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