Pauli Exclusion Principle - academic.uprm.eduacademic.uprm.edu/pcaceres/Courses/Smart/SMD-2B.pdf · Pauli Exclusion Principle Electrons in a single atom occupy discrete levels of

Pauli Exclusion PrincipleElectrons in a single atom occupy discrete levels of energy. No two “energy levels” or “states” in an atom can have the same energy.Each energy level can contain at most two electrons -- one with “clockwise spin” and one with “counterclockwise spin”. If two or more atoms are brought together, their outer (i.e., valence) energy levels must shift slightly so they will be different from one another.If many (e.g., N) atoms are brought together to form a solid, the Pauli Exclusion Principle still requires that only two electrons in the entire solid have the same energy. There will be N distinct, but only slightly different valence energy levels, forming a valence band.

When a solid is formed, the different split energy levels of electrons come together to form continuous bands of energies (electron energy band).

The extent of splitting depends on the interatomic separation and begins with the outer most electron shells as they are the first to be perturbed as the atoms coalesce.

Band Theory: Two Approaches• There are two approaches to finding the electron energies associated with

atoms in a periodic lattice.• Approach #1: “Bound” Electron Approach (Solve single atom energies!)

– Isolated atoms brought close together to form a solid.• Approach #2: “Unbound” or Free Electron Approach (E = p2/2m)

– Free electrons modified by a periodic potential (i.e. lattice ions).

• Both approaches result in grouped energy levels with allowed and forbidden energy regions.

– Energy bands overlap for metals.– Energy bands do not overlap (or have a “gap”) for semiconductors.

Solid of N atomsTwo atoms Six atoms

Band Theory: “Bound” Electron Approach• For the total number N of atoms in a solid (1023 cm–3), N energy levels

split apart within a width ΔE.– Leads to a continuous band of energies for each initial atomic

energy level (e.g. 1s energy band for 1s energy level).

The electrical properties of a solid material are a consequence of its electron band structure, that is, the arrangement of the outermost electron bands and the way in which they are filled with electrons.

Four different types of band structure are possible at 0oK:

Outermost band is only partially filled with electrons (eg, Na, Cu) (Cu has a single 4s valence electron, however for a solid comprised of N atoms, the 4s band is capable of accommodating 2N electrons. Thus, only half of the 4s band is filled.)

Overlap of an empty band and a filled band (eg. Mg). (An isolated Mg atom has 2 electrons in its 3s level. When a solid is formed, the 3s and 3p energy levels overlap.)

A completely filled valence band is separated from the conduction band by a relatively wide energy band gap (insulators).

A completely filled valence band is separated from the conduction band by a relative narrow energy band gap (semiconductors)

States Filled with Electrons

Empty States

“Freedom”E

lect

ron

Ene

rgy

Distance

Energy Band

Ef

For a Li crystal with N atoms there are 3N electrons. The 1s band is filled and the 2s band is half-filled.

The Fermi Level corresponds to the Highest Occupied Molecular Orbital (HOMO).

The energy difference between adjacent states is infinitesimally small. The electrons near the Fermi level can move from filled states to empty states with no activation energy (metals).

Ele

ctro

n E

nerg

y

“Conduction Band”Empty

“Valence Band”Filled with Electrons

“Forbidden”EnergyGap

Distance

If “valence” band is filled, no empty space are available above the filled states. Electrons can be promoted to the “conduction band “Lowest Unoccupied Molecular Orbital –LUMO” with an activation energy > energy gap (band gap). Difference between semiconductors and insulators is due to the size of the bandgap.

When the temperature of the metal increases, some electrons gain energy and are excited into the empty energy levels in a valence band. This condition creates an equal number of empty energy levels, or holes, vacated by the excited electrons. Only a small increase in energy is required to cause excitation of electrons.

Both the excited electrons (free electrons) and the newly created holes can then carry an electrical charge.

Temperature Effect

Fermi Energy is the energy corresponding to the highest filled state at 0oK.

Only electrons above the Fermi energy can be affected by an electric field (free electrons).

