Magnetism and multiferroic properties at MnTiO3 surfaces_ A DFT study

The present study illustrates how density functional theory calculations can rationalize the surface structure and magnetism for the low-index (1 1 0), (1 0 1), (1 0 0), (0 0 1), (1 1 1), and (0 1 2) surfaces of MnTiO3. A simple procedure, without surface reconstructions or chemical adsorptions in which the stability, magnetism and the morphological transformations is presented in detail to clarify the control of their multiferroic nature. The surface stability was found to be controlled by the octahedral [MnO6] and [TiO6] clusters formed by the Mn and Ti cations i.e., their local coordination at the surfaces, respectivelywith nonpolar (1 1 0) being the most stable. Enhanced superficial magnetism was found for (0 1 2), (0 0 1), and (1 1 1) surfaces in agreement with the more undercoordinated [TiOn]′ and [MnOn] complex clusters at the surface plane. Our calculation suggests the existence of magnetic [TiOn]′ species for unstable (0 0 1) and (1 1 1) surfaces, explained by the unusual crystalfield associated with the surface environment. The crystal morphology has been predicted to determine the most likely terminations to be present as well as the intrinsic magnetization density associated with morphologies. Moreover, the (0 0 1) surface plane plays a key role in the enhancement of the magnetic properties for shapeoriented MnTiO3 nanoparticles, suggesting a superior magnetoelectric coupling due to the presence of uncompensated spins and polar distortions perpendicular to the surface plane.


Introduction
Polar oxides, such as ferroelectric materials, have been widely investigated in recent years owing to the potential technological applications in the development of advanced devices [1]. In addition, several studies have exploited the electrical field at the ferroelectric surfaces, indicating that the surface polarity can be switched by an external electric field that controls the bulk spontaneous polarization [1][2][3]. In this case, the chemical and physical properties associated with the surfaces can be controlled, resulting in several interesting applications-for instance, the dynamic control of catalysis, photocatalysis and artificial photosynthesis, and switchable chemical sensors [3][4][5][6][7][8][9][10][11].
The manipulation of multiferroic surfaces-widespread candidates for spintronic-based technologies-becomes more intriguing owing to the coupling between magnetic and ferroelectric orders [12]. In this context, a plethora of theoretical and experimental efforts have been devoted to investigating a wide range of multiferroic materials, with BiFeO 3 being the most reported candidate owing to the large ferroelectric polarization associated with G-type antiferromagnetic order [13]. The first study on the properties of BiFeO 3 surfaces was developed by Zhu et al., in which the stability and electronic structure of a polar (1 1 1) surface was resolved from density functional theory (DFT) calculations, showing marked differences with respect to the bulk [14]. More recently, Shimada et al. reported the chemical nature of nonpolar (1 1 0) surfaces of BiFeO 3 through DFT+U calculations, suggesting that the unique surface magnetoelectric response is associated with the distinct rotation pattern of ferroelectric polarization from the breaking symmetry addressed to the surfaces [15]. Further, Dai et al. performed systematic DFT+U calculations to study the nature, thermodynamic stability, and ferroelectric-induced Pd adsorption on the BiFeO 3 (0 0 0 1) polar surfaces. The main observations of these authors are focused on the different chemical natures of positive and negative surface terminations, which exhibit an enhanced relation between spontaneous polarization and weak ferromagnetism as well as distinct behavior as a substrate for metal adsorption, suggesting the intriguing catalytic properties of the metal/BiFeO 3 interface [16][17][18].
Transition metal titanates ATiO 3 (A = Mn, Fe, Ni), with a LiNbO 3type (R3c) structure, are among the most promising multiferroic materials owing to their remarkable magnetoelectric coupling. These compounds are structurally isomorphic to BiFeO 3 except that the positions of A and B cations are exchanged [19,20]. In the past few years, the scientific interest in such materials has increased owing to the possibility of controlling the magnetism from antisymmetric Dzyaloshinskii-Moriya interaction [20][21][22][23][24]. In a previous work, we successfully described the multiferroic properties of such materials using DFT calculations [25]. The R3c structure consists of oxygen layers in a distorted hexagonal close-packed configuration, in which the octahedral local coordinations-i.e., [AO 6 ] and [TiO 6 ] clusters-are equally occupied by A and Ti cations, following the order Ti-vac-A-Ti-vac-A, as depicted in Fig. 1. Below the Curie temperature, the A cations are displaced from the central positions of oxygen octahedral cages-i.e., [AO 6 ] cluster-resulting in a spontaneous polarization along the (0 0 1) direction [10,19]. This distortion also induces a singular crystal-field splitting, especially for A-site cations, owing to the existence of a trigonal prismatic arrangement (D 3h ) that splits the t 2g levels into nondegenerated a 1g and double-degenerated e g π states, whereas e g becomes e g σ . In contrast, the smallest distortion degree associated with [TiO 6 ] clusters maintain the octahedral crystal-field distribution (O h ). Despite the current theoretical studies developed for ATiO 3 (A = Mn, Fe, Ni) materials, the intrinsic physical and chemical properties of the surfaces remain unclear [20][21][22][23][24]26,27].
