The complete elastic tensors of SmScO3 and NdScO3 were measured using resonant ultrasound spectroscopy (RUS) in combination with ab-initio calculations. Measurement of the elastic tensor of these recently synthesized single crystal RE scandates is essential for understanding dynamic lattice applications including phonon confinement, strain induced thin film growth and superlattice construction. On average, the experimental elastic constants differed by less than 5% of the theoretical values, further validating the accuracy of modern ab-initio calculations as a means of estimating the initial elastic constants used in RUS measurements

Page, Y. Le

Pestka, K. A., II

Scott, E. S.

English

Rollins College: Rollins Scholarship Online (RSO)

Rollins CollegeRollins Scholarship OnlineFaculty Publications9-8-2011Measurement of the elastic tensor of SmScO3 andNdSc03 using resonant ultrasound spectroscopywith ab initio calculationsK. A. Pestka IIRollins College, kpestka@rollins.eduE. S. ScottY. Le PageFollow this and additional works at: http://scholarship.rollins.edu/as_facpubPart of the Physical Sciences and Mathematics CommonsThis Article is brought to you for free and open access by Rollins Scholarship Online. It has been accepted for inclusion in Faculty Publications by anauthorized administrator of Rollins Scholarship Online. For more information, please contact rwalton@rollins.edu.Published InPestka, K. A., Scott, E. S., & Page, Y. L. (2011). Measurement of the elastic tensor of SmScO3 and NdScO3 using resonant ultrasoundspectroscopy with ab initio calculations. AIP Advances, 1(3). DOI: 10.1063/1.3641248.Measurement of the elastic tensor of SmScO3 and NdScO3 using resonantultrasound spectroscopy with ab initio calculationsK. A. Pestka, E. S. Scott, and Y. Le Page  Citation: AIP Advances 1, 032154 (2011); doi: 10.1063/1.3641248 View online: http://dx.doi.org/10.1063/1.3641248 View Table of Contents: http://aipadvances.aip.org/resource/1/AAIDBI/v1/i3 Published by the American Institute of Physics.  Related ArticlesEffect of Pd on the Ni2Si stress relaxation during the Ni-silicide formation at low temperature Appl. Phys. Lett. 101, 111910 (2012) Effects of stress on the dielectric function of strained pseudomorphic Si1−xGex alloys from 0 to 75% Ge grownon Si (001) J. Appl. Phys. 112, 053519 (2012) Morphology dependence of radial elasticity in multiwalled boron nitride nanotubes Appl. Phys. Lett. 101, 103109 (2012) Mechanical properties of metal-organic frameworks: An indentation study on epitaxial thin films Appl. Phys. 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Le Page21Rollins College, Winter Park, FL 32789, USA2ICPET, National Research Council of Canada, Ottawa, Ontario K1A 0R6, Canada(Received 9 June 2011; accepted 25 August 2011; published online 8 September 2011)The complete elastic tensors of SmScO3 and NdScO3 were measured using resonantultrasound spectroscopy (RUS) in combination with ab-initio calculations. Measure-ment of the elastic tensor of these recently synthesized single crystal RE scandatesis essential for understanding dynamic lattice applications including phonon con-finement, strain induced thin film growth and superlattice construction. On average,the experimental elastic constants differed by less than 5% of the theoretical val-ues, further validating the accuracy of modern ab-initio calculations as a means ofestimating the initial elastic constants used in RUS measurements. Copyright 2011Author(s). This article is distributed under a Creative Commons Attribution 3.0 Un-ported License. [doi:10.1063/1.3641248]Rare-earth (RE) scandates possess high band gaps (Eg≈ 5.75 eV),1–4 high dielectric constants(K=20-40, depending on direction)5 and thermal stability in contact with silicon.6 Thus, these ma-terials are suited for electronic applications including gate dielectrics and as an alternative to SiO2in silicon-based metal-oxide semiconductor field-effect transistors (MOSFETs).4, 7–9 RE scandateshave been exploited as substrates to produce strain induced growth effects such as dramatically mod-ified ferroelectric transitions in BaTiO3 thin films10, 11 and thin film superlattices with unprecedentedstructural uniformity.12 In similar oxide superlattices, phonon confinement has been demonstrated13with potential applications in acoustic lasers.14 In all of these instances knowledge of elastic con-stants is essential. Thus, the complete elastic tensors of SmScO3 and NdScO3, which both exhibitorthorhombic crystal symmetry, were found using resonant ultrasound spectroscopy (RUS) withab-initio calculations.The material sample’s elastic constants