Theoretical Structure Determination of a Complex Material: κ-Al2O3

Theoretical Structure Determination of a Complex Material: k-Al2O3

Yashar Yourdshahyan,† Carlo Ruberto,† Mats Halvarsson,‡ Lennart Bengtsson,†Vratislav Langer,§ and Bengt I. Lundqvist†

Departments of Applied Physics, Experimental Physics, and Environmental Inorganic Chemistry,Chalmers University of Technology and Goteborg University, SE-412 96 Goteborg, Sweden

Sakari Ruppi

Seco Tools AB, SE-737 82 Fagersta, Sweden

Ulf Rolander

Sandvik Coromant AB, SE-126 80 Stockholm, Sweden

The atomic and electronic structures ofk-Al2O3 are deter-mined using theoretical first-principles techniques based ondensity-functional theory (DFT), plane waves, and pseudo-potentials. The obtained structure is confirmed by analysisof powder X-ray diffraction data. The structure is ortho-rhombic with oxygen ions in close-packed ABAC stackingand aluminum ions occupying both tetrahedral (1/4) andoctahedral (3/4) interstitial sites. A growth model forchemical vapor deposition ofk-Al2O3 is proposed based onthe atomic structure. Calculated electronic structure andcharge density yield a band gap of 5.3 eV and a high ioniccharacter of the bonds. The study shows the applicability ofDFT-based methods to complex and metastable materials.

I. Introduction

PREDICTING and understanding the properties of technologi-cally important materials, even of as-yet unsynthesized

ones, by theoretical means is a valuable complement to tradi-

tional empirical methods. In many cases, money and time canbe saved by conducting theoretical simulations on a materialbefore testing it in the laboratory. Many questions might bevery difficult to study by empirical methods alone because ofpractical problems involved with the experimental details.

In recent years, incredibly rapid advancements have beenmade in the field of computational materials physics. It is be-coming possible to apply the laws of quantum mechanics to thestudy of macroscopic properties of real materials from theatomic level, using the computer to perform the complex cal-culations involved. This results from improving theoreticalmethods, more efficient computational algorithms, and rapidlygrowing computer power, including parallelization.

In particular, first-principles methods, which do not rely onmodel interatomic potentials and empirically fitted parameters,are gaining more and more applicability. In many cases, thecomplexity of the atomic interactions in a system is such thatmodel potentials can prove insufficient in accuracy to yielduseful results, such as when studying the small energy differ-ences connected with the different structural modifications of acomplex crystal.

Alumina (Al2O3) is a material of high technological signifi-cance because of its hardness, abrasion resistance, mechanicalstrength, corrosion resistance, and good electrical insulation.Apart from the stablea phase, it manifests itself in a variety ofmetastable structures, such asg, h, u, k, andx aluminas. Thek phase can be produced by heat treatments of hydrated alu-minas1,2 but is technologically most often produced by chemi-cal vapor deposition (CVD). The most important application ofk-Al2O3 is as wear-resistant coatings on cemented-carbide cut-ting tools. The advantages of thek phase overa-Al2O3 are itssmaller grain size, lower pore density, and epitaxial growth

R. H. French—contributing editor

Manuscript No. 189886. Received September 9, 1998; approved March 18, 1999.Theoretical work supported by the Swedish Foundation for Strategic Research

(SFS) via Materials Consortium No. 9. Experimental work supported by the SwedishNational Board for Industrial and Technical Development (NUTEK) and the SwedishResearch Council for Engineering Sciences (TFR).

†Department of Applied Physics.‡Department of Experimental Physics.§Department of Environmental Inorganic Chemistry.

J. Am. Ceram. Soc., 82 [6] 1365–80 (1999)Journal

1365

when produced using CVD.3,4 Although metastable,k-Al2O3appears to be very stable, maintaining its structural identity upto temperatures of∼1200 K, where it transforms toa-Al2O3.5–8

The atomic structure ofk-Al2O3 has been studied experi-mentally for a long time, but there have been many problemsand controversies on the subject over the years. Theoretically,challenges are provided by the complexity of the interatomicinteractions in Al2O3 (Ref. 9) and the relatively large unit cellof k-Al2O3.

The purpose of this study is twofold: on one hand, we wishto resolve the uncertainties regarding the structure ofk-Al2O3,thus opening the way for further investigations on the material;on the other hand, we wish to demonstrate that the predictivepower of first-principles methods based on density-functionaltheory (DFT) is adequate for handling complex materials.

Starting with the experimental information known at thebeginning of this study, we use a first-principles method basedon DFT, pseudopotentials, and plane waves to investigate theatomic structure ofk-Al2O3. Our theoretical results are com-bined with experimental observations, performing new analy-ses on our recent powder X-ray diffraction (XRD) data.10 Atheoretical structure model fork-Al2O3 is obtained that agreeswell with the presently existing experimental information onthe material.

Our investigation is motivated by the fact that, althoughsome experimental studies have very recently been publishedpointing toward a specific structure,11,12until now no study hasbeen performed that rules out other possible alternatives for thestructure ofk-Al2O3. At the same time, to our knowledge, thisis the first time that first-principles methods have been appliedto the structure determination of such a complex system. Thestudy is a continuation of our previous first-principles theoret-ical investigations,13,14 where some of the possible structurecandidates fork-Al2O3 are considered.

The organization of the article is as follows. In Section II wegive a historical background of the experimental attempts todetermine the structure ofk-Al2O3. Section III describes ourmethod of calculation and the details of the XRD experiments.Then, in Section IV, the structure models fork-Al2O3 that arepossible, given the initially known experimental information,are derived. In Section V the results from our calculations arepresented and compared with the XRD results. Section VIpresents and describes the atomic structure that we conclude isvalid for k-Al2O3. A model for the CVD growth mechanism isproposed in Section VII. Calculated electronic structure andbond character of the crystal are discussed in Sections VIII andIX, respectively. The article concludes with a commented sum-mary of our results in Section X.

II. Historical Background

The structure of the stablea-Al2O3 phase has been well-known for a long time.15–17The smallest unit cell is rhombo-hedral with 10 atoms, but the structure is often described interms of a trigonal unit cell composed of 30 atoms. The struc-ture belongs to the space groupR3c and can be considered asan almost close-packed ABAB stacking of oxygen ions inplanes perpendicular to the [0001] direction, with aluminumions occupying 2/3 of the octahedral positions between theoxygen planes.

The experimental determination of the structure ofk-Al2O3has been hampered by the poor degree of crystallinity, thedifficulty to obtain significant amounts of phase-pure samples,and the metastability. The earliest work onk-Al2O3 was per-formed during a study of the transition aluminas resulting fromheat treatments of various alumina hydrates.1,2 Under certainconditions, as a step in the dehydration process leading even-tually to the formation ofa-Al2O3, a transition alumina, whichwas arbitrarily given the namek-Al2O3, was observed. How-ever, the crystal structure was not determined.

It was later reported by several authors that the unit cell ofk-Al2O3 is hexagonal,5,18 with a close-packed ABAC stacking

of the oxygen ions.5 It was not until the late 1980s that it wasshown that the structure is instead primitive orthorhombic.19

The previous confusion had been caused by twinning of thek-Al2O3 grains. The twin-related domains make up a false,almost hexagonal, superlattice. Thus, the powder XRD diffrac-tograms are similar (but not identical) to patterns from a hex-agonal structure.

In a subsequent study, using convergent beam electron dif-fraction (CBED), Liu and Skogsmo20 determined the point andspace groups fork-Al2O3 to bemm2 andPna21, respectively.They also proposed an atomic model for the structure ofk-Al2O3, with aluminum ions in both octahedral and tetrahe-dral interstitial positions. However, the model is incorrect, be-cause it does not satisfy the symmetry of the space groupPna21.

It has been shown in more-recent studies21 that many mod-els, consistent with thePna21 symmetry, are possible fork-Al2O3. These models differ in the positions of the aluminumions, but have in common an almost close-packed ABACstacking of the oxygen ions along thec-axis. Thereafter, someof these structure models have been studied more closely withthe aid of DFT-based total-energy calculations.13,14

More recently, during the course of this study, a structuremodel based on high-resolution electron microscopy (HREM)was proposed.11 Another study,12 based on XRD, TEM (trans-mission electron microscopy), and NMR (nuclear magneticresonance), also suggests the same structure, consisting of alu-minum ions in both octahedral and tetrahedral positions.

However, there is a need for further investigations, where allpossible structure models are carefully examined.

