Physica B 407 (2012) 4615–4621

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Physica B

0921-45

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n Corr

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Electronic structure and optical properties of ordered compounds potassiumtantalate and potassium niobate and their disordered alloys

T.P. Sinha a, Alo Dutta a, Sonali Saha b, Kartick Tarafder c, Biplab Sanyal c,Olle Eriksson c, Abhijit Mookerjee d,n

a Department of Physics, Bose Institute, 93/1, Acharya Prafulla Chandra Road, Kolkata 700009, Indiab Department of Physics, Sarojini Naidu College for Women, Kolkata 700028, Indiac Division of Materials Theory, Department of Physics and Materials Science, Uppsala University, Box 530, 751 21 Uppsala, Swedend Department of Condensed Matter and Materials Sciences, S.N. Bose National Centre for Basic Sciences, JD Block, Sector 3, Salt Lake City, Kolkata 700091, India

a r t i c l e i n f o

Article history:

Received 30 April 2012

Received in revised form

29 July 2012

Accepted 7 September 2012Available online 13 September 2012

Keywords:

Optical properties

Disordered alloys

26/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.physb.2012.09.011

esponding author. Tel.: þ91 33 2335 5706; fa

ail address: abhijit@bose.res.in (A. Mookerjee

a b s t r a c t

The electronic energy band structure, site and angular momentum decomposed density of states (DOS)

of cubic perovskite oxides KNbO3 and KTaO3 have been obtained from a first principles density

functional based full potential linearized augmented plane wave (FLAPW) method within a generalized

gradient approximation (GGA). The total DOS in valence region is compared with the experimental

photo-emission spectra (PES). The calculated DOS is in good agreement with the experimental energy

spectra and the features in the spectra are interpreted by comparison with the projected density of

states (PDOS). The valence band PES is mainly composed of Nb-4d/Ta-5d and O 2p states in KNbO3 and

KTaO3, respectively. Using the PDOS and the band structure we have analyzed the inter-band

contribution to the optical properties of these materials. The real and imaginary parts of the dielectric

function have been calculated and compared with experimental data. They are found to be in a

reasonable agreement. The role of band structure on the optical properties have been discussed.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

Ferro-electric and related perovskite compounds having che-mical formulae ABO3 are the subject of extensive investigation,both because of their technical importance and because offundamental interest in the physics of their phase transitions [1].Within this family of materials, one finds transitions to a widevariety of low symmetry phases, ranging from non-polar anti-ferro-distortive (AFD) to polar ferro-electric (FE) and antiferro-electric (AFE) transitions. The ideal ordered structure is a cubiclattice with the A atoms sitting at the cube corners and B(typically transition metal) atoms sitting at the cube centers.The oxygen atoms sit at the cube faces, forming an octahedralcage surrounding the B cation. The generic compound is ABO3. Atypical example, studied in this paper is shown in Fig. 1.

We can think of this as three interpenetrating sublattices: twoof them simple cubic one displaced from the other along the(1 1 1) direction by half the body-diagonal distance. These areoccupied by the A and B atoms, respectively. The third is a latticeof O octahedra with a B atom at the octahedron centers.

ll rights reserved.

x: þ91 33 2335 3477.

).

The ideal perovskite structure displays a wide variety ofstructural instabilities in the various materials. These may involverotations and distortions of the O octahedra as well as displace-ments of the atoms from their ideal sites. The interplay of theseinstabilities accounts for the rich variety of FE and AFE behaviors.

Potassium tantalate (KTaO3), a so called incipient ferro-elec-tric, has been the subject of ongoing experimental and theoreticalstudies because of its unusual structural property which deviatesfrom other systems of the perovskite family of ferro-electricmaterials. Its failure to transform into a ferro-electric phase withlowering of temperature has been interpreted as a quantum para-electric state in which the long range ordering of the dipolemoments in ferro-electric phase is suppressed by the stabilizedpara-electric phase arising from the quantum fluctuations of theatomic positions at very low temperatures [2,3]. It does notundergo any phase transition, remaining in the cubic perovskitestructure over the whole temperature range [1].

