Density-functional study of Mn monosilicide on the Si ( 111 ) surface : Film formation versus island nucleation

The stability of thin films and of small crystallites of Mn monosilicide MnSi on the Si 111 surface is investigated by density-functional theory calculations. Extending previous studies of MnSi /Si 001 , our calculations indicate that MnSi films on Si 111 have similar electronic and magnetic properties, i.e., large magnetic moments at the Mn atoms near the surfaces and interfaces and a high degree of spin polarization at the Fermi level. Hence, such MnSi films could be interesting as a spintronics material compatible with silicon. Moreover, from our calculated total energies we conclude that the Si 111 substrate should be more suitable to grow MnSi layers than the Si 001 substrate. This result is obtained by analyzing the conditions for the formation of three-dimensional 3D MnSi islands, either in the B20 crystal structure or as pseudomorphic islands in the B2 structure: On Si 001 , 3D islands, even if they are just a few lattice constants wide, are found to be already more stable than a homogeneous MnSi film. A bipyramidal “iceberg” island consisting of MnSi in the B20 structure on the Si 001 substrate is found to be most stable among the structures investigated. For MnSi on Si 111 , however, our calculations show that the nucleus for forming a 3D island is larger. Therefore, Mn deposition initially leads to the formation of flat 2D islands. On Si 111 , the lowest-energy structure for such islands is found to be similar to the B20 structure of bulk MnSi, whereas on Si 001 this structure is incompatible with the substrate lattice. Our results are in agreement with the experimental observations, formation of an almost closed film with 3 3 structure on Si 111 , and 3D island formation on Si 001 .


I. INTRODUCTION
The injection of a spin-polarized electric current from a ferromagnet into a semiconductor is one of the key challenges in present-day spintronics research.2][3][4] However, on the most common semiconductor substrate silicon, growth of magnetic films of Fe or Co with the quality required for efficient spin injectors has not been achieved up to now.In most cases, a transition metal silicide is formed instead of the desired metal film.This is in contrast to Fe on GaAs, where metallic Fe films were obtained.The silicides formed on Si, e.g., bulk FeSi and CoSi 2 , are nonmagnetic and thus useless for building spintronics devices.Interestingly, some monosilicides, such as CoSi and MnSi, are ferromagnetic as thin films in the pseudomorphic B2 ͑cesium-chloride͒ crystal structure, as we showed recently by density-functional calculations. 5,6Moreover, MnSi in the hypothetical zincblende structure is predicted to be ferromagnetic by DFT calculations. 7While calculations for MnSi films on Si͑001͒ have been presented recently, 5,8 the present paper deals with thin films and islands of MnSi on Si͑111͒.Our present calculations suggest that the Si͑111͒ substrate is more suitable than Si͑001͒ for growing MnSi overlayers.Results for thin layers with either the ground-state crystal structure of MnSi ͑the so-called B20 structure͒, or the B2 structure are compared.Moreover, special attention is given to the possibility of three-dimensional island formation.While we find the tendency to island formation to be very strong for MnSi/ Si͑001͒, the three-dimensional islands on Si͑111͒ only become stable above a certain size, according to our calcu-lations.Thus, the preferred substrate for growing layers of MnSi should be Si͑111͒.The previously studied system MnSi/ Si͑001͒ could be an interesting system for preparing and studying ferromagnetic or superparamagnetic MnSi nanostructures.
By depositing Mn either on Si͑111͒ or Si͑001͒, silicides with varying stoichiometry have been obtained experimentally, such as MnSi 1.7 , 9 MnSi, 10 or Mn 5 Si 3 , 11 depending on the growth conditions.In this study, we restrict ourselves to the epitaxial growth of Mn monosilicide ͑MnSi͒, and investigate which substrate orientation of silicon is promising to grow thin films of MnSi.2][13][14][15][16][17] Surface structural phase transitions, 18 as well as a nonmetal-to-metal phase transition in 5-10 monolayers ͑ML͒ thick films 19 have been reported.Evans, Glueckstein, and Nogami observed by low-energy electron diffraction ͑LEED͒ and by scanning-tunneling microscopy ͑STM͒ that deposition of Mn on Si͑111͒ first leads to formation of large, flat islands ͑diameter several tens of nanometer͒ displaying a ͑ ͱ 3 ϫ ͱ 3͒ reconstruction. 12Above about four monolayers of Mn, these islands almost completely cover the surface.These findings were confirmed by Kumar et al. 20 using similar techniques.Likewise, flat-top islands with ͑ ͱ 3 ϫ ͱ 3͒ struc- ture were observed by Azatyan, Iwami, and Lifshits after depositing submonolayers of Mn on Si͑111͒͑7 ϫ 7͒ and annealing to 450 °C. 21Moreover, this characteristic ͑ ͱ 3 ϫ ͱ 3͒ pattern has been used to identify side facets of MnSi islands on Si͑001͒ as MnSi͑111͒ facets. 22So far, no atomistic model of the ͑ ͱ 3 ϫ ͱ 3͒ reconstruction has been presented in the literature, and even its relation to the bulk crystal structure of MnSi has remained unclear.In this paper, we suggest an atomistic model for the ͑ ͱ 3 ϫ ͱ 3͒ structure for thin MnSi films derived from the B20 structure of bulk MnSi.The magnetic properties are studied both for these B20-like films and for films with a simpler, B2-like structure on Si͑111͒ in order to make contact to the previously studied B2-like films on Si͑001͒.
