Tailoring the formation of metastable Mg through interfacial engineering: A phase stability analysis

Highlights

• Metastable bcc Mg and hcp Nb can be stabilized under Mg/Nb multilayers at ambient conditions.

• A classical thermodynamic approach was used for predicting the bi-phase diagram, which agrees well with the experimental observation.

• The design of metastable phases can be performed through interfacial engineering in thin films.

Abstract

A classical thermodynamic approach has been used for describing the pseudomorphic growth in Mg/Nb multilayer films. The bi-phase diagram of these films has been predicted theoretically and the predictions were verified experimentally by growing multiple Mg/Nb thin films that were characterized through high resolution transmission electron micrograph (HRTEM) and X-ray diffraction (XRD) measurement. The good agreement between predictions and experiments shows that the stability of multilayer films can be explained through simple thermodynamic analysis on the competition between bulk and interfacial energies.

Introduction

On the basis of the underlying thermodynamic constraints, phase stability of constituents in metallic multilayer thin films can be markedly different from the stability of the same material in bulk form [1]. In the absence of size effects, metastable (and even unstable) phases can only be stabilized in bulk form under further constraints, such as elevated-pressures and temperatures. On the other hand, interfacial effects resulting from the growth of nanolayer systems has been shown to stabilize these metastable phases under ambient conditions, albeit under coherency constraints arising from the epitaxial growth of these films. The stabilization of these so-called pseudomorphic phases results from the competition between bulk and interfacial contributions to the free energies of the multilayer systems [2].

Over the past few decades, the stabilization of metastable phases has been observed in nanolayer thin films consisting of at least two layers of dissimilar materials. For example, the metastable face-center-cubic (fcc) structure of Ti was observed in Ti/Ag thin films [3], Ti/Al multilayers [4], [5], and Ti thin films on a (100) Si substrate [6]. The metastable tetragonal Ta phase was formed in Ta/Cu multilayers [7]. The metastable rhombohedral Fe phase was observed in the Fe/Sb multilayers [8]. Examples of stabilization are not limited to metastable phases. In fact, thin film conditions have been shown to stabilize structures that are mechanically unstable under bulk conditions. For example, bcc Zr has been shown to be stable in Nb/Zr multilayers [9], [10]. Although, bulk bcc Zr is dynamically unstable, with the high-symmetry phonon branch [$\xi \xi \xi$] (with $\xi =\frac{2}{3}$) exhibiting pronounced softening at low temperatures. The displacement pattern generated by this phonon mode corresponds to the $\mathit{bcc}\to \omega$ transformation [11]. The epitaxial conditions arising from the growth of Zr on Nb suppresses this unstable mode and stabilizes the bcc structure. Similarly, the fcc Nb structure has been observed to become stable in Cu/Nb multilayers [12]. Many studies reported the formation of the unstable bcc Co or the metastable hcp Cr in Co/Cr multilayers [13], [14], [15].

In the past few decades, there have been significant efforts to investigate the effect of epitaxial constraints on the phase stability of alloys. Zunger and collaborators [16], [17], [18], for example, have considered the effect of epitaxial constraints by constraining the lattice parameters of the alloy system along the epitaxial plane and have used cluster expansion techniques to predict theoretical phase diagrams of alloy systems. The theory considers long-range elastic interactions resulting from biaxially deforming a structure from its own ground state lattice parameters on the plane of the substrate. Formally, they used a mixed basis approach through the use of the so-called Constituent Strain Energy (CSE) in which the long-range elastic interactions are defined in reciprocal space, while the chemical configurational contributions are treated in the real space. This approach has already yielded important theoretical results, such as the stabilization of solid solutions—and even ordered phases—in systems, such as (In,Ga)N [17], that in the bulk state exhibit phase separation behavior.

