Band Theory of Solids.pdf

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1. The Band Theory of Solids
Dr. Md. Arzu Miah
Professor
Department of Chemistry
Jahangirnagar University
Savar, Dhaka-1342
Bangladesh
Cell: 01732150725.
Email: [email protected]
[email protected]

2. What is Band Theory of Solids?
Thistheoryexplainsthequantumstatethatanelectron
takesinsidemetalsolid.Everymoleculecomprisesof
variousdiscreteenergylevels.Thewayelectronsbehave
inside a molecule is well explained through this theory.
Inatoms,electronsarefilledinrespectiveenergyorbits
following Pauli’s exclusion principle.
Inmolecules,Twoatomicorbitalscombinetogetherto
form a molecular orbital with two distinct energy levels.
Insolids,10
23
stackeduplinesconfinedinatinyspace
wouldlooklikeaband.Therebyformingenergy
continuum called energy bands.
Thistheoryhelpstovisualisethedifferencebetween
conductor,semiconductorandaninsulatorby
plotting available energies for an electron in a material.

3. ConsideraSodiumaatom.Itcomprisesof11electrons.Theyfill
up energy level following Pauli’s exclusion principle
Energy Bands In Inside an atom

4. Whathappenswhentwosodiumatomsverycloseto
eachotheralmostformingamolecule?Noweach
atomcannothavetheconfigurationasitfollowedas
anindividualatomIftheydotheywillbeviolating
Pauli’sexclusionprincipleandendupwithalotof
electrons of the same energy levels.
Whentwoatomscomeveryclosetoeachother,What
isgoingtohappentothissystem?Theansweris,their
respectiveenergybandsaregoingtooverlaponeach
otherandtransformintowhatwecallasMolecular
orbital.Thatisthe1sorbitofindividualSodiumatom
combinestoform1smolecularorbital.Astwoatomic
orbitalsareoverlapping,themolecularorbitendsup
havingtwodiscreteenergylevels.WheretheLower
energyleveliscalledbondingorbitalandHigher
energyleveliscalledanti-bondingorbital.Thiswill
repeatforallorbits
Energylevelsinsideamoleculemade
up of two atoms

6. Energy levels inside a molecule made up of
three atoms
Nowtrytopicturise,whatisgoingtohappenif
weaddathirdsodiumatomtothemix?Well,
accordingtothetheorywelearnedjustnow.here
threeatomicorbitalswillbeoverlappingforming
singlemolecularorbitalwiththreediscrete
energylevels.Eachmolecularorbitalherewill
inheritthreeenergylevels.Ingeneral,themore
weaddatoms,moreenergylevelsthemolecular
orbit going to have.

8. Energy levels inside a solid made up of
Avogadro number of atoms
Eventually,ifwehaveanentiresolid,whichis
madeofsodiumwithsomethinglike10
23
atoms
packedtogether,Eachmolecularorbitalofthis
solidwillhavenow10
23
discreteenergylevels.
Forbetterunderstandingpurpose,Thinkabout
drawing1sorbitalofSodiumsolidblock,draw
lowerenergylevelandupperenergylevelandin
betweenstackitwith1023energylevels!The
gapsbetweenthemwillbeextremelysmallsuch
that,nolongerwecannoticeindividualenergy
levels.Asaresult,Itisconvenienttothinkofit
ascontinuousenergyorenergycontinuum.
Whenwethinkinthisway,wecancallthemas
energy band instead of molecular orbit.

10. Energy levels inside a solid made up of n-
number of atoms
Ingeneral,Iftherearen-numberofatoms,thentherewillbe
ndiscreteenergylevelsineachenergyband.Insuchasystem
ofnnumberofatoms,themolecularorbitalsarecalledas
energybands.Single1sorbitaland2sorbitalcanfit2
electronseach.thus,thetotalnumberofelectronsa1sand2s
energybandcanfitis2n.Asingle2plevelcanfit6electrons
so 2p energy band can fit is 6n electron so on and so forth.
Asatomscomeclosetoeachotherandeventuallyformasolid,
Theyendupformingenergycontinuumandwenamethat
continuumasbands.Withinthebands,energylevelswhichare
availablearecontinuous.Thus,thenameofthistheorywithout
any surprise is, “The band theory of solids”
Usingthistheory,wecanunderstandhowfreeelectronsare
generatedandwhycertainmaterialreadilyhavefreeelectron
available making them a conductor and why some others don’t?

12. Atomshavebeenseentohave
discrete energy levels.
Whenahugenumberofatomsare
combinedtoformasolidhowever,
thesediscreteenergylevelsare
replacedbydiscreterangesof
energy,orenergybands,within
whichtherearesomanyindividual
allowedenergyvaluesthatwithin
thebandsthedistributioncanbe
consideredtobecontinuous.This
idea is seen in the following figure:

25. Intrinsic Semiconductor Extrinsic Semiconductor
Pure semiconductor Impure semiconductor
Density of electrons is equal to
the density of holes
Density of electrons is not equal
to the density of holes
Electrical conductivity is low Electrical conductivity is high
Dependence on temperature only
Dependence on temperature as
well as on the amount of
impurity
No impurities
Trivalent impurity, pentavalent
impurity
Difference between Intrinsic and Extrinsic Semiconductors

31. What is Fermi Energy?
Named after the Physicist, Enrico Fermi, a Fermi
level is the measure of the energy of least tightly
held electrons within a solid. It is important in
determining the thermal and electrical properties of
solids. It can be defined as:
The Fermi energy is a concept in quantum
mechanics usually referring to the energy difference
between the highest and lowest occupied single-
particle states in a quantum system of non-
interacting fermions at absolute zero temperature.
The value of the Fermi level at absolute zero
temperature (−273.15 °C) is known as the Fermi
energy. It is also the maximum kinetic energy an
electron can attain at 0K. Fermi energy is constant
for each solid.

32. What is Fermi Level?

33. What is Quasi-fermi energy level?
Quasi-fermi energy level is defined as the
change in the level of Fermi level as the charge
carriers are added excessively to the
semiconductor

34. Element
Fermi Energy
eV
Li 4.74
K 2.12
Na 3.24
Cs 1.59
Rb 1.85
Ag 5.49
Cu 7.00
Be 14.3
Au 5.53
Ca 4.69
Mg 7.08
Ba 3.64
Sr 3.93
Fe 11.1
Nb 5.32
Zn 9.47
Mn 10.9
Hg 7.13
Cd 7.47
Al 11.7
Ga 10.4
In 8.63
Tl 8.15
Sn 10.2
Pb 9.47
Bi 9.90
Sb 10.9
Value of Fermi energy for
different elements

35. Applications of Fermi Energy
Itisoneoftheimportantconceptsinquantum
mechanicsandcondensedmatterphysics.Somefermi
energy applications are given in the points below.
It is used in semiconductors and insulators.
Itisusedtodescribeinsulators,metals,and
semiconductors.
Fermienergyisappliedindeterminingtheelectrical
and thermal characteristics of the solids.
Itisalsoimportantinnuclearphysicstounderstand
thestabilityofwhitedwarfs.Whitedwarfsarestars
thathaveamasscomparabletotheSunbuthave
about a hundredth of its radius.