[1]Ali Tehranchi

[1]Siyuan Zhang

1] Max-Planck-Institut für Eisenforschung GmbH, Düsseldorf, D-40237, Germany
2] BAM Federal Institute for Materials Research and Testing, Berlin, D-12489,
Germany

#  Metastable defect phase diagrams as a tool to describe chemically driven
defect formation: Application to planar defects

<EMAIL_ADDRESS><EMAIL_ADDRESS>Ali Zendegani  Christina Scheu
<EMAIL_ADDRESS>Tilmann Hickel<EMAIL_ADDRESS>Jörg Neugebauer
<EMAIL_ADDRESS>[  [

###### Abstract

Thermodynamic bulk phase diagrams have become the roadmap used by researchers
to identify alloy compositions and process conditions that result in novel
materials with tailored microstructures. Recent experimental studies show that
changes in the alloy composition can drive not only transitions in the bulk
phases present in a material, but also in the concentration and type of
defects they contain. Defect phase diagrams in combination with density
functional theory provide a natural route to study these chemically driven
defects. Our results show, however, that direct application of thermodynamic
approaches can fail to reproduce the experimentally observed defect formation.
Therefore, we extend the concept to metastable defect phase diagrams to
account for kinetic limitations that prevent the system from reaching
equilibrium. We successfully applied this concept to explain the formation of
large concentrations of planar defects in supersaturated Fe-Nb solid solutions
and to identify in a joint study with experiments conditions in Mg-Al-Ca
alloys for defect phase occurrence. The concept offers new avenues for
designing materials with tailored defect structures.

###### keywords:

Metastable defect phase diagram, Ab initio, chemical potential, Laves phases,
transmission electron microscopy

##  1 Introduction

The microstructure of a material is often characterized by complex patterns
resulting from the presence of several bulk phases and precipitates. Notable
examples of this are bainite or pearlite steels djaziri2016deformation . In
order to design materials with these intricate microstructures, it is
necessary to accurately predict the alloy compositions and process
temperatures at which the desired phases will form. To meet this need, modern
computational tools such as CALPHAD (Calculation of Phase Diagrams) have been
developed saunders1998calphad ; sundman2007computational ; liu2009first .
Recent advances in experimental techniques have enabled researchers to study
materials at the atomic scale, providing detailed information about the
chemical and structural nature of bulk phases as well as extended defects such
as grain boundaries and dislocations zhang2022atomistic ;
meiners2020observations ; cantwell2014grain ; rhode2013mg . These high-
resolution studies have allowed for the characterization of defects in terms
of their local composition, size, and structure as a function of alloy
composition and temperature, leading to the development of defect or
complexion phase diagrams.

It is generally assumed that the bulk phase diagram, which represents the vast
majority of the atoms in a material, accurately predicts the experimentally
observed bulk phases. However, a recent study on Fe-Nb found that in a region
of the phase diagram where the Fe2Nb-C14 Laves phase and the
Fe7Nb6-$\mu$-phase were expected to coexist, solely the C14 phase with a large
number of Nb-rich planar defects were observed instead vslapakova2020atomic .
Thus, the supersaturated Nb-rich Fe2Nb phase formed after cooling did not lead
to the formation of $\mu$-phase precipitates indicated by the binary Fe-Nb
bulk phase diagram. Instead, the excess Nb resulted in the formation of planar
defects and a microstructure that differed from what would be expected based
on the bulk phase diagram. This is highly significant because the
microstructure of a material can strongly influence its mechanical or
corrosion properties korte2022defect . In order to effectively use this
mechanism for a wide range of materials systems, it is essential to have a
computational approach that is able to treat both bulk and chemically driven
defect phases on the same level.

In this letter we first outline a general approach that combines density
functional theory (DFT) calculations with grand-canonical thermodynamic
concepts. As will be shown for the Fe-Nb system, assuming thermodynamic
equilibrium fails to reproduce the experimentally observed deviation from the
bulk phase diagram. We therefore explicitly include in this approach the
presence of kinetic barriers and show that they are crucial to faithfully
reproduce experimental observations. Having demonstrated the validity of our
approach for the Fe-Nb system, we employ it for a model Mg-Al-Ca alloy to
identify conditions for defect phase formation. To further validate this
approach, we use the identified alloy and process conditions to cast the
corresponding Mg alloy and perform high resolution transmission electron
microscopy measurements to show the predicted defect formation.

##  2 Metastable defect phase diagrams

In our proposed approach, we will use the chemical potential of the solute
species as a key quantity to understand and quantify the reaction that
converts excess solutes into either bulk precipitates (a secondary bulk phase
forms according to the traditional bulk phase diagram) or into a solute-rich
defect phase. In order to compare the two qualitatively very different
reactions on equal footing, the chemical potential of the resulting (reactant)
phase has to include both the energy required to form the new lattice/defect
structure and the energy gained during its formation in this phase (i.e., the
segregation energy). We note that this choice is very different from the
often-considered case of conventional segregation, where defects are already
present. In this case, the energy cost for defect formation is not needed
resulting in much lower chemical potentials. However, for the case we are
interested in – solute-induced defect formation – this contribution is crucial
weissmuller1993alloy ; kirchheim2007reducing . In fact, it is exactly the
competition between endothermic defect formation and exothermic segregation
that controls solute-driven defect formation nazarov2010first ; nazarov2014ab
. This approach, which includes both contributions, is also consistent with
the treatment of bulk phases.

