Supplementary Materials to
“Surface polar states and pyroelectricity in ferroelastics induced by flexo-roto field”
A.N. Morozovska1,2, E.A. Eliseev2, S.V. Kalinin3, Long Qing Chen4 and Venkatraman Gopalan4
1 Institute of Semiconductor Physics, NationalAcademy of Science of Ukraine,
41, pr. Nauki, 03028 Kiev, Ukraine
2 Institute for Problems of Materials Science, NationalAcademy of Science of Ukraine,
3, Krjijanovskogo, 03142 Kiev, Ukraine
3 Center for Nanophase Materials Science, Oak Ridge National Laboratory,
Oak Ridge, TN, 37831
4Department of Materials Science and Engineering, PennsylvaniaStateUniversity,
University Park, Pennsylvania16802, USA
S.1. Equations of state
Free energy function has the form:
(S.1a)
(S.1b)
Polarization-dependent energy is
, (S.2a)
structural order parameter dependent energy is
, (S.2b)
gradient energy is
, (S.2c)
elastic strain energy is
, (S.2d)
electro-striction and roto-striction energy is
, (S.2e)
biquadratic coupling term is [[1], [2], [3]]
. (S.2f)
Summation is performed over all repeated indices.
It should be noted that the flexoelectric coupling energycould be written in the different forms, namely it is either theform of Lifshitz invariant [[4], [5], [6]]
, (S.2g)
or “short” form . Despite both forms give the same equations of state, they have crucial implications on the boundary conditions [4]. Moreover if the Lifshitz invariant is used, one should include squares of the strain gradients in the free energy (S.2c), , where the tensor is positively defined in order to provide the system stability at arbitrary stresses. Since the strength of the material tensor is typically unknown and regarded very small,most of the authors (see e.g. [[7], [8]])avoided its inclusion and used two different coefficients before and (recent review and relations between these variants could be found in [[9]]). We should stress that any of the form could be safely used till the bulk system is considered and boundary conditions (surface terms) are not important.
At fixed values of conductors potentials(e.g. for short-circuited system placed between conducting electrodes) the energy of electric field could be written as (see e.g. Ref.[9, [10], [11]]):
(S.3a)
At fixed values of conductors charges(e.g. for open system without electrode(s)) the electric field energy (S.3a) should be modified by adding the term (see e.g. Ref.[[12]]):
(S.3b)
Here we introduced the electrostatic displacement and the electrostatic field potential as . It is seen that the energy at fixed values of conductors charges(S.3b) differ from Eq.(S.3a) only in the surface energy term, proportional to the surface free charges. Note that since , expression (S.3b), i.e. , looks consistent with corresponding expressions in [9,[13]]. As anticipated, the variation of both Eqs.(S.3a) and (S.3b) gives the same contribution into the equation of state for polarization.
The conditions of the free energy (S.1) extremumare given by zero variation:
, , . (S.4a)
Here is the variational derivative. Equations (S.4a) can be written in the evident form:
, , (S.4b)
. (S.4c)
Here are the stress tensor components. In the case of inhomogeneous distributions of strains one should not regard them as independent ones, since compatibility relations should be satisfied.
The explicit form of equations (S.4) is
, (S.5a)
(S.5b)
(S.5c)
Mechanical equilibrium equation , with determined byEq.(S.4c) gives:
. (S.6)
In the dielectric limit depolarization electric field components (if any exist) are determined self-consistently from Maxwell equation
, (S.7)
where 0=8.8510−12F/m is the universal dielectric constant, is the “background” isotropic lattice permittivity, different from the ferroelectric soft mode permittivity [[14], [15], [16]].
S.2. Boundary conditions
Equations (S.5) should be supplemented with the boundary conditions (BC) at for the structural order parameter, polarization and stress components. As we mentioned above, the form of boundary conditions essentially depends on whether the higher order displacement derivatives are included in the free energy. Below we consider the cases of “short” formand Lifshitz invariant formfor flexoelectric coupling energy.
For the structural order parameter BC is the same for both forms of the flexoelectric coupling. They were obtained by the variation of the surface energyand the gradient energy on components :
, (S.8a)
Here are the components of outer normal to the surface, in the considered case only is relevant. No summation over “i” is performed in the term . The first term in Eq.(S.8a) comes from the variation of the surface energy, while the second is obtained after integration in parts of the bulk energy gradients variations.
BC for elastic stress on the mechanically free flat surface has the form [5]:
(S.8b)
BC for polarization were obtained by the variation of the surface energy, the gradient energy and the flexoelectric energy on .In the case of Lifshitz invariant form of flexoelectric coupling (S.2g) the higher order derivatives of displacement should be included into the free energy and the BC for polarization are
(S.8c)
No summation over “i” is performed in the term .
For the “short” form of the flexoelectric energyand variation gives:
, (S.8d)
Note that Eq.(S.8c) and (S.8d) differ in factor ½ in the flexoelectric term (see also [4]).
Also additional boundary conditions should be considered for the case[4]:
(S.9)
The latter comes from the facts that the variation of the free energy on displacement vector besides (S.4d) leads to the following BC and and Lame type equation. Note that Eq.(S.7d) loses its sense in the limit, ifone regard the surface energy independent on the stress and strain. Otherwise it simply relates the unknown surface energy coefficients and .
S.3. Evident form of equations of state and elastic solution
For the parent phase of m3m symmetry Eqs.(S.5) have the form:
(S.10a)
(S.10b)
(S.10c)
Equations of state for polarization components could be written as
(S.10d)
(S.10e)
(S.10f)
For considered geometry(,,) only three equations remained:
(S.11a)
(S.11b)
(S.11c)
Note that for considered case of 1D distribution of order parameters and strains along x3 axis, the second and the third terms in Eq.(S.8b) are cancelled out so that Eq.(S.8b) reduces to the conventional BC in the form .
