Tetragonal ba tio3_v8

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Tetragonal ba tio3_v8

  1. 1. First principles computation of second-order elastic constants and equations of state for tetragonal BaT iO3 Ghous Narejo and W. F. Perger Electrical Engineering Department Michigan Tech UniversityAbstractFirst-principles computational techniques are employed for computing the second-order elastic constants (SOEC) and equations of state for the tetragonal phase ofBaT iO3 . The bulk modulus is computed by two independent methods. A varietyof potentials and basis sets are used. The impact of the potentials, the basis setsand the sensitivity of the crystalline geometry optimization on the computationalvalues is discussed.Key words: Elastic constant, Bulk modulus1 IntroductionThe perovskites are an important class of materials which have potential tofind applications in the fields of memory, logic design and switching. However,the theoretical exploration of these materials is not on par with their tech-nological importance. Crystalline BaT iO3 possesses a perovskite geometrywhich may occur in cubic, tetragonal, rhombohedral and orthorhombic crys-talline phase. There is a general lack of the data on the mechanical propertiesof all phases of BaT iO3 . However, there have been attempts [?] to understandthe electronic and mechanical properties of this material. The computations ofelastic constants and bulk modulus were done by Piskunov et al [?] by ab-initocomputational method for cubic BaT iO3 . Email address: wfp@mtu.edu (W. F. Perger). URL: http://www.ee.mtu.edu/faculty/wfp.html (W. F. Perger).Preprint submitted to Elsevier 7 February 2011
  2. 2. At room temperature, BaT iO3 has tetragonal crystalline geometry. In thepast, the elastic constants and bulk modulus of tetragonal BaT iO3 could notbe computed due to the complexity of these computations. In this paper, thecomputations of elastic constants and the bulk modulus of tetragonal BaT iO3are done with ab-initio Hartee Fock (HF), density functional theory (DFT)and hybrid potentials. The ab-initio computational techniques are employedeffectively due to recent advancements in basis sets, geometry optimization,and computational power.In this work, the computations of the tetragonal phase of BaT iO3 are madewith the CRYSTAL09 code using a variety of basis sets. The CRYSTAL09code has the unique characteristic to employ the Hartree-Fock (HF), exchangecorrelation potentials and the effective combination of the former with thelatter. In addition, the availability of a variety of basis sets and geometryoptimization techniques further enhances the efficiency of these computations.An important factor that affects the efficiency of computation is obtaining theoptimized geometry of the crystalline structure. The geometry optimizationof the tetragonal phase of BaT iO3 is an essential step during computationsof elastic constants and bulk modulus. The total geometry optimization isdone to optimize the atomic positions and lattice parameters of the unit cellof tetragonal BaT iO3 . These special features of the geometry optimization inthe CRYSTAL09 code [?] makes it an efficient program to achieve the optimumcomputational results.The computations of the bulk modulus are done independently with ELAST-CON and EOS programs. These programs employ the geometry optimizationtechniques available in the CRYSTAL09 code. The ELASTCON algorithmdetermines the number of crystalline deformations based upon the crystallinegeometry of tetragonal BaT iO3 . Geometry optimization of the crystallinestructure is carried out after each deformation. The analytic first derivative,numerical second derivative and Levenberg Marquardt (LM) curve fitting isdone in sequence to compute elastic constants for the tetragonal BaT iO3 . Adetailed discussion of the ELASTCON program can be seen in ref. [?].The equation of state algorithm is utilized for computation of bulk modulusfrom a pressure vs. volume curve. The range of volumes around the optimizedequilibrium may be selected by the input parameters available in the EOSprogram. The computation of total energy is then done for this range of vol-umes. The geometry optimization is done for each volume while keeping thevolume constant. The computation of elastic constants and bulk modulus oftetragonal BaT iO3 is complicated due to its crystalline geometry and the po-sition of the atoms in the perovskite crystalline structure. With the lowering ofcrystalline symmetry the number of elastic constants increases and geometryoptimization steps are increased. 2
