450 CHAPTER 9. FIELD EFFECT TRANSISTORS: MOSFET
whereWmaxis defined by equation 9.3.13. At the onset of strong inversion, free electrons begin
to collect in the inversion channel and the depletion width remains unchanged with bias. The low
frequency capacitance of the semiconductor again increases since a small change inψscauses a
large change inQs. The capacitance of the MOS device thus returns toward the value ofCox:
Cmos(inv)=Cox=ox
dox(9.4.6)
Another important point on the C-V characteristics is the point where the bands become flat. The
flat band capacitance of the MOS device is (see example 9.5)
Cmos(fb)=ox
dox+oxs√
kBT
es
eNa(9.4.7)
One must now ask as to where the electrons come from when the device is in inversion. The
excess electrons needed are introduced into the channel bye-hgeneration, by thermal generation
processes, or by diffusion of the minority carriers from thep-type substrate.Sincethegeneration
processtakesafinitetime,theinversionsheetchargecanfollowthevoltageonlyifthevoltage
variationsareslow. If the variations are fast, the capacitance due to the free electrons makes
no contribution and the capacitance is dominated by the depletion capacitance. Thus, under
high-frequency measurements, the capacitance does not show a turnaround and remains at the
valueCmos(min), as shown in figure 9.12. The capacitance in the inversion regime starts to
decrease even at frequencies of 10 Hz and at 10^4 Hz it reaches the low value ofCmos(min).In
the MOSFET this is not an issue since electrons can be rapidly supplied by the ohmic contacts.
The presence of the fixed charge simply causes a voltage drop across the oxide given by
ΔVfb=ΔV=−Qss
Cox(9.4.8)
whereQssis the fixed charge density (cm−^2 ) in the oxide. As a result, ifQssis positive the
entire C-V curve shifts to a more negative value. Since the chargeQssis independent of the gate
bias, the entire C-V curve shifts as shown schematically in figure 9.13a. The value ofQsscan be
obtained by measuring the shift as compared with the calculated ideal curve. Such measurements
are very important for characterizing the quality of MOS devices.
The interface charge,Qis, has a somewhat different effect on the C-V characteristics. In an
ideal system, there are no allowed electron states in the bandgap of a semiconductor. However,
since the Si-SiO 2 interface is not ideal, a certain density of interface states are produced that lie
in the bandgap region.
In contrast to the fixed charge, electrons can flow into and out of these interface states de-
pending upon the position of the Fermi level. The character of the interface states is defined as
“acceptor-like” and “donor-like.” An acceptor state is neutral if the Fermi level is below the state
(i.e., the state is unoccupied) and becomes negatively charged if the Fermi level is above it (i.e.,
the state is occupied). The donor state is neutral if the Fermi level is above it (i.e., the state is
occupied) and positively charged when it is empty. As a result, when the position of the Fermi
level is altered, the charge at the interface changes.