# Mechanistically-guided materials chemistry: synthesis of new ternary
nitrides, CaZrN2 and CaHfN2

Christopher L. Rom  [  Andrew Novick  [  Matthew J. McDermott  Materials
Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA
Andrey A. Yakovenko  [  Jessica R. Gallawa  [  Gia Thinh Tran  [  Dominic C.
Asebiah  [  Emily N. Storck  [  Brennan C. McBride  [  Rebecca C. Miller  [
Amy L. Prieto  [  Kristin A. Persson  Molecular Foundry, Lawrence Berkeley
National Laboratory, Berkeley, CA, USA  Eric Toberer  [  Vladan Stevanović  [
Andriy Zakutayev  [  James R. Neilson  [<EMAIL_ADDRESS>
###### Abstract

Recent computational studies have predicted many new ternary nitrides,
revealing synthetic opportunities in this underexplored phase space. However,
synthesizing new ternary nitrides is difficult, in part because intermediate
and product phases often have high cohesive energies that inhibit diffusion.
Here, we report the synthesis of two new phases, calcium zirconium nitride
(CaZrN2) and calcium hafnium nitride (CaHfN2), by solid state metathesis
reactions between Ca3N2 and $M$Cl4 ($M$ = Zr, Hf). Although the reaction
nominally proceeds to the target phases in a 1:1 ratio of the precursors via
Ca3N2 + $M$Cl4 -¿ Ca$M$N2 + 2 CaCl2, reactions prepared this way result in Ca-
poor materials (Ca_$x$ $M$_${2-x}$N2, $x<1$). A small excess of Ca3N2 (ca. 20
mol%) is needed to yield stoichiometric Ca$M$N2, as confirmed by high-
resolution synchrotron powder X-ray diffraction. In situ synchrotron X-ray
diffraction studies reveal that nominally stoichiometric reactions produce
Zr^3+ intermediates early in the reaction pathway, and the excess Ca3N2 is
needed to reoxidize Zr^3+ intermediates back to the Zr^4+ oxidation state of
CaZrN2. Analysis of computationally-derived chemical potential diagrams
rationalizes this synthetic approach and its contrast from the synthesis of
MgZrN2. These findings additionally highlight the utility of in situ
diffraction studies and computational thermochemistry to provide mechanistic
guidance for synthesis.

Colorado State University]Department of Chemistry, Colorado State University,
Fort Collins, CO, USA [National Renewable Energy Laboratory]Materials Science
Center, National Renewable Energy Laboratory, Golden, CO, USA Colorado School
of Mines]Department of Physics, Colorado School of Mines, Golden, CO, USA
Department of Materials Science and Engineering, University of California,
Berkeley, CA, USA Argonne National Laboratory]X-ray Science Division, Advanced
Photon Source, Argonne National Laboratory, Lemont, IL, USA Colorado State
University]Department of Chemistry, Colorado State University, Fort Collins,
CO, USA Colorado State University]Department of Chemistry, Colorado State
University, Fort Collins, CO, USA Colorado State University]Department of
Chemistry, Colorado State University, Fort Collins, CO, USA Colorado State
University]Department of Chemistry, Colorado State University, Fort Collins,
CO, USA Colorado State University]Department of Chemistry, Colorado State
University, Fort Collins, CO, USA Colorado State University]Analytical
Resources Core, Colorado State University, Fort Collins, CO, USA Colorado
State University]Department of Chemistry, Colorado State University, Fort
Collins, CO, USA Department of Materials Science and Engineering, University
of California, Berkeley, CA, USA Colorado School of Mines]Department of
Physics, Colorado School of Mines, Golden, CO, USA Colorado School of
Mines]Department of Physics, Colorado School of Mines, Golden, CO, USA
National Renewable Energy Laboratory]Materials Science Center, National
Renewable Energy Laboratory, Golden, CO, USA School of Advanced Materials
Discovery, Colorado State University, Fort Collins, CO, USA Colorado State
University]Department of Chemistry, Colorado State University, Fort Collins,
CO, USA

##  1 Introduction

Predictive synthesis remains an important goal for materials chemists.1, 2, 3
While many joint computational-experimental synthesis efforts have been
reported in literature,4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 trial-and-
error approaches remain the dominant paradigm. Within this current paradigm,
in situ X-ray diffraction (XRD) is a powerful technique for materials
discovery.17, 18, 15, 19, 20 Sometimes called “panoramic synthesis,” 17, 18,
19 in situ X-ray diffraction allows for observation of solid-state reactivity
as it happens. Short-lived intermediate species and subtle crystallographic
changes can be detected which would otherwise be missed by ex situ
experiments, accelerating materials discovery. In a move towards predictive
synthesis, in situ XRD has been used in concert with computationally-generated
chemical potential diagrams21 to rationalize polymorph selectivity in the
synthesis of ternary oxides22 However, this specific combined approach of in
situ XRD and chemical potential diagrams has yet to be used to discover new
materials.

Nitrides are a compelling class of materials for melding computational
guidance with in situ X-ray diffraction (XRD) studies because they are
difficult to synthesize. High-temperatures are often needed to overcome the
relative inertness of N2 (941 kJ/mol dissociation energy)23 and drive solid
state diffusion.24, 25 However, elevated temperatures increase the entropic
favorability of N2 gas, which leads many nitrides to decompose at relatively
low temperatures.26, 27 Therefore, successful synthesises may be limited to
narrow temperature windows, which in situ XRD can help identify.28 Recent
computational studies have predicted numerous stable phases in previously
uncharted spaces29, 11, 30, 31, 4, accelerating the discovery of new phases
(e.g., MgSnN232, 33, CaSnN234, and MgZrN235, 36). Yet many predicted phases
remain undiscovered, providing copious targets for synthesis.

All of the $AB$N2 phases ($A$ = Mg, Ca, Sr, Ba; $B$ = Ti, Zr, Hf) have been
synthesized previously35, 37, 38, 39, 40, with the exception of CaZrN2 and
CaHfN226. As summarized in Figure 1, these phases crystallize with several
structure types: rocksalt ($Fm\overline{3}m$), KCoO2-type ($P4/nmm$), and
$\alpha$-NaFeO2 ($R\overline{3}m$). Given the intermediate size of Ca^2+
relative to the other alkali earths (Figure 1)41, CaZrN2 and CaHfN2 were
predicted to crystallize in the $\alpha$-NaFeO2 structure type42. However,
syntheses of CaZrN2 and CaHfN2 have not been previously reported, although one
group described some synthetic attempts towards these ternaries that yielded
only ZrN and HfN43.

