Diffusion kinetics in thin film structures

It is necessary to point out that the real driving force for diffusion is not the concentration gradient, nor the mobility of a certain phase, but rather the chemical potential gradient between the phases in contact with each other. Components always diffuse down the chemical potential gradient which in some cases could actually be against the concentration gradient (uphill diffusion), for example in phases exhibiting a miscibility gap. Therefore, diffusion kinetics based on Fick’s laws can sometimes be limiting in some applications. The interconnection between thermodynamics and kinetics is always strongly evident in chemical reactions, either in diffusion couples or thin films. The diffusion kinetics in thin films is however more complicated since the mass transport can be heavily influenced by other microstructural factors such as lattice/topographic defects, interfacial transport and stresses.

As mentioned above in section 1.3, according to Zotov et al. [12], vacancy diffusion best describes the atomic mechanism of mass transport in a wide range of miscible metallic multilayer thin films.

Therefore, any regions with high vacancy concentrations mediate diffusion to reach higher rates, such as GBs, voids, and topographic defects, as well as phase boundary interfaces which have strong disturbance in the lattice symmetry and can act as sink for diffusion [46]. Moreover, interfaces and lattice defects act as sources and sinks for vacancies which intensifies their contributions in the mass transport [30].

1.7.1. Interdiffusion in bilayer films

In the continuum approach of treating diffusion, the interdiffusion coefficient between two metals with a wide homogeneity range can be calculated from the compositional profile across the phase local thermal equilibrium of vacancies (sinks and sources) across the interface. On the other hand, the interdiffusion coefficient, , based on the Nerst-Planck (same as the Nazorov-Gurov method), offers a more detailed description on the mesoscopic scale for the interdiffusion, which can be influenced by non-equilibrium vacancy distribution across the interface that affects the mobility of both phases [29]. It is important to note that in the case when the mobility of both phases in contact vary significantly, leading to the formation of asymmetric concentration profiles. In these conditions, the balance in the effective sources and sinks created across the interface, which assist in the migration of atoms, rapidly vanishes, and in some diffusion couples, it results to the appearance of voids close to the interface. This is known as the Kirkendall effect [28]. Strong variations in the mobility of reactant phases together, as well as their diffusion rates in the compound phases plays a

vital role in the concentration gradients between the phases [54] as well as the growth rate of intermetallic phases.

As mentioned in sections 1.3 and 1.4.2, in thin films, the slow atomic exchange mechanism of lattice diffusion can sometimes be considered almost ‘frozen-in’, where lattice defects in the microstructure dominate the interdiffusion between two phases. The contribution of the transport along fast-diffusion paths is highly dependent on the microstructure, and as discussed in section 1.6, it can be also dependent on the period thickness in a multilayer structure.

1.7.2. Grain boundary diffusion and kinetic regimes

In polycrystalline materials, a hierarchy of fast diffusion paths can be found in the microstructure which plays a key role in interdiffusion of thin films. Activation energies for GB diffusion, QGB, are typically about 0.4-0.6 the activation energies for volume diffusion (lattice), QV, leading to GB diffusion rates reaching four to six orders of magnitude higher compared to volume diffusion [28].

Diffusion rates in low-angle grain boundaries (LAGB) are normally slightly slower than GB diffusion, due to their more constricted open structure, which can be similar to diffusion rates found at dislocations. Diffusivities in high angle GBs can also vary according to the structural characteristics of the GBs (misorientation and tilt angles). Thus, random GB networks in thin films can exhibit fast and slow diffusion pathways, which was recently proved experimentally [11]. The fastest type of GBs diffusion is along triple junctions (TJs), where higher vacancy concentrations exist due to their more open structure, where diffusivities can exceed high angle GBs by up to three orders of magnitude [55]. TJs therefore serve as a special type of GB diffusion. Moreover, regarding fast diffusion paths, depending on the roughness in thin films, especially multilayers, topographic defects can build up (discussed in section 1.3), which are principally agglomerations of voids within the structure. The diffusion in these pathways can be comparable that to surface diffusion (or slightly less), in which activation energies for surface diffusion, Q_s, are about 0.1-0.3 activation energies for volume diffusion [28]. Hence, the diffusion rates in topographic defects (D_TD) are higher than diffusion rates in any lattice defects within the microstructure. The hierarchy of diffusion rates is:

(31)

GB diffusion can exhibit different regimes where the ratio between the contribution of the lattice and GB diffusion can vary. A certain regime can dominate depending on the annealing temperature,

annealing time, grain size, as well as the characteristics of the lattice and GBs [23]. The dependency of the regime on the grain size is a very important aspect, which is of great interest in the work here, since grain sizes vary with the period thickness in multilayer thin film structures which should therefore influence the GB kinetic regime. The different GB kinetic regimes, according to Harrison’s classification [23], [56], can be defined as follows:

i. Type A diffusion

In this regime, lattice diffusion has a significant contribution in the effective diffusion, and the diffusion front appears almost planar with slight penetrations at the GB regions. This regime is generally observed at high annealing temperatures, and/or long annealing times, and/or materials with fine grains. The diffusion length for lattice diffusion (D·t^1/2) is usually larger than or equal to the spacing between GBs (d). Since GBs exhibit segregation of the solute elements during diffusion, the actual thickness is therefore defined by the product of the segregation factor and the initial GB thickness, which equal, s·δ. The actual thickness of the segregated GBs is much smaller than the lattice diffusion length, hence, in type A boundary into the interior of the grains, where the GB acts as a source for such diffusion path.

This regime is usually observed after annealing at low temperatures, and/or short annealing

In this regime, lattice diffusion is theoretically absent, where diffusion only proceeds along the GBs, without leakage of atoms into the grain interiors. Therefore, no GB diffusion tails exist. This regime is usually observed at low temperature anneals, and/or very short annealing times, and/or nanostructured crystalline materials. The lattice diffusion length is therefore smaller than the segregated GB thickness:

(34)

Since the diffusion here is only localized at the GBs (δ ~ 0.5 nm), diffusion studies using tracer elements or radioactive isotopes are very difficult to be carried out (limited diffuser amount to be detected). However, this regime is very common in thin film structures [39], which emphasizes the key role of GBs in the interdiffusion in thin films. The different types of GB regimes are illustrated in Figure 10.

Figure 10 Illustration for the main diffusion regimes according to Harrison’s classification [23] in five grains positioned parallel to the diffusion direction. The red color represents the diffuser element.

1.7.3. Effective diffusivity

The volume fraction of GBs in the microstructure therefore influences the contribution of the GB diffusion to the total diffusivity in the material, which is referred to as the effective diffusivity. The effective diffusivity for solute atoms can be estimated by the Hart-Morlock relationship [57] for a simple microstructure with GBs in parallel direction with the diffuser (as seen in Figure 10) as:

(35)

where s is the segregation factor for solute atoms and X_GB is the volume fraction of GBs.