In contrast, potassium niobate (KNbO3) crystallizes in thesimple cubic perovskite structure at high temperatures (above710 K for KNbO3), and undergoes three ferro-electric phasetransitions at lower temperatures, resulting in a series of dis-torted perovskite structures: the tetragonal phase, the orthor-hombic phase and the ground state rhombohedral phase [1]. Theground state structure is derived from the cubic perovskitestructure by opposing shifts of the Nb and O atoms, the Nb atom

K

O

Nb/Ta

K K

K

KK

K

K

O

O

O

O

Fig. 1. The structure of undistorted perovskite ABO3 compounds, showing the

inter-penetrating A and B cubic sublattices with the latter surrounded with O

octahedra.

T.P. Sinha et al. / Physica B 407 (2012) 4615–46214616

along a cubic axis and the large O shifts roughly along the sameaxis. There exists a controversy concerning the mechanism bywhich the ferro-electric ordering is established: whether it is anorder-disorder transition or a displacive one, involving softeningof a zone-center transverse optical phonon mode, since theexperimental evidence favors soft-mode damping [4,5] as wellas an order-disorder character of the ferro-electric transition forKNbO3 has been presented [6,7].

The electronic band structure of KTaO3 and KNbO3 and thezone center phonon frequencies of KTaO3 have been calculated bySingh [8] using local density approximation. Also the densityfunctional studies of the electronic structure of KTaO3 and KNbO3

[9,10] as well as generalized gradient approximation (GGA)studies of KNbO3 has already been done [11]. Though theelectronic structures and the ground state properties of KTaO3

[12–14] as well as the high temperature phases and the relatedcharacter of the phase transitions of KNbO3 [8,12–20], have beenstudied, no serious attempt has been made to calculate the opticalproperties of these materials and compare the results withexperimental data using modern first-principles calculations.The only optical study reported in literature in the case of KTaO3

is the self-consistent tight-binding calculation by Castet-Mejeanand Michel-Calendini [21], where the real and imaginary parts ofthe dielectric constant and the related optical constants such asthe refractive index, the reflectivity and absorption coefficientshave been calculated and compared with experimental data.

Experimental studies of the optical properties of KTaO3 andKNbO3 are available [22–27]. These include reflectivity studies[22,23] which give us information about the general features ofthe optical spectra in KTaO3 and KNbO3 and their relation to theelectronic band structure ; X-ray photo-electron and X-rayfluorescence studies [24,25], where the nature of the spectrawere explained on the basis of first-principles band structurecalculations, with the dipole transition matrix elements takeninto account ; photo-electron spectroscopy (PES) experimentshave been performed [13] and the measured spectra of KTaO3 andKNbO3 were compared with the calculated density of states (DOS)and X-ray photo-electron spectroscopy and X-ray absorption andemission spectroscopy experiments have been carried out tostudy the density of occupied and empty states of KNbO3 [26,27].

In the present study, the electronic structure and opticalproperties of ordered cubic perovskites KTaO3 and KNbO3 werecalculated using a density functional calculation within the GGAon the basis of the full-potential linearized augmented planewave method (FP-LAPW) using Wien-2K code [28]. Since bothTantalum and Niobium are heavy elements relativistic correctionsare necessary. As in an earlier work [12], with which we wish tocompare, we have included the scalar relativistic or Darwincorrection. In a later work we will include both the Darwin and

spin–orbit terms in our Hamiltonian. Wien-2K supports suchcorrections. However, the ASR codes will require extensive revi-sions which we propose to carry out. Our density of states resultsmatch those of Neumann et al. [12] closely.

