Transcript Lecture 4 - web page for staff

ENE 311
Lecture 4
Quantum numbers
• As Pauli exclusion principle stating that no two electrons
in an interacting system can have the same set of
quantum numbers n, l, m, s.
• Only two electrons can have the same three quantum
numbers n, l, m, and those two must have opposite spin.
These can be summarized as
n = 1, 2, 3, …
l = 0, 1, 2,…, (n-1)
m = -l, …, -1, 0, 1, …, +l
s =  1/2
Quantum numbers
• The quantum states shown in the table are used
to indicate the electronic configurations for
atoms in the lowest energy state.
Quantum numbers
n
l
m
s
Allowable
states in
subshell
1
0
0
1/2
2
0
0
1/2
2
-1
1/2
0
1/2
1
1/2
0
1/2
-1
1/2
0
1/2
1
1/2
-2
1/2
-1
1/2
0
1/2
1
1/2
2
1/2
2
1
0
1
3
2
6
Allowable
states in
complete
shell
2
8
2
6
18
10
Quantum numbers
Shell(n)
K1
L2
M3
N4
0
0
1
0
1
2
0
1
2
3
s
s
p
s
p
d
s
p
d
f
2
2
6
2
6
10
2
6
10
14
Subshell (l)
# of electrons
2
8
18
32
Quantum numbers
• There is a simple shorthand notation for
electronic structures that is the naming of l
values expressed as
l = 0, 1, 2, 3, 4
s, p, d, f, g
• These s, p, d, f stand for sharp, principal,
diffuse, and fundamental. The rest will be
written in alphabetical order beyond f.
Quantum numbers
• For example, Si (atomic number = 14): 1s22s22p63s23p2
Bonds
Bonds
a b
E (r )   m  n
r
r
where r = interatomic distance
a = attraction constant
b = repulsion constant
m,n = constant of characteristic of each type
of bond or structure
b
a
• Therefore,  m and r n are attraction and
r
repulsion energy, respectively.
Bonds
We may conclude that
• E  0 at r   : Zero energy as the energy in
the absence of interaction.
• At r > r0, atoms attract each other from r   to
r  r0.
• At r < r0, atoms repel each other up to the point
r0 .
• At r0, equilibrium position occurs. It is where
the attraction energy and repulsion energy
balance each other.
Types of bonds
• Bonds may be classified into 4 types as
1. ionic bond: non-directional
2. metallic bond: non-directional
3. covalent bond: directional
4. van der Waals bond.
Ionic bond
• This happens from electrostatic attraction
between ions with different charges such as
NaCl or LiF. The cohesive energy, Ec, the energy
needed to take the crystal apart, may be
written as
Me2
b
Ec  
 n
4 0r r
where M = Madelung constant

e2
4 0 r
= Coulomb electrostatic
attraction energy between 2 ions.
Metallic bond
• Metallic bond is similar to the ionic bond as
electrostatic forces play big part on it, but this
electrostatic forces are everywhere and come
from all directions.
• In metals, the negative charges are highly
mobile, electrons act like a glue to hold the
lattice together.
• The cohesive forces in metals are very strong
and hard to break.
Covalent bond
• This bond happens from the sharing of electrons
between two atoms.
• The simplest example of covalent bond is shown
by hydrogen atom.
• Hydrogen atom needs another electron to fill its
1s shell.
• It would find that extra electron from another
hydrogen atom as they both finally share their
electrons.
Covalent bond
• In covalent bond, all electrons pair up and orbit
around a pair of atoms, so more of them can
wander away to conduct electricity.
• In case of carbon, it acts like an insulator, but
this bond in silicon or germanium is weaker.
• Some of electrons in the latter case might be
shaken off and able to conduct electricity, so
we call them “semiconductors”.
Covalent bond
(a) A broken bond at Position A, resulting in a conduction
electron and a hole.
(b) A broken bond at position B.
The van der Waals bond
• This is like a secondary bond since its force is
very weak.
• This bond can be seen in atoms that their outer
shell is fully filled.
• Consider atom A has a dipole moment then it
will induce an opposite dipole moment on atom
B.
• This attraction force is called “van der Waals
bond”.
Energy Bands
• Condiser two identical atoms, when they are far
apart, the allowed energy levels for a given
principal quantum number (n = 1) consist of one
doubly degenerate level (both atoms have
exactly the same energy).
• When they are brought closer, the doubly
degenerate energy levels will spilt into two
levels by the ineraction between the atoms.
Energy Bands
• N isolated atoms are brought together to form a
solid, the orbits of the outer electrons of
different atoms overlap and interact with each
other.
• This causes a shift in the energy levels and N
separate closely spaced levels are formed.
Energy Bands
Energy Bands
• Consider isolated silicon atom, 10 of the 14
electrons occupy energy levels whose orbital
radius is much smaller than the interatomic
separation in the crystal.
• The four remaining valence electrons are
relatively weakly bound and can be involved in
chemical reactions.
• Therefore, the valence electrons are the ones
that will be considered.
• The two inner shells are completely full and
tightly bound to the nucleus.
Energy Bands
• As the interatomic
distance decreases, the 3s
and 3p subshell of the N
silicon stoms will interact
and overlap.
• At the equilibrium
interatomic distance, the
bands will again split with
four quantum states per
atom in the lower band
(valence band) and four
quantum states per atom
in the upper band
(conduction band).
Energy Bands
• At absolute zero temperature (T = 0 K),
electrons occupy the lowest energy states, so
that all states in the lower band will be full and
all states in the upper band will be empty.
• The bottom of the conduction band is called Ec,
and the top of the valence band is called Ev.
• The bandgap energy Eg is the width of the
forbidden energy level between the bottom of
the conduction band and the top of the valence
band.
Energy Bands
• The bandgap energy is the energy required to
break a bond in the semiconductors to free and
electron to the conductgion band and leave a
hole in the valence band.
Energy Bands
Energy Bands
The energy-momentum diagram
• The energy E of a free electron is given by
p2
E
2m0
where p is the momentum
m0 is the free-electron mass
(1)
Energy Bands
• In a semiconductor,
an electron in the
conduction band is
similar to a free
electron in that it is
free to move about
inside the crystal as
shown in the right
figure.
Energy Bands
• However, the above equation for E can not be
used due to the periodic potential of the
nucleus.
• Anyway, if replacing m0 with an effective mass,
in an equation (1), it yields the energy E of an
electron as
p2
E
2me*
Energy Bands
• The effective mass in a solid is a result of
charged particle moving under nucleus of
applied electric field in presence of a periodic
potential.
• This differs from the mass in free space.
• The electron effective mass depends on the
properties of the semiconductor.
Energy Bands
In quantum-mechanic, the velocity of electron is
described by its group
d
vg 
dk
E  h  
1 E
vg 
k
Energy Bands
The acceleration can be obtained by
d  1 E 
a
 
