GaAs and GaP make solid solutions that have the same crystal structure as the parent materials, with As and Prandomly distributed throughout the crystal. GaP As \(_{1-x}\) 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 GaP \(_{0.5} \mathrm{As}_{0.5}\) . (GaAs and GaP band gaps are 1.43 \(\mathrm{eV}\) and 2.26 \(\mathrm{eV}\) , respectively.) What wavelength of light does this correspond to?

The band gap of GaP\(_{0.5}\)As\(_{0.5}\) can be calculated as: $$ \text{Band Gap}_{\text{GaP}_{0.5}\text{As}_{0.5}} = (1-0.5) \times 1.43\,\text{eV} + 0.5 \times 2.26\,\text{eV} = 1.845\,\text{eV} $$ To find the corresponding wavelength of light, we first convert the energy to Joules: $$ E = 1.845\,\text{eV} \times 1.602\times10^{-19}\,\text{J/eV} = 2.958\times10^{-19}\,\text{J} $$ Now, using the energy-wavelength relationship, we can calculate the wavelength: $$ \lambda = \dfrac{hc}{E} = \dfrac{6.626\times10^{-34}\,\text{J s} \times 3\times10^{8}\,\text{m/s}}{2.958\times10^{-19}\,\text{J}} = 6.714\times10^{-7}\,\text{m} $$ Therefore, the band gap of GaP\(_{0.5}\)As\(_{0.5}\) is approximately 1.845 eV, and the corresponding wavelength of light is approximately \(6.714\times10^{-7}\) m, or 671.4 nm.