Strain-related Tensorial Properties: Elasticity, Piezoelectricity and Photoelasticity

Strain-related Tensorial Properties: Elasticity, Piezoelectricity and Photoelasticity Torino, Italy, September 4-9, 2016 Alessandro Erba Dipartimento di Chimica, Università di Torino (Italy) alessandro.erba@unito.it

Outline Tensorial Properties of Crystals with CRYSTAL14 and CRYSTAL17 Main Tensors related to Crystal Strain Elastic Fourth-rank Tensor and Related Properties Equation-of-State (EOS) Approach Piezoelectric Tensors Photoelastic and Piezo-optic Tensors DOI: 10.1002/qua.24658

Tensors in Crystals Many properties of anisotropic crystalline materials can be represented in terms of Cartesian tensors of different ranks.

Tensors with CRYSTAL14

Tensors with CRYSTAL14 Geometry Optimization

Tensors with CRYSTAL14 Dielectric Response

Tensors with CRYSTAL14 Spectroscopic Properties

Tensors with CRYSTAL14 Strain-related Properties

New Tensors in CRYSTAL17 - Piezo-optic Tensor (fourth-rank) - Force-responce Internal-strain tensor (second-rank) (mixed Hessian) - Second Harmonic Generation and Pockel's Tensors (third-rank) (electric field frequency-dependent first-hyper-polarizability tensor)

Crystal Strain Several properties of crystals can be computed by finite differences over strained configurations. The second-rank strain tensor e can be decomposed into two parts: An antisymmetric part w that corresponds to a pure rotation A symmetric part ε that corresponds to a pure strain Then, if we are interested only in the pure strain part of the transformation, we can refer to the following pure strain tensor:

Crystal Strain When the strain tensor acts on the lattice parameters, they are transformed as: As only six components of the symmetric matrix are independent, a more compact oneindex notation can be adopted, according to Voigt s proposal. In this convention the strain tensor becomes: 1 = xx 2 = yy 3 = zz 4 = yz 5 = xz 6 = xy i,j,k = 1,...,3 Cartesian indices v,u = 1,...,6 Voigt's indices

Crystal Strain-related Properties Elastic Tensor 4 Piezoelectric Tensor 3 Photoelastic Tensor 4 Order of the Tensors Second derivatives of the total energy E with respect to a pair of strains, for a 3D crystal First derivative of the polarization P (computed through the Berry phase approach) with respect to the strain First derivative of the inverse dielectric tensor (difference with respect to the unstrained configuration) with respect to strain

Crystal Strain-related Properties Elastic Tensor Geometry definition ELASTCON [Optional keywords] END END Basis set definition END Comput. Parameters END Piezoelectric Tensor Photoelastic Tensor Geometry definition PIEZOCON [Optional keywords] END END Basis set definition END Comput. Parameters END Geometry definition PHOTOELA [Optional keywords] END END Basis set definition END Comput. Parameters END

Hooke's Law Stress-strain relation: 81 components 36 as both stress and strain tensors are symmetric 21 as C itself is symmetric (second energy derivatives wrt strain)

Efect of Symmetry Fourth Rank Elastic Tensor Triclinic Cubic Hexagonal J. F. Nye, Oxford University Press, (1985)

Electronic and Nuclear Terms Strain-induced tensorial properties of solids can be formally decomposed into a purely electronic clamped-nuclei term and into a nuclear-relaxation term due to the rearrangment of atomic positions upon strain. The evaluation of the latter is generally much more computationally expensive than that of the former and can be achieved following two alternative approaches: (i) performing numerical geometry optimizations to relax atomic positions at actual strained lattice configurations; (ii) evaluating in a more analytical fashion the internal-strain tensor of energy secondderivatives with respect to atomic displacements and lattice deformations, as combined with the interatomic force constant Hessian matrix.

