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SSPD_Chapter_2.2.6. Theoretical
Formulation of thermal equilibrium
values of majority and minority
carriers in Semi-conductors.
∗
Bijay_Kumar Sharma
This work is produced by OpenStax-CNX and licensed under the
†
Creative Commons Attribution License 3.0
Abstract
SSPD_Chapter 2.2.6 gives the theoretical formula for determining the majority and minority carrier
density.
2.2.6. Theoretical Formulation of thermal equilibrium values of majority and minority
carriers in Semi-conductors.
The broad view of Silicon(intrinsic or doped) is that there is thermal generation of EHP(electron-hole
pair), EHP are recombining and there is the ionization of the dopent.At any given temperature we have a
thermal equilibrium established. At this thermal equilibrium, the concentrations achieved are time invariant
and are referred to as thermal equilibrium concentrations.
In intrinsic Silicon: electron thermal equilibrium concentration = hole thermal equilibrium concentration
= intrinsic concentration.
Figure 1
The bar indicates thermal equilibrium value.
In N-Type Silicon we have:
∗
†
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Figure 2
In P-Type Silicon we have:
Figure 3
By Law of Mass Action in each case:
Figure 4
2.2.6.1. Majority and Minority Carrrier thermal equilibrium concentrations as a function
of Temperature in Kelvin.
In intrinsic Silicon, thermal equilibrium concentrations of electrons and holes are equal and are called
intrinsic concentration. In Figure 2.2.26. the Intrinsic Concentration of Silicon as a function of Temperature(K) is given. It should be noted that conducting electron and holes in intrinsic Si is only due to thermal
generation of EHP. Hence below 155K(liquid Nitrogen temperature), thermal generation stops and intrinsic
Si becomes a perfect insulator.
In Figure 2.2.29. Fermi-Dirac distribution is given at T=155K and at T=300K. At T= 300K, Fermi-Dirac
distribution is more skewed hence the tail end of the distribution has a greater distribution as compared to
that at 155K.
At 300K, ni = 4×109 and P(EC ) = 4×10-10 ;
At 155K, ni = 6.55 and P(EC ) = 6.55×10-19 ;
For all practical purposes, thermal generation of EHP has altogether stopped and the sample freezes out.
It becomes an insulator.
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Figure 5
In Extrinsic Silicon, below 155K, there is no thermal generation of EHP as well as there is no ionization
of the dopent. Hence there are no charge carriers whatsoever in Si below 155K.
From 155K to 200K, there is thermal generation of EHP but there is no ionization because the thermal
lattice vibration does not provide enough energy for ionization. Above 200K the thermal energy is kT=17meV
whereas ionization energy is only 10meV hence practically all dopent atoms get ionized and majority and
minority carriers remain constant inspite of the fact that EHP thermal generation is exponentially increasing.
This increase is felt only above 473K when intrinsic carrier concentration starts dominating the dopent
concentration.
In extrinsic Si, dopent concentration is in the range 1012 /cc to 1015 /cc whereas the intrinsic concentration
at 473K and above is above 1013 /cc.
Well above 473K depending on the dopent concentration, the sample becomes intrinsic type.
The thermal equilibrium values of majority and minority carriers in N-Type Silicon with a donor doping
density of 1013 /cc as a function of temperature is given in Figure 2.2.30.
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Figure 6
There are four distinct regions of Doped Silicon:
Freeze out region (0K to 155K) - neither there is EHP thermal generation nor there is the ionization of
the donor atoms.
Intrinsic Semiconductor (155K to 200K) - thermal generation of EHP starts but there is no ionization of
Donor Atoms.
Extrinsic Semiconductor (200K to 473K) Ionization of Donor atoms is 100% hence majority carriers
thermal equilibrium density is decided by Donor Density hence it is constant at 1013 /cc. Thermal generation
of EHP is taking place but it is many orders of magnitude less than the majority carrier hence from 200K
to 473K resistivity of the sample is relatively constant.
Intrinsic Carrier Concentration dominates over the doping density(473K and beyond) at 473K , thermal
generation of EHP becomes comparable to doping density. At 600K it becomes one order of magnitude
greater than the doping density. So in this region it again behaves as Intrinsic Semiconductor.
2.2.6.2.Theoretical Formulation of Majority Carriers and Minority Carriers.
Because of Pauli-Exclusion Principle, we have a term called Density of States N(E). This essentially
means the number of electrons which can be accommodated per unit volume per unit eV in permissible
energy states.
