Control of Impurity Phase Segregation in a PdCrO₂/CuCrO₂ Heterostructure

A PdCrO₂ layer with thickness of approximately 10 nm was grown on a one-monolayer ($\sim$0.38 nm) CuCrO₂ buffer layer on an Al₂O₃ substrate via pulsed laser deposition using polycrystalline targets. Before the film growth, commercially available Al₂O₃ (0001) substrates (CrysTec, Germany) were annealed at 1100 ^∘C for 1 h to achieve atomically flat surfaces with step-terrace structure. For PdCrO₂ films, the growth conditions were widely varied: temperature ($T$) was 500–800 ^∘C, and oxygen partial pressure ($P_{\mathrm{O_{2}}}$) was 10–500 mTorr. The repetition rate and fluence of KrF excimer laser ($\uplambda$ = 248 nm) were fixed at 5 Hz and 1.5 J/cm², respectively. The cross-sectional STEM specimens were prepared using low-energy ion milling at LN₂ temperature after mechanical polishing. High-angle annular dark field (HAADF) STEM measurements were performed on a Nion UltraSTEM200 operated at 200 kV. The microscope is equipped with a cold-field emission gun and a third- and fifth-order aberration corrector for sub-angstrom resolution. The convergence half-angle of 30 mrad was used, and the inner angle of the HAADF STEM was approximately 65 mrad.

Density functional theory (DFT) implemented in the VASP package [28] was used to understand the energetics of competing phases during the experimental growth process. The Perdew–Burke–Ernzerhof (PBE)+$U$ method [29, 30] was used. The Hubbard $U$ correction was applied to the 3$d$ shell of the Cr atoms. The $U$ value was 3.3 eV, which was optimized compared with the results of the HSE06 functional [31], as described in the Supporting Information. The core electrons were replaced with pseudopotentials made by the projector-augmented wave method accompanied by the VASP code [32, 33, 34]. The cutoff energy was 520 eV, and $k$-spacing was 0.30 Å^-1, which converged the Cr vacancy formation energy in CuCrO₂ within 2 meV. Experimental lattice vectors for CuCrO₂ [35], PdCrO₂ [36], and Al₂O₃ were used ¹¹1The room temperature structure of Ref. [47] was used. The lattice vectors reported in the Materials project [38] were used for chromium oxides and chromium metal ²²2The database ID for each material: Cr = mp-90, CrO = mp-19091, Cr₃O₄ = mp-756253, and Cr₂O₃ = mp-19399.. The atomic coordinates were relaxed for the functional. The convergence criteria for the self-consistent field and ionic cycles were $1.0\times 10^{-7}$ eV and $1.0\times 10^{-6}$ eV, respectively.

Impurity phases including Cu_xPd_1-x, Cr₂O₃, and Cr₃O₄ have been observed experimentally [40]. In Figure 1 we show in addition to 2$\uptheta$–$\uptheta$ XRD spectrum, the intensity of the XRD data as a function of the growth conditions. As reported in Ref. [22], the high-quality PdCrO₂ films can be achieved only within a relatively narrow growth window. Outside the growth window, the metallic properties are severely deteriorated by the impurity formation. The resistance could not be measured because of the high resistivity. The rectangular boxes in Figure 1 highlight the main impurities observed in XRD: Cr₃O₄ and Cr₂O₃. The bottom two panels of Figure 1 map the XRD intensities of Cr₃O₄ and Cr₂O₃ for temperature and oxygen partial pressure. The relative abundances between Cr₃O₄ and Cr₂O₃ are difficult to assess quantitatively using the XRD intensities because the XRD reflectivity varies with substances and angles. However, we use the intensities to assess qualitatively how the formation of each substance is affected by growth conditions. The XRD intensity of Cr₃O₄ strongly positively correlates with temperature (correlation coefficient ³³3 The correlation coefficient $r$ indicates the strength of correlation between two data-series, $\left\{X_{i}\right\}_{i=1}^{n}$ and $\left\{Y_{i}\right\}_{i=1}^{n}$. $r$ can be from -1 to +1. $r$ = +1 (-1) indicates the complete positive (negative) correlation between $\left\{X_{i}\right\}_{i=1}^{n}$ and $\left\{Y_{i}\right\}_{i=1}^{n}$. The correlation coefficient is defined by $$r=\frac{\upsigma_{X,Y}}{\upsigma_{X}\upsigma_{Y}}.$$ (1) Here, $\upsigma_{X}$ and $\upsigma_{Y}$ are the standard deviations of $\left\{X_{i}\right\}_{i=1}^{n}$ and $\left\{X_{i}\right\}_{i=1}^{n}$. $\upsigma_{X,Y}$ is the covariance of the two data-series: $$\upsigma_{XY}=\frac{1}{n}\sum_{i=1}^{n}(X_{i}-\bar{X})(Y_{i}-\bar{Y}).$$ (2) $\uprho=+0.82$), whereas the correlation between the XRD intensity of Cr₂O₃ and temperature is weak ($\uprho=+0.19$). Moreover, Cr₃O₄’s peak strength does not depend on oxygen partial pressure ($\uprho=+0.01$), but Cr₂O₃’s peak strength negatively depends on oxygen partial pressure ($\uprho=-0.48$). These results are compared with our calculations in the last paragraph of Section III.6.