Recently, MnTiO 3 , as a representative antiferromagnetic semiconductor of the ilmenite family, is the main subject in the multiferroics area owing to the potential magnetoelectric effect [24,26]. MnTiO 3 exhibits a weak ferromagnetism below the Néel temperature, T N = 28 K, which is perpendicular to the electrical polarization resulting from cation displacement enabling the switches of two ferroic properties. Furthermore, magnetodielectric measurements indicate a dielectric anomaly at T N when a magnetic field is applied along the c axis, giving rise to a linear magnetoelectric coupling [24,26]. A note of caution is mandatory here, multiferroic materials such as MnTiO 3 would exhibit non-collinear spin ordering at the surface, probably associated with the formation of magnetic domains. The description of such effects s beyond the scope of the present paper.
In this study, we seek to fulfill a twofold objective. First, we provide detailed information on the surface energies of the (1 0 0), (1 1 0), (1 0 1), (0 0 1), (1 1 1), and (0 1 2) surfaces of MnTiO 3 material obtained from DFT calculations. Second, a microscopic interpretation of the changes in the values of the surface energies are related with the number of unpaired electrons per surface in order to determine the magnetization of different morphologies. We believe that these novel results can arouse enough interest because they contribute to broadening the fundamental knowledge on the multiferroic nature of the MnTiO 3 associated with polar and nonpolar surfaces.
The rest of the paper is divided into three sections. In the next section, we describe in detail the computational methodology. Section 3 contains the results and discussion divided into three main subsections. The first subsection addresses the surface energies and relaxations. The second presents our analysis of the surface electronic and spin properties, whereas the superficial magnetism and crystal morphology are presented in the third subsection. The paper ends with the main conclusions of our work.

Computational methodology
To investigate the multiferroic properties associated with the surfaces of MnTiO 3 , a slab construction model was employed. In this case, the chemical structure of the surface was described by a two-dimensional periodic film formed by atomic layers parallel to the (hkl) crystalline plane of interest, cut from the optimized bulk geometry [25]. Here, (1 0 0), (1 1 0), (1 0 1), (0 0 1), (1 1 1), and (0 1 2) surface planes were considered, enabling the structure of polar and nonpolar surfaces and its effects on the multiferroic properties to be explored.
The purpose of this paper is to introduce in the simplest possible terms the apparent difficulties associated with defining polarization in bulk solids [28]. Upon examining the atomic layers of the planes, it was observed that (1 1 0) is non-polar (µ z = 0), whereas (1 0 0), (0 0 1), (1 0 1), (1 1 1) and (0 1 2) generates polar surfaces due to the ferroelectric polarization containing two slab terminations, which cannot be made equivalent. The purpose of this paper is to introduce in the simplest possible terms the apparent difficulties associated with defining polarization in bulk solids. Then, we use a theoretical procedure to explore the surface energies without surface reconstructions or chemical adsorptions commonly used to cancel the macroscopic dipole and stabilize the Type-3 Tasker's surface, as proposed by Dai and co-authors [16][17][18]. First, we introduce the unrelaxed cleavage energy (E cleav unrlx ) of the complementary terminations (Z + and Z − ), as the required energy to cut the crystal into two unrelaxed complementary terminations, as follows: Here, E slab unrlx and E bulk correspond to the total energies for the unrelaxed slab model and the bulk unit, whereas n and A represent the number of bulk units used in the slab construction and the surface area, respectively.