  3. 3. The crystalline geometry of the tetragonal BaT iO3 must be fully optimizedduring the computation of the elastic constants and bulk modulus. This ispossible only if the tetragonal crystalline system is relaxed during each in-dividual computation. The task of crystalline geometry optimization of theperovskite BaT iO3 has remained a challenge as the crystalline system un-dergoes a sudden decrease in energy when it is deformed. The ELASTCONalgorithm has detected this sudden decrease in the strain vs. energy computa-tions as reported earlier [?,?,?]. Detailed energy vs. strain computations werecarried out to explore the sudden decrease in energy. The ELASTCON algo-rithm issues a warning message if there is an abrupt and unexpected changein the energy during optimization which helps to understand the optimizationprocess further.2 ELASTCON and EOS programs in CRYSTAL09 codeThe computation of elastic constants and bulk modulus is an automated pro-cess in ELASTCON program and is determined from the crystalline symmetryof tetragonal BaT iO3 . Deformations are applied depending upon the tetrago-nal crystalline geometry. The analytic first derivative and the numerical secondderivative of total energy are computed for each deformation. The LevenbergMarquardt (LM) curve fitting [?] is used to compute the elastic constants andbulk modulus.The linear deformation of solids is expressed by Hooke’s law as σij = Cijkl kl (1) klwhere i, j, k, l = 1, 2, 3, σi,j,k,l , k,l and Ci,j,k,l are stress, strain and second-orderelastic constant tensors.The second-order elastic constants may be computed with different tech-niques. The molecular dynamics and ab-initio computational techniques aretwo prominent methods to compute the elastic constants and bulk modulus.The ab-initio computational techniques compute the second-order elastic con-stant (SOEC) from the total energy. The elastic constants can be computedfrom the Taylor series expansion of the total energy with respect to the appliedstrains, as shown in Eq. (??). The Taylor series terms up to the second-ordermay be utilized for the estimation of the elastic constants and bulk modulusif the strains are very small and the higher order terms have negligible effects 3
  4. 4. on the computational results. V E (V, ) = E(V0 ) + σα α + Cαβ α β + ..... (2) α 2 αβThe terms α, β = 1, .., 6 express the elastic constants in Voigt notation andV0 is the equilibrium volume. In Eq. (??), the non-zero stress term may beignored if the crystalline geometry of the system is fully optimized. The thirdterm in Eq. (??) can be rewritten to express the elastic constant as the secondderivative of the total energy with respect to the applied strain in a crystallinedirection 1 ∂ 2E Cαβ = . (3) V ∂ α∂ β 0In Eq. (??), the terms Cαβ , E, and V express the elastic constant tensor,energy and volume of the crystalline structure, respectively.The ab-initio computations are done by calculating the analytic first deriva-tive of the total energy and numerical second derivative with respect to theapplied strain. The appropriate number of strains are applied in a systematicmanner, the elastic constants are calculated, and the compliance coefficientsare computed from Eq. (??). The compliance coefficients are then utilized forthe computation of the bulk modulus as shown in Eq. (??). [S] = [C]−1 (4) B = 1/(S11 + S22 + S33 + 2(S12 + S13 + S23 )). (5)The terms Sij and B in Eq. (??) express the compliance tensor