2.4.1 Modeling of MOSFETs with Nondegenerate Gate

An obvious approach to include the non-equipotential gate in a device simulator like MINIMOS [415][160] is to solve numerically the discretized continuity equations with the Poisson equation in the polysilicon gate, simultaneously with the silicon bulk [163]. The simulation structure is displayed in Figure 2.14. An enlarged mesh, due to resolving the gate/oxide interface, and the space charge in the gate do not pose additional difficulties in solving the Poisson equation, which is a quite stable numerical problem. In contrast, solving the continuity equations in the gate area significantly increases the computational burden, since these equations consume the largest portion of the total calculation time. Therefore, this approach is interesting solely for transient conditions, where the currents become nonzero in the gate. For the steady-state a much more efficient approach may be proposed as is introduced later, because the currents as well as the net generation-recombination vanish in the gate.

However, for a moment let us consider transient effects that can occur in nondegenerate gates. In MOSFETs, source and drain are efficient suppliers of minority carriers for the channel. Quasi-static conditions can be maintained up to a high frequency (MHz or more) in the channel, which depends in the first place on the channel length . There is no external supply of minorities in the gate. Thereby, transient effects occurring in MOS capacitors due to finite minority carrier response may be expected to occur in nondegenerate gates:

1)

thermal generation-recombination over interface traps (and direct) [331];
many interface traps can reside at the poor gate/oxide interface,

2)

generation-recombination over traps at grain boundaries in polysilicon [316][222][153],

3)

impact ionization, if the gate is forced in deep-depletion by a suddenly increasing gate bias. All phenomena noted here can trigger the avalanche multiplication.

4)

band-to-band tunneling (BBT) can easily take place in deep-depletion in the gate. In an -type gate electrons tunnel from the valence band close to the gate/oxide interface towards the conduction band remote from the interface, while holes are left at the interface contributing in the formation of the inversion layer. Due to high doping in the gate, the depletion region is shallow even for large band-bending. For example: if of only is achieved by pulsing the gate into deep-depletion, the minimal tunneling length becomes and the surface field for doping of . Assuming a doping of , it follows and . Since the BBT rate can be very large, we expect that this mechanism may clamp further increasing in when forcing the gate into deep-depletion. To prove this claim, however, selfconsistent transient simulation is necessary (Section 3.2), including BBT (Section 4.2) in the gate.

5)

injection of minorities into the oxide from the gate-side can occur in deep-depletion, particularly electrons in devices with -type gate because of a lower barrier for electrons than for holes.

Steady-state two-dimensional numerical model:

Henceforward we assume steady-state conditions. Due to quasi-equilibrium in the gate, as already pointed out in Section 2.2, carrier concentrations and may be calculated analytically as a function of the local potential , instead of solving the continuity equations, particularly when the latter must be discretized accounting for FD statistics [166][163]. For and the relationships 2.10 and 2.11 are implemented in a two-dimensional numerical model. Since solving the continuity equations is restricted to bulk, this approach demands nearly the same computational effort as the conventional MOSFET simulation with the equipotential gate. As boundary conditions at the ohmic gate/polysilicon interface (gate-contact) the potential is given by 2.12, whereas neutrality is assumed for the carrier concentrations, leading to equations 2.13 and 2.14. The approximations to the Fermi integral and its inverse are clarified in Appendix A. Note that in this approach we still need to increase the mesh-size due to a finite thickness of the gate if we are particularly interested in the gate-side or gate-corner effects, like fringing capacitances and corner fields. For most analyses this is not necessary and the gate depletion can be implemented in an even simpler and more efficient way than this, as explained below.

Quasi-two-dimensional approach:

Remember that denotes the difference between the potential at the gate/oxide interface and the gate-contact potential (determined by ). In MOSFETs varies along the gate/oxide interface. Note that and are the coordinates parallel and perpendicular to the interface, respectively. It is obvious that the value of can range from roughly in strong accumulation to in strong inversion assuming an -type gate. As a consequence the maximum difference between the potential at the drain-side and the source-side of the gate is limited to about , whereas the corresponding difference in the channel is roughly . Therefore, the gate may be considered to be an MOS device with a moderate drain-source bias in the direction. In these devices the gradual-channel-approximation (G.C.A.) and the capacitance relation are reasonable assumptions near the interface. To estimate the lateral field at the interface in the gate , remark that relationships 2.4 and 2.6 are valid in general ( is the transversal component of the surface field). Applying on these formulas with benefit of , we derive

which is valid in general. is the lateral field at the interface in the bulk. If G.C.A. is valid in the bulk is always positive and follows from 2.28. Because of in the bulk and the relation between and given by 2.2, is valid (rigorously, always when no interface charge is presented). The preceding analysis may be applied to the complete channel if the MOSFET is biased in linear (triode) region. For MOSFET is in saturation, the G.C.A. holds in the so-called source channel-part ([162][161] and references cited therein). Moreover, it is valid in the corresponding part of the gate as well - just on the other side of the oxide. The G.C.A. does not hold in the drain channel-part after the pinch-off point and probably vis-à-vis in the gate too. However, this fact is irrelevant, since the gate has lost its influence on the bulk in the drain-channel part. Consequently, where the gate controls the channel, the G.C.A. holds in the gate. We may conclude that, unlike for the bulk, two-dimensional effects are of second order importance in the gate, except at the corners.

According to the preceding discussion one can assume the problem to be one-dimensional in the gate area. The surface potential depends only on the local perpendicular field in the polysilicon at the interface and can be calculated by solving equation 2.16 for FD statistics or A.14 for MB statistics. Furthermore, if one has no interest in the gate-side effects, the actual gate thickness can be neglected. The model reduces to the conventional planar MOSFET simulation, but with the gate-contact potential variable along the coordinate: . In MINIMOS, this is implemented by updating the gate potential boundary condition after each Poisson/continuity equation solution-step by calculating for a given local and an interface charge , up to selfconsistency. This calculation is very fast compared to two-dimensional solution of the Poisson equation in the gate. Moreover, the additional and grid lines necessary to resolve the gate/oxide interface, as well as grid adaptation are avoided. In this approach, the damping of the potential changes applied to the gate may be employed to accelerate the convergence of the simulation [160].

The quasi-two-dimensional model has been verified by comparison with the full two-dimensional approach. We compared the drain current in the linear and the saturation region, the transconductance in saturation, the lateral and the transversal field in the bulk near the gate corner, the potential distribution along the gate/oxide interface and the QS gate-bottom capacitance for short-channel MOSFET with and . With respect to these quantities, only negligible differences between the two approaches have been observed.