Elastic and piezoelectric tensorial properties Michele Catti Dipartimento di Scienza dei Materiali Università di Milano Bicocca, Italy (catti@mater.unimib.it) 1
Tensorial physical properties of crystals Tensor of rank n: set of 3 n coefficients with n subscripts, associated with a given Cartesian basis, which transform according to the formula: y' ihkl.. =.. T ip T hq T kr T ls...y pqrs.., when the basis is transformed into a new one by action of the T matrix 3 1 p,q,r,s 2
Tensor of first order: vectorial quantity (3 vector components) electric polarization intensity: P i (i=1,2,3) magnetic polarization intensity Tensor of second order: linerar relationship between two vectorial quantities strain 3 j= 1 s ij (i,j=1,2,3): u i = x i -x i = s ij x j symmetrical strain: ε ij = ½(s ij +s ji ) = ε ji Voigt notation: ε ij ε h ε = [ε 1 ε 2 ε 3 ε 4 ε 5 ε 6 ] (11 1, 22 2, 33 3, 23 4, 13 5, 12 6) 3
stress τ ij (i,j=1,2,3): p i = τ ij n j symmetrical stress: τ ij = τ ji Voigt notation: τ ij τ h τ = [τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 ] (11 1, 22 2, 33 3, 23 4, 13 5, 12 6) Tensor of fourth order: linerar relationship between two second-order tensorial quantities 3 j= 1 elasticity tensor c ijpq (i,j,p,q=1,2,3): τ ij = pq c ijpq ε pq 3 1 Voigt notation: c ijpq c hk (h,k=1,2,3,4,5,6) τ h = c hk ε k 6 k= 1 4
Tensor of third order: linerar relationship between a first- and a second-order tensorial quantity piezoelectricity tensor 3 1 e ijp (i,j,p=1,2,3): P i = jp e ijp ε jp Voigt notation: e ijp e ih (i=1,2,3; h=1,2,3,4,5,6) P i = e ih ε h 6 h= 1 5
Elastic constants c hk = c kh : linear coefficients relating stress components to the ensuing strain components 6 h= 1 Mechanical work (linear régime): W = V τ h ε h 6 1 Elastic energy: W = ½(V hk c hkε h ε k ) 6
Suitable strains ε = [ε 1 ε 2 ε 3 ε 4 ε 5 ε 6 ] have to be selected, so as to obtain a quadratic dependence of the elastic energy on the amplitude of the deformation Example: CaF 2 (cubic Fm-3m) three independent elastic constants (c 11, c 12, c 44 ) c c c 11 12 12 c c c 12 11 12 c c c 12 12 11 c 44 c 44 c 44 c 12 =c 13 =c 23 c 44 =c 55 =c 66 7
1) ε = [ε ε ] a/a = b/b = ε strain amplitude (the symmetry is lowered to tetragonal I4/mmm) W = V(c 11 +c 12 )ε 2 c 11 +c 12 coefficient of the parabolic W(ε) numerical fit 2) ε = [ε ε-2ε ] a/a = b/b = -2 c/c = ε tetragonal strain amplitude W = 3V(c 11 -c 12 )ε 2 c 11 -c 12 coefficient of the parabolic W(ε) numerical fit 8
3) ε = [ ε εε] ε cosα=cosβ=cosγ strain amplitude (the symmetry is lowered to rombohedral R-3m) The F atom acquires a degree of freedom along the trigonal axis, which has to be relaxed W = 3/2(Vc 44 )ε 2 c 44 coefficient of the parabolic W(ε) numerical fit 9
Symmetry and physical properties of crystals Neumann s principle: The symmetry of a matter ( intrinsic ) tensorial physical property can not be lower than the point group symmetry of the crystal 1
Ferroelectricity, pyroelectricity: spontaneous electric dipole moment per unit volume P Polar vector with symmetry m only subgroups of m are allowed as point groups of ferroelectric crystals (non-centrosymmetrical polar groups): 1, 2, 3, 4, 6, m, mm2, 3m, 4mm, 6mm 11
BaTiO 3 - P4mm ferroelectric phase at RT O2 O1 O1 Ti O1 O1 P Ba O2 12
Piezoelectricity - in ferroelectric crystals: the spontaneous polarization P changes under strain all ferroelectric crystals are also piezoelectric - in non-ferroelectric crystals: a non-spontaneous polarization P arises under strain, which lowers the point symmetry from non-polar to polar only non-centrosymmetrical non-polar point groups are allowed for piezoelectric non-ferroelectric crystals: -4, -6, 222, 32, 422, 622, 23, -42m, -6m2, -43m 13
