elastic and piezoelectric tensorial properties · symmetry and physical properties of crystals •...
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ElasticElastic and and piezoelectricpiezoelectric tensorialtensorialpropertiesproperties
Michele CattiDipartimento di Scienza dei MaterialiUniversità di Milano Bicocca, Italy
Michele Catti - ASCS2006, Spokane
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Tensorial physical properties of crystals
Tensor of rank n:
set of 3n coefficients with n subscripts, associated with a given Cartesian basis, which transform according to the formula:
y'ihkl.. = .. TipThqTkrTls....ypqrs.. ,
when the basis is transformed into a new one by action of the
T matrix
Michele Catti - ASCS2006, Spokane
p,q,r,s1
3∑
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Tensor of first order: vectorial quantity (3 vector components)electric polarization intensity: Pi (i=1,2,3)magnetic polarization intensity
Tensor of second order:linerar relationship between two vectorial quantities
strain sij (i,j=1,2,3): ui = xi’-xi = sijxj
symmetrical strain: εij = ½(sij+sji) = εji
Voigt notation: εij → εh → ε = [ε1 ε2 ε3 ε4 ε5 ε6] (11→1, 22→2, 33→3, 23→4, 13→5, 12→6)Michele Catti - ASCS2006, Spokane
∑3
1j=
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stress τij (i,j=1,2,3): pi = τijnj
symmetrical stress: τij = τ ji
Voigt notation: τij → τh → τ = [τ1 τ2 τ3 τ4 τ5 τ6] (11→1, 22→2, 33→3, 23→4, 13→5, 12→6)
Tensor of fourth order:linerar relationship between two second-order tensorialquantities
elasticity tensor cijpq (i,j,p,q=1,2,3): τij = pq cijpqεpq
Voigt notation:cijpq → chk (h,k=1,2,3,4,5,6) τh = chkεk
Michele Catti - ASCS2006, Spokane
∑3
1j=
∑3
1
∑6
1=k
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Tensor of third order:
linerar relationship between a first- and a second-ordertensorial quantity
piezoelectricity tensor eijp (i,j,p=1,2,3): Pi = jp eijpεjp
Voigt notation:eijp → eih (i=1,2,3; h=1,2,3,4,5,6) Pi = eihεh
Michele Catti - ASCS2006, Spokane
∑3
1
∑6
1=h
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Elastic constants chk = ckh :
linear coefficients relating stress components to the ensuingstrain components
Mechanical work (linear régime): W = V τhεh
Elastic energy: W = ½(V hkchkεhεk)
Michele Catti - ASCS2006, Spokane
∑6
1=h
∑6
1
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Suitable strains ε = [ε1 ε2 ε3 ε4 ε5 ε6] have to be selected, so as to obtain a quadratic dependence of the elastic energy on the ‘amplitude’ of the deformation
Example: CaF2 (cubic Fm-3m)three independent elastic constants (c11, c12, c44)
Michele Catti - ASCS2006, Spokane
44
44
44
111212
121112
121211
000000000000000000000000
cc
cccccccccc c12=c13=c23
c44=c55=c66
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1) ε = [ε ε 0 0 0 0] → ∆a/a = ∆b/b = ε strain amplitude(the symmetry is lowered to tetragonal I4/mmm)
W = V(c11+c12)ε2 →c11+c12 coefficient of the parabolic W(ε) numerical fit
2) ε = [ε ε -2ε 0 0 0] → ∆a/a = ∆b/b = -2∆c/c = εtetragonal strain amplitude
W = 3V(c11-c12)ε2 →c11-c12 coefficient of the parabolic W(ε) numerical fit
Michele Catti - ASCS2006, Spokane
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3) ε = [0 0 0 ε ε ε] → ε ∝ cosα=cosβ=cosγ strain amplitude(the symmetry is lowered to rombohedral R-3m)
The F atom acquires a degree of freedom along thetrigonal axis, which has to be relaxed
W = 3/2(Vc44)ε2 →c44 coefficient of the parabolic W(ε) numerical fit
Michele Catti - ASCS2006, Spokane
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SymmetrySymmetry and and physicalphysical propertiesproperties of of crystalscrystals
• Neumann’s principle:
The symmetry of a matter (‘intrinsic’) tensorial
physical property can not be lower than the point
group symmetry of the crystal
Michele Catti - ASCS2006, Spokane