Elastic Tensor: Numerical Approach

Independent Strains C11 C12 = C13 First deformation Cubic Lattice

Independent Strains C11 C12 = C13 First deformation C22 = C33 and C21 = C13 Cubic Lattice

Independent Strains C11 C12 = C13 First deformation C22 = C33 and C21 = C13 Fourth deformation C44 Cubic Lattice

Independent Strains C11 C12 = C13 First deformation C22 = C33 and C21 = C13 Fourth deformation C44 C66 = C55 = C44 Cubic Lattice

Elastic Tensor: Numerical Approach

Elastic Tensor: Internal-strain Approach The force-responce internal-strain tensor is defined as (second energy derivative wrt an atomic Cartesian displacement and a Cartesian lattice vector deformation): Let us express it in terms of the strain tensor components: By adopting Voigt's notation:

Elastic Tensor: Internal-strain Approach Let us introduce the displacement-responce internal-strain tensor, which describes firstorder atomic displacements as induced by a first-order strain: which combines the force-response internal-strain tensor with the interatomic forceconstant Hessian matrix:

Elastic Tensor: Internal-strain Approach The elastic tensor can be written as: Electronic Nuclear The diagonalization of the mass-weighted interatomic force-constant matrix allows for a physically meaningful partition of the nuclear relaxation contribution in terms of phonon normal modes p=1,...,3n: Vibrational frequencies

Elastic Tensor: Internal-strain Approach

Geometry Optimizer Vs. Internal Strain ZnO Parallel mode on 16 CPUs Convergence tolerance of the geometry optimizer on the gradient

Elastic Tensor: The Evolution of the Implementation CRYSTAL03 and CRYSTAL06 CRYSTAL09 CRYSTAL14 CRYSTAL17 Many separate SCF calculations at strained configurations Results of different runs analysed by a script Elastic constants as numerical energy second-derivatives Geometry optimizations for nuclear-relaxed terms Fully-automated implementation (single calculation) Elastic constants as analytical gradients finite differences Generalization to 1D and 2D systems Elastic tensor under pressure via analytical stress-tensor Nuclear-relaxation term from analytical internal-strain tensor

Silicate Garnets X3Y2(SiO4)3 Silicate garnets are among the most important rockforming minerals of the Earth s lower crust, upper mantle and transition zone. They belong to the cubic space group Ia3d. Pyralspite Mg3Al2(SiO4)3 Pyrope Fe3Al2(SiO4)3 Mn3Al2(SiO4)3 Almandine Spessartine Ugrandite Ca3Al2(SiO4)3 Ca3Fe2(SiO4)3 Ca3Cr2(SiO4)3 Grossular Andradite Uvarovite 80 percell) cell (80atoms atoms per

Elastic Constants at P = 0

Elastic Constants at P = 0 EXPERIMENTAL THEORETICAL THIS STUDY

Elastic Properties at P = 0 Many elastic properties of isotropic polycrystalline aggregates can be computed from the elastic (C) and compliance (S=C -1) constants via the Voigt-Reuss-Hill averaging scheme (expressions for cubic crystals): Bulk modulus Shear modulus Young modulus Poisson's ratio Anisotropy index The average values of transverse (shear), vs, and longitudinal, vp, seismic wave velocities, for isotropic polycrystalline aggregates, can be computed as:

Elastic Properties at P = 0 Pyrope Almandine Spessartine Grossular Andradite Uvarovite

Seismic Wave Velocities at P = 0 From the elastic constants, through Christoffel's equation, seismic wave velocities can be computed: Vp Andradite Vs2 Vs1 Experimental data from single-crystal Brillouin scattering measurements by Jiang et al., J. Phys.: Condens. Matter, 16, S1041 (2004)

Seismic Wave Velocities at P = 0 Pyrope Almandine Spessartine Grossular Andradite Uvarovite

Seismic Wave Velocities at P = 0 Pyrope Almandine Elastic Anisotropy Spessartine Grossular Seismic wave velocity Andradite Uvarovite

