session 3: solid state physics chemical bonds energy bands...
TRANSCRIPT
Chemical Bonds
Energy Bands (E-K diagram)
Session 3: Solid State Physics
1
1. Bonding
2. Energy Bands
3. e/h CurrentOutline
� A� B
� C
� D
� E
� F� G
� H
� I
� J
2
1. Bonding
2. Energy Bands
3. e/h CurrentAtomic Bonding
3
Atoms vibrate with small amplitudes about fixed
equilibrium positions. We assume that atoms are
fixed, unless phonons are considered.
Atoms look like outer valence electrons orbiting
around the core. Core consists of nucleus plus
inner core electrons.
valence electrons
nucleus +
core
electrons
Ionic bond: Na+Cl-
Covalent bond: sharing e- to complete an octetH need only one atom to complete the octet and
therefore we only have H2. Silicon needs 4 e- and so can
bond to four other Si atoms, forming a crystal.
Metallic bond:
Van derWaals:
1. Bonding
2. Energy Bands
3. e/h CurrentIonic Bonding
4
Complete transfer of electrons from one
atom (usually a metal) to another (non
metal ion) (compounds only, not elemental
solids). Bond comes from electrostatic
attraction between ions.
Na + Cl � Na+ + Cl− � NaCl
All ionic compounds have a degree of covalent bonding. The larger the
difference in electronegativity between two atoms, the more ionic the bond is.
• Bond is strong (high melting point, large elastic modulus)
• Not directional (high density, high coordination number)
• Compounds only
• Good insulators (except near melting point)
• Transparent up to UV (strong bonds �electrons need a lot of energy to
become free)
Mathematical form: Energy ~ 1/r , Example: Sodium Chloride
1. Bonding
2. Energy Bands
3. e/h CurrentIonic Bonding
5
Coulomb force:
� � ��4����
Energy needed to separate charges e and -e
� ��. � � ���4����
�
�.���� 7��
1. Bonding
2. Energy Bands
3. e/h CurrentPolar Bonds
6
Partly covalent and partly ionic. The more electronegative element will have
more negative charge.
1. Bonding
2. Energy Bands
3. e/h CurrentCovalent Bonding
7
Equal sharing of electrons between atoms � both atoms have full shells
(Example: Diamond, Silicon)
Note continuum of behavior, ionic � covalent (e.g. III-V compounds GaAs,
InSb, are partially covalent and partially ionic.)
• Bond is strong (high melting point, large elastic modulus)
• Directional (from orientation of QM orbitals) � low density
• Saturable (limited number of bonds per atom) ↑• Good insulators
1. Bonding
2. Energy Bands
3. e/h CurrentCovalent Bonding: Square well potential
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����
��� � �0∞
insidecubeoutsidecube
� '�2)�
*�Ψ � Ψ
Ψ � 2 2������ sin
,��-�� sin ,��.
�� sin ,��/��
�0�1�2 � '���2)
,����� 3 ,��
��� 3 ,�����
1. Bonding
2. Energy Bands
3. e/h CurrentCovalent Bonding: Square well potential
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� 35�8)7�
�0�1�2 � '���2)
,����� 3 ,��
��� 3 ,�����
Energy of a particle
confined to a cube L×L×L
Energy of a particle
confined to a cube L×L×2L � 95�
32)7�
Decrease in energy: � 35�16)7�
For L= 0.2 nm: ∆ �14eV
1. Bonding
2. Energy Bands
3. e/h CurrentSilicon
10
<
Generation / Recombination
14P
14P 14P 14P
14P
14P
14P 14P 14P
14P
Free electron
hole
2D representation
An electron shared
by two neighboring
atoms to form a
covalent bond. This
way an atom can
have a stable
structure with
eight valence band
electrons.
When an electron
breaks loose and
becomes a
conduction
electron, a hole is
also created.
=
<=
1. Bonding
2. Energy Bands
3. e/h CurrentMetallic Bonding
11
Positive ions plus gas (sea) of electrons. Think of this as the
limiting case of ionic bonding in which the negative ions are
electrons. (BUT electrons can’t be forced to sit at lattice points
from Uncertainty Principle: ∆>∆- ? 5/2 as for electrons ) is
small so the zero point energy ∆ � ∆>�/2)is very large;
the electrons would
shake themselves free and are therefore delocalized)
• Bonds are non directional (high coordination number, high
density, malleable and ductile)
• Variable strength
• Free electrons � high electrical conductivity, shiny (Electric
field associated with incident light makes free electrons at
surface move back and forth, re-radiating the light, as a
reflected beam)
1. Bonding
2. Energy Bands
3. e/h CurrentMetallic Bonding
12
The electron wave functions spread out over the entire crystal.