In the next step, the relaxation of the complementary terminations (Z + and Z − ) was performed, considering that only the outer MnTiO 3 layers were allowed to relax while the inner positions were clamped to reproduce the bulk [16][17][18]. The unrelaxed cleavage energy (E cleav relax ) was then computed as: These approaches have been successfully applied to investigate the (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) surface properties for many kinds of materials [29][30][31][32][33][34][35]. Here, it is important to note that cleavage energy is helpful to understand the energy cost to cut a singular surface focusing on the cleaved bonds and superficial metal coordination; the relaxed surface energies account for the local relaxations that contribute to increasing the surface stability. The convergence test for the values of the surface energies with slab thickness was performed for all surfaces. After the corresponding optimization process and thickness convergence tests,  (1 1 1) with surface terminations giving rise to a minimal dipole moment, although different surface terminations may occur. These numbers of layers were found to be sufficient for a convergence of the results. In the course of the geometry optimization, we do not place any constraints on the atoms except for the conservation of the original crystal symmetry in the two dimensions parallel to the surface. MnTiO 3 crystallizes in the acentric LiNbO 3 -type structure, showing a G-type antiferromagnetic order. In our previous work [25], the bulk structure was represented by two collinear magnetic configurations using the primitive (10-atoms) unit cell: (i) ferromagnetic (FEM), where the spins for all neighbors are parallel ordered; (ii) antiferromagnetic (AFM) for which the spins on the nearest neighbors are antiparallel ordered to each other. Here, we consider the FEM structure containing two spin-up symmetry related Mn atoms in the primitive cell in order to reduce the computational cost for the slab optimization process. In addition, single-point calculations based on the optimized FEM structure for the slab were performed to describe the experimental AFM ordering.
All calculations were performed using the CRYSTAL14 [36] code with PBE0 [37] hybrid functional. Mn, Ti, and O centers were described from all-electron atomic basis sets composed of Gaussian type functions denominated 86-411d41G, 86-51(3d)G, and 8-411, respectively [38][39][40]. It is important to recognize that the electron correlation effects are important in manganite materials, but we are confident in our results because the computed bulk structural properties and bandgaps of MnTiO 3 are in agreement with experimental results [25]. Diagonalization of the Fock matrix was performed at adequate k-point grids (Pack-Monkhorst) in the reciprocal space [41]. The thresholds controlling the accuracy of the Coulomb and exchange integral calculations were controlled by five thresholds set to 7, 7, 7, 7, and 14. The irreducible Brillouin zone (IBZ) was represented by a number of sampling points, which are chosen as (8 × 8 × 8) and (4 × 4) for the bulk and slab, respectively. As a result, there are 65 k -points in the bulk IBZ and 4 k -points in the slab IBZ. The convergence criteria for mono-and bielectronic integrals were set to 10 −8 Hartree, and the RMS gradient, RMS displacement, maximum gradient, and maximum displacement were set to 3 × 10 −5 , 1.2 × 10 −4 , 4.5 × 10 −5 , and 1.8 × 10 −4 a.u., respectively.

Surface energies and relaxations
The values of the unrelaxed calculated cleavage, E cleav unrlx , their decreasing after optimization process, E relax , and the relaxed calculated cleavage, E cleav relax for each surface are given in Table 1.
Considering the values for E cleav unrlx , the observed stability order is  1 1). An analysis of the results reported in Table 1  however, there is not a linear relationship between these values and the number of MeO breaking bonds for the investigated surfaces [42][43][44], indicating that the nature of cleaved bonds is also important to deeply understand the relative stability of both E cleav unrlx and E cleav relax [45,46].
To clarify the surface structure upon relaxation, the relation among atomic displacements, undercoordinated cations, bond distances (Supporting Information), and surface energies (Table 1) was investigated. The relaxed surface structures are shown in Fig. 2(a-f). Here, the comparison was based on bulk values reported in our previous work [25].
First, we considered the nonpolar (1 1 0) surface. In this case, the complementary surface terminations were equal, eliminating the resulting dipole moment perpendicular to the slab, according Type-2 Tasker's classification. The surface arrangement was described by a local coordination of fivefold Mn/Ti cations-i.e., undercoordinated [MnO 5 ] and [TiO 5 ] clusters, respectively. Upon the relaxation, the topmost and outermost oxygen anions relaxed toward the Mn and Ti cations to stabilize the fivefold atomic configuration through a slight shortening of Ti-O and MneO bond distances.
