QUANTUM MECHANICAL MODELLING OF PURE AND DEFECTIVEstachio/2000-Kluwer... · 2006. 2. 15. · t exp...

QUANTUMMECHANICALMODELLING OF PURE AND DEFECTIVE

KNbO� PEROVSKITES

N� E� CHRISTENSEN

Institute of Physics and Astronomy� University of Aarhus�

Aarhus C� DK�� Denmark

E� A� KOTOMIN� R� I� EGLITIS

Institute of Solid State Physics� University of Latvia�

� Kengaraga� Riga LV�� Latvia

A� V� POSTNIKOV� G� BORSTEL

Universit�at Osnabr�uck � Fachbereich Physik�

D�� Osnabr�uck� Germany

D� L� NOVIKOV

Arthur D� Little� Inc� Acorn Park� Cambridge�

MA �� USA

S� TINTE� M�G� STACHIOTTI

Instituto de F �sica Rosario� Universidad Nacional de Rosario�

�� Rosario� Argentina

AND

C�O� RODRIGUEZ

IFLYSIB� Grupo de F �sica S olido� �� La Plata� Argentina

Abstract� Ab initio electronic structure calculations using the density�functional theory �DFT� are performed for KNbO� with and without de�fects� Ferroelectric distortive transitions involve very small changes in en�ergies and are therefore sensitive to DFT�approximations� This is discussedby comparing results obtained with the local density approximation �LDA�to those where generalized gradient approximations �GGA� are used� Theresults of ab initio calculations for F �type centers and bound hole polaronsare compared to those obtained by a semiempirical method of the Interme�diate Neglect of the Di�erential Overlap �INDO�� based on the Hartree�Fock formalism� Supercells with �� and � atoms were used in these twoapproaches� respectively� The relevant experimental data are discussed�

"Defects and Surface-Induced Effects in Advances Perovskites", Kluwer Academic Publishers, The Netherlands, pag.3-16, (2000).

N� E� CHRISTENSEN ET AL�

�� Introduction

KNbO� is one of the oldest known room�temperature ferroelectric mate�rials� that undergoes a sequence of ferroelectric transitions from the high�temperature cubic perovskite structure �� The simplicity of crystal struc�ture makes it appealing for ab initio theory simulations� whereas interestingphotorefractive characteristics and low cost keep KNbO� in a list of seriouscandidates for practical applications in electrooptical devices even with theadvent of new ferroelectric substances� �From the point of view of applica�tions� KNbO� always contains defects � either intentionally introduced ones�like substitutional impurities introduced in the search for particular op�tical characteristics�� or those immanently present even in nominally puresamples �vacancies on di�erent sublattices�� In the latter case� the identi��cation of defects accounting for �sometimes indesirable� absorption bandsmay present a problem�

The ab initio microscopic simulations of electronic structure� lattice dis�tortions and related optical excitations in the presence of defects may meetthe practical needs of defects� identi�cation quite e�ciently� However� thetheory faces certain problems already in the description of ideal defect�freeKNbO�� For one thing� a small magnitude of atomic displacements andcorrespondingly small �of the order of � mRy per formula unit� energygains related to ferroelectric transition set the standards of accuracy for anunderlying calculation quite high� The experience of precision total�energycalculations based on the local density approximation �LDA� within thedensity functional theory �DFT� have shown that the equilibrium volume�although only slightly �by �� underestimated with respect to the exper�imental one� is nevertheless almost too small for the ferroelectric instabilityto become qualitatively possible � � This problem may be solved on thepath of constructing better approximations for the exchange�correlationenergy within the DFT than the LDA� When turning to the simulationsof electronic excitations in defect systems� the DFT�based methods are notalways fully satisfactory� As a simple alternative� the Hartree�Fock schemesmay prove to be useful�

In the present paper� we summarize essential results of our recent stud�ies devoted to di�erent aspects of ab initio electronic structure simulationsin perovskites� We addressed the problem of tuning the generalized gradi�ent approximation �GGA� for KNbO�� among other perovskites� in Ref� � �Two di�erent vacancy�type defects� namely F �centers associated with an Ovacancy� and hole polarons� bound assumedly on a K vacancy� have beenstudied respectively in Refs� �� and �� In the following� we brie�y discussthe technical side of the calculations and concentrate on three abovemen�tioned problems� refering to related experimental information�


