Computer Physics Communications 185 (2014) 330338Contents lists available at ScienceDirect

Computer Physics Communications

journal homepage: www.elsevier.com/locate/cpc

Automated procedure to determine the thermodynamic stability of amaterial and the range of chemical potentials necessary for itsformation relative to competing phases and compounds

J. Buckeridge a,, D.O. Scanlon a,b, A. Walsh c, C.R.A. Catlow aa University College London, Kathleen Lonsdale Materials Chemistry, Department of Chemistry, 20 Gordon Street, London WC1H 0AJ, United Kingdomb Diamond Light Source Ltd., Diamond House, Harwell Science and Innovation Campus, Didcot, Oxfordshire OX11 0DE, United Kingdomc Centre for Sustainable Chemical Technologies and Department of Chemistry, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom

a r t i c l e i n f o

Article history:Received 6 June 2013Received in revised form20 August 2013Accepted 28 August 2013Available online 9 September 2013

Keywords:Thermodynamic stabilityChemical potentialMaterials designDefect formation analysis

a b s t r a c t

We present a simple and fast algorithm to test the thermodynamic stability and determine the necessarychemical environment for the production of a multiternary material, relative to competing phases andcompounds formed from the constituent elements. If the material is found to be stable, the region ofstability, in terms of the constituent elemental chemical potentials, is determined from the intersectionpoints of hypersurfaces in an (n 1)-dimensional chemical potential space, where n is the number ofatomic species in the material. The input required is the free energy of formation of the material itself,and that of all competing phases. Output consists of the result of the test of stability, the intersectionpoints in the chemical potential space and the competing phase to which they relate, and, for two- andthree-dimensional spaces, a file which may be used for visualization of the stability region. We specifythe use of the program by applying it both to a ternary system and to a quaternary system. The algorithmautomates essential analysis of the thermodynamic stability of a material. This analysis consists of aprocess which is lengthy for ternary materials, and becomes much more complicated when studyingmaterials of four or more constituent elements, which have become of increased interest in recent yearsfor technological applications such as energy harvesting and optoelectronics. The algorithmwill thereforebe of great benefit to the theoretical and computational study of such materials.

Program summary

Program title: CPLAP

Catalogue identifier: AEQO_v1_0

Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEQO_v1_0.html

Program obtainable from: CPC Program Library, Queens University, Belfast, N. Ireland

Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html

No. of lines in distributed program, including test data, etc.: 4301

No. of bytes in distributed program, including test data, etc.: 28851

Distribution format: tar.gz

Programming language: FORTRAN 90.

Computer: Any computer with a FORTRAN 90 compiler.

Operating system: Any OS with a FORTRAN 90 compiler.

RAM: 2 megabytes

Classification: 16.1, 23.

This paper and its associated computer program are available via the Computer Physics Communication homepage on ScienceDirect (http://www.sciencedirect.com/science/journal/00104655). Corresponding author. Tel.: +44 2076790486.

E-mail address: j.buckeridge@ucl.ac.uk (J. Buckeridge).0010-4655/$ see front matter 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.cpc.2013.08.026

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rJ. Buckeridge et al. / Computer Physics

Nature of problem:To test the thermodynamic sthe constituent atomic speciits synthesis.Solution method:Assume that the formation ostandard states of the constion the elemental chemical pn unknowns, where m > ncompatible with the conditiothe compatible results definchemical potentials.Restrictions:The material growth environAdditional comments:For two- and three-dimensvisualization of the stabilityRunning time:Less than one second.

1. Introduction

Over the past few decades, there has been considerable growthin the development of advanced materials for energy harvestingand transparent electronics applications [15]. At present, twoof the greatest challenges facing the optoelectronics industry arethe production of stable and economically viable p-type materials[6,7], and the replacement of rare or inaccessible components suchas indium with more earth-abundant elements [811]. This hasled to increased interest in more exotic materials, consisting ofternary [1214], quaternary [1517], and quinternary [1820] sys-tems. Thesematerials are also of increased interest for applicationsin batteries [21] and solid-state electrochemistry [22]. Having alarge number of elements in a compound offers a greater degree ofchemical freedom, where the tuning of properties of interest, suchas band gaps, can be performed by varying the composition.

Instrumental in this research is the theoretical prediction of

material properties, using various computational approaches, forexample, density functional theory (DFT) and methods based oninteratomic potentials [23]. A key consideration when predict-ing materials appropriate for particular applications is the ther-modynamical stability of the system, as stable materials presentfar fewer technological challenges when incorporated into de-vices [24,25]. It is of great interest to predict the range of chemicalpotentials of the component elemental species over which the tar-get phase is stable, rather than the elemental species themselvesor competing phases, as this gives an indication the chemical envi-ronment necessary for the synthesis of that phase. Indeed, in or-der to predict the stability of a material, one needs to compareits free energy with that of all competing phases, including thoseconsisting of subsets of the elemental species in the material [26].The standard procedure [26,27] is to calculate all relevant freeenergies at the athermal limit, under the assumption of thermody-namic equilibrium. Assuming that the material is thermodynam-ically stable, rather than the competing phases, leads to a set ofconditions on the elemental chemical potentials, from which onecan work out the stability range (if it exists). For binary systems,where thenumber of independent variables is one, the procedure istrivial. For ternary systems, though the calculation is still straight-forward, if there are many competing phases, the exercise canbecome tedious. For quaternary or higher-order systems, the cal-culation of the stability region becomes quite involved, as thereommunications 185 (2014) 330338 331

ability of amaterial with respect to competing phases and standard states ofs and, if stable, determine the range of chemical potentials consistent with

f the material of interest occurs, rather than that of competing phases anduent elemental species. From this assumption derive a series of conditionsotentials. Convert these conditions to a system of m linear equations withSolve all combinations of n linear equations, and test which solutions arens on the chemical potentials. If none are, the system is unstable. Otherwise,e boundary points of the stability region within the space spanned by the

ment is assumed to be in thermal and diffusive equilibrium.

ional spaces spanned by the chemical potentials, files are produced foregion (if it exists).