This paper is arranged as follows.Section II provides the technical details of the method used, mainly intended for those who want to reproduce the calculations.In Sec.III, we discuss the structure, the thermodynamic stability and the magnetic properties of MnSi films on Si͑111͒.Formation of crystallites with certain shapes ͑pyramid or bipyramid͒ and sizes is considered in Sec.IV.A summary and conclusions follow in Sec.V.

II. CALCULATIONS
In this work, we calculate the atomic structure, the stability, and the electronic and magnetic properties of thin manganese silicide films using density-functional theory ͑DFT͒.The electronic exchange and correlation is treated within the generalized gradient approximation ͑GGA-PBE͒. 23We perform all-electron calculations using the full-potential augmented plane wave plus local orbital ͑FP-APW+ LO͒ method implemented in the WIEN2K code. 24In all calculations, the muffin-tin sphere radius is chosen to be 1.11 Å for both Mn and Si.Inside the muffin-tin spheres, wave functions, electron density and potential are expanded in spherical harmonics up to angular momentum quantum numbers l max WF = 12 and l max pot = 6, respectively.In the interstitial region a plane-wave expansion with a cutoff energy of E cut WF = 13.8Ry is used.Repeated slab geometries are used to describe the surfaces and interfaces in this study.The lateral dimensions of the unit cell are determined by the bulk lattice constant of silicon, calculated to be 5.47 Å within the GGA-PBE functional.For Si͑111͒ ͑1 ϫ 1͒, the Brillouin zone sampling is done by a set of 12 k points in the irreducible part of the Brillouin zone, derived from a 9 ϫ 9 ϫ 1 k-point mesh.
For the larger Si͑111͒ ͑ ͱ 3 ϫ ͱ 3͒ unit cell, a coarser k-point mesh of 6 ϫ 6 ϫ 1 k points is used.For calculations for Si͑001͒ surfaces and thin films, the quality of Brillouin sampling corresponds to 8 ϫ 8 ϫ 1 k points in the Brillouin zone of the ͑1 ϫ 1͒ real-space unit cell.The entire slab ͑which contains the Si substrate plus a manganese silicide film on both sides of the slab͒ was allowed to relax until the calculated forces on each atom were smaller than 0.03 eV/ Å.Three types of slab systems are studied.
͑i͒ Clean silicon slabs with ten layers for the ͑111͒, or eight layers for the ͑001͒ surface, in order to obtain the surface energy of these silicon surfaces.
͑ii͒ Slabs of manganese monosilicide MnSi, both in its ground state ͑the B20 phase͒ and in the cesium-chloride structure ͑B2 phase͒.Surface energies were calculated for the ͑111͒ and ͑1 ¯1 ¯1 ¯͒ surfaces of the B20 phase, and for both the ͑001͒ and ͑111͒ surfaces of the B2 phase.Inversion symmetry for the B2 structure ensures that the surface energies for its ͑111͒ and ͑1 ¯1 ¯1 ¯͒ surfaces are equal.For the B20 structure, we use a slab with ͑111͒ surface normal and inequivalent top and bottom surfaces of the slab.Thus, the average of the ͑111͒ and ͑1 ¯1 ¯1 ¯͒ surface energies is obtained, 25 which turns out to be sufficient for the considerations in this work.
͑iii͒ Slab models of the Si͑111͒ surface covered by thin films of MnSi of various thickness, either in the B20 structure or the B2 structure.Specifically, we use threedimensional periodic supercells made of ten Si layers and up to 12 layers of alternating Mn and Si atomic planes along the ͑111͒ direction on both sides of the slab.For details concerning the calculations of MnSi thin films on Si͑001͒, we refer to our earlier published work. 5,8ests performed for the thickness of the supercell proved that the considered number of layers was sufficiently large to reproduce bulklike behavior in the central layers.In order to compare the structural stability of the clean surfaces and of different films, surface energies ͑for the bare surfaces͒ or formation energies ͑for thin films͒ are calculated by ab initio thermodynamics. 26In this approach, surface energies are calculated under the assumption that the surface is in thermodynamic equilibrium with the respective bulk material.For the surfaces of elemental silicon, the surface energy ͑at zero temperature and pressure͒ is thus independent of the chemical potential, and only depends on the surface orientation where E tot is the total energy of the slab ͑relative to isolated atoms͒ Si =−E Si coh , the cohesive energy of Si in the diamond structure, N Si is the number of silicon atoms in the supercell, and A is the area of the surface unit cell.
