Measurement of the elastic tensor of SmScO 3 and NdSc 03 using resonant ultrasound spectroscopy with ab initio calculations

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.

The complete elastic tensors of SmScO 3 and NdScO 3 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.Copyright 2011 Author(s).This article is distributed under a Creative Commons Attribution 3.0 Unported License.[doi:10.1063/1.3641248]Rare-earth (RE) scandates possess high band gaps (E g ≈ 5.75 eV), 1-4 high dielectric constants (K=20-40, depending on direction) 5 and thermal stability in contact with silicon. 6Thus, these materials are suited for electronic applications including gate dielectrics and as an alternative to SiO 2 in silicon-based metal-oxide semiconductor field-effect transistors (MOSFETs). 4, 7-9RE scandates have been exploited as substrates to produce strain induced growth effects such as dramatically modified ferroelectric transitions in BaTiO 3 thin films 10,11 and thin film superlattices with unprecedented structural uniformity. 12In similar oxide superlattices, phonon confinement has been demonstrated 13 with potential applications in acoustic lasers. 14In all of these instances knowledge of elastic constants is essential.Thus, the complete elastic tensors of SmScO 3 and NdScO 3 , which both exhibit orthorhombic crystal symmetry, were found using resonant ultrasound spectroscopy (RUS) with ab-initio calculations.
6][17][18][19][20][21] The theoretical frequencies are calculated using the sample geometry, density and an assumed set of elastic constants.The assumed elastic constants are then iteratively adjusted in a computer program until the minimum between the two spectra is found thus recovering the true elastic constants.The RUS method requires at least as many experimentally measured frequencies as unknown elastic constants.Typically more than the minimum number of frequencies are used because not every frequency necessarily couples to all the unknown elastic constants.In order to ensure that adequate coupling exists between the measured frequencies, f n , and the elastic constants, c i j , the derivative, ∂ f n /∂c i j , are also calculated.
An essential component of the RUS method is reasonable starting values for the initial estimates of the elastic constant.Relatively recent improvements in ab-initio calculations used to produce theoretical elastic constants have enabled RUS to be utilized as the sole experimental source for determination of elastic constants. 22For ab-initio calculations, the cell parameters of NdScO 3 23 and SmScO 3 24 were taken from corresponding crystal structure reports. 23,24 ccepting Nigglis a<b<c ranking of cell edges placed all five isostructural materials in a same Pbnm setting of space group Pnma (#62).The XYZ axes of the Cartesian reference system were then selected respectively along the xyz crystallographic axes of this Pbnm setting.A thorough ab-initio optimization of the atom coordinates in fixed cell parameters was then performed, followed by ab-initio calculation of elastic coefficients according to the symmetrygeneral scheme. 25ienna ab-initio simulation package (VASP) 26,27 was used for all computations.All VASP input files were generated and interpreted by Materials Toolkit. 28GGA PAW (Generalized gradient approximation, projector augmented wave) potentials 29 were used for all calculations.Wave functions were iteratively optimized with the Davidson blocked scheme 30 combined with reciprocal space projectors down to an electronic convergence of 1×10 −7 eV.Reciprocal space integration was performed using a 5×5×5 k-mesh and a Monkhorst-Pack scheme. 31Energy corrections were implemented as Methfessel-Paxton smearing of order 1 and width 0.2 eV. 32The convergence criterion for forces was 1×10 −4 eV/Å.Spin polarization corrections were performed according to the scheme of Vosko, Wilk and Nusair 36 as implemented in VASP.The results for NdScO3 and SmScO3 are summarized in Table I.
The experimental resonant frequencies were determined using a Quasar RUSpec 4000.Polished rectangular parallelepiped samples were placed between two transducers while contact was maintained 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 at the other transducer.
The samples used in this experiment were single crystals of SmScO 3 and NdScO 3 , grown by the Czochralski method, which are available commercially via CrysTec GmbH. 33The samples were cut and polished into rectangular parallelepipeds with the X, Y, and Z sample axes oriented perpendicular to the (1 10), (110), and (001) crystal planes, respectively.The standard RUS algorithm 18 uses the measured frequencies to determine the elastic tensor (c i jkl ) defined relative to the sample coordinate system (X,Y,Z).Whereas the crystal elastic tensor (c i jkl ), which in this case is not aligned with the sample coordinate system, is defined with respect to the crystal axes (x,y,z).In order to recover the crystal elastic tensor using the standard RUS algorithm, the crystal coordinates were rotated into the sample coordinates via where R i j (α, β, γ ) , is the rotation matrix and α, β and γ are the Euler angles. 21For this experiment α=45 • , β=0 • , and γ =0 • .The experimentally measured sample dimensions and densities are shown in Table II.The number of frequencies used in the resonant spectrum fits were 17 and 18 for SmScO 3 and NdScO 3 respectively and ranged from 280 kHz to 1.6 MHz.A complete list of the experimental frequencies and the relative strength of their associated derivatives, which were used to ensure adequate coupling between the measured frequencies and the unknown elastic properties, can be found in the supplemental document. 34The complete elastic tensors (all nine experimentally determined elastic constants for each sample) are given in Table I.
In summary, the results shown in Table I illustrate the robust nature of the RUS method in combination with modern ab-initio calculations.On average, the experimental elastic constants differed by less than 5% of the theoretical values.These findings further confirm the accuracy of Materials Toolkit as a means of determining elastic constants, 28 and the capability of the RUS method for accurately determining elastic constants without prior external experimental information.

TABLE II .
Sample dimensions and density.