Chapter 5
Semiconductor in Equilibrium
 

 

 

 

 

 



 
 

Equilibrium Distribution of Electrons and Holes :


* The distribution of electrons in the conduction band is given by the
    density of allowed quantum states times the probability that a
    state will be occupied.

The thermal equilibrium conc. of electrons no is given by

 

* Similarly, the distribution of holes in the valence band is given by the
   density of allowed quantum states times the probability that a state will
   not be occupied by an electron.

* And the thermal equilibrium conc. Of holes po is given by

 

 



 
 
 
 
 

Equilibrium Distribution of Electrons and Holes


Density of states functions Fermi-Derac probability function   and areas representing electrons and holes concentrations for the case when Ef is near the midgab energy

 


 
 
 
 
 
 
 

 

The no and po eqs.


* Recall the thermal equilibrium conc. of electrons

*Assume that the Fermi energy is within the bandgap. For electrons in
  the conduction band, if Ec-EF >>kT, then E-EF>>kT, so the Fermi
  probability function reduces to the Boltzmann approximation,


* We may define
                          
,(at T=300K, Nc ~1019 cm-3), which is called the effective density of states
 function in the conduction band

 


 

 

Geometry


* The thermal equilibrium conc. of holes in the valence band
is given by

* For energy states in the valence band, E<Ev. If (EF-Ev)>>kT,

* Then

 

* we may define
                              
 

,(at T=300K, Nc ~1019 cm-3), which is called the effective density of states
 function in the valence  band

 

 

 

nopo product


* The product of the general expressions for no and po are given by

for a semiconductor in thermal equilibrium, the product of no and po is
always a constant for a given material and at a given temp.

* Effective Density of States Function

 
 

 

 

Intrinsic Carrier Concentration


* For an intrinsic semiconductor, the conc. of electrons in the conduction
   band, n
i , is equal to the conc. of holes in the valence band, pi.

* The Fermi energy level for the intrinsic semiconductor is called the
    intrinsic Fermi energy, EFi.
* For an intrinsic semiconductor ,


 

* For an given semiconductor at a constant temperature, the value of ni is
   constant, and independent of the Fermi energy.

 

 

 

Intrinsic Carrier Conc.


Commonly accepted values of ni
at T = 300 K

Silicon n
i = 1.5x1010 cm-3
GaAs n
i   = 1.8x106 cm-3
Germanium n
i = 1.4x1013 cm-3
 


the intrinsic carrier concentration

 

 

 

 

Intrinsic Fermi-Level Position


* For an intrinsic semiconductor, ni = pi,


 

*  Emidgap =(Ec+Ev)/2: is called the midgap energy.

*  IF mp*  = mn*  , then EFi = Emidgap (exactly in the center of the bandgap)
*  IF mp*  > mn*  , then EFi > Emidgap (above the center of the bandgap)
*  IF mp*  < mn*  , then EFi < Emidgap ( below the center of the bandgap)


 

 
 
 
Back to the beginning of Semiconductor in Equilibrium .

 

 

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
electron into the conduction band. 

 

 

 

 

Back to the beginning of Semiconductor in Equilibrium .
 

 

Extrinsic Semiconductor


● Adding donor or acceptor impurity atoms to a
semiconductor will change the distribution of electrons
and holes in the material, and therefore,
the Fermi energy position will change correspondingly.

Recall

 

 

Extrinsic Semiconductor


● When the donor impurity atoms are added,
the density of electrons is greater
than the density of holes,
 (no > po)
 --> n-type; EF > EFi

When the acceptor impurity atoms are added,
 the density of electrons is less than the density of holes,
 ( no < po)
--> p-type; EF < EFi)

 

 


 

 

 

Degenerate and Nondegenerate


● If the conc. of dopant atoms added is small compared to the density of
  the host atoms, then the impurity are far apart so that there is no
  interaction between donor electrons, for example, in an n-material.


    
-->  nondegenerate semiconductor

● If the conc. of dopant atoms added increases such that the distance
   between the impurity atoms decreases and the donor electrons begin to
   interact with each other, then the single discrete donor energy will split
   into a band of energies.
 --> EF move toward Ec

● The widen of the band of donor states may overlap the bottom of the
   conduction band. This occurs when the donor conc. becomes
   comparable with the effective density of states, EF
 Ec

  
 --> degenerate semiconductor

 

 

 

 

Degenerate and Nondegenerate


 

Simplified energy band diagram for degeneratey   doped 
(a) n- type semi-conductor   (b) p- type semi-conductor

 
Back to the beginning of Semiconductor in Equilibrium .

 

 

Statistics of Donors and Acceptors


●  The probability of electrons occupying the donor
 energy state was given by


●  where Nd is the conc. of donor atoms, nd is the density of electrons
   occupying the donor level and Ed is the energy of the donor level.
   g = 2 since each donor level has two spin orientation, thus each donor level
   has two quantum states.

●  Therefore the conc. of ionized donors Nd+ = Nd nd
    
Similarly, the conc. of ionized acceptors Na- =  Na pa, where
                                   
g = 4 for the acceptor level in Si and GaAs

 

 

Complete Ionization


●  If we assume Ed-EF >>  kT or EF-Ea >> kT ( e.g. T= 300 K), then

 




that is, the do
nor/acceptor states are almost
completely ionized and all the donor/acceptor
impurity atoms have donated an electron/hole
to the conduction/valence band.

 

 

 

Freeze-out


At T = 0K, no electrons from the donor state are thermally elevated
into the conduction band; this effect is called freeze-out.

At T = 0K, all electrons are in their lowest possible energy state; that is
for an n-type semiconductor, each donor state must contain an electron,
therefore, nd =Nd or Nd+=0, which means that the Fermi level must be
above the donor level.


   
    

 

 

 

 

Charge Neutrality


* In thermal equilibrium, the semiconductor is electrically neutral.
   The electrons distributing among the various energy states creating
   negative and positive charges, but the net charge density is zero.

* Compensated Semiconductors: is one that contains both donor and
   acceptor impurity atoms in the same region.
   A n-type compensated semiconductor occurs when Nd > Na
and a p-type semiconductor
   occurs when Na > Nd.

* The charge neutrality condition is expressed by
                                                             
* where no and po are the thermal equilibrium conc.
  of e- and h+ in the conduction band and valence band, respectively.
  Nd+ is the conc. Of positively charged
  donor states and Na - is the conc. of negatively
  charged acceptor states.

 

 

 

Compensated Semiconductor



Energy band diagram of compansated semiconductor
showing ionized and un-ionized donor and acceptor

 

 

 

Compensated Semiconductor


* If we assume complete ionization, Nd+ = Nd and Na- = Na, then


* If Na = Nd = 0, ( for the intrinsic case),  no =po

* If Nd >> Na, no = N

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  
Back to the beginning of Semiconductor in Equilibrium .
 

 

Position of Fermi Level


* The position of Fermi level is a function of the doping concentration
    and a function of temperature, EF(n, p, T).

* Assume Boltzmann approximation is valid, we have


 

 

 

 

EF(n, p, T)


 

 

 

 

 

 

position of Fermi level as a function of:
donor concentration  ( n- type )
acceptor concentration (p - type )
















position of Fermi level for an :
(a) n - type Nd > Na

(b) p - type Na > Nd

 
 

 

EF(n, p, T)


      

    

The Fermi- leveles of :
(a)  Material A in thermal Equilibrium
(b)  Material B in thermal Equilibrium
(c)  Materials A and B at the instance that they are place in contact
(d)  Materials A and B in contact at thermal Equilibrium

 
 
Back to the beginning of Semiconductor in Equilibrium .