Redshift Academy


Intrinsic and Extrinsic Semiconductors
--------------------------------------

Electron and Hole Densities
---------------------------

Since the E-k curve is parabolic near the band edges we can 
use the free electron theory to calculate the density of 
electrons near the bottom of the conduction band and holes
near the top of the valence band. 

The density of states function is given by:

g(E) = (1/2π²h³)(2m)^3/2E^1/2

The number of electrons/m³ with energies between E and E + dE 
at the bottom of the conduction band is:

n = ∫[1/2π²h³)(2m_e^*)^3/2E^1/2][1/(1 + exp(E_g/kT)]dE

Where m_e^* is effective mass of the electron.

Set the zeo energy point at E_c.  Replace E by E - E_c 

E
^
| ////////// CB
  ---------- E_c
   ^
  .|........ E_f (Fermi energy)
   | E_g
   v
  ---------- E_v
  /////////  VB

n = ∫[1/2π²h³)(2m_e^*)^3/2(E - E_c)^1/2][1/(1 + exp(E - E_f))/kT]dE

E_g for Silicon ~ 0.6eV.  If we evaluate this integral under 
the assumption that E_f is at mid gap we get such that 
exp(E - E_f)/kT >> 1 we get:

n_e = 2(m_e^*KT/2h²π)^3/2exp(-(E_c - E_f)/KT)
 
  = N_cexp(-(E_c - E_f)/KT)

Where N_c is the effective density of states in the conduction 
band.

W can regard a hole as the absence of an electron.  Therefore,
the Fermi statistic becomes:

F(E)_h = 1 - F(E)_e

By performing a similar procedure to the above we arrive at:

n_h = 2(m_h^*KT/2h²π)^3/2exp(-(E_f - E_v)/KT)
 
  = N_vexp(-(E_c - E_f)/KT)

Where N_v is the effective density of states in the valence 
band.

Intrinsic versus Extrinsic
--------------------------

In an intrinsic semiconductor n_e = n_h.  It is possible to 
increase the conductivity of Germanium or Silicon by adding 
small amounts of certain elements.  P, As or Sb have 5 valence
electrons (Group V). The impurity atom substitutes for a Ge or 
Si atom in the lattice. 4 of the electrons satisfy the covalent 
bonding requirements while the 5th is localized but easily 
removed.  The energy of this electron is just below the CB edge.  
The net result is that these electrons in the CB dominate the 
conduction process.  The impurities are refered to as DONORS
and the semiconductor is said to be n-type with electrons as
the majority carriers.

Conversely, there are certain elements that behave in the opposite
way.  B, Ga and In have 3 valence electrons (Group III).  These
atoms trap electrons from the VB in order to fullfill the covalent
bonding requirements.  In the process they leave behind holes.  
The energy of these holes is just above the VB.  The net result 
is that these holes in the VB dominate the conduction process.  
The impurities are refered to as ACCEPTORS and the semiconductor 
is said to be p-type with holes as the majority carriers.

  ////////// CB
  ---------- E_c
  ---------- E_d
 
  
  ---------- E_a 
  ---------- E_v
  /////////  VB

We can use the formulas from above to calculate the number of 
electrons/holes in at E_d and E_a respectively.

n_d  = N_dexp(-(E_d - E_f)/KT)

and

n_a  = N_aexp(-(E_f - E_a)/KT) 

Where N_d and N_a are the density of states at E_d and E_a.

The condition for charge neutrality is:

p + (N_d - n_d) = n + n_a

# of holes + # of ionized donors 
                      = # of electrons + # of ionized acceptors

Fermi Level
-----------

In an intrinsic semiconductor the Fermi level is at midgap.
In n type semiconductors the Fermi level is shifted towards
the CB reflecting the increased probability of finding electrons 
in the CB.  In p type semiconductors the Fermi level is shifted 
towards the VB reflecting the increased probability of finding 
holes in the VB. 

Hall Effect
-----------

The Hall effect can be used to measure the majority carrier 
concentration.  A current is passed through a semiconductor
and a magnetic field, B, is applied in a perpendicular fashion.
The magnetic field deflects the carriers and causes a build 
up of charge on one side.



            ............ V_H ..........
           :                        :
           :            ^ I         :
           :      ------|-------    :
           :   l/    A  |     /|    :
           :    ---------------|    :
           :...|        ^    | .....:
               |        |v   | |
               |             | |
       B      d| <- F_m   F_e -> | |      B 
 (out of page) |             | |   (out of page)
               |       w     |/ 
                -------------- 
                        ^
                        | I

Let n = number of charges/unit volume, A = area of a conductor
and d = length.  The mobile charge, Q, in d is:

Q = neAd

Now t = d/v where v is the drift velocity of the charge.  
Therefore,

I = Q/t = neAd/(d/v) = neAv or v = I/neA
                       
Now F_m = e(v x B) and F_E = eE_H = V_H/w

At equilibrium F_m = F_E.  Therefore,

V_H/w = IB/neA

V_H = IBw/neA

But A = lw.  Therefore,

V_H = IB/nel

We can now experimentally measure V_H, B, I and the sample 
dimensions to arrive at a value for n.  The sign of V_H is 
used to establish whether the sample is p or n type.