Figure 1: The $AB$N2 ternaries with larger alkali earth cations (Sr and Ba)
crystallize in cation-ordered phases of the KCoO2-type and
$\alpha$-NaFeO2-type structures, while Mg$B$N2 phases take the rocksalt (RS)
structure. This work shows that CaZrN2 and CaHfN2 crystallize in the cation-
disordered RS structure. Radii are sourced from literature41.

Here, we report the discovery of CaZrN2 and CaHfN2 in the disordered rocksalt
(RS) structure via a metathesis approach. Heating Ca3N2 and ZrCl4 or HfCl4 up
to a high-temperature dwell (ca. 1000 °C) for a brief period of time (10 min)
yields CaZrN2 and CaHfN2 as fine powders (along with a byproduct CaCl2).
Importantly, a slight excess of Ca3N2 is needed, a fact we rationalize based
on our in situ synchrotron X-ray diffraction studies and computational
thermodynamic analysis of the reaction pathway that show reduced Zr^3+
intermediates. Chemical potential diagrams rationalize the thermodynamics
behind the observed reactions, accurately predicting differences in the
reaction pathway between the reactions targeting CaZrN2 vs. MgZrN2 vs. ZnZrN2
(which has not yet been synthesized in bulk). Additional thermodynamic
calculations show that these disordered RS ternaries are metastable (relative
to the cation-ordered polymorph) even at the high synthesis temperatures. In
sum, these findings show how chemical potential diagrams can augment in situ
XRD experiments and improve our ability to predict synthesis pathways.

##  2 Results and Discussion

###  2.1 Structural and compositional analysis

Figure 2:  SXRD data of a) CaZrN2 as synthesized and b) washed along with c)
CaHfN2 as synthesized and d) washed. Simulated patterns for relevant phases
are shown for reference (top). The predicted, ordered structure
($R\overline{3}m$) of Ca$M$N2 was not observed. A single impurity peak appears
after washing (marked by $*$), possibly a disordered Ca4Cl6O or calcium
methoxide. Dashed gray vertical lines guide the eye to the (111), (200), and
(220) reflections of Ca$M$N2 $Fm\overline{3}m$. Full diffraction patterns are
shown in Figure S2, and refinement results are in Tables S1-S2 and Figure S3.

High-resolution synchrotron powder X-ray diffraction (SXRD) measurements show
that metathesis reactions between Ca3N2 and $M$Cl4 ($M$ = Zr, Hf) yielded
CaZrN2 and CaHfN2 in the RS structure ($Fm\bar{3}m$, Figure 2). These samples
(1.21 Ca3N2 + ZrCl4 and 1.16 Ca3N2 + HfCl4) were heated at +5 °C/min to 1100
°C, dwelled for 10 min, and then air-quenched to yield Ca$M$N2 along with an
array of side products (CaCl2, Ca2NCl, Ca4Cl6O, and in the case of $M$ = Hf,
trace HfN). Recovered samples were black, red, or green depending on precise
synthesis conditions (Figure S1). Washing the samples with anhydrous methanol
removes the chloride-containing byproducts (CaCl2, Ca2NCl, Ca4Cl6O). However,
one unindexed diffraction peak is present at $Q=0.8$ Å-1. This peak matches
both the (100) reflection from Ca4Cl6O and the most intense reflection in a
prior report of calcium methoxide44. Unfortunately, the presence of the
impurity phase inhibits accurate compositional analysis by Scanning Electron
Microscopy (SEM) with Energy Dispersive X-ray Spectroscopy (EDS), rigorously
air-free X-ray Photoelectron Spectroscopy (XPS), and combustion analysis
(further described in the Supporting Information, Figures S5-S8 and Tables
S3-S4).45, 46, 47, 48 Consequently, our characterization efforts centered on
diffraction.

Rietveld analysis of the RS products show these phases are consistent with
cation-disordered CaZrN2 and CaHfN2. The metal site occupancy ($x$ in Ca_$x$
$M$_$2-x$N2) refined to 1.00(2) for the washed CaZrN2 and 1.00(1) for the
washed CaHfN2 sample. Washing does not substantially change the refined metal
site occupancy or lattice parameter compared to the samples as synthesized
(Figure S3). Lattice parameters of the washed Ca$M$N2 phases refined to
substantially larger unit values ($a=4.7486(3)$ Å and $4.7224(4)$ Å for CaZrN2
and CaHfN2, respectively) than their isostructural binaries ($a=4.58$ Å and
$4.52$ Å for ZrN and HfN, respectively)49, 50. These unit cell values are
close to the Ca-N-Zr distance in the computationally-predicted, cation-ordered
CaZrN2 structure (4.744 Å), which serves as an estimate for the theoretical
disordered RS lattice parameter (Figure S4). We focused most of our work on
the Zr analog, since Zr and Hf exhibit analogous chemistry in this case.

Figure 3: a) The precursor ratio (expressed as $n$ in $n$Ca3N2 + ZrCl4)
impacted the lattice parameter of the RS phase and b) the distribution of
byproduct phases, as determined by Rietveld analysis. Approximately 20% mole
excess ($n=1.2$) Ca3N2 was necessary to yield stoichiometric CaZrN2, as
indicated by the estimated value of $a=4.744$ Å from the computationally-
predicted structure (Figure S4). However, some of the excess Ca3N2 also
yielded a Ca2NCl byproduct by reacting with CaCl2 (bottom). The byproduct
relative mol% of CaCl2, Ca2NCl, and Ca4Cl6O sum to 100% (i.e.,
Ca_${x}$Zr_${2-x}$N2 is not included owing to uncertainty in $x$). These data
were from samples held in alumina crucibles sealed in ampules under vacuum and
heated at 5 °C/min to 1000 °C, dwelled for 10 min, then quenched by removing
the ampule from the furnace and placing on the benchtop.