Because the system segregates oxygen-deficient oxides Cr₂O₃ or Cr₃O₄, the segregation process should be accompanied by the appearance or disappearance of point defects. The ratio of O to Cr (O/Cr) in Cr₂O₃ is $\mathrm{O}/\mathrm{Cr}=1.5$. In Cr₃O₄, $\mathrm{O}/\mathrm{Cr}$ is about $1.3$. In the delafossite materials, $\mathrm{O}/\mathrm{Cr}$ is 2.0. Therefore, the formation energies of multiple types of point self-defects in bulk Al₂O₃, CuCrO₂, and PdCrO₂ were calculated. To simplify the problem, defects in the bulk were calculated, even though samples with different thickness of PdCrO₂ and CuCrO₂ films have been grown in this and other works [40, 22].

The following describes the notation used for the point defects and explains how to evaluate the formation energies. $\mathrm{V_{Cr},V_{Cu},V_{Pd},and\,V_{O}}$ indicate vacancies in the Cr, Cu, Pd, and O sites. The Cu (Pd) replacement defects in the Cr sites, or antisite defects, are indicated by Cu_Cr (Pd_Cr). Larger defect complexes such as Cu_Cr&$\mathrm{V_{Cu}}$ (Pd_Cr&$\mathrm{V_{Pd}}$) can form when Cu (Pd) atoms move to a preformed $\mathrm{V_{Cr}}$, leaving the $\mathrm{V_{Cr\,(Pd)}}$. The formation energies of these defects are given as follows: for CuCrO₂,

	$\displaystyle\Delta E\left(\mathrm{V_{\upalpha}}\right)$	$\displaystyle=$	$\displaystyle E\left(\mathrm{CuCrO_{2}}\right)_{\mathrm{V_{\upalpha}}}-E\left(\mathrm{CuCrO_{2}}\right)_{\mathrm{bulk}}$		(3)
		$\displaystyle+$	$\displaystyle\upmu_{\mathrm{\upalpha}},\;\;\left(\upalpha=\mathrm{Cu,\,Cr,\,or\,O}\right),$
	$\displaystyle\Delta E\left(\mathrm{Cu_{Cr}\&V_{Cu}}\right)$	$\displaystyle=$	$\displaystyle E\left(\mathrm{CuCrO_{2}}\right)_{\mathrm{Cu_{Cr}\&V_{Cu}}}-E\left(\mathrm{CuCrO_{2}}\right)_{\mathrm{bulk}}$		(4)
		$\displaystyle+$	$\displaystyle\upmu_{\mathrm{Cr}},$

and for PdCrO₂,

	$\displaystyle\Delta E\left(\mathrm{V_{\upalpha}}\right)$	$\displaystyle=$	$\displaystyle E\left(\mathrm{PdCrO_{2}}\right)_{\mathrm{V_{\upalpha}}}-E\left(\mathrm{PdCrO_{2}}\right)_{\mathrm{bulk}}$		(5)
		$\displaystyle+$	$\displaystyle\upmu_{\mathrm{\upalpha}},\;\;\left(\upalpha=\mathrm{Pd,\,Cr\,or\,O}\right),$
	$\displaystyle\Delta E\left(\mathrm{Pd_{Cr}\&V_{Pd}}\right)$	$\displaystyle=$	$\displaystyle E\left(\mathrm{PdCrO_{2}}\right)_{\mathrm{Pd_{Cr}\&V_{Pd}}}-E\left(\mathrm{PdCrO_{2}}\right)_{\mathrm{bulk}}$		(6)
		$\displaystyle+$	$\displaystyle\upmu_{\mathrm{Cr}}.$

Here, $E$(CuCrO₂)_bulk and $E$(PdCrO₂)_bulk are the total energies of pristine delafossite structures. $E$(CuCrO₂)_X and $E$(PdCrO₂)_X are the total energies of structures with type-$X$ defects. The chemical potential of atomic species $\upalpha$ is $\upmu_{\upalpha}$. The oxygen vacancy formation energy in the Al₂O₃ substrate was also calculated by

$\displaystyle\Delta E\left(\mathrm{V_{O}}\right)$

$\displaystyle=$

$\displaystyle E\left(\mathrm{Al_{2}O_{3}}\right)_{\mathrm{V_{O}}}-E\left(\mathrm{Al_{2}O_{3}}\right)_{\mathrm{bulk}}+\upmu_{\mathrm{O}}.$

(7)