While ab initio based microscopic approaches have been shown to be very powerful to investigate the effect of small scales (and coherency) on the phase stability of systems, other approaches based on classical (continuum) thermodynamic approaches have been shown to be rather useful. The so-called nano-CALPHAD (CALculation of PHAse Diagram) approach [19] proposes to use the number of atoms in the system as another thermodynamic variable. This approach results in the re-definition of thermodynamic concepts, such as the Gibbs phase rule, tie-lines, etc. Perhaps one of the salient aspects of this recent work is the realization that concepts used in bulk thermodynamics are sometimes not adequate to represent equilibria (i.e. phase diagrams) in nano-systems.

In the very specific case of nanolayer films, a formalism to understand the phase stability observed in experiments was developed by Dregia et al. [1]. Classical thermodynamics was applied to systematically understand equilibria in metallic multilayers. Specifically, the unit bilayer consisting of component A and B in their strain-free, stable crystal structures, is a reference state. Two independent thermodynamic degrees of freedom are used to completely define a thermodynamic state. The first degree of freedom is the bilayer thickness (λ), which is defined as the summation of the thicknesses of the individual component layers A and B ($\lambda =h_A+h_B$). The other degree of freedom corresponds to the overall composition of the bilayer and this is expressed in terms of the volume fractions of component A (f_A) and component B (f_B), satisfying $f_A+f_B=1$. The free energy of the reference bilayer is denoted by G. If at least one of the components of the bilayer is transformed to a metastable (or unstable) structure, the total free energy of the bilayer system is $G\text{'}$

$$G=N_A{G}_A+N_B{G}_B+2\gamma S$$

$$G\text{'}=N_A{G}_A^{\text{'}}+N_B{G}_B^{\text{'}}+2\gamma \text{'}S$$

where N_i and G_i are the number of atoms and chemical potential of species i (i=A and B), respectively. S and γ stand for the area of interface and the interfacial energy, respectively. Assuming that the interface area between layers A and B is invariant during the transformation, the specific free energy of formation of the bilayer—normalized per interfacial area—$\mathrm{\Delta }g$, is given by

$$\mathrm{\Delta }g=2\mathrm{\Delta }\gamma +[\mathrm{\Delta }{G}_A{f}_A+\mathrm{\Delta }{G}_B{f}_B]\lambda$$

In this equation, $\mathrm{\Delta }\gamma$ is the change in the interfacial energy upon transformation, $\mathrm{\Delta }{G}_i$ is the allotropic free energy change (normalized per volume) of the reference component, i. Note that $\mathrm{\Delta }\gamma$ is affected by chemical and structural interfacial effects and that $\mathrm{\Delta }{G}_i$ may include not only the difference in the thermodynamic bulk free energies of the phases but also include other volume effects, such as those associated to strain energy.

If one considers that $1/\lambda$ and f_i are the thermodynamic degrees of freedom in the system, it is then possible to determine the thermodynamic state of the system as a function of $1/\lambda$ and f_i and encode the results in so-called bi-phase diagrams. In these diagrams, the transitions from different configurations in the $A/B$ multilayers can be expressed in terms of so-called coexistence lines.

A simplification can be made if one assumes that $\mathrm{\Delta }\gamma$ and $\mathrm{\Delta }{G}_i$ are independent of the layer thickness. Along the coexistence lines, $\mathrm{\Delta }g$ is equal to zero, in an analogous fashion to the conditions for equilibrium for coexistence lines in single-component, multi-phase systems. From this condition, a (linear) function relating $1/\lambda$ and f_i can be derived. The slope and intercepts of each of the bi-phase boundary lines can be calculated from $\mathrm{\Delta }{G}_i$, $\mathrm{\Delta }\gamma$, and the constrained coordinates (f_i, $1/\lambda$) such as the (0,0) and the (1,0) points. This approach was used to predict the bi-phase diagram of Co/Cr and Al/Ti multilayer thin-films and compared to experimental results. The stabilization of the metastable phases was described by the decrease of the interfacial energy ($\mathrm{\Delta }\gamma$) [1], [20].

