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50 years Journey of IC Technology_Appendix IV_Band Gap Narrowing and its effect on Emitter Injection Efficiency

Module by: Bijay_Kumar Sharma. E-mail the author

Summary: Appendix IV describes the effect of band gap narrowing in degenerate semiconductor.

Appendix IV

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

In homojunction:

Figure 1
Figure 1 (graphics1.png)
Figure 2
Figure 2 (graphics2.png)

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

Figure 3
Figure 3 (graphics3.png)

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

Figure 4
Figure 4 (Picture 1.png)

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

Ncrit = 1/Vd= 1/(2rd)3 = 1.7×1019/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 5
Figure 5 (Picture 2.png)

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

Figure 6
Figure 6 (graphics4.png)

IV.1

Figure 7
Figure 7 (graphics5.png)

Figure 8
Figure 8 (graphics6.png)
Figure 9
Figure 9 (graphics7.png)

IV.2

With BGN the intrinsic carrier concentration increases

Figure 10
Figure 10 (graphics8.png)

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

Figure 11
Figure 11 (graphics9.png)

IV.3

Therefore

Figure 12
Figure 12 (graphics10.png)

IV.4

The empirical relationship for BGN is:

Figure 13
Figure 13 (graphics11.png)

Where Eref = 0.009eV and Nref = 1017/cc for typical cases.

Table IV.1. ∆Eg vs ND .

Table 1
Emitter Doping DensityND (number per cc) BGN∆Eg (eV)
1020 0.12466
1019 0.083
1018 0.0424
1017 6.36×10-3
1016 1meV

Table IV.2. NDeff vs ND.

Table 2
ND (number per cc) ∆Eg (eV) NDeff (number per cc)
1020 0.12466 8.258×1017
1019 0.083 4×1017
1018 0.0424 1.96×1017
1017 6.36×10-3 8×1016
1016 1meV 1×1016

For a doping level of ND = 1020/cc, ∆Eg=0.12eV.

This gives an effective doping level of NDeff = 1018/cc.

The classical formula for current gain without BGN is:

Figure 14
Figure 14 (graphics12.png)

If we take the following data: (ND)E = 1020/cc, (NA)B = 1017/cc, WE = WB = 1µm and DpE=1.25 (cm)2/sec and DnB=20 (cm)2/sec then βF = 1.6×104 ;

But taking into account of BGN, βF = 160.

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