Band Diagram: Fermi-Dirac “Filling” FunctionThe Fermi-Dirac function gives the fraction of allowed states, fFD(E), at an energy level E, that are populated at a given temperature.

where the Fermi Energy, EF, is defined as the energy where fFD(EF) = 1/2. That is to say one half of the available states are occupied. T is the temperature (in K) and kB is the Boltzman constant (kB = 8.62 x10-5 eV/K)

( ) ( )1

1F

FD E EkT

f E

e−

=

+

→ Step function behavior “smears” out at higher temperatures.

( ) ( )1

1F

FD E EkT

f E

e−

=

+

Probability of electrons (fermions) to be found at various energy levels.

Temperature dependence of Fermi-Dirac function shown as follows:

For E – EF = 0.05 eV ⇒ f(E) = 0.12For E – EF = 7.5 eV ⇒ f(E) = 10 –129

Exponential dependence has HUGE effect!

FermiFermi--Dirac Function Dirac Function Metals and SemiconductorsMetals and Semiconductors

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

f(E) for a semiconductor

Velocity of the electronsIn metals, the Fermi energy gives us information about the velocities of the electrons which participate in ordinary electrical conduction. The amount of energy which can be given to an electron in such conduction processes is on the order of micro-electron volts (see copper wire example), so only those electrons very close to the Fermi energy can participate. The Fermi velocity of these conduction electrons can be calculated from the Fermi energy.

Speed of light: 3x108m.s-1

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/ohmmic.html#c2




Fermi Energies, Fermi Temperatures, and Fermi Velocities

Element Fermi Energy eV Fermi Temperature x10^4 K Fermi Velocity 10^6 m/s

Li 4.74 5.51 1.29Na 3.24 3.77 1.07K 2.12 2.46 0.86Rb 1.85 2.15 0.81Cs 1.59 1.84 0.75Cu 7.00 8.16 1.57Ag 5.49 6.38 1.39Au 5.53 6.42 1.40Be 14.3 16.6 2.25Mg 7.08 8.23 1.58from N. W. Ashcroft and N. D. Mermin,

The Fermi temperature is the temperature associated with the Fermi energy by solving

EF = kB TF

for , where m is the particle mass and kB is Boltzmann's constant.

http://hyperphysics.phy-astr.gsu.edu/hbase/solids/solref.html#c1

http://scienceworld.wolfram.com/physics/FermiEnergy.html

http://scienceworld.wolfram.com/physics/BoltzmannsConstant.html

Electrical Conductivity (σ)Conductivity is the “ease of conduction”. Ranges over 27 orders of magnitude!

( ) 11 −−Ω= mρ

σ

Metals 107 (Ω.cm)-1

Semiconductors 10-6 - 104 (Ω.cm)-1

Insulators 10-10 -10-20 (Ω.cm)-1

(a) Charge carriers, such as electrons, are deflected by atoms or defects and take an irregular path through a conductor.

(b) Valence electrons in the metallic bond move easily. (c) Covalent bonds must be broken in semiconductors and

insulators for an electron to be able to move. (d) Entire ions must diffuse to carry charge in many ionically

bonded materials.

• Electronic conduction:– Flow of electrons, e- and electron holes, h+

• Ionic conduction– Flow of charged ions, Ag+

Microscopic ConductivityMicroscopic ConductivityWe can relate the conductivity of a material to microscopic parameters that describe the motion of the electrons (or other charge carrying particles such as holes or ions). From the equations

If J=Current density (I/A) Ampere/m2; and ξ=electric field intensity (V/L) then

We can also determine that the current density is

Where n is the number of charge carriers (carriers/cm3); q is the charge on each carrier (1.6x10-19C); and ν is the average drift velocity (cm/s) at which the charge carriers move.Therefore

ρAI

LV

LRAd ρIR anV =⇒==

σξ=J

νnqJ =

( ) 11 −−Ω= mρ

σ

μσ nqV.scmmobilityμ

ξν

ξνnqthen σσξ νnqJ

=

⎟⎟⎠

⎞⎜⎜⎝

⎛==

===

2

The charge q is a constant. Electrons are the charge carriers in metals. Electrons and holes are both carriers of electricity in semiconductors. Electrons that “hop” from one defect to another or the movement of

ions are both the carriers of electricity in ceramics.

