How can we make a mechanically unstable cubic system, stable

I have to calculate the elastic constants of a Cubic system using a Density functional theory-based software. A cubic system will be mechanically stable if it satisfies the Born-stability condition (eq. 6 from ref),

$$C_{11}−C_{12}>0 ;C_{11}+2C_{12}>0 ;C_{44}>0.\tag{1}$$

But in my case, the system failed to satisfy the first condition ($$C_{11}-C_{44}$$ should be > 0 but I am getting a negative value).

How I can make this system stable, like applying stress, strain or something else? My aim is to predict if this material is useful in photovoltaic cells (up to ~2eV) and in thermometric devices (up to 800 Kelvin).

• Did you do a full geometry optimization (relaxation) before calculating the elastic properties?
– Camps
Nov 26, 2020 at 12:08
• Yes, I did the full geometry optimisation. Nov 26, 2020 at 13:23
• I gave you a +1 but please consider my edit, it's important. Nov 27, 2020 at 4:38
• If your aim is to predict a material with a band gap of about 2 eV, why does it matter which specific structure it has? Can you not distort the crystal to remove the instability? It will change the symmetry of the system, but it may keep the 2 eV gap that is after all what you are really interested in... Nov 27, 2020 at 21:44

Depending on your calculation and how much time you want to invest in getting the best property you can, I suggest two different approaches to make this system stable:

1. Fully relax the geometry of the structure. Some comments below your question suggest that you tried to fully relax your structure, but the structure is probably not fully relaxed if it is still mechanically unstable. In your software of choice, you may have to turn off the symmetry or break the symmetry "by hand" of the structure so that the relaxation does not constrain the relaxation to preserve the cubic symmetry (and therefore mechanical instability) of your structure.
2. Use the inflection detection method. Some relatively recent work proposes to compute the properties of mechanically unstable structures at the limit of stability [1] - at the inflection point of the energy. Some examples of applications to pure elements are laid out by van de Walle et al. [2] in a later paper with energies for mechanically unstable phases in good agreement with CALPHAD lattice stabilities. However, this method is considerably more computationally demanding than a geometry relaxation. It requires determining the softest phonon mode for several candidate geometries (with broken symmetry) between the mechanically unstable and fully relaxed states to find the geometry where the softest phonon mode has an eigenvalue of zero.

[1] van de Walle, A., Hong, Q., Kadkhodaei, S., Sun, R., 2015. The free energy of mechanically unstable phases. Nat Commun 6, 7559. https://doi.org/10.1038/ncomms8559

[2] van de Walle, A., 2018. Reconciling SGTE and ab initio enthalpies of the elements. Calphad 60, 1–6. https://doi.org/10.1016/j.calphad.2017.10.008