Seismic Wave Velocities (AWESoMe) The AWESoMe program has been merged into CRYSTAL17 and can now be used as a keyword (AWESOME) of the ELASTCON input block. The code computes the phase and group velocities for all the possible propagation directions, as well as some related parameters such as the polarization vectors, the power flow angle and the enhancement factor. D. Munoz-Santiburcio, A. Hernandez-Laguna and J.I. Soto, Comp. Phys. Commun., 192, 272-277 (2015)

Seismic Wave Velocities (AWESoMe) The AWESoMe program has been merged into CRYSTAL17 and can now be used as a keyword (AWESOME) of the ELASTCON input block. D. Munoz-Santiburcio, A. Hernandez-Laguna and J.I. Soto, Comp. Phys. Commun., 192, 272-277 (2015)

Towards Geophysical Conditions 2900 km depth In The Earth's Mantle: Pressures up to ~140 GPa Temperatures between ~800 K and 1200 K

Towards Geophysical Conditions Ab initio characterization of structural and elastic properties of minerals at geophysical conditions. Compositional models for the Earth's deep interior are based on seismic information obtained by analysing earthquakes. In order to correctly interpret seismological data, the elastic response properties of the constituents of the mantle have to be fully characterized in terms of: - elastic constants - bulk modulus - seismic wave propagation velocity - elastic anisotropy - P,T dependence

Elastic Tensor under Pressure Elastic constants computed at V(P) Eulerian strain correction The P-V relation must be determined (EoS approach, analytical stress tensor approach)

P-V Relation An analytical approach is used to determine the P-V relation that is based on the stress tensor: exprerssed in terms of analytical cell gradients, wrt deformed lattice parameters: By adding a pre-stress in the form of a hydrostatic pressure the expression of the pressure-constrained cell gradients is obtained:

P-V Relation

Elastic Constants at P > 0 A. Erba, A. Mahmoud, D. Belmonte and R. Dovesi, J. Chem. Phys., 140, 124703 (2014)

Directional Seismic Velocities under Pressure

Directional Seismic Velocities under Pressure Forsterite Mg2SiO4

Equation of State It is a pressure-volume (or energy-volume) relation describing the behavior of a solid under compression or expansion. It is given an analytical (approximated) expression. Many functional forms have been proposed. Universal expressions are not specifically parametrized for a given system or family of systems. It embodies information on the bulk modulus and its dependence on pressure. In Solid State Physics and Chemistry, it is commonly used to extrapolate at high pressure from low pressure data. In Solid State Quantum Chemistry, E-V data are usually fitted to get the P-V relation.

Bulk Modulus Energy Volume K >> K <<

Equation of State at Work The energy of the system is computed at different volumes (compression and expansion). For each volume, the most stable configuration is found by performing a V-constrained optimization. Energy E-V points are fitted to a particular EOS Murnaghan, order 3 (1944): By taking the derivative wrt the volume, one gets the pressure: Volume

Equation of State at Work Vinet exponential EOS: T The exponential "universal" Vinet's equation of state, published in 1987, reads:

Equation of State at Work Poirier-Tarantola logarithmic EOS: The third-order Poirier-Tarantola logarithmic equation of state (derived from the natural strain), proposed in 1998, is:

Equation of State at Work In CRYSTAL, one can compute the EOS with a fully-automated procedure: In the output:

Equation of State at Work

Piezoelectric Efects

Efect of Symmetry Third Rank Piezoelectric Tensor Triclinic Hexagonal Cubic J. F. Nye, Oxford University Press, (1985)

Piezoelectric Tensors in CRYSTAL In the CRYSTAL program, the direct piezoelectric tensor is explicitly computed, whereas the converse piezoelectric tensor is obtained by combining the direct one with the elastic compliance tensor: In CRYSTAL, there are two possible approaches to the calculation of the direct piezoelectric tensor: NUMERICAL (from CRYSTAL06) (QUASI)ANALYTICAL (from CRYSTAL17) Berry-phase approach for electronic term CPHF/KS approach for electronic term Geometry optimizations for nuclear-relaxation term Internal-strain tensor approach for nuclear-relaxation term

