Table of Contents - Glossary - Study Aids - ¬ ®

In this Section

- Introduction
- 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

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
the depletion
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 derive the 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.

In accumulation there is no depletion layer. The remaining capacitor is the oxide capacitance, so that the capcitance equals:

- (mc11)

- (mc12)

- (mc2)

- (mc8)

- (mc13)

mosexact.xls - moslfcap.gif

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:

- (mc16)

- (mc17)

- (mc18)

- (mc19)

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 *x*_{d,dd}.
This yields the following equation:

- (mc14)

- (mc15)

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 (1 ms) 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.

cv1.gif

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.

cv2.gif

**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
temperatures provide
the surface state density throughout the bandgap.

© Bart J. Van Zeghbroeck, 1996, 1997