aarXiv:0704.2323v2 [physics.optics] 19 Apr 2007 pplied Physics A manuscript No. (will be inserted by the editor) Bandstructure and optical properties of α − LiI O crystal Yung-mau Nie
Institute of Physics, Academia Sinica, 128 Sec. 2, Academia Rd, Nankang, Taipei 115, Taiwan, R.O.C., e-mail: [email protected]
Received: 26 February 2007/ Revised version: date
Abstract
The bandstructure was calculated by thefull-potential linearized augmented plane wave method.The result reveals two important insights to the novelsecond harmonic generation (SHG) of alpha-phase lithiumiodate ( α − LiIO ) crystal: the existence of finite intra-band momentum matrix elements due to the non-inversionsymmetry of the crystal illuminating the potential ofthe intra-band transition, and the strong covalent bond-ing between the I -atoms and the ligand O -atoms re-sulting the condition of the double-resonance. An inter-band transition scenario in SHG as α − LiIO in nano-structure is proposed. The optical properties were cal-culated within the theoretical framework of the time-dependent perturbation of the independent-particle model.The dielectric tensors and the refractive index were eval-uated. Comparisons between the predictions and the re- ⋆ Present address: Computational Materials Science Cen-ter, National Institute for Materials Science, Sengen 1-2-1 Tsukuba, Ibaraki, Japan, 305-0047, Tel/Fax:+81-29-858-8003 sults were made: the x-ray near edge absorption spec-tra; the refractive index at the static limit, and at finitefrequencies. Possible factors attributing the calculationerrors is discussed.
PACS
The alpha-phase lithium iodate ( α − LiIO ), in a hexag-onal crystal as depicted in Fig. 1 [1,2], has been inten-sively studied in the past years due to its novel nonlinearoptical [3], piezoelectric [4,5], and acousto-optic [5,6,7]properties. Its nonlinear optical property, especially inthe second harmonic generation (SHG), always receivesthe attentions from the electro-optical technology, whichis recently rising up due to the all-optical informationprocessing for modern telecommunications systems. Fur-thermore, very recently the development of α − LiIO -based nano-structural systems for the nonlinear optical andstructure and optical properties of α − LiIO crystal 3 waveguide is very promising because of the preservationfor the novel nonlinear optical functions in the bulk. [8,9]In addition, the developments of photorefractive devicesin real-time holography, and image processing, also ur-gently demand the ultraviolet photorefractive propertyof α − LiIO to improve resolution and storage capacity.[10] With the analogous characteristics: high refractiveindex, excellent optical transmittance in the visible andnear-infrared region, and high dielectric constant, thenewly invented room-temperature ferromagnetic semi-conductor, T iO in anatase structure under a Co -doping[11], has inspired the spintronic technology toward themagneto-optic and the optoelectronic applications. Thuspresumably, the α − LiIO crystal should have the poten-tial to be developed as magnetically doped reformationsor a associated substrate material in the same way. Inspite of this material being invented very early, the the-oretical study at First-principles level on the electronicstructure and the optical transition is surprisingly verylimited. The resulting characters of bandstructure canfurther guide the searching and the tailoring of new non-linear optical materials.The present First-principles calculations were per-formed by the full-potential linearized augmented planewave (FLAPW) method of density functional theory de-fined by the local density approximation (DFT-LDA).[12,13,14,15,16,17] Testing by the f-sum rule [18,19], theeigen-values and the momentum-matrix elements wasverified to be valid to produce correct evaluations of op- tical properties. By virtue of the analysis of the band-structure, the insights for the intra-band and inter-bandtransitions of the remarkable SHG property are discov-ered. The near-edge states dominating the optical tran-sition were further analyzed. The conclusion of it sup-plies an useful scenario in the classical dipole-oscillationframework to understand the dielectric response of the α − LiIO crystal. In order to comparing with the exper-imental results about the bandstructure, a simulation ofthe x-ray near edge absorption spectra was incorporatedhere.On the calculation of optical functions, the time-dependent perturbation of independent particle model atthe long-wavelength limit was applied, in which the eigenstates of quasi-particle were approximated as those givenby the DFT-LDA. In addition, in a non-perturbative waythe time-dependent DFT , allowing for the ab-initio cal-culations on electron in external electromagnetic fields,also solves the analogous problem, so it is able to cap-ture the strong dynamic effect of system illuminated inthe high power. [21,20] On the other hand, the recentlywell-developed Berry phase method [22] can also directlysimulate the macroscopic polarization at the ab-initiolevel to the external field effect. Due to a greater de-mand of the computation, so far they are more spec-ified to the nonlinear optical simulations involved sig-nificant dynamic effect and the polarization in ferroelec-tric systems, respectively. However, the relation betweenthe bandstructure information and the optical transi- Yung-mau Nie tion mechanisms, directly inspiring the tailoring the elec-tronic structure of materials, can be feasibly accessed bythe perturbation method. Hence, plenty previous worksapplied it to calculate the dielectric and SHG tensors oftransparent insulators.[23,24,25,26,27,28,29,30,31,32,33]So far comparing with experimental results, such a methodmakes good agreement for the dielectric response calcu-lations.The present article is outlined as follows. In Sec. 2,the implementation of the FLAPW method and the for-malism for the dielectric response will be illustrated. InSec. 3, the resulting bandstructure and the dielectricfunctions will be exhibited. The insights for the novelSHG property and the optical transition structure willbe discussed. In Sec. 4, the comparisons with the experi-ments on the x-ray near edge absorption spectra, the re-fractive index at the static limit and the finite frequencywill be given therein. Finally, a brief summary will begiven in Sec. 5.
