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Giant piezooptics effect in the ZnO–Er3+ crystalline films deposited on the glasses
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Optics & Laser Technology 36 (2004) 173 – 180 www.elsevier.com/locate/optlastec
Giant piezooptics e%ect in the ZnO–Er 3+ crystalline )lms deposited on the glasses J. Ebothea , W. Gruhnb , A. Elhichoua , I.V. Kitykb;∗ , R. Douniaa , M. Addoua a Universite
de Reims, UFR Sciences, Unite de Thermique et d’Analyse, Physique-LMET UPRES EA no 2061, 21 rue Clement Ader, 51865 Reims Cedex 02, France b Institute of Physics, WSP, Al. Armii Krajowej 13/15, Cze 1 stochowa 42-200, Poland Received 28 March 2003; received in revised form 22 July 2003; accepted 29 July 2003
Abstract An appearance of the giant photoelastic e%ect (up to 17 × 10−13 m2 =N) (for the wavelength of about 450 nm) in the ZnO )lms doped by erbium ions is found. For a description of the observed phenomenon, a complex approach including self-consistent band structure calculations together with the appropriate molecular dynamics simulations of the interfaces was used. The origin of the observed e%ect is caused by the appearance of substantial charge density redistribution within the crystalline ZnO–Er 3+ )lms deposited on the bare glass substrate as well as due to additional charge density polarization by the Er. A possibility of enhancement of the photoelastic coe@cients 2222 at = 600 nm was proved for the )rst time by the simultaneous use of interface charge density redistribution and cationic doping of the )lms. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Piezooptics, Photoinduced e%ects; Elastooptics
1. Introduction The wurtzite-like ZnO single crystals possess hexagonal symmetry (sp. group C46v ) and they may promise to have good properties described by the fourth-order polar tensors. Particularly, in Ref. [1] it was reported that their large electromechanical coupling bears a large piezoelectric e%ect, that is just used in industry. From general phenomenological considerations one can expect that the mentioned materials should also possess good third-order non-linear optical properties, to which also belongs the piezooptical or photoelastic e%ect (PEE). Of particular interest is the PEE described by the fourth-order polar tensor ijkl . Use of )lm interfaces as well as appropriate doping is a promising way for enhancement of the PEE. To perform a search of an appropriate dopant, we have applied a complex approach including theoretical band structure (BS) and molecular dynamics (MD) simulations. To evaluate the third-order optical susceptibilities, particularly the ijkl tensor components of the PEE and inJuence of the interface and doping on them, we perform ∗
Corresponding author. E-mail address: [email protected] (I.V. Kityk).
0030-3992/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2003.07.004
the self-consistent BS calculations of the perfect and doped ZnO–Er 3+ crystalline )lms. A separate topic presents MD structure interface optimization. The technique of sample preparation measurements is presented in Section 2. In Section 3 we present results of the BS calculations together with the MD dynamics simulations of interface structures. The results of the BS simulations together with the evaluated Pockels coe@cients are given in Section 4. We show that the large Pockels e%ect is caused both by ZnO–Er 3+ glass interfaces as well as by doping. 2. Experimental methods 2.1. Sample preparation The single ZnO crystalline )lms were deposited by the spray deposition technique from an aqueous solution of ZnCl2 (Zn2+ concentration = 0:05 M) for a pure ZnO sample. For the doped specimens the appropriate solution was made of ZnCl2 + NH4 F (same Zn2+ conc.) for ZnO–Er 3+ 1%, 2%, 3%, 5%, 10% and 15% in weight. Substrate temperature was about T = 723 K. The investigated thin )lms possess prevailingly a hexagonal-like
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J. Ebothe et al. / Optics & Laser Technology 36 (2004) 173 – 180
structure and the preferred growth orientation was along the [002] axis which is normal to the bare glass substrate plane. The particular electron microscopy investigations have shown that doping does not change this orientation substantially. The crystallinity is very well de)ned due to the relatively high substrate temperature favoring the formation of pure material. 2.2. Measurement technique The photoelastic e%ect described by the fourth rank tensor is determined by the second-order microscopic hyperpolarizability. From the general phenomenological considerations, one can expect an occurrence of large second-order hyperpolarizabilities on the borders separating the chromophore and glass. For such kinds of molecules external mechanical stress may shift energies of the trapping levels originating from the interfaces and change the values of the corresponding transition dipole moments. The piezooptic e%ect is described by the fourth rank tensor ijkl that relates the second-order mechanical tensor component kl with the refractive index coe@cient’s nij . It could be expressed by the equation
1 n2ij
=
3
ijkl kl :
(1)
k;l=1
The typical dependence is presented in Fig. 1. The measurements of the PEE tensor coe@cients were conducted using the standard Senarmont method [2]. A light from a grating monochromator with spectral range 380 – 900 nm and power of about 8–14 mW has been used as a light source. The diameter of the light beam was equal to about 2:5 nm. The precision of the birefringence evaluation was about 10−5 . This allows us to measure the PEE tensor component with an accuracy up to 5 × 10−14 m2 =N. The grating monochromator’s spectral resolution was about 0:3 nm=mm.
Fig. 1. Typical dependence of the transmitted light intensity. The squares indicate the data for the Er-doped )lms and ×—the data corresponding to the pure )lms.