The surface energies of a compound material, such as MnSi, depend on an additional variable, namely, the chemical potential of manganese Mn : where MnSi =−E MnSi coh and N Mn is the number of manganese atoms in the slab.In practice, the value of Mn will depend on the chemical environment in which the surface is prepared.However, there are limits for the value of Mn under thermodynamic equilibrium conditions.In one limiting case, the surface is in equilibrium with a reservoir of bulk manganese Mn ഛ − E Mn coh .͑3͒ For the cohesive energy of bulk Mn E Mn coh , we substitute the energy of the ground state ␣-Mn, obtained by applying a correction of −0.07 eV per Mn atom 27 to the cohesive energy of ␥-Mn ͑fcc structure͒ calculated by WIEN2K.In the other limit, the surface is in equilibrium with a reservoir of bulk silicon.This leads to an estimate for the minimum value of Mn , For films on a substrate, we define the formation energy as the energy required to form the film from elemental reservoirs of its constituents on the clean ͑but possibly recon-structed͒ surface of the substrate Here E tot , N i and i refer to total energy of the slab per ͑1 ϫ 1͒ supercell, the number of atoms of species i in the ͑1 ϫ 1͒ supercell ͑i =Mn,Si͒, and chemical potential of these atomic species.In the last term, ␥ substrate ͑j͒ represents the surface energy of either the Si͑001͒ or the Si͑111͒ surface, depending on the substrate orientation used to deposit the thin film, and A is the area of the respective ͑1 ϫ 1͒ nonreconstructed unit cell.For formation of the film from elemental reservoirs ͑Mn and Si͒, the chemical potentials are set equal to the cohesive energies of the respective bulk materials Si =−E Si coh , Mn =−E Mn coh .In the following, this way of preparing the films, under excess of elemental Mn, is referred to as "Mn-rich regime."Alternatively, we discuss a situation where the thin film is formed from reservoirs of bulk Si and bulk MnSi.In this case, referred to as the "Mn-poor regime," the values to be inserted in Eq.
. We note that the formation energy of the thin film contains energetic contributions from both the surface of the film, its interface with the substrate, and the strain energy due to lattice mismatch between the film material and the substrate.Both for the B2 phase and for the B20 phase of MnSi, the calculated lattice mismatch with silicon is 2.2 and 4.9% ͓experimental value: 3.2% ͑Ref.28͔͒, respectively.For the films considered here, with a coverage of up to three monolayers ͑ML͒, the energy due to the elastic distortion of the epitaxial films are small.We estimate this elastic energy by performing DFT calculations of appropriately distorted bulk unit cells of MnSi, and find the elastic energies to be less than 2 meV/ Å 3 for the largest mismatch, 4.9% for B20/ Si͑111͒, and smaller for the other film-substrate combinations.
Finally, we define the interface energy ␥ interface between the thin films and the substrate, using the formation energy and the surface energy introduced above.Since the epitaxial strain contributions are small, we can neglect them here, and express the interface energy approximately as Since all quantities in this expression depend on the chemical potential of manganese, it is important that both E form and ␥ surface MnSi are evaluated for the same value of Mn .In the following, surface and interface energies of films with B2-like structure are quoted for Mn =−E MnSi,B2 coh + E Si coh , i.e., with respect to a reservoir of ͑hypothetical͒ bulk MnSi in the B2 crystal structure.For films with a B20-like structure ␥ surface and ␥ interface are quoted with Mn determined by a reservoir of the B20 phase of bulk MnSi.

A. Films with B20-like structure
Manganese monosilicide is known to crystallize in a cubic crystal structure with four Mn and four Si atoms per unit cell, 29 the B20 structure.The point group of this crystal structure is P2 1 3, i.e., it possesses four threefold rotational axes, pointing in the cubic ͑111͒, ͑11 ¯1 ¯͒, ͑1 ¯11 ¯͒, and ͑1 ¯1 ¯1͒ directions.A ͑111͒ film of MnSi can be matched epitaxially with the Si͑111͒ surface.The ͑1 ϫ 1͒ surface unit cell of MnSi͑111͒ is similar in size to the ͑ ͱ 3 ϫ ͱ 3͒ unit cell of Si͑111͒: The mismatch amounts to 3.2% for the experimental, or 4.9% for our calculated lattice constants of the two bulk materials.A ͑111͒ slab of MnSi in the B20 structure consists of alternating layers of Mn and Si atoms with a repetition period of 12 layers.Groups of four layers follow each other in an ABCABC. . .stacking sequence.Each such four-layer group consists of a "dense" Mn layer, a "sparse" Si layer, a "sparse" Mn layer, and a "dense" Si layer.Dense and sparse layers have three and one atom per ͑1 ϫ 1͒ unit cell of MnSi͑111͒, respectively.
We performed total-energy DFT calculations for several thin film structures of MnSi on Si͑111͒.The lateral size of the unit cell used for the Si substrate was ͱ 3 ϫ ͱ 3 in terms of the ideal Si͑111͒ surface unit mesh.Since the B20 structure of MnSi is more close packed than the diamond structure of silicon, we define one monolayer of manganese by having two Mn atoms per surface Si atom.The film structures investigated contained one dense layer of Mn with three Mn atoms ͑corresponding to 0.5 ML͒, or two dense plus two sparse layers of Mn with a total of eight Mn atoms ͑4 / 3 ML͒.In the latter case, the interface is formed by a dense layer of Si, where each Si atom makes one vertical bond to a Si substrate atom.In both cases, the surface is terminated by a dense layer of threefold-coordinated Si atoms.This termination is supported by the analysis of x-ray photoemission experiments 20 indicating the presence of surface Si atoms.The atomic structure of this film is shown in Fig. 1͑c͒ in top view and in Fig. 1͑d͒ in side view.
