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3.23 Electrical, Optical, and Magnetic Properties of MaterialsFall 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
1. Hamiltonian in a periodic potential, translation operators
2 BlBloch’s ththeorem (f(from common eiigensttattes of H, TTR))2. h’ f H
3. n, k quantum numbers
4. Born-von Karman boundary conditions
5. Explicity proof of Bloch’s theorem
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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•
Study
• ChapChap. 3 Singleton3 Singleton
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Bloch Theorem (in two equiv forms)
rr r r r r rΨ =r ik r exp u( ) ( ) ( )nk nk
rr r r r
r rΨ + = exp ( ik R )( ) ( ) Ψ ( )rr Rnk nk
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
⎛ ⎞h2 −− +⎜ ⎟( )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)
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⎛ ⎞
Explicit solution for the Bloch orbitals
⎛ ⎞⎜ ⎟
( )q G′ − +′ ′∑V ′h2 − 2
E Cq G G− −′ G Cq G− ′′ = 0⎜ ⎟ ′′2m⎝ ⎠ G ⎝ ⎠
h2 2⎛ ⎞( )− 2 V V V⎜ ⎟ ⎜ ⎟
q G G G2 3G V 42m − − − − G
h2 ⎛ ⎞⎜ ⎟2 C C⎛ ⎞V q( )− V 2 2q GG V V q G− −G G 2 3− G⎜ ⎟2m − − G ⎜ ⎟ ⎜ ⎟⎜ ⎟ C C 2 ⎜ ⎟q G ⎜ q G ⎟− − ⎜ ⎟h 2 ⎜ ⎟ ⎜ ⎟ ) q q = EV V ((q) V V C C2G G − −G 2G⎜ ⎟⎜ ⎟2 ⎜ ⎟⎜ ⎟ ⎜⎜ ⎟⎟2m⎜ ⎟ C C
2 ⎜ ⎟q G ⎜ q G ⎟⎜ ⎟+ +
⎟h 2V V V q + ⎜ ⎟( )G V C C⎜⎜ ⎟3 2 G 2m −G + +2 2⎝ ⎠G G q G G⎝ ⎠ q⎜ ⎟ ⎜ ⎟V V V V h
2
( )q + 2G 2⎜ ⎟4 3 2 GG G G 2m⎝ ⎠
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Free, free, set me free
h2 2⎛ ⎞− V V V⎜ ⎟( )q G G G2 3 V− G2 − − − G 42m⎜ ⎟ h2 2⎜ ⎟
⎜ ⎟ 2 2G GV V G C Cq G q G− −V q( )− G − −2 3V− G ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ 2 ⎜ ⎟ ⎜ ⎟
2m C Cq G q G⎜ ⎟ − − ⎜ ⎟h 2 ⎜ ⎟C C= E ⎜ ⎟V V2 ( )q V G V q qG G − −2G⎜ ⎟⎜ ⎟ ⎜ ⎟2m⎜ ⎟ C C
2 ⎜ ⎟q G ⎜ q G ⎟⎜ ⎟+ +
⎜ ⎟⎜ ⎟ ⎜⎜ ⎟⎟ V V V G G + G) V 2 2GV V V q G C C 3 2 G h ((q + G)2 V− C Cq G+ +q⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠⎜ ⎟2m⎜ ⎟2⎜ ⎟V V V V q + 2G⎜ ⎟4 3G G 2G G 2
h
m( )2
⎝ ⎠
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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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
5
Figure by MIT OpenCourseWare.
Band Structures: Free Electron Gas, Silicon
Band Structure of Diamond
6
Figure by MIT OpenCourseWare.
Courtesy L. J. Terminello and F. J. Himpsel. Used with permission.
Band Edge h2 2⎛ ⎞( )− 2 V V V ⎜ ⎟
⎜ ⎟2mq G G G2 3G V−4G− − −
h2 2⎜ ⎟ C C⎜ ⎟
C CG G ⎛ ⎞⎜ ⎟V GG V V ) V V 2 3V ⎛ ⎞⎛ ⎞q 2 22 2G ⎛ ⎞ q G− −VG G ((q − ) − − G V − G2m ⎜ ⎟ ⎜ ⎟⎜ ⎟ C Cq G q G− −2 ⎜ ⎟ ⎜ ⎟⎜ ⎟h 2 C C⎜ ⎟V ⎜ ⎟ = EV V ( )q V q q2G G G 22m − − G ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ C C2 q G+ +⎜ q G⎜ ⎟ ⎟h 2 ⎜ ⎟ ⎜⎜ ⎟V V V q + G V C C ⎟ 3 2G G G ( ) q G q G2m −G + +2 2⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎜ ⎟
⎜ ⎟V V V V h2
( )q + 2G 2 ⎜ ⎟4 3 2GG G G 2m⎝ ⎠
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)
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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
-0.939
-0.539
X W L
Γ12
Γ25'
Γ1
Γ Κ
∆1
Ζ3
Ζ2
Α1
Σ1
Σ2 K2
K3
K1K1
K4Σ4 Σ1Σ3
Σ1
Α1Α3
Α3
Α1
Ζ1Ζ3Ζ4
Ζ1
∆2
∆5∆2'
∆1
5 3
333 2'
2'
1 1
1
1'2
Q_
Q+
Q+
Q+
Q_
Q_
Copper
9
( )2
( )r f rn k, ,n kn k,
rρ = Ψ∑ r rrr
Figure by MIT OpenCourseWare.
The Fermi surface
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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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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.
http://www.phys.ufl.edu/fermisurface/jpg/K.jpghttp://www.phys.ufl.edu/fermisurface/jpg/Cu.jpghttp://www.phys.ufl.edu/fermisurface/