The change of the total energy of the system per excess solute, $E_{\rm
solute}$, versus different configurations is sketched schematically in Fig.
1a. At casting conditions, i.e., close to the melting temperature, the system
is in a single-phase solute solution containing defects. When the system is
quenched to a lower temperature, where the solute concentration is above the
solubility limit, at this temperature, the chemical potential of the initial
supersaturated phase is $\mu_{\rm initial}$. This phase is thermodynamically
unstable against the formation of precipitates. The resulting precipitates
provide an energetically favorable configuration for the excess solutes. The
chemical potential for the precipitate is $\mu_{\rm equilibrium}$ and well
below the initial chemical potential. The (positive) difference between the
two potentials $\mu_{\rm initial}-\mu_{\rm equilibrium}$ drives the
partitioning and precipitate formation.

As outlined above and implied by the experimental observations an alternative
way to accommodate the excess solutes in the initial solid solution phase is
the formation of defects that provide energetically more favorable sites than
the original matrix. The chemical potential to form and occupy the defect is
$\mu_{\rm defect}^{\rm equilibrium}$.

Figure 1: Schematic representation of the equilibration process (a) and the
metastable defect phase diagram (b) as discussed in the text. Non-equilibrium
(excess) solutes are marked by red dots. Energetically favorable solute
configurations, e.g., in the bulk-like precipitates or at defects are shown as
blue dots. The chemical potentials related to the various configurations
sketched in the gray boxes are given by the dashed horizontal lines in (a) and
vertical lines in (b).

In the absence of any reaction barriers, the reaction with the lowest chemical
potential in the final state would dominate. In many cases, $\mu_{\rm
equilibrium}<\mu^{\rm equilibrium}_{\rm defect}$. As a consequence, solute-
driven exothermic defect formation should not occur. Since this is in obvious
contradiction to the experimentally observed presence of these defects, the
assumption of barrier-less transitions does not hold. We therefore include in
the following discussion also kinetics. For the formation of precipitates, the
nucleation barrier has to be overcome. This barrier can be rather large. The
chemical potential for a solute in such a barrier configuration is $\mu_{\rm
barrier}$. The presence of this barrier has an important consequence: As
schematically shown in Fig. 1a the presence of this barrier makes the
thermodynamically unstable (against the precipitate bulk phase) defects
metastable. Thus, as long as the chemical potential is below the barrier
configuration, the dominant reaction will be defect rather than precipitate
formation. Since metastability plays a key role in enabling and driving this
unconventional solute-driven defect formation that even outperforms bulk
formation we call the formalism metastable defect phase diagram (MDPD).

Based on the above discussion, the formalism can be schematically summarized
in Fig. 1b. The central quantities are the defect formation energies as a
function of the solute chemical potential. This representation has been
originally developed to describe point defects in semiconductors
zhang1991chemical  and has been later extended to surface phase diagrams
freysoldt2014first . An important advantage of this approach is that it can be
readily connected to ab initio calculations since it requires only DFT
computed energies and the number of atoms for each species as input

$\displaystyle E_{\rm f}^{\rm defect}=E_{\rm tot}^{\rm defect}-E_{\rm
tot}^{\rm bulk}-n_{\rm solute}\mu_{\rm solute},$ (1)

where $E_{\rm tot}^{\rm defect}$, $E_{\rm tot}^{\rm bulk}$, $n_{\rm solute}$,
and $\mu_{\rm solute}$ are the total energy of the supercells containing
defects and defect-free bulk, the number of excess solute atoms segregated to
the defects, and the chemical potential of solute atoms, respectively. Fig. 1b
shows exemplarily the formation energy of two defect states (orange and green
solid lines). The different slopes reflect the different coverages of solute
atoms at the defect. We use coverage, as commonly used in surface science,
instead of concentration to reflect that the excess solute concentration at a
planar defect is referenced to an area and not to the bulk volume.

The mechanism that leads to the experimentally observed formation of solute-
rich defects can be described as follows (see Fig. 1b): If the initial
chemical potential of the supersaturated solid solution is less than the
hypothetical chemical potential solute atoms would have in the nucleation
barrier $\mu_{\rm barrier}$, the formation of the bulk precipitates ($\mu_{\rm
equilibrium}$), though thermodynamically preferred, is kinetically prohibited.
An alternative reaction path to lower the energy of the excess solutes is the
formation of extended defects, which provide for the solutes binding sites
that are energetically much more favorable than the substitutional sites in
the solid solution. The final chemical potential is found in most cases above
the bulk equilibrium one ($\mu_{\rm defect}^{\rm equilibrium}>\mu_{\rm
equilibrium}$), i.e., the defect phases are thermodynamically metastable.
However, since barriers to form the lower dimensional defects are typically
lower than the bulk nucleation ones, defect formation may take precedence over
bulk nucleation.

The small offset between $\mu_{\rm final}$ and $\mu_{\rm defect}^{\rm
equilibrium}$ shown in Fig. 1b is given by the defect nucleation barrier. It
arises, since a defect in contrast to a bulk precipitate can only host a
finite number of solutes. Thus, once a defect becomes completely filled a new
one needs to be created. Once the chemical potential gets below this barrier,
the formation of defects ceases and a metastable equilibrium is reached.

##  3 Competition between bulk precipitates and Nb-rich planar defects

To demonstrate the applicability and the predictive power of this approach in
connection with modern DFT calculation we first consider the Fe-Nb system. In
this material system, the presence of the C14 Laves phase has been a topic of
many theoretical and experimental studies liu2012ab ; voss2011phase ;
stein2015phase . The reason is that Laves phases in steels are often brittle
at ambient temperature, but their high melting temperature makes them very
stable and a suitable constituent in (ferritic) steels serving at elevated
temperatures knezevic2002martensitic ; falat2005mechanical ;
aghajani2009effect .

