to transfer in UHV to the STM or AES for analysis.

Ex situ analyses are carried out with a Bragg-Brentano $θ−2θ$ XRD system from Rigaku with a Cu $Kα$ x-ray source. Bulk compositional measurements are taken with a Rutherford backscattering spectrometer (RBS) system using a 4.5-MV tandem accelerator with measurements made using 2.2 and 3.035 MeV alpha beams. Vibrating sample magnetometry (VSM) measurements are carried out at room temperature with a max field of $±16kOe$⁠. The specimens are cut such that the center (⁠$∼0.5×∼0.5cm2$⁠) of the as-grown films is used in VSM measurements.

The error in the sample temperature during MBE growth is determined from calibration curves of a thermocouple mounted behind the substrate and a pyrometer focused on the front of the sample. The fluxes are determined using a quartz thickness monitor with errors in the fluxes determined from the standard deviation of several flux measurements for a particular temperature. The N flux is estimated using the crossover point between Ga- and N-rich growth conditions for a GaN film.²⁴ 

The DFT calculations are preformed using spin-polarized first principles and are carried out with the PWscf code of the Quantum ESPRESSO package.²⁵ The electronic states are expanded in delocalized plane waves with a kinetic energy cutoff of 30 Ry. The charge density expansion is truncated at 240 Ry. The nonclassical electronic interactions are treated with the generalized gradient approximation, as implemented in the Perdew–Burke–Ernzerhof parametrization.²⁶ Core electrons are treated using ultrasoft pseudopotentials.²⁷ To evaluate the Brillouin zone integrations, we use a k-point mesh of $6×6×1$⁠.²⁸ Every interface model is fully relaxed until the force and energy criteria of 0.001 Ry/a.u. and 0.0001 Ry are accomplished, respectively. Such criteria are sufficient to account for different magnetic alignments at the interface.

In Fig. 2(a), a plot of the in-plane lattice spacing versus thickness for the MnGa film is shown, determined from RHEED spacings. The bulk in-plane lattice constant of $L10−MnGa$ is $3.88Å$⁠, while MnGa initially nucleates commensurate to the $η⊥−Mn3N2$ with a larger than bulk in-plane lattice constant until a certain critical thickness where the MnGa lattice suddenly contracts to reduce the lattice strain resulting in an in-plane lattice spacing closer to the bulk. We observe the sudden contraction, or critical thickness, at $1.05±0.65nm$ as seen by the step in Fig. 2(a). Coincidentally, this marks the separation between the commensurate layer growth mode and the beginning of the island + layer mode. The width of the transition region in the step is used as the error in the critical thickness. After the critical thickness, the MnGa relaxes to a slightly larger than bulk in-plane spacing by dislocation incorporation into the film.²⁹ Reflection high energy diffraction measurements probe the topmost atomic layers. As such, a further relaxed in-plane lattice is expected for the complete film. The RHEED measurements are an excellent method to probe the critical thickness transition as this typically occurs after only several atomic layers.

The evolution of the [100] RHEED direction from commensurate layers to the Stranski-Krastanov (layer + island growth) and finally to layer-by-layer growth is presented as a series of [100] patterns shown in Fig. 2(b), as taken along the data in Fig. 2(a).³⁰ The transition from the layer + island to strictly the layer growth happens via the islands growing larger laterally and vertically until merging together. As the film thickens, less RHEED beam is transmitted through the islands, resulting in fainter fractional dotted patterns. The transition from the layer + island $→$ the layer growth mode is continuous/smooth, unlike the transition from the commensurate $→$ the layer + island mode, which is stepwise/abrupt. This qualitative evolution of the RHEED pattern indicates that the $L10−MnGa$ growth undergoes a transition from layer + island growth to a strictly layer growth after a thickness of $55.0±5.0nm$⁠. Figures 2(c) and 2(d) show the RHEED patterns of the $73.0±5.0nm$ MnGa film along the [100] and [110] MgO directions at the growth temperature. From the primary RHEED streaks along the [100] and [110] directions, the ratio of the streak spacing of [100] to [110] directions is equal to $2$⁠, indicating that the MnGa grows without rotation with respect to the $Mn3N2$ substrate resulting in the epitaxial relation [100] MnGa $∥$ [100] $Mn3N2$⁠.