The modified FLAPW method, ’APW+lo’ [16], was ap-plied via the implementation of the WIEN2K code. [13,14,15,16] The exchange-correlation potential functionalwas defined by the generalized gradient approximation(GGA) parameterized by Perdew and Wang [34]. Thecore and the valence states were respectively calculated relativistically and semi-relativistically. The muffin-tinradii were set to be 1 .
8, 1 .
83, 1 . Li , the I ,and the O -atom, respectively. The expansions of associ-ated Legendre polynomials for spherical harmonics of thewave function and of the non-spherical full-potential ex-pansion were truncated at l = 10 and l = 4, respectively.The parameter RK max was set to be 8 .
5. Additional lo-cal orbitals were added to incorporate low-lying valencestates in the semi-core regime: 4 d -states of the I -atoms,2 s -states of the Li and of the O -atoms. The lattice con-stants a and c at energy minimum, 5 .
574 and 5 . Li (0 . , . , . I (0 . , . , . O (0 . , . , . In the independent particle model, the optical conduc-tivity function σ is expressed as [35] σ aa ( ω ) = 2 πωΩ Z d k π X n,m | p anm | δ ( ω n − ω m − ω ); (1) δ ( x ) ≡ √ πΓ e − ( xΓ ) , (2)where ω n is the energy of the n -th band; Ω denotes thevolume of unit cell; ω is the photon energy, and p anm isthe a -component of the momentum matrix element inthe Cartesian coordinate. Herein, the subscript n ( m ) andstructure and optical properties of α − LiIO crystal 5 symbolizes a conduction (valance) band. The δ func-tion was defined as a smooth Gaussian distribution withthe Γ of 0 .
35 eV. Thus, the imaginary part of dielec-tric tensor ε can be obtained by the relation: ε ( ω ) =(2 π/ω ) σ ( ω ). By means of Kramers-Kronig transform,the real part ε and the refractive index n ( ω ) can beobtained as follows ε ( ω ) = 1 + 8 P Z ∞ ε ( ω ′ ) ω ′ − ω dω ′ , (3) n ( ω ) = ( p ε ( ω ) + ε ( ω ) + ε ( ω )2 ) / . (4)The Brillouin-zone integration are achieved by the ir-reducible points of special-point sampling [36]. The con-vergence of the integration was tested. The error rangewas estimated to be at least less 0 . × × × ×
21 mesh, respectively.
Firstly, the remarkable feature, all resulting bands withinthe range − O − O and I − O bonding. With respect tothis point, according to the ’ k · p ’ method, the band dis-persion about the Γ -point can be approximated to the second order in ∆ k as[37] E m ( k + ∆ k ) = E m ( k ) + ¯ hm ∆ k · p mm + ¯ h ( ∆ k ) m + ¯ h m X n = m ( ∆ k · p mn )( ∆ k · p nm ) E m ( k ) − E n ( k ) . (5)In a state with inversion symmetry, the parity of any itsphysical expectation is even, so the intra-band momen-tum matrix element p mm vanishes due to the odd parityof the momentum operator. Contrarily, in a state with-out the inversion symmetry, a finite p mm exists for theopposite situation. The former causes the first order termin the expansion to vanish so to give a perfect parabolicband curvature; however, the later results a non-zero lin-ear dependent energy-split added to the parabolic band.The scale of p mm should be much less than the ¯ h∆ k ,deduced from the resulting energy-split appearing to bevery narrow. Then such a fine structure character can beviewed as the finger-print of the structural non-inversionsymmetry of the α − LiIO crystal. In fact, about theSHG the existence of finite p mm subjects the intra-bandtransitions [28,29], though prohibited in the optical tran-sition, so the present work reveals the potential of thistransition mechanism to generate certain contribution.Secondly, there is a large energy-split around 10eVresulted by the covalent bonding between the I -atomsand the ligand O-atoms according to the resulting LDOS,implies an extremely strong interaction associated withthe bond forming. Actually it is even greater than themagnitude of the strong on-site Coulomb repulsion U inmost of transition-metal or rare-earth atoms in the per- Yung-mau Nie ovskite crystals [38,39], also widely applied as nonlinearoptical materials [3]. Furthermore, the derived double-gap feature exhibited in the Fig. 2, separating the I − O bonding and anti-bonding states, as well as the inter-vening states localized on the O -atom near the Fermi-surface, naturally meets the isometric inter-band spac-ing condition for the double-resonance of the inter-bandtransition in SHG. However, the Li -atom was determinedto make only little contribution to the aforementionedstates.Predictably in the nano-structure system some sur-plus bands were induced within the gap because of theincorporation of surface localized states. According tothe double-resonance scenario of the visual hole or thevisual electron mechanism [23] illustrated in Fig. 4, theoriginal novel SHG in the bulk can be still preserved;however, those bands of surface localized states, unlessthey near the gap-edge, would be hard to give a signif-icant impact. On the other hand, after all presumablythe number of the bands from surface localized statesis much