ConductivityConductivitySemiconductors & Semiconductors &

MetalsMetals

Chemistry 754Chemistry 754Solid State Chemistry Solid State Chemistry

Lecture #20Lecture #20May 14, 2003May 14, 2003

References – ConductivityReferences – Conductivity

There are many references that describe There are many references that describe electronic conductivity in metals and electronic conductivity in metals and semiconductors. I used primarily the following semiconductors. I used primarily the following texts to develop this lecture.texts to develop this lecture.

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

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

““Solid State Physics”Solid State Physics”H. Ibach and H. Luth, Springer-Verlag, Berlin (1991).H. Ibach and H. Luth, Springer-Verlag, Berlin (1991).

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

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

Resistivities of Real Resistivities of Real MaterialsMaterials

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

Ca 3.9 10- 6 Si ~ 0.1

Ti 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 1019

Most semiconductors in their pure form are not good Most semiconductors in their pure form are not good conductors, they need to be doped to become conductors, they need to be doped to become

conducting. conducting.

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

Microscopic ConductivityMicroscopic Conductivity

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

= ne(e= ne(e/m*)/m*)

= e= e/m*/m*

= ne= newherewhere

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

e = the charge of an electron = 1.602 e = the charge of an electron = 1.602 10 10-19-19 C C

= the relaxation time (s) {the time between = the relaxation time (s) {the time between collisions}collisions}

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

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

EF

DOS

En

erg

y

MetalMetal

EF

DOS

En

erg

y

SemimetalSemimetal

EF

DOS

En

erg

y

SemiconductSemiconductoror

/Insulator/InsulatorIn a metal the Fermi level cuts through a band to produce a partially In a metal the Fermi level cuts through a band to produce a partially filled band. In a semiconductor/insulator there is an energy gap between filled band. In a semiconductor/insulator there is an energy gap between

the filled bands and the empty bands. The distinction between a the filled bands and the empty bands. The distinction between a semiconductor and an insulator is artificial, but as the gap becomes semiconductor and an insulator is artificial, but as the gap becomes

large the material usually becomes a poor conductor of electricity. A large the material usually becomes a poor conductor of electricity. A semimetal results when the band gap goes to zero.semimetal results when the band gap goes to zero.

Conduction Conduction BandBand

Valence Valence BandBand

Metals, Semiconductors & Metals, Semiconductors & InsulatorsInsulators

Resistivity and Carrier Resistivity and Carrier ConcentrationConcentration

The resistivities of real materials span nearly 25 orders of magnitude. The resistivities 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 (). ). Let’s first consider carrier concentration.Let’s first consider carrier concentration.

•The carrier concentration only includes electrons which can easily be The carrier concentration only includes electrons which can easily be excited from occupied states into empty states. The remaining excited from occupied states into empty states. The remaining electrons are localized.electrons are localized.

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

•Only electrons whose energies are within a few kT of EOnly electrons whose energies are within a few kT of EFF can contribute can contribute to the electrical conductivity. to the electrical conductivity.

Generally this means that EGenerally this means that EFF should cut a band to should cut a band to achieve appreciable carrier concentration. achieve appreciable carrier concentration.

Alternatively impurities/defects are introduced to Alternatively impurities/defects are introduced to partially populate a band.partially populate a band.

Fermi-Dirac FunctionFermi-Dirac FunctionThe Fermi-Dirac function gives the fraction of allowed states, f(E), at The Fermi-Dirac function gives the fraction of allowed states, f(E), at an 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 the Fermi Energy, Ewhere the Fermi Energy, 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 the That is to say one half of the available states are occupied. T is the temperature (in K) and k is the Boltzman constant (k = 8.62 temperature (in K) and k is the Boltzman constant (k = 8.62 10 10-5-5 eV/K)eV/K)

As an example consider f(E) for T = 300 K and a state 0.1 eV above As an example consider f(E) for T = 300 K and a state 0.1 eV above EEFF::

f(E) = 1/[1 + exp{(0.1 eV)/((300K)(8.62 f(E) = 1/[1 + exp{(0.1 eV)/((300K)(8.62 10 10-5-5 eV/K)}] eV/K)}]f(E) = 0.02 = 2%f(E) = 0.02 = 2%

Consider a band gap of 1 eV.Consider a band gap of 1 eV.f(1 eV) = 1.6 f(1 eV) = 1.6 10 10-17-17

See that for even a moderate band gap (Silicon has a band gap of 1.1 See that for even a moderate band gap (Silicon has a band gap of 1.1 eV) the intrinsic concentration of electrons that can be thermally eV) the intrinsic concentration of electrons that can be thermally

excited to move about the crystal is tiny. Thus pure Silicon (if you excited 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 Dirac Function Fermi Dirac Function Metals and SemiconductorsMetals and Semiconductors

f(E) as determined experimentally for Ru metal (note the energy scale)

f(E) for a semiconductor

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

= e= e/m*/m*where,where,

ee = electronic charge = electronic chargem*m* = the effective mass = the effective mass = 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 depends upon the band width, which in turn

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

mobility?mobility?• A defect or impurity (A defect or impurity ( increases as purity increases) increases as purity increases)• Lattice vibrations, phonons (Lattice vibrations, phonons ( 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 structure In our development of the electronic band structure from a linear combination of atomic orbitals the from a linear combination of atomic orbitals the variable k was used to determine the phase of the variable k was used to determine the phase of the orbitals. What exactly is k?orbitals. What exactly is k?

