Towards predictive modelling of near-edge structures in electron energy loss spectra of AlN based ternary alloys

1.2 MB·English

by  David Holec

Towards predictive modelling of near-edge structures in electron energy loss spectra
of AlN based ternary alloys
D. Holec,1,2,∗ R. Rachbauer,1 D. Kiener,3,4 P.D. Cherns,2
P.M.F.J. Costa,2,5 C. McAleese,2 P.H. Mayrhofer,1 and C.J. Humphreys2
1Department of Physical Metallurgy and Materials Testing,
Montanuniversit¨at Leoben, Franz-Josef-Strasse 18, A-8700 Leoben, Austria
2Department of Materials Science and Metallurgy, University of Cambridge,
Pembroke Street,Cambridge CB2 3QZ, United Kingdom
2 3Erich Schmid Institute of Materials Science, Austrian Academy of Sciences, Jahnstrasse 12, A-8700 Leoben, Austria
1 4Department of Materials Physics, Montanuniversit¨at Leoben, Franz-Josef-Strasse 18, A-8700 Leoben, Austria
0 5CICECO, Department of Ceramics and Glass Engineering, University of Aveiro, 3810-193 Aveiro, Portugal
2 (Dated: January 16, 2012)

n
Although electron energy loss near edge structure analysis provides a tool for experimentally
a
probing unoccupied density of states, a detailed comparison with simulations is necessary in order
J
to understand the origin of individual peaks. This paper presents a density functional theory
3
based techniquefor predicting theN K-edge for ternary (quasi-binary) nitrogen alloys byadopting
1
a core hole approach, a methodology that has been successful for binary nitride compounds. It is

demonstratedthatusingthespectraofbinarycompoundsforoptimisingthecoreholecharge(0.35e
]
i for cubic Ti1−xAlxN and 0.45e for wurtzite AlxGa1−xN), the predicted spectra evolutions of the
c ternary alloys agree well with theexperiments. Thespectral features are subsequentlydiscussed in
s
terms of theelectronic structureand bondingof thealloys.
-
l
r
t PACSnumbers: 61.66.Dk,71.15.Mb,71.20.Be,71.20.Nr,79.20.Uv,81.15.Cd,81.15.Gh
m Keywords: Density functional theory (DFT); Electron energy loss near edge structure (ELNES); Alloys;
TiAlN;AlGaN
.
t
a
m
I. INTRODUCTION luted alloys8. Therefore,the here proposedmethodology
- is highly relevant also for XANES.
d
n Wurtzite aluminium nitride (w-AlN), gallium nitride Recent developments of theoretical methods for solid
o (w-GaN) and their ternary alloy w-AlxGa1−xN are im- state physics have provided the EELS community with
c
portantmaterialsfordevicessuchaslightemittingdiodes increasingly reliable and comprehensive tools to simu-
[
(LEDs) and laser diodes (LDs)1. Cubic titanium ni- late ELNES. A particularly important example is the

1 tride (c-TiN) and in particular its alloy with AlN, cu- TelnesprogramwhichisdistributedasapartofWien2k9,
2v bic Ti1−xAlxN, are widely used hard coatings due to anall-electronfull-potentiallinearisedaugmentedplane-
their high hardness, corrosionand oxidation resistance2. wave (FP-LAPW) code. It has been suggested in the
4
In both ternary alloys, a crucial requirement for getting literature10 that a calculation including a core hole pro-
8
2 theoptimalapplication-tailoredpropertiesisanaccurate vides a better description of the excitation process by
. control of both the composition and structure of the al- means of the standard (ground state) density functional
1
loy. theory (DFT). To create a core hole, one takes an elec-
0
tron(orafractionofit)fromitsgroundstateposition(N
2 Electron energy loss spectroscopy (EELS) is a pow-
1 erful technique to microanalyse compositional features3. 1s-stateinthecaseofNK-edge)andputsitinthelowest
v: The spectra can be recorded with very high spatial res- unoccupied state abovethe Fermi level(N 2p)or adds it
asabackgroundchargetokeepthetotalchargeofthecell
i olution, thus taking advantage of the high sensitivity
X neutral10. This can be easily realised in Wien2k where
of EELS to local changes in the electronic structure of
r materials4,5. Its subset, electron energy loss near edge all electrons are accessible and explicitly treated. Re-
a cently, the core hole calculations have become routinely
structure (ELNES) reflects the density of unoccupied
available also for pseudopotential codes11,12.
states and provides thus an experimental probe for this
partoftheelectronicstructureofmaterials. However,the A considerable effort has been spent on studying
interaction of high energy electrons with lattice atoms ELNES of binary nitrides with respect to their crystal
does not always have a straightforward interpretation. structures (wurtzite, rock salt, zinc blende)13–18, polari-
In order to understand the experimental data, measured sationeffects13,14,18–20,doping21,andstoichiometry22–24.