elements andbulk modulus respectively.The EOS algorithm in CRYSTAL09 computes the energy for a range ofvolumes around the optimized equilibrium volume. The Crystal09 code isequipped with a wide variety of equations of state such as Birch Murnaghan,3rd order Birch Murnaghan, logarithmic, Vinet and polynomial. In this pa-per, the 3rd order Birch Murnaghan equation of state algorithm was utilizedfor computing the bulk modulus from the energy vs. volume computations asexpressed in Eq. (??): B −1 1 Vo V 1 E(V ) = Bo Vo + − + Eo . (6) B (B − 1) V B Vo B − 1In Eq. (??), V0 represents the volume at the lowest energy, B0 is the bulk 4
  5. 5. modulus at pressure P = 0, B is the derivative of bulk modulus B at P = 0and E0 is the minimum energy. The optimization of crystalline geometry ateach step is done during energy-volume (E-V) calculations. The bulk modulusresults are obtained with Levenberg-Marquardt curve fitting of the E vs. Vcomputations.3 Choice of basis setsThe selection of basis sets [?,?] vs. [?,?] affects the computations of the elasticconstants and bulk modulus. The selection of adequate basis sets is an impor-tant factor in the computational physics for examining the complex materialphysics of the perovskite BaT iO3 . We have chosen 6-31d1 and 8-411d11 basissets for Oxygen and HAYWSC ECP basis set for Ba and Ti atoms from theCrystal06 basis set library [?]. We have named the combination of basis sets8-411d11 and HAYWSC ECP as basis set 1 and 6-31d1 and HAYWSC ECPas basis set 2. The computational efficiency of the results with basis set 1 isalso reported earlier for cubic BaT iO3 in ref. [?].The variation in the values of elastic constants and bulk modulus confirm thefact that the improper choice of basis sets can severely affect the computations.The efficient vs. inefficient usage of basis sets hint at a trade-off betweenthe efficiency of the results and the computational time. Our approach is tocheck the credibility of the results by employing the basis sets in two differentalgorithms and comparing the results attained by each of these basis sets.4 Geometry optimizationFor temperatures above 393◦ K, BaT iO3 has a stable cubic crystalline struc-ture. However, as the temperature is lowered, the BaT iO3 crystalline structurechanges into tetragonal, orthorhombic and rhombohedral phases at 393◦ K,278◦ K and 183◦ K [?]. This paper reports the computations of the elastic con-stants and bulk modulus of the tetragonal P 4mm BaT iO3 . The experimentalvalues [?] of the lattice constants and the atomic positions for tetragonalBaT iO3 are used as an input to the Crystal09 code.Geometry optimization is an essential step for the computations of the elasticconstants and bulk modulus. It relaxes the crystalline structure and removesthe effect of the strain due to the non-optimized crystalline geometry. Theeffect of the non-optimized crystalline geometry would appear as the secondterm on the right hand side of Eq. (??) not being zero if the geometry opti-mization is not employed in the computations. The geometry optimization of 5
  6. 6. the atomic positions and cell parameters of a tetragonal BaT iO3 crystallinesystem was done with Hartree Fock, density functional theory (DFT) poten-tials and hybrid mixing of the former with the latter.In the ELASTCON program [?], the computation of the bulk modulus requiresa series of deformations along the crystalline axes and optimization of the in-ternal co-ordinates of atomic positions and cell parameters for each of thesedeformations. This step is followed by determining analytic first-derivativesand numerical second-derivatives of the total energy with respect to applieddeformations resulting in elastic constants. The elastic constants are then em-ployed for computation of the bulk modulus as in Eq. (??).In the EOS algorithm [?], a series of volumes is chosen around the equilibriumand optimization of the internal co-ordinates is performed holding the volumeconstant. The energy vs. volume curve is fitted to an equation of state and thebulk modulus is computed. (See ref. [?] for the detailed discussion about thetypes of geometry optimization implemented in Crystal06 code). The values ofthe optimized lattice constants along with the energy and volume are reportedin Tables ?? and ??.5 Selection of Hartree Fock, DFT and hybrid potentialsThe computations of elastic constants and bulk modulus were performed withthe potentials HF, DFT and hybrid mixing of the former with the latter. TheDFT and hybrid exchange correlation potentials are employed due to the lackof correlation of HF as reported in [?]