6.2 Modeling BioFETs

There exist two main approaches for the simulation in such devices. In the first approach the system is described via a set of differential equations and proper boundary conditions, while in the second one every molecule in the solute is described separately and is thus free to move within the solute, attempting to minimize acting forces (the energy of the system). This process is simulated via a stochastic Monte Carlo process [214]. The Monte Carlo approach allows a relatively easy description of the system via the fundamental interactions between single molecules/atoms, but needs a high amount of memory and offers an accuracy of the results of only $\sim\frac{1}{\sqrt{N}}$ ($N$ is the sample size). The high amount of memory is caused by the vast amount of molecules/atoms within the solute. For instance, $1\,\mathrm{ml}$ water contains about $\approx3.35\,\times10^{22}$ of water molecules. Even, restricting to the simulation of the macromolecules and describing the water molecules via a permittivity coefficient of $\sim80$, the memory consumption remains on a high level, since macromolecules easily contain several thousand atoms. There are further approaches to reduce the memory need, but the overall memory consumption abides on a high level and the simulation domain is restricted to small volumes. Also the time scale such simulations can handle are quite small ( $\sim\mathrm{ps}$, [215,216]) and not comparable to time scales needed in real world applications (seconds to hours).

The differential equation approach on the contrary, is less memory and time consuming, but treats the quantities as continuous. This can cause misleading results for low buffer concentrations as will be shown later.

The difference between these two approaches originate from the different scales of the macromolecule and the FET device. Chemical reactions take place at the Angstrom length-scale, while the ISFET channel gate length is in the micrometer regime. Therefore, depending on the results to achieve one has to decide which approach is the more promising to conquer the multi-scale problem for his needs.

In this work the differential equation approach was chosen, because the emphasis was laid on the device behavior in conjunction with the prefered usage by engineers.

The simulation domain is split into several parts. There are the semiconducting region, the dielectric region, the Stern layer, the zone where the macromolecules are held, and the region containing the buffer (shown in Fig. 6.2). The device is in the micrometer regime and it is therefore possible to model transport in the semiconducting part via the drift-diffusion model [217,218]. The dielectric is modeled with the Laplace equation, assuming that there is no charge in the oxide. The Stern layer, ensuring a minimal distance of the charged zone containing the macromolecules to the oxide interface, is modeled with the Laplace equation and a relative permittivity of $\varepsilon_{\mathrm{sol}}\approx80$. The presence of charge at the dielectric-electrolyte interface depends on the preparation of the device. Often the surface is passivated before the macromolecules are attached to it. The reasons for this are manifold: a need to suppress unwanted charge accumulation at the open oxide sites which could mask the charges from the macromolecules during detection, to avoid an unwanted pH dependence, and to prepare the surface in a way that the macromolecules can be linked (attached) to the surface. The charge density in the zone containing the macromolecules can either be determined by experimental data or must be estimated from the macromolecules structure itself derived from a protein data bank (e.g. [2]). This zone and the remaining electrolytic region are modeled with different approaches, described in the following sections.

Figure 6.2: BioFET: different simulation zones.