Physica B 406 (2011) 1961–1965

Contents lists available at ScienceDirect

Physica B

0921-45

doi:10.1

n Corr

E-m

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

Ab-initio calculations of electronic structure and optical properties ofTiAl alloy

Altaf Hussain a, Sardar Sikandar Hayat b,n, M.A. Choudhry a

a Department of Physics, The Islamia University of Bahawalpur, Bahawalpur 63120, Pakistanb Department of Physics, Hazara University, Mansehra 21300, Pakistan

a r t i c l e i n f o

Article history:

Received 19 January 2011

Accepted 25 February 2011Available online 2 March 2011

Keywords:

Density functional

Electronic structure

Optical properties

Titanium aluminides

Intermetallic compounds

26/$ - see front matter & 2011 Elsevier B.V. A

016/j.physb.2011.02.065

esponding author. Tel.: þ92 321 6823467; fa

ail address: sikandariub@yahoo.com (S. Sikan

a b s t r a c t

The electronic structures and optical properties of TiAl intermetallic alloy system are studied by the

first-principle orthogonalized linear combination of atomic orbitals method. Results on the band

structure, total and partial density of states, localization index, effective atomic charges, and optical

conductivity are presented and discussed in detail. Total density of states spectra reveal that (near the

Fermi level) the majority of the contribution is from Ti-3d states. The effective charge calculations show

an average charge transfer of 0.52 electrons from Ti to Al in primitive cell calculations of TiAl alloy. On

the other hand, calculations using supercell approach reveal an average charge transfer of 0.48

electrons from Ti to Al. The localization index calculations, of primitive cell as well as of supercell,

show the presence of relatively localized states even above the Fermi level for this alloy. The calculated

optical conductivity spectra of TiAl alloy are rich in structures, showing the highest peak at 5.73 eV for

supercell calculations. Calculations of the imaginary part of the linear dielectric function show a

prominent peak at 5.71 eV and a plateau in the range 1.1–3.5 eV.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

Ab initio calculations based on density functional theory (DFT)have become the center of attention among the researchersworking in the field of computational materials science. In therecent past, these types of calculations were performed toinvestigate various interesting properties of many importantsystems and materials [1–10]. The term ‘‘intermetallic alloys’’,in a narrow sense, refers to partially ordered stoichiometric alloysand to ordered non-stoichiometric alloys. Strength at high tem-peratures and superior oxidation resistance make intermetallicmaterials exceptional candidates for use in component design,where long service life in a hostile environment is required.Promising applications include heat treating fixtures, transferrolls for hot metal processing, forging dies, radiant burner tubes,or pyrolyzer parts [11–14]. TiAl, Ti3Al, Ni3Al, NiAl, Ni3Si, Fe3Al,FeAl, and MoSi2 are among the most technologically importantintermetallic alloys.

Due to their attractive mechanical properties, especially atelevated temperatures, TiAl based alloys are promising candidatesfor high temperature structural applications [15]. In 2005, Sot and

ll rights reserved.

x: þ92 062 925 5519.

dar Hayat).

Kurzydlowski [16] calculated elastic properties of Ni3Al and TiAlintermetallic alloys under pressure using first-principle ab initio

technique. The problem of brittleness at room temperatureoffered by titanium aluminides has limited their use as structuralmaterials [17]. It was pointed out in the previous studies [18,19]that brittleness problem can be addressed starting from theresults of ab initio self-consistent electronic structure calculations.Following this idea, Matar and Etourneau [20] investigated theelectronic structure of carbon-containing TiAl alloy using the self-consistent augmented spherical wave (ASW) method. To the bestof our knowledge, no detailed studies have been performed onelectronic structures and optical properties of this technologicallyvery important intermetallic alloy.