3. Molecular dynamics interface optimization on the ZnO–Er3+ crystal–glass interfaces Geometry optimization of the ZnO )lm–glass interfaces plays a key role in understanding the origin of the PEE e%ect described by third-order polar tensors. The structure and coordination bonding of a bare glass sheet touching the ZnO polycrystalline )lm were optimized using Monte-Carlo and Langevin molecular dynamics structure calculations. MD reconstruction of the glass Si–O sp3 -bonds touching the ZnO wurtzite )lms was performed. The geometry optimization of the reconstructed ZnO surfaces was started from the interface between the bare glass substrate and the ZnO–Er 3+ )lm sheet background. The optimization was started from the )fth neighboring layers (two from the crystallite side and three in the direction of the bare glass substrate). About 90 –120 atoms from both sides of the interface have been included. The MD procedure was carried out until the total energy minimum value for the next cluster was the same as for the whole cluster (re-normalization per one ion). At the next step, the next layer was considered, and the procedure was repeated for the new total energy per ion with a precision up to 0:01 eV. The iteration process was repeated until the relative displacement of the ions for two S The latter value was successive layers was less than 0:32 A. limited within the framework of the adopted model. More details are presented in Ref. [3]. Afterwards, the structure of the surrounding (near-theinterface ZnO) sheets was additionally optimized using ab initio Car–Parrinello molecular dynamics (CPMD) [4] within a density functional theory (DFT) pseudopotential description. The functional used in the CPMD approach was of the Kleinman–Bylander type [5]. The atom core positions and the plane wave (PW) expansion coe@cients were treated as simultaneous dynamic variables. Simulations in this approach were performed using super cells containing 242 atoms with a PW cuto% of 56 Ry. Martins– Trouiller pseudopotentials [6] in the Kleinmann–Bylander form [5] were used to represent the Zn and O ion cores. This pseudopotential [7] was veri)ed for accuracy against the well-known molecules. At a PW cuto% of 56 Ry, the computed bond lengths fell within 3% of the observed experimental values. The so-called “liquid quench” method was employed to produce the )nal network at each bare glass con)guration. The relative stability of the reconstructed interfaces was studied, in order to address the discrepancy between the experiment and existing DFT-LDA calculations. Prevailing evaluations have shown that the local density approximation (LDA) approach underestimates the ZnO bulk lattice constant by 2.68% and the pseudopotential approach overestimates it by 0.16%. The more preferable geometry S corresponds to the lattice constant of about a = 3:254 A, S c = 5:2103 A. The performed calculations have shown that the near-the-interface ZnO region may be considered as a
J. Ebothe et al. / Optics & Laser Technology 36 (2004) 173 – 180
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Table 1 Changes of e%ective structural parameters of the ZnO–Er 3+ reconstructed surfaces while moving from the interface (bare glass-ZnO crystallite) towards the deeper crystalline layers
Distance d from the glass ZnO–Er 3+ (7%) substrate (nm)
a (nm), ZnO–Er 3+ (7%)
c (nm)
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
2.863(4) 2.899(3) 2.901(1) 2.942(5) 2.961(4) 2.990(2) 2.992(4) 3.002(1) 3.021(6) 3.002(1)
5.507(1) 5.476(3) 5.442(2) 5.407(3) 5.371(2) 5.259(4) 5.128(3) 5.203(4) 5.205(7) 5.201(6)
(2.869) (2.906) (2.913) (2.949) (2.968) (2.998) (3.189) (3.242) (3.245) (3.246)
(5.517) (5.482) (5.451) (5.407) (5.372) (5.261) (5.729) (5.201) (5.209) (5.203)
In the brackets the same parameters for the Er-doped ZnO crystallites are given.
reconstructed ZnO quantum con)ned surfaces deposited on the bare glass. The optimized structural ZnO data are presented in Table 1. Because they are dependent on the distance from the ZnO layer–glass interface, the appropriate results are presented versus the distance d between the interface and actually optimized ZnO layer. One can see that the e%ective distance d for the Zn-O chemical bond shows considerable changes while moving from the ZnO–glass interfaces towards more deeper ZnO layers. It is crucial that the ratio of the c=a increases with penetration into the interface. An additional increase is also observed for the Er 3+ -doped crystals. For the distances higher than 1:6 nm the reconstruction of the ZnO wurtzite structure disappears and the bulk-like crystalline structure begins to dominate. In Figs. 2–4 di%erential local charge density distribution versus the distance d obtained by the method described above are presented. One can see substantial charge density redistribution determining the electronic structure of the reconstructed ZnO surface. Because the thickness of the ZnO crystallines is relatively large (about 1000 nm) in this case one can propose a structural model similar to the ZnO wurtzite-like crystallites disturbed by the bare glass. In Fig. 2 fragments of the local Zn–O charge density distribution presented for near-the-surface crystallites without doping (see Fig. 2) as well as after doping by Er 3+ (see Fig. 2b and c) are shown. First of all one can see that the interface causes substantial redistribution of the charge density electrostatic potential. The distances Zn–O–Zn show only little charge density redistribution near the glass–)lm interface. Such disturbances in the structure indicate two key contributions to the charge density non-centrosymmetry. The )rst one is connected with the presence of the ZnO– Er 3+ interface. The second one is caused by involvement of the Er 3+ ions favoring additional enhancement of dipole moments. The output e%ect will be dependent on a sign of the particular contributions.
4. Band structure calculations 4.1. Criterion of choosing the calculation technique From the reconstructed ZnO structural fragments, one can expect an appearance of the ZnO BS with a substantially modi)ed k-dispersion. Due to varied e%ective lattice constants (see former Section) one can expect the existence of a modi)ed BS dispersion and increasing e%ective energy gap value. There exist several works devoted to the BS calculations of the perfect bulk ZnO crystals [3]. The choice of the BS calculation technique depends on the kind of properties we would like to simulate. Because we are interested in simulations of the optical behaviors, a main criterion consists in a maximal agreement of experimental optical parameters with the theoretically simulated tones. So we have applied several BS approaches, particularly, a self-consistent full linear augmented PW approach within the local density approximation (FLAPW-LDA) [8,9], modi)ed norm-conserving pseudopotential (NCPP) [10,11], semi-empirical pseudopotential methods (SEPM) [12] etc. Among the BS calculation techniques these methods seem to be more appropriate to interpret the optical data. Thus we will brieJy compare the obtained data with the experimental ones. 4.2. Norm-conserving pseudopotential method The total energy of a crystalline system is expressed within a local density functional approximation (LDFA) with respect to the charge density (r). Electrostatic and exchange-correlation e%ects were taken into account. A non-linear extrapolation procedure was carried out for evaluation of the weighting coe@cients of the corresponding pseudo-wavefunctions, as well as for corresponding derivatives, in a manner convenient for analytic evaluations of
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Fig. 2. (a) Distribution of the electrostatic potential for the ZnO pure )lms, (distance 0:2 nm near the bare glass interface) before erbium introduction. (b) Changes of local charge density distribution after introducing Er 3+ for the deep ZnO crystallines (larger than 2 m). (c) Disturbances of the local Zn–O distribution due to the inJuence of Er 3+ for the distance of about 0:2 nm from the surface.
secular equation matrix elements, particularly (l; r; ) = an r n exp[ − n(l; ) r n ];
(2)
n
where denotes atoms’ kind, l is the corresponding angular momentum, n determines the level of precision of the non-linear )tting evaluated and is varied from 1 to 5. The coe@cients a and n(l; ) are )tting parameters determined using a non-linear )tting procedure with an accuracy of about 0:03 eV. The pseudopotential was chosen in the form Vps(; l) =
3 i=1
Fig. 3. Band structure of the ZnO polycrystalline )lms disturbed by the interface and Er 3+ (squares) calculated by the FLAPW method.
