conductivity lect20

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ConductivityConductivitySemiconductors & MetalsSemiconductors & Metals

Chemistry 754Chemistry 754

Solid State ChemistrySolid State Chemistry

Lecture #20Lecture #20

May 14, 2003May 14, 2003

References – ConductivityReferences – Conductivity

There are many references that describe electronicThere are many references that describe electronicconductivity in metals and semiconductors. I usedconductivity in metals and semiconductors. I usedprimarily the following texts to develop this lecture.primarily the following texts to develop this lecture.

“The Electronic Structure and Chemistry of“The Electronic Structure and Chemistry ofSolids”Solids”

P.A. Cox, Oxford University Press, Oxford (1987).P.A. Cox, Oxford University Press, Oxford (1987).

“Solid State Physics”“Solid State Physics”H.H. IbachIbach and H.and H. LuthLuth, Springer-, Springer-VerlagVerlag, Berlin (1991)., Berlin (1991).

“Physical Properties of Semiconductors”“Physical Properties of Semiconductors”C.M. Wolfe, N.C.M. Wolfe, N. HolonyakHolonyak, Jr., G.E., Jr., G.E. StillmanStillman, Prentice Hall,, Prentice Hall,

Englewood Cliffs, NJ (1989).Englewood Cliffs, NJ (1989).



ResistivitiesResistivities of Real Materialsof Real Materials

Compound Resistivity (ΩΩΩΩ-cm) Compound Resistivity (ΩΩΩΩ-cm)

Ca 3.9 ×××× 10-6 Si ~ 0.1Ti 42 ×××× 10-6 Ge ~ 0.05

Mn 185 ×××× 10-6 ReO3 36 ×××× 10-6

Zn 5.9 ×××× 10-6 Fe3O4 52 ×××× 10-6

Cu 1.7 ×××× 10-6 TiO2 9 ×××× 104

Ag 1.6 ×××× 10-6 ZrO2 1 ×××× 109

Pb 21 ×××× 10

-6

Al2O3 1 ×××× 10

19

Most semiconductors in their pure form are not goodMost semiconductors in their pure form are not good

conductors, they need to be doped to become conducting.conductors, they need to be doped to become conducting.

Not all so called “ionic” materials like oxides are insulators.Not all so called “ionic” materials like oxides are insulators.

Microscopic ConductivityMicroscopic Conductivity

We can relate the conductivity,We can relate the conductivity, σσσσσσσσ, of a material to microscopic, of a material to microscopicparameters that describe the motion of the electrons (or otherparameters that describe the motion of the electrons (or othercharge carrying particles such as holes or ions).charge carrying particles such as holes or ions).

σσσσσσσσ == nene(e(eττττττττ/m*)/m*)

µµµµµµµµ = e= eττττττττ/m*/m*

σσσσσσσσ == neneµµµµµµµµwherewhere

n = the carrier concentration (cmn = the carrier concentration (cm-3-3))

e = the charge of an electron = 1.602e = the charge of an electron = 1.602 ×× 1010-19-19 CC

ττττττττ = the relaxation time (s) the time between collisions= the relaxation time (s) the time between collisions

m* = the effective mass of the electron (kg)m* = the effective mass of the electron (kg)

µµµµµµµµ = the electron mobility (cm= the electron mobility (cm22/V-s)/V-s)



EF

DOS

E n e r g

y

MetalMetal

EF

DOS

E n e r g

y

SemimetalSemimetal

EF

DOS

E n e r g

y

SemiconductorSemiconductor

/Insulator/Insulator

In a metal theIn a metal the FermiFermi level cuts through a band to produce a partially filledlevel cuts through a band to produce a partially filledband. In a semiconductor/insulator there is an energy gap between the filledband. In a semiconductor/insulator there is an energy gap between the filledbands and the empty bands. The distinction between a semiconductor and anbands and the empty bands. The distinction between a semiconductor and an

insulator is artificial, but as the gap becomes large the material usuallyinsulator is artificial, but as the gap becomes large the material usuallybecomes a poor conductor of electricity. A semimetal results when the bandbecomes a poor conductor of electricity. A semimetal results when the band

gap goes to zero.gap goes to zero.