A slight excess of Ca3N2 (ca. 20 mol%) is required to yield stoichiometric
CaZrN2 (Figure 3), as measured by the product lattice parameter. Although the
nominal balanced equation for the reaction is Ca3N2 + ZrCl4 -¿ CaZrN2 + 2
CaCl2, a 1:1 precursor ratio yields a lattice parameter of 4.72 Å, trending
towards ZrN and suggesting a Ca-poor phase. With an additional 20 mol% Ca3N2
(i.e., $n=1.20$), the RS lattice parameter is closer to the 4.744 Å value
estimated from the computationally-predicted structure (Figure S4). The
continued growth in the RS lattice parameter above $n=1.20$ could suggest the
formation of Ca-interstitials, but characterizing such defects is beyond the
scope of this study. This trend contrasts with our prior synthesis of MgZrN2,
where the stoichiometric reaction 2.0 Mg2NCl + ZrCl4 -¿ MgZrN2 + 3 MgCl2
produced the targeted ternary (RS MgZrN2), and additional Mg2NCl did not
change the RS lattice parameter or increase Mg content (as measured by SXRD,
Inductively Coupled Plasma Atomic Emission Spectroscopy, and Energy Dispersive
X-ray Spectroscopy).36 Some of the excess Ca3N2 reacts with the CaCl2
byproduct to make Ca2NCl, which is observed in an increasing mol% with
increasing Ca3N2 content. The need of excess Ca3N2 is not attributable to the
oxide impurity in the Ca3N2 precursor, which is estimated at $<5$ wt% of our
precursor by quantitative Rietveld analysis (below the detection limit of
laboratory PXRD). We also note that increasing dwell time from 10 min to 4 h
did not substantially change the RS lattice parameter, size, or strain for Ca-
rich reactions (2.0 Ca3N2 + ZrCl4, $T_{\mathrm{rxn}}=1000$ °C, Figure S10).
Increasing temperature decreases the lattice parameter but increases the
crystallite size (1.2 Ca3N2 + ZrCl4, dwell time = 10 min, Figure S11).
Furthermore, increasing dwell times for Ca-poor reactions decreases the
rocksalt lattice parameter (dwell temperature = 900 °C, Figure S12). The key
observation is that some excess Ca3N2 is necessary. This result is explained
by observations from our in situ SXRD measurements, as discussed in the next
section.

Using metathesis (ion exchange) reactions is also a key synthetic choice for
the synthesis of CaZrN2, as this process circumvents the solid-state diffusion
challenges common to traditional ceramic syntheses51, 52. A control reaction
between Ca3N2 and Zr under flowing N2 at 1000 °C for 10 h produces only ZrN
alongside unreacted Ca3N2, indicating that Ca diffusion into the ZrN is slow
(Figure S13). Literature also shows the difficulty of making CaZrN2 and
CaHfN2. In their report on the synthesis of Ca4TiN4 and Ca5NbN5 with the
assistance of a Li3N flux, Hunting, et al., noted that similar reactions
between Ca3N2 and Zr or Hf powders in Li3N flux only yielded the ZrN and
HfN43. And although many closely related compounds have been synthesized by
ceramic techniques (i.e., CaTiN2, SrZrN2, BaZrN2)37, 39, 53, the absence of
CaZrN2 and CaHfN2 from the literature implies that a synthetic strategy had
previously been elusive.

We acknowledge that the perfectly stoichiometric compounds CaZrN2 and CaHfN2
may not be synthesized from the reactions, 1.21 Ca3N2 + ZrCl4 and 1.16 Ca3N2 +
HfCl4; however, these were the optimal reaction conditions identified in this
study. These synthesis conditions produced RS structures most consistent with
the title compounds, given our structural analysis of lattice parameter and
cation site occupancy. Detailed compositional analysis was not possible with
our current samples given an unidentified impurity phase (See the Supporting
Information, Figures S5-S8 and Tables S3-S4). Precise analysis of composition
(including defects) will be essential for better understanding the potential
applications of these new semiconductors, but such analysis is beyond the
scope of this study. Instead, we focus on the mechanisms of this synthesis,
with the goal of showing how in situ SXRD studies and thermodynamic
calculations can guide metathesis reactions for the synthesis of elusive new
materials.

###  2.2 In situ synchrotron powder X-ray diffraction

Figure 4: Phase contributions to the in situ SXRD data as a function of
temperature as indicated by the weighted scale factors (W.S.F.) for reactants,
intermediates, and products upon heating precursor mixtures of a) Ca3N2 +
ZrCl4, b) Mg3N2 + ZrCl4, c) 2 Ca2NCl + ZrCl4, and d) 2 Mg2NCl + ZrCl4. In
reactions with Mg3N2, Ca3N2, and Ca2NCl, trivalent Zr species were observed
(ZrCl3 and Zr6NCl15), indicating reduction of Zr^4+ to Zr^3+ and loss of N2.
Only the reaction using Mg2NCl maintained the tetravalent oxidation state of
Zr. The ternary phases Ca_$x$Zr_${2-x}$N2 and Mg_$x$Zr_${2-x}$N2 are denoted
as “RS”. The grey highlighted region of (d) shows the temperature range used
for step 1 of the two-step synthesis of MgZrN2 detailed in prior reports36,
54.  Figure 6: Chemical potential diagrams for a) the Ca-Zr-N-Cl chemical
system and b) the Mg-Zr-N-Cl chemical system calculated for 300 °C. Focused
view highlighting the interfaces between ZrNCl and c) Ca2NCl or d) Mg2NCl.
Select nitrogen-containing phases in the 4-component system are illustrated
via their intersection with the 3-dimensional
$\mu_{\mathrm{Ca}}-\mu_{\mathrm{Zr}}-\mu_{\mathrm{Cl}}$ or
$\mu_{\mathrm{Mg}}-\mu_{\mathrm{Zr}}-\mu_{\mathrm{Cl}}$ spaces. These
intersections may appear with complicated volumes due to the lower dimensional
intersection (e.g., CaZrN2, ZrN). Full quaternary diagrams are shown in Figure
S22. Curved arrows qualitatively illustrate the experimentally observed
reaction pathways.

