PHYSICAL REVIEW B 15 AUGUST 2000-IVOLUME 62, NUMBER 7

Equilibrium and adhesion of NbÕsapphire: The effect of oxygen partial pressure

Iskander G. Batyrev,1,2 Ali Alavi, 1 and Michael W. Finnis11Atomistic Simulation Group, School of Mathematics and Physics, The Queen’s University of Belfast, Belfast BT7 1NN,

Northern Ireland, United Kingdom2The University of Texas at Austin, Department of Physics, Austin, Texas 78712-1081

~Received 14 January 2000!

We derive a formula, useful for first-principles calculations, which relates the free energy of an oxide/metalinterface to the free energies of surfaces and the work of separation of the interface. We distinguish the lattermechanicalquantity from thethermodynamicwork of adhesion, and we describe explicitly how both may becalculated. Our formulas for interfacial and surface energies are cast in terms of quantities which can becalculated or looked up in tables, and include as additional parameters the ambient temperature and partialpressure of oxygenPO2

. From total-energy calculations for the Nb(111)/a-Al2O3 ~0001! interface, free Nb andAl2O3 surfaces, we obtain firstly numerical estimates of the works of separation, which are independent ofPO2

.We then obtain surface energies, interfacial energies, and the equilibrium work of adhesion as a function ofPO2

.

elcihecao

bes.

tste

pins

iourd

fuly-teude

a

tlep

I

rly

lu-

t-ry

ffect

ingheofto

-hed-of

r-

then-

ofsedon

avet ofandthe

o-atasto

tedecialce

I. INTRODUCTION

Oxide-metal interfaces continue to be studied intensivbecause of the many ways in which they are of commerand scientific importance. Applications range from tnanoscale in microelectronics packaging to the macrosengineering of thermal barrier coatings or the formationprotective scales. The science of these interfaces hasaddressed in volumes of conference papers and review1–4

There are also reviews in the literature5,6 that specificallyaddress the theoretical questions about the nature ofbonding at these interfaces, such as what determines thepreference of metal atoms on the oxide surface; whetherbonding can be thought of as predominantly covalent or mtallic and how to quantify these concepts; whether a simclassical image model can be used to interpret the bondwhat is the strength of adhesion of metal to oxide. The bafor answering these questions is to have reliable calculatof the electronic structure and total energy of particular sfaces and interfaces. Such calculations came of age ovepast ten years or so with the use of first-principles methoThese mainly apply density-functional theory~DFT! and thelocal-density approximation~LDA !,7,8 which are the basis othe calculations we shall report here. Hartree-Fock calctions are also feasible and have been applied effectivelthe Ag/MgO interface,9 although they tend to be more expensive than DFT for larger systems. Since the reviews cithere have been numerous applications of DFT to stbonding in the initial stages of deposition of metal on oxidwith cluster or multilayer geometries, notably on MgO,10–21

but to a lesser extent on more complex oxides suchTiO2,22 MgAl2O4,23 anda-Al2O3.24–27

The bonding of Nb toa-Al2O3 has long been a subjecfor experimental work, because besides its relevance to etronic components it offers practical advantages for sampreparation: the two materials bond strongly28 ~anomalouslystrongly according to a recent study29!, do not react chemi-cally and have similar coefficients of thermal expansion.the orientation Nb~111!/Al 2O3 ~0001! there is a lattice mis-

PRB 620163-1829/2000/62~7!/4698~9!/$15.00

yal

lefen

heite

he-

leg;isnsr-thes.

a-to

d,y,

s

c-le

n

match of less than 2%, allowing the preparation of a neacoherent interface~using molecular beam epitaxy!, theatomic structure of which has been studied by high resotion transmission electron microscopy~HRTEM! and ana-lyzed in detail.30–34 This interface was the subject of firsprinciples calculations which used periodic boundaconditions, making the reasonable assumption that the eof misfit dislocation can be neglected.25–27Our recent work26

analyzed the nature of the bonding in detail by calculatMulliken populations and bond orders, concluding that tbonding across the interface is strongly ionic. The workseparationWsep of the interface was calculated, and foundbe very high: of order 10 J m22 when niobium was bonded tothe oxygen-terminated Al2O3 surface. Lower energy pathways for the cleavage of this interface would be within tNb metal or the oxide itself. Two other interfaces were stuied corresponding to the two other possible terminationsbulk Al2O3 ~0001!; namely the stoichiometric, aluminum temination ~one layer of aluminum! and the aluminum-richtermination ~two layers of aluminum!. HRTEM could notdistinguish between the stoichiometric termination andoxygen termination; however, evidence from electroenergy-loss spectroscopy35 ~EELS! favored the oxygen ter-mination.

We point out here that as far as we know the questionwhich termination is more stable has not yet been addresin all the theoretical work which has been published so farany oxide-metal interfaces. The structural predictions hbeen confined to the question of the relative displacementhe crystals, parallel and perpendicular to the interface,the local relaxations of atoms at the interface, as well asenergy needed to separate the crystalsWsep. This has beendone for interfaces with different terminations or local stichiometry; all calculations were carried out with atomsrest (T50 K! and minima in the total energy were locateda function of atomic positions. However, the question aswhether the oxygen-terminated or the aluminum-terminainterface is more stable was not discussed. There are spdifficulties associated with calculating the absolute interfa

4698 ©2000 The American Physical Society

an

ob

yemalth

atanciibn

esst

ilitrep-relcufo

herbo

mEsanith

om

en

nch.a

ot oenrieeuw

re

ycelllle-rgyer-

ell

theoaryhanro-

ll,ner-

fd in

of

ederur--tem,ionetheankxy-

r itsthewoin

uidhicer-but-

edweui--

PRB 62 4699EQUILIBRIUM AND ADHESION OF Nb/SAPPHIRE: . . .

energy for non-stoichiometric interfaces, a problem that wfirst addressed in a first-principles manner by Chetty aMartin.36 These authors dealt with a part of the general prlem, which is to formulate the total energy as alocal quan-tity; it can therefore be used in the definition of thermodnamic excesses. A second important part of the problwhich they did not treat, is to bring the chemical potentiof the components into the formalism. These aspects ofproblem were married in Ref. 26. It is now well known ththe question of which termination is stable can only beswered with respect to the chemical potentials of the spein the environment with which the interfaces are in equilrium, which is normally characterized by temperature apartial pressure of oxygen,37 and the difficulty of relating thequantities accessible to a first principles calculation to thparameters may have been a reason for leaving this queto one side.

