GaAs and GaP make solid solutions that have the same crystal structure as the parent materials, with As and P randomly distributed throughout the crystal. GaPxAs1 - x exists for any value of x. If we assume that the band gap varies linearly with composition between x = 0 and x = 1, estimate the band gap for GaP0.5As0.5. (GaAs and GaP band gaps are 1.43 eV and 2.26 eV, respectively.) What wavelength of light does this correspond to?

Verified step by step guidance

Identify the given band gaps for GaAs and GaP: GaAs has a band gap of 1.43 eV and GaP has a band gap of 2.26 eV.

Recognize that the problem states the band gap varies linearly with composition. This means we can use a linear interpolation formula to find the band gap for GaP_{0.5}As_{0.5}.

Apply the linear interpolation formula: E_{gap}(GaP_{0.5}As_{0.5}) = (1-x) * E_{gap}(GaAs) + x * E_{gap}(GaP), where x = 0.5.

Substitute the known values into the formula: E_{gap}(GaP_{0.5}As_{0.5}) = (1-0.5) * 1.43 eV + 0.5 * 2.26 eV.

To find the corresponding wavelength of light, use the equation \( \lambda = \frac{hc}{E} \), where \( h \) is Planck's constant, \( c \) is the speed of light, and \( E \) is the energy in electron volts.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Band Gap Energy

The band gap energy is the energy difference between the top of the valence band and the bottom of the conduction band in a semiconductor. It determines the electrical conductivity and optical properties of the material. In this context, GaAs and GaP have specific band gap energies (1.43 eV and 2.26 eV, respectively), which influence the behavior of the solid solution GaPxAs1-x.

Linear Interpolation

Linear interpolation is a mathematical method used to estimate unknown values that fall within a range defined by known values. In this case, it is applied to estimate the band gap of GaP0.5As0.5 by taking the average of the band gaps of GaAs and GaP, weighted by their respective compositions. This approach assumes a linear relationship between composition and band gap energy.

Wavelength of Light

The wavelength of light is inversely related to its energy, described by the equation E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. By calculating the band gap energy for GaP0.5As0.5, one can determine the corresponding wavelength of light emitted or absorbed by the material, which is crucial for applications in optoelectronics.