# N-carriers, number density, energy gap E_g, temperature

• Yroyathon
In summary, the conversation discusses a problem related to the number densities of n-carriers in silicon at 100K and 300K. The individual has researched and used formulas to solve the problem, but has not been able to get the correct answer. They question whether the mistake is in how the energy gap changes with temperature or if there is another factor at play. Another person suggests taking into account the hole's effective mass and double checking the calculations. Eventually, the individual discovers that the professor had changed the answers and their previous submission was actually correct. They also discuss the relationship between the effective masses and N_v and N_c. Overall, the conversation provides insight into the process of solving this type of problem and the importance of double checking calculations

#### Yroyathon

hi folks, I've gotten most of this problem but for one part. I've learned quite a bit reading about the physics involved here on the web, but for all I've learned when I apply it, it doesn't work (ie give the right answer). So, I'm missing something.

## Homework Statement

In silicon, E_g = 1.12 eV and the effective mass of the n-carriers is m* = 0.31*m_e, where m_e is the electron mass. Find the number densities of n-carriers at 100 K and at 300 K.

ans = _____ (n-carriers/cm^3)

## Homework Equations

n = N_c * e^-(E_c - E_f)/(kT)
or
n = N_c * e^-E_g/(2kT) (is this ok to use?)

N_c = 2 * ((m* * k * T)/(2*Pi*h-bar^2)) ^ (3/2)
not a typo here, it's m-star times k times T, etc.

## The Attempt at a Solution

I can get the answer at 300 K, because the energy gap E_g given works for room temperature, ie 300 K. what I can't get is the answer for when T = 100 K, where in my understanding the value of E_g is slightly larger. I've researched and found a Varshni empirical formula, three constants which are material properties specific to silicon. but this didn't end up helping, that is, my slightly larger value for E_g did not yield the correct answer.

my question is, am I right in thinking the central part of my mistake in the T = 100 K case is how E_g changes with temperature? or does it not change in this problem? Or is there another thing going on here that I'm ignoring/misinterpreting?

Tips or suggestions are very much appreciated on this one.
Thanks.

,Yroyathon

You can use 100K in that formula. Just because E_g is larger will only mean that there will be very few conduction electrons found in the system (in fact exponentially smaller than at 300K). Once the temperature becomes greater than the gap, then the carrier density shoots up.

nickjer, I guess I'm not sure I understand.

i've tried just using the above formula (for n, in terms of N_c and E_g) with T=100, and it doesn't work.

I don't doubt the possible truth of what you wrote, but I'm not sure how it explains why just using T=100 does not yield the proper result.

What result are you getting, and how do you know it is not correct?

the result I'm getting is, n=5.10299*10^(-11), but it should be 5.5*10^(-11). it's off by a little bit, and I can't explain how. I have the answer for this textbook problem, but I have to figure out How to get it.

Are you using:

$$n_i = \sqrt{N_v N_c} e^{-E_g/2kT}$$

You need to take into account the hole's effective mass as well which is slightly larger.

ok, but how do i calculate N_v without information about m*_p? the effective mass of hole. you say slightly larger, but, how do I calculate it with what's given in the problem?

You can't calculate $$m_p^*$$ based on what is given in the problem. Unless you look it up in a table somewhere. Besides, both the electron and hole effective masses are roughly the same for silicon, so you should get the roughly the same answer. Just double check your work and units.

well, you were right about them being roughly the same answer. the prof changed the answers on us, so that my previous submission (where I just used T=100 and the effective mass of the n-carrier) that had been marked incorrect is now correct.

thanks!

Good to hear that.

does that mean that ((m_n*m_p)/me^2) = (0.31)^2 since we know that m_n & m_p are almost equal to each other?

I'm not sure but possibly.

I am more certain that Sqrt[N_v * N_c] is approximately N_c.

## 1. What are N-carriers?

N-carriers, also known as charge carriers, are particles that carry an electric charge, either positive or negative, in a material. They are responsible for the flow of electric current in a material.

## 2. What is number density?

Number density refers to the concentration of N-carriers in a material, measured as the number of carriers per unit volume. It is an important factor in determining the electrical properties of a material.

## 3. How is energy gap E_g related to N-carriers?

Energy gap E_g is a measure of the energy difference between the valence band and the conduction band in a material. It determines the number of N-carriers that can be excited from the valence band to the conduction band, and therefore affects the electrical conductivity of the material.

## 4. How does temperature affect N-carriers?

At higher temperatures, the number of N-carriers increases due to thermal excitation. This leads to an increase in the electrical conductivity of the material.

## 5. How are N-carriers and energy gap E_g related to the band structure of a material?

The band structure of a material determines the energy levels available for N-carriers and the energy gap E_g between the valence band and the conduction band. This band structure directly affects the electrical properties of the material, such as its conductivity and resistivity.