Metastable phases, such as bcc Co or Ni and hcp Fe or Ni, reportedly possess extraordinary magnetic properties for epitaxial ultra-thin films. To understand phase stability of these epitaxy-oriented phases upon substrate lattices, we calculated novel phase diagrams of Co, Fe, and Ni ultrathin films by considering the chemical free energy, elastic strain energy, and surface energy. Verified by experimental data in the literatures, the stable epitaxy-oriented phases are readily identified from the phase diagrams. The stabilization of these metastable phases is determined by the interplay between orientation-dependent elastic strain energy and surface energy.

Magnetic properties and performances of metallic thin film magnets such as magnetic anisotropy, magnetization reversal processes, and domain wall magneto-resistance are crucial for designing spintronic devices.1–4 In particular, magnetic ultra-thin films made of 3d transition metals, e.g., cobalt (Co),5,6 iron (Fe),7,8 and nickel (Ni),9,10 have been investigated extensively in order to obtain the desirable characteristics for specific applications. Compared with the bulk, the magnetic moment and anisotropies of ultrathin films are remarkably different and sensitively dependent on the crystal structure and thermal stability of the film.11–13 However, massive studies of the epitaxial Fe, Co, and Ni ultrathin films suggest that the differences in crystal structures between film and substrate can lead to unpredictable fundamental magnetic properties. For instance, it has been proved experimentally that metastable bcc structure of Co14–16 or Ni17,18 and the hcp structure of Fe19,20 or Ni21–23 can be stabilized in ultrathin films, which are, however, difficult to achieve in bulk materials. The unstable crystal structures provide an extra degree of freedom to tailor film morphology and thus to improve the physical property.24 Strain-engineering, i.e., a technique that makes use of film-substrate misfit strain to grow epitaxial film, has recently been employed to improve the properties25 of materials such as ferroics,26 semiconductors,27 quantum dots,28 as well as other devices.29 Under biaxial epitaxial strain, the mobility of transistors, catalytic activity, band structure, and transition temperatures can be altered at will. It is anticipated that the crystal structures and magnetic properties of the Fe, Co, and Ni ultrathin films could be manipulated and purposely designed via strain-engineering through precise control of biaxial strain.24 

Biaxial strain arises from the difference in the lattice constants between the epitaxial film and substrate. Without relaxation, the lattice of ultrathin film is completely constrained by that of the substrate, mostly resulting in the change of crystallographic structure and orientation.30 By tuning the lattice constants of the substrate artificially, the elastic field and the thermodynamic state of the film could be correspondingly manipulated. However, a comprehensive model that is able to predict phase stability and crystallographic structure and orientation as functions of substrate lattice constants seems to be still lacking.24 In this letter, we present a theoretical model that is capable of calculating total free energy of the epitaxial film, including chemical free energy, elastic strain energy, and surface energy, and predicting the stable crystallographic structures and orientations. Phase diagrams of Fe, Co, and Ni ultrathin films are established to describe relationships between thermodynamically stable phases and such corresponding conditions as crystallographic orientations, temperature, and film thickness.

Assuming the thickness of epitaxial film is far below the critical thickness and thus film is coherent with the substrate, the total free energy of the system can be formulated as the sum of the chemical Gibbs free energy, magnetic excess energy, elastic strain energy, and surface energy,

F tot = G chem + G mag + E elastic + E surface ,
(1)

where Gchem is the chemical free energy that can be directly calculated by the molar Gibbs free energy of the bulk, Gm, e.g., Gchem = VfGm, with Vf is the volume of the film, Gmag is the magnetic excess energy of the ferromagnetic phase, Eelastic is the coherency strain energy that can be calculated by the hook’s law, e.g., E elastic = 1 / 2 V f C ijkl ε ij ε kl , with Cijkl and εij being the elastic constant and misfit strain, respectively. The misfit strain of epitaxial film can be calculated by ε ij = δ ij a f a s / a s , where δij is the Kronecker delta, af and as are the lattice constants of the film and substrate, respectively. Temperature dependence of both elastic constants and lattice constants should be considered under variable temperature conditions.31 The surface energy of the film, Esurface, can be obtained by the bond-cutting theory,32 e.g., E surface = A 1 w / u Δ H v 0 N 0 2 / 3 ,33,34 where A is the surface area, u is the number of nearest neighbors of an atom in the bulk, and w is the number of neighbors of an atom on the film surface. Therefore, the term (1 − w/u) means the number of dangling bonds of an atom on the surface. ΔHv0 is the enthalpy of evaporation of the bulk material, and N0 is the number of atoms per unit volume.

