Metals, Semiconductors, and Insulators

Metals have free electrons and partially filled valence bands, therefore they are highly conductive (a).

Semimetals have their highest band filled. This filled band, however, overlaps with the next higher band, therefore they are conductive but with slightly higher resistivity than normal metals (b). Examples: arsenic, bismuth, and antimony.

Insulators have filled valence bands and empty conduction bands, separated by a large band gap Eg(typically >4eV), they have high resistivity (c ).

Semiconductors have similar band structure as insulators but with a much smaller band gap. Some electrons can jump to the empty conduction band by thermal or optical excitation (d). Eg=1.1 eV for Si, 0.67 eV for Ge and 1.43 eV for GaAs

Every solid has its own characteristic energy band structure.In order for a material to be conductive, both free electrons and empty states must be available.

An energy band is a range of allowed electron energies.The energy band in a metal is only partially filled with electrons.Metals have overlapping valence and conduction bands

Metals

Conduction in Terms of Band

Drude Model of Electrical Conduction in Metals

Conduction of electrons in metals – A Classical Approach:In the absence of an applied electric field (ξ) the electrons move in random directions colliding with random impurities and/or lattice imperfections in the crystal arising from thermal motion of ions about their equilibrium positions. The frequency of electron-lattice imperfection collisions can be described by a mean free path λ -- the average distance an electron travels between collisions. When an electric field is applied the electron drift (on average) in the direction opposite to that of the field with drift velocityThe drift velocity is much less than the effective instantaneous speed (v) of the random motion

v

In copper while where

The drift speed can be calculated in terms of the applied electric field ξ and of v and λWhen an electric field is applied to an electron in the metal it experiences a force qξresulting in acceleration (a)

Then the electron collides with a lattice imperfection and changes its direction randomly. The mean time between collisions is

The drift velocity is

If n is the number of conduction electrons per unit volume and J is the current density

Combining with the definition of resistivity gives

1210 −−≈ scmv . 1810 −≈ scmv . Tkvm Be 23

21 2 =

emqa ξ=

vλτ =

vmq

mqav

ee ⋅⋅⋅

=⋅⋅

=⋅=λξτξτ

σξ νnqJ ==

vmqne ⋅⋅⋅

=λσ

2

ee mq

vmq τλμ ⋅=⋅⋅

=

q=1.6x10-19C

For an electron to become free to conduct, it must be promoted into an empty available energy stateFor metals, these empty states are adjacent to the filled statesGenerally, energy supplied by an electric field is enough to stimulate electrons into an empty state

States Filled with Electrons

Empty States

“Freedom”

Ele

ctro

n E

nerg

y

Distance

Energy Band

At T = 0, all levels in conduction band below the Fermi energy EF are filledwith electrons, while all levels above EF are empty.Electrons are free to move into “empty” states of conduction band with only a small electric field E, leading to high electrical conductivity!At T > 0, electrons have a probability to be thermally “excited” from below the Fermi energy to above it.

Band Diagram: Metal

EF

EC

Conduction band(Partially Filled)

T > 0

Fermi “filling”function

Energy band to be “filled”

E = 0

Resistivity (ρ) in MetalsResistivity typically increases linearly with temperature:

ρt = ρo + αTPhonons scatter electrons. Where ρo and α are constants for an

specific materialImpurities tend to increase resistivity: Impurities scatter electrons in metalsPlastic Deformation tends to raise resistivity dislocations scatter electrons

The electrical conductivity is controlled by controlling the number of charge carriers in the material (n) and the mobility or “ease of movement” of the charge carriers (μ)

μρ

σ nq== 1

Temperature Dependence, MetalsThere are three contributions to ρ:ρt due to phonons (thermal)ρi due to impuritiesρd due to deformation

ρ = ρt + ρi+ ρd The number of electrons in the conduction band does not vary with temperature.

