Figure 4.5: Bader charges qi (black solid line) for each atom of the DFT model of the oxidized titanium surface along with the classical charges calculated using the electroneg-ativity equalization method EEM (red dashed line).
second term the electrostatic interaction between different atomic sites. While the deter-mination of the values for the parameters χi andJi will be addressed in the next section, we anticipate here the good agreement between the Bader and EEM charges (shown by the dashed red line in Fig. 4.5). Although the modulus of the EEM charges is slightly smaller compared to the Bader charges, the charge distributions differ only by a constant of proportionality, revealing a good consistency between classical and quantum mechan-ical charges. This could have been indeed expected for fully oxidized TiO2 allomorphes, which present a strong ionic character, but was not foreseeable a priori for the thin oxide layer including variable oxidation states of the Ti species.
calcu-lations (for instance within the LOTF formalism [47]). This poses the problem of keeping the analytical potential form as simple as possible, while at the same time ensuring a good degree of transferability of the same form between different environments, such as between different crystal structures or amorphous oxides, and systems including variable oxidation states. In particular the variation of the atomic charges in the thin oxide layer will be a crucial point of our development, as described below.
4.2.1 Definition of the Potential Form
The hcp metallic region underneath the superficial oxide layer can be treated using a manybody potential suitable for metallic Ti. In this work we use a Finnis-Sinclair po-tential as proposed by Ackland [2]. Albeit simple, this FS popo-tential is able to reproduce structural and mechanical properties of bulk Ti in a satisfactory way. In the Ackland approach, the energy of each metal atom is given by
Eimet= 1 2
j
Vmet(r)fcmet(qi)fcmet(qj)−
˜
ρi (4.2)
where the sum runs over all atom indexes and
˜
ρi =
j
ρ(r)fcmet(qi)fcmet(qj). (4.3) The pair potential Vmet(r) and the density ρ(r) are parametrized using cubic splines:
Vmet(r) =
k
ak(rk−r)3Θ(rk−r) ; (4.4)
ρ(r) =
k
Ak(Rk−r)3Θ(Rk−r), (4.5) where Θ(x) is the step function. In Eqs. (4.2) and (4.3) we have introduced cut-off functions
fcmet(q) =
1, q < Qmet−Δ
1−(q−Qmet+ Δ)/2Δ, Qmet−Δ< q < Qmet+ Δ
0, q > Qmet+ Δ
(4.6)
that depend linearly on the atomic chargesq. These cut-off functions ensure that the Ti-Ti FS interaction continuously decreases and is eventually switched off when the charges on the titanium atoms increase (i.e. by entering in the oxide region of a Ti/TiOx interface).
As far as the interactions within the oxide layer are concerned, the basis for their definition is the agreement between quantum mechanical charges and classical charges obtained by purely electrostatic arguments, as described in the previous Section of this paper. After replacement of the partial charges of common ionic force fields by these EEM charges, our finding suggests that Ti oxides, including non-stoichiometric ultrathin oxide layers, can be well described by using only electrostatic (Coulomb) interactions between the atoms, together with a short ranged two-body potential:
Vijox(r) = 1 2
i=j
qiqj
r + 1 2
i=j
VijSR(r)·fcox(qi, qj), (4.7) where r is the distance between atoms i and j and
VijSR(r) =Aijexp(−r/ρij)−Cij
r6 (4.8)
is a Buckingham potential including dispersion and repulsive interactions. This is the same potential form that has been used by of Matsui and Akaogi (MA) to model TiO2
polymorphs,[119] except for the cut-off function
fcox(qi, qj) = 1−fcmet(qi)fcmet(qj), (4.9) if i andj both correspond to Ti atoms, whereas fcox = 1 otherwise. This factor gradually switches off the oxide-related Ti-Ti interactions while entering in the metallic region of the system. Thus, our combined potential (equations 4.2 and 4.7) continuously changes from the metal region to the oxide layer and reduces to the Ackland potential or the MA potential form in the case of bulk Ti or bulk TiO2, respectively.
