Theoretical description

The present model was developed and the calculations were done by Hassan Chaib, a guest scientist in our group. The application of the model to a variety of materials is published in detail in [18, 88, 89, 90], for example. The two most important ingredients of the model are the electronic polarizabilities of all ions as a function of the local electric field on the one hand, and the effective charges and positions of the ions on the other hand, as illustrated in Fig. 7.1. The polarizabilitiy of each individual ion under the influence of the local electric fieldE~ is derived by quantum mechanical perturbation theory. Since the

perturbation theory

influence of local electric field for each constituent ion

structural data

ion positions effectiv ionic charges

dipole-dipole interaction

numerical calculation (until self-consistency is reached)

results

electronic polarizabilities net polarization optical properties

Fig. 7.1 Steps for modelling the optical properties of BaTiO₃ domain structures.

7.2. Theoretical description 77

Hamiltonians of the present ions are too complex, the so-called “orbital approximation” is applied. This approximation considers an outer electron orbiting at the distanceRaround the core of the ion, which consists of the nucleus and the inner electrons with an effective chargeZ, and leads to the HamiltonianH0 at zero field as noted in Eq. 7.1. The influence of the electric fieldE~ is introduced by H₁ (Eq. 7.2), leading to the resulting Hamiltonian H (Eq. 7.3) for the core-shell system in an external electric field.

H0= p² 2m− Z

Re², (7.1)

H1=e ~E·R ,~ (7.2)

H=H₀+H₁ (7.3)

In this theory, the perturbation, that means the influence of the local electric field, is introduced by a variational parameterλ. Therefore, it is assumed that the resulting wave functionψ in the presence of an external field can be written as

ψ(λ) =

1 +λ ~E·R~

ψ0, (7.4)

whereψ₀ is the wavefunction for zero external field. The variational principle now claims that the expectation value of the energy hψ|H|ψi must be a minimum with respect to the variational parameterλ. This determines not only the parameterλ, but also the resulting wave functionψ.

From the wavefunctionψ_rof an orbitalrin the presence of a local electric fieldE, we may~ deduce the resulting dipole moment~p_r^eand the electronic polarizabilityα_kl,r according to:

~

We find then the polarizabilityα_kl(j) of the whole ioniby summing up the contributions of all orbitals r. Since the polarizabilities have to match the experimental value for the free ion at zero field, an anisotropic effective ionic charge of the outer shells of the ion is assumed and fitted to fulfill this condition. This determines the coefficients α^∗_k,r,θ^∗_r and leads to the final expression for the polarizability of a constituent ion in the presence of a local electric field E~ [22]:

αkl(i) =α^exp(i)

δkl−θk(i) E²(i)δkl+ 2Ek(i)El(i)

. (7.7)

The next step is to calculate the local electric field E~ at the position of each ion i. This field depends on the properties and the arrangement of all other ions of the crystal and can be expressed as a superposition of the fields produced by all dipoles~pj in the crystal [6]:

E(i) =~ E~^ext+X The position ~r_i of each ion is calculated using structural data of the unit cell, including its size and the position of the ion within the unit cell, as well as the spontaneous ionic shifts s_k⁰ in the ferroelectric phase. These numbers are taken from the literature. Across the domain wall, the ionic shift makes a transition from positive to negative values. We assume that this transition can be described by the following function [19]:

s_k⁰( ˜m, j) =s⁰_k⁰(j)·tanh domain boundary. The parameter rc characterizes the thickness of the domain wall and s⁰_k0(j) is the ionic shift far away from the boundary. The calculation of the local electric field is iterated until a self-consistent solution is found. As the final result, the polarization Pkis calculated by averaging the dipole moment across one unit cell of volumev, and the optical coefficientsε^opt_kl0 are calculated as follows [19]:

P_k= 1

Figure 7.2 shows the results for the refractive-index profiles across a domain wall for different domain wall thicknessesr_c. For all these cases, the crystal shows three different refractive indices within the domain wall, that means the crystal gets biaxial while it is uniaxial far away from the domain wall. At the domain wall, the refractive index n_c, i.e.

for light polarized parallel to the spontaneous polarization, exhibits the biggest change, followed by n_b. The refractive index n_a is left almost unchanged. The shape of the refractive-index profiles is symmetric with respect to the center of the domain wall and depends strongly on its width rc. For the smaller values ofrc, the maximum of n_b at the center is flanked by two minima. With increasing rc, the profile acquires a Gaussian-like shape.

7.2. Theoretical description 79

Unit cell center x-position (Å) 2.26

refractive index n a, n b, n c

Fig. 7.2 Modelled refractive index profiles for a 180° domain wall of BaTiO₃ for different values of the domain wall thickness parameter rc. The index nc experiences the biggest change, followed by n_b, while na re-mains almost unchanged. In case of smaller values of r_c, the profiles of nc and nb show a small decrease at the periphery of the wall, be-side the stronger increase at the domain wall center. For larger rc, the profiles get Gaussian-like.

The refractive-index profiles, can be approximated by continuous functions, which will be used for estimations of experimental quantities in the following section. For this, the profiles of n²_k were fitted with up to two Gaussian function of different amplitude and width as stated in equation 7.14. The resulting functions are plotted in Fig. 7.2 and the corresponding fit parameters are listed in table 7.1.

n²_k=n²_k,bulk+A_k,1∗exp(−2x²/w²_k,1) +A_k,2∗exp(−2x²/w²_k,2) (7.14) Table 7.1 Parameters of fit functions describing the refractive index-profiles for

different values of the domain wall width rc. The parameters corre-spond to Eq. 7.14.

r_c [˚A] n_k A_k,1 w_k,1 [˚A] A_k,2 w_k,2 [˚A]

5 n_a -0.0083 16.16 -

-5 nb 0.0762 13.35 -0.0282 51.20 5 nc 0.4622 13.49 -0.1617 51.28 10 na -0.0092 24.53 0.0106 59.34 10 n_b 0.0648 24.88 -0.0211 61.94 10 n_c 0.3991 25.12 -0.1228 61.25 15 n_a -0.0063 28.38 0.0151 79.70 15 nb 0.0519 34.21 -0.0111 71.38 15 n_c 0.3233 34.55 -0.0647 68.83 20 na -0.0051 38.99 0.0209 95.10

20 n_b 0.0393 42.91 -

-20 nc 0.2439 42.23 -