Our main interest is to study the effect of alloying KTaO3 withKNbO3. In the alloy, the K sublattice and the O octahedra remainunchanged, while the other sublattice is randomly occupied byeither Nb or Ta atoms. The alloy can be generically written asABxB01�xO3. Nb and Ta have almost the same atomic radii: 145 pmand 146 pm, respectively, and covalent radii: 137 pm and 138 pm,respectively. Therefore, they should alloy without any accompa-nying lattice strain. They also have the same number of extendedvalence electrons. Their electronic configurations are [Xe 4f14]4d45s1 and [Kr] 5d36s2, respectively. On alloying them, therefore,large charge transfer is not likely. In this communication we shallpropose the augmented space recursion formalism (ASR), intro-duced by one of us [29,30] to calculate the configuration averagedoptical response function in this disordered alloy. The ASR goesbeyond the single site mean-field approximations like the coher-ent potential approximation (CPA). Optical response is related tothe current-current correlation function. The representation ofthe current operator is off-diagonal in real space, as such opticalresponse in a disordered alloy is dominated by off-diagonaldisorder. The single site CPA cannot accurately deal with it andthe strength of the ASR becomes important. We report anapplication to the x¼50 alloy.

2. Computational details

We shall first study the electronic and optical properties of theordered compounds KNbO3 and KTaO3. Having gained an insight,we shall go on to study the disordered alloy.

2.1. The ordered compounds

The cubic phase of both the materials belong to the spacegroup Pm3m and contain one molecule per unit cell with the Ksitting at the origin (0, 0, 0)a, the Ta or Nb at the body-center (1/2,1/2, 1/2)a and the three oxygen atoms at the three face centers (1/2, 1/2, 0)a, (0, 1/2, 1/2)a and (1/2, 0, 1/2)a. The experimentallattice constants: 7.54 a.u. for KTaO3 and 7.553 a.u. for KNbO3

were input into the FP-LAPW. We have chosen the muffin-tinradii for K, Ti, Nb and O to be 2.0, 1.8, 1.8 and 1.6 a.u. respectively.XC effects have been treated within the GGA [31]. The integralsover the Brillouin zone are performed up to 20k points in theirreducible Brillouin zone, using the special k-points approach.

The linear response of the system to an external electromag-netic field with a small wave vector is reflected in the complexdielectric function EðoÞ. The frequencies of interest to us will bewell above those of phonons, so that we shall consider theelectronic excitations alone. The cubic nature of the materialsleads to a diagonal and isotropic dielectric tensor. The imaginarypart of the dielectric function E00ðoÞ is given by

E00ðoÞ ¼ Ve2

2p_m2o2

!Zd3kXnn0

9/kn9p9kn0S92f ðknÞ

ð1�f ðkn0ÞÞdðEkn�Ekn0�_oÞ ð1Þ

Here o is the energy of the incident photon, p is themomentum operator ð_=iÞ@=@z, 9kn4 is a crystal wave-functionand f(kn) is the Fermi function. Other symbols have their usualmeanings. The real part of EðoÞ can be derived from the imaginarypart using the Kramers–Kronig relations

E0ðoÞ ¼ 1þ2

p P

Z 10

o0E00ðo0Þo02�o2

do0 ð2Þ

T.P. Sinha et al. / Physica B 407 (2012) 4615–4621 4617

where P implies the principal value of the integral. The knowledgeof both the real and imaginary parts of the dielectric functionallows the calculation of important optical functions. If weassume orientation of the crystal surface parallel to the opticalaxis, the reflectivity RðoÞ follows directly from Fresnel’s formula:

