Question

GaAs and GaP \(\left(E_{\mathrm{g}}=2.26 \mathrm{eV}\right)\) make solid solutions that have the same crystal structure as the parent materials, with As and Prandomly distributed throughout the crystal. \(\mathrm{GaP}_{x}\) 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 \(\mathrm{GaP}_{0.5} \mathrm{As}_{0.5} .\) What wavelength of light does this correspond to?

Short answer

The estimated band gap for GaP\(_{0.5}\)As\(_{0.5}\) is \(1.845 \ \mathrm{eV}\), and this corresponds to a light wavelength of approximately \(675 \ nm\).

Step-by-step solution

Linear interpolation of the bandgap energy

To estimate the band gap energy for the mixture of \(\mathrm{GaP}_{0.5} \mathrm{As}_{0.5},\) we can use linear interpolation. We are given that the variation of bandgap energy with composition is linear. Therefore, the energy change in the bandgap can be written as: \(E(x) = E_{\mathrm{GaAs}} + (E_{\mathrm{GaP}} - E_{\mathrm{GaAs}})x\) For this exercise, we only have the bandgap energy value for GaP (\(E_{\mathrm{GaP}}=2.26 \mathrm{eV}\)). In order to find the value of \(E_{\mathrm{GaAs}}\) we can use the data from any reference table or textbook. Commonly, the bandgap energy for GaAs is around \(1.43 eV\). Now we can estimate the bandgap of the GaP\(_{0.5}\)As\(_{0.5}\) mixture:

Estimate the bandgap energy

Replace the given values and \(x=0.5\) in the formula: \(E(0.5) = 1.43 + (2.26 - 1.43)(0.5)\) \(E(0.5) = 1.43 + (0.83)(0.5)\) \(E(0.5) = 1.43 + 0.415\) \(E(0.5) = 1.845 \mathrm{eV}\) The estimated bandgap for GaP\(_{0.5}\)As\(_{0.5}\) is approximately \(1.845 \mathrm{eV}\).

Calculate the corresponding wavelength

To find the corresponding wavelength of light, we can use the energy-wavelength relation: \(\lambda = \frac{hc}{E}\) Where \(h = 4.135 \times 10^{-15} eV \cdot s\) is the Planck's constant, \(c = 3 \times 10^8 m/s\) is the speed of light, and \(E\) is the energy in electron volts. Plug the values and calculate the wavelength: \(\lambda = \frac{(4.135 \times 10^{-15} eV \cdot s)(3 \times 10^8 m/s)}{1.845eV}\) \(\lambda = 6.75 \times 10^{-7} m\) The corresponding wavelength of light is approximately \(675 \ nm\). In conclusion, the estimated band gap for GaP\(_{0.5}\)As\(_{0.5}\) is \(1.845 \ \mathrm{eV}\), and this corresponds to a light wavelength of approximately \(675 \ nm\).

Key concepts

Linear Interpolation in Chemistry

Linear interpolation is a method used to estimate intermediate values between two known values on a line or curve. In the context of chemistry, specifically when analyzing solid solutions such as semiconductors, linear interpolation can prove invaluable.

For instance, when we deal with mixtures of two substances, like GaAs and GaP, that form solid solutions, understanding how the properties - such as band gap energy - change with composition is crucial. The assumption that the property varies linearly with composition simplifies calculations; it allows us to use the linear interpolation formula. By taking the band gap energies of the pure substances and the proportion of the mixture, we can estimate the band gap energy of the intermediate composition.

Practically, if you know the end points, you can draw a straight line between them and pick the value which falls on that line for your given mixture ratio. This simplicity makes linear interpolation a vital tool in materials science, where it's often necessary to predict the properties of new material combinations quickly and without extensive experimentation.

Semiconductor Bandgap Energy

Understanding the concept of semiconductor bandgap energy is pivotal in the field of materials science and electronic engineering. The bandgap represents the energy difference between the top of the valence band and the bottom of the conduction band in a semiconductor.

This energy gap is essential because it determines a semiconductor's electrical conductivity and its response to various wavelengths of light. Essentially, electrons must overcome this gap to move from being bound within an atom to free-flowing and capable of carrying a charge. In semiconductors like GaAs and GaP, the bandgap energy changes depending on the materials' composition, which affects their electronic and optical properties.

The ability to tailor the bandgap through methods such as solid solution formation, allows for the creation of semiconductors with specific properties for different applications, from solar cells to lasers and LEDs. By estimating the bandgap energy, as done through linear interpolation, engineers and scientists can design materials that respond to the precise wavelengths of light necessary for their applications.

Wavelength-Energy Relation

The relationship between the wavelength of light and its energy is fundamental in physics and chemistry, governed by the equation \( \lambda = \frac{hc}{E} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant, \( c \) is the speed of light, and \( E \) is the energy of a photon.

The equation demonstrates that the energy of light is inversely proportional to its wavelength: shorter wavelengths correspond to higher energy photons and vice versa. This wavelength-energy relation is particularly significant when dealing with semiconductors. The bandgap energy of a semiconductor directly determines the wavelength of light that the material can absorb or emit.

For example, by calculating the bandgap energy of a semiconductor, you can deduce the specific wavelength of light that will be absorbed or released when an electron transitions across that gap. This calculation is crucial not only in understanding the fundamental aspects of light-matter interaction but also in designing optical devices, such as photodetectors and photovoltaic cells, which depend on such characteristics for their operation.