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matériaux ultra-durs dans le ternaire B C N.
Maurizio Mattesini
To cite this version:
Maurizio Mattesini. Proposition et modélisation ab initio de nouveaux matériaux ultra-durs dans le
ternaire B C N.. Other. Université Sciences et Technologies - Bordeaux I, 2001. English. �tel-00003143�
N d'ordre: 2429 TH ESE PR ESENT EE A L'UNIVERSIT E DE BORDEAUX I
ECOLE DOCTORALE DES SCIENCES CHIMIQUES
Par
Maurizio MATTESINI
POUR OBTENIR LE GRADE DE
DOCTEUR
Spe ialite: PHYSICO-CHIMIE DE LA MATI
ERECONDENS EE PROPOSITION ET MOD
ELISATION AB INITIO DE NOUVEAUX
MAT
ERIAUX ULTRA-DURS DANS LE TERNAIRE BCN
A soutenirle 23Novembre2001 apresavisde:
M. P. MOHN Professeur Rapporteurs
M. S.CSILLAG Professeur
Devant laCommissiond'examenformee de:
M. J.ETOURNEAU Professeur President
M. P. MOHN Professeur Rapporteurs
M. S.CSILLAG Professeur
M. H.DREYSSE Professeur Examinateurs
M. R.AHUJA Do teur
Due to the te hnologi al importan ebehind the possibility to dis over novel lasses of
hardmaterialsan enormousresear he orthasbeendire tedduringthelastde ades
to-wards the synthesis and hara terisationof promising arbon-based ompoundssu has
arbonnitridesand boron arbonnitrides. However, despitemanyattemptsofsynthesis
andtheindisputableprogressesmadeinthe eld,amorphous sampleswithun lear
rys-tallographi datahavebeenoftenobtainedinmanyresear h laboratories. In parti ular,
severalproblemsarisefrom thefa tthatmostofthesamplesareofpolymorphi nature,
thusleadingto a diÆ ultand un ertainspe tros opi hara terisation.
A general understanding of the relations between omposition and the ele troni
stru turepropertieshasthereforebeenprovidedtheoreti allyinthisThesistogetfurther
insightinto the hara teristi s of pure rystallineforms. Asone might expe tthiswork
hassuddenlybeenturnedoutintoa ompli ate and hallengingtaskbe ause ofthela k
ofreliableexperimental rystalstru turesto beusedasreferen esforthe omputational
inputs. Therefore itbe ame essentialto proposehypotheti albi- and three-dimensional
model phasesto obtain trendson the relative stability, ele troni and me hani al
prop-erties of arbon- and boron arbon-nitrides. So far as that is on erned, a systemati
study of pure rystalline CN x
(where x=0.36 and 1.33) and BC
2
N systems has been
proposed as an important omplement to the experimental knowledge. Thanks to the
progress in modern omputer te hnology it has also been possible to ompute su h an
investigationviaab-initio( rst-prin iples)methodsbytestingandprobingdi erentsolid
state al ulationalapproa hes. Infa t,one ofthe rstobje tivesofthisproje thasbeen
the sear h of a valid omputational density-fun tional-based s heme able to reprodu e
and/orpredi t thehardnessand stabilityof awide varietyofultra-hard materials.
Cal ulations of the ohesive properties and standard enthalpies of formation have
been arriedouttoaddressthethermodynami stabilityofdi erentisoele troni
ompo-sitions,namelyC 3 N 4 ,C 11 N 4 andBC 2
N.Thehardnesshasalsobeenstudiedbymeansof
theanalysisofthe al ulatedelasti andbulkmoduli. Theinvestigationoftheele troni
propertieshasbeena hievedwiththe al ulationofthedensityofstates,bandstru ture,
ele tron density maps and rystal orbital overlap population analysis. For some of the
studiedmole ular lusters,the 13
CNMRshiftshavebeenevaluatedtoprovidea
spe tro-s opi dis riminationbetweensystems withvery similarstru tural hara teristi s. This
is the ase of the hexagonal and orthorhombi modelsof the graphiti -like C 3
N 4
form.
Finally, the determination of the ele tron-energy loss near edge stru tures of C, B and
N K ionisation edges has been omputed in order to provide referen e spe tra of pure
rystallinematerials,likelyto allow adis riminationofpolymorphi samples.
Results are presented to demonstrate that arbon nitrides are ultra-hard systems
with outstanding me hani al properties. In parti ular, the arbon ri h omposition,
C 11
N 4
analogueC 3
N 4
. However,thepossibilitytodepositsinglephasesamplesshouldbehighly
hampered inbothstoi hiometriesbytheirlargepositiveenthalpiesof formation.
The introdu tion of boron atoms (boron arbon nitrides) has displayed a slight
de- reasing inthemagnitudes of theelasti and bulkmoduli,though the al ulated values
arestillhigherthanthatof ubi boronnitride(i.e. these ondhardestknownmaterial).
Nevertheless, three-dimensional BC 2
N phases have also shown exothermi enthalpiesof
formation whi h point to an easier deposition of the \BCN" materials with respe t to
arbonnitrides. Therefore,by onsideringthewholesetoftheinvestigatedmodelphases,
sp 3
-bondedboron arbon nitridesresult asthebest andidatesfornovel ultra-hard
ma-terialswhi h ould, inprin iple,be synthesisedwiththe a tualte hniques. Very re ent
Compte tenu des enjeux te hnologiques qui sous-tendent la de ouverte de nouvelles lasses
de materiaux ultra-durs, des e orts de re her he onsiderables ont ete destines durant les
deuxdernieres de ades ala synthese et a la ara terisation de omposes legers prometteurs
tels queles nitrures et boronitrures de arbone.
Cependant, malgre de nombreuses tentatives de synthese et les progres indis utables
realises dans e domaine, seuls des e hantillons amorphes (mal ara terises du point de la
ristallographie)ontpu^etreobtenusdansdi erentslaboratoiresdere her he.Enparti ulier,
plusieurs problemes sont souleves de par la nature polymorphe des e hantillons produits,
onduisantde e faitaune ara terisation spe tros opique peupre ise.
Par onsequent l'etablissement de relations entre omposition et proprietes destru ture
ele tronique est fourni sur une base theorique dans ette These a n d'approfondir les
a-ra teristiques des formes ristallines des materiaux. Comme on pouvait s'y attendre ette
t^a he omplexeest vitedevenueunde omptetenudu manquededonneesexperimentales
pourles stru tures ristallinessus eptiblesdeservir depoint dedepart aux al uls.
Il devintalors essentielde proposer des phasesmodeles (hypothetiques) auxe helles
bi-et tri-dimensionnelles pour etablir des tendan es omparatives sur les stabilites, proprietes
ele troniquesetme aniquesdesnitruresetboronitruresde arbone.Enparti ulier,lesetudes
systematiquesdessystemes ristallinsbinairesCN x
(o ux=0,36et1,33)d'unepartetternaires
BC 2
Nd'autrepartontetemeneesetpresentees ommeunefor edepropositionvisavisdes
experimentateurs.
Gr^a e aux enormes progres de la te hnologie moderne des ordinateurs, il a ete possible
demener esetudes au moyende methodes ab initio (des le depart) en testantet sondant
di erentesappro hesdel'etudedusolide.Enfait,l'undespremiersobje tifsdemontravailde
Theseaetedevaliderlemeilleurs hema al ulationnelauseindelatheoriedelafon tionnelle
densite,DFT,sus eptibledereproduireet/oudepredireladurete etlastabilited'unegrande
variete demateriauxultra-durs.
Les al uls des proprietes de ohesion et les enthalpies standard de formation ont ete
entreprises a n d'expliquer la stabilite thermodynamique des di erentes ompositions
iso-ele troniques, nommement C 3 N 4 , C 11 N 4 et BC 2
N. La durete a ete egalement etudiee au
moyen de l'analyse des modules d'elasti ite et de ompressibilite. L'examen des proprietes
de stru ture ele tronique a ete realise par le al ul des densites d'etats, de la stru ture de
bandes d'energie, des artes de densite ele tronique et des populations de re ouvrement.
L'etude des depla ements himiques par RMNdu 13
C de lusters mole ulaires a permis de
fournir un moyen de dis rimination entre systemes ayant des ara teristiques stru turales
tres voisines. C'est notamment le asdes stru tures hexagonaleet orthorhombique deC 3
N 4
graphitique. En n, les seuils d'ionisation K de C, B et N ont ete al ules (spe tros opie
ele tronique par perte d'energie \EELS") pour les di erentes stru tures ristallines a n de
e hantillonspolymorphes.
Lesresultatsdemontrentquelesnitruresde arboneetudiessontdesmateriauxultra-durs
ayant desproprietes me aniques ex eptionnelles.En parti ulier,les phasesdela omposition
ri he en arbone, C 11
N 4
, montrent des energies de ohesion superieures et se presentent
omme plusduresque l'analogueiso-ele troniqueC 3
N 4
. Neanmoins lapossibilitededeposer
desstoe hiometriesmonophasiques seraitpenalisee pourles deux ompositions omptetenu
deleurs energies deformation fortement positives.
