Elementary question about current and temperature in doped Si.

Hi!

I'm solving an excercise from Razavi's book (Fundamentals of microelectronics).

The problem is:
"A n-type piece of silicon with a lenght of 0.1 um and a cross area section of 0.05 um x 0.05 um sustains a voltage difference of 1V.
Doping level is 10^17 cm^-3.
(a) Calculate the total current flowing through the device at T = 300K."

I calculated from equation:

Jtot = un * E * n * q + up * E * p * q
from ni^2 = n*p I found p = 1,17 * 10^3 cm^-3, so part up* E * p * q is neglible

Jtot ≈ un * E * n * q = 5,4 uA

Then, there is:
"(b) Repeat (a) for T = 400K assuming for simplicity that mobility does not change with temperature. (This is not a goog assumption)"

But whats the difference? for T = 400K, p = 1,38 * 10^8 so its still neglible when n = 10^17

Probably im doing something wrong - but what? Please, give me some clues

If mobility is assumed to be independent of temperature - the current should be independent of temperature as well.

Your expression for the total current should include a factor of cross-sectional area (it seems you multiplied by the current density by the area - but your formula does not show that).

Also, 1V across 0.1 um give a large electric field - 100 kV/cm - so the mobility should be much lower due to this high longitudinal electric field, as compared to a low-field case.