3.2 The New Discretization Procedure

The expression (2.18,2.19) result from solving the implicit expression

$$\mathbf{J}^0 = \mathbf{J} + \mu^* \mathbf{J} \times \mathbf{B}$$ (3.1)

where $\mathbf{J}$ is the current density including the galvanomagnetic effects, and $\mathbf{J}^0$ is the current density for zero magnetic field.

The previous equation can be written in matrix notation

$$\mathbf{J}^0 = \mathbf{A}\mathbf{J}$$ (3.2)

with $\mathbf{A}$ being an operator in the three-dimensional space. Inverting the previous expression one gets

$$\mathbf{J} = \mathbf{A}^{-1} \mathbf{J}^0$$ (3.3)

thus $\mathbf{A}$ is an operator whose inverse applied to the zero magnetic field current $\mathbf{J}^0$ gives the deflected current $\mathbf{J}$. In spatial components, $\mathbf{A}$ is formed with the product between the magnetic vector and the Hall mobilities $\mu^*\mathbf{B}$. The symbol $\beta$ will be used to represent this dimensionless product, so $\mathbf{A}$ reads

$$\mathbf{A} = \begin{pmatrix}1 & \beta_z & -\beta_y \\ -\beta_z & 1 & \beta_x \\ \beta_y & -\beta_x & 1 \end{pmatrix}$$ (3.4)

$\mathbf{A}^{-1}$ can be expressed as

$$\mathbf{A}^{-1} = \frac{\mathbf{I}+\beta\beta^T+\beta\times}{1+\vert\beta\vert^2}$$ (3.5)

where $\mathbf{I}$ is the identity matrix, $\vert\beta\vert^2 = \beta_x^2 + \beta_y^2 + \beta_z^2$, and $\beta\times$ as shown in (2.6).

The explicit components of each vector in (3.3) must be replaced by the current projections, because this is the only available information that any discretization procedure provides. So, $\mathbf{J}^0$ is transformed to $\mathbf{J}^0_k$ where the subindex $k$ is the index of a simplex, a grid point which includes its neighboring points (See Figure 3.2). $\mathbf{J}^0_k$ has all the current projections from a $k$ simplex, from the point $i$ to the neighboring points $j$.

$$\mathbf{J}_k = \mathbf{A}^{-1} \mathbf{J}^0_k$$ (3.6)

Figure 3.2: Cubic representation of a
$k$-simplex with its Voronoi volume.

Because a $k$ simplex could have any number of neighboring points, $\mathbf{J}^0_k$ must hold the same number of current projections too. So, (3.3) must be modified accordingly. Expression (3.5) gives a 3 $\times$ 3 matrix unless an operation is introduced in such a way that it gives a square matrix with the same number of current projections of a $k$ simplex. This is done by introducing a transformation matrix $\mathbf{N}$ composed of the unitary vectors which are computed from the point $i$ to every neighboring point $j$.

$$\mathbf{N} = \sum_j n_{i,j}$$ (3.7)

Introducing the transformation matrix $\mathbf{N}$ in (3.5), and replacing in (3.6) one gets

$$\mathbf{J}_k = \frac{(\mathbf{I}+\mathbf{N}^T (\beta\beta^T+\beta\times) \mathbf{N}^{-T})\mathbf{J}^0}{1+\vert\beta\vert^2}$$ (3.8)

The above expression determines a matrix which combines $n$ current projections with the components of a magnetic field. However, the neighboring points $j$ of the point $i$ for every simplex $k$ can be arbitrarily placed in the simulation domain. Care must be taken in this step to obtain the proper weight of current projections with the magnetic field. This is done by means of a matrix multiplication with the $\mathbf{s}_k$ vector whose components are the inverse of the distances between the point $i$ and the neighboring points $j$.

$$\mathbf{s}_k = \sum_j \frac{1}{d_{i,j}}$$ (3.9)

Finally, the discretization procedure taking into account the magnetic field reads

$$\mathbf{J}_k = \frac{s_k^T (\mathbf{I} + \mathbf{N}^T(\beta\beta^T + \beta_\times) \mathbf{N}^{-T})\mathbf{J}^0}{1+\beta^2}$$ (3.10)

This implementation introduces extra entries in the Jacobian matrix, but it guarantees the existence of a solution if the magnetic field times the carrier mobility is much less than one. Also, the grid plays an important role in computing the solution, because the weights of the current projections with the magnetic field can generate large numbers which impact the properties of the Jacobian matrix. This discretization scheme was successfully implemented in MINIMOS-NT [19].