6.2.3 Examples and Results

As example a typical on-chip spiral inductor structure as discussed in [113] is investigated. The simulation domain consists of a transparent insulating rectangular brick over an opaque substrate brick as shown in Fig. <6.12>. The aluminum inductor is placed in the insulating environment about $5 {\mu}$ m above the substrate area. The substrate is modelled as region with a constant relative low resistivity of $10 {\Omega} cm$ . Thus the induced electric voltage in this region causes relative law electric current. The cross-section of the conductor is $20 {\mu}$ m $\times 1.2 {\mu}$ m. The horizontal distance between the winding wires is $10 {\mu}$ m. The outer dimensions of the inductor are $300 {\mu}$ m $\times 300 {\mu}$ m. The inductor is completely surrounded by the dielectric environment, except of the two small delimiting faces which lie directly in the boundary planes of the simulation domain. The conductor area, the dielectric, and the substrate area close to the conductor are discretized much finer then the remaining simulation domain. This is shown in Fig. <6.13> where a part of the dielectric environment is removed to visualize in detail the generated mesh inside the simulation domain. The variation of the fields in the finer discretized areas is expected to be much higher than in the coarser discretized domain. This special discretization reduces the number of generated nodes and edges, and the number of the linear equations respectively, even for big simulation environments which have to be used to satisfy the assumption of homogeneous Neumann boundary conditions (the tangential component of $\vec{E}$ on $\mathcal{A}_{N1}$ and normal component of $\vec{B}$ on $\mathcal{A}_{N2}$ are zero). Of coarse such a discretization is only possible, if an unstructured mesh is used.

The current density distribution depends heavily on the operating frequency in the analyzed frequency domain. It is unknown and arises from the simulation. At the beginning of the simulation only the total current in the inductor is known. As mentioned above it is set by the Dirichlet boundary condition for $\vec{H}_1$ which is given by the tangential component of the magnetic field $H_t$ on the element edges, surrounding one of the conductor faces lying on the outer bound of the simulation domain.

Table 6.2: Calculated inductance and resistance.

$f$ [GHz]	$L$ [nH] without	$L$ [nH] with	$R [\Omega]$
	substrate	substrate
0.001	2.6887	2.6881	3.127
0.01	2.6887	2.6877	3.127
0.1	2.688	2.688	3.132
1	2.6516	2.6514	3.463
10	2.5501	2.5493	5.396
100	2.5458	2.5457	13.156

The resistance and inductance values of the structure of interest are calculated numerically at different frequencies with and without the substrate influence. The corresponding results are presented in Table 6.2. While the inductance decreases slowly with increasing operating frequency, the resistance rises dramatically, which matches well the observed current density distribution and the skin effect, respectively. A surface view of the current density distribution in the conductor is shown in Fig. <6.14> and Fig. <6.15> for $100$ MHz and $10$ GHz, respectively. At $100$ MHz the skin depth is about $6 {\mu}$ m and nearly the whole conductor cross-section is filled up by the current. At $10$ GHz the skin depth is about $0.6 {\mu}$ m and the current is concentrated at the vertical side walls of the conductor. Fig. <6.16> shows the spatial current density distribution at $1$ GHz as directed cones placed in the discretization nodes inside of the conductor area. The cone's size and darkness are proportional to the field strength. Fig. <6.17> depicts the corresponding spatial distribution of the magnetic field inside the dielectric environment around the inductor. Fig. <6.18> and Fig. <6.19> show the very different current density distribution close to the small via for $100$ MHz and $10$ GHz, respectively. Fig. <6.20> and Fig. <6.21> show the current density distribution and the corresponding magnetic field in the substrate at $1$ GHz. The underlying substrate does not influence the inductance and the resistance of the inductor, because of the relative high substrate resistivity. The values of the current density in the substrate (Fig. <6.20>) differ vastly to those in the inductor (Fig. <6.14> and Fig. <6.15>). As shown in Table 6.2 the calculated inductance taking into account the influence of the substrate does not differ from the inductance without substrate influence.

As the Q-factor of an inductor is inversely proportional to its resistance, making the inductor wire thicker might decrease the resistance and increase the Q-factor. However, as the examples show this is not the case for high frequencies at which the skin effect is noticeable. In these cases the current flows only in the area very close to the vertical surface and a wider transversal conductor cross section would not change the situation. For the visualization VTK [114,115] is used.

Figure 6.12: The simulation domain.

Figure 6.13: The generated mesh.

Figure 6.14: Surface view of the current density [A/m$^2$ ] distribution at $100$ MHz.

Figure 6.15: Surface view of the current density [A/m$^2$ ] distribution at $10$ GHz.

Figure 6.16: Current density distribution at $1$ GHz.

Figure 6.17: Magnetic field distribution [A/m] at $1$ GHz.

Figure 6.18: Current density distribution in the via at $100$ MHz.

Figure 6.19: Current density distribution in the via at $10$ GHz.

Figure 6.20: Current density [A/m$^2$ ] distribution in the substrate at $1$ GHz.

Figure 6.21: Magnetic field [A/m] in the substrate at $1$ GHz.