2013 Elsevier B.V. All rights reserved.

are typically a large number of competing phases to consider, andthree or more independent variables. It is evident that an auto-mated process to perform these tasks would be of great benefit totheoreticians working on these problems.

Consideration of the chemical potential landscapewithinwhichamaterial forms is also crucialwhenpredicting the nature and con-centration of defects. The synthesis of a material in different con-ditions can mean that the formation of different defects becomesfavorable. Calculations of defect formation energies, which dependon the chemical potentials, provide useful information to ex-perimentalists wishing to produce a material with a particulardefect-related property. For example, to produce a material withsignificant concentrations of a p-type donor incorporated dur-ing the growth process, it is necessary to know which chem-ical environment favors the formation of that particular donordefect. Knowledge of the full range of elemental chemical poten-tials within which the material is stable is required, in order to

predict where in that range the formation of the p-type donor de-fect is favored. It is therefore necessary to work out accurately thestability region in the chemical potential space; not carrying outthis procedure correctly can lead to unphysical predictions of de-fect formation energies [28,29]. We stress that this type of analysisis limited to growth conditions where the assumption of thermo-dynamic equilibrium is reasonable.

In this paper, we present a simple, fast, and effective algorithmto determine the range of the elemental chemical potentialswithinwhich the formation of a stoichiometric material will be favorable,in comparison to the formation of competing phases. If there is norange, then the material is not thermodynamically stable withinthe specified environment. The algorithm works by first readingin the free energy of formation of the material itself and that ofthe competing phases, which are provided by the user. Setting thecondition that the material is, in principle, stable constrains thevalues of the elemental chemical potentials, effectively reducingthe number of independent variables by one, meaning that thespace spanned by the elemental chemical potentials is (n 1)-dimensional, where n is the total number of elements in thematerial. The condition that the competing phases do not formpro-vides further conditional relations among the independent vari-ables. A set of linear equations, corresponding to the set of allconditions on the independent variables, is constructed. All pos-sible combinations of the linear equations in the set are solved in

C332 J. Buckeridge et al. / Computer Physics

order to find their intersection points. The intersection points arethen checked to determine which ones satisfy every condition (ifnone do, the system is not thermodynamically stable). Those thatdo form the corner points of the region of stability in the chem-ical potential space. The algorithm is based on the fact that eachcompeting phase and standard state effectively defines a hyper-surface in the elemental chemical potential space, and the regionbounded by these hypersurfaces corresponds to the region of val-ues of chemical potentials in which the material will be stable. Theelemental chemical potentials are given with respect to their stan-dard states, i.e.we set the energy that the element has (per atom) inits standard state as the zero of chemical potential for that element.The algorithm requires that the energy of formation of thematerialand each competing phase is calculated (ormeasured) prior to exe-cution. For an in silico study, it is therefore of great importance thatthe user searches the chemical databases (such as the InorganicCrystal Structure Database [30]) extensively, and calculates the en-ergy of all phases and limiting compounds using the same level oftheory [31,32,27,26,1]. We have incorporated the algorithm in aFORTRAN program called the Chemical Potential Limits AnalysisProgram (CPLAP), which we have made available online [33,34].For convenience, if the chemical potential space is twodimensionalor three dimensional, the program produces files that can be usedas input to GNUPLOT [35] and MATHEMATICA [36], to visualize theregion of stability. Anoption to fix the value of a particular chemicalpotential is available, which effectively reduces the dimensionalityby one.

The rest of the paper is structured as follows. In Section 2, wediscuss the relevant theory on which the algorithm is based; inSection 3, we present the algorithm; in Section 4, we demonstratehow the program works using a ternary system and a quaternarysystem as examples; and in Section 5 we summarize our work.The sorting routine used by the main algorithm is described in theAppendix. All the figures in this work, apart from the flowcharts,have been produced using GNUPLOT, from the output from CPLAP.

2. Theory

The fundamental assumption, uponwhich analysis of the chem-ical potential landscape in which a material forms is based, is thatthe combined system in the growth environment is in thermody-namic equilibrium. To illustrate the necessary theory, we considera binary system AmBn, which forms via the reaction

mA+ nB AmBn, (1)at constant pressure and temperature. The formation of AmBn com-petes with the phase ApBq. The procedure is then to assume thatAmBn forms, rather than ApBq or the standard states of A and B, andsee if this leads to a contradiction.

We recall that the chemical potential of species or com-pound is defined as

=GN

p,T, (2)

where G is the Gibbs free energy of the system (G = UTS+pV ,Uis the internal energy, T is the temperature, S is the entropy, p is thepressure, and V is the volume) and N is the number of particles ofspecies or compound .

We first consider the chemical potential of individual speciesin the compound AmBn (i.e. A and B). We denote the chemicalpotential of species in its standard state as S . We wouldnow like to refer the elemental chemical potentials to theirrespective S , i.e. we set = T S, (3)ommunications 185 (2014) 330338

where T is the chemical potential of species that sharesa common reference with S . We do this for convenience; bydetermining the S in a consistent manner, we will automaticallyobtain a common reference for all elemental chemical potentials.We note that, when calculating formation energies that depend onthe chemical potentials,T = S+ should be used. In order toavoid formation of the standard states of A and B, we must have

0, (4)placing an upper bound on each elemental chemical potential.

