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

In summary, the problem involves finding the temperature coefficient of resistivity for silicon at T=300 Kelvin, assuming that the mean time between collisions of charge carriers is independent of temperature. The key equations needed are the temperature coefficient of resistivity, resistivity, occupancy probability, density of states, and density of occupied states. Different approaches were attempted to find n, the number of charge carriers per unit volume, but the simplest approach involves differentiating the integral for n.
  • #1
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
 
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  • #2
You don't need to find n, only ##\frac 1n\frac{dn}{dT}##. So try differentiating the integral you have for n wrt T.
 

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

1. What is the temperature coefficient of resistivity?

The temperature coefficient of resistivity is a measure of how much the resistance of a material changes with temperature. It is expressed as a percentage change in resistance per degree Celsius (or Kelvin) of temperature change.

2. How is the temperature coefficient of resistivity calculated?

The temperature coefficient of resistivity is calculated by taking the change in resistivity (∆𝜌) and dividing it by the initial resistivity (𝜌₀) and the change in temperature (∆T). This formula can be expressed as 𝛼 = (∆𝜌/𝜌₀)/∆T.

3. Why is it important to know the temperature coefficient of resistivity for pure silicon at T=300K?

The temperature coefficient of resistivity is important because it allows us to predict how the resistance of a material will change with temperature. In the case of pure silicon, it is a crucial factor in understanding and designing electronic devices such as transistors and diodes.

4. How does the temperature coefficient of resistivity affect the performance of electronic devices?

The temperature coefficient of resistivity can have a significant impact on the performance of electronic devices. As the temperature changes, the resistance of the material also changes, which can affect the functionality and efficiency of the device. It is important to consider the temperature coefficient of resistivity when designing and using electronic devices to ensure optimal performance.

5. How does the temperature coefficient of resistivity vary for different materials?

The temperature coefficient of resistivity can vary greatly among different materials. Some materials, like metals, have a positive temperature coefficient, meaning their resistance increases with temperature. Other materials, like semiconductors, have a negative temperature coefficient, meaning their resistance decreases with temperature. The magnitude of the temperature coefficient can also vary depending on the composition and purity of the material.

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