In situ synchrotron powder X-ray diffraction studies (SXRD) at the Advanced
Photon Source (17-BM) reveals the pathway through which the metathesis
reactions proceed. We examined reactions in sealed quartz capillaries
targeting CaZrN2 (and MgZrN2, for comparison) starting from reagent mixtures
for the ideal reactions: $A$3N2 + ZrCl4 -¿ $A$ZrN2 + 2 $A$Cl2 and 2 $A$2NCl +
ZrCl4 -¿ $A$ZrN2 + 3 $A$Cl2 ($A$ = Mg, Ca). These ratios are calcium-deficient
compared to the optimized synthesis of CaZrN2 discussed previously, yet are
nonetheless illuminating. They show how ostensibly similar precursors ($A$3N2
and $A$2NCl) proceed via different reaction pathways. These differences in
reaction pathway ultimately required different synthetic conditions for the
optimized syntheses of MgZrN2 and CaZrN2. The relative amounts of crystalline
phases present during the reaction (determined by quantitative phase analysis
using the Rietveld method of in situ diffraction data) are shown in Figure 4,
with the raw diffraction patterns shown in Figures S17 and S18. The observed
Zr-containing intermediates are summarized in Table
LABEL:tab:insitu_Zr_summary.

Quantitative phase analysis of in situ SXRD data from the reaction between
Ca3N2 + ZrCl4 show that reduced Zr^3+ species form at low reaction
temperatures (Figure 4a). ZrCl3 is the first intermediate to form, appearing
by 200 °C. Subsequently, ZrCl4 disappears from the patterns as it sublimes
($T_{\mathrm{sub}}$ = 331 °C), at which point CaCl2 begins to grow in. Near
400 °C, the ZrCl3 converts to a nitride chloride, Zr6NCl15. Throughout this
process, the Ca3N2 precursor steadily declines. Ca3N2 is fully consumed by 600
°C, with some of it reacting with the CaCl2 byproduct to make Ca2NCl (Ca3N2 +
CaCl2 -¿ 2 Ca2NCl). Ca2NCl persists up to 700 °C. By ca. 550 °C, the RS phase
Ca_$x$Zr_${2-x}$N2 begins to grow in (initially as ZrN, as will be detailed
subsequently). A minor phase of Ca4Cl6O also grows in, indicating a small
degree of oxygen impurity. The reaction of 2 Ca2NCl + ZrCl4 proceeds in a
similar fashion as Ca3N2 + ZrCl4. However, using Ca2NCl also includes a ZrNCl-
like intermediate, which is not detected in the Ca3N2 reaction (Figure S17).
In sum, the initial reaction between Ca3N2 + ZrCl4 proceeds through Zr^3+
intermediates via the following balanced equations:

The final step of this process involves the reaction of the Ca2NCl
intermediate with the ZrN rocksalt phase:

$\ce{4\ Ca2NCl+6\ ZrN+N2->6\ CaZrN2+2\ CaCl2}.$ (1)

In contrast, the analysis of in situ SXRD data in the analogous Mg system
shows reactivity differences between Mg3N2 and Mg2NCl, with Mg2NCl conserving
the tetravalent state of Zr^4+ throughout the reaction pathway (Figure 4b and
d, Table LABEL:tab:insitu_Zr_summary). The observed reaction pathways are
consistent with our prior (lower resolution) in situ study on the synthesis of
MgZrN2.36 Both the reaction of Mg3N2 + ZrCl4 and the reaction of 2 Mg2NCl +
ZrCl4 begin with ZrCl4 sublimation, followed by the formation of a small
amount of MgZrCl6 near 300 °C. However, the reaction with Mg3N2 goes on to
produce Zr6NCl15 near 430 °C, which indicates the reduction of Zr^4+ to Zr^3+,
likely paired with the production of N2. Furthermore, the amount of Mg3N2 only
decreases slightly by 600 °C (at which point, the measurement was stopped).
This result suggests that Mg3N2 is a kinetically slow precursor. On the other
hand, the Mg2NCl reaction does not produce any distinct Zr^3+ intermediates.
Instead, the amount of Mg2NCl decreases in a stepwise fashion near 440 °C
coincident with the formation of ZrNCl and the rapid increase of MgCl2. A
second stepwise decrease in Mg2NCl occurs near 600 °C, coincident with the
consumption of the ZrNCl and the formation of the RS phase
(Mg_$x$Zr_${2-x}$N2). Therefore, the synthesis of MgZrN2 appears to proceed
through two reaction steps:

The chemical potential diagrams shown in Figure 4 describe the energetic
landscape through which these metathesis reactions proceed. These models are
computed using calculated phase energies from the Materials Project database56
(see Methods). Each visible facet is a stable phase in the $A$-Zr-Cl system;
larger facets indicate phases with deeper stability on the convex hull of a
compositional phase diagram relative to competing phases. Nitrogen-containing
phases are illustrated by 3-dimensional ($A$-Zr-Cl) polyhedral slices of the
full 4-dimensional polytopes. Intersection points, edges, or faces between
these regions of chemical potential space indicate which phases share stable
interfaces. It has previously been observed that a chemical reaction
proceeding under local equilibrium (i.e., diffusion-controlled) conditions
will follow a pathway between adjacent phases22, 2.

Figure 4a and c show that no Zr^4+-containing species in the Ca-Zr-N-Cl phase
space shares a stable interface with the CaZrN2 product, indicating that
Zr^3+-containing intermediates must form if the system proceeds through local
equilibrium at interfaces. Specifically, the region of stability for ZrN
connects both the Zr6NCl15 and ZrNCl spaces to the CaZrN2 space, indicating
that ZrN should form along the pathway to CaZrN2. In reality, due to the
accommodation of off-stoichiometry, there is likely no sharp boundary
(discontinuity) between ZrN and CaZrN2.21 Rather, this sharp transition is due
to the limitations of our thermodynamic data, as only ordered, stoichiometric
compounds are permitted in the DFT calculations of the Materials Project
database. We discuss the stability of cation-disordered CaZrN2 further in the
next section and in the Supporting Information (Figure S21).

