2.2.2 Model for the Moderately Doped Gate

As mentioned at the beginning, the single-crystal band model will be assumed for the polysilicon gate in this section. In other words we did not account for the grain-structure of polysilicon. The validity of such an approach depends on the ratio between the activated dopant concentration in the gate and the equivalent volume trap density in the polysilicon . The densities near the gate/oxide interface are particularly important. is given by the quotient of the surface-trap density at grain boundaries and the average grain size [422]. Two conditions can occur:

• If holds, the charge trapped at the grain boundaries is much smaller than the free charge of the majority carriers within the grains. Even if the grains are depleted in some part of the gate due to the terminal bias, the space charge descending from the doping is dominant over the space charge coming from an eventual trapped charge. The trapped charge has minor impact on the neutral Fermi level and an average space charge density. Therefore, it may be omitted from the Poisson equation in a first approximation. Note that this is not the case for the transport problem in polysilicon - low barriers at the grain boundaries also affect the current if holds [516][298][247][235]. We consider, however, a pure electrostatic problem in this study.
• When is comparable to or less than the trap-effect is dominant and dictates all electrical properties of the polysilicon. The Fermi level position, potential drop in the gate and especially, the gate capacitance become sensitive functions of the model parameters such as density of the traps at grain boundaries, their energy distribution and the average grain size.

The rigid-parabolic-band model is assumed to be valid in the gate for dopant concentrations of interest [296][295][294][292]. The rigid-band model means that the dispersion relation does not change with respect to pure silicon, but only shifts on the energy scale due to the heavy-doping effects. As a consequence, the effective densities of states for the conduction and valence band ( and , respectively) are invariant of spatial coordinates. In this concept we account for the states in the band tails [239] by additionally increasing the rigid shift of the conduction and valence band caused by the many-body effects [267]. The rigid-parabolic-band model is not only simple and convenient for analytical and numerical handling, but also supported by experiments: assuming the bands in heavily doped silicon to be parabolic close to band edges, but only shifted in the energy level, the experimental data of differential tunneling conductance [2] and radiative recombination [110], both at very low temperature ( and ), have been successfully explained. It is commonly accepted that the rigid-parabolic-band model is a reasonable approach to describe the carrier transport in heavily doped quasi-neutral regions [358]. The same approach is often employed to model the current and the recombination of minority carriers injected into heavily doped quasi-neutral emitter [327][307][190][94] and base [470][438] of bipolar transistors. However, regarding modeling heavily doped space-charge regions, there are likely to be strong limitations to this approach, because of appearing deep band-tails due to lack of screening [280][61]. A detailed discussion of this topic is given in Section 2.3 after demonstrating that the rigid-parabolic-band approach fails to fit accurately the experimental - data of devices with moderately doped gates.

In the rigid-parabolic-band model the carrier concentrations in the gate can be simply written

with the Fermi integral of order . In this work Fermi integrals are defined in the normalized form

with being the Gamma function. The band edges in the gate are given by

The spatial variations of the band edges in the ideal silicon band and are due to inhomogeneity in the dopant concentration and the electric-field penetration into the gate. The band edges and vary in addition because of spatially variable band-gap narrowing and which represents the shift of conduction band downward and the valence band upward, as introduced in [358]. Let the potential in the gate be defined as the potential difference between the intrinsic Fermi level in the ideal silicon band and the reference Fermi level in the source of MOS device. The carrier concentrations in the gate become

with the potential at the gate-contact

representing the boundary condition. From the neutral carrier concentration at the gate-contact it follows

The neutral electron concentration equals to the activated impurity concentration at the gate-contact in -type gate. For -type of gate, holds. in 2.13 is the inverse Fermi integral. With or calculated by 2.13, the second quantity follows from

where is the ideal silicon band gap and denotes the total band-gap narrowing at the gate-contact. The Fermi barrier in the gate occurring in the voltage conversation 2.6 may be related to with

It holds that , as is evident from Figure 2.3.

Previous relationships account properly for a position-dependent band-gap narrowing in the gate. They are implemented in a two-dimensional numerical model of MOSFETs, as introduced in Section 2.4. To simplify the expressions for analytical modeling we adopt a constant doping approximation in the gate near the gate/oxide interface as schematically shown in Figure 2.3. The necessary width of this uniform region with concentration is only a few extrinsic Debye length wide. For a minimal the maximal width of about may be estimated, which is much shorter than the polysilicon thickness. The remaining part of the gate is in quasi-neutral condition because of heavy doping and the impurity profile in this region is not important for our study. Consequently, the quantity does not vary from the gate-contact until the region close to the gate/oxide interface where the potential begins to change due to electric-field penetration. Thereby, both, and may be determined with respect to the uniform concentration .

The surface potential is obtained after integration of the Poisson equation in the homogeneous part of the gate, using the conditions and for and and at . Practically, the field decreases from to a negligible value after a few from the gate/oxide interface. Employing the well known integration technique shown in Appendix A the relationship between the surface field and the surface potential follows

Note that and vanish due to the constant doping assumed. in 2.16 is the Fermi integral of order . Consistent with Figure 2.2 and the definition of , it follows for , while holds for . At the flat-band in the gate and . The three terms at the left-hand-side in 2.16 model accumulation, depletion and inversion in the -polysilicon gate, respectively.