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Lecture11:ContinuetheBandTheoryofSolidsBandStructures

TwoDimensionalBrillouinZone:

WeconstructthefirstBrillouinzonefromtheshortestlatticevector𝐺!asfollows.WeconstructthesecondBrillouinzonefromthenextshortestvector𝐺!andsoon.BrillouinZones‐2D

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BZconstruction• Reciprocallattice• Bisectvectorstothenearestneighbors• Areadefinedbybisectinglinesrepresents1BZ

ThreeDimensionalBrillouinZones:• A3‐dimensionalBrillouinzonecanbeconstructedinasimilarwaybybisectingalllatticevectorsandplacingplanesperpendiculartothesepointsofbisection.

• ThisissimilartotheWignerSeitzcellinthereallattice.

WignerSeitzCell:

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• Aprimitiveunitcellwhichshowsthecubicsymmetryofthelattice(forthecubicsystem).(reallattice)

• TheFirstBrillouinzoneistheWignerSeitzcellinthereciprocallattice

LetsStudyTheseFigures:1. *FirstBrillouinzoneofthebccstructure2. ⇒Freeelectronbandsforbccstructure3. *FirstBrillouinzoneofthefccstructure4. ⇒Freeelectronbandsforfccstructure

Explanationofthesesymbols:LookbetweenthegraphofbandsandthefirstBrillouinzone,youwillfind:

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Γ: center of the Brillouin zoneΧ: 100 interceptΚ: 110 interceptL: 111 interceptΓ− Χ:path ΔΓ− L:path ΛΓ− Κ:path Σ

DRAWINGNextFigure:BandstructureofAl(fcc)• Notetheparabolashapebands:

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Γ− Χ     Γ− 𝐿  Γ− Κ‐ Comparethisgraphwiththefreeelectronbandsoffcc⇒LookscloseorsimilarwhichsuggeststhatelectronsinAlbehavelikefreeelectrons.

Important:Therearesomebandgapsbetweenpoints

(𝑋!!  ,𝑋!)(𝑊! ,𝑊!′)

Buttheindividualenergybandsoverlapindifferentdirections

⇒Nobandgapexistsasawhole

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BandStructureofCu:• Thecloselyspacedbandsareduetothe3d‐bands• 4sbands:theheavilymarked4s,3dbandsoverlap• Nobandgapexists

BandstructureofSi:• Bandgapexistsof“1eV”• Thezeropointofenergyscaleisplacedundertheenergygap

• Note:theindirectbandgapinsilicon(Illexplainitlateron)

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BandstructureofGaAs:• Notetheenergygapandnoteitisdirectgap

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Directandindirectbandgaps:1. Direct:

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• Directbandgap:Themaximumofvalencebandandthemaximumofconductionbandhavethesamek.vector

2. Indirect:

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• Indirectbandgap:Themaximumofvalencebandandthemaximumofconductionbandhavedifferentk.vectors

Δ𝑘 ≠ 0Whatistheimplication??

Westillneedtounderstandmoreabouttheshapeofbandstructure.Todothat,weneedtounderstandtheeffectivemass:

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Effectivemassofelectron:m*Themassofanelectroninasolidisdeviatedfromthefreeelectronmassduetointeractionsofelectron‐electron,andelectrons‐ions.

!∗

!couldbegreaterthan1orsmallerthan1

Let’sderivem*Thegroupvelocityv!:

v! =𝑑𝜔𝑑𝑘

ω = 2πυ        k =2πλ

=d(2πυ)dk

= d2πE/hdk

v! =1ℏ𝑑𝐸𝑑k ⇒

Acceleration(a)=!!!!"

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=1ℏ𝑑!𝐸𝑑k!

∙𝑑k𝑑t      𝐸𝑞.𝟏

Let’sfind!!!!

𝑃 = ℏ𝑘 

!"!"= ℏ !"

dt      𝐸𝑞 𝟐inEq1,

a =1ℏ!d!EdΚ!

dPdt

= !

ℏ!!!!!!!

⋅ !(!")!"

whoLaw?

𝑎 =1ℏ!d!Edk!

𝐹

a =Fm

𝑚∗ = ℏ!d!Edk!

!!

m*isinverselyrelatedtothecurvatureofE(K)Ifthecurvatureof𝐸 = 𝑓 (𝐾)atagivenpointislarge,theeffectivemassissmallandviceversa.

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Lookbackintothebandstructures:• Someregionshavehighcurvature,nearthecenteroftheboundaryofaBrillouinzone.⇒Effectivemassisreduced(sometimesuptolessthan1%ofm)‐ Atpointswheretherearemorethanoneband,thanoneeffectivemass

• Anegativemassmeanselectrontravelin• • oppositedirectionstoanelectricfield(electronhole).

• Holesappearnearthetopofvalenceband.• Gobacktothebandstructures;findthevalenceandconductionbandsandthelightandheavymass.

ItreachesinfinityatK = π

2a

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Memory Aid !a hairpin is lighter than a frying pan"

light m*

(larger d2E/dK2) heavy m*

(smaller d2E/dK2)