8.29 Sspd_chapter 6_part 9_appendix device physics  (Page 4/8)

Therefore in a classical BJT the dc current gain is as given below:

But as lateral scaling and vertical scaling have scaled down in the process of evolution from SSI to MSI to LSI, base width have been scaled down from 1000nm to 100nm and now they are being scaled down to 10nm. Emitter Bulk Width (W _E ) have scaled down from 2 microns to a fraction of micron. Hence diffusion length of holes [ L _P =√(D _P τ _P ) ] in Emitter has become comparable to Emitter Bulk Width and carrier concentration profile is akin to that of a narrow base diode. This means two things:

1. In the Expression II, Base Recombination Current becomes negligible compared to hole diffusion current in Emitter;
2. Hole Diffusion current becomes a function of surface recombination velocity at the emitter contact.

At a metal Emitter Contact surface recombination velocity is very high. Hence dc current gain drastically deteriorates.

In Equation XVI , diffusion length is replaced by emitter bulk width and base recombination current is neglected.

Where

Therefore

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. For homogenous devices we assume Band-gap to be uniform everywhere hence

E _gE = E _gB = 1.12 eV.

Therefore β _F = 1.6×10 ⁴ ;

As is evident from Equation XVIII, 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 VII. 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 Figure VIII.

Heavy doping of emitter causes Band Gap Narrowing in Emitter as shown in Figure VIII.

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

There is Band Gap Narrowing in Emitter and Band Gap in Base remains E _g0 therefore we get the following expressions

Substituting Equation XIX in Equation XVIII we get the DC current gain as:

Therefore

The empirical relationship for BGN is:

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

Table I. ∆E g vs N D .

Table II. N Deff vs N D .