Question

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?

Short answer

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.

Step-by-step solution

Calculate GaP\(_{0.5}\)As\(_{0.5}\) band gap

As it is assumed that the band gap varies linearly with composition between \(x=0\) and \(x=1\), we can use the following equation to calculate the band gap of GaP\(_{0.5}\)As\(_{0.5}\): $$ \text{Band Gap}_{\text{GaP}_{0.5}\text{As}_{0.5}} = (1-x) \times \text{Band Gap}_{\text{GaAs}} + x \times \text{Band Gap}_{\text{GaP}} $$ where \(x=0.5\) since we are considering GaP\(_{0.5}\)As\(_{0.5}\). Replacing the given values of the band gaps for GaAs and GaP: $$ \text{Band Gap}_{\text{GaP}_{0.5}\text{As}_{0.5}} = (1-0.5) \times 1.43\,\text{eV} + 0.5 \times 2.26\,\text{eV} $$ By calculating the above expression, we will obtain the band gap of GaP\(_{0.5}\)As\(_{0.5}\).

Find the corresponding wavelength of light

We will now calculate the wavelength of light corresponding to the obtained band gap using the energy-wavelength relationship: $$ E = \dfrac{hc}{\lambda} $$ where \(E\) is the energy (band gap), \(h\) is the Planck constant (\(6.626\times10^{-34}\,\text{J s}\)), \(c\) is the speed of light (\(3\times10^{8}\,\text{m/s}\)), and \(\lambda\) is the wavelength of light. We will rearrange the equation to solve for \(\lambda\): $$ \lambda = \dfrac{hc}{E} $$ First, we need to convert the band gap energy from eV to Joules. The conversion factor is \(1.602\times10^{-19}\,\text{J/eV}\). Now, we can substitute the Planck constant, the speed of light, and the energy (in Joules) into the equation and calculate the corresponding wavelength of light.

Key concepts

GaAs

Gallium Arsenide (GaAs) is a compound semiconductor primarily used in electronics and optoelectronics due to its direct band gap and high electron mobility. This property makes GaAs an excellent material for solar cells, laser diodes, and other high-speed electronics. Different from silicon, GaAs can efficiently emit and absorb light, which explains its prevalence in devices requiring efficient optical performance. Understanding GaAs is crucial when studying band gap values, as it serves as a benchmark for GaAs-based solid solutions.

GaP

Gallium Phosphide (GaP) is another semiconductor with unique properties, primarily used to produce LED lights and other optoelectronic devices. Unlike GaAs, GaP possesses an indirect band gap, which makes it less efficient for light emission. However, its robustness and thermal stability make GaP ideal for high-power and high-temperature applications. By understanding the differences in band gaps between GaP and GaAs, students can better grasp how properties vary within semiconductor solid solutions.

Solid Solutions

Solid solutions involve mixing different elements at an atomic level to create a mixture that retains a uniform crystal structure. In the case of GaP extsubscript{0.5}As extsubscript{0.5}, the arsenic and phosphorus atoms randomly distribute over the lattice sites, maintaining the zinc-blende crystal structure. Solid solutions allow for the tuning of properties such as band gap by altering the composition, making them versatile for optimizing performance in various applications. This concept is essential for students to understand the adjustment of semiconductor properties without altering the crystal structure.

Crystal Structure

The crystal structure of mixed semiconductor alloys like GaAs and GaP plays a crucial role in determining their properties. Both materials share the zinc-blende structure, a common arrangement for many semiconductors. This structure consists of a cubic lattice with each gallium atom coordinating with four arsenic or phosphorus atoms. Maintaining the crystal structure while creating solid solutions ensures that the fundamental electronic properties remain intact, allowing for predictable modifications in properties like the band gap.

Energy-Wavelength Relationship

The energy-wavelength relationship is a fundamental concept in physics, critical for understanding how a material's band gap determines its optical properties. This relationship is described by the formula:

• \( E = \dfrac{hc}{\lambda} \)

where \( E \) is the energy in joules, \( h \) is the Planck constant, and \( \lambda \) is the wavelength in meters. Knowing that the speed of light is a constant, this relationship allows conversion of energy values into corresponding wavelengths, facilitating the analysis of material properties in optoelectronics. By mastering this relationship, students can predict how different band gaps influence light absorption or emission, crucial for designing effective semiconductor devices.