Silicon (50 g) has been doped with 27 mg of antimony (Sb). Determine the concentration of free charge carriers (carriers/cm3) at room temperature in this material.
Calculate the absorption edge of this material. Express your answer as a wavenumber in units of inverse meters
9.1E21
wrong
a) 6.22*10^18
b)wavenumber= Eg / hc
Eg(Silicon band energy)= 1.11 eV (to joules)
h=6.6*10^-34
c=3*10^8
wavenumber= Eg*1.6*10^-19 / hc = 896969.69
To determine the concentration of free charge carriers in the doped silicon material, you need to know the number of free carriers per unit volume. This can be calculated using the following formula:
n = N / V,
where:
- n is the concentration of free charge carriers (carriers/cm³),
- N is the number of dopant atoms, and
- V is the volume of the material.
First, we need to convert the mass of silicon and antimony to moles. The molar mass of silicon (Si) is approximately 28.09 g/mol, while the molar mass of antimony (Sb) is approximately 121.76 g/mol.
Calculating the number of moles of silicon:
Number of moles of silicon (Si) = mass of silicon / molar mass of silicon
= 50 g / 28.09 g/mol
Calculating the number of moles of antimony:
Number of moles of antimony (Sb) = mass of antimony / molar mass of antimony
= 27 mg / 121.76 g/mol
Since the dopant (antimony) substitutes for silicon in the silicon lattice, the number of doping atoms is equal to the number of moles of antimony.
Next, calculate the concentration of free charge carriers (n):
n = N / V
To calculate the volume (V) of the material, we need to know its density. The density of silicon is approximately 2.33 g/cm³.
Calculating the volume of the material:
V = mass of silicon / density of silicon = 50 g / 2.33 g/cm³
Now that you have the concentration of free charge carriers (n), you can proceed to calculate the absorption edge of the material.
The absorption edge is the energy level at which photons with a lower energy cannot be absorbed by the material. It is often represented as a wavenumber in units of inverse meters (cm⁻¹).
The energy of a photon can be calculated using the formula:
E = hc / λ,
where:
- E is the energy of a photon (in joules),
- h is the Planck's constant (approximately 6.626 x 10⁻³⁴ J·s),
- c is the speed of light (approximately 3.0 x 10⁸ m/s), and
- λ is the wavelength of light (in meters).
The absorption edge occurs when the photon energy is equal to or slightly higher than the energy required to excite an electron across the band gap.
To calculate the absorption edge, we need the band gap energy of silicon.
The band gap energy of silicon is approximately 1.12 eV (electron volts).
Converting electron volts to joules:
1 eV = 1.602 x 10⁻¹⁹ J.
The absorption edge energy can be calculated using the formula:
E_edge = E_g (band gap energy) + E_photon,
where E_photon is the energy of the photon. E_photon is calculated using the formula mentioned earlier.
Finally, you can convert the energy calculated above to a wavenumber using the formula:
wavenumber (cm⁻¹) = 1 / λ (in meters) * 10⁻².
By following these steps, you can determine the concentration of free charge carriers and calculate the absorption edge of the doped silicon material.