4.2.2 Definition of the Potential Parameters
The original MA parametrization was developed to model bulk TiO2 polymorphs and includes partial charges qT i = 2.196 and qO = −1.098, the modulus of the latter being only slightly smaller than the Bader charges on O atoms as computed in the previous Section. Taking into account the previously highlighted correspondence between quantum and classical charges, we can adapt the MA parameters to model oxidized Ti surfaces.
Consistency with the original MA parametrization can be ensured by choosing the atomic parameters χi and Ji in Eq. 4.1 so as to obtain the qT i and qO MA charges when minimizing the EEM electrostatic energy of the bulk TiO2 rutile structure (see
Table 4.1: Values of the atomic electronegativity parameter χi and the atomic hardness parameter Ji used in the electronegativity equalization method.
Ti O
χI [V] 0.0 8.729 JI [V/e] 12.864 17.197
Table 4.1). With this values of χi and Ji we can now compute the atomic charges for any arbitrary oxide structure, in particular containing multiple formal Ti valencies. As already mentioned in section 4.1.3 the EEM charges obtained for the oxidized Ti(0001) surface show very good proportionality to the corresponding Bader charges (see Fig. 4.5).
Although the EEM parameters are commonly fit to atomic ionization energies rather than to ionic charges, the fact that the resulting charges are consistent with the Bader charges of the oxidized surface justifies our way of determining them, and enables us to use the original MA set of parameters also for the short-term potential ( Eq. 4.8).
We have also tested an alternative parametrization, in which we choose not to include any dispersion interaction in the Buckingham potential (Cij = 0 for both Ti and O) and to remove all non-Coulomb Ti-Ti interactions in the oxide (AT i−T i = 0). The remaining parameters are obtained by a fit to the lattice and elastic constants of bulk rutile, again keeping the original MA values for qT i and qO. In this case, the value of AT i−O are significantly higher than in the previous parametrization (see Table 4.2). This leads to steeper Ti-O repulsion and thus to a sharper potential minimum than in the previous case, as visible from the corresponding Vox(r) curves for 3 different charges qT i = 2.196, qT i = 1.5 andqT i = 0.5 (at constant oxygen charge qO =−1.098) (Figure 4.6).
The interactions in the metallic region were again described by the FS-potential of Ackland, after adjusting the values Qmet and Δ to reproduce the DFT Ti-Ti and Ti-O bond lengths across the Ti/TiOx interface. As we will see below, with this additional parametrization the structure of the oxide layer obtained by FPMD is slightly better reproduced than in the previously described “MA-like” parametrization, and has the advantage of avoiding the calculation of a number of interactions.
Table 4.2: The two parameter sets for our Ti-O potential, and the parameter set for the FS Ti potential.[2] The lattice parameter for titanium was set to a= 2.94 ˚A. The values marked with ∗ are the original MA parameters.[119]
Buckingham Potential
“MA-like” parameters Alternative parameters
A˜IJ [eV] ρIJ[˚A] CIJ[eV] A˜IJ[eV] ρIJ[˚A] CIJ[eV]
Ti-Ti 31120.429 ∗ 0.154 ∗ 5.247 ∗ 0.000 0.000 0.000
O-O 11782.847 ∗ 0.234 ∗ 30.222 ∗ 85.164 0.489 0.000
Ti-O 16957.656 ∗ 0.194 ∗ 12.593 ∗ 7211646.2 0.115 0.000 Charge dependencies
Qmet [e] 0.8 1.45
Δ [e] 0.1 0.25
Finnis-Sinclair Ti Potential
k Ak[eV a−32−3/2] Rk[a21/2] ak[eV a−32−3/2] rk[a21/2]
1 39.795927 1.22 -57.099097 1.22
2 -40.061305 1.05 80.735598 1.20
3 -21.761468 1.12
4 -10.396479 0.95
5 74.515028 0.80
6 35.921024 0.707107