Thermodynamic stability.In Table I, the formation energy of the B20 film corresponding to a Mn coverage of =4/ 3 ML, as well as the interfacial, surface and central layer͑s͒ spin magnetic moments inside of muffin-tin sphere 25  formation energy of −76 meV/ Å 2 , is most stable among the structures studied.The minus sign indicates that formation of such a film from elemental Mn and Si is exothermic.This is in line with the experimental observation of ͱ 3 ϫ ͱ 3-reconstructed islands 21 or films 12,20 indicative of MnSi in the B20 structure.In these experiments, flat extended islands several tens of nm in diameter and about 4, 8, or 12 Å in height ͑Ref.12͒, or flat, about 10 nm wide islands near step bunches ͑Ref.21͒ had been observed in STM images after depositing Mn on Si͑111͒ and annealing to 350-450 °C.Furthermore, our calculations clearly show that formation of a pure Mn film would be energetically very unfavorable, with a formation energy of 292 meV/ Å 2 , similar to our previous findings for Mn/ Si͑001͒. 5 Despite the lattice mismatch between the B20 structure and silicon, transmission electron microscopy images show that thin films of MnSi ͑Ͻ16 ML of Mn͒ can be grown epitaxially. 16For similar films grown with Bi as a surfactant, a hexagonal superstructure with a period of about 100 Å superimposed on the ͱ 3 ϫ ͱ 3 reconstruction was detected by SPA-LEED and STM. 17 For thicker films, it is observed experimentally that the lattice mismatch between the B20 structure and silicon induces a long-range dislocation network. 20ince we are interested in very thin films, our calculations assume a perfect epitaxial relationship, and ignore imperfections of the interface related to these long-range structures.With these limitations in mind, we believe that the atomic structure of the ͱ 3 ϫ ͱ 3-reconstructed islands observed in Refs.12, 20, and 21 is similar to our calculated structure, Fig. 1͑c͒.The top-most layer in our structural model consists of threefold coordinated Si atoms with one dangling bond each.As we have shown recently by comparing simulated and measured STM images for Mn/ Si͑001͒, 30 filled-state STM images in this system map the dangling bonds of surface Si atoms, while subsurface Mn atoms remain invisible and are detectable only indirectly by their effect on the electronic structure.In the experimental STM images, a single structural feature with ͱ 3 ϫ ͱ 3 periodicity was observed. 12is could be tentatively ascribed to the triangle of the three surface Si atoms centered at the corner of the unit cell in Fig. 1͑c͒.However, we cannot be conclusive about the structure of the ͑dense or sparse͒ Mn subsurface layers, since there is no experimental data that would allow to compare the position of subsurface atoms between our calculations and experiment.
Magnetic properties.For the magnetic structure of the B20-type film corresponding to 4 / 3 ML Mn coverage, we find a structure with all spin magnetic moments of the Mn atoms aligned in parallel and a structure with partial compensation of the spin moments to be almost energetically degenerate within the accuracy of our calculations.In the calculation with all spin magnetic moments of the Mn atoms aligned in parallel, these moments ͑inside the muffin-tin sphere of the Mn atoms͒ are found to be 2.3, 0.9, 0.02, and 3.4 B per Mn atom at the surface, the two central, and the interface layers, respectively.For the dense layers of the B20-type film, the quoted numbers are averages of the three Mn atoms per dense layer.The Si atoms in MnSi are found to have small induced magnetic moments aligned antiparallel to the Mn magnetic moments, with magnitudes between −0.1 B and −0.02 B .This observation points to the fact that the magnetic coupling in MnSi films originates from sp-d exchange mediated by bands with silicon sp-orbital character, as described previously. 5,31The total magnetic moment of the film at 4 / 3 ML coverage amounts to 12.5 B per ͱ 3 ϫ ͱ 3 unit cell.The state with antiparallel alignment of the Mn spin magnetic moments, being only slightly higher in energy, has similar magnitude of the magnetic moments at the surface and interface Mn atoms, but very small magnetic moments ͑ഛ0.1 B ͒ in the central layers.For both types of magnetic ordering, the magnetic moments at the surface and interface are considerably higher than the spin magnetic moments of Mn atoms in bulk MnSi, which were reported to be ϳ0.4B from experiments. 32The spin polarization at the Fermi level, calculated from the density of states, is about 50%.