Quantum Mechanical Modelling��

�� Methods

Our ab initio DFT calculations in all three cases used linear band struc�ture methods �� the full�potential linear augmented�plane�wave �FLAPW�method �� for pure KNbO� and the linear mu�n�tin orbital �LMTO�scheme� also in a full�potential implementation �� for the study of de�fects �in the supercell approach�� The �� supercells were chosen� i�e�the distance between repeated point defects was �� A� As a consequenceof the large number of eigenstates per k�point in a reduced Brillouin zone�BZ� of the supercell and of the metallicity of the doped system� it wasessential to maintain a dense mesh for the k�integration by the tetrahedronmethod over the BZ� Speci�cally� clear trends in the total energy as func�tion of atomic displacements were only established at �� divisionsof the BZ �i�e�� irreducible k�points for a one�site polaron��

In the study on pure KNbO�� we concentrated on the accuracy of totalenergy evaluations obtained with di�erent approximation schemes in theDFT� The LDA in general leads to overbinding when applied to solids�and this error is particularly serious for perovskites where predictions offerroelectric properties are incorrect if calculated at too small volumes� Asimple way to improve accuracy simply consists in performing the calcula�tions for the experimental volume� but this still leaves the question openabout additional �LDA errors�� Some improvement over the LDA has beenobtained by using gradient approximations for better descriptions of theinhomogeneous electron gas� In the Kohn�Sham density functional theoryonly the exchange�correlation energy EXC �EX�EC which is a functionalof the electron spin densities must be approximated� and for slowly varyingdensities� n� it can be expressed as the volume integrals of n times �unifxc

in the LDA case and f�n��n��rn��rn�� for the GGA case� For practicalcalculations the exchange�correlation energy density of a uniform electrongas� �unifxc �n��n�� and f must be parametrized� The form of �unifxc is nowwell established but which is the best choice of f is still under debate�

The GGA version suggested by Perdew� Burke and Ernzerhof �PBE�� has been very useful in several cases� but some uncertainty in its use isrelated to the choice of the parameter �� in the enhancement factor FX�s�which is directly associated to the degree of localization of the exchange�correlation hole� �Here the variable s is a measure of the relative densitygradient� s � jrnj�kFn� kF giving the Fermi wavenumber of an electrongas of density n�� In their original work PBE proposed FX�s� � � � � ��s�� which satis�ed the inequality FX � �� with � � �� andwith the value of � �� Zhang and Yang found �� that the resultsfor several atoms and molecules were improved by increasing � beyond theoriginally proposed value of �� But� this does not hold for all types of


� N� E� CHRISTENSEN ET AL�

bonds �� and it may well happen that applications to solids appear to beimproved when smaller values of � are used� The parameter � might be aweak function of the reduced Laplacian� � � g�r�n��kF��n �

The motivation for using the DFT�based and Hartree�Fock �HF� �basedcalculation methods in parallel� when applied to defects� is to combinestrong sides of both in a single study� The DFT is expected to be ableto provide good description of the ground state� i�e� to deliver reasonablerelaxation energies and ground�state geometry� In the HF approach� therelaxation energies are generally less accurate because of the omission ofcorrelation e�ects� On the other hand� the HF formalism is well suited forthe evaluation of excitation energies� because the total energies can be cal�culated for any �ground�state or excited� electronic con�guration on equalfooting� This is generally not the case in the DFT� Practical experienceshows that HF and DFT results often exhibit similar qualitative trends inthe description of dielectric properties but quantitatively lie on oppositesides of experimental data� thus e�ectively setting error bars for a theo�retical prediction �� For a HF calculation scheme in a present study� thesemi�empirical Intermediate Neglect of the Di�erential Overlap �INDO� �� method� modi�ed for ionic and partly ionic solids �� has been used�The supercells of the same size as with the FP�LMTO method� i�e� ��were used for the study of F and F� centers� and a larger supercell� �� atoms�� for the more recent calculation of a hole polaron� The e�ectof the supercell size can bee seen from the comparison of the present data�reproduced from Ref� �� with those of Ref� �� where a small ��at� su�percell has been used for the hole polaron as well� In the supercell INDOcalculations� the Brillouin zone summation was restricted to the zone centeronly �in the appropriately reduced zone�� This introduced a certain error�especially large for small supercells� A discussion on possible magnitude ofsuch error is given in Ref� �� The parametrization of the INDO methodfor the calculations on KNbO� has been done in Ref� ��

�� Pure KNbO�� GGA vs� LDA

The accuracy of the LDA when applied to the perovskites� as well as thee�ects of introducing the GGA improvements� are illustrated in the follow�ing by the calculations of� �� the energy�volume curves from which latticeparameters and bulk moduli are derived� � �� phonon modes for the cubicstructure� and � the tendency to undergo a ferroelectric transition whenthe atoms are displaced according to the soft�mode displacement pattern�

In that connection the sensitivity to the choice of � in the PBE GGAis examined� In Fig� � the values of V�V� �V� is the corresponding exper�imental value� are plotted for four perovskites over the range of � values



0.30.99

1.00

1.01

1.02

1.03

1.04

0.4 0.5

κ0.6 0.7 0.8

KNbO3

BaTiO3

KTaO3

SrTiO3

V/V

0

C-178-247

Figure �� Equilibrium volumes calculated for KNbO�� BaTiO�� SrTiO� and KTaO� asfunctions of the PBE�GGA parameter �� V� are the experimental equilibrium volumes�

TABLE �� Frequencies of the �� modes �in cm�� lattice parametera �in �A� and bulk modulus B �in GPa� in KNbO� as calculated in theLDA and GGA approximations� i in the frequency values indicate thatthese are imaginary� i�e� soft mode� Note that all phonon frequenciesare calculated at the experimental equilibrium volume� In parantheses�bulk modulus at the experimental equilibrium volume�

LDA GGA �� GGA ��eq� Expt

Frequencies ��i ��i ��i soft

��

��

a ��

B ��

in the PBE�GGA functional varying from �� to �� As can be seen�both BaTiO� and KNbO� would give perfect determination of the latticeparameter �V�V�� for � � �� In the case of SrTiO� and KTaO� thevalue should be further reduced to � � �� The need of varying � from onesystem to another re�ects the fact that the localization of the exchange�correlation hole is system�dependent�

In order to clarify the e�ect of the GGA functional on the phonon en�



0.0 0.1 0.2

-2

-1

0

1 KNbO3

LDA GGA K=0.8 GGA K=0.6

Ene

rgy

(mR

y/ce

ll)

Displacement (Å)

Figure �� Total energy as a function of the ferroelectric displacement of the Nb atomrelative to the center�of�mass of the O�octahedron in KNbO�� The calculations wereperformed for the experimental equilibrium volumes� and the straight LDA �full linesand crosses� are compared to PBE�GGA results for two choices of � �� and ��

ergies� we have performed frozen phonon calculations for KNbO� at itsexperimental lattice constant� and examined the e�ects of choosing dif�ferent exchange�correlation approximations� i�e� LDA and PBE�GGA withdi�erent � values� The calculated frequencies of the �� modes and the ex�perimental values are shown in Table �� It is seen that the GGA hardensthe phonon frequencies �as compared to the LDA results�� This hardeningproduces a slight reduction of the errors� since the LDA provides phononfrequencies which are understimated by �� as compared with experi�ments� The second conclusion concerns the parameter �� It is evident fromTable � that the e�ect introduced on the GGA phonon frequencies by themodi�cation of � is negligible� In addition� we found � that the eigenvec�tors are practically unchanged�

Finally� to test the sensitivity of the energetics involved in the ferro�electric instabilities when the di�erent exchange�correlation functionals areused� we performed total energy calculations as a function of the o��centerdisplacement of Nb atom� In Fig� � we show the energy as a function ofsuch displacement along the �� Both LDA and GGA �with ��and �� yield a clear ferroelectric instability with similar energetics anddisplacements� and with an energy gain of � �� mRy�cell� The similarobservation has been done by Singh �� in his GGA study of KNbO��



�� Simulation of defects

It is well understood now that point defects play an important role in theelectro�optical and non�linear optical applications of KNbO� and relatedferroelectric materials �� The prospects of the use of KNbO� for the lightfrequency doubling are seriously a�ected by the presence of unidenti�eddefects responsible for induced infrared absorption �� The photorefractivee�ect� important in particular for holographic storage� is also well knownto depend on the presence of impurities and defects� Most of as�grownABO� perovskite crystals are non�stoichiometric and contain considerableamounts of vacancies�The electron F and F� centers �an O vacancy� VO� which traps two orone electron� respectively� �� belong to the most common defectsin oxide crystals� In electron�irradiated KNbO�� a broad absorption bandobserved around �� eV at room temperature has been tentatively ascribedto the F �type centers � � These two defects were the subject of recentab initio LDA and semiempirical calculations �� A transient opticalabsorption band at �� eV has been associated recently �� in analogywith other perovskites� with a hole polaron �a hole bound� probably� to aK vacancy�� The ESR study of KNbO� doped with Ti�� gives a proof thatholes could be trapped by such negatively charged defects �� For example�in BaTiO�� the hole polarons bound to Na and K alkali ions replacing Baand thus forming a negatively charged site attracting a hole �� have alsobeen found� Cation vacancies are the most likely candidates for bindinghole polarons� In irradiated MgO� they are known to trap one or two holesgiving rise to the V� and V� centers �� which are in their nature boundhole polaron and bipolaron� respectively� The results of the experimentalstudies of hole polarons in alkali halides and ferroelectric perovskites revealtwo di�erent forms of atomic structure of polarons� atomic one �one�site��when a hole is localized on a single atom� and molecular�type �two�site��when a hole is shared by two atoms forming a quasi�molecule �� In the present study� we simulate both electon centers and hole polaronsassociated with a K vacancy in KNbO��

�� F �TYPE CENTERS

In the cubic KNbO� all O atoms are equivalent and have the local sym�metry C�v �due to which the excited state of the F �type centers could besplit into a nondegenerate and a doubly�degenerate levels�� The optimizedatomic relaxation around the F center as done by the LMTO shows thatthe Nb neighbours to the O vacancy are displaced outwards by �� a� Theassociated lattice relaxation energy is shown in Table �

The optimized Nb relaxation found in the INDO simulations was ��



TABLE � Absorption �Eabs� and lattice relaxationenergies �Erel� for the electron centers and hole po�larons �relatively to the perfect crystal with a Kvacancy� �in eV�� calculated by LMTO and INDOmethods�

INDO LMTO INDO

Eabs Erel Erel

F �center ��

F��center ��

one�site polaron ��

two�site polaron ��

0

0.4

0.8

1.2

KNbO3LDAF-centre

40 at. cell

(a)

(b)

1.6

2

DO

S (

stat

es/e

V c

ell)

C-178-209/211

-100

0.4

0.8

1.2

KNbO3F-centreNb-d upshifted

1.6

-8 -6 -4 -2ENERGY (eV)

0 2 4

DO

S (

stat

es/e

V)

Figure �� Local density of states of the F �center �left panel� and of the Nb atom nearestto it as calculated by the LMTO method�



i�e� very close to the ab initio calculations� The outward relaxation of nearestK atoms and inward displacements of O atoms are much smaller� Theycontribute �� of the net relaxation energy of �� eV� The F center localenergy level lies �� eV above the top of the valence band� Its molecularorbital contains primarily the contribution from the atomic orbitals of thetwo nearest Nb atoms� Only �� e resides at the orbitals centered at thevacancy site! hence the electron localization inside vacancy is much weakerthan for F centers in ionic oxides where typically �� of the electrondensity is localized in the ground state � � The symmetry analysis of theground�state wave function associated with the F center� done by the TB�LMTO method with the use of the LDA�U formalism �� and by INDO�revealed the same result� namely that the major contribution comes fromthe eg states centered at Nb neighbors �more speci�cally� it is essentiallythe z��r� component� with z in the direction towards the F center�� Thepartial densities of states from the LMTO calculation are shown in Fig� �

For the F� center the relaxation energy of � eV and the Nb displace�ments of �� of a are larger than those for the F center due to a strongerCoulomb repulsion between unscreened O vacancy and Nb atoms� a shareof the electron density inside the O vacancy decreases to �� e� The opticalabsorption energies calculated by means of the "SCF method �the di�er�ence of total energies in relaxed ground state and excited state� for the F�

and F centers are given in Table � Both defects are predicted to have oneof the bands around �� eV� which was observed experimentally � �

�� HOLE POLARONS

In the K vacancy�containing supercell� the relaxation of either one �for theone�site polaron� or two neighboring �for the two�site polaron� O atoms�amongst twelve closest to the K vacancy� has been allowed for� and thechanges in the total energy �as compared to the unrelaxed perovskite struc�ture with a K atom removed� have been analyzed� Also� we studied the fullysymmetric relaxation pattern �breathing of twelve O atoms� around the va�cancy�

The removal from the supercell of a K atom with its � electrons con�tributing to the valence band �VB� produces slightly di�erent e�ects onthe electronic structure� as described within the DFT and in the HF for�malism� Acording to the LMTO result� the Fermi energy lowers� and thesystem becomes metallic �remaining non�magnetic�� Therefore� no speci�coccupied localized state is associated with the vacancy� The local densityof states �DOS� at the sites of interest is shown in Fig� �� As is typicalfor LDA calculations� the one�electron band gap in KNbO� comes out un�derestimated �� eV� as compared to the experimental optical gap ��


�� N� E� CHRISTENSEN ET AL�

−5 0 5 10Energy (eV)

1

2

DO

S (

stat

es/e

V) 1

2

3

1.0

0.5

K−vacancy

O2p

O2p

Nb4d

Figure �� Local DOS at the K vacancy site and at the adjacent oxygen atom �top panel�and at Nb and O sites in perfect KNbO� �bottom panel�� as calculated by LMTO�

eV�� The removal of a K�s electron amounts to adding a hole which formsa localized state at �� eV above the Fermi level� i�e� above the unoccu�pied Nb�d band� In the p�DOS of O atoms neighboring the vacancy� aquasi�local state �that e�ectively screens the hole� is visible just below theFermi level� Apart from that� the Op�DOS is largely una�ected by thepresence of vacancy� and the changes in the DOS of more distant sites �K�Nb� are negligible as compared with those in the perfect crystal� As thecubic symmetry is lifted by allowing a non�uniform relaxation of O atoms�the �screening� quasi�local state is clearly localized at the atom closest tothe vacancy� At the same time� the hole state becomes smeared out in en�ergy� This amounts to the bonding being established between the hole andthe screening charge on one of its neighbors�

In the case of one�site polaron� a single O� ion is displaced towards theK vacancy by �� of the lattice constant �LMTO� or by � �INDO� � seeFig� �� The INDO calculations show that simultaneously� �� other nearestoxygens surrounding the vacancy tend to be slightly displaced outwards



h3 21

VB CBν

Figure �� Sketch of the polaron optical transition from the quasi�local state � near thetop of the valence band to the hole state � below the conduction band bottom� � indicatesthe level of an unpaired electron�

the vacancy� In the two�site �molecular� con�guration� a hole is shared bythe two O atoms which approach each other � by �� LMTO� or ��INDO� � and both shift towards a vacancy � by �� LMTO� or ��INDO�� The lattice relaxation energies �which could be associated with theexperimentally measurable hole thermal ionization energies� are presentedin Table � In both methods the two�site con�guration of a polaron is lowerin energy�

In the INDO treatment� the one�electron optical gap is overestimated�as is typical for the HF calculations �� eV �� but the "SCF gap for thetriplet state is �� eV� close to the experiment� The quasi�local screeningstate is described by a wide band close to the VB top� This is consistent withthe LDA description� The INDO calculation also suggests� and this di�ersfrom the LDA� that the removal of an electron leaves an unpaired electronstate split�o� at ��eV above the VB band top� In case of asymmetricalO relaxation� the molecular orbital associated with this state is centeredat the displaced O atom� on which about �� of unpaired spin densityis localized� The same applies qualitatively to the two�site polaron� withthe only di�erence that the localized state is formed from the p orbitalsof both O atoms approaching the vacancy� with a corresponding symmetrylowering� The localized hole state is also present in the HF description butlies much lower than the corresponding state in the LDA� forming a ��eV �wide band located � �� eV below the conduction band bottom �seeFig� �� In agreement with the general theory of small�radius polarons inionic solids �� the optical absorption corresponds to a hole transferto the state delocalized over nearest oxygens� The absorption energies dueto the electron transition from the quasi�local states near the VB top ��Fig� �� into the vacant polaron band �� Fig� �� for one�site and two�sitepolarons are close �Table �� and both are twice smaller than the experi�mental value for a hole polaron trapped by the Ti impurity �� This showsthat the optical absorption energy of small bound polarons can be stronglydependent on the defect involved�



0 1 2 3 4O displacement (% of a)

−0.4

−0.3

−0.2

−0.1

0.0

Rel

axat

ion

ener

gy (

eV)

LMTO

INDO

0.0 0.5 1.0 1.5 2.0 2.5

O-O center-of-mass displ. (% of a)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

sho

rten

ing

of

O-O

dis

tan

ce (

% o

f a)

�a� �b�

Figure �� Relaxation energy for one�site polaron according to LMTO and INDO cal�culations vs the displacement of one oxygen atom towards the vacancy �a�� relaxationenergy for two�site polaron as function of the center�of�mass displacement of the O�Opair and of the O�O distance according to the INDO calculation �b��

In spite of generally observed considerable degree of covalency in KNbO�

and contrary to a delocalized character of the F center state �� theone�site polaron state remains well localized at the displaced O atom� withonly a small contribution from atomic orbitals of other O ions but nonefrom K or Nb ions� Although there are some di�erences in the descriptionof the �one�particle� electronic structure within the DFT� and HF�basedmethods� the trends in the total energy driving the structure optimizationremain essentially the same� In both approaches� both one�site and two�sitecon�gurations of the hole polaron are much more energetically favorablethan the fully symmetric �breathing mode� relaxation of twelve O atomsaround the K vacancy� This is in line with what is known about small�radius polarons in other ionic solids �� and is caused by the fact thatthe lattice polarization induced by a point charge is much larger than thatdue to a delocalized charge�

�� Conclusions

We emphasize that when dealing with tiny �on the energy scale� e�ectsrelated to the o��center instabilities in ferroelectric perovskites� one shouldbe particularly careful to the details of the method used� like e�g� the treat�ment of exchange�correlation in a DFT�based ab initio scheme� In the studyof charged defects where characteristic energies are much larger� the use ofappropriately tuned semiempirical methods may provide reliable results�



Especially for the study of optical excitations� the use of a HF�type schemein parallel with a DFT�based analysis turns out to be useful�

We demonstrated that the F �type centers could be responsible for theoptical absorption observed around �� eV � � The calculated polaronabsorption �� eV� is close to the observed short�lived absorption bandenergy �� ! hence this band could indeed arise due to a hole polaron boundto a cation vacancy�

�� Acknowledgments

This study was partly supported by the NATO Program for science se�nior visitors �through the Danish Research Agency grant No �� toE� A� K�� as well as by DFG to A� P� and G� B� �the SFB �� and theVolkswagen Foundation �grant to R� E�� Authors are greatly indebted toL� Grigorjeva� D� Millers� and A� I� Popov for fruitful discussions�

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�� N� E� CHRISTENSEN ET AL�

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�� O� F� Schirmer� P� Koidl� and H� G� Reik �� Bound Small Polaron OpticalAbsorption in V� � Type Centres in MgO� phys� status solidi �b �� pp� ��O� F� Schirmer �� Optical Absorption of Small Polarons Bound in OctahedralSymmetry� V� Type Centers in Alkaline Earth Oxides� Z� Phys� B �� pp� ��


QUANTUM MECHANICAL MODELLING OF PURE AND DEFECTIVEstachio/2000-Kluwer... · 2006. 2. 15. · t exp...

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Transcript of QUANTUM MECHANICAL MODELLING OF PURE AND DEFECTIVEstachio/2000-Kluwer... · 2006. 2. 15. · t exp...