ConductionConductionBandBand

ValenceValenceBandBand

Metals, Semiconductors & InsulatorsMetals, Semiconductors & Insulators

ResistivityResistivity and Carrier Concentrationand Carrier ConcentrationTheThe resistivitiesresistivities of real materials span nearly 25 orders of magnitude.of real materials span nearly 25 orders of magnitude.This is due to differences in carrier concentration (n) and mobility (This is due to differences in carrier concentration (n) and mobility (µµ).).Let’s first consider carrier concentration.Let’s first consider carrier concentration.

••The carrier concentration only includes electrons which can easily beThe carrier concentration only includes electrons which can easily beexcited from occupied states into empty states. The remaining electronsexcited from occupied states into empty states. The remaining electronsare localized.are localized.

••In the absence of external excitations (light, voltage, etc.) theIn the absence of external excitations (light, voltage, etc.) theexcitation must be thermal, this is on the order ofexcitation must be thermal, this is on the order of kT kT (~ 0.03(~ 0.03 eVeV at RT)at RT)

••Only electrons whose energies are within a fewOnly electrons whose energies are within a few kT kT of Eof EFF can contributecan contributeto the electrical conductivity.to the electrical conductivity.

Generally this means that EGenerally this means that EFF should cut a band toshould cut a band toachieve appreciable carrier concentration. Alternativelyachieve appreciable carrier concentration. Alternativelyimpurities/defects are introduced to partially populate aimpurities/defects are introduced to partially populate a

band.band.



FermiFermi--DiracDirac FunctionFunctionTheThe FermiFermi--DiracDirac function gives the fraction of allowed states, f(E), atfunction gives the fraction of allowed states, f(E), atan energy level E, that are populated at a given temperature.an energy level E, that are populated at a given temperature.

f(E) = 1/[1 + exp(E-Ef(E) = 1/[1 + exp(E-EFF)/)/kT kT ] ]where thewhere the FermiFermi Energy, EEnergy, EFF, is defined as the energy where f(E) = 1/2., is defined as the energy where f(E) = 1/2.That is to say one half of the available states are occupied. T is theThat is to say one half of the available states are occupied. T is thetemperature (in K) and k is thetemperature (in K) and k is the BoltzmanBoltzman constant (k = 8.62constant (k = 8.62 ×× 1010-5-5

eVeV/K)/K)

As an example consider f(E) for T = 300 K and a state 0.1As an example consider f(E) for T = 300 K and a state 0.1 eVeV above Eabove EFF::

f(E) = 1/[1 + exp(0.1f(E) = 1/[1 + exp(0.1 eVeV)/((300K)(8.62)/((300K)(8.62 ×× 1010-5-5 eVeV/K)]/K)]f(E) = 0.02 = 2%f(E) = 0.02 = 2%

Consider a band gap of 1Consider a band gap of 1 eVeV..

f(1f(1 eVeV) = 1.6) = 1.6 ×× 1010-17-17

See that for even a moderate band gap (Silicon has a band gap of 1.1See that for even a moderate band gap (Silicon has a band gap of 1.1eVeV) the intrinsic concentration of electrons that can be thermally) the intrinsic concentration of electrons that can be thermallyexcited to move about the crystal is tiny. Thus pure Silicon (if youexcited to move about the crystal is tiny. Thus pure Silicon (if you

could make it) would be quite insulating.could make it) would be quite insulating.

Fermi DiracFermi Dirac FunctionFunctionMetals and SemiconductorsMetals and Semiconductors

f(E) as determinedexperimentally forRu metal (note theenergy scale)

f(E) for asemiconductor



Carrier MobilityCarrier MobilityRecall the expression for carrier mobility:Recall the expression for carrier mobility:

µµµµµµµµ = e= eττττττττ/m*/m*

where,where,ee = electronic charge= electronic chargem*m* = the effective mass= the effective massττττττττ = the relaxation time between scattering events= the relaxation time between scattering events

What factors determine the effective mass?What factors determine the effective mass?•• mm** depends upon the band width, which in turn dependsdepends upon the band width, which in turn depends

upon orbital overlap.upon orbital overlap.What entities scatter the carriers and reduce theWhat entities scatter the carriers and reduce the

mobility?mobility?•• A defect or impurity (A defect or impurity (ττττττττ increases as purity increases)increases as purity increases)•• Lattice vibrations, phonons (Lattice vibrations, phonons (ττττττττ decreases as temp.decreases as temp.

increases)increases)

What is the meaning of k?What is the meaning of k?

