How Is Maximum Resistivity Calculated from Intrinsic Resistivity in Silicon?

[roeb]

Homework Statement

Assume the mobility ratio u_n/u_p = b in Si is a constant independent of impurity concentration. Find the maximum resistivity $$\rho_m$$ in terms of the intrinsic resistivity $$\rho_i$$. If b = 3 and the hole mobility of intrinsic Si is 450 cm^2/Vs calculate $$\rho_i$$ and $$\rho_m$$

Homework Equations

The Attempt at a Solution

Well I know that conductivity is $$\sigma$$ = q(u_n * n + u_p * p)
For the intrinsic conductivity $$\sigma_i$$ = q(u_p *b * n_i + u_p * n_i) = q u_p n_i (b + 1).

Now I'm having a bit of trouble expressing $$\sigma_m$$ where this is a minimum conductivity (or maximum resistivity)...
$$\sigma_m$$ = q ( b * u_p * n + u_p * p)
To get a minimum conductivity I would think that n should be extremely low but I fail to see how I can express this in terms of the intrinsic conductivity. I know that n_i at 300 K is 10^10/cm^3. Does anyone have any hints?

 

[Jhero]

First, let's define some variables:
- u_n: electron mobility
- u_p: hole mobility
- b: mobility ratio
- n: electron concentration
- p: hole concentration
- n_i: intrinsic carrier concentration
- \rho_i: intrinsic resistivity
- \rho_m: maximum resistivity

To find the maximum resistivity, we need to find the minimum conductivity. This occurs when n and p are at their minimum values, which is when the material is intrinsic (no impurities).

So, for the minimum conductivity, we have n = n_i and p = n_i. Substituting this into the equation for conductivity, we get:

\sigma_m = q ( b * u_p * n_i + u_p * n_i)
= q u_p n_i (b + 1)

Now, we know that for intrinsic Si, the hole mobility is 450 cm^2/Vs and the intrinsic carrier concentration is 10^10/cm^3. So, substituting these values into the equation above, we get:

\sigma_m = (1.602 * 10^-19 C) * (450 cm^2/Vs) * (10^10/cm^3) * (3 + 1)
= 2.41 * 10^-4 (1/Ohm * cm)

Since \rho_m is the reciprocal of conductivity, we have:

\rho_m = (1 / 2.41 * 10^-4) Ohm * cm = 4.15 * 10^3 Ohm * cm

So, the maximum resistivity in terms of the intrinsic resistivity is:

\rho_m = 4.15 * 10^3 * \rho_i