Our experiments observed the Cu-Pd alloy and Cr oxide impurity phases on the composite sample of Al₂O₃, CuCrO₂, and PdCrO₂ (see § III.1). The defect formation energies should be evaluated for the experimental conditions: the chemical equilibrium states consisting of Al₂O₃, CuCrO₂, PdCrO₂, Cu_xPd_1-x, and a chromium oxide. The exact value of $x$ in Cu_xPd_1-x is not known experimentally. The ratio $x$ potentially depends on the volume comparison of CuCrO₂ and PdCrO₂. However, the change in the results is negligible when $x$ changes from 0.5 to 0.25 or 0.75 (variations of only 0.33 eV were observed), as described in the Supporting Information. Therefore, the results reported below assumed $x=0.5$. Solving the following equations yields the chemical potentials. For example, if Al₂O₃, CuCrO₂, PdCrO₂, CuPd, and Cr₃O₄ coexist, then

	$\displaystyle 2\upmu_{\mathrm{Al}}+3\upmu_{\mathrm{O}}$	$\displaystyle=$	$\displaystyle E\left(\mathrm{Al_{2}O_{3}}\right),$		(8)
	$\displaystyle\upmu_{\mathrm{Cu}}+\upmu_{\mathrm{Cr}}+2\upmu_{\mathrm{O}}$	$\displaystyle=$	$\displaystyle E\left(\mathrm{CuCrO_{2}}\right),$		(9)
	$\displaystyle\upmu_{\mathrm{Pd}}+\upmu_{\mathrm{Cr}}+2\upmu_{\mathrm{O}}$	$\displaystyle=$	$\displaystyle E\left(\mathrm{PdCrO_{2}}\right),$		(10)
	$\displaystyle\upmu_{\mathrm{Cu}}+\upmu_{\mathrm{Pd}}$	$\displaystyle=$	$\displaystyle E\left(\mathrm{CuPd}\right),$		(11)
	$\displaystyle 3\upmu_{\mathrm{Cr}}+4\upmu_{\mathrm{O}}$	$\displaystyle=$	$\displaystyle E\left(\mathrm{Cr_{3}O_{4}}\right).$		(12)

There exist as many independent linear equations as unknown chemical potentials, so the chemical potentials are trivially determined.

The formation energies of point defects in Al₂O₃, CuCrO₂, and PdCrO₂ were calculated for different chromium oxides, as described in Section III.2. We also considered the Cr metal as the Cr source of the Cr-rich limit. The results are summarized in Figure 2. The point-defect formation energies are all positive, so CuCrO₂ and PdCrO₂ are thermodynamically stable and stoichiometric under the considered chemical conditions. For low values of the oxygen chemical potential ($\lesssim-8.6$ eV), the $\mathrm{V_{O}}$ in CuCrO₂ and PdCrO₂ have the lowest formation energies; the $\mathrm{V_{O}}$ in Al₂O₃ is much higher. As soon as the oxygen chemical potential increases, the $\mathrm{V_{Cu}}$ and $\mathrm{V_{Pd}}$ become the lowest formation energy defects. The Cr vacancies, by contrast, have much higher formation energies. The experimental chemical potentials are not well defined because the system is out of equilibrium, as described in Section III.2. However, each element’s stability corresponds to anywhere between the vertical lines that correspond to the oxygen chemical potentials with $\mathrm{Cr_{3}O_{4}}$ and $\mathrm{Cr_{2}O_{3}}$. The $\mathrm{V_{Cr}}$, $\mathrm{Cu_{Cr}}$&$\mathrm{V_{Cu}}$, and $\mathrm{Pd_{Cr}}$&$\mathrm{V_{Pd}}$ are all Cr-deficient point defects. For CuCrO₂, the formation energy of $\mathrm{Cu_{Cr}}$&$\mathrm{V_{Cu}}$ is lower than that of $\mathrm{V_{Cr}}$. Therefore, the Cr site does not have a vacancy because a neighboring Cu occupies the Cr site by forming $\mathrm{V_{Cu}}$ next to $\mathrm{Cu_{Cr}}$.

Experiments found the segregation of impurity phases of Cu_xPd_1-x, Cr₂O₃, and Cr₃O₄ on a 10 nm PdCrO₂ layer with a one-monolayer CuCrO₂ buffer layer on an Al₂O₃ substrate. The samples were grown under low oxygen partial pressures. The simultaneous presence of seven compunds (CuCrO₂, PdCrO₂, Cu_xPd_1-x, Cr₂O₃, Cr₃O₄, O₂, and Al₂O₃) but only five chemical elements complicates the theoretical analysis. Finding a solution for the chemical potential equations is impossible when the equations outnumber the independent variables. In this case, the system is out of equilibrium. The chemical potentials may not be uniform throughuot the sample. For instance, near the surface, the oxygen chemical potential may be a function of temperature and oxygen partial pressure. By contrast, near the regions where Cr₂O₃ and Cr₃O₄ coexist, the chemical potentials of Cr and O are uniquely determined by the formation energies of the two solids. Alternatively, near the Al₂O₃ substrate, the oxygen chemical potential may be determined by temperature and the concentration of oxygen vacancies in Al₂O₃.