[Ai + r 2 Ai+3 exp(−l(l; ) r 2 )];
(3)
where Ai ; Ai+3 ; l(l; ) are the evaluated )tting coe@cients.
J. Ebothe et al. / Optics & Laser Technology 36 (2004) 173 – 180
for satisfying self-consistency: m − m−1 ¡ ": m
Fig. 4. Dispersion of ordinary refractive indices for pure ZnO and doped Zn–Er crystalline )lms.
This calculation technique is described in detail elsewhere (see for example Ref. [10,11]). A secular equation was expressed in the form 2 ˝ (k + Gn; n )2 − E(k) n; n 2m
+
V (Gn − Gn )S (Gn − Gn ) = 0;
(4)
where E(k) is the eigenenergy for the k-point in the Brillouin zone (BZ); Gn ; Gn are basis wavevectors of interacting PW. Structural form-factors for the atoms of kind were expressed as follows: g() S (Gn − Gn ) = exp(−t(Gn − Gn ) ); (5) N ;
where g() are weighting factors determining partial contributions of the particular structural components (for every ZnO layers) to the total potential, positions of the ; atoms. A similar approach for di%erent structural fragments has been successfully applied for binary solid alloys [13], glasses and organic materials [13]. One can expect that this approach may also be appropriate for di%erent disordered and partially ordered solids including the investigated ZnO and ZnO–Er 3+ )lm interfaces. As a consequence the reconstructed interface contribution is directly taken into account within the mentioned approach. The special Chadhi–Cohen [14] point method was applied for calculations of spatial electron charge density distribution. Secular equation’s diagonalization procedure was carried out at special weighting points of the BZ for each structural type. Acceleration of the iteration convergence was achieved by transferring 77% of the (m − 1)th iteration result to the mth iteration. The following condition was taken as a criterion
177
(6)
During the self-consistent calculations, a level of calculation error (") was better than 0.18%. As a ZnO basic BZ the wurtzite-like hexagonal structure was chosen because it corresponds to the major structural component of the investigated )lms. Varying structural factors for the di%erent layers and from the structural data obtained using the MD geometry optimization (see Table 1) the appropriate atomic positions were put in Eq. (5). To make the core-like state contributions more realistic, the NCPP wavefunctions were modi)ed additionally through their orthogonalization with respect to the core-like LCAO wavefunctions as described in Refs. [10,11,13]. As a result, the total energy deviation during the self-consistent procedure with respect to the energy cuto% and the Perdew– Adler screening parameter was stabilized within the 0:31 eV. 4.3. Local-density-derived SEPM As a second method for calculations of the mentioned structures we have applied a modi)ed SEPM approach introduced in Ref. [12]. This method was proposed explicitly for the quantum dots, wires, wells, and )lms with typical linear dimensions of 1–5 nm. In our case the thickness of the reconstructed ZnO–Er 3+ layers is of the same order. The SEPM approach is much faster than the self-consistent )rst-principle methods used for bulk solids. Indeed, the SchrVodinger equation is solved and an e@cient diagonalization method providing energy levels in a )xed “energy window” is available and appropriate. Unlike e%ective-mass-band approaches [15], this method uses explicit and variation Jexible basis functions, permitting a direct comparison of the wavefunctions with the LDF studies when available. However, contrary to the LDF approach, the current method provides only BS information, transition probabilities, and wave functions, but not the ground-state properties such as equilibrium structural geometry, which have to be assumed at the outset. The general mathematical formalism is similar to that described in Section 4.2. In the scalar local-density approximation (SLDA) the scalar-relativistic atomic pseudopotential is obtained using the Troullier–Martins [6,16] procedure to perform the core-like orthogonalized corrections. A PW basis with a kinetic energy cuto% of 32:5 Ry was used. The non-local part of the ionic pseudopotential V was obtained numerically. It was kept unchanged while moving from the LDA to the SLDA. The “small box” implementation was applied to handle the non-local part, and uses the PW basis set with a large momentum to expand it in Fourier space. Because the LDF band gaps are usually underestimated, it is adjusted to reproduce the experimentally observed BS using self-energy corrections [15]. The changes in the SLDA potentials are small compared with the LDA results. The
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performed calculations have shown that su@cient sizes of the secular matrix were achieved for the matrix sizes varying within the range 140 –290. Since the )nal pseudopotential was rather smooth, a rapidly converged PW expansion was possible. In fact using the e@cient diagonalization method and the full semi-empirical pseudopotential, the electronic structure of up to the 180-atom ZnO–Er supercluster was calculated. 4.4. FLAPW-LDA approach Among the di%erent BS calculation methods, the augmented plane wave (APW) methods are very popular. One of the well known ones is a computer version WIEN97 [7]. In the FLAPW-LDA and in accordance with Ref. [8], the exchange-correlation Hedin–Lundquist [7] parameterization was performed. The ZnO PW basis set was taken for the energy cuto%s of about 17 eV and the potential representation was about 82 eV. The expansion by angular-dependent spherical harmonics was carried out up to l ¡ 12. About 36 states have been taken into account including the six empty ones with a criterion of the self-consistent eigenenergy value stabilization of about 0:016 eV. Summation within the BZ was performed at 36 special points of the BZ [14]. All the core states (1sO, 1sZn, 2sZn, 2pZn, 3sZn, 3pZn, 3dZn, 2sO, 2pO) were recalculated for each iteration step. The Thomas– Fermi screening potential was applied at the starting step and afterwards it was successfully replaced by the Engel screened functions [7]. Additional corrections using the generalized gradient approximation (GGA) [17] to prevent energy gap underestimation were undertaken. The PW basis set was limited by wave vectors less than 9=Rm (Rm —mu@n-tin sphere radius). The pure ZnO crystallines require 2100 PW and ZnO–Er 3+ structural type needs 820 ones. The radius of the Zn sphere was varied within the range RZn =1:08−1:21RO . The interacting spheres were chosen to be non-overlapping (only touching). As core states, 1sO wavefunctions and 1, 2s, 2p Zn were taken. All the remaining orbitals were considered as valence states. For numerical calculations of the total and partial density of states (DOS) numerical simulations by the tetrahedral method [7] using 218 k points with an energy resolution of 0:13 eV were performed. The Zn, O atoms are relatively light and the spin–orbit interaction was neglected during the consideration. For Er 3+ ions we considered only the local part. For the classi)cation of the BZ the notations presented in Ref. [12] are given. 4.5. Results of the BS calculations As an important criterion of the applicability of the concrete method, the structural deviation of the calculated (see Table 2) parameters for the experiment is used. The maximal agreement between experimental and theoretically calculated data gives the modi)ed NCPP approach. The SEPM
