REVISIÓN DE LOS CONTENIDOS DEL EOT DE ACUERDO A LO DISPUESTO EN LA LEY

As demonstrated later in this chapter a multitude of doping additions have been included in a HfO2-based material and several have been found to stabilise metastable crystalline

phases. There is some pattern to the successful stabilisers: Ce, Dy, Er and Gd are all lanthanides; Zr is a so-called chemical twin with Hf; finally Ba, Sr and Y surround Hf in the periodic table. This isn’t a precise grouping of dopants, and other subtle factors have to be taken into consideration, but considering dopant additions in this way helps to explain the solid solution phase changes. First-principles computational results by Chen et al. [19] indicate that Y doping in a HfO2 lattice provides better stability of

the cubic phase by decreasing the energy difference between the material supporting the monoclinic and cubic phases. The addition was also found to reduce the transition pressure between phases compared to the undoped material. The computational work by Lee et al. [20] develops a discussion of ‘type I’ and ‘type II’ dopants in a HfO2 matrix.

The paper discusses addition elements in the context of their ionic radii compared to the ionic radius of hafnium. A general trend was noted in that the smaller ionic radius type

Chapter 2. Dielectric materials 31

Figure 2.3: Schematic diagram of the energy paths for equilibrium and non-

equilibrium phase changes in a Ti0.5Hf0.5O2 thin film solid solution.

I additions (Si, Ge, Sn, P, Al and Ti in the paper) energetically favoured the tetragonal form of metastable HfO2while the relatively larger ionic radius type II additions (Y, Gd

and Sc) favoured the cubic phase. The following discussion in the paper related these discoveries to the difference in bond lengths between an addition’s oxide and the bonds it would have to form as part of a distorted HfO2 lattice. From this treatment Ce, one

of the dopants looked at in this thesis, would be expected to be a type II addition and so forming a metastable cubic phase would be expected.

Additionally the kinetics of nucleation have to be considered in a discussion on metastable phase change/retention, especially where an amorphous phase is retained making the reaction mechanism less transparent. Figure 2.3 shows a schematic diagram of the equilibrium and non-equilibrium reaction paths for a TiHfO4 alloy - adapted from

[21] to be relevant for a discussion of the materials systems examined in chapter 6. The diagram represents the post-deposition annealing of a Ti0.5Hf0.5O2 ALD film. The

Q maxima in the curves represent energy barriers to nucleation of a new phase while the ∆G values are free energy gains from each reaction. The asterisk is the reaction start-point while the dots represent potential phases formed in the reaction. For a 50:50 TiO2:HfO2 alloy the expected equilibrium phases are orthorhombic TiHfO4 with HfO2.

Chapter 2. Dielectric materials 32

The non-equilibrium phase depicted is an amorphous TiHfO4. The grey curve follows

the equilibrium reaction with an energy barrier Q1 to overcome before forming the equilibrium products with a net energy gain ∆G1. The energy change ∆G2 is less than ∆G1 by definition as the latter is the favoured phase energetically. The energy barrier Q1 is greater than Q2 as a comparatively higher nucleation rate would be expected for the equilibrium phase formation. Following the black non-equilibrium reaction path the curve has a second energy barrier, Q3, preventing the reaction continuing on to form the equilibrium phases. This barrier would have to be as high or higher than Q2 to prevent the reaction continuing to form the equilibrium phases.

One of the conditions for amorphous phase nucleation in solid solutions is that one of the elements is a fast diffuser compared to the other and the negative heat of mixing of the amorphous alloy formed [22, 23]. A difference in relative diffusion speed between two atomic species is often correlated with a difference in atomic radius between the two; the smaller atom being the fast diffuser. Table 2.1 presents ionic radii for relevant

Table 2.1: Effective ionic radii for Hf, O and relevant addition species. Effective ionic

radii are calculated from r3 vs V plots. From [24].

Ion Electron config. Coord. Crystal radius Effective ionic radius

Hf4+ 4f14 VIII 0.97 0.83 O2− 2p6 IV 1.24 1.38 Ce3+ 6s1 VIII 1.283 1.143 Ce4+ 5p6 VIII 1.11 0.97 Er3+ 4f11 VIII 1.144 1.004 Gd3+ 4f7 VIII 1.193 1.053 Ti4+ 3p6 VIII 0.88 0.74

species taken from the ‘revised effective ionic radii’ work by Shannon [24]. Depending which values for atomic/ionic radii are used in a treatment on lattice substitutions the numbers can vary significantly, but the proportions between the species we consider here are always within a few percent between the quoted sets of values. In table 2.1 the first two rows are the data for the matrix material HfO2, where Hf has VIII coordination and

O has IV coordination. The four ions in the middle block are ones that have been shown to substitute for Hf in the lattice to cause the retention of cubic HfO2 - we will assume

cubic rather than tetragonal, despite the structural ambiguities, due to the average κ

Chapter 2. Dielectric materials 33

Figure 2.4: Schematic diagram of a network former (a) and a network modifier (b)

in a HfO2 matrix. Bond lengths/angles are not drawn to scale.

ion of ∼17% difference in ionic radius. The last ion in the table is Ti4+ which has a smaller effective ionic radius compared to Hf4+. Although a smaller ion can potentially substitute in the lattice it might not promote the retention of a high-κphase as a result. Also energetically such an inclusion is balanced against Ti’s fast diffusive properties due to its size and hence network-modifying capabilities. Figure 2.4a shows schematically the substitution of Ce in a HfO2 lattice. Figure 2.4b demonstrates the effect of adding

a network-modifying addition in a high enough proportion to disturb the long range periodicity of the lattice. Additionally the thickness of the films is thought to be a factor when preserving an amorphous phase in a solid state reaction. At some critical film thickness the nucleation of the equilibrium phase is favoured [25]. The thin films described in the results chapters are in the thickness range to potentially promote the retention of such a non-equilibrium amorphous phase.

To summarise, in the addition of a species to produce a ternary oxide with HfO2

as the starting material the film thickness, size and valance of addition ion and rela- tive proportion are key parameters. Deposition temperature and post-deposition heat treatments are then of importance in influencing the type of material formed.

Chapter 2. Dielectric materials 34