Voß et al. voss2011phase  determined experimentally the C14-NbFe2 Laves phase
on both sides of its stoichiometric 33.3 at.% Nb composition, i.e., between
25.1 at.% and 37.6 at.% Nb. According to the Fe-Nb system phase diagram
he2017thermodynamic  as shown in Fig. 2a, an alloy with an overall composition
between 35 at.% and 37.5 at.% Nb is in the single-phase region at high
temperature, yet at lower temperatures it enters the two-phase region of
C14+$\upmu$ phases (see vertical dashed line in Fig. 2a). Therefore, when
cooling down from high temperatures to room temperature bulk thermodynamics
predicts the precipitation of the Nb-rich $\upmu$ phase (Nb6Fe7).

Šlapáková et al. vslapakova2020atomic  performed a transmission electron
microscopy (TEM) study on single-phase stoichiometric (33 at.% Nb) as well as
Nb-rich (35 at.% Nb) NbFe2 Laves phase specimens to characterize their
microstructures. In their work, representative microstructures of the
stoichiometric and the Nb-rich alloy were shown by bright-field TEM images.
The stoichiometric sample only reveals the presence of isolated dislocations
with a comparatively low defect density (see Fig. 2b). However, for Nb-rich
alloys bright-field TEM did not detect precipitates but a large increase
compared to the stoichiometric samples of extended and confined planar defects
(Fig. 2c). An energy dispersive X-ray spectroscopy (EDS) analysis performed in
a scanning TEM (STEM) showed that the Nb concentration at these planar defects
is largely increased compared to its bulk concentration. These experimental
observations are in obvious contrast to the above-mentioned prediction of the
bulk phase diagram, according to which the 2 at.% excess Nb content should
result in the formation of $\upmu$-precipitates.

Figure 2: (a) The Fe-Nb phase diagram, reprinted with permission from Ref.
he2017thermodynamic . Bright field-TEM images of (b) a stoichiometric alloy
with 33 at.% Nb and (c) a Nb-rich alloy with 35 at.%, reprinted with
permission from Ref. vslapakova2020atomic . Elsevier 2020.

Fig. 3 shows the calculated energies of formation for the planar defects as a
function of the chemical potential of Nb, $\Delta\mu_{\rm Nb}$. Here
$\Delta\mu_{\rm Nb}=\mu_{\rm Nb}-\mu_{\rm Nb}^{0}$ where $\mu_{\rm Nb}^{0}$ is
the chemical potential of Nb atoms in bulk Nb.

Figure 3: The energies of the formation of structures with and without defects
are plotted as a function of the Nb chemical potential $\mu_{\rm Nb}$
according to Eq. (1). The thermodynamically (meta)stable bulk phase at a given
chemical potential domain is given in the white box at the bottom. The color
coding labels the nature of defect formation: In the orange region the
undecorated defect state is dominant. In the light green and green regions,
the decorated defect state is endothermic and exothermic, respectively. The
atomic structures of extended basal stacking fault (EBSF), pyramidal stacking
fault (PSF), confined basal stacking fault (CBSF), and basal synchroshear-
formed stacking fault (BSSF), are sketched in the right panel. The blue, and
green spheres denote Nb, and Fe atoms, respectively. The planar defects are
highlighted by the gray regions.

The slope of the line describing the formation energy of a given defect is
proportional to the amount of the excess Nb atoms of this defect phase. The
excess is defined with respect to the composition of the C14-NbFe2 Laves
phase. Consequently, the defect with zero excess Nb atoms (BSSF), i.e. the
defect-free C14-NbFe2 Laves phase, has zero slope.

In the region of chemical potentials with $\Delta\mu_{\rm Nb}\leq\mu_{\rm
equil.}=-0.07$ eV the energetically most stable bulk phase is the C14-NbFe2
Laves phase. If the Nb chemical potential exceeds this potential $\mu_{\rm
equil.}$, the $\upmu$ phase becomes the most stable phase. The positive defect
formation energies in Fig. 3 for $\Delta\mu_{\rm Nb}\leq-0.07$ eV imply that
none of the experimentally detected defect structures is thermodynamically
stable: From a thermodynamic perspective a phase separation into the $\upmu$
phase and C14 Laves phase should occur. The defects are even unstable against
the formation of undecorated basal synchroshear-formed stacking faults (BSSF).
We can therefore conclude that the defect phases observed experimentally
cannot be explained when assuming that the system has reached thermodynamic
equilibrium. As a consequence, the bulk phase diagram alone is unable to tell
us into which phases the excess solutes partition or what are the actual
(i.e., non-equilibrium) chemical potentials in the sample.

As outlined in the discussion of Fig. 1a the existence of a large nucleation
barrier prevents the formation of the bulk precipitates, resulting in a
supersaturated solid solution with a chemical potential for the solutes well
above that at which the bulk precipitates could form thermodynamically. An
upper limit of the solute chemical potential is given by the chemical
potential $\mu_{\rm barrier}$ the solutes have in the precipitate phase. Once
the potential reaches this value precipitates form spontaneously without
having to overcome a barrier. The chemical potential $\mu_{\rm barrier}$ is
thus a key quantity to extend defect phase diagrams to supersaturated
conditions not covered by bulk phase diagrams.