EELS spectra need to be compared in detail with their Theliteratureisvastandprovidesagoodbackgroundfor
calculated counterparts. understanding the origin of peaks in ELNES in terms of
bonding, and thus establishes a solid basis for a finger-
X-ray absorbtion near-edge structure (XANES) is a
print identification of materials and their properties.
closely related technique to ELNES. The edge shapes
are very similar5,6, it also allows for studying polarisa- Nonetheless,thereareonlyafewreportsonthecompo-
tion effects7, and it has been successfully applied to di- sitional dependence of ELNES of ternary alloys. Keast
2
structure a [˚A] c [˚A] B0 [GPa]
TiN B1 4.256 292
Bk 3.524 4.227 238
AlN B1 4.070 253
B4 3.129 5.016 197
GaN B4 3.216 5.238 169
TABLE I. Optimised lattice parameters and bulk moduli of
thebinary compounds used in this work.
alloywereoptimisedwithrespecttovolume(lattice con-
stants)aswellastointernal(local)relaxations,forwhich
we used the VASP code30,31 together with the projector
FIG.1. Crystalstructuresinvestigatedinthiswork: (a)cubic augmented wave pseudopotentials32 employing the gen-
B1 (NaCl prototype), (b) wurtzite B4 (ZnS prototype), and eralised gradient approximation (GGA) as parametrised
(c) five-coordinated hexagonal Bk structure (BN prototype). by33. We used 500eV for the plane-wave cut-off energy
and a minimum of ≈ 600k-points·atom (usually more).
Such parameters guarantee the calculation accuracy in
et al.25 measured N K-edge ELNES of InxGa1−xN al- the order of meV/atom. The obtained equilibrium lat-
loys and correlated their findings with DFT calculations
tice parameters (see Tab. I) and energies are in good
using, however, extremely small supercells. MacKenzie agreement with those previously published34–36.
et al.26 used ordered structures and averaging of the
TiN is mechanically unstable in the four co-ordinated
boundary binaries spectra to get a guess on the shape of
wurtzite (B4) structure (Fig. 1b) and relaxes into a five-
Ti1−xAlxN N K-edge. Holec et al.27 showed that using coordinated B structure (Fig. 1c). The reason for this
k
small ordered cells failed to reproduce the experimental
is that the presence of d electrons favoursa different hy-
spectrum of an AlxGa1−xN alloy. To date, a systematic bridisation scheme (sp3d) than the tetrahedral sp3 (see
studyshowingtheeffectofalloyingontheELNESshape
Refs. 27 and 37). Consequently, the wurtzite variant of
(“evolution” of the edge), as well as discussing the com-
the Ti1−xAlxN alloy becomes unstable around x ≈ 0.6
putational methodology in a recipe-like form, is missing. whereas for x / 0.6 the Bk structure is obtained27,38.
The present paper is aiming to fill this gap by facilitat-
This is, however, the composition where also the phase
ing a semi-empirical approach in which the calculation
transition to a lower energy, experimentally observed
parameters (e.g. the core hole charge) are first adjusted cubic variant happens34. Therefore, in the following
toreproducethespectraoftheboundarybinarysystems,
we present only results for the high Al containing w-
andsubsequently used to predict the N K-edgeevolution
Ti1−xAlxN.
of ternary alloys.
As for the AlxGa1−xN alloy, we optimised the crys-
tal lattices only for the boundary binary compounds
(Tab. I), AlN and GaN, and then used Vegard’s rule to
II. METHODOLOGY
obtain the lattice constants for the intermediate compo-
sition. This is justified by the workofDridi et al.39 who
A. Calculation details showed for this alloy that the lattice parameters (unlike
the band gap) exhibit a linear dependence on the com-
The individual structures are modelled with super- position.
cells constructed using a special quasi-random structure ElectronicpropertiesandtheELNESspectrawerecal-
(SQS) approach28. All alloys considered in this paper culated using the Wien2k code9 employing the GGA–
are quasi-binary which means that mixing of elements PBE parametrisation40 of the exchange-correlation po-
(either Ti and Al or Al and Ga) takes place only on one tential. An equivalent of ≈ 900 k-points within the
sublattice (bigger atoms in Fig. 1); the other sublattice whole first Brillouin zone of the unit cell, the expansion
is fully occupied with N atoms. 3×3 ×2 (36 atoms) of the spherical harmonics up to l = 10 inside the non-
and2×2×2 (32 atoms)supercells were used for the cu- overlapping muffin tin (MT) spheres, and R k = 7
MT max
bic B1 and wurtzite B4 modifications, respectively. The were used41. The MT radii were automatically set by
short range order parameters were optimised for pairs structGen (a part of the Wien2k package) to values
up to the fourthorder,triplets up to the third orderand ≈ 1.70–1.80, ≈ 1.95–2.00, ≈ 1.85–1.95 and ≈ 1.90–
quadruplets up to the second order28,29. More details 1.95a.u. for N, Ti, Al, and Ga atom, respectively. The
about the cells and the process of their generation can spinpolarisationeffectswerenottakenintoaccount. The
be found in Ref. 29. core holes were implemented by reducing the N 1s core
The cubic and wurtzite structures of the Ti1−xAlxN level occupation on a specific site and putting the corre-
3
sponding charge in the background in order to keep the III. RESULTS
cell neutral10. ELNES was calculated using the Telnes
program, a part of Wien2k. This was repeated for all A. Binary compounds
N sites in the supercell. The spectrum representing the
particular alloy composition was obtained by averaging
The strategy adopted in this paper is to find calcula-
this set of N K-edges.
