2.2.8 Mechanical Subsystem

This thesis deals with mechanical phenomena mainly caused by electro-thermal stress conditions. Since the electrical burden produces heat and the heat non-negligible volume expansion, the mechanical part has to be considered as well. The basic equation used for TCAD purposes is HOOKE's^2.27 law which has been originally introduced by the words ``Ut tensio sic vis''^2.28. The corresponding formula reads

$$F {=}c u,$$ (2.107)

where the absolute value of the applied force $F$ to a body is proportional to its elongation $u$ . Here the constant $c$ determines the stiffness of the body. More generally, HOOKE's law can be formulated for local quantities in a body where the local stress tensor ${\tilde{{\mathbf{{\sigma}}}}^{\mathrm{mech}}}$ is associated to the GREEN^2.29 tensor (local strain tensor) ${\tilde{{\mathbf{{\varepsilon}}}}^{\mathrm{mech}}}$ for a given body by

$${\tilde{{\mathbf{{\sigma}}}}^{\mathrm{mech}}}{=}\tilde{C} \cdot {\tilde{{\mathbf{{\varepsilon}}}}^{\mathrm{mech}}},$$ (2.108)

$${\sigma^{\mathrm{mech}}}_{ij} {=}\sum_{k,l} C_{ijkl} {\varepsilon^{\mathrm{mech}}}_{kl},$$ (2.109)

where the proportionality factor is determined by the $4^{\mathrm{th}}$ -rank stiffness tensor $\tilde{C}$ and the strain is defined according to CAUCHY^2.30via local displacements

$${\varepsilon^{\mathrm{mech}}}_{ij} {=}{{\frac{1}{2}}}\left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right),$$ (2.110)

where ${\mathbf{{u}}}$ is the displacement or deformation vector and ${\mathbf{{x}}}$ the local position.

Using the VOIGT^2.31 notation [96,97,98], the ranks of the tensors involved in (2.108) can be reduced due to the symmetry of the material and due to the symmetry according to energy conservation laws [59]. Thus, the number of independent tensor entities reduces from $81$ to $36$ by material symmetry and further reduces to $21$ mutual independent tensor entities due to energy conservation [96,59]. Therefore, equation (2.108) can be expressed as

$${\widehat{{\mathbf{{\sigma}}}} ^ {\mathrm{mech}}}{=}\hat{C} \cdot {\widehat{\varepsilon} ^ {\mathrm{mech}}},$$ (2.111)

where ${\widehat{{\mathbf{{\sigma}}}} ^ {\mathrm{mech}}}$ and ${\widehat{\varepsilon} ^ {\mathrm{mech}}}$ are the vector-valued quantities for the mechanical stress and strain in the VOIGT notation, respectively. Furthermore, $\hat{C}$ represents the stiffness matrix of $2^{\mathrm{nd}}$ rank also in the VOIGT notation.

In TCAD applications of modern devices it is often sufficient to deal with static stresses, only. In that cases, the speed of involved particles can be neglected [99]. The mechanical equations have to fulfill general conservations laws [96,97] for energy, momentum, angular momentum, and mass. Thus, the mechanical subsystem can be described by the local conservation laws of energy, momentum, and mass

$${{\partial}_{t}}{w} + {{{{\boldmath {\nabla}}}}\cdot}{{\mathbf{{q}}}} {=}{\mathbf{{S}}}^{\mathrm{mech}},$$ (2.112)

$${{\partial}_{t}}{{\mathbf{{g}}}} + {{{{\boldmath {\nabla}}}}\cdot}{{\mathbf{{{p^{\mathrm{mech}}}}}}} {=}{\mathbf{{f}}},$$ (2.113)

$${{\partial}_{t}}{{\varrho^{\mathrm{mass}}}} + {{{{\boldmath {\nabla}}}}\cdot}{\left( {\mathbf{{v}}} {\varrho^{\mathrm{mass}}}\right)} {=}0.$$ (2.114)

In (2.112) the local energy density is denoted by $w$ , the energy flux density by ${\mathbf{{q}}}$ , and ${\mathbf{{S}}}^{\mathrm{mech}}$ represents the mechanical power density. The latter equation represents the mechanical analogon of the POYNTING vector. Equation (2.113) is the momentum conservation equation, where ${\mathbf{{g}}}$ is the momentum density, ${\mathbf{{f}}}$ is the local force density, ${\mathbf{{v}}}$ the velocity of the moving particles, and ${\tilde{p}^{\mathrm{mech}}}$ is the momentum flux density which is often called pressure tensor. Equation (2.114) presents the local mass continuity equation, where the specific mass density is denoted as ${\varrho^{\mathrm{mass}}}$ . If mass fluxes have to be considered, for instance in electro-migration analysis, the kinetic pressure tensor ${\tilde{p}^{\mathrm{mech}}}$ becomes

$${\tilde{p}^{\mathrm{mech}}}{=}{\varrho^{\mathrm{mass}}}{\mathbf{{v}}} \otimes {\mathbf{{v}}} - {\tilde{{\mathbf{{\sigma}}}}^{\mathrm{mech}}},$$ (2.115)

where the specific mass density is denoted by ${\varrho^{\mathrm{mass}}}$ , ${\tilde{{\mathbf{{\sigma}}}}^{\mathrm{mech}}}$ is the stress tensor, and ${\mathbf{{v}}}$ is the speed of the moving particles.

Later on, ${\tilde{p}^{\mathrm{mech}}}$ can be also used as a scalar-valued quantity ${p^{\mathrm{mech}}}$ when the simplified VOIGT notation is used:

$${p^{\mathrm{mech}}}{=}{{\mathop\mathrm{trace}}({\varrho^{\mathrm{... ...{v}}} \otimes {\mathbf{{v}}} - {\tilde{{\mathbf{{\sigma}}}}^{\mathrm{mech}}})}.$$ (2.116)

If the flux of mass has not to be considered, the associated velocity of the particles becomes ${\mathbf{{v}}}{=}0$ and the hydrostatic pressure can be determined by

$${p^{\mathrm{mech}}}{=}- {{\mathop\mathrm{trace}}({\tilde{{\mathbf{{\sigma}}}}^{\mathrm{mech}}})}.$$ (2.117)

The definitions of hydrostatic pressure ${p^{\mathrm{mech}}}$ in (2.116) and (2.117) can be used as a metric which provides a possibility to visualize, to compare within measurements, or to define a figure of merit in an optimization loop [100].

If moving particles are considered, the mechanical analogon to the electrical continuity equation is the mass continuity equation (2.114) and can be treated with the EULERian^2.32 continuity equation [96]

$${{\partial}_{t}}{{\varrho^{\mathrm{mass}}}} + {\varrho^{\mathrm{m... ... + {\mathbf{{v}}} \cdot {{\boldmath {\nabla}}}{{\varrho^{\mathrm{mass}}}} {=}0,$$ (2.118)

which is the mass conservation equation (2.114). A mathematical coupling between the mass flux and the mechanical stress can be obtain by using the first law for continuity mechanics from CAUCHY

$${\varrho^{\mathrm{mass}}}\left( {{\partial}_{t}}{{\mathbf{{v}}}} ... ...abla}}}}\cdot}{{\tilde{{\mathbf{{\sigma}}}}^{\mathrm{mech}}}} + {\mathbf{{f}}},$$ (2.119)

where the ${\mathbf{{f}}}$ represents the externally applied force density.