The paper is aimed to present results of theoretical calculationson electronic structure and optical properties of ordered TiAlintermetallic alloy system using the first-principle orthogonalizedlinear combination of atomic orbitals (OLCAO) method within thelocal density approximation (LDA) of the density functional theory(DFT) [21]. No attempt is made to go beyond the LDA theory foroptical properties calculations although we are aware of recentdevelopments such as time dependent DFT and more rigoroustheories that account for many electron effects, which have beenapplied to simple crystals. We also calculated the localizationindex (LI) of the electron states and effective charges (Qn) on Ti andAl, thereby providing additional understanding of the alloy and thecharge transfer effects that influence its energetics [22].

A. Hussain et al. / Physica B 406 (2011) 1961–19651962

The next part of the paper is organized as follows. Section 2 isfocused on method of calculation. The results on the electronicstructure are presented in Section 3. In Section 4 the results ofoptical properties are reported. The paper is ended by describingthe major conclusions of the work presented.

Ti1

Ti2

Al

Fig. 1. Unit cell representation of TiAl alloy.

2. Method of calculation

In the OLCAO method, the basis functions used for theconstruction of Bloch functions are atomic orbitals. This is awell-established all-electron method that has been used success-fully to study many crystal systems and various proper-ties [23–26]. The atomic orbital basis is most useful for theinterpretation of results, especially in the form of effectivecharges (Qn) on each atom, and the atom-, orbital-, or spin-resolved partial density of states (PDOS). Different types of basisfunctions are used in the OLCAO method. A full basis (FB) – whichconsists of the core orbitals, the valence shell orbitals, and oneadditional shell of atomic orbitals for each atom – is used mainlyfor the self-consistent field calculation of the crystal potential.A minimal basis (MB) set is used for Qn calculations usingthe Mulliken population analysis scheme [27], which is mosteffective when the basis functions are more localized. As anexample, the MB set of Ti consists of the 1s, 2s, 2p, 3s, 3p, 3d,and 4s orbitals and the FB has 4p, 4d, and 5s added to it. For theelectronic structure and optical properties calculations, we usedthe FB set. Self-consistency is achieved by an iterative processwith energy convergence up to 0.0001 eV in less than 20 itera-tions. We used 512 k-points in the irreducible portion of theBrillouin zone (BZ) for primitive cell calculations of TiAl. Toperform our calculations on 5�5�5 supercell, we used 27k-points. We used experimental lattice constants: a¼b¼4.000and c¼4.075 A [28] to perform our calculations.

The LI (Lm) for the state m is an approximate measure of thespread of the wave function of the electron state and is particu-larly useful for identifying the degree of disorder in amorphoussystems. Mathematically, LI is calculated as

Lm ¼Xi,a½rm

ia�2,

where rmia is the fractional electron charge assigned to the ath

atom and is calculated from the resulting eigenfunctions as

rmia ¼

Xj,b

Cm�

ia CmjbSia,jb,

where Cmjb are the eigenvector coefficients, Sia,jb are the overlap

integrals, m is the band index, i and j are the atomic orbitals, and aand b are the atomic labels. Lm lies between 1/N for a completelydelocalized state and 1 for a completely localized state in whichcharge is confined to a single orbital (N is the dimension of theSecular equation, or the total number of states of the system).Further details of the OLCAO method have been previouslypublished [21,27].

The effective charge Q�a based on the Mulliken scheme [29] ofmolecular orbital theory is defined as the valence electronsassociated with an atom a in the crystal and is calculated as

Q�a ¼X

i

Xn,occ

Xj,b

C�nia CnjbSia,jb:

Q�a also provides information about charge transfer. This numberis always positive and should not be confused with the valencecharge state.

The interband optical conductivity s is an important physicalobservable which can be directly compared with experimentaldata. In the OLCAO approach, the real part of the interbandoptical conductivity sð_oÞ is calculated in the random phase

approximation (RPA) using the Kubo–Greenwood formula [30]

sð_oÞ ¼ e2

4p2mo

Zd k!X

n,l

9/cnð k!

, r!Þ9 P!

9clð k!