A three dimensional potential square well is a simple model
for a metal.
�0�1�2 � '���2)
,����� 3 ,��
��� 3 ,�����
Energy mostly determines by Electrostatic force!
1. Bonding
2. Energy Bands
3. e/h CurrentVan der Waals Bond
13
Even a neutral atom with a full shell, can, at a given instant, have a dipole moment (i.e.
one side of the atom more positive than the other) This instantaneous dipole will
induce a dipole in a neighboring atom, and the resulting dipole-dipole interaction is the
origin of the van der Waals bond. Although the original dipole time-averages to zero,
the interaction does not – it is always attractive. Energy ~ 1/r6
• Bond is weak (�low melting point, large expansion coefficient)
• Non directional so high coordination number BUT
• Long bond lengths (�low density)
Examples: Solid inert gases (Argon, Neon), molecular solids (solid Oxygen)
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
1. Bonding
2. Energy Bands
3. e/h CurrentHydrogen Bonding
14
Hydrogen loses its electron and becomes positively charged
particularly easily. Therefore the region of a molecule around
a hydrogen atom is often quite positive, and this allows an
electrostatic bond to form between it and negative parts of
neighboring molecules.
Example: ice – the strength of the hydrogen bond explains the
anomalously high melting point of ice
1. Bonding
2. Energy Bands
3. e/h CurrentComparing Bonds
15
Bond Energy (GPa) Example of Bond
Covalent 1,000 Diamond
Ionic 30 – 100 Salt and Ceramics
Metallic 30 – 100 Metals
Hydrogen 8 Ice
Van der Waals 2 Polythene
1. Bonding
2. Energy Bands
3. e/h CurrentEnergy Bands
16
Always simplify!Simplest state: Isolated atom
∞Periodic atoms:
Single isolated atom:
H:
Si:
Z =14
01s22s22p63s23p2
1. Bonding
2. Energy Bands
3. e/h CurrentBohr Atomic Model
17
Energy Bands:
wave-particle duality A � 5 >⁄
de Broglie standing wave
)C � ,'
D
�
EF
1. Bonding
2. Energy Bands
3. e/h CurrentBohr Atomic Model
18
Single atom:
D
�
EF
Pauli exclusion principle
D�
��
ED
DD
�D
E�
2 atoms: N atoms:
D�DD
D�⋮
���D
��⋮
2N
ele
ctro
ns
forbidden
energies
all
ow
ed
en
erg
ies
1. Bonding
2. Energy Bands
3. e/h CurrentElectrons in Solids
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Formation of
energy bands
as a diamond
lattice crystal is
formed by
bringing
isolated silicon
atoms together
In a solid, there are so many electrons with energies very near each other
that ‘bands’ of states develop.
Lattice spacing
3s
3p
2N electrons2N states
2N electrons6N states
5 A
Electron energy
4N electrons4N states
-
Valance band
Conduction band
<=
H
All we draw is the
“band diagram”
solid Isolated atoms
1. Bonding
2. Energy Bands
3. e/h CurrentMaterials
20
IJΩL)MInsulator
EG (Si) = 1.1eV
EG (Ge) = 0.7eV
EG (SiO2) = 9eV
Conductor Semi-conductor10N� 10O
2N
sta
tes
Valance band
Conduction band
N electrons
H
empty seat / filled seat
Conduction band
Valance band
1. Bonding
2. Energy Bands
3. e/h CurrentEnergy Diagrams
21
Energy band diagram shows the bottom edge of conduction band, Ec ,
and top edge of valence band, Ev .
Ec and Ev are separated by the band gap energy, Eg .
Electrons and holes tend to seek their lowest energy positions, electrons
tend to fall in energy band diagram, holes float up like bubbles in water.
Valance band
H
Conduction band <
=
-
Incr
ea
sin
g e
lect
ron
en
erg
y
Incr
ea
sin
g h
ole
en
erg
y
PQ
R
K.E.
K.E.