L'introdu tiond'atomesdebore(boronitruesde arbone) onduitaunelegerediminution
desamplitudesdesmodulesd'elasti iteetde ompressibilite.Maislesvaleurs al uleesrestent
superieures a elles de BN ubique, le se ond meilleur materiau ultra-dur onnu apres le
diamant.Neanmoinsles phasestri-dimensionnellesBC 2
Nanalyseespresententdesenthalpies
deformation nettementexothermiques, e quiestenfaveurd'une preparation (pardep^otde
ou hes min es par exemple) plus aisee de phases \BCN" par rapport aux nitrures binaires
CN x
pourlesquelsH 0
f
>0.Par onsequenten onsiderantl'ensembledessystemesmodeles,
lesphases\BCN"a liaisonshybridees essentiellementsp 3
(tri-dimensionnelles) se presentent
omme les meilleurs andidats pour de nouveaux materiaux ultra-durs a base d'elements
legers sus eptiblesd'^etresynthetises parles moyensa tuels. Ces observationssontappuyees
This Thesis illustrates the work that I arried out between 1998 and 2001 at the
In-stitut de Chimie de la Matiere Condensee de Bordeaux (ICMCB-CNRS), University of
Bordeaux I. The purposeof my resear h withintheEuropean Trainingand Mobilityof
Resear hers (TMR) Network
1
has been the hara terisation of the properties of
di er-ent arbon-andboron arbon-nitride ompoundsbyattested, highlya urateele troni
stru ture al ulations. Inparti ular,themodellingofnovelpotentialhardmaterialslike
binaryCN x and ternaryB x C y N z
have beenaddressed.
When I started my work in November 1998 there were already several published
s ienti papers(boththeoreti alandexperimental)dealingwiththedistin tfeaturesof
novel ompounds,quiteoften alledsuper-orultra-hardmaterials,that ouldinprin iple
ompete with the hardness of the onventional diamond. However, one of the greatest
attra tions of this subje t that has always appeared important to me is the lose link
existing between hardness and phase stability on the one hand and the bonding and
stru ture of the material on the other. The onne tion between these two aspe ts has
beento some degree proved inthisThesisto be one of theessentialprin ipleson whi h
thedevelopment of thenext generation's hardmaterialsshouldbebased.
Althoughmostoftheinvestigationswereperformedat thesolidstatelevel,thestudy
ofsome mole ular lustershasalso beensu essfullyintegratedfortheevaluationofthe
13
CNMR hemi alshifts. Thelargestpartofthe al ulationshavebeena hievedbyusing
the omputationalfa ilitiesof theintensive entre of al ulation \p^o leModelisation
Mi- ros opiqueetMesos opiqueenPhysique,dans l'EnvironnementetenChimie"(M3PEC)
of theUniversity of Bordeaux I. The resultsobtained have beenwell re eived in an
ex- hange ofinformationwiththe otherpartners of theEuropean ommission.
Thepresentmanus riptshowsanintrodu torypartintendedtoexplainsome spe i
on eptsabouthardmaterialsandto overthebasi ideasbehindtheemployed
theoret-i almethods. These ondpartisspe i allydedi atedtothethoroughdes riptionofthe
resultsobtainedduringthestudyof arbon nitrideand boron arbon nitridesystems.
Bordeaux,September2001
Maurizio Mattesini
1
Frequentlyused abbreviations:
APW Augmentedplane wave
ASA Atomi sphereapproximation
ASW Augmentedspheri al wave
b Body entered ubi
COOP Crystalorbitaloverlap population
CVD Chemi alvapordeposition
DFT Di erent densityfun tionaltheory
DOS Densityof states
EELS Ele tron energylossspe tros opy
E F Fermi energy E g Band gap
ELNES Energylossnear edgestru ture
f Fa e entered ubi
FFT Fast fouriestransforms
FP-LAPW Full-potentiallinearizedaugmentedplane wave
GGA Generalizedgradient approximation
h p Hexagonal losepa ked
ICOOP Integrated rystal orbitaloverlappopulation
KS orbitals Kohn-sham orbitals
LAPW Linearizedaugmented planewave
LDA Lo aldensityapproximation
LMTO LinearmuÆn tinorbital
NMR Nu lear magneti resonan e
PP Pseudo-potential
PVD Physi alvapordeposition
R mt MuÆn-tinradius sp 2 , sp 3
Ele tron orbitalhybridization
US-PP Ultra-soft pseudo-potential
, Bondingtypes
1. Relative stabilities, bulk moduli and ele troni stru ture properties of
di erent ultra hard materials investigated within the lo al spin density
fun tionalapproximation,M. Mattesini, S.F.Matar,A.Snis,J.Etourneau,A.
Mavromaras, J.Mater. Chem., 9, 3151 (1999).
2. Stability and ele troni properties investigations of the graphiti C 3
N 4
systemshowingan orthorhombi unit ell, M. Mattesini,S.F.Matar andJ.
Etourneau, J.Mater. Chem., 10, 709-713 (2000).
3. First-prin iples hara terisation of new ternary heterodiamond BC
2 N
phases,M. MattesiniandS.F.Matar, Comput. Mat. S i.,20/1,107-119 (2001).
4. Abinitio sear h of arbon nitrides, isoele troni with diamond, likely to
lead to new ultra-hard materials, S.F. Matar and M. Mattesini, C.R. A ad.
S i. Paris, 4,255 (2001).
5. Sear h for ultra-hard materials: theoreti al hara terisation of novel
orthorhombi BC
2
N rystals, M. Mattesini and S. F. Matar, Int. Jour. of
Inorgani Materials, inpress (2001).
6. DFTinvestigationofhardness,stabilityandele tron-energy-lossspe tra
of arbonnitridesintheC 11
N 4
Stoi hiometry,M. MattesiniandS.F.Matar,
TitlePage . . . i
Abstra t . . . ii
Resume . . . iv
Prefa e . . . vi
Nomen lature . . . vii
PublishedPapers . . . viii
Tableof Contents. . . xi
Listof Figures . . . xv
Listof Tables . . . xix
1 Introdu tion 1 1.1 The interestinnovel ultra-hardmaterials . . . 1
1.2 Aimsof theThesis . . . 2
1.3 Outlineof theThesis . . . 3
2 The Hardness and Covalen y 5 2.1 Firsttheoreti al propositionof CarbonNitrides asnovelhardmaterials . 5 2.2 Ele tron ount onsiderations . . . 6
3 The on ept of Hardness 9 3.1 Introdu tion. . . 9
3.1.1 Measure oftheresistan e uponvolume hange insolids . . . 10
3.1.2 Resistan e to reversible deformationuponshape hange . . . 11
4 Density Fun tional Theory 13 4.1 Introdu tion. . . 13
4.2 The basi prin iplesof themethod . . . 14
4.3 Singleparti leKohn-Shamequations . . . 16
4.3.1 The basissets . . . 19
5 Planewave Pseudo-Potential methods 21
5.1 Introdu tion. . . 21
5.2 Blo h's Theoremand Planewaves . . . 22
5.3 GeneralApproximations . . . 23
5.4 Pseudo-Potentials. . . 23
5.4.1 Norm onserving pseudo-potentials . . . 23
5.4.2 Ultrasoft Pseudo-Potentials (US-PP) . . . 24
5.4.3 Generation of theUS-PP . . . 26
6 The Full Potential LAPW method 29 6.1 Introdu tion. . . 29
6.2 The LAPWbasis . . . 31
7 The ASW method 33 7.1 Aboutlinearmethods . . . 33
7.1.1 ASW and LMTOmethods . . . 33
7.1.2 The ASAand its impli ations . . . 34
7.1.3 Solution of thewave fun tion . . . 34
7.2 Furtherformalismwith theASWmethod . . . 34
7.2.1 The augmentationpro ess . . . 35
7.2.2 The variationalmethod of Rayleigh-Ritz . . . 38
8 Carbon Nitrides 40 8.1 Introdu tion. . . 40 8.2 Study ofthe C 3 N 4 stoi hiometry . . . 42
8.2.1 Methodsand omputationaldetails. . . 42
8.2.2 Stru tural modelsfortheC 3 N 4 stoi hiometry . . . 43
8.2.3 Relativestabilityof variuos C 3 N 4 phases. . . 44
8.2.4 Hardness . . . 51
8.2.5 Hexagonal and Orthorhombi graphiti -C 3 N 4 . . . 55 8.2.6 Cal ulation ofthe 13 CNMR hemi alshifts . . . 66 8.2.7 Con lusions . . . 70 8.3 The isoele troni C 11 N 4 model system . . . 72 8.3.1 Introdu tion . . . 72
8.3.2 Methodsand omputationaldetails. . . 73
8.3.3 The analysed rystallinestru tures . . . 73
8.3.4 Relativestabilityand phasetransitions . . . 76
8.3.5 Cal ulationsof theelasti and bulkmoduli . . . 83
8.3.6 Ele troni stru ture . . . 89
9 Boron Carbon Nitrides 98
9.1 Ternary BCN ompounds . . . 98
9.2 Settingup novelthree-dimensionalBC 2 Nphases . . . 100
9.2.1 Cubi andhexagonal diamond . . . 100