are determined using RUS by minimizing the leastsquares difference between the experimentally measured natural frequencies of a sample and a set oftheoretically calculated frequencies.15–21 The theoretical frequencies are calculated using the samplegeometry, density and an assumed set of elastic constants. The assumed elastic constants are theniteratively adjusted in a computer program until the minimum between the two spectra is foundthus recovering the true elastic constants. The RUS method requires at least as many experimentallymeasured frequencies as unknown elastic constants. Typically more than the minimum number offrequencies are used because not every frequency necessarily couples to all the unknown elasticconstants. In order to ensure that adequate coupling exists between the measured frequencies, fn ,and the elastic constants, ci j , the derivative, ∂ fn/∂ci j , are also calculated.An essential component of the RUS method is reasonable starting values for the initial estimatesof the elastic constant. Relatively recent improvements in ab-initio calculations used to producetheoretical elastic constants have enabled RUS to be utilized as the sole experimental source fordetermination of elastic constants.22 For ab-initio calculations, the cell parameters of NdScO323 andSmScO324 were taken from corresponding crystal structure reports.23, 24 Accepting Nigglis a<b<cranking of cell edges placed all five isostructural materials in a same Pbnm setting of space groupPnma (#62). The XYZ axes of the Cartesian reference system were then selected respectively alongthe xyz crystallographic axes of this Pbnm setting.2158-3226/2011/1(3)/032154/3 C© Author(s) 20111, 032154-1Downloaded 14 Sep 2012 to 66.195.118.22. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license.See: http://creativecommons.org/licenses/by/3.0/032154-2 K. A. Pestka II, E. S. Scott, and Y. Le Page AIP Advances 1, 032154 (2011)TABLE I. A comparison of the theoretical and RUS experimental elastic constants of SmScO3 and NdScO3.SmScO3 c11 c22 c33 c44 c55 c66 c23 c12 c13Experimental (GPa) 287.0 302.1 266.1 117.2 80.2 85.6 122.0 128.7 116.3Theoretical (GPa) 284.7 299.5 248.1 100.9 81.5 83.7 122.7 133.3 115.4% difference 0.81 0.87 7.00 14.93 1.59 2.29 0.54 3.45 0.78NdScO3 c11 c22 c33 c44 c55 c66 c23 c12 c13Experimental (GPa) 307.9 307 281.5 107.8 89.5 81.2 130.3 146.7 129.7Theoretical (GPa) 297.9 296.9 282.5 102.1 85.8 86.6 128.7 148.9 128.2% difference 3.31 3.34 0.34 5.38 4.23 6.41 1.24 1.52 1.16A thorough ab-initio optimization of the atom coordinates in fixed cell parameters was thenperformed, followed by ab-initio calculation of elastic coefficients according to the symmetry-general scheme.25Vienna ab-initio simulation package (VASP)26, 27 was used for all computations. All VASPinput files were generated and interpreted by Materials Toolkit.28 GGA PAW (Generalized gradi-ent approximation, projector augmented wave) potentials29 were used for all calculations. Wavefunctions were iteratively optimized with the Davidson blocked scheme30 combined with reciprocalspace projectors down to an electronic convergence of 1×10−7 eV. Reciprocal space integration wasperformed using a 5×5×5 k-mesh and a Monkhorst-Pack scheme.31 Energy corrections were im-plemented as Methfessel-Paxton smearing of order 1 and width 0.2 eV.32 The convergence criterionfor forces was 1×10−4 eV/Å. Spin polarization corrections were performed according to the schemeof Vosko, Wilk and Nusair 36 as implemented in VASP. The results for NdScO3 and SmScO3 aresummarized in Table I.The experimental resonant frequencies were determined using a Quasar RUSpec 4000. Pol-ished rectangular parallelepiped samples were placed between two transducers while contact wasmaintained at body diagonal corners. One transducer received a signal from the function generator,which swept through a range of ultrasonic frequencies, and the sample’s response was measured atthe other transducer.The samples used in this experiment were single crystals of SmScO3 and NdScO3, grown bythe Czochralski method, which are available commercially via CrysTec GmbH.33 The samples werecut and polished into rectangular parallelepipeds with the X, Y, and Z sample axes oriented per-pendicular to the (1¯10), (110), and (001) crystal planes, respectively. The standard RUS algorithm18uses the measured frequencies to determine the elastic tensor (c′i jkl) defined relative to the samplecoordinate system (X,Y,Z). Whereas the crystal elastic tensor (ci jkl), which in this case is not alignedwith the sample coordinate system, is defined with respect to the crystal axes (x,y,z). In order