We now consider all species involved in the reaction given inEq. (1), so = A, B, AmBn, and follow the analysis given in Ref. [37].Under the assumption of constant p and T , the differential dG in theGibbs free energy is given by

dG =

TdN. (5)

As dN is proportional to the coefficient i in the reaction givenby Eq. (1) (i = m for = A, i = n for = B, i = 1 for = AmBn), it can be written as dN = idN , where dN is thenumber of occurrences of the reaction in Eq. (1). We can thereforewrite

dG =

iT

dN. (6)

At equilibrium,1 dG = 0, implying that

iT = 0, (7)

from which we obtain (remembering that iAmBn = 1)mA + nB = AmBn = 1Gf [AmBn]; (8)here, 1Gf [X] = 1Hf [X] T1S is the Gibbs free energy of for-mation of compound X with respect to the standard states of itsconstituent elements, Hf [X] is the enthalpy of formation of X , and1S is the change in entropy. For crystalline systems with low lev-els of disorder, a good approximation is to set 1S = 0, so that1Gf [X] = 1Hf [X]. Under this approximation, we can set thechemical potentials of A and B in their standard states equal to thetotal energy (per atom) of the standard states. Calculating all totalenergies, including those required to determine 1Hf [X], in aconsistent manner ensures that all chemical potentials have acommon reference. Although it is possible to include vibrationalentropic effects using, for example, the quasi-harmonic ap-proximation, and configurational entropic effects for disorderedsystems, in the remainder of this paper we assume that the ap-proximation1Gf [X] = 1Hf [X] applies. Eq. (8) now becomesmA + nB = AmBn = 1Hf [AmBn], (9)effectively constraining our mathematical problem, so that onechemical potential can be written in terms of the other, i.e. thenumber of independent variables is one. For a binary system, there-fore, the chemical potential space is one dimensional, spanned bythe one independent variable.

Combining Eqs. (4) and (9) and takingA to be the independentvariable, we find that

1Hf [AmBn]m

A 0, (10)

1 Once equilibrium is reached, the reaction will not proceed further; thereforethere will not be any further change in the thermal average values of theconcentrations. This implies that, given the volume at equilibrium, Eq. (7) will be

valid when V and T are specified instead of p and T , as was our initial assumption.See Ref. [37].

CJ. Buckeridge et al. / Computer Physics

with B being determined for each value of A from Eq. (9). Itfollows then that the boundary A = 0 corresponds to A-rich/B-poor growth conditions, and the boundary A = 1Hf [AmBn]/mcorresponds to B-rich/A-poor growth conditions. Eq. (10) definesthe stability region (a line segment) in the one-dimensional (1D)chemical potential space spanned by A.

We now include in our calculation the competing phase ApBq.The assumption that ApBq does not form leads to the followingcondition:

pA + qB ApBq = 1Hf [ApBq]. (11)Combining this with Eq. (9) provides the following limits:p qm

n

A 1Hf [ApBq] qn1Hf [AmBn];

q pnm

B 1Hf [ApBq] pm1Hf [AmBn]. (12)

If these limits are inconsistent with Eq. (4), then AmBn is unstablewith respect to the formation of ApBq. If they are consistent, thenthey effectively reduce the range given in Eq. (10), i.e. they reducethe extent of the stability region. The addition of more competingphases will further restrict the stability region, which (if it exists)will consist of a line segment in the 1D space spanned by A, withcorresponding values of B derived from Eq. (9). This solves thecase of a binary system.

We now consider a ternary system, to demonstrate the gener-alization of the process as one increases the dimensionality of thechemical potential space. We consider the system AmBnCp, whoseformation competes with the phases AqBr and AsBtCv .

Corresponding to Eq. (9), the assumption that AmBnCp forms inan equilibrium reaction with the constituent elements standardphases provides the constraint

mA + nB + pC = AmBnCp = 1Hf [AmBnCp], (13)allowing us to express one of the chemical potentials, say C , interms of the other two, leaving two independent variables A andB spanning a two-dimensional (2D) chemical potential space.Allowing C to adopt its maximum bounded value of zero (seeEq. (4)) gives the following condition on A and B:

mA + nB 1Hf [AmBnCp]. (14)Combining Eqs. (4) and (13) gives the following conditions on thechemical potentials:

i 1Hf [AmBnCp]/i, (15)where i stands for eitherm, n, or p, whichever is appropriate.

Assuming that the competing phases do not form leads to theconditions

qA + rB AqBr = 1Hf [AqBr ]; (16)sA + tB + vC AsBtCv = 1Hf [AsBtCv]. (17)Using Eq. (13) to eliminate C from Eq. (17), we see that Eqs.(4), (14), (15), (16), and (17) define conditional relations on a 2Dplane formed by A and B. If there does not exist a region inthe 2D plane that conforms to every condition, then the systemis not thermodynamically stable. Otherwise, we have a region ofstability. One method of determining if this is the case is to set theinequality signs in Eqs. (4), (14), (15), (16), and (17) to equalitysigns, giving a series of linear equations with two unknowns.These linear equations define lines on the 2D plane formed by Aand B. Their intersection points can be determined by solvingthe appropriate combinations of the linear equations. Those thatthen simultaneously satisfy the conditions given by Eqs. (4) and

(14)(17) (if any) will bound the region of stability. The resultwill be a 2D stability region in the plane defined by A and B,ommunications 185 (2014) 330338 333

with the corresponding value of C at each point in the stabilityregion determined from Eq. (13). Graphically, one can display thissolution as a 2D plot in the space spanned by A and B, with thecorresponding values of C given at points of interest.