9.2.2 Carbonsubstitution . . . 101
9.3 Computational details . . . 105
9.4 BC 2 N phasesand theirrelativestability . . . 106
9.4.1 Enthalpyof formation . . . 108
9.4.2 Dis ussionof theresults . . . 111
9.5 Theoreti alestimation ofhardness . . . 114
9.5.1 Me hani al stability . . . 118
9.6 Ele troni densityof statesand bandstru ture . . . 120
9.6.1 The orthorhombi phases(I and II) . . . 120
9.6.2 The trigonalmodel stru ture(III-BC 2 N) . . . 127
9.7 Theoreti alELNES forBC 2 N model systems . . . 127 9.7.1 The layered BC 2 N model system . . . 131 9.8 Con lusions . . . 135
10Summary and Outlook 141 10.1 Carbon Nitrides. . . 141
10.2 Boron CarbonNitrides . . . 142
10.3 Prospe tive studies and \what'sleft" . . . 143
11Con lusions 144 11.1 Nitruresde Carbone . . . 144
11.2 Boronitruresde Carbone . . . 145
11.3 Prospe tives et \ e qui reste a faire" . . . 146
Bibliography 158
2.1 S hemati ternary omposition diagram indi ating di erent "hard"
stoi- hiometries. . . 8
4.1 S hemati representationofvariousDFT-based methodsof al ulation. . 17
4.2 Flow- hartforself- onsistentdensityfun tional al ulations.. . . 18
5.1 Illustrationdiagramof therepla ementof the"all-ele tron"wavefun tion
and orepotential bya pseudo-wavefun tionand pseudo-potential. . . 24
6.1 Adaptation of the basis set by dividing the unit ell into atomi spheres
and interstitialregions.. . . 30
7.1 Comparison between the augmented plane (APW) and spheri al (ASW)
waves.ThisFigurehasbeentakenfromtheoriginalworkofA.R.Williams,
J.Kublerand C. D.Jr. Gelatt [74 ℄.. . . 36
8.1 -C
3 N
4
model system. Carbon and Nitrogen are depi ted in grey and
white,respe tively. This olors heme iskept throughoutall theThesis. . 45
8.2 One layerof thehexagonal graphiti -C 3
N 4
model.. . . 45
8.3 One layerof theorthorhombi graphiti -C 3
N 4
phase. . . 46
8.4 Relativestabilitybetweendi erentC 3
N 4
phasesbyusingdi erentmethod
of al ulations. . . 46
8.5 Energy dependen e of the unit ell volume for ubi -C 3
N 4
as a fun tion
of threedi erent al ulationalmethods. Datapoint have been tted with
theBir h type EOS. . . 53
8.6 FigurefromR.Riedel[149℄ showingthes atteringoftheVi kershardness
forhardmaterialswhen ompared withbulkand shearmoduli. . . 54
8.7 Ele tron ir ulationinthehexagonal graphiti -C 3
N 4
model. . . 57
8.8 Ele tron ir ulationintheorthorhombi graphiti -C 3
N 4
model. . . 57
8.9 The above gure shows the general di eren es in the ring's geometry for
8.10 Total COOPforthehexagonal and theorthorhombi phases(ASW). . . . 63
8.11 IntegratedCOOPforthehexagonal and theorthorhombi systems (ASW). 63
8.12 TotalCOOPfortheorthorhombi phase(ASW).For larityea h
nitrogen- arbonintera tionshavebeenshiftedalongtheverti alaxis.ThelabelsB
and ABde nethebonding andthe antibondingregion, respe tively. . . . 64
8.13 Siteproje tedDOSplotfortheAAAorthorhombi graphiti phase(ASW).
The energy referen e along the x-axis is taken with respe t to the Fermi
level;the y-axisgives theDOSperatom and unitenergy. . . 65
8.14 Site proje ted DOS forthehexagonal graphiti model system(ASW). . . 66
8.15 Valen eele tron densitymap fortheorthorhombi graphiti -C 3
N 4
model
system(FP-LAPW). . . 67
8.16 Total DOS for the orthorhombi phase (FP-LAPW). Noti e the absen e
of energygap at thetopof theVB. . . 68
8.17 Total DOS forthehexagonal phase (FP-LAPW). . . 68
8.18 Mole ular lusterrelative to thehexagonal graphiti -C 3
N 4
. . . 69
8.19 Mole ular lusterrelative to theorthorhombi graphiti -C 3
N 4
.. . . 70
8.20 Ballandsti kmodelofthebl-C 3
N 4
stru ture.Figureshowstheproje tion
of theatomsalong the[001℄ plane. . . 75
8.21 One layerof thegraphiti -C 11
N 4
model phase.. . . 75
8.22 Crystalstru tureof thetetragonal C 11
N 4
. Proje tion along the[100℄
plane exhibitingthe \nitrogen-hole". . . 77
8.23 Proje tionoftheorthorhombi C 11
N 4
rystalstru turealongthe[010℄
plane. . . 77
8.24 Free energies (eV/atom) versus atomi volumes ( A 3
/atom) for various
C 3 N 4 and C 11 N 4 phases(US-PP). . . 80
8.25 Front view of the\ arbon-hole" in -C 11
N 4
. . . 81
8.26 The al ulatedDOS forthebl-C 3
N 4
phase(FP-LAPW). . . 90
8.27 The al ulatedtotal DOS forthebl-C 3 N 4 and -, C 11 N 4 (FP-LAPW). 92
8.28 Theoreti alC KELNESof diamondand graphite(FP-LAPW). . . 94
8.29 Theoreti al C K ELNES of various phases (FP-LAPW). The spe tra for
theinequivalentatomspositionshavebeen al ulatedseparatelyand
weigh-tedinthepresentFigure. . . 95
8.30 Theoreti al N K ELNES of various phases (FP-LAPW). As in Fig. 8.29
inequivalent atoms have been al ulated separately and weighted in the
present spe tra. . . 96
9.1 Unit ell of ubi diamond. This stru ture was rst determined in 1913
by W. H. and W. L. Bragg [195 ℄. That was also the rst time that the
stru tureofanelementwasdeterminedbytheuseofX-raydi ra tion[196℄.100
9.3 Crystalstru tureoftheorthorhombi I-BC 2
N.Carbon,nitrogenandboron
atomsare depi tedinbla k,white and grey, respe tively. . . 104
9.4 Crystalstru tureof theorthorhombi II-BC 2
N. . . 105
9.5 Crystalstru tureof thetrigonalIII-BC 2
N phase. . . 106
9.6 Cohesiveenergies(eV/atom)asafun tionoftheatomi volume( A 3
/atom)
forthestartingmaterialsandBC 2
Nstru tures.The urvesweregenerated
withtheUS-PP/LDA method. . . 109
9.7 Crystalstru tureof theorthorhombi graphiti -BC 2
N modelphase.. . . . 110
9.8 Crystalstru tureof theh-BN.. . . 112
9.9 Unit ellofthe -BN. . . 113
9.10 Energyversus pressurefordi erent BC 2
N phases(US-PP). . . 114
9.11 Idealised hemi alenvironment around theB/N sitein -BN and various
BC 2
N phases.Part(a)ofthes heme referstotheorthorhombi phases(I
and II)while,part (b) on ernsthelo al hemi albondingofthephase III.115
9.12 Valen e ele tron densitymap showingthe polarisation of the C-C bonds
inI-BC 2
N.. . . 117
9.13 The al ulatedpartialdensityof statesof I-BC 2
N. . . 121
9.14 Band stru ture of I-BC 2
N along the symmetry linesof theorthorhombi
Brillouinzone:X=(0 1 2 0)! =(000)!Z=(00 1 2 )!U=(0 1 2 1 2 )!R=( 1 2 1 2 1 2 ) ! S=( 1 2 1 2 0) ! Y=( 1 2 00) ! =(000). . . 122
9.15 The al ulatedpartialdensityof statesof II-BC 2
N. . . 124
9.16 Bandstru tureof II-BC 2
N alongthesymmetry linesoftheorthorhombi
Brillouinzone:X=(0 1 2 0)! =(000)!Z=(00 1 2 )!U=(0 1 2 1 2 )!R=( 1 2 1 2 1 2 ) ! S=( 1 2 1 2 0) ! Y=( 1 2 00) ! =(000). . . 125
9.17 Total DOS fortheorthorhombi phasesIand IIin arbitraryunits. . . 126
9.18 The al ulatedpartialdensityof statesof III-BC 2
N. . . 128
9.19 Band stru ture of III-BC 2
N along the symmetry lines of the hexagonal
Brillouinzone: =(000)!M=(0 1 2 0)!K=( 1 3 2 3 0)! =(000)!A=(00 1 2 ) ! L=(0 1 2 1 2 ) ! H=( 1 3 2 3 1 2 ) ! A=(00 1 2 ). . . 129
9.20 Theoreti alCKELNESofvariousphases(FP-LAPW).Theexperimental
CVDdiamondspe tra[204℄hasbeenshiftedby+1.05eValongtheenergy
axisinorder to alignits rst
peakwith theone ofthe theoreti al urve.132
9.21 Theoreti al N K ELNES of various phases (FP-LAPW). The spe tra of
thehigh pressuresynthesised -BN[205℄ hasbeenmovedby+2.05 eV to
mat h the rsttheoreti al
peak. . . 133
9.22 Theoreti alBKELNESofvariousphases(FP-LAPW).Thehighpressure
synthesised -BN spe tra [205 ℄ has beenshifted by +3.15 eV inorder to
alignthe rst
peakwith thetheoreti al urve. . . 134
9.24 Theoreti alN KELNESof graphiti -BC 2
N in omparisonto h-BN. . . 138
9.25 Theoreti alBK ELNESofgraphiti -BC 2 N in omparisonto h-BN. . . 139 10.1 The three-dimensionalC 7 N 4
model system.Ongoing al ulationsseemto
indi ate the same general tenden y found for the C 3 N 4 and C 11 N 4 om-positions. . . 143