The impurity phases are oxygen deficient (i.e., chromium rich) relative to CuCrO₂ and PdCrO₂: the O/Cr ratios of Cr₂O₃ (1.5) and Cr₃O₄ ($\sim$1.3) are smaller than that of CuCrO₂ and PdCrO₂ (2.0). To elucidate the possible cause of impurity phase segregation, several different possible reactions originating from out-of-equilibrium states were considered. Then their potential to destabilize CuCrO₂ and PdCrO₂ was examined. The analysis revealed that low oxygen partial pressures and high temperatures could explain the segregation of Cr₂O₃ and Cu_xPd_1-x. Preexisting defects as energetic as oxygen vacancies in Al₂O₃ could enhance the segregation of Cr₃O₄ and Cu_xPd_1-x.

To simplify the analysis, the Cu_xPd_1-x alloy is assumed to be CuPd, as described in the last paragraph of Section III.2. The theoretical approach shows that the combination of CuCrO₂ and PdCrO₂ is stable against the CrO₂ and CuPd impurity phase segregation, which is a stoichiometric process. The thermochemical equation of this segregation is

	$\displaystyle E(\mathrm{CuCrO_{2}})+E(\mathrm{PdCrO_{2}})$	$\displaystyle=$	$\displaystyle E(\mathrm{CuPd})+2E(\mathrm{CrO_{2}})$		(13)
		$\displaystyle+$	$\displaystyle Q\left(\mathrm{CrO_{2}}\right),$

where $Q\left(\mathrm{CrO_{2}}\right)$ is the energy gained, or lost if negative, to form CrO₂. The value of $Q\left(\mathrm{CrO_{2}}\right)$ was calculated to be $-1.102$ eV per two formula units of CrO₂, so this reaction is endothermic.

By contrast, the Cr/O ratios of CuCrO₂ and PdCrO₂ vs. Cr₂O₃ or Cr₃O₄ are different. Therefore, the impurity phase segregation may be caused by an impurity-absorbing defect. For Cr₂O₃+CuPd, the impurity phase segregation may be caused and promoted by an oxygen-adsorbent mechanism because the Cr/O ratios of CuCrO₂ and PdCrO₂ (1/2) and Cr₂O₃ (2/3) are different. This oxygen deficiency may be the result of low environmental oxygen concentration relative to chromium from either (i) defective CuCrO₂, PdCrO₂, or Al₂O₃ or (ii) low oxygen content in the vacuum growth chamber ⁴⁴4The samples were grown under $<1000$ mTorr oxygen partial pressure.. For mechanism (i), preexisting V_O in CuCrO₂, PdCrO₂, or Al₂O₃ and formation of Cr-deficient defects such as V_Cr in CuCrO₂ or PdCrO₂ were considered to keep the Cr/O ratios constant before and after the process ⁵⁵5 Preexisting point defects with extra Cr, such as $\mathrm{Cr_{Pd}}$ and $\mathrm{Cr_{Cu}}$, were not considered because their existence explains the appearance of Cr_xO_1-x but not Cu_yPd_1-y without involving multiple types defects to balance the chemical reaction. .

Therefore, the energy gain obtained by the (dis)appearance of point defects in CuCrO₂, PdCrO₂, and Al₂O₃ was compared with the release of oxygen molecules into the oxygen gas in the growth chamber. All these possibilities were considered as particle exchanges with a particle bath.

The energy cost of taking an atom ($\upalpha=$ O or Cr) from one of these particle baths is defined as

$\displaystyle\upnu_{\upalpha}\equiv E(\mathrm{\mathrm{bath}})_{\mathrm{bulk}}-E(\mathrm{\mathrm{bath}})_{\upalpha}.$

(14)

Here, $E(\mathrm{\mathrm{bath}})_{\mathrm{bulk}}$ and $E(\mathrm{\mathrm{bath}})_{\alpha}$ are the energies of the particle bath without defects and with an $\upalpha=$ O or Cr vacancy, respectively ⁶⁶6$\upnu_{\upalpha}$ is conceptually similar to the chemical potential of $\upalpha$, but chemical potentials are not defined for a nonequilibrium process..

