6.4 electronic band structures
DESCRIPTION
6.4 ELECTRONIC BAND STRUCTURES. Dongwoo , Shin. Contents. 6.4.1 . Reciprocal Lattices and the First Brillouin Zone 6.4.2 . Bloch’s Theorem 6.4.3 . Band Structures of Metals and Semiconductors. FT for time. Reciprocal lattice. FT for space. - PowerPoint PPT PresentationTRANSCRIPT
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6.4 ELECTRONIC BAND STRUC-TURES
Dongwoo, Shin
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Contents
6.4.1. Reciprocal Lattices and the First Brillouin Zone
6.4.2. Bloch’s Theorem
6.4.3. Band Structures of Metals and Semiconduc-tors
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6.4.1. Reciprocal Lattices and the First Brillouin Zone
• Reciprocal Lattice – Crystal is a periodic array of lattices
Performing a spatial Fourier transform
– Reciprocal Lattice • Expression of crystal lattice in fourier space
t
Reciprocal lat-tice
FT for time
FT for space
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Reciprocal Lattice
• Primitive Vector for a simple orthorhombic lattice
• Reciprocal primitive vectors
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Reciprocal Lattice
*a
b-c plane
Next lattice plane
d*aa
* b ca
a b c
* 1| |a
d
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First Brillouin Zone
• The smallest of a Wigner-Seitz cell in the reciprocal lattice
The reciprocal lat-tices (dots) and cor-responding first Bril-louin zones of (a) square lattice and (b) hexagonal lattice.
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The first Brillouin zone of a FCC structure
Face-centered cubic
K Middle of an edge joining two hexagonal faces
L Center of a hexagonal face
UMiddle of an edge joining a hexagonal and a square face
W Corner point
X Center of a square face
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The first Brillouin zone of a FCC structure
-Primitive basis vectors of the face-centered cubic lat-tice
- Primitive vectors :
b
c
a
ˆ ˆ( )2
ˆ ˆ( )2
ˆ ˆ( )2
aa y z
ab x z
ac x y
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The first Brillouin zone of a FCC structure
- Reciprocal primitive vec-tors :
General reciprocal lattice vector:
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The first Brillouin zone of a FCC structure
2ˆ ˆ ˆ( )( )x y z
a
2 2 2ˆ ˆ ˆ( )( 2 ) ; ( )( 2 ) ; ( )( 2 )x y z
a a a
• 1st Brillouin zone : the short-est
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6.4.2 Bloch’s Theorem
• Hamiltonian Operator (for the one-electron model)From (3.68),
“The one-electron Schrödinger equa-tion”
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Bloch’s Theorem
- The periodicity of the lattice structure :
( ) ( )
U r U r R
- can be expanded as a fourier series :( )
U r
𝑈 ( �⃗� )=∑⃗𝐺
𝑈�⃗�𝑒𝑖 �⃗�∙ �⃗� h𝑊 𝑒𝑟𝑒𝐺𝑖𝑠𝑔𝑒𝑛𝑒𝑟𝑎𝑙 𝑟𝑒𝑐𝑖𝑝𝑟𝑜𝑐𝑎𝑙𝑙𝑎𝑡𝑡𝑖𝑐𝑒𝑣𝑒𝑐𝑡𝑜𝑟
- The solution of the Schrödinger equation for a peri-odic potential must be a special form :
Where is a periodic function with the periodicity of the lattices
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Central Equation
• The wavefunction can be expressed as a Fourier se-ries:
From the one-electron Schrödinger equation, the coeffi-cients of each Fourier component must be equal on both sides of the equation.
2 2
02
k G k G
Ge
kE C U C
m
: Central equa-
tion
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Central Equation for 1-D
• From the first Brillouin zone 22
k G G 2 2 2 2 21 1 1( ) ; ( ) ( ) ( )2 2 2
k G k G G G G
/ 2 /k G a
0
0
( ) 0
( ) 0
E E C UC
E E C UC
2 2
01,
2 2
e
kG E
m where
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Central Equation for 1-D
0
0
( )0
( )
E E U
U E E
20 ( / )
2 e
aE E U U
m
- Near the zone boundary
0
0
( ) 0
( ) 0
k k k G
k G k G k
E E C UC
E E C UC
- Nontrivial solutions for the two coefficient
1/ 2
0 0 0 0 2 21 1( ) ( ) ( )
2 4k k G k k GE k E E E E U
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Standing Waves
- Wave function at the zone edge
( / ) ( / )1( )
2i x a i x ae e
L
( ) 2 / cos( / )
( ) 2 / sin( / )
x L x a
x i L x a
0E E U
Forming 2 standing waves
h𝑤 𝑒𝑟𝑒𝐿𝑖𝑠 h𝑙𝑒𝑛𝑔𝑡 𝑜𝑓 h𝑡 𝑒𝑐𝑟𝑦𝑠𝑡𝑎𝑙
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Electron band structure
(a) The extended-zone scheme (b)The reduced –zone scheme
- Representation of the electronic band struc-ture
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6.4.3 Band Structures of Metals and Semiconductors
• Band Structures of Metals– Copper outermost configuration :– Electron in the s band can be easily excited from
below the to above the “CONDUCTOR”– Interband transition The absorption of photons will cause the electrons in thes band to reach a higher level within the same band.
1 104 3s d
- Calculated energy band structure of copper
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Band structure of semiconductor
Calculated energy band structure of Sili-con
Calculated energy band structure of GaAs
- Interband transitions : The excitation or relaxation of electrons between subbands- Indirect gap : The bottom of the conduction band and the top of the va-lence band do not occur at the same k
- Direct gap : The bottom of the conduction band and the top of the va-lence band occur at the same k
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Band structure of semiconductor
2 2 2 2
* *( ) ( )2 2e C h V
e h
k kE k E and E k E
m m
- Energy versus wavevector relations for the car-riers
,g
d Ev
dk
- Effective mass1
g
dEv
dk