Department of EECS University of California, Berkeley

Prof. Ali M. Niknejad

Lecture 6: Semiconductor Conduction

EE 105 Fall 2016 Prof. A. M. Niknejad

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Lecture Outline

l Physics of conductionl Semiconductors

– Si Diamond Structure– Bond Model

l Intrinsic Carrier Concentration– Doping by Ion Implantation

l Drift– Velocity Saturation

l Diffusion

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EE 105 Fall 2016 Prof. A. M. Niknejad

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Physics of Conduction

EE 105 Fall 2016 Prof. A. M. Niknejad

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Ohm’s Lawl One of the first things we learn as EECS majors is:

l Is this trivial? Maybe what’s really going on is the following:

l In the above Taylor exansion, if the voltage is zero for zero current, then this is generally valid

l The range of validity (radius of convergence) is the important question. It turns out to be VERY large!

RIV ´=

IfIfIffIfV )0('...2/)0('')0(')0()( 2 »+++==

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EE 105 Fall 2016 Prof. A. M. Niknejad

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Ohm’s Law Revisitedl In Physics we learned:

l Is this also trivial? Well, it’s the same as Ohm’s law, so the questions are related. For a rectangular solid:

l Isn’t it strange that current (velocity) is proportional to Force?

l Where does conductivity come from?

EJ s=

IRIALV

LV

AIJ ====

ss

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EE 105 Fall 2016 Prof. A. M. Niknejad

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Conductivity of a Gasl Electrical conduction is due to the motion of

positive and negative chargesl For water with pH=7, the concentration of

hydrogen H+ ions (and OH-) is:

l Typically, the concentration of charged carriers is much smaller than the concentration of neutral molecules

l The motion of the charged carriers (electrons, ions, molecules) gives rise to electrical conduction

313323-103-107 cm106cm1002.610mole/cm10mole/L10 --- ´=´´==

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EE 105 Fall 2016 Prof. A. M. Niknejad

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Collisions in Gasl At a temperate T, each charged carrier will move in

a random direction and velocity until it encounters a neutral molecule or another charged carrier

l Since the concentration of charged carriers is much less than molecules, it will most likely encounter a molecule

l For a gas, the molecules are widely separated (~ 10 molecular diameters)

l After colliding with the molecule, there is some energy exchange and the charge carrier will come out with a new velocity and new direction

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EE 105 Fall 2016 Prof. A. M. Niknejad

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Memory Loss in Collisionsl Schematically

l Key Point: The initial velocity and direction is lost (randomized) after a few collisions

NeutralMolecule

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EE 105 Fall 2016 Prof. A. M. Niknejad

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Application of Fieldl When we apply an electric field, during each “free

flight”, the carriers will gain a momentum ofl Therefore, after t seconds, the momentum is given

by:

l If we take the average momentum of all particles at any given time, we have:

qtE

qtM Eu +

( )å +=j

jj qtMN

M Euu 1

Number ofCarriers

Initial Momentum

Before Collision

MomentumGained from

Field

Time from Last Collision

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EE 105 Fall 2016 Prof. A. M. Niknejad

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Random Things Sum to Zero!l When we sum over all the random velocities of the

particles, we are averaging over a large number of random variables with zero mean, the average is zero

( )å +=j

jj qtMN

M Euu 1

tqqtN

Mj

j EEu == å1

AverageTime

BetweenCollisions

EEEuJ stt==÷

øö

çèæ==

MNq

MqNqNq 2

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EE 105 Fall 2016 Prof. A. M. Niknejad

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Negative and Positive Carriersl Since current is contributed by positive and

negative charge carriers:

EJJJ ÷÷ø

öççè

æ --=-= -

--

+

++-+

MeN

MeNe tt

÷÷ø

öççè

æ+= -

--

+

++

MN

MNe tts 2

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EE 105 Fall 2016 Prof. A. M. Niknejad

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Conduction in Metalsl High conductivity of metals is due to large

concentration of free electrons l These electrons are not attached to the solid but are

free to move about the solidl In metal sodium, each atom contributes a free

electron: l From the measured value of conductivity (easy to

do), we can back calculate the mean free time:

322atoms/cm105.2 ´=N

( )( )( )( ) sec103

1023105.2109109.1 14

2022

2817

2-

-

-

´=´´´´

==Nemst

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EE 105 Fall 2016 Prof. A. M. Niknejad

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A Deep Puzzlel This value of mean free time is surprisingly longl The mean velocity for an electron at room temperature is

about:

l At this speed, the electron travels

l The molecular spacing between adjacent ions is only

l Why is it that the electron is on average zooming by 10 positively charged ions?

sec/cm10323

27

2

´== vkTmv

cm103 7-´=tv

cm108.3 8-´

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EE 105 Fall 2016 Prof. A. M. Niknejad

