6.6 The MOS Capacitance
Table of Contents -
Study Aids -
In this Section
- Simple capacitance model
- The flatband capacitance
- Exact analysis
- Deep depletion capacitance
- Experimental results and comparison with theory
- Non-Ideal effect in MOS capacitors
Next: 6.7 The exact analytical solution
Capacitance voltage measurements of MOS capacitor structure provide a
wealth of information about the structure which is of direct interest
when one evaluates an MOS process. Since the MOS structure is simple
to fabricate the technique is widely used.
To understand capacitance-voltage measurements one must first be familiar
with the frequency dependence of the measurement. This frequency dependence
occurs primarily in inversion since a certain time is needed to generate
the minority carriers in the inversion layer. Thermal equilibrium is
therefore not obtained immediately.
The low frequency or quasi-static measurement
maintains thermal equilibrium at all times. This capacitance is the difference
in charge divided by the difference in gate voltage while the capacitor
is in equilibrium at each voltage. A typical measurement is performed with
an electrometer which measured the charge added per unit time as one slowly
varies the applied gate voltage.
The high frequency capacitance is obtained
from a small signal capacitance measurement at high frequency. The gate
voltage is varied slowly to obtain the capacitance versus voltage. Under such
conditions one finds that the charge in the inversion layer does not change
from the equilibrium value corresponding to the applied DC voltage. The
high frequency capacitance therefore reflects the charge variation in
layer and the (rather small) movement of the inversion layer charge.
In this section we first derive the
simple capacitance model which
is based on the full depletion approximations and our basic assumption.
The comparison with the exact low frequency capacitance
reveals that the largest error occurs at the flatband voltage. We therefore
exact flatband capacitance using the linearized Poisson's equation. Then we
discuss the full
exact analysis followed by a discussion of
deep depletion as well as the
non-ideal effects in MOS capacitors.
6.6.2 Simple capacitance model
The capacitance of an MOS capacitor is obtained using the same
assumptions as in the analysis in section
6.5. The MOS structure is treated
as consisting of a series connection of two capacitors: the capacitance
of the oxide and the capacitance of the depletion layer.
In accumulation there is no depletion layer. The remaining capacitor is the
oxide capacitance, so that the capcitance equals:
In depletion the MOS capacitance
is obtained from the series connection of
the oxide capacitance and the capacitance
of the depletion layer, or:
where xd is the variable
depletion layer width which is
In order to find the capacitance corresponding to a specific
value of the gate voltage we also need to use the relation between
the potential across the depletion region and the gate voltage,
In inversion the capacitance becomes independent of the gate voltage.
The low frequency capacitance equals the oxide capacitance since charge
is added to and from the inversion layer in a low frequency measurement.
The high frequency capacitance is obtained from the series connection of the
oxide capacitance and the capacitance of the depletion layer having its
maximum width, xd,max. The
capacitances are given by:
The capacitance of an MOS capacitor as calculated using the
simple model is shown in the figure below. The dotted lines
represent the simple model while the solid line corresponds
to the low frequency capacitance as obtained from the
mosexact.xls - moslfcap.gif
Fig. 6.6.1 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.
6.6.3 Flat band capacitance
The simple model predicts that the flatband capacitance equals
the oxide capacitance. However, the comparison with the exact solution
of the low frequency capacitance as shown in the above figure
reveals that the error can
be substancial. The reason for this
is that we have ignored any charge variation in the semiconductor.
We will therefore now derive the exact flatband capacitance.
To derive the flatband capacitance including the charge
variation in the semiconductor we first linearize Poisson's
equation. Since the potential across the semiconductor
at flatband is zero, we expect the potential to be small as
we vary the gate voltage around the flatband voltage. Poisson's
equation can then be simplified to:
The solution to this equation is:
where LD is called the
Debye length. The solution of the
potential enables the derivation of the capacitance of the
semiconductor under flatband conditions, or:
The flatband capacitance of the MOS structure at flatband is
obtained by calculating the series connection of the oxide
capacitance and the capacitance of the semiconductor, yielding:
6.6.4 Exact analysis
For a description of the derivation of the MOS capacitance
using the exact analysis we refer the reader to that
6.6.5 Deep depletion capacitance
Deep depletion occurs in an MOS capacitor when measuring the high-frequency
capacitance while sweeping the gate voltage "quickly". Quickly here means that
the gate voltage must be changed fast enough so that the structure is not in
thermal equilibrium. One then observes that when ramping the voltage from
flatband to threshold and beyond the inversion layer is not or only partially formed
as the generation of minority carriers can not keep up with the amount needed
to form the inversion layer. The depletion layer therefore keeps
increasing beyond its maximum thermal equilibrium value,
in a capacitance which further decreases with voltage.
The time required to reach thermal equilibrium when abruptly
biasing the MOS capacitor at a voltage larger then the threshold voltage
can be estimated by taking the ratio of the total charge
in the inversion layer to the thermal generation rate of minority carriers. A
complete analysis should include both a surface generation rate as well as
generation in the depletion layer and the quasi-neutral region. A good
approximation is obtained by considering only the generation rate in the
depletion region xd,dd.