The chemical potential diagram for the Mg-Zr-N-Cl system (Figure 4b) appears
similar to the Ca system but with one key difference: ZrNCl shares a stable
interface with both the precursor ZrCl4 and the product MgZrN2 (Figure 4d).
This shared boundary suggests that the intermediate ZrNCl (a Zr^4+-containing
phase) facilitates the formation of MgZrN2 at the ZrCl4$|$Mg2NCl interface
while preventing the system from undergoing Zr and N-based reduction and
oxidation.

Reaction network analysis57 can quantify these differences between the Ca- and
Mg-Zr-N-Cl systems. The method calculates a distance in chemical potential
space, simplifying these geometric representations into numeric values. The
chemical potential distance for the reaction of Ca2NCl + ZrNCl -¿ CaZrN2 +
CaCl2 is 0.733 eV/atom, where the non-zero number means that the reactants and
products do not share stable interfaces (as visualized in Figure 4c). In
contrast, Mg2NCl + ZrNCl -¿ MgZrN2 + MgCl2 (Figure 4d) has a chemical
potential distance of 0.033 eV/atom, which is only non-zero because of how
metastable phases like Mg2NCl are calculated. Further details are in the
Supporting Information (Tables S5-S13).

The chemical potential diagram boundaries are consistent with the in situ SXRD
measurements. For the Ca-system, the annotated reaction arrows in Figure 4a
correspond to the intermediates observed to form by in situ SXRD (Figure 4),
and the progression from ZrN to CaZrN2 via a Ca_$x$Zr_2-$x$N2 solid solution
(Figure LABEL:fig:wsf_v_lattice): arrow (i) represents Equation 2.2, arrow
(ii) Equation 2.2, arrow (iii) Equation 2.2, and arrow (iv) Equation 1 (or an
alternative hypothesis, Equations S1, S2). Similarly, the chemical potential
diagram for the Mg-system also matches the observed behavior of the Mg2NCl +
ZrCl4 reaction: arrow (v) represents Equation LABEL:eq:Mg2NCl_ZrNCl and arrow
(vi) Equation LABEL:eq:ZrNCl_MgZrN2.

This mechanistic guidance provided the key insight that ultimately led to our
successful syntheses. As the Ca-Zr-N-Cl chemical potential diagram (Figure 4a,
c) shows that ZrN formation is unavoidable along the path to CaZrN2, we
changed our synthesis conditions (relative to the two-step synthesis of
MgZrN2)36 to deal with the refractory ZrN intermediate. Specifically, we used
Ca3N2 in slight excess to provide an oxidant, and we increased our reaction
temperature to increase diffusion. We also used only one heating step, as a
low temperature step would not avoid Zr^4+ reduction (whereas in the case of
MgZrN2, the two-step process maintained the oxidation state of Zr^4+)36. These
changes led to the successful syntheses shown in Figure 2.

Importantly, these chemical potential diagrams can also be used to identify
thermodynamically disfavored reactions, allowing researchers to quickly move
beyond systems that are unlikely to work. To demonstrate this point, we also
conducted metathesis syntheses and thermodynamic calculations targeting ZnZrN2
from 1.2 Zn3N2 + ZrCl4 and 2.4 Zn2NCl + ZrCl4 (Figures S19 and S20). ZnZrN2
has been synthesized via combinatorial sputtering of thin films14 but has not
yet been synthesized in bulk. Our in situ SXRD experiments and thermodynamic
analysis reveal that the phase is unlikely to be synthesized via metathesis
without significantly increasing $\mu_{\mathrm{N}}$. The chemical potential
diagrams calculated as a function of temperature show that while ZnZrN2 is
predicted to be stable at room temperature, it becomes destabilized at the
more synthetically relevant temperature of 300 °C (Figure S19). Instead, ZrNCl
and ZrN are calculated to be the main thermodynamic products. In situ SXRD
measurements confirm these predictions (Figure S20). Although these
calculations do not rule-out the possibility of ZnZrN2 forming as a kinetic
product, they do show that reactions proceeding through local thermodynamic
equilibrium at elevated temperature and ambient N2 pressure will not yield the
targeted phase. These chemical potential diagrams therefore serve as powerful
tools to accelerate materials discovery via mechanistically guided synthesis.

Several shortcomings limit the predictive ability of chemical potential
diagrams. First, chemical potential diagrams with $>3$ dimensions may obscure
certain phases: the 4-dimensional chemical potential diagrams shown here are
only visualized in 3-dimensions of chemical potential ($\mu_{A}$,
$\mu_{\mathrm{Zr}}$, and $\mu_{\mathrm{Cl}}$) with select N-containing phases
shown as intersections with the 3-dimensional space. Fortunately, quaternary
compositional phase diagrams provide a complementary and complete, but
complicated, visualization (Figure S22). The geometric distances for
$n$-dimensional chemical spaces can also be calculated numerically using
reaction network analysis57 as further discussed in the Supporting Information
(Tables S5-S14). Chemical potential diagrams also lack true kinetic
information; they can only illustrate effective kinetic barriers provided by
the need of a reaction to form an additional intermediate phase to establish
local equilibrium2. Lastly, they are limited by the phases in the
computational databases (e.g., the Materials Project). These databases, while
large and growing larger, are incomplete4. They also only contain ordered
phases (not the disordered structures we observe).

###  2.4 Metastability of cation disorder in CaZrN2

Figure 7: Simulated diffraction patterns for the thermodynamic ground state of
CaZrN2, calculated as a function of temperature from the ensemble average. The
supercell reflection at $Q=1.1$ Å corresponding to the (003) plane of the
fully ordered ($R\bar{3}m$) structure persists even up to 3,000 K.

The observed disorder in CaZrN2 is counter-intuitive. The large contrast in
atomic radii between Ca^2+ and Zr^4+ (1.00 Å and 0.72 Å, respectively)41
should favor cation ordering as observed in CaTiN2, SrZrN2, and BaZrN2.37, 39,
53 Indeed, through the use of DFT and cluster expansion, we found that the
ground state configuration is strongly favored at 1,300 K (Figure S21). If
CaZrN2 were in thermodynamic equilibrium, there would be an insignificant
degree of cation disorder.