The main purpose of our present paper is to show howfact we are already able to make predictions of the stabof different interfaces when they differ not only in structubut also in composition. With certain simplifying assumtions we show how this can now be done with little moeffort than the calculations which need to be done to calate the work of separation, and we present first resultsthe Nb/Al2O3 interface. The ingredients of the theory are tworks of adhesion and surface energies. For these we dupon the results reported briefly in Ref. 52 supplementedsome further calculations to discuss the case of oxygenthe Nb surface. The basic theory is outlined in Sec. II. Wderive the equations for a generalAmOn oxide in contactwith a metalB; it would be a short step to generalize thestill further to an interface between arbitrary compounds.sentially the same thermodynamics was applied by Wet al.38 in calculations of the surface energy of oxides wdifferent terminations, over a range of chemical potentialsoxygen; our theory makes the further connection to the teperature and in particular thepressureof oxygen, which arethe parameters directly under the control of the experimtalist. A detailed study of the Al2O3 surface is reportedelsewhere.39

The plan of the rest of the paper is as follows. SectioIII–V cover aspects of our total energy calculations whiwere not dealt with, or dealt with only briefly, in our paper52

In Sec. III our method of total-energy calculation is summrized. In Sec. IV we describe the atomic relaxationsparallelto the interfaces which are generally not commented upAlthough this structural aspect is not central to the thrusour paper, it turned out that lateral relaxations also havstrong part to play in determining the interplanar relaxatioand energies reported previously, and we therefore descthem for completeness. In Sec. V we describe and common the results for the work of separation on different planand with different terminations of the interface. Our calclated interfacial free energies are presented in Sec. VI andconclude in Sec. VII.

II. PRINCIPLES OF CALCULATING INTERFACIAL ANDSURFACE ENERGIES

Let us consider the interface between metalB and an ox-ide of metalA in equilibrium at temperature and pressu

sd-

-,

se

-es-d

eion

iny

-r

awyn

e

-g

f-

-

s

-

n.fasbents-

e

(T,P). The stoichiometric composition of theA oxide isAmOn . We obtain the definitions of interfacial quantities breferring to the contents of a periodically repeated superof areaS parallel to the interfaces which it may contain. Aextensive thermodynamic quantities in the following will rfer to the contents of such a supercell. The interfacial eneper unit area, counting the two interfaces within each supcell, is given by:40

g int5@Gint~T,P!2NAmA~T,P!2NOmO~T,P!

2NBmB~T,P!#/2S, ~1!

whereGint is the Gibbs energy of the contents of a superccontaining two interfaces,mA , mB , andmO are the chemicalpotentials of the three components, andNA , NB , andNO arethe numbers of atoms of the three components withinsupercell. The denominator 2S occurs because there are twinterfaces in the supercell, as required by periodic boundconditions. Chemical potentials here are per atom rather tper mole, which would be the usual convention for macscopic thermodynamics. Special cases of Eq.~1! are wheneither the metalB or the oxide is absent from the supercein which cases we recover expressions for the surface egies of the oxidegAO and the metalgB , respectively,

gAO5@GSAO~T,P!2NAmA~T,P!2NOmO~T,P!#/2S,~2!

gB5@GSB~T,P!2NBmB~T,P!#/2S. ~3!

The quantitiesGSAO and GSB are the Gibbs energies oslabs of oxide and of metal, with free surfaces separatetheir respective supercells by an adequate thicknessvacuum. We have assumed in Eq.~3! that the metal surfaceis clean; this will suffice for a calculation of the energy of thinterfacediscussed below. However, we can easily consifor example adding a monolayer of oxygen to the metal sface in the calculation ofGSB(T,P). Since there is no separate oxide phase, the number of oxygen atoms in the sysNO, now resides on the metal surface. The contribut2NOmO(T,P) must be subtracted in the calculation of thcorresponding surface energy just as in the calculation ofinterface energy. We shall in fact make this calculation of‘‘oxidized’’ Nb surface in the course of obtaining the worof separation of an interface by a pathway which leaves ogen on the exposed metal surface.

The motivation for calculatingg int is as follows. An in-terface between two crystals requires five parameters fomacroscopic specification, for example three to specifyrelative crystallographic orientation of the materials and tmore to specify the orientation of the interface. We notepassing that a free surface in contact with vapor or liqonly requires two parameters to specify its crystallograporientation. There is always a large set of hypothetical intfaces that have the same five macroscopic parameterswhich differ in their atomic structure and local stoichiometry. The member of this set which minimizesg int for givenchemical potentials is the equilibrium interface. So providwe know the chemical potentials of the components,could in principle predict the atomistic structure of the eqlibrium interface, including its local stoichiometry, by evalu

set,x

ten

rubowt

ke

iom

f

gh

erosafaFi

wooes

all

ole

nra

retoityaaalan-leideh

et-

d innr-toto

r

ect

ce

retonrcellno

theas

lorn-the

ndcalheofay.extifi-ichto-

ins

4700 PRB 62BATYREV, ALAVI, AND FINNIS

ating g int for each member of the set. In practice, of courwe can only calculateg int for a small subset of the entire seand rely on our experience and intuition, together with eperimental information, to ensure that we have not omitan important structure. Prior to the present work we aothers have calculated total energies for a number of sttures at 0 K, in which the atomic positions are relaxedenergy minimization. The equations to be derived belshow how to go the two important steps further, namelycorrect the 0 K information to finite temperature and to taaccount of local stoichiometry.

