3.
Current-Voltage Relationships
3.1 Influence of Drain-Source Voltage
Discussion of the operation of the MOS transistor up to now has
assumed that the drain-source voltage, VDS, was equal to zero. This
meant that conditions in the capacitive layer were controlled solely by
the gate-source voltage, VGS. It is also possible to vary the drainsource voltage, VDS, so that when a conducting channel has been
formed, current can flow through this channel. The conditions in the
channel and the current which flows through it are also dependent on
the value of the drain-source voltage applied.
Zero Drain-Source Voltage with Channel Formed, VGS  VT, VDS = 0
When the gate-source voltage is greater than the threshold
voltage, a conducting channel is formed. The voltage VGS - VT can be
thought of as the channel-forming voltage. If the drain-source voltage
is zero this channel will be uniform along its length as can be seen in
Fig. 3.1(a). The depletion region underneath the channel will also be
uniform along the channel between source and drain.
(i)
Non-Saturation Region, VGS  VT, 0  VDS  VGS - VT
When the drain-source voltage is increased above zero an electric
field now exists along the length of the channel, acting from drain to
source which causes electrons in the channel to flow from source to
drain. This gives rise to a conventional current flowing in the opposite
direction from drain to source, referred to as the drain current, iD, of
the transistor. This electric field also causes the conducting channel to
taper becoming narrower at the drain end as shown in Fig. 3.1(b). The
depletion region also becomes wider around the drain. This tapering of
the channel becomes more extensive as VDS is increased but is
maintained less than the value VGS - VT. This region of operation of the
transistor is known as the non-saturation region (often referred to a
little confusingly as the linear region). The drain current that flows
also increases with increasing VDS, for a given value of VGS, but not in a
linear fashion as shall be seen.
(ii)
(iii) Saturation Region, VGS  VT, VDS  VGS - VT
If the drain-source voltage is further increased, it will be
accompanied by an increase in drain current. Eventually, however, it
will become equal to the channel forming voltage applied to the gate,
VDS = VGS - VT. When this occurs, the tapering of the channel becomes
complete to the point that the channel becomes closed off at the drain
1
VDS = 0V
VSB = 0V
VGS >VT
G
B
D
S
n+
n+
pFig. 3.1(a) Uniform Channel Formed in MOSFET with VGS > VT, VDS = 0
0<VDS<VGS-VT
VSB = 0V
VGS >VT
G
B
D
S
n+
n+
pFig. 3.1(b) MOSFET in Non-Saturation Mode, VGS >VT, 0 <VDS < VGS -VT
2
end as shown in Fig. 3.1(c). This condition is referred to as ‘pinch-off’
of the channel. It should be noted that effectively what has happened
here is that the voltage at the drain, relative to the source, has become
equal to the channel forming voltage VGS - VT so that the channel
cannot remain established at the drain. Once this happens the drain
current becomes limited to its value at pinch-off and further increase
in the drain-source voltage brings only a slight increase in drain
current, i.e. the drain current saturates. Consequently, this region of
operation where VDS > VGS - VT is called the saturation region. Most
analogue circuits operate entirely in this region.
(iv) Channel Length Modulation
In older devices, where the length of the channel is greater than
the minimum technology dimension (long-channel devices), the
conditions in the channel remain at those of pinch-off once the
saturation region is entered. Hence the value of current for a given
gate-source voltage remains constant at the pinch-off value when
operating in the saturation region. In more modern devices, where
channel lengths are much shorter (short-channel devices), the pinchoff condition extends along the channel towards the source when VDS
> VGS - VT, as shown in Fig. 3.1(d). This effectively shortens the
channel slightly as VDS increases, a phenomenon referred to as channel
length modulation. This, in effect, means that the depletion region
begins to extend back towards the source. However, current flow is
maintained by acceleration of electrons through this region in a thin
film close to the surface of the semiconductor under the influence of
the very high electric field here. Indeed, in saturation, the increase in
VDS above the channel forming voltage is developed almost entirely
across the depletion region at the shortened end of the channel. This
shortening of the channel also allows the drain current to increase
slightly when operating in the saturation region.
3
VDS = VGS-VT
VSB = 0V
channel
pinched
off at
drain end
VGS >VT
G
B
D
S
n+
n+
pFig. 3.1(c)
MOSFET at Channel Pinch-Off, VGS > VT, VDS = VGS -VT
VDS > VGS-VT
VSB = 0V
VGS >VT
shortened or
modulated
channel
G
B
D
S
n+
n+
pFig. 3.1(d) Channel Length Modulation in MOSFET, VGS >VT, VDS >VGS-VT
4
3.2 Current-Voltage Relationships
(i) Non Saturation Region, VGS  VT, 0  VDS  VGS - VT
Fig. 3.2 shows a diagram representing the channel induced in the
MOS Transistor underneath the oxide when operating in the nonsaturation region. It can be seen to be wedge-shaped due to the
influence of the drain-source voltage. The x-direction will be taken as
the vertical direction under the oxide while the y-direction will be
taken as horizontal distance along the channel as shown. It is
assumed, for simplicity of analysis, that the threshold voltage is
constant along the length of the channel and that the distribution of
charge is uniform in any elemental section dy, having a depth xc. As
already outlined, when the drain-source voltage is zero, VDS = 0, the
effective voltage forming the channel is the gate-source voltage over
and above the threshold voltage, VGS - VT. When the drain-source
voltage is not zero, the effective voltage forming the channel is
reduced by an opposing contribution from the drain-source voltage Vy
and is modified to VGS - VT – Vy. This channel voltage varies as a
function of y depending on location in the channel with Vy = 0 at y = 0,
the source end and Vy = VDS at y = L, the drain end.
The total charge induced in the channel per unit area of oxide at
the point y, the location of the elemental section dy, is then given as:
Qy  COX VGS  VT  Vy 
The charge per unit volume in this section is then given as:
ρy 
Qy
xc
It will be recalled that the drift current for electrons caused by an
electric field E is given as:
Idrift  JA  ρμnEA
where the direction of current flow is the same as the direction of the
electric field. Applied to the elemental section of the channel, dy, this
gives for the drain current:
ID  ρyμnEy Wxc
5
direction of electric field
0
x
y
y=L
E
L
Source
W
Drain
xc
elemental
section of
channel
dy
Fig. 3.2
inverted
Channel
VDS < VGS - VT
Elemental Section of Channel with 0<VDS < VGS - VT
6
since the electric field is acting in the negative y direction. Substituting
for y and taking the electric field as constant along the elemental
length, E y   dVy dy gives:
Qy
ID 
xc
μn
dVy
Wxc  QyμnW
dy
dVy
dy
This expression applying to the elemental length dy can now be
integrated along the length of the channel from source, y = 0 to drain,
y = L so that:
L