The (0 1 2) surface was also terminated by fivefold Mn and Ti cations-i.e., undercoordinated [MnO 5 ] and [TiO 5 ] clusters-but in opposite terminations due to the polarity on the surface. In both terminations, the O anions moved toward the metal cations to stabilize the dangling bond effect. Despite the distinct chemical character associated with MneO and TieO bonds, the cleaved bonds resulted in a similar relaxation mechanism, as highlighted in Table 1 Table 1), exhibited Mn 5c , Ti 5c , Ti 3c , and Mn 2c cations for the positive termination, whereas along the opposite direction, we can sense the presence of Mn 5c , Ti 5c , Ti 3c , and Mn 3c cations. These arrangements require the largest cationic and anionic displacements to reduce the dangling bond effect.

Electronic and spin properties of surfaces
In this section, we discuss the electronic properties of the investigated surfaces from spin populations, charge density maps, and density-of-states projections. Table 2 summarizes the numbers of local coordination for the Mn and Ti cations at each surface and value of the magnetic moments. In general, it was observed that the magnetic moment followed the  dangling bond effect, once the surfaces with lowest coordination number for the exposed cations showed an unusual spin population compared to the bulk value (4.8) [25], as reported for other materials For instance, the nonpolar (1 1 0) and polar (0 1 2), (1 0 1), and (1 0 0) surfaces displayed the smallest deviation for the Mn magnetic moment compared to the bulk, showing that fiveand fourfold configurations can stabilize the magnetic properties through a charge density rearrangement associated with the anionic displacement and shrinkage of MneO bond distances against the bulk (Supporting Information).
In contrast, for the polar (0 0 1) and (1 1 1) surfaces, an unusual spin population was observed for both positive and negative terminations, which can be directly associated with the lowest coordination number of the exposed metal cations. In the (0 0 1), the positive termination exhibited a magnetic Ti 3c , indicating the existence of magnetic Ti species. On the other hand, the negative termination showed a reduced magnetic moment on the Mn 3c center, showing the presence of high undercoordinated centers that are capable of modifying the charge density distribution along the plane. Similar behavior was observed for the (1 1 1) surface, where the existence of fivefold and threefold Mn/Ti sites resulted in a reduced spin population for Mn cations, whereas the exposed undercoordinated Ti centers became magnetic. From the viewpoint of technological applications, this mechanism is very interesting because of the possibility to induce superficial magnetism localized on nonmagnetic cations.
To clarify the electronic features associated with magnetic Ti species along (0 0 1) and (1 1 1) surfaces, the density-of-states (DOS) projections and spin isosurfaces were analyzed, and are depicted in Fig. 3. Both surfaces exhibited an unusual electronic density distribution close to the valence band maximum (VBM). In both cases, the largest contribution of 3d orbitals for exposed Mn/Ti species mixed with 2p (O) atomic orbitals was observed in the valence band (VB). Compared with the reported DOS profile for bulk MnTiO 3 [25], the major differences are associated with the presence of occupied 3d Ti orbitals, confirming the existence of reduced Ti species. This mechanism is attributed to the crystal field effects associated with the undercoordinated cations, because the missing oxygen atoms along the surface plane induced a perturbation on the 3d orbital degeneracy, resulting in a pseudo-atomic orbital configuration, which enabled the location of unpaired electrons on these orbitals. In a similar way, the reduced coordination for Mn modified the t 2g and e g energies, mainly for the orbitals pointing toward or closer to the z-axis. Furthermore, we can argue that the existence of reduced Ti species suggested an enhancement of superficial magnetism for (0 0 1) and (1 1 1) surfaces added to the reduced bandgap, compared to the bulk, indicating a surface metallization mechanism. The reordering of surface electronic density at (0 0 1) and (1 1 1) surfaces due to Mn and Ti undercoordination suggested a highly reactivity character against the other surfaces. In addition, similar results have been reported for other solid state materials, indicating that the dangling bond effect is the key to induce superficial magnetism from spin transition at magnetic and nonmagnetic cations [57,58].
Moreover, the observed DOS results were confirmed by the spin isosurfaces (Fig. 3b, d), indicating that threefold Ti cations exhibited a spin population that was fundamental to increase the magnetic moment perpendicular to the surface plane. In addition, it was observed that fivefold Ti cations behaved as 3d 0 bulk cations, showing no spin population, proving that the dangling bonds were the main effects of the unusual Ti magnetism.