The $η⊥−Mn3N2$ surface is morphologically made up of nanopyramid shapes with three unique chemical terraces.^10,19,31 For growth along the [001] direction, MnGa has two possible chemical layers, either an Mn or a Ga layer. In order to understand the interfacing between MnGa and $Mn3N2$⁠, theory is needed to untangle how the MnGa is stable atop of the $η⊥−Mn3N2$⁠. The three possible different $η⊥−Mn3N2$ terminations are an Mn-MnN-Mn-MnN terminated surface (model A1), an Mn-MnN-MnN terminated surface (model B), and an Mn-MnN terminated structure (model C).³¹ Models A1, B, and C are composed of 9, 8, and 7 monolayers, respectively. The MnGa films which are deposited on top of each substrate (A1, B, and C) are (x-MnGa-y, in which, $x=interface$ layer, and $y=surface$ layer) Mn-MnGa-Mn is labeled model 1, Ga-MnGa-Ga which is model 2, Ga-MnGa-Mn is model 3, and Mn-MnGa-Ga which is model 4. Therefore, the interfaces are combinations of A1, B, and C with x-MnGa-y (e.g., A1-1 is a bilayer between Mn-MnN-Mn-MnN and Mn-MnGa-Mn). Also, we are taking advantage of the inversion symmetry, so then, two equivalent surface terminations and two interfaces are present.³² Shown in Fig. 3(d) the MnGa preferentially begins growth with Mn layers atop of the nanopyramids and then begins to order in alternating Ga and Mn layers.

In order to describe the thermodynamic stability of the interfaces, we use the interface formation energy formalism.^32,33 Such formalism has been developed in detail for magnetic interfaces in the work by Guerrero-Sanchez and Takeuchi.³² The interface energy [$Λ(MnGa)/(Mn3N2)$] as a function of $Δμ(1)=3μMn−2μN$ and $Δμ(2)=μMn−μGa$ can be written as

where $ΔEInt$ is the total energy of the interface, $EMn3N2Slab$ and $EMnGaSlab$ are the total energies of the $Mn3N2$ and MnGa surfaces respectively, A is the surface area in each case, $μMn3N2bulk=3μMnbulk+2μNbulk$ and $μMnGabulk=μMnbulk+μGabulk$ are the chemical potentials of the bulk $Mn3N2$ and MnGa respectively, and finally, $ni$ is the number of atoms of the $ith$ species (Mn, N, and Ga). The $Λ(MnGa)/(Mn3N2)$ can be plotted only in the following limits: $−ΔHfMn3N2≤Δμ(1)≤ΔHfMn3N2$ and $−ΔHfMnGa≤Δμ(2)≤ΔHfMnGa$⁠, where $ΔHfxy$ is the formation enthalpy of each compound (x and y). The upper and lower limits correspond to Mn-rich and N-rich or Ga-rich conditions, respectively.

As mentioned before, the interface models are constructed by combining the substrates (A1, B, and C) with all the possible films (x-MnGa-y). For example, A1-1 is an interface formed by an A1 substrate and the Mn-MnGa-Mn film; the film is Mn initiated, has alternating Mn and Ga layers, and then is Mn terminated. After fully relaxing all the possible combinations between substrate-film models, we have found only three stable interfaces labeled A1-4, B-4, and C-4, as shown in Fig. 3(b). They are the combinations of A1, B, and C substrates with film 4 (the Mn-MnGa-Ga film). The A1-4 interface is stable for very Mn-rich conditions of $Δμ(1)$⁠, while the C-4 and B-4 interfaces are stable for progressively lower Mn chemical potential as seen in Fig. 3(b). It is important to mention that all stable models are favorable for all ranges of $Δμ(2)$ if we only consider $1×1$ surface structures, see Fig. 3(b).

