Exam Results:
Exam high = 76, distribution below:
Also a reminder, next HW is due Thursday this week (22nd).
Recall: Electron bands,
v Can re-­‐express states by replacing k by k ± G.
v Saw before, crystal symmetry means k in single B.Z. conserved, not overall momentum. [“Crystal momentum”]
v Bloch theorem also shows k ± G equivalence, we’ll see.
Free-­‐electron states
• Note, each state is displayed multiple times in the figure.
• States appear once in each B. Zone. [Twice with spin.]
Recall: Electron bands,
v Can re-­‐express states by replacing k by k ± G.
v Saw before, crystal symmetry means k in single B.Z. conserved, not overall momentum. [“Crystal momentum”]
v Bloch theorem also shows k ± G equivalence, we’ll see.
Free-­‐electron states
Free-­‐electron states folded into 1BZ.
• Note, each state is displayed multiple times in the figure.
• States appear once in each B. Zone. [Twice with spin.]
Electrons with a crystal potential:
› Electron energies (and wavefunctions) no longer same as simple plane wave states; strongest effects at zone boundaries.
› Shown is case of “nearly free electron model”.
Free-­‐electron states folded into 1BZ.
Electrons with a crystal potential:
› Electron energies (and wavefunctions) modified; strongest effects at zone boundaries.
› Shown is case of “nearly free electron model”.
Free-­‐electron states folded into 1BZ.
A metal
εF
2N states per band; N = # cells in crystal.
Electrons bands in 2D:
o Free electron Fermi surfaces are circles, area ∝ number of electrons.
square lattice Brillouin zones
iucr.org
Electrons bands in 2D:
o Free electron Fermi surfaces are circles, area ∝ number of electrons.
Free-­‐electron Fermi surfaces
(2 electrons/
cell)
Folded gives electron and hole pockets.
iucr.org
Bloch theorem:
• Hamiltonian including periodic crystal potential
!
"2 ! 2
H =−
∇ + U (r )
2m
← Actually summed over coordinates for each electron; solution is product wavefunction.
(Assuming no e-­‐e interactions.)
• Potential has Bravais lattice translation symmetry.
!
! !
U (r ) = U (r + R)
• Bloch theorem: Solutions of the above consist of plane wave
multiplied by function with lattice symmetry.
(× a spin state; assume no spin-orbit coupling.)
!
! ik!⋅r!
ψ ( r ) = u ( r )e
!
! !
u (r ) = u (r + R)
!
k by periodic B.C.
Bloch theorem:
Assume N1N2N3 cells.
!
ni !
k = ∑ bi
Ni
Most general function with periodic B.C.: ψ = ∑ α k e
k
! !
!
iG ⋅ r
Potential has symmetry: U (r ) = ∑ U G e
!!
ik ⋅ r
Fourier theorem
G
!
"2 ! 2
Hψ = −
∇ ψ + U (r )ψ = Eψ
2m
Expand:
Bloch theorem:
Assume N1N2N3 cells.
!
ni !
k = ∑ bi
Ni
Most general function with periodic B.C.: ψ = ∑ α k e
k
! !
!
iG ⋅ r
Potential has symmetry: U (r ) = ∑ U G e
!!
ik ⋅ r
Fourier theorem
G
!
"2 ! 2
Hψ = −
∇ ψ + U (r )ψ = Eψ
2m
Expand:
!!
!!
! ! !
"2
2 ik ⋅ r
i ( G + k )⋅ r
ik ⋅ r
α
k
e
+
U
α
e
=
E
α
e
∑
∑
∑
k
G k
k
2m k
k ,G
k
Bloch theorem:
!
"2 ! 2
Hψ = −
∇ ψ + U (r )ψ = Eψ
2m
Expand:
!!
!!
! ! !
"2
2 ik ⋅ r
i ( G + k )⋅ r
ik ⋅ r
α
k
e
+
U
α
e
=
E
α
e
∑
∑
∑
k
G k
k
2m k
k ,G
k
∑U
k ,G
α k ′−G e
G
! !
i ( k ′ )⋅ r
k, k’ dummy indices
Bloch theorem:
!
"2 ! 2
Hψ = −
∇ ψ + U (r )ψ = Eψ
2m
Expand:
!!
!!
! ! !
"2
2 ik ⋅ r
i ( G + k )⋅ r
ik ⋅ r
α
k
e
+
U
α
e
=
E
α
e
∑
∑
∑
k
G k
k
2m k
k ,G
k
∑U
α k ′−G e
! !
i ( k ′ )⋅ r
k, k’ dummy indices
G
k ,G
∑e
k
""
ik ⋅ r
⎧⎛ ! 2 k 2
⎫
⎞
− E ⎟⎟α k + ∑ U Gα k −G ⎬ = 0
⎨⎜⎜
G
⎠
⎩⎝ 2m
⎭
o To be true all r, curly bracket term must = 0 all k.
o Result couples states k to k ± G only.
Bloch theorem:
⎧⎛ ! 2 k 2
⎫
⎞
− E ⎟⎟α k + ∑ U Gα k −G ⎬ = 0
⎨⎜⎜
G
⎠
⎩⎝ 2m
⎭
o Result couples states k to k ± G only.
o Solution is a sum of k ± G states:
!!
! !
⎛
⎞
ψ = ⎜ ∑ α k , G e iG ⋅ r ⎟ e ik ⋅ r
⎝ G
⎠
!
≡ u (r )
Lattice
symmetry
Note, can always re-configure k to k + G/
Bloch theorem:
⎧⎛ ! 2 k 2
⎫
⎞
− E ⎟⎟α k + ∑ U Gα k −G ⎬ = 0
⎨⎜⎜
G
⎠
⎩⎝ 2m
⎭
o Result couples states k to k ± G only.
o Solution,
!!
! !
⎛
⎞
ψ = ⎜ ∑ α k , G e iG ⋅ r ⎟ e ik ⋅ r
⎝ G
⎠
!
≡ u (r )
!
! ik!⋅r!
ψ ( r ) = u ( r )e
Lattice
symmetry
Electrons bands in 2D:
o Free electron Fermi surfaces are circles, area ∝ number of electrons.
iucr.org
Fermi surfaces in “repeated-­‐
zone scheme”