In order to practically apply our results to the fabrication of ultra-thin films, we here focus on several common crystal structures and planes of pure Co, Fe, and Ni, such as the bcc and fcc phases with family planes of {100}, {110}, and {111}, and the hcp phase with family planes of {0001} and { 10 1 ̄ 0 } . Figure 1 shows the selections of reference frame and lattice parameters of different family planes in the bcc, fcc, and hcp phases, as well as those in substrate. Considering the diversity of substrate crystal structures, substrates having rectangular lattice surfaces were considered for simplicity but without losing generality. The elastic constants for specific crystal orientations were transformed by the rotation matrices, e.g., C ijkl = T ip T iq T im T in C ijkl , where Tip, Tiq, Tim, and Tin are the rotation matrices. The chemical free energies used in the present calculation were taken from SGTE Database35 and shown in Table I. The ΔHv0 of the pure Co Fe and Ni were set to be 428, 415, and 430 kJ/mol, respectively.36 The thickness-dependence Curie temperatures of pure Co, Fe, and Ni film were calculated by using the model developed by Zhang and Willis.37 The magnetic excess energies, dependent on both film thickness and Curie temperature, were calculated by the model of Hillert and Jarl,38 which has been successfully implemented in the CALPHAD (CALculation of PHAse Diagrams) method. The lattice constants used in the present calculations are listed in Table II.39 The elastic constants can be acquired from experimental measurement or first principle calculations.39 In order to simplify the model, the epitaxial film was assumed to be fully coherent with the substrate lattice, meaning that the film thickness is far below the critical thickness and no relaxation or misfit dislocation exists.

FIG. 1.

The coordinate selections and lattice parameters of different family planes in the bcc, fcc, and hcp phases and substrate.

FIG. 1.

The coordinate selections and lattice parameters of different family planes in the bcc, fcc, and hcp phases and substrate.

Close modal
TABLE I.

Gibbs free energies used in the present calculations.35 

Elements Structure Gibbs free energy expressions (J/mol)
Co  Bcc  737.832 + 132.750 762 T − 25.086 1 Tln(T) − 2.654 739 × 10−3T2 − 0.173 48 × 10−6T3 + 72 527 T−1 
  Fcc  3 248.241 + 132.652 21 T − 25.086 1 Tln(T) − 2.654 739 × 10−3T2 − 0.173 48 × 10−6T3 + 72 527 T−1 
  Hcp  310.241 + 133.366 01 T − 25.086 1 Tln(T) − 2.654 739 × 10−3T2 − 0.173 48 × 10−6T3 + 72 527 T−1 
Fe  Bcc  1 225.7 + 124.134 T − 23.514 3 Tln(T) − 4.397 52 × 10−3T2 − 0.058 927 × 10−6T3 + 77 359 T−1 
  Fcc  −236.7 + 132.416 T − 24.664 3 Tln(T) − 3.757 52 × 10−3T2 − 0.058 927 × 10−6T3 + 77 359 T−1 
  Hcp  −2 480.08 + 136.725 T − 24.664 3 Tln(T) − 3.757 52 × 10−3T2 − 0.058 927 × 10−6T3 + 77 359 T−1 
Ni  Bcc  3 535.925 + 114.298 T − 22.096 Tln(T) − 4.840 7 × 10−3T2 
  Fcc  −5 179.159 + 114.298 T − 22.096 Tln(T) − 4.840 7 × 10−3T2 
  Hcp  −4 133.159 + 119.109 2 T − 22.096 Tln(T) − 4.840 7 × 10−3T2 
Elements Structure Gibbs free energy expressions (J/mol)
Co  Bcc  737.832 + 132.750 762 T − 25.086 1 Tln(T) − 2.654 739 × 10−3T2 − 0.173 48 × 10−6T3 + 72 527 T−1 
  Fcc  3 248.241 + 132.652 21 T − 25.086 1 Tln(T) − 2.654 739 × 10−3T2 − 0.173 48 × 10−6T3 + 72 527 T−1 
  Hcp  310.241 + 133.366 01 T − 25.086 1 Tln(T) − 2.654 739 × 10−3T2 − 0.173 48 × 10−6T3 + 72 527 T−1 
Fe  Bcc  1 225.7 + 124.134 T − 23.514 3 Tln(T) − 4.397 52 × 10−3T2 − 0.058 927 × 10−6T3 + 77 359 T−1 
  Fcc  −236.7 + 132.416 T − 24.664 3 Tln(T) − 3.757 52 × 10−3T2 − 0.058 927 × 10−6T3 + 77 359 T−1 
  Hcp  −2 480.08 + 136.725 T − 24.664 3 Tln(T) − 3.757 52 × 10−3T2 − 0.058 927 × 10−6T3 + 77 359 T−1 
Ni  Bcc  3 535.925 + 114.298 T − 22.096 Tln(T) − 4.840 7 × 10−3T2 
  Fcc  −5 179.159 + 114.298 T − 22.096 Tln(T) − 4.840 7 × 10−3T2 
  Hcp  −4 133.159 + 119.109 2 T − 22.096 Tln(T) − 4.840 7 × 10−3T2 
TABLE II.