2.1 Hardness of minerals and some syntheti erami s a ording to F. Mohs.
Forsyntheti materialsmi ro-hardnessvaluesaregiven inunitsof Knoop
s ale. Valuesareshown as ompiledbyR. RiedelinRef.[25 ℄. . . 7
5.1 Parametersdeterminingtheultra-softpseudo-potentialusedinthisThesis.
ARCrepresentstheatomi referen e on gurationand r ;l
(where l=s, p,
d) the ut-o radii inatomi units. . . 27
8.1 Total energiesand densities fordi erent C 3
N 4
phases. Energy valuesare
expressed in eV/C 3
N 4
unit and are s aled with respe t to the stable
graphiti -C 3
N 4
form. Pseudo-potential al ulations refer to the work of
D. M. Teter and R.J.Hemley[29 ℄. . . 47
8.2 Cohesiveenergies(eV/atom) ofdi erent C 3
N 4
modelsystems.Valuesare
onfronted with those of the starting materials : diamond/graphite and
N 2
. For the al ulations of the nitrogen dimer it has been used a simple
ubi ell (a=10
A) with atoms displa ed along the diagonal dire tion.
It should be noted that an overbinding of more than 1 eV/atom is not
unusualinlo al-density al ulationsforse ond-periodelementalsolids,as
forexamplediamond[136 , 137 ℄. . . 49
8.3 Cal ulatedenthalpyofformation,H o
f;0
(kJ/mol),fordi erent
ex hange- orrelation fun tionals.Theabovetable showsonlyvaluesrepresentatives
forthelayeredgraphiti -C3
4
Nandthethree-dimensionalbl-C3
4
N.The
om-pletelistof enthalpiesisgiven inTab.8.18of Se tion8.3, p.72. . . 50
8.4 Equilibriumlatti e onstants(a o
)fortheinvestigatedmodelsystems.The
energyvs. volumedata were tted witha thirdorder Bir h equation.. . . 51
8.5 Bulkmodulus,B (GPa) and its pressurederivatives, B 0
(valuesin
paren-thesis)forvariousC 3
N 4
phasesand diamond. . . 52
8.6 Cal ulatedelasti onstants( ij
inGPa),atomi densities(ing/ m 3
)and
isotropi shear moduli(G in GPa) for ve di erent C 3
N 4
phases. Values
8.7 Cal ulated elasti onstants (GPa) and bulk moduli (GPa) for diamond
as a fun tion of di erent ex hange- orrelation methods : Perdew-Wang
91 (PW91) [47 ℄, Perdew-Be ke (PB) [46 ℄, Perdew-Wang 86 (PW86) [57 ℄,
Langreth-Mehl-Hu (LM) [45 ℄. The subs ript \relaxed" and \frozen"
de-notes values al ulated withorwithouttherelaxation of theatomi
posi-tions.. . . 56
8.8 Cal ulated elasti onstants(GPa) and bulkmoduli(GPa)forbl-C 3
N 4
as
a fun tionofdi erent ex hange- orrelation fun tionals.. . . 56
8.9 Stru turalparameters fortheorthorhombi stru turewithAAAsta king
order. . . 59
8.10 Bondlengthsbeforeandaftertheoptimisationoftheorthorhombi stru ture. 60
8.11 Angles before and after the optimisation of the orthorhombi stru ture.
The notationprimerefers to atomsbelongingtheadia ent unit ell. . . . 60
8.12 FP-LAPWand US-PP ohesive energies(eV/atom) fortheorthorhombi
and hexagonal latti es. . . 62
8.13 Cal ulationsof the 13
CNMR hemi alshift(ppm)for thetwo
graphiti -like phases. For thereferen e TMS it hasbeenestimated, with the same
omputationalapproa h,a hemi al shiftof184.4 ppm. . . 71
8.14 Optimised parameters for the bl-C 3 N 4 and the -, -C 11 N 4 phases. The
tableshows rystalsystem,spa egroup,atomsunit ell 1
andtheatomi
positions. Cell onstants areexpressed in unitof
A and theangles , ,
indegrees.. . . 76
8.15 Optimisedparameters forthegraphiti -C 11
N 4
phase. . . 78
8.16 Cohesiveenergy, E(eV/atom), forvariousCN x
phases. Free energy
va-lues are s aled with respe t to the stable graphiti -C 11
N 4
stru ture. The
ratio of the numberof hemi al bondsperunit ell, R
(C C=C N)
, is also
shown. . . 79
8.17 Cal ulatedFP-LAPW ohesiveenergiesofgraphiti - and -C 11
N 4
.Values
aregiven ineV/atom. . . 82
8.18 Computedstandardmolarenthalpyofformation(H o
f;0
)forthetwoCN x
stoi hiometries (x=1.33 and 0.36) by usingthe ohesive energies seen in
Tabs. 8.2, 8.12 and 8.17. Values in parenthesis orrespond to the use of
graphite asastartingmaterial. . . 82
8.19 Strains andelasti moduliforthe orthorhombi phase. . . 84
8.20 Strains andelasti moduliforthe tetragonalphase. . . 85
8.21 Strains and elasti moduli for a ubi system. By al ulating the
tetra-gonal shear onstant, C 0 = 1 2 ( 11 12
) , and the bulk modulus, B =
1 ( 11 +2 12 ), itispossibleto extra t 11 and 12 . . . 86
8.22 Theoreti al values of the elasti onstants ( ij
in GPa), isotropi shear
modulus (G in GPa), bulk modulus (B in GPa), its pressure derivative
(B 0 ), atomi volume (V o in A 3
/atom), ohesive energy (E o
in eV/atom)
and atomi densities ( ing/ m 3 ) of bl-C 3 N 4 and -, -C 11 N 4 . Values in
roundbra ketsrefertotheworkofA.Y.LiuandR.M.Wentz ovit h[114 ℄
whereasthose insquarebra kets on ernthebulkmodulus al ulatedby
ombining theelasti onstants. . . 87
8.23 Table shows the al ulated B/G ratio, Young'smodulus(GPa) and
Pois-son'sratio(dimensionless)ofbl-C 3 N 4 and -, -C 11 N 4
. Diamondhasalso
beenlistedas a referen e material.Values inround bra kets on ern the
propertiesof CVDdiamondas ompiledinRef. [178 ℄. . . 89
8.24 Positions of peaks I-V in the spe tra of Fig. 8.28. All the positions are
s aledwith respe tto the main
peak II. Values areinunits of eV.(y)
Valuesas ompiledinRef.[181℄. . . 93
9.1 Crystal stru ture data for ubi and hexagonal diamond. Cell onstants
valuesareexpressedin unitof
A. . . 102
9.2 Substitutionofthe arbon atomsinthef diamond.. . . 102
9.3 OptimisedparametersforheterodiamondBC 2
Nstru tures.Cell onstants
and bonddistan esare giveninunit of
A. . . 103
9.4 Substitutionofthe arbon atomsinthehexagonal diamond.. . . 104
9.5 Stru tural and ohesive properties of various phases : atomi volume V o
( A 3
),bulkmodulusB(GPa), pressurederivativesB 0
and ohesiveenergy
E oh:
(eV/atom).Thelattervalueshavebeenobtainedbytakingthe
di e-ren ebetweenthetotal energyofthesolidsand theground-stateenergies
of the spheri al non spin-polarised atoms. No orre tion for zero-point
motionhas beenmade.. . . 107
9.6 Cal ulated energy di eren e, E (eV/atom), for various phases relative
to thegraphiti -BC 2
Nform. . . 108
9.7 Optimisedparameters forthegraphiti -BC 2
Nmodelphase. . . 111
9.8 Cal ulated ohesive energies(E oh:
ineV/atom) forvariousBC 2
N phases
and some of the starting materials as a fun tion of di erent
ex hange- orrelation fun tionals. . . 116
9.9 Cal ulated standard enthalpyof formation. Values in parenthesis
orres-pondtotheformationenergyofgraphiti -BC 2
Nwhengraphiteistaken as
a startingmaterial. . . 116
9.10 Cal ulatedenthalpyofformationfortherea tion: -BN ( ) +2C ( ) !BC 2 N ( ) .
Valuesinparenthesis orrespondtotheformationenthalpyofthe
9.11 Strainsandelasti moduliforthetrigonalphase.UnlistedÆ ij
aresetequal
to zero. . . 118
9.12 Independent elasti onstants, ij
, and isotropi shear modulifor BC 2
N,
diamond,lonsdaleiteand -BN. Valuesare expressedinunitsof GPa. . . 119
9.13 Theabove tableshows the al ulatedB/G ratio,Young'smodulus(GPa)
and Poisson's ratioofthestudiedBC 2
Nphases. Diamondand -BNhave
also been listed as referen e materials. Numbers given within bra kets
orrespondtotheuseoftheexperimentalBand GvaluesofTabs.9.5and
9.12. y
Bulkmodulusfromthe ombinationofthevariouselasti onstants.
