Question

GaAs and GaP make solid solutions that have the same crystal structure as the parent materials, with \(\mathrm{As}\) and \(\mathrm{P}\) randomly distributed throughout the crystal. \(\mathrm{GaP}_{x} \mathrm{~A} s_{1-x}\) 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 \(\mathrm{GaP}_{0.5} \mathrm{As}_{05}\). (See the previous two exercises for GaAs and GaP band gaps.) What wavelength of light does this correspond to?

Short answer

The band gap for \(\mathrm{GaP}_{0.5} \mathrm{As}_{0.5}\) is \(1.84\, \mathrm{eV}\), and the corresponding wavelength of light is approximately \(673.9\,\mathrm{nm}\).

Step-by-step solution

1. Recall the band gap of GaAs and GaP

According to the previous exercises, the band gaps of \(\mathrm{GaAs}\) and \(\mathrm{GaP}\) are: \(E_g (\mathrm{GaAs}) = 1.42\,\mathrm{eV}\) \(E_g (\mathrm{GaP}) = 2.26\,\mathrm{eV}\)

2. Find the band gap of GaP\(_{0.5}\)As\(_{0.5}\) using the linearity rule

We are given the information that the band gap varies linearly with composition. Let's denote the composition as \(x\), where \(x=0\) corresponds to GaAs and \(x=1\) corresponds to GaP. We can write the linear relation for the band gap as follows: \(E_g (x) = E_g (\mathrm{GaAs}) + x \cdot (E_g (\mathrm{GaP}) - E_g (\mathrm{GaAs}))\) Now we can substitute \(x=0.5\) to find the band gap of \(\mathrm{GaP}_{0.5} \mathrm{As}_{0.5}\): \(E_g (0.5) = 1.42\, \mathrm{eV} + 0.5 \cdot (2.26\,\mathrm{eV} - 1.42\,\mathrm{eV})\) \(E_g (0.5) = 1.42\,\mathrm{eV} + 0.5 \cdot 0.84\,\mathrm{eV}\) \(E_g (0.5) = 1.42\,\mathrm{eV} + 0.42\,\mathrm{eV}\) \(E_g (0.5) = 1.84\, \mathrm{eV}\) Therefore, the band gap of \(\mathrm{GaP}_{0.5} \mathrm{As}_{0.5}\) is \(1.84\, \mathrm{eV}\).

3. Calculate the wavelength of light corresponding to the band gap

To find the corresponding wavelength of light, we will use the energy-wavelength relation: \(E_g = \frac{hc}{\lambda}\) Rearranging for \(\lambda\) gives us: \(\lambda = \frac{hc}{E_g}\) where: \(h\) is Planck's constant (\(6.626\cdot 10^{-34}\,\mathrm{Js}\)), \(c\) is the speed of light (\(3.0\cdot 10^8\,\mathrm{m/s}\)), \(E_g\) is the band gap energy found in step 2 (in Joules). First, we need to convert the band gap energy to Joules. \(1\, \mathrm{eV} = 1.602\cdot 10^{-19}\,\mathrm{J}\), so: \(E_g (0.5) = 1.84\,\mathrm{eV} \cdot 1.602\cdot 10^{-19}\,\mathrm{J/eV} = 2.950\cdot 10^{-19}\,\mathrm{J}\) Now we can calculate the wavelength: \(\lambda = \frac{6.626\cdot 10^{-34}\,\mathrm{Js} \cdot 3.0\cdot 10^8\,\mathrm{m/s}}{2.950\cdot 10^{-19}\,\mathrm{J}}\) \(\lambda = 6.739\cdot 10^{-7}\,\mathrm{m}\) \(\lambda = 673.9\,\mathrm{nm}\) Hence, the wavelength of light corresponding to the band gap of \(\mathrm{GaP}_{0.5} \mathrm{As}_{0.5}\) is approximately \(673.9\,\mathrm{nm}\).

Key concepts

Understanding Solid Solutions

Solid solutions are mixtures of two or more elements that form a single continuous crystal structure. They are very important in materials science because they can combine properties of different elements to create materials with unique characteristics. This is exactly what happens when you create a solid solution from GaAs and GaP; you mix arsenic (As) and phosphorus (P) throughout a crystal lattice of gallium (Ga), the same structure found in the parent materials. This allows the fabrication of semiconductors with specific properties tailored for various applications, such as in light-emitting diodes (LEDs) and laser diodes.

In our case, GaAs and GaP create a solid solution where composition can change continuously from one endmember to the other, denoted as GaP_xAs_1-x for a value of x between 0 and 1. Such ability to form solid solutions is crucial in the semiconductor industry because it permits gradual tuning of material properties, such as the band gap.

Deciphering Crystal Structure

The crystal structure describes the orderly arrangement of atoms in a solid. GaAs and GaP, which are used to form the solid solution we're discussing, each have a specific crystal structure known as the zinc blende (or sphalerite) structure. It is a type of cubic crystal system characterized by a lattice where each gallium atom is surrounded by four arsenic or phosphorus atoms at the corners of a tetrahedron, and vice versa.

The regularity of such structures determines the electronic properties of the material, including the band gap. For semiconductors, the uniformity of the crystal structure is key because it ensures consistent behavior of the electrons that move through the lattice, which in turn affects how the semiconductor will perform in an electronic device.

Linear Variation of Band Gap

The concept of linear variation of band gap in solid solutions is vital to material engineering. This idea posits that the band gap of a solid solution can be predicted based on the concentrations of its components. The band gap is the energy difference between the highest valence band and the lowest conduction band in a semiconductor, which determines what wavelength of light the material can absorb or emit.

According to Vegard's Law, the linear relation suggests that the band gap of the mixed material (GaP_xAs_1-x) will change uniformly between the known band gaps of the pure components (GaAs and GaP). When x equals 0.5, implying an equal mixture of GaAs and GaP, the band gap will be halfway between the band gaps of GaAs and GaP. This predictable change allows scientists to design semiconductors for specific purposes by carefully choosing the ratio of materials in their solid solution.

Relationship Between Energy and Wavelength

In the realm of semiconductors and, indeed, physics in general, the energy-wavelength relation holds great importance. This relationship is captured by the equation E = hc/λ, where E stands for the energy of a photon, h is Planck's constant, c is the speed of light, and λ is the wavelength. By understanding this equation, one can calculate the wavelength of light that corresponds to a particular band gap energy of a semiconductor.

This relationship is utilized when designing optical devices like LEDs, where you want to create light of a specific color, which corresponds to a specific wavelength. By adjusting the semiconductor's band gap through solid solutions, as with GaP_xAs_1-x, the emitted light’s wavelength can be precisely controlled to meet the required specifications of the application.