III. Method

(1) Computational DetailsOur calculations are based on DFT.22,23Within the adiabatic

approximation, DFT exactly maps the many-body problem of astrongly interacting electron system onto Schro¨dinger-like one-electron equations, thus introducing an effective exchange-correlation (XC) potential,vXC(r ), to account for the quantum–mechanical interactions between the electrons. Given anexpression forvXC, the one-particle Schro¨dinger-like equationscan be solved to self-consistency, yielding the electron densityand the total energy of the ground state of the system.

For an inhomogeneous electron system,vXC is not knownand must be calculated in some approximate way. The simplestmethod is the local-density approximation (LDA),23 in whichvXC(r ) is given by the XC potential of the homogeneous systemhaving the same electron density as the actual density atr .LDA is valid for slowly varying electron densities. An im-provement can be obtained using a generalized-gradient ap-proximation (GGA), in which the first-order gradient of theelectron density is added to the evaluation ofvXC. Our resultsare based on self-consistent LDA and “post-LDA GGA,” i.e.,non-self-consistent GGA based on the electron density fromself-consistent LDA. Post-LDA GGA has been shown in sev-eral cases to yield energies of practically the same accuracy asself-consistent GGA.24–26The Vosko–Wilk–Nusair LDA27 and1991 Perdew–Wang GGA25 potentials are used.

In the implementation of DFT used here,28 the one-particleorbitals are expanded in a plane-wave basis set, containingplane waves up to a chosen cutoff value for the kinetic energy.The true ionic potentials of the crystal are replaced by pseudo-potentials to keep the number of plane waves small. Theseexactly mimic the influence of the true ionic potentials on thevalence electrons outside the core shells but replace thestrongly attractive potential near the atomic nuclei with a muchweaker one, thus eliminating the strong oscillations of the va-lence orbitals within the core regions. The pseudopotentialsused here are the norm-conserving ones of Troullier and Mar-tins29 for oxygen and Bacheletet al.30 for aluminum.

The calculations are performed on a unit cell with 40 atoms

1366 Journal of the American Ceramic Society—Yourdshahyan et al. Vol. 82, No. 6

and periodic boundary conditions. The Brillouin zone issampled with a Monkhorst–Pack sampling scheme.31 We havepreviously performed convergence tests with respect to theplane-wave cutoff energy and the number ofk sampling pointsneeded to reach sufficient accuracy for bulkk-Al2O3 (see TableI in Ref. 13). These tests show that the total-energy differencesbetween different structure models are converged to an accu-racy of 0.01 eV/Al2O3 when the irreducible part of the Bril-louin zone is sampled at eightk points and the plane-waveexpansion is cut off at 650 eV, and to 0.03 eV/Al2O3 whenusing only onek point and 650 eV. All our final results arefrom calculations with eightk points and 650 eV cutoff, al-though onek point has been used for the unrelaxed total ener-gies and for the first part of the relaxation procedure—seeSection V for more details.

The forces acting on each ion in the unit cell are derivedfrom the calculated LDA charge densities. These forces areused to relax the positions of the ions within the unit cell. Thisis done by a modified-velocity–Verlet algorithm, where, ateach step, only the part of the ion displacement velocity that isparallel to the force on the ion is maintained. We use effectivemasses of 26mp and 16mp for aluminum and oxygen, respec-tively, wheremp is the proton mass. The time step is succes-sively decreased during the relaxation procedure, starting at 1.0fs in the first steps and ending at 0.1 fs in the last steps.

More details on the methods of calculation can be found inRef. 14.

We previously performed calculations on thea-Al2O3 struc-ture, using the same technique, to compare our results fork-Al2O3 with the a phase and also to test the accuracy of ourmethod.14 The results from such an optimization are summa-rized in Table I and compared with experimental results.17

Table I shows that the agreement between the DFT–LDA cal-culations and the experiments is very good, DFT–LDA under-estimating the experimental lattice parameter by <1%.

(2) Experimental DetailsThe procedure for obtaining thek-Al2O3 powder used in the

XRD experiments is as follows. First, 4mm thick k-Al2O3coatings were formed on TiC-coated cemented-carbide insertsusing CVD. The deposition was conducted in a computer-controlled hot-wall CVD reactor. The details of the CVD pro-cesses are reported elsewhere.32 The specimens then were cutparallel to the coating and the substrate removed by polishingand chemical etching. Finally, the remainingk-Al2O3 pieceswere crushed to a fine powder in a mortar, and thek-Al2O3powder was packed into a capillary glass tube 0.3 mm in di-ameter. See Ref. 10 for more details.

The XRD data were collected with au–u diffractometer(Model D5000, Siemens, Karlsruhe, Germany) using CuKaradiation and a scintillation detector with a diffracted-beamnickel filter. The 2u scan range was 3°–80°. The X-ray sourcewas operated at 45 kV and 40 mA. The mean temperature ofthe measurement was 22°C, and SiO2 was used as an externalstandard to calibrate the instrument.

Simulations of powder XRD patterns for different theoreticalk-Al2O3 models were performed with the computer softwareTEXSAN (Molecular Structure Corp., The Woodlands, TX,1993) and compared with the experimental XRD data.

A Patterson function,33 which is a map in vector space,

P(u,v,w), was deduced from powder XRD data by theexpression

P~u,v,w! =1

V (h,k,l

|F~h,k,l!|2 cos@2p~hu + kv + lw!#

whereF is the structure factor,V the unit-cell volume, andu,v, andw the vector-space coordinates. The Patterson functionalso can be described as

P~u,v,w! = *V

r~x,y,z! r~x + u, y + v, z + w! dV

wherer is the electron density. Thus, the Patterson function isa convolution of the electron density at (x, y, z) with the elec-tron density at (x + u, y + v, z + w). This means that peaks inthe Patterson map at (u, v, w) correspond to the interatomicvectors [u, v, w] in k-Al2O3. The Patterson-function maps wereobtained using the computer softwareSHELXTL (Siemens Indus-trial Automation, Inc., Karlsruhe, Germany, 1994).

IV. Structure Models

(1) Experimental Structure ModelAccording to the experimental results mentioned in Section

II, it is known that the unit cell ofk-Al2O3 belongs to theorthorhombic class, with the space groupPna21. Based on themeasured lattice parameters, it has been proposed that the oxy-gen ions are arranged in four close-packed planes in an ABACstacking sequence along thec-axis, with six ions per plane.However, to our knowledge, no Patterson function map hasbeen published explicitly proving the close packing. Figure 1shows the Patterson function map forw 4 0 obtained from ourpowder XRD data. The map describes the interatomic vectorsthat have aw component equal to zero. Figure 1 shows that thePatterson function has maxima of the types [0 1/3 0], [1/21/2 0], [1/2 1/6 0], and [1 0 0]. This Patterson map has peaksthat form a close-packed network; this means that the atoms inthe (001) planes ofk-Al2O3 have a close-packed arrangement.

Therefore, the aluminum ions are positioned ideally in theinterstitial sites between the oxygen layers, forming four alu-minum planes. Because of the stoichiometry ofk-Al2O3, eachaluminum plane consists of four ions. The aluminum intersticesare of two types: octahedral and tetrahedral, depending on theircoordination to the neighboring oxygen ions (see Fig. 2). Theunit cell of k-Al2O3 thus consists of 40 atoms total: 24 oxygenand 16 aluminum.

Because of thePna21 space group, there are four symmetry-related positions within the unit cell: (x, y, z), (1/2 − x,1/2 +y, 1/2 +z), (1/2 +x, 1/2 −y, z), and (−x, −y, 1/2 +z).34 Ink-Al2O3 this means that each position in an atomic layern hasone symmetry-related position in the same layer and two morein layer n + 2. The easiest way to visualize this is to think of

Table I. Crystallographic Specifications fora-Al2O3: Calculated and Experimental†

ParameterOur calculations

(Ref. 14)Experimental

(Ref. 17)

a (Å) 5.091 5.1284a (deg) 55.33 55.28w 0.3522 0.3520u 0.5562 0.556

†Specifications are for the rhombohedral unit cell, as definedin Ref. 17.

Fig. 1. Patterson function map forw 4 0 obtained from our powderXRD data.

June 1999 Theoretical Structure Determination of a Complex Material:k-Al2O3 1367

each plane as made up of pairs of atomic positions, instead ofindividual ones, and that each pair in planen is related toanother pair in planen + 2. Only 10 independent atomic posi-tions are thus needed to describe the unit cell.