z
Measuredelasti modulusfrom nanoindentationsofpoly rystalline -BN
bulksamples[203 ℄. . . 120
9.14 PositionsofthepeaksA-Erelativetothespe trashowninFigs.9.20,9.21
and 9.22. All the positions are s aled with respe t to the main
peak
Aand refer to thebroadenedspe tra.Valuesareexpressed inunitsof eV
withan estimated errorof 0.25eV. . . 130
9.15 PositionsofthepeaksA-Grelativetothespe trashowninFigs.9.23,9.24
and 9.25. All the energies are s aled with respe t to the rst
peak B
and refer to thebroadenedspe tra. Theestimated erroris 0.25eV. . . 135
9.16 Valuesoftheseparationbetween the rst
and
peaks (0.5 eV)for
theKedges of graphiti -BC 2
Introdu tion
1.1 The interest in novel ultra-hard materials
The possibility to synthesise new materials with hardness 1
similar or even larger than
diamond has be ome of fundamental and te hnologi al interest for hemists, physi ists
and in parti ular for the whole materials s ientists ommunity. It was in the middle
of the last entury when most of the known ultra-hard materials (i.e. diamond and
ubi boron nitride) were synthesised and manufa tured with high pressure and high
temperaturepro esses[1 ,2,3℄. The ontinueresear honthe eldhasre entlypermitted
tosynthesiseorredis oversuperhard ompoundssu hasSiO 2 -stishovite[4 ℄, ubi -Si 3 N 4 [5 ℄ and ubi -BC 2
N [6 ℄. The onstant growing interest inthisdomain isalso dueto the
development(1980's)ofnewvapordepositionte hniques(CVD,PVDandlaserablation),
whi h allow thedepositionof hardmaterials lmsat lowtemperature and pressures on
di erentsubstrates [7 , 8 ,9 ,10 ,11 ℄.
Diamond exhibits ex ellent me hani al, hemi aland physi al properties and
nowa-days remains thehardest known material. However, it is wellknown that it annot be
used in utting tools for steel owing to a ertain instabilityat high temperatures. As
a matter of fa t, its stability drasti ally de reases in the presen e of oxygen at even
moderate temperature (873 K). Itis also nota very suitableabrasive for utting and
polishing ferrous alloys sin eit tends to rea t and form iron arbides. Furthermore, its
super abrasive performan e is somehow limited. For these reasons and be ause of the
needtosubstituteexpensivediamondinmanyotherappli ations,newhardmaterialsare
required. Itismostlythestrongindustrialdemandsofwearresistant oatingsfor utting
and forming tools whi h has driven the sear h of novel hard materials. Common hard
1
hardness(h ard 0
n
is), n. [AS.heardness.℄ 1. The qualityorstateof beinghard, literallyor
gura-tively. Sour e: The Ameri anHeritage Di tionaryofthe EnglishLanguage, Fourth EditionCopyright
solids are usually lassi ed into ompounds with metalli (TiN or WC), ioni (Al 2
O 3
)
or ovalentbonding(diamond,Si 3
N 4
et ..). Transitionmetalnitridesand arbides(TiN
and TiC)have beenlargelyused as oatings forwear prote tive appli ationsinthelast
de ades. However, arbon based materials su h as arti ially grown diamond and
hy-drogenated arbon ompoundshavebe ome avalidalternative. These materialspossess
good prote tive properties and low fri tion oeÆ ient, thus open the possibilityto use
the oatings as solid lubri ants. Another important lass of materials is represented
by arbon nitrides ompounds with general formula CN x
. The growing resear h
inter-est arose from the theoreti al work of A. Y. Liuand M. L. Cohen [12 ℄ whi h predi ted
for -C 3
N 4
a hardness omparable to that of diamond. Despite the synthesis of pure
rystalline and stoi hiometri C 3
N 4
has been found extremely diÆ ult, some non
stoi- hiometri arbon nitrides have eviden ed interesting properties su h as high hardness
and elasti ity, and lowfri tion. These ompounds arethuspromising andidatesforthe
nextgeneration's wearprote tive oating. However, thefundamentalproblem withsu h
materialsremains theextreme diÆ ultyfoundin growing pure rystalline nitrogen-ri h
samples. Espe iallywith thin lmte hnology various depositionte hniquesand growth
onditionshave beentestedwithoutgreatsu ess: non- rystallineandnitrogen-de ient
lmsarealways obtained.
Theintrodu tionofboronatomsinto arbonnitridesleadstothepossibilitytoobtain
new hardmaterialswith general formula B x
C y
N z
. Withsu ha boron-based ompound
thelowoxidationresistan eofdiamondmightbeimprovedthusremovingtheproblemof
usinghardmaterialsathightemperaturesinair. There entinterestinboron arbon
ni-trideshasbeenmostlyfo ussedontheBC 2
Nstoi hiometry,whi hisaphaseisoele troni
with thewellknown C 3
N 4
. The rst eviden e of the graphiti BC 2
N dates ba k to the
synthesisofKouvetakisetal. [13 ,14 ℄,where hemi alvapordepositionmethodwasused
withBCl 3
and CH
3
CNasstartingmaterials. Several e ortshave beenmadeinorder to
modifythesegraphiti BC 2
Nsystemsintohighlydensethree-dimensionalphasesbut
un-fortunately,despitetheuseofhigh-pressureandhigh-temperaturemethods,no ommon
results were found in the last de ade. Some resear hers foundproblems witha ertain
limited solubility [15 , 16 ℄, while others laimed a segregation in a mixture of diamond
and ubi boron nitride( -BN) [17 , 18, 19℄. Nevertheless, early theoreti al al ulations
[20 ,21 ,22 ℄havesuggestedthatthese ompoundsshouldpossessanintermediatehardness
betweendiamondand -BN.
1.2 Aims of the Thesis
It is ertain that despite the initial s ienti enthusiasm, the synthesis of arbon
ni-tridesand boron arbonnitrides hassuddenlyturnedoutina very diÆ ulttask. Many
hara terise polymorphi samples. The sear h of a pure rystalline material and its
subsequent spe tros opi hara terisation remains nowadays the main topi for all the
resear hers workingon CN x and B x C y N z ompounds.
Giventhe ostand the omplexityofthesynthesis/ hara terisationpro edure,
om-puter modelling investigation has here been used to dis over new possible rystalline
models and to predi t theirmaterial properties in a faster and heaper way. The
om-putationalmethodshave already beenappliedto diamondand ubi boronnitride(i.e.
the hardest known solids) with great su ess, provoking a onsiderable interest in
in-vestigating other hypotheti al materials. The rst goal of my resear h has been the
determination of an eÆ ient omputational approa h for simulating the relative
stabil-ity and thehardness of some potentialphases thathave re ently beenproposed forthe
C 3
N 4
stoi hiometry. In parti ular, several Density Fun tional Theory (DFT) methods
have been tested, among the various simulation s hemes available inour laboratory, in
orderto inspe ttheirpe uliarreliabilityandusefulness. Subsequently, themost
promis-ing rst-prin iplesmethodshavebeenemployed intherestoftheThesisto al ulatethe
ohesiveproperties,bulkandelasti moduliofdi erentkindsof arbonnitrideandboron
arbonnitridemodelstru tures. Ele troni propertieshavealsobeenstudiedbymeansof
densityof statesand bandstru ture analysis. In addition,thein uen eofhybridisation
on the hemi al bonding and stability hasbeendis ussed interms of the siteproje ted
densities of states as well as the rystal orbital overlap population. Finally, sin e the
hara terisation of arbon nitridesand boron arbonnitridesis mostlyrestri ted bythe
problem of obtaining pure rystallinesamples, the al ulation of the theoreti al energy
lossnearedge stru turehasbeenshowninorder to providereferen e spe tra.
A large part of this work has also been oriented to the theoreti al proposition and
hara terisation of novel model systems isoele troni with diamond and ubi boron
nitride. I have in my resear h fo used most of the attention on the rystal engineering
ofthe C-B-N networksbyproposingvariousbinary (C 11 N 4 )and ternary (BC 2 N)model
ompounds. Theirele troni ,me hani alandspe tros opi hara terisationgiveninthis
Thesisshouldprovidea pre ioustool fortheinterpretation of theexperimental results.
