Lecture 34 - Intrinsic Semiconductors - PHYSICS OF SEMICONDUCTOR DEVICES

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 Page 1


 Module 6 : PHYSICS OF SEMICONDUCTOR DEVICES
 Lecture 34 : Intrinsic Semiconductors
 
Objectives
 
In this course you will learn the following
Intrinsic and extrinsic semiconductors.
Fermi level in a semiconductor.
p-type and n-type semiconductors.
Compensated semiconductors.
Charge neutrality and law of mass action.
 
Intrinsic Semiconductors
 
An intrinsic semiconductor is a pure semiconductor, i.e., a sample without any impurity. At absolute zero
it is essentially an insulator, though with a much smaller band gap. However, at any finite temperature
there are some charge carriers are thermally excited, contributing to conductivity. Semiconductors such
as silicon and germanium, which belong to Group IV of the periodic table are covalently bonded with
each atom of Si(or Ge) sharing an electron with four neighbours of the same specis. A bond picture of
silicon is shown in the figure where a silicon atom and its neighbour share a pair of electrons in covalent
bonding.
 
 
 
 
Gallium belongs to Group III and bonds with arsenic which belongs to Group V to give a III-V
semiconductor. In GaAs, the bonding is partly covalent and partly ionic. Other commonly known III-V
semiconductors are GaN, GaP, InSb etc. Like the III-V compounds, Group II elements combine with
Group VI elements to give semiconductors like CdTe, CdS, ZnS etc. Several industrially useful
semiconductors are alloys such as Al GA As.
Page 2


 Module 6 : PHYSICS OF SEMICONDUCTOR DEVICES
 Lecture 34 : Intrinsic Semiconductors
 
Objectives
 
In this course you will learn the following
Intrinsic and extrinsic semiconductors.
Fermi level in a semiconductor.
p-type and n-type semiconductors.
Compensated semiconductors.
Charge neutrality and law of mass action.
 
Intrinsic Semiconductors
 
An intrinsic semiconductor is a pure semiconductor, i.e., a sample without any impurity. At absolute zero
it is essentially an insulator, though with a much smaller band gap. However, at any finite temperature
there are some charge carriers are thermally excited, contributing to conductivity. Semiconductors such
as silicon and germanium, which belong to Group IV of the periodic table are covalently bonded with
each atom of Si(or Ge) sharing an electron with four neighbours of the same specis. A bond picture of
silicon is shown in the figure where a silicon atom and its neighbour share a pair of electrons in covalent
bonding.
 
 
 
 
Gallium belongs to Group III and bonds with arsenic which belongs to Group V to give a III-V
semiconductor. In GaAs, the bonding is partly covalent and partly ionic. Other commonly known III-V
semiconductors are GaN, GaP, InSb etc. Like the III-V compounds, Group II elements combine with
Group VI elements to give semiconductors like CdTe, CdS, ZnS etc. Several industrially useful
semiconductors are alloys such as Al GA As.
 
The number of carriers in a band at finite temperatures is given by , where 
 is the density of state and  is the Fermi function which gives the
thermal probability. If , we may ignore the term 1 in the denominator of the Fermi
function and approximate it as
 
 Using this the density of electrons in the conduction band ( ) may be written as follows.
 
 
where we have substituted
 
 
The integral  is a gamma function  whose value is . Substituting this
value, we get for the density of electrons in the conduction band
 
 
where
 
 
One can in a similar fashion one can calculate the number density of holes, , by evaluating the
expression
 
Page 3


 Module 6 : PHYSICS OF SEMICONDUCTOR DEVICES
 Lecture 34 : Intrinsic Semiconductors
 
Objectives
 
In this course you will learn the following
Intrinsic and extrinsic semiconductors.
Fermi level in a semiconductor.
p-type and n-type semiconductors.
Compensated semiconductors.
Charge neutrality and law of mass action.
 
Intrinsic Semiconductors
 
An intrinsic semiconductor is a pure semiconductor, i.e., a sample without any impurity. At absolute zero
it is essentially an insulator, though with a much smaller band gap. However, at any finite temperature
there are some charge carriers are thermally excited, contributing to conductivity. Semiconductors such
as silicon and germanium, which belong to Group IV of the periodic table are covalently bonded with
each atom of Si(or Ge) sharing an electron with four neighbours of the same specis. A bond picture of
silicon is shown in the figure where a silicon atom and its neighbour share a pair of electrons in covalent
bonding.
 
 
 
 
Gallium belongs to Group III and bonds with arsenic which belongs to Group V to give a III-V
semiconductor. In GaAs, the bonding is partly covalent and partly ionic. Other commonly known III-V
semiconductors are GaN, GaP, InSb etc. Like the III-V compounds, Group II elements combine with
Group VI elements to give semiconductors like CdTe, CdS, ZnS etc. Several industrially useful
semiconductors are alloys such as Al GA As.
 
