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Semiconductor

Nov 1, 2013 (4 years and 8 months ago)

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SEMICONDUCTORS

Semiconductors

Semiconductor devices

Electronic Properties

Robert M Rose, Lawrence A Shepart, John Wulff

Wiley Eastern Limited, New Delhi (1987)

Energy gap in solids

In the free electron theory a constant potential was assumed inside the solid

In reality the presence of the positive ion cores gives rise to a varying

potential field

The travelling electron wave interacts with this periodic potential

(for a crystalline solid)

The electron wave can be Bragg diffracted

Bragg diffraction from a 1D solid

n

= 2d

n

= 2d Sin

1D

=90
o

The Velocity of electrons for the above values of
k

are zero

These values of
k

and the corresponding E are forbidden in the solid

The waveform of the electron wave is two standing waves

The standing waves have a periodic variation in amplitude and hence the

electron probability density in the crystal

The potential energy of the electron becomes a function of its position

(cannot be assumed to be constant (and zero) as was done in the

free electron model)

k

E

Band gap

The magnitude of the Energy gap between two bands is the difference

in the potential energy of two electron locations

k

E

K.E of the electron increasing

Decreasing velocity of the electron

ve effective mass (m
*
) of the electron

Within a band

Effective energy gap
→ Forbidden gap →
Band gap

k

E

E

[100]

[110]

k

Effective gap

The effective gap for all directions of motion is called the forbidden gap

There is no forbidden gap if the maximum of a band for one direction of

motion is higher than the minimum for the higher band for another

direction of motion

this happens if the potential energy of the electron

is not a strong function of the position in the crystal

Energy band diagram: METALS

Monovalent metals

Divalent metals

Monovalent metals: Ag, Cu, Au → 1 e

in the outermost orbital

outermost energy band is only half filled

Divalent metals: Mg, Be → overlapping conduction and valence bands

they conduct even if the valence band is full

Trivalent metals: Al → similar to monovalent metals
!!!

outermost energy band is only half filled
!!!

Energy band diagram: SEMICONDUCTORS

2
-
3 eV

Elements of the 4
th

column (C, Si, Ge, Sn, Pb) → valence band full but no

overlap of valence and conduction bands

Diamond → PE as strong function of the position in the crystal

Band gap is 5.4 eV

Down the 4
th

column the outermost orbital is farther away from the nucleus

and less bound

the electron is less strong a function of the position

in the crystal

reducing band gap down the column

Energy band diagram: INSULATORS

> 3 eV

P(E)

E

E
g

E
g
/2

Intrinsic semiconductors

At zero K very high field strengths (~ 1010 V/m) are required to move an

electron from the top of the valence band to the bottom of the

conduction band

Thermal excitation is an easier route

T > 0 K

n
e

→ Number of electrons promoted

across the gap

(= no. of holes in the valence band)

N → Number of electrons available

at the top of the valance band

for excitation

Unity in denominator can be ignored

Under applied field the electrons
(thermally excited into the conduction
band)

can move using the vacant sites in the conduction band

Holes move in the opposite direction in the valence band

The conductivity of a semiconductor depends on the concentration of
these charge carriers (
n
e

&
n
h
)

Similar to drift velocity of electrons under an applied field in metals in
semiconductors the concept of mobility is used to calculate conductivity

Conduction in an intrinsic semiconductor

Mobility of electrons and holes in Si & Ge (at room temperature)

Species

Mobility (m
2

/ V / s)

Si

Ge

Electrons

0.14

0.39

Holes

0.05

0.19

Conductivity as a function of temperature

Ln(

)

1/T (/K)

Extrinsic semiconductors

The addition of doping elements significantly increases the conductivity

of a semiconductor

Doping of Si

V column element
(P, As, Sb)

→ the extra unbonded electron

is practically free
(with a radius of motion of ~ 80 Å)

Energy level near the conduction band

n
-

type semiconductor

III column element
(Al, Ga, In)

→ the extra electron for bonding

supplied by a neighbouring Si atom → leaves a hole in Si.

Energy level near the valence band

p
-

type semiconductor

E
g

Donor level

n
-
type

Ionization Energy

Energy required to promote an

electron from the Donor level to

conduction band

E
Ionization

< E
g

even at RT large fraction of

the donor electrons are exited

into the conduction band

Electrons in the conduction band are the majority charge carriers

The fraction of the donor level electrons excited into the conduction band

is much larger than the number of electrons excited from the valence band

Law of mass action:

(n
e
)
conduction band

x (n
h
)
valence band

= Constant

The number of holes is very small in an n
-
type semiconductor

Number of electrons ≠ Number of holes

E
Ionization

Acceptor level

E
g

p
-
type

At zero K the holes are bound to the dopant atom

As T↑ the holes gain thermal energy and break away from the dopant atom

available for conduction

The level of the bound holes are called the acceptor level
(which can accept

and electron)

and acceptor level is close to the valance band

Holes are the majority charge carriers

Intrinsically excited electrons are small in number

Number of electrons ≠ Number of holes

E
Ionization

Ionization energies for dopants in Si & Ge (eV)

Type

Element

In
Si

In
Ge

n
-
type

P

0.044

0.012

As

0.049

0.013

Sb

0.039

0.010

p
-
type

B

0.045

0.010

Al

0.057

0.010

Ga

0.065

0.011

In

0.16

0.011

(/ Ohm / K
)

1/T (/K)

0.02

0.04

0.06

0.08

0.1

Intrinsic

Exhaustion

Exponential

function

Slope can be used

for the calculation

of E
Ionization

10 K

50 K

+ve slope due to

Temperature dependent

mobility term

All dopant atoms have been excited

slope

Semiconductor device

chose the flat region where the conductivity does

not change much with temperature

Thermistor
(for measuring temperature)

maximum sensitivity is

required