The relationship between local stoichiometry andg int iswell known in thermodynamics as the Gibbs adsorptequation, in which local stoichiometry is measured in terof excessesG i of one or more components labeledi. Ourfinal version of Eq.~1! will be in terms of the excess ooxygen at the interface with respect to the metalA, per unitsurface area, which is defined as

GO5S NO2n

mNAD Y 2S. ~4!

This choice of componenti is arbitrary; we could equallywell work in terms ofGAl , because they are related throu

mGO1nGAl50. ~5!

We note three further points in connection with excessFirstly, each further component in the system would intduce another excess, each excess being referred to thedesignated component. Secondly, a stoichiometric interis by definition one for which all the excesses vanish.nally, since one of the phases is the pure metalB, there canbe no excess of the metalB. In particular, ifB were in factalso Al, thereby reducing the number of components to tthe interface could not be described as having an excessor Al. For more discussion of the thermodynamics of excquantities the reader is referred to Ref. 40.

A difficulty up to now has been to calculate the chemicpotentials involved in these equations and, more specificato relate them to given experimental conditions. In the flowing we show how Eq.~1! can be reformulated to relatg int to the partial pressure of oxygenPO2

.

First we define thework of separation Wsep of the inter-face. It does not refer to chemical equilibrium states atherefore does not involve chemical potentials of the sepacomponents:

Wsep5~GSAO1GSB2Gint!/2S5gAO1gB2g int . ~6!

For brevity we do not explicitly indicate the temperatuand pressure dependence of all the terms unless it needsemphasized. An important point to note about this quantwhich makes it relatively straightforward to calculate, is ththe separate slabs of metal and oxide have exactly the scomposition as the two slabs which are joined to forminterface. This would not in general be the case if these sand the interface were in equilibrium with a given enviroment ~constantm i), because one would expect for exampsome loss or gain of surface oxygen or metal from the oxto the vapor phase when the surfaces are created. If thterface as well as the exposed surfaces are the ones whic

,

-ddc-y

o

ns

s.-me

ce-

,f Os

ly,-

dte

be,tmenbs

ein-are

in chemical equilibrium~which we denote by superfixeq),an equation similar to Eq.~6! defines thework of adhesion:

Wad5gAOeq 1gB

eq2g inteq . ~7!

This is the quantity of relevance to contact angles and wting for example, and unlikeWsep it is not obtainable by asimple comparison of three total energies.

Calculations ofWsep for Nb/Al2O3 were reported in ourpaper52 and these have been extended here, as describethe following sections.Wsep is probably more relevant thaWad in formulating a fracture criterion, when internal sufaces are formed which are not in equilibrium, but in orderpredict the equilibrium structure of interfaces we also needbe able to evaluate Eqs.~1!–~3!.

We now introduce the quantitygAO, the Gibbs energy peformula unit of bulkAmOn in equilibrium with metalA andoxygen in vapor form,

gAO~T,P!5mmA~T,P!1nmO~T,P!, ~8!

so that

GAO5~NA /m!gAO ~9!

is the Gibbs energy of a stoichiometric cell containingNAatoms ofA. Inserting Eqs.~8! and ~4! into Eq. ~2! gives thesurface energy of the oxide in a form which makes the effof the excess oxygen explicit,

gAO5~GSAO2GAO!/2S2GOmO. ~10!

Consider now how to go about calculating the two surfaenergies from Eqs.~3! and ~10!, which we will eventuallycombine withWsep in Eq. ~6! to give usg int . The Gibbsenergy of all slabs can be calculated atT50 K and P50from first principles; it is just the total energy. If the slabs abulk pure material, their Gibbs energy can be correctedtemperatureT by using experimental specific heat data. Othe other hand, when the slabs are separated in the supeby a layer of vacuum to represent free surfaces, there issuch experimental data and the correction to finiteT could bedone by calculating the phonon spectrum and usingquasiharmonic approximation for the free energy. This hbeen done previously for classical ionic models by Tayand co-workers,41 in order to obtain the temperature depedence of their surface energy, but we have not yet madeequivalent calculation with ourab initio code. For a metalslab ~Ag!, the quasiharmonic free energy based onab initiophonon frequencies was recently calculated by Xie aco-workers.42 In the case of the pure metal slab, the chemipotentialmB is the Gibbs energy per atom of a bulk slab. Tsurface energy ofB is therefore obtained from the resultstwo supercell total energy calculations in the standard wThe main present issue, which is less familiar in the contof total energy calculations, is how to calculate the signcant term due to the chemical potential of oxygen, whmust be included when the surface of the oxide is nonsichiometric (GOÞ0).

The chemical potential of oxygen is well describedterms of its partial pressurePO2

by the standard ideal gaexpression

n-

uayte

ityiromfo

T

phinyictia

iis

rg

ia

.as

xith

ion

tryized

rgydi-

to-10

i-itralga-tsor-ted

n-sure.de at

PRB 62 4701EQUILIBRIUM AND ADHESION OF Nb/SAPPHIRE: . . .

mO5mO0 1 1

2 kT log~PO2/P0!. ~11!

In Eq. ~11!, mO0 is the oxygen chemical potential in its sta

dard state~STP! at T05298.15 K,P051 at. Chemists wouldsetmO

0 to zero by definition, but we cannot do that since ozero of energy is already defined as the energy of separions and electrons atT50 K. On the other hand, the energof oxygen molecules is not something we want to calculasince there are well-known problems in using densfunctional theory for this system. Fortunately, we can ccumvent the problem by using a thermodynamic cycle. Frthe defining equation for the standard Gibbs energy ofmationDGAO

0 ,

gAO0 5mmA

01nmO0 1DGAO

0 , ~12!

we obtain the troublesome oxygen chemical potential at Sin terms ofgAO

0 andDGAO0 . The quantitiesgAO

0 andmA0 are

things we can calculate accurately, and we can look uDGAO

0 in tables of thermodynamic data. Once we have tvalue, it could be used in principle for calculations in amaterial containing oxygen. Since, however, the numervalue ofgAO

0 is strongly dependent on which pseudopotenis used,mO

0 obtained by this procedure should only be usedthe same pseudopotentials are used throughout, otherwmust be recalculated.