The thermochemical equations for the segregation of Cr₂O₃ when introducing O to or removing Cr from the particle bath are given as follows:

	$\displaystyle E(\mathrm{CuCrO_{2}})$	$\displaystyle+$	$\displaystyle E(\mathrm{PdCrO_{2}})=E(\mathrm{CuPd})+E(\mathrm{Cr_{2}O_{3}})$		(15)
		$\displaystyle+$	$\displaystyle\upnu_{\mathrm{O}}+Q\left(\mathrm{Cr_{2}O_{3},V_{O}^{rem}}\right),$

	$\displaystyle E(\mathrm{CuCrO_{2}})$	$\displaystyle+$	$\displaystyle E(\mathrm{PdCrO_{2}})=E(\mathrm{CuPd})+(4/3)E(\mathrm{Cr_{2}O_{3}})$		(16)
		$\displaystyle-$	$\displaystyle(2/3)\upnu_{\mathrm{Cr}}+Q\left(\mathrm{Cr_{2}O_{3},V_{Cr}^{int}}\right).$

In equation (15), the term $\upnu_{\mathrm{O}}$ takes into account the effect of removing an oxygen vacancy in the particle bath, and $-(2/3)\upnu_{\mathrm{Cr}}$ considers the effect of creating a fraction of Cr vacancies in the particle bath.

Similarly, the segregation of Cr₃O₄ could be explained by the following reactions:

	$\displaystyle E(\mathrm{CuCrO_{2}})$	$\displaystyle+$	$\displaystyle E(\mathrm{PdCrO_{2}})=E(\mathrm{CuPd})+(2/3)E(\mathrm{Cr_{3}O_{4}})$		(17)
		$\displaystyle+$	$\displaystyle(4/3)\upnu_{\mathrm{O}}+Q\left(\mathrm{Cr_{3}O_{4},V_{O}^{rem}}\right),$

	$\displaystyle E(\mathrm{CuCrO_{2}})$	$\displaystyle+$	$\displaystyle E(\mathrm{PdCrO_{2}})=E(\mathrm{CuPd})+E(\mathrm{Cr_{3}O_{4}})$		(18)
		$\displaystyle-$	$\displaystyle\upnu_{\mathrm{Cr}}+Q\left(\mathrm{Cr_{3}O_{4},V_{Cr}^{int}}\right).$

Their derivations are described in detail in Appendix A.

The exothermic energies, $Q$, are shown in eqs (15)–(18), for different values of $\upnu_{\mathrm{O}}$ and $\upnu_{\mathrm{Cr}}$, depending on the particle baths in Table 1. The table shows that only $Q\left(\mathrm{Cr_{2}O_{3},V_{O}^{rem}}\right)$ and $Q\left(\mathrm{Cr_{3}O_{4},V_{O}^{rem}}\right)$ can be positive (i.e., exothermic reaction), whereas the reactions involving the formation of Cr-deficient defects are always endothermic. Therefore, the preexisting oxygen vacancies could explain the spontaneous segregation of Cr₂O₃, Cr₃O₄, and CuPd impurity phases.

Figure 3 shows $Q\left(\mathrm{Cr_{2}O_{3},V_{O}^{rem}}\right)$ and $Q\left(\mathrm{Cr_{3}O_{4},V_{O}^{rem}}\right)$ in Table 1 for different $\upnu_{\mathrm{O}}$ (i.e., different particle bath). The stability of oxygen atoms in each particle bath negatively correlates with $\upnu_{\mathrm{O}}$: oxygen atoms are the most (least) stable in $\mathrm{Al_{2}O_{3}}$ (O₂ gas). $Q\left(\mathrm{Cr_{2}O_{3},V_{O}^{rem}}\right)$ changes depending on $-\upnu_{\mathrm{O}}$, as given in eq (15). Similarly, $Q\left(\mathrm{Cr_{3}O_{4},V_{O}^{rem}}\right)$ changes depending on $-(4/3)\upnu_{\mathrm{O}}$, as given in eq (17). The impurity phase segregation is endothermic when O₂ gas is the particle bath and exothermic for the other particle baths. The energetically favored chromium oxide changes from $\mathrm{Cr_{2}O_{3}}$ to $\mathrm{Cr_{3}O_{4}}$ with $\nu_{\mathrm{O}}$ decreasing from $\mathrm{PdCrO_{2}}$ to $\mathrm{Al_{2}O_{3}}$.

In this section, entropy contributions to the positive $Q\left(\mathrm{Cr_{2}O_{3},V_{O}^{rem}}\right)$ and $Q\left(\mathrm{Cr_{3}O_{4},V_{O}^{rem}}\right)$ are considered. For convenience, $Q(\mathrm{Cr_{2}O_{3}})\equiv Q\left(\mathrm{Cr_{2}O_{3},V_{O}^{rem}}\right)$ and $Q(\mathrm{Cr_{3}O_{4}})\equiv Q\left(\mathrm{Cr_{3}O_{4},V_{O}^{rem}}\right)$.

