4.3.10 Examples

This section depicts several examples to show the usability of the introduced device simulator approach. The examples are chosen to depict the multi-dimensional support as well as the device template mechanism, the different simulation problems, and the stepping facility. Implementation details as well as simulation results are shown.

One-Dimensional Capacitor

This section discusses a one-dimensional capacitor device, solving the Laplace problem (Section 4.3.6). This particular case has been chosen to depict the support for one-dimensional devices, usually required for developing and debugging more advanced models. Therefore, this rather trivial device is required to be supported by every device simulator, before delving into more complicated models.

The device consists of five segments; two metal contact segments are attached to either side of a silicon dioxide-silicon-silicon dioxide (SiO-Si-SiO) structure, both assigned as Dirichlet contacts. As the implementation of the Laplace problem⁴ keeps the permittivity on the left side of the equation (Section 4.3.6), the potential reflects the transition between the materials, as shown in Figure 4.16. The potential drops more significantly in the oxide segments than in the middle semiconductor segment.

Figure 4.16: The potential (V) over the spatial dimension (m) of a one-dimensional capacitor is depicted. Dirichlet boundary conditions have been applied to the left (V) and the right (V) metal contact. Note the potential transitions at the material interfaces due to the different relative permittivities of SiO and Si.

Two-Dimensional PN Diode

This section shows the simulation of a two-dimensional pn-junction diode. The DD problem (Section 4.3.6) is solved for a set of contact potentials. This particular example has been chosen to depict the support for two-dimensional devices as well as the evaluation of device characteristics.

The device consists of four segments, where two metal contact segments are attached to either side of a p-Si-n-Si structure (Figure 4.17). The p-Si offers a constant donor and acceptor doping of cm and cm, respectively. The n-Si offers a constant donor and acceptor doping of cm and cm, respectively.

The device characteristics is computed by applying a constant cathode contact potential by simultaneously varying the anode contact potential, ranging from V to V, with a stepsize of V (Figure 4.18). In forward and reverse operation a maximum current of A and nA is computed, respectively. Figure 4.19 depicts the computed potential distributions for the reverse, equilibrium, and forward case. In the forward case, the polarity of the anode contact is switched. Figure 4.20 depicts the computed electron concentration distributions for the reverse, equilibrium, and forward case. Where in the reverse case, the electrons retract toward the cathode contact, in the forward case the electrons are distributed over the entire device. Figure 4.21 depicts the computed hole concentration distributions for the reverse, equilibrium, and forward case. Where in the reverse case, the holes retract toward the anode contact, in the forward case the holes are distributed over the entire device.

Figure 4.17: The setup of the two-dimensional pn-junction diode; Each color denotes a different device segment. The p-Si and n-Si offer a constant acceptor and donor doping of cm, respectively.

Figure 4.18: The IV-characteristics of a two-dimensional pn-junction diode; the contact potential of the anode contact has been gradually increased from negative to positive voltages relative to a constant cathode potential (V). For forward bias (positive potential values) the diode is conductive, whereas for negative bias (negative potential values) the diode is non-conductive. Note the current saturation (V) induced by high injection effects.

(a) Reverse (b) Equilibrium (c) Forward

Figure 4.20: The electron distributions (cm) in reverse (left), equilibrium (middle), and forward (right) mode of a two-dimensional pn-junction diode are shown. The contact segments have been removed to ensure proper color mapping. Where in reverse mode the electrons retreat towards the cathode, in forward mode the electrons populate the entire device.

(a) Reverse (b) Equilibrium (c) Forward

Figure 4.21: The hole distributions (cm) in reverse (left), equilibrium (middle), and forward (right) mode of a two-dimensional pn-junction diode are shown. The contact segments have been removed to ensure proper color mapping. Where in reverse mode the holes retreat towards the anode, in forward mode the holes populate the entire device.

Three-Dimensional FinFET

This section shows the simulation of a three-dimensional symmetrically sliced Si-based FinFET device, based on solving the DD problem (Section 4.3.6). This particular example has been chosen to depict the support for three-dimensional devices.

Figure 4.22 depicts the device setup. The source and drain region are set at a constant donor doping of cm, whereas the bulk region is set at a constant acceptor doping of cm.

The device has been simulated in its active state, by setting the gate and drain contact potential to V as well as the source and bulk contact potential to V (Figure 4.23). As can be seen from the results, the electrons gather primarily under the gate contact, forming a conducting channel from the source to the drain contact.

Figure 4.22: The setup of the three-dimensional FinFET device; Each color denotes a different device segment. S, D, B, and G refer to the source, drain, bulk, and gate contacts. The source (blue) and drain (green) region offer a constant donor doping of cm. The bulk (brown) region offers a constant acceptor doping of cm.

(a) Potential (V) (b) Electron Concentration (cm) (c) Hole Concentration (cm)

Figure 4.23: The potential, electron, and hole distributions of an active FinFET device; The gate and drain contact potential is set to V, whereas the source and bulk contact potential is set to V. The contact, oxide, and bulk segments have been removed for the sake of improved visualization. Iso-surfaces have been added to depict the behavior in the interior of the device. A conducting channel is formed under the gate as can be seen from the increased electron concentrations.