As previously established in detail, $L10−MnGa$(001) has two possible terminations, a Ga-terminated $1×1$ and a $1×2$ with Mn substituting Ga at 1/4 of the surface Ga sites.^32,34 To verify that the $1×2$ Mn/Ga substituted $1×1$ Ga surface is stable, we compare the energy of the C-4 interface having a terminating $1×2$ surface structure to that of a simple $1×1$ Ga termination. After full relaxation of the two surfaces, we see that the C-4 $1×2$ model is more stable than the C-4 $1×1$ model for Mn-rich conditions of $Δμ(2)$⁠, as seen in Fig. 3(c).

These results demonstrate that MnGa can directly grow on $Mn3N2$⁠, where the interfaces are formed by an MnN layer of the substrate and an Mn layer of the film, thus solidifying the multifaceted epitaxial relationship and formation of this bilayer system.

The X-ray diffraction spectrum shows the MnGa 001 peak at $2θ=23.98±0.01°$⁠, the MnGa 002 peak at $49.01±0.01°$⁠, and the MnGa 004 peak at $112.29±0.05°$⁠, corresponding to the 001 direction with an out-of-plane lattice constant (c) equal to $3.71±0.01Å$ [Fig. 4(a)]. The error in c is determined from the error in the peak positions in $2θ$ of the 001 family of peaks using Braggs law. Rutherford backscattering confirms a stoichiometric phase with a compositional ratio $Mn:Ga=1.01±0.15$⁠, with the error in the composition determined from the goodness of the fit to the Rutherford backscattering spectrometer (RBS) intensity-energy profile. We rule out $D022−Mn3Ga$ since (1) the composition of this phase (Mn:Ga of 3:1) deviates too much from our measured value and (2) the observed second order diffraction peak falls within the $2θ$ range for $L10$ (49.0–49.6 °), not the $2θ$ range for $D022$ (50.9–51.4 °).

The observation of 002 and 004 higher order reflections is indicative of the layering superstructure of alternating Mn and Ga layers in a pure phase growth of $L10−MnGa$⁠.⁵ However, it should be noted that growth of a stoichiometric MnGa film is possible with a so-called disordered phase lacking the alternating Mn and Ga planes of atoms while retaining the overall tetragonal structure.^5,29,35,36 In our XRD spectrum, we lack the observation of the 003 peak which suggests that we do not have a highly ordered layered structure, but instead a mixture between well-ordered layers and some layer disordering.⁵ 

Our calculations of the interface formation guide our understanding of the layer disordering. The $η⊥−Mn3N2$ forms a nanopyramidal morphology with a terrace width of approximately 10–20 nm and a nominal step height of $2.0Å$⁠.³⁷ In Fig. 5(a), the three possible $Mn3N2$ terraces are shown along the [110] with alternating Mn and Ga layers atop of it in accordance with our theory calculations. Going across a step edge of A1 to B, B to C, and C to A1, it is clear that the second layer (Ga) from the A1 surface is leveled with the third (Mn) layer from the B surface and with the fourth (Ga) layer from C and so on.

In the $L10−MnGa$ structure, the Ga layer is geometrically the same as the Mn layer but with an apparent lateral shift of $12120$ relative to the Mn layer. When stepping from one $Mn3N2$ terrace down (or up) to the next, the relative positions of the N atoms in the MnN terminating layer can be viewed as changing from corner atoms to edge-centered atoms, as shown in Fig. 5(b). This change in the relative positions of N atoms going across an $Mn3N2$ step is crucial, as this causes an effective $12120$ shift relative to the adjacent terraces. The relative $12120$ shift in N positions allows for an Mn layer to geometrically match with a Ga layer across an $Mn3N2$ step edge as shown in Fig. 5(c) causing only a chemical disordering.