To relate these predictions to experimental observations, we simulated PXRD
patterns across all 24-atom configurations and ensemble averaged them for the
temperatures of interest (Figure 7). As expected for increasing disorder, the
reflection from the (003) plane in the ordered structure (arising from cation
layering) decays in intensity with increasing temperature. Temperatures
generated from DFT can be offset a couple hundred degrees from experiment58,
but the range from 300 to 3,000 K safely encompasses the synthesis temperature
of 1,300K. The (003) peak in the simulated patterns is significant even at
3,000 K, suggesting that the ordering in CaZrN2 would be observed in the XRD
pattern if the system were fully in thermodynamic equilibrium.

The SXRD pattern of the experimentally synthesized CaZrN2 does not show this
low-angle peak (Figure 2), indicating that this cation-disordered CaZrN2 is
far from thermodynamic equilibrium. This hypothesis can be supported due to
the short synthesis times relative to the slow rate of diffusion in
nitrides.59 Here we demonstrate an instance of a short metathesis reaction
kinetically trapping disorder in a ternary nitride. Further annealing or
higher synthesis temperatures could produce cation ordering in CaZrN2, as seen
in ZnGeN2 and ZnGeP2.60, 61

This finding highlights that accounting for metastability is an area of
further growth for predictive synthesis. As with thin film sputtering,14 bulk
syntheses involve myriad thermodynamic and kinetic factors that influence
final product formation. While computational techniques to account for
temperature effects on enthalpy have been developed,62 entropic contributions
(e.g., compositional and configurational disorder) are not yet included in
these chemical potential diagrams as they are computationally expensive to
calculate. This gap motivates further work aligning computational and
synthetic techniques to bring about a truly predictive synthesis paradigm.

##  3 Conclusion

We report the discovery of two new ternary nitrides, CaZrN2 and CaHfN2, via
mechanistically-guided metathesis synthesis of bulk powders. These phases
crystallize in the cation-disordered RS structure ($Fm\overline{3}m$), as
opposed to the computationally-predicted $\alpha$-NaFeO2 structure type
($R\bar{3}m$). In situ synchrotron powder X-ray diffraction analysis shows how
Zr^3+ intermediates form early on in the reaction process, which rationalizes
why excess of the Ca3N2 reagent is synthetically necessary to generate a
higher chemical potential of nitrogen. These findings stand in contrast to our
prior synthesis of MgZrN2, which proceeds stoichiometrically from 2 Mg2NCl +
ZrCl4 to MgZrN2 + 2 MgCl2 via a ZrNCl intermediate, as to avoid any N2(g)
linked redox chemistry. The observed synthetic pathways match well with the
predictions generated from chemical potential diagrams. Additional
thermodynamic calculations show that the observed $Fm\bar{3}m$ structure is
metastable with respect to the ordered $R\bar{3}m$ structure, suggesting that
the $R\bar{3}m$ structure may be achievable under different synthetic
conditions. In sum, these findings demonstrate a generalizable strategy for
predicting and conducting metathesis reactions with the goal of synthesizing
new ternary nitrides: combine in situ SXRD studies with thermodynamic
calculations for mechanistically-guided materials chemistry.

##  4 Methods

###  4.1 Synthesis

Caution: N2 gas formation within sealed ampules can cause explosions at high
temperatures, and appropriate precautions should be taken to prevent equipment
damage and unsafe conditions.  All reagents are moisture-sensitive, and were
therefore handled in an argon-filled glovebox unless otherwise noted (O2 $<$
0.1 ppm, H2O $<$ 0.5 ppm). Ca3N2 (Chem Cruz or Alfa Aesar, $>$98 % metals
basis), Mg3N2 (Alfa Aesar, $>$98% metals basis), and MgCl2 (Sigma Aldrich,
anhydrous, $>$98% metals basis) were used as received. ZrCl4 (Acros, 98%) and
HfCl4 (Sigma Aldrich, 99% metals basis, except for Zr, with $<2.7$%) were
purified by heating approximately 4 g in a sealed quartz ampule (10 mm inner
diameter, 12 mm outer diameter, ca. 30 cm long) in a 3-zone horizontal tube
furnace to transport ZrCl4 from the hot zone (400 °C) to the colder zone (300
°C), leaving behind less volatile, oxide-based impurities (e.g., ZrO2). CaCl2
(anhydrous, Alfa Aesar, 98%) was dried under flowing Ar for 24 h at 300 °C.

Ca2NCl was synthesized following a method adapted from literature63.
Stoichiometric amounts of Ca3N2 and CaCl2 were combined in an agate mortar and
pestle and ground into a homogeneous pink powder (ca. 2 g). The powder was
then cold pressed (P $\approx 300$ MPa) into a dense pellet (diameter = 0.25
in) and placed in a niobium tube (10 cm long, 0.375 inch outer diameter, 0.015
inch wall thickness). Prior to use, the surface oxide of the niobium was
removed by scrubbing with an abrasive scour pad in the glovebox. The ends of
the tube were crimped down using the arbor press. This ampule was then brought
out of the glovebox and quickly sealed under an argon atmosphere via arc-
melting of the crimped ends of the niobium. The sealed metal ampule was then
sealed in a quartz ampule under vacuum and heated at 5 °C/min in a muffle
furnace to a set point of 740 °C. The reaction was held at temperature for 50
h, and then allowed to cool to room temperature before opening the ampule in
the glovebox.

Mg2NCl was synthesized following a method adapted from literature64, as we
described previously36. Stoichiometric amounts of Mg3N2 (ca. 3 g) and MgCl2
(ca. 3 g) were combined in an agate mortar and pestle and ground into a
homogeneous tan powder. The powder was then cold pressed ($P\approx 80$ MPa)
into a dense pellet (diameter = 0.5 in) and placed in a quartz ampule (14 mm
i.d., 16 mm o.d.). This ampule was then brought out of the glovebox and
quickly sealed under vacuum ($\leq 20$ mTorr, as determined by a Pirani gauge)
by using an oxygen/methane torch. The sealed ampule (ca. 15 cm3 internal
volume) was then heated at 10 °C/min in a muffle furnace to a set point of 550
°C. The reaction was held at temperature for 5 days and then allowed to cool
to room temperature before opening the ampule in the glovebox.