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 ⋅

⋅⋅=

λσ2

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

Conduction in Materials --Classical approach

1. Drude Model

2. Temperature‐dependent conductivity

3. Matthissen’s Rule

4. Hall effect

5. Skin effect (HF resistance of a conductor)

1. Heat Capacity and thermal conductivity

Classic Model (Drude Model)electron act as a particle

• Collision: the scattering of an electron by (and only by) an ion core

• Between collision: electrons do not interact with each other or with ions

• An electron suffers a collision with probability per unit time τ-1, (τ-1 scattering rate).

• Electrons achieve thermal equilibrium with their surrounding only through collisions

Vibrating Cu+ ions

uΔx

Ex

V

(a) (b)

Fig. 2.2 (a): A conduction electron in the electron gas moves about randomly in ametal (with a mean speed u) being frequently and randomly scattered by bythermal vibrations of the atoms. In the absence of an applied field there is no netdrift in any direction. (b): In the presence of an applied field, Ex, there is a netdrift along the x-direction. This net drift along the force of the field issuperimposed on the random motion of the electron. After many scattering eventsthe electron has been displaced by a net distance, Δx, from its initial positiontoward the positive terminal

From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Relaxation time approximation and MobilityA

Jx

Δx

vd x

Fig. 2.1: Drift of electrons in a conductor in the presence of anapplied electric field. Electrons drift with an average velocity vdx inthe x-direction.(Ex is the electric field.)


Ex

timet2 t

Electron 2

vx1-ux1

time

vx1-ux1

t1 t

Last collision

Free time

Electron 1

Present timeVelocity gained along x

timet3 t

Electron 3

vx1-ux1


Fig. 2.3: Velocity gained in the x-direction at time t from theelectric field (Ex) for three electrons. There will be N electrons toconsider in the metal.

e

x

xdx

mene

EEenJEv

τμ

μσσμ

μ

=

===

=

(Ohm’s law)

Temperature dependence of Conductivity for Metal

a

u

Electron

S = π a2

l=uτ

Fig. 2.4: Scattering of an electron from the thermal vibrations of theatoms. The electron travels a mean distance l = u τ betweencollisions. Since the scattering cross sectional area is S, in the volumeSl there must be at least one scatterer, Ns(Suτ) = 1.

Conduction by free electrons and scattered by lattice vibration

Resistivity (ρ) in Metals

Resistivity typically increases linearly with temperature:ρt = ρo + αT

Phonons scatter electrons. Where ρo and α are constants for an specific material

Impurities 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

Matthiessen’s Rule

τI

Strained region by impurity exerts ascattering force F = - d(PE) /dx

Fig. 2.5: Two different types of scattering processes involving scatteringfrom impurities alone and thermal vibrations alone.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

τΤ

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

Fig. 2.6: The resistivity of various metals as a function of temperatureabove 0 °C. Tin melts at 505 K whereas nickel and iron go through amagnetic to non-magnetic (Curie) transformations at about 627 K and1043 K respectively. The theoretical behavior (ρ ~ T) is shown forreference.[Data selectively extracted from various sources including sections inMetals Handbook, 10th Edition, Volumes 2 and 3 (ASM, MetalsPark, Ohio, 1991)]

ρ ∝ T

Tungsten

Silver

Copper

Iron

Nickel

Platinum

NiCr Heating Wire

Tin

Monel-400

Inconel-825

10

100

1000

2000

100 1000 10000Temperature (K)

Resis

tivity

(nΩ

m)


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

0.00001

0.0001

0.001

0.01

0.1

1

10

100

1 10 100 1000 10000Temperature (K)

00.5

11.5

22.5

33.5

0 20 40 60 80 100T (K)

ρ ∝ T

ρ ∝ T5

ρ = ρRρ = ρR

ρ ∝ T5

ρ ∝ T

ρ (nΩ m)

Fig.2.7: The resistivity of copper from lowest to highest temperatures(near melting temperature, 1358 K) on a log-log plot. Above about100 K, ρ ∝ T, whereas at low temperatures, ρ ∝ T 5 and at thelowest temperatures ρ approaches the residual resistivity ρR . Theinset shows the ρ vs. T behavior below 100 K on a linear plot ( ρRis too small on this scale).From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Resis

tivity

(nΩ

m)

