2p
E2p
EMPTY
ET
solid(N)
2s
FULL
Electron Energy in the System
of N Li Atoms
System of N Li Atoms
EB
solid(1)
1s
a
Solid
E2s
SYSTEM
N Li Atoms
N Electrons
N Orbitals
2N States
E1s
Interatomic
Separation (R)

Isolated Atoms
The formation of a 2s-energy band from the 2s-orbitals when N Li atoms
come together to form the Li solid. The are N 2s-electrons but 2N states in
the band. The 2s-band therefore is only half full. The atomic 1s orbital is
close to the Li nucleus and remains undisturbed in the solid. Thus each Li
atom has a closed K-shell (full 1s orbital).
From Principles of Electronic Materials and
Devices, Third Edition, S.O. Kasap (©
McGraw-Hill, 2005)
Fig 4.8
EMPTY
E3s
E 2p
E2s
FULL
Electron energy
Free electron
E = 0 (Vacuum Level)
E 1s
R=a
The Solid
Interatomic
Separation (R)
R=
Isolated Atoms
As solid atoms are brought together from infinity, the atomic orbitals
overlap and give rise to bands. Outer orbitals overlap first. The 3s
orbitals give rise to the 3s band, 2p orbitals to the 2p band and so on.
The various bands overlap to produce a single band in which the energy
is nearly continuous.
From Principles of Electronic Materials and
Devices, Third Edition, S.O. Kasap (©
McGraw-Hill, 2005)
Fig 4.9
E
Overlapping
energy bands
Free electron
E=0
Vacuum level
E lectron E nergy
3s
2p
3p
3s
2p
2s
Electrons
2s
1s
1s
Solid
Atom
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.
From Principles of Electronic Materials and
Devices, Third Edition, S.O. Kasap (©
McGraw-Hill, 2005)
Fig 4.10
Electron outside
the metal
Electron Energy
Vacuum
Level
Electron inside the 
metal
EF0
0
7.2 eV
-2.5 eV
4.7 eV
-7.2 eV
0
EF0
EB
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).
From Principles of Electronic Materials and
Devices, Third Edition, S.O. Kasap (©
McGraw-Hill, 2005)
Fig 4.11
From Principles of Electronic
Materials and Devices, Third
Edition, S.O. Kasap (©
E
E
Empty states
b
FO
E
Lattice
scattering
FO
a
E
a
b
Electrons
px
p-x
-x
0
(a)
pav = 0
x
(b)
p-x
px
pav > 0
(c)
(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.
From Principles of Electronic Materials and
Devices, Third Edition, S.O. Kasap (©
McGraw-Hill, 2005)
Fig 4.12
V
x
V(x)
x
EFO
EB
Em
pt y
Ele
leve
ls
ctro
ns
Energy band diagram
EFO - eV
EB - eV
Conduction in a metal is due to the drift of electrons around the Fermi
level. When a voltage is applied, the energy band is bent to be lower at the
positive terminal so that the electron's potential energy decreases as it
moves towards the positive terminal.
From Principles of Electronic Materials and
Devices, Third Edition, S.O. Kasap (©
McGraw-Hill, 2005)
Fig 4.13
Electron energy
CB
Ec
Thermal
excitation
Eg
Ev
VB
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, Third Edition, S.O. Kasap (©
McGraw-Hill, 2005)
Fig 4.18
Ex
Ex
CRYSTAL
VACUUM
e-
a=
Fext
F int
Fext
a=
me
Fext
m e*
x
x
(a)
(b)
(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, Third Edition, S.O. Kasap (©
McGraw-Hill, 2005)
Fig 4.19
hyb orbitals
Si crystal in 2-D
Electron energy
Ec+
Valence
electron
CONDUCTION
BAND (CB)
Empty of electrons
at 0 K.
Ec
B
Bandgap = Eg
Ev
VALENCE BAND
(VB)
Full of electrons
at 0 K.
Si ion core
(+4e)
0
(a)
(b)
(c)
(a) A simplified two dimensional illustration of a Si atom with four hybrid
orbitals, hyb. Each orbital has one electron. (b) A simplified two dimensional
view of a region of the Si crystal showing covalent bonds. (c) The energy band
diagram at absolute zero of temperature.
From Principles of Electronic Materials and
Devices, Third Edition, S.O. Kasap (©
McGraw-Hill, 2005)
Fig 5.1
Ex
Hole energy
(a)
Electron Energy
V(x)
Electrostatic PE(x)
Ex
CB
CB
(b)
VB
VB
x
x=0
x=L
When an electric field is applied, electrons in the CB and holes in the
VB can drift and contribute to the conductivity. (a) A simplified
illustration of drift in Ex. (b) Applied field bends the energy bands
since the electrostatic PE of the electron is -eV(x) and V(x) decreases in
the direction of Ex whereas PE increases.
From Principles of Electronic Materials and
Devices, Third Edition, S.O. Kasap (©
McGraw-Hill, 2005)
Fig 5.6
V(x)
x
Electrostatic PE(x) = -eV
Electron Energy
Ex
Ec
Ed
EF
EFi
Ev
n-Type Semiconductor
A
B
V
Energy band diagram of an n-type semiconductor connected to a
voltage supply of V volts. The whole energy diagram tilts because the
electron now has an electrostatic potential energy as well
From Principles of Electronic Materials and
Devices, Third Edition, S.O. Kasap (©
McGraw-Hill, 2005)
Fig 5.13
The E-k Diagram
Ek
The Energy Band
Diagram
CB
CB
Empty k
e-
e-
Ec
Ec
h
Eg
h+
VB
h
Ev
Ev
Occupied k
h+
VB
-/a
/a
k
The E-k diagram of a direct bandgap semiconductor such as GaAs. The E-k
curve consists of many discrete points each point corresponding to a possible
state, wavefunction y k(x), that is allowed to exist in the crystal. The points
are so close that we normally draw the E-k relationship as a continuous curve.
In the energy range Ev to Ec there are no points (yk(x) solutions).
From Principles of Electronic Materials and
Devices, Third Edition, S.O. Kasap (©
McGraw-Hill, 2005)
Fig 5.49
a
b
Minority Carrier
Concentration
E o +E
Neutral n-region
M
e(Vo +Vr)
H ole P E (x)
Neutral p-region
Thermall
y
generated
EHP
eVo
Holes
Electrons
pno
npo
x
Wo
Wo
x
W(V = -Vr )
W
Diffusion
Drift
V
r
Reverse biased pn junction. (a) Minority carrier profiles and the origin of the reverse
current. (b) Hole PE across the junction under reverse bias
From Principles of Electronic Materials and
Devices, Third Edition, S.O. Kasap (©
McGraw-Hill, 2005)
Fig 6.8
Eo
p
E
(a)
n
M
E
c
(b)
Eo-E
c
eVo
E
E
c
E
Fp
E
Fn
v
E
E
eV
Fp
E
c
E
Fn
v
E
p
E
n
v
e(Vo -V)
p
v
n
I
V
Energy band diagrams for a pn junction under (a) open circuit and (b) forward bias
From Principles of Electronic Materials and
Devices, Third Edition, S.O. Kasap (©
McGraw-Hill, 2005)
Fig 6.11