Brillouin Zones:
•
•
•
Mth zone ≡ region having origin as Mth nearest K point.
Equivalent definition: Region reached from origin by crossing (M - 1)
perpendicular bisector planes.
Each zone contains N “allowed k points” (e.g. as defined with periodic boundary
conditions), where N = number of Bravais-lattice cells in crystal.
! ! !
! ! !
b1 ⋅ b2 × b3 = (2π )3 / a1 ⋅ a2 × a3
•
All zones have same total volume; can “fold” zones into 1st zone by translation
through K vectors.
2D case
3D Brillouin Zone examples:
FCC first Brillouin zone (image from Wikipedia). Note, Brillouin zones rely only on
Bravais lattice; applies to FCC metals, silicon, GaAs, etc.
Labels: center (k = 0) is always Γ, other labels by historical convention (L, X, etc.).
BCC first Brillouin zone.
Cubic structures , first 4 zones in extended
space: (image from MathWorld)
Body centered tetragonal, 1st zone
with KFe2Se2 superconductor Fermi
surface (Liu et al. Physica B 407, 1139, 2012)
Nearly-­‐Free electron model:
Recall,
!
U (r ) = ∑ U K! e
K
! !
iK ⋅ r
!!
! !
! ik!⋅r! ⎛
⎞
ψ i = u ( r ) e = ⎜ ∑ α k , K e iK ⋅ r ⎟ e ik ⋅ r
⎠
Bloch states ⎝ K
Assume U weak; wavefunctions ≈ plane waves,
Perturbation through 2nd order:
2k 2
ε o (k) ≡
2m
2
UK
"k
!
! !
ε=
+∑
2m
K ε o (k ) − ε o (k − K )
2
2
q Uses result from before, U joins only states k ± K.
q Small energy changes occur except when states are degenerate.
q Degeneracies: Bands always cross on Brillouin Zone Boundaries.
q (also sometimes accidental crossings).
Nearly-­‐Free electron model:
Recall,
!
U (r ) = ∑ U K! e
K
! !
iK ⋅ r
!!
! !
! ik!⋅r! ⎛
⎞
ψ i = u ( r ) e = ⎜ ∑ α k , K e iK ⋅ r ⎟ e ik ⋅ r
⎠
Bloch states ⎝ K
Assume U weak; wavefunctions ≈ plane waves,
2k 2
ε o (k) ≡
2m
Degenerate case: assume situation where only 2 states cross; ! 2 k 2 ! 2 (k − K ) 2
≈
so can show, diagonalize 2×2 system:
2m
2m
⎛ ! 2 k 2 2m
⎞ ⎛ αk ⎞ ⎛ αk ⎞
UK
⎜
⎟ ⋅ ⎜⎜
⎟⎟ = ε ⎜⎜
⎟⎟
2
2
⎜ U
⎟
! ( k − K ) 2m ⎠ ⎝ α k − K ⎠ ⎝ α k − K ⎠
−K
⎝
Results: States always have avoided crossing, gap = 2|UK|.
General solution shown in Mathematica notebook.
Nearly-­‐free electrons:
Free-­‐electron states, Extended zone scheme
2|UK|
folded into 1BZ.