A Williamson–Hall analysis of the 001, 002, and 004 peaks, consisting of plotting a factor $βcos⁡(θ)$ versus $4sin⁡(θ)$⁠, is presented in Fig. 4(b), where $β$ is the integral broadening of the diffraction peak and $θ$ is Bragg angle.³⁸ By plotting $βcos⁡(θ)$ versus $4sin⁡(θ)$⁠, the slope is proportional to the out-of-plane coherence length $d⊥$ and the intercept is the residual strain.³⁹ The Williamson–Hall analysis reveals tall columnar grains with an out-of-plane coherence length of $d⊥=67±17nm$ and a residual strain $ϵ⊥=2.38±0.26(×10−3)$⁠.^38,40 Errors in the $d⊥$ and strain are determined from the errors of linear regression. It is interesting to notice that the $d⊥$ is in good agreement with the thickness of the deposited film. This indicates that the column of MnGa grown atop a terrace of $Mn3N2$ is a relatively unperturbed crystallite with ordered layering and one crystallographic orientation.

Zheng et al. investigated the out-of-plane lattice parameter for $L10−MnGa$ as a function of film thickness over 5–100 nm, with a corresponding variation of 3.32–3.71 Å in c.¹¹ This thickness trend is in good agreement with our out-of-plane spacing of $3.71±0.01Å$⁠.

Displayed in Fig. 6(a) is a large-scale STM image showing the surface morphology of the MnGa surface. Atomically smooth, flat terraces dominate the surface which is consistent with the layer-by-layer growth suggested by the streaky RHEED pattern during the growth, particularly during the final stages of growth. Hundreds of screw dislocations are evident in Fig. 6(a) manifested as small folds on the surface where several selected screw dislocations are indicated with a red-dashed circle in Fig. 6(a). A high-resolution image corresponding to a single screw dislocation is shown in the inset of Fig. 6(c).

Screw dislocations have been observed in other $L10$-ordered films. For the case of TiAl, Feng et al. and Jiao et al. showed that the $1/2⟨110$] screw dislocation was energetically most favorable, with screw dislocations able to cross slip into the (001) plane.^41,42 However, unlike TiAl, the elastic constants of MnGa crystals are unknown which leaves dislocations in $L10−MnGa$ based only on the empirical evidence, as we are unable to comment on the energetics of [001] screw dislocations. For consistency with other $L10$ dislocation analyses, the Burgers vector notation $⟨hkl]$ means that permutations of h and k are allowed, but not l as indicated by the two different brackets.³⁵ 

The topographical AFM images shown in the work of Zheng et al. also revealed screw dislocations along [001], although they did not comment on these features in their report.¹¹ They observed a buckled surface with screw dislocations over a thickness range of 30–70 nm. At 100 nm thickness, their surface was no longer buckled and qualitatively agreed with our STM observations of surface morphology.¹¹ However, our surface exhibits smaller terrace widths, far more screw dislocations, and without the pile-up of $1/2⟨110]$ edge dislocations, which appear as steps of $∼4nm$ height extending across the surface.

To understand the distribution of screw dislocations, as well as estimate the average distance between neighboring dislocations, a radial distribution plot (RDP) of screw dislocations is generated and shown in Fig. 6(b). To create the RDP, we follow the method developed by Crocker et al. by counting the number of screw dislocations in a loop with thickness dR, at a distance R away from a reference screw dislocation;⁴³ we sum over X number of reference screw dislocations with an $Rmax=100nm$ and a $dR=10nm$⁠.

The radial distribution plot of screw dislocations reveals a short range order with an average nearest neighbor distance of 15 nm. The short range order of screw dislocations extends to approximately 40 nm, but larger than this, the random distribution of screws becomes equally likely. Error bars of the RDP are propagated from error in the loop area and error in the areal density of screw dislocations.

From the average separation between dislocations, an estimation of the total dislocation density (⁠$ρ$⁠) can be made via the formula:

where d is the average distance between nearest dislocations and the factor of 2 comes from the relation of a surface dislocation density to a total bulk dislocation density.²⁹ As seen in Fig. 6(b), the RDP for screw dislocations has the first maximum at 15 nm which is used as an estimate for d which results in $ρ=8.88±2.9(×1011)/cm2$ with an estimated 33% error in $ρ$⁠. The errors in $ρ$ are known to be large, where a best estimate in precision is 33%.