The generalization of this procedure to systems with largernumber of constituent elements is as follows. For a system withn constituent elements, we will have n 1 independent variables.The higher-dimensional analogues of Eqs. (4), (14), and (15) pro-vide 2n 1 linear equations (which correspond to hypersurfacesin the (n 1)-dimensional space), and each competing phase pro-vides an additional linear equation. We therefore have a minimumof 2n1 linear equationswithn1unknowns.Mathematically, thesolution is trivial, as it only involves solving different combinationsof the linear equations and checking which solutions are compati-ble with a series of conditional statements. In practice, however, ifwe have m competing phases there are 2n+m1Cn1 combinationsto consider, and carrying out the procedure can be quite time con-suming and error prone. This is the reason we have developed aprogram to automate it.

3. Algorithm

Input to the program consists of the number of species in thecompound of interest, the names and stoichiometry of the species,and the free energy of formation of the compound. One must alsoinput the total number (if any) of competing phases, and, for eachone, the number of species, the names and stoichiometry of eachspecies, and the free energy of formation of that competing phase.The input can be provided via a file, or interactively while runningthe program.

The user must specify which elemental chemical potential is tobe set as the dependent variable. We note here that the procedurecarried out by the program can, in principle, be performedwithoutany dependent variable set. If this is done, however, only theintersection points with the hypersurface corresponding to thecompound of interest are viable solutions, since, by not settinga dependent variable, the constraint given by Eq. (9) or (13) (orthe higher-dimensional analogue) is assumed no longer to apply,and instead effectively the equality sign is replaced by a greaterthan or equals to sign (i.e. the assumption that the reactionin Eq. (1) is in equilibrium no longer holds). Only those resultsthat are consistent with the constraint are actual solutions of theproblemat hand. So, thoughmore intersectionpointsmaybe foundwhen no dependent variable is set, only those that intersect thehypersurface corresponding to the compound of interest are actualsolutions. If no dependent variable is set, the program warns theuser of this fact, and how to interpret the results. It is alwayspreferable to set a dependent variable.

After reading in the input, the main algorithm begins (seeFig. 1). If the system is binary, the solution is relatively trivial. Theprogram carries out the procedure as described in Section 2 forbinary systems, which is to check that the limits imposed by thecompeting phases (Eq. (12)) are consistent with Eq. (4) and theconstraint (Eq. (9)), and, if they are, to return the line segment thatdefines the region of stability. The constraint is also returned asoutput. Note that this is a separate procedure from that used whenthe number of species is greater than two.

For ternary and higher-order systems, a more complex algo-rithm is used. From the input, the program constructs a matrix oflinear equations with n 1 unknowns, where n is the number ofspecies in the system. The compound of interest itself provides onelinear equation (Eq. (14), or its higher-dimensional analogue). Eachindependent variable then contributes two linear equations: onegiven by Eq. (4), and the other given by Eq. (15), which means that

there are, at a minimum, 2n1 linear equations in the matrix. Ad-ditional equations are provided by the competing phases: one per

C334 J. Buckeridge et al. / Computer Physics

Fig. 1. Flowchart of main algorithm.

phase. If there arem competing phases, we therefore have, in total,2n+m1 linear equations. Once thematrix has been constructed,it is passed to a sorting routine which extracts every possible com-bination of n 1 equations from the 2n + m 1 total. This sort-ing routine is described in the Appendix. For each combination, then 1 equations are solved using a standard LU decomposition andback-substitutionmethod [38], if a solution exists. In this way a se-ries of intersection points are found (redundancies are checked for,and removed). Each intersection point is tested to see if it obeyssimultaneously all the conditions on the elemental chemical po-tentials (Eqs. (4), (14), (15), (16), and (17) or higher-dimensionalanalogues). If none do, the system is not thermodynamically stable.Otherwise, those that do correspond to corner points in the stabil-ity region. The output is then sent to file, consisting of the limitingconditions applied, the resulting intersection points (with, for eachone, the corresponding value of the dependent variable), and a listcomposed of each competing phase, with its corresponding linearequation and intersection points (if any).

An option is provided to print to file a grid of points withinthe stability region, with the grid density provided by the user.Such a grid of valuesmay be useful for demonstrating the variationof the formation energy of a particular defect as the elementalchemical potentials are varied; for this, the user would be requiredto calculate the formation energy at each grid point. If the chemicalpotential space is two dimensional or three dimensional, theprogramoutputs a filewhichmaybe loadeddirectly intoGNUPLOT,and text which may be pasted into a notebook in MATHEMATICA,to produce a plot of the stability region, which will be usefulfor visualization of results. In addition, for 2D chemical potentialspaces, a text file is produced which contains the necessary data to

plot the lines in the chemical potential space corresponding to thematerial of interest and its competing phases.ommunications 185 (2014) 330338

It is possible to restart a run fromaprevious calculation. Optionsare then available to set a different chemical potential as thedependent variable, to provide additional competing phases notconsidered in the original run, or to set a chemical potential toa particular value (effectively reducing the dimensionality of thechemical potential space by one). The latter option is not availablefor binary systems, as the solution is trivial.

We note that, in principle, the procedure could be extended toarbitrary pressure and temperature ranges by including ther-modynamic potentials either from computations (using phononfrequency calculations and/or statistical mechanics) or thermo-chemical data. Such an extension is beyond the immediate scopeof the present work.

Our approach should be compared with that of the CALPHADcode [3941], which is widely used in modeling phase diagramsof alloys over a range of temperature, pressure, and composition.CALPHAD uses a model with adjustable parameters to describe thethermodynamic properties of each phase of a material, fitting theparameters to results from thermochemical and thermophysicalstudies stored in databases, and determines a consistent phasediagram using a wide range of data. The aim of our approach isdifferent: it identifies the range of elemental chemical potentialsover which a specified phase is stable.