Reaction mixtures targeting CaZrN2 and CaHfN2 were prepared by combining the
desired reagents in specific mole ratios (e.g., 1.17 Ca3N2 + ZrCl4, as
specified in the text) and homogenizing with an agate mortar and pestle.
Reaction scales ranged from ca. 50 mg to ca. 500 mg total charge of reactants.
Reaction mixtures were loaded into crucibles as loose powders or as 0.25 inch
diameter pellets (cold pressed at 300 MPa) and sealed under vacuum in quartz
ampules ($<30$ mTorr). Many syntheses were conducted using alumina crucibles,
but optimized syntheses used homemade stainless steel crucibles to minimize
oxygen contamination. These steel crucibles were made from stainless steel
tubes (3/8 in outer diameter, 0.020 in wall thickness, TP-304L), welded closed
on one end and left open on the other. The optimized synthesis involved
pelletized reaction mixtures (ca. 500 mg) held in the steel crucibles nested
inside quartz ampules along with a 1 g graphite rod as an oxygen getter,
physically separated from the pellet. These ampules were transferred from the
glovebox to a vacuum manifold using a custom air-free transfer valve, and
sealed flame-sealed under vacuum ($<30$ mTorr) using an oxygen/methane torch.
Optimized reactions were heated in a muffle furnace at $+5$ °C/min to the
specified dwell temperature, allowed to thermally equilibrate for 10 min, and
then air-quenched by removing the ampule from the furnace and placing it in an
insulating brick holder on the benchtop. Other synthesis conditions are
specified in the text where relevant. The products are moisture sensitive. To
remove byproduct CaCl2 from the target compounds, samples were washed with
anhydrous methanol (dried over molecular sieves for at least five days)65 in
an argon glovebox.

###  4.2 X-ray diffraction experiments

The products of all reactions were characterized by powder X-ray diffraction
(PXRD). PXRD measurements were performed by using a Bruker DaVinci
diffractometer with Cu K$\alpha$ X-ray radiation. Prior to collecting PXRD,
silicon powder was ground in with the reaction products as an internal
standard for lattice parameters. All samples were prepared for PXRD from
within the glovebox by placing powder on off-axis cut silicon single crystal
wafers to reduce the background and then covered with polyimide tape to slow
exposure to the atmosphere.

High-resolution synchrotron powder X-ray diffraction (SXRD) measurements on
select samples were conducted via the mail-in program at the 11-BM-B end
station of the Advanced Photon Source at Argonne National Laboratory66.
Samples were prepared loading reaction products (from optimized synthetic
conditions) into extruded quartz capillaries (0.69 mm inner diameter, 0.7 mm
outer diameter), which were subsequently nested inside kapton capillaries for
measurement. Data were collected at $\lambda\approx 0.459$Å. SXRD patterns are
shown in $Q$-space to account for small difference in wavelength between
measurements ($Q=4\pi\sin(\theta)/\lambda$).

In situ SXRD were conducted at the 17-BM-B end station of the Advanced Photon
Source at Argonne National Laboratory ($\lambda=0.24101$ Å). Samples were
prepared by loading reaction mixtures into extruded quartz capillaries (0.9 mm
inner diameter, 1.1 mm outer diameter) and flame-sealing under vacuum ($<30$
mTorr). Capillaries were loaded into a flow-cell apparatus67 and heated at 5
°C/min to the specified temperature. Caution: N2 gas formation within the
capillary can cause it to break at high temperatures. Diffraction pattern
images were collected using a PerkinElmer plate detector positioned 700 mm
away from the sample. Images were collected every 30 s by summing 20 exposures
of 0.5 s each, followed by 20 s of deadtime. Images collected from the plate
detector were radially integrated using GSAS-II and calibrated using a silicon
standard.

###  4.3 X-ray diffraction analysis

Quantitative phase analysis of PXRD and in situ SXRD data was conducted using
the Rietveld method as implemented in TOPAS v647. For laboratory-diffraction
PXRD experiments (Bruker system, Cu K$\alpha$ radiation), the sample
displacement was first refined against the silicon standard. The sample
displacement was then fixed, and the relevant phases were then refined. For
each phase, lattice parameters, size broadening, and atomic thermal parameters
were refined. A 10-term polynomial was used to fit the background.

High-resolution SXRD datasets were analyzed in a similar way to identify the
cation occupancy for the Ca_$x$ $M$_$2-x$N2 phase in each sample ($M$ = Zr,
Hf). Cubic ZrN ($Fm\bar{3}m$) was used as a starting model, with Ca and Zr or
Hf on the cation site with 50% occupancy each. Cation occupancy ($x$) was
constrained to equal anion occupancy (i.e., Ca + $M$ = N) during refinements.
The anion site was fixed at full occupancy. The atomic displacement parameters
were refined isotropically. Other relevant phases were also added to the
model, and their lattice parameters, size broadening (Lorentzian), and atomic
thermal parameters were refined. A broad background peak from the quartz
capillary was modeled with a Lorentzian function, and the remaining background
was modeled with a 10-term polynomial. The datasets were collected without an
internal standard, and sample displacement was not refined during analysis.

Due to the number and positional overlap of intermediates during the
sequential refinements, several variables were fixed to better compare phase
fractions. Thermal displacement parameters were fixed at 1 Å2 for each phase
to better account for changes in peak intensity during the reaction. Atoms
were fixed at full occupancy for each site. RS phases were modeled as ZrN
(i.e., metal site mixing was not refined). Crystalline size domain was fixed
at 200 nm (as modeled using a Lorentzian polynomial) for most phases, except
the ZrNCl and RS phases, which exhibited noticeable peak broadening. ZrNCl was
refined using thermal displacement parameters were fixed at 20 Å2 and
anisotropic size broadening. Size broadening for RS phases was manually
refined from the final scan in the sequence and fixed for all other scans
(crystallite size ca. 7 nm). In order to compare the relative fractions of
phases determined from Rietveld calculations, a weighted scale factor (W.S.F.)
is defined as $Q_{p}=S_{p}\cdot V_{p}\cdot M_{p}$ where $Q_{p}$ = weighted
scale factor of phase $p$, $S_{p}$ is scale factor calculated from Rietveld,
$V_{p}$ is the volume of the unit cell, and $M_{p}$ is the atomic mass of the
unit cell. It should be noted that we omit the Brindley coefficient for
microabsorption correction in our calculation of W.S.F. due to the unreliable
refinement of particle sizes for individual phases. Amorphous material and
product lost as vapor are not accounted for in the sequential refinement,
hence we use the W.S.F. instead of relative wt% or mol%. We reference all
phases by their nominal stoichiometric formula; however, the actual chemical
formula may be distinct from the written formula as XRD data alone cannot
typically resolve nonstoichiometric compounds.