Entropy contributions depend on the temperature, point-defect densities, and oxygen partial pressure. The energies $E$ were replaced by Helmholtz free energies $F(T)$ in eqs (15) and (17). For bulk structures, $F(T)$ was evaluated by

$\displaystyle F(T)=E+F_{\mathrm{vib}}(T).$

(19)

Here, $F_{\mathrm{vib}}(T)$ is the vibrational free energy. For $\upnu_{\mathrm{O}}$ in a defective solid, the vacancy configurational entropy contribution was considered in addition to $F_{\mathrm{vib}}(T)$. If the vacancy density is $\mathrm{c_{v}}$, then the free energy change when removing one vacancy is

$\displaystyle\Delta F_{\mathrm{config}}(T,c_{\mathrm{v}})=k_{\mathrm{B}}T[-\ln{(c_{\mathrm{v}})}+\ln{(1-c_{\mathrm{v}})}]$

(20)

(details in Appendix B). Here, $k_{\mathrm{B}}$ is the Boltzmann constant. Therefore, $Q(\mathrm{Cr_{2}O_{3}})$ and $Q(\mathrm{Cr_{3}O_{4}})$ depend on vacancy density and temperature when the bath location is a defective solid.

When considering the case of O₂ released into the growth chamber, because the experimental oxygen partial pressure is very low and the temperature is high, the translational entropy contribution could significantly stabilize the oxygen gas. This stabilization may change $Q(\mathrm{Cr_{2}O_{3}})$ and $Q(\mathrm{Cr_{3}O_{4}})$ from negative to positive. Without entropy contributions, they are negative, as shown in Table 1. The Helmholtz free energy $F(T)$ of the oxygen gas per molecule is defined as

$\displaystyle F(T,P_{\mathrm{O_{2}}})=E(\mathrm{O_{2}})+F_{\mathrm{vib}}(T)+F_{\mathrm{rot}}(T)+F_{\mathrm{trans}}(T,P_{\mathrm{O_{2}}})$

(21)

Here, $E(\mathrm{O_{2}})$ is the energy of an isolated oxygen molecule and $F_{\mathrm{vib}}(T)$, $F_{\mathrm{rot}}(T)$, and $F_{\mathrm{trans}}(T,P_{\mathrm{O_{2}}})$ are free energies by vibrational, rotational, and translational entropies, respectively. Then $F_{\mathrm{rot}}(T)$ and $F_{\mathrm{trans}}(T,P_{\mathrm{O_{2}}})$ [45] are given by

	$\displaystyle F_{\mathrm{rot}}(T)$	$\displaystyle=$	$\displaystyle-k_{\mathrm{B}}T\left(1+\ln{\frac{8\pi^{2}Ik_{\mathrm{B}}T}{2h^{2}}}\right),$		(22)
	$\displaystyle F_{\mathrm{trans}}(T,P_{\mathrm{O_{2}}})$	$\displaystyle=$	$\displaystyle-k_{\mathrm{B}}T\ln{\frac{k_{\mathrm{B}}T}{P_{\mathrm{O_{2}}}\Lambda^{3}}},$		(23)
	$\displaystyle\Lambda$	$\displaystyle\equiv$	$\displaystyle\frac{h}{\sqrt{2\pi mk_{\mathrm{B}}T}}$		(24)

Here, $h$ is the Planck constant, and $I$ is the moment of inertia of an oxygen molecule. Therefore, $Q(\mathrm{Cr_{2}O_{3}})$ and $Q(\mathrm{Cr_{3}O_{4}})$ depend on oxygen partial pressure and temperature when the bath location is the dilute oxygen gas.

The entropy contributions in eqs (15) and (17) yield $Q(\mathrm{Cr_{2}O_{3}})$ and $Q(\mathrm{Cr_{3}O_{4}})$ for different bath locations and conditions. In these equations, the bulk free energies depend on only the temperature. When the bath is a defected crystal, $\upnu_{\mathrm{O}}$ depends on temperature and vacancy density. When the bath is the oxygen gas, $\upnu_{\mathrm{O}}$ depends on temperature and oxygen partial pressure. When $\upnu_{\mathrm{O}}$’s entropy contributions are ignored, $Q(\mathrm{Cr_{2}O_{3}})$ and $Q(\mathrm{Cr_{3}O_{4}})$ barely depend on the temperature: $Q(\mathrm{Cr_{2}O_{3}})$ and $Q(\mathrm{Cr_{3}O_{4}})$ do not change more than 60 meV from 600 to 1000 K, and $Q(\mathrm{Cr_{2}O_{3}})-Q(\mathrm{Cr_{3}O_{4}})$ does not change more than 11 meV from 600 to 1000 K. Therefore, the dependence of $Q(\mathrm{Cr_{2}O_{3}})$ and $Q(\mathrm{Cr_{3}O_{4}})$ on the conditions is almost equivalent to that of $\upnu_{\mathrm{O}}$.