tionparametersthatreproducetheNK-edgeELNESfor
the binary compounds as closely as possible, and subse-
quently use these settings for predicting the ELNES evo-
lutionofthealloys. Figure2showshowtheedgeshapeof
cubicandwurtzite/hexagonalAlNandTiNchangeswith
B. Experiment increasingcore hole chargefrom0e (groundstate) to 1e
(finalexcitedstate–fullcorehole). Theeffectisstronger
forAlNthanforTiN.Thisisduetofastcoreholescreen-
In order to experimentally confirm the ab initio pre- ing in TiN originatingfrom its metallic character5. Nev-
dictedELNESspectra,twomaterialsystems,Ti1−xAlxN ertheless, some small changes can still be observed, e.g.
and AlxGa1−xN were investigated. thepeakbroadeningforthecubicmodificationorthedis-
The Ti1−xAlxN samples were deposited in Leoben us- appearing high-energyshoulder of the main peak for the
ing the plasma-assisted unbalanced magnetron sputter- hexagonal TiN with increasing core hole charge. Lazar
ing technique42. The variation of the Al mole fraction et al.23 arrived at the same conclusion for c-TiN based
x in Ti1−xAlxN was achieved by using powder metal- onthecomparisonoftheircalculationswithexperimental
lurgically produced targets (PLANSEE AG, 99% pu- measurements.
rity), with Ti/Al ratios of 1, 0.5 and 0.33, and man- The strong effect of the core hole charge on the AlN
ual placing of additional Ti or Al platelets (∅5×3mm) NK-edgeshape hasbeendiscussedinthe literature12,27.
in the racetrack of the targets, respectively. TEM sam- A detailed analysis of the relative peak positions and in-
ple preparation was performed by Ar-ion thinning in a tensities for the wurtzite AlN revealed that a core hole
Gatan precision ion polishing system (PIPS) at 4 and charge ≈ 0.5–0.6e reproduces the experimental ELNES
2.2keVinplanview. TheEELSmeasurementswerecar- the best29. Such a comparison could still be misleading
riedoutonaCscorrectedJeol2100Foperatedat200kV as the experimental results also depend strongly on the
and equipped with a Gatan Tridiem GIF camera using acquisition conditions20. Since the spectra around 0.5e
nanobeam diffraction mode. This ensures high signal to (Slater’stransitionstate)foreachallotropelookakinand
noise ratiosand makes it possible to acquireinformation are almost equally resembling the experiment, another
from individual grains, necessary for the investigation criterionwasadoptedhere. Theedgeonset,measuredas
of the polycrystalline Ti1−xAlxN films. Thus, several the energy between the initial core state and the lowest
different grains were measured for each alloy composi- unoccupiedstate5,isplottedasafunctionofthecorehole
tion to rule out possible orientation effects. The spectra charge in Fig. 3. This dependence is rather strong, and
were recorded with a dispersion of 0.3eV/channel and using the experimental value for the edge onset allows
theenergyresolution,measuredbythefull-widthathalf- anoptimalvalueofthe corehole chargeto be estimated.
maximumofthezero-losspeak,was1.5–1.8eV. Thecon- Taking 402eV for w-AlN15,45 (or this work) and 397eV
vergence and collection semi-angles during analysis were for c-TiN46,47 (or this work) yields 0.45e and 0.35e, re-
5mrad and >8mrad, respectively. spectively, which are the values used in this work.
