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3.23 Electrical, Optical, and Magnetic Properties of Materials
Fall 2007
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3.23 Fall 2007 – Lecture 11
FERMI
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Last time
1.
2
2.
3.
4.
5.
6.
Explicit solution of the Bloch equation, energy bands
Brill
illouiin zone, Fermii surfface
Energy of molecules and solids
Mean field approaches – Hartree and Hartree-Fock
Spin-statistics, Slater determinant, Pauli principle
Huckel approach (LCAO for aromatic compounds)
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
1
Study
• Chap.
Chap 5 Singleton
• Read Chap. 6 Singleton
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Tight-binding (LCAO for solids)
r
• Hamiltonian Hˆ = Hˆ at + ∆U
∆Û ( r )
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
2
Tight-binding (LCAO for solids)
• Bloch eigenstates of an ATOMIC CRYSTAL
r r
r
r r
r
Ψ nk ( r ) = ∑
exp ik R ψ n r − R
r
R
(
) (
)
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Tight-binding (LCAO for solids)
• Bloch eigenstates of a REAL CRYSTAL
r r
r
r r
r
Ψ nk ( r ) = ∑
exp ik R φ r − R
r
R
r
(
) (
)
r
φ ( r ) = ∑ bnψ n ( r )
n
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
3
Some despicable algebraic workout
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
More workout
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
4
More
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
More
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
5
From s level to s bands
r
ε k = Es − β −
( )
r r
r
γ R cos k r
( )
∑ ( )
nearest
neighb.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
From s level to s bands
Conduction band
4
6
Eg
p
s
2
4
Valence band
Figure by MIT OpenCourseWare.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
6
Tight-binding vs. empirical psp
Image removed due to copyright restrictions. Please see: Fig. 2.24 in
Yu, Peter Y., and Manuel, Cardona. Fundamentals of Semiconductors: Physics and Materials Properties. New York, NY: Springer, 2001.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Tight-binding vs. empirical psp
Xc
12
8
L2'c
L3c
Γ2'c
L1c
L3'v
Valence
4
Γ15c
0
X1c
Γ25'v
Σ1
X4v
-4
-8
L1v
L2'v
L
Γ
X K,U
(a) LCAO bands
Γ
Band energy [eV]
Gap Conduction
Lc
L
Γ1
Λ
Γ
∆
X K,U
Σ
Γ
L
(b) True bands
Γ
X K,U
-12
Γ
(c) Free-electron bands
A comparison of the band structure of Ge as calculated by (a) the tight-binding, (b) the empirical pseudopotential, and (c) the nearly
free electron methods.
Figure by MIT OpenCourseWare.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
7
Bands in Ge
Image removed due to copyright restrictions. Please see http://www.ioffe.ru/SVA/NSM/Semicond/Ge/Figs/221.gif
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Ferroelectric perovskites
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
8
Ferroelectric perovskites
Image removed due to copyright restrictions. Please see: Fig. 3 in
King-Smith, R. D., and David Vanderbilt. "First-principles Investigation of Ferroelectricity
in Perovskite Compounds." Physical Review B 49 (March 1994): 5828-5844.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Ferroelectric perovskites
Image removed due to copyright restrictions. Please see Fig. 4 in
King-Smith, R. D., and David Vanderbilt. "First-principles Investigation of Ferroelectricity
in Perovskite Compounds." Physical Review B 49 (March 1994): 5828-5844.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
9