RðoÞ ¼ffiffiffiffiffiffiffiffiffiffiEðoÞ

p�1ffiffiffiffiffiffiffiffiffiffi

EðoÞp

þ1ð3Þ

Related optical response functions are refractive index andextinction coefficient

nðoÞ ¼ 1ffiffiffi2p ½

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE0ðoÞ2þE00ðoÞ2

qþE0ðoÞ�1=2

kðoÞ ¼ 1ffiffiffiffiffiffiffi2op ½

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE0ðoÞ2þE00ðoÞ2

q�E0ðoÞ�1=2 ð4Þ

It is to be noted that for the interpretation of the opticalspectra of systems, it does not seem realistic to give a singletransition assignment to the peaks present in a crystal reflectingspectrum since many transitions (direct to indirect) may be foundin the band structure with an energy corresponding to the peakand since states outside the lines and point of symmetry couldcontribute to the reflectivity. Therefore, the symmetry-allowedtransition energies lead to an incomplete description of theoptical spectrum. We have generated theoretical curves usingthe customary approximation made to interpret the opticalspectra from band structure calculations, namely by consideringthe imaginary part E00ðoÞ of the optical dielectric constant EðoÞ asproportional to the joint density of states weighted by o�2. Thetheoretical results are compared with the observed experimentaldata.

2.2. The disordered alloy

As we stated earlier, it would be interesting to study theoptical properties of the disordered alloy of KTaO3 and KNbO3. Inthis alloy, one sublattice is occupied by K atoms, while the other isoccupied randomly with a transition metal Ta or Nb. Thesetransition metal atoms are then surrounded by oxygen octahedra.The type of disorder exhibited by such alloys is called ‘‘partial’’ or‘‘sublattice’’ disorder. Partial,because the disorder exists only inone sublattice between Ta and Nb. The other K sublattice and theO octahedra remain as in the pristine compounds.

We shall propose a methodology for the study of opticalresponse in such partially disordered systems. Our methodologywill involve two distinct parts: first, disorder will be treatedthrough the augmented space formalism (AS) [29] and secondlythe generalized recursion (GR) method [32] for obtaining correla-tion functions related to optical response.

The augmented space formalism was introduced by Mookerjee[29] to calculate configuration averages beyond the standardsingle-site mean-field CPA. It is capable of addressing the problemof off-diagonal disorder successfully and accurately. Off-diagonaldisorder in the current terms is central to our problem of opticalresponse. The formalism has been described in detail in a review[30]. Here we shall introduce only those specific points which arerelevant to its application to the computation of configurationaveraged current-current correlation functions related to opticalresponse.

The binary substitutionally disordered alloy Hamiltonianinvolves random local ‘occupation’ parameters fnRg (where R

labels sites in the disordered sublattice). nR takes the values0 or 1 according to whether the site R in the sublattice {R} isoccupied by a Nb or Ta atom. The AS approach associates witheach random parameter nR an operator NR such that its eigenva-lues are the values taken by nR and its spectral density is theprobability density of nR. The idea of associating operators with

observables is taken from measurement theory familiar in thebasics of quantum mechanics.

If starting with the function f ðnRÞ we construct an operator~f ð ~NRÞ by replacing the variables nR by the corresponding operator~NR, then the average is simply the matrix element of this newoperator between the ‘average state’. This is the essence of theaugmented space theorem [29].

If we wish to calculate averages involving functions of arandom Hamiltonian HðfnRgÞAH, the first step is to constructthe augmented Hamiltonian bHAH�F¼C by replacing therandom parameters with the associated operators. The AS theo-rem [29] states that the matrix element with respect to theaverage state (one without any configuration fluctuations) is theconfiguration average of that function

0F½HðfnRgÞ�T¼/f|g9 ~F ½bHðf ~NRgÞ�9f|gS ð5Þ

The fluctuation-dissipation theorem relates the imaginary partof the Laplace transform of the generalized susceptibility to theLaplace transform of the current–current correlation function

