# Variation in a BJT's forward current gain compared to variation in saturation current

(Questions summarized at the bottom of the post.) For discrete BJT circuits, a biasing scheme like the one below is often described which helps reduce the sensitivity of $$\I_C\$$ to variations in the BJT's $$\\beta\$$ and the divider resistances $$\R_{1,2}\$$. (This figure is from Razavi's Fundamentals of Microelectronics, 2nd edition, p. 186).

Analysis of the above circuit shows that if the voltage divider carries sufficient current compared to the base current, the influence of the relation $$\I_C=\beta I_B\$$ is reduced compared to $$\I_C=I_S(e^{V_{BE}/V_T}-1)\$$, and so changes in $$\\beta\$$ are not as significant as they would be if, for example, the biasing omitted $$\R_2, R_E\$$.

However, I'm wondering about the practical variations observed in $$\\beta\$$ compared to those in $$\I_S\$$, which basic device physics seems to suggest are highly related parameters. It seems that the same physical processes which lead to variations in $$\\beta\$$ would also lead to similar variation in $$\I_S\$$, which would then translate directly to variations in $$\I_C\$$ since $$\I_C \propto I_S \$$. From Chenming Hu's book Modern Semiconductor Devices for Integrated Circuits (publicly available online), we have that

$$\beta = \frac{D_B W_E N_E n_{iB}^2}{D_E W_B N_B n_{iE}^2}$$

$$I_S = A_E q \frac{D_B}{W_B} \frac{n_{iB}^2}{N_B}$$

(Here $$\D\$$ is the diffusion constant in cm^2/s in the base or emitter; $$\W\$$ is the width in cm of the base or emitter; $$\N\$$ is the doping concentration in the base or emitter in 1/cm^3; $$\n_i\$$ is the intrinsic carrier concentration in the base or emitter in 1/cm^3.) On a surface level it seems that variations in $$\\beta\$$ are likely to directly translate to variations in $$\I_S\$$.

To summarize, is it the case that in real BJTs, $$\\beta\$$ tends to vary much more than $$\I_S\$$? Is there information available on the statistical distribution in the variation of both parameters in some example devices? (I haven't been able to find info comparing the two.) Is there a physical reason for why one would vary much more than the other? If they both vary, does the biasing strategy above actually reduce variations in $$\I_S\$$ as well?

Is is a device characteristic, so the the circuit it is connected in will have no influence on it.

You can get some idea of Is variation from Vbe variation- as it shows up in a log term a small variation of Vbe represents a large variation in Is. A couple transistors from the same lot at the same temperature might typically match within 5 or 10mV, on the same die within 1mV or better. Just rough back of envelope numbers. Beta might match within 10-20% for transistors from the same lot. Not sure about adjacent parts on the same die.

Is is not that directly useful in transistor calculations. Reflecting that lack of utility is the fact it's not specified on any transistor or diode datasheet I can recall. Of course, it's part of SPICE models.

However, I'm wondering about the practical variations observed in $$\\beta\$$ compared to those in $$\I_S\$$, which basic device physics seems to suggest are highly related parameters. It seems that the same physical processes which lead to variations in $$\\beta\$$ would also lead to similar variation in $$\I_S\$$

$$\beta = \frac{D_B W_E N_E n_{iB}^2}{D_E W_B N_B n_{iE}^2}$$
$$I_S = A_E q \frac{D_B}{W_B} \frac{n_{iB}^2}{N_B}$$
The ratio between $$\\beta\$$ and $$\I_S\$$ is given by:
$$\frac{\beta}{I_S} = \frac{\frac{D_B W_E N_E n_{iB}^2}{D_E W_B N_B n_{iE}^2}}{A_E q \frac{D_B}{W_B} \frac{n_{iB}^2}{N_B}} =\frac{W_E N_E }{D_E n_{iE}^2 A_E q}$$
From this it follows that changing any one of the multiple emitter parameters on the right hand side of the above equation will change the ratio between $$\\beta\$$ and $$\I_S\$$.