WavevectorWavevector – It tells us the how the phases of the – It tells us the how the phases of the orbitals change when translational symmetry is orbitals change when translational symmetry is applied.applied.

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

Crystal MomentumCrystal Momentum – In free electron theory k is – In free electron theory k is proportional to the momentum of the electron in the proportional to the momentum of the electron in the kkthth wavefunction. wavefunction.

To better understand the meaning of k, consider an electron To better understand the meaning of k, consider an electron at the outer edge of the Brillouin zone, where k = at the outer edge of the Brillouin zone, where k = /a. The /a. The phase of the electronic wavefunction changes sign every unit phase of the electronic wavefunction changes sign every unit cell (similar to a p-orbital changing phase at its nodal plane)cell (similar to a p-orbital changing phase at its nodal plane)

= 2a = 2a a = a = /2/2k = k = /a /a a = a = /k/k

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

/2 = /2 = /k/kk = 2k = 2//22/k/k

The wavelength of the wavefunction is The wavelength of the wavefunction is inversely proportional to k.inversely proportional to k.

Crystal MomentumCrystal Momentum

a

Now consider the DeBroglie relationship (wave-particle duality Now consider the DeBroglie relationship (wave-particle duality of matter)of matter)

= h/p= h/p

p = h/p = h/

p = hk/2p = hk/2

where.,where.,• p is the momentum of the wavepacket,p is the momentum of the wavepacket,• h is Planck’s constant, 6.626 h is Planck’s constant, 6.626 10 10-34-34 J-s J-s

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

k is a measure of the “crystal” momentum of an k is a measure of the “crystal” momentum of an electron in the electron in the KK wavefunction. wavefunction.

Crystal MomentumCrystal Momentum

From the ideas on the previous 2 slides one can derive the From the ideas on the previous 2 slides one can derive the following relationships to describe the properties of a following relationships to describe the properties of a

conduction electron:conduction electron:

Velocity Velocity v = hk/2v = hk/2m = (2m = (2/h)(dE/dk)/h)(dE/dk)Energy Energy E = (h/2E = (h/2)(k)(k22/2m/2m**))

Effective Mass Effective Mass mm** = (2 = (2/h)/h)22 (1/{d (1/{d22E/dkE/dk22})})

dE/dk dE/dk 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 Band dE/dkdE/dk Large Large Small Small HighHigh Low LowVelocityVelocity Fast Fast Slow Slow m*m* Light Light Heavy Heavy

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

Bandstructure & DOS for CuBandstructure & DOS for Cu

EEFF cuts the very wide (disperse) s band, giving cuts the very wide (disperse) s band, giving rise to a large carrier concentration, along with rise to a large carrier concentration, along with

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

Temperature Dependence-Temperature Dependence-MetalsMetals

Recall that Recall that

= ne= ne22/m*/m*

In MetalsIn Metals– The carrier concentration, n, changes very The carrier concentration, n, changes very

slowly with temperature.slowly with temperature.– is inversely proportional to temperature is inversely proportional to temperature

( ( 1/T), due to scattering by lattice 1/T), due to scattering by lattice vibrations (phonons).vibrations (phonons).

– Therefore, a plot of Therefore, a plot of vs. 1/T (or vs. 1/T (or vs. T) is vs. T) is essentially linear.essentially linear.

– Conductivity goes down as Conductivity goes down as temperature increasestemperature increases..

Scattering by Impurities and Scattering by Impurities and PhononsPhonons

Phonon scattering

•Proportional to temperature

Impurity scattering

•Independent of temperature

•Proportional to impurity concentration

Bandstructure for GeBandstructure for Ge

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

Ge is an indirect gap semiconductor, because the uppermost Ge is an indirect gap semiconductor, because the uppermost VB energy and the lowest CB energy occur at different VB energy and the lowest CB energy occur at different

locations in k-space.locations in k-space.

p-bandsp-bands

s-bands-band

No No mixing mixing at at ..