The radial distribution of screw dislocations and Williamson–Hall analysis allow us to construct a model of the crystallite structure of the film, consisting of a collection of tall and thin $(15×15×67nm3)$ crystallites with a 001 orientation. In conjunction, the RHEED, the Williamson–Hall, and RDP analyses verify the incorporation of dislocations to relax the misfit between MnGa and $Mn3N2$⁠. Although the dislocation density is high $∼1012/cm2$⁠, so is the lattice misfit of 8%.⁴⁰ Furthermore, the results are consistent with the findings of Zheng et al. who found comparable FWHM values in XRD spectra of their MBE grown MnGa films.¹¹ 

The MnGa(001) surface structures have been studied in detail experimentally and theoretically in our previous work.^32,34 To recap the results of our previous studies as they are pertinent here, there are two lowest energy surface formations for the $L10−MnGa$(001) surface: an ideal $1×1$ Ga termination and an Mn/Ga substitution in the $1×1$ Ga termination forming a $1×2$ row structure under very slightly Mn-rich conditions. Models of the two surface structures are shown in Fig. 7(a).

This alternating stacking of Mn and Ga layers creates a puzzle as one would expect an alternating composition across monolayer-height atomic steps and to observe terrace-specific surface structures and terrace-specific composition. However, we do not find evidence for this experimentally in the STM images. Theoretically, the Mn-terminations of the $L10−MnGa$(001) surface are highly unfavorable energetically compared to the Ga terminations.^32,34 The surface terminates in the lowest energy structure (Ga-terminations) via the screw dislocations. With such a high density of screw dislocations, it is possible for the surface to maintain the same Ga-termination while containing multiple single height atomic steps. It is even possible to traverse a 600 nm path across the surface [see Fig. 6(a)] without ever crossing a single step due to the high density of screw dislocations creating an Escheresque surface.⁴⁴ 

An STM conductance mapping is sensitive to the electronic density of states (DOS) which is dependent on the surface composition.^19,45 Checking the DOS across a step edge can be an additional verification of any compositional variations. Conductance mapping is performed utilizing a lock-in amplifier technique.⁴⁵ We found no changes in the DOS across the MnGa terrace steps, revealing no step-alternating compositional changes as shown in Fig. 6(d).

Edge dislocations are visible in the STM images as kinks in the $1×2$ surface structure as shown in Fig. 7(b). Edge dislocations similar to those observed in face-centered crystals are expected in $L10$-ordered materials which have c:a $∼$1:1 which is fairly close for the MnGa case $(c:a=0.93)$⁠.^35,46 The energy of an edge dislocation is proportional to the square of the magnitude of the Burgers vector. With the $a/2⟨110⟩$ being the shortest lattice translation vector, one expects $L10−MnGa$ films to be populated with $a/2⟨110]$ edge dislocations.^29,35 Indeed, Zheng et al. observed via AFM a cross-hatch network in 100 nm thick $L10−Mn1.4Ga$ films which they attributed to a pile-up of $1/2⟨110]$ edge dislocations.

We observe atomic-level edge dislocations of type $1/2⟨110]$ as an angled kink in the row structure extending across dozens of rows and even monoatomic step edges, as indicated by the arrows in Fig. 7(b). To create an atomic surface model of the $1/2⟨110]$ type dislocation, one has to keep in mind that we are observing the effect of the edge dislocation perturbing the $1×2$ Mn/Ga substituted surface structure. In part 1 of Fig. 7(c), an atomic model of a single row of Mn/Ga substitutions is displayed, where the Mn atoms are represented in red and Ga in blue. If the MnGa crystal is displaced by a $a/2⟨110]$ edge dislocation through the row (as indicated by the dashed line) then the dislocated Mn atoms (orange colored) would occupy a nonequivalent position thus breaking symmetry of the $1×2$ Mn/Ga substituted structure, as seen in part 2 of Fig. 7(c). As such, the position of the Mn substitution is located at the next-nearest equivalent position that would maintain the $1×2$ surface structure. The incorporation into the next-nearest equivalent positions gives an additional shift in the position of the Mn-substituted atom, as seen in part 3 of Fig. 7(c). Therefore, the kink in the row structure is the fingerprint of the $1/2⟨110]$ type edge dislocation. In Fig. 7(d), a derivative mode STM image is shown with our model of the $1×2$ perturbed by a $1/2⟨110]$ edge dislocated surface overlayed along the rows containing a kink, giving good agreement with our experiment.