1.3 Outline of the Thesis
The rst Chaptersare mostly on erning a generalintrodu tionto thedomainof
ultra-hard materials (Chapters 2 and 3) and to the employed omputational methods. In
parti ular,Chapter4resumesthebasi ideasbehindtheDFT,whileChapters5,6and7
ontainabriefdes riptionofthevariousmethodof al ulations. InChapter8adetailed
investigation of the CN x
systems is presented by paying most of the attention to the
di eren es between the C 3 N 4 and C 11 N 4
stoi hiometries. The study of boron arbon
intheirme hani al and ele troni properties. The on lusions are drawnin Chapter 10
The Hardness and Covalen y
2.1 Firsttheoreti al propositionof CarbonNitrides as novel
hard materials
It was in 1985 that M. L. Cohen[23 ℄ proposed an empiri alrelation between the bulk
modulus, B (volumetri ompressibilityor bulk modulus), and the rystalline solids of
elementsoftheIII,IVandV olumnoftheperiodi table. Inthefree-ele trongasmodel,
the ase of metals, the expressionof the B modulus (GPa) s ales as the Fermi energy,
E F
, andtheele tron on entration,n,
B= 2 3 nE F : (2.1)
StartingfromthemodelofPhillips-VanVe hten[24 ℄itispossibletoextendtheexpression
of B to semi ondu tors. The bond geometry of ovalent bonds is roughly represented
with a ylindri alshape with volumes (2a B
) 2
d, where a B
is theBohr radiusand d
(
A)thelengthof the ylinder. Usingthisapproximation we obtain,
B=45:6E h d 1 (2.2) where E h
(eV) representsthe homopolar ontribution of theopti gap, E g (E 2 g =E 2 h + E 2 ioni
). Using the s aling of Phillips (E h
/ d
2:5
) for the dependen e of E h
on d for
tetrahedral ompoundssharing eight valen eele tronsperatom pair,we obtain
B=1761d
3:5
; (2.3)
wherethenearest-neighbordisagainin
AandBinGPa. Theintrodu tionoftheioni ity
parameter, ,permitsto onsider theioni hara terof thebonding:
B =(1971 220)d
3:5
Thisempiri alrelationresultsappropriateforthegroup-IV(=0),III-V(=1)and
II-VI(=2) semi ondu tors. Furthermore,inordertoa ount foradi erent oordination
number (di erent from 4 of the tetrahedral site), M. L. Cohenintrodu edthe variables
N
, whi h represents the mean oordination number. The nal version of the equation
takes thefollowingform:
B = N 4 (1971 220)d 3:5 : (2.5)
Theaboveequationgivesana urateB valuefordiamondandforsemi ondu torswitha
zin -blendestru ture. Thevolumetri ompressibilityBin reases withtheloweringof d
and. Thehardestmaterialsarethusthosethatshowlowerioni ityandstrongerbonds.
Diamond responds to these hara teristi s; indeedit shows N
=4, =0 and d=1.54
A.
The bulk modulus al ulated for diamond with the Eq. 2.5 is 435 GPa, whi h is very
lose to the experimental one of 443 GPa. In the ase of arbon nitrides with formula
C 3
N 4
themean oordinationnumber(N
)is 24
7 1
whi hislowerthanthatofdiamond,4.
Takinginto a ount thesmallele tronegativitydi eren ebetween arbon andnitrogen,
we assumetheC-N bond to be slightlyioni with= 1
2
. From thevaluesofthe ovalent
radius(r C =0.77 Aandr N =0.75
A)wede neaC-Nbondlengthof1.52
A.Theinsertion
oftheseparametersinEq. 2.5providesaB valueof430 GPa. Therefore, arbonnitrides
withformulaC 3
N 4
shouldexhibitabulkmodulus omparable to thatof diamond.
Thiswasthe rsttheoreti alindi ationofthepossibilityto ndnewpromising lasses
of arbonbasedhardmaterials. Inparti ular,thelargebulkmodulus al ulatedfromthe
simpleempiri alrelationofM. L. CohenwassuÆ ient enoughto provoke inthemiddle
ofthe1980's anoutstandings ienti enthusiasmwhi his,nowadays,stillnotvanished.
2.2 Ele tron ount onsiderations
Thede nitionof"ultra-hard" materialsisusuallyemployedtodes ribeallthe ompounds
that have shown hardness values omparable to that of diamond. Generally speaking,
thesematerialsaresolidswithanhardnessinbetween8-10Mohss ale(Tab. 2.2). Sin e
diamond, ubi boronnitride( -BN) and boron arbides (B 13 C 2 -B 12 C 3
)arethehardest
materialsknown,it anreasonablybeexpe tedthatnovelultra-hardsolidswillbefound
inthe same B-C-N ternary ompositiondiagram (see Fig. 2.1). However, asone might
anti ipate many ombination of C, B and N atoms are, in prin iple, possible and an
huge amount ofdi erentstoi hiometriesandstru tures anrapidlybeimaginedforboth
binary and ternary ompounds. Therefore, the proposition of novel hard phases has
1
Carbonhasfourvalen eele trons([He℄2s 2
2p 2
)and anformone ovalentbondwithfournitrogen
atoms,whereasnitrogenpossesses vevalen eele trons ([He℄ 2s 2
2p 3
) and anonlyhave one ovalent
bondwiththreeatomsof arbon. ForthisreasonN =
(34) +(34)
Mineralsor Formula Mohs Knoop100
Syntheti Materials (GPa)
Tal um Mg 3 [( OH) 2 =Si 4 O 10 ℄ 1
Hexagonal Boron Nitride y h BN 0.15-0.30 Gypsum CaSO 4 2H 2 O 2 Cal ite CaCO 3 3 Fluorite CaF 2 4 Apatite Ca 5 [( F;OH)=( PO 4 )℄ 5 Feldspar K[AlSi 3 O 8 ℄ 6 Quartz SiO 2 7 Topaz Al 2 [F 2 =SiO 4 ℄ 8 -Sili onNitride y -Si 3 N 4 17 Corundum x -TitaniumNitride y Al 2 O 3 TiN 9 21 Sili onCarbide y SiC 26 -Sili onNitride y -Si 3 N 4 26-35 TitaniumCrabide y TiC 28 Boron Carbide y -TitaniumDiboride y B 4 C TiB 2 30 Boron suboxides B n O 30-59 Stishovite y SiO 2 33
Cubi Boron Nitride y
BN 45
Diamond x
C 10 75-100
[ y℄ Syntheti material. [x℄ Syntheti materialornatural mineral.
Table 2.1: Hardness ofminerals andsome syntheti erami s a ordingto F. Mohs. For
syntheti materials mi ro-hardnessvaluesare given inunits of Knoop s ale. Valuesare
shownas ompiledbyR.Riedel inRef. [25 ℄.
generallybeenrestri tedinthisThesisbytheadoptionoftheso- alledele tron ounting
rule. A systemati investigation of the various stoi hiometries be omes thus possible
thanksto thelimited numberof allowed atomi ombinations.
If we look, for example, at the building up of the two-dimensional arbon nitride
ompounds,one ould rstlyenvisagearandomrepla ementofCbyN withinthelayers
ofgraphite. However, thisresultsinanunstable ele troni stru ture on guration. This
isdueto theadditionalele tronsofthenitrogenatomswhi hhave tobea ommodated
in energeti ally unfavourable ele troni bands. But if ompounds are designed to be
isoele troni to diamond and graphite the stability and the ele troni stru tures are
Figure 2.1: S hemati ternary omposition diagram indi ating di erent "hard"
stoi- hiometries.
2s states are in luded. Distributing the ele trons on eight sites gives four ele trons on
ea hsite whi his isoele troni withdiamondand graphite. Theeighth siteisa va an y
(C 3 2 1 N 4
)andthelonepairsofthree ofthenitrogenatomsarepointingtowardthishole.
From this, graphiti C 3
N 4
should have a similar band stru ture at the Fermi level as
graphite, and C 3
N 4
with a three-dimensional network is also expe ted to have a band
gap similarto diamond. Consequentlya seriesof di erent ombinations of C, B and N
an be investigatedfor the sear h of new hard ompounds, provided that thefollowing
simple onditionisrespe ted:
pZ V ( B)+mZ V (C)+lZ V (N)=4n (2.6)
The values p, m, l and n are integers and Z V
(B), Z V
(C) and Z
V
(N) are the atomi
valen e states (2s and 2p) for boron, arbon and nitrogen, respe tively. Examples are
representedbythesystemsC 3 N 4 , C 11 N 4 , BN, B 4 C, BC 2 N et ...
The attention has therefore beenrestri ted onlyto those ompositions that are
iso-ele troni to arbon,i.e.,diamond. Thisparti ular hoi e alsoderivesfromthefa tthat
all the substan es obeying this ruleshould likely posses the same attra ting properties
The on ept of Hardness
3.1 Introdu tion
From the me hani al point of view we usually de ne the hardness as the resistan e
of the material to deformations. This property strongly depends on many parameters
like pressure, temperature, porosity, impurities, dislo ations and defe ts. It is usually
orrelatedtovariousotherphysi alproperties(ioni ity,meltingpoint,bandgap, ohesive
energy, et ...) and an thus be studied indire tly. The hardness for a given sample
is usually determined by empiri al methods su h as the s rat h test (Mohs s ale) or
indentationbydroppingaweightonthesample. Theresultsarevery usefulbutdiÆ ult
to interpret and they often dependent on the sample and its state of purity. In the
Vi kers test the hardness is estimated by measuring the indentation left by a diamond
stylusundera xedload. Thistest and thes rat h test (irreversiblemethods)arequite
oftenemployed experimentallyto lassify thehardness ofthevarious ompounds.
Many theoreti al predi tions on the hard materials have been made in the last two
de ades by looking at the magnitude of the bulk modulus, B, [26 , 27 , 23 , 28 , 12 , 29 ℄.
However, in1977A.P.Gerk[30 ℄hasalreadysuggestedthattheshearmodulus,G,whi h
de nes the resistan e to reversible deformation upon shape hange, might be a better
predi torofthehardness. Morere ently, D.M.Teter[22 ℄showedthatforawidevariety
of materials the shear modulus is really more orrelated to the Vi kers hardness than
the bulk modulus (further details are given in Se tion 8.2.4, p. 53). The hardness of
rystallinematerialsthusbe omesbetter de ned bytaking into a ount thedislo ation
theory, i.e., by measuringhow readilya largenumberof dislo ationsare generated and
areable to move throughoutthe solidinresponseto theshear stresses.