All the observed temperature dependence of σ in metals arise from changes in μ

Scattering by Impurities and PhononsScattering by Impurities and Phonons

Thermal: Phonon scatteringProportional to temperature

Impurity or Composition scatteringIndependent of temperatureProportional to impurity concentration

Solid SolutionTwo Phase

Deformation

Taot += ρρ

)1( iii cAc −=ρ

ββαα ρρρ VVt +=

determinedallyexperimentbemust=dρ

InsulatorE

lect

ron

Ene

rgy

“Conduction Band”Empty

“Valence Band”Filled with Electrons

“Forbidden” EnergyGap

Distance

The valence band and conduction band are separated by a large (> 4eV) energy gap, which is a “forbidden” range of energies. Electrons must be promoted across the energy gap to conduct, but the energy gap is large. Energy gap º Eg

Band Diagram: Insulator

At T = 0, lower valence band is filled with electrons and upper conduction band is empty, leading to zero conductivity.Fermi energy EF is at midpoint of large energy gap (2-10 eV) between conduction and valence bands.At T > 0, electrons are usually NOT thermally “excited” from valence to conduction band, leading to zero conductivity.

EF

EC

EV

Conduction band(Empty)

Valence band(Filled)

Egap

T > 0

Conduction in Ionic Materials (Insulators)Conduction by electrons (Electronic Conduction): In a ceramic, all the outer (valence) electrons are involved in ionic or covalent bonds and thus they are restricted to an ambit of one or two atoms.

If Eg is the energy gap, the fraction of electrons in the conduction band is:Tk

E

B

g

e 2−

A good insulator will have a band gap >>5eV and kBT~0.025eV at room temperature

As a result of thermal excitation, the fraction of electrons in the conduction band is

~e-200 or 10-80.There are other ways of changing the electrical conductivity in the ceramic which have a far greater effect than temperature.

•Doping with an element whose valence is different from the atom it replaces. The doping levels in an insulator are generally greater than the ones used in semiconductors. Turning it around, material purity is important in making a good insulator.

•If the valence of an ion can be variable (like iron), “hoping” of conduction can occur, also known as “polaron” conduction. Transition elements.

•Transition elements: Empty or partially filled d or f orbitals can overlap providing a conduction network throughout the solid.

Conduction by Ions: ionic conductionIt often occurs by movement of entire ions, since the energy gap is too large for electrons to enter the conduction band.

The mobility of the ions (charge carriers) is given by:

Where q is the electronic charge ; D is the diffusion coefficient ; kB is Boltzmann’s constant, T is the absolute temperature and Z is the valence of the ion.

The mobility of the ions is many orders of magnitude lower than the mobility of the electrons, hence the conductivity is very small:

TkDqZ

B ...

=μ

μσ ... qZn=Example:Suppose that the electrical conductivity of MgO is determined primarily by the diffusion of Mg2+ ions. Estimate the mobility of Mg2+ ions and calculate the electrical conductivity of MgO at 1800oC.Data: Diffusion Constant of Mg in MgO = 0.0249cm2/s ; lattice parameter of MgO a=0.396x10-7cm ; Activation Energy for the Diffusion of Mg2+ in MgO = 79,000cal/mol ; kB=1.987cal/K=k-mol; For MgO Z=2/ion; q=1.6x10-19C; kB=1.38x10-23J/K-mol

First, we need to calculate the diffusion coefficient D

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

−=⎟

⎠⎞

⎜⎝⎛ −=

KKxmolcalmolcal

scm

kTQDD Do )(/.

/exp.exp27318009871

79000023902

D=1.119x10-10cm2/s

Next, we need to find the mobility

sJcmCCioncarriers

TkDqZ

B ...

))(.().)(.)(/(

... 29

23

1019

10121273180010381

101110612 −−

−−

×=+×

××==μ

C ~ Amp . sec ; J ~ Amp . sec .Volt μ=1.12x10-9 cm2/V.s

MgO has the NaCl structure (with 4 Mg2+and 4O2- per cell)

Thus, the Mg2+ ions per cubic cm is:

32237

2

1046103960

4 cmionscmcellionsMgn /.

).(/

×=×

= −+

sVcmcmC

nZq

....

).)(.)()(.(

3

26

91922

109422

10121106121046

−

−−

×=

×××==

σ

μσ

C ~ Amp.sec ; V ~ Amp.Ω σ = 2.294 x 10-5 (Ω.cm)-1

Example:

The soda silicate glass of composition 20%Na2O-80%SiO2 and a density of approximately 2.4g.cm-3 has a conductivity of 8.25x10-6 (Ω-m)-1 at 150oC. If the conduction occurs by the diffusion of Na+ ions, what is their drift mobility?

Data: Atomic masses of Na, O and Si are 23, 16 and 28.1 respectively

Solution:

We can calculate the drift mobility (μ) of the Na+ ions from the conductivity expression

ii qn μσ ××=Where ni is the concentration of Na+ ions in the structure.