B. Films with B2-like structure
It is well known that epitaxial thin films sometimes grow pseudomorphically, i.e., with a crystal structure different from the bulk.Due to epitaxial strain and/or contributions from the interface and surface free energy, thin films with an unusual crystal structure may become lower in energy and thus be more stable than films with the known bulklike structure.As we proposed previously, 5,8 MnSi could possibly be grown pseudomorphically on Si͑001͒ in the cesium-chloride structure ͑B2 structure͒.Experimentally, such a structure has been stabilized for Fe and Co monosilicide by epitaxial growth on Si͑111͒, using molecular beam epitaxy ͑MBE͒. 33n addition to the B20 structure, we therefore investigate the structural stability of the B2 structure for epitaxial films of MnSi on Si͑111͒.We note that the B2 structure can be matched epitaxially with both the Si͑111͒ and the Si͑001͒ surface.As reported before for epitaxial MnSi films on Si͑001͒, 5,6,8 matching is achieved by rotating the cubic unit cell of the B2 structure by 45°with respect to the Si͑001͒ ͑1 ϫ 1͒ surface unit cell.The ͑111͒ plane of the B2 structure can be matched directly with the Si͑111͒ surface using the ͑1 ϫ 1͒ surface unit cells of both materials.In the B2͑111͒ / Si͑111͒ films, the Mn and Si layers of the film have the same atomic density as the substrate.Figure 1͑a͒ and 1͑b͒ show an example for 1.5 ML coverage.Following our previous results obtained from DFT calculations for MnSi/ Si͑001͒, 5 where a capping layer of ͑fourfold coordi- nated͒ Si adatoms was found to increase the stability of the films, all structures are considered to be covered with a capping layer of Si ͑threefold coordinated͒.Thermodynamic stability.We compare the formation energy of the ultrathin films with B2 and B20 structure as a function of coverage up to 1.5 ML in Fig. 2. The films with B2 structure have generally a higher formation energy than those with the B20 structure, i.e., they are only metastable.With respect to bulk MnSi, the formation energies of all film are higher than the energy of the bare Si͑111͒ surface.This implies that there is no wetting of the Si͑111͒ surface by MnSi; instead, growth via formation of MnSi islands is to be expected.
Magnetic properties.The values of the surface and interface spin magnetic moments are found to be somewhat lower in the film with B2-like structure compared to the B20-like structure at similar coverage ͑see Table I͒.For the B2-like film, a considerable magnetic moment ͑0.7 B ͒ is also found for the Mn atom in the central layer.Comparing to the ideal cubic B2 bulk structure, which is found to be nonmagnetic in our calculations, 34,35 such an increased magnetic moment may ͑at least partially͒ be attributed to the effect of the tetragonal lattice distortion in the film on the electronic band structure close to the Fermi energy: The vertical distance between layers is about 0.84 Å which is 5% larger than for the B2 bulk phase.Moreover, the magnetic moments of individual atoms are related to their coordination and bond length: The Mn atoms in the surface and interface layer make seven bonds with Si atoms at distances between 2.38 and 2.5 Å, while the Mn atoms in the central layer have eight bonds to Si atoms.The bonds between the Si substrate and Mn interface layer are slightly shorter than the Mn-Si bonds in the central layers.Consequently, there is stronger orbital mixing between electronic states at the interface, consistent with the smaller spin magnetic moment of the Mn at the interface ͑Mn I in Table I͒ compared to the surface ͑M S in Table I͒.For the B2-like film at 1.5 ML coverage, the total magnetic moment amounts to 4.5 B per 1 ϫ 1 unit cell.The spin polarization at the Fermi level, calculated from the density of states, is 59% for this Mn coverage.
In order to check if these magnetic properties persist for thicker films, we calculated MnSi films with B2-like structure corresponding to = 3 ML coverage.It is found that the magnetic moments of the Mn atoms at the surface and interface are still large ͑1.8 B and −2.0 B , respectively͒, but now antiparallel alignment between the surface and interface magnetic moments is energetically preferred.This finding is analogous to our previous results for 3 ML of MnSi on Si͑001͒. 5 In the 3 ML-thick B2-like film on Si͑111͒, the magnetic moment of the Mn atoms gradually changes from layer to layer when going from the interface to the surface.In the two central layers, where the magnetic moment changes sign, its absolute value is only about 0.2 B .However, from analyzing the partial density of states, we find that the 3 ML film still has a rather high spin polarization at the Fermi level for both the interface and surface layer of 70 and 84%, respectively.In our previous study of a 3 ML MnSi film on Si͑001͒, this quantity had been found to be 27% only. 8n summary, we find that a homogenous MnSi film on the Si substrate is thermodynamically metastable, since it is the most stable film structure, but still higher in energy than large three-dimensional ͑3D͒ islands with MnSi bulk structure.From the structures studied here, we conclude that islands with a ͑ ͱ 3 ϫ ͱ 3͒R30°surface reconstruction are most favorable.For such islands, considerable magnetic moments at the interface and surface Mn atoms and a high degree of spin polarization at the Fermi level are predicted.Both for the films with the B2 and B20 structure with coverage ഛ1.5 ML, ferromagnetic ordering is found to be ͑slightly͒ more favorable than other magnetic structures that we have tested in our calculations.Experimentally, there is no clear evidence yet for ferromagnetism in MnSi films.However, magneto-optic Kerr effect measurements 16 have found deviations from a Curie-Weiss law of the magnetic susceptibility.For the applicability of this material in spintronics devices, a key quantity is the interface spin polarization at the Fermi level.For the films predicted to be ferromagnetic, the values are comparable to the spin polarization of, e.g., bulk iron of about 60%. 36