Table 2 Lattice constants for the 4H- and 3C-polytypes calculated by the di%erent methods
NCPP modi)ed as in the present work FLAPW-LDA SEPM Experimental obtained from experimental di%raction [15]
S ZnO (A)
S ZnO–Er 3+ (A)
3.258
3.265
3.241 3.233 a = 3:259 c = 5:2081
3.252 3.242 a = 3:263 c = 5:2162
and FLAPW methods give results substantially di%erent from the experimental ones. Typical fragments of the BS dispersions calculated for the ZnO wurtzite structure with and without Er 3+ modi)cation by near-the-interface structure are shown in Fig. 3. One can see that for the both methods taking into account the reconstructed surfaces leads to the substantial modi)cation of the band energy dispersions. Moreover in the case of the NCPP method one can see a substantial shift of the e%ective valence band top and conduction band bottom (indicated by solid line). The presentation is limited only by the NCPP as well as FLAPW approaches because the SEPM data give results more di%erent from experimental data. One can see (see Fig. 3) an appearance of substantial deviations of the k-space BS dispersion from the BS for the pure ZnO wurtzite-like crystals. For convenience we show only deviations for the top of the valence band and the bottom of the conduction band. The more Jat modi)ed bands indicate on important role of the reconstructed near-the-interface structure. This fact con)rms structural reconstruction for the near-the-interface state and a drawback of the semi-empirical method for describing the excited states. E%ective deviation for the NCPP is in a general agreement with the experimental data on the photoelectron spectroscopy. Thus this method was chosen as a basis for the simulations of optical properties. 5. Dispersion of PEE coe.cients At the beginning for simulation of the linear and PEE properties, calculations of imaginary part of dielectric susceptibility optical tensor were carried out. The general formalism [17] was applied by taking into account the calculated matrix dipole moments obtained by the NCPP method. Additionally, the performed numerical veri)cation has shown that non-local e%ects change the average oscillator strength up to 13%. The plasmon peak in the energy loss spectra is shifted towards higher energies up to 2.8–3:6 eV. As a consequence, all the calculations were performed with inclusion of the non-local contributions. The real parts of the dielectric tensors were evaluated using the Kramers– Kronig dispersion relations. The reJectivity and energy loss functions were obtained from Fresnel’s formula.
J. Ebothe et al. / Optics & Laser Technology 36 (2004) 173 – 180
A crucial point in the calculations of the "2 (E) was found by precision of the k-space integration. A linear tetrahedral 1 method for the 526 k points in the 96 irreducible part of the BZ was applied. In the present paper during the BS summation, 72 conduction bands were included in order to achieve su@ciently converged results. The calculated data allow to estimate changes of refractive indices which correspond to the real part of "(E). From the general phenomenology one can also expect substantial enhancement of the PEE tensor components. The results calculated by this method, ijkl data, are presented in Table 3. From Table 3 one can see substantial anisotropy in the tensor component distribution. This reJects the simultaneous speci)c contribution of interface and cationic polarization. From Fig. 4 one can see that doping by erbium ions favors increasing refractive indices connected with the additional anisotropy in the space charge density distribution. This fact reJects the crucial role played by Er 3+ ions in the behavior of the refractive index perpendicular to the optical axes. The performed spectral measurements of PEE coe@cients (see Fig. 5) con)rm theoretical predictions (Table 3) that doping by Er 3+ leads to enhanced PEE coe@cients 2222 . Table 3 Evaluated values of the PEE tensor components for di%erent wavelengths
(nm)
2222 × 10−13 m2 = N
2233 × 10−13 m2 = N
600 650 700 750 800 850 900
5.75 5.12 5.01 4.99 4.97 4.96 4.95
1.15 1.10 1.09 1.03 1.09 1.08 1.00
179
For comparison we have included the data calculated by the method described above for the ZnO–Er samples. One can see a good agreement between the experimental and theoretically calculated results, which indicates a good level of the theoretical approach. Moreover for the wavelengths shorter than 600 nm an increase of PEE coe@cients 2222 values is up to the 5:8 × 10−12 m2 =N. Such a large increase of the PEE coe@cient may open a new stage in the design of thin-like electrooptics light modulators for the lasers. This fact also indicates the particular role of the Er 3+ ions in the observed spectral dependences of PEE e%ect. The observed anisotropy in the PEE tensor components con)rms the substantial role of simultaneous anisotropy of the interface and Er 3+ contributions. 6. Conclusion We have theoretically predicted and experimentally observed a large increase (up to one order) of the photoelastic e%ect for ZnO wurtzite )lms doped by Er 3+ . Using di%erent BS calculations as well as MD interface optimization we have predicted that due to insertion of Er 3+ ions one can expect a substantial increase of the dipole moments on the interface between the bare glass substrate and the appropriate crystalline )lm. So we demonstrate a possibility of simultaneous manipulation by interface charge density distribution together with appropriate doping in order to enhance the piezooptical tensor coe@cients. The discovered phenomenon may open a new stage for creation of the competitive miniaturized PEE devices for the lasers. The proposed theoretical approach may be used for the searching of new )lm-like crystalline materials possessing the enhanced second-order non-linear optical properties. Acknowledgements Financial support from the State Committee for Scienti)c Research (KBN), through University Grant (I.K., W.G.) is gratefully acknowledged. References
Fig. 5. Spectral dependences of the 2222 ×10−12 m2 =N for di%erent )lms deposited on a bare glass substrate ×—2233 ; ◦—theoretically calculated 2222 .
