Chap. 41: Chap. 41: Conduction of electricity Conduction of electricity

in solidsin solids

Hyun-Woo LeeHyun-Woo Lee

41-1 What Is Physics?41-1 What Is Physics?

Q: Why certain materials conduct electricitQ: Why certain materials conduct electricity?y?

Q: Why certain materials do NOT conduct Q: Why certain materials do NOT conduct electricity?electricity?

Solid materialSolid material Many many electrons and atomsMany many electrons and atoms

Solid state physicsSolid state physics Application of quantum physics to solidsApplication of quantum physics to solids

41-2 Electrical Properties of 41-2 Electrical Properties of SolidsSolids

Crystalline solidsCrystalline solids Lattice structureLattice structure

• Repetition of unit cellsRepetition of unit cells

Classification criteriaClassification criteria Resistivity Resistivity at room temperature ( at room temperature (m)m) Temperature coefficient of resistivity Temperature coefficient of resistivity (K (K-1-1))

Number density of charge carriers Number density of charge carriers nn (m (m-3-3))• Can be found from Hall effect measurementCan be found from Hall effect measurement

Metals, semiconductors, insulatorsMetals, semiconductors, insulators

)/)(/1( dTd

Insulators, semiconductors Insulators, semiconductors & metals& metals

InsulatorsInsulators Extremely large Extremely large Ex: Diamond Ex: Diamond diamonddiamond//coppercopper~10~102424

Semiconductors vs MetalsSemiconductors vs Metals insulatorinsulator >> >> semiconductorsemiconductor >> >> metalmetal

siliconsilicon==3310103 3 m, m, coppercopper=2=21010-8 -8 mm

semiconductorsemiconductor <0, <0, metalmetal>0>0siliconsilicon= -70 = -70 1010-3 -3 KK-1-1, , CopperCopper= +4= +41010-3-3 K K-1-1

nnsemiconductorsemiconductor << << nnmetalmetal

• nnsiliconsilicon=1=110101616 m m-3-3, , n ncoppercopper=9=910102828 m m-3-3

41-3 Energy Levels 41-3 Energy Levels in a Crystalline Solidsin a Crystalline Solids Single atom (Ex: Cu Single atom (Ex: Cu ZZ=29)=29)

11ss22 2 2ss22 2 2pp66 3 3ss22 3 3pp66 3 3dd1010 4 4ss11

Two atomsTwo atoms Tunneling between two atomsTunneling between two atoms

Three atomsThree atoms More tunnelingMore tunneling

Tunneling effectsTunneling effects

Two wellsTwo wells

Level splitting into two levelsLevel splitting into two levels

Tunneling effects inTunneling effects in solidssolids

NN(() wells) wells Energy level splitting into Energy level splitting into NN levels levels Energy bands & energy gapsEnergy bands & energy gaps

41-4 Insulators41-4 Insulators

No partially filled bandsNo partially filled bands

For a current to exist,For a current to exist, Kinetic energy must increaseKinetic energy must increase

Electrons must move to higher-energy levelsElectrons must move to higher-energy levels Pauli exclusion principlePauli exclusion principle

• Transition to filled state is prohibitedTransition to filled state is prohibited

Energy gap (Ex: Energy gap (Ex: EEgg=5.5 eV in diamond)=5.5 eV in diamond) Large energy supply neededLarge energy supply needed

Current flow strongly suppressedCurrent flow strongly suppressed

Thermal fluctuation effectsThermal fluctuation effects

Thermal excitationsThermal excitations Finite probability to jump Finite probability to jump EEgg

Probability Probability PP for the jump for the jump For For EEgg=5.5 eV, =5.5 eV, TT=300K=300K

• cf: # of electron in 1 cmcf: # of electron in 1 cm33 ~ 10 ~ 102323

93213 103)/exp(~ ekTEP g

41-5 Metals41-5 Metals

Partially filled bandsPartially filled bands Easy to induce energy “jump”Easy to induce energy “jump”

Fermi level Fermi level EEFF

Highest occupied level at Highest occupied level at TT=0K=0K Ex: Ex: EEFF=7.0 eV for copper=7.0 eV for copper

Fermi speed Fermi speed vvFF

Electron speed at Electron speed at EEFF

Ex: Ex: vvFF=1.6=1.6101066 m/s for copper m/s for copper

No relaxation of No relaxation of vvFF due to Pauli exclusion princip due to Pauli exclusion princip

lele

How Many Conduction How Many Conduction Electrons Are There?Electrons Are There?