Based on the pioneering work of Dregia et al., there were some studies including the effect of coherency strains in the theoretical models. For example, Banerjee et al. investigated the pseudomorphic growth of fcc Ti in Ti/Al multilayers. The strain energy (W_el) term, which is derived from a biaxial elastic modulus (Y_i) and a coherency strain of component i (ε_i), is included into Eq. (3). The predicted bi-phase diagram of Ti/Al multilayers compared well to experimental data. Thompson et al. studied the Nb/Zr multilayers [10] and proposed the bi-phase diagram by using the classical thermodynamic approach, including both the coherency effect and the chemical component of the interfacial energy. For the structural component of $\mathrm{\Delta }\gamma$, a misfit dislocation network model, evaluated from Bollman's O-lattice [21] formalism, was used for a semi-coherent interface.

The surface and interfacial energies of the solid are difficult to determine by using experimental or theoretical means [22], [23]. An alternative method for estimating interfacial free energies is to first consider that a liquid can be thought of as a solid with a high concentration of dislocation cores. At the melting point, a solid–solid interfacial energy (γ_ss) is shown to be approximately twice of the solid–liquid interfacial energy (γ_sl) [24], [25]. This concept was used for estimating the first order approximation of γ_ss from mean values of both constituent substances in multilayers. It was used to approximate interfacial energies of various interfaces such as Cu/Mo, Cu/W, Ni/Mo, Ni/W, Cu/Ni and Mo/W systems, and these approximated values provided a good agreement with other works [25]. Recently, Li et al. applied this method for estimating an incoherent interfacial free energy (γ_inc) expressed in Eq. (4) [2]

$${\gamma }_{\mathit{inc}}\approx 2{\gamma }_{\mathit{sl}}=\frac{4\overline{d}{\overline{S}}_{\mathit{vib}}{\overline{H}}_m}{3\overline{V}R}$$

Properties present in Eq. (4) are the mean values of both constituent elements. ${\overline{S}}_{\mathit{vib}}$ and ${\overline{H}}_m$ are the mean values of the vibrational melting entropy and the melting enthalpy, respectively. $\overline{d}$ and $\overline{V}$ denote the mean values of the atomic diameter and molar volume, respectively. The R is the universal gas constant.

Li et al. estimated the interfacial energy in coherent or semi-coherent interfaces with a small interface misfit (γ_c) from ${\gamma }_c={\mathit{Nu}}_d/S$. N and S denote the number of dislocations and the interfacial area between the film and the substrate. u_d defines the misfit energy of a single dislocation. This u_d can be derived for an edge dislocation without dislocation core energy [26]

$$u_d\approx \frac{{\mu }_A{\mu }_B{b}^{2}l}{2\pi ({\mu }_A+{\mu }_B)(1+\nu )}\left[\ln \frac{\delta }{b}+1\right]$$

where the Burger vector $b=\overline{d}$ and l is the dislocation length. μ and ν are the shear modulus and the Poisson ratio, respectively. δ represents a half of the distance between two adjacent dislocations

$$\frac{\delta }{\overline{d}}=\frac{d_A}{2|d_A-d_B|}$$

and the total number of dislocations on the interface is $N=S/\delta$. After substituting $\delta /\overline{d}$ and N into Eq. (5), γ_c can be found from

$${\gamma }_c=\frac{{\mu }_A{\mu }_B\overline{d}|d_A-d_B|}{\pi ({\mu }_A+{\mu }_B)(1+\nu )d_A}\left[\ln \frac{d_A}{2|d_A-d_B|}+1\right]$$

The expressions for γ_inc and γ_c can be used to obtain the change in interfacial energy ($\mathrm{\Delta }\gamma$) according to the phase transformation in multilayers. Finally, taking into account the coexistence condition $\mathrm{\Delta }g=0$, one can obtain expressions for each slope of the boundary lines in a typical bi-phase diagram in which constituents A and B have different ground state crystal structures [2]. This approach was used for investigating the bi-phase diagrams of the Co/Cr, the Ti/Nb, the Zr/Nb and the Ti/Al multilayers [2] and it has recently been used to explain the observation of fcc Nb in Cu/Nb multilayers [12].

