What is lattice mismatch in hetero-structural growth?

Introduction

The hetero-structural growth is the growth of A material on B material having different lattice constants. The different lattice constant results in structural deformation which leads to defects in the grown A material. The discrepancy of the A material quality (in terms of crystal perfection) can be easily understood by the lattice mismatch percentage. Lattice mismatch is a very important terminology to understand hetero-structural growth. In hetero-structural growth, the mismatch percentages play an important role to decide the performance of the proposed structure. The researcher can judge their expectation with the reality of the mismatch. This is very popular in device physics to judge defects or defect-free growth (strain film less defect), structural deformation for a new role(tetragonal to ferroelectric orthorhombic phase change leading to new properties of the material in new structure, etc.). 

How to calculate lattice mismatch?

The mismatch is nothing but a mismatched strain in the film against the substrate is used. For example, if we consider Si substrate and HfZrO₂ film of different crystal structures as shown in table I. Then the mismatch can be defined in terms of percentage as follow:

                            Lattice mismatch strain= [a(film)-a(substrate)]/a[substrate]  -----------(1)

Table I: Si and HfZrO2 (HZO) lattice constant

Using equation (1), we can have the lattice mismatch strain in the film. Table II shows the mismatch percentages between cubic Si and HfZrO2 corresponding to their lattice constants executed from the relation. 

Table II: Lattice mismatch percentage between Si and HZO

Using equation (1), we can have the lattice mismatch strain in the film. Table II shows the mismatch percentages between cubic Si and HfZrO2 corresponding to their lattice constants executed from the relation. The sign of the strain can be negative or positioned. The negative sign indicates compressive stress which means the lattice constant of the film is smaller than the substrate. More elaboratively, the compressive stress exists in the film (film lattice smaller than substrate), therefore tensile stress is necessary to accommodate the film lattice with the substrate lattice in the in-plane direction (figure 1). 

Figure 1: Lattice mismatch in-plane lead strain in the film

What is the significance of the mismatch sign?

Fundamentally, the strain is nothing but a ratio of the change in length, radius, area, or volume with respect to its original value and stress in the amount of force applied to or exerted by a body to its surrounding. Please review basic mechanics to understand the detail of the strain and stress phenomenon.

 The opposite is the case for a positive sign. The positive sign of the mismatches indicates tensile stress exists in the film. This tensile stress may lead to the structural change of the film which results in new phenomena in the grown film. Please find the lattice calculation here.