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Giant piezooptics e%ect in the ZnO–Er 3+ crystalline )lms deposited on the glasses J. Ebothea , W. Gruhnb , A. Elhichoua , I.V. Kitykb;∗ , R. Douniaa , M. Addoua a Universite
de Reims, UFR Sciences, Unite de Thermique et d’Analyse, Physique-LMET UPRES EA no 2061, 21 rue Clement Ader, 51865 Reims Cedex 02, France b Institute of Physics, WSP, Al. Armii Krajowej 13/15, Cze 1 stochowa 42-200, Poland Received 28 March 2003; received in revised form 22 July 2003; accepted 29 July 2003
Abstract An appearance of the giant photoelastic e%ect (up to 17 × 10−13 m2 =N) (for the wavelength of about 450 nm) in the ZnO )lms doped by erbium ions is found. For a description of the observed phenomenon, a complex approach including self-consistent band structure calculations together with the appropriate molecular dynamics simulations of the interfaces was used. The origin of the observed e%ect is caused by the appearance of substantial charge density redistribution within the crystalline ZnO–Er 3+ )lms deposited on the bare glass substrate as well as due to additional charge density polarization by the Er. A possibility of enhancement of the photoelastic coe@cients 2222 at = 600 nm was proved for the )rst time by the simultaneous use of interface charge density redistribution and cationic doping of the )lms. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Piezooptics, Photoinduced e%ects; Elastooptics
1. Introduction The wurtzite-like ZnO single crystals possess hexagonal symmetry (sp. group C46v ) and they may promise to have good properties described by the fourth-order polar tensors. Particularly, in Ref. [1] it was reported that their large electromechanical coupling bears a large piezoelectric e%ect, that is just used in industry. From general phenomenological considerations one can expect that the mentioned materials should also possess good third-order non-linear optical properties, to which also belongs the piezooptical or photoelastic e%ect (PEE). Of particular interest is the PEE described by the fourth-order polar tensor ijkl . Use of )lm interfaces as well as appropriate doping is a promising way for enhancement of the PEE. To perform a search of an appropriate dopant, we have applied a complex approach including theoretical band structure (BS) and molecular dynamics (MD) simulations. To evaluate the third-order optical susceptibilities, particularly the ijkl tensor components of the PEE and inJuence of the interface and doping on them, we perform ∗
Corresponding author. E-mail address: [email protected] (I.V. Kityk).
0030-3992/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2003.07.004
the self-consistent BS calculations of the perfect and doped ZnO–Er 3+ crystalline )lms. A separate topic presents MD structure interface optimization. The technique of sample preparation measurements is presented in Section 2. In Section 3 we present results of the BS calculations together with the MD dynamics simulations of interface structures. The results of the BS simulations together with the evaluated Pockels coe@cients are given in Section 4. We show that the large Pockels e%ect is caused both by ZnO–Er 3+ glass interfaces as well as by doping. 2. Experimental methods 2.1. Sample preparation The single ZnO crystalline )lms were deposited by the spray deposition technique from an aqueous solution of ZnCl2 (Zn2+ concentration = 0:05 M) for a pure ZnO sample. For the doped specimens the appropriate solution was made of ZnCl2 + NH4 F (same Zn2+ conc.) for ZnO–Er 3+ 1%, 2%, 3%, 5%, 10% and 15% in weight. Substrate temperature was about T = 723 K. The investigated thin )lms possess prevailingly a hexagonal-like
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structure and the preferred growth orientation was along the [002] axis which is normal to the bare glass substrate plane. The particular electron microscopy investigations have shown that doping does not change this orientation substantially. The crystallinity is very well de)ned due to the relatively high substrate temperature favoring the formation of pure material. 2.2. Measurement technique The photoelastic e%ect described by the fourth rank tensor is determined by the second-order microscopic hyperpolarizability. From the general phenomenological considerations, one can expect an occurrence of large second-order hyperpolarizabilities on the borders separating the chromophore and glass. For such kinds of molecules external mechanical stress may shift energies of the trapping levels originating from the interfaces and change the values of the corresponding transition dipole moments. The piezooptic e%ect is described by the fourth rank tensor ijkl that relates the second-order mechanical tensor component kl with the refractive index coe@cient’s nij . It could be expressed by the equation
1 n2ij
=
3
ijkl kl :
(1)
k;l=1
The typical dependence is presented in Fig. 1. The measurements of the PEE tensor coe@cients were conducted using the standard Senarmont method [2]. A light from a grating monochromator with spectral range 380 – 900 nm and power of about 8–14 mW has been used as a light source. The diameter of the light beam was equal to about 2:5 nm. The precision of the birefringence evaluation was about 10−5 . This allows us to measure the PEE tensor component with an accuracy up to 5 × 10−14 m2 =N. The grating monochromator’s spectral resolution was about 0:3 nm=mm.
Fig. 1. Typical dependence of the transmitted light intensity. The squares indicate the data for the Er-doped )lms and ×—the data corresponding to the pure )lms.
