# Find the temperature coefficient of resistivity for pure silicon at T=300K

1. May 6, 2008

### opticaltempest

1. The problem statement, all variables and given/known data

I need to find the temperature coefficient of resistivity $$\alpha$$ for silicon at the temperature of $$T=300$$ Kelvin. I am supposed to assume that $$\tau$$, the mean time between collisions of charge carriers, is independent of temperature.

2. Relevant equations

Temperature Coefficient of Resistivity
The temperature coefficient of resistivity $$\alpha$$ is the fractional change in resistivity per unit change in temperature. It is given below.
$$\alpha = \frac{1}{\rho} \cdot \frac{d \rho}{dT}$$, where $$\rho$$ is the resistivity of the material, and $$T$$ is the temperature in Kelvin.

Resistivity
The resistivity of a material $$\rho$$ is
$$\rho = \frac{m}{e^2 n \tau}$$, where $$m$$ is the electron mass, $$e$$ is the fundamental charge, $$n$$ is the number of charge carriers per unit volume, and $$\tau$$ is the mean time between collisions of the charge carriers.

Occupancy Probability
The occupancy probability $$P(E)$$ - the probability that an available level at energy $$E$$ is occupied by an electron is

$$P(E) = \frac{1}{e^{\frac{E-E_F}{kT}}+1}$$,

where $$E_F$$ is the Fermi energy and $$k$$ is Boltzmann's constant.

Density of States
The density of states at energy level $$E$$ is
$$N(E)\frac{8 \sqrt{2} \pi m^{3/2}}{h^3}E^{1/2}$$,

where $$h$$ is Planck's constant.

Density of Occupied States
The density of occupied states $$N_o(E)$$ is given by

$$N_o=N(E) \cdot P(E)$$

3. The attempt at a solution

I have been working on this problem for 4 hours with various approaches. I will list one of the more simple approaches below.

$$\alpha = \frac{e^2 n \tau}{m} \cdot \frac{d}{dT}\bigg[\frac{m}{e^2 n \tau}\bigg]$$

Since, we are assuming $$\tau$$ is independent of temperature, we can simplify the above equation to

$$\alpha = n \cdot \frac{d}{dT}\bigg[\frac{1}{n}\bigg]$$

I'm not quite sure how to calculate the number of charge carriers per unit volume $$n$$ for pure silicon at 300 Kelvin. I am assuming the expression for $$n$$ will depend on the temperature.

Can I find an expression for n by integration $$N_{o}(E)$$ over some range of energy levels?

By book states:

Suppose we add up (via integration) the number of occupied states per unit volume at T=0K for all energies between $$E=0$$ and $$E=E_F$$. The result must equal $$n$$, the number of conduction electrons per unit volume for the metal.

They then list the corresponding integral below the paragraph:

$$n = \int_{0}^{E_F}{N_o(E)dE}$$

The various other approaches I tried for finding n always ended up with me getting stuck with formulas involving electron mobility and effective mass - topics I have not yet covered. I'm hoping to stay away from more complicated calculations that involve the carrier mobility and effective mass if it's possible.

Thanks