Electronic Supplementary Material (ESI) for Physical Chemistry Chemical Physics. This journal is © the Owner Societies 2020

Electronic Supplementary Information Tuning metal oxides defect chemistry by thermochemical quenching Shehab Shousha a,b, Sarah Khalil a, Mostafa Youssef b,* a Department of Nuclear and Radiation Engineering, Alexandria University, Alexandria, Egypt b Department of Mechanical Engineering, The American University in Cairo, AUC Avenue, P.O. Box 74, New Cairo 11835, Egypt *Corresponding author: [email protected]

Derivation of the ratio between the concentrations of the charge states of a quenched defect

To compute the ratios of the charge states of a certain ionic defect at the quenched state, we consider, without loss of generality, the example of oxygen vacancies domination regime where they are charge balanced by electrons. It is important to note that the combination of charged oxygen vacancies that gives the minimum free energy will be achieved while maintaining the quenching constraint that the total number of oxygen vacancies remains constant.

The difference in between the quenched defective crystal and the perfect ideal crystal (∆퐺 ) at the final temperature ( T = 300 K in this study) is given by

× × ∙ ∙ ∙∙ ∙∙ ∆푮 = [푽푶]푬풇(푽푶) + [푽푶]푬풇(푽푶) + [푽푶]푬풇(푽푶) + 풏풄푬풇(풏풄) − 푻∆푺ퟏ − 푻∆푺ퟐ − 푻∆푺ퟑ − 푻∆푺풆풍풆풄 (ퟏ)

× ∙ ∙∙ Where, 퐸푓(푉푂 ), 퐸푓(푉푂), and 퐸푓(푉푂) represent the equilibrium formation energies of neutral, (1+) and (2+) oxygen vacancies respectively. While,∆푆1, ∆푆2 , and ∆푆3 are the increase in due to configurational disorder introduced to the crystal by the 3 oxygen vacancies charge states. Similarly 퐸푓(푛푐) and ∆푆푒푙푒푐 are the formation energy of electrons and the change in the configurational

electronic entropy. nc is the concentration of electrons.

By applying the for the entropy of a disordered system: 푺 = 풌 퐥퐧 푾 (ퟐ)

Where k is the and W is the number of ways of distributing n defects over N lattice sites: 푵! 푾 = (ퟑ) (푵 − 풏)! 풏! Accordingly, 푵! 푺 = 풌 풍풏 ≈ 풌{푵풍풏(푵) − (푵 − 풏) 퐥퐧(푵 − 풏) − 풏풍풏(풏)} (ퟒ) (푵 − 풏)! 풏! Where in the latter equation we used the standard Stirling approximation. By substituting the entropy expression in equation (1)

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∙ ∙ ∙∙ ∙∙ ∆푮 = [푽푶]푬풇(푽푶) + [푽푶]푬풇(푽푶) + 풏풄푬풇(풏풄) ∙ ∙ ∙ ∙ − 풌푻 {푵풍풏푵 − (푵 − [푽푶])풍풏(푵 − [푽푶]) − [푽푶]퐥 퐧([푽푶])} ∙∙ ∙∙ ∙∙ ∙∙ − 풌푻 {푵풍풏푵 − (푵 − [푽푶])풍풏(푵 − [푽푶]) − [푽푶]퐥 퐧([푽푶])} − 풌푻{푵풄풍풏푵풄 − (푵풄 − 풏풄)풍풏(푵풄 − 풏풄) − 풏풄퐥 퐧(풏풄)} (ퟓ)

In the latter equation Nc is the effective at the conduction band. Note that in this derivation we appeal to this simplified picture of representing the electronic density of stat. More rigorously this should be an integration of the density of states as in the manuscript. Here we aim at simplifying the notation.

For the quenched state:

× ∙ ∙∙ [푽푶] + [푽푶] + [푽푶] = 풄풐풏풔풕풂풏풕 = 풏풗 (ퟔ),

Where nv is the total number of vacancies frozen from high temperature. Thus,

× ∙ ∙∙ [푽푶] = 풏풗 − [푽푶] − [푽푶] (ퟕ) And from the charge neutrality

∙ ∙∙ 풏풄 = [푽푶] + ퟐ[푽푶] (ퟖ) Therefore by substituting back in equation 5, ∙ ∙ ∙∙ ∙∙ ∙ ∙∙ × ∙ ∙∙ ∆푮 = [푽푶]푬풇(푽푶) + [푽푶]푬풇(푽푶) + (풏풗 − [푽푶] − [푽푶] )푬풇(푽푶) + ([푽푶] + ퟐ[푽푶])푬풇(풏풄) ∙ ∙ ∙ ∙ − 풌푻 {푵풍풏푵 − (푵 − [푽푶])풍풏(푵 − [푽푶]) − [푽푶]퐥 퐧([푽푶])} ∙∙ ∙∙ ∙∙ ∙∙ − 풌푻 {푵풍풏푵 − (푵 − [푽푶])풍풏(푵 − [푽푶]) − [푽푶]퐥 퐧([푽푶])} ∙ ∙∙ ∙ ∙∙ − 풌푻 {푵풍풏푵 − (푵 − 풏풗 + [푽푶] + [푽푶])풍풏(푵 − 풏풗 + [푽푶] + [푽푶]) ∙ ∙∙ ∙ ∙∙ − (풏풗 − [푽푶] − [푽푶])퐥 퐧(풏풗 − [푽푶] − [푽푶])} ∙ ∙∙ ∙ ∙∙ − 풌푻{푵풄풍풏푵풄 − (푵풄 − [푽푶] − ퟐ[푽푶])풍풏(푵풄 − [푽푶] − ퟐ[푽푶]) ∙ ∙∙ ∙ ∙∙ − ([푽푶] + ퟐ[푽푶])퐥 퐧([푽푶] + ퟐ[푽푶])} (ퟗ)

Subject to the constraints (equations 6 and 8) which are fixed total number of oxygen vacancies and charge neutrality, we minimize the Gibbs free energy by requiring null partial derivatives as in equations 10 and 17 below.