0w00ðoÞT¼ ð1=2_Þð1�exp�b_oÞ0SðoÞT ð6Þ

where

0SðoÞT¼ limd-0þ

Z 10

dt eiðoþ idÞt Tr0ðJmðtÞ Jmð0ÞÞT ð7Þ

0SðtÞT¼/R� f|g9bJðtÞbJð0Þ9R� f|gS¼ S½bHðf ~NRgÞ� ð8Þ

where the augmented space Hamiltonian bHAC and the currentoperators bJðtÞAC are constructed by replacing every randomvariable nR by the corresponding operator ~NRAF. The subsequentrecursive procedure has been described in detail by Tarafder andMookerjee [33]. The recursion process generates an orthogonalbasis f9f kSg

(a)

The initial conditions are

9f�1S¼ 0, 9f 0S¼bJ9R� |S

(b)

Further members are generated by a three term recurrence

9f kþ1S¼ bH9f kS�ak9f kS�b2k9f k�1S, kZ0

where mutual orthogonality gives,

ak ¼/f k9H9f kS/f k9f kS

, b2k ¼

/f k9f kS/f k�19f k�1S

ð9Þ

The recursion coefficients now enter the equations

ðz�akÞdkðzÞ�idk,0 ¼ dk�1ðzÞþb2kþ1dkþ1ðzÞ, kZ0 ð10Þ

This set of equations can be solved for d0ðzÞ as a continuedfraction representation:

d0ðzÞ ¼i

z�a0�b2

1

z�a1�b2

2

z�a2� � � �

ð11Þ

Then 0SðoÞT, which is the Laplace transform of the correlationfunction, can be obtained from the above

0SðoÞT¼ limd-0

2 R e d0ðoþ idÞ ð12Þ

The expression for the current operator may be obtained usingthe prescription of Hobbs [34].

T.P. Sinha et al. / Physica B 407 (2012) 4615–46214618

Once the augmented current operators are set up, we use it toconstruct the starting state and perform the recursion in aug-mented space to calculate the configuration averaged correlationfunction 0SðoÞT. Finally the imaginary part of the dielectricfunction is related to this correlation function through

E00ðoÞ ¼ 4po2

0SðoÞT ð13Þ

3. Results and discussion

3.1. The ordered compounds

We have calculated the band structure for KTaO3 and KNbO3

along the high symmetry directions in the Brillouin zone andshown them in Fig. 2. This is to provide an understanding of thelink between the optical properties and band structure of thesematerials. Our band structure are found a good agreement withthe previous work [8,12]. Nine bands are mainly derived fromO 2p orbitals in the valence region which is produced by the crystalfield and the electrostatic interaction between O 2p orbitals. It isclear that the band gap appears between the top-most valenceband (VB) at R-point and the bottom most conduction band (CB) atG-point. Our calculated value of the indirect band gap is � 2:1 eV(FLAPW) for KTaO3 where the experimental band gap is 3.8 eV. Theindirect band gap for KNbO3 is � 1:5 eV (FLAPW) as compared tothe experimentally reported value of 3.1 eV.

In both cases the band gaps are underestimated grossly. This ischaracteristic of all DFT calculations which underestimate bandgaps. In our case we have corrected this by the simple minded‘‘scissors operator’’ technique [35,36], which adds an empiricalenergy to all excited states equal to the difference between theexact and the DFT band gaps. The band gap problem in thedensity functional theory and its local density approximation iswell known. It has often been argued that this is the underlyingreason for disagreement between theory and experiment [37].A very simple, although slightly unsatisfactory, method of cor-recting this problem is the scissors operator. The origin of thescissors shift is partially justified [38]. The ionization energy isgiven by the Kohn–Sham energy I¼�ENðNÞ and the electronaffinity A¼�ENþ1ðNþ1Þ. The band gap Eg is

Eg ¼ I�A¼ ENþ1ðNþ1Þ�ENðNÞ ¼ fENþ1ðNÞ�ENðNÞg

þfENþ1ðNþ1Þ�ENþ1ðNÞg ¼ egþDxc ð14Þ

Here EkðNÞ is the kth Kohn–Sham energy for a N electron system.The shift Dxc is the familiar derivative discontinuity. The simplify-ing assumption is that this Dxc is both k and E independent for theunoccupied bands and zero for the occupied ones. This correctionto the DFT spectrum has been recently calculated within the