(2) Structure NotationIt is necessary to introduce a notation that can distinguish

between the structure alternatives in order to study all structurecandidates that are possible given the above experimental re-quirements. Both the oxygen and the aluminum positions canbe described by their type of stacking. The stacking letters usedfor the oxygen ions also can be used for the aluminum inter-stitial sites. Capital letters (A, B, and C) denote the oxygenions, and small letters (a, b, and c) aluminum ions.

For each stacking letter, there are three different ways toposition a pair of ions in a close-packed layer because of thesymmetry relationships. We label thesea, b, andg in such away that the two ideal positions for ana pair are aty 4 0 and

y 4 1/2, while those forb andg are at 0 <y < 1/2 and 1/2 <y < 1, respectively, as shown in Fig. 2.

Each symmetry-related pair of positions then can beuniquely labeled by two letters: stacking and pair type, such as,for example, Aa, bg, and cb. The relationships between layersn and n + 2 become easy to visualize with this notation. Anoxygen (aluminum) pair with stacking letter A (a), B (b), or C(c) in one layer is related to a pair in the other layer withstacking letter A (a), C (c), or B (b), respectively. In the sameway, a pair of typea, b, or g is related to a pair of typea, g,or b, respectively.

Each oxygen layer is composed of six ions per unit cell.Therefore, there are no vacancies in the oxygen layers, and theoxygens can be labeled only with their stacking letter. On theother hand, the aluminum layers are composed of four ions,that is, two ion pairs. Each structure candidate then can belabeled with a notation of the type AcbbgBcacgAbgcbCbabb,where each capital letter denotes an oxygen plane by its stack-

Panel A. Density Functional Theory

The density-functional theory (DFT) of Hohenberg andKohn (HK)22 and Kohn and Sham (KS)23 has revolution-ized our possibilities to calculate properties of extendedelectron systems by allowing many-electron properties to becalculated in a one-electron approach. It is the key motiva-tion for the Nobel Prize given to Walter Kohn in 1998. HKfirst showed that the ground state of a system is uniquelydefined by the electron-density distributionn(r ) that mini-mizes the total energy and that all other ground-state prop-erties of a system also are functionals of the ground-stateelectron density. KS then introduced the foundation for theone-electron approximation, by separating the ground-stateenergy functional into one part with the same form as fornoninteracting electrons with the density distributionn(r )and a correction term. The advantage of this is that thekinetic-energy functionalTs[n] can be treated exactly, thetotal-energy DFT functional being written as

E@n# = Ts@n~r !# + *d3r n~r !S1

2F~r ! + Vext~r !D

+ EXC@n~r !# (1)

where the second term represents the interaction energy ofelectrons in the classical Coulomb potentialF and the ex-ternal potentialVext. The final term, the exchange-correlation (XC) energyEXC term, captures all many-electron interactions and is the only part of the KS equationthat cannot be solved exactly.

KS have shown that solving theSchrodinger equationforthe ground-state energy is equivalent to minimizing the en-ergy functional,E[n], with respect to an orthonormal set ofsingle-particle wavefunctionsci(r ),

S−1

2=2 + Veff~r !Dci(r ) = «ici~r ! (2)

The effective potential in this equation,

Veff~r ! = Vext~r ! + e2*d3r8n~r 8!

|r − r 8| +­EXC@n~r !#

dn~r !(3)

consists of an external (Vext), a Hartree (4 CoulombF),and an XC-potential term, derived from the exchange-correlation functional. These KS equations are solved self-consistently, in the sense that the wavefunctions must pro-duce the same density as the one that is used to construct theHartree and XC potentials.

From the solution of this system of equations, the chargedensity is calculated from

n~r ! = (i

N

|ci~r !|2

whereN is the number of electrons.The formalism described above is essentially a way of

concentrating all nontrivial electronic interactions into theXC-energy functionalEXC. Another virtue of DFT is thefact that, even if the exact functional form ofEXC[n] is notknown, the XC functional can be expressed exactly in termsof the so-called exchange-correlation hole,nXC(r ,r 8),through44,45

EXC@n# =e2

2 *d3r n~r ! *d3r81

|r − r 8| nXC~r ,r 8! (5)

i.e., just an electrostatic interaction between an electron atrand its XC hole. Thisadiabatic-connection formula, forwhich there is a precise prescription for calculating the XChole,44,45

nXC~r ,r 8! = n~r 8! *0

1dl ~gn~r ,r 8; l! − 1! (6)

wheregn(r ,r 8) is the pair-correlation function of the systemwith density n(r ) and coupling constantle2 makesEXCavailable for analysis and construction of successively im-proved approximations.46

It is reasonable to believe that an approximateEXC[n]might yield ground-state energies in good agreement withexact results, provided that the XC hole is accounted foraccurately. There are some minimum requirements, such ascharge conservation (hole normalized to unity).44 Also,there are some relieving features, such as Eq. (5) demandingonly a spherical average of the XC hole.44 In the simplest ofapproximations, the local-density approximation (LDA),23

Eqs. (5) and (6) are evaluated as for a locally homogeneouselectron liquid, for which the XC energy can be calculatednumerically very exactly.46 In improved XC approxima-tions, such as the very successful generalized-gradient ap-proximation (GGA),25 features of the response of the ho-mogeneous electron liquid to charge perturbations areutilized.45,47 Strict limits for the accuracy and applicabilityof XC approximations are difficult to set from internal cri-teria, however. Sometimes approximate XC functionals ap-pear to be even more favorable than expected. In practice,the accuracy has to be assessed by calibration against ex-perimental results for systems similar to the one studied (seeRefs. 48 and 49).

1368 Journal of the American Ceramic Society—Yourdshahyan et al. Vol. 82, No. 6

ing type, and each small letter with a Greek subscript denotesan occupied aluminum pair position between the oxygenplanes.

(3) Structure CandidatesWe now derive the number of structure candidates that are

possible given the requirements above. These candidates arethe starting point for our calculations. Because of the ABACstacking sequence, the ideal positions of the oxygen ions areknown. However, the number of interstitial sites for the alu-minum ions is far greater than the number of available alumi-num ions. There are nine pair positions in each aluminum layer,as shown in Fig. 2. However, each aluminum layer is com-posed of four aluminum ions, that is, two pairs, that then can bearranged, in principle, in 9 × 8 4 72 different ways.

If we consider the energy disadvantage of both aluminumpairs being in immediately adjacent sites (such as, for example,aa and bb), this number reduces to 15.¶ We are able to verifythis assumption a posteriori, on the basis of the results from ourcalculations (see Section V(1)).

There are only two independent aluminum layers in the unitcell, giving a total of 152 4 225 structure candidates fork-Al2O3. However, many of these are the same structure withdifferent choices of coordinate system: a rotation by 180° of theunit cell around [010] (Ref. 21) and the translation (x + 1/2,y + 1/2, z) (Ref. 14) reduce the number of candidates to 60independent ones.

The aluminum layers can be categorized by the type of co-ordination of their aluminum pairs:O (all aluminum ions inoctahedral sites),T (all in tetrahedral), andM (mixed, i.e., onepair in octahedral and the other pair in tetrahedral). The 60independent structure candidates then can be divided into six

groups:OO, OT, MO, MM, MT, and TT, depending on thecoordination type of the first two aluminum layers.

In our calculations we start with these 60 independent struc-ture candidates. We use as lattice parameters those determinedfrom XRD experiments at room temperature35 (see Table II).Because one of the purposes of this work is to put the methodunder test and to evaluate to what extent it is capable of dis-tinguishing different levels of stability in a general structure-determination context, we set up the structures with no precon-ceived notion, such as, Pauling’s structural rules or previousobservations on other oxide structures. Therefore, we placedthe atomic layers equidistant from each other, with all the ionsin the same layer at the same height. We assumed a geometri-cally regular close-packed arrangement within the layers, as inFig. 2. These “ideal” structures are then relaxed in search of thetrue lowest-energy configuration.

V. Structure Determination

(1) Theoretical ResultsFigure 3 shows the results from our calculations. We start

with all 60 ideal independent structure candidates fork-Al2O3described above. First, the total energies for these unrelaxedstructures are calculated, using onek sampling point and a 650eV cutoff energy. These energies are shown as open symbols inFig. 3.

The ideal structures group themselves energetically accord-ing to the number of aluminum ions that are in tetrahedral sites:the OO structures have the lowest energies, followed, in in-creasing energy order, byMO (where 1/4 of all aluminum ionsare in tetrahedral sites),MM andOT (2/4), MT (3/4), andTT(4/4).13 The energy differences between the groups are ratherlarge. Apparently, the tetrahedral aluminum positions are veryunfavorable in the ideal structures.

