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opticaltempest
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Homework Statement
I need to find the temperature coefficient of resistivity [tex]\alpha[/tex] for silicon at the temperature of [tex]T=300[/tex] Kelvin. I am supposed to assume that [tex]\tau[/tex], the mean time between collisions of charge carriers, is independent of temperature.
Homework Equations
Temperature Coefficient of Resistivity
The temperature coefficient of resistivity [tex]\alpha[/tex] is the fractional change in resistivity per unit change in temperature. It is given below.
[tex]\alpha = \frac{1}{\rho} \cdot \frac{d \rho}{dT}[/tex], where [tex]\rho[/tex] is the resistivity of the material, and [tex]T[/tex] is the temperature in Kelvin.
Resistivity
The resistivity of a material [tex]\rho[/tex] is
[tex]\rho = \frac{m}{e^2 n \tau}[/tex], where [tex]m[/tex] is the electron mass, [tex]e[/tex] is the fundamental charge, [tex]n[/tex] is the number of charge carriers per unit volume, and [tex]\tau[/tex] is the mean time between collisions of the charge carriers.
Occupancy Probability
The occupancy probability [tex]P(E)[/tex] - the probability that an available level at energy [tex]E[/tex] is occupied by an electron is
[tex]P(E) = \frac{1}{e^{\frac{E-E_F}{kT}}+1}[/tex],
where [tex]E_F[/tex] is the Fermi energy and [tex]k[/tex] is Boltzmann's constant.
Density of States
The density of states at energy level [tex]E[/tex] is
[tex]N(E)\frac{8 \sqrt{2} \pi m^{3/2}}{h^3}E^{1/2}[/tex],
where [tex]h[/tex] is Planck's constant.
Density of Occupied States
The density of occupied states [tex]N_o(E)[/tex] is given by
[tex]N_o=N(E) \cdot P(E)[/tex]
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.
[tex]\alpha = \frac{e^2 n \tau}{m} \cdot \frac{d}{dT}\bigg[\frac{m}{e^2 n \tau}\bigg] [/tex]
Since, we are assuming [tex]\tau[/tex] is independent of temperature, we can simplify the above equation to
[tex]\alpha = n \cdot \frac{d}{dT}\bigg[\frac{1}{n}\bigg][/tex]
I'm not quite sure how to calculate the number of charge carriers per unit volume [tex]n[/tex] for pure silicon at 300 Kelvin. I am assuming the expression for [tex]n[/tex] will depend on the temperature.
Can I find an expression for n by integration [tex]N_{o}(E)[/tex] 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 [tex]E=0[/tex] and [tex]E=E_F[/tex]. The result must equal [tex]n[/tex], the number of conduction electrons per unit volume for the metal.
They then list the corresponding integral below the paragraph:
[tex]n = \int_{0}^{E_F}{N_o(E)dE}[/tex]
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