Microelectronics Processing Technology
6.152J / 3.155J
Fall 2001
Metal Oxide Semiconductor (MOS) Capacitor
The MOS capacitor structure is shown in Figure 1. The “metal” plate is a heavily doped p+ poly-silicon layer which behaves as a metal. The insulating layer is silicon dioxide and the other
plate of the capacitor is the semiconductor layer which in our case is n-type silicon whose
resistivity is 1-10 Ω-cm corresponding to a doping of 1015 cm-3.
The capacitance of the MOS structure depends on the voltage (bias) on the gate. For the purposes
of this discussion, we shall refer to the contact to the semiconductor as the body (B) while the
poly-silicon is called the gate (G). Typically a voltage is applied to the gate while the body is
grounded and the applied voltage is VG but more accurately VGB. The two (VG & VGB) will be
used interchangeably in this document.
Gate
p+ Poly-Si
Oxide
n-Si
Body
Figure 1: The MOS Capacitor structure. The substrate (body) is grounded and a voltage VGB is
applied to the poly-silicon gate.
The capacitance depends on the voltage that is applied to the gate (with respect to the body). The
dependence is shown in Figure 2 and there are roughly three regimes of operation separated by
two voltages. The regimes are described by what is happening to the semiconductor surface.
These are (1) Accumulation in which carriers of the same type as the body accumulates at the
surface (2) Depletion in which the surface is devoid of any carriers leaving only a space charge
or depletion layer, and (3) Inversion in which carriers of the opposite type from the body
aggregate at the surface to “invert” the conductivity type. The two voltages that demarcate the
three regimes are (a) Flatband Voltage (VFB) which separates the accumulation regime from the
depletion regimes and (b) the Threshold Voltage (VT) which demarcates the depletion regime
from the inversion regime.
Let us now look at our particular device MOS capacitor with an n-type body / substrate.
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C MOS
Cmax
CQS
C HF
Cmax=Cox
C min
C MOS(VT)=Cmin
VFB
VT
Inversion
Depletion
VGB
Accumulation
Figure 2: Capacitance per unit area vs. Gate Voltage (CV) diagram of a MOS Capacitor. The
flatband voltage (VFB) separates the Accumulation region from the Depletion regime. The
threshold voltage (VT) separates the depletion regime from the inversion regime. CHF is high
frequency capacitance while CQS is quasi-static or low frequency capacitance.
Surface Accumulation (VGB > VFB)
An applied positive gate voltage larger than the flatband voltage (which will be defined shortly)
(VGB > VFB) induces positive charge on the “metal” gate and negative charge in the
semiconductor. The only negative charges available are electrons and they accumulate at the
surface. The electron concentration at the surface is above the bulk value, thus leading to a
condition that is called surface accumulation. The charge distribution and equivalent circuit is
shown in Figure 3. The flatband voltage (VFB) is the voltage at which there is no charge on the
plates of the capacitor and hence there is no electric field across the oxide. It’s numerical value
depends on the doping of the semiconductor and on any residual interface charge that may exist
at the interface between the semiconductor and the insulator. When the surface of the
semiconductor is accumulated, a plot of the charge per unit area (QN) at the semiconductor /
oxide interface versus the applied voltage (VGB) is linear and the slope is the oxide capacitance
per unit area., Cox., which is given by
ε
C MOS ,accumulation = C max = C ox = ox
t ox
where εox is the permittivity of the oxide and it is 3.9εo. εo is the permittivity of free space or air.
εo = 8.854x10-14 Fcm-1. The unit for Cox is Fcm-2. Figure 4 is a plot of the charge per unit area
(QN) as a function of the applied voltage (VGB).
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charge density
ρ
+ + + + +
- - - - -
Oxide
Silicon
Oxide
0
- - - - -
G
+ + + + +
Poly-Si
-tox
B
x
Cox
Figure 3: Charge distribution in a MOS Capacitor biased into accumulation. The electron
distribution at the Si/SiO2 interface could be approximated as a δ-distribution.
Figure 4: Accumulation charge density as a function of the applied voltage. The slope of the line
is the oxide capacitance per unit area, Cox.