###  4.4 Thermodynamic analysis

Thermodynamic calculations were performed using DFT-computed values and tools
from the Materials Project (v2022.10.28) and pymatgen (v2023.9.10)56, 68.
Reaction energies were calculated from the Gibbs formation energies as a
function of temperature, using the method of Bartel, et al.62 Chemical
potential diagrams were generated in the method of Yokokawa21 as detailed by
Todd, et al.22 and implemented in pymatgen. Mg2NCl is calculated to be +0.02
eV/atom metastable relative to the binaries, which is within DFT error of the
convex hull. We therefore applied a small correction to bring the phase to the
hull, allowing for visualization in the chemical potential diagrams. All other
phases shown in Figures 4 and S21-S23 are calculated to be stable.

###  4.5 Thermodynamic analysis of disorder in CaZrN2

To produce a training set for the cluster expansion, a number of DFT
calculations were conducted using VASP 69. The PBE functional was used within
the projector-augmented wave method 70. A planewave cutoff of 400 eV was used.
K-point densities were used such that the total energies were converged within
1 meV/atom. All structural degrees of freedom were allowed to be optimized
within the structural relaxation (ie. volume, cell shape, atom positions).
Structures were generated using the pylada software 71. To produce symmetry-
inequivalent supercells, Hermite Normal Form transformation matrices were
constructed using an algorithm developed by Hart and Forcade 72. The
transformation matrices were applied to the RS primitive cell to produce
supercells with varying dimensions. The supercells were then decorated with
different atomic orderings to make varying CaZrN2 configurations.

The Alloy Theoretic Automated Toolkit (ATAT) was employed to fit a cluster
expansion 73, 74. To ensure that the cluster expansion was properly
representing the highly ordered configurations, we exhaustively enumerated all
8-atom and 12-atom CaZrN2 configurations and included them in the training
set. In total, the training set was made up of 70 symmetry inequivalent CaZrN2
configurations, ranging in size from 8 to 24-atoms. A cross validation score
of 10.4 meV/atom was obtained. Considering that the training set had an energy
range of 324 meV/atom, we deemed a 10.4 meV/atom cross validation score to be
sufficiently small. The energies of all 37,883 24-atom CaZrN2 configurations
were predicted using the fitted cluster expansion.

To ensemble average the XRD patterns, first the partition function, $Z$, was
calculated using all 37,883 24-atom configurations

$Z(T)=\sum_{i}^{37,883}e^{-E_{i}/k_{B}T}.$ (8)

Here, $i$ is the index of the configuration, $E_{i}$ is the total energy of
the $i$th configuration, $k_{B}$ is Boltzmann’s constant, and T is the
absolute temperature. The temperature-dependent probability of each
configuration, $P_{i}(T)$, was then determined using the following equation

$P_{i}(T)=\frac{e^{-E_{i}/k_{B}T}}{Z(T)}.$ (9)

Finally, the intensity of the ensemble-averaged XRD, $I$, was calculated as a
function of temperature and wavenumber, $Q$:

$I(Q,T)=\sum_{i}^{n}(I_{i}(Q)\cdot P_{i}(T)).$ (10)

For a justification of ensemble averaging XRD patterns in the above way, see
Jones et. al 75. The powder XRD pattern for each structure was generated with
a Cu Kα wavelength using pymatgen68.

##  5 Acknowledgements

The authors thank Paul Todd and Akira Miura for helpful discussions. This work
was supported primarily by the National Science Foundation (DMR-2210780). The
authors thank the Analytical Resources Core at Colorado State University for
instrument access and training (RRID: SCR_021758). Use of the Advanced Photon
Source at Argonne National Laboratory was supported by the U. S. Department of
Energy, Office of Science, Office of Basic Energy Sciences, under Contract No.
DE-AC02-06CH11357. Work conducted at Lawrence Berkeley National Laboratory was
supported as part of GENESIS: A Next Generation Synthesis Center, an Energy
Frontier Research Center funded by the U.S. Department of Energy, Office of
Science, Basic Energy Sciences under Award Number DE-SC0019212. This research
used resources of the National Energy Research Scientific Computing Center
(NERSC), a U.S. Department of Energy Office of Science User Facility operated
under Contract No. DE-AC02-05CH11231. This work was authored in part at the
National Renewable Energy Laboratory (NREL), operated by Alliance for
Sustainable Energy, LLC, for the U.S. Department of Energy (DOE) under
Contract No. DE-AC36-08GO28308. Funding for work conducted at NREL was
provided by DOE Basic Energy Sciences Early Career Award “Kinetic Synthesis of
Metastable Nitrides”. The views expressed in the article do not necessarily
represent the views of the DOE or the U.S. Government. The U.S. Government
retains and the publisher, by accepting the article for publication,
acknowledges that the U.S. Government retains a nonexclusive, paid-up,
irrevocable, worldwide license to publish or reproduce the published form of
this work, or allow others to do so, for U.S. Government purposes.

##  6 Author Contributions

C.L.R. and J.R.N. conceptualized the project. C.L.R., E.N.S., B.C.M., and
G.T.T. conducted syntheses and measured PXRD. D.C.A. conducted optical
property measurements. J.R.G. conducted X-ray photoelectron spectroscopy
measurements with support from A.L.P. The remote experiments were set up by
beamline scientist A.A.Y., enabling C.L.R. to remotely control the data
collection and conduct analysis. R.C.M. and C.L.R. conducted scanning electron
microscopy and energy dispersive X-ray spectroscopy measurements. C.L.R. and
M.J.M. conducted thermodynamic reaction calculations with support from K.A.P.
and J.R.N. A.N. conducted thermodynamic modeling of cation disorder with
support from E.T. and V.S., A.L.P., A.Z., and J.R.N. supervised and supported
the project.

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