![Page 1: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/1.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Group Applications to Band Theory
Andrew Galatas
Group Theory, Spring 2015
![Page 2: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/2.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Outline
Introduction to CrystalsBravais LatticesBrillouin Zones
Groups and Basic Band TheorySpace Group RepresentationsSimple Cubic Example
Perturbation on Bandsk · p Perturbation TheorySelection Rules in Action
![Page 3: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/3.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Outline
Introduction to CrystalsBravais LatticesBrillouin Zones
Groups and Basic Band TheorySpace Group RepresentationsSimple Cubic Example
Perturbation on Bandsk · p Perturbation TheorySelection Rules in Action
![Page 4: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/4.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Space Groups
I Crystals are characterized by space groups
I Symmetry under translations and point groupoperations
I 73 total symmorphic (simple) space groups
![Page 5: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/5.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Bravais Lattices
I Only 14 Bravais lattices
http://www.seas.upenn.edu/
![Page 6: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/6.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Outline
Introduction to CrystalsBravais LatticesBrillouin Zones
Groups and Basic Band TheorySpace Group RepresentationsSimple Cubic Example
Perturbation on Bandsk · p Perturbation TheorySelection Rules in Action
![Page 7: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/7.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Reciprocal Lattice Vectors
I Solutions to e iK·r = 1
I Typically form their own Bravais lattice
I Is essentially the Fourier space equivalent of the realspace crystal
I Will be related to periodicity of electron plane waves
![Page 8: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/8.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Brillouin Zones
I Smallest repeating unit in reciprocal space
I Collection of reciprocal vectors that are closest to the 0vector
I Behavior of electrons in periodic potential characterizedby their properties in the first B-Z
I Second, third, etc. Brillouin zones also exist (thisperiodicity gives rise to bands)
![Page 9: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/9.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Brillouin Zones
Wikipedia, ”Brillouin Zone”
![Page 10: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/10.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Bloch’s Theorem
I Ψk(r) = e ik·ruk(r)
I Crucially, uk(r) has (at least some of) the same spacegroup symmetries of the overall reciprocal lattice
I This leads to Ψk(r + T) = Ψk(r)e ik·T
I For nearly free electrons (NFE), uk(r) ≈ 1√Ωr
I IMPORTANTLY, Ek+K = Ek
![Page 11: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/11.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Outline
Introduction to CrystalsBravais LatticesBrillouin Zones
Groups and Basic Band TheorySpace Group RepresentationsSimple Cubic Example
Perturbation on Bandsk · p Perturbation TheorySelection Rules in Action
![Page 12: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/12.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Space Groups on Reciprocal Vectors
I Ti ·Kj = 2πNij
I OTi ·K′j = 2πNij ⇒ K′j = O−1Kj
I The group of all space group transformations whichsend k to k + K is the space group of the wavevector
I For a general Bloch wavefunction, EOk 6= Ek, BUT if kis a point of symmetry, all equivalent vectors aredegenerate
![Page 13: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/13.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Representations of Point Groups
I Classes of a point group are equivalent transformations,such as 4-fold rotations or inversions
I Irreducible representations are primarily derived fromthe Great Orthogonality Theorem
I Irreps of one point group might be decomposable inanother related point group
I For a given irrep and Bloch eigenfunction,
ORkΨ
(i)kλ(r) =
∑µ
Ψ(i)kµ(r)D(i)(Rk)µλ
![Page 14: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/14.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Character Table for Oh
Dresselhaus
![Page 15: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/15.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Compatibility Relations
I Band degeneracies are lifted in moving from a wavevector of high symmetry to one with lower
I The manner in which bands are split is characterized bycompatibility relations, illustrating irreduciblecomponents of now-reducible representations
![Page 16: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/16.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Outline
Introduction to CrystalsBravais LatticesBrillouin Zones
Groups and Basic Band TheorySpace Group RepresentationsSimple Cubic Example
Perturbation on Bandsk · p Perturbation TheorySelection Rules in Action
![Page 17: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/17.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
An Example with a Simple Lattice
![Page 18: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/18.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
An Example with a Simple Lattice
![Page 19: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/19.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
An Example with a Simple Lattice
![Page 20: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/20.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Outline
Introduction to CrystalsBravais LatticesBrillouin Zones
Groups and Basic Band TheorySpace Group RepresentationsSimple Cubic Example
Perturbation on Bandsk · p Perturbation TheorySelection Rules in Action
![Page 21: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/21.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Perturbation Theory
[ p2
2m+ V (r) +
~k · pm
+~2k2
2m
]un,k(r) = En(k)un,k(r)
I Energy is assumed known at some k0, and the wavevector has some symmetry Γi
I Then Schrodinger becomes Hk0u(Γi )n,k0
(r) = εn(k0)u(Γi )n,k0
(r)
I For a small perturbation,(Hk0 + ~κ·p
m
)u
(Γi )n,k0+κ(r) = εn(k0 + κ)u
(Γi )n,k0+κ(r)
![Page 22: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/22.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Selection Rules
I Suppose we want to calculate the matrix element⟨Φ(i)|H ′|Φ(j)
⟩I Under space group operations which respect the
Hamiltonian, this matrix element must either beconstant or trivially zero, depending on whether thewavefunctions provided transform like equivalentrepresentations
I More concretely, if the matrix element transforms asΓi ⊗ Γn ⊗ Γj , then either (Γn ⊗ Γj) is orthogonal to Γi ,or the direct product is a constant.
