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# Control of Impurity Phase Segregation in a PdCrO2/CuCrO2 Heterostructure

Tom Ichibha1<EMAIL_ADDRESS>Sangmoon Yoon1,2 Jong Mok Ok1,3 Mina Yoon1 Ho
Nyung Lee1 Fernando A. Reboredo1<EMAIL_ADDRESS>1Materials Science and
Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
2Department of Physics, Gachon University, Seongnam 13306, Republic of Korea
3Department of Physics, Pusan National University, Pusan 46241, Republic of
Korea

###### Abstract

PdCrO2 films are synthesized on CuCrO2 buffer layers on Al2O3 substrates. This
synthesis is accompanied by impurity phase segregation, which hampers the
synthesis of high quality PdCrO2 films. The potential causes of impurity phase
segregation were studied by using a combination of experiments and ab initio
calculations. X-ray diffraction and scanning transmission electron microscopy
experiments revealed impurity phases of CuxPd1-x alloy and chromium oxides,
Cr2O3 and Cr3O4, in PdCrO2. Calculations determined that oxygen deficiency can
cause the impurity phase segregation. Therefore, preventing oxygen release
from delafossites could suppress the impurity phase segregation. The amounts
of Cr2O3 and Cr3O4 depend differently on temperature and oxygen partial
pressure. A reasonable theory-based explanation for this experimental
observation is provided.

##  I Introduction

Delafossites are intriguing materials that can combine 2D electronic
conductivity in the cation $A$ layers and magnetism in slightly distorted
octahedra in the $B$O6 layers, which stack alternately [1, 2, 3]. The abundant
possible choices of monovalent $A$ and trivalent $B$ cations lead to a number
of delafossite materials with diverse physical properties [4, 5]. The $AB$O2
delafossites were first reported in 1971 by a group of the DuPont Experimental
Station [1, 2, 3, 4]. A quarter of century after delafossites were first
reported, they received renewed attention when the transparent $p$-type
semiconductor CuAlO2 was discovered [6, 7]. Simultaneously, Tanaka et al.[8]
reported the strong anisotropy of electronic conduction for the metallic
PdCoO2 single crystals [3]. One decade later, Takatsu and Maeno et al.,
working on PdCoO2 and PdCrO2, reported the growth of single crystals of PdCrO2
[9]. These single crystals exhibit intriguing phenomena[4] such as the
unconventional anomalous Hall effect in PdCrO2[10] and anomalous temperature
dependence of specific heat and electrical resistivity that are driven by
high-frequency phonons in PdCoO2[10]. Their seminal work originated the
continuous study of delafossite metals to this day.

Delafossite metals have electronic conductivity comparable with the most
conductive pure metals [3, 11, 8, 4] owing to their remarkably long electronic
mean free paths of up to 20 $\upmu$m [4, 12, 13]. Among delafossite metals,
PdCrO2 is especially interesting because it coexists with a layerwise non-
collinear spin state [14, 15, 16, 17, 18] and exhibits high electronic
conductivity [16]. Its topological properties, primarily caused by spin–orbit
coupling in Pd, allow for the observation of an unconventional anomalous Hall
effect [10, 19] in bulk PdCrO2. Additionally, PdCrO2 films and surfaces have
been studied. Angle-resolved photoemission spectroscopy experiments showed
that Pd-terminated PdCrO2 has surface ferromagnetism, which may originate from
the Stoner-like instability[20, 21]. Experimental studies of PdCrO2 films
established that the antiferromagnetic spin state remains stable down to a
thickness of 3.6 nm [22].

Hybrid layered heterostructures, composed of PdCrO2 and other delafossite
materials, could exhibit interesting and different phenomena than their parent
compounds [23]. However, despite the interest in the material, the epitaxial
growth of PdCrO2 films has not been widely studied [24, 25, 26, 27, 22]. The
growth of PdCrO2 films on Al2O3 is sometimes accompanied by impurity phases
(i.e., CuxPd1-x alloy and chromium oxides) [22]. Recent research discovered
that a one-monolayer buffer layer of CuCrO2 on an Al2O3 substrate suppresses
this instability [22]. However, a nonnegligible amount of impurity phase is
still formed. Understanding the mechanism of the impurity phase segregation
and how to suppress it is highly desired for the growth of heterostructures
containing PdCrO2 or other Pd-based delafossites.

In this work, the mechanism of impurity phase segregation of a heterostructure
of a PdCrO2 layer with a CuCrO2 buffer layer on an Al2O3 substrate was studied
using a combination of experiments and ab initio calculations. X-ray
diffraction (XRD) and scanning transmission electron microscopy (STEM)
experiments were performed, and the segregation of CuxPd1-x alloy and chromium
oxide (Cr2O3 and Cr3O4) impurity phases was observed. These experiments
revealed that the formation of Cr2O3 negatively correlates with oxygen partial
pressure, whereas the formation of Cr3O4 does not correlate with oxygen
partial pressure. Moreover, the Cr2O3 (Cr3O4) formation weakly (strongly)
positively correlates with temperature. The segregation of CuxPd1-x alloy and
chromium oxide impurity phases must be accompanied by the appearance or
disappearance of point defects because the segregation processes are not
stoichiometric. In this scenario, calculations revealed that oxygen vacancies
can cause the impurity phase segregation. Calculations also revealed that the
segregation of Cr2O3 or Cr3O4 is energetically the most favorable among the
chromium oxides, agreeing with the experiments described in Section III.1.
Finally, the calculations also revealed that the formation of Cr2O3 and Cr3O4
depends on temperature and oxygen partial pressure.