4. Examples

4.1. Ternary system

As our first example of the application of our program, we con-sider the systemBaSnO3 [42], an indium-free transparent conduct-ing oxide (TCO). The formation of BaSnO3 (in the cubic perovskitestructure) occurs in competition with the phases BaO, SnO, SnO2,and Ba2SnO4, as determined by searching the Inorganic CrystalStructure Database [30] for systems consisting of combinations ofthe elements Ba, Sn, and O. Our aim here is to derive the rangesof chemical potentials in which stoichiometric BaSnO3 will form,using our program. The enthalpies of formation of the competingphases and thematerial itself have been calculated previously [42],using DFT with the PBE0 [43,44] hybrid functional (at the ather-mal limit). The values are presented in Table 1. These, and thestoichiometries of the relevant compounds, form the input to ourprogram. The constraint on the elemental chemical potentials is(see Eq. (13))

Ba + Sn + 3O = BaSnO3 = 11.46 eV. (18)We set the chemical potential of O, O, as the dependent variable.

After running the program, we find that the system is thermo-dynamically stable. Given that BaSnO3 forms, the limiting condi-tions that apply to the two independent variables Ba and Sn are(energies in eV)

Ba + Sn 11.46,2Ba Sn 5.52,2Ba Sn 3.95,Ba + 2Sn 3.85,Ba 0,Sn 0,Ba 11.46,Sn 11.46.

(19)

We present the resulting intersection points bounding the stabilityregion in Table 2, wherewe give the corresponding value of the de-

pendent variableO, and the competing phases towhich the inter-section points correspond. The stability region is plotted in Fig. 2.

CJ. Buckeridge et al. / Computer Physics

Table 1Enthalpies of formation (1Hf ) of BaSnO3 and its relevant competing phases. Thevalues, which are taken from Ref. [42], were determined using DFT with the PBE0hybrid functional.

System 1Hf (eV) System 1Hf (eV)

BaSnO3 11.46 Ba2SnO4 17.13BaO 5.14 SnO 2.54SnO2 5.29

Table 2Intersection points bounding the stability region in the 2D chemical potential spacespanned by the independent variables Ba and Sn . The corresponding values ofthe dependent variable O , and the relevant competing phases, are also given. Allenergies are in eV.

Ba Sn O Competing phases

A 5.66 5.80 0.00 Ba2SnO4 , BaSnO3B 6.18 5.29 0.00 SnO2 , BaSnO3C 2.76 0.00 2.90 Ba2SnO4D 3.53 0.00 2.64 SnO2

Fig. 2. (Color online) Region of stability (shaded) for BaSnO3 in the 2D spacespanned by Ba and Sn . The (colored) lines indicate the limits imposed by thecompeting phases and the compound of interest.

We note that, if we change which chemical potential is set as thedependent variable, we obtain the same results (as we must). Theonly differencewill be in the appearance of the figure, as one of theaxes will be changed to that of the new independent variable.

It is worth noting that, if one of the competing phases, sayBa2SnO4 (which could easily be overlooked), is not included in thecalculation, the resulting stability region (see Fig. 3) is approxi-mately twice as extensive as that shown in Fig. 2, indicating theimportance of taking into account all relevant competing phases.If one or more is left out, the analysis may be incorrect. Similarly,when calculating the total energies of the standard phases of theconstituent elements, the correct ground state of O2 (triplet spinconfiguration) must be used, as well as sufficient k-point samplingfor the metallic standard phases.

As is discussed in Ref. [42], the most stable n-type intrinsicdefect in BaSnO3 is the O vacancy (VO). The formation enthalpy1Hf [VO] of the (neutral) defect is determined from the reaction

OO VO + 12O2 (20)according to

1Hf [VO] = (ED EH)+ EO2 + O, (21)

where EH is the energy of the stoichiometric host supercell, EDis the energy of a supercell containing the defect, and EO2 is theommunications 185 (2014) 330338 335

Fig. 3. (Color online) Region of stability (shaded) for BaSnO3 in the 2D spacespanned by Ba and Sn when the competing phase Ba2SnO4 is not taken into ac-count .

Fig. 4. (Color online) Variation in VO formation enthalpy as a function of chemicalpotential, shown within the stability region for the formation of BaSnO3 . The(colored) lines indicate the limits imposed by competing phases.

energy per atom of O in its elemental (O2 gas) form, which wehave set as the chemical potential of O in its standard state,SO (seeSection 2). As can be seen, 1Hf [VO] depends on O. By printing agrid of points contained within the stability region, one obtains alist of points at which 1Hf [VO] can be determined, which in turncan be used to demonstrate how the defect formation energy variesat the different possible growth conditions. We show the resultsof such a calculation in Fig. 4, where the variation in 1Hf [VO] isshown within the stability region in the chemical potential space.We see that, unsurprisingly, Ba- and Sn-rich conditions favor itsformation. It should be remembered that the defect concentrationdepends exponentially on this quantity.

4.2. Quaternary system

We now discuss the application of CPLAP to the quaternarysystem LaCuOSe. This layered oxyselenide is a promising degen-erate p-type wide band-gap semiconductor [4547,16]. With fourspecies in the compound, we have a three-dimensional (3D) chem-ical potential space. There are a large number of competing phases(22) to be taken into consideration, as determined by searching theInorganic Crystal Structure Database [30] for systems consistingof combinations of the elements La, Cu, O, and Se. We therefore

have a much more complicated problem than for the ternary sys-tem BaSnO3, discussed in Section 4.1. This example demonstrates

C336 J. Buckeridge et al. / Computer Physics

Table 3Enthalpies of formation (1Hf ) of LaCuOSe and its relevant competing phases. Thevalues were determined using DFT with the HSE06 hybrid functional.