P i = P i + k e ik ε k electric dipole moment per unit volume P : spontaneous polarization vector - present in ferroelectric and pyroelectric crystals, with polar symmetry - absent in other (non-centrosymmetrical but nonpolar) piezoelectric crystals Warning: The absolute macroscopic polarization P of a crystal cannot be measured as a bulk property, independent of sample termination 14
finite polarization changes P between two different crystal states (ferroelectricity), or polarization derivatives P/ f with respect to a physical effect f (piezoelectricity, pyroelectricity) are the measurable observables div(j + P/ t) = P = - Jdt the polarization change is equal to the integrated macroscopic current density J passing through the crystal while the perturbation is switched on 15
Quantum-mechanical mechanical calculations of spontaneous polarization and/or piezoelectric properties P = P (2) - P (1) = - Jdt - the current density depends on the phase of the wave function in addition to its modulus - the charge density ρ(x) = Ψ(x) 2 does not determine P uniquely R.D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993) R. Resta, Rev. Mod. Phys. 66, 899 (1994) R. Resta, in Quantum-mechanical ab initio calculation of the properties of crystalline materials (Ed. by C. Pisani), pp. 273-288, Springer (1996) D. Vanderbilt, J. Phys. Chem. Solids 61, 147 (2) 16
P nucl (α) = (e/v) s Z s x s P el (α) = (e/8π 3 ) n <f nk (α) -i K f nk (α) >dk α denotes the crystal structure configuration (positions of nuclei) P (α) = P nucl (α) + P el (α) ; P = P (2) - P (1) < f nk -i K f nk > = -i f nk (x) * K f nk (x)dx; f nk (x) = ψ nk (x) e -ik x ψ nk (x): eigenfunctions of the one-electron Hamiltonian H f nk (x): cell-periodic Bloch functions; f nk (x + l) = f nk (x) 17
Berry phases Berry phases are related to the crystallographic components of the P vector, and can be computed quantum-mechanically: ϕ i = 2π(V/e)P a i* = (V/4π 2 ) n <f nk (α) -ia i* K f nk (α) >dk Derivatives ( ϕ i / ε k ) are invariant with respect to (e/v)l vectors added to P: ϕ i (P)/ ε k = ϕ i [P+ (e/v)l]/ ε k 18
Berry phases and piezoelectric constants The derivatives ϕ i / ε k are computed numerically for a given strain [ε 1 ε 2 ε 3 ε 4 ε 5 ε 6 ] Then phase derivatives are transformed into polarization derivatives (piezoelectric constants): e ik =( P i / ε k ) E = (e/2πv) h a ih ( ϕ h / ε k ) ε= = e ik, where the a ih quantities are Cartesian components of the direct unit-cell vectors 19
Wurtzite and zinc blende phases of ZnO and ZnS M. Catti, Y. Noel, R. Dovesi, J. Phys. Chem. Solids 11, 2183 (23) wurtzite: 6mm zinc blende: -43m d 15 d 14 d 15 d 14 d 31 d 31 d 33 d 14 e 15 e 14 e 15 e 14 e 31 e 31 e 33 e 14 2
Ab initio (Hartree-Fock and DFT-GGA) and experimental structural data for wurtzite and zinc blende ZnO and ZnS ZnO ZnS HF GGA exp. HF GGA exp. Wurtzite a (Å) 3.286 3.275 3.25 3.982 3.875 3.82 c (Å) 5.241 5.251 5.2 6.5 6.37 6.25 u.383.382.38.377.377.378 Zinc blende a (Å) 4.619 4.61 --- 5.627 5.511 5.413 21
Wurtzite and zinc blende phases of ZnO and ZnS Strains chosen appropriately for each piezoelectric constant wurtzite: 6mm zinc blende: -43m e 33 : [ ε ] e 14 : [ ε εε] e 31 : [ε ε ] e 15 : [ ε 3ε ] Structural optimization at each value of ε for every strain 22
Wurtzite a 1 = ( 3/2)a i 1 - (1/2)i 2 ; a 2 = a i 2 ; a 3 = c i 3 ε = [ ε ] ε = [εε ] ε = [ ε 3ε ] e 33 = (e/2πv)c( ϕ 3 / ε) e 31 = (e/2πv)c/2( ϕ 3 / ε) e 15 = (e/2πv)a/2( ϕ 1 / ε) Zinc blende a 1 = (a/2)i 2 + (a/2)i 3 ; a 2 = (a/2)i 1 + (a/2)i 3 ; a 3 = (a/2)i 1 + (a/2)i 2 ε = [ 2ε 2ε 2ε] e 14 = (e/2πv)a/2( ϕ 1 / ε) 23
ZnO wurtzite: Berry phases along a and c vs. the ε 4 strain e 15 piezoelectric constant.4.16.2 ZnO e 15.12 e 15 φ 1 (rad). φ 3 (rad).8 -.2.4 ZnO -.4 -.4 -.2..2.4 ε 4. -.4 -.2..2.4 ε 4 24