-8

-4

0

4

8

12

Ener

gy (e

V)

KTaO3

R X MZ ΓΣΛ Γ Δ

EF

1525

112

15

25

1225

15

25

12

Fig. 2. Calculated DFT energy-band structures for (top) KTaO3 and (bottom) KNbO3. The

in the dielectric function. Those at the G and R points are clearly marked. Energy is ex

Harbola–Sahni formalism [39]. The comparison between the DFTand the Harbola–Sahni band structures for Si is shown in Fig. 6.The comparison shows up the inadequacy of the scissors operatorfor a large energy range. However,for energies near the bottom ofthe conduction band,this approximation is roughly valid. For bandgap calculations the scissors operation is not bad, however forfull-fledged optical properties we need to look at excited stateswith higher energies above the Fermi level. In that case, we haveto take recourse to much more expensive techniques like theexact exchange, GW or the recently proposed Harbola–Sahni.

The nine valence bands at G-point are the three triplydegenerate levels (G15,G25 and G15) separated by energies of1.1 eV (G25�G15) and 0.5 eV (G15�G25). These splittings areproduced by the crystal field and the electrostatic interactionbetween O 2p orbitals. There are six bands in the CB regionrepresented by G250 , G1 and G12 at G-point. The triply (G250 ) anddoubly (G12) degenerate levels mainly represent t2g and eg statesof Ta-4d for KTaO3 and Nb-4d for KNbO3 orbitals separated byenergy of 4.2 eV for Ta-5d and 4 eV for Nb-4d. The single band(G1) arises from the K 4s orbital.

The site and angular-momentum-resolved PDOS for both thematerials are shown in Fig. 3 top panel, as obtained from FLAPWcalculations. The structures in the PDOS are very similar to eachother. Their positions with respect to the Fermi energy do vary alittle, but overall similarity is clear and reflects similar chemistryin the two compounds. Fig. 3, bottom panel shows a comparisonof the experimental PES results [13] with our calculated DOS inthe valence band (VB) region. The calculated band width (� 7 eV)for KTaO3 in both the method is in good agreement with theexperimental band width (� 7 eV) in Fig. 3 (bottom, left). Simi-larly the band width of KNbO3 is also � 7 eV is in good agreementwith the experimental result in Fig. 3 (bottom, right). In the VBmain contribution comes from O-2p orbitals, showing the non-bonding states at the top of the VB. As it is observed from the DOSspectra that there is also a small contribution of t2g and asignificant contribution of eg states of Ta-d (or Nb-d) states inthe middle and lower parts of the VB along with very smallcontributions of Ta-s and Ta-p (or Nb-s and Nb-p) orbitals suggestthat a strong hybridization of Ta with O orbitals (or Nb with O)exist.

The calculated E00ðoÞ spectra are compared with experimentaldata in Fig. 4 [23]. The calculated (FLAPW) imaginary part E00ðoÞ(left panel of Fig. 4) of the complex dielectric constant EðoÞ inKTaO3 shows mainly four peaks and is in good agreement withthe experimental peak structure. The first peak at 5.8 eV isattributed to G15 to G250 transition. The second peak at 6.6 eVand third peak at 10.1 eV come from the X50 to X5 and G15 to G12

transitions, respectively. The fourth peak at 11.14 eV originatesfrom transitions M1 to M5.

-8

-4

0

4

8

12

Ener

gy (e

V)

KNbO3

R X MZ ΓΣΛ Γ Δ

EF

152515

112

25

12

25

15

25

12

se band structures indicate the possible optical transitions which lead to structure

pressed in eV and k!

in a�1 where a is the lattice parameter.