In the following we will therefore introduce two models. Together with DFT
simulations they will allow us to estimate values for $\mu_{\rm barrier}$. For
substitutional solutes as considered here, the initial nucleus will be a
coherent agglomeration of solutes in a sphere, with a solute concentration
close or equal to that of the bulk precipitate. Since we are interested in a
lower limit of the barrier chemical potential, we can safely neglect the
interface energy between nucleus and matrix. The barrier configuration is then
described by the precipitate bulk system, except that the lattice constant is
not that of the plastically relaxed precipitate, which would have the chemical
potential $\mu_{\rm equillibrium}$ of the bulk phase. Rather, the lattice
constant is dictated by the coherency to the bulk resulting in significantly
higher chemical potentials.

For homogeneous nucleation coherence extends in all three spatial directions.
The chemical potential for this barrier (nucleation) configuration is
$\mu_{\rm barrier}^{\rm iso}=\mu_{\rm equil.}+(E_{\rm sol}^{\rm DFT}(a_{\rm
coh})-E_{\rm sol}^{\rm DFT}(a_{\rm eq}))/n_{\rm sol}$ and can be directly
obtained via DFT calculations. $E_{\rm sol}^{\rm DFT}(a)$ is the total energy
of the precipitate phase at the coherency lattice constant $a_{\rm coh}$ and
the equilibrium lattice constant $a_{\rm eq}$. $n_{\rm sol}$ is the number of
solute atoms in the corresponding supercell.

In case of nucleation at basal defects, isotropy is broken. The coherency
constrain may thus involve only the two lateral lattice constants, with the
lattice constant normal to the basal plane being allowed to relax. The
corresponding chemical potential is $\mu_{\rm barrier}^{\rm lateral}=\mu_{\rm
equil.}+(E_{\rm sol}^{\rm DFT}(a^{\parallel}_{\rm coh},a^{\perp}_{\rm
eq})-E_{\rm sol}^{\rm DFT}(a^{\parallel}_{\rm eq},a^{\perp}_{\rm eq}))/n_{\rm
sol}$. The values of $\mu_{\rm barrier}^{\rm lateral}$ and $\mu_{\rm
barrier}^{\rm iso}$ are 0.01 eV and 0.1 eV, respectively. These values are
substantially larger than the maximum Nb chemical potential that can be
reached in the C14 phase in thermodynamic equilibrium. These values are
highlighted in the metastable phase diagram which is displayed in the white
box at the bottom of Fig. 3.

As shown in Fig. 3 at such high chemical potentials defect states become
thermodynamically favored that were absent in the equilibrium case.
Specifically, a chemically ordered extended basal stacking fault EBSF becomes
stable. This defect is for intermediate values of the Nb chemical potential
endothermic (light green), i.e., this defect phase will form only at existing
basal stacking faults. At even higher Nb chemical potentials defect formation
becomes exothermic: The system will create such defects spontaneously. Since
the creation of these defects takes solute atoms out of the host matrix the Nb
chemical potential will decrease until defect formation becomes endothermic.
At this point the driving force for spontaneous defect formation vanishes and
the formation of further defects stops.

The mechanism outlined here is to a large part very similar to lowering the
solute chemical potential via bulk-like precipitates. The main differences are
that precipitate formation is fully covered by the bulk phase diagram, i.e.,
an oversaturated solid solution partitions into precipitates and the host
matrix. Like precipitates, exothermic defects coexist with the host matrix and
remove large amounts of solutes from the surrounding host matrix. In this
scenario the defects can no longer be regarded as low-dimensional objects
having an infinitely small volume fraction but control like bulk precipitates
phase equilibration and coexistence.

The MDPD, i.e., the defect phase diagram in combination with the upper limit
for the chemical potential due to the nucleation barrier shown in Fig. 3,
provides a direct quantitative approach to describe phase coexistence between
defects and bulk. It is a natural extension of both defect phase and bulk
phase diagrams and explains the experimentally observed formation of large
concentrations of Nb rich planar defects in the C14 phase. The theoretical
approach of MDPDs outlined here is general and not restricted to the specific
Fe-Nb alloy system considered in this section. In the next section we will use
it to identify conditions where exothermic defect formation can be induced in
Mg-based alloys.

##  4 Using Metastable defect phase diagrams to induce planar defects in
ternary Mg-Al-Ca alloys

To demonstrate the exciting opportunities opened by ab initio computed MDPDs
in materials design we will use them in the following to identify suitable
process conditions where chemically driven defect formation takes precedence
over bulk precipitates. As prototype system we will consider Mg-alloys, which
are highly attractive for, e.g., automotive applications due to their low cost
and weight zubair2021co ; guenole2021exploring . We will focus on Mg alloyed
with Al and Ca, since adding small amounts of these elements largely improves
the ductility of Mg sandlobes2017rare ; pei2015rapid . Going above the
solubility limit, Al2-xCaMgx Laves phases can be formed, where the Al- ($x$=0)
and Mg-rich ($x$=1) stable Laves phases are Al2Ca-C15 and CaMg2-C14,
respectively.

Platelets of C15-like atomic arrangements are known to form along the Mg basal
plane (0002)Suzuki2008 ; zubair2022laves . The small lattice mismatch between
the in-plane directions of Al2Ca(111) and Mg(0002), makes the C15 phase also
an interesting candidate for forming planar defect structures within the Mg
matrix. Experiments of Kashiwase et al. Kashiwase2019 ; Kashiwase2020  using
high-resolution TEM measurements show that the platelets of C15 can be thinner
than the bulk unit cell of C15. However, the atomic structure and chemical
distribution of elements of such C15-like planar defects remained unclear. To
explore whether such defects consisting of a few atomic layers can be
exothermically formed we study their formation energy. Specifically, we
consider a series of defects starting from a single Al layer and sequentially
adding further layers. The first six defect structures are shown in Fig. 4
(right side) with the last one representing a single-layer C15 structure
embedded in the Mg matrix.