System 1Hf (eV) System 1Hf (eV)

LaCuOSe 9.55 CuSe2 1.16La2CuO4 19.94 CuSe2O5 6.54CuLaO2 10.60 La2SeO2 16.27La2O3 17.70 La2(SeO3)3 27.94La3Se4 15.77 La4Se3O4 33.16LaCuSe2 6.96 LaCu2 2.13LaSe2 5.64 LaCu5 4.52LaSe 4.41 La(CuO2)2 13.07Ce2Se 1.95 LaCuO3 10.61CuSe 1.17 Se2O5 3.37Cu3Se2 3.55 SeO2 1.95La2Cu(SeO3)4 32.78

Table 4Intersection points bounding the stability region in the 3D chemical potential spacespanned by the independent variables La , Cu , and O . The corresponding valuesof the dependent variable Se , and the relevant competing phases, are also given.All energies are in eV.

La Cu O Se Competing phases

A 5.78 1.18 2.59 0.00 LaCuSe2 , Cu3Se2 , LaCuOSeB 5.70 1.26 2.59 0.00 LaCuSe2 , La4Se3O4 , LaCuOSeC 6.77 1.18 1.60 0.00 Cu3Se2 , La2(SeO3)3 , LaCuOSeD 6.67 1.26 1.62 0.00 La2(SeO3)3 , La4Se3O4 , LaCuOSeE 6.62 1.00 1.49 0.44 CuLaO2 , La2O3 , La2(SeO3)3F 3.95 0.11 3.27 2.22 CuLaO2 , La2O3 , LaCu5G 5.67 0.36 2.28 1.24 CuLaO2 , Cu2Se, Cu3Se2H 4.60 0.00 3.00 1.95 CuLaO2 , Cu2SeI 6.77 0.91 1.46 0.42 CuLaO2 , Cu3Se2 , La2(SeO3)3J 4.52 0.00 3.04 1.99 CuLaO2 , LaCu5K 5.77 1.11 2.05 0.62 La2O3 , La2SeO2 , La4Se3O4L 3.23 0.26 3.75 2.32 La2O3 , La2SeO2 , LaCu5M 6.51 1.19 1.56 0.29 La2O3 , La2(SeO3)3 , La4Se3O4N 2.96 0.56 4.31 1.72 La3Se4 , LaCuSe2 , La2SeO2O 2.61 0.38 4.57 1.98 La3Se4 , LaCuSe2 , LaCu5P 2.54 0.40 4.58 2.04 La3Se4 , La2SeO2 , LaCu5Q 4.12 0.36 3.83 1.24 LaCuSe2 , Cu2Se, Cu3Se2R 3.60 0.18 4.18 1.59 LaCuSe2 , Cu2Se, LaCu5S 4.61 1.11 3.21 0.62 LaCuSe2 , La2SeO2 , La4Se3O4T 4.52 0.00 3.08 1.95 Cu2Se, LaCu5

well the power of our program in analyzing the chemical potentialranges.

We have calculated the enthalpy of formation of the compoundand its competing phases using DFT with the HSE06 [48] hybridfunctional. Our purpose here is to discuss the ranges of chemicalpotentials consistent with the growth of the material, which cansupport future studies of its defect and materials physics. Thecalculated enthalpies of formation are shown in Table 3. These,along with the stoichiometries of the compounds, form the inputto CPLAP.

The constraint on the chemical potentials is

La + Cu + O + Se = LaCuOSe = 9.55 eV. (22)We choose Se as the dependent variable. Running the program,we find that the system is thermodynamically stable. As there are29 limiting conditions on the independent variables, we do notlist them here. We find 20 intersection points in the 3D chemicalpotential space spanned by La, Cu, and O. They are presented,along with the corresponding values of Se and the relevantcompeting phases, in Table 4. The 3D stability region is shown inFig. 5. The relevant competing phases describe 2D surfaces in the3D space, which are shown using colors in Fig. 5 (we note that,because GNUPLOT cannot plot surfaces parallel to the z-axis, wemust represent such surfaces by placing a cross at their midpoint,as we do for the competing phase LaCu5).If we are interested in, for example, O-poor conditions, we canset O to a low value, say 4 eV (which is close to, but a littleommunications 185 (2014) 330338

Fig. 5. (Color online) Region of stability for LaCuOSe in the 3D space spannedbyLa ,Cu , andO . The thick (blue) lines indicate the boundary provided by the compoundof interest (LaCuOSe). The (colored) surfaces indicate the limits imposed by thecompeting phases and the compound of interest. Surfaces parallel to the z-axis arerepresented by a cross at their midpoint.

Fig. 6. (Color online) Region of stability for LaCuOSe in O-poor growth conditions(O = 4 eV). The chemical potential space is two dimensional, spanned by Laand Cu . The region is effectively a slice taken from the 3D stability region shownin Fig. 5. The (colored) lines indicate the limits imposed by the relevant competingphases and the compound of interest.

above, its minimum value of 4.58 eVsee Table 4). Doing thisreduces the dimensionality of the problem by one. The resultingstability region is a 2D slice taken from the 3D stability regionshown in Fig. 5. We present this 2D stability region in Fig. 6, alongwith the relevant competing phases, which describe lines in the 2Dchemical potential space.

Other sections of the stability region that may be of interestcan be extracted easily by setting chemical potentials to particularvalues. The visualization of the resulting regions can be furthermodified by changing the dependent variable. This demonstratesthe versatility of CPLAP in exploring the region of stability in thechemical potential space. To carry out these types of manipulation,in particular for a quaternary systemsuch as LaCuOSe, by hand canbe quite time consuming and error prone. Once the calculation isset up, the region of stability can be explored easily and accurately,with visualization possible when the system is two dimensionalor three dimensional. For quinternary (or higher-order) systems,one has to set a chemical potential to a particular value beforethe stability region can be visualized (in three dimensions). Theease with which one can systematically explore the stability

region using CPLAP will be of great benefit to the theoretical andcomputational study of systems consisting of four or more species.