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• Ferroelectricity, pyroelectricity: “spontaneous” electric dipole moment per unit volume P0
• 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
Michele Catti - ASCS2006, Spokane
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BaTiO3 - P4mm – ferroelectric phase at RT
Michele Catti - ASCS2006, Spokane
P0
Ti
Ba
O1
O1
O1O1
O2
O2
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• Piezoelectricity
- in ferroelectric crystals: the spontaneous polarization Pchanges under strain
⇓all ferroelectric crystals are also piezoelectric
- in non-ferroelectric crystals: a non-spontaneous polarizationP arises under strain, which lowers the point symmetry fromnon-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 Michele Catti - ASCS2006, Spokane
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• Pi = P0i + ∑keikεk electric dipole moment per unit volume
• P0 : spontaneous polarization vector- present in ferroelectric and pyroelectric crystals,
with polar symmetry- absent in other (non-centrosymmetrical but non-
polar) piezoelectric crystals
• Warning:
The absolute macroscopic polarization P of a crystal cannot be measured as a bulk property, independent of sample termination
Michele Catti - ASCS2006, Spokane
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• 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) = 0 → ∆P = -∫ Jdt
⇓
the polarization change is equal to the integrated macroscopic current density J passing through the crystal while the perturbation is switched on
Michele Catti - ASCS2006, Spokane
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QuantumQuantum--mechanical calculations of spontaneous mechanical calculations of spontaneous polarization and/or piezoelectric propertiespolarization 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 ∆Puniquely
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 (2000)
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Pnucl(α) = (e/V) ∑s Zsxs
Pel(α) = (e/8π3) ∑n ∫ < fnK
(α)-i∇KfnK(α) >dK
α denotes the crystal structure configuration (positions of nuclei)
P (α) = Pnucl(α) + Pel
(α); ∆P = P (2) - P (1)
< fnK-i∇KfnK > = -i ∫ fnK(x)*∇KfnK(x)dx;
fnK(x) = ψnK(x) e-iK⋅ x
ψnK(x): eigenfunctions of the one-electron Hamiltonian H
fnK(x): cell-periodic Bloch functions; fnK(x + l) = fnK(x)Michele Catti - ASCS2006, Spokane
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BerryBerry phasesphases
Berry phases are related to the crystallographic components of the P vector, and can be computed quantum-mechanically:
ϕi = 2π(V/e)P ⋅ ai* = (V/4π2) ∑n ∫ < fnK
(α)-iai* ⋅ ∇KfnK
(α) >dK
Derivatives (∂ϕi/∂εk) are invariant with respect to (e/V)lvectors added to P:
∂ϕi(P)/∂εk = ∂ϕi[P+ (e/V)l]/∂εk
Michele Catti - ASCS2006, Spokane
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BerryBerry phasesphases and and piezoelectricpiezoelectric constantsconstants
• The derivatives ∂ϕi/∂εk are computed numerically for a given strain [ε1 ε2 ε3 ε4 ε5 ε6]
• Then phase derivatives are transformed into polarization derivatives (piezoelectric constants):
eik = (∂Pi/∂εk)E = (e/2πV)∑h aih(∂ϕh/∂εk)ε=0 = eik,
where the aih quantities are Cartesian components of the direct unit-cell vectors
Michele Catti - ASCS2006, Spokane
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WurtziteWurtzite and and zinczinc blende blende phasesphases of of ZnOZnO and and ZnSZnS
M. Catti, Y. Noel, R. Dovesi, J. Phys. Chem. Solids 11, 2183 (2003)
wurtzite: 6mm zinc blende: -43m
0 0 0 0 d15 0 0 0 0 d14 0 0 0 0 0 d15 0 0 0 0 0 0 d14 0 d31 d31 d33 0 0 0 0 0 0 0 0 d14
0 0 0 0 e15 0 0 0 0 e14 0 0 0 0 0 e15 0 0 0 0 0 0 e14 0 e31 e31 e33 0 0 0 0 0 0 0 0 e14
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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.