20%Na2O-80%SiO2 can be written as(Na2O)0.2-(SiO2)0.8 . Its mass can be calculated as:

14860

162128118016123220−=

+×++×=

molgMM

At

At

..))().((.))()((.

The number of (Na2O)0.2-(SiO2)0.8 units per unit volume can be found from the density

3802202

22

1

1233

10392

486010023642

−

−

−−

−×=

×=

×=

cmunitsSiOONanmolg

molxcmgM

NnAt

A

.. )()(...

).()..(ρ

The concentration of Na+ ions (ni) can be obtained from the concentration of (Na2O)0.2-(SiO2)0.8 units

32122 101831039221801220

220 −×=××⎥⎦

⎤⎢⎣

⎡+×++×

×= cmni ..)(.)(.

.

And μi

11214

362119

116

10621

101018631060110258

−−−

−−

−−−

×=

××××Ω×

=×

=

sVmmC

mnq

i

ii

.).().(

).(

μ

σμ

This is a very small mobility compared to semiconductors and metals

Electrical BreakdownAt a certain voltage gradient (field) an insulator will break down.

There is a catastrophic flow of electrons and the insulator is fragmented.

Breakdown is microstructure controlled rather than bonding controlled.

The presence of heterogeneities in an insulator reduces its breakdown field strength from its theoretical maximum of ~109Vm-1 to practical values of 107V.m-1

Energy Bands in SemiconductorsEnergy Levels and Energy Gap in a Pure Semiconductor.The energy gap is < 2 eV. Energy gap º Eg

Semiconductors have resistivities in between those of metals and insulators.Elemental semiconductors (Si, Ge) are perfectly covalent; by symmetry electrons shared between two atoms are to be found with equal probability in each atom.Compound semiconductors (GaAs, CdSe) always have some degree of ionicity. In III-V compounds, eg. Ga+3As+5, the five-valent As atoms retains slightly more charge than is necessary to compensate for the positive As+5 charge of the ion core, while the charge of Ga+3 is not entirely compensated. Sharing of electrons occurs still less fairly between the ions Cd+2 and Se+6 in the II-VI compund CdSe.

Elec

tron E

nerg

y

“Conduction Band” (Nearly) Empty – Free electrons

“Valence Band” (Nearly) Filled with Electrons – Bonding electrons

“Forbidden” Energy Gap

Semiconductor MaterialsSemiconductor Bandgap Energy EG (eV)Carbon (Diamond) 5.47Silicon 1.12 Germanium 0.66Tin 0.082Gallium Arsenide 1.42Indium Phosphide 1.35Silicon Carbide 3.00Cadmium Selenide 1.70Boron Nitride 7.50 Aluminum Nitride 6.20Gallium Nitride 3.40Indium Nitride 1.90

IIIA IV A V A V IA

10.8115

BBo ro n

12.011156

CC a rb o n

14.00677

NN itro g e n

15.99948

OO xyg e n

IIB

26.981513

AlA lum inum

28.08614

SiSilic o n

30.973815

PPho sp ho rus

32.06416

SSulfur

65.3730

ZnZinc

69.7231

G aG a llium

72.5932

G eG e rm a nium

74.92233

AsA rse nic

78.9634

SeSe le nium

112.4048

C dC a d m ium

114.8249

InInd ium

118.6950

SnTin

121.7551

SbA ntim o ny

127.6052

TeTe llurium

200.5980

HgM e rc ury

204.3781

TiTha llium

207.1982

PbLe a d

208.98083

BiBism uth

(210)84

PoPo lo nium

Portion of the Periodic Table Including the Most Important Semiconductor Elements

Band Diagram: Semiconductor with No Doping

At T = 0, lower valence band is filled with electrons and upper conduction band is empty, leading to zero conductivity.Fermi energy EF is at midpoint of small energy gap ( 0, electrons thermally “excited” from valence to conduction band, leading to measurable conductivity.

EFECEV

Conduction band(Partially Filled)

Valence band(Partially Empty)

T > 0

Semi-conductors (intrinsic - ideal)Perfectly crystalline (no perturbations in the periodic lattice).Perfectly pure – no foreign atoms and no surface effectsAt higher temperatures, e.g., room temperature (T @ 300 K), some electrons are thermally excited from the valence band into the conduction band where they are free to move.“Holes” are left behind in the valence band. These holes behave like mobilepositive charges.