The AlxGa1−xN films were epitaxially grownat Cam-
bridge by 6×2-inchThomasSwanClose-CoupledShow- B. Ti1−xAlxN alloy
erhead metalorganic vapour-phase epitaxy (MOVPE)
system43. A standard two-stepgrowthmethod was used
The calculated evolution of the N K-edge for
to deposit a 5µm thick GaN pseudo-substrate on (0001)
sapphire44 on which the AlxGa1−xN layers were grown Tani1d−wxAurlxtzNiteismshoodwifincabtyiosnosl.idTlhineersaiwn FEiLgN. 4ESfowrathsebrcouabdic-
◦
at 1020 C. The different compositions were obtained
ened with a Gaussian having 1eV FWHM. Moreover,
by varying the flow rate for Al and Ga precursors. The
thecurvesofpureAlN(x=1)wereshiftedalongtheen-
EELS spectra were obtained on a FEI Tecnai F20 mi-
ergyaxis(aslabelledinFig.4)toaccountfortheabrupt
croscope equipped with a Schottky FEG source, Gatan
change in the Fermi energy due to the development of
Imaging Filter and operated at 200kV. The electron
the band gap (no Ti d-states present).
beam was parallel to the h0001i direction.
Thecompositionalstepusedforthecubicalloyis∆x=
Priortothecomparisonwiththeabinitio calculations, 0.167 (Fig. 4a). Three developments of the main peaks
all measuredspectra were corrected for the dark current are predicted: (i) the double-maxima at 0–5eV above
and the channel-to-channelgain variation. The pre-edge E disappearswithincreasingAlNcontent,(ii)thepeak
F
backgroundwasextrapolatedusingapower-lawfunction at ≈ 13eV for TiN gradually moves to ≈ 9eV for AlN
and subtracted from the original data3. and at the same time its intensity increases, and (iii) a
4
(a) AlN cubic (B1) (b) AlN wurtzite (B4) (c) TiN cubic (B1) (d) TiN hexagonal (B)
k
s]
nit
arb.u 1e 1e 1e 1e
y [
sit
n
e
nt
S i
E
N 0.5e
EL 0.5e 0.5e 0.5e
e
g
d
e
K-
N
0e
0e 0e 0e
0 10 20 0 10 20 0 10 20 0 10 20 30
Energy above E [eV] Energy above E [eV] Energy above E [eV] Energy above E [eV]
F F F F
FIG. 2. Calculated N K-edgeELNES as a function of thecore hole charge changing from a ground state calculation (0e) to a
full core hole (1e): (a) cubic B1 AlN,(b) wurtzite B4 AlN,(c) cubicB1 TiN, and (d) hexagonal Bk TiN.
405 acteristic triple-peak character of the w-AlN N K-edge
440 is levelled out with the addition of only 0.125 mole frac-
tion of TiN. Additionally, the spectra do not show any
430 clear trends in the peak development as in the cubic
400
case, apart from the peak at ≈ 20eV, whose position
is not influenced by the composition. This is most likely
V]420 connected with local relaxations taking place similar to
e
et [ 395 those reported for Nb1−xAlxN48. In particular, N sites
ns410 0.3 0.4 0.5 nearAltendtohavethefour-coordinated(wurtzite-like)
o
e neighbourhood, while in the vicinity of Ti atoms, five-
g
d
E400 coordinatedlocalneighbourhoodsare preferred(hexago-
AlN B1 nalBk-like). Thestructureisthereforemuchmoresensi-
AlN B4 tive to the actual arrangement of atoms in the supercell
390
TiN B1 whichisreflected,e.g. inmuchbiggerscatterofthe“op-
TiN B timised” lattice constants for different arrangements of
k
380 atoms in the SQS (with a constant composition x)38.
0 0.2 0.4 0.6 0.8 1
Core hole charge [e]
FIG.3. NK-edgeonset(energydifferencebetweentheinitial
core state and lowest unoccupied state) as a function of the To compare the calculated and measured N K-edge
core hole charge for thetwo allotropes of each AlN and TiN. evolutions (Fig. 4c), a larger spectrometer broadening
Theinset showszoomed-in region aroundtheexperimentally parameter of 1.5eV was used. As a consequence, the
observed N K-edgeonset energy. double-maximum of the first peak in c-Ti1−xAlxN at
≈ 0–5eV above the E “smears out” and the measured
F
shape is obtained. It is therefore concluded that the fine
small hump at ≈ 32eV for TiN broadens to an almost double-maximum character is not resolved due to ex-
undetectable background at x = 0.5. At the same time, perimental limitations. The experimental spectra were
a small hump develops at ≈ 26eV with increasing AlN smoothedandnormalisedtofitthe intensityofthe high-
content. The origins of these composition-induced peak estpeakofthesimulatedpatternineachindividualcase;
variationscanbetrackeddowntothechangesinbonding no other operation was performed on them. The theo-
in the alloy as is discussed later in section IVB. retical spectra were, on the other hand, shifted by the
The situation is more complicated for the w- calculated energy of the core-holed core level. The spec-
Ti1−xAlxN alloy (Fig. 4b, compositional step ∆x = trathusobtainedexhibitaverygoodcorrelationbetween
0.125). The calculated spectra suggest that the char- experiment and theory.