CJ. Buckeridge et al. / Computer Physics

5. Conclusion

In summary, we have described a simple and effective algo-rithm to determine the thermodynamical stability and range ofchemical potentials consistent with the formation of a particu-lar compound of interest, in comparison with the formation ofcompeting phases and elemental forms of the constituent species.By assuming that the compound of interest forms in equilibrium,rather than competing phases and standard states, a set of condi-tions on the chemical potentials can be derived. These conditionscan be interpreted as defining a region bounded by hypersurfacesin an (n 1)-dimensional chemical potential space, where n is thenumber of species in the system. Determining this region of sta-bility gives the chemical potential landscape consistent with theproduction of the compound of interest. The algorithm works byreading in the energies of formation of the compound itself, andall competing phases, then constructing a matrix of linear equa-tions, solving all possible combinations of the equations, and find-ing which solutions (if any) obey the conditions on the chemicalpotentials. We have incorporated the algorithm in a FORTRAN pro-gram (CPLAP). Options are available to set a chemical potential to aparticular value, and to print a grid of pointswithin the stability re-gion. For 2D and 3D systems, files are produced to allow visualiza-tion of the results. We have demonstrated the effectiveness of theprogram using a ternary system and a quaternary system.We havealso demonstrated the flexibility with which the program may beused to explore a region of stability in the chemical potential space.

This program will be of benefit to the theoretical and compu-tational study of materials with three or more constituent species,particularly for the design of novel functional materials that arethermodynamically stable, and the generality of the present ap-proach has clear advantages.

Acknowledgments

The authors acknowledge funding from EPSRC grant EP/IO1330X/1. D.O.S. is grateful to the Ramsay memorial trust andUniversity College London for the provision of a Ramsay Fellow-ship. The authors also acknowledge the use of the UCL LegionHigh Performance Computing Facility (Legion@UCL) and associ-ated support services, the IRIDIS cluster provided by the EPSRCfunded Centre for Innovation (EP/K000144/1 and EP/K000136/1),and the HECToR supercomputer through membership of the UKsHPC Materials Chemistry Consortium, which is funded by EPSRCgrant EP/F067496. A.W. and D.O.S. acknowledge membership ofthe Materials Design Network. We would like to thank M.R. Far-row and A.A. Sokol for useful discussions.

Appendix. Sorting routine

In this appendix, we describe the sorting algorithm used to ex-tract all appropriate combinations from the set of linear equationsderived from the conditions on the chemical potentials (see Sec-tion 3). We assume that there are n unknowns, andm linear equa-tions, where m n. The aim of the sorting algorithm is to extractall combinations of n equations from the totalm (there will be mCncombinations).

The input to the routine is the matrixMij, which ism (n+ 1)dimensional. Each row corresponds to a linear equation; the firstn columns are the coefficients of the n unknowns, and the n+ 1thcolumn is the right-hand side of the linear equation. The outputfrom the routine will be the mCn matrices Sij, which are the n n-dimensional matrices of coefficients, and the vectors vi, which arethe n-dimensional corresponding vectors consisting of the right-

hand sides of the appropriate equations. S and v can then be used ina standard LU decomposition and back-substitution approach [38]ommunications 185 (2014) 330338 337

Fig. A.7. Flowchart of sorting routine.

to determine the unknowns (i.e. to find the intersection points ofthe linear equations, if they exist).

The routine works by creating an array ival, which is n dimen-sional. The elements in the array are initially set as the integers1, 2, . . . , n. The array is then used to construct S and v. By sequen-tially changing the arrangement of the elements of the array (andallowing the array elements to adopt values up tom), all mCn com-binations are extracted from thematrixM . The algorithm is shownin detail in Fig. A.7.

References

[1] A.Walsh, S. Chen, S.-H.Wei, X.-G. Gong, Adv. EnergyMater. 2 (2012) 400409.[2] D.M. Powell, M.T. Winkler, H.J. Choi, C.B. Simmons, D.B. Needleman, T.

Buonassisi, Energy Environ. Sci. 5 (2012) 58745883.[3] K. Ellmer, Nat. Photonics 6 (2012) 809812.[4] A.Walsh, A.B. Kehoe, D.J. Temple, G.W.Watson, D.O. Scanlon, Chem. Commun.

49 (2013) 448450.[5] A.Walsh, A.A. Sokol, C.R.A. Catlow (Eds.), Computational Approaches to Energy

Materials, John Wiley and sons, West Sussex, United Kingdom, ISBN: 978-1-119-95093-6, 2013.

[6] G. Trimarchi, H. Peng, J. Im, A.J. Freeman, V. Cloet, A. Raw, K.R. Poeppelmeier,K. Biswas, S. Lany, A. Zunger, Phys. Rev. B 84 (2011) 165116.

[7] D.O. Scanlon, G.W. Watson, J. Mater. Chem. 22 (2012) 2523625245.[8] D.O. Scanlon, A. Walsh, Appl. Phys. Lett. 100 (2012) 251911.[9] L.A. Burton, A. Walsh, J. Phys. Chem. C 116 (2012) 2426224267.

[10] L. Lahourcade, N.C. Coronel, K.T. Delaney, S.K. Shukla, N.A. Spaldin, H.A.Atwater, Adv. Mater. 25 (2013) 25622566. http://dx.doi.org/10.1002/adma.201204718.

[11] J.P. Bosco, S.B. Demers, G.M. Kimball, N.S. Lewis, H.A. Atwater, J. Appl. Phys. 112(2012) 093703.

[12] N. Feldberg, B. Keen, J.D. Aldous, D.O. Scanlon, P.A. Stampe, R.J. Kennedy, R.J.Reeves, T.D. Veal, S.M. Durbin, in: 2012 38th IEEE Photovoltaic SpecialistsConference, PVSC, pp. 002524002527.