Wurtzitea (Å) 3.286 3.275 3.25 3.982 3.875 3.82c (Å) 5.241 5.251 5.20 6.500 6.307 6.25u 0.383 0.382 0.380 0.377 0.377 0.378
Zinc blendea (Å) 4.619 4.610 --- 5.627 5.511 5.413
_________________________________________________
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WurtziteWurtzite and and zinczinc blende blende phasesphases of of ZnOZnO and and ZnSZnS
• Strains chosen appropriately for each piezoelectric constant
wurtzite: 6mm zinc blende: -43m
e33: [0 0 ε 0 0 0] e14: [0 0 0 ε ε ε ]
e31: [ε ε 0 0 0 0]
e15: [0 0 0 ε √3ε 0]
• Structural optimization at each value of ε for every strain
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Wurtzite
a1 = (√3/2)a i1 - (1/2)i2; a2 = a i2; a3 = c i3
ε = [0 0 ε 0 0 0] e33 = (e/2πV)c(∂ϕ3/∂ε)
ε = [ε ε 0 0 0 0] e31 = (e/2πV)c/2(∂ϕ3/∂ε)
ε = [0 0 0 ε √3ε 0] e15 = (e/2πV)a/2(∂ϕ1/∂ε)
Zinc blende
a1 = (a/2)i2 + (a/2)i3; a2 = (a/2)i1 + (a/2)i3; a3 = (a/2)i1 + (a/2)i2
ε = [0 0 0 2ε 2ε 2ε] e14 = (e/2πV)a/2(∂ϕ1/∂ε)
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ZnO wurtzite: Berry phases along a and c vs. the ε4 strain⇒ e15 piezoelectric constant
-0.04 -0.02 0.00 0.02 0.04ε4
-0.40
-0.20
0.00
0.20
0.40
φ 1 (r
ad)
ZnOe15
-0.04 -0.02 0.00 0.02 0.04
ε4
0.000
0.004
0.008
0.012
0.016
φ 3 (r
ad)
ZnO
e15
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Ab initio and experimental proper piezoelectric constants eik (Cm-2) for wurtzite and zinc blende ZnO and ZnS
____________________________________________________Wurtzite Zinc blende
________________________ __________e33 e31 e15 e14
ZnOHF 1.19 -0.55 -0.46 0.69GGA 1.20 -0.59exp.a 0.96 -0.62 -0.37 ---
ZnSHF 0.18 -0.13 -0.13 0.11GGA 0.21 -0.16exp.b 0.34 -0.10 -0.08 0.14____________________________________________________
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ExternalExternal-- (clamped(clamped--ion) and internalion) and internal--strain strain components of piezoelectric constantscomponents of piezoelectric constants
eik = ∂Pi/∂εjk = (∂Pi/∂εjk)u + (∂Pi/∂u)ε=0(∂u/∂εjk)ε=0 = eik(0) + eik
int
• Clamped-ion component eik(0) = (∂Pi/∂εjk)u:
atomic fractional coordinates are kept fixed along the strain (homogeneous deformation)
• Internal-strain component eikint = (∂Pi/∂u)ε=0(∂u/∂εjk)ε=0:
atomic fractional coordinates are relaxed at fixed strain
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Rigid-ion point-charge model,and pure ionic (nuclear) contribution to piezoelectricity:
vanishing clamped-ion component of the proper piezoelectric constant ⇓
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 topiezoelectricity)
⇓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
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[001] fractional component of Zn-X spacing vs. ε3strain in wurtzite
-0.04 -0.02 0.00 0.02 0.04ε3
0.370
0.375
0.380
0.385
0.390
0.395
u
ZnO
ZnS
e33
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[110] fractional component of Zn-X spacing vs. ε4strain in wurtzite
-0.04 -0.02 0.00 0.02 0.04ε4
-0.015
-0.010
-0.005
0.000
0.005
0.010
0.015