1. Bonding
2. Energy Bands
3. e/h CurrentMeasuring EGby Light Absorption
22
Bandgap energies of selected semiconductors
Valance band
H
Conduction band <
=
photon energy:
5C S H
EG can be determined
from the minimum
energy (5C) of photons
that are absorbed by
the semiconductor.
Semi
conductorInSb Ge Si GaAs GaP ZnSe Diamond
EG(eV) 0.18 0.67 1.12 1.42 2.25 2.7 6
1. Bonding
2. Energy Bands
3. e/h CurrentElectron / Hole Current
23
TUV � W �X CYUV
<
=
In semiconductor, electrons that are in the
conduction band move by applying the electric field
ZCY TUV
As number or electrons in conduction band is much
less than that in conductors hence I[\�Y<]�Q S I<]�Q
More realistic picture
could be:
<=
ZTUV
1. Bonding
2. Energy Bands
3. e/h CurrentElectron / Hole Current
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TUV � W �X CY^V
� W �X CYPY__\Q
� W �X CY\�`a�
� W XCY\�`a�
<
=
For each electron in CB there is a hole in VB
(thermal excitation), Now applying an electric field
will for electrons in VB to fill the empty location,
hence “hole” is moving in direction of electric field!
Z
CYTUV
Therefore “hole” can be considered as a positively
charged particle (or an electron with negative mass!)
1. Bonding
2. Energy Bands
3. e/h CurrentAnalogy
25
<=
Z
T � 0
Electric field � gravitational field
<=
<=
Z
T � bZ
electron � droplet
hole � bubble
1. Bonding
2. Energy Bands
3. e/h CurrentE-K Diagram
26
A � 5 >⁄ → > � 'd � 5e d
Consider a free electron with mass m,
dYdYf
Wave-particle duality:
� >�2) � '�d�
2) → ∝ d�
Available states: 1022/cm3
Number of e- & h+ s: 1010/cm3� Freely moving electrons
1. Bonding
2. Energy Bands
3. e/h CurrentEffective Mass
27
d
The electron wave function is the solution of the three dimensional Schrodinger
wave equation
effective mass =
�hYi∙kThe solution is of the form
acceleration = conduction band
�'�2) *�l 3 �mnl � l�'�2) *�l 3 �mnl � l
Where k = wave vector = 2π/electron wavelength
For each k, there is a corresponding E.
valence band
H
�XZ'�
� d� � �
)�XZ'�
� d� � �
)
'� � d�⁄ '� � d�⁄
1. Bonding
2. Energy Bands
3. e/h CurrentEffective Mass
28
In 3-D crystals the electron acceleration will not be colinear. Thus, in general
we have an effective mass tensor as
electron and hole effective masses at 300K
In an electric field, Z, an electron or a hole accelerates.electrons o � �XZ
)�o � �XZ
)�
1)∗Yq
� 1'�
r�rdYrdq
1)∗Yq
� 1'�
r�rdYrdq
holes o � XZ)`
o � XZ)`
Si Ge GaAs
mn / m0 0.26 0.12 0.067
mp / m0 0.34 0.21 0.34
Si Ge GaAs
mn / m0 1.1 0.55 0.067
mp / m0 0.81 .37 0.45
for density of states calculations for conductivity calculations
1. Bonding
2. Energy Bands
3. e/h CurrentMeasuring Effective Mass
29
Cyclotron Resonance Technique
Centripetal force = Lorentzian force
)�C� � XCs)�C� � XCs
• tUu is the Cyclotron resonance frequency.
• It is independent of C and .
• Electrons strongly absorb microwaves of that
frequency.
• By measuring tUu, )� can be found.
ss
C � Xs)�
C � Xs)�
tUu � C2� � Xs
2�)�tUu � C
2� � Xs2�)�
1. Bonding
2. Energy Bands
3. e/h CurrentExample: Effective Mass
30
d
A schematic energy-
momentum diagram for
a special semiconductor
with )� � 0.25)�and
)` � )�.
1. Bonding
2. Energy Bands
3. e/h CurrentDirect / Indirect
31
d
Direct Materials Indirect Materials
GaAs Si , Ge
d
Conservations of:
1. Energy
2. Momentum
Applications: LEDs, Lasers
� 5e
1. Bonding
2. Energy Bands
3. e/h CurrentDirect / Indirect
32
GaAsSi