3.23 electrical, optical, and magnetic properties of materials · ferroelectric perovskites 3.23...
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3.23 Electrical, Optical, and Magnetic Properties of Materials 2007
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Fall
3.23 Fall 2007 – Lecture 11
FERMI
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Last time
1. Explicit solution of the Bloch equation, energy bands
2 BBrillillouiin zone, FFermii surfface2.
3. Energy of molecules and solids
4. Mean field approaches – Hartree and Hartree-Fock
5. Spin-statistics, Slater determinant, Pauli principle
6. Huckel approach (LCAO for aromatic compounds)
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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•
•
Study
• ChapChap. 5 Singleton5 Singleton
• Read Chap. 6 Singleton
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Tight-binding (LCAO for solids)
• Hamiltonian H H= + ∆U rr r
Hamiltonian H Hat (( ))ˆ ˆ= + ∆U
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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•
•
∑
Tight-binding (LCAO for solids)
• Bloch eigenstates of an ATOMIC CRYSTAL Bloch eigenstates of an ATOMIC CRYSTAL
r r r Ψ =∑ k
rr ( )r ( )r iexp ( )R ψ r − Rnk r n
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 Bloch eigenstates of a REAL CRYSTAL r r rr r Ψ = φr ( )r iexp kR r − Rnk ∑r
( ) ( )R
r r (r b (φ ψ( )) =∑ n n (r ))n n n
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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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)
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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)
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r rr
From s level to s bands
r r r) = − − ( r )ε ((k E) β γ (R))cos ((k r ) s ∑∑ nearest neighb .
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
From s level to s bands
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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Conduction band
Valence band
p
sEg
4
4
6
2
Figure by MIT OpenCourseWare.
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
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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Figure by MIT OpenCourseWare.
L X K,UΓ Γ
Vale
nce
Gap
Con
duct
ion
(a) LCAO bandsL X K,UΓ Γ
(c) Free-electron bands(b) True bands
Γ1
Γ15cΓ2'c
Γ25'v
L2'v
L1v
L3'v
L1c
L3c
L2'c
LcXc
X1c
X4vΣ1
L X K,UΓ Γ∆Λ Σ-12
-8
-4
0
4
8
12
Ban
d en
ergy
[eV
]
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.
Bands in Ge
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)
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Image removed due to copyright restrictions. Please see http://www.ioffe.ru/SVA/NSM/Semicond/Ge/Figs/221.gif
Ferroelectric perovskites
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)
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Image removed due to copyright restrictions. Please see: Fig. 3 in King-Smith, R. D., and David Vanderbilt. "First-principles Investigation of Ferroelectricityin Perovskite Compounds." Physical Review B 49 (March 1994): 5828-5844.
Image removed due to copyright restrictions. Please see Fig. 4 inKing-Smith, R. D., and David Vanderbilt. "First-principles Investigation of Ferroelectricityin Perovskite Compounds." Physical Review B 49 (March 1994): 5828-5844.