5
5eV (a) 5eV (b) (c)
cubic Ti1−xAlxN wurtzite Ti1−xAlxN
x=1
x=1 x=1
s]
nit x=0.875
u
b. x=0.833
ar
y [ x=0.75
nsit x=0.667
nte x=0.625 x=0.61
S i
NE x=0.5
L x=0.44
E
e
g
d x=333
e
K- x=0.26
N
x=0.167
x=0
x=0
0 10 20 30 40 0 10 20 30 40 400 410 420 430
Energy above the Fermi level [eV] Energy above the Fermi level [eV] Electron energy loss [eV]
FIG. 4. Calculated N K-edge evolution in the (a) cubic and (b) wurtzite phase of Ti1−xAlxN alloy. The spectra of pure AlN
(x=1) areshifted in ordertocorrect fortheopened bandgap. Thedashed linecorresponds toalinear interpolation between
thebinarycompoundsspectra(seesectionIVA).(c)Comparisonofthecalculated(solidblackline)andexperimentalNK-edge
ELNES of Ti1−xAlxN.Note that the structurechanges from cubic(x<0.7) to wurtzite (x>0.7).
C. AlxGa1−xN alloy IV. DISCUSSION
A. Shape and evolution of the N K-edge ELNES
The spectra of the binary TiN, AlN and GaN systems
have been extensively discussed in the literature both
from the experimental and theoretical perspective. De-
spitethat,severalissuesremainunclear,inparticularthe
edgeonsetenergy: itsvaluevariesintheliteraturewithin
Asanotherexample,thesemiconductingwurtzitesolid
the range of several eVs12,16,17,21,26,27,45. Consequently,
solution of AlN and GaN is chosen to demonstrate the
we used our own measurements to calibrate the calcula-
ability of the current approach to predict ELNES. In
tions instead of taking spectra from the literature. The
contrast to the meta-stable Ti1−xAlxN, the wurtzite
lineshapes of binary w-AlN, c-TiN, and w-GaNresemble
AlxGa1−xN mixture is stable in this modification for all
those previously published for the same materials.
concentrationsx. Therefore,nolocaldistortionsasinthe
caseofthew-Ti1−xAlxNalloyareexpected,whichresults The edge evolutionfor c-Ti1−xAlxN exhibits the same
inagradualNK-edgeevolutionasshowninFig.5a. The trends as the one previously published by MacKenzie
intensities of the first and third peaks of the triple-peak et al.26 (which,however,providedonlyoneintermediate
shape characteristic for w-AlN decrease with decreasing composition). Gagoetal.50 usedXANEStomeasurethe
AlN mole fraction, and become shoulders around a cen- N K-edge of Ti1−xAlxN experimentally. Their XANES
tral peak, a shape typical for GaN. The peak at ≈24eV spectra have all the main features of our experimental
abovetheE inthe w-AlNspectrumgraduallymovesto as well as calculated N K-edge ELNES. Also in the case
F
≈ 26eV for w-GaN with increasing GaN content. The of w-AlxGa1−xN, the calculated evolution of N K-edge
edge onset moves towards the Fermi level reflecting the ELNES correlates with the here presented experimental
narrowingbandgapfrom4.2eV(AlN)to1.7eV(GaN)49. data as well with those published previously27,5152.
Thisgradualtranformationofthespectrashapeistraced MacKenzieet al.26 triedto modelthe edge witha lin-
down to the changes in electronic structure, see section earinterpolationoftheboundarybinaryspectra. Having
IVB. The predicted evolution of the N K-edge is again inmindtheproblemswiththeaccurateedgeonsetdeter-
confirmed by the experimental observations (Fig. 5b). minationandthe lackofcubicAlN (inthe B1structure)
6
(a) wurtzite AlGa N (b)
x 1−x
x=1 x=1
s]
nit x=0.875
u
b.
y [ar x=0.75 x=0.73
sit
n
e x=0.625
nt
S i
E
N x=0.5
L
E x=0.40
ge x=0.375
d
e
K-
N x=0.25
x=0.125
x=0 x=0
0 10 20 30 40400 410 420 430 440
Energy above the Fermi level [eV] Electron energy loss [eV]
FIG.5. (a) Calculated evolution ofNK-edgeof thewurtziteAlxGa1−xNalloy and(b)acomparison ofcalculated (solid black
line) with experimental N K-edge ELNES shapes. The dashed lines correspond to linearly interpolated boundary spectra of
AlN (x=1) and GaN (x=0).
for getting a reliable binary spectrum, this approach is not necessary.
questionable. To demonstrate this further, Figs. 4a and On the other hand, when the spectrum evolution is
5a include the linear interpolations of the binary spectra pronounced, small ordered structures do not provide re-
(dashed lines). Although this may serve as a first (and liable predictions. This has been shown by MacKenzie
quick) guess on what the evolution should look like, in etal.26 forthecaseofc-Ti1−xAlxNandbyHolecetal.27
manycasestherelativeintensitiesand/orpositionsofthe for w-AlxGa1−xN. In such cases, the approach adopted
peaks are not predicted correctly. here is essential.
