Chapter
5
Semiconductor in Equilibrium 



Equilibrium Distribution of Electrons and Holes : 



*
The distribution of electrons in the conduction
band is given by the
The thermal equilibrium conc. of electrons n_{o} is given by
*
Similarly, the distribution of holes in the valence band is given by the * And the thermal equilibrium conc. Of holes p_{o} is given by 




Equilibrium Distribution of Electrons and Holes  


Density of states functions FermiDerac probability function and areas representing electrons and holes concentrations for the case when E_{f} is near the midgab energy 



The n_{o} and p_{o} eqs. 



* Recall the thermal equilibrium conc. of electrons *Assume
that the Fermi energy is within the bandgap. For electrons in
*
We may define




Geometry 



*
The thermal equilibrium conc. of holes in the valence band
* For energy states in the valence band, E<E_{v}. If (E_{F}E_{v})>>kT, * Then
*
we may define ,(at
T=300K, N_{c} ~10^{19} cm^{3}), which is called the
effective density of states 



n_{o}p_{o} product 



* The product of the general expressions for n_{o} and p_{o} are given by for
a semiconductor in thermal equilibrium, the product of n_{o} and p_{o}
is * Effective Density of States Function 


Intrinsic Carrier Concentration 



*
For an intrinsic semiconductor, the conc. of electrons in the conduction
*
For an given semiconductor at a constant temperature, the value of n_{i}
is 



Intrinsic Carrier Conc. 



Commonly
accepted values of n_{i}





Intrinsic FermiLevel Position 



* For an intrinsic semiconductor, n_{i} = p_{i},
*
E_{midgap} =(E_{c}+E_{v})/2: is called the midgap
energy.


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Dopant and Energy Levels 






Two dimensional representation of silicon lattice doped with phosphorus  The
energy band diagram showing (a) discrete donor energy state (b) The effect of donor state being ionized 



Acceptors and Energy Levels 




Ionization Energy 



●
Ionization energy is the energy required to elevate the donor




 
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Extrinsic Semiconductor 



●
Adding donor or acceptor impurity atoms to a Recall 



Extrinsic Semiconductor 



●
When the donor impurity atoms are added, When the
acceptor impurity atoms are added,






Degenerate and Nondegenerate 



●
If the conc. of dopant atoms added is small compared to the density of ●
If the conc. of dopant atoms added increases such that the distance ●
The widen of the band of donor states may overlap the bottom of the 



Degenerate and Nondegenerate 



Simplified
energy band diagram for degeneratey doped 

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Statistics of Donors and Acceptors 



●
The probability of electrons occupying the donor
●
where N_{d} is the conc. of donor atoms, n_{d} is the density
of electrons ●
Therefore the conc. of ionized donors N_{d}^{+} = N_{d}
– n_{d
}Similarly, the conc. of ionized acceptors N_{a}^{}
= N_{a} – p_{a}, where




Complete Ionization 



● If we assume E_{d}E_{F }>> kT or E_{F}E_{a} >> kT ( e.g. T= 300 K), then 





Freezeout 



At
T = 0K, no electrons from the donor state are thermally elevated At
T = 0K, all electrons are in their lowest possible energy state; that is 



Charge Neutrality 



*
In thermal equilibrium, the semiconductor is electrically neutral. * Compensated
Semiconductors: is one that contains both donor and *
The charge neutrality condition is expressed by 



Compensated Semiconductor 







Compensated Semiconductor 



* If we assume complete ionization, N_{d}^{+} = N_{d} and N_{a}^{} = N_{a}, then
* If N_{a} = N_{d} = 0, ( for the intrinsic case), n_{o} =p_{o} * If N_{d} >> N_{a}, n_{o} = N_{d } is used to calculate the conc. of holes in valence band




Compensated Semiconductor 



Electron concentration versus temperature showing the 3 regions Partial Ionization Extrinsic intrinsic


Energy band diagram showing the redistribution of electrons when donor are added  
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Position of Fermi Level 



*
The position of Fermi level is a function of the doping concentration * Assume Boltzmann approximation is valid, we have




E_{F}(n, p, T) 



position
of Fermi level as a function of: 


E_{F}(n, p, T) 



The Fermi leveles of :


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