The cutting process linked to the surface structures generated different kinds of undercoordinated centers from the existence of dangling bonds-an inherent process associated with the symmetry-adapted cutting strategy without any changes to the crystal stoichiometry. In this case, the exposed local geometries were notably different compared to the bulk. In this study, the investigated surfaces could exhibit three kinds of undercoordinated clusters: [MO 5 ], [MO 4 ], and [MO 3 ] (M = Mn, Ti). Considering the electron density reorganization on the cut surfaces, the metal undercoordinated degree modified the energylevel distribution in the VB, as presented in Fig. 3. Therefore, the intervalence charge-transfer that is responsible for the enhanced surface magnetism and magnetic Ti species could be understood from the electronic disorder associated with oxygen valence orbitals linked to x bond path. Fig. 4 schematically represents the existence of one, two, and three dangling bonds in the investigated surfaces, which induces the perturbation on the energy levels of the remaining atoms. The combination of DOS (Fig. 3) and spin-population analysis ( Table 2 and Fig. 3) enables us to note that the existence of dangling bonds in the investigated surfaces created intermediary levels in the vicinity of the bandgap region compared to the bulk. Thus, local disorder associated with undercoordinated cations increased the VB energy through changes in Mn (3d) crystal-field energy splitting summed to the electronic reordering with increased energy of remaining O (2p) orbitals. With increasing number of dangling bonds, the upper part of VB drastically changed, mainly because the energy of occupied 2p states increased, enabling the enhancement of the bond path, which stabilized the occupation of Ti (3d) orbitals. In addition, by analyzing the local environment of reduced Ti clusters, it was observed that oxygen atoms from subsurface layers moved toward the exposed Ti atom to redistribute the electron density along the surface. On the other hand, the local arrangement for subsuperficial (n = 4 , 5, 6) clusters using the Kroger-Vink notation [59]. In this formalism, the charge accumulation mechanism could be described using the neutral [MO n ] x , positively charged [MO n • V O ] • , and negatively charged [MO n ]′ cluster notation. In this case, the charge-transfer mechanism was as follows: Therefore, the perturbation on the VBM was attributed to the increase of oxygen (2p) energy levels, which behaved as positivelycharged vacancies in subsuperficial [MnO n ] clusters that became that enabled a charge accumulation on superficial [TiO n ] ' clusters for (0 0 1) and (1 1 1) surfaces. Furthermore, the obtained results suggest that the creation of oxygen vacancies in these surfaces could also contribute to enhance the magnetic moment along the planes, owing to the combination of the dangling bond effect and electron trapping from missing atoms, as reported for other materials [58,60]. This topic will be investigated in forthcoming studies.

Superficial magnetism and crystal morphology
In this section, we investigate the MnTiO 3 crystal morphology by employing the Wulff construction [61], in which the values of E surf for each surface determine the final morphology. The Wulff proposal refers to a simple relation between E surf and the distance in the normal direction from the center of the crystallite, which also allows this ideal morphology to be modified by tuning the surface energies of the different facets [62,63]. In recent years, this method was successfully applied for different kinds of materials, showing excellent agreement between theoretical predicted and experimental reported morphologies [45,[64][65][66][67].
Recently, we proposed a theoretical procedure to rationalize the relation between superficial magnetism and crystal morphologies by combining the Wulff construction model with spin analysis along the surface planes, providing a different perspective to understand the uncompensated spins responsible for unusual magnetic properties of Co 3 O 4 [68]. Here, this method was employed to investigate the magnetic properties of shape-controlled MnTiO 3 material.
First, the complete set of available morphologies for the multiferroic MnTiO 3 material are summarized in Fig. 5, in which the transformations were obtained by tuning the surface energies of the different facets. It is important to note that the E surf values used to obtain the different morphologies correspond to a mean value between the complementary terminations, E cleav relax , presented in Table 1.
The vacuum ideal morphology proposed by MnTiO 3 exhibited a corner-truncated cylindrical shape that predominantly exposed the (1 1 0) surface and, to a minor extent, the (0 1 2) and (0 0 1) surfaces. Despite the higher value of E surf for (0 0 1) surface, the ideal morphology of MnTiO 3 showed an extent of this plane, suggesting the existence of intriguing magnetic behavior due to the exposure of unusual magnetic Ti species, as previously discussed. In addition, it was observed that different shapes were obtained by tuning E surf for MnTiO 3 , such as truncated and nontruncated cylindrical, octahedral, rhombic, and monoclinic shapes. However, no research has reported the synthetic control of crystal morphology for MnTiO 3 , making difficult the deeper comparison between theoretical and experimental morphologies.