The number of carriers in a band at finite temperatures is given by , where 
 is the density of state and  is the Fermi function which gives the
thermal probability. If , we may ignore the term 1 in the denominator of the Fermi
function and approximate it as
 
 Using this the density of electrons in the conduction band ( ) may be written as follows.
 
 
where we have substituted
 
 
The integral  is a gamma function  whose value is . Substituting this
value, we get for the density of electrons in the conduction band
 
 
where
 
 
One can in a similar fashion one can calculate the number density of holes, , by evaluating the
expression
 
 
where  is the Fermi function for the occupancy of holes which is the same as
the probability that an electron state at energy  is unoccupied. For , the density
of holes is given by
 
 
where
 
 
The following table gives generally accepted values of some of the quantities associated with the three
most common semiconductors at room temperature (300 K).
 
 
  
 in eV
/m /m /m 
Si 1.12 1.08 0.56
       
Ge 0.66 0.55 0.37
       
GaAs 1.4 0.04 0.48
       
 
Exercise 1
 
Derive expression (B).
 
For an intrinsic semiconductor the number of electrons in the conduction band is equal to the number of
holes in the valence band since a hole is left in the valence band only when an electron makes a
transition to the conduction band,
 
 
Using this and assuming that the effective masses of the electrons and holes are the same one gets
 
 
Page 4


 Module 6 : PHYSICS OF SEMICONDUCTOR DEVICES
 Lecture 34 : Intrinsic Semiconductors
 
Objectives
 
In this course you will learn the following
Intrinsic and extrinsic semiconductors.
Fermi level in a semiconductor.
p-type and n-type semiconductors.
Compensated semiconductors.
Charge neutrality and law of mass action.
 
Intrinsic Semiconductors
 
An intrinsic semiconductor is a pure semiconductor, i.e., a sample without any impurity. At absolute zero
it is essentially an insulator, though with a much smaller band gap. However, at any finite temperature
there are some charge carriers are thermally excited, contributing to conductivity. Semiconductors such
as silicon and germanium, which belong to Group IV of the periodic table are covalently bonded with
each atom of Si(or Ge) sharing an electron with four neighbours of the same specis. A bond picture of
silicon is shown in the figure where a silicon atom and its neighbour share a pair of electrons in covalent
bonding.
 
 
 
 
Gallium belongs to Group III and bonds with arsenic which belongs to Group V to give a III-V
semiconductor. In GaAs, the bonding is partly covalent and partly ionic. Other commonly known III-V
semiconductors are GaN, GaP, InSb etc. Like the III-V compounds, Group II elements combine with
Group VI elements to give semiconductors like CdTe, CdS, ZnS etc. Several industrially useful
semiconductors are alloys such as Al GA As.
 
The number of carriers in a band at finite temperatures is given by , where 
 is the density of state and  is the Fermi function which gives the
thermal probability. If , we may ignore the term 1 in the denominator of the Fermi
function and approximate it as
 
 Using this the density of electrons in the conduction band ( ) may be written as follows.
 
 
where we have substituted
 
 
The integral  is a gamma function  whose value is . Substituting this
value, we get for the density of electrons in the conduction band
 
 
where
 
 
One can in a similar fashion one can calculate the number density of holes, , by evaluating the
expression
 
 
where  is the Fermi function for the occupancy of holes which is the same as
the probability that an electron state at energy  is unoccupied. For , the density
of holes is given by
 
 
where
 
 
The following table gives generally accepted values of some of the quantities associated with the three
most common semiconductors at room temperature (300 K).
 
 
  
 in eV
/m /m /m 
Si 1.12 1.08 0.56
       
Ge 0.66 0.55 0.37
       
GaAs 1.4 0.04 0.48
       
 
Exercise 1
 
Derive expression (B).
 
For an intrinsic semiconductor the number of electrons in the conduction band is equal to the number of
holes in the valence band since a hole is left in the valence band only when an electron makes a
transition to the conduction band,
 
 
Using this and assuming that the effective masses of the electrons and holes are the same one gets
 
 
 
 
giving
 
 
i.e. the Fermi level lies in the middle of the forbidden gap . Note that there is no contradiction with
the fact that no state exists in the gap as  is only an energy level and not a state.
 
By substituting the above expression for Fermi energy in (A) or (B), we obtain an expression for the
number density of electrons or holes ( )
 
 where  is the width of the gap.
 
Exercise 2
 
Derive the expression (D).
 
Exercise 3
 
For a two band model of silicon, the band gap is 1.11 eV. Taking the effective masses of electrons and
holes as  and , calculate the intrinsic carrier concentration in silicon
at 300 K.
(Ans.  m .)
 