Since the planar defects coexist with bulk Mg, we impose $\Delta\mu_{\rm
Mg}=0$, i.e., the chemical potential of Mg atoms is the same as their chemical
potential in bulk Mg. To maximize the amount of Ca in the matrix, as needed to
drive the formation of these defects, we further consider Ca-rich conditions.
These conditions are realized at or above the Ca solubility limit, i.e., the
chemical potential of Ca is $\Delta\mu_{\rm Ca}=E_{\rm f}^{\rm
C14-CaMg_{2}}-2\Delta\mu_{\rm Mg}=-0.378$ eV. Hence, the chemical potential of
Al becomes the only free parameter. The reason we chose Al rather than Ca is
that the solubility of Al in Mg is much higher than that of Ca.

Figure 4:  (a) Formation energies (Eq. (1)) as a function of the Al chemical
potential for various planar structures with and without defects. The
thermodynamically (meta)stable bulk phase for a given chemical potential are
given in the bottom white box. The dark green region denotes the exothermic
formation of the R3 defect. The light green regions at left and right of it
denote the defect-free and the exothermic R6 formation regions, respectively.
(b) Variation of the temperature of solid solution versus chemical potential
for various concentrations of Al. The melting temperature for each alloy is
highlighted by the $\otimes$ symbol. The experimental conditions of Ref.
Kashiwase2019  and the present work are highlighted by hollow and filled
triangles, respectively. The atomic structures of the different planar defects
are given in the right panel. The blue, green and gray spheres denote Mg, Ca,
and Al atoms, respectively.

The corresponding MDPD is shown in Fig. 4a. In thermodynamic equilibrium,
i.e., using the conventional bulk phase diagram, the upper limit of the Al
chemical potential in hcp Mg is $\Delta\mu_{\rm Al}=\mu_{\rm equil.}=-0.30$
eV. In the thermodynamic stability domain of hcp Mg, defect formation becomes
exothermic at $\Delta\mu_{\rm Al}=-0.315$ eV, which is just below the
$\mu_{\rm equil.}$.

To study whether defect formation can be kinetically driven, we computed again
the nucleation barrier. For the barrier structure we considered the bulk
Al2Ca-C15 Laves phase, but fixed its lattice constants to those of pure Mg.
The Al chemical potential of this highly strained phase is $\mu_{\rm
barrier}=-0.014$ eV. This value, which sets another limit for the Al chemical
potential, is substantially higher than the limit set by bulk equilibrium
($\mu_{\rm equil.}=-0.30$ eV). In this metastable hcp Mg region defect
formation becomes highly exothermic (see Fig. 4a).

The MDPD shown in Fig. 4a allows us to extract suitable experimental process
and alloying conditions to stimulate formation of defects. The thermodynamic
window of Al chemical potentials at which the formation of the R3 defect
dominates the process is in the range of $-0.315~{}{\rm eV}\leq\mu_{\rm
initial}\leq-0.16~{}{\rm eV}$. Since this window is well below the chemical
potential of the barrier configuration ($\mu_{\rm barrier,C15}$) R3 planar
defects rather than bulk C15 precipitates are expected to form. The upper
bound of this interval is given by the onset of R6 formation. However, even
for chemical potentials larger than $-0.16~{}{\rm eV}$ the formation of R6
will lead to a decrease of the chemical potential and thus R6 will decompose
into R3.

To translate the chemical potential into experimentally controllable
quantities we use the fact that the relevant bulk Al concentrations are in the
dilute limit. In this limit the chemical potential is given by $\mu_{\rm
Al}=\mu^{0}_{\rm Al}+k_{\rm B}T\ln{c_{\rm Al}}$ where $\mu^{0}_{\rm Al}$ is
the solution enthalpy and $c_{\rm Al}$ is the concentration of Al in the pure
Mg matrix, $T$ is the temperature and $k_{\rm B}$ is the Boltzmann constant.
We retrieve the solution enthalpy as $\mu^{0}_{\rm Al}=0.045$ eV and the
concentration $c_{\rm Al}$ from the binary Al-Mg phase diagram murray1982mg .
To this end, we impose the condition that the chemical potential of Al atoms
in the solid solution and in the intermetallic Mg17Al12 phase should be equal
for the concentration and temperature pairs lying on the equilibrium line of
solid solution and intermetallic compound in the phase diagram.

Fig. 4b contains the variation of temperature of solid solution versus
chemical potential for various $c_{\rm Al}$. The upper limit for each line is
set to the melting temperature of the solution. The low concentrations such as
1.0 at. % cannot provide enough driving force for formation of the defects. On
the other hand, the chemical potential of high concentration solid solutions
e.g. 11.5 at.%, will exceed the chemical potential needed for the domination
of R3. But solid solutions with $\approx$ 4 to 5 at.% near their melting
temperature will fall in the suitable window of chemical potentials. Thus, we
predict that in samples with these concentrations, the formation of planar
defects in as cast condition is possible. We note that the casts used in
Kashiwase et al. Kashiwase2019 ; Kashiwase2020  experiments in which the
platelet C15 structures are detected fulfill the set forth conditions ($T$ =
796 K and $c_{\rm Al}$=4.55 at.%). The nominal composition of our cast
designed to capture the planar defects is chosen to be Mg-4wt.%Al-4wt.%Ca with
a composition of 1.1 at.% Al and 0.2 at.% Ca in the Mg matrix zubair2022laves
.