In the Appendix of this Chapter I will show that:
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Figure 7
(2.2.6.2.1)
N(E)dE=number of energy states per m3 between energy level E and E+dE.
Therefore actual number of electrons present in conduction band between energy level E and E+dE are
= P(E)N(E)dE
Where P(E) = 1/[1+Exp[(E-EF )/kT];
But at T = 0K, P(E)= 1 upto E = EF hence
actual number of electrons present in conduction band between energy level E and E+dE are = (1)N(E)dE
In Copper we know that electrons are lled up to the Fermi-Level and that the Fermi-level is at 7.05eV
as shown in Figure 2.2.31.
Figure 8
Hence number
electrons in the conduction band from the bottom of the conduction band
R of conducting
√
to Fermi-level= (1)K EdE,{E,0,7.05×1.6×10-19 }
Here K is a constant = 1.06216771234466×1056 .
Using Mathematica:
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Figure 9
(2.2.6.2.2)
Conduction Band conducting electron density = n= 8.5×1028 /m3 =8.5×1022 /cc.
This also implies that Copper atom packing density in Copper metal is 8.5×1028 /m3 =8.5×1022 /cc.
You can see normally the packing density is of this order i.e.1022 /cc.
Exactly the same procedure is applied to determine the majority carrier thermal equilibrium density.
In N-Type Silicon:
R
Number of electrons actually present in the conduction band at 300K= P(E)N(E)dE .
This is a denite integral between the lower limit EC and the upper limit innity as shown below:
Figure 10
(2.2.6.2.3)
It will be shown in the Appendix of this Section that this integral comes out to be:
Electron density= nn = NC Exp[-(EC -EF )/(kT)]_____________(2.2.6.2.4)
Where NC =2[(2π me *kT)/h2 ]3/2 =2.82×1025 /m3 = 2.82×1019 /cc.
Here me *= eective mass of the electron at the bottom of the conduction band.
For non-degenerate semiconductors where EF is several kT below EC , in such cases
Fermi-Dirac Statistics(P(E) = 1/[1+Exp[(E-EF )/kT]) approximates Boltzmann-Maxwell Statistics namely
P(E) = Exp[-(E-EF )/kT].
At E = EC this reduces to P(EC )= Exp[-(EC -EF )/kT].
Hence in Equation 2.2.6.2.4, if Exp[-(EC -EF )/kT] is the probability of occupancy at the lower edge of
the conduction band and nn is the conduction electron density in conduction band then NC is the eective
density of states at EC .
We follow the same procedure for holes concentration calculation:
R
Number of holes actually present in the valence band at 300K= (1-P(E))N(E)dE
This is a denite integral between the lower limit EV and the upper limit (- innity) as shown below:
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Figure 11
(2.2.6.2.5)
As will be shown in the Appendix this reduces to:
Hole density= pn = NV Exp[-(EF -EV )/(kT)] _____________(2.2.6.2.6)
Where NV =2[(2π mh *kT)/h2 ]3/2 =1.83×1025 /m3 = 1.83×1019 /cc.
Since Exp[-(EF -EV )/(kT)] = probability of non-occupancy at EV = probability of occurrence of hole at
the upper edge of the valenvce band hence NV = is the eective density of states at the upper edge of the
valence band.
NV = NC this identity does not hold since the eective masses are not the same as seen in Table 2.2.6.1.
By Law of Mass Action: nn × pn =ni 2 =NC NV Exp[- EG /(kT)] where EG =EC -EV ;
Hence
√ intrinsic carrier concentration =
ni = (NC NV ) ×Exp[- EG /(2kT)]_________________________(2.2.6.2.7)
In Table 2.2.5.1. a comparative study of the band-gap, eective mass of electron and hole, eective
density of states and intrinsic carrier concentration is made for Ge,Si and GaAs.By the inspection of Table
2.2.5.1, it is evident that density of states are in the range of 1019 per cc for all three semi-conductors..
Table 2.2.6.1. Band-gap, eective mass , eective densiy of states and intrinsic carrier
concentration at 300K.
All parameters at 300K
EG (eV)
(eecteive mass)me * (×me )
(eecteive mass)mh * (×me )
NC (per cc)
NV (per cc)
ni (per cc)
Ge
Si
GaAs
0.66
1.12
1.424
0.56
1.08
0.067
0.29
0.81
0.47
1.05×1019
3.92×1018
1.83×1013
Table 1
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2.82×1019
1.83×1019
8.81×109
4.37×1017
8.68×1018
2.03×106