0
IDdy 

VDS
0
QyμnWdVy
Substituting for Qy from above gives:
L

0
IDdy  μnWCOX 
VDS
V
GS
0
 VT  Vy dVy
so that:
0 IDdy  μnWCOX 0
L
VDS
VGS  VT  dVy  0
VDS
VydVy 

Integrating then gives:
2


VDS
IDL  μnWCOX VGS  VT VDS 

2 

and finally:
ID 

1
W
2
μnCOX
2VGS  VT VDS  VDS
2
L

The prefix constant term is referred to as the transconductance
parameter for the MOSFET
Kn 
1
W
μnCOX
2
L
and has dimensions of AV-2 so that
2
ID  K n[ 2VGS  VT VDS  VDS
]
It can be seen from this that the level of current flow in the MOS
transistor can be controlled by the physical dimensions of length, L
and width, W in its fabrication. In fact it is these dimensions which are
the principal tool at the disposal of the circuit design engineer.
7
ID
in
A
Boundary
VDS =VGS - VT
Non-saturation
(linear) region
Saturation region
VGS = V5
VGS = V4
VGS = V3
VGS = V2
VGS =V1
0
VDS in Volts
Fig. 3.3(a) Drain Current vs Drain-Source Voltage
in a Long-Channel MOSFET
8
Fig. 3.3(a) shows a plot of the drain current vs drain-source voltage
with the gate source voltage as an individual parameter. It can be
seen that these curves have a hyperbolic shape. However, it is
important to remember that this relationship is valid only for operation
in the non-saturation region where VDS  VGS - VT.
Of interest is the point where the maximum occurs on the curves. This
can be found by differentiating the above expression for drain current
with respect to VDS and equating to zero:
ID
 Kn 2VGS  VT   2VDS   0
VDS
which is satisfied for VDS = VGS - VT.
This shows that the drain current reaches a maximum value at the
boundary between the non-saturation and the saturation region as
shown in Fig. 3.3(a).
(ii) Saturation Region
For long-channel devices it has been seen that the drain current
saturates at its pinch-off value at VDS = VGS - VT and remains at this
value in the saturation region. Hence, an expression for the drain
current in saturation can be obtained by simply substituting the
boundary condition into the expression obtained for operation in the
non-saturation region. This gives:
ID  Kn[2VGS  VT VGS  VT   VGS  VT  ]
2
which gives:
ID  Kn VGS  VT 
2
This relationship is shown in Figure 3.3(a).
9
(iii) Channel Length Modulation
For short-channel devices, the effect of channel length modulation
must be accounted for. It can be seen in Fig. 3.3(b) that the drain
current in these devices increases slightly in the saturation region as
the drain-source voltage is increased. However, it can also be seen
that the slope in this region is constant indicating a linear dependency.
This can be accounted for relatively simply by including the channel
length modulation factor,  , which is applied to the drain current in
the saturation region. The expression for the drain current is then
modified to become:
ID  K n VGS  VT  1  λV
2
DS

where  has dimensions of V-1 and has a typical value of 0.005 to 0.05.
10
ID
in
A
Boundary
VDS =VGS - VT
Non-saturation
(linear) region
Saturation region
VGS = V5
VGS = V4
VGS = V3
VGS = V2
VGS = V1
VDS in Volts
0
Fig. 3.3(b) Drain Current vs Drain-Source Voltage
for Short-Channel MOSFET
11