3. Molecular dynamics interface optimization on the ZnO–Er3+ crystal–glass interfaces Geometry optimization of the ZnO )lm–glass interfaces plays a key role in understanding the origin of the PEE e%ect described by third-order polar tensors. The structure and coordination bonding of a bare glass sheet touching the ZnO polycrystalline )lm were optimized using Monte-Carlo and Langevin molecular dynamics structure calculations. MD reconstruction of the glass Si–O sp3 -bonds touching the ZnO wurtzite )lms was performed. The geometry optimization of the reconstructed ZnO surfaces was started from the interface between the bare glass substrate and the ZnO–Er 3+ )lm sheet background. The optimization was started from the )fth neighboring layers (two from the crystallite side and three in the direction of the bare glass substrate). About 90 –120 atoms from both sides of the interface have been included. The MD procedure was carried out until the total energy minimum value for the next cluster was the same as for the whole cluster (re-normalization per one ion). At the next step, the next layer was considered, and the procedure was repeated for the new total energy per ion with a precision up to 0:01 eV. The iteration process was repeated until the relative displacement of the ions for two S The latter value was successive layers was less than 0:32 A. limited within the framework of the adopted model. More details are presented in Ref. [3]. Afterwards, the structure of the surrounding (near-theinterface ZnO) sheets was additionally optimized using ab initio Car–Parrinello molecular dynamics (CPMD) [4] within a density functional theory (DFT) pseudopotential description. The functional used in the CPMD approach was of the Kleinman–Bylander type [5]. The atom core positions and the plane wave (PW) expansion coe@cients were treated as simultaneous dynamic variables. Simulations in this approach were performed using super cells containing 242 atoms with a PW cuto% of 56 Ry. Martins– Trouiller pseudopotentials [6] in the Kleinmann–Bylander form [5] were used to represent the Zn and O ion cores. This pseudopotential [7] was veri)ed for accuracy against the well-known molecules. At a PW cuto% of 56 Ry, the computed bond lengths fell within 3% of the observed experimental values. The so-called “liquid quench” method was employed to produce the )nal network at each bare glass con)guration. The relative stability of the reconstructed interfaces was studied, in order to address the discrepancy between the experiment and existing DFT-LDA calculations. Prevailing evaluations have shown that the local density approximation (LDA) approach underestimates the ZnO bulk lattice constant by 2.68% and the pseudopotential approach overestimates it by 0.16%. The more preferable geometry S corresponds to the lattice constant of about a = 3:254 A, S c = 5:2103 A. The performed calculations have shown that the near-the-interface ZnO region may be considered as a
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Table 1 Changes of e%ective structural parameters of the ZnO–Er 3+ reconstructed surfaces while moving from the interface (bare glass-ZnO crystallite) towards the deeper crystalline layers
Distance d from the glass ZnO–Er 3+ (7%) substrate (nm)
a (nm), ZnO–Er 3+ (7%)
c (nm)
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
2.863(4) 2.899(3) 2.901(1) 2.942(5) 2.961(4) 2.990(2) 2.992(4) 3.002(1) 3.021(6) 3.002(1)
5.507(1) 5.476(3) 5.442(2) 5.407(3) 5.371(2) 5.259(4) 5.128(3) 5.203(4) 5.205(7) 5.201(6)
(2.869) (2.906) (2.913) (2.949) (2.968) (2.998) (3.189) (3.242) (3.245) (3.246)
(5.517) (5.482) (5.451) (5.407) (5.372) (5.261) (5.729) (5.201) (5.209) (5.203)
In the brackets the same parameters for the Er-doped ZnO crystallites are given.
reconstructed ZnO quantum con)ned surfaces deposited on the bare glass. The optimized structural ZnO data are presented in Table 1. Because they are dependent on the distance from the ZnO layer–glass interface, the appropriate results are presented versus the distance d between the interface and actually optimized ZnO layer. One can see that the e%ective distance d for the Zn-O chemical bond shows considerable changes while moving from the ZnO–glass interfaces towards more deeper ZnO layers. It is crucial that the ratio of the c=a increases with penetration into the interface. An additional increase is also observed for the Er 3+ -doped crystals. For the distances higher than 1:6 nm the reconstruction of the ZnO wurtzite structure disappears and the bulk-like crystalline structure begins to dominate. In Figs. 2–4 di%erential local charge density distribution versus the distance d obtained by the method described above are presented. One can see substantial charge density redistribution determining the electronic structure of the reconstructed ZnO surface. Because the thickness of the ZnO crystallines is relatively large (about 1000 nm) in this case one can propose a structural model similar to the ZnO wurtzite-like crystallites disturbed by the bare glass. In Fig. 2 fragments of the local Zn–O charge density distribution presented for near-the-surface crystallites without doping (see Fig. 2) as well as after doping by Er 3+ (see Fig. 2b and c) are shown. First of all one can see that the interface causes substantial redistribution of the charge density electrostatic potential. The distances Zn–O–Zn show only little charge density redistribution near the glass–)lm interface. Such disturbances in the structure indicate two key contributions to the charge density non-centrosymmetry. The )rst one is connected with the presence of the ZnO– Er 3+ interface. The second one is caused by involvement of the Er 3+ ions favoring additional enhancement of dipole moments. The output e%ect will be dependent on a sign of the particular contributions.
4. Band structure calculations 4.1. Criterion of choosing the calculation technique From the reconstructed ZnO structural fragments, one can expect an appearance of the ZnO BS with a substantially modi)ed k-dispersion. Due to varied e%ective lattice constants (see former Section) one can expect the existence of a modi)ed BS dispersion and increasing e%ective energy gap value. There exist several works devoted to the BS calculations of the perfect bulk ZnO crystals [3]. The choice of the BS calculation technique depends on the kind of properties we would like to simulate. Because we are interested in simulations of the optical behaviors, a main criterion consists in a maximal agreement of experimental optical parameters with the theoretically simulated tones. So we have applied several BS approaches, particularly, a self-consistent full linear augmented PW approach within the local density approximation (FLAPW-LDA) [8,9], modi)ed norm-conserving pseudopotential (NCPP) [10,11], semi-empirical pseudopotential methods (SEPM) [12] etc. Among the BS calculation techniques these methods seem to be more appropriate to interpret the optical data. Thus we will brieJy compare the obtained data with the experimental ones. 4.2. Norm-conserving pseudopotential method The total energy of a crystalline system is expressed within a local density functional approximation (LDFA) with respect to the charge density (r). Electrostatic and exchange-correlation e%ects were taken into account. A non-linear extrapolation procedure was carried out for evaluation of the weighting coe@cients of the corresponding pseudo-wavefunctions, as well as for corresponding derivatives, in a manner convenient for analytic evaluations of
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Fig. 2. (a) Distribution of the electrostatic potential for the ZnO pure )lms, (distance 0:2 nm near the bare glass interface) before erbium introduction. (b) Changes of local charge density distribution after introducing Er 3+ for the deep ZnO crystallines (larger than 2 m). (c) Disturbances of the local Zn–O distribution due to the inJuence of Er 3+ for the distance of about 0:2 nm from the surface.
secular equation matrix elements, particularly (l; r; ) = an r n exp[ − n(l; ) r n ];
(2)
n
where denotes atoms’ kind, l is the corresponding angular momentum, n determines the level of precision of the non-linear )tting evaluated and is varied from 1 to 5. The coe@cients a and n(l; ) are )tting parameters determined using a non-linear )tting procedure with an accuracy of about 0:03 eV. The pseudopotential was chosen in the form Vps(; l) =
3 i=1
Fig. 3. Band structure of the ZnO polycrystalline )lms disturbed by the interface and Er 3+ (squares) calculated by the FLAPW method.
