MSE/ECE 310
Elec Props of Matls
Linear Combination of Atomic Orbitals (LCAO)
Form Molecular Orbitals (MO) = Hybridization
H
rA
e-
rB
e-
H
Two hydrogen atoms
approaching each other.
ψ1s(rA)
ψ1s(rB)
r
R=∞
A
B
Bonding Molecular Orbital
ψσ = ψ1s(rA) + ψ1s(rB)
r
a
r
ψσ* = ψ1s(rA) - ψ1s(rB)
Antibonding Molecular Orbital
Fig. 4.1: Formation of molecular orbitals, bonding and antibonding
( ψσ and ψσ∗ ) when two H atoms approach each other. The two
electrons pair their spins and occupy the bonding orbital ψσ.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Knowlton
MSE/ECE 310
Elec Props of Matls
1
Linear Combination of Atomic Orbitals (LCAO)
Form Molecular Orbitals (MO) = Hybridization
H
(a)
H
H
|ψσ
|2
H
|ψσ∗ |2
(b)
Fig. 4.2: (a) Electron probability distributions for bonding
and antibonding orbitals, ψσ and ψσ*. (b) Lines represent
contors of constant probability (darker lines represent
greater relative probability).
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Knowlton
2
1
MSE/ECE 310
Elec Props of Matls
Linear Combination of Atomic Orbitals (LCAO)
Form Molecular Orbitals (MO) = Hybridization
E
ψσ∗
Eσ∗(R)
(a)
0
Bonding
Energy
E1s ψ 1 s
Eσ(R)
Eσ(a)
ψσ
0
SYSTEM
2 H-Atoms
2 Electrons
1 Electron/Atom
1 Orbital/Atom
R, Interatomic
R=∞
a
Separation
Eσ∗
(b)
E1s
E1s
ΔE = Bonding
Energy
Eσ
H -atom
H2
H -atom
Fig. 4.3: Electron energy in the system comprising two hydrogen
atoms. (a) Energy of ψσ and ψσ∗ vs. the interatomic separation, R.
(b) Schematic diagram showing the changes in the electron energy
as two isolated H atoms, far left and far right, come to form a
hydrogen molecule.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Knowlton
3
MSE/ECE 310
Elec Props of Matls
HF – SP Hybridization
Half-Full ψ1s
Full pz
Half-Full px
Bonding Orbital, ψσ
py
py
px
ψ1s
px
ψ1s
px
pz
pz
Full py
H
px
F
H-F
Fig. 4.6: H has one half empty ψ1s orbital. F has one half empty px
orbital but full py and pz orbitals. The overlap between ψ1s and px
produces a bonding orbital and an antibonding orbital. The two
electrons fill the bonding orbital and thereby form a covalent bond
between H and F.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
S. Zuhmdahl, Chemistry (Heath & Co., 1986)
Knowlton
4
2
MSE/ECE 310
Elec Props of Matls
Electron Energy
Isolated Silicon
3p
3s
2p
2s
1s
Fig. 4.15: The electronic structure of Si.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Knowlton
5
MSE/ECE 310
Elec Props of Matls
(c)
(b)
(a)
ψ
ψ
A
(d)
CONDUCTION BAND
hyb
ψ
A
3p
Energy gap, Eg
3s
ψ
B
Si ATOM
ψ
hyb
ψ
VALENCE BAND
B
Si CRYSTAL
Fig. 4.17: (a) Formation of energy bands in the Si crystal first involves hybridization
of 3s and 3p orbitals to four identical ψhyb orbitals which make 109.5° with each
other as shown in (b). (c) ψhyb orbitals on two neighboring Si atoms can overlap to
form ψB or ψA. The first is a bonding orbital (full) and the second is an antibondiong
orbital (empty). In the crystal ψB overlap to give the valence band (full) and ψA
overlap to give the conduction band (empty).
Knowlton
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
6
3
Linear Combination of Atomic Orbitals (LCAO)
Form Molecular Orbitals (MO)
TiO2
Ti
Ti
O
MSE/ECE 310
Elec Props of Matls
PRB
Imada et al., Review of Modern Physics (1998)
Lucovsky, SEMATECH High-k Workshop, 2004
Knowlton
7
MSE/ECE 310
Elec Props of Matls
Electron energy
CB
Ec
Thermal
excitation
Eg
Ev
VB
Fig. 4.18: Energy band diagram of a semiconductor. CB is the
conduction band and VB is the valence band. At 0 K, the VB is full
with all the valence electrons.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Knowlton
8
4
MSE/ECE 310
Elec Props of Matls
Ex
VACUUM
ea=
Fext
Fext
me
x
(a)
CRYSTAL
Fint
a=
Fext
me *
x
(b)
Fig. 4.19: (a) An external force Fext applied to an electron in
vacuum results in an acceleration avac = Fext / me . (b) An external
force Fext applied to an electron in a crystal results in an
acceleration acryst = Fext / me*. (Ex is the electric field.)
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Knowlton
9
5