Chap-4 Modern theory of solids.pptx

Chapter - 3
Modern Theory of Solids

Hydrogen Molecule:
Molecular Orbital Theory of Bonding

r
 = 1s(rA) + 1s(rB)
* = 1s(rA) -1s(rB)
Bonding Molecular Orbital
Antibonding Molecular Orbital
a
r
H
r
R = 
Two hydrogen atoms
approaching each other.
1s(rA)
A B
H
rA e-
1s(rB)
rB e-
Fig. 4.1: Formation of molecular orbitals, bonding and antibonding
( and  ) when two H atoms approach each other. The two
electrons pair their spins and occupy the bonding orbital .
Hydrogen Molecule:
Molecular Orbital Theory of Bonding

Bonding formation in Molecules (II)
Linear Combination of Atomic Orbitals (LCAO)

Energy Diagram for H2
H -atom H -atom
H2
E
E
E = Bonding
Energy
E1s
E1s
(b)

E(R)
1s
E1s

E
(a)
E
(R)
a
0
E
SYSTEM
2 H-Atoms
2 Electrons
1 Electron/Atom
1 Orbital/Atom
R, Interatomic
Separation
0 R= 
Bonding
Energy
(a)
Fig. 4.3: Electron energy in the system comprising two hydrogen
atoms. (a) Energy of  and  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.

Band Theory of Solids
Lithium
N electrons fill
all the levels
up to N/2

2s
2p
3s
Overlapping energy
bands
Electrons
Vacuum level
2s
2p
3s
3p
1s
1s
Solid Atom
E = 0
Free electron
E
Electron
Energy
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Fig. 4.10: In a metal the various energy bands overlap to give a
single band of energies that is only partially full of electrons. There
are states with energies up to the vacuum level where the electron is
free.
Near Free electron model for metal
( potential ~ 0 or small)
V ~= 0
What is Fermi level ?
What is the workfunction ??

Energy band diagram of Metal
Electron Energy
Vacuum
Level
EF0
EB
EF0

Electron inside
the metal
Electron outside
the metal
0
-2.5 eV
-7.2 eV 0
4.7 eV
7.2 eV
Fig. 4.11: Typical electron energy band diagram for a metal All
the valence electrons are in an energy band which they only
partially fill. The top of the band is the vacuum level where the
electron is free from the solid (PE = 0).
Empty levels
Levels occupied
by electrons
E
px
Lattice
scattering
-x x
pav > 0
E
p-x
EFO
a
E
px
p-x
E
pav = 0
a
b
b
EFO
Electrons
Empty states
0
(b)
Fig. 4.12 (a) Energy band diagram of a metal. (b) In the absence of
a field, there are as many electrons moving right as there are
moving left. The motions of two electrons at each energy cancel
each other as for a and b. (c) In the presence of a field in the -x
direction, the electron a accelerates and gains energy to a where it
is scattered to an empty state near EFO but moving in the x
direction. The average of all momenta values is along the +x
direction and results in a net electrical current.
(c)
(a)
Typical energy band of free electron
V=0
No energy gap between CBM and
VBM in metal
Only electrons near the Fermi
level contribute to conduction

Periodic Potential for Semiconductors
r
PE(r)
x
V(x)
x = L
x = 0 a 2a 3a
0
a
a
Surface Surface
Crystal
PE of the electron
around an isolated
atom
When N atoms are
arranged to form the
crystal then there is an
overlap of individual
electron PE functions.
PE of the electron,
V(x), inside the crystal
is periodic with a
period a.
Fig. 5.48: The electron PE, V(x), inside the crsytal is periodic with
the same periodicity as that of the crystal, a. Far away outside the
crsytal, by choice, V = 0 (the electron is free and PE = 0).
V(x)=V(x+ma)
The solution of
Schrodinger equation
can be described by
Bloch function.
)
(
)
(
)
exp(
)
(
x
U
a
x
U
ikx
x
U
k
k
k
k




e
)
exp(
)
(
)
( ikx
x
U
x k
k 

)
(
)
( a
x
U
x
U k
k 


Common semiconductor
crystal structures
The zinc-blende crystal structure of GaAs
and InP
The diamond lattice of silicon and
germanium

Electronic Structure of Si atom

Band Gap Formation in Semiconductor
Si ATOM

B

B

A

A

hyb
CONDUCTION BAND
VALENCE BAND
Energygap, Eg
(a) (b) (c) (d)
3p
3s
Si CRYSTAL

hyb
Fig. 4.17: (a) Formation of energy bands in the Si crystal first involves hybridization
of 3s and 3p orbitals to four identical hyb orbitals which make 109.5° with each
other as shown in (b). (c) hyb orbitals on two neighboring Si atoms can overlap to
form B or A. The first is a bonding orbital (full) and the second is an antibondiong
orbital (empty). In the crystal B overlap to give the valence band (full) and A
overlap to give the conduction band (empty).