γ_inc and γ_c, which are expressed in Eqs. (4), (7), are structural contributions to the total interfacial energy (γ^struc). Another component of the total interfacial energy is a chemical contribution (γ^chem) that is related to the chemical effects that arise as atomic bonds are broken/modified as interfaces are created. Thompson et al. estimated γ^chem for bcc Nb/bcc Zr and hcp Nb/hcp Zr multilayers by using a simple nearest neighbor bond model [10]. In this approximation, the binary solution thermodynamics interaction energy (ω), which is given in Eq. (8), can be used for estimating γ^chem

$$\omega =\frac{\Omega }{Z}$$

where $\Omega$ is the regular solution parameter and Z is the coordination number at the interface [10]. The interaction energy between constituent atoms (${\omega }_{(A-B)}$) in a particular structure can be calculated from the bond energy (ϕ) as shown in the following equation:

$${\omega }_{A-B}={\varphi }_{A-B}-\frac{1}{2}({\varphi }_{A-A}+{\varphi }_{B-B})$$

For instance, ${\omega }_{A-B}^{\mathit{bcc}}={\varphi }_{A-B}^{\mathit{bcc}}-1/2({\varphi }_{A-A}^{\mathit{bcc}}+{\varphi }_{B-B}^{\mathit{bcc}}$) can be used for calculating the interaction energy in a bcc system. The chemical contribution to the interfacial energy per atom for bcc system (γ^chem_bcc) can be calculated from the excess energy per atom on the A side ($\mathrm{\Delta }{E}_{\mathit{atom}}^A$) and the excess energy per atom on the B side ($\mathrm{\Delta }{E}_{\mathit{atom}}^B$), which is expressed in the following equation:

$${\gamma }_{\mathit{bcc}A/\mathit{bcc}B}^{\mathit{chem}}=\frac{1}{2}(\mathrm{\Delta }{E}_{\mathit{atom}}^A+\mathrm{\Delta }{E}_{\mathit{atom}}^B)=2[{\varphi }_{A-B}^{\mathit{bcc}}-\frac{1}{2}({\varphi }_{A-A}^{\mathit{bcc}}+{\varphi }_{B-B}^{\mathit{bcc}}\left)\right]=2{\omega }_{A-B}^{\mathit{bcc}}=0.25{\Omega }_{A-B}$$

This approach was used for estimating the magnitude of γ^chem of bcc Nb/bcc Zr (${\gamma }_{\mathit{bcc}\mathit{Nb}/\mathit{bcc}\mathit{Zr}}^{\mathit{chem}}~91\mathrm{mJ}/{\mathrm{m}}^2$) and hcp Nb/hcp Zr (${\gamma }_{\mathit{hcp}\mathit{Nb}/\mathit{hcp}\mathit{Zr}}^{\mathit{chem}}~44\mathrm{mJ}/{\mathrm{m}}^2$) [10]. However, γ^chem of bcc Nb/hcp Zr cannot be estimated by using this approximation due to the difference in the number of out-of-plane bonds between bcc Nb (110) and hcp Zr (0001) planes. In the section below we present the application of this model to obtain the chemical contributions to the interfacial energy in bcc/bcc Mg–Nb multi-layers as a means to estimate the likely magnitude of chemical effects.

In this work, we use the ideas for the establishment of equilibrium states in multilayer thin films, initiated by Dregia [1] and modified by Li [2], to investigate the bi-phase diagram in Mg/Nb multilayers. Since the standard crystal structures of Mg and Nb are hcp and bcc, respectively, the bi-phase diagram of Mg/Nb thin films necessarily requires for describing the stabilization of metastable bcc Mg and metastable hcp Nb. From the literature, the bcc Mg has been observed at very high pressure at room temperature [27]. This bcc phase is actually unstable under normal conditions but the isotropic constraints stabilize it against transformation into the stable hcp crystal structure. Recently, Ham et al. observed the pseudomorphic growth of bcc Mg in Mg/Nb multilayered thin films in an ambient condition [28]. The epitaxial growth of bcc Mg about 1 nm thick at the interface was shown in the HRTEM and the orientation of the bcc Mg and bcc Nb was confirmed by the fast Fourier transform (FFT). The origins of instability of bulk bcc Mg were further investigated by using Density Functional Theory (DFT) calculations [29].