00.020.04

00.05

0.1

012

PDO

S (e

V-1)

00.020.04

-5 0 5 10E-EF (eV)

0

1

Nb-s

Nb-p

Nb-d

K-s

O-p

00.020.04

0

0.1

012

PDO

S (e

V-1)

00.020.04

-5 0 5 10E-EF (eV)

0

1

Ta-s

Ta-p

Ta-d

K-s

O-p

-8 -6 -4 -2 0Relative BE (eV)

ExperimentalFP-LAPW

KTaO3

-8 -6 -4 -2 0Relative BE (eV)

FP-LAPWExperimental

KNbO3

Fig. 3. (Top) The angular momentum and site-projected DOS calculated by FLAPW are shown (left) for KNbO3 and (right) for KTaO3. Dashed line shows the position of

Fermi level. (Bottom) The calculated convoluted DOS curve in valence band region is compared with the PES spectrum (left) for KTaO3, (right) for KNbO3. Dashed lines

show experimental data, continuous lines shows theory. Energy is given in eV and PDOS and DOS in eV�1.

Fig. 4. The calculated imaginary parts of complex dielectric functions (E00ðoÞ) of (top panel) KTaO3 and (bottom panel) KNbO3 are in reasonable agreement with

experiment. The theoretical dielectric functions are shown in arbitrary units with prefactors taken as unity, while the experimental functions are scaled so that the maxima

coincide. Photon energies are in eV.

T.P. Sinha et al. / Physica B 407 (2012) 4615–4621 4619

For KNbO3 in the right panel of Fig. 4 there are four main peaksin calculated E00ðoÞ. The peak at 4.7 eV attributed to X50 to X5

transition. The second peak at 6.3 eV arises from G25 to G12 andX40 to X5 transitions. The third peak at 7.42 eV and fourth peak at9.07 eV are due to X5 to X40 transition and M2 to M3 transition,respectively.

Fig. 5 shows the related optical responses: the refractive indexand the extinction coefficient associated with transmission andabsorption, respectively, for both KTaO3 and KNbO3.

The effective number of valence electrons per unit cell con-tributing in the inter-band transitions can be calculated by meansof the sum rule

neff ðEmÞ ¼2m

Ne2h2

Z Em

0E00ðEÞ dE ð15Þ

where Em denotes the upper limit of integration. The quantities m

and e are the electron mass and charge, respectively. N stands forthe electron density. The results are shown in Fig. 7 for KTaO3 and

0 3 6 9 120

1

2

3

4

n(w

)

0 3 6 9 120

1

2

3

4

0 3 6 9 12photon energy

0

0.2

0.4

0.6

0.8

k(w

)

0 3 6 9 120

0.2

0.4

0.6

0.8

KNbO3KTaO3

KTaO3 KNbO3

Fig. 5. The related optical responses (top) refractive index (bottom) extinction

coefficient for (left) KTaO3 and (right) KNbO3. These response functions show

similar behavior for both the compounds. This is related to the similarities in their

band structures. The response functions are given in arbitrary units, with

prefactors taken as one. Photon energies are given in eV.

L G X W

-0.5

0

0.5

1

Ener

gy (R

y)

Si

Fig. 6. Comparison between LDA and Harbola–Sahni band structures for Si. The

occupied bands almost coincide, while the unoccupied bands are shifted by a

k-dependent factor.

0 5 10 15 20Energy (eV)

0

10

20

30

40

n eff(

E)

KTaO3

KNbO3

Fig. 7. The calculated effective number of electrons (neff ) participating in the

inter-band optical transitions of KTaO3 and KNbO3. The neff ðEmÞ reaches a

saturation value above 20 eV for both compounds showing that the deep-lying

valence orbitals do not participate in the inter-band transition.

