The 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.
Originally Written Article here.The author Jimmy Lee is involved in article writing, publishing, and website design on a freelance basis amid a daytime job as an electrical engineer. His favourite works can be found at http://flashgor.blogspot.com/ and http://diypc.wordpress.com/
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