InsertingmO0 from Eq.~12! into Eq.~11! and Eq.~11! into

Eq. ~10! gives us our final expression for the surface eneof the oxide:

gAO5S GSAO~T,P!2NA

mgAO~T,P! D Y 2S

2GO~gAO0 2mmA

02DGAO0 !/n2GO

12 kT log~PO2

/P0!

~13!

from which we obtain the final expression for the interfacenergy by substituting Eq.~13! into Eq. ~6!,

g int5gB~T,P!2Wsep~T,P!

1FGSAO~T,P!2NA

mgAO~T,P!GY2S

2GO~gAO0 2mmA

02DGAO0 !/n

2GO12 kT log~PO2

/P0!. ~14!

The quantitiesgAO0 andmA

0 entering the third line of Eq.~14!are well described byT50 K quantities which we calculateIt can be verified that correcting them to standard state hnegligible effect on the surface energy.

The minimum physically meaningful value ofPO2, which

we denotePO2

min , is set by the condition that ifPO2<PO2

min the

oxide would spontaneously decompose into metal and ogen. Neglecting the small variation in solid energies wtemperature by comparison withDGAO

0 this condition is

log~PO2

min/P0!52

nkTDGAO

0 . ~15!

rted

,--

r-

P

s

allfe it

y

l

a

y-

Similarly, the maximum physically meaningful value ofPO2

is defined by the lowest standard Gibbs energy of formatof a metalB oxide DGBO

O ,

log~PO2

max/P0!52

n8kTDGBO

0 , ~16!

where the first oxide to form would have the stoichiomeBm8On8 . The thermodynamic data used here are summarin Table I.

III. METHOD OF TOTAL-ENERGY CALCULATION

For the interface calculations we use the total-eneplane-wave pseudopotential method based on Lanczosagonalization of the Kohn-Sham density matrix.43 The super-cell has the form of a rhombohedral prism and in the sichiometric slab it contains 45 atoms: 14 Al, 21 O, andNb atoms~see Fig. 1!. By stripping off the outer plane of Alfrom each interface we obtain an interface which is O termnated with an O excessGOS511.5 atoms per surface uncell. By adding the surface plane of Al atoms to the neutinterface we make an oxygen poor interface, with the netive O excessGOS521.5. The total energy of the contenof a supercell is minimized with respect to the atomic codinates by the quasi-Newton method with Hessian updausing the Broyden-Fletcher-Goldfarb-Shano~BFGS!

TABLE I. Thermodynamic data used for calculating the depedence of surface and interfacial energies on oxygen partial presThe pressure represents the dissociation pressure of the oxi1500 K. P0 is one standard atmosphere.

a-Al2O3 NbO

DG0 ~kJ/mol!a 21582.3 2391.9log10(PO2

/P0) 236.8 227.3

aReference 51.

FIG. 1. Side view of the Nb(111)/Al2O3(0001) interface, show-ing labeling of the layers.

r--

veve

ofa

es

-o

threonchnt

Th

o

nelhaaxu

eae

ionou

s,

ur

pe

tieO

rsne

er-Al

talay

inl-onsnaraseAl

edge.

ro-is

-fsub-

4702 PRB 62BATYREV, ALAVI, AND FINNIS

method. The pseudopotential for Nb was of TroullieMartins form,44 with s andd nonlocality. The pseudopotential for O was of optimized form,45 with an s nonlocality.The pseudopotential for Al was of Gonze type46 with s non-locality.

All calculations were made with twok points in the irre-ducible wedge of the Brillouin zone, and with a plane-wacutoff of 40 Ry. The effect of increasing the plane-wacutoff from 40 to 60 Ry was to reduceWsep by 3.3% for theNb/Al interface, which we take as a satisfactory indicationthe basis-set convergence. For the neutral 45-atom interwe have made test calculations with sixk points which re-sults in a decrease of total energy by about 4 mRy and vsmall (,1023 nm! changes of relaxed positions of atomcompared with twok-point calculations. The effect of increasing thek-point sampling from two points to nine is tchangeWsep by less than 1%.

By doubling the original unit cell in thex-y plane weobtained a 180 atom cell, with which we recalculatedwave functions at the gamma point with the previouslylaxed atomic coordinates. The gamma point wave functiin this cell were used for Mulliken population analysis whiwas made by projecting the optimized wave functions othe pseudoatomic orbitalsuf ia& ~i labels site,a labels orbit-als! according to the procedure suggested in Ref. 47.‘‘spillage’’ of each occupied orbitalc was less than 1.5%.

IV. RELAXATION OF THE INTERFACE

The slab with which the Nb~111!/Al 2O3 ~0001! interfacewas modeled is shown in Fig. 1. The interlayer relaxationthe interface has been reported previously,52 and we refer tothat paper for results. Here we mention a feature that haspreviously been discussed, namely the relaxations parallthe interface, which we refer to as in-plane relaxations. Itbeen found that to make a calculation of the interlayer relation of the alumina surface one needs to take into accothe in-plane relaxations of the oxygen atoms,26 which wereneglected in some earlier work.25 The present results showthat in-plane relaxation of the oxygen ions is a general fture of the structure near the interface. The geometry of threlaxations is described by two parameters; the rotationaand the bond length extensiondr of the equilateral triangleof oxygen atoms in a plane, illustrated in Fig. 2. The rotatand dilation of this equilateral triangle does not break asymmetry, preserving for example the threefold axis abthe center of the triangle concerned.