To understand the conditions under which different oxides might be generated experimentally, different baths for exchanging oxygen were systematically considered. When the bath is a defected crystal, $\upnu_{\mathrm{O}}$ was calculated for vacancy densities in the range of $10^{-8}$–$10^{-1}$ per site and temperatures in the range of 600–1000 K. When the bath is the oxygen gas, $\upnu_{\mathrm{O}}$ was calculated for oxygen partial pressures in the range of $10^{-6}$–$10^{0}$ atm and temperatures in the range of 600–1000 K. Figure 4 shows a map of the $\upnu_{\mathrm{O}}$ calculated for different bath locations. The vertical width of each area indicates the variation width corresponding to vacancy densities in the range of $10^{-8}$–$10^{-1}$ per site or oxygen partial pressures in the range of $10^{-6}$–$10^{0}$ atm. Figure 4 is divided into the three regions I–III according to the corresponding $Q(\mathrm{Cr_{2}O_{3}})$ and $Q(\mathrm{Cr_{3}O_{4}})$ values. The region I is $Q(\mathrm{Cr_{2}O_{3},Cr_{3}O_{4}})<0$: the impurity phase segregation does not proceed spontaneously. The region II is $Q(\mathrm{Cr_{2}O_{3}})>0$ and $Q(\mathrm{Cr_{2}O_{3}})>Q(\mathrm{Cr_{3}O_{4}})$: $\mathrm{Cr_{2}O_{3}+Cu_{x}Pd_{1-x}}$ is spontaneously predominantly formed. The region III is $Q(\mathrm{Cr_{3}O_{4}})>Q(\mathrm{Cr_{2}O_{3}})>0$: $\mathrm{Cr_{3}O_{4}+Cu_{x}Pd_{1-x}}$ is spontaneously predominantly formed. Therefore, when the particle bath is oxygen gas, CuCrO₂, or PdCrO₂, the majority of chromium oxide is $\mathrm{Cr_{2}O_{3}}$. When the particle bath is $\mathrm{Al_{2}O_{3}}$, the majority of chromium oxide is $\mathrm{Cr_{3}O_{4}}$.

Furthermore, other bath locations than the above listed are realistically possible. For example, an oxygen-terminated PdCrO₂ surface could lead to very high $\upnu_{\mathrm{O}}$. By contrast, a Pd-terminated surface could lead to very low $\upnu_{\mathrm{O}}$. Investigation of such further complicated mechanisms is a possible future work for theory and experiments.

These calculations revealed that $\upnu_{\mathrm{O}}$ in O₂ gas decreases with decreasing oxygen partial pressure, and $\upnu_{\mathrm{O}}$ in defected crystals decreases with increasing $c_{\mathrm{v}}$ (details in supporting information). This result is not surprising because it indicates that gaseous oxygen molecules are more stable under oxygen-poor conditions. In reality, $c_{\mathrm{v}}$ would negatively correlates with oxygen partial pressure, so $\upnu_{\mathrm{O}}$ positively correlates with oxygen partial pressure in every bath location: lower oxygen partial pressure facilitates the segregation of impurity phases. This analysis agrees with the experimental finding described in Section III.1: $\mathrm{Cr_{2}O_{3}}$ formation negatively correlates with oxygen partial pressure.

However, this analysis does not explain the independence of $\mathrm{Cr_{3}O_{4}}$ formation on oxygen partial pressure. Rather, $\mathrm{Cr_{3}O_{4}}$ formation strongly depends on temperature, unlike $\mathrm{Cr_{2}O_{3}}$. Some hypotheses are considered to explain the $\mathrm{Cr_{3}O_{4}}$ experimental results. (i) Most of the bath locations belong to region II in Figure 4, and the temperature determines how much the metastable Cr₃O₄ is segregated. (ii) Some of the bath locations belong to region III, but high barrier energy is required to transfer oxygen atoms, moving oxygen vacancies, so the temperature determines the oxygen exchange rate.

For example, the barrier energy required to transfer oxygen atoms from Al₂O₃ to the surface would be higher than from PdCrO₂ to the surface. To verify the hypotheses, saddle state analyses by methods such as the nudged elastic band method, molecular dynamics, and/or modeling the sample’s surface should be applied in future works.

For ease of explanation, let the particle bath be PdCrO₂. Consider the following thermochemical equations for the segregation of Cr₂O₃ by removing preexisting O vacancies or creating Cr vacancies.

			$\displaystyle E(\mathrm{CuCrO_{2}})_{\mathrm{bulk}}^{(n)}+E(\mathrm{PdCrO_{2}})_{\mathrm{V_{O}}}^{(n)}$		(25)
		$\displaystyle=$	$\displaystyle E(\mathrm{CuCrO_{2}})_{\mathrm{bulk}}^{(n-1)}+E(\mathrm{PdCrO_{2}})_{\mathrm{bulk}}^{(n-1)}$
		$\displaystyle+$	$\displaystyle E(\mathrm{CuPd})+E(\mathrm{Cr_{2}O_{3}})+Q\left(\mathrm{Cr_{2}O_{3},V_{O}^{rem}}\right)$

and

			$\displaystyle E(\mathrm{CuCrO_{2}})_{\mathrm{bulk}}^{(n)}+E(\mathrm{PdCrO_{2}})_{\mathrm{bulk}}^{(n)}$		(26)
		$\displaystyle=$	$\displaystyle E(\mathrm{CuCrO_{2}})_{\mathrm{bulk}}^{(n-1)}+E(\mathrm{PdCrO_{2}})_{\mathrm{(2/3)V_{Cr}}}^{(n-1)}$
		$\displaystyle+$	$\displaystyle E(\mathrm{CuPd})+(4/3)E(\mathrm{Cr_{2}O_{3}})+Q\left(\mathrm{Cr_{2}O_{3},V_{Cr}^{int}}\right).$