As shown in Fig. 4a, the initial Al chemical potential of our cast $\mu_{\rm
initial}=-0.19$ eV is well in the range of chemical potentials suitable for an
exothermic formation of R3. By forming the planar R3 defects the solute
concentration decreases, resulting in a monotonous decrease of the solute
chemical potential until at a chemical potential $\mu_{\rm final}$ the
chemical driving force is no longer sufficient to overcome the nucleation
barrier for forming new defects. According to the Al concentration in defect-
free bulk like regions measured by STEM-EDS the final solute chemical
potential $\mu_{\rm final}=-0.27$ eV. As shown in Fig. 4 this value is
slightly (0.05 eV) above the onset of exothermic defect formation ($\mu_{\rm
equilibrium,PD}$). We note that this small positive value is not the actual
nucleation barrier of the defect. This is given by the number of solute atoms
needed to form a critical nucleus of the defect times this value.

We imaged the planar defects in the Mg matrix in the as cast condition by
atomic resolution STEM using the high angle annular dark field (HAADF)
detector to provide Z(atomic number) contrast. As shown in Fig. 5c the R3
atomic structure matches perfectly the defect observed in our HAADF-STEM
images (Fig. 5b). As shown in Fig. 5a, the planar defects are closely spaced
by a few tens of nm. Their brighter contrast is consistent with the enrichment
in the heavier elements Al and Ca, which are also confirmed by the EDS maps.
Atomically resolved STEM images (Fig. 5b,c) further reveal the orientation
relationship between the C15-like planar defect and the Mg matrix: Al2Ca(111)
// Mg(0002), and Al2Ca[11-20] // Mg[1-100]. Note that this orientation
relationship has also been observed between bulk-phase C15 and Mg.Suzuki2008 ;
zubair2022laves  Nevertheless, it is clearly shown in Fig. 5b that only a
fraction of a C15 unit cell is sandwiched between the Mg(0002) planes. The
C15-like planar defect is symmetric with respect to the middle plane with
alternating brighter and weaker contrast, corresponding to atomic columns with
double Al and single Al occupancy. The next plane has bright contrast above
the single Al occupancy column, corresponding to the Ca columns. The plane
further above shows similar contrast to the Mg columns in the matrix and is
believed to be either Mg or Al, which has a similar Z contrast.

It is important to note that the Ca columns in the structure are prone to
electron beam damage. As shown in Fig. 5c, the bright contrast in the Ca
column positions disappeared after prolonged image in the area. This could
have hampered the structural analysis of this planar defect in earlier works.
Our low electron dose images clearly identified the position of the Ca
columns.

Figure 5: (a) HAADF-STEM image of planar defects in the Mg matrix of an as-
cast Mg-4wt.%Al-4wt.%Ca alloy with corresponding EDS maps in the region
highlighted by the yellow box. The EDS maps have been analized with the
principal component analysis code from Zhang2018 . Atomic resolution HAADF-
STEM images of the planar defects are acquired along Mg[1-100] (b) before and
(c) after the Ca column was damaged by electron beam radiation. The
corresponding atomic structures are overlaid, the blue, green, and gray
spheres denote Mg, Ca, and Al atoms.

The above discussion demonstrates the power of the MDPD concept in predicting
formation and structure of defects under non-equilibrium conditions as well as
designing experiments to activate certain defects. It faithfully reproduces
the various experimentally available observables (HR-TEM defect structures,
concentrations) and allows for a quantitative description of the reaction
paths qualitatively sketched in Fig. 1.

##  5 Conclusions

Inspired by experimental observations showing that supersaturated alloys
partition into defects rather than bulk-like precipitates, contrary to what
would be expected based on their bulk thermodynamic phase diagrams, we have
introduced the concept of metastable defect phase diagrams (MDPD). MDPDs
provide an intuitive connection between defect phase diagrams, equilibrium
bulk phase diagrams and bulk nucleation energies, all of which can be
straightforwardly computed using DFT. We have applied this concept to two
chemically and structurally very different materials systems - intermetallic
Fe-Nb alloys and Mg alloyed with small concentrations of Al and Ca - and have
demonstrated that MDPDs accurately predict the atomic structure and process
conditions for their formation. In both cases, the presence of excess solutes
resulted in the formation of planar defects that are confined in thickness and
represent local energy minima. Consequently, these defect structures form
long-lived metastable defect phases and cannot serve as nuclei for the
thermodynamically stable bulk precipitates because further growth would
increase their energy. Therefore, the presence of these defect states does not
reduce the nucleation barrier of the bulk precipitate phases. In contrast, the
formation of these defects substantially lowers the solute chemical potential,
effectively reducing or even eliminating the chemical driving force required
for the formation of the bulk precipitates.

In addition to the theoretical framework, our proposed concept also has
practical implications and provides a direct link to experimental
observations. Specifically, the initial and final chemical potentials of the
supersaturated host matrix can be determined based on the net-alloy
composition and the alloy composition in defect-free regions of the host,
respectively. These compositions can be easily obtained through techniques
such as electron backscatter diffraction (EBSD) or atom probe tomography (APT)
measurements, while the structure and solute excess can be measured using HR-
TEM or APT. It is important to note that the concept is not limited to the two
prototype material systems studied here and can be readily applied to other
systems as well. This opens up a new avenue for materials designers to control
microstructure and properties through partitioning into defects, in the same
way that precipitate formation is commonly used. By providing materials
scientists with an effective tool to exploit these opportunities, MDPDs offer
exciting new possibilities for the design of advanced materials.

