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Department of EECS University of California, Berkeley
Prof. Ali M. NiknejadProf. Rikky Muller
Module 2.2:IC Resistors and Capacitors
EE 105 Fall 2016 Prof. A. M. Niknejad
2
IC Fabrication: Si Substratel Pure Si crystal is starting material (wafer)l The Si wafer is extremely pure (~1 part in a billion
impurities) l Why so pure?
– Si density is about 5 10^22 atoms/cm^3– Desire intentional doping from 10^14 – 10^18– Want unintentional dopants to be about 1-2 orders of
magnitude less dense ~ 10^12 l Si wafers are polished to about 700 µm thick
(mirror finish)l The Si forms the substrate for the IC
University of California, Berkeley
EE 105 Fall 2016 Prof. A. M. Niknejad
3
IC Fabrication: Oxidel Si has a native oxide: SiO2
l SiO2 (Quartz) is extremely stable and very convenient for fabrication
l It’s an insulators so it can be used for house interconnection
l It can also be used for selective dopingl SiO2 windows are etched using photolithographyl These openings allow ion implantation into selected
regionsl SiO2 can block ion implantation in other areas
University of California, Berkeley
EE 105 Fall 2016 Prof. A. M. Niknejad
4
IC Fabrication: Ion Implantation
l Si substrate (p-type)l Grow oxide (thermally)l Add photoresistl Expose (visible or UV source)l Etch (chemical such as HF)l Ion implantation (inject dopants)l Diffuse (increase temperature and allow dopants to diffuse)
University of California, Berkeley
P-type Si Substrate
oxide
N-type diffusion region
EE 105 Fall 2016 Prof. A. M. Niknejad
5
“Diffusion” Resistor
l Using ion implantation/diffusion, the thickness and dopant concentration of resistor is set by process
l Shape of the resistor is set by design (layout)l Metal contacts are connected to ends of the resistorl Resistor is capacitively isolation from substrate
– Reverse Bias PN Junction!University of California, Berkeley
P-type Si Substrate
N-type Diffusion RegionOxide
EE 105 Fall 2016 Prof. A. M. Niknejad
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Poly Film Resistor
l To lower the capacitive parasitics, we should build the resistor further away from substrate
l We can deposit a thin film of “poly” Si (heavily doped) material on top of the oxide
l The poly will have a certain resistance (say 10 Ohms/sq)
University of California, Berkeley
OxidePolysilicon Film (N+ or P+ type)
P-type Si Substrate
EE 105 Fall 2016 Prof. A. M. Niknejad
7
Ohm’s Lawl Current I in terms of Jnl Voltage V in terms of electric field
– Result for R
University of California, Berkeley
IRV =
JtWJAI ==
LVE /=
EWtJtWJAI s===
VLWtJtWJAI s
===
tWLRs1
=tW
LR r=
EE 105 Fall 2016 Prof. A. M. Niknejad
8
Sheet Resistance (Rs)
l IC resistors have a specified thickness – not under the control of the circuit designer
l Eliminate t by absorbing it into a new parameter: the sheet resistance (Rs)
University of California, Berkeley
÷øö
çèæ=÷
øö
çèæ÷øö
çèæ==
WLR
WL
tWtLR sq
rr
“Number of Squares”
EE 105 Fall 2016 Prof. A. M. Niknejad
9
Using Sheet Resistance (Rs)
l Ion-implanted (or “diffused”) IC resistor
University of California, Berkeley
EE 105 Fall 2016 Prof. A. M. Niknejad
10
Idealizations
l Why does current density Jn “turn”?l What is the thickness of the resistor?l What is the effect of the contact regions?
University of California, Berkeley
EE 105 Fall 2016 Prof. A. M. Niknejad
11
Electrostatics Review (1)l Electric field go from positive charge to negative
charge (by convention)
l Electric field lines diverge on charge
l In words, if the electric field changes magnitude, there has to be charge involved!
l Result: In a charge free region, the electric field must be constant!
University of California, Berkeley
+++++++++++++++++++++
− − − − − − − − − − − − − − −
er
=×Ñ E
EE 105 Fall 2016 Prof. A. M. Niknejad
12
Electrostatics Review (2)l Gauss’ Law equivalently says that if there is a net
electric field leaving a region, there has to be positive charge in that region:
University of California, Berkeley
+++++++++++++++++++++
− − − − − − − − − − − − − − −
Electric Fields are Leaving This Box!