Second, the ionic positions of the 60 structures are allowedto relax within the fixed dimensions of the unit cell given byexperiments, again with onek sampling point and 650 eVcutoff. The relaxation is performed until the mean-square sumsof the remaining forces on the ions are <0.1 eV/Å. Almost allstructures transform into one of eight structures during therelaxation, shown as solid symbols in Fig. 3.

The relaxation of the structures can be noted by the following.(i) The layer-by-layer structure and the oxygen ABAC

stacking sequence are very stable, with only three of our start-ing configurations losing such an arrangement.

(ii) Most of the aluminum ions that are in tetrahedral po-sitions move toward the geometric centers of the tetrahedraduring the first few steps of the relaxation (i.e., at 1/4 of thetetrahedral height (see Fig. 4(a))).

(iii) The transformation from one structure to another isachieved by the aluminum ions moving to immediately adja-cent sites, and/or by the oxygen layers moving to another stack-ing letter.

(iv) The aluminum ions do not move to sites that are im-mediately adjacent to other aluminum ions, which, in hind-sight, motivates our choice at the beginning not to take intoaccount configurations with immediately adjacent aluminumions.

(v) The octahedral sites are relatively stable; i.e., alumi-num ions in octahedral positions do not move to other sites.

¶In previous papers,13,14 the number of different arrangements was given as 13,then judging it energetically unreasonable to arrange an aluminum layer by mixingtwo tetrahedral aluminum pairs with different stacking letters.

Fig. 2. Aluminum pair positions: different stacking and pair types ofthe aluminum positions in the unit cell are shown between two oxygenlayers in A and B stacking. Unit cell is drawn with solid lines.

Table II. Lattice Parameters for the k-Al2O3 StructureExperimental

(room temperature)Experimental

(extrapolated to 0 K)Calculated (0 K)

(this study)(Ref. 35) (Ref. 12) (From Ref. 35) (From Ref. 12)

a (Å) 4.8351(3) 4.8437(2) 4.831 4.840 4.8041b (Å) 8.3109(5) 8.3300(3) 8.300 8.321 8.2543c (Å) 8.9363(3) 8.9547(4) 8.923 8.945 8.8785V (Å3) 359.09(6) 361.3044 357.9 360.2 352.07

June 1999 Theoretical Structure Determination of a Complex Material:k-Al2O3 1369

The aluminum ions in tetrahedral sites in some of the struc-tures do not move toward the geometric centers of the tetrahe-dra during the first relaxation steps. Therefore, it is instructiveto examine the energies of the 60 structures with all ions inideal positions, but where the tetrahedral aluminum ions arepositioned at the geometric centers of the tetrahedra (see Fig.4(a)). The calculated energies of these “modified ideal” struc-tures are shown in Fig. 3 as dots, with dotted lines connectingthem to their respective ideal configurations.

There is considerable energy decrease for many structureswhen the tetrahedral aluminum ions are positioned in the tet-rahedral centers. The largest energy decreases are obtained forthe structures in which the tetrahedra do not share faces (seeFig. 4(b)). However, it is energetically unfavorable for a num-

ber of structures to position the aluminum ions in the tetrahe-dral centers. These are the structures in which the number offace sharings between aluminum tetrahedra is greatest (theseare presented in Fig. 3 directly above the ideal ones, withoutconnecting lines). It is commonly accepted, on the basis ofobservations on known crystal structures, that the presence offace sharings between cation polyhedra (and especially be-tween ones corresponding to low coordination number, in ourcase the tetrahedra) decreases the stability of a structure (Paul-ing’s third structural rule for ionic crystals36). The simple elec-trostatic explanation is that, when polyhedra share faces, theirgeometric centers are closer; thus, aluminum ions placed in thecenters of face-sharing polyhedra experience a stronger electro-static repulsion, which increases the energy of the structure.36

However, the results from our quantum-mechanic calcula-tions also show this trend: If we consider the energy order ofthe structures, when care is taken to position the aluminum ionsin the tetrahedral centers, we see that all 23 modified idealstructures with lowest energies (that is, below −1425.5 eV/Al2O3) have no face sharing or areOO structures andMOstructures with face sharings only between octahedra. Amongthese structures, the most stable are no longer those in theOOgroup, presumably because of the polyhedral face sharings intheOO structures. The most stable modified ideal structure is,instead, anMO structure, followed, in increasing energy order,by anOT, anMT, aTT, and only thereafter by anOOstructure.There is no face sharing between polyhedra in the first fourstructures.

We may conclude that the degree of stability of each non-OO structure candidate is determined to a large extent by theinterplay between the Pauli repulsion pushing the tetrahedralaluminum ions toward the tetrahedral centers and the electro-static repulsion occurring when the tetrahedral aluminum ionsshare faces. For many structures, the most favorable position ofthe aluminum ions inside the tetrahedra is somewhere betweenthe tetrahedral center and the middle point between the oxygenplanes. We observe that the paths many tetrahedral aluminumions have followed during the relaxation process from the ideal

Fig. 3. Calculated total energies of the 60 structure candidates fork-Al2O3. Open symbols correspond to the energy values of the unrelaxed “ideal”structures, grouped according to the types of the first two aluminum layers. The eight “stable” relaxed structures are shown by solid symbols. Dottedlines connect the unrelaxed structures to the structures they have transformed to. Dots show the energies of the “modified ideal” structures.

Fig. 4. Aluminum tetrahedra: (a) tetrahedral aluminum ions in“ideal” and in “modified ideal” positions; (b) two face-sharing alumi-num tetrahedra.

1370 Journal of the American Ceramic Society—Yourdshahyan et al. Vol. 82, No. 6

structures are initially toward the geometric centers but deviatevery soon toward other directions.

A crucial point here is what is the “right” choice of an idealstructure from which to start the relaxation process. The goal isto minimize the calculation time necessary to relax the struc-tures and be sure that the relaxations choose no “wrong” paths.We have noted that, when starting from the ideal configura-tions, the movement of the tetrahedral aluminum ions towardthe tetrahedral centers occurs very “fast,” in only a small frac-tion of the total number of relaxation steps needed to reach thefinal stable configuration. Also, as noted above, the tetrahedralcenter is not always the most favorable position within thetetrahedral site. A test relaxation performed on a structure withtetrahedral face sharing, starting from the modified ideal con-figuration, yields such great forces on the ions that they destroythe close-packed arrangement of the oxygen layers. This doesnot occur if the relaxation is started from the ideal structure,showing that it is not obvious that the modified ideal configu-ration is the most favorable starting point for the relaxation. Onthe other hand, for the three structures that lose the close-packed stacking when started from the ideal configurations, nodifferent result is obtained when starting from the modifiedideal configurations. We conclude that starting from the idealconfigurations yields the most reliable results.

As mentioned above, only 8 of the 60 structure candidatesare “stable.” This is indicated in Fig. 3 by the lines connectingthe ideal structures (open symbols) and the stable relaxed ones(solid symbols). The eight stable structures are the fourOOstructures, theMO, OT, and TT structures mentioned above,and anotherOT structure with no polyhedral face sharing,calledOT-I in Fig. 3. These structures are relaxed more care-fully using eight k points and 650 eV cutoff, and their cellvolumes optimized by minimizing their total energies. The cellvolumes of theTT, the OT-I, and theOT structures increaseduring this optimization to values of∼20%, 10%, and 5%larger than the experimental room-temperature cell volume(359.09 Å3), respectively. Our calculations correspond to 0 Ktemperature, whereas the experiments were performed at roomtemperature; therefore, these three structures are incompatiblewith the experimental results and can be ruled out as possiblecandidates fork-Al2O3. Because of the very low total energy ofthe OT structure, however, we keep this structure for furtherevaluation.

For the remaining six stable structures, the relaxation andthe optimization ofa and of the ratiosb/a andc/a of the lat-tice parameters are performed until the mean-square sumsof the remaining forces on the ions are <0.01 eV/Å. The resultsare shown in Table III.

In addition to theOT structure, theOO-I and theOO-IIIstructures show larger cell volumes than the experimentallyobserved one and thus can be ruled out. Further investigationshows that the optimizedOO-I and OO-III structures have alower symmetry andOO-IV a higher symmetry than the oneobserved (Pna21) experimentally.