##  II Experimental and calculation details

###  II.1 Experimental details

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.

###  II.2 Calculation details

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.

##  III Results and Discussions

###  III.1 Segregation of impurity phases

Impurity phases including CuxPd1-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.

Figure 1:  XRD spectrum and mapping of XRD peak strengths of Cr3O4 and Cr2O3
for temperature and oxygen partial pressure. This measurement was performed
for an approximately 6 nm PdCrO2 film grown on a one-monolayer CuCrO2 buffer
layer.

###  III.2 Point-defect formation energy

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 CuCr (PdCr). Larger defect complexes such as
CuCr&$\mathrm{V_{Cu}}$ (PdCr&$\mathrm{V_{Pd}}$) 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, CuxPd1-x, and
a chromium oxide. The exact value of $x$ in CuxPd1-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.

###  III.3 Formation energies of defects as a function of the chemical
potentials

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 $\mathrm{Cu_{Cr}}$.

Figure 2:  Formation energies of intrinsic point defects in CuCrO2, PdCrO2,
and Al2O3 calculated by the PBE+$U$ method as a function of the oxygen
chemical potential. The chemical potentials are calculated for CuCrO2, PdCrO2,
Al2O3, CuPd, and different chromium oxides.

###  III.4 Instability of CuCrO2 and PdCrO2 for oxygen-deficient samples

Experiments found the segregation of impurity phases of CuxPd1-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, CuxPd1-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 CuxPd1-x. Preexisting defects as energetic as oxygen
vacancies in Al2O3 could enhance the segregation of Cr3O4 and CuxPd1-x.

###  III.5 Thermochemical reactions

To simplify the analysis, the CuxPd1-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
VO in CuCrO2, PdCrO2, or Al2O3 and formation of Cr-deficient defects such as
VCr 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 CrxO1-x but not CuyPd1-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 $\mathrm{Al_{2}O_{3}}$.

Table 1:  Exothermic energies, $Q$ in eqs (15)–(18), for the formation of Cr2O3 or Cr3O4 and CuPd accompanied by $\mathrm{V_{O}^{rem}}$’s removal from or $\mathrm{V_{Cr}^{int}}$’s introduction to different bath locations.  Bath location | $Q\left(\mathrm{Cr_{2}O_{3},V_{O}^{rem}}\right)$ | $Q\left(\mathrm{Cr_{2}O_{3},V_{Cr}^{int}}\right)$ | $Q\left(\mathrm{Cr_{3}O_{4},V_{O}^{rem}}\right)$ | $Q\left(\mathrm{Cr_{3}O_{4},V_{Cr}^{int}}\right)$
---|---|---|---|---
PdCrO2 | +2.703 |  $-$0.990 | +2.466 |  $-$3.192
CuCrO2 | +3.308 |  $-$0.989 | +3.273 |  $-$3.189
Al2O3 | +5.666 | – | +6.417 | –
O2 in vacuum ($T=0$) |  $-$1.619 | – |  $–$3.297 | –
Figure 3:  Plots of $Q\left(\mathrm{Cr_{2}O_{3},V_{O}^{rem}}\right)$ and
$Q\left(\mathrm{Cr_{3}O_{4},V_{O}^{rem}}\right)$ shown in Table 1 for
different $\upnu_{\mathrm{O}}$ (i.e., different O particle baths).

###  III.6 Entropy contributions to the formation of
Cr2O3 and Cr3O4

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.

Figure 4:  Plots of the energy of the oxygen sink ($\nu_{\mathrm{O}}$) using
eq (14) for different possible locations in the experimental range of
temperatures and estimated concentration. The upper and lower edges of
$\mathrm{V_{O}^{rem}}$ in O2 gas are $P_{\mathrm{O_{2}}}$ = 1 and 10-6 atm.
The upper and lower edges of the other areas are $c_{\mathrm{v}}$ = $10^{-1}$
and $10^{-8}$ per site. The dark green area between the blue and green areas
is the overlap of the blue and green areas.

##  IV Conclusion

The mechanism of impurity phase segregation with the epitaxial growth of a
PdCrO2 layer on a CuCrO2 buffer layer on an Al2O3 substrate was investigated
via a combination of experiments and ab initio calculations. XRD experiments
revealed the formation of CuxPd1-x alloy and chromium oxide (Cr2O3 and Cr3O4)
impurity phases. Consequently, the impurity phase segregation should be
involved with appearance or disappearance of point defects or oxygen migration
because the possible segregation processes are not stoichiometric. In this
scenario, several possible mechanisms of impurity phase segregation were
considered with oxygen vacancy disappearance or chromium vacancy appearance
into different particle baths: Al2O3, CuCrO2, PdCrO2, and the dilute oxygen
gas. Calculations established that the oxygen vacancy consumption processes
are energetically favorable and supported experimental evidence that Cr2O3 or
Cr3O4 are the predominant chromium oxide impurity phases. Specifically,
preventing the release of oxygen atoms from delafossite materials could
suppress the impurity phase segregation.

## Appendix

###  IV.1 Derivation of eqs (15) and (16)

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).

###  IV.2 Configurational entropy of removing a vacancy.

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)

##  V Acknowledgments

We acknowledge E. Heinrich for valuable help with manuscript preparation. This
work was supported by the US Department of Energy, Office of Science, Basic
Energy Sciences, Materials Sciences and Engineering Division (theory and
synthesis) and as part of the Computational Materials Sciences Program and
Center for Predictive Simulation of Functional Materials (structural
characterization). VESTA [46] was used to draw the crystal structures.

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