Number density Number density nn

Ex: Magnesium w/ volume 2.00Ex: Magnesium w/ volume 2.001010-6-6 m m33

BivalentBivalent

atomper electrons

valenceofnumber

samplein

atoms ofnumber

samplein electrons

conduction ofnumber

Vn

volumesample

samplein electrons conduction ofnumber

221061.8samplein

atoms ofnumber

231072.1samplein electrons

conduction ofnumber

Conductivity Above Absolutely Conductivity Above Absolutely ZeroZero

Ex: Ex: TT=1000 K=1000 K kTkT=0.086 eV=0.086 eV cf: cf: EEFF=7.0 eV in copper=7.0 eV in copper

# of charge carriers extremely insensitiv# of charge carriers extremely insensitive to e to TT

41-6 Semiconductors41-6 Semiconductors

No partially filled bandsNo partially filled bands But small energy gapBut small energy gap

Ex: Ex: EEgg=1.1 eV for silicon=1.1 eV for silicon

cf: cf: EEgg=5.5 eV for diamond=5.5 eV for diamond

Valence bandValence band Highest filled bandHighest filled band

Conduction bandConduction band Lowest vacant bandLowest vacant band

Number Density of Charge Number Density of Charge CarriersCarriers

Probability Probability PP for jump for jump

Charge carriersCharge carriers ElectronsElectrons

• Conduction bandConduction band HolesHoles

• Valence bandValence band # of charge carriers extremely sensitive to # of charge carriers extremely sensitive to TT

196.42 104)/exp(~ ekTEP g

Motion of charge Motion of charge carrierscarriers

Electrons in conduction bandElectrons in conduction band

Holes in valence bandHoles in valence band

Efficient description in terms of holesEfficient description in terms of holes Effective charge of hole: +Effective charge of hole: +ee

E

E

Resistivity Resistivity siliconsilicon / / coppercopper = 1.5 = 1.510101111

Classical estimationClassical estimation

Difference between Difference between siliconsilicon and and coppercopper mainly mainly

from carrier density from carrier density nn

nem 2/

Temperature Coefficient of ResistiviTemperature Coefficient of Resistivity ty

::

Temperature dependenceTemperature dependence Classical estimationClassical estimation

Semiconductor (Ex: silicon)Semiconductor (Ex: silicon) nn increases as increases as TT increases increases < 0 < 0

Metal (Ex: copper)Metal (Ex: copper) decreases as decreases as T T increases increases > 0 > 0

dT

d

1

nem 2/

More about metalsMore about metals

How Many Quantum States Are How Many Quantum States Are There?There?

Too many states to list all statesToo many states to list all states Density of states Density of states NN((EE))

NN((EE))dEdE : # of states between : # of states between EE and and EE++dEdE per v per volumeolume

Near lower edge of partially filled bandNear lower edge of partially filled band

)Jm(28

)( 1-3-2/13

2/3

Eh

mEN

How Many Quantum States Are How Many Quantum States Are There ? There ? (continued)(continued)

Ex: Metal w/ Ex: Metal w/ VV=2=21010-9-9 m m33 at at EE=7 eV=7 eV

1-19 eV104eV 7at eVper

states of #

17101eV 7at eV 0.003 rangein

states of #

N

The Occupancy The Occupancy Probability Probability PP((EE))

Maxwell distributionMaxwell distribution Not applicable due to Pauli exclusion principleNot applicable due to Pauli exclusion principle

Fermi-Dirac statisticsFermi-Dirac statistics

At At EE==EEFF PP((EE)=1/2 regardless of )=1/2 regardless of TT Useful way to define Useful way to define EEFF at at TT>0>0

1

1)( /)( kTEE Fe

EP

How Many Occupied States Are How Many Occupied States Are There?There?

Density of occupied states Density of occupied states NN00((EE)) NN00((EE)=)=NN((EE))PP((EE))

Calculating the Fermi EnergyCalculating the Fermi Energy

At At TT=0,=0, Due to Pauli exclusion principleDue to Pauli exclusion principle

With With NN((EE) ) EE1/21/2

F

0

)(E

dEENn

3

228 2/3F

3

2/3 E

h

mn

3/22

3/223/2

F

121.0

216

3n

m

hn

m

hE

More about semiconductorsMore about semiconductors

41-7 Doped 41-7 Doped SemiconductorsSemiconductors

DopingDoping Introducing a small number of replacement Introducing a small number of replacement

atoms (impurities) into semiconductor latticeatoms (impurities) into semiconductor lattice

~ 1 out of 10~ 1 out of 1077 atoms replaced atoms replaced

nn-Type Semiconductors-Type Semiconductors Pure silicon: Si (Pure silicon: Si (ZZ=14) 1=14) 1ss22 2 2ss22 3 3pp66 3 3ss22 3 3pp22

Valence number: 4Valence number: 4

Doping by P (Doping by P (ZZ=15, valence=5)=15, valence=5) One extra el.One extra el. nn(egative)(egative)-type-type 5th el. in the “conduction band”5th el. in the “conduction band”

Extra electron & protonExtra electron & proton w/o extra protonw/o extra proton

w/ extra protonw/ extra proton

Weakly bound donor levels Weakly bound donor levels

At room temperatureAt room temperature

Thermal excitationsThermal excitations EEdd =0.045 eV for phosphorous doping =0.045 eV for phosphorous doping

cf: cf: EEgg=1.1 eV=1.1 eV Excitations from donor levels to conduction Excitations from donor levels to conduction

band much easierband much easier

Majority carriersMajority carriers Electrons in conduction bandElectrons in conduction band