Craven47 showedusingseveralbinarytransitionmetal
nitrides (TMN) that the spacing between the double-
B. Electronic origin of the peaks
maximumpeaks increaseswith increasinglattice param-
eter. This is not predicted for the c-Ti1−xAlxN al-
loy (Fig. 4) where the lattice parameter decreases from Thereexistsextensiveliteratureonthe originofpeaks
4.25˚A for TiN (x = 0) to 4.07˚A for AlN (x = 1)34, but for semiconducting III-N binaries. As summarised by
the peak spacing is almost unaffected. The reason for Mizoguchi45 using the overlap population analysis, the
thisisthatthebondingofvariousTMNissimilarandthe mainpeakstructure(upto≈10eVabovetheedgeonset)
peaks follow small shifts of the density of states associ- reflects the anti-bonding N–cation interaction while the
atedwiththe varyingvalenceconfiguration. Onthecon- later peak (20–30eV above the edge onset) corresponds
trary, the peak shifts in the Ti1−xAlxN evolution result tocation–cation(mostlyanti-bonding)interactions. The
in the first place from the changing character of bond- difference betweenthe AlN and GaNELNESshapes can
ing (see section IVB). Consequently, when the ELNES betraceddowntothepresenceofthevalenced-electrons
ofthe twoboundarybinarysystemsaresimilar,the sim- inGaNwhichcause (slight)redistributionofthe valence
pleapproachofinterpolatingbetweenthebinaryELNES density of states, and consequently also the unoccupied
spectra26 is expected to give acceptable results, see e.g. density of states. The electronic structure of the valence
Ti1−xVxN53 or In1−xGaxN19. The extremely small cells band of InN is similar to that of GaN thus resulting in a
(1 In and 1 Ga atom for In Ga N) used in the latter similar ELNES spectrum.
0.5 0.5
reference satisfactorily reproduced the N K-edge evolu- The ground state projected density of states (PDOS)
tion, and a much more computationally expensive ap- in Fig. 6 helps to understand the meaning and evolution
proach (using the averagingof several N core-holed sites of peaks in ELNES of the c-Ti1−xAlxN alloy. The site
in SQSs as inthe presentpaper) employedby Holec29 is and symmetry projected DOS were obtained by averag-
7
(a) x=0 (b) x=0.167 (c) x=0.333 (d) x=0.5 (e) x=0.667
s]
nit N K-edge
u
b.
ar
y [
sit
n N p-PDOS
e
nt
S i
E
N
L
E
S /
O Ti d-PDOS
D
P
s p
Al PDOS
0 10 20 0 10 20 0 10 20 0 10 20 0 10 20 30
Energy above Fermi level [eV]
FIG. 6. PDOS/ELNES spectra for c-Ti1−xAlxN with (a) x=0, (b) x=0.167, (c) x=0.333, (d) x=0.5, and (e) x=0.667.
The peaks in N K-edge ELNES closely reflect the distribution of the unoccupied N p states. The overlap of the individual
PDOSes suggest that the double peak at ≈2–4eV above EF arises from hybridisation of N p and Ti d states while the peak
at ≈10eV can beascribed primarily to N p states hybridisedwith Al p states.
ing the corresponding quantities over all sites occupied
with the same specie. The final states of the N K-edge 2.6
transition are unoccupied N p-states which clearly cor-
relate with the ELNES. Based on the PDOS overlaps
it can be concluded that the double-maximum structure e]2.4 N − charge accumulation
justabovethe edgeonsetarisesfromtheNp–Tid-states ρ∆| [ ATil −− cchhaarrggee lloossss
interaction which agrees with the findings of Tsujimoto e |2.2
et al.54 and Lazar et al.23 for binary c-TiN. The second g
ar
peak at around 10–12eV above E have the strongest h
F c
contribution from the N p–Al p-states interaction, the ed 2.0
only exception being pure TiN (no Al present) where a err
sf
small peak in Ti d-states at the same position can be n
a
detected. The different interactions contributing to this Tr1.8
delayedpeakareresponsibleforasharpermaximumwith
clearshoulders inthe case ofpure TiN while resulting in
1.6
a rather broad (and nearly symmetric) shape when Al
is present, see Fig. 6. The peak position changes by al-
0 0.2 0.4 0.6 0.8 1
most 1eV upon adding x = 0.167 mole fraction of AlN AlN mole fraction x
to TiN, while further increase of AlN content results in
only a small shift of the peak (0.5eV for increasing x FIG. 7. Bader analysis of a charge transfer (absolute values)
from 0.167 to 0.667). This can serve as an example why on atoms of individual species.
thesimpleinterpolationbetweenthepropertiesofbinary
compounds (as suggested by MacKenzie et al.26) does
not work.