Lattice parameters used in the present calculations.39 

Elements Structure Lattice parameters (Å)
Co  bcc  a = 2.82 
  fcc  a = 3.53 
  hcp  a = 2.50c = 4.06 
Fe  bcc  a = 2.84 
  fcc  a = 3.45 
  hcp  a = 2.46c = 3.90 
Ni  bcc  a = 2.80 
  fcc  a = 3.52 
  hcp  a = 2.48c = 4.09 
Elements Structure Lattice parameters (Å)
Co  bcc  a = 2.82 
  fcc  a = 3.53 
  hcp  a = 2.50c = 4.06 
Fe  bcc  a = 2.84 
  fcc  a = 3.45 
  hcp  a = 2.46c = 3.90 
Ni  bcc  a = 2.80 
  fcc  a = 3.52 
  hcp  a = 2.48c = 4.09 

Figure 2 shows the calculated phase diagram of pure Co film with different crystal structures and crystal orientations, where different colors indicate specific stable phase. The results were calculated for 300 K with film thickness of 10 nm. It can be clearly seen that the bcc, fcc, and hcp phases with different crystal planes are stabilized by the constraint of substrate, depending on lattice parameters of the substrate. Notice that the phase diagram is symmetrical with respect to the line along x = y since two axes are exchangeable. The solid and dash lines define the critical misfit between film and substrate, i.e., 5% and 10%, respectively. The metastable phases in bulk Co, i.e., bcc and fcc structure phases, appear and extend in the phase diagram. In contrast, the hcp phase in bulk is stabilized within a small and limited area in the phase diagram. For most structure-oriented phases, the stable phase regions locate around the zero misfit point except the fcc {111} phase. The similar lattice parameters for fcc {111}, hcp {0001}, and hcp { 10 1 ̄ 0 } phases lead to the overlap of their low elastic constrained region. The hcp {0001} phase region occupies the lower misfit area leading to the separation of the fcc {111} phase region. The region of bcc {111} phase with low elastic strain energy disappears in the calculated phase diagrams when the lattice parameter of the substrate ranges in 2 ∼ 5 Å. The predictions agree well with the experimental observations14,40–48 available for bcc, fcc, and hcp structured Co films grown on different substrates with specific crystal planes.

FIG. 2.

Calculated phase diagrams of epitaxy and lattice structure for ultra-thin epitaxial Co films as a function of substrate lattice parameter. Substrates having rectangular lattice surfaces and varying lattice parameters were considered.

FIG. 2.

Calculated phase diagrams of epitaxy and lattice structure for ultra-thin epitaxial Co films as a function of substrate lattice parameter. Substrates having rectangular lattice surfaces and varying lattice parameters were considered.

Close modal

The calculated epitaxy-oriented phase diagram of pure Fe film with different crystal structures and crystal orientations is shown in Fig. 3. Three metastable phases of epitaxial Fe film with various crystal orientations are also stabilized by the misfit strain of the substrate. Compared with the Co phase diagram in Fig. 2, bcc {110} phase of Fe film has larger stable phase region than that of Co film, whereas the fcc {100} phase of Fe film almost reduces to the 10% misfit region. It can be seen that the hcp { 10 1 ̄ 0 } phase occupies the lower misfit region. Additional calculation indicates that the hcp { 10 1 ̄ 0 } phase region decreases with the increasing of film thickness, while the hcp {0001} phase is confined within the large misfit region by the substrate lattice. The calculated results are consistent with the reported experimental data.19,49–54

FIG. 3.

Calculated phase diagrams of epitaxy and lattice structure for ultra-thin epitaxial Fe films as a function of substrate lattice parameter. Substrates having rectangular lattice surfaces and varying lattice parameters were considered.

FIG. 3.