In our development of the electronic band structureIn our development of the electronic band structurefrom a linear combination of atomicfrom a linear combination of atomic orbitalsorbitals the variablethe variablek was used to determine the phase of thek was used to determine the phase of the orbitalsorbitals..What exactly is k?What exactly is k?

WavevectorWavevector – It tells us the how the phases of the– It tells us the how the phases of theorbitalsorbitals change whenchange when translationaltranslational symmetry is applied.symmetry is applied.

Quantum NumberQuantum Number – Identifies a particular electronic– Identifies a particular electronicwavefunctionwavefunction (that can hold 2 electrons with opposite(that can hold 2 electrons with oppositespin).spin).

Crystal MomentumCrystal Momentum – In free electron theory k is– In free electron theory k isproportional to the momentum of the electron in theproportional to the momentum of the electron in the kkthth

wavefunctionwavefunction..



To better understand the meaning of k, consider an electron atTo better understand the meaning of k, consider an electron atthe outer edge of thethe outer edge of the BrillouinBrillouin zone, where k =zone, where k = ππ/a. The phase of/a. The phase of

the electronicthe electronic wavefunctionwavefunction changes sign every unit cell (similar tochanges sign every unit cell (similar toa p-orbital changing phase at its nodal plane)a p-orbital changing phase at its nodal plane)

λ λλ λ λ λλ λ = 2a= 2a →→→→→→→→ a =a = λ λλ λ λ λλ λ /2/2k =k = ππππππππ/a/a →→→→→→→→ a =a = ππππππππ/k/k

Combining these two relationships gives:Combining these two relationships gives:

λ λλ λ λ λλ λ /2/2 == ππππππππ/k/k

k = 2k = 2ππππππππ//λ λλ λ λ λλ λ

λ =λ =λ =λ =λ =λ =λ =λ =

22ππππππππ/k/k

The wavelength of theThe wavelength of the wavefunctionwavefunction is inverselyis inverselyproportional to k.proportional to k.

Crystal MomentumCrystal Momentum

λ

a

Now consider theNow consider the DeBroglieDeBroglie relationship (wave-particle duality ofrelationship (wave-particle duality ofmatter)matter)

λ λλ λ λ λλ λ = h/p= h/p

p = h/p = h/λ λλ λ λ λλ λ

p =p = hkhk/2/2ππππππππ

where.,where.,•• p is the momentum of thep is the momentum of the wavepacketwavepacket,,

•• h is Planck’s constant, 6.626h is Planck’s constant, 6.626 ×× 1010-34-34 J-sJ-s

The momentum of an electron is directly proportional to k.The momentum of an electron is directly proportional to k.

k is a measure of the “crystal” momentum of an electron in thek is a measure of the “crystal” momentum of an electron in theψ ψψ ψ ψ ψψ ψ KK wavefunctionwavefunction..

Crystal MomentumCrystal Momentum



From the ideas on the previous 2 slides one can derive theFrom the ideas on the previous 2 slides one can derive thefollowing relationships to describe the properties of a conductionfollowing relationships to describe the properties of a conduction

electron:electron:

VelocityVelocity →→→→→→→→ v =v = hkhk/2/2ππππππππm = (m = (22ππππππππ/h)(/h)(dEdE//dkdk))

EnergyEnergy →→→→→→→→ E = (h/E = (h/22ππππππππ)(k)(k22/2m/2m**))Effective MassEffective Mass →→→→→→→→ mm** = (= (22ππππππππ/h)/h)22 (1/d(1/d22E/dkE/dk22 ) )

dEdE//dkdk →→ The first derivative of the E vs. k curve.The first derivative of the E vs. k curve.dd22E/dkE/dk22 →→ The second derivative of the E vs. k curve.The second derivative of the E vs. k curve.