Thus, the calculations result in only two stable structures,MO andOO-II, that have all the necessary requirements to beconsidered as final candidates for thek-Al2O3 structure at 0 K.The calculated energy order of the structures points toward theoptimizedMO structure (shown in Fig. 5) as being the most

favorable one. The energy difference between the two struc-tures is of the order of that observed in LDA–DFT calculationsbetween the two different phasesa andu of Al2O3.9

There is a cell-volume increase associated with the presenceof aluminum ions in tetrahedral sites. This is possibly a reasonfor the instability of many of the structure candidates during therelaxation. The relaxation has been conducted with the latticeparameters fixed at the experimental values, which could betoo small for many of the structures with tetrahedral aluminumions to be stable.

(2) Comparison with Experimental ResultsTheMO, OT, andOO-II structures are lowest in total energy

according to our calculations. In order to evaluate these struc-tures, powder XRD patterns are simulated for theMO, OT, andOO-II structures and compared with our experimental data.

An experimental powder XRD pattern is shown in Fig. 6(a)with diffraction indexes for some selected peaks. Simulatedpatterns for the optimizedMO, OT, andOO-II structures areshown in Figs. 6(b), (c), and (d), respectively. The simulatedpatterns have not been refined to fit the experimental data.Only the optimized atomic coordinates resulting from our theo-retical calculations have been used. Therefore, little attentionshould be given to the exact heights of individual peaks, but onthe appearance of the pattern as a whole.

Figure 6(d) shows that the simulated pattern for theOO-IIstructure has the largest deviation from the experimental pat-tern, with numerous strong peaks not present in the experimen-

Table III. Calculated Total Energies and Unit-Cell Volumes of the Six Optimized“Stable” Structure Candidates

Structure Etot (eV/Al2O3)

Type Notation Post-LDA GGA LDA V (Å3)

MO AcbbgBcacgAbgcbCbabb −1431.466 −1424.716 352.07OT AcacbBbbagAbabgCcgab −1431.335 −1424.342 376.67OO-II AcbcgBcacgAbbbgCbabb −1430.990 −1424.279 350.16OO-IV AcacgBcacbAbabbCbabg −1430.726 −1424.029 350.16OO-I AcbcgBcbcgAbbbgCbbbg −1430.360 −1423.567 366.27OO-III Ac acgBcacgAbabbCbabb −1429.351 −1422.511 365.91

Fig. 5. Three-dimensional picture of the calculatedk-Al2O3 unit cell(MO). Large balls are oxygen ions, and small ones are aluminum.Sticks show the Al–O bonds. Only the parts of the sticks lying insidethe unit cell are drawn.

June 1999 Theoretical Structure Determination of a Complex Material:k-Al2O3 1371

tal data. The deviation is smaller forOT, but many simulatedpeaks cannot be matched with experimental ones. On the otherhand, there is a close resemblance between the experimentaland the simulatedMO powder diffractograms.

TheMO structure, with the lowest total energy in the abovecalculations, shows the best agreement with the XRD experi-ments. We conclude that the optimizedMO structure is theright one fork-Al2O3.

The experimental results published during the course of thisstudy11,12 agree with this conclusion. However, we note thatwe have now thoroughly examined and ruled out all otherpossible structure candidates. Furthermore, our results showthat it is possible to use theoretical methods to determine thestructure of a complex and metastable crystal.

VI. Structure Analysis

(1) Analysis ofk-Al2O3 StructureThe obtainedk-Al2O3 structure is presented in Tables IV and

II, which list the calculated positions of the ten independentatoms and the calculated lattice parameters (at 0 K), comparedwith the experimental ones (at room temperature and extrapo-lated to 0 K using the thermal expansion coefficients of Ref.35), respectively. Our calculated values for the lattice param-eters are in very good agreement with the experimental ones,underestimating them by <1%. The crystallographic data ofTable IV also can be compared with the experimental results ofOllivier et al.,12 their values differing by no more than ±0.02(or, in absolute values, ±0.1 Å) from ours.

Figure 7 shows the structure layer by layer. The aluminum

ions are positioned in zigzag lines along the [100] direction.These lines are equidistant in the first and third layers (whereevery other line consists of tetrahedral aluminum ions), while,in the second and fourth layers, they result in vacancy lines.Thus, there is a large anisotropy in the structure; the aluminumions are more closely packed in the [100] direction than in the[010]. Indeed, an anisotropy of this type has been observed inmeasurements of the thermal expansion ofk-Al2O3.35

Figure 8 shows the structure projected on the (130) plane,with polyhedra drawn to show the sites of the aluminum ions.Only a slice of the structure is shown, one aluminum polyhe-dron thick. The picture clearly shows a distortion of the alu-minum octahedra, with the ions moving toward the vacanciesand away from the aluminum tetrahedra.

The structure can be described utilizing the symmetry asconsisting of three aluminum octahedra and one tetrahedron.

Fig. 6. Powder XRD patterns: (a) from experimental data; and (b), (c), and (d) simulated patterns for the optimizedMO, OT, andOO-II structures,respectively.

Table IV. Calculated Crystallographical Data for thek-Al2O3 Structure

Atom x y z

O(1) 0.01662 0.32374 0.99841O(2) 0.47610 0.48864 0.00405O(3) 0.15831 0.16861 0.24329O(4) 0.15967 0.49910 0.27374O(5) 0.48201 0.16871 0.50994O(6) 0.34057 0.33891 0.76765Al(1) 0.31585 0.34539 0.15648Al(2) 0.32935 0.03082 0.36827Al(3) 0.32362 0.34772 0.56882Al(4) 0.18098 0.15993 0.87799

1372 Journal of the American Ceramic Society—Yourdshahyan et al. Vol. 82, No. 6

The Al–O bond lengths in the octahedra vary between 1.79 and2.20 Å, with averages for each octahedron between 1.90 and1.94 Å. The Al–O lengths for the tetrahedron are between 1.73and 1.77 Å, with an average of 1.75 Å. These average Al–Obond lengths agree well with the empirical ionic radii of Shan-non,37 which give, depending on the coordination of the oxy-gen ions, values between 1.895 and 1.925 Å for the octahedralaluminum ions, and 1.75–1.77 Å for the tetrahedral ions.

The distortion of the polyhedra can be quantified with theexpression

D =1

N (i=1

N SRi − R

RD2

where N is the number of corners of the polyhedron,Ri anindividual Al–O bond length, andR the average Al–O bondlength in the polyhedron.37 We then obtained values ofD 455 × 10−4 (R 4 1.94 Å) for the octahedra in the first layer(aluminum pair cb) and D 4 51 × 10−4 (R 4 1.93 Å), andD 4 7.2 × 10−4 (R 4 1.90 Å) for the aluminum octahe-dron pairs ca and cg, respectively, in the second layer (the sameapplies to the corresponding octahedra in layers three and four,because of the symmetry). The larger distortion is observed in

the octahedra that are directly above or below the aluminumvacancies. By contrast, the distortion of the tetrahedron is onlyD 4 0.79 × 10−4 (R 4 1.75 Å). The average bond lengths arerelatively constant for similar polyhedra. The large variationof R values in the octahedra is thus due only to their largedistortion.

The octahedra thus appear to be “softer” than the tetrahedra,i.e., more capable of distortion, because the volumes of theoctahedra are approximately three times larger than those of thetetrahedra. In the final, optimized,MO structure, the volume ofthe tetrahedra is 2.7 Å3, which is 9% larger than in the startingideal structure. The volumes of the octahedra are between 9.0and 9.3 Å3, 6%–10% smaller than the ideal ones. Again, thevolume expansion is associated with the tetrahedral aluminumions. This volume expansion in the finalMO structure is pos-sible within the experimental lattice parameters because of thepresence of aluminum vacancies near the tetrahedra, and thesoftness of the alumina octahedra. The observed instability orlarge cell-volume increase of many of the other structure can-didates then can be explained by the fact that the tetrahedra aremore “crammed,” such as when tetrahedra share faces, and/orthe structure consists mainly of tetrahedra.

The O–O distances in the final structure vary between 2.52and 3.00 Å. According to the empirical ionic radii of Pauling36

and Shannon,37 the O–O bonds should have a length of 2.80 Å.However, as noted by Pauling, edges that are shared by twopolyhedra should be shorter (because edge sharing reduces thedistance between like-charged cations and, thus, increases theirrepulsion), with a length of∼2.50 Å. This agrees with ourcalculated values fork-Al2O3, shared edges lying between2.52 and 2.69 Å, with an average of 2.61 Å, and unshared oneshaving lengths of 2.64–3.00 Å, with an average of 2.80 Å.