Decoupling on polarization vector in Eqs.(S.11) and neglecting the strain gradient in (S.5c) leads to the following equations:
, (S.12a)
, (S.12b)
. (S.12c)
For considered geometry (see Fig.1a) elastic solution for strain tensor in Voigt notations (11=1, 22=2, 33=3, 23=4, 13=5, 12=6) has the form:
, , (S.13a)
, (S.13b)
, , . (S.13c)
Stress tensor components in Voigt notations has the form
, (S.13d)
, (S.13e)
. (S.13f)
Substituting Eqs.(S.10, 11) in Eq.(S.5) we get coupled system of equations for structural order parameters and polarization. Polarization satisfies the following equation:
(S.14)
Here we introduced the expansion coefficients at given stress. Namely, for coupling coefficients
, (S.15a)
, (S.15b)
. (S.15c)
And for nonlinearity coefficients
, (S.16a)
, (S.16b)
S.4. SrTiO3 material parameters
Table S.I. SrTiO3 material parameters included intoLGD free energy (S.1)
Parameter
/ SI units / Value / Source and notesb / dimensionless / 43 / [a, b]
T / 106m/(FK) / 0.75 / [c, d]
/ K / 30 / ibidem
/ K / 54 / ibidem
aij / 109m5/(C2F) / =2.025, =1.215,
=1.724, =1.396 / ibidem
calculated from
qij / 1010 m/F / q11=1.251, q12=0.108, q44=0.243 / [c]
gijkl / 10-11Vm3/C / g11=g44=1, g12=0.5 / Estimation based on Ref. [e]
T / 1026J/(m5K) / 9.1 / [c]
TS / K / 105 / [c]
/ K / 145 / [c]
bij / 1050J/m7 / =1.94, =3.96,
=1.69, =3.88 / [c]
calculated from
rij / 1030J/(m5) / r11=1.3, r12=2.5, r44=-2.3 / [c]
ijkl / 1029 (Fm)-1 / =-3.366, =0.135, =6.3
=-2.095, =0.849, =5.860 / [c]
calculated from
vijkl / 1010J/m3 / v11=0.28, v12=7.34, v44=7.11 / [d, f]
cij / 1011J/m3 / c11=3.36, c12=1.07, c44=1.27 / [c, d]
sij / 10-12m3/J / s11=3.52, s12=0.85, s44=7.87 / calculated from cij
/ V / =3.24, =1.44, =1.08 / Recalculated from Ref.[g] at given stress
aG. Rupprecht and R.O. Bell, Phys. Rev. 135, A748 (1964).
bG.A. Smolenskii, V.A. Bokov, V.A. Isupov, N.N Krainik, R.E. Pasynkov, A.I. Sokolov, Ferroelectrics and Related Materials (Gordon and Breach, New York, 1984). P. 421
cN. A. Pertsev, A. K. Tagantsev, and N. Setter, Phys. Rev. B 61, R825 (2000).
d A.K. Tagantsev, E. Courtens and L. Arzel, Phys. Rev. B, 64, 224107 (2001).
e J. Hlinka and P. Marton, Phys. Rev. B 74, 104104 (2006).
f W. Cao and R. Barsch, Phys. Rev. B, 41, 4334 (1990)
gP. Zubko, G. Catalan, A. Buckley, P.R. L. Welche, J. F. Scott. Phys. Rev. Lett. 99, 167601 (2007).
1
[1] M. J. Haun, E. Furman, T. R. Halemane and L. E. Cross, Ferroelectrics, 99, 55 (1989), ibidem p.13.
[2] A.K. Tagantsev, E. Courtens and L. Arzel, Phys. Rev. B, 64, 224107 (2001).
[3]. B. Houchmanzadeh, J. Lajzerowicz and E Salje, J. Phys.: Condens. Matter 3, 5163 (1991)
[4]E.A.Eliseev, A.N.Morozovska, M.D. Glinchuk, and R. Blinc. Phys. Rev. B.79, 165433,(2009).
[5]A.S. Yurkov, JETP Letters, 94, 455 (2011).
[6] A. K. Tagantsev, A. S. Yurkov, Flexoelectric effect in finite samples, arXiv:1110.0380v1 [cond-mat.mtrl-sci] 3 Oct 2011
[7]M. S. Majdoub, R. Maranganti, and P. Sharma, Phys. Rev. B 79, 115412 (2009).
[8] Yue Zheng, Biao Wang, C. H. Woo. Appl. Phys. Lett. 89, 062904 (2006).
[9] N.D. Sharma, C.M. Landis, and P. Sharma, J. Appl. Phys. 108, 024304 (2010).
[10] M. Marvan and J. Fousek, Phys. Stat. Sol. (b) 208, 523 (1998).
[11] A.N. Morozovska, E.A. Eliseev, G.S. Svechnikov, and S.V. Kalinin. Phys. Rev. B 84, 045402 (2011).
[12] L.D. Landau, E.M. Lifshitz, and L.P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Butterworth-Heinenann, Oxford, 1984) Vol. 8.
[13] A.K. Tagantsev, L.E. Cross, and J. Fousek: "Domains in Ferroic Crystals and Thin Films" Springer, 2010; Section 2.3 Thermodynamic approach, pgs. 81-84; and section 8.2 Basics of Domain State Reorientation, pgs. 354-360.
[14] A.K. Tagantsev, and G. Gerra, J. Appl. Phys. 100, 051607 (2006).
[15] C.H. Woo and Yue Zheng, Appl. Phys. A 91, 59 (2007)
[16] G. Rupprecht and R.O. Bell, Phys. Rev. 135, A748 (1964).