IV. FORMATION OF 3D ISLANDS
As discussed above, thin MnSi films tend to transform to 3D structures.However, formation of side facets at 3D islands costs additional surface energy.Therefore, 3D islands will be energetically favorable only if the islands exceed a certain size, corresponding to a certain coverage of manganese ͑at a given island density͒.In the following, we give an estimate for the minimum size of a stable island of MnSi on both the Si͑001͒ and the Si͑111͒ substrates.This analysis follows thermodynamic arguments in the spirit of the thermodynamic nucleation theory of Gibbs and Volmer and Becker and Döring. 37,38It is expected to apply to situations close to thermodynamic equilibrium, where the critical nucleus consists of a large number of atoms.For example, the critical nucleus for Si homoepitaxy on Si͑001͒ was found to consist of ϳ650 dimers at the typical growth temperature Bare surface FIG. 2. Formation energy of ultrathin films with B2 ͑dashed lines, filled symbols͒ and B20 ͑solid lines, hollow symbols͒ structure of Mn monosilicide on Si͑111͒ as a function of the deposited amount of Mn.The energy of film formation is calculated by assuming equilibrium either with the B20 bulk structure of Mn monosilicide ͑squares͒, or the B2 structure of Mn monosilicide ͑tri-angles͒, or Mn bulk ͑circles͒.The surface energy of the bare, unreconstructed Si͑111͒͑1 ϫ 1͒ surface ͑dotted line͒ is chosen as energy zero.
of 650 °C ͑Ref.39͒.This regime is in contrast to nucleation far from equilibrium: experiments performed under such conditions suggest that nucleation proceeds via small clusters 21 whose size and shape are closely related to the Si͑111͒ ͑7 ϫ 7͒ reconstruction.
For simplicity, we restrict our analysis to structures with ͑001͒ or ͑111͒ surfaces or interfaces.Realistic island shapes are more complex ͑see, e.g., Refs.21 and 22͒, and have additional facets.However, already the assumed simple island shape allows for a preliminary estimate of the differences in the stable island size on Si͑001͒ and Si͑111͒.Specifically, we consider pyramid-shaped islands of MnSi with a square base ͓for Si͑001͒ substrate͔, or triangular base ͓for Si͑111͒ substrate͔, both with base length a. Figure 3 schematically illustrates island formation on both substrates.It is supposed that the island forming on Si͑001͒ has two ͕111͖ and two ͕1 ¯1 ¯1 ¯͖ facets, and the tetrahedral island on Si͑111͒ has three ͕001͖ facets, cf.Fig. 3.For Si͑001͒, we additionally consider bipyramids ͑with an inverted pyramid extending into the substrate͒, in the following called "iceberg," see Fig. 3͑d͒, with both ͕111͖ and ͕1 ¯1 ¯1 ¯͖ interfaces between the island and the substrate.In general, the material in the island is assumed to have the B2 crystal structure, whose ͑001͒ and ͑111͒ surface and interface energies we have calculated in this work.For the iceberg island on Si͑001͒ with only ͕111͖ and ͕1 ¯1 ¯1 ¯͖ surfaces and interfaces, we are in position to consider the B20 structure for the island material, too.
Next, we calculate the energy difference ⌬E between the MnSi island ͑on an otherwise clean, reconstructed substrate surface͒ and a hypothetical homogeneous film of MnSi containing the same amount of Mn.The thickness of such a film is denoted by d, see Fig. 3͑a͒.We are primarily interested in the stability of films with the B2 structure with respect to island formation.Therefore this structure is assumed for the film.⌬E can be written as where E film is the formation energy of a film covering an area A film , E island is the energy associated with formation of an island of volume V, and the last term ⑀V accounts for energy changes associated with a possibly different crystal structure of the film and the island.This term is important if the B2-like film transforms into an island of B20 structure.In this case, ⑀ = 11 meV/ Å 3 is the difference in the specific formation enthalpy of the two structures per molar volume of MnSi.Moreover, also the strain state of the film and the island will generally differ.However, due to the good lattice match of both the B2 and the B20 structure with Si, the contribution from strain is small ⑀ Ͻ 2 meV/ Å 3 and therefore neglected in the following.The first two contributions to ⌬E in Eq. ͑7͒ are given by Here ␥ film is the formation energy of the film per unit area, as defined in Eq. ͑5͒.N is the number of facets of the island and A facet is their area.These two quantities follow from geometrical considerations, and are given in Table II.Likewise, the area A interface of the interface between substrate and island is determined geometrically, and specified in Table III.
␥ substrate and ␥ surface MnSi are the surface energy of the bare substrate and of the island facets.Both quantities are obtained from our DFT calculations according to Eq. ͑1͒ and ͑2͒.Their values are summarized in Table IV.Note that ␥ form and ␥ interface are evaluated with respect to being in equilibrium with the MnSi structure of the corresponding island, i.e., Mn =−E MnSi,j coh + E Si coh , where j = ͑B2,B20͒, depending on the crystal structure of MnSi in question.In general, Eq. ͑9͒ gives a lower bound for the energy of island formation, since the additional energy cost associated with the formation of edges and/or with imperfect surface reconstructions on very small facets has been omitted.However, for larger islands, these contributions are negligible compared to the surface contribution ͑see, e.g., the discussion in Ref. 40 for InAs islands͒.