(2) Comparison witha-Al2O3

We now compare the structure ofk-Al2O3 with a-Al2O3 inmore detail. The structure fora-Al2O3 can be described as

Acbg(I)Bcbg(II)Acbg(III) Bcbg(I)Acbg(II)Bcbg(III)

using a notation similar to ours. The subscriptbg indicates thatthe vacancies are ina positions, thus forming a hexagonalnetwork, while (I), (II), and (III) indicate that there are threehexagonal networks, one at each height, which are slightlydisplaced with respect to each other. All aluminum ions are inc stacking between the A and B oxygen planes, that is, they arein octahedral interstices. Therefore, there are three key contri-butions to the stability ofa-Al2O3: (i) close packing with oxy-gen ions in ABAB stacking; (ii) hexagonal aluminum net-works; and (iii) octahedral aluminum positions.

Fig. 7. Bulk structure ofk-Al2O3 shown layer by layer ((a), (b), (c), and (d) depict oxygen layers A, B, A, and C with aluminum overlayers,respectively). Large open circles represent oxygen ions, and small circles show the aluminum ions in octahedral (black) and tetrahedral (gray)positions. Unit cell is drawn with solid lines in each layer.

Fig. 8. Projection on (130) of a slice of the bulk structure ofk-Al2O3.Large and small circles represent oxygen and aluminum ions, respec-tively. Shaded oxygen ions lie behind the “empty” ones in a samelayer. Dotted lines indicate the “ideal” heights of the oxygen planes.Solid lines show the octahedra and the tetrahedra in which the alumi-num ions are positioned. Slice shown is one polyhedron thick. Allatomic displacements from the ideal positions have been exaggeratedby a factor of 2 to evidence the distortion of the polyhedra.

June 1999 Theoretical Structure Determination of a Complex Material:k-Al2O3 1373

Although k-Al2O3 is a metastable structure, it is virtuallystable up to temperatures of∼1200 K. This indicates that thestructure ofk-Al2O3 should be relatively similar toa-Al2O3,but should include some “instability,” causing it to transform athigher temperatures. The smallest difference from the stablea-Al2O3 would be fork-Al2O3 to haveeithersome aluminumvacancies not forming a hexagonal networkor one (indepen-dent) aluminum ion in a tetrahedral position. However,k-Al2O3 has both nonhexagonal vacancy networks and onealuminum-ion pair in tetrahedral position. Also, the stackingsequence of the oxygen ions is ABAC ink-Al2O3, as opposedto ABAB in a-Al2O3.

It is shown in the discussion below on the CVD growth ofk-Al2O3 that one instability, e.g., the tetrahedral aluminumions, stabilizes the other instability, the vacancy lines, and viceversa, in a way that makes it possible for the crystal to grow ina very stable manner up to 8mm high. Apparently, the reasonfor the stability of bulkk-Al2O3 is based on an interplay be-tween the three instabilities: vacancy lines, tetrahedral alumi-num ions, and ABAC stacking sequence. Also, the presence oftetrahedrally coordinated aluminum ions ink-Al2O3 makes itpossible for the crystal to avoid energetically unfavorable facesharings between the aluminum polyhedra. If all the aluminumions are in octahedral sites, such as in theOO structures, or inthe a phase, octahedral face sharing is unavoidable.

VII. Structural Implications on CVD Growth

It is not fully understood why it is possible to grow themetastablek-Al2O3 with CVD. Little has been known aboutthe mechanisms behind the growth processes of thea and kphases. The nucleation surface is a very important factor for thephase composition (a/k) of the Al2O3 layer.38 Non-oxidized(111) facets of TiC and TiN, typical substrates, have proved tofavor the growth ofk-Al2O3 instead of the stablea-Al2O3. Thishas been explained by a layer-by-layer growth and a lowerlattice mismatch fork-Al2O3 than fora-Al2O3.

It also has been observed that, although the domains ofk-Al2O3 typically have a width of∼50 nm, they can extend upto 8 mm into the coating. This means that, oncek-Al2O3 hasbeen nucleated, it can be grown as thick as 10 000 unit cells.8

Because the atomic structure ofk-Al2O3 has been determined,a possible mechanism for this type of epitaxial growth of CVDk-Al2O3 coatings can now be proposed.

A crucial step in growingk-Al2O3 with CVD is the nucle-ation on the (111) TiC (or TiN) surface. If the surface is tita-nium terminated,39 the first Al2O3 layer should consist of oxy-gen ions, whereas, if the surface is carbon (or nitrogen)terminated, the first layer should be composed of aluminumions. In both cases, it is the nucleation of the first aluminumlayer that is important for the choice of Al2O3 polymorph. Thereason is that the oxygen stacking in botha-Al2O3 andk-Al2O3is . . . AB . . . in thefirst two layers, whereas the differencebetween the two phases lies in the aluminum positions:a-Al2O3 has a hexagonal network of aluminum ions, whereask-Al2O3 has two zigzag lines of aluminum ions (see Fig. 7).

If the first Al2O3 layer is a close-packed oxygen layer, it canbe arbitrarily denoted A. The first aluminum ions should thenposition themselves in hollow sites on the oxygen layer, be-cause there is no mechanism available to stabilize bridge or toppositions. Obviously, there are several different ways for thealuminum ions to position. The most favorable one should besuch that the distances between the aluminum ions are maxi-mized, to reduce the electrostatic repulsion.

Each aluminum ion ina-Al2O3 has three nearest neighborsin the same aluminum plane, all (ideally) at the same distance.On the other hand, for the cbbg arrangement of the first alu-minum layer ofk-Al2O3 (see Fig. 7(a)), the shortest distancebetween aluminum ions in neighboring zigzag lines is∼15%longer. Thus, the electrostatic repulsion can be reduced byplacing the aluminum ions as a cbbg arrangement.

If the first aluminum layer is grown in this way, then the rest

of the crystal will prefer to grow ask-Al2O3 in a very stablemanner. A complete dynamic investigation is necessary todraw conclusive results, but, if the temperature of the CVDreaction is high enough to overcome diffusion barriers, then itis plausible that each layer forms a configuration that mini-mizes its energy.

The second oxygen layer cannot be positioned directly abovethe first oxygen layer. The second oxygen layer then can haveB or C stacking. However, independent of B or C stacking, halfof the aluminum ions in the first layer are now trapped intetrahedral positions. In fact, both choices lead to the samestructure, because they are equivalent with respect to a rotationof the unit cell,21 and we can here choose stacking letter B forthe second oxygen layer.

The aluminum ions in the second layer must position them-selves in the other four hollow sites, cacg, to avoid face sharingwith the octahedral aluminum pair in the first layer, as shownin Fig. 7(b). In a way, the tetrahedral aluminum ions in layerone stabilize the aluminum vacancy lines of the next layer.

In the next (third) oxygen layer, the oxygen ions avoid oc-cupying positions that are on top of the aluminum and theoxygen ions of the second layer; therefore, they are assignedstacking letter A. Now we have the same situation as in thebeginning, with aluminum layer No. 1; therefore, the thirdaluminum layer should be the same as the first one, cbbg. Also,this causes no face sharing between aluminum polyhedra,which should be energetically favorable.

The fourth oxygen layer becomes either B or C stacking. Thethird oxygen and aluminum layers do not favor B or C stack-ings, because the oxygen layer is in A stacking and the alumi-num layer is half in b and half in c stacking. However, thesecond oxygen layer is in B stacking, and the second aluminumlayer is in c stacking. The ionic charges from these layers arescreened only partly by the ions of the third layers. Therefore,the oxygen ions of the fourth layer can feel the electrostaticattraction from the aluminum ions in c stacking in the secondlayer and the repulsion from the oxygen ions in B stacking inthe second oxygen layer. It is then plausible that the fourthoxygen layer is assigned stacking letter C.

If similar arguments as above are used, the fourth aluminumlayer is babb and the fifth oxygen layer is exactly as the firstone (A). We thus find that, once the first layer of aluminum hasbeen nucleated, thek-Al2O3 crystal can grow in a stable man-ner in the [001] direction. This is, as mentioned above, exactlywhat has been observed. Also, our description requires no im-purity to stabilize the growth of thek phase, although impuri-ties may be important in the nucleation of the Al2O3 layer.

An important corollary of this discussion is that the growthdirection ofk-Al2O3 can be along only thepositivec-direction.If growth along the negative direction is to occur, the tetrahe-dral aluminum ions have to position themselves in top sitesabove the oxygen ions during the growth, which we find to beimplausible.

