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

[opticaltempest]

Homework Statement

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.

Homework 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)

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

 

[haruspex]

You don't need to find n, only $\frac 1n\frac{dn}{dT}$. So try differentiating the integral you have for n wrt T.