CB minimumCB minimum

VB maximumVB maximum

Direct & Indirect Gap Direct & Indirect Gap SemiconductorsSemiconductors

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

(h(h-E-Egg))1/21/2

Indirect Gap SemiconductorIndirect Gap Semiconductor: Maximum of the valence band and : Maximum of the valence band and minimum of the conduction band fall different points in k-space. A lattice minimum of the conduction band fall different points in k-space. A lattice vibration (phonon) is involved in electronic excitations, this decreases the vibration (phonon) is involved in electronic excitations, this decreases the absorption efficiency. absorption efficiency. (h(h-E-Egg))22

GeGe SiSi GaAsGaAs

Figure taken from “Fundamentals of

Semiconductor Theory and Device

Physics”, by S. Wang

Doping SemiconductorsDoping SemiconductorsThe Fermi-Dirac function shows that a pure semiconductor with The Fermi-Dirac function shows that a pure semiconductor with

a band gap of more than a few tenths of an eV would have a a band gap of more than a few tenths of an eV would have a very small concentration of carriers. Therefore, impurities are very small concentration of carriers. Therefore, impurities are

added to introduce carriers.added to introduce carriers.

n-dopingn-doping Replacing a lattice Replacing a lattice atom with an impurity (donor) atom with an impurity (donor) atom that contains 1 additional atom that contains 1 additional valence electron (i.e. P in Si). This valence electron (i.e. P in Si). This ee-- can easily be donated to the can easily be donated to the conduction band.conduction band.

p-dopingp-doping Replacing a lattice Replacing a lattice atom with an impurity (acceptor) atom with an impurity (acceptor) atom that contains 1 less valence atom that contains 1 less valence electron (i.e. Al in Si). This atom electron (i.e. Al in Si). This atom can easily accept an ecan easily accept an e-- from the VB from the VB creating a hole.creating a hole.

ee--

Valence BandValence Band

Conduction Conduction BandBandee--

Valence BandValence Band

Conduction Conduction BandBand

EEFF EEFF

Common Semiconductor Common Semiconductor StructuresStructures

DiamondDiamondFd-3m Fd-3m (Z=8)(Z=8)

C, Si, Ge, C, Si, Ge, SnSn

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

GaAs, ZnS, InSbGaAs, ZnS, InSb

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

CuFeSCuFeS22, ZnSiAs, ZnSiAs22

Properties of Properties of SemiconductorsSemiconductors

CompoundCompound StructureStructure Bandgap Bandgap (eV)(eV)

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

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

SiSi DiamondDiamond 1.11 (I)1.11 (I) 1,3501,350 480480

GeGe DiamondDiamond 0.67 (I)0.67 (I) 3,9003,900 1,9001,900

AlPAlP SphaleriteSphalerite 2.43 (I)2.43 (I) 8080 ------

GaAsGaAs SphaleriteSphalerite 1.43 (D)1.43 (D) 8,5008,500 400400

InSbInSb SphaleriteSphalerite 0.18 (D)0.18 (D) 100,000100,000 1,7001,700

AlAsAlAs SphaleriteSphalerite 2.16 (I)2.16 (I) 1,0001,000 180180

GaNGaN WurtziteWurtzite 3.4 (D)3.4 (D) 300300 ------

Temperature Dependence-Temperature Dependence-SemiconductorsSemiconductors

Recall that Recall that

= ne= ne22/m*/m*

In SemiconductorsIn Semiconductors– The carrier concentration increases as temperature The carrier concentration increases as temperature

goes up, due to excitations across the band gap, Egoes up, due to excitations across the band gap, Egg. .

– n is proportional to exp{-En is proportional to exp{-Egg/2kT}./2kT}.

– is inversely proportional to temperatureis inversely proportional to temperature– The exponential dependence of n dominates, The exponential dependence of n dominates,

therefore, a plot of ln therefore, a plot of ln vs. 1/T is essentially linear. vs. 1/T is essentially linear.– Conductivity increases as temperature Conductivity increases as temperature

increasesincreases..

When a p-type and an n-type semiconductor are brought into When a p-type and an n-type semiconductor are brought into contact electrons flow from the n-doped semiconductor into the p-contact electrons flow from the n-doped semiconductor into the p-

doped semiconductor until the Fermi levels equalize (like two doped semiconductor until the Fermi levels equalize (like two reservoirs of water coming into equilibrium). This causes the reservoirs of water coming into equilibrium). This causes the

conduction and valence bands to bend as shown above.conduction and valence bands to bend as shown above.

In the middle of the In the middle of the junction Ejunction EFF falls midway falls midway between the VB & CB as between the VB & CB as it would in an intrinsic it would in an intrinsic

semiconductor. semiconductor.

p-n Junctionsp-n Junctions

Applications of p-n JunctionsApplications of p-n Junctions

MOSFET MOSFET TransistorTransistor

Photovoltaic CellPhotovoltaic Cell

LEDLEDRectifier:Rectifier:Reverse Reverse

BiasBias