In thefollowingsubse tions weshowhowto des ribethehardness ofsolidswiththe
3.1.1 Measure of the resistan e upon volume hange in solids
The bulkmodulusmeasuresthe resistan e to thevolume hange insolids and provides
anestimationof theelasti response ofthematerialto an externalhydrostati pressure.
TheB(V) valueis relatedto the urvature ofE(V),
B(V)= V P V =V 2 E V 2 (3.1)
whereV isthevolumeoftheunit ell,E(V) istheenergyperunit ellatvolumeV, and
P(V) is the pressurerequired to keep the unit ell at volumeV. Sin e the al ulations
an only providea restri ted set of energiesE(V i
), the se ond derivative, 2 E V 2 , must be
approximated. The least squares t of the urves E vs. V has been performed in this
Thesisbyusingthe rstthree terms of theBir h equation[31 ℄:
E(V)=E(V o )+ 9 8 V o B " V o V 2 3 1 # 2 + 9 16 B B 0 4 V o " V o V 2 3 1 # 3 + N X n=4 n " V o V 2 3 1 # n ; (3.2) whereE o , V o , B and B 0
aretheequilibriumenergy, volume, bulkmodulusand pressure
derivatives of the bulk modulus, respe tively. In the above summation the n
symbol
representsthe total ontra tion terms [32 ℄, whilstthe maximumorder of the t is
sym-bolised by the N index. The Eq. 3.2 is normally employed by assuming the following
trend: the larger the value of B, the harder is the material. The magnitude of B 0
is
generallyutilisedtodes ribethevariationofthehardnesswithrespe tto agiven hange
ofthe pressure(P).
Di erent semiempiri al relations su h as nite stress-strain have been proposed to
des ribetheso- alledEquation ofState(EOS)(seeRef. [33℄and Refs. therein). S aling
experimental ompression data for measured isotherms of di erent sorts of solids the
EOS is known. The above Bir h type equation of state is a well tested tting form
able to des ribe the P, V, T data for a wide variety of solids. The main assumption
made in its utilisation is that no phase transition o urs duringthe ompressionof the
material. Despitetheexisten e ofdi erent varietiesof EOS,the al ulationsof thebulk
modulus have mostly been performed in this Thesis by using the Bir h type equation.
Sin e su h a tting form provides good results for systems like diamond and -BN I
thought worthwhileto use the same equation for the investigation of new hypotheti al
phasesforwhi htheexperimentaldataarenotyetavailable. Furthermore,bydoingthis
a homogeneous analysis of the results be omes possible with respe t to the previously
3.1.2 Resistan e to reversible deformation upon shape hange
Inthestudyofme hani alstrengththeelasti ityofsolids,i.e.,theresponseofamaterial
toappliedfor es, mustbetaken into a ount. Thefor esaredes ribedbytensors alled
stresses whi h determine thedire tion of thefor e and the plane to whi h it is applied.
Theresponses intermsof relative hanges indimensionsorshapeare alledstrainsand
theyarealsogivenbytensors. Theratiostress/strainis alledelasti modulus. Forsmall
stressesthemodulusis onstantandthematerialbehaveselasti allysothatitreturnsto
theoriginal onditionwhenthestress isremoved. Forlargestress thesample undergoes
apermanentorplasti deformation. Whenthefor ea tsonlyinonedimensionthestress
is alled ompressional,andwhenita tsinalldire tionsthestressishydrostati . Inthe
shearingstress, for es a tto move parallel planesofthesolid sothatat themi ros opi
levelthesestresses ausetheglidingofplanesofatomsoverea hother. Thisistheeasiest
wayfor asolid to hange its shape and the for e needed (hardness)depends very mu h
on the presen e of rystaldefe ts. Edge and s rew dislo ationsare the mostimportant
defe tsforglidingmotion. Anappliedshearingstress will ausethedislo ationstomove
throughoutthe rystal.
A ording to the nding of A. P. Gerk and D. M. Teter, the hardness of the solids
has mostly been investigated in this Thesis by omputing the value of the isotropi
shear modulus. This magnitude an be expressed as a linear ombination of a set of
elasti onstants, ij
, and is onsidered nowadays as the best hardness predi tor for
solids. The ij
onstants determine the response of the rystal to external for es and
providesinformationaboutthebonding hara teristi sbetweenadja ent atomi planes,
anisotropi hara terofthebondingandstru turalstability. Ea hoftheelasti onstants
isa measure ofhardness foraparti ular kindofunit elldeformation.
Cal ulation of the elasti onstants: ubi system asa simple example
Thebasi problem in al ulatingelasti onstantsfromabinitiomethodsisnotonlythe
demandofa urate al ulationals hemesforevaluatingthetotalenergyofthesolidbut
alsothemassiveand onerous omputations impliedintheestimationof theentiresetof
the inequivalent ij
. For instan e, when the symmetry of the system is de reased, the
number of independent elasti onstants expands and a larger number of distortions is
ne essaryto omputethefullsetof ij
[34 ℄. These onstants anbededu edbyapplying
smallstrainsto theequilibriumlatti e and thendeterminingtheresulting hangeinthe
total energy. In parti ular we al ulate thelinear ombinations of theelasti onstants
by straining the latti e ve tors R a ording to the rule ~
R = R D. The matrix D
representsthesymmetri distortionmatrixwhi h ontainsthestrain omponentsand ~ R
is thematrix that ontains the omponents of the distortedlatti e ve tors. In order to
(e.g. strains within1.5 %).
In ubi materialsthereareonlythreeinequivalentelasti onstants: 11 , 12 and 44 .
Thesevalues anbeestimatedby al ulatingthetotalenergyofthesystemasafun tion
oftheshearsdes ribedbelow[35℄. For 11
and 12
thefollowingshear,D 1 , is onsidered, D 1 = 0 B 1+Æ 0 0 0 1+Æ 0 0 0 1 (1+Æ) 2 1 C A (3.3)
wherethezaxisismodi edandthexandyaxesarekeptthesameinavolume onserving
way. The variationofthe strainenergy density(U =Energy=Volume)asa fun tionof
theshear Æ is des ribedwiththefollowing equation,
U =6C 0 Æ 2 +O(Æ 3 ) (3.4) with C 0 = 1 2 ( 11 12
). From the al ulation of C 0
and the bulk modulus, B =
1 3 ( 11 +2 12
), one an evaluate the rst two elasti onstants. With the same
pro e-dure,but onsideringthe followingshear,
D 2 = 0 B 1 Æ 0 Æ 1 0 0 0 1 ( 1 Æ 2 ) 1 C A (3.5) the 44
onstant an be al ulated fromtheequation,
U =2 44 Æ 2 +O(Æ 4 ): (3.6)
Isotropi shear modulus The isotropi shear modulus, G
Iso
, was rstly expressed
byA.Reussaslongago asin1929 [36 ℄. IntheVoigt's approximation theequation takes
thefollowingform:
G Iso = 1 15 [( 11 + 22 + 33 ) ( 23 + 31 + 12 )+3( 44 + 55 + 66 )℄ (3.7)
Forthespe ial ase ofa ubi symmetrytheabove relationtranslatesinto theform of
G = 1 15 (3 11 3 12 +9 44 ): (3.8)
Therefore, after having a omplished the al ulation of the whole set of single rystal
elasti onstants, itispossibleto estimate(forallthematerials)theelasti shear moduli
fora poly rystalline 1
solid bysimply applying theabove relation (Eq. 3.7). A ording
to the nding of A. P. Gerk [30 ℄ and D. M. Teter [22 ℄, the larger is the value of the
al ulatedG, thehardershouldbethematerial.
1
Ingeneral,asingle rystalismorediÆ ulttoprepare thanapoly rystallinematerial. Asamatter
Density Fun tional Theory
4.1 Introdu tion
Condensedmatter physi s andmaterialss ien earebasi allyrelatedto the
understand-ingandexploitingthepropertiesofsystemsofintera tingele tronsandatomi nu lei. In
prin iple,all theproperties of materials an be addressed given suitable omputational
toolsforsolvingthisquantumme hani sproblem. Infa t,throughtheknowledgeofthe
ele troni properties itis possibleto obtaininformationon stru tural,me hani al,
ele -tri al,vibrational,thermaland opti properties. However, the ele tronsand nu leithat
omposematerials onstituteastronglyintera tingmanybodysystemandunfortunately
thismakes thedire t solutionof the S hrodinger's equation an impra ti al proposition.
As stated by Dira inthe far 1929 [37 ℄, progress depends mostly on the elaboration of
suÆ ientlya urateand approximatete hniques.
Thedevelopmentofdensityfun tionaltheoryandthedemonstrationofthe
tra tabil-ity and a ura y of the Lo al Density Approximation (LDA) represents an important
milestone in ondensed matter physi s. The DFT of Hohenberg and Kohn [38 ℄ was
adopted bythe LDA whi h was rstly developed and applied by Slater [39 ℄ and his
o-workers [40 ℄. First prin iplesquantum me hani al al ulations based on theLDA have
be ome one of themostfrequentlyusedtheoreti al tools inmaterials s ien e.
