Question

The energy gap of AlAs is \(2.3 \mathrm{eV},\) and the energy gap of \(\mathrm{AlSb}\) is 1.7 eV \([9,\) p. 19\(]\). Energy gaps of materials of composition \(A l A s_{x} S b_{1-x}\) with \(0 \leq x \leq 1\) vary approximately linearly between these values \([9, \mathrm{p} .19]\). Suppose you would like to make a semiconductor laser from a material of composition AlAs \(_{x} \mathrm{Sb}_{1-x}\). Find the value of \(x\) that specifies the composition of a material which emits light at wavelength \(\lambda=640 \mathrm{nm}\)

Short answer

For emission at 640 nm, \(x \approx 0.767\) in the composition AlAs\(_x\)Sb\(_{1-x}\).

Step-by-step solution

Convert Wavelength to Energy

The energy of a photon is related to its wavelength by the equation \(E = \frac{hc}{\lambda}\), where \(h\) is Planck's constant \(6.626 \times 10^{-34} \text{ J·s}\) and \(c\) is the speed of light \(3 \times 10^8 \text{ m/s}\). Given \(\lambda = 640 \text{ nm} = 640 \times 10^{-9} \text{ m}\), convert this wavelength to energy in electronvolts using \(1 \text{ eV} = 1.602 \times 10^{-19} \text{ J}\).

Calculation of Energy in eV

Calculate the energy using:\[E = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{640 \times 10^{-9}} \text{ J}\]Convert Joules to electronvolts:\[E = \frac{1.986 \times 10^{-19} \text{ J}}{1.602 \times 10^{-19} \text{ J/eV}} \approx 1.24 \text{ eV}\]The light emission should correspond to an energy gap of \(1.24 \text{ eV}\).

Interpolation Using Linear Variation

The energy gap of the alloy varies linearly between \(2.3 \text{ eV}\) for \(x = 1\) and \(1.7 \text{ eV}\) for \(x = 0\). Set up the linear interpolation equation:\[E_g(x) = 2.3x + 1.7(1-x)\]Find \(x\) where \(E_g(x) = 1.24 \text{ eV}\).

Solve the Linear Equation for x

Substitute the energy value \(1.24 \text{ eV}\) into the linear equation:\[1.24 = 2.3x + 1.7 - 1.7x\]Simplify and solve for \(x\):\[1.24 = 0.6x + 1.7\]\[0.6x = 1.7 - 1.24\]\[0.6x = 0.46\]\[x = \frac{0.46}{0.6} \approx 0.767\]

Conclusion

The value of \(x\) for which the material AlAs\(_x\)Sb\(_{1-x}\) emits light at a wavelength of 640 nm is approximately \(x = 0.767\).

Key concepts

Energy Gap Calculation

The energy gap, often called the bandgap, is a key property of semiconductor materials that determines their ability to conduct electricity and emit light. It represents the energy difference between the valence band, which is filled with electrons, and the conduction band, where electrons can move freely. In semiconductors like AlAs and AlSb, the energy gap is crucial to designing devices such as lasers and LEDs. For a material to emit light at a particular wavelength, the energy gap must match the energy of the photons corresponding to that wavelength. Understanding how to calculate the energy gap allows engineers and scientists to tailor semiconductor materials to specific applications. By knowing the energy gap, we can predict the electrical and optical properties of a material.

Wavelength Conversion

Wavelength conversion involves translating the wavelength of light into its corresponding photon energy. This is essential in designing semiconductor lasers, where the emitted light's wavelength is a critical parameter. Using the relationship \(E = \frac{hc}{\lambda}\), where \(E\) is the energy in Joules, \(h\) is Planck's constant, and \(c\) is the speed of light, we can convert between wavelength and energy. For practical applications, energy is often expressed in electronvolts (eV), a more convenient unit for semiconductor physics. For example, to find the energy corresponding to a 640 nm wavelength, the wavelength is converted to meters, the energy calculated using the formula, and then converted to eV. This energy should match the material's energy gap for the material to emit light at this wavelength.

Linear Interpolation

Linear interpolation is a mathematical method used to estimate unknown values by using two known values. It comes in handy when dealing with semiconductor materials, where properties often change in a linear fashion between two extremes.The energy gap for the alloy AlAs\(_{x}\)Sb\(_{1-x}\) can be calculated using linear interpolation between the energy gaps of AlAs and AlSb. Given the energy gaps of these materials, AlAs (2.3 eV) and AlSb (1.7 eV), the energy gap of the alloy can be expressed as:\[E_g(x) = 2.3x + 1.7(1-x)\]This equation allows for the determination of the alloy composition (value of \(x\)) that results in a specific energy gap, which in our case should match the photon energy corresponding to the desired emission wavelength.

Semiconductor Materials

Semiconductor materials are the building blocks of modern electronic devices. They have a unique ability to conduct electricity better than insulators but not as well as conductors. This property makes them indispensable in the electronics industry. Common semiconductor materials include silicon (Si), germanium (Ge), and compound semiconductors like Gallium Arsenide (GaAs), Aluminum Arsenide (AlAs), and Aluminum Antimonide (AlSb). Compound semiconductors are formed by combining elements from columns III and V of the periodic table, creating materials with specific electrical and optical properties. These materials are crucial for making devices such as transistors, diodes, solar cells, and lasers. By carefully controlling their composition and structure, engineers can design materials with precisely tuned energy gaps, enabling the creation of devices that operate efficiently at specified wavelengths and frequencies.