This yields the following equation:
where the generation in the depletion layer was assumed to be constant.
The rate of change required to observe deep depletion is then obtained
This equation enables to predict that deep depletion is less likely at
ambient temperature since the intrinsic concentration
exponentially with temperature, while it is more likely to occur in MOS
structures made with wide bandgap
materials (for instance SiC for which Eg = 3 eV)
intrinsic concentration decreases exponentially
with the value of the energy bandgap.
In silicon MOS structures one finds that the occurance of deep depletion
can be linked to the minority carrier lifetime: while structures with a long (0.1 ms)
lifetime require a few seconds to reach thermal equilibrium which
results in a pronounced deep depletion effect at room temperature
, structures with a short
lifetime do not show this effect.
Carrier generation due to light
will increase the generation rate beyond the thermal
generation rate which
we assumed above and reduce the time needed to reach
equilibrium. Deep depletion measurements are therefore done in the dark.
6.6.6 Experimental results and comparison with theory
As an example we show below the measured low frequency (quasi-static) and high frequency
capacitance-voltage curves of an MOS capacitor. The capacitance was
measured in the presence of ambient light as well as in the dark as explained
in the figure caption.
Fig. 6.6.2 Low frequency (quasi-static) and high frequency
capacitance of an MOS
capacitor. Shown are, from top to bottom, the low frequency capacitance
measured in the presence of ambient light (top curve), the low frequency capacitance
measured in the dark, the high frequency capacitance measured in the
presence of ambient light and the high frequency capacitance measured in
the dark (bottom curve). All curves were measured from left to right.
The MOS parameters are Na = 4 x 1015 cm-3
and tox = 80 nm. The device area is 0.0007 cm2
The figure illustrates some of the issues when measuring the capacitance of an
MOS capacitance. First of all one should measure the devices in the dark; the presence
of light causes carrier generation in the capacitor which affects the measured capacitance. In
addition one must avoid the deep depletion effects such as the initial linearly varying capacitance
of the high frequency capacitance measured in the dark on the above figure (bottom curve). The
larger the carrier lifetime, the slower the voltage is to be changed to avoid deep depletion.
The low frequency measured is compared to the theorical value in the figure below. The high frequency
capacitance measured in the presence of light is also shown on the figure. The figure
illustrates the agreement between experiment and theory. A
comparison of the experimental low (rather than high) frequency
capacitance with theory is somewhat easier to carry out since the theoretical expression is easier
to calculate while the low frequency measurement tends to be less sensitive to deep depletion effects.
Fig. 6.6.3 Comparison of the theoretical low
frequency capacitance (solid line) and the experimental data (open squares) obtained
in the dark. Also
shown is the high frequency measurement in the presence of light
of the MOS capacitor (filled squares) and the low and high frequency capacitance
obtained with the simple model (dotted lines). Fitting parameters
are Na = 3.95 x 1015 cm-3
and tox = 80 nm.
6.6.7 Non-Ideal effects in MOS capacitors
Non-ideal effects in MOS capacitors include fixed charge, mobile charge
and charge in surface states. All three types of charge can be identified
by performing a capacitance-voltage measurement.
Fixed charge in the oxide simply shifts the measured curve.
A positive fixed charge at the oxide-semiconductor interface shifts
the flatband voltage by an amount which equals the charge divided by the
oxide capacitance. The shift reduces linearly as one reduces the position
of the charge relative to the gate electrode and becomes zero if the
charge is located at the metal-oxide interface. A fixed charge is caused by
ions which are incorporated in the oxide during growth or deposition.
The flatband voltage shift due to mobile charge is described by the
same equation as that due to fixed charge. However the measured curves
differ since a positive gate voltage causes mobile charge to move away
from the gate electrode, while a negative voltage attracts the charge
towards the gate. This causes the curve to shift towards the
applied voltage. One can recognize mobile charge by the hysteresis in the
high frequency capacitance curve when sweeping the gate voltage back and
forth. Sodium ions incorporated in the oxide of silicon MOS capacitors
are known to yield mobile charge. It is because of the high sensitivity
of MOS structures to a variety of impurities that the industry carefully
controls the purity of the water and the chemicals used.
Charge due to electrons occupying surface states also yields a shift in
flatband voltage. However as the applied voltage is varied, the fermi
energy at the oxide-semiconductor interface changes also and affects the
occupancy of the surface states. The interface states cause the transition
in the capacitance measurement to be less abrupt. The combination of
the low frequency and high frequency capacitance allows to calculate the
surface state density. This method provides the surface state density
over a limited (but highly relevant) range of energies within the bandgap.
Measurements on n-type and p-type capacitors at different
the surface state density throughout the bandgap.
© Bart J. Van Zeghbroeck, 1996, 1997