We determine that some of these kinks are due to bulk edge dislocations because the edge dislocation extends across a step edge as indicated in Fig. 7(b) by the white arrows. If these kinks were purely surface effects, one would expect the dislocation to terminate an atomic step edge.

An edge dislocation in the $⟨100]$ direction is observed as a distortion of the row structure along the length of the row, as seen in Fig. 7(b). The $⟨100]$ edge dislocation is observed much less frequently than the $1/2⟨110]$ dislocation; this is expected since the $⟨100]$ dislocation has twice the energy of the lowest energy $1/2⟨110]$ edge dislocation.

In addition to edge and screw dislocations, the surface is densely packed with small pits, where edge dislocations can be seen to terminate or connect as shown in Fig. 7(b). Zheng et al. found the formation of large pits (hundreds of nm in diameter) at a film thickness of $∼30nm$ in $L10−Mn1.4Ga$ films and also observed a competition between pit and dislocation formation as a function of thickness.¹¹ For our $73±5nm$ films, $∼5nm$ diameter pits were observed on the surface over a wide range. A large number of pits are observable as tiny black dots in the large-scale STM image, as seen in Fig. 6(a).

The pit diameter is significantly smaller than Zheng et al. reported for their films investigated via AFM. On the other hand, Zheng et al. observed that the pit formation diminished as the films thickened above 30 nm and gave rise to an accumulation of edge dislocations in thick films of $∼100nm$⁠.¹¹ Given that our films are $73±5nm$ in thickness, an intermediate combination of edge dislocations and pits would be expected.¹¹ 

To understand the ordering of the pit formation, an RDP of the pits is generated from the STM image shown in Fig. 6(a). From this RDP, we determined that the pits develop across the surface of the sample randomly, as demonstrated by the essentially flat line with a value of 1 (to within error) across all measured distances. This indicates that there is no short or long range order, with a pit equally likely to be encountered anywhere on the surface, as seen in Fig. 6(b).

A magnetic force microscopy (MFM) image showing the domain structure of the as-grown bilayer is presented in Fig. 8(a), where the stripe-bubble pattern indicates an out-of-plane anisotropic behavior.⁴⁷ Observed in various $L10$-ordered magnetic films is a thickness dependence of the hysteresis shape. In Fig. 8(b), the out-of-plane hysteresis as measured by VSM has an interesting sheared and pinched shape which is known to occur for sufficiently thick $L10$ ferromagnetic films.^48,49 The sheared-pinched feature in the hysteresis is understood from micromagnetic theory to be related to the competing formation of bubble and stripe domains.^48–52 Initially, when removing the saturating magnetic field, the creation of antiparallel bubble domains is energetically more favorable, until the bubble domains coalesce and form a network of stripes in the demagnetized state.^48,52 We observed a network of stripe domains, with some lingering bubble domains, as seen in Fig. 8(a) as anticipated from the work of Cape.^48,52 The transition from bubble to stripe domain patterns in PMA films has been well documented for the multilayer system Co/Pt, where the domain patterns are mapped to the hysteresis.⁵² 

From the magnetic hysteresis loops [see Fig. 8(b)], the magnetic contributions per atom in a given unit cell is calculated from the M$s$ and the unit cell volume (⁠$Vunitcell$⁠) via the relation:

Kim et al. investigated the impact of uniform lattice strain on the magnetic moment per unit cell in the $δ−Mnx−Ga1−x$ structures, resulting in a theoretical maximum of $9.88μb$ per unit cell (or $2.47μb$ per Mn atom) for the stoichiometric case.⁵³ Note: Kim et al. used a conventional unit cell of the $L10−MnGa$ (two primitive stoichiometric unit cells stacked along the c-axis) for direct comparison to the off stoichiometric $δ−Mnx−Ga1−x$ cases. They found that strains present in the lattice can lower the theoretical limit from $9.88μb$ per conventional unit cell to $9.77μb$ with reasonably linear trends of $munitcell$ versus strain.⁵³ 