torecover the crystal elastic tensor using the standard RUS algorithm, the crystal coordinates wererotated into the sample coordinates viac′i jkl = Rip R jq Rkr Rlscpqrs (1)where Ri j (α, β, γ ) , is the rotation matrix and α, β and γ are the Euler angles.21 For this experimentα=45◦, β=0◦, and γ=0◦.The experimentally measured sample dimensions and densities are shown in Table II. Thenumber of frequencies used in the resonant spectrum fits were 17 and 18 for SmScO3 and NdScO3respectively and ranged from 280 kHz to 1.6 MHz. A complete list of the experimental frequenciesand the relative strength of their associated derivatives, which were used to ensure adequate cou-pling between the measured frequencies and the unknown elastic properties, can be found in thesupplemental document.34 The complete elastic tensors (all nine experimentally determined elasticconstants for each sample) are given in Table I.In summary, the results shown in Table I illustrate the robust nature of the RUS method incombination with modern ab-initio calculations. On average, the experimental elastic constantsdiffered by less than 5% of the theoretical values. These findings further confirm the accuracy ofDownloaded 14 Sep 2012 to 66.195.118.22. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license.See: http://creativecommons.org/licenses/by/3.0/032154-3 K. A. Pestka II, E. S. Scott, and Y. Le Page AIP Advances 1, 032154 (2011)TABLE II. Sample dimensions and density.Sample dimensions (mm) X Y ZSmScO3 (ρ = 6325kg/m3) 0.508 mm 2.651 mm 2.179 mmNdScO3 (ρ = 5987kg/m3) 0.517 mm 2.716 mm 2.448 mmMaterials Toolkit as a means of determining elastic constants,28 and the capability of the RUS methodfor accurately determining elastic constants without prior external experimental information.1 S-G. Lim, S. Kriventsov, T. N. Jackson, J. H. Haeni, D. G. Schlom, A. M. Balbashov, R. Uecker, P. Reiche, J. L. Freeouf,and G. Lucovsky, J. Appl. Phys. 91, 4500 (2002).2 V. V. Afanasev, A. Stesmans, C. Zhao, M. Caymax, T. Heeg, J. Schubert, Y. Jia, D. G. Schlom, and G. Lucovsky, Appl.Phys. Lett. 85, 5917 (2004).3 T. Heeg, M. Wagner, J. Schubert, Ch. Buchal, M. Boese, M. Luysberg, E. Cicerrella, and J. L. Freeouf, Micr. Eng. 80, 150(2005).4 H. M. Christen, G. E. Jellison Jr., I. Ohkubo, S. Huang, Me. E. Reeves, E. Cicerrella, J. L. Freeouf, Y. Jia, D. G. Schlom,Appl. Phys. Lett. 88, 262906 (2006).5 J. Haeni, Nanoengineering of Ruddlesden-Popper Phases using Molecular Beam Epitaxy, (PhD Thesis, The PennsylvaniaState University, 2002).6 D. G. Schlom and J. H. Haeni, MRS Bulletin 27, 198 (2002).7 C. Zhao, T. Witters, B. Brijs, H. Bender, O. Richard, M. Caymax, T. Heeg, J. Schubert, V. V. Afanasev, A. Stesmans, andD. G. Schlom, Appl. Phys. Lett. 86, 132903 (2005).8 M. Wagner, T. Heeg, J. Schubert, St. Lenk, S. Mantl, C. Zhao, M. Caymax, and S. De Gendt, Appl. Phys. Lett. 88, 172901(2006).9 K. H. Kim, D. B. Farmer, J.-S.M. Lehn, P. Venkateswara Rao, and R. G. Gordon, Appl. Phys. Lett. 89, 133512 (2006).10 J. Schubert, O. Trithaveesak, A. Petraru, C. L. Jia, R. Uecker, P. Reiche, and D. G. Schlom, Appl. Phys. Lett. 82, 3460(2003).11 K. J. Choi, M. Biegalski, Y. L. Li, A. Sharan, J. Schubert, R. Uecker, P. Reiche, Y. B. Chen, X. Q. Pan, V. Gopalan,L.-Q. Chen, D. G. Schlom, C. B. Eom, Science 306, 1005 (2004).12 A. Soukiassian, W. Tian, V. Vaithyanathan, J. H. Haeni, L. Q. Chen, X. X. Xi, D. G. Schlom, D. A. Tenne, N. D. Lanzillotti-Kimura, A. Bruchhausen, A. Fainstein, H. P. Sun, X. Q. Pan, K. J. Choi, D. M. Kim, C. B. Eom, Y. L. Li, Q. X. Jia,C. Constantin, R. M. Feenstra, M. Bernhagen, P. Reiche, and R. Uecker, J. Mater. Res. 23, 1417 (2008).13 A. Soukiassian, W. Tian, D. A. Tenne, X. X. Xi, D. G. Schlom, N. D. Lanzillotti-Kimura, A. Bruchhausen, A. Fainstein,H. P. Sun, X. Q. Pan, A. Cros, and A. Cantarero, Appl. Phys. Lett. 90, 42909 (2007).14 P. M. Walker, A. J. Kent, M. Henini, B. A. Glavin, V. A. Kochelap, and T. L. Linnik, Phys. Rev. B 79, 245313 (2009).15 I. Ohno, J. Phys. Earth 24, 355 (1976).16 W. Visscher, A. Migliori, T. M. Bell, and R. A. Reinert, J. Acoust. Soc. Am. 90, 5668 (1991).17 J. D. Maynard, J. Acoust. Soc. Am. 91, 1754 (1992).18 J. D. Maynard, Phys. Today 49, 26 (1996).19 A. Migliori, J. L. 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Measurement of the elastic tensor of SmScO3 and NdSc03 using resonant ultrasound spectroscopy with ab initio calculations

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Measurement of the elastic tensor of SmScO3 and NdSc03 using resonant ultrasound spectroscopy with ab initio calculations

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