## Acknowledgement

The authors acknowledge financial support by the Deutsche
Forschungsgemeinschaft (DFG) through the projects A06, B01, and C05 of the
CRC1394 “Structural and Chemical Atomic Complexity – From Defect Phase
Diagrams to Material Properties”, project ID 409476157.

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# The geometry of Bloch space in the context of quantum random access codes

###### Abstract

We study the communication protocol known as a Quantum Random Access Code
(QRAC) which encodes $n$ classical bits into $m$ qubits ($m<n$) with a
probability of recovering any of the initial $n$ bits of at least
$p>\tfrac{1}{2}$. Such a code is denoted by $(n,m,p)$-QRAC. If cooperation is
allowed through a shared random string we call it a QRAC with shared
randomness. We prove that for any $(n,m,p)$-QRAC with shared randomness the
parameter $p$ is upper bounded by
$\tfrac{1}{2}+\tfrac{1}{2}\sqrt{\tfrac{2^{m-1}}{n}}$. For $m=2$ this gives a
new bound of $p\leq\tfrac{1}{2}+\tfrac{1}{\sqrt{2n}}$ confirming a conjecture
by Imamichi and Raymond (AQIS’18). Our bound implies that the previously known
analytical constructions of $(3,2,\tfrac{1}{2}+\tfrac{1}{\sqrt{6}})$\- ,
$(4,2,\tfrac{1}{2}+\tfrac{1}{2\sqrt{2}})$\- and
$(6,2,\tfrac{1}{2}+\tfrac{1}{2\sqrt{3}})$-QRACs are optimal. To obtain our
bound we investigate the geometry of quantum states in the Bloch vector
representation and make use of a geometric interpretation of the fact that any
two quantum states have a non-negative overlap.

##  1 Introduction

Introduction/Introduction

##  2 Preliminaries

Preliminaries/IntroPre

###  2.1 Definition of a QRAC

Preliminaries/Definition_of_QRAC

###  2.2 Bloch Vector Representation of Quantum States

Preliminaries/Bloch_vector_representation

##  3 Geometry of Bloch space

Geometry_of_Bloch_space/GeomIntro

Geometry_of_Bloch_space/LemmaHyperPlane

Geometry_of_Bloch_space/LemmaUpperOnComponent

##  4 Upper Bounds on QRACs

Upper_bounds_on_QRACs/POVMlemma

Upper_bounds_on_QRACs/LemmaUnitVectors

Upper_bounds_on_QRACs/Main_theorem

##  5 Discussion

Discussion/Discussion

## Acknowledgements

This paper is based on S. Storgaard’s bachelor’s thesis. L. Mančinska is
supported by the Villum Fonden via the QMATH Centre of Excellence (Grant No.
10059).

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## Appendix A: Proof of Observation LABEL:obs1

We list the eigenvalues of $\boldsymbol{\rho}(\beta)$ as
$\lambda_{\text{max}}\geq\cdots\geq\lambda_{\text{min}}\geq 0$. Then the
eigenvalues of
$\tfrac{1}{2}\sum_{i=1}^{N^{2}-1}\beta_{i}\boldsymbol{\sigma}_{i}$ can be
listed as

$\lambda_{\max}-\tfrac{1}{N}\geq\cdots\geq\lambda_{\min}-\tfrac{1}{N}.$ (1)

Let $\beta^{\prime}=\gamma\beta$ for some $\gamma\in\mathbb{R}$. If
$\gamma\geq 0$ The eigenvalues of $\boldsymbol{\rho}(\gamma\beta)$ can be
listed as

$\tfrac{1}{N}(1-\gamma)+\gamma\lambda_{\max}\geq\cdots\geq\tfrac{1}{N}(1-\gamma)+\gamma\lambda_{\min}\geq
0$ (2)

and hence $\beta^{\prime}\in\partial\boldsymbol{\beta}(Q_{N})$ is in the
boundary of Bloch space if $\gamma=\tfrac{1}{1-N\lambda_{\min}}$. Similarly if
$\gamma<0$ then $\beta^{\prime}\in\partial\boldsymbol{\beta}(Q_{N})$ if
$\gamma=\tfrac{1}{1-N\lambda_{\max}}$.

## Appendix B: Proof of Lemma LABEL:mancinskalemma

The inequality in (LABEL:upperboundonunitvectors) holds if all the $\mu_{i}$’s
are 0, so we assume they are not all 0. We can interpret the sum on the left
hand side of (LABEL:upperboundonunitvectors) as the inner product of

$y_{1}=(1,...,1)\in\mathbb{R}^{\\{0,1\\}^{n}}\cong\mathbb{R}^{2^{n}}$ (3)

and a vector $y_{2}\in\mathbb{R}^{\\{0,1\\}^{n}}$, whose entry corresponding
to bit string $x\in\mathbb{R}^{\\{0,1\\}^{n}}$ is given by

$(y_{2})_{x}=\mathinner{\\!\left\lVert\sum_{i\in[n]}(-1)^{x_{i}}\mu_{i}\right\rVert}.$
(4)

Applying the Cauchy-Schwarz inequality, we get that the left hand side of
(LABEL:upperboundonunitvectors) is upper bounded by