The calculated values ofa anddr are shown in Table IIfor the terminating and second layer oxygen planes~denotedby subscripts 1 and 2! in five cases. The first two caselabeled N(b)/A(O) and N(b)/A(Al) are the O- and Al-terminated bulk Nb/Al2O3 interfaces. Cases three and foare pure alumina surfaces, labeledA(O) andA(Al) to indi-cate that they are oxygen and aluminum terminated, restively. Case five, labeledN(m)/A(O) is a Nb monolayer onan oxygen-terminated alumina surface. One can see thain-plane relaxation is a feature of all the systems studFrom the evidence of the first two layers, the rotation oftriangles and the increase of the O-O bond lengths appeabe localized near the surface of the alumina. The Nb molayer on the O-terminated surface of alumina shows this

fce

ry

e-s

o

e

f

ottos-nt

-se

nyt

c-

thed.

too-f-

fect most strongly, which is quite surprising, since the intplanar relaxation in this case is much less than that of thelayer for which the Nb substitutes. There is experimenconfirmation of the effect, obtained by small-angle x-rdiffraction,48 in the case of theA(Al) surface. In this case theexperimental results aredr 154.5% anda156.7°, comparedwith our calculated results ofdr 153.2% anda153.1°. Theagreement is only qualitative.

V. WORK OF SEPARATION

All our results for the calculated work of separationWsepof different interfaces and cleavage planes are shownTable III. The column of ‘‘unrelaxed’’ results refers to vaues obtained by assuming bulk unrelaxed atomic positiboth at the interface and for the free surfaces. The interplaspacing between O and Nb across the interface in this cwas simply taken as the bulk spacing between O andplanes. The results in the ‘‘relaxed’’ column are calculatwith atomic positions relaxed both before and after cleava

FIG. 2. Plane view of the~0001! surface of neutral aluminashowing the lateral relaxation within the topmost O plane. Thetation and expansion of the O triangle below the surface Al atomindicated by arrows.

TABLE II. In-plane relaxation of O- and Al-terminated interfaces and surfaces of alumina.a anddr correspond to the angle orotation and bond-length increase of O triangles; surface andsurface layers are indicated by the subscripts.

N(b)/A(0) N(b)/A(Al) A(O) A(Al) N(m)/A(O)

a1 4.1 4.2 4.1 3.1 8.4a2 0.1 0.2 0.5 0.2 0.4dr 1 4.2 4.3 4.3 3.2 8.4dr 2 0.2 0.2 0.2 0.5 0.6

eenou

, a

d

wn

efthf

i-alb

isnt

cegenc.ee

re

ce,area-enof

oflue

sosi-enthe

Alenthe

K;

eio--lop-

esility

-

pa-

ing

c-

l.

as

ial

cerre-

PRB 62 4703EQUILIBRIUM AND ADHESION OF Nb/SAPPHIRE: . . .

The effect of relaxations onWsepnaturally depends on thinterface. It is most pronounced when an Al-terminatAl2O3 surface is exposed, because of the large relaxatiothis surface, which lowers its surface energy by ab1.5 J m22. On the other hand, if the relaxation of theinter-facedominates the energy balance in Eq.~6! then the relaxedvalue ofWsep can even be larger than the unrelaxed valuein the case of cleavage between Nb and O at theN(b)/A(O)interface.

The lowest value ofWsep, 2.7 J m22, is found for thecleavage of bulk Nb from the stoichiometric Al-terminatealumina. The highest values are found for the cleavagebulk Nb from the O-terminated alumina surface. Indeedcan deduce from Table III that this interface would be ulikely to separate between Nb and O planes, but would prto separate inside the Nb, leaving a monolayer of Nb onsurface, or even between O and Al, leaving a monolayer oon the Nb surface.

The highest value~relaxed! of all in Table I is for thecleavage of pure Al2O3 between O and Al planes. Expermentally,a-Al2O3 does not cleave on the basal plane atbut its lowest-energy cleavage on this plane would clearlybetween Al planes. This is what one expects on the bascharge neutrality arguments, because by cleaving betweeplanes two identical, neutral surfaces are created. Onother hand, by cleaving between O and Al, different surfaare created which, in order to be neutral, require the oxyor aluminum at the surface to be in an unfavorable valestate, hence this is a final state of especially high energy

The above interpretations are supported by the Mullikpopulations shown in Table IV for the three interfaces btween bulk Nb and Al2O3. We make the usual caveat he

TABLE III. Wsep~in J/m2) for both unrelaxed and relaxed strutures.

Interface Cleavage plane Unrel. Re

N(b)/N(b) . . . Nb-Nb-Nb / Nb-Nb-Nb . . . 4.9 4.2A(Al)/ A(Al) . . . Al-O-Al / Al-O-Al . . . 7.0 3.9A(Al)/ A(O) . . . Al-Al-O / Al-Al-O . . . 13.3 12.7N(m)/A(O) Nb / O-Al-Al . . . 10.9 10.8N(b)/A(O) . . . Nb-Nb-Nb / O-Al-Al-O . . . 9.3 9.8N(b)/A(Al) . . . Nb-Nb-Nb / Al-O-Al-Al . . . 4.2 2.7N(b)/Nb-A(Al) . . . Nb-Nb-Nb / Nb-O-Al-Al . . . 4.0 3.8N(b)/Al-A(Al) . . . Nb-Nb-Nb / Al-Al-O . . . 4.4 2.8N(b)-O/Al-A(Al) . . . Nb-Nb-O / Al-Al-O . . . 7.3 4.9

TABLE IV. Mulliken charges for atoms near the interface asfunction of the excess of O. The notation for atomic planes is aFig. 1.