Minority carriersMinority carriers Holes in valence bandHoles in valence band

Doping levelDoping level

Pure siliconPure silicon # density of conduction el. at room temp# density of conduction el. at room temp

• ((nn00))no-dopingno-doping ~ 10 ~ 101616 m m-3-3

Q: Doping for (Q: Doping for (nn00))dopingdoping=10=1066 ((nn00))no-dopingno-doping

((nn00))dopingdoping= = ((nn00))no-doping no-doping + + nnPP

nnP P 10 102222 m m-3-3

cf: cf: nnSi Si 5 510102828 m m-3-3

6Si

P

105

1

n

n

pp-Type Semiconductors-Type Semiconductors Doping by Al (Doping by Al (ZZ=13)=13)

One missing el One missing el pp(ositive)-type(ositive)-type Missing el in “valence band”Missing el in “valence band”

w/ missing protonw/ missing proton Weakly bound acceptor levelsWeakly bound acceptor levels

At room temperatureAt room temperature

Thermal excitationsThermal excitations EEdd =0.067 eV for aluminium doping =0.067 eV for aluminium doping

cf: cf: EEgg=1.1 eV=1.1 eV Excitations from valence band to acceptor lExcitations from valence band to acceptor l

evels much easierevels much easier

Majority carriersMajority carriers Holes in valence bandHoles in valence band

Minority carriersMinority carriers Electrons in conduction bandElectrons in conduction band

41-8 The 41-8 The pp--nn Junction Junction Junction of Junction of pp-type and -type and nn-type semicond.-type semicond.

Upon contact, …(no bias yet)Upon contact, …(no bias yet)

Junction planeJunction plane

Motions of the Majority Motions of the Majority CarriersCarriers

DiffusionDiffusion Diffusion current Diffusion current IIdiffdiff

Space chargeSpace charge

-e

+eDepletion Depletion zonezoneContact potential Contact potential difference difference VV00

Idiff

Idiff = 0

Motions of the Minority CarriersMotions of the Minority Carriers

Minority carriersMinority carriers Drift current Drift current IIdriftdrift

Space charge somewhat relaxedSpace charge somewhat relaxed

Majority & minority carriersMajority & minority carriers Balance of Balance of IIdiffdiff & & IIdriftdrift

Idrift

41-9 The Junction Rectifier41-9 The Junction Rectifier II vs. vs. VV

pp--nn junction as a rectifier junction as a rectifier AC AC DC conversion DC conversion

Forward biasForward bias

Reduce Reduce VV00

Reduce V0

Narrower depletion zone

Backward biasBackward bias

Enhance Enhance VV00

Enhance V0

Wider depletion zone

41-10 The Light-Emitting Diode (LED)41-10 The Light-Emitting Diode (LED) LEDLED

Light emission from Light emission from pp--nn junction junction

Photon or lattice vibration Forward bias

pp--nn junction as LED junction as LED Forward biased Forward biased pp--nn junction junction

Photon wavelengthPhoton wavelength

Commercial LEDsCommercial LEDs in visible rangein visible range Ex: Gallium (valence 3) doped with arsenic (valEx: Gallium (valence 3) doped with arsenic (val

ence 5, 60%) and phosphorous (valence 5, 4ence 5, 60%) and phosphorous (valence 5, 40%) atoms0%) atoms

• EEgg=1.8 eV (red color)=1.8 eV (red color)

gg E

hc

hE

c

f

c

/

The Photo-DiodeThe Photo-Diode Photo-diode = (LED)Photo-diode = (LED)-1-1

Photon Photon Current Current

Ex: TV remote controlEx: TV remote control• Remote control : LEDRemote control : LED

Generate a certain sequence of infrared photonsGenerate a certain sequence of infrared photons

• TV : Photo-diodeTV : Photo-diode Photon detection Photon detection Electric signal Electric signal

Photon-induced transition

The Junction LaserThe Junction Laser

Stimulated emission inStimulated emission in p p--nn junction junction

Junction laserJunction laser• Ex: Laser head in compact disc (CD) playersEx: Laser head in compact disc (CD) players

MirrorMirror

41-11 The Transistor41-11 The Transistor

TransistorTransistor Intentional control of on-offIntentional control of on-off Application: AmplifierApplication: Amplifier

FET (Field Effect Transistor)FET (Field Effect Transistor) Integrated circuitsIntegrated circuits

TransistorsTransistors CapacitorsCapacitors Resistors etc.Resistors etc.

Intel Pentium chip (w/ ~7 million transistors)

MOSFETMOSFET(Metal-Oxide-Semiconductor-FET)(Metal-Oxide-Semiconductor-FET)

MOSFETMOSFET High speed on-offHigh speed on-off ~500 nm in length~500 nm in length

Gate voltage Gate voltage VVGSGS

Negatively charge gateNegatively charge gate Repel el. in Repel el. in nn-channel down into substrate-channel down into substrate Wider depletion zone between Wider depletion zone between pp and and nn nn-channel width reduced -channel width reduced Larger resistance (off realized)Larger resistance (off realized)

The EndThe End