Strained Tetragonal States and Bain Paths in Metals

Abstract

Paths of tetragonal states between two phases of a material, such as bcc and fcc, are called Bain paths. Two simple Bain paths can be defined in terms of special imposed stresses, one of which applies directly to strained epitaxial films. Each path goes far into the range of nonlinear elasticity and reaches a range of structural parameters in which the structure is inherently unstable. In this Letter we identify and analyze the general properties of these paths by density functional theory. Special examples include vanadium, cobalt, and copper, and the epitaxial path is used to identify an epitaxial film as related uniquely to a bulk phase. [S0031-9007(97)03190-6] PACS numbers: 64.70. Kb, 61.50.Ks, 62.20.Dc Pseudomorphic epitaxy of a cubic or tetragonal (001) film typically results in a strained tetragonal structure. If the stresses on the tetragonal state vanish and also the state corresponds to a local minimum of energy with respect to tetragonal deformations, the structure will be called a tetragonal phase. Such a phase will be stable or metastable, depending on whether it has the lowest energy compared to other minima. Frequently metals have two tetragonal phases; sometimes both are cubic, e.g., bcc and fcc Na and Rb [1,2]; sometimes one is cubic and the other phase is tetragonal, e.g., Ti and V [2,3]. They can, of course, also have phases with other structures; e.g., Ti also has a hcp phase.Many paths can go from one tetragonal phase to the other. If the geometries along such a path have tetragonal symmetry and if they connect bcc and fcc phases, the paths have been called Bain paths [4]. A purpose of the present work is to define and discuss a particular Bain path which will be called the epitaxial Bain path (EBP). The EBP is produced by isotropic stress or strain in the (001) plane of tetragonal phases accompanied by vanishing stress perpendicular to the plane, such as pseudomorphic epitaxy produces on an (001) cubic or tetragonal film. Epitaxy provides a valuable means of stabilizing metastable phases and of putting phases under very large strains, both tensile and compressive in the plane of the epitaxy. The EBP of a material identifies the phase that has been strained, checks quantitatively the elastic behavior, which can be highly nonlinear, and predicts which phase of the material will form on a given substrate. Thus, in order to understand the properties of epitaxial films and new materials, the knowledge of the EBP is indispensable.A different Bain path has long been discussed, particularly by Milstein [1], in which uniaxial stress is applied to a tetragonal state along the ͓001͔ axis accompanied by zero stress in the (001) plane; this path is conveniently called the uniaxial Bain path (UBP). We compare the two paths, EBP and UBP, which are both physically realizable. We show that both have the same lowest possible maximum energy or barrier energy of all Bain paths between the two tetragonal phases. However, the EBP has a special value in relating strained tetragonal structures produced by epita...

Effects of anharmonic strain on the phase stability of epitaxial films and superlattices: Applications to noble metals

“…As noted in Ref. 65, the epitaxial path crosses all extremal points of E(γ, V ) because Eq. (23) is satisfied where conditions Eq.…”

“…64,65 (ii) Bain path E(γ): A more specific function E(γ) ≡ E(γ, V )| V =const is defined by the tetragonal Bain path, 63 connecting fcc and bcc structures. The Bain path is obtained by changing the c/a ratio while keeping V = ca 2 constant.…”

“…The energy as a function of γ must have extremal points at both γ values corresponding to the cubic symmetry fcc (γ fcc = 1) and bcc (γ bcc = 2 − 1 3 ) states, as well as at least another bct point γ bct with a zero derivative E ′ (γ) = 0. 66,67 Usually, [64][65][66][68][69][70] for fcc stable elements the bcc lattice is unstable [i.e., E(γ) has a local maximum at γ bcc ] and the bct state (a local minimum) occurs for γ bct < γ bcc .…”

Ab initiocalculation of phase boundaries in iron along the bcc-fcc transformation path and magnetism of iron overlayers

“…However, other extrema may occur that are not dictated by symmetry and reflect properties of the specific material. Configurations corresponding to energy minima at the transformation paths represent stable or metastable structures and may mimic atomic arrangements that could be encountered when investigating thin films 14 and extended defects such as interfaces and dislocations. 15,16 We start with the bcc structure and consider it as a tetragonal one with the c/a ratio equal to 1.…”

Strain-induced change in the elastically soft direction of epitaxially grown face-centered-cubic metals

“…For noble metals having the fcc structure (c/a = √ 2) at equilibrium, this deformation path contains the bcc (c/a = 1) saddle point, and the bct (c/a = 0.96 in the case of Cu) local minimum. 14,15,16 The low amplitude of the epitaxial 001 Bain path relative to the hydrostatic path (Fig. 1) defines the softness of q(a s , [001]) via Eq.…”