[Ai + r 2 Ai+3 exp(−l(l; ) r 2 )];
(3)
where Ai ; Ai+3 ; l(l; ) are the evaluated )tting coe@cients.
J. Ebothe et al. / Optics & Laser Technology 36 (2004) 173 – 180
for satisfying self-consistency: m − m−1 ¡ ": m
Fig. 4. Dispersion of ordinary refractive indices for pure ZnO and doped Zn–Er crystalline )lms.
This calculation technique is described in detail elsewhere (see for example Ref. [10,11]). A secular equation was expressed in the form 2 ˝ (k + Gn; n )2 − E(k) n; n 2m
+
V (Gn − Gn )S (Gn − Gn ) = 0;
(4)
where E(k) is the eigenenergy for the k-point in the Brillouin zone (BZ); Gn ; Gn are basis wavevectors of interacting PW. Structural form-factors for the atoms of kind were expressed as follows: g() S (Gn − Gn ) = exp(−t(Gn − Gn ) ); (5) N ;
where g() are weighting factors determining partial contributions of the particular structural components (for every ZnO layers) to the total potential, positions of the ; atoms. A similar approach for di%erent structural fragments has been successfully applied for binary solid alloys [13], glasses and organic materials [13]. One can expect that this approach may also be appropriate for di%erent disordered and partially ordered solids including the investigated ZnO and ZnO–Er 3+ )lm interfaces. As a consequence the reconstructed interface contribution is directly taken into account within the mentioned approach. The special Chadhi–Cohen [14] point method was applied for calculations of spatial electron charge density distribution. Secular equation’s diagonalization procedure was carried out at special weighting points of the BZ for each structural type. Acceleration of the iteration convergence was achieved by transferring 77% of the (m − 1)th iteration result to the mth iteration. The following condition was taken as a criterion
177
(6)
During the self-consistent calculations, a level of calculation error (") was better than 0.18%. As a ZnO basic BZ the wurtzite-like hexagonal structure was chosen because it corresponds to the major structural component of the investigated )lms. Varying structural factors for the di%erent layers and from the structural data obtained using the MD geometry optimization (see Table 1) the appropriate atomic positions were put in Eq. (5). To make the core-like state contributions more realistic, the NCPP wavefunctions were modi)ed additionally through their orthogonalization with respect to the core-like LCAO wavefunctions as described in Refs. [10,11,13]. As a result, the total energy deviation during the self-consistent procedure with respect to the energy cuto% and the Perdew– Adler screening parameter was stabilized within the 0:31 eV. 4.3. Local-density-derived SEPM As a second method for calculations of the mentioned structures we have applied a modi)ed SEPM approach introduced in Ref. [12]. This method was proposed explicitly for the quantum dots, wires, wells, and )lms with typical linear dimensions of 1–5 nm. In our case the thickness of the reconstructed ZnO–Er 3+ layers is of the same order. The SEPM approach is much faster than the self-consistent )rst-principle methods used for bulk solids. Indeed, the SchrVodinger equation is solved and an e@cient diagonalization method providing energy levels in a )xed “energy window” is available and appropriate. Unlike e%ective-mass-band approaches [15], this method uses explicit and variation Jexible basis functions, permitting a direct comparison of the wavefunctions with the LDF studies when available. However, contrary to the LDF approach, the current method provides only BS information, transition probabilities, and wave functions, but not the ground-state properties such as equilibrium structural geometry, which have to be assumed at the outset. The general mathematical formalism is similar to that described in Section 4.2. In the scalar local-density approximation (SLDA) the scalar-relativistic atomic pseudopotential is obtained using the Troullier–Martins [6,16] procedure to perform the core-like orthogonalized corrections. A PW basis with a kinetic energy cuto% of 32:5 Ry was used. The non-local part of the ionic pseudopotential V was obtained numerically. It was kept unchanged while moving from the LDA to the SLDA. The “small box” implementation was applied to handle the non-local part, and uses the PW basis set with a large momentum to expand it in Fourier space. Because the LDF band gaps are usually underestimated, it is adjusted to reproduce the experimentally observed BS using self-energy corrections [15]. The changes in the SLDA potentials are small compared with the LDA results. The
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performed calculations have shown that su@cient sizes of the secular matrix were achieved for the matrix sizes varying within the range 140 –290. Since the )nal pseudopotential was rather smooth, a rapidly converged PW expansion was possible. In fact using the e@cient diagonalization method and the full semi-empirical pseudopotential, the electronic structure of up to the 180-atom ZnO–Er supercluster was calculated. 4.4. FLAPW-LDA approach Among the di%erent BS calculation methods, the augmented plane wave (APW) methods are very popular. One of the well known ones is a computer version WIEN97 [7]. In the FLAPW-LDA and in accordance with Ref. [8], the exchange-correlation Hedin–Lundquist [7] parameterization was performed. The ZnO PW basis set was taken for the energy cuto%s of about 17 eV and the potential representation was about 82 eV. The expansion by angular-dependent spherical harmonics was carried out up to l ¡ 12. About 36 states have been taken into account including the six empty ones with a criterion of the self-consistent eigenenergy value stabilization of about 0:016 eV. Summation within the BZ was performed at 36 special points of the BZ [14]. All the core states (1sO, 1sZn, 2sZn, 2pZn, 3sZn, 3pZn, 3dZn, 2sO, 2pO) were recalculated for each iteration step. The Thomas– Fermi screening potential was applied at the starting step and afterwards it was successfully replaced by the Engel screened functions [7]. Additional corrections using the generalized gradient approximation (GGA) [17] to prevent energy gap underestimation were undertaken. The PW basis set was limited by wave vectors less than 9=Rm (Rm —mu@n-tin sphere radius). The pure ZnO crystallines require 2100 PW and ZnO–Er 3+ structural type needs 820 ones. The radius of the Zn sphere was varied within the range RZn =1:08−1:21RO . The interacting spheres were chosen to be non-overlapping (only touching). As core states, 1sO wavefunctions and 1, 2s, 2p Zn were taken. All the remaining orbitals were considered as valence states. For numerical calculations of the total and partial density of states (DOS) numerical simulations by the tetrahedral method [7] using 218 k points with an energy resolution of 0:13 eV were performed. The Zn, O atoms are relatively light and the spin–orbit interaction was neglected during the consideration. For Er 3+ ions we considered only the local part. For the classi)cation of the BZ the notations presented in Ref. [12] are given. 4.5. Results of the BS calculations As an important criterion of the applicability of the concrete method, the structural deviation of the calculated (see Table 2) parameters for the experiment is used. The maximal agreement between experimental and theoretically calculated data gives the modi)ed NCPP approach. The SEPM