, r!ÞS92

�flð k!Þ½1�fnð k

!Þ�dðEnð k

!Þ�Elð k

!Þ�_oÞ,

where E¼ _o is the photon energy, f ð k!Þ is the Fermi distribution

function, and l labels an occupied state and n an unoccupied state./cnð k

!, r!Þ9 P!

9clð k!

, r!ÞS is the momentum matrix element (MME).

3. Results and discussion

3.1. Structural setup

TiAl crystallizes in the CuAu tetragonal structure. Its Pearsonsymbol is tP4 and space group is P4/mmm with number 123.Strukturbericht designation for this intermetallic alloy is L10. TiAlstructure can be considered as a succession of Ti and Al layersperpendicular to the c axis (Fig. 1). Ti atoms are present on cubecorners as well as on top and bottom faces, while Al atoms arepresent on remaining four faces. In this way there is a fifty–fiftypercent contribution from both elements Ti and Al in TiAlintermetallic alloy system.

3.2. Electronic structure calculations

Fig. 2 shows our calculated band structure of TiAl intermetallicalloy system along high-symmetry directions. Four bands 6–9 cutthe Fermi level EF. For our calculations, we have set the energy ofthe top of the valence band (i.e. the Fermi level EF) equal to 0 eV.

The calculated total density of states (TDOS) and atomresolved partial density of states (PDOS) for primitive cell andsupercell of TiAl alloy are shown in Fig. 3(a) and (b), respectively.Both the spectra are rich in structures arising from variousnonequivalent sites present in the alloy system. There are twononequivalent sites of Ti (Ti1 and Ti2) in TiAl alloy. Both the sitescarry the same density of states (DOS) spectra. Ti sites dominatethe Al sites in TDOS spectra. Al DOS spectra carry a major peakstructure at �1.605 eV below the Fermi level (EF). In the case ofTi1 and Ti2, the major peak lies at 1.81 eV above the Fermi level.There are also three other peak structures clearly visible in case ofTi. Out of these three, two lie at �1.605 and �0.337 eV below theFermi level. The third one lies at 3.895 eV above the Fermi level.Two shoulder like structures occurring at �0.665 and �2.260 eVbelow the Fermi level can be observed in the DOS spectra of Ti1and Ti2. The Fermi level is on the steep side of Ti1 and Ti2 andalso for Al. There is also a shoulder present at �2.321 eV in AlDOS spectra. The general shapes of the TDOS and PDOS spectracalculated using the primitive cell and supercell approaches are

A. Hussain et al. / Physica B 406 (2011) 1961–1965 1963

very similar. The similarity exists both in the valence band (VB)and the conduction band (CB) regions. However, the supercellspectra are smoother and broadened. Primitive cell calculationsshow that the total DOS at EF, N(EF), are 3.43 states/(eV formulaunit), mostly from Ti (2.56 states/(eV formula unit)) and less fromAl (0.87 states/(eV formula unit)). However calculations usingsupercells show the total DOS at EF, N(EF), are 3.48 states/(eV for-mula unit), mostly from Ti (2.66 states/(eV formula unit)) and lessfrom Al (0.84 states/(eV formula unit)).

The calculated Mulliken effective charges Qn on the Ti and Alatoms in TiAl intermetallic alloy system are tabulated in Table 1.Primitive cell as well as the supercell approach was used to obtain

0

2

4

6

8

1

2

3

4

1

2

3

4

-10 -8 -6 -4 -2 0 2 4 6 8 10

0

1

2

3

4

Total

Ti1

Ti2

DO

S (S

tate

s/(e

V f

.u.)

)

Al

Energy (eV)

Fig. 3. Total and partial DOS plots for TiAl using (a) primitive cell and (b) s

-10

-8

-6

-4

-2

0

2

4

6

8

Wave vector

Z P X XM

Ene

rgy

(eV

)

TiAl

Γ Γ

Fig. 2. Band structure of L12 TiAl in CuAu structure along high-symmetry lines.