What is difference between Metal and
Semiconductor ??
• No energy gap in metal.
• There is an energy gap in semiconductor.
• Origin of Energy gap– electron diffraction
at certain wavevector k .

Origin of Energy gap (I)
Forward wave
B'
a
A
A'
Reflected waves
C'
B C
 k
Backward wave
-k
x
Fig. 4.48: An electron wave propagation through a linear lattice.
For certain k values the reflected waves at successive atomic
planes reinforce each other to give rise to a reflected wave
travelling in the backward direction. The electron then cannot
propagate through the crystal.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 200
c

Energy = Ec
Energy = Es
k = ±  / a
s

Fig. 4.49: Forward and backward waves in the crystal with k = ±
 / a give rise to two possible standing waves, c and s. Their
probability density distributions, cand s , have maxima
either at the ions or between the ions.

Origin of Energy gap (II)
a
O a
a
a
2V1
Ec
Es
First Brillouin Zone
Second
Brillouin
Zone
Second
Brillouin
Zone
2V2
k
Energy gap
Energy
gap
Band
Band
Band
Allowed
energies
Forbidden
energies
-k
E
Fig. 4.50: The energy of the electron as a function of its wavevector k
inside a one-dimensional crystal. There are discontinuities in the energy
at k = ±n/a values where the waves suffer Bragg reflections in the
Energy
Metal- potential is small or 0
Semiconductor—potential large - energy splitting is large

Electron

k
k1
k2
Diffracted electron
(11) Planes
(01) Planes
(10) Planes

k3
[01]
[10]
y
x
[11]
45°
45°
45°
a
a
a/
Fig. 4.51: Diffraction of the electron in a two dimensional cubic
crystal. Diffraction occurs whenever k has a component satisfying
k1 = ±n/a, k2 = ±n/a or k3 = ±n/a . In general terms, when
ksin = n/a.


a
E
[11]
k3
E
k1
[10]
Band
Band
Energy gap
Energy gap
Band
Band
First
Brillouin Zone
Second Brillouin
Zone
Second
Brillouin Zone
First
Brillouin Zone
a

Fig. 4.52: The E-k behavior for the electron along different
directions in the two dimensional crystal. The energy gap along
[10] is at /a whereas it is at  along [11].
a
/
2

Energy gap
[11]
[10]
Bands overlap
energy gaps
Energy gap
1st BZ
band
2nd BZ
&
1st BZ
overlapped
band
2nd BZ
band
(a) Metal
Energy gap
[11]
[10] Overlapped
energy gaps
Energy gap = Eg
1st BZ
band
2nd BZ
band
(b) Semiconductor and insulator

Figure 2.19. Schematic energy band representations of (a) 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), (b)
a semiconductor, and (c) an insulator.
Metal, Semiconductor, Insulator

Electron Excitation in Semiconductor

Ec
Ev
CB
VB
Eg
Thermal
excitation
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.
Electron
energy
Thermal excitation in Intrinsic Semiconductor

E-k diagram
Conduction band:
• Similar features with set of sub-bands. However,minima in different directions.
• Ge has eight equivalent conduction-band minima in the <111> directions.
• Si has six equivalent conduction-band minima in the <100> directions.
• GaAs has minimum at the zone center. Small separation to the L-valley.
Valence band:
1. Valence band minimum at k = 0
2. The valence band has 3 sub-bands
a. Heavy-hole band
b. Light hole band
c. Split-off band
3. Near k = 0 the curvature of the
sub-bands is essentially
orientation independent
Ge Si GaAs

Direct and indirect bandgap semiconductors
E
CB
VB
k
-k
Direct Band Gap
(a) GaAs
Eg Photon
Ec
Ev
E
CB
VB
Indirect Band Gap, Eg
k
-k
kcb
(b) Si
Ec
Ev
kvb
E
k
-k
Phonons
(c) Si with a recombination center
VB
CB
Er Ec
Ev
Fig. 5.50: (a) In GaAs the minimum of the CB is directly above the
maximum of the VB. GaAs is therefore a direct band gap
semiconductor. (b) In Si, the minimum of the CB is displaced from
the maximum of the VB and Si is an indirect band gap
semiconductor. (c) Recombination of an electron and a hole in Si
involves a recombination center.

Electron Effective Mass
Fext
VACUUM
e-
x
Fext
me
a =
(a)
Fint
CRYSTAL
x
Fext
me
*
a =
(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.)
Ex
e
ext
vac
m
F
a 
s
*
int
e
ext
e
ext
crystal
m
F
m
F
F
a 



A schematic energy-
momentum diagram for a
special semiconductor with mn
= 0.25 m0 and mp = m0.
The parabolic energy (E) vs.
momentum (p) curve for a free
electron.