VIII. Electronic Structure

Figures 9 and 10 show the calculated band structure for theobtainedk-Al 2O3 structure and the corresponding orbital-resolved partial density of states (PDOS). The lower valenceband (LVB) has a width of 3.4 eV and consists mainly of O 2sorbitals. The LVB is separated by a large gap of 9.0 eV fromthe upper valence band (UVB), which consists mainly of O 2porbitals and has a width of 7.1 eV. The conduction band (CB)consists mainly of Als and Al p orbitals. The occupied bandsare clearly dominated by the oxygen orbitals, implying a highlypolar character of the Al–O bonds. The bottom of the CB liesat theG point, whereas the top of the valence band (VB) is ata point of 1/6 ofGX from G. This indirect band gap is verylarge, 5.344 eV. However, the VB is very flat; therefore, thedirect band gap atG is only slightly higher, 5.348 eV. Forpractical purposes, we can considerk-Al2O3 to be a directband-gap insulator.

1374 Journal of the American Ceramic Society—Yourdshahyan et al. Vol. 82, No. 6

As far as we know, there are no experimental results on theelectronic structure ofk-Al2O3 for comparison. DFT–LDA un-derestimates the band gap.40 A comparison of our results canbe made witha-Al2O3, where DFT–LDA calculations give adirect band gap of 6.6 eV atG,14 whereas the experimentalvalue is 8.8 eV.41 Therefore, our results indicate thatk-Al2O3has a smaller band gap than doesa-Al2O3.

We obtain from the curvature of the bands the effectiveelectron masses of the CB at theG point to be 0.40me in allthree directions (G → Z, G → X, andG → Y), whereme is theelectron mass. The values of the effective electron mass aresimilar to those of most semiconductors. The top of the VB,however, is very flat, giving very large values for the effective

hole masses. This flatness indicates that the correspondingelectrons are tightly bound to the host atoms.

IX. Bond Character

The distribution of the electron density around the atoms ina material reflects the type of bonding in this material. A co-valent bond is associated with a strongly direction-dependentaggregation of charge between the neighboring atoms. On theother hand, if the bond is of ionic character, the charge-densitydistribution resembles a superposition of spherically symmetricatomic charge densities. In this case, the total amount of chargearound an atom minus the charge of the nucleus is a measureof the ionicity.

Figure 11 shows the calculated valence-charge density for aplane cut through a (001) layer of oxygen atoms ink-Al2O3.Figure 11 shows that the charge distribution is almost sphericalaround the oxygen atoms. Figure 12 shows a (110) slice con-taining both oxygen and aluminum atoms. The contour linesrepresent the valence-charge density; i.e., inner shells do notcontribute. The distance between successive contour lines ismore or less uniform along a line joining an oxygen and analuminum atom; therefore, the valence-charge density decaysapproximately exponentially away from the oxygen ions. Theexponential tails reach the aluminum ions, where the valenceelectronic density has local minima. The charge density aroundthe aluminum atoms is almost zero; therefore, we conclude thatthe aluminum atoms have given away most of their valenceelectrons to the oxygen atoms. No trace of covalent bonds isshown in Fig. 12.

The ionicity of the material can be determined by summingup the charges around each atom. A straightforward method isto divide the unit cell into Voronoi cells (a generalization ofWigner–Seitz cells) and sum up the charges in each cell. How-ever, this procedure is inadequate in our case because oxygenand aluminum have different ionic radii. A better method is thefollowing: Assume that atomi with positionRi has ionic radiusbi. Then, an arbitrary pointr belongs to the Voronoi-like cellaround atomj, if

|r − Rj |bj

#|r − Ri |

bi

for all atomsi Þ j. We choose the “crystal radii” suggested byShannon37 (see his Table I), i.e., between 1.22 and 1.26 Å foroxygen, depending on the coordination number; 0.53 Å fortetrahedral aluminum; and 0.675 Å for octahedral aluminum.

The results are an ionicity of +2.79 for the octahedral alu-minum ions, and of +2.92 for the tetrahedral ones. The oxygenions have charge −1.92 or −1.80, depending on whether they

Fig. 9. Calculated band structure ofk-Al2O3 along the symmetrylines shown in the Brillouin zone (inset). Energies are relative to theFermi level calculated at the electronic temperature of 0.1 eV.

Fig. 10. Calculated orbital-resolved partial density of states (PDOS)of k-Al2O3. Energies are relative to the Fermi level calculated at theelectronic temperature of 0.1 eV.

Fig. 11. Contour plot of the valence-charge density across a (001)plane cut through a layer of oxygen ions. Contour scale is logarithmicwith a ratio between adjacent contour levels of 1.5. Dark shades cor-respond to low density.

June 1999 Theoretical Structure Determination of a Complex Material:k-Al2O3 1375

have a neighboring tetrahedral aluminum ion. The higher ionic-ity of the tetrahedral aluminum ions can probably be explainedby the relatively short distance to the neighboring atoms. Fora-Al2O3, we obtain a value for the ionicity of the aluminumions of +2.63, in good agreement with the value obtained byChing and Xu.42 Thus, we can conclude thatk-Al2O3 is evenmore ionic thana-Al2O3.

X. Conclusions

This study investigates two issues: demonstration of the ap-plicability and predictive power of DFT-based first-principlesmethods to the ground-state properties of complex materials;and the determination of the atomic structure ofk-Al2O3.

With the experimental information available at the beginningof this study as input, a first-principles method based on DFT,pseudopotentials, and plane waves has been successfully ap-plied to the identification of the atomic structure ofk-Al2O3.

This crystal, with its metastability, large unit cell, and mul-titude of structural candidates, sets high demands on compu-tation materials physics as a structure-determination method. Inparticular, it is necessary to calculate energy differences with a“chemical” accuracy (10 meV corresponding to∼100 K).Another important aspect is the relaxation of the atomic posi-tions and of the lattice parameters, which must be done verycarefully.

Based on the theoretical calculations and experimental re-sults, we conclude that the structure ofk-Al2O3 is given by thestacking sequence

AcbbgBcacgAbgcbCbabb

The notation is explained in Section IV(2). The structure isorthorhombic, with eight molecular units in the unit cell. Theoxygen ions are arranged in layers in an ABAC stacking se-quence with the aluminum ions occupying both octahedral andtetrahedral interstitial positions. One-fourth of the aluminumions are in tetrahedral coordination. The calculated lattice pa-rameters and atomic coordinates are given in Tables II and IVand the structure is shown in Figs. 5, 7, and 8.

Compared with the stablea phase,k-Al2O3 distinguishes

Panel B. Structure Determination Method

First-principles quantum-mechanical computationalmethods can be successfully used to determine theatomic structure of complex materials, including meta-stable materials. The following describes the generalprocedure that would be used for determining the crystalstructure of the metastable phases of Al2O3. The esti-mated computation times (CT, in CPU time) for con-ducting each step of the procedure for one structure can-didate ofk-Al2O3 are given in parentheses. The CTs arebased on running a fully tested and developed modernstate-of-the-art DFT–plane-wave–pseudopotentialcode43 on one (300 MHz) processor (Model Ultra 60,Sun Microsystems, Mountain View, CA).

● Obtain the symmetry group, the unit cell, and thelattice parameters of the crystal. These can be foundfrom experimental results, such as XRD experiments.

● Determine the atomic positions from experimentalinformation when possible. The Al2O3 phases are char-acterized by being composed of almost close-packedstackings of oxygen atoms, albeit in different stackingsequences. The experimental information can be used toderive the oxygen stacking sequence. The uncertaintylies in the position of the aluminum atoms, which, be-cause of the stoichiometry, are outnumbered by theavailable interstitial sites.

● List all possible aluminum sublattice configura-tions, considering the symmetry group of the crystal.

● Reduce the number of candidates by excluding theelectrostatically unfavorable cases of structures with alu-minum atoms occupying immediately adjacent sites inthe same layer and/or having many aluminum atoms inface-sharing tetrahedra.

● Perform first-principles total-energy calculationson the possible configurations. Use the experimentallydetermined lattice parameters (CT≈ 10 min/structure).Relax the atomic positions of each structure, keeping thelattice parameters constant at their experimental values(CT ≈ 20 h/structure). The relaxations show which of thestructure candidates are stable with respect to the experi-mental lattice parameters.

● Optimize the lattice parameters and atomic posi-tions more accurately for the remaining candidates (CT≈70 h/structure).

● Rule out candidates that show larger cell volumethan the one determined experimentally. Also, investi-gate whether the symmetry is conserved.