compound. The influence of Al on bonding in the alloy
The bonding of the c-Ti1−xAlxN alloy consists of a was discussed by several authors34,35,56, generally show-
mixture of covalent and ionic type. The bonding of TiN ingagradualweakeningofthesp3d2hybridisation(which
has been discussed many times in the literature55. It is reflectedby the decreasedintensity ofthe firstdouble-
was concluded that the covalent part is realised by the maximumpeak in the N K-edge ELNES,see Figs. 4 and
sp3d2 orbitals (interaction of Ti 4s and 3d-states with N 6). Additionally, the Bader analysis57 as implemented
2p-states). Additionally, the interaction between Ti 3d in Wien2k shows that there is a significantly increased
orbitalswiththet symmetrycausesanon-zeroDOSat charge transfer from metallic sites to N resulting in a
2g
the Fermi level resulting in the metallic character of the strongerionicbondingwithincreasingAlNmolefraction
8
(seeFig. 7). Itisinterestingtonote thatthe differentAl D. Energy of the edge onset
sites“provide”onaveragealwaysalmostthesamecharge
tobetransferredonNsites(practicallynoscatteraround
The edge onset energy is another important feature
themeanvaluesshownbythetrianglesinFig.7),butthe of the edge; for example, Mizoguchi et al.15 predicted
chargetransferredfrom Ti sites is much more influenced
a range of ≈ 2eV for the N K-edge onset of AlN de-
by the alloy composition. This is likely to be due to dif-
pending on the crystal structure thus having a potential
ferent degrees of hybridisation between Ti and N atoms
to distinguish between these allotropes. It is, however,
depending on the actual neighbourhood of N atoms (i.e.
not straightforward to define the edge energy, in partic-
second-order neighbours of Ti sites).
ulardue to the ambiguityin the backgroundsubtraction
as well as due to the background noise itself. Conse-
quently, we have chosen the energy of the first inflection
C. Influence of the local environment point above the edge threshold for the comparison be-
tween experiment and theory (the edge onset is about
There is some controversyin the literature on how big 2–4eV below).
the supercells should be in order to suppress the mu- It is not surprising that we obtained excellent agree-
tual interactions between core holes. For example, Mi- ment for the binary systems (< 0.1eV for GaN and
zoguchi et al.16 and Tanaka and Mizoguchi58 claimed ≈0.35eVforAlN)sinceforthesesystemstheedgeonset
that cells with more than 100 atoms are needed while energy was used as a fitting parameter for the core hole
32 atom cells were found sufficient by Lazar et al.59 for charge(see sectionIIIA).However,both the experiment
GaNandbyHolecet al.27 forw-AlN.Toaddressthis is- and the theory suggest that the energy of the inflection
sue we plotted the individual spectra resultingfrom core pointdoesnotvarytoomuchwiththecomposition. The
hole being placed on various N sites, and sorted them variationswithin0.2eV(theory)and0.4eV(experiment)
according to the number of the nearest neighbours of can be regardedas the accuracy of the present approach
each specie (in total 4 for the tetrahedrally coordinated (due to, e.g. the background subtraction on the experi-
wurtzite structure (Fig. 8a) and 6 for the octahedrally mentalsideorthesupercelldesign/sizeonthetheoretical
coordinated cubic structure (Fig. 8b,c)) surrounding the side).
particular N site with the core hole. The numbers of A common method calculating the excitation energy
spectra corresponding to individual local environments is to calculate the difference between total energies of
(i.e. the nearest neighbour configuration) results from the initial ground state and the final state with a full
their real counts in the used supercells. Although con- corehole15,46,58 which,inprinciple,followstheexcitation
figurations of the nearest neighbours of some N sites are process60. Thisresultsinvaluesof384.5eVand368.2eV
the same, the higher-order nearest neighbour configura- for w-AlN and c-TiN, respectively, which are hugely un-
tions differ which is why small variations between indi- derestimated as compared with the experimental values
vidual spectra labelled with the same local environment of 402eV and 397eV. The reasonfor this is that the ex-
are obtained. The thick lines on top of each panels in citedelectronwasputasthebackgroundcharge(Fig.9b)
Fig. 8 were obtained by averaging all the curves under- rather than in the unoccupied states (Fig. 9c). When
neath(theirnumberisthesameasthenumberofNsites excited to the unoccupied states values of 406.4eV and
inthe supercell),andthus accountforthe statisticaldis- 404.6eV for the w-AlN and c-TiN are obtained which
tribution of various local environments of N atoms. are much closer to the experimental values. Rashkova
The graphs clearly demonstrate the huge differences et al.46 showedthatafurtherimprovement(towardsthe
between spectra depending on the local environment of experimental values) could be obtained by performing
the N site where the excitation takes place. At the spinpolarisedcalculations. Nevertheless,theedgeshapes
same time one can see that almost doubling the num- aswellastheenergiesoftheinitialcorelevelsarealmost
ber of atomsin the supercell(from36in Fig. 8bto 64in identical using both approaches (background charge vs.