Dotted line represents the Fermi level (EF¼0 eV).

effective charges Qn data. Primitive cell calculations give effectivecharges Qn of 3.948 and 3.052 electrons for Ti and Al, respectively.These results indicate that Ti atoms have a charge losing behavior(�0.52 electrons), while Al atoms show a charge gaining behavior(þ0.52 electrons). Similar trend can be observed for supercellcalculations but with a slight variation in calculated Qn values. Itshould be emphasized that the Qn calculation for a metallicsystem is used only for a qualitative estimation. For metallicsystems the electron wave functions are quite extended, while Qn

calculations based on the Mulliken scheme use a MB set, that isrelatively localized.

The localization index (LI) is calculated in a way as discussed inSection 2. The LI results calculated for TiAl alloy using primitive aswell as supercell approach are plotted in Figs. 4(a) and (b),respectively. The generally accepted notion regarding electronlocalization in a disordered solid such as an amorphous semicon-ductor is that the states at the band edges are localized and thoseat the center of the bands are delocalized. In metallic systems thisnotion is less clear since electrons in metals are supposed to bemobile (i.e. totally delocalized), although d-orbitals are usuallymore localized than s- or p-orbitals. In the context of the presentwork, we will focus on the electron states within a few eV of theFermi level. Highly localized states have been observed for super-cell calculations in the range 1–4 eV above the Fermi level. States

0

2

4

6

8

1

2

3

4

1

2

3

4

-10 -8 -6 -4 -2 0 2 4 6 8 10

1

2

3

4

Total

Ti1

Ti2

DO

S (S

tate

s/(e

V f

.u.)

)

Energy (eV)

Al

upercell calculations. The Fermi level is taken as the zero energy level.

Table 1Calculated results of the electronic structure of TiAl intermetallic alloy.

Element Effective charges Qn

(electrons)Charge transfer DQn

(electrons)

Primitive cell Ti 3.948 �0.52

Al 3.052 þ0.52

Supercell (500

atoms)

Ti 3.952 �0.48

Al 3.048 þ0.48

-10 -8 -6 -4 -2 0 2 4 6 8 10

0.000

0.001

0.002

0.003

0.004

0.005

0.006

LI

Energy (eV)

-10 -8 -6 -4 -2 0 2 4 6 8 10

0.0

0.1

0.2

0.3

0.4

0.5

LI

Energy (eV)

Fig. 4. Localization index (LI) plot for TiAl using (a) primitive cell and (b) supercell approach.

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

F

E D/

D

A

B

C

Energy (eV)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

BD

D/

A

E

C

Opt

. Con

d. (

1015

sec

-1)

Opt

. Con

d. (

1015

sec

-1)

Energy (eV)

Fig. 5. Optical conductivity spectrum of TiAl using (a) primitive cell and (b) supercell approach.

A. Hussain et al. / Physica B 406 (2011) 1961–19651964

below the Fermi level 0�1 eV are also highly localized in supercellcalculation. This is due to the fact that in TiAl alloy the d-orbitalsof Ti (xx, yy, zz, x2

�y2, and r2�3z2) with specific interactions with

the nearest neighbor Al atoms form symmetry related bands(see Fig. 2), some of which are relatively more localized (flatbands). The flat bands present between the ranges 2 and 4 eVabove the Fermi level and also below the Fermi level areresponsible for this type of behavior. On the other hand, the statesabove 4 eV are completely delocalized, since their wave functionsare composed mostly of s- and p-orbitals. Calculations of LI revealthe same trend for primitive cell but with fewer states.

3.3. Optical properties

The calculated optical conductivity (s) spectra of TiAl alloyusing both primitive and supercell approaches are shown inFig. 5(a) and (b), respectively. The calculated s shows threeprominent peaks A, B, and C at 0.5, 3.02, and 5.73 eV, respectively,and two less prominent features D on the left of peak B and D0 onthe right side of the peak C at 2.10 and 6.48 eV, respectively.There is also a shoulder like structure E present at 4.77 eV. Thehighest peak C seems to be split in the case of primitive cell. Alsothe peak F at 1.39 eV in primitive cell s spectrum is not present inthe case of supercell. On the average, the two spectra seem to besimilar with minute differences.