Numerical Approach In CRYSTAL, the direct piezoelectric effect is computed: In the theory of polarization of 3D crystals, the Berry Phase along the l-th crystallographic axis, for a given strain state of the system, is given by: So that the polarization (by inverting the previous expression) reads: and the piezoelectric constants can be expressed as:

Numerical Approach: The Algorithm Geometry optimization and calculation of the cell gradients of the reference structure Berry phase calculation Full symmetry analysis and definition of minimal set of strains Application of each strain, geometry optimization of atomic positions and calculation of cell gradients and Berry phase for different strain amplitudes Piezoelectric constants are obtained by numerical fitting with respect to the strain

Analytical Approach In CRYSTAL17 we have recently developed an analytical approach to the calculation of the direct piezoelectric tensor of 3D periodic systems, based on the Coupled-Perturbed-Hartree-Fock/Kohn-Sham (CPHF/KS) scheme and on the internal-strain tensor: where Electronic Nuclear

Numerical vs. Analytical Approach We already discussed the comparison for the nuclear-relaxation term: ZnO Parallel mode on 16 CPUs Convergence tolerance of the geometry optimizer on the gradient

Numerical vs. Analytical Approach Now for the electronic term:

Piezoelectricity of 1D and 2D Systems The numerical approach to the evaluation of the piezoelectric response has also been generalized to low-dimensional 1D and 2D systems. Induced piezoelectricity in Graphene: The in-plane response is dominated by the electronic term and tends to a common limit value as the defect concentration decreases Kh. E. El-Kelany, Ph. Carbonnière, A. Erba and M. Rérat, J. Phys. Chem. C 119, 8966-8973 (2015)

Piezoelectricity of 1D and 2D Systems The numerical approach to the evaluation of the piezoelectric response has also been generalized to low-dimensional 1D and 2D systems. Induced piezoelectricity in Graphene: The out-of-plane response is dominated by the nuclear term, tends to zero as the defect concentration decreases and is large when a soft IR-active phonon mode is found. Kh. E. El-Kelany, Ph. Carbonnière, A. Erba, J.-M. Sotiropoulos and M. Rérat, J. Phys. Chem. C, 120, 7795-7803 (2016)

Photo-elastic and Piezo-optic Tensors With the CRYSTAL program, the photoelastic tensor is computed numerically by evaluating the dielectric tensor (as computed via the CPHF/KS approach) at different strained configurations: The piezo-optic tensor can then be computed by combining the photo-elastic one with the elastic compliance one:

Photoelastic Tensor: The Algorithm Geometry optimization and calculation of the cell gradients of the reference structure Dielectric tensor calculation through CPHF/KS Full symmetry analysis and definition of minimal set of strains Application of each strain, relaxation of atomic positions and calculation of cell gradients and the dielectric tensor for different strain amplitudes Photoelastic constants are obtained by numerical fitting with respect to the strain

Photoelastic Tensor: Validation We have computed the photoelastic properties of several simple crystals for which accurate experimental data exist: MgO, NaCL, LiF, KCl, Silicon, Diamond, Quartz, Rutile.

Photoelastic Tensor: Frequency dependence We have computed the photoelastic properties of several simple crystals for which accurate experimental data exist: MgO, NaCL, LiF, KCl, Silicon, Diamond, Quartz, Rutile.

Piezo-Optic Tensor 57 measurements on 16 properly cut samples To determine the 36 independent piezo-optic constants of a triclinic system

Piezo-Optic Tensor

Piezo-Optic Tensor of CaWO4

Piezo-Optic Tensor of PbMoO4

Thank you for your kind attention Alessandro Erba Dipartimento di Chimica, Università di Torino, Italy E-mail: alessandro.erba@unito.it