Ab initio and experimental proper piezoelectric constants e ik (Cm -2 ) for wurtzite and zinc blende ZnO and ZnS Wurtzite Zinc blende e 33 e 31 e 15 e 14 ZnO HF 1.19 -.55 -.46.69 GGA 1.2 -.59 exp. a.96 -.62 -.37 --- ZnS HF.18 -.13 -.13.11 GGA.21 -.16 exp. b.34 -.1 -.8.14 25
External- (clamped-ion) and internal-strain components of piezoelectric constants e ik = P i / ε jk = ( P i / ε jk ) u + ( P i / u) ε= ( u/ ε jk ) ε= = e ik () + e ik int Clamped-ion component e ik () = ( P i / ε jk ) u : atomic fractional coordinates are kept fixed along the strain (homogeneous deformation) Internal-strain component e ik int = ( P i / u) ε= ( u/ ε jk ) ε= : atomic fractional coordinates are relaxed at fixed strain 26
Rigid-ion point-charge model, and pure ionic (nuclear) contribution to piezoelectricity: vanishing clamped-ion component of the proper piezoelectric constan Clamped-ion component from ab initio calculations: a measure of the deviation of the electronic behaviour from the simple point-charge RI model (pure electronic contribution to piezoelectricity) a pure homogeneous strain yet induces polarization effects on the atomic electron distributions, which give rise to non-vanishing clamped-ion terms of the piezoelectric constants 27
[1] fractional component of Zn-X spacing vs. ε 3 strain in wurtzite.395.39 e 33 ZnO u.385.38 ZnS.375.37 -.4 -.2..2.4 ε 3 28
[11] fractional component of Zn-X spacing vs. ε 4 strain in wurtzite.15.1 ZnO e 15 x'(zn)-x'(x).5. ZnS -.5 -.1 -.15 -.4 -.2..2.4 ε 4 29
[111] fractional component of Zn-X spacing vs. ε 4 strain in zinc blende.27.26 ZnO e 14 cubic u.25 ZnS.24.23 -.4 -.2..2.4 ε 4 3
Ab initio, classical (rigid ion model) and experimental proper piezoelectric constants e ik (Cm -2 ) for ZnO and ZnS e 33 (H) e 31 (H) e 15 (H) e 14 (C) ZnO ZnS ZnO ZnS ZnO (H) ZnS ZnO ZnS HF 1.19.18 -.54 -.13 -.46 -.13.69.11 HF(ext.) -.45 -.59.22.29.22.28 -.45 -.53 HF(int.) 1.63.74 -.76 -.42 -.68 -.41 1.14.64 HF(ext.)/HF(int.) 28% 8% 29% 69% 33% 69% 4% 83% du/dε (HF) -.228 -.147.215.16.32.27 -.372 -.29 exp. a.96.34 -.62 -.1 -.37 -.8 ---.14 31
Analysis of ab initio results on piezoelectric constants of ZnO and ZnS Signs and values of the e ik constants and of their e ik () and e ik int components - the internal-strain term e ik int is always dominant, and has thus the same sign as the total value e ik - the sign is opposite to that of du/dε, because the Zn to X inter-layer distance is (1/2-u)c (W) or (1/3-u)c (ZB) - the clamped-ion term e ik () has opposite sign to that of e ik int and e ik 32
ZnO/ZnS ZnS hexagonal ε 3 strain e 33 > 33
ZnO/ZnS ZnS hexagonal ε 1 =ε 2 strain e 31 < 34
Internal-strain effect: - major source of piezoelectric response - quantum-mechanical results are similar to those by classical RI model based on ab initio values of u/ ε relaxation terms Clamped-ion effect: reduces the piezoelectric response, and corresponds to effects of electronic polarization opposing the ionic polarization caused by structural relaxation 35
ZnO/ZnS ZnS hexagonal ε 4 = ε 5 strain e 15 < 36
ZnO/ZnS ZnS hexagonal ε 4 = ε 5 strain e 15 < 37
ZnO/ZnS ZnS cubic ε 4 = ε 5 =ε 6 strain e 14 > 38
SUMMARY Elastic and piezoelectric constants can be computed reliably for any crystal by periodic quantum-mechanical methods Complexity of the crystal structure and/or low symmetry may increase the computational cost, because of the full structural relaxation needed Two goals can be achieved: - prediction of elastic and piezoelectric properties not yet measured or not measurable - physical insight into the microscopic origin of elastic and piezoelectric behaviour of crystals 39