Derived from the work of Kim et al. one arrives at an empirical equation:

where $mideal$ is the theoretical limit of the moment per unit cell set by the stoichiometry and $Δmstrain=mstrain−mideal$ with $mstrain$ being the theoretically calculated moment per unit cell under a given lattice distortion.⁵³ 

For the MnGa films investigated in this work, an $munitcell$ was determined to be $5.7±0.2μb$ from our hysteresis measurements, substantially lower than the theoretical limit of $9.88μb$⁠. Even when accounting for the lattice distortion, an $mpredicted=9.74μb$ is expected based on a linear extrapolation of measured moment versus lattice volume.

However, $5.7μb$ is significantly lower than the strain-accounted case.⁵³ Zheng et al. investigated the effects of growth parameters (annealing temperature and thickness) on MBE grown $L10−Mn1.4Ga$ films.¹¹ Using trends in the work of Zheng et al. for FWHM of XRD 002 peaks and $Ms$ versus annealing temperature, we make a trend for $Δmρ$ versus $ρ$⁠,¹¹ where $Δmρ=mρ−mideal$ and $mρ$ is the moment per unit cell at a given $ρ$ (where $ρ$ is the film dislocation density). We determine this trend for $Δmρ$ versus $ρ$ by processing the M$s$ and FWHM data as follows: first, we calculate $munitcell$ from M$s$ via Eq. (3), giving us a plot of $munitcell$ versus annealing temperature. Then, we use the Debye–Scherrer formula to estimate crystallite size from FWHM, and finally, we convert crystallite size to $ρ$ following Bindu et al. resulting in a $ρ$ versus annealing temperature plot.^39,54 Since both $munitcell$ and $ρ$ are plotted versus same annealing temperatures, we can finally plot $munitcell$ versus $ρ$⁠. Finally, with $mρ$ versus $ρ$ we take it one step further and plot $Δmρ$ versus $ρ$⁠. The final plot is roughly a linear relation, see Fig. 8(c). This result leads one to an additional empirical term in Eq. (4) resulting in a final form:

We can then obtain $Δmρ$ from the straight line fit in Fig. 8(c) using the estimated $ρ=8.88±2.9(×1011)/cm2$ for our film, which results in $Δmρ=−3.92±0.55μb$⁠. Combining this with the $Δmstrain=−0.15±0.16μb$ it is found to be $mpredicted=5.82±0.57μb$⁠, which is in very good agreement with the measured value of $5.7±0.2μb$⁠.

How the quality of the crystal can impact the magnetism of the film can be understood by realizing that the Mn magnetic moment in the $L10$ structure depends significantly on what atomic sites the Mn occupy and by having the alternating layers of Mn and Ga. If a given sample does not have high-quality layering, but is still stoichiometric, the Mn atoms which adopted irregular $L10$ sites can couple antiferromagnetically, acquiring a ferrimagnetic behavior. We know from the XRD spectrum that we have reasonably high quality layering, but the lack of the 003 peak suggests the lateral layer discontinuity presented in Fig. 5.

Note that although the films in the work of Zheng et al. were not stoichiometric, we assumed that the change in moment versus $ρ$ is sufficiently similar to our stoichiometric case. This assumption is motived by the work of Kim et al. who showed for the different concentrations of Mn in the $δ−Mnx−Ga1−x$ structure that the change in moment versus strain was the same despite different compositions.

Additionally, the role of oxidation could potentially play a role in altering the magnetization. However, reports show that capped samples and uncapped samples are comparable,⁴ and even in the work of Mandru et al. uncapped ultrathin samples showed no evidence of oxygen degradation, having very large moment per Mn atom of $4.84±0.52μb/Mn$⁠.¹⁰ 