[13] H.J. Kim, U.K.M. Kim, T.H. Kim, H.S. Mun, B.-G. Jeon, K.T. Hong, W.-J. Lee, C. Ju,K.H. Kim, K. Char, Appl. Phys. Express 5 (2012) 061102.

[14] D.O. Scanlon, P.D.C. King, R.P. Singh, A. de la Torre, S. McKeown Walker, G.Balakrishnan, F. Baumberger, C.R.A. Catlow, Adv. Mater. 24 (2012) 21542158.

[15] S. Chen, X.G. Gong, A. Walsh, S.-H. Wei, Appl. Phys. Lett. 94 (2009) 041903.[16] H. Hiramatsu, T. Kamiya, T. Tohei, E. Ikenaga, T. Mizoguchi, Y. Ikuhara, K.Kobayashi, H. Hosono, J. Am. Chem. Soc. 132 (2010) 1506015067.[17] A. Zakutayev, J. Tate, H.A.S. Platt, D.A. Keszler, C. Hein, T. Mayer, A. Klein, W.

Jaegermann, J. Appl. Phys. 107 (2010) 103713.

C338 J. Buckeridge et al. / Computer Physics

[18] D.C. Green, S. Glatzel, A.M. Collins, A.J. Patil, S.R. Hall, Adv. Mater. 24 (2012)57675772.

[19] D.O. Scanlon, G.W. Watson, Chem. Mater. 21 (2009) 54355442.[20] B.C. Melot, J.-M. Tarascon, Acc. Chem. Res. 46 (2013) 12261238. http://dx.doi.

org/10.1021/ar300088q.[21] R.Marom, S.F. Amalraj, N. Leifer, D. Jacob, D. Aurbach, J.Mater. Chem. 21 (2011)

99389954.[22] K.C. Wincewicz, J.S. Cooper, J. Power Sources 1 (2005) 280296.[23] C.R.A. Catlow, Z.X. Guo, M. Miskufova, S.A. Shevlin, A.G.H. Smith, A.A. Sokol,

A. Walsh, D.J. Wilson, S.M. Woodley, Phil. Trans. R. Soc. A 368 (2010)33793456.

[24] S.M. Woodley, C.R.A. Catlow, Nature Mater. 7 (2008) 937946.[25] S. Ping Ong, L. Wang, B. Kang, G. Ceder, Chem. Mater. 20 (2008)

17981807.[26] C. Persson, Y.-J. Zhao, S. Lany, A. Zunger, Phys. Rev. B 72 (2005) 035211.[27] C.G.V. de Walle, J. Neugebauer, J. Appl. Phys. 95 (2004) 3851.[28] D.O. Scanlon, G.W. Watson, J. Mater. Chem. 21 (2011) 36553663.[29] D.O. Scanlon, G.W. Watson, J. Phys. Chem. Lett. 1 (2010) 31953199.[30] G. Bergerhoff, I.D. Brown, in: F.H. Allen, et al. (Eds.), CrystallographicDatabases,

International Union of Crystallography, Chester, 1987 (Hrsg.).[31] S.B. Zhang, J.E. Northrup, Phys. Rev. Lett. 67 (1991) 23392342.[32] K. Reuter, M. Scheffler, Phys. Rev. B 65 (2001) 035406.[33] https://sourceforge.net/projects/cplap.[34] https://github.com/jbuckeridge/cplap.git.ommunications 185 (2014) 330338

[35] T.Williams, C. Kelley, et al. Gnuplot 4.4: an interactive plotting program, 2010.http://gnuplot.sourceforge.net.

[36] Mathematica, Version 8.0, Wolfram Research, Inc., Champaign, IL, USA, 2010.[37] C. Kittel, H. Kroemer, Thermal Physics, second ed., W.H. Freeman and

Company, 1980 (Chapter 9).[38] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes:

The Art of Scientific Computing, third ed., Cambridge University Press, NewYork, NY, USA, 2007.

[39] L. Kaufman, H. Bernstein, Computer Calculation of Phase Diagrams withSpecial Reference to Refractory Metals, Academic Press, New York, NY, USA,1970.

[40] N. Saunders, A. Miodownik (Eds.), CALPHAD (Calculation of Phase Diagrams):A Comprehensive Guide, Elsevier Science Inc., New York, NY, USA, 1998.

[41] H. Lukas, S.G. Fries, B. Sundman, Computational Thermodynamics, TheCalphadMethod, Cambridge University Press, New York, NY, USA, 2007.

[42] D.O. Scanlon, Phys. Rev. B 87 (2013) 161201.[43] M. Ernzerhof, G.E. Scuseria, J. Chem. Phys. 110 (1999) 50295036.[44] C. Adamo, V. Barone, J. Chem. Phys. 110 (1999) 61586170.[45] K. Ueda, H. Hosono, J. Appl. Phys. 91 (2002) 47684770.[46] H. Hiramatsu, H. Kamioka, K. Ueda, H. Ohta, T. Kamiya, M. Hirano, H. Hosono,

Phys. Status Solidi A 203 (2006) 28002811.[47] H. Hiramatsu, K. Ueda, H. Ohta, M. Hirano, M. Kikuchi, H. Yanagi, T. Kamiya, H.

Hosono, Appl. Phys. Lett. 91 (2007) 012104.[48] J. Heyd, G.E. Scuseria, M. Ernzerhof, J. Chem. Phys. 124 (2006) 219906.

Automated procedure to determine the thermodynamic stability of a material and the range of chemical potentials necessary for its formation relative to competing phases and compoundsIntroductionTheoryAlgorithmExamplesTernary systemQuaternary system

ConclusionAcknowledgmentsSorting routineReferences