Nonethe-less, the great ontribution of the lo al density approximation al ulations remained
limited untilthelate 1970's when several workshave demonstrated thea ura y of the
approa hin determiningproperties of solids[41 , 42 ,43 , 44 ℄. Even thoughit hasbeena
great dealto state whytheLDA shouldorshouldnotbe adequatefor al ulating
prop-erties of materials, there is however no doubtthat themost onvin ingarguments have
beenderivedfromthedire t omparisonof al ulationswithexperiments. In parti ular,
despiteitssimpli itythelo aldensityapproximationhasbeenvery su essfulin
arealso situationswheretheaboveapproa h donotleadto suÆ ientlya urateresults.
This an bethe asewhen thedi eren es inthetotal energy, whi h areusuallyrelevant
in al ulatingstru turalpropertiesand binding,areto beestimatedvery a urately. As
a matter of fa t, smallina ura ies may have here a dramati e e ts. In general, LDA
su er from more or less well-known failures and therefore there have during the last
de adebeenseveralattemptsto go beyondthislo alapproximation byin ludinge e ts
dependingon thevariationof theele tron density.
Nowadays,improvedtheoreti als hemesandtherapidgrowthin omputingfa ilities
have ausedmanytypesofsystemsandpropertiestobestudiessu essfullywithdensity
fun tional methods. In the next following Se tions we brie y resume the fundamental
on eptswhi hareat the baseof thisimportant and fas inatingtheory.
4.2 The basi prin iples of the method
ThetheoremofHohenbergandKohnisatthebaseoftheDFTandstatesthatthetotal
energy, E,ofanon-spin-polarisedsystemofintera tingele tronsinanexternalpotential
isgiven exa tlyasa fun tionalof thegroundstate ele troni density, .
E = E[℄ (4.1)
They further showed that the true ground state density is the density that minimises
E[℄and that theother groundstate propertiesarealso fun tionalsof thegroundstate
density. The extension to spin-polarisedsystems is also possible whereE and the other
groundstate propertiesbe ome fun tionalsof boththeupand downspindensities.
E = E[
" ;
#
℄ (4.2)
TheHohnenberg-Kohntheoremprovidesnoguidan etotheformofE[℄,thustheutility
ofDFTdependson thedis overyof suÆ ientlya urate approximations. In orderto do
this,the unknownfun tionalE[℄ is rewrittenasthe Hartree total energy plusanother
smallerunknownfun tional alledex hange- orrelation(x ) fun tional,E x [ ℄. E[℄ = T s [℄+E ei [℄+E H [℄+E ii [℄+E x [℄ (4.3) In Eq. 4.3 T s
[ ℄ represents the single parti le kineti energy while E ei
[℄ denotes the
Coulombintera tionenergybetweentheele tronsand thenu lei. ThetermE ii
[℄ arises
from the intera tionof the nu leiwith ea h other and E H
[℄ is the Hartree omponent
ofthe ele tron-ele tronenergy.
E H [℄ = e 2 Z d 3 rd 3 r 0 ( r)( r 0 ) 0 (4.4)
IntheLDA, E x [ ℄is writtenas E x [℄ = Z d 3 r(r)" x (( r)) (4.5) where" x
()isapproximatedbyalo alfun tionofthedensity,whi husuallyreprodu es
theknownenergy ofthe uniformele tron gas. Re nement of theLDA are theso- alled
generalisedgradientapproximation(GGA)andtheweightedapproximation(WDA).An
expression similar to Eq. 4.5 is used in the GGA where the " x
( ) is repla ed by a
lo al fun tion of the density and the magnitude of its gradient, " x
(;jrj). From
the in orporationof theadditional information ontained inthe lo al gradient a better
des riptionof thesystem isexpe ted [45 , 46 , 47 ℄. Several di erent parameterisationsof
theGGA fun tional have beenproposed [47 ℄ and testedon a widevariety of materials.
The GGA improve signi antly the ground state properties of light atoms, mole ules
andsolidsandgenerallytendstoprodu elargerequilibriumlatti eparametersandband
gapswithrespe tto the LDA.
A more sophisti ated approa h is the WDA that in orporates true non-lo al
infor-mationthroughCoulomb integrals ofthedensitywithmodel ex hange orrelation holes
[48 , 49 ,50 ℄. Itameliorates greatly theenergiesof atomsand forthediamondstru tures
of Si and Ge yields bulk properties that are mu h improved as well. Nonetheless, the
WDA ismore demanding omputationallythan theLDA or GGA, and a ordinglyfew
WDA studies have beenreportedforsolids.
Following the Kohn and Sham indi ations [51 ℄, the ele tron density an be written
as a sum of single parti le densities. Given the fun tional E x
the ground state energy
and density an be obtained by the self- onsistent solution of a set of single parti le
S hrodinger-likeequations,knownastheKohn-Shamequationswithadensitydependent
potential, (T +V ei ( r)+V H (r)+V x (r) )' i (r)= i ' i (r) (4.6)
wherethedensityisgiven byaFermi sum over theo upiedorbitals.
( r)= X o ' i (r)' i ( r) (4.7) The ' i
aresingle parti leorbitals, i
arethe orresponding eigenvalues, T isthekineti
energy operator, V ei
is the Coulomb potential due to the nu lei, V H
is the Hartree
potential and V x
is the ex hange orrelation potential. V H and V x depend on as follows: V H (r) =e 2 Z d 3 r 0 (r) jr r 0 j (4.8)
and V x (r)= ÆE x [℄ Æ(r) (4.9)
Inthisframework,a al ulationrequirestheself- onsistentsolutionofequations4.6and
4.7. This meansthat a ertain densityhas to be found su h that it yields an e e tive
potential that, inserted into the S hrodinger-like equations, yields orbitals that an
re-produ eit. Forthisreason,insteadoffa ing-upwiththeproblemofsolvingamany-body
S hrodingerequation,usingDFTwe annowhave theeasierproblemofdeterminingthe
self- onsistentsolutiontoa seriesofsingle parti leequations. Insolids,afurther
simpli- ationthat fa ilitatesDFT al ulationsis providedbythe Blo h'stheorem, wherethe
harge densityand thesingleparti leKSHamiltonianhave theperiodi ityofthelatti e.
ThusKS orbitalswith di erent Blo h momenta are oupledonlyindire tlythroughthe
density dependent potential. Therefore, in DFT based al ulations, the single parti le
KS equations may be solved separately on a grid of sampling points in the symmetry
irredu iblewedge of the Brillouin zone and the resulting orbitals used to onstru t the
harge density(thisisnotthe ase, forexample, inHartree-Fo kmethods).
Asalready mentionedthegreat advantageofthe densityfun tionalapproa h isthat
theresultingsingle-parti leequationsare omputationallysimplertosolvethenthe
equiv-alent Hartree-Fo k equations. This makes possible to onsider systems that are more
omplex(i.e. largersizeor ompli atestru ture)thenthosetreatedbytheHartree-Fo k
derived methods.
4.3 Single parti le Kohn-Sham equations
Dependingontherepresentationsthatareusedfordensity,potentialandKSorbitals,
dif-ferent DFT basedele troni stru turemethods an be lassi ed. Manydi erent hoi es
aremadeinorderto minimisethe omputationaland human ostsof al ulations, while
maintaining suÆ ient a ura y. A briefsummary of themany possibilitiesto solve the
S hrodinger'sequation isgiven inFig. 4.1. InthisThesis al ulationshave beenmostly
on ernedwithtwoparti ularapproa hesnamely, planewavePseudo-Potential(PP)and
theLinearizedAugmentedPlane-Wave(LAPW).Othersimplerandfastermethods,su h
asAugmentedSpheri alWave(ASW)andtheLinearMuÆnTinOrbital(LMTO),have
alsobeenemployedinthestudyof arbonbasedhardmaterials. However, these
ompu-tational approa hes are usually reliableonly when applied to rystalline materialswith
highsymmetry andlarge ompa tness.
The expli it use of a basis an be avoided in onstru ting the KS orbitals by
nu-meri ally solving the di erential equations on grids. However, it is important to note
Figure 4.1: S hemati representationof variousDFT-based methodsof al ulation.
LAPW methods, do rely on a basis set expansion forthe KS orbitals. Be ause of this,
thedis ussionis here on ned to methodsthat do use a basisin whi h the KSorbitals
are: ' i ( r)= X C i ( r) (4.10) wherethe
(r)arethebasisfun tionsandtheC i
aretheexpansion oeÆ ients. Given
a hoi e ofbasis, the oeÆ ientsaretheonlyvariablesintheproblem,sin ethedensity
dependsonlyontheKSorbitals. Sin ethetotalenergyinDFTisvariational,thesolution
oftheself- onsistentKSequationspermitstodeterminetheC i
fortheo upiedorbitals
that minimisethetotal energy. In order to eliminatethe unknown fun tionalT s
[℄ the
total energy an be rewrittenusingthesingle parti leeigenvalues:
E[℄=E ii [ ℄+ X o i +E x [℄ Z d 3 r(r) V x ( r)+ 1 2 V H (r) (4.11)
where the sum is over the o upied orbitals and , V H
and V
x
are given by Eqs. 4.7,
4.8and 4.9, respe tively.
Densityfun tional al ulations requirethe optimisationof the C i
and the
determi-nationof the harge density(Fig. 4.2). Thispro edure is usuallyperformed separately
and hierar hi ally. Using standardmatrix te hniques it is possibleto repeatedly