![Page 23: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/23.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Outline
Introduction to CrystalsBravais LatticesBrillouin Zones
Groups and Basic Band TheorySpace Group RepresentationsSimple Cubic Example
Perturbation on Bandsk · p Perturbation TheorySelection Rules in Action
![Page 24: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/24.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Selection Rules in Action
I We can see these selection rules in action by returningto our band perturbation theory
I Let us look at the Γ+1 band from the 0 vector in a
simple cubic lattice
ε(Γ1)n (κ) = En(0) +
⟨uΓ1n,0|H
′|uΓ1n,0
⟩
+∑n′ 6=n
⟨uΓ1n,0|H
′|uΓin′,0
⟩⟨uΓin′,0|H
′|uΓ1n,0
⟩EΓ1n (0)− EΓi
n′ (0)
![Page 25: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/25.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Selection Rules in Action
I We can see these selection rules in action by returningto our band perturbation theory
I Let us look at the Γ+1 band from the 0 vector in a
simple cubic lattice
ε(Γ1)n (κ) = En(0) +
⟨uΓ1n,0|H
′|uΓ1n,0
⟩
+∑n′ 6=n
⟨uΓ1n,0|H
′|uΓin′,0
⟩⟨uΓin′,0|H
′|uΓ1n,0
⟩EΓ1n (0)− EΓi
n′ (0)
![Page 26: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/26.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Selection Rules in Action
I H ′ = ~κ·pm is a vector, and so transforms like the Γ−15
representation of Oh
I This means that the linear term vanishes, asΓ−15 ⊗ Γ+
1 = Γ−15 6= Γ+1
I Looking at the second order terms, we realize the
expression simplifies to
⟨u
Γ+1
n,0 |H′|u
Γ−15
n′,0
⟩⟨u
Γ−15
n′,0|H′|u
Γ+1
n,0
⟩E
Γ+1
n (0)−EΓ−
15n′ (0)
![Page 27: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/27.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Selection Rules in Action
I H ′ = ~κ·pm is a vector, and so transforms like the Γ−15
representation of Oh
I This means that the linear term vanishes, asΓ−15 ⊗ Γ+
1 = Γ−15 6= Γ+1
I Looking at the second order terms, we realize the
expression simplifies to
⟨u
Γ+1
n,0 |H′|u
Γ−15
n′,0
⟩⟨u
Γ−15
n′,0|H′|u
Γ+1
n,0
⟩E
Γ+1
n (0)−EΓ−
15n′ (0)
![Page 28: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/28.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Selection Rules in Action
I H ′ = ~κ·pm is a vector, and so transforms like the Γ−15
representation of Oh
I This means that the linear term vanishes, asΓ−15 ⊗ Γ+
1 = Γ−15 6= Γ+1
I Looking at the second order terms, we realize the
expression simplifies to
⟨u
Γ+1
n,0 |H′|u
Γ−15
n′,0
⟩⟨u
Γ−15
n′,0|H′|u
Γ+1
n,0
⟩E
Γ+1
n (0)−EΓ−
15n′ (0)
![Page 29: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/29.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Selection Rules in Action
I After some simplification with appropriate basisfunctions, we get
EΓ+
1n (κ) = E
Γ+1
n (0) +~2κ2
2m
+~2κ2
m2
∑n 6=n′
⟨S |p|P
⟩2
EΓ+
1n (0)− E
Γ−15n′ (0
![Page 30: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/30.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Selection Rules in Action
I After some simplification with appropriate basisfunctions, we get
EΓ+
1n (κ) = E
Γ+1
n (0) +~2κ2
2m
+~2κ2
m2
∑n 6=n′
⟨S |p|P
⟩2
EΓ+
1n (0)− E
Γ−15n′ (0
![Page 31: Group Applications to Band Theory - Welcome to SCIPPscipp.ucsc.edu/.../physics251_15/GroupTheoryBands.pdf · Group Applications to Band Theory Andrew Galatas Introduction to Crystals](https://reader034.vdocument.in/reader034/viewer/2022042312/5edb930cad6a402d6665dc4b/html5/thumbnails/31.jpg)
Group Applicationsto Band Theory
Andrew Galatas
Introduction toCrystals
Bravais Lattices
Brillouin Zones
Groups and BasicBand Theory
Space GroupRepresentations
Simple Cubic Example
Perturbation onBands
k · p PerturbationTheory
Selection Rules inAction
Summary
Summary
I A working knowledge of the group theory of point andspace groups is incredibly useful in the field oftheoretical solid state physics
I One particular application is in the analysis of bandsaround points of high symmetry in the reciprocal lattice
I We looked at how these tools can be used to predictband energies close to symmetric points in a particularperturbation model
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Group Applicationsto Band Theory
Andrew Galatas
Bibliography I
D. BishopGroup Theory and Chemistry
M. S. Dresselhaus.Applications of Group Theory to the Physics of Solids.
V. HeineGroup Theory in Quantum Mechanics
M. P. MarderCondensed Matter Physics