Electron conduction in nonmetals

10610310010-310-610-910-1210-1510-18 109

Semiconductors Conductors

1012

Conductivity (Ωm)-1

AgGraphite NiCrTeIntrinsic Si

DegeneratelyDoped Si

Insulators

Diamond

SiO2

Superconductors

PETPVDF

AmorphousAs2Se3

Mica

Alumina

Borosilicate Pure SnO2

Inorganic Glasses

Alloys

Intrinsic GaAs

Soda silica glass

Many ceramics

MetalsPolypropylene

Figure 2.24: Range of conductivites exhibited by various materialsFrom Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Control of Electrical ConductivityBy controlling the number of charge

carriers in the material (n)

By controlling the mobility or “ease of movement” of the charge carriers (μ)

μσ nq=

Example: Calculation of Drift velocity of electrons in copper (valence =1)Assuming that all of the valence electrons contribute to current flow, (a) calculate the mobility of an electron in copper and (b) calculate the average drift velocity for electrons in a 100cm copper wire when

10V are applied.Data: conductivity of copper = 5.98 x 105(Ω.cm)-1 ; q = 1.6x10-19C ; lattice

parameter of copper = 0.36151x10-7cm ; copper has a FCC structure

Number of carriers (n): The valence of copper is 1. Therefore, the number of valence electrons equals the number of copper atoms in the material.

μσ nq=The equation that we need to apply is:

n is the number of carriers. q is the charge . μ is the mobility

32237 104668

1036151014 cmelectrons

cmvolumecellatomselectronscellatomsn /.

).(_)/)(/(

×=××

== −

Part (b)

The electric field intensity is: 110

10010 −=== cmV

cmV

LV ..ξ

The equation we need to use is ξμν =

scmcmV

sVcmv /.).)(

..( 42410244

2

==

Part (a)

sVcm

Ccm

Ccmelectronscm

nq ..

..

).)(/.(.. 22

19322

115

2442441061104668

10985=

Ω=

××Ω×

== −

−−σμ

ondCoulombampNote sec/: 11 =

What does it mean?

The drift velocity is the average velocity that a particle, such as an electron, attains, due to an electric field.

The electron moves at the Fermi speed and it has only a tiny drift velocity superimposed by the applied electric field.

In metals, the Fermi energy gives us information about the velocity of the electrons which participate in ordinary electrical conduction. The amount of energy which can be given to an electron in such conduction processes is on the order of micro-electron volts, so only those electrons very close to the Fermi energy (EF) can participate.

mEv F

F2

=

Element Fermi EnergyeV

Fermi Temperaturex 10^4 K

Fermi Velocityx 10^6 m/s

Li 4.74 5.51 1.29

Na 3.24 3.77 1.07

K 2.12 2.46 0.86

Rb 1.85 2.15 0.81

Cs 1.59 1.84 0.75

Cu 7.00 8.16 1.57

Ag 5.49 6.38 1.39

Au 5.53 6.42 1.40

Be 14.3 16.6 2.25

Mg 7.08 8.23 1.58

Ca 4.69 5.44 1.28

Sr 3.93 4.57 1.18

Ba 3.64 4.23 1.13

Nb 5.32 6.18 1.37

Fe 11.1 13.0 1.98

Mn 10.9 12.7 1.96

Zn 9.47 11.0 1.83

Cd 7.47 8.68 1.62

Hg 7.13 8.29 1.58

Al 11.7 13.6 2.03

Ga 10.4 12.1 1.92

In 8.63 10.0 1.74

Tl 8.15 9.46 1.69

Sn 10.2 11.8 1.90

Pb 9.47 11.0 1.83

Bi 9.90 11.5 1.87

Sb 10 9 12 7 1 96

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: TkE

B

g

e 2−

A good insulator will have a band gap >>5eV and 2kBT~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

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

−=⎟

⎠⎞

⎜⎝⎛ −

=KKxmolcal

molcals

cmkTQDD D

o )(/./exp.exp

273180098717900002390

2

D=1.119x10-10cm2/s

Next, we need to find the mobility

sJcmCCioncarriers

TkDqZ

B ...