흏(∆푮) ∙ = ퟎ (ퟏퟎ) 흏([푽푶]) ∙ × ∙ ∙ ퟎ = 푬풇(푽푶) − 푬풇(푽푶) + 푬풇(풏풄) − 풌푻{ퟏ + 풍풏(푵 − [푽푶]) − ퟏ − 퐥 퐧([푽푶])} ∙ ∙∙ ∙ ∙∙ − 풌푻{−ퟏ − 풍풏(푵 − 풏풗 + [푽푶] + [푽푶]) + ퟏ + 퐥 퐧(풏풗 − [푽푶] − [푽푶])} ∙ ∙∙ ∙ ∙∙ − 풌푻{ퟏ + 풍풏(푵풄 − [푽푶] − ퟐ[푽푶]) − ퟏ − 퐥 퐧([푽푶] + ퟐ[푽푶])} (ퟏퟏ) Therefore, by rearrangement, ( ∙ )( ∙ ∙∙ )( ∙ ∙∙ ) ∙ × 푵 − [푽푶] 풏풗 − [푽푶] − [푽푶] 푵풄 − [푽푶] − ퟐ[푽푶] 푬풇(푽푶) − 푬풇(푽푶) + 푬풇(풏풄) = 풌푻 퐥퐧 ∙ ∙ ∙∙ ∙ ∙∙ (ퟏퟐ) [푽푶](푵 − 풏풗 + [푽푶] + [푽푶])([푽푶] + ퟐ[푽푶])

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But in the dilute limit,

∙ ∙ ∙∙ (푵 − [푽푶]) ≈ (푵 − 풏풗 + [푽푶] + [푽푶]) ≈ 푵 (ퟏퟑ) ∙ ∙∙ ∙ ∙∙ × And from equations 6 and 8 we have ([푉푂] + 2[푉푂]) = 푛푐 , (푛푣 − [푉푂] − [푉푂]) = [푉푂 ] Thus,

× ∙ × [푽푶](푵풄−풏풄) 푬풇(푽푶) − 푬풇(푽푶) + 푬풇(풏풄) = 풌푻 퐥퐧 ∙ (ퟏퟒ) [푽푶]풏풄 × −푬풇(푽푶) × 풆풙풑 [ ] [푽 ] 풏풄 푬풇(풏풄) 풌푻 푶 = 풆풙풑 [ ] (ퟏퟓ) [푽∙ ] (푵 − 풏 ) 풌푻 −푬 (푽∙ ) 푶 풄 풄 풆풙풑 [ 풇 푶 ] 풌푻

−퐸 (푛 ) And since 푛 = 푁 푒푥푝 [ 푓 푐 ] 푐 푐 푘푇 Then, × −푬풇(푽푶) × 풆풙풑 [ ] [푽푶] 풌푻 = (16) [푽∙ ] −푬 (푽∙ ) 푶 풆풙풑 [ 풇 푶 ] 풌푻

Similarly,

흏(∆푮) ∙∙ = ퟎ (ퟏퟕ) 흏([푽푶]) ∙∙ × ∙∙ ∙ ퟎ = 푬풇(푽푶) − 푬풇(푽푶) + 푬풇(풏풄) − 풌푻{ퟏ + 풍풏(푵 − [푽푶]) − ퟏ − 퐥 퐧([푽푶])} ∙ ∙∙ ∙ ∙∙ − 풌푻{−ퟏ − 풍풏(푵 − 풏풗 + [푽푶] + [푽푶]) + ퟏ + 퐥 퐧(풏풗 − [푽푶] − [푽푶])} ∙ ∙∙ ∙ ∙∙ − 풌푻{ퟐ + 풍풏(푵풄 − [푽푶] − ퟐ[푽푶]) − ퟐ − ퟐ퐥 퐧([푽푶] + ퟐ[푽푶])} (ퟏퟖ) ∙ By simplifying analogously to the earlier differentiation with respect to [푉푂] :

[푽×](푵 − 풏 )ퟐ ( ∙∙ ) ( ×) ( ) 푶 풄 풄 푬풇 푽푶 − 푬풇 푽푶 + ퟐ 푬풇 풏풄 = 풌푻 퐥퐧 ∙∙ ퟐ (ퟏퟗ) [푽푶]풏풄 Then,

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−푬 (푽×) 풆풙풑 [ 풇 푶 ] [푽×] 풌푻 푶 = (ퟐퟎ) [푽∙∙ ] −푬 (푽∙∙ ) 푶 풆풙풑 [ 풇 푶 ] 풌푻

From equations 16 and 20, it is clear that the ratio between the 3 charged states of oxygen vacancy at quenched state is:

× ∙ ∙∙ −푬풇(푽푶) −푬풇(푽푶) −푬풇(푽푶) [푽×]: [푽∙ ] ∶ [푽∙∙ ] = 풆풙풑 [ ] ∶ 풆풙풑 [ ] ∶ 풆풙풑 [ ] (ퟐퟏ) 푶 푶 푶 풌푻 풌푻 풌푻

Which is the same equilibrium ratio we get by using the Boltzmann expression:

풒 −푬풇(푽 ) [푽풒 ] = 푵 퐞퐱 퐩 [ 푶 ] (ퟐퟐ) 푶 푶 풌푻

This derivation can be applied to any other charge compensation regime and will lead to the same conclusion. Independent of the number of charge states included, this derivation will always give the same expression for the concentration ratios.

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