CB electrons and VB holes can move around (carriers).

At edges of band the kinetic energy of the carriers is nearly zero. The electron energy increases upwards. The hole energy increases downwards.

Si Si Si Si Si Si Si

Si Si Si Si Si Si Si

Si Si Si Si Si Si Si

free electron

free hole

Si Si Si Si Si Si Si

Si Si Si Si Si Si Si

Si Si Si Si Si Si Si

positive ion core

valence electron

Semiconductors in Group IVCarbonSiliconGermaniumTinEach has 4 valence Electrons.Covalent bond

Generation of Free Electrons and HolesIn an intrinsic semiconductor, the number of free electrons equals the number of holes.Thermal : The concentration of free electrons and holes increases with increasing temperature.Thermal : At a fixed temperature, an intrinsic semiconductor with a large energy gap has smaller free electron and hole concentrations than a semiconductor with a small energy gap.Optical: Light can also generate free electrons and holes in a semiconductor.Optical: The energy of the photons (hν) must equal or exceed the energy gap of the semiconductor (Eg) . If hν > Eg , a photon can be absorbed, creating a free electron and a free hole.This absorption process underlies the operation of photoconductive light detectors, photodiodes, photovoltaic (solar) cells, and solid state camera “chips”.

UV 100-400 nm 12.4-3.10 eVViolet 400-425 nm 3.10-2.92 eVBlue 425-492 nm 2.92-2.52 eVGreen 492-575 nm 2.52-2.15 eVYellow 575-585 nm 2.15-2.12 eVOrange 585-647 nm 2.12-1.92 eVRed 647-700 nm 1.92-1.77 eVNear IR 10,000-700 nm 1.77-0.12 eV

RedRed

OrangeOrange

YellowYellow GreenGreen

BlueBlue

VioletViolet

Photoconductivity

Eg ω � Eg

Conductivity is dependent on the intensity of the incident electromagnetic radiation

E = hν = hc/λ, c = λ(m)ν(sec -1)

hν ≥ Eg

Band Gaps: Si - 1.11 eV (Infra red)Ge 0.66 eV (Infra red)GaAs 1.42 eV (Visible red)ZnSe 2.70 eV (Visible yellow)SiC 2.86 eV (Visible blue)GaN 3.40eV (Blue)AlN 6.20eV (Blue-UV)BN 7.50eV (UV)

Total conductivity σ = σe + σh = nqμe + pqμhFor intrinsic semiconductors: n = p & σ = nq(μe + μh)

As T ↑ then ni ↑As Eg↑ then ni ↓

What is the detailed form of these dependencies?We will use analogies to chemical reactions. The electron-hole formation can be though of as a chemical reaction……..Similar to the chemical reaction………

Question: How many electrons and holes are there in an intrinsic semiconductor in thermal equilibrium? Define:no equilibrium (free) electron concentration in conduction band [cm-3]po equilibrium hole concentration in valence band [cm-3]Certainly in intrinsic semiconductor: no = po = nini intrinsic carrier concentration [cm-3]

+− +⇔ hebond−+ +⇔ )(OHHOH2

iOO npn ==

⎟⎠⎞

⎜⎝⎛−≈=

−+

kTE

OHOHHK exp

][]][[

2

The Law-of-Mass-Action relates concentration of reactants and reaction products. For water……Where E is the energy released or consumed during the reaction………….

This is a thermally activated process, where the rate of the reaction is limited by the need to overcome an energy barrier (activation energy).

By analogy, for electron-hole formation:

Where [bonds] is the concentration of unbroken bonds and Eg is the activation energy

In general, relative few bonds are broken to form an electron-hole and therefore the number of bonds are approximately constant.

2iOO npn =×

⎟⎟⎠

⎞⎜⎜⎝

⎛−≈=

kTE

bondspnK goo exp

][]][[

tcons[bonds] ,pn[bonds] oo

tan=>>

⎟⎟⎠

⎞⎜⎜⎝

⎛−≈

kTE

pn goo exp

⎟⎟⎠

⎞⎜⎜⎝

⎛−≈

kTE

n gi 2exp

Two important results:

1)……………………..

2)………………………………………….……..