Some MnGa films are known to exhibit PMA.⁴ To quantify the PMA in our MnGa films, the uniaxial anisotropy constant $Ku$ is determined from the area between the one side (e.g., the right side) of the out-of-plane and in-plane hysteresis loops plus the shape anistropy term via the approximate form:¹¹ 

where $Heff$ can be estimated at the crossing point of the out-of-plane and in-plane loops.¹¹ In Fig. 8(b), the out-of-plane and in-plane VSM loops are plotted. The saturated magnetization $Ms$ is determined from the out-of-plane loop and is measured from Fig. 8(b) to be $435±1emu/cc$⁠. $Heff$ is estimated to be $18721.0±3Oe$⁠. With the measured $Ms$ and $Heff$⁠, $Ku$ is calculated to be $5.27±0.01Merg/cc$⁠; although large, this value is only about 1/3 – 1/4 of values reported for 73 nm thick L1₀-MnGa.^5,11 This reduced Ku may be due to the crystalline defects/ dislocations and layer intermixing present in our film.

The angle of the easy axis (⁠$Θ$⁠) relative to the normal can be computed from the remnant magnetizations of the out-of-plane (⁠$Mr⊥=34.1emu/cm3$⁠) and in-plane (⁠$Mr∥=25.6emu/cm3$⁠) hysteresis loops via the formula:

which results in $Θ=36.8±0.1°$⁠. An out-of-plane easy axis is expected for MnGa thin films. The squareness and coercivity of the out-of-plane hysteresis loops depend on the thickness of the film.^4,55 The coercivity decreases dramatically as a function of increasing thickness and the $Mr⊥$ no longer equals the $Ms$ for films thicker than $∼60nm$⁠, as also seen in our out-of-plane hysteresis in Fig. 8(b), thereby tilting the easy axis slightly in-plane.

Additionally, $η⊥−Mn3N2$ is an antiferromagnet with a Neél point $TN=624°C$⁠, while $L10−MnGa$ has a $TC=354°C$⁠, well below the Neél point creating the possibility of an unconventional EB system.^16–18 Strikingly the in-plane loop is found to be shifted to right by $66±15Oe$ with an in-plane coercivity of $Hc=760±25Oe$⁠, as shown in Fig. 8(d). This is a strong experimental indication of possible EB in qualitative agreement with our interface calculations which found an antiferromagnetic coupling between $Mn3N2$ and MnGa. We compute $Hc$ and $He$ from the left and right x-intercepts of the hysteresis (⁠$Hc,L=−694Oe$⁠, $HC,R=+826Oe$⁠) by fitting straight lines to the low applied field regime of each branch. The coercivity is defined as the magnitude of the H_C,L and H_C,R, while the exchange bias field is the difference between $Hc$ and one of the loop x-intercepts, i.e., $He=HC,L+Hc$⁠.

This exciting observation opens new possibilities in these materials. Notably, Feng et al. found unconventional EB using a nonstoichiometric MnGa and MnO in the nanoparticle state.² Recently, other reports found EB with the $θ$-phase of the $MnxNy$ system²⁰ which encouraged the exploration of $L10−MnGa$ on $Mn3N2$ as a potential EB system.² Although this measurement is a first indicator of exchange bias, a more complete exchange bias study is needed. Detailed explorations of the EB effects in this system are left to a future report.

In conclusion, we revealed an epitaxial growth process resulting in tall columnar crystallites of $L10−MnGa$⁠. Evidence of edge dislocations of the type $1/2⟨110]$ and $⟨100]$ and screw dislocations along the [001] were observed by STM. The magnetic moment per unit cell (⁠$munitcell$⁠) was found to depend significantly on the dislocation density (⁠$ρ$⁠). We observed $munitcell$ lower by $−3.92±0.55μb$⁠. This work demonstrates the influence of the interface and of layering disorder on the resulting structural and magnetic properties of a thin bi-layer film grown by MBE.

This research was supported by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award No. DE-FG02-06ER46317. Also, the authors acknowledge the use of WSxM software for image processing and Fityk for curve fitting. The authors thank the DGAPA-UNAM project IN100516 and the CONACYT-Mexico Project 281052 for partial financial support. The calculations were performed in the DGCTIC-UNAM supercomputing center project LANCAD-UNAM-DGTIC-051 and Laboratorio Nacional de Supercómputo del Sureste de México, CONACYT member of the network of national laboratories.