-6 -3 0 3 6 9 120

1

2

3

-6 -3 0 3 6 9 120123456

-3 0 3 6 9E-EF

0.5

1

1.5

2PDO

S (e

V-1)

-6 -3 0 3 6 9 12E-EF

0

5

10

15

20

K O

Nb/Ta DOS (per formula unit)

KTaxNb1-xO3

Fig. 8. Partial or component projected densities of states and the total density of

states for the disordered alloy. Since the PDOS of the constituents are similar, the

disordered alloy also has similar PDOS, but smoothed with the life-time correction

due to disorder scattering. Energy is given in eV and DOS in eV�1.

T.P. Sinha et al. / Physica B 407 (2012) 4615–46214620

KNbO3, respectively. The neff ðEmÞ reaches a saturation value above20 eV for both the materials. This shows that the deep-lyingvalence orbitals do not participate in the inter-band transition.

3.2. The disordered alloy

We shall begin by looking at the electronic structure of thedisordered alloy KTaxNb1�xO with x¼0.5. For this we shall turn tothe DFT self-consistent augmented space recursion (ASR) packagedeveloped by us [40]. The ASR package is capable of addressingsuch partially disordered solids.

The Nb and Ta partial densities of states resemble each other.Nb and Ta are isoelectronic, so that there is little charge transfereffects on alloying. The 5d electrons of Ta and more localized thanthe 4d electrons of Nb. Fig. 8 shows the partial or componentprojected DOS for the 50–50 disordered alloy. Disorder effect onthe ordered sublattices is small arising simply out of the couplingof atoms in this sublattices to the disordered (Ta, Nb) sublattice. Asimple minded virtual crystal approximation (VCA) in which theHamiltonian is itself averaged, can adequately describe thesePDOS. The disorder effects are more prominent in the (Ta/Nb)sublattice. Disorder induced scattering leads to finite life-timeeffects in the electronic structure. States spill into the gap andmost of the Bloch structure in the PDOS is smoothed out.

Fig. 9 shows imaginary parts of the complex dielectric functionfor the ordered compounds and the disordered alloy. The virtualcrystal (VCA) as well as the ASR results are shown for comparison.The VCA has no self-energy or life-time effects, however the ASRaccurately reflects such effects. We note that apart from

Fig. 9. Imaginary parts of the complex dielectric function for the disordered alloy

K(Ta,Nb)O3 with x¼0.5. Dashed lines show the same quantities for the constituent

compounds for comparison. Alloying leads to both a shift in the peak and

broadening due to lifetime effects. Energy is given in eV and DOS in eV�1.

T.P. Sinha et al. / Physica B 407 (2012) 4615–4621 4621

smoothing out fine structures, there is a shift in weight away fromthe central peak around 5 eV and shrinking of the gap (as seen inthe DOS). The predominant effect of alloying is this life-timeeffects arising from scattering due to configuration fluctuationscharacteristic of disorder.

4. Conclusions

As a starting point in understanding the effect of alloying onoptical properties, we have made a detailed investigation of theelectronic structure and optical properties of pristine KNbO3 andKTaO3 using the FLAPW method. The total DOS in VB regionobtained from our first principles calculations are compare wellwith the PES experimental results. Our calculations show that thefundamental gaps are indirect in these materials. Our calculatedDFT gaps are small compared with the experimental data. Wehave corrected this with the simplest method of scissorsoperators.

After a comprehensive understanding of the constituent com-pounds, we have looked at their disordered alloys and haveillustrated the disorder scattering contributions leading to energyshifts and life-time dependent broadening in the PDOS as well asoptical response functions. We have suggested the ASR formalismto study the effect of alloying the two compounds. We haveapplied this ASR technique in the TB-LMTO basis to the 50–50disordered alloy and have illustrated the energy shift and life-time broadening due to disorder induced configuration fluctua-tion scattering. This is well reflected in the ASR calculations.

Acknowledgments

This work was done under the HYDRA collaboration betweenour groups. We thank Dr. Mukul Kabir for critical perusal of thepaper prior to submission.

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