The equilibrium np product in a semiconductor at a certain temperature is a constant specific to the semiconductor.

Effect of Temperature on Intrinsic SemiconductivityThe concentration of electrons with sufficient thermal energy to enter the conduction band (and thus creating the same concentration of holes in the valence band) ni is given by

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−≈

TkEn

Bi exp

For intrinsic semiconductor, the energy is half way across the gap, so that

⎟⎟⎠

⎞⎜⎜⎝

⎛ −≈

TkE

nB

gi 2

exp

Since the electrical conductivity σ is proportional to the concentration of electrical charge carriers, then

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

TkE

B

gO 2

expσσ

Example

Calculate the number of Si atoms per cubic meter. The density of silicon is 2.33g.cm-3and its atomic mass is 28.03g.mol-1.

Then, calculate the electrical resistivity of intrinsic silicon at 300K. For Si at 300K ni=1.5x1016carriers.m-3, q=1.60x10-19C, μe=0.135m2(V.s)-1 and μh=0.048m2.(V.s)-1

Solution

32810005 −−×=×= matomsSiA

NnSi

SiASi ..

ρ

( )myresistivit

mqn hei−Ω×==

−Ω×=+××= −−

3

13

10282

1043850

.)(.

ρ

μμσ

ExampleThe electrical resistivity of pure silicon is 2.3x103Ω-m at room temperature (27oC ~ 300K). Calculate its electrical conductivity at 200oC (473K). Assume that the Eg of Si is 1.1eV ; kB =8.62x10-5eV/K

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

TkE

CB

g

2exp.σ ⎟⎟

⎠

⎞⎜⎜⎝

⎛ −=

)(exp.

4732473 Bg

kE

Cσ ⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

)(exp.

3002300 Bg

kE

Cσ

13300473

300

473

5300

473

300

473

04123851032

12385

7777

4731

3001

10628211

4731

3001

230024732

30024732

−

−

Ω=×

==

=⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎠⎞

⎜⎝⎛ −

×=⎟

⎠⎞

⎜⎝⎛ −=+

−=⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−

−=

).(.)(.

)(

.ln

).(.

)()(ln

)()(exp

m

eVk

Ek

Ek

E

kE

kE

B

g

B

g

B

g

B

g

B

g

σσ

σσ

σσ

σσ

Example: For germanium at 25oC estimate (a) the number of charge carriers, (b) the fraction of total electrons in the valence band that are excited into the conduction

band and (c) the constant A in the expression when E=Eg/2

Data: Ge has a diamond cubic structure with 8 atoms per cell and valence of 4 ; a=0.56575nm ; Eg for Ge = 0.67eV ; μe = 3900cm2/V.s ; μh = 1900cm2/V.s ; ρ = 43Ω-cm ; kB=8.63x10-5eV/K

eVKeVTkCT

B

o

0514025273106382225

5 .))(/.)(( =+×==

−

313

19 10521900390010610230

cmelectrons

qn

he

×=+×

=+

= − .)(..

)( μμσ

There are 2.5x1013 electrons/cm3 and 2.5x1013 holes/cm3 helping to conduct a charge in germanium at room temperature.

(a) Number of carriers

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

TkE

AnB

g

2exp

b) the fraction of total electrons in the valence band that are excited into the conduction band

The total number of electrons in the valence band of germanium is :

371056575048

).()/)(/(

cmxatomselectronsvalencecellatoms

electronsValence −−=−

32310771 cmelectronselectronsvalenceTotal /. ×=−−

1023

13

3

3

10411107711052 −×=××

=−−−−

=− ...

//cmelectronsvalenceTotalcmelectronsexcitednumberexcitedFraction

(c) the constant A

319

05140670

13

2

101411052 cmcarriersee

nATk

E

B

g/..

..

−

⎟⎠⎞

⎜⎝⎛ −

⎟⎟⎠

⎞⎜⎜⎝

⎛ −×=

×==

Direct and Indirect SemiconductorsThe real band structure in 3D is calculated with various numerical methods, plotted as E vs k. k is called wave vectorFor electron transition, both E and p (k) must be conserved.

momentum is pkp =

A semiconductor is indirect if the …do not have the same k valueDirect semiconductors are suitable for making light-emitting devices, whereas the indirect semiconductors are not.

A semiconductor is direct if the maximum of the conduction band and the minimum of the valence band has the same k value