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Semiconductors - Inversion Layer CapacitanceThe essence of this paper is to argue that in a semiconductor a precise equality exists between the bulk capacitance and inversion-layer capacitance at the threshold voltage. To start, the semiconductor threshold condition is defined as that for which surface potential is equal in magnitude and opposite in sign to the bulk potential, with the Fermi level taken as potential reference. Also, during threshold, the volumetric electron density at the surface equals the volumetric ionic density at the surface of the semiconductor
To explain the precise equality, the MOS capacitor is utilized, such that the problem can be made one-dimensional to any degree of accuracy by increasing its area. The assumptions utilized include uniform substrate doping, complete ionization of donors/acceptors at room temperature, and approximations involving Boltzmann statistics, band symmetry, and the equivalent densities of states.
Using the MOS capacitor, a series of volumetric-charge density profiles was determined for progressively increasing surface potential. As seen from this profile, there is a range of distances for which the charge density is constant. Correspondingly, the electric field is linear within this range, given the constant charge density. From the electric-field profile in the neighborhood of the depletion-layer boundary, we can define the position of the abrupt space-charge boundary by extrapolating the linear field profile to the x-axis, provided the electric field from the depletion approximation is equal to the actual situation. When that is satisfied, we see that beyond the constant space charge density range, the charge density for the depletion-approximation profile and the actual profile are the same.
This position is now taken as the spatial origin, as it allows us to write very simplified and accurate forms of analytical expressions for the asymptotic behavior of potential, field and other functions that enter the surface problem. When using the depletion approximation model, this spatial origin is useful for modeling of the surface, junction and the device, as the profiles for charge, field and potential are unchanged at the depletion-layer edge. When we consider the surface potential to be an independent variable, and the surface position as a function of the surface potential, we see that the silicon crystal surface varies in distance relative to the spatial origin, depending on the surface potential. With this approach, finding the ionic charge density in the bulk silicon can be accurately found, which is independent of the formulation used, be it the actual or the depletion-approximation model of the semiconductor.
A slight increase in surface potential at threshold will lead to equal amounts of inversion-layer charge and bulk charge in the added few monolayers of silicon at the surface layer, located at the place where the inversion-layer charge density and the bulk charge density functions intersect. Extending this result to actual semiconductor devices, the bulk charge and inversion-layer charge increments are the same in the actual MOS capacitor given a potential increment at threshold. Since capacitance is defined as the ratio of the charge stored over the potential, the inversion-layer and bulk capacitance are precisely equal at threshold.
This equality does not hold for modern semiconductor capacitors as most of them are short channel devices, designed in order to achieve improvements in packing density, speed, and power. Given that the channel length is the same order of magnitude as the depletion-layer widths, we see short-channel effects such as threshold voltage roll-off, and drain induced barrier lowering. Comparing the long channel MOS transistor, and the short channel MOS transistor, we see that in the case of the former, the depletion is only due to the electric field created by the gate voltage, whereas in the latter case, there is an added contribution by the depletion charge near the heavily doped regions at the source and drain. Hence, in the short channel device, the bulk depletion charge is smaller than expected in the long channel device, and this result in lower threshold voltage for conduction to take place. Hence, for a given applied voltage beyond the threshold, the inversion layer charge is not equal to the bulk depletion charge, and the equality of the inversion-layer capacitance and the bulk capacitance fails.
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