Lecture 6: Integrated Circuit Resistorsee105/fa03/handouts/lectures/Lecture6.pdfDepartment of EECS...
Transcript of Lecture 6: Integrated Circuit Resistorsee105/fa03/handouts/lectures/Lecture6.pdfDepartment of EECS...
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6
Lecture 6: Integrated Circuit Resistors
Prof. Niknejad
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
Lecture Outline
Semiconductors Si Diamond StructureBond Model Intrinsic Carrier ConcentrationDoping by Ion ImplantationDriftVelocity Saturation IC Process Flow Resistor LayoutDiffusion
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
Resistivity for a Few Materials
Pure copper, 273K 1.56×10-6 ohm-cmPure copper, 373 K 2.24×10-6 ohm-cmPure germanium, 273 K 200 ohm-cmPure germanium, 500 K .12 ohm-cmPure water, 291 K 2.5×107 ohm-cmSeawater 25 ohm-cm
What gives rise to this enormous range?Why are some materials semi-conductive?Why the strong temp dependence?
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
Electronic Properties of Silicon
Silicon is in Group IV – Atom electronic structure: 1s22s22p63s23p2
– Crystal electronic structure: 1s22s22p63(sp)4
– Diamond lattice, with 0.235 nm bond length
Very poor conductor at room temperature: why?
(1s)2
(2s)2
(2p)6 (3sp)4
Hybridized State
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
Periodic Table of Elements
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
The Diamond Structure
3sp tetrahedral bond
o
A43.5
o
A35.2
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
States of an Atom
Quantum Mechanics: The allowed energy levels for an atom are discrete (2 electrons can occupy a state since with opposite spin)When atoms are brought into close contact, these energy levels splitIf there are a large number of atoms, the discrete energy levels form a “continuous” band
Ene
rgy
E1
E2
...E3
Forbidden Band Gap
AllowedEnergyLevels
Lattice ConstantAtomic Spacing
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
Energy Band Diagram
The gap between the conduction and valence band determines the conductive properties of the materialMetal– negligible band gap or overlap
Insulator – large band gap, ~ 8 eV
Semiconductor– medium sized gap, ~ 1 eV
Valence Band
Conduction Band
Valence Band
Conduction Band
e-Electrons can gain energy from lattice (phonon) or photon to become “free”
band gap
e-
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
Model for Good ConductorThe atoms are all ionized and a “sea” of electrons can wander about crystal:The electrons are the “glue” that holds the solid togetherSince they are “free”, they respond to applied fields and give rise to conductions
+ + + + + + + +
+ + + + + + + +
+ + + + + + + +
On time scale of electrons, lattice looks stationary…
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
Bond Model for Silicon (T=0K)Silicon Ion (+4 q)
Four Valence ElectronsContributed by each ion (-4 q)
2 electrons in each bond
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
Bond Model for Silicon (T>0K)
Some bond are broken: free electronLeave behind a positive ion or trap (a hole)
+
-
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
Holes?
Notice that the vacancy (hole) left behind can be filled by a neighboring electronIt looks like there is a positive charge traveling around!Treat holes as legitimate particles.
+-
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
Yes, Holes!The hole represents the void after a bond is brokenSince it is energetically favorable for nearby electrons to fill this void, the hole is quickly filledBut this leaves a new void since it is more likely that a valence band electron fills the void (much larger density that conduction band electrons)The net motion of many electrons in the valence band can be equivalently represented as the motion of a hole
∑ ∑∑ −−−=−=BandFilled StatesEmpty
iivb
ivb vqvqvqJ )()()(
∑∑ =−−=StatesEmpty
iStatesEmpty
ivb qvvqJ )(
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
More About Holes
When a conduction band electron encounters a hole, the process is called recombinationThe electron and hole annihilate one another thus depleting the supply of carriersIn thermal equilibrium, a generation process counterbalances to produce a steady stream of carriers
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
Thermal Equilibrium (Pure Si)
Balance between generation and recombination determines no = po
Strong function of temperature: T = 300 oK
optth GTGG += )(
)( pnkR ×=
RG =)()( TGpnk th=×
)(/)( 2 TnkTGpn ith ==×
K300atcm10)( 310 −≅Tni
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
Doping with Group V Elements
P, As (group 5): extra bonding electron … lost to crystal at room temperature
+
ImmobileCharge
Left Behind
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
Donor Accounting
Each ionized donor will contribute an extra “free”electronThe material is charge neutral, so the total charge concentration must sum to zero:
By Mass-Action Law:
000 =++−= dqNqpqnρ
Free Electrons
Free Holes
Ions(Immobile)
)(2 Tnpn i=×
00
2
0 =++− di qN
nnqqn
0022
0 =++− nqNqnqn di
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
Donor Accounting (cont)
Solve quadratic:
Only positive root is physically valid:
For most practical situations:
24
022
0
20
20
idd
id
nNNn
nnNn
+±=
=−−
24 22
0idd nNN
n++
=
id nN >>
ddd
idd
NNNNnNN
n =+≈
++
=222
412
0
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
Doping with Group III ElementsBoron: 3 bonding electrons one bond is unsaturatedOnly free hole … negative ion is immobile!
-
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
Mass Action Law
Balance between generation and recombination:
2ioo nnp =⋅
• N-type case:
• P-type case:
)cm10,K300( 310 −== inT
dd NNn ≅= +0
aa NNp ≅= −0
d
i
Nnn
2
0 ≅
a
i
Nnp
2
0 ≅
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
CompensationDope with both donors and acceptors: – Create free electron and hole!
+
-
-
+
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
Compensation (cont.)
More donors than acceptors: Nd > Na
iado nNNn >>−=
• More acceptors than donors: Na > Nd
ado NN
np i
−=
2
idao nNNp >>−=da
o NNn
n i
−=
2
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
Thermal Equilibrium
Rapid, random motion of holes and electrons at “thermal velocity” vth = 107 cm/s with collisions every τc = 10-13 s.
Apply an electric field E and charge carriers accelerate … for τc seconds
zero E field
vth
positive E
vth
aτ c
(hole case)
x
kTvm thn 212*
21 =
cthv τλ =
cm1010/cm10 6137 −− =×= ssλ
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
Drift Velocity and Mobility
Ev pdr µ=
Emq
mqE
mFav
p
cc
pc
p
ecdr
=
=
=⋅=
ττττ
For electrons:
Emq
mqE
mFav
p
cc
pc
p
ecdr
−=
−=
=⋅=
ττττ
For holes:
Ev ndr µ−=
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 6 Prof. A. Niknejad
“default” values:
Mobility vs. Doping in Silicon at 300 oK
1000=nµ 400=pµ