Fig. 8c) does not alter the N K-edge significantly. The valence band) thus yielding comparable results (except
small changes might be due to having insufficiently big fortheedgeonset),whichisinagreementwithH´ebert10.
cell in the case of 36 atoms, but also could be due to When the edge onset is calculated as the total energy
a non-representative (i.e. not SQS-like) cell for the big- difference between the ground and full core hole states,
ger structure which is suggested, e.g. by not having the its value is a given number without any degree of free-
2Al, 4Ti local environment present. In summary, Fig. 8 dom for adjustments. The corresponding ELNES shape
demonstrates that (i) the local environment influences then should be calculated with exactly 0.5e core hole61.
thefinalshapeoftheedgemuchmoredrasticallythanthe This could be useful when estimating, e.g. ELNES of
actual cell size (provided the cell is big enough to model experimentally inaccessible phases. However, when one
the “randomness” of an alloy), and (ii) the similarity of uses the core hole charge as a fitting parameter (as in
the curves from Al-, Ti-, Ga-rich local neighbourhoods this paper) then it is well justified that also the edge
to those of pure AlN, TiN and GaN, respectively, gives onset is not a unique number but instead a function of
some groundsfor the interpolationapproach(see section the core hole charge. For evaluation of the energy differ-
IVA). ence between the core state and the lowest unoccupied
9
(a) Al Ga N (b) Ti Al N (c) Ti Al N
0.5 0.5 0.5 0.5 0.5 0.5
36 atoms 64 atoms
s]
nit
u 1Al - 5Ti
b.
ar
y [
sit
n
e
nt
S i 1Al - 5Ti
E 0Al - 4Ga
N
L
E 2Al - 4Ti 3Al - 3Ti
e 1Al - 3Ga
g
d
e
K-
N 2Al - 2Ga 3Al - 3Ti
3Al - 1Ga 4Al - 2Ti
5Al - 1Ti
4Al - 0Ga 5Al - 1Ti
0 10 20 30 0 10 20 30 0 10 20 30
Energy above the Fermi level [eV] Energy above the Fermi level [eV] Energy above the Fermi level [eV]
FIG.8. Theresultingspectrarepresentingthealloys (thickcurveson top ofeach panel) andspectracalculated byplacingthe
core hole on individual N sites and sorted accorting to the nearest neighbour configurations. (a) Wurtzite Al0.5Ga0.5N, (b)
cubic Ti0.5Al0.5N supercells with 36 atoms, and (c) cubic Ti0.5Al0.5N supercells with 64 atoms.
(a) (b) (c)
ground state background charge valence band
Fermi
d d d level
n n n
a a a
b b b
e e e
c c c
n n n
e e e
al y al y al y
v g v g v g
er er er
n n n
e e e
et et et
s s s
n n n
o o o
e e e
g g g
d d d
e e e
1s
1s 1s
FIG.9. (a)Agroundstateandanexcitedstatewithafullcorehole: (b)theexcitedelectronisputasthe“backgroundcharge”
(i.e. effectively removed from thesystem) while (c) it is put in thevalence band (intoprior theexcitation unoccupied states).
The scheme corresponds to a metallic system.
state, however, the approach with background charge is V. CONCLUSIONS
more appropriate since, for example, in the case of the
conductive TiN it allows to get the energy of the lowest
(originally) unoccupied state (compare Figs. 9b and c). In this paper we have demonstrated a semi-empirical
approach for predictive calculations of the N K-edge
ELNES of various classes of alloys (cubic vs. wurtzite,
metallic vs. semiconducting). We used fractional core
holes with charges, carefully adjusted according to the
10
edge shape and the onset energy, to reproduce experi- sityofthe NK-edgestructure≈3eVabovethe edgeon-
mental spectra. Subsequently we utilised these to model set reflects a weakening of the sp3d2 hybridisation with
the ELNES spectra of alloys using the special quasi- increasing Al content in Ti1−xAlxN.
random supercells. We introduced core holes on all
individual N sites, and by averaging these spectra we
achieved a representative alloy spectrum. A comparison
with the experimental measurements (for c-Ti1−xAlxN VI. ACKNOWLEDGEMENTS
and w-AlxGa1−xN systems) yielded an excellent agree-
ment on both the edge shapes (including peak positions
and relative intensities) as well as the edge onset ener- Financial support by the START Program (Y371) of
gies. Finally, we related the individual peaks in the N the Austrian Science Fund (FWF) and by the UK En-
K-edge ELNES to various interactions between cations gineering and Physical Sciences Research Council (EP-
and N atoms demonstrating that the decrease in inten- SRC) is greatly acknowledged.
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