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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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• 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
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