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Wave Nature of Electron

l The free carrier can penetrate right through positively charged host atoms!

l Quantum mechanics explains this! (Take Physics 117A/B)l For a periodic arrangement of potential functions, the

electron does not scatter. The influence of the crystal is that it will travel freely with an effective mass.

l So why does it scatter at all?EECS

EE 105 Fall 2016 Prof. A. M. Niknejad

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Scattering in Metalsl At temperature T, the atoms are in random motion

and thus upset periodicity

l Even at extremely low temperatures, the presence of an impurity upsets periodicity

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EE 105 Fall 2016 Prof. A. M. Niknejad

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Summary of Conduction

Conductivity determined by:l Density of free charge carriers (both positive and negative)l Charge of carrier (usually just e)l Effective mass of carrier (different inside solid)l Mean relaxation time (time for memory loss … usually the

time between collisions)– This is determined by several mechanisms, e.g.:

l Scattering by impuritiesl Scattering due to vibrations in crystal

s = e2 N +t +

M + + N−t −

M −

⎛⎝⎜

⎞⎠⎟

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Semiconductors

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EE 105 Fall 2016 Prof. A. M. Niknejad

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Resistivity for a Few Materialsl Pure copper, 273K 1.56×10-6 ohm-cml Pure copper, 373 K 2.24×10-6 ohm-cml Pure germanium, 273 K 200 ohm-cml Pure germanium, 500 K .12 ohm-cml Pure water, 291 K 2.5×107 ohm-cml Seawater 25 ohm-cm

What gives rise to this enormous range?Why are some materials semi-conductive?Why the strong temp dependence?

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Electronic Properties of Silicon

l Silicon is in Group IV – Atom electronic structure: 1s22s22p63s23p2

– Crystal electronic structure: 1s22s22p63(sp)4

– Diamond lattice, with 0.235 nm bond lengthl Very poor conductor at room temperature:

why?

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(1s)2

(2s)2

(2p)6 (3sp)4

Hybridized State

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Periodic Table of Elements

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The Diamond Structure

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3sp tetrahedral bond

!

A43.5

!

A35.2

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States of an Atom

l Quantum Mechanics: The allowed energy levels for an atom are discrete (2 electrons can occupy a state since with opposite spin)

l When atoms are brought into close contact, these energy levels split

l If there are a large number of atoms, the discrete energy levels form a “continuous” band

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Ener

gy

E1

E2

...E3

Forbidden Band Gap

AllowedEnergyLevels

Lattice ConstantAtomic Spacing

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Energy Band Diagraml The gap between the conduction and valence band

determines the conductive properties of the materiall Metal

– Partially filled band

l Insulator – large band gap, ~ 8 eV

l Semiconductor– medium sized gap, ~ 1 eV

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Valence Band

Conduction Band

e- Electrons can gain energy from lattice (phonon) or photon to become “free”

band gap

e-

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Model for Good Conductorl The atoms are all ionized and a “sea” of electrons can

wander about crystal:l The electrons are the “glue” that holds the solid togetherl Since they are “free”, they respond to applied fields and

give rise to conductions

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+ + + + + + + +

+ + + + + + + +

+ + + + + + + +

On time scale of electrons, lattice looks stationary…

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Bond Model for Silicon (T=0K)

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Silicon Ion (+4 q)

Four Valence ElectronsContributed by each ion (-4 q)

2 electrons in each bond

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Bond Model for Silicon (T>0K)l Some bond are broken: free electronl Leave behind a positive ion or trap (a hole)

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+

-

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Holes?

l Notice that the vacancy (hole) left behind can be filled by a neighboring electron

l It looks like there is a positive charge traveling around!l Treat holes as legitimate particles.

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+-

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Yes, Holes!l The hole represents the void after a bond is brokenl Since it is energetically favorable for nearby

electrons to fill this void, the hole is quickly filledl But 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)

l The net motion of many electrons in the valence band can be equivalently represented as the motion of a hole

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å åå ---=-=BandFilled StatesEmpty

iivb

ivb vqvqvqJ )()()(

åå =--=StatesEmpty

iStatesEmpty

ivb qvvqJ )(

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More About Holesl When a conduction band electron encounters a

hole, the process is called recombinationl The electron and hole annihilate one another thus

depleting the supply of carriersl In thermal equilibrium, a generation process

counterbalances to produce a steady stream of carriers

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Department of EECS University of California, Berkeley