Here, for example, $E(\mathrm{CuCrO_{2}})_{\mathrm{bulk}}^{(n)}$ is the energy of $n$ f.u. bulk CuCrO₂, $E(\mathrm{PdCrO_{2}})_{\mathrm{V_{O}}}^{(n)}$ is the energy of $n$ f.u. PdCrO₂ with an oxygen vacancy, $E(\mathrm{PdCrO_{2}})_{(2/3)\mathrm{V_{Cr}}}^{(n-1)}$ is the energy of $(n-1)$ f.u. PdCrO₂ with a fraction of $2/3$ chromium vacancies, and $E(\mathrm{CuPd})$ is the energy of 1 f.u. bulk CuPd. For the bulk, the following relationships hold according to the definitions.

	$\displaystyle E(\mathrm{CuCrO_{2}})_{\mathrm{bulk}}^{(n)}$	$\displaystyle=$	$\displaystyle n\,E(\mathrm{CuCrO_{2}}),$		(27)
	$\displaystyle E(\mathrm{PdCrO_{2}})_{\mathrm{bulk}}^{(n)}$	$\displaystyle=$	$\displaystyle n\,E(\mathrm{PdCrO_{2}}).$		(28)

Define the energy gain from removing $m$ oxygen or chromium vacancies in $n\mathrm{PdCrO_{2}}$ as follows:

	$\displaystyle\nu_{\mathrm{O}}(n,m)$	$\displaystyle\equiv$	$\displaystyle E(\mathrm{PdCrO_{2}})_{\mathrm{bulk}}^{(n)}-E(\mathrm{PdCrO_{2}})_{m\,\mathrm{V_{O}}}^{(n)},$		(29)
	$\displaystyle\upnu_{\mathrm{Cr}}(n,m)$	$\displaystyle\equiv$	$\displaystyle E(\mathrm{PdCrO_{2}})_{\mathrm{bulk}}^{(n)}-E(\mathrm{PdCrO_{2}})_{m\,\mathrm{V_{Cr}}}^{(n)}.$		(30)

In the thermodynamic limit ($n\rightarrow\infty$), the following relationships should hold:

	$\displaystyle\nu_{\mathrm{O,Cr}}(n,m)$	$\displaystyle\simeq$	$\displaystyle\upnu_{\mathrm{O,Cr}}(n-1,m),$		(31)
	$\displaystyle\nu_{\mathrm{O,Cr}}(n,m)$	$\displaystyle\simeq$	$\displaystyle m\,\upnu_{\mathrm{O,Cr}}(n,1),$		(32)

Moreover, define

$$\upnu_{\mathrm{O,Cr}}\equiv\upnu_{\mathrm{O,Cr}}(n,1)$$

(33)

These are the $\upnu_{\upalpha}$ defined in eq (14). Applying eqs (27)–(33) to eqs (25) and (26), yields eqs (15) and (16).

When $n$ vacancies exist in $N$ sites, the configurational entropy is

$\displaystyle S(N,n)=k_{\mathrm{B}}\ln{\frac{N!}{(N-n)!n!}}.$

(34)

The entropy change achieved by adding one vacancy is given by

	$\displaystyle\Delta S(N,n)$	$\displaystyle\equiv$	$\displaystyle S(N,n+1)-S(N,n)$		(35)
		$\displaystyle=$	$\displaystyle k_{\mathrm{B}}\left[-\ln{(c_{\mathrm{v}}+1/N)}+\ln{(1-c_{\mathrm{v}})}\right],$
	$\displaystyle c_{\mathrm{v}}$	$\displaystyle\equiv$	$\displaystyle n/N.$		(36)

For the limit of $N\to\infty$ with fixed $c_{v}$,

$\displaystyle\Delta S(N,n)\to\Delta S(c_{\mathrm{v}})=k_{\mathrm{B}}\left[-\ln{(c_{\mathrm{v}})}+\ln{(1-c_{\mathrm{v}})}\right]$

(37)

Therefore, the free energy change achieved by removing one vacancy is given by

	$\displaystyle\Delta F(T,c_{v})$	$\displaystyle=$	$\displaystyle-T(-S(c_{\mathrm{v}}))$		(38)
		$\displaystyle=$	$\displaystyle k_{\mathrm{B}}T\left[-\ln{(c_{\mathrm{v}})}+\ln{(1-c_{\mathrm{v}})}\right].$		(39)