IEEE Solid-State Circuits Magazine - Fall 2015 - 88

For a single-transistor amplifier, the
GBW can be substituted by (6), giving rise to the second expression.
It follows thus the same curve as
g m /I DS in Figures 2, 5, and 6. The
FOM is largest in weak inversion and
decreases toward strong inversion.
For the example above, at IC =
0.064, the FOM is 4,370 MHzpF/mA
but decreases to 3,900 at IC = 0.1
and 2,880 at IC = 1. For IC = 10 it is
only 1,380 MHz, about one third of
the value at deep weak inversion.

1
nUT

gm
IDS

1/2

λc
nUT

-1

IC

IDS

gmnUT

1
gmnUT

λc

1/2

IC

IDSsp
-1

gmnUT

gmnUTλc

-1/2
2

1

Future CMos Technologies
For
smaller
channel
lengths,
1/m 2c = (L/20 nm) 2 decreases as well
as it becomes unity for L = L sat c
20 nm. This situation is depicted (in
red) in Figure 7 [7]. For even smaller
channel lengths, 1/m 2c becomes
smaller than unity as shown (in
green). The black curves are taken
from Figure 2. They are valid for a
channel length larger than 20 nm,
such as, for example, 65 nm.
The three regions of operation
are easily identified. Weak inversion
occurs for IC < 1. Velocity saturation
occurs for IC > 1/m 2c = (L/20 nm) 2
and strong inversion is in the middle. The values of g m are easily
derived from (3). For channel lengths
of 20 nm and below, the stronginversion region has vanished. Moreover, the values in weak inversion

IC

1/λc

Figure 6: Normalized (a) g m /I DS, (b) I DS, and (c) I DSsp or W/L versus IC.

IC = 1 is about the minimum value,
below which the current I DS hardly
decreases. For high-frequency applications, IC = 1/m 2c = (L/20 nm) 2 is
the maximum value. For two-stage
amplifiers, a similar design procedure can be developed [7].
The performance of amplifiers
is compared by means of a figure
of merit (FOM) given by the first
expression in

length. Then the IC must be reduced
to lower values, closer to IC min .
The curves of Figure 4 can also
be derived in asymptotic form, from
Figure 2, leading to similar conclusions. They are shown in Figure 6.
They are derived for constant
g m, obtained from the GBW and C L
as given by (6). Parameter IC is thus
the only design variable. Figure 6(a)
is taken from Figure 2. For constant
g m, the current I DS is the inverse.
Figure 6(c) is I DSsp, which is equal
to I DS /IC. Finally W/L is part of I DSsp
as given by (3). It is clear again, that

FOM = GBW.C L
IB
g m1
=
.
2r I DS

L < 20 nm

gm

Gsp

L > 20 nmt

1/2
Gsp IC

gm
IDS

Gsp
λc

1

1

nUT

nUT √ IC

= WCoxvsat

1
(a)

fa l l 2 0 15

fTsp IC

λc

=

vsat

2πL

1
IC

fT gm
IDS

1/λc2

IC

IEEE SOLID-STATE CIRCUITS MAGAZINE

fTsp

1

nUT

Figure 7: (a) g m, (b) g m /I DS, (c) fT , and (d) fT x g m /I DS versus IC.

88

fTsp

1/2

fTsp IC

-1
1/λc2

L > 20 nmt

fTsp √ IC

2
vsat

1
nUTλcIC

λc
nUT

L = 20 nmt
fTsp

IC

1

L < 20 nm

fT

L = 20 nm
Gsp √ IC

(9)

2

1/λc

nUT

1

2π nUT 2µUTIC
24 THz/V
≈
-1
IC
2

(b)

1/λc

IC
Table of Contents for the Digital Edition of IEEE Solid-State Circuits Magazine - Fall 2015
