Body Effect for MOS Transistors
David Johns
University of Toronto
david.johns@utoronto.ca
Body, or Bulk (or Substrate)
So far, we have assumed a 3 terminal MOSFET
Actually, a MOSFET is a 4 terminal device
D D
G G B
S S
3 terminal 4 terminal
B ⇒ Body or Bulk (or Substrate)
2/19
Body, or Bulk (or Substrate)
S G D B
n+ n+ p+
p-
p+ is used to connect metal to p- body
If p+ is not used, metal direct to p- would result in a Schottky diode
3/19
Schottky Diode
p- metal
on voltage ≈ 0.3 V
We DO NOT want a Schottky diode (current only one direction
and a voltage drop)
By using metal to p+ to p-, we have an ohmic connection
4/19
Body Effect - Large Signal
VSB effect
D
G B
VSB
S
Effect of VSB can be modelled as changing the threshold voltage,
Vt . p p
Vt = Vt0 + γ[ 2Φf + VSB − 2Φf ] (1)
where
− Vt0 is the threshold voltage with VSB = 0
− 2Φf ≈ 0.6V (surface potential)
p
− γ = 2qNA S /COX
(NA - doping concentration of p-; s - permittivity of silicon; COX is
gate oxide capacitance per unit area)
5/19
Body Effect - Large Signal
VSB effect
− As VSB ⇑ then Vt ⇑
− In other words, if the source voltage is greater than the bulk voltage,
the threshold is increased.
Vt increase due to body effect
− Will reduce available signal swing especially for source-follower
amps
Bulk connection acts like another ”gate” if the source is held
constant
− If VBS ⇑ then Vt ⇓ and ID ⇑
For DC analysis, Vt depends on VSB and VSB may depend on Vt
− Hand analysis requires an iterative approach
− Best left for simulation
6/19
Body Effect - Small Signal
Recall the definition for gm which relates the change in drain
current to the change in VGS
∂ID ∂(0.5µn Cox (W /L)(VGS − Vt )2 )
gm ≡ =
∂VGS ∂VGS
(2)
gm = µn Cox (W /L)(VGS − Vt )
Since the body also ”controls” the drain current, we can also find
gmb
7/19
Body Effect - Small Signal
We define
∂ID
gmb ≡ (3)
∂VBS
∂(0.5µn Cox (W /L)(VGS − Vt )2 )
gmb =
∂VBS
∂Vt ∂VSB
= µn Cox (W /L)(VGS − Vt )(−1)
∂VSB ∂VBS
(4)
∂Vt
= µn Cox (W /L)(VGS − Vt )
∂VSB
∂Vt
= gm
∂VSB
8/19
Body Effect - Small Signal
Using (1) and defining χ as
∂Vt
χ≡
∂VSB
γ (5)
= p
2 2φf + VSB
We have that gmb is related to gm as
gmb = χgm (6)
Typical values for χ are 0.1 to 0.3
9/19
Body Effect - Small-Signal Model
D
G B
VBS
S
G B
vgs ro vbs
gm vgs gmb vbs
gmb = χgm
S
10/19
Body Effect Small-Signal Model
In the case where the bulk is a small-signal ground
G D
vgs ro
gm vgs gmb vsb
gmb = χgm
S
This is common in integrated circuits
Let’s look at 3 amps with body at small-signal ground
− Common-gate
− Common-drain
− Common-source
11/19
Common-Gate
vo
0
Rout
G B
0 = (1 + χ)g
gm
vi m
0 r )
0
Rout = ro 0 = (1+gm o 0 = (1+g 0 r )v
voc
isc ro vi m o i
All results same as 3 terminal device except that gm increased by
(1 + χ)
Due to vsg = vsb = vs since vg = vb = 0
12/19
Common-Drain
RD
vi
0
Rout
0 = (1 + χ)g
gm m
vo
0 ro +RD 0 = gm r o 0 = gm ro
Rout = 1+gm0 r isc ro +RD vi voc 1+gm0 r vi
o
o
Rout same as 3 terminal device except larger gm
isc same as 3 terminal device since output shorted so vsb = 0
13/19
Common-Drain
For gm ro 1
0 gm 1
voc ≈ 0 vi = vi (7)
gm 1+χ
voc is reduced
Rout also reduced but usually overall gain is reduced by body
effect.
14/19
Common-Source
vo
0
Rout
vi
RS 0 = (1 + χ)g
gm m
0 =r +(1+g 0 r )R
Rout 0 =
isc −gm ro
vi 0 = −g r v
voc
o m o S r 0
o +(1+gm ro )RS
m o i
Rout same as 3 terminal device but larger gm
voc same as 3 terminal device since when drain open, no current
through RS so vs = 0 so vsb = 0
15/19
Common-Source
For gm ro 1
0 −gm ro vi −gm vi
isc ≈ 0 r R = 1 + g0 R
ro + g m o S m S
(8)
−vi
≈
(1/gm ) + (1 + χ)RS
Body effect:
− isc reduced
− Rout increased
16/19
Example 1
Common-drain
VDD gm1 = 1mA/V
ro1 = 20kΩ
RD gm2 = 2mA/V
100Ω ro2 = 10kΩ
χ = 0.2
M1 0 = 1.2mA/V
gm1
vi
vo
Rout
M2 RL
VB
5kΩ
M1 has body effect since VB 6= VS
Rout = RS1 ||RD2
Ro = Rout ||RL
ro1 +RD
RD2 = ro2 = 10kΩ; RS1 = (1+gm10 r ) = 804Ω
o1
17/19
Example 3
Rout = RS1 ||RD2 = 744Ω
Ro = Rout ||RL = 647Ω
For isc we have isc = Gm vi where
Gm = (gm1 ro1 )/(ro1 + RD ) = 995µA/V
vo /vi = Gm × Ro = 0.644V/V
Without body effect
− vo /vi = Gm × Ro = 0.74V/V
− A gain reduction of 13% when body effect included
18/19
Example 3 - Approx Solution
0 ) + R /(g 0 r ) = 838Ω
RS1 = (1/gm1 D m1 o1
RD2 = ro2 = 10kΩ
1
voc = 1+χ vi = 0.833vi
vo node is a resistor divider node
(RD2 ||RL ) 3.33k
vo = (RD2 ||RL )+RS1 voc = 3.33k+838 (0.833)vi
vo/vi = 0.665V/V
19/19