To rationalize the superficial magnetism in such morphologies, we performed a summation between the magnetic moments along the surface planes, respecting the experimental G-type antiferromagnetic ordering. The magnetization density (D µ ) index of a given surface was calculated (Supporting Information), considering the magnetic moment (µ B ) per unit cell area (A): Further, the magnetization density (M) index of a given morphology was obtained by combining the polyhedron composition (c hkl ) and the magnetization density (D µ ): The total magnetization density (M) for the different shapes reported in Fig. 4 are summarized in Table 3, where the contribution of each surface (%) on M is listed. Regarding the superficial magnetism for the different morphologies predicted for MnTiO 3 (Fig. 5), it was observed that all M values were obtained by combining the uncompensated spins of (0 1 2), (0 0 1), and (1 1 1) surfaces, in agreement with the spin populations presented in Table 2. In addition, the increased superficial magnetism of some polyhedrons (higher M values) was attributed to the contribution of the (0 0 1) surface due to the higher magnetic moment along the surface plane. From a technological point of view, this result is exciting because the control of (0 0 1) surface properties of many perovskite (ABO 3 ) materials has been intensively explored during recent years. Moreover, some reports for multiferroic BiFeO 3 based-material show the existence of a (0 0 1) surface plane in the reported morphologies, and the z-cut of LiNbO 3 (R3c) exhibits an intriguing mechanism associated with the dependence of ferroelectric ordering [10,[69][70][71].
The precise control of superficial magnetism for antiferromagnetic multiferroic materials is important to increase the magnetoelectric coupling in such compounds. Here, we argue that the (0 0 1) surface orientation, which exhibits a large ferroelectric tensor, shows an enhanced magnetic moment. Thereby, this result allows us to predict that polar distortions along the (0 0 1) direction can control the superficial magnetism through the exposure of undercoordinated cations, once the ferroelectric distortion involves the cationic displacement of A and Ti cations along this direction. By combining the values for the surface energy (E surf ) and total magnetization density (M), it is possible to obtain a morphology map that summarizes the highest values of M following the percentage of (0 0 1) surface area composition, as depicted in Fig. 6.
In this way, the shape-oriented multiferroic nanoparticles could present a superior magnetoelectric effect from the reorientation of the crystalline structure, which induces a different spin distribution along the material surfaces.

Conclusions
In this work, first-principles DFT calculations have been performed to investigate the stoichiometric polar and nonpolar surfaces for the R3c multiferroic MnTiO 3 . The surface stability, in order of increasing surface energy, for the low index surfaces of MnTiO 3 was found to be (1 1 0) > (0 1 2) > (1 0 1) > (1 0 0) > (0 0 1) > (1 1 1), with the most stable surfaces having both high Ti and Mn surface coordination. Electronic and magnetic properties of the different surfaces were investigated by a simple procedure, without surface reconstructions or chemical adsorptions showing evidence for the presence of an enhanced magnetic moment at cleaved surfaces that is capable to clarify their multiferroic nature. A deep analysis of the superficial uncompensated spin, especially for unstable (0 0 1) and (1 1 1) surfaces, shows the presence of magnetic Ti species. This intriguing result was explained rationally by the crystal-field energy scales in the cleaved surface environments associated with the presence of different local Ti and Mn coordinations. To the best of our knowledge, we report the first study about the morphological modulations of multiferroic material. The calculated equilibrium morphology is elongated along the c-axis, showing a hexagonal corner-truncated cylindrical shape containing nonpolar (1 1 0) and polar (0 1 2) and (0 0 1) surfaces. Furthermore, by controlling the ratio between the calculated surface energies, a morphological map for MnTiO 3 was predicted. By combining the morphological map with the singular spin densities associated with the surfaces, it was found that the exposure of the (0 0 1) surface plane is mandatory to increase the superficial magnetism and the magnetoelectric coupling for MnTiO 3 nanoparticles, enabling us to conclude that superior multiferroic properties and exciting applications can be obtained from the morphological control of MnTiO 3 .

Author contributions
The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.  These authors contributed equally.