Surface Depletion (VT < VGB < VFB)
If the applied gate voltage is brought below the flat band voltage (remember the flat band voltage
is the gate voltage at which there is no charge in the MOS capacitor), a negative charge is
induced at the interface between the poly-silicon gate and the oxide. This leads to a positive
charge being induced at the other interface i.e. the oxide / semiconductor interface. This could
only be accomplished by “pushing” all the mobile negative carriers (electrons) away and
exposing the fixed positive charge from the donors. Hence the surface of the semiconductor is
depleted of mobile carriers leaving behind a positive space charge. Figure 5 shows the charge
distribution under these circumstances. The space charge layer resulting behaves also like a
capacitor having a capacitance per unit area (CD), which depends on VGB and is given by
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C D (VGB ) =
ε Si
x d (VGB )
where εSi is the permittivity of the silicon and it is 11.7εo. εo is the permittivity of free space or
air. εo = 8.854x10-14 Fcm-2. xd is the depleted silicon layer thickness. The unit of CD is Fcm-2.
From Figure 5, we observe that the oxide capacitance per unit area (Cox) and depleted silicon
capacitance per unit area (CD) are connected in series. Thus the capacitance of the MOS
structure when it is in the surface depletion regime is given by
1
1
1
=
+
C depletion C ox C D
⇒ C MOS ,depletion = C depletion =
C ox C D
C ox + C D
The unit for Cdepletion is Fcm-2. The silicon depletion layer thickness increases as the gate voltage
is decreased because more electrons are pushed away exposing more fixed positive ionized
dopants leading to thicker space charge layer. The capacitance of the depleted silicon decreases
and hence the MOS capacitance decreases as the gate voltage is decreased.
charge density
Poly-Si
Mobile electrons
Ionized
Donors
G
Oxide
- - - - + + + + + Ionized Donors
n-Si
B
Oxide
Mobile
Electrons
- - - - -
p+-poly
ρ
-tox
+ +
+ + +
xd
0
x
Silicon
Cox
CD
Figure 5: Charge distribution in a MOS Capacitor biased into depletion and the equivalent
circuit diagram. xd is the depleted silicon layer thickness.
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Surface Inversion (VGB < VT)
If the applied gate voltage is lowered below the threshold voltage (VT), the semiconductor
surface inverts its conduction type from n-type to p-type in our particular situation. It is natural
to ask why such a thing would occur? Before answer the question let us define the threshold
voltage (VT) as the gate voltage at which the conductivity type of the surface layer changes from
n-type to p-type because more holes have been attracted to the surface that the number of
electrons that existed at the surface at flatband. It demarcates the depletion region from the
inversion region. We shall now proceed to answer the question about why the surface inverts.
Starting from flatband, as the gate voltage is lowered negative mobile carriers (electrons) are
“pushed away” from the Si/SiO2 interface, a positive space charge is exposed. We approximate
this as a depletion layer in which we make the assumption that the layer is devoid of all mobile
carriers. However, this is only an approximation. What happens in reality is that the density of
electrons decreases exponentially from the surface going into the bulk. An important fact
pertinent to this discussion is that we assumed that system is in quasi-equilibrium, hence the law
of mass action p o n o = n i2 is still valid. Thus at the surface, the number of electrons decreases as
the applied voltage decreases. Correspondingly, the number of holes at the surface increases as
the applied gate voltage decreases. This is depicted in Figure 6. At a particular voltage called the
threshold voltage (VT) the concentration of holes at the surface exceeds the concentration of
electrons in the bulk. The conductivity type of the silicon surface is inverted. Figure 7 shows the
charge distribution of the MOS capacitor in inversion. There is a mobile charge delta-distribution
at the silicon / silicon dioxide interface. Additional increases in the applied gate voltage only
leads to a linear increase in the charge per unit area of the inversion layer. Figure 8 shows the
dependence of the charge density on the applied gate voltage. The inversion layer charge density
is given by
Q P = −C ox [VGB − VT ] (Ccm −2 )
The above expression for the hole density in the inversion layer will be used when considering
the p-MOSFET.
An important fact we need to state is that once the inversion layer forms, the depletion layer
thickness reaches a maximum. The total voltage drop across the semiconductor is “pinned” at a
maximum value. Increases in the gate voltage applied to the structure is dropped mostly across
the oxide layer as reflected by the expression for inversion layer charge. Thus when the gate
voltage is equal to the threshold voltage, the depleted layer capacitance per unit area reaches a
minimum CDmin and likewise the MOS capacitance. This capacitance is Cmin and it is given by
C MOS (VT ) = C min = C depletion,min =
C ox C D min
C ox + C D min
where
C D min =
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ε Si
x d max
MOS Capacitor
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Figure 6: Semilog plots of the carrier concentration distribution for the MOS capacitor in (a)
accumulation, (b) flatband, (c) depletion and (d) inversion.