dt dt  k 
dvg
1  2 E k
a
k 2 t
(2)
Energy Bands
• For classical part, it expresses dE as the work
done by a particle traveling a distance vgdt
under the influence of a force eE. It yields
dE  F .dx
 F .(vg dt )
 1 E 
dE  F .
 dt
 k 
Energy Bands
This leads to
dk F

dt
Substituting (3) into (2)
1 2E  F 
a

2 
k  
1 2E
a 2 2 F
k
(3)
Energy Bands
From F = ma, we have
1
 1  E
 E
m  2 2   2 
 k 
 p 
2
2
1
*
e
A similar expression can be written for holes with
effective mass m.h*
Energy Bands
• A schematic energymomentum diagram for a
special semiconductor
with me* = 0.25 m0 and
mh* = m0.
• The electron energy is
measured upward and hole
energy is measured
downward.
• This energy-momentum
relationship is called
“energy-band diagram”.
Energy Bands
• Energy band diagram may
be classified
semiconductors into 2
groups as direct
semiconductors and
indirect semiconductors.
• Energy band structures of
Si and GaAs. Circles (º)
indicate holes in the
valence bands and dots
(•) indicate electrons in
the conduction bands.
Energy Bands
• Let us consider the figure,
GaAs is a direct s/c with a
direct bandgap since it does
not require a change in
momentum for an electron
transition from the valence
band to the conduction
band.
• Unlike in the case of Si, an
electron transition from the
valence band to the
conduction band requires not
only an energy change but
also momentum change
(called indirect s/c).
Energy Bands
• This difference between direct and indirect
bandgap is crucial for making the light sources
such as LEDs or LASERs.
• These light sources require direct
semiconductors for efficient generation of
photons.
Energy Bands
Conduction in Metals, S/C, and Insulators
• The electrical conductivity of metals,
semiconductors, and insulators could be
explained by their energy bands.
• These can be done by considering the highest
two bands, valence and conduction bands, of
the materials.
• Electron occupation of the conduction band
determines the conductivity of a solid.
Energy Bands
• (left) a conductor with two possibilities (either the
partially filled conduction band shown at the upper
portion or the overlapping bands shown at the lower
portion)
• (middle) a semiconductor
• (right) an insulator.
Energy Bands
• Metals: Highest allowed occupied band or conduction
band is partially filled (such as Cu) or overlaps the valence
band (such as Zn or Pb). Therefore, electrons are free to
move to the next energy level with only a small applied
field.
Energy Bands
• Insulators: The valence electrons
form strong bonds between their
neighboring atoms. These bonds are
difficult to break.
• Therefore, valence band is fully
filled and the conduction band is
totally empty.
• Also, these two bands are separated
by a wide bandgap.
• Thermal energy or the energy from
applied electric field is not enough
to raise the uppermost electron in
the valence band up to the
conduction band.
• Therefore, there is no conductivity.
Energy Bands
• Semiconductor: This is similar to
the insulators, but the bandgap is
much smaller than in the case of
insulators.
• At T = 0 K, all electrons are in the
valence band and no electron in the
conduction band.
• Therefore, semiconductors are poor
conductors at low temperatures. At
room temperature, some electrons
are thermally excited from the
valence band to the conduction
band.
• Also, it needs just small applied
electric field to move these
electrons and that results in
conductivity.