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I have successfully completed the SCF of doped system Ni doped ZnO using GGA+U. I have checked the SCF.out file and found the energy value : E= -6112.0389 Ry.

Hwo to calculate the formation energy of doped system using this $formula^1$ ?

$E_f = E(doped_{ZnO}) -E_{ZnO}+2(\mu_{Zn}+\mu_{Ni}) $

Where I can find the chemical value of Zn and Ni ($\mu_{Zn},\mu_{Ni}$) in my outputfil.

Also I would like to know what is the minimum formation energy for a crystal to be stable?

Your help will be appreciated

(1). Effect of two identical 3d transition‐metal atoms M doping (M = V, Cr, Mn, Fe, Co, and Ni) on the structural, electronic, and magnetic properties of ZnO

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Based on reference paper given above, the reference system of interest here is $\ce{Zn_{22}M_{2}O_{24}}$. As two $\ce{Zn}$ atoms were substituted by transition metal ($M=\ce{V}, \ce{Cr}, \ce{Mn}, \ce{Fe}, \ce{Co}, \ce{Ni}).$ Formation energy for doping is given as $E_f = E(doped_{\ce{ZnO}}) -E_{\ce{ZnO}}+2(\mu_{\ce{Zn}}-\mu_{\ce{Ni}}) $ where $ E(doped_{\ce{ZnO}})$ is total energy of $\ce{Zn_{22}M_{2}O_{24}}$ and $E_{\ce{ZnO}}$ is energy of $\ce{Zn_{24}O_{24}}$. The chemical potentials of the $\ce{V}$, $\ce{Cr}$, $\ce{Mn}$, or $\ce{Fe}$ are energies of single atom in the body centered cubic structure, where as in case of $\ce{Co}$ and $\ce{Ni}$, hexagonal crystal structure and face centered cubic structure were considered respectively. For $\ce{Zn}$ they have considered considered two state, one $\ce{Zn}$ as its elementary substance and second as $\mu_{\ce{Zn}}=E_{\ce{ZnO}}-\mu_{\ce{O}}$ where $\mu_{\ce{O}}$ is energy of oxygen as elementary substance. For stability of any structure you can do mechanical(elastic constants) and dynamic stability(phonon) calculations. Based on formation energy one can say that if it is negative, such doping are more favorable.

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  • $\begingroup$ +1 But please take a look at my edit where I fixed the molecular formula for $\ce{Zn22M2O24}$ and try yourself to fix the rest of the atoms/chemical-formulas! $\endgroup$
    – Nike Dattani
    May 12, 2021 at 17:00
  • $\begingroup$ @PranavKumar The last part of the formation energy should be $-\mu_{Ni}$ since it's the impurity atom being added. $\endgroup$
    – Nana Kofi Boakye
    Aug 18 at 22:30
  • $\begingroup$ @NanaKofiBoakye Thanks for pointing out. Fixed $\endgroup$
    – Pranav kumar
    Aug 20 at 15:55
  • $\begingroup$ Shouldn't $\mu_{i}$ include ZPE and TS of the considered crystal? $\endgroup$
    – Okano
    Aug 20 at 20:09
  • $\begingroup$ @Okano The $\mu_i$ here is the energy per atom of the reference material. Also the energy is calculated at T=0K so the entropy part is zero $\endgroup$
    – Nana Kofi Boakye
    Aug 20 at 22:16

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