The real difference can be envisioned near the low energy end(�0 eV) of the two spectra. Supercell s spectrum shows a sharpincrease of conductivity corresponding to intraband transitions inmetals. Such transitions are possible only in sufficiently large

supercells due to the band folding taking place there. The intrabandtransitions in metals are usually approximated by the Drude model

sðoÞ ¼ s0

1þ iot

� �,

where t is the relaxation time. In this work, it is actually calculatedusing ab initio wave functions. From Fig. 5, we can also conclude thatinterband transitions in TiAl start at about 0.5 eV. We have alsochecked that the use of a finer mesh of more k-points, as comparedto 27 k-points for supercell calculations, modifies only the inten-sities of the peaks slightly but not their positions.

The imaginary part of the linear dielectric function (LDF) e2(o)has been calculated from sl(o) using the relation e2(o)¼(4p/o)sl(o) and is presented in Fig. 6 using the supercell approach.The e2(o) curve shows different spectral features. The curve hasa prominent peak A at 5.71 eV and a plateau in the range of1.1–3.5 eV. There is a deep valley like structure observed at 1 eVand a kink B present at 0.47 eV. The curve ascends sharply forenergy values below 0.45 eV. No interesting feature is observedabove energy value of 6.5 eV.

4. Conclusions

We have performed electronic structure and optical conduc-tivity calculations of the L12 TiAl intermetallic alloy system usingthe first-principle OLCAO method considering primitive as well assupercell approach. Our calculations show that the major con-tribution to the TDOS spectra for both the primitive cell andsupercell approaches is from the Ti-3d orbitals with the Fermi

0

0

10

20

30

40

50

B

A

Eps

ilon2

Energy (eV)

Supercell

1 2 3 4 5 6 7 8 9 10

Fig. 6. Calculated imaginary part of the dielectric function using supercell

approach for TiAl alloy.

A. Hussain et al. / Physica B 406 (2011) 1961–1965 1965

level cuts at the steeper side of the TDOS spectra. The PDOSspectra in Al contain fewer structures than those in the Ti and Ti2.LI shows that the states are highly localized above the Fermi levelup to 4.0 eV, while the higher conduction band states (above4.0 eV) are completely delocalized. The calculation of effectivecharges shows a net charge transfer from Ti to Al for bothprimitive as well as the supercell phases.

The optical conductivity calculation of TiAl shows that sincreases rapidly as the transition energy approaches 0 eV insupercell. This is purely a characteristic of metallic systemsinvolving intraband transitions. This is achieved by using asufficiently large supercell. In most calculations, this part of theoptical conductivity is always approximated by the Drude termwith an adjustable parameter to fit the experimental data ifavailable. In the present work, we demonstrate that the intrabandoptical conductivities can be calculated using ab initio method inthe supercell approach. The success of this approach criticallydepends on the efficiency and accuracy of the underlying methodfor the electronic structure theory such that the computations arenot prohibitive when large supercells are used.

Finally, we conclude that this work demonstrates the impor-tance of detailed computational studies of TiAl alloy using primi-tive and supercell approaches wherever they are required. Asmentioned in the introduction, there are major applications forTiAl alloys at high temperatures; fundamental understanding oftheir electronic structure and optical properties can acceleratetheir use in potential new applications. The present study does notinclude the spin-dependent magnetic ordering in the calculation

due to prohibitive computational costs for large supercells. Futureinvestigations will extend to such studies. It is also possible tostudy the compositional dependence of the electronic structureand optical properties of non-stoichiometric compositions of theTiAl alloy using the same supercell approach.

Acknowledgements

Altaf Hussain is thankful to the Department of Physics, Uni-versity of Missouri—Kansas City (UMKC), for providing access tothe computational resources and providing local hospitality. He isalso thankful to the Higher Education Commission (HEC) ofPakistan, for supporting financially and making it possible forhim to work at UMKC (USA).

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