GOS 1.5 0 21.5

Nb2 0.27 0.37 20.13Nb1/Al1 0.77 0.73 0.36

O1 20.86 20.93 missingAl2 1.49 1.50 0.15Al3 1.52 1.51 0.95O2 21.00 21.00 20.99

doft

s

ofe-ereO

l,eofAlhesne

n-

that Mulliken charges do not have absolute significansince they depend on the choice of basis set, but theynevertheless a useful indicator of trends in ionicity or covlency. The interfaces in Table III are labeled by their oxygexcesses, to highlight certain trends with the stoichiometrythe interface. Bulk oxygen carries a Mulliken charge21, and for the oxygen plane nearest the interface this vais reduced to20.99,20.93, and20.86 in turn as the excesof oxygen at the interface is increased from negative to ptive. The change is rather modest, indicating that oxygdoes not readily alter its valence state. The charge oninterfacial oxygen is provided by the terminating layer ofin the case of Al termination, or in the case of the oxygterminated interface the electrons are provided mainly byfirst two layers of Nb.

VI. INTERFACIAL FREE ENERGY AND OXYGENPRESSURE

Five surface energies are shown in Fig. 3~a! as a functionof PO2

. Thex axis is appropriate to a temperature of 1500to obtain the results at temperatureT the numbers on thexaxis should be multiplied by 1500/T. The O-rich (GOS51.5) and O-poor (GO S521.5) alumina surfaces havnegative and positive slopes, respectively, while the stoichmetric Al2O3 and pure Nb~111! surface energies are constant, and by chance nearly equal. The most negatively sing surface energy we have plotted here refers to the Nb~111!surface with an attached monolayer of oxygen. It becomnegative at an oxygen pressure inside the regime of stabof NbO.

The interfacial free energy from Eq.~1! is shown as afunction of PO2

in Fig. 3~c! for three interfaces, O rich, sto

ichiometric, and O-poor (GOS51.5,0,21.5). To generate theinterfacial free energies one has to subtract the work of seration shown for convenience in Fig. 3~b!, from the sum ofthe equilibrium surface energies of the two correspond

in

FIG. 3. ~a! Surface energies as a function of oxygen partpressure at 1500 K.~b! Works of separationWsep ~see Table III!.~c! Interfacial energies.~d! Work of adhesionWad , obtained bysubtracting equilibrium interfacial energy from equilibrium surfaenergies. The region to the left between the vertical lines cosponds to the possible equilibrium states of the interface.

blin

ceinfigo

nd

en.

oo

ge

ane

thAc

yicabth

th

m

anin

oglvnetoinrnioich-

otur

nio

o-e

era-ebut

tial;f allure.by

ecta

rk.allwesent

-hemo-pen-andDA

Al-

toweatble

dedichoneri-

-dies

ble

he

enhe

face

heee is

owAl-tthel-

LSsting

4704 PRB 62BATYREV, ALAVI, AND FINNIS

free surfaces. With increasingPO2the O-rich interface be-

comes increasingly stable, the Al-rich interface less staand the free energy of the stoichiometric interface remaconstant, exactly parallel to the behavior of the free surfaThe interfaces can only be in thermodynamic equilibriumthe range of oxygen pressure which is indicated on theure. At 1500 K this range is as given in Table I; at valuesPO2

above this range, the Nb would oxidize to NbO, abelow it the alumina would decompose.

The work of adhesionat a givenPO2can also be esti-

mated from the results on this graph using Eq.~7!. The freesurface and interface energies should be those with lowfree energy at the given oxygen pressure, and these caread off from Figs. 3~a! and 3~c!. The result is plotted in Fig3~d!.

VII. DISCUSSION AND CONCLUSIONS

We have made a careful distinction between a workseparation, a mechanically defined quantity, and a workadhesion, a thermodynamic quantity, focusing on how toabout calculating these quantities within an atomistic modWe particularly consider the interface between a metalan oxide, since it is of practical importance and since oxygis a troublesome component for which to calculatechemical potential, a key quantity in interfacial energies.useful practical equation for the free energy of an interfainvolving oxygen has been derived, namely Eq.~7!, whichgets around the previous difficulty by using a thermodnamic cycle to express the result in terms of quantities whcan be readily calculated, namely the total energies of sland quantities which can be obtained from tables, namelystandard Gibbs energy of formation of the oxide andGibbs energies of the bulk materials relative to theirT50 Kvalues.

To illustrate and apply the method we have made a nuber of first-principles calculations for Nb~111!/Al 2O3 ~0001!interfaces, oxygen rich, oxygen poor, and stoichiometric,for several surfaces. We fully relax the atomic positionssupercells using a plane wave, pseudopotential methodolThe relaxations are significant, and in all cases they invoin-plane as well as interlayer relaxations of the oxygen ioResults on the work of separation of these interfaces wgiven in a paper52 recently, and we have extended theminclude the possibility of a cleavage of the O-terminatedterface which leaves the Nb coated with oxygen. This tuout indeed to be a lower-energy mode of separat(4.9 J m22) than the alternative which leaves an oxygen-rAl2O3 surface behind (9.8 J m22), because the favorable degree of ionicity of oxygen is thereby preserved as it is in bbulk alumina and its stoichiometric surface. Considering fther the strongly bound O-terminated Nb/Al2O3 interface, itturns out that the hypothetical processes of~i! cleavagewithin bulk Nb (4.2 J m22), or ~ii ! leaving a monolayer ofNb on the oxide surface (3.8 J m22), or even~iii ! cleavagewithin bulk Al2O3 (3.9 J m22) are all marginally of lowerenergy than the cleavage which takes oxygen with thebium.

By combining the results of our calculations with thermdynamic data we obtain surface energies and interfacial

e,ss.

-f

stbe

ff

ol.dne

e

-hs,e

e

-

d

y.es.re

-sn

h-

-

n-

ergies as a function of oxygen partial pressure and tempture. An approximation we make here is in omitting thtemperature dependence of the solid-state free energy,we include thekT log(PO2

/P0) term which describes the

temperature dependence of the oxygen chemical potenthis is also the term which describes the dependence othe interfacial and surface free energies on oxygen pressIt is clear how a more accurate calculation could be madeimplementing the quasiharmonic approximation to corrsolid surface free energies, and it will probably becomeroutine matter to include such a correction in future woAnother approximation is made by considering only a smset of possible interface and surface compositions whichthink are representative. Nevertheless, despite the presimplifications, some clear results have emerged.