Intrinsic Carrier Concentrationand Doping

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Thermal Equilibrium (Pure Si)

l Balance between generation and recombination determines no = po

l Strong function of temperature: T = 300 oK

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optth GTGG += )(

)( pnkR ´=

RG =)()( TGpnk th=´

)(/)( 2 TnkTGpn ith ==´

K300atcm10)( 310 -@Tni

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Doping with Group V Elements

l P, As (group 5): extra bonding electron … lost to crystal at room temperature

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ImmobileCharge

Left Behind

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Donor Accountingl Each ionized donor will contribute an extra “free”

electronl The material is charge neutral, so the total charge

concentration must sum to zero:

l By Mass-Action Law:

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000 =++-= dqNqpqnr

Free Electrons

Free Holes

Ions(Immobile)

)(2 Tnpn i=´

00

2

0 =++- di qNnnqqn

0022

0 =++- nqNqnqn di

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Donor Accounting (cont)l Solve quadratic:

l Only positive root is physically valid:

l For most practical situations:

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022

0

20

20

idd

id

nNNn

nnNn

+±=

=--

24 22

0idd nNN

n++

=

id nN >>

ddd

idd

NNNNnNN

n =+»÷øö

çèæ++

=222

412

0

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Doping with Group III Elementsl Boron: 3 bonding electrons à one bond is

unsaturatedl Only free hole … negative ion is immobile!

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Mass Action Law

l Balance between generation and recombination:

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2ioo nnp =×

• N-type case:

• P-type case:

)cm10,K300( 310 -== inT

dd NNn @= +0

aa NNp @= -0

d

i

Nnn2

0 @

a

i

Nnp2

0 @

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Compensationl Dope with both donors and acceptors:

– Create free electron and hole!

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+

-

-

+

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Compensation (cont.)

l More donors than acceptors: Nd > Na

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iado nNNn >>-=

• More acceptors than donors: Na > Nd

ado NN

np i

-=

2

idao nNNp >>-=da

o NNn

n i

-=

2

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Drift Currents

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Thermal Equilibrium

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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

a c

(hole case)

x

kTvm thn 212*

21 =

cthv tl =

cm1010/cm10 6137 -- =´= ssl

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Drift Velocity and Mobility

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Ev pdr µ=

Emq

mqE

mFav

p

cc

pc

p

ecdr ÷

÷ø

öççè

æ=÷

÷ø

öççè

æ=÷

÷ø

öççè

æ=×=

tttt

For electrons:

Emq

mqE

mFav

p

cc

pc

p

ecdr ÷

÷ø

öççè

æ-=÷

÷ø

öççè

æ -=÷

÷ø

öççè

æ=×=

tttt

For holes:

Ev ndr µ-=

EE 105 Fall 2016 Prof. A. M. Niknejad

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Mobility vs. Doping in Silicon at 300 oK

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Typical values: 1000=nµ 400=pµ

Department of EECS University of California, Berkeley

Diffusion Currents

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Diffusionl Diffusion occurs when there exists a concentration

gradientl In the figure below, imagine that we fill the left

chamber with a gas at temperate Tl If we suddenly remove the divider, what happens?l The gas will fill the entire volume of the new

chamber. How does this occur?

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Diffusion (cont)l The net motion of gas molecules to the right

chamber was due to the concentration gradientl If each particle moves on average left or right then

eventually half will be in the right chamberl If the molecules were charged (or electrons), then

there would be a net current flowl The diffusion current flows from high

concentration to low concentration:

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Diffusion Equationsl Assume that the mean free path is λl Find flux of carriers crossing x=0 plane

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)(ln)0(n

)( l-n

0l- l

thvn )(21 lthvn )(

21 l-

( ))()(21 ll nnvF th --=

÷÷ø

öççè

æúûù

êëé +-úû

ùêëé -=

dxdnn

dxdnnvF th ll )0()0(

21

dxdnvF thl-=

dxdnqvqFJ thl=-=

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Einstein Relationl The thermal velocity is given by kT

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kTvm thn 212*

21 =

cthv tl =Mean Free Time

dxdn

qkTq

dxdnqvJ nth ÷÷

ø

öççè

æ== µl

nn qkTD µ÷÷

ø

öççè

æ=

**2

n

c

n

ccthth m

qqkT

mkTvv tttl ===

EE 105 Fall 2016 Prof. A. M. Niknejad

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Total Current and Boundary Conditions

l When both drift and diffusion are present, the total current is given by the sum:

l In resistors, the carrier is approximately uniform and the second term is nearly zero

l For currents flowing uniformly through an interface (no charge accumulation), the field is discontinous

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dxdnqDnEqJJJ nndiffdrift +=+= µ

21 JJ =

2211 EE ss =

1

2

2

1

ss

=EE

)( 11 sJ

)( 22 sJ

EE 105 Fall 2016 Prof. A. M. Niknejad

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Referencel Edward Purcell, Electricity and Magnetism,

Berkeley Physics Course Volume 2 (2nd Edition), pages 133-142.

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