. Moreover, the aim of employing HF,DFT and hybrid potentials is to verify and confirm the computational merit ofthese results. The computational results of the optimized lattice constants ob-tained with basis set 1 and exchange correlation potential PWGGA and hybridexchange correlation potentials B3LYP and B3PW have shown an improve-ment with respect to experiment. Therefore the employment of HF, DFT andhybrid exchange correlation potentials have essentially reinforced the compu-tational merit of PWGGA and hybrid potentials. The computational valuesof the optimized lattice constants, elastic constants and bulk modulus resultsobtained with HF, DFT and hybrid potentials are reported in Tables ?? -??. 6
  7. 7. Table 1 The values of relaxed lattice constants (in ˚), ambient volume (in ˚3 ), A Aand total energy, E (in a.u.), for tetragonal BaT iO3 . The computationswere done by using Hartree-Fock, DFT-BLYP, DFT-B3LYP and DFT-PWGGA potentials. The basis sets employed for Ba, Ti and O atoms arecollectively named as basis set 1 (see section ??). a c c/a Vol. E HF 3.96 4.26 1.07 67.17 -307.5388 LDA 3.93 3.93 1.00 61.12 -307.8828 PWGGA 4.00 4.03 1.00 64.75 -309.5928 BLYP 4.05 4.14 1.02 68.12 -309.4749 B3LYP 4.01 4.10 1.02 66.15 -309.4084 B3PW 3.98 4.02 1.01 63.75 -309.5360 Exp. [?,?] 3.99 4.03 1.01 64.16 -Table 2The values of relaxed lattice constants (in ˚), ambient volume (in ˚3 ), A Aand total energy, E (in a.u.), for tetragonal BaT iO3 . The values are com-puted with Hartree-Fock, DFT-PWGGA, DFT-BLYP and DFT-B3LYPpotentials. The basis set used for Ba, Ti and O atoms are collectivelynamed as basis set 2 (see section ??). a c c/a Vol. E HF 3.98 3.98 1.00 63.23 -307.4627 PWGGA 3.95 3.95 1.00 61.94 -309.4862 BLYP 4.00 4.00 1.00 64.06 -309.3636 B3LYP 3.97 3.97 1.00 62.66 -309.3082 Exp. [?,?] 3.99 4.03 1.01 64.16 - 7
  8. 8. Table 3 The elastic constants and bulk modulus computational results using theHF and DFT LDA, PWGGA, BLYP, and B3LYP and B3PW potentialswith basis set 1 (see section ??). All values are in GPa. C11 C12 C13 C33 C44 C66 B HF 308.16 127.89 88.00 66.09 120.55 162.87 61.64 LDA 443.63 93.05 93.08 443.54 224.297 224.31 209.91 PWGGA 353.79 76.01 56.46 227.22 174.97 193.53 138.64 BLYP 284.53 82.10 58.05 146.28 73.33 160.04 109.82 B3LYP 315.30 94.53 64.13 166.22 76.47 176.58 123.31 B3PW 370.94 88.85 61.25 228.04 147.91 203.69 145.11 Exp.[?] 211+-6 107+-5 114+-8 160 +-11 56.2+-1.7 127+-4 125-141[?] [?] 242.7 128.3 122.6 147.9 54.9 120.1 - [?] 275.1 178.9 151.55 164.8 54.3 113.1 - [?] 222.9 - 147.0 240.0 61.7 133.7 -Table 4The elastic constants and bulk modulus computations using the HF, DFTLDA, PWGGA, BLYP, B3LYP and B3PW potentials with basis set 2(see section ??). All values are in GPa. C11 C12 C13 C33 C44 C66 B HF 408.03 154.19 154.19 408.02 184.72 184.72 238.80 PWGGA 444.52 107.43 107.42 444.52 107.42 213.71 219.79 BLYP 381.51 111.14 111.15 381.52 184.79 184.79 201.27 B3LYP 414.21 122.55 122.54 414.21 200.01 200.01 219.77 Exp.[?] 211+-6 107+-5 114+-8 160 +-11 56.2+-1.7 127+-4 125-141[?] [?] 242.7 128.3 122.6 147.9 54.9 120.1 - [?] 275.1 178.9 151.55 164.8 54.3 113.1 - [?] 222.9 - 147.0 240.0 61.7 133.7 - 8
