6.7 Analytic Solution for the MOS Capacitor
Table of Contents -
Glossary -
Study Aids -
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In this Section
- Introduction
- Electric field versus surface potential
- Inversion layer charge
- Low frequency capacitance
- Full text
Next: Chapter 7. The MOSFET
6.7.1 Introduction
An exact analytic solution can be obtained for the MOS capacitance as long
as
surface electron concentration is not degenerate. The non-linear
second-order
differential equation can then be solved yielding first the
electric field as
a function of the potential in the semiconductor. A solution for the
electric
field and/or the potential as a function of the position can not
be
obtained analytically. This requires a numeric integration. Combining the
electrical field and the surface potential yields the gate voltage, since
the
field in the semiconductor and that in the oxide are related by their
respective
dielectric constants. The same approach also yields a good approximation
for the charge in
the depletion layer, the inversion layer or
the accumulation layer. The derivative of
the charge with the applied voltage equals the capacitance of the MOS structure.
The calculation of the low-frequency or quasi-static capacitance is relative
straight forward, while the calculation of the high-frequency capacitance
requires an additional numeric integration. A detailed derivation of the items
mentioned above as well as the deep depletion
capacitance and an approximate expression for the high-frequency capacitance
can be found in the full derivation.
The solution for the electric field is obtained
by solving Poisson's equation while including the charge due to electrons,
holes and the ionized donors and acceptors. This solution provides the
relation between the electric field at the surface of the semiconductor and
the surface potential. The absolute value of the field is shown in the figure
below. This figure was obtained for a substrate with an acceptor
concentration, Na = 1017 cm-3,
and an oxide thickness, tox = 20 nm.
mosexact.xls - mosfield.gif
Fig.6.7.1 Electric field at the surface of the semiconductor as a function
of the potential across the semiconductor. Shown are the electric field (solid line) and
the field due to the inversion layer charge only (black dotted line) The red vertical line
indicates the threshold voltage. Na = 1017 cm-3
and tox = 20 nm.
When applying a positive potential (which can be done by applying a positive
gate voltage) the surface of the silicon is first depleted. This causes an electric
field
which varies as the square root of the surface potential. At higher positive
potential the surface inverts which results in a sharp rise of the electric field since
the inversion layer charge increases exponentially with the surface potential. The vertical
dotted
line on the figure indicates the threshold voltage or the onset of strong inversion. The other
dotted line represents the fraction of the surface field which is due to the electrons in
the inversion layer. It is calculated from the ratio of the inversion layer charge and the dielectric
constant of the semiconductor.
When applying a negative surface potential, the holes accumulate at the surface, yielding
an exponential rise of the electric field with decreasing potential.
The above figure is an active figure which can be further explored by the
reader. An MOS structure with a n-type
substrate can also be analyzed by entering a negative doping density.
The total charge in the inversion layer can also be calculated with this method.
It is obtained by substracting the charge in the depletion layer from the total charge for the
same surface potential. The details can be found in the full derivation.
The gate voltage is obtained by adding the flat band voltage, the surface potential and
the voltage across the oxide. The resulting charge density is plotted versus the gate voltage
in the figure below. This figure was calculated for an oxide thickness of 20 nm. The doping density is
also 1017 cm-3 as before.
mosexact.xls - moscharg.gif
Fig. 6.7.2 Charge density due to electrons
in the inversion layer of an MOS capacitor. Compared are
the analytic solution (solid line) and our basic assumption (dotted
line). Na = 1017 cm-3
and tox = 20 nm.
The dotted line on the figure represents the standard approximation for the inversion layer charge:
it implies that the charge is simply proportional to the gate oxide capacitance and the gate voltage
minus the threshold voltage. For voltages below the threshold voltage, there is no inversion layer and
therefore no inversion layer charge. While not exact, the standard approximation is very good.
The low frequency or quasi-static capacitance can be obtained by taking
the derivative of the charge in the semiconductor with respect to the potential
across the semiconductor. Since this derivative represents the change
between two thermal equilibrium situations, this capacitance is also to be
measured while maintaining equlibrium conditions at all times. The
low-frequency or quasi-static measurement is typically obtained by measuring
the current with a sensitive electrometer while varying the applied gate
voltage.
The expected behavior of such measurement is shown in the figure below: The
capacitance is close to the oxide capacitance except for a gate voltage
between the flat band voltage and the threshold voltage, as charge is then
added deeper into the semiconductor at the edge of the depletion layer,
rather than at the oxide-silicon interface. This results in the
characteristic dip in the capacitance curve.
mosexact.xls - moslfcap.gif
Fig. 6.7.3 Low frequency capacitance of an MOS
capacitor. Shown are the exact solution for the low frequency
capacitance (solid line) and the low and high frequency capacitance
obtained with the simple model (dotted lines). The red square indicates the
flatband voltage and capacitance, while the green square indicates
the threshold voltage and capacitance. Na = 1017 cm-3
and tox = 20 nm.
This figure was calculated using an oxide thickness of 20 nm and an acceptor
concentration of 1017 cm-3.
This is an active figure. The dotted lines
indicate the high- and low-frequency capacitance as obtained using the full
depletion approximation. It is clear from the figure that the approximation
is rather crude when it comes to describing the full behavior, but it is
sufficient to extract the oxide thickness and substrate doping
concentration from a measured curve.
The complete derivation contains a step by
step derivation of the equations used to generate the above figures.
6.6
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© Bart J. Van Zeghbroeck, 1996, 1997