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3.23 Electrical, Optical, and Magnetic Properties of Materials
Fall 2007
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3.23 Fall 2007 – Lecture 9
BAND STRUCTURE
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Last time
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Hamiltonian in a periodic potential, translation operators
Bloch’s
Bl
h’ th
theorem (f
(from common eiigensttattes off H
H, TR)
n, k quantum numbers
Born-von Karman boundary conditions
Explicity proof of Bloch’s theorem
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
1
Study
• Chap
Chap. 3 Singleton
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Bloch Theorem (in two equiv forms)
r r
r
r
r
r
Ψ nk ( r ) = exp ik r unk ( r )
(
Ψ nkr
(
)
r r
r r
r
r + R = exp ik R Ψ nkr ( r )
)
(
)
2
Hamiltonian in the Bloch representation
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Explicit solution for the Bloch orbitals
⎛ h 2 ( q − G′ )2
⎞
− E ⎟ Cq −G′ + ∑ VG′′−G′Cq −G′′ = 0
⎜
⎜
⎟
2m
G ′′
⎝
⎠
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
3
Explicit solution for the Bloch orbitals
⎛ h 2 ( q − G′ )2
⎞
− E ⎟ Cq −G′ + ∑ VG′′−G′Cq −G′′ = 0
⎜
⎟
⎜
2m
G ′′
⎝
⎠
⎛ h2
2
( q − 2G )
⎜
2m
⎜
⎜
VG
⎜
⎜
⎜
V2G
⎜
⎜
⎜
V3G
⎜
⎜
⎜
V4G
⎜
⎝
V− G
V−2G
V−3G
h2
2
(q − G)
2m
V−G
V−2G
VG
h2
2
(q)
2m
V− G
V2G
VG
h2
2
(q + G)
2m
V3G
V2G
VG
⎞
⎟
⎟
⎟⎛C
⎞
⎛ Cq − 2 G ⎞
V−3G
⎟ ⎜ q − 2G ⎟
⎟
⎜
⎟ ⎜ Cq − G ⎟
⎜ Cq − G ⎟
⎟⎜ C ⎟ = E ⎜ C ⎟
V−2G
q
⎟⎜ q ⎟
⎜
⎟
⎟ ⎜ Cq + G ⎟
C
⎜ q +G ⎟
⎟⎜
⎟
⎟
⎜C
V−G
⎝ q + 2G ⎠
⎟ ⎝ Cq + 2 G ⎠
⎟
h2
2
( q + 2G ) ⎟⎟
2m
⎠
V−4G
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Free, free, set me free
⎛ h2
2
( q − 2G )
⎜
⎜ 2m
⎜
VG
⎜
⎜
⎜
V2G
⎜
⎜
⎜
V3G
⎜
⎜
⎜
V4G
⎜
⎝
V− G
V−2G
V−3G
h2
2
(q − G)
2m
V−G
V−2G
VG
h2
2
(q)
2m
V− G
V2G
VG
h2
2
(q + G)
2m
V3G
V2G
VG
⎞
⎟
⎟
⎟⎛C
⎞
⎛ Cq − 2 G ⎞
V−3G
⎟ ⎜ q − 2G ⎟
⎜
⎟
⎟ ⎜ Cq − G ⎟
⎜ Cq − G ⎟
⎟⎜ C ⎟ = E ⎜ C ⎟
V−2G
q
⎟⎜ q ⎟
⎜
⎟
⎟ ⎜ Cq + G ⎟
C
⎜ q +G ⎟
⎟⎜
⎟
⎜C
⎟
V−G
⎝ q + 2G ⎠
⎟ ⎝ Cq + 2 G ⎠
⎟
h2
2
( q + 2G ) ⎟⎟
2m
⎠
V−4G
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
4
Free electron dispersions, 1-d
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Band Structures: Free Electron Gas, Silicon
Figure by MIT OpenCourseWare.
5
Band Structures: Free Electron Gas, Silicon
Figure by MIT OpenCourseWare.
Band Structure of Diamond
Courtesy L. J. Terminello and F. J. Himpsel. Used with permission.
6
Band Edge
⎛ h2
2
( q − 2G )
⎜
⎜ 2m
⎜
VG
⎜
⎜
⎜
V
2G
⎜
⎜
⎜
V3G
⎜
⎜
⎜
V4G
⎜
⎝
V−G
V−2G
V−3G
h
2
(q − G)
2m
V−G
V−2G
VG
h2
2
(q)
2m
V−G
V2G
VG
h2
2
(q + G)
2m
V3G
V2G
VG
2
⎞
⎟
⎟
⎟⎛C
⎞
⎛ Cq −22G ⎞
V−3G
⎟ ⎜ q −22G ⎟
⎜
⎟
⎟ ⎜ Cq −G ⎟
⎜ Cq −G ⎟
⎟⎜ C ⎟ = E ⎜ C ⎟
V−2G
q
⎟⎜ q ⎟
⎜
⎟
⎟ ⎜ Cq + G ⎟
⎜ Cq + G ⎟
⎟⎜
⎟
⎜
⎟
V−G
⎝ Cq + 2 G ⎠
⎟⎝ Cq + 2G ⎠
⎟
h2
2
( q + 2G ) ⎟⎟
2m
⎠
V−4G
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Band Edge
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
7
Ψnk (r) is not a momentum eigenstate
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Group velocity
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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Effective mass
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Fermi energy
Copper
r
r
ρ (r ) = ∑r f n ,kr Ψ n ,kr ( r )
2
Α1
-0.939
Ζ3
n,k
Q_
Σ1
∆1
-0.539
Γ12
Γ25'
∆2
5
2
Ζ2 1'
1
Ζ3 Ζ1
3
Ζ4
3
2'
1 Ζ
1
∆5
∆2'
2'
Q+
Q_
3
Q+
Q_
Q+
Α1
Σ2
Α3
3 Α
3
Σ3
1
Σ1
K2
K4
K3
K1
K1
Σ1
Α1
∆1
Σ4
Γ1
X
W
L
Γ
Κ
Figure by MIT OpenCourseWare.
9
The Fermi surface
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Images from the Fermi Surface Database. Used with permission.
Please see: http://www.phys.ufl.edu/fermisurface/jpg/K.jpg,
http://www.phys.ufl.edu/fermisurface/jpg/Cu.jpg.
The Fermi surface
http://www phys ufl edu/fermisurface/
http://www.phys.ufl.edu/fermisurface/
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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