1.13 50 years journey of ic technology_appendix iv_band gap narrowing

Appendix IV

Band Gap Narrowing(BGN) and its effect on Emitter Injection Efficiency.

In homojunction:

To obtain high gain, Emitter needs to be heavily doped and we need thin Base Region.

Due to heavy doping in Emitter, degeneracy is introduced which leads to Band Gap Narrowing(BGN) by ∆E _g . The radius of the fifth electron or the donor electron orbiting donor atom is 13×10 ^-8 cm = 13Angstrom = 13Aº.

Figure IV.1.Calculation of critical packing density where donor electrons wave function of donor atoms start to overlap.

N _crit = 1/V _d = 1/(2r _d ) ³ = 1.7×10 ¹⁹ /cc.

When the doping level approaches this critical level, energy level corresponding to donor atoms no more remain DISCRETE but they become a continuum in effect extending the conduction band and narrowing the Forbidden Energy Band Gap which is known as Band Gap Narrowing(BGN). This is a degenerate semiconductor and the Fermi-level lies at the edge of the conduction band or within the conduction band as it is in metal. Hence a degenerate semi-conductor is semi-metal. The energy band diagram of degenerate and non-degenerate semiconductor is given in Fig.IV.2.

Figure IV.2. The Energy Band Diagram of Pure, doped and heavily doped semiconductor.

IV.1

IV.2

With BGN the intrinsic carrier concentration increases

Substituting Eq.IV.2 in Eq.IV.1 we get

IV.3

Therefore

IV.4

The empirical relationship for BGN is:

Where E _ref = 0.009eV and N _ref = 10 ¹⁷ /cc for typical cases.

Table IV.1. ∆E _g vs N _D .

Table IV.2. N _Deff vs N _D .

For a doping level of N _D = 10 ²⁰ /cc, ∆E _g =0.12eV.

This gives an effective doping level of N _Deff = 10 ¹⁸ /cc.

The classical formula for current gain without BGN is:

If we take the following data: (N _D ) _E = 10 ²⁰ /cc, (N _A ) _B = 10 ¹⁷ /cc, W _E = W _B = 1µm and D _pE =1.25 (cm) ² /sec and D _nB =20 (cm) ² /sec then β _F = 1.6×10 ⁴ ;

But taking into account of BGN, β _F = 160.