Due to mass conservation, the island base a, film thickness d, and film area A film , are related by the equation FIG. 3. Schematic illustration of ͑a͒ formation of a film with thickness d and ͑b͒ island with island base length a.On the Si͑111͒ substrate, tetrahedral islands with B2͑001͒ facets ͑b͒ will form, whereas pyramid islands ͑c͒ with B2͑111͒ facets will form on the Si͑001͒ substrate.The "iceberg" island ͑d͒ with B2 structure can form on both Si substrates.
Here, ␣ is the angle between the island facets and the substrate and v mol island and v mol film are the molecular volumes of MnSi in the crystal structure of the island and film, respectively.The calculated volume per MnSi formula unit in the B20 and B2 crystal structures is 56.08 and 43.68 Å 3 , respectively.
Solving the above equation for A film and inserting the resulting expression into Eq.͑8͒ allows us to eliminate this quantity.The total energy difference between film and island per unit volume can be expressed as

͑11͒
A positive value of ⌬E indicates that MnSi film formation is energetically favorable compared to island formation for the specified substrate orientation and Mn coverage, while a negative value indicates the stability of the islands.The condition ⌬E = 0, i.e., setting the right-hand side of Eq. ͑11͒ to zero, defines the volume V 0 for which the island first becomes stable.As an example, we calculate this island volume and the corresponding base length a 0 explicitly for deposition of 3 ML of manganese.
For clarity, the steps of this calculation are described in the following.First, we need to calculate the surface energy of both the Si substrates and the MnSi surfaces that appear as island side facets.The numerical results are collected in Table IV.According to our GGA-PBE calculations, the surface energy ␥ substrate of the Si͑111͒ surface is less than that of Si͑001͒, both in the reconstructed and in the unreconstructed case.This is in agreement with the results of previous LDA calculations. 41Similarly, the ͑111͒ surface of MnSi films with B2 structure is found to be somewhat more stable than the ͑001͒ surface.The average surface energy of the ͑111͒ and ͑1 ¯1 ¯1 ¯͒ surfaces of MnSi in the B20 phase is comparable to the same surfaces of the B2 phase.The lower surface energies of the ͑111͒-oriented surfaces compared to the ͑001͒ surface may be related to the surface termination with threefold or fourfold coordinated Si adatoms, respectively.Whereas threefold coordinated Si adatoms are low-energy motifs ͓see the Si͑111͒ ͑7 ϫ 7͒ reconstruction͔, Si atoms with four bonds all pointing into the same half-space are energetically more costly.Generally, the surface energies for MnSi, for the specified value of Mn , are higher than those of the ͑reconstructed͒ Si surfaces of the same orientation.Under these conditions, MnSi will not form a wetting layer on silicon; we rather expect island growth in the Volmer-Weber mode for sufficiently large coverage of manganese.
Next, the values collected in Tables II and III are to be inserted into Eq.͑11͒.Table III contains the formulas for calculating the interface area and the island volume of various cases.Moreover, the interface energy, and the final result, the base length a 0 of the smallest stable island, are given.For Si͑001͒, we consider three possible kinds of islands, a pyramid with four ͑001͒ facets with B2 structure, an iceberg island with B2 structure and an iceberg island with B20 structure.For the pyramid, the interface is between the B2͑001͒ and the bare Si͑001͒ surface.For the iceberg island, there are two ͑111͒ and two ͑1 ¯1 ¯1 ¯͒ interfacial facets between the B2 ͑or B20͒ structure of MnSi and Si͑111͒.Because the B20 structure is incompatible with ͑001͒ lattice planes of Si, the energy of such an interface is not calculated, and the formation of an iceberg island with B20 structure on Si͑111͒ is discarded.The interface of a pyramid on Si͑001͒ is a square with a 2 , while icebergs have four interfacial facets with an area of ͑ ͱ 3 / 4͒ a 2 each.The interface of a pyramid island on Si͑111͒ has a triangular base with an area of ͑ ͱ 3 / 4͒ a 2 , while in case of the iceberg the interface area consists of three such equilateral triangles.From the minimum stable island size a 0 ͑right column of Table III͒ the following conclusions can be drawn: On Si͑001͒, the stable island size is generally smaller than on Si͑111͒.Therefore Volmer-Weber growth is to be expected for MnSi on Si͑001͒, as observed experimentally. 22,42In contrast, a ͑metastable͒ film of MnSi on Si͑111͒ will only decay into islands if fluctuations during growth overcome some threshold value.As one can see from the values of the stable island size a 0 , iceberg islands with B20 structure on Si͑001͒ are the smallest islands to become stable.In general, the iceberg variants of the islands are found to be more favorable than a pyramid shape on both substrates.On Si͑001͒, the icebergs having B20 structure are stabilized additionally due to the transition from the B2 film to the B20 island structure.This is consistent with the observation of a ͑ ͱ 3 ϫ ͱ 3͒ struc- ture on the side facets of MnSi islands on Si͑001͒, 22 indicating that the B20 structure has indeed formed in these islands.The base length of a stable island of the B20 structure on Si͑001͒ with iceberg shape is 15 Å, which amounts to only about four surface lattice constants of Si͑001͒, or of MnSi ͑in the B20 structure͒.In contrast, stable pyramid islands on Si͑111͒ are much larger than the ͑7 ϫ 7͒ unit mesh of the surface reconstruction.Their base length is about 30 times the lattice constant of MnSi in the B20 structure.
