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How Germanium Actually Makes Your Transistor Better. . .
by Nick Smears, STMicroelectronics

This section attempts to explain, using a few equations, the principal effects of germanium and what those effects mean for basic transistor performance.

The best way to understand the effect of Germanium when one puts it into a transistor is to start by looking at the compromises that people designing classical bipolar junction transistors (BJT's) are forced into.

Let's first look the gain of the transistor, being the most fundamental characteristic. As we know this is given by:

Gain, which, to first order is

Where N_E is the emitter dopant concentration and N_B is that of the base dopant.

Putting that to one side for later, we switch to the next characteristic, the cut-off frequency or f_T . This is defined as the frequency at which the current gain decays to unity or 0dB. The physical parts of an npn transistor can be simplified to the diagram below, with a flow of electrons (e-) as shown

Now it turns out, not too surprisingly, that the f_T is a function of the length of time these electrons take to pass through the various parts of the transistor. The actual equation is:

Where

• The τ terms are the times taken to transit the emitter, base and base-collector depletion regions.
• The C terms are the capacitances of the emitter-base junction and collector-substrate junctions
• k is Boltzmann's constant, T the temperature, q the charge on an electron
• the R terms are the real physical resistances of the emitter and collector. This is not the resistivity of the semiconductor layer but the actual effective resistance that the section in question presents that is, it is also a function of geometry and operating conditions (which we'll get to shortly).

Before we go further and get too focused on the f_T as the only measure of frequency performance, we should look at the parameter f_MAX, which is given by

This is defined as the frequency where the power gain falls to unity or 0dB. This is a good measure of in-circuit performance because it includes two very important parasitic elements, the base resistance R_b and the collector junction capacitance, C_JC

In real life, as the numbers work out, the dominant first order factor is the τ_B, the base transit time, which is itself a function of the base width W_B. So, from all of this we see quite easily that we choose a narrow base, which has a low resistance.

Before we get too confident and go further, we should look at this base resistance term. Though this isn't simply the resistivity of the base layer, it does depend heavily on it. In fact, the relationship is roughly:

(W_B and N_B we have seen before)

This might seem puzzling until one realizes that this resistance term is NOT the series resistance across the base (vertically) but the effective resistance the base presents in the transistor as circuit. It is mainly a lateral element between the active bit of the base (under the emitter) and the base contact, which is to one side. At this point, we should remember that this base resistance is one of the primary factors in determining the noise figure.

What about the collector and its doping level? Well, the term τ_BC, which is the transit time of the electrons in the base-collector depletion region depends on where N_C is the dopant concentration in the collector.

So let's increase the collector doping. But if we do another effect comes into play - modulation of the base width. The higher the ratio N_C/N_B the more the base-collector depletion region eats into the base under the typical reverse bias between base and collector. This in turn has an impact on the collector current - the Early effect.

So now we find ourselves in the frustrating situation where:

• We increase base doping to improve the noise factor but this degrades the gain.
• We narrow the base to improve the f_T and this degrades the noise factor and the f_MAX.
• We increase collector doping to improve the f_T and we reduce the Early voltage and, if we go far enough, degrade f_MAX.

The third point is interesting because it is saying that f_MAX does not always follow f_T, indeed it can be difficult to improve f_T and f_MAX at the same time. If this seems surprising in the light of the fact that these two seem to track, one needs to recall that the process designers have been playing with all the other variables and not just collector doping; it's question of where the curves cross.

In short, we must trade one thing against another. Indeed, this problem is increasingly evident with recent generations of BJT with their f_T's in the 20-40GHz region and Early voltages (V_A) as low as 20V (a level once considered too low).

So how does Germanium help? As said in the main article, its direct effect is to permit a very large improvement in the gain for given conditions of doping and geometry. To see how this happens, we have to look into the structure of the conduction and valence bands

Turning a classical BJT transistor on its side and simplifying, we can draw the structure of the conduction and valence bands as in Figure 6.

Figure 6: Simplified bipolar transistor cross section
and structure of bands
Click for entire image (24KB)

The energy bandgap (E_G) between conduction and valence bands in silicon is 1.1eV. Now that of germanium is 0.7eV. Quite simply, the addition of germanium reduces the band gap (Figure 7).

Figure 7

Up to concentrations of about 30%, the reduction in band gap is given by the relation known as Vegard's law :

where [x] is the germanium concentration.

So with say 10% Germanium (not far off typical) , we arrive at the bandgap reduction of ~0.07eV which seems hardly anything until we consider the relationship which relates collector current to bandgap :

Compare the equations for with and without Ge and we get :

Plugging in the numbers, we find a 10% addition of Ge multiplies the collector current by 30. However the base current has not changed because it is not affected by this phenomenon so the gain leaps by factor of 30. This is the principal effect of the germanium but not the only one.

By grading the concentration of germanium in the base (Figure 8), increasing it from emitter toward collector, we increase the at the collector end. This has two nice effects:

• Of creating a pseudo field which accelerates the electrons across the base and therefore increasing f_T and f_MAX
• Of reducing the effect of collector bias on base narrowing; classical BJT's of f_T's in the 25GHz often have Early voltages around 20 whereas SiGe HBT's with higher f_T's reach values well over 60.

Figure 8

We can now sacrifice this gain increase:

• We make the base doping higher and this improves the noise factor
• We have a narrow base which gives us a high f_T by reducing transit time
• The higher doped base then gives us a high f_MAX

We take advantage of the accelerating field which further reduces the transit time, allowing us a slightly wider base and thus giving as a further improvement in f_MAX and noise figure

As mentioned earlier, the gain increase (and what it can be traded for) is not the only benefit. The bandgap structure can be arranged such that the charge storage effects that occur at base-collector and base-emitter interfaces are reduced. This is thought to be behind the improved linearity seen with SiGe technologies. Finally the bandgap reduction also gives a large reduction of the Early effect.

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