Of the free surfaces of Al2O3, the stoichiometric one, terminated by a single layer of Al, is the most stable over twhole range of oxygen partial pressure up to over one atsphere. It may be that a treatment of the temperature dedence of the energy of the slabs could modify the upperlower bounds on pressure somewhat. Correction of the Lerror is also likely to lower surface energies by 10–20 %@I.G. Batyrev,~Ref. 49!#. For example, work of Sims50 withclassical potentials indicates that the surface energies ofterminated Al2O3 can be reduced by up to 0.2–0.3 J m22 at1500–2000 K. We expect an oxygen-terminated surfacebe stable at a pressure not too far from atmospheric, butcannot unfortunately be more quantitative in the predictionthis stage. At very low oxygen pressures it is also reasonathat the experimentally observed Al-rich (A313A31) struc-ture is stable; we cannot model a supercell of the size neeto calculate this. Instead we modeled a much simpler Al-rinterface, which is predicted to become the most stablejust abovePO2

min where Al2O3 decomposes. Since the expe

mental A313A31 is a very Al-rich surface (GOS527.5 in the present notation!, the slope of its surface energy versus log10(PO2

/P0) is correspondingly very steep anpositive, and it must intersect all the other surface energjust abovePO2

min .

The Nb free surface should obviously become unstawith respect to some adsorption of oxygen whenPO2

.PO2

max, the pressure at which NbO begins to form. T

particular configuration and concentration of an oxygmonolayer which we have calculated is not likely to be toptimum configuration of the first oxygen-covered Nb~111!surface, but it does become more stable than the free surat pressures somewhat abovePO2

max @Fig. 3~a!#.

A significant new result is the theoretical analysis of tthermodynamicstability of the O-terminated interface, thstrong bonding of which we discussed above. No interfacthermodynamically stable above the~very low! oxygen pres-sure at which NbO forms, but over most of the range belthis the O-terminated interface is less stable than theterminated one@Fig. 3~c!#, despite its strong bonding. In facat the very lowest pressure of oxygen, as would pertain inpresence of pure aluminum, our prediction is of an Aenriched interface. The experimental indications from EE~Ref. 35! show no evidence for Al-Nb bonding, and suggerather the existence of the O-terminated interface. Accord

y-f-

h

e

roba

ba-ocpnsth

ulnre

rib-to

nates ofheg anse

a-ingNo.an-andhel-

PRB 62 4705EQUILIBRIUM AND ADHESION OF Nb/SAPPHIRE: . . .

to our analysis this could only be marginally in thermodnamic equilibrium if the oxygen pressure is being ‘‘bufered’’ by Nb/NbO and lies close toPO2

max, which does not

seem unreasonable.We have not included in our comparison interfaces wit

different macroscopic orientation such as the Nb~110!/Al 2O3~0001! interface.30 Although this interface is believed to bthermodynamically more stable than the Nb~111!/Al 2O3~0001! interface, the kinetic barrier to changing the macscopic orientation is presumably much greater than theriers to changing the local interface structure.

The present work shows how vibrational entropy mayincluded with nonstoichiometry in the calculation of interfcial free energy. The formalism included the contributionpoint defects if these are regularly arranged at the interfasuch as the oxygen vacancies treated here. Another asarises if the nonstoichiometry is in the form of a high cocentration of randomly distributed point defects. In this caan ensemble of interfaces would have to be calculated. Ifconcentration of defects is low, on the other hand, it wobe adequate to include their individual formation energy aconfigurational entropy as additional terms in the above fenergy.

a-

a,

h,

vo

ta

r.

ch

ys

a

.

a

-r-

e

fe,ect-ee

dde

Although the formalism has been developed for descing metal-oxide bonding, there are obvious applicationssystems in which water or other substances may contamisurfaces or interfaces. The comparison of the energeticinterfaces with differing amounts of segregation follows tsame lines. The application of the present formalism usinthermodynamic cycle to avoid the most difficult calculatiomay be fruitful in other situations in the field of interfacchemistry.

ACKNOWLEDGMENTS

We thank J. Hutter for technical help with the calcultions. This work has been supported by the UK Engineerand Physical Sciences Research Council under GrantsGR/L08380 and No. GR/M01753, and by the EuropeCommunities HCM Network ‘‘Electronic Structure Calculations of Materials Properties and Processes for IndustryBasic Science’’ under Grant No. ERBFMRXCT980178. TCenter for Supercomputing in Ireland is gratefully acknowedged for computer resources.

.

er-

.

hys.

ev.

.

A.

e-t.

1Metal-Ceramic Interfaces, edited by M. Ru¨hle, A. G. Evans, M.F. Ashby, and J. P. Hirth~Pergamon Press, Oxford, 1990!.

2M. Ruhle, A. H. Heuer, A. G. Evans, and M. F. Ashby, Interntional Symposium on Metal-Ceramic Interfaces@Acta Metall.Mater.40 ~1992!#.