In a situation where Gibbs' theory of nucleation is applicable, we expect that islands nucleate more easily on the Si͑001͒ compared to the Si͑111͒ substrate.For an island size larger than the stable value a 0 , ⌬E becomes negative, indicating that further growth of the islands is exothermic.This is in agreement with the recently observed Ostwald ripening of MnSi islands on Si͑001͒. 42,43 SUMMARY In summary, we have investigated the morphology of manganese monosilicide films on the Si͑111͒ substrate by means of DFT calculations.The Si͑111͒ substrate is compatible with Mn monosilicide in both the B20 and B2 structures, while the formerly studied Si͑001͒ substrate only matches with the ͑so far not yet synthesized͒ B2 structure of MnSi.We find that the thermodynamic stability of MnSi films on Si͑111͒ is generally higher than on Si͑001͒.On Si͑111͒, films with the B20 structure are energetically more favorable than those with B2 structure.A Si-capping layer is found to enhance the stability of all MnSi films investigated.Moreover, we calculate the stable island size of pyramidal or bipyramidal MnSi islands on both Si͑001͒ and Si͑111͒.While already very small islands with a base length of a few lattice constants ͑if edge energy contributions are neglected͒ are found to be more stable than a homogeneous film on Si͑001͒, much large nuclei are required on Si͑111͒ to destabilize a MnSi film.Both the MnSi films on Si͑001͒ and Si͑111͒ show large magnetic moments at the Mn atoms near the surfaces and interfaces.The values for the spin polarization at the Fermi level in MnSi/ Si͑111͒ are even higher than those found in our previous calculations for Mn/ Si͑001͒.Therefore we conclude that experimental preparation of an MnSi film on Si͑111͒ is probably easier to achieve than on Si͑001͒, and that such films could even be more useful from the viewpoint of spintronics applications.

FIG. 1 .
FIG. 1. ͑Color online͒ MnSi film with B2 structure ͑a͒ in top view and ͑b͒ in side view, and with B20 structure ͑c͒ in top view and ͑d͒ in side view.In the top views, the ͱ 3 ϫ ͱ 3 cell is indicated by dashed ͑red͒ lines.Small balls represent Si atoms ͑with top-layer atoms shaded͒, and large dark balls Mn atoms.The hexagonal bonding pattern in ͑c͒ displays the Si substrate lattice.

TABLE I .
Formation energy E form with respect to reservoirs of elemental Mn and Si, and average spin magnetic moment per Mn atom in the interface ͑M I ͒, surface ͑M S ͒ and central layers ͑M C ͒ of MnSi epitaxial films on Si͑111͒ with B 2 structure ͑1.5 ML Mn cov-erage͒ and B20 structure ͑4 / 3 ML Mn coverage͒.

TABLE II .
Film thickness d ͑Å͒ for an MnSi film formed after deposition of 3 ML of manganese, facet area A facet , energy of film formation per area ␥ film ͑meV/ Å 2 ͒, and surface energy of the bare surface ␥ substrate ͑meV/ Å 2 ͒.For Si, reconstructed surfaces are considered, while for MnSi relaxed ͑1 ϫ 1͒ surfaces are considered, ignoring possible ͑hitherto unknown͒ reconstructions.

TABLE III .
Interface area A interface , volume of the islands V island , interface energy per area ␥ interface , and stable island size a 0 .Bulk MnSi in the B2 structure is considered as reservoir for calculating the chemical potential of Mn.

TABLE IV .
Surface energy ␥ surface ͑meV/ Å 2 ͒ of the nonreconstructed Si͑001͒ and Si͑111͒ surfaces and of the stable reconstructed surfaces Si͑001͒͑2 ϫ 2͒ and Si͑111͒͑7 ϫ 7͒.The GGA-PBE value for the surface energy of the reconstructed Si͑111͒͑7 ϫ 7͒ is extrapolated from LDA calculations using ultrasoft pseudopotentials ͑Ref.41͒ by comparing the surface energies of smaller cells in both methods.Moreover, surface energies of MnSi in the B2 structure are given for the unreconstructed ͑001͒ and ͑111͒ ͑1 ϫ 1͒ surfaces.For MnSi in the B20 structure, the quoted number is the average surface energy of the ͑111͒ and the ͑1 ¯1 ¯1 ¯͒ surface.Bulk MnSi in the B2 structure is considered as a reservoir for the chemical potential of Mn.