Table 2 Lattice constants for the 4H- and 3C-polytypes calculated by the di%erent methods
NCPP modi)ed as in the present work FLAPW-LDA SEPM Experimental obtained from experimental di%raction [15]
S ZnO (A)
S ZnO–Er 3+ (A)
3.258
3.265
3.241 3.233 a = 3:259 c = 5:2081
3.252 3.242 a = 3:263 c = 5:2162
and FLAPW methods give results substantially di%erent from the experimental ones. Typical fragments of the BS dispersions calculated for the ZnO wurtzite structure with and without Er 3+ modi)cation by near-the-interface structure are shown in Fig. 3. One can see that for the both methods taking into account the reconstructed surfaces leads to the substantial modi)cation of the band energy dispersions. Moreover in the case of the NCPP method one can see a substantial shift of the e%ective valence band top and conduction band bottom (indicated by solid line). The presentation is limited only by the NCPP as well as FLAPW approaches because the SEPM data give results more di%erent from experimental data. One can see (see Fig. 3) an appearance of substantial deviations of the k-space BS dispersion from the BS for the pure ZnO wurtzite-like crystals. For convenience we show only deviations for the top of the valence band and the bottom of the conduction band. The more Jat modi)ed bands indicate on important role of the reconstructed near-the-interface structure. This fact con)rms structural reconstruction for the near-the-interface state and a drawback of the semi-empirical method for describing the excited states. E%ective deviation for the NCPP is in a general agreement with the experimental data on the photoelectron spectroscopy. Thus this method was chosen as a basis for the simulations of optical properties. 5. Dispersion of PEE coe.cients At the beginning for simulation of the linear and PEE properties, calculations of imaginary part of dielectric susceptibility optical tensor were carried out. The general formalism [17] was applied by taking into account the calculated matrix dipole moments obtained by the NCPP method. Additionally, the performed numerical veri)cation has shown that non-local e%ects change the average oscillator strength up to 13%. The plasmon peak in the energy loss spectra is shifted towards higher energies up to 2.8–3:6 eV. As a consequence, all the calculations were performed with inclusion of the non-local contributions. The real parts of the dielectric tensors were evaluated using the Kramers– Kronig dispersion relations. The reJectivity and energy loss functions were obtained from Fresnel’s formula.
J. Ebothe et al. / Optics & Laser Technology 36 (2004) 173 – 180
A crucial point in the calculations of the "2 (E) was found by precision of the k-space integration. A linear tetrahedral 1 method for the 526 k points in the 96 irreducible part of the BZ was applied. In the present paper during the BS summation, 72 conduction bands were included in order to achieve su@ciently converged results. The calculated data allow to estimate changes of refractive indices which correspond to the real part of "(E). From the general phenomenology one can also expect substantial enhancement of the PEE tensor components. The results calculated by this method, ijkl data, are presented in Table 3. From Table 3 one can see substantial anisotropy in the tensor component distribution. This reJects the simultaneous speci)c contribution of interface and cationic polarization. From Fig. 4 one can see that doping by erbium ions favors increasing refractive indices connected with the additional anisotropy in the space charge density distribution. This fact reJects the crucial role played by Er 3+ ions in the behavior of the refractive index perpendicular to the optical axes. The performed spectral measurements of PEE coe@cients (see Fig. 5) con)rm theoretical predictions (Table 3) that doping by Er 3+ leads to enhanced PEE coe@cients 2222 . Table 3 Evaluated values of the PEE tensor components for di%erent wavelengths
(nm)
2222 × 10−13 m2 = N
2233 × 10−13 m2 = N
600 650 700 750 800 850 900
5.75 5.12 5.01 4.99 4.97 4.96 4.95
1.15 1.10 1.09 1.03 1.09 1.08 1.00
179
For comparison we have included the data calculated by the method described above for the ZnO–Er samples. One can see a good agreement between the experimental and theoretically calculated results, which indicates a good level of the theoretical approach. Moreover for the wavelengths shorter than 600 nm an increase of PEE coe@cients 2222 values is up to the 5:8 × 10−12 m2 =N. Such a large increase of the PEE coe@cient may open a new stage in the design of thin-like electrooptics light modulators for the lasers. This fact also indicates the particular role of the Er 3+ ions in the observed spectral dependences of PEE e%ect. The observed anisotropy in the PEE tensor components con)rms the substantial role of simultaneous anisotropy of the interface and Er 3+ contributions. 6. Conclusion We have theoretically predicted and experimentally observed a large increase (up to one order) of the photoelastic e%ect for ZnO wurtzite )lms doped by Er 3+ . Using di%erent BS calculations as well as MD interface optimization we have predicted that due to insertion of Er 3+ ions one can expect a substantial increase of the dipole moments on the interface between the bare glass substrate and the appropriate crystalline )lm. So we demonstrate a possibility of simultaneous manipulation by interface charge density distribution together with appropriate doping in order to enhance the piezooptical tensor coe@cients. The discovered phenomenon may open a new stage for creation of the competitive miniaturized PEE devices for the lasers. The proposed theoretical approach may be used for the searching of new )lm-like crystalline materials possessing the enhanced second-order non-linear optical properties. Acknowledgements Financial support from the State Committee for Scienti)c Research (KBN), through University Grant (I.K., W.G.) is gratefully acknowledged. References
Fig. 5. Spectral dependences of the 2222 ×10−12 m2 =N for di%erent )lms deposited on a bare glass substrate ×—2233 ; ◦—theoretically calculated 2222 .
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