Exercise 4
Show that, if the effective masses of electrons and holes are not equal, the position of the Fermi energy
for an intrinsic semiconductor is given by
Page 5


 Module 6 : PHYSICS OF SEMICONDUCTOR DEVICES
 Lecture 34 : Intrinsic Semiconductors
 
Objectives
 
In this course you will learn the following
Intrinsic and extrinsic semiconductors.
Fermi level in a semiconductor.
p-type and n-type semiconductors.
Compensated semiconductors.
Charge neutrality and law of mass action.
 
Intrinsic Semiconductors
 
An intrinsic semiconductor is a pure semiconductor, i.e., a sample without any impurity. At absolute zero
it is essentially an insulator, though with a much smaller band gap. However, at any finite temperature
there are some charge carriers are thermally excited, contributing to conductivity. Semiconductors such
as silicon and germanium, which belong to Group IV of the periodic table are covalently bonded with
each atom of Si(or Ge) sharing an electron with four neighbours of the same specis. A bond picture of
silicon is shown in the figure where a silicon atom and its neighbour share a pair of electrons in covalent
bonding.
 
 
 
 
Gallium belongs to Group III and bonds with arsenic which belongs to Group V to give a III-V
semiconductor. In GaAs, the bonding is partly covalent and partly ionic. Other commonly known III-V
semiconductors are GaN, GaP, InSb etc. Like the III-V compounds, Group II elements combine with
Group VI elements to give semiconductors like CdTe, CdS, ZnS etc. Several industrially useful
semiconductors are alloys such as Al GA As.
 
The number of carriers in a band at finite temperatures is given by , where 
 is the density of state and  is the Fermi function which gives the
thermal probability. If , we may ignore the term 1 in the denominator of the Fermi
function and approximate it as
 
 Using this the density of electrons in the conduction band ( ) may be written as follows.
 
 
where we have substituted
 
 
The integral  is a gamma function  whose value is . Substituting this
value, we get for the density of electrons in the conduction band
 
 
where
 
 
One can in a similar fashion one can calculate the number density of holes, , by evaluating the
expression
 
 
where  is the Fermi function for the occupancy of holes which is the same as
the probability that an electron state at energy  is unoccupied. For , the density
of holes is given by
 
 
where
 
 
The following table gives generally accepted values of some of the quantities associated with the three
most common semiconductors at room temperature (300 K).
 
 
  
 in eV
/m /m /m 
Si 1.12 1.08 0.56
       
Ge 0.66 0.55 0.37
       
GaAs 1.4 0.04 0.48
       
 
Exercise 1
 
Derive expression (B).
 
For an intrinsic semiconductor the number of electrons in the conduction band is equal to the number of
holes in the valence band since a hole is left in the valence band only when an electron makes a
transition to the conduction band,
 
 
Using this and assuming that the effective masses of the electrons and holes are the same one gets
 
 
 
 
giving
 
 
i.e. the Fermi level lies in the middle of the forbidden gap . Note that there is no contradiction with
the fact that no state exists in the gap as  is only an energy level and not a state.
 
By substituting the above expression for Fermi energy in (A) or (B), we obtain an expression for the
number density of electrons or holes ( )
 
 where  is the width of the gap.
 
Exercise 2
 
Derive the expression (D).
 
Exercise 3
 
For a two band model of silicon, the band gap is 1.11 eV. Taking the effective masses of electrons and
holes as  and , calculate the intrinsic carrier concentration in silicon
at 300 K.
(Ans.  m .)
 
Exercise 4
Show that, if the effective masses of electrons and holes are not equal, the position of the Fermi energy
for an intrinsic semiconductor is given by
 
 
Current in an intrinsic semiconductor
 
For semiconductors both electrons and holes contribute to electric current. Because of their opposite
charge, their contribution to the current add up. For an intrinsic semiconductor with a single valence
band and a conduction band, the current density is given by
 
 
where  and  are respectively the electron density and speed while  and  are the hole density
and speed. Using  and  and the fact that , we have
 
 
which gives the conductivity as
 
 
Example 7
 
Estimate the electrical conductivity of intrinsic silicon at 300 K, given that the electron and hole
mobilities are  m /V-s and  m /V-s.
 Solution
 The conductivity arises due to both electrons and holes
 
 
The intrinsic carrier concentration  was calculated to be  at 300 K. Thus
 
 
 
 
Exercise 5
 
A sample of an intrinsic semiconductor has a band gap of 0.7 eV, assumed independent of temperature.
Taking  and , find the relationship between the conductivity at 200 K and
300 K.
(Ans. ratio of conductivity = 2014.6,  eV )
Extrinsic Semiconductors
An extrinsic semiconductor is formed by adding impurities, called dopants to an intrinsic semiconductor
to modify the former's electrical properties. There are two types of such impurities - those which provide
electrons as majority carriers are known as n-type and those which provide holes as majority carriers
are known as p-type .
Using this and assuming that the effective masses of the electrons and holes are the same one gets
n- type Semiconductors
Consider a matrix of silicon where the atoms are covalently bonded.
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