x'(Z
n)-x
'(X)
ZnO
ZnS
e15
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[111] fractional component of Zn-X spacing vs. ε4strain in zinc blende
-0.04 -0.02 0.00 0.02 0.04ε4
0.230
0.240
0.250
0.260
0.270
uZnO
ZnS
e14 cubic
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Ab initio, classical (rigid ion model) and experimental proper piezoelectric constants eik (Cm-2) for ZnO and ZnS___________________________________________________________________________
e33 (H) e31 (H) e15 (H) e14 (C)___________ ____________ _____________ ____________ZnO ZnS ZnO ZnS ZnO (H) ZnS ZnO ZnS
HF 1.19 0.18 -0.54 -0.13 -0.46 -0.13 0.69 0.11
HF(ext.) -0.45 -0.59 0.22 0.29 0.22 0.28 -0.45 -0.53
HF(int.) 1.63 0.74 -0.76 -0.42 -0.68 -0.41 1.14 0.64
HF(ext.)/HF(int.) 28% 80% 29% 69% 33% 69% 40% 83%
du/dε (HF) -0.228 -0.147 0.215 0.160 0.320 0.270 -0.372 -0.290
exp.a 0.96 0.34 -0.62 -0.10 -0.37 -0.08 --- 0.14
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Analysis of Analysis of abab initioinitio results on piezoelectric results on piezoelectric constants of constants of ZnOZnO and and ZnSZnS
• Signs and values of the eik constants and of their eik(0) and
eikint components
⇓- the internal-strain term eik
int is always dominant, and has thus the same sign as the total value eik
- the sign is opposite to that of du/dε, because the Zn to Xinter-layer distance is (1/2-u)c (W) or (1/3-u)c (ZB)
- the clamped-ion term eik(0) has opposite sign to that of
eikint and eik
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ZnOZnO//ZnSZnS hexagonalhexagonal •• εε33 strainstrain •• ee33 33 > 0> 0
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ZnOZnO//ZnSZnS hexagonalhexagonal •• εε11==εε22 strainstrain •• ee31 31 < 0< 0
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• Internal-strain effect:
- major source of piezoelectric response
- quantum-mechanical results are similar to those byclassical RI model based on ab initio values of ∂u/∂εrelaxation terms
• Clamped-ion effect:
reduces the piezoelectric response, and corresponds to effects of electronic polarization opposing the ionicpolarization caused by structural relaxation
Michele Catti - ASCS2006, Spokane
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ZnOZnO//ZnSZnS hexagonalhexagonal •• εε44 = = εε55 strainstrain •• ee15 15 < 0< 0
Michele Catti - ASCS2006, Spokane
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ZnOZnO//ZnSZnS hexagonalhexagonal •• εε44 = = εε55 strainstrain •• ee15 15 < 0< 0
Michele Catti - ASCS2006, Spokane
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ZnOZnO//ZnSZnS cubiccubic •• εε44 = = εε5 5 ==εε6 6 strainstrain •• ee14 14 > 0> 0
Michele Catti - ASCS2006, Spokane
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SUMMARY SUMMARY
♦ Elastic and piezoelectric constants can becomputed reliably for any crystal by periodicquantum-mechanical methods
♦ Complexity of the crystal structure and/or lowsymmetry 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
Michele Catti - ASCS2006, Spokane