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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