Theoretical models developed and applied to evaluate resis-

The measured resistance data correlate with the volume fraction of the formed mag-nesiumdihydride. As discussed in section, 2.1.3 MgH2 is (in bulk) an insulator with a much higher resistance. The specific resistance ρ_{M gH2} ≈ 10²µΩm of hydrided MgH2 thin films [104] was found to be much lower than that of grown MgH2 thin films (ρ > 10⁻¹⁰µΩm [101]). Yet, the specific resistance of hydrided thin films is

still higher than the specific resistanceρ_{M g} = 65µΩmof Mg thin films [104]. There-fore the increasing resistance found in magnesium films during hydrogen loading is assumed to originate from the forming hydride. Using the nucleation and growth model introduced by H. Uchida [44] it can be assumed that after the initial nucle-ation a closed blocking layer forms which grows further into the film. This means three layers exist above the silicon substrate: magnesium of decreasing thickness d_{M g}, magnesiumdihydride of increasing thickness d_{M gH2} and palladium of a fixed thickness d_{M g}. The overall thickness is d=d_{P d} +d_{M gH2}+d_{M g}. It is assumed that these three layers are contacted in parallel by the four-point measurement. Even if the actual connection is only on the Pd layer this can be assumed, because the distance between the four contacts of the setup is much larger than the distance through the layers. This means that the resistance between the contacts should be much larger that the resistance connecting the three layers.

The overall resistanceRas a combination of resistances of the three layers in parallel contact is:

The resistance can be expressed as R = ρ_A^l using the specific resistance ρ, the distance between the contacts l and the area A through which the current travels.

The specific resistance of the palladium is equal to ρP d ≈ 150 Ωnm [207]. For the simple geometry applied here the area can be simplified as the film thicknessd and the distance between the contacts should be almost equal for all resistors in equation 3.11. This allows the following formulation of equation 3.11:

d

Before loading the equation simplifies to:

d₀

Combining equation 3.12 and 3.13 leads to:

R

As discussed in chapter 2.1.2 the MgH2 expands in out-of-plane film direction as its unit cell is larger than the unit cell of the Mg. Therefore,d≥d₀ is true for all times.

A is one at timet = 0and afterwards increases. The volume fraction φ_{M gH2} of the magnesiumdihydride at a given time is defined in the following, as:

φM gH2 = d_{M gH2}

The final inequality is true because of the expansion of d with time as discussed above. As A is not known at a given time the volume fraction φ_{M gH2} is calculated with A = 1, which underestimates the true volume fraction.

A more realistic volume fraction of the MgH2 would be φ^real_{M gH2} =d_{M gH2}/(d−d_{P d}). However, this can not be calculated from the measurements in this work as dis not known at each point of time. Only d₀ (the initial thickness) and d∞ (the thickness after loading) are known. However, because d ≥ d₀, φ_{M gH2} will become smaller than φ^real_{M gH2} at some point in time ². It should be noted that this is only true because ρ_{M g} is similar to ρ_{P d}. As discussed above the volume fraction φ_{M gH2} is underestimated in this work because A = 1 is assumed for calculations. Using φM gH2 instead of φ^real_{M gH2} only enhances this difference. This is important to keep in mind, but does not hinder the data evaluation. First of all, the volume fraction is studied to evaluate the limiting kinetic process during hydrogen loading of the Mg films (see also section 2.3). Here, the shape of the curve is important, which is distorted by the increasing underestimation of the true volume fraction. However, the shape of curves that follow different limiting processes differ strongly, so that they can be separated under non-ideal conditions. It may be difficult to extract true kinetic parameters as e.g. the diffusion coefficient, but this is not the goal in this step. The second important use of the volume fraction is the diffusion length, which is calculated from it by multiplying the volume fraction with dV F (see equation 3.14). This means that the diffusion length is also underestimated leading to an underestimation of the resulting diffusion coefficient. This is included in the error of the diffusion coefficients by using an error of the diffusion length of25 %. This error includes the general underestimation, as well as the fact that the model assumes a perfect straight layer. In reality, the layer thickness probably has some dependency on the position in the film. In general, all diffusion coefficients in this work may be slightly higher than the calculated values.

Giebels et al. introduced a model to calculate the time dependent electrical proper-ties [104] in a mixture of two materials. It bases on the Bruggeman effective medium approximation and was also applied by Qu et al. [140]. The basic equation is given by Qu et al:

Figure 3.8:

U is a geometrical factor and equals 1/3 for nucleation of spherical hydrides. This can be transformed to: Because ρM gH2 >> ρM g the equation further simplifies to:

φ_{M gH2} =U ρ

ρ_{M gH2} + (1−U)ρ_{M g}

ρ + (1−U). (3.16)

ρ ≈ R/d can be estimated from the measured resistance. d is the thickness of the combined Mg and MgH2 layer. This assumes that the lengthl between the contacts is similar to width b. The width forms, together with the thickness d, the area A = b ·d through which the current has to travel. This model does not include the Pd capping layer. However, the measured resistance may be corrected for this by using equation 3.11 and R_{P d} ≈ ρ_{P d}/d_{P d}. The resistance contribution of the palladium layer can be subtracted and afterwardsρ of the combined Mg and MgH2

layer can be calculated.

Figure 3.8 shows a comparison of the volume fraction φ_{M gH2} calculated by the three-layer model introduced above and by the model of Giebels et al. (including the correction for the Pd capping layer). Because of the approximations done above, the model of Giebels et al. gives a smaller volume fraction φ_{M gH2} than the three-layer model for all times. This is an important result, because, as discussed above, the three-layer model already underestimates the true volume fraction. Therefore, the three-layer model seems to be a better choice to calculate the volume fraction of MgH2 in this work. In the following all volume fractions φ_{M gH2} are therefore calculated by the three-layer model outlined above.