Density of States
-n1
-n2
n1
n2
n1
2
+ n2
2
= n'
2
n1
= 1
n2
= 3
0
1
2
3
4
5
1 2 3 4 5 6
n1
= 2, n2
= 2
Fig. 4.21: Each state, electron wavefunction in the crystal, can be
represented by a box at n1,n2.
n2
In here n1
2
+n2
2
+n3
2
 n'2
n'
Vol. = 1/8(4/3 n'3
)
n3
n1
Fig. 4.22: In three dimensions, the volume defined by a sphere of
radius n' and the positive axes n1, n2 and n3, is all the possible
combinations of poisitive n1, n2 and n3, values which satisfy
n1
2+n2
2+n3
2  n'2.

Density of States in an Energy Band

Carrier distribution function –
Fermi-Dirac Distribution
• Electrons are fermions which follow Fermi-
Dirac distribution function.
E
EF
0 1
/
2
1
f(E)
T1
T = 0
T2
> T1
Fig. 4.26: The Fermi-Dirac function, f(E), describes the
statistics of electrons in a solid. The electrons interact with
each other and the environment so that they obey the Pauili
Exclusion Principle.
Further reading:
Maxwell-Boltzmann
Bose-Einstein distribution

Schematic Illustration of F-D distribution
Eight of the 24 possible configurations in which 20 electrons can be
placed having a total energy of 106 eV

Different Distribution Function
Fermi-Dirac
Bose-Einstein
Maxwell-Boltzmann
electron

Quantum theory of metal
Carrier
density

Seebeck Effect and Thermalcouple
E
1
EF
H
E
f(E)
1
EF
C
Conductor
0 0
Hot Cold
Hot Cold
-
-
-
-
+
+
+
+
Temperature, T
f(E)
Voltage, V
-
+
Fig.4.30: The Seebeck effect: A temperature gradient along a
conductor gives rise to a potential difference.
S at 0 C
V K-1
)
S at 27 C
V K-1
)
EF
(eV)
x
Al –1.6 –1.8 11.6 2.78
Au +1.79 +1.94 5.5 –1.48
Cu +1.70 +1.84 7.0 –1.79
K -12.5 2.0 3.8
Li +14 4.7 –9.7
Mg –1.3 7.1 1.38
Na –5 3.1 2.2
Pd –9.00 –9.99
Pt –4.45 –5.28

 
H C
Energy
x
 
H C
Energy
x
(b) S positive
(a) S negative
Fig. 4.31: Consider two neighboring regions H (hot) and C (cold)
with widths corresponding to the mean free paths  and  in H and
C. Half the electrons in H would be moving in +x direction and the
other half in -x direction. Half of the electrons in H therefore cross
into C, and half in C cross into H.

emf
(mV)
0
10
20
30
40
50
60
70
80
0 200 400 600 800 1000
Temperature (°C)
E-Type
J-Type
S-Type
T-Type
K-Type
Fig. 4.33: Output emf vs temperature (°C) for various termocouples
between 0 to 1000°C.
Material emf, mV
At 100 C
emf, mV
At 200 C
Copper, Cu 0.76 1.83
Aluminum, Al 0.42 1.06
Nickel, Ni –1.48 –3.10
Palladium, Pd –0.57 –1.23
Platinum, Pt 0 0
Silver, Ag 0.74 1.77
Alumel –1.29 –2.17
Chromel 2.81 5.96
Constantan –3.51 –7.45
Iron, Fe 1.89 3.54
90%Pt-10%Rh
(Platinum-Rhodium)
0.643 1.44

N-type
semiconductor
P-type
semiconductor

e
e
e e
e
e
e
e
e e
e
e
e e
e e
e
e
Electron emission in solids
E-field
Solid
vacuum e (h)
Thermal emission
Secondary emission
(Photoemission) Field emission

Image PE
0
EF
+ 
x
x
Net PE
x
EF
+ 
Applied PE
EF
+ eff
(a) (b) (c)
Fig. 4.36: (a) PE of the electron near the surface of a conductor, (b)
Electron PE due to an applied field e.g. between cathode and anode
(c) The overall PE is the sum.

PE(x)
x
EF
+ eff
xF
Metal Vacuum
EF
0
0
Vo
e-
x = 0 x = xF
EF
(a)
(b)
Fig. 4.37 (a) Field emission is the tunneling of an electron at an
energy EF through the narrow PE barrier induced by a large applied
field. (b) For simplicity we take the barrier to be rectangular. (c) A
sharp point cathode has the maximumfield at the tip where the field-
emission of electrons occurs.
E
Cathode
Grid or Anode
HV V
(c)

Spindt-type Mo tip
1.2m
Nanotube tip
(Samsung)
Nanostructured diamond
(Lucent Technology)
Basic structure of field emission display
Nanotip

Application for Field Emission Display (FED)
FED
Mo Tip
CNT

Chap-4 Modern theory of solids.pptx

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