IEEE Solid-State Circuits Magazine - Fall 2015 - 87

use it to make an improved choice.
For reasons of parasitic poles, the
GBW cannot be much larger than
1/10 of the fT [5]. Moreover, we
assume that we end up in deep weak
inversion and that IC < 1. This is
very reasonable as the value of fT
is usually quite small with respect
to fTsp, given in (1). In this region. fT
equals IC.fTspec as shown in Figure 2.
In deep weak inversion the minimum value of the IC is thus given by

W/L
W/L
W
IDS µA
fT GHz
1 00E+02
1.00E+02
ICminn = 0.0064

0.01

0.1

(7)

For example, to design an amplifier
with a GBW of 50 MHz in 65-nm CMOS
(with an fTsp of 78 GHz), the IC min
is about 0.0064, which is clearly
a lot smaller than unity. If we take
IC min = 0.1, the corresponding GBW
is 780 MHz. This example shows
that, for smaller and smaller channel lengths, even real-high-speed
amplifiers end up in weak inversion.
It doesn't make much sense to
use such small values of the IC as
the current consumption doesn't
decrease any more for values
below unity. This is also shown in a
straight calculation of the I DS, W/L,
and fT versus the IC [7], shown in
Figure 4. In this example L = 65 nm,
fTsp = 78 GHz (n = 400 cm 2 /Vs) . For
GBW = 50 MHz, IC min = 6.410 -3 . For
C L = 2 pF, g m = 0.63 mS.
Figure 4 shows the resulting values. For IC = IC min = 0.0064, I DS =
23 nA, W/L = 1,860, C GS = 0.2 pF,
fT = 480 MHz. For IC = 0.1, I DS = 25 nA
is somewhat larger but W/L = 134 is a
lot smaller, leading to C GS = 15 fF and
fT = 6.7 GHz, which is a lot larger as
well. It is clear that for very small values of the IC, the transistor size W/L
blows up, reducing fT considerably.
Thus, choosing very low values of the
IC should be avoided. The values of fT
are reduced considerably without giving much lower current consumption.
For IC = 1, the full expressions
must be used [7]. The current will
increase a bit more but the W/L will
be even smaller and fT much larger.
To explore this intermediate region

1.00E+01
1
00E+01

1

10

IC
Figure 4: Values of fT , W/L, and I DS versus IC for constant g m = 0.63 mS [7].

around IC = 1 into more detail, the
normalized g m /I DS curve, shown
in Figure 2, is repeated in Figure 5.
Actual values are shown now instead
of asymptotic ones. Velocity saturation is omitted.
This curve is easily derived from
(3) and given by
gm
nU T = 1 - e
I DS
IC

IC

.

must be used to extract the value of
the current I DS from the value of g m .
For the above example, I DS = 34 nA,
which is larger indeed than 25 at
IC = 0.1. Now W/L = 18 is again much
smaller, leading to C GS = 2 fF and
fT = 49 GHz.
For values of VGS - VT c 0.2, which
correspond to IC = 10, I DS = 72 nA
and W/L = 3.8, leading to an fT of
237 GHz. This is clearly quite high.
Parasitic capacitances and resistances
will reduce this value to less than half!
Whatever choice of the IC is taken,
the resulting value of fT may have to
be verified. This shown with a dotted line in Figure 3. If a value of
the IC is chosen too high, then the
resulting value of fT is too high with
respect to fTsp . This means that such
a high value of fT cannot actually
be reached for this specific channel

(8)

Its value is clearly unity for a small
IC. Some more values show that this
function changes very slowly with
the IC. At IC = 0.1, used in the above
example, the current is already
wrong with a difference of 0.14. Real
weak inversion is thus reached only
for values of the IC below 0.01! An
error of only 5% is negligible indeed!
For values of the IC around one
and higher, this full expression (8)

gm
nUT
IDS

IC min . 10 GBW .
fTsp

1.00E+03

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.001

0.95
0.86
0.63

0.30

0.01

0.1

1

10

100

IC
Figure 5: Normalized g m /I DS versus IC for weak and strong inversion.

IEEE SOLID-STATE CIRCUITS MAGAZINE

fa l l 2 0 15

87



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