ò =×eQdSE
òò ==×ÑVV
QdVdVE eer /
eQdSEdVE
SVòò =×=×Ñ
Recall:
EE 105 Fall 2016 Prof. A. M. Niknejad
13
Electrostatics in 1Dl Everything simplifies in 1-D
l Consider a uniform charge distribution
University of California, Berkeley
er
==×ÑdxdEE dxdE
er
=
')'()()(0
0 dxxxExEx
xò+=
er
)(xr
xxdxxxE
x
er
er 0
0
')'()( == ò
Zero fieldboundarycondition
1x
0r
1x
)(xE1
0 xer
EE 105 Fall 2016 Prof. A. M. Niknejad
14
Electrostatic Potentiall The electric field (force) is related to the potential
(energy):
l Negative sign says that field lines go from high potential points to lower potential points (negative slope)
l Note: An electron should “float” to a high potential point:
University of California, Berkeley
dxdE f
-=
dxdeqEFef
-==1f
2f
dxdeFef
-=e
EE 105 Fall 2016 Prof. A. M. Niknejad
15
More Potentiall Integrating this basic relation, we have that the
potential is the integral of the field:
l In 1D, this is a simple integral:
l Going the other way, we have Poisson’s equation in 1D:
University of California, Berkeley
ò ×-=-C
ldExx!
)()( 0ff
)(xf
)( 0xf E
ld!
ò-=-x
xdxxExx
0
')'()()( 0ff
erf )()(
2
2 xdxxd
-=
EE 105 Fall 2016 Prof. A. M. Niknejad
16
Boundary Conditionsl Potential must be a continuous function. If not, the fields
(forces) would be infinite l Electric fields need not be continuous. We have already
seen that the electric fields diverge on charges. In fact, across an interface we have:
l Field discontiuity implies charge density at surface!
University of California, Berkeley
)( 11 eE
)( 22 eE
ò =+-=× insideQSESEdSE 2211 eeexD
00¾¾ ®¾ ®DxinsideQ
02211 =+- SESE ee
1
2
2
1
ee
=EES
EE 105 Fall 2016 Prof. A. M. Niknejad
17
IC MIM Capacitor
l By forming a thin oxide and metal (or polysilicon) plates, a capacitor is formed
l Contacts are made to top and bottom platel Parasitic capacitance exists between bottom plate and
substrate
University of California, Berkeley
Top PlateBottom Plate Bottom Plate
Contacts
CVQ =
Thin Oxide
EE 105 Fall 2016 Prof. A. M. Niknejad
18
Review of Capacitors
l For an ideal metal, all charge must be at surfacel Gauss’ law: Surface integral of electric field over
closed surface equals charge inside volume
University of California, Berkeley
+++++++++++++++++++++
− − − − − − − − − − − − − − −
+− Vs
ò =×eQdSE
ò -=×eQdSE
sox VtEdlE ==×ò 0ox
s
tVE =0
ò ==×eQAEdSE 0 e
QAtV
ox
s =
sCVQ =
oxtAC e
=
EE 105 Fall 2016 Prof. A. M. Niknejad
19
Capacitor Q-V Relation
l Total charge is linearly related to voltagel Charge density is a delta function at surface (for
perfect metals)
University of California, Berkeley
sCVQ =
sV
Q
y
)(yQ
y+++++++++++++++++++++
− − − − − − − − − − − − − − −
EE 105 Fall 2016 Prof. A. M. Niknejad
20
A Non-Linear Capacitor
l We’ll soon meet capacitors that have a non-linear Q-V relationship
l If plates are not ideal metal, the charge density can penetrate into surface
University of California, Berkeley
)( sVfQ =
sV
Q
y
)(yQ
y+++++++++++++++++++++
− − − − − − − − − − − − − − −
EE 105 Fall 2016 Prof. A. M. Niknejad
21
What’s the Capacitance?l For a non-linear capacitor, we have
l We can’t identify a capacitancel Imagine we apply a small signal on top of a bias
voltage:
l The incremental charge is therefore:
University of California, Berkeley
ss CVVfQ ¹= )(
sVV
sss vdVVdfVfvVfQ
s=
+»+=)()()(
Constant charge
sVV
s vdVVdfVfqQQ
s=
+»+=)()(0
EE 105 Fall 2016 Prof. A. M. Niknejad
22
Small Signal Capacitancel Break the equation for total charge into two terms:
University of California, Berkeley
sVV
s vdVVdfVfqQQ
s=
+»+=)()(0
ConstantCharge
IncrementalCharge
ssVV
vCvdVVdfq
s
===
)(
sVVdVVdfC
=
º)(
EE 105 Fall 2016 Prof. A. M. Niknejad
23
Example of Non-Linear Capacitorl We’ll see that for a PN junction, the charge is a
function of the reverse bias:
l Small signal capacitance:
University of California, Berkeley
bpaj
VxqNVQf
--= 1)(
ConstantsCharge At N Side of Junction
Voltage Across NPJunction
b
j
b
b
pajj V
CV
xqNdVdQ
VC
fff
-=
-==
11
12
)( 0