less than that from near-edge valence states inthe bulk, also make it not be a major role in SHG. Suchconcepts should be useful to figure out the novel nonlin-ear optical properties in the bulk still preserving in thenano-structured systems. [8,9]The present calculation indicates the band gap as thetype of allowed indirect transition. Based on the result-ing LDOS, the permitted transition between the gap-edge bands indeed occurs within each atomic muffin-tin sphere, according to the angular momentum and parityselection rules of the atomic spectroscopy. The obtainedindirect type agrees with the previous experimental con-clusion [40]. It is deduced for the sake of the direction-ality of the near-edge p-states tending to maximize theband dispersion at the Γ -point, and to minimize it atthe zone corner.[41] The obtained gap value, 3 . ≃ R ∞ Ωπ σ ( ω ) dω = Σ i f i = N eff , was performed to exam the obtained band-structure and the momentum matrix elements. For thevery less contribution, the d -electrons of the I -atomsshould be excluded, so the effective valence electronsis 52, being compatible with the results of N eff : 56 . .
66 for the xx and the zz -component, respectively.This implies the obtained bandstructure and momentummatrix elements to be amenable to the following opticalcalculations. The resulting components of the dielectric tensor are ex-hibited in Fig. 5. In fact, the resulting yy -component isidentity to the xx -one, consisting with the experimentalobservations. The significant optical anisotropy behavesas the obviously different dispersions between the xx -and the zz -component. It is actually dominated by the andstructure and optical properties of α − LiIO crystal 7 discrepancy of respective strength component, | p anm | , inthe Eqn.( 1). Though the information of bandstrucure ishard to directly access the insight of this quantity, itstill supplies the optical transition knowledge: the ini-tial state almost localizing on the O -atoms and the fi-nal state mainly derived from the I − O anti-bondingstates. In fact, the absorption resonance-edge of the re-sulting imaginary part given by the gap separating theabove two classes states, and the location of the absorp-tion peak, 7eV, equivalent to the energy-split betweenthem, further solidate the aforementioned state informa-tion in the transition. Taking advantage of the classicallyelectromagnetic radiation concept, the dynamical distri-bution of the charge-density in transition like a dipolerapidly oscillating out of phase along the I − O bond,is useful to figure out the resulting optical anisotropy.It indeed gives an isotropic radiation on the xy -plane asthe calculated result. The resulting X-ray absorption near edge spectra of the I -atom at the L I , and the L III -edge are shown in Fig. 6.The used parameters of resolution identity to experimen-tal values: 0 .
75, and 0 .
66 eV [46]; the inserted values ofatomic natural widths are 3 .
46, and 3 .
08 eV [47] for the L I - and the L III -edge, respectively. The present calcu-lation agrees well with the previous measurements andtheir associated calculations of the multiple scatteringtheory [46], especially in the near edge regime. Such as the experimentally observed white line feature at the L I -edge, as well as the previously discovered pre-edgestructure at the L III -edge were reproduced. However,the discrepancy at the L I -edge might be due to the re-sulting 5 p -hybridization to be under-estimated.The resulting refractive indices and the correspond-ing experimental results are listed in Table 1. In fact,the adopted formula all are approximated at the long-wavelength limit, so the most adequate comparison tothe experimental data, to drastically rule out the lo-cal field and the dynamic effects, should be right at thestatic limit. The constant, measured from a number ofindependent experiments [48,49,50], is reproducible inthe current calculation. Besides, the remarkable exper-imental negative birefringence of this crystal is also re-produced here. Since a slightly over-estimated ordinaryand extra-ordinary component was produced here, thiscauses the disagreement with experimental values in thenegative birefringence. Previous publications suggestedthe causes of discrepancy originating from the defect indetails of the bandstructure [53,54], so the unincorpo-rated non-local effect of the exact density functional [55]is deduced to make certain influence to them.The resulting dispersion of the ordinary n o and theextra-ordinary refractive index n e at finite frequenciesare shown in the Fig. 7. Generally, the present resultsperfectly match the frequency-dependent trend and givea consistent deviation with respect to the experimentalmeasurements in the transparent regime. Some degree of Yung-mau Nie discrepancy from the experiments might be the result ofun-incorporations of the aforementioned non-local effectand the dynamic many-body effect, and the local fieldeffect.