QuantityQuantity Wide BandWide Band Narrow BandNarrow BanddEdE//dkdk LargeLarge SmallSmall

µµµµµµµµ HighHigh LowLowVelocityVelocity FastFast SlowSlow

m*m* LightLight HeavyHeavy

Wide (disperse) bands are better for conductivityWide (disperse) bands are better for conductivity..

BandstructureBandstructure & DOS for Cu& DOS for Cu

EEFF cuts the very wide (disperse) s band, giving rise to acuts the very wide (disperse) s band, giving rise to alarge carrier concentration, along with high mobility.large carrier concentration, along with high mobility.

This combination gives rise to high conductivity.This combination gives rise to high conductivity.



Temperature Dependence-MetalsTemperature Dependence-Metals

Recall thatRecall that

σσσσσσσσ = ne= ne22

ττττττττ/m*/m*In MetalsIn Metals

–– The carrier concentration, n, changes veryThe carrier concentration, n, changes veryslowly with temperature.slowly with temperature.

–– ττ is inversely proportional to temperatureis inversely proportional to temperature((τ ατ α 1/T), due to scattering by lattice vibrations1/T), due to scattering by lattice vibrations(phonons).(phonons).

–– Therefore, a plot ofTherefore, a plot of σσ vs. 1/T (orvs. 1/T (or ρρ vs. T) isvs. T) isessentially linear.essentially linear.

–– Conductivity goes down as temperatureConductivity goes down as temperatureincreasesincreases..

Scattering by Impurities and PhononsScattering by Impurities and Phonons

Phonon scattering

•Proportional to temperature

Impurity scattering

•Independent of temperature•Proportional to impurityconcentration



BandstructureBandstructure forfor GeGe

EEFF falls in the (0.67falls in the (0.67 eVeV) band gap. Carrier concentration and) band gap. Carrier concentration andconductivity are small.conductivity are small.

GeGe is an indirect gap semiconductor, because the uppermost VB energyis an indirect gap semiconductor, because the uppermost VB energyand the lowest CB energy occur at different locations in k-space.and the lowest CB energy occur at different locations in k-space.

p-bandsp-bands

s-bands-band

No mixingNo mixingatat Γ ΓΓ Γ Γ ΓΓ Γ ..

CB minimumCB minimum

VB maximumVB maximum

Direct & Indirect Gap SemiconductorsDirect & Indirect Gap Semiconductors

Direct Gap SemiconductorDirect Gap Semiconductor: Maximum of the valence band and minimum of the: Maximum of the valence band and minimum of theconduction band fall at the same place in k-space.conduction band fall at the same place in k-space.

αααααααα αα ((hh ν νν ν ν νν ν--EEgg))1/21/2

Indirect Gap SemiconductorIndirect Gap Semiconductor: Maximum of the valence band and minimum of: Maximum of the valence band and minimum ofthe conduction band fall different points in k-space. A lattice vibrationthe conduction band fall different points in k-space. A lattice vibration(phonon) is involved in electronic excitations, this decreases the absorption(phonon) is involved in electronic excitations, this decreases the absorptionefficiency.efficiency. αααααααα αα ((hh ν νν ν ν νν ν--EEgg))

22

GeGe SiSi GaAsGaAs

Figure taken from

“Fundamentals ofSemiconductorTheory and Device

Physics”, by S. Wang



Doping SemiconductorsDoping SemiconductorsTheThe FermiFermi--DiracDirac function shows that a pure semiconductor with afunction shows that a pure semiconductor with a

band gap of more than a few tenths of anband gap of more than a few tenths of an eVeV would have a very smallwould have a very smallconcentration of carriers. Therefore, impurities are added toconcentration of carriers. Therefore, impurities are added to

introduce carriers.introduce carriers.

