Calculate Elastic Strain Energy in Solid State Physics

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Homework Statement

A think film of ErAs(rock salt structure, a=0.574nm) is grown on top of a thick GaAs (Zn-blend structure, a=0.565nm) substrate. The substrate orientation and the film growth direction are both <001>. For very thin films, the ErAs is tetragonally distorted such that its in-plane lattice constants match that of the substrate. This occurs in order to minimize the interfacial energy. Of course, there is also a price paid in terms of elastic strain energy.

Assume that the GaAs substrate is perfectly rigid and that the elastic constants (assume isotropic medium) of ErAs are C11=13*10E11[erg/cm^3] and C12=1.6*10E11[erg/cm^3]. Because there is no shear, the elastic energy density of the film is given by:

Homework Equations

U=1/2{C11[εxx^2+εyy^2+εzz^2]+2C12[εxxεyy+εxxεzz+εyyεzz]}

a) what is the elastic strain energy(in eV) per Er atom?

b) What is te value of the in-plane stress?

The Attempt at a Solution

At first, I thought it would be a simple plug-in-and-calculate question, but then I realize that I have no idea how to calculate all the ε's or where and how to find them. Nor do I know the units on those constants. Also, I can't imagine how ErAs "tetragonally distorted" looks like. Would someone help me please?

 

[mark726]

Thank you for your question. it is important to have a clear understanding of the context and parameters of a problem before attempting to solve it. Let's start by breaking down the information given in the forum post.

The problem involves a thin film of ErAs (erbium arsenide) grown on a GaAs (gallium arsenide) substrate. Both the substrate and film have a <001> orientation, meaning that their crystal axes are aligned in the same direction. This is important because it allows for the film to be tetragonally distorted, meaning that its lattice structure is stretched or compressed in one direction in order to match that of the substrate.

The elastic constants given (C11 and C12) are measures of the stiffness of a material. In this case, they represent the stiffness of ErAs and are given in units of erg/cm^3. These constants are used to calculate the elastic energy density (U) of the film, which is a measure of the energy stored in the film due to its deformation.

To answer part (a) of the question, you will need to calculate the elastic energy per ErAs atom. This can be done by dividing the elastic energy density (U) by the volume of one ErAs atom. The volume of an atom can be calculated using its lattice constant (a) and assuming a cubic unit cell (a^3). You will need to convert the units of the elastic constants from erg/cm^3 to eV/atom in order to get the final answer in eV.

For part (b) of the question, you will need to use the given formula for the elastic energy density (U) and solve for the in-plane stress (σ). This can be done by setting the derivatives of U with respect to εxx and εyy equal to zero and solving for σ. Again, you will need to convert the units of the elastic constants to get the final answer in eV.

As for the "tetragonally distorted" structure of ErAs, it means that the lattice structure of the film is stretched or compressed in one direction in order to match that of the substrate. This can be visualized as a square being stretched or compressed into a rectangle, with the other two sides remaining the same length.

I hope this helps you approach the problem with a better understanding. Remember to always break down the information given and use your knowledge of relevant concepts to guide your solution.