  9. 9. 6 Computation of bulk modulus by equation of state (EOS)The EOS algorithm utilizes the systematic changes [?] in the volume aroundthe optimized equilibrium state of a crystalline structure. The EOS calcula-tions are carried out by selecting a range of volumes around a minimum totalenergy at an equilibrium state of the tetragonal BaT iO3 crystalline structure.The EOS algorithm has permits selection of a range of volumes and a numberof volumes within that range. At each of the volumes in the range, the con-stant volume optimization is carried out. The energy vs. volume results arecurve-fitted to an EOS such as Murnaghan EOS [?] as shown in Eq. (??).The Crystal09 code [?] can accomplish an optimization of the internal co-ordinates and lattice parameters while keeping the volume constant (see ref. [?]for the detailed description of CVOLOPT option of geometry optimization).Table ?? and ?? show the results of EOS algorithm for the calculation of aseries of total energies at the chosen volumes and fitted to Eq. (??) using aLevenburg-Marquardt [?] curve fitting algorithm.7 Discussion of computational results[?] We have observed the deviation in values of optimized lattice constants aand c of tetragonal BaT iO3 as compared with the experimental values [?,?].The main sources of these deviations are basis sets and potentials and thesensitivity of each due to the displacive nature of tetragonal BaT iO3 [?]. Itis generally observed that local DFT (LDFT) exchange correlation potentialsunderestimate and nonlocal DFT (NLDFT) exchange correlation potentialsoverestimate the lattice constants. The comparison of lattice constants a, cand the ratio c/a with experimental values is shown in Tables ?? and ??.The values of optimized lattice constants have shown a better agreement withthe experimental values for basis set 1. The computational values of a and c areslightly higher than the experimental values for DFT-PWGGA, DFT-BLYP,DFT-B3LYP and DFT-B3PW exchange correlation potentials employed withbasis set 1. However, the optimized lattice constants a and c computed with HFand DFT-LDA exchange correlation potential employed with basis set 1 show adifferent trend. The optimized lattice constants a and c, computed with DFT-LDA exchange correlation potential, are lower than the experimental latticeconstants. On the other hand, the optimized lattice constant a has decreasedand c has increased from the experimental value for the computations donewith HF potential. Surprisingly, the optimized values of lattice constants a andc computed with basis set 2 show large deviations from the experimental valuesof 3.99 and 4.03. The values of the optimized lattice constants a and c are equal 9
  10. 10. for Hartree Fock, exchange correlation and hybrid potentials employing basisset 2 as shown in Table ??.The computational values of bulk modulus, done with ELASTCON and EOSmethods employing HF, DFT and hybrid potentials with basis set 1, are re-ported in Tables ?? and ??. The DFT-LDA have shown higher bulk mod-ulus values due to underestimation of the optimized lattice constant values.Whereas, the computations done with DFT-PWGGA and DFT-BLYP haveshown lower values of bulk modulus due to the overestimation of optimizedlattice constants. The computations done with hybrid potentials have shownthe optimum values of bulk modulus as a consequence of the fact that the op-timized lattice constants are comparatively close to the experimental values.In contrast with these results, the computational values of bulk modulus arehigher with the basis set 2 as reported in Tables ?? and ??. The large increasein computational values of bulk modulus for the basis set 2 is attributed tothe deviation in optimized lattice constants from the experimental values.The order -disorder nature [?] of perovskite BaT iO3 is another importantfactor that makes the computation of optimum values of the bulk modulusproblematic, requiring accurate basis sets and potentials. It has been observedthat the computation of the bulk modulus for tetragonal BaT iO3 is coupledintimately with the geometry optimization of its complex crystalline structure.Slight deviations in the values of optimized lattice constants a and c haveshown large effects on the computational values of bulk moduli.The bulk moduli