3J. M. Howe, Int. Mater. Rev.38, 233 ~1993!.4J. M. Howe, Int. Mater. Rev.38, 257 ~1993!.5C. Noguera and G. Bordier, J. Phys. III4, 1851~1994!.6M. W. Finnis, J. Phys.: Condens. Matter8, 5811~1996!.7P. Hohenberg and W. Kohn, Phys. Rev.136, B864 ~1964!.8W. Kohn and L.-J. Sham, Phys. Rev.140, A1133 ~1965!.9E. Heifets, Y. F. Zhukovskii, E. A. Kotomin, and M. Caus

Chem. Phys. Lett.283, 395 ~1998!.10J. Goniakowski, Phys. Rev. B59, 11 047~1999!.11J. Goniakowski and C. Noguera, Phys. Rev. B60, 16 120~1999!.12A. V. Matveev, K. M. Neyman, G. Pacchioni, and N. Rosc

Chem. Phys. Lett.299, 603 ~1999!.13V. Musolino, A. DalCorso, and A. Selloni, Phys. Rev. Lett.83,

2761 ~1999!.14K. M. Neyman, S. Vent, G. Pacchioni, and N. Rosch, Nuo

Cimento D19, 1743~1997!.15K. M. Neyman, S. Vent, N. Rosch, and G. Pacchioni, Top. Ca

9, 153 ~1999!.16G. Pacchioni and N. Rosch, J. Chem. Phys.104, 7329~1996!.17I. Tanaka, M. Mizuno, S. Nakajyo, and H. Adachi, Acta Mate

46, 6511~1998!.18I. V. Yudanov, S. Vent, K. Neyman, G. Pacchioni, and N. Ros

Chem. Phys. Lett.275, 245 ~1997!.19I. Yudanov, G. Pacchioni, K. Neyman, and N. Rosch, J. Ph

Chem. B101, 2786~1997!.20R. Benedek, D. N. Seidman, and L. H. Yang, Microsc. Microan

3, 333 ~1997!.21R. Benedek, D. N. Seidman, M. Minkoff, L. H. Yang, and A

Alavi, Phys. Rev. B60, 16 094~1999!.

l.

,

.

l.

22L. Thien-Nga and A. T. Paxton, Phys. Rev. B58, 13 233~1998!.23S. Kostlmeier, C. Elsa¨sser, B. Meyer, and M. W. Finnis, Phys

Status Solidi A166, 417 ~1998!.24G. L. Zhao, J. R. Smith, J. Raynolds, and D. J. Srolovitz, Int

face Sci.3, 289 ~1996!.25C. Kruse, M. W. Finnis, J. S. Lin, M. C. Payne, V. Y. Milman, A

De Vita, and M. J. Gillan, Philos. Mag. Lett.73, 377 ~1996!.26M. W. Finnis, Phys. Status Solidi A166, 397 ~1998!.27C. Verdozzi, D. R. Jennison, P. A. Schultz, and M. P. Sears, P

Rev. Lett.82, 799 ~1999!.28D. Korn, G. Elssner, H. F. Fischmeister, and M. Ru¨hle, Acta

Metall. Mater.40, S355~1992!.29G. Song, A. Remhof, K. Theis-Brohl, and H. Zabel, Phys. R

Lett. 79, 5062~1997!.30J. Mayer, C. P. Flynn, and M. Ru¨hle, Ultramicroscopy33, 51

~1990!.31J. Mayer, J. A. Dura, C. P. Flynn, and M. Ru¨hle, Surf. Coat.

Technol.43Õ44, 199 ~1990!.32G. Gutekunst, J. Mayer, and M. Ru¨hle, Scr. Metall. Mater.31,

1097 ~1994!.33G. Gutekunst, J. Mayer, and M. Ru¨hle, Philos. Mag. A75, 1329

~1997!.34G. Gutekunst, J. Mayer, V. Vitek, and M. Ru¨hle, Philos. Mag. A

75, 1357~1997!.35J. Bruley, R. Brydson, H. Mu¨llejans, J. Mayer, G. Gutekunst, W

Mader, D. Knauss, and M. Ru¨hle, J. Mater. Res.9, 2574~1994!.36N. Chetty and R. M. Martin, Phys. Rev. B45, 6074~1992!.37M. L. Shalz, B. J. Dalgleish, A. P. Tomsia, R. M. Cannon, and

M. Glaeser, J. Mater. Sci.29, 3678~1994!.38X. G. Wang, W. Weiss, S. K. Shaikhutdinov, M. Ritter, M. P

tersen, F. Wagner, R. Schlo¨gl, and M. Scheffler, Phys. Rev. Let81, 1038~1998!.

39I. G. Batyrev, A. Alavi, and M. W. Finnis, Faraday Discuss.114,33 ~2000!.

,

.

ev

v.

.

om

rf.

n

o-rd.y,

4706 PRB 62BATYREV, ALAVI, AND FINNIS

40J. W. Cahn, inInterfacial Segregation, edited by W. C. Johnsonand J. M. Blakely~American Society for Metals, Metals Park1977!, pp. 3–23.

41M. B. Taylor, C. E. Sims, G. D. Barrera, N. L. Allan, and W. CMackrodt, Phys. Rev. B59, 6742~1999!.

42J. J. Xie, S. de Gironcoli, S. Baroni, and M. Scheffler, Phys. RB 59, 970 ~1999!.

43A. Alavi, J. Kohanoff, M. Parrinello, and D. Frenkel, Phys. ReLett. 73, 2599~1994!.

44N. Troullier and J. L. Martins, Phys. Rev. B43, 1993~1991!.45M.–H. Lee, Ph.D. thesis, The University of Cambridge, 199146X. Gonze, R. Stumpf, and M. Scheffler, Phys. Rev. B44, 8503

~1991!.47D. Sanchez-Portal, E. Artacho, and J. M. Soler, Solid State C

.

-

mun.95, 685 ~1995!.48P. Guenard, G. Renaud, A. Barbier, and M. Gautier-Soyer, Su

Rev. Lett.5, 321 ~1998!.49I.G. Batyrev~unpublished!.50C. E. Sims, Ph.D. thesis, University of Bristol, 1999; N. L. Alla

and W. C. Mackrodt~private communication!.51H. Y. Affeefy, J. F. Liebman, and S. E. Stein, ‘‘Neutal Therm

chemical Data’’ inNIST Chemistry WebBook, NIST StandaReference Database Number 69, edited by W. G. Mallard and PJ. Linstrom ~National Institute of Standards and TechnologGaithersburg, MD! ~http://webbook.nist.gov!.

52I. G. Batyrev, A. Alavi, and M. W. Finnis, Phys. Rev. Lett.82,1510 ~1999!.