properties Michele Catti Dipartimento di Scienza dei Materiali Università di Milano Bicocca, Italy

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1 Elastic and piezoelectric tensorial properties Michele Catti Dipartimento di Scienza dei Materiali Università di Milano Bicocca, Italy 1

2 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

3 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

4 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

5 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

6 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

7 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 c c c c c c c 44 c 44 c 44 c 12 =c 13 =c 23 c 44 =c 55 =c 66 7

8 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

9 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

10 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

11 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

12 BaTiO 3 - P4mm ferroelectric phase at RT O2 O1 O1 Ti O1 O1 P Ba O2 12

13 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

14 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

15 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

16 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 , Springer (1996) D. Vanderbilt, J. Phys. Chem. Solids 61, 147 (2) 16

17 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

18 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

19 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

20 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

21 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) c (Å) u Zinc blende a (Å)

22 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

23 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

24 ZnO wurtzite: Berry phases along a and c vs. the ε 4 strain e 15 piezoelectric constant ZnO e e 15 φ 1 (rad). φ 3 (rad) ZnO ε ε 4 24

25 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 GGA exp. a ZnS HF GGA exp. b

26 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

27 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

28 [1] fractional component of Zn-X spacing vs. ε 3 strain in wurtzite e 33 ZnO u ZnS ε 3 28

29 [11] fractional component of Zn-X spacing vs. ε 4 strain in wurtzite.15.1 ZnO e 15 x'(zn)-x'(x).5. ZnS ε 4 29

30 [111] fractional component of Zn-X spacing vs. ε 4 strain in zinc blende ZnO e 14 cubic u.25 ZnS ε 4 3

31 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 HF(ext.) HF(int.) HF(ext.)/HF(int.) 28% 8% 29% 69% 33% 69% 4% 83% du/dε (HF) exp. a

32 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

33 ZnO/ZnS ZnS hexagonal ε 3 strain e 33 > 33

34 ZnO/ZnS ZnS hexagonal ε 1 =ε 2 strain e 31 < 34

35 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

36 ZnO/ZnS ZnS hexagonal ε 4 = ε 5 strain e 15 < 36

37 ZnO/ZnS ZnS hexagonal ε 4 = ε 5 strain e 15 < 37

38 ZnO/ZnS ZnS cubic ε 4 = ε 5 =ε 6 strain e 14 > 38

39 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

Tensorial and physical properties of crystals

Tensorial and physical properties of crystals Michele Catti Dipartimento di Scienza dei Materiali, Universita di Milano Bicocca, Milano, Italy (catti@mater.unimib.it) MaThCryst Nancy 2005 International