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

... 2

923

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

Conduction in ionic crystal and glasses


Interstitial cation diffuses

E

Vacancy aids the diffusion of positive ion

Na+

(a) (b)

E

Anion vacancyacts as a donor

Fig. 2.27: Possible contributions to the conductivity of ceramic andglass insulators (a) Possible mobile charges in a ceramic (b) A Na+ion in the glass structure diffuses and therefore drifts in thedirection of the field. (E is the electric field.)

1×10-15

1×10-13

1×10-11

1×10-7

1×10-5

1×10-3

1×10-1

1.2 1.6 2 2.4 2.8 3.2 3.6 4103/T (1/K)

SiO2

Pyrex

12%Na2O-88%SiO2

24%Na2O-76%SiO2

PVAc

PVC

As3.0Te3.0Si1.2Ge1.0 glass

1×10-9

Fig. 2.28: Conductivity vs reciprocal temperature for various lowconductivity solids. (PVC = Polyvinyl chloride; PVAc = Polyvinylacetate.) Data selectively combined from numerous sources.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Con

duct

ivity

1/(Ω

m)

⎟⎠⎞

⎜⎝⎛−=

kTE0

0 expσσ

Mobile charges contribute to conduction

Hopping process

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 (<1 eV) between conduction and valence bands.At T > 0, electrons thermally “excited” from valence to conduction band, leading to measurable conductivity.

EF

EC

EV

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



free electron

free hole




positive ion core

valence electron

Semiconductors in Group IVCarbon, Silicon, Germanium, TinEach has 4 valence Electrons.Covalent bond

The III-V semiconductors are prominent for applications in optoelectronics with particular importance for areas such as wireless communications because they have a potential for higher speed operation than silicon semiconductors. The compound semiconductors have a crystal lattice constructed from atomic elements in different groups of the periodic chart. Each Group III atom is bound to four Group V atoms, and each GroupV atom is bound to four Group III atoms, giving the general arrangement shown in Figure. The bonds are produced by sharing of electrons such that each atom has a filled (8 electron) valence band. The bonding is largely covalent, though the shift of valence charge from the Group V atoms to the Group III atoms induces a component of ionic bonding to the crystal (in contrast to the elemental semiconductors which have purely covalent bonds).

Compound Semiconductors

(Group III and Group V)

Representative III-V compound semiconductors are GaP, GaAs, GaSb, InP, InAs, and InSb.GaAs is probably the most familiar example of III-V compound semiconductors, used for both high speed electronics and for optoelectronic devices. Optoelectronics has taken advantage of ternary and quaternary III-V semiconductors to establish optical wavelengths and to achieve a variety of novel device structures. The ternary semiconductors have the general form (Ax;A’1-x)B (with two group III atoms used to filled the group III atom positions in the lattice) or A(Bx;B’1-x) (using two group V atoms in the Group V atomic positions in the lattice). The quaternary semiconductors use two Group III atomic elements and two Group V atomic elements, yielding the general form (Ax;A’1-x)(By;B’1-y). In such constructions, 0<x,y<1. Such ternary and quaternary versions are important since the mixing factors (x and y) allow the band gap to be adjusted to lie between the band gaps of the simple compound crystals with only one type of Group III and one type of Group V atomic element. The adjustment of wavelength allows the material to be tailored for particular optical wavelengths, since the wavelength ¸ of light is related to energy (in this case the gap energy Eg) by ¸ λ= hc/Eg, where h is Plank's constant and c is the speed of light.

Hall effect and Hall devices

B

F = qv×B

v

B

F = qv×B

v

B

q = +e q = -e

(a) (b)

Fig. 2.16 A moving charge experiences a Lorentz force in a magneticfield. (a) A positive charge moving in the x direction experiences aforce downwards. (b) A negative charge moving in the -x directionalso experiences a force downwards.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Hall effect in Semiconductor

Fig. 2.26: Hall effect for ambipolar conduction as in asemiconductor where there are both electrons and holes. Themagnetic field Bz is out from the plane of the paper. Both electronsand holes are deflected toward the bottom surface of the conductorand consequently the Hall voltage depends on the relative mobilitiesand concentrations of electrons and holes.(E is the electric field.)