The truth of the bandstructure and the validity of themomentum matrix elements given by First-principles cal-culations are respectively verified by the comparison withthe previous experimental result of X-ray absorption nearedge spectra, and the test of the f-sum rule. The exis-tence of finite intra-band momentum matrix elementsdue to the non-inverse symmetry of the crystal is re-vealed by the resulting bandstructure, which dominatesthe intra-band transition in SHG. In addition, the energy-split of the I − O bonding results the condition of thedouble-resonance of the inter-band transition in SHG.The suggested scenario of the double-resonance in SHGas α − LiIO in the nano-structure is an useful refer-ence to the tailoring of a same type of nonlinear opticalsystems. The present simulation, basing on the time-dependent perturbation in independent-particle model,can well capture the features of the linear dielectric re-sponse of the α − LiIO crystal. The discrepancies fromthe comparison with experiments are deduced from theignored non-local effect in calculating the bandstructure,and the unincorporated many-body dynamic and localfield effects in evaluating the dielectric functions. Acknowledgements
The author acknowledges Prof. Ding-shengWang for his advisements to start the study. The presentwork has been financially supported by National Science Coun-cil, R. O. C. (Project No. NSC92-2811-M-002-041 and NSC93-2811-M-001-065).
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The refractive indices of α − LiIO crystal at thestatic limit. The ’ n o ’, the ’ n e ’ and the ’B’ label the ordi-nary, the extra-ordinary component, and the birefringence,respectively. The λ labels the operating wavelength.Result n o n e B λ (nm)Present 2.1127 1.7940 -0.3187 -Exp1 [48] 1.8385 1.7050 -0.1275 2249.3Exp2 [49] 1.860 1.719 -0.141 1060Exp3 [50] 1.7940 1.6783 -0.1157 500055. W. G. Aulbur, L. J¨onsson, and J. W. Wilkins, Phys. Rev.B (1996) 8540. Fig. 1
The unit cell structure of the α − LiIO crystal. Inthe bonding sketch, each I -atom (purple color) locates atthe top of pyramid based by three O -atoms (red color). The Li -atoms (grey color)reside the sites on the edge.andstructure and optical properties of α − LiIO crystal 11 K Γ A −20−18−16−14−12−10−8−6−4−2 0 2 4 6 8 10 12 14 16 18 20 M Γ E ( e V ) Fig. 2
The bandstructure of the α − LiIO crystal. Here Γ : (0 , , M : 2 π/a (1 / , , K : 2 π/a (1 / , / , A : 2 π/c (0 , , / DO S ( / e V ) eV ’px’’py’’pz’’s’ 0 0.02 0.04 0.06 0.08−12 −8 −4 0 4 8’px+py’’pz’’s’ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−12−11−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 DO S ( / e V ) eV Fig. 3
The LDOS for the s- and p-states of the I -atom(top) and the O -atom (bottom). The inset of the top paneldepicts the result of d-states of I -atom.2 Yung-mau Nie Visual Electron
Conduct. BandsFermi LevelValence Bands
Visual Hole
Surface Localized StateConduct. BandsFermi LevelValence BandsSurface Localized State photonphoton
Fig. 4
The visual hole and the visual electron mechanimsof the inter-band transition in SHG. The blue arrow depictsthe original transition in the bulk, and the red specifies tothe case for the bands of surface localized states only. εεε −4−2 0 2 4 6 8 10 0 5 10 15 20 25 30 R e I m A b s Photon energy (eV)
Fig. 5
The dispersion of the imaginary part, the real part,and the absolution of the dielectric tensor (in the unit of( eV · sec ) − ). The black and the red lines depict the ε xx - andthe ε zz -components of the dielectric function, respectively.andstructure and optical properties of α − LiIO crystal 13 L I −edge−edgeL III −20 −10 10 20 30 40 50
E(eV) N o r m a li ze d A b s . ( a r b . un it s ) E(eV) −20 −10 0 10 20 30 40 N o r m a li ze d A b s . ( a r b . un it s ) Fig. 6