2.3 Material Properties

The bulk properties of materials used in today's Si-based semiconductor devices like conductivities and permittivities are well known. Due to the steady enhancements over the last decades and because the material behavior has changed in terms of purity, doping, and microstructure, the previously well known values of the material properties have to be reconsidered to appropriately account the recently observed discrepancies. However, the values of the bulk properties are still a good approximation of the real values and are thus often used in material models. Once the materials become thinner than a certain threshold size the behavior changes and the material can no longer be considered a macroscopic object. In this case new models have to be applied, which differ from the well known standard bulk models. When the material dimensions are above the threshold value, the new models are required to converge to the well known bulk models.

For anisotropic materials the current density ${\mathbf{{J}}}$ is proportional to the electric field ${\mathbf{{E}}}$ by a tensor-valued quantity as

$${\mathbf{{J}}} {=}{{\tilde{\sigma}}} \cdot {\mathbf{{E}}},$$ (2.120)

where ${{\tilde{\sigma}}}$ is the electrical conductivity tensor. In most cases the material can be assumed to be isotropic and so the conductivity can be assumed scalar-valued. Hence, (2.120) reads

$${\mathbf{{J}}} {=}\sigma {\mathbf{{E}}}.$$ (2.121)

This assumption is valid as long as the material has no priority directions as can be observed for instance in cubic-crystallized materials. The historical definition of $\sigma$ (cf. [58,112]) includes averaged values as

$$\sigma {=}\frac{n_{\mathrm{e}} {q}^2{\lambda_{\mathrm{MFP}}}}{2m_{\mathrm{e}}v_{\mathrm{m}}}$$ (2.122)

where the density of free electrons $n_{\mathrm{e}}$ , the length of the mean free paths of electrons ${\lambda_{\mathrm{MFP}}}$ , and the average mass of the electrons $m_{\mathrm{e}}$ are material-dependent. Furthermore, $q$ denotes the elementary charge and $v_{\mathrm{m}}$ the thermal velocity of the electrons. Moreover, it is obvious that the average values of the length of the mean free paths, the thermal velocity, and the density of free electrons are temperature-dependent due to the thermal expansions of the material and the resulting mechanical stresses as well as thermally induced diffusion. Therefore, the commonly given values for the electrical conductivity are only valid within a certain temperature range.

When the layer thickness of the material is reduced to a thin film of only several atomic layers, the electrical conductivity becomes anisotropic. This also happens when grains determine the current flow inside polycrystalline materials. Due to considerable research efforts in material science and in semiconductor process technology [113,31,114,115] most of the materials can be appropriately deposited in such a way that the anisotropic part of the tensors becomes negligible for many applications [116].

To deepen the understanding of the material behavior and to improve reliability, accelerating tests [117,118,119,33] have been introduced, which stress the materials at high temperatures, causing aging effects within a very short time period. This procedure enables life-time tests at the very beginning of the life cycle and at very low costs costs compared to field experiments.