n-dopingn-doping →→ Replacing a lattice atomReplacing a lattice atomwith an impurity (donor) atom thatwith an impurity (donor) atom thatcontains 1 additional valence electroncontains 1 additional valence electron(i.e. P in(i.e. P in SiSi). This e). This e-- can easily becan easily bedonated to the conduction band.donated to the conduction band.

p-dopingp-doping →→ Replacing a lattice atomReplacing a lattice atomwith an impurity (acceptor) atom thatwith an impurity (acceptor) atom thatcontains 1 less valence electron (i.e. Alcontains 1 less valence electron (i.e. Alinin SiSi). This atom can easily accept an). This atom can easily accept anee-- from the VB creating a hole.from the VB creating a hole.

ee--

Valence BandValence Band

Conduction BandConduction Bandee--

Valence BandValence Band

Conduction BandConduction Band

EEFF EEFF

Common Semiconductor StructuresCommon Semiconductor Structures

DiamondDiamondFdFd-3m (Z=8)-3m (Z=8)C,C, SiSi,, GeGe,, SnSn

SphaleriteSphaleriteF-43m (Z=4)F-43m (Z=4)

GaAsGaAs,, ZnSZnS,, InSbInSb

ChalcopyriteChalcopyriteI-42d (Z=4)I-42d (Z=4)

CuFeSCuFeS22, ZnSiAs, ZnSiAs22



Properties of SemiconductorsProperties of Semiconductors

4004008,5008,5001.43 (D)1.43 (D)SphaleriteSphaleriteGaAsGaAs

------3003003.4 (D)3.4 (D)WurtziteWurtziteGaNGaN

180180

1,0001,000

2.16 (I)2.16 (I)

SphaleriteSphalerite

AlAsAlAs

1,7001,700100,000100,0000.18 (D)0.18 (D)SphaleriteSphaleriteInSbInSb

------80802.43 (I)2.43 (I)SphaleriteSphaleriteAlPAlP

1,9001,9003,9003,9000.67 (I)0.67 (I)DiamondDiamondGeGe

4804801,3501,3501.11 (I)1.11 (I)DiamondDiamondSiSi

hh++ mobilitymobility(cm(cm22/V-s)/V-s)

ee-- mobilitymobility(cm(cm22/V-s)/V-s)

BandgapBandgap((eVeV))

StructureStructureCompoundCompound

Temperature Dependence-SemiconductorsTemperature Dependence-Semiconductors

Recall thatRecall that

σσσσσσσσ = ne= ne22ττττττττ/m*/m*

In SemiconductorsIn Semiconductors

–– The carrier concentration increases as temperatureThe carrier concentration increases as temperaturegoes up, due to excitations across the band gap,goes up, due to excitations across the band gap, EEgg..

–– n is proportional to exp-n is proportional to exp-EEgg/2kT./2kT.

–– ττ is inversely proportional to temperatureis inversely proportional to temperature

–– The exponential dependence of n dominates,The exponential dependence of n dominates,therefore, a plot oftherefore, a plot of lnln σσ vs. 1/T is essentially linear.vs. 1/T is essentially linear.

–– Conductivity increases as temperature increasesConductivity increases as temperature increases..



When a p-type and an n-type semiconductor are brought into contactWhen a p-type and an n-type semiconductor are brought into contact

electrons flow from the n-doped semiconductor into the p-dopedelectrons flow from the n-doped semiconductor into the p-dopedsemiconductor until thesemiconductor until the FermiFermi levels equalize (like two reservoirs oflevels equalize (like two reservoirs of

water coming into equilibrium). This causes the conduction and valencewater coming into equilibrium). This causes the conduction and valencebands to bend as shown above.bands to bend as shown above.

In the middle of theIn the middle of the junction E junction EFF falls midwayfalls midwaybetween the VB & CB as itbetween the VB & CB as it

would in an intrinsicwould in an intrinsicsemiconductor.semiconductor.

p-n Junctionsp-n Junctions

Applications of p-n JunctionsApplications of p-n Junctions

MOSFET TransistorMOSFET Transistor

Photovoltaic CellPhotovoltaic Cell

LEDLEDRectifier:Rectifier:

Reverse BiasReverse Bias

conductivity lect20

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