At what temperature will ni=10^14/cm^3 in silicon? And ni=10^16/cm^3?
ni is the intrinsic carrier (electron and hole) concentration. You can find the ni(T)equation at this website:
http://pvcdrom.pveducation.org/SEMICON/NI.HTM
It is based on experimental data obtained up to 375 K. You may have to extrapolate to higher temperatures to get ni = 10^16
The site that I referenced has a calculation tool that says that
ni = 10^14 cm^-3 at T = 470 K , and
ni = 10^16 at T = 645 K.
Just be aware that you are extrapolating beyond the experimental data base.
To find the temperature at which the intrinsic carrier concentration (ni) equals a certain value in silicon, we need to use a formula called the intrinsic carrier equation. The intrinsic carrier equation relates ni to the temperature (T) using the following expression:
ni = A * T^3/2 * exp(-Eg / (2 * k * T))
where:
- ni is the intrinsic carrier concentration,
- A is a material-dependent constant,
- T is the temperature in Kelvin,
- Eg is the bandgap energy of silicon (1.12 eV),
- k is the Boltzmann constant.
Let's start by solving for the temperature at which ni equals 10^14/cm^3.
1. Rearrange the equation to solve for T:
T = (2 * k * ni / (A * exp(-Eg / (2 * k * T))))^(2/3)
2. Plug in the given values:
ni = 10^14/cm^3
3. Convert ni to SI units (m^-3):
ni_SI = ni * (1 cm / 100)^3 = 10^14 * (1/100)^3 m^-3
4. Use the value of ni_SI in the equation:
T = (2 * k * ni_SI / (A * exp(-Eg / (2 * k * T))))^(2/3)
5. Iterate using numerical methods to solve for T. In this case, there isn't a straightforward equation to solve it explicitly, so you can use numerical methods like the Newton-Raphson method or use a scientific computing software like MATLAB or Python to find the solution.
Repeat the same process for ni = 10^16/cm^3 to find the temperature at which ni equals that value.
Note: The intrinsic carrier concentration in silicon can vary depending on the impurity doping levels, so these values are approximations for undoped or lightly doped silicon.