$\mathinner{\\!\left\lVert y_{1}\right\rVert}\mathinner{\\!\left\lVert
y_{2}\right\rVert}=\sqrt{2^{n}}\sqrt{\sum_{x\in\\{0,1\\}^{n}}\mathinner{\\!\left\lVert\sum_{i\in[n]}(-1)^{x_{i}}\mu_{i}\right\rVert}^{2}}.$
(5)

We claim now that

$\sum_{x\in\\{0,1\\}^{n}}\mathinner{\\!\left\lVert\sum_{i\in[n]}(-1)^{x_{i}}\mu_{i}\right\rVert}^{2}=2^{n}\sum_{i\in[n]}\mathinner{\\!\left\lVert\mu_{i}\right\rVert}^{2}.$
(6)

Maybe we can refer to Lemma 5 in [1]. Afterall the proofs are essentially the
same… This can be proved by induction on $n$. First, Eq. (6) holds for $n=1$
since

$\mathinner{\\!\left\lVert\mu_{1}\right\rVert}^{2}+\mathinner{\\!\left\lVert-\mu_{1}\right\rVert}^{2}=2\mathinner{\\!\left\lVert\mu_{1}\right\rVert}^{2}.$
(7)

Assume that Eq. (6) holds for $n=k$ and consider the case when $n=k+1$. By
explicitly carrying out the sum over $x_{k+1}\in\\{0,1\\}$ on the left hand
side of Eq. (6) we get

$\sum_{x\in\\{0,1\\}^{k}}\Big{[}\mathinner{\\!\left\lVert\big{(}(-1)^{x_{1}}\mu_{1}+...+(-1)^{x_{k}}\mu_{k}\big{)}+\mu_{k+1}\right\rVert}^{2}+\mathinner{\\!\left\lVert\big{(}(-1)^{x_{1}}\mu_{1}+...+(-1)^{x_{k}}\mu_{k}\big{)}-\mu_{k+1}\right\rVert}^{2}\Big{]}.$
(8)

Applying the parallelogram identity, i.e.

$\mathinner{\\!\left\lVert
u_{1}+u_{2}\right\rVert}^{2}+\mathinner{\\!\left\lVert
u_{1}-u_{2}\right\rVert}^{2}=2(\mathinner{\\!\left\lVert
u_{1}\right\rVert}^{2}+\mathinner{\\!\left\lVert u_{2}\right\rVert}^{2}),$ (9)

the expression in (8) equals

$2\sum_{x\in\\{0,1\\}^{k}}\big{(}\mathinner{\\!\left\lVert\textstyle\sum_{i\in[k]}(-1)^{x_{i}}\mu_{i}\right\rVert}^{2}+\mathinner{\\!\left\lVert\mu_{k+1}\right\rVert}^{2}\big{)}.$
(10)

Finally, applying the induction hypothesis, we complete the inductive step as
follows:

$2\big{(}2^{k}\sum_{i\in[k]}\mathinner{\\!\left\lVert\mu_{i}\right\rVert}^{2}+2^{k}\mathinner{\\!\left\lVert\mu_{k+1}\right\rVert}^{2}\big{)}=2^{k+1}\sum_{i\in[k+1]}\mathinner{\\!\left\lVert\mu_{i}\right\rVert}^{2}.$
(11)

Now, by inserting (6) in (5) we can conclude (LABEL:upperboundonunitvectors).

Recall that $\braket{y_{1},y_{2}}=\mathinner{\\!\left\lVert
y_{1}\right\rVert}\mathinner{\\!\left\lVert y_{2}\right\rVert}$ if and only if
$y_{2}=ky_{1}$ for some $k\in\mathbb{R}$. Hence, the bound in
(LABEL:upperboundonunitvectors) holds with equality if and only if the
quantity $(y_{2})_{x}$ in Eq. (4) is equal to some constant $c$ independent of
$x\in\\{0,1\\}^{n}$. In other words, for all $x\in\\{0,1\\}^{n}$ we must have
that

$\sum_{i\neq
j\in[n]}(-1)^{x_{i}+x_{j}}\braket{\mu_{i},\mu_{j}}=c^{2}-\sum_{i\in[n]}\mathinner{\\!\left\lVert\mu_{i}\right\rVert}^{2},$
(12)

where the right hand side is constant. Now, fix $m\in[n]$. The left hand side
of (12) can be rewritten as

$\sum_{i\neq
j\in[n]\backslash\\{m\\}}(-1)^{x_{i}+x_{j}}\braket{\mu_{i},\mu_{j}}+2\sum_{i\in[n]\backslash\\{m\\}}(-1)^{x_{m}+x_{i}}\braket{\mu_{m},\mu_{i}}$
(13)

This must be invariant upon the interchange
$x_{m}\rightarrow\overline{x_{m}}$. Since
$(-1)^{x_{m}}=-(-1)^{\overline{x_{m}}}$ we have that

$\sum_{i\in[n]\backslash\\{m\\}}(-1)^{x_{m}+x_{i}}\braket{\mu_{m},\mu_{i}}=0.$
(14)

Now fix $m^{\prime}\in[n]\backslash\\{m\\}$ and rewrite (14) as

$(-1)^{x_{m}+x_{m^{\prime}}}\braket{\mu_{m},\mu_{m^{\prime}}}+\sum_{i\in[n]\backslash\\{m,m^{\prime}\\}}(-1)^{x_{m}+x_{i}}\braket{\mu_{m},\mu_{i}}=0.$
(15)

This must be invariant upon the interchange
$x_{m^{\prime}}\rightarrow\overline{x_{m^{\prime}}}$ so we get
$\braket{\mu_{m},\mu_{m^{\prime}}}=0$. This completes the proof.