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charge density
ρ
p+-poly
G
- - - - + + + + +
+ + + + + Ionized Donors
n-Si
Poly-Si
Oxide
Oxide
Holes
Mobile
Electrons
B
- - - - -
Mobile electrons
-tox
Holes at interface
+ + + + +
Poly-Si
0
Ionized
Donors
+ +
+ + +
xdmax
x
Silicon
Cox
CDmin
Figure 7: Charge distribution in a MOS Capacitor biased into inversion. The hole distribution at
the Si/SiO2 interface could be approximated as a δ-distribution.
Figure 8: Inversion layer hole charge density as a function of the applied voltage (low frequency
& quasi-static situation). The slope of the line is the oxide capacitance per unit area, Cox.
Where do the holes that form the inversion layer come from? In a MOS capacitor in
depletion or inversion, the holes and electrons are generated in the depleted silicon surface
region. The holes are attracted to the Si/SiO2 interface while the electrons are “pushed” into the
substrate. However the holes could also come from a p-doped region that is in close proximity to
the MOS capacitor such as the source/drain region of a p-MOSFET.
What are the expressions for xd and xdmax? I will not try to derive the equations that give you
xd and xdmax because they are rather involved, but I will have give you the expressions. If you
will like to derive it, I will refer you to 6.012 Text. Microelectronics: An Integrated Approach,
Howe and Sodini.
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x d (VGB ) =
ε Si
C ox
2

(φ B + VGB ) 
2C ox
− 1
 1+
ε Si qN D


where
φ B = (φ m − φ n ) = φ p + − φ n
φ p + = −0.55 V
φn =
N
kT
• ln D
q
ni
x d (VT ) = x d max =
2ε Si • 2φ n
qN D
The next question to be asked is that why are there two values of capacitance at any voltage
in the inversion regime?
The capacitance of the structure is usually measured by imposing a DC bias voltage which in our
case will be VGB superposed on the DC bias is a small signal which is an alternating current (ac).
This could be a high frequency ac signal (f=1 MHz) or low frequency signal (f<1 kHz). The
frequency of the signal affects the capacitance versus voltage (CV) curve for an MOS capacitor.
The capacitance depends on the measurement frequency and what other structures are connected
to the basic MOS capacitor. At very low frequencies (sometimes referred to as quasi-static
conditions), the generation rate of holes (and electrons) in the depleted silicon surface layer is
fast enough and hence holes are swept to the Si/SiO2 interface where the thin layer holes forms a
sheet of charge. Thus the inversion layer capacitance per unit area under quasi-static conditions
is given
C MOS ,inversion,QS =
ε ox
= C max
t ox
At high frequencies, the generation rate is not fast enough to allow the formation of a hole charge
density at the Si/SiO2 interface. In this case the silicon surface depletion layer thickness is still at
it’s maximum value xdmax and the corresponding inversion layer capacitance per unit area at high
frequency is Cmin and it is given by
C MOS ,inversion,HF = C min = C depletion,min =
C ox C D min
C ox + C D min
The inversion layer capacitance of a p-MOSFET even at high frequencies has the same value as
the quasi-static MOS capacitor (Cmax) because there is a ready supply of holes coming from the
p-type source/drain regions that are in close proximity.
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What are the flatband and threshold voltages?
Flatband Voltage
In theory, the flatband voltage should be
(
VFB =− φ B = −(φ m − φ n ) = − φ p + − φ n
)
φ p + = −0.55 V
φn =
N
kT
• ln D
q
ni
In real devices, there is a positive charge located at the Si/SiO2 interface and it modifies the
equation thus
VFB =− (φ m − φ n ) − Q I
t ox
ε ox
where QI is the Si/SiO2 interface charge density (# cm-2)
Threshold Voltage
VT =VFB − 2φ n −
t ox
ε ox
2ε Si qN D (2φ n )
φm = φp+ is the gate (metal) potential
φn is the potential of n-Si substrate
ni is the intrinsic carrier concentration of Si
tox is oxide thickness
εox , εSi the oxide & Si dielectric constants
ND is the substrate doping
QI is the oxide charge density at the interface
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