Jx

vex

eEy

Jy = 0

xz

y

evexBz

Bz

V

Bz

A

Jx

EyEx

vhx

evhxBz

+

++

+ ++


eEy

Thermal Conduction

• Metal

Vibrating Cu+ ions Electron Gas

HOT COLDHEAT


Fig. 2.18: Thermal conduction in a metal involves transferringenergy from the hot region to the cold region by conductionelectrons. More energetic electrons (shown with longer velocityvectors) from the hotter regions arrive at cooler regions and collidethere with lattice vibrations and transfer their energy. Lengths ofarrowed lines on atoms represent the magnitudes of atomicvibrations.

δT

δx

HOT COLD

dQdtHEAT A

Fig. 2.19: Heat flow in a metal rod heated at one end. Consider therate of heat flow, dQ/dt, across a thin section δ x of the rod. The rateof heat flow is proportional to the temperature gradient δ T/δ x and thecross sectional area A.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

The failure of Drude model

• It can explain the electric conduction, not thermal conductivity and heat capacity

• Electronic heat capacity – (Drude)

– Experimental result

Need more sophisticated model (Sommerfeld or Quantum mechanics)

RCel 23

=

3TTCV αγ +=

0

100

200

300

400

450

0 10 20 30 40 50 60 70Electrical conductivity, σ, 106 Ω-1 m-1

Al

Ag

Au

Cu

Brass (Cu-30Zn)

Be

Pd-40Ag

Bronze (95Cu-5Sn)

Ag-3Cu

Ag-20Cu

WMo

Hg

Mg

Steel (1080)

Ni

κσ

= T CWFL

Ther

mal

cond

uctiv

ity, κ

(W

K-1

m-1

)

Fig. 2.20: Thermal conductivity, κ vs. electrical conductivity σ forvarious metals (elements and alloys) at 20 °C. The solid linerepresents the WFL law with CWFL ≈ 2.44×108 W Ω K-2. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Copper

Brass (70Cu-30Zn)

Al-14%Mg

Aluminum

10

100

1000

10000

50000

1 10 100 1000Temperature (K)

Ther

mal

con

duct

ivity

, κ (

W K

-1 m

-1)

Fig. 2.21: Thermal conductivity vs. temperature for two pure metals(Cu and Al) and two alloys (brass and Al-14%Mg). Data extracted fromThermophysical Properties of Matter, Vol. 1: Thermal Conductivity,Metallic Elements and Alloys, Y.S. Touloukian et. al (Plenum, NewYork, 1970).From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Phonon• Non-metal

Energetic atomic vibrations

Hot Cold

Equilibrium

Fig. 2.22: Conduction of heat in insulators involves thegeneration and propogation of atomic vibrations through thebonds that couple the atoms. (An intuitive figure.)

Diamond—Good thermal conductor

Polymer- -Bad thermal conductor

Thermal conductivity and

resistanceFig. 2.23: Conduction of heat through a component in (a)can be modeled as a thermal resistance θ shown in (b) whereQ′ = ΔT/θ.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Hot

L

ΔTΔT

Q′

(a) (b)

θ

Q′ = ΔT/θ

Q′

Cold

AQ′

Pure metal Nb Fe Zn W Al Cu Ag

κ (W m-1 K-

1)

52 80 113 178 250 390 420

Metal alloys Stainless Steel 55Cu-45Ni 70Ni-30Cu 1080

Steel

Bronze

(95Cu-

5Sn)

Brass (63Cu-

37Zn)

Dural (95Al-

4Cu-1Mg)

κ (W m-1 K-

1)

12 - 16 19.5 25 50 80 125 147

Ceramics

and glasses

Glass-

borosilicate

Silica-fused

(SiO2)

S3N4 Alumina

(Al2O3)

Saphire

(Al2O3)

Beryllia

(BeO)

Diamond

κ (W m-1 K-

1)

0.75 1.5 20 30 37 260 ~1000

Polymers Polypropylene PVC Polycarbonate Nylon

6,6

Teflon Polyethylene

low density

Polyethylene

high density

κ (W m-1 K-

1)

0.12 0.17 0.22 0.24 0.25 0.3 0.5

Pauli Exclusion Principle - academic.uprm.eduacademic.uprm.edu/pcaceres/Courses/Smart/SMD-2B.pdf · Pauli Exclusion Principle Electrons in a single atom occupy discrete levels of

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