obtained with PWGGA and hybrid potentials employingbasis set 1 are computationally accurate. This analysis is based upon thecomputational results obtained with these potentials and basis sets for cu-bic BaT iO3 [?] and rutile T iO2 [?] crystalline systems. Moreover, the valuesof bulk modulus computed with ELASTCON and EOS algorithms are suffi-ciently close to each other for the same potential and basis set. These factspoint to the relative merit of the approach for the selection of a variety ofbasis sets, potentials and algorithms. Therefore, the computational values ofelastic constants computed with hybrid potentials are reliable if we ignore thevariations in computations originating due to the order-disorder nature of thetetragonal BaT iO3 . 10
  11. 11. Table 5Equation of state results for tetragonal BaT iO3 with the Birch Mur-naghan 3rd order equation. The energy-volume curve was fitted witheleven points and the range of volume around equilibrium was chosen as±10%. The basis sets employed for for Ba, Ti and O atoms are collectivelynamed as basis set 1 (see section ??). ˚ B(GPa) Vo (A3 ) E0 (a.u.) HF 61.71 67.41 -307.5386 LDA 197.72 61.16 -307.8827 PWGGA 144.34 64.75 -309.5928 BLYP 96.38 68.09 -309.4749 B3LYP 103.51 66.07 -309.4083 B3PW 149.14 63.82 -309.5360 Exp.[?] 125-141 63.75 -Table 6Equation of state data for BaT iO3 using the 3rd-order Birch Murnaghanequation. Eleven points in the energy-volume curve were used and therange of volumes used around equilibrium was ±10%. The basis sets usedfor Ba, Ti and O atoms are collectively named as basis set 2 (see section??). ˚ B(GPa) Vo (A3 ) E0 (a.u.) HF 202.41 64.23 -307.4678 LDA 256.96 59.07 -307.7672 PWGGA 220.55 61.93 -309.4862 BLYP 202.76 64.09 -309.3636 B3LYP 229.02 62.77 -309.3082 B3PW 229.68 61.41 -309.4395 Exp. [?] 125-141 63.75 - 11
  12. 12. Table 7Equation of state data for BaT iO3 using the 3rd-order Birch Murnaghanequation. Eleven points in the energy-volume curve were used and therange of volumes used around equilibrium was ±10%. The basis sets usedfor Ba, Ti and O atoms are collectively named as basis set 2 (see section??). B(GP a) B(GP a) HF 61.71 61.64 LDA 197.72 209.91 PWGGA 144.34 138.64 BLYP 96.38 109.82 B3LYP 103.51 123.31 B3PW 149.14 145.11 Exp. [] 125-141Table 8Equation of state data for BaT iO3 using the 3rd-order Birch Murnaghanequation. Eleven points in the energy-volume curve were used and therange of volumes used around equilibrium was ±10%. The basis sets usedfor Ba, Ti and O atoms are collectively named as basis set 2 (see section??). B(GP a) B(GP a) HF 202.41 238.80 PWGGA 220.55 201.27 BLYP 202.76 219.77 B3LYP 229.02 219.77 Exp. [?] 125-141 12
  13. 13. 8 ConclusionsThe second order elastic constants and equations of state computations weresuccessfully for tetragonal BaT iO3 . We have observed a close agreement inthe values of the bulk modulus computed with ELASTCON and EOS meth-ods. There have been attempts [?,?,?] to compute the different properties oftetragonal BaT iO3 . However, the experimental and computational values ofelastic constants and bulk modulus of tetragonal BaT iO3 are not available inliterature. We have further verified the computational accuracy of the resultsby implementing different algorithms, potentials and basis sets on the rutileT iO2 [?] and cubic BaT iO3 [?]. The computational results obtained with thesecrystalline systems have provided additional evidence for the accuracy of thecomputational results for tetragonal BaT iO3 . It has been observed that thecrystalline structure and position of atoms in tetragonal phase [?,?] has in-creased the complexity of our computations. In our opinion the agreement incomputational values of bulk modulus provided an opportunity to verify thecomputational accuracy of elastic constants of tetragonal BaT iO3 .AcknowledgementsOne of the authors (WFP) gratefully acknowledges the support of the Officeof Naval Research Grant N00014-01-1-0802 through the MURI program. 13

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