Calculated phase diagrams of epitaxy and lattice structure for ultra-thin epitaxial Fe films as a function of substrate lattice parameter. Substrates having rectangular lattice surfaces and varying lattice parameters were considered.

Close modal

Figure 4 shows the calculated epitaxy-oriented phase diagram of Ni varying with the lattice parameters of the substrate. It is obvious that the fcc {100} phase of Ni film has an extended phase region while the hcp { 10 1 ̄ 0 } phase sustains in a small phase region. It shows that the fcc {111} phase instead of hcp {0001} phase exists in the small misfit region whereas the larger misfit regions favor the fcc {110} phase, which are in good accordance with the experimental observation.55–60 

FIG. 4.

Calculated phase diagrams of epitaxy and lattice structure for ultra-thin epitaxial Ni films as a function of substrate lattice parameter. Substrates having rectangular lattice surfaces and varying lattice parameters were considered.

FIG. 4.

Calculated phase diagrams of epitaxy and lattice structure for ultra-thin epitaxial Ni films as a function of substrate lattice parameter. Substrates having rectangular lattice surfaces and varying lattice parameters were considered.

Close modal

Stabilization of the epitaxy-oriented phases in substrate constrained films is readily recognized in our calculated results. By tuning lattice parameters of the substrate, the metastable phases in the bulk materials, such as bcc phase in Co or Ni and hcp phase in Fe or Ni, could be stabilized by the misfit strain. In general, the stabilization of oriented epitaxial phases is ascribed to the energy minimum of the system. For epitaxial ultra-thin films, chemical free energy, elastic strain energy, and surface energy could individually dominate the energy state of the system. For instance, the hcp phase of Co is stable at low temperature while fcc phase presents at high temperature, thus resulting in the extended phase regions in the calculated epitaxy-oriented phase diagram, as shown in Fig. 2, which indicates that the chemical free energy dominates the total system energy. Because of the lattice similarity of the fcc {111} and hcp {0001} planes, their corresponding phase regions shown in the phase diagrams are next to each other or even overlapped. The interplay between orientation-dependent elastic strain energy and surface energy determines the area of phase region, especially for those metastable phases in bulk materials.

As for hcp {0001} and hcp { 10 1 ̄ 0 } phases, the rectangular lattices were used to fit the substrate lattice. It is apparently due to the ratios of c/a for Co, Fe, and Ni being close to the ideal value 1.633, which leads to the overlap of the low misfit region. Consequently, the stabilities of the hcp {0001} and hcp { 10 1 ̄ 0 } phases tend to be governed by the surface energy. Closer inspections at higher temperature and larger film thickness show their insignificant effect on the phase diagrams of the film, therefore implying further increase in the chemical free energy with temperature fails to prevail over the elastic constraint energy between the film and the substrate. However, the large misfit strain and film thickness may cause great lattice distortion and generate misfit dislocations. According to the prediction of Zhang and Willis37 of the thickness dependence of the Curie temperature, the epitaxy-oriented phases considered in our work are all under Tc and thus present ferromagnetism. Hence, the ferromagnetic excess energy was considered in the calculation, but it hardly changes the epitaxy-oriented phase diagrams.

Most of the experimental date shown in Figs. 2-4 were determined by the reflection high-energy electron-diffraction or low energy electron diffraction, which are generally reliable for validating the calculated results. In addition, the transmission electron microscopy images and convergent beam electron diffraction patterns reported by Mangan et al.14 clearly indicate the growth of bcc Co (110) film on GaAs (110) substrate, consistent with the present results. The reported scanning tunneling microscopy investigations of fcc Ni film growth on Cu (100) substrate57 and hexagonal structure of Ni film growth on the Ru(0001) substrate60 are also in accordance with the calculated results shown in Fig. 4.

In summary, epitaxy-oriented phase diagrams of ultra-thin epitaxial Co, Fe, and Ni films were calculated using a theoretical model by considering the chemical free energy, elastic strain energy, and surface energy. By tuning the two lattice parameters of the substrate, the stabilization of the epitaxy-oriented phases in substrate constrained films is readily distinguished from the phase diagrams. Such novel epitaxy-oriented phase diagrams are expected to shed light on utilizing strain-engineering to fabricate desirable stable magnetic ultra-thin films with specific crystal orientations.

This work was supported by the National Natural Science Foundation (Grant No. 51301146) of China (Y. Lu), the National Key Basic Research Program (973 Program) (Grant No. 2012CB825700), Ministry of Science and Technology of China (Grant No. 2014DFA53040), and the National Natural Science Foundation (Grant No. 51571168) of China.

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