● Consider the set of (hopefully) few structure can-didates that remains: The differences between the calcu-lated total energies of these structures can then be usedas a final criterion, with a comparison of simulated XRDdiffractograms for these candidates with an experimen-tally obtained one.

The total CT for conducting the structure determina-tion of k-Al2O3 would not exceed 2.5 months of CPUtime considering current resources. The present workwas conducted on massively parallelized supercomput-ers (Model Origin 2000, SGI, and Model 10000, SunEnterprise, at Chalmers; and Model SP, IBM, at the Cen-ter for Parallel Computers (PDC) at the Royal Institute ofTechnology (KTH), Stockholm, Sweden). If 12 proces-sors running continuously on such machines are used,the total wall-clock CT would be∼5 d, not consideringthe time needed to set up the structures and analyze theoutput data (which could, however, be automated in thecomputer to a large extent).

Fig. 12. Contour plot of the valence-charge density across a (110)plane containing both aluminum and oxygen ions. Aluminum ions intetrahedral coordination are marked with “t”. Contour scale is loga-rithmic with a ratio between adjacent contour levels of 1.5. Darkshades correspond to low density.

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itself by the ABAC oxygen stacking sequence, the arrangementof the aluminum ions in zigzag lines, and the presence ofaluminum ions in tetrahedral coordination. Because of the pres-ence of tetrahedral aluminum ions ink-Al2O3, there are no facesharings between the aluminum polyhedra, which, instead, oc-cur in a-Al2O3. The arrangement of the aluminum ions inzigzag lines makes thek-Al2O3 crystal anisotropic, the alumi-num ions more closely packed in the [100] direction than in the[010] direction.

Some structural implications on the CVDk-Al2O3 growthare drawn from the determined structure. The very stablegrowth of the crystal in the [001] direction can be explained byconsidering the obtained structure, without the need to resort toimpurities. The crucial step is the nucleation of the first alu-minum layer on the substrate. Also, we find that, because of thestructural condition, the growth process is not possible alongthe [001] direction.

Calculated electronic structure and charge density for theobtained structure show a higher ionic character of the Al–Obonds, especially for the tetrahedral aluminum ions, and alower band gap than fora-Al2O3.

The study shows that computational methods based on first-principles quantum-mechanic techniques are powerful tools inthe prediction of properties of technologically important com-plex materials. The advantage of computational methods is thatthey provide full control over many of the “experimental” con-ditions. Currently, DFT is known to have great predictivepower, with an accuracy often of the order of 10 meV or 100K. The present determination of thek-Al2O3 structure and itsexperimental confirmation adds to the support for the applica-bility of DFT. The energy differences between the six struc-tures in Table III are only of the order of 0.1 eV/Al2O3, i.e.,∼20 meV per atom, and there is good hope for calculating, e.g.,energy barriers for deformation from one such structure toanother.

The energy difference between the two final structure can-didates fork-Al2O3 obtained from “post-LDA–GGA” is 0.471eV/Al2O3. However, the value obtained from self-consistentLDA is 0.436 eV/Al2O3. Thus, the accuracy of LDA is suffi-cient to determine the stability of different structures.

We hope that we have shown that it is possible to predict andunderstand properties of technologically important materials bytheoretical means, as a valuable complement to the traditionalempirical methods, and to deduce important implications onthe technological processing of materials. This should providenew opportunities for further investigations on materials andallow savings in money and time in the materials laboratory.

Acknowledgment: The allocation of computer time at the UNICC fa-cility at Goteborg University and Chalmers is gratefully acknowledged.

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Yashar Yourdshahyan has extensive experience in the computational design andcharacterization of materials. He received his M.Sc. degree in engineering physics atthe Chalmers University of Technology, Go¨teborg, Sweden, in 1995. He then joinedthe Materials and Surface Physics Theory group of the Department of Applied Phys-ics at Chalmers. Dr. Yourdshahyan earned his Ph.D. in theoretical physics in 1999,under the supervision of Professor Bengt I. Lundqvist. His Ph.D. thesis presentstheoretical density-functional studies of Al2O3 and the oxidation process of alumi-num. He received the 1998 Sandvik Coromant Materials Award for his determinationof the structure and properties ofk-Al2O3 and for showing that theoretical methodsare powerful tools in the development of industrial processes. Dr. Yourdshahyancurrently holds a postdoctoral position in the Chemistry Department of the Universityof Pennsylvania, performing research with Professor Andrew M. Rappe. His currentresearch interests include catalysis, oxidation mechanisms, ceramic oxides, andoxide–metal interfaces.

Carlo Ruberto is currently a Ph.D. student in the Materials and Surface PhysicsTheory group of the Department of Applied Physics at Chalmers University ofTechnology in Go¨teborg, Sweden, under the supervision of Professor Bengt I.Lundqvist. He joined the group in 1997, after receiving his M.Sc. degree in engi-neering physics at Chalmers. He obtained his Licentiate of Engineering degree inphysics in 1998, with a thesis based on the theoretical determination of the bulk andsurface structures ofk-Al2O3. He is currently completing work on the surface struc-ture of k-Al2O3 and is in the process of starting work on cemented-carbide/aluminainterfaces.

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Mats Halvarsson works as a researcher in the Department for Microscopy and Mi-croanalysis at Chalmers University of Technology/Go¨teborg University, Go¨teborg,Sweden. At Chalmers he received his M.Sc. in engineering physics in 1990 and hisPh.D. in physics in 1994. Dr. Halvarsson works mainly with electron microscopy andrelated techniques in the area of materials science. His research concerns mainlyCVD coatings and materials for high-temperature applications.

Lennart Bengtsson received his Ph.D. in theoretical physics at Chalmers Universityof Technology in 1999. His thesis, “Efficient Density-Functional-Based CalculationalMethods for Surfaces,” describes how efficient implementation of density-functionaltheory can be used for studying technologically interesting surface processes, such ashydrogen adsorption and crystal growth. He also has a M.Sc. degree in engineeringphysics from Chalmers.

Vratislav Langer received a M.Sc. degree in chemical physics from Charles Univer-sity in 1972 and a Ph.D. in physics and mathematics from Charles University in 1978.He has been Associate Professor in the Structural Chemistry Department at Go¨teborgUniversity since 1991. Relationships between structure and properties are among hismain research interests.

June 1999 Theoretical Structure Determination of a Complex Material:k-Al2O3 1379

Bengt I. Lundqvist received his Ph.D. in 1969 in theoretical physics at ChalmersUniversity of Technology. He has held many research and teaching positions, in-cluding professorships at Aarhus University, University of Linko¨ping, and, since1978, in mathematical physics at Go¨teborg University and Chalmers. He has heldvisiting positions at Cornell University, IBM, Rutgers University, NORDITA, andthe Technical University of Denmark. He has had many academic commitments atdifferent levels, including department chair, vice-dean, and board member for Chalm-ers, dean of the physics faculty, chair for physics and Chalmers computer services,member of the boards of the Surface and Interface Section and the Condensed MatterDivision of the European Physical Society, chair of the Condensed Matter Divisionof the Swedish Physical Society, and currently is the leader of the Swedish MaterialsConsortium No. 9. He has about 40 years of experience of university teaching at mostlevels and forms, supervised more than 20 graduate students to Ph.D. exams, andpublished about 130 papers. His research is in theoretical surfaces, materials, andcondensed-matter physics.

Sakari Ruppi is a Senior Company Scientist at Seco Tools AB, Fagersta, Sweden. Hereceived his M.Sc. degree in materials science and process metallurgy from HelsinkiUniversity of Technology, Finland, in 1977, and his Ph.D. in applied physics from theTechnical University of Denmark in 1982. Dr. Ruppi was a Senior Scientist at theAcademy of Finland from 1995 to 1998 and has been Docent at Helsinki Universityof Technology since 1986. He joined Seco Tools AB in 1988 and is currentlyresponsible for research and development of CVD processes and materials and holdsmore than 25 patents in this area. Dr. Ruppi’s current research focuses on deposition,crystal structures, phase transformations, and wear mechanisms of CVD and PVDcoatings.

Ulf Rolander is currently Manager of the Raw Materials and Powder TechnologyDepartment at AB Sandvik Coromant R&D facilities in Stockholm, Sweden. Hereceived his Ph.D. in physics at Chalmers University of Technology in 1991 andjoined AB Sandvik Coromant in 1992. Since then, Dr. Rolander has been involved inseveral research and product-development projects concerning materials and pro-cesses for cutting-tool applications.

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