Variation of Fermi Level in Intrinsic and Extrinsic Semiconductors

Variation of Fermi level with temperature and carrier concentration in intrinsic and extrinsic semiconductors Consider an intrinsic semi-conductor in thermal equilibrium at temperature T K. Let n be the number density of electron-hole pairs in the semiconductor. For the simplicity let us assume that (i) all conduction electrons have energy equal to EC and all valence electrons have energy equal to EV (Fig.1) (ii) the electrons in conduction band behaves as free with an effective mass m*e

Fig.1

The electron concentration(number of electrons per unit volume) in the conduction band in thermal equilibrium is given by

 2  me* k T  ne=2   h2  

3/ 2

e

( E F  EC ) kT

-----------------------------(1):

Similarly, for hole concentration nh we assume that (i) the holes near the top of valence band behave as if they are free particles with an effective mass mh*. Then the density of holes in the valence band is given by

 2  m h* k T  nh=2   h2  

3/ 2

e

( EV  E F ) kT

--------------------------------------(2)

For an intrinsic semiconductor ne=nh = ni (say) where ni is the density of electron –hole pairs in an intrinsic semiconductor is Multiplying eqn (1) and (2), we get

 2  kT  ne nh =4  (me* mh* ) 3 / 2 e 2   h  3

( EV  EC ) kT

2

3

=A T e



Eg kT

---------------------(3)

where Eg = (EC –EV) is the width of forbidden energy gap between conduction and valence bands.

32  3 k 3 * * 3 / 2  2 k  A2 =4  2  (me* mh* ) 3 / 2  (me mh ) -----------------------------(4) h6  h  3

Equation (3) shows that the product of hole and electron densities at thermal equilibrium is independent of the position of Fermi level and depends on the forbidden energy gap Eg and on the absolute temperature.



ni = (nenh)1/2 =A T3/2 e  2  kT  = 2 2   h 

Eg 2kT

-----------------------------------------------------------------------------(5)

3/ 2 * e

* 3/ 4 h

(m m )



e

Eg 2k T

----------------------------------------------------(6)

The experimental value of A is 9.64x1015

As ne = nh Equating (1) and (2), we get

 2  me* k T  2  h2  

e e e

( E F  EC ) kT ( EV  E F ) kT

2 E F  EC  EV kT

3/ 2

e

 m h*  = *   me 

( E F  EC ) kT

 2  m h* k T  =2   2  

3/ 2

e

( EV  E F ) kT

3/ 2

 m*  =  h*   me 

3/ 2

Taking log on both the sides, we get 2 EF  EC  EV 3  log e kT 2

EF 

mh* me*

 m*  EC  EV 3  k T log e  h*  --------------------------------------------------(7) 2 4  me 

This equation gives the Fermi level and its variation with temperature for intrinsic semiconductor. Case 1: At T=0K

EF 

EC  EV 2

Case 2: When me* = mh* then E F 

EC  EV 2

Case 3: When me* is not equal to mh* and T≠0K then Ef is slightly raised towards conduction band. Strictly speaking the effective mass of electron me* is different than that of a hole mh* but to a good approximation me and mh* may be assumed to be equal and so eqn. (7)

EF 

EC  EV 2

-------------------------------------------------------------------------(8)

This shows that in an intrinsic semi-conductor, the Fermi-level essentially lies at the centre of the forbidden energy gap Eg.

Energy Band Diagram and Fermi-Level in N-type semiconductor The impurity introduces new energy levels into the energy band picture. The location of this new level, donor level is slightly below the bottom of the conduction band. At 0 K all allowed energy levels in the valence band are filled by electrons. All donor levels are filled by unbound electrons. The conduction band is free. So charge carriers do not exist, and the semiconductor behaves as an insulator. At 0 K the Fermi level is between the donor levels and the bottom of the conduction band as shown below. The energy required to move an electron from a donor impurity level into the conduction band is of the order of 0.01eV and since at ambient temperature the thermal energy is considered to be about 0.02eV, it is concluded that almost all the electrons are detached from the donor atoms and have conduction band energies. If we assume that all the donor atoms are ionised, the donor electrons will occupy the states near the bottom of the conduction band. Density of electrons in the conduction band will be approximately equal to the density of donor atoms i.e., ne ≈ Nd where Nd is the density of donor atoms

 2  me* k T  Nd = ne= 2   h2   = NC e

( E F  EC ) kT

Nc e Nd

3/ 2

e

( E F  EC ) kT

[from eqn. (1)]

 2  me* k T   where NC=2  h2  

( E F  EC ) kT

3/ 2

Taking logarithms on both the sides

ln

Nc ( E  EC )  F Nd kT

 N  E F  EC  k T log e  C   Nd  This shows that in N-type semiconductor the Fermi-level lies below the bottom of the conduction band. As temperature rises, the Fermi level goes on falling below EC. As temperature is sufficiently raised, the electrons and holes generated due to thermal agitation increase significantly and at a stage intrinsic become fully dominant over the extrinsic carriers. Then the Fermi level approaches the middle of forbidden energy gap.

Energy Band Diagram and Fermi-Level in P-type semiconductor The energy band diagram of a P-type semiconductor is shown in Fig. The acceptor level is shown by EA near the top of the valence band, the Fermi level by EF. At absolute zero, all the holes are in acceptor levels, but as the temperature rises, the electrons from valence band jump into acceptor level on the absorption of energy (EA-EV) by each electron. As a result, these electrons are trapped in the acceptor levels and an equal number of holes are created in the valence band. These holes provide conduction currents. At the room temperature, almost all acceptor atoms trap electrons and thus the number of holes available in the valence band is almost equal to the number of impurity atoms added. In a P-type crystal the concentration of holes in the valence band is more than the concentration of electrons in the conduction band, therefore the Fermi level is shifted from the middle position of forbidden gap towards the upper edge of the valence band by an amount of ΔE´. If we assume that only acceptor atoms are present and all are ionised, we have  2  mh* k T  n h = Na = 2   h2  

3/ 2

e

( EV  E F ) kT

 NV e

( EV  E F ) kT

 2  mh* k T  where NV = 2   h2    NV e Na

3/ 2

( EV  E F ) kT

Taking logarithms on both the sides

ln

NV (E  EF )  V Na kT

 N  E F  EV  k T log e  V   Na  This shows that Fermi level lies above the top of the valence band. The position of Fermi level depends upon the temperature and the number of impurity atoms. As the temperature is sufficiently increased, electrons from the valence band are excited to the conduction band and finally the P-type crystal will start behaving like an intrinsic semi-conductor when the number of electrons in the conduction band will be nearly equal to the valence holes. Thus at extremely high temperatures the Fermi level shifts towards the middle of forbidden energy gap. Effect of Temperature If we increase the temperature of an n-type semiconductor. Since all the donors have already donated their free electrons at room temperature, the additional thermal energy will only increase the generation of electron-hole pairs. A temperature is ultimately reached when the number of covalent bonds broken is very large such that the number of holes and electrons is almost equal. The extrinsic semiconductor then behaves like an intrinsic semiconductor, although its conductivity is higher. This critical temperature is 850 C for germanium and 200C for silicon. The same arrangement can be put forward for the p-type semiconductor. Thus with an increase in the temperature of an extrinsic semiconductor, it behaves almost intrinsically.