Problem 80 - Chemistry: The Central Science in SI Units, Global Edition

Question

Orange light-emitting diodes are made from GaAs and GaP solid solutions, \(\mathrm{GaP}_{x} \mathrm{As}_{1-x}\) (see Exercise 12.79). The original orange LEDs emitted light with a wavelength of \(590 \mathrm{nm}\). If we assume that the band gap varies linearly with composition between \(x=0\) and \(x=1\), estimate the composition (the value of \(x\) ) that is used in these LEDs.

Step-by-Step Solution

Answer

The composition used in orange LEDs made from GaAs and GaP solid solutions is approximately \(x \approx 0.62\). This means that the GaP solid solution contains 62% GaP and 38% GaAs.

Step 1: Convert the wavelength to energy using Planck's formula

Planck's formula states that the energy E of a photon can be expressed in terms of its wavelength \(\lambda\) and Planck's constant \(h\): \(E = \dfrac{h c}{\lambda}\) Here, \(c\) is the speed of light. Given that the wavelength of the emitted light is \(590\) nm, let's find the energy E. \(E = \dfrac{6.626 \times 10^{-34} Js \times 3 \times 10^8 m/s}{590 \times 10^{-9} m}\) \(E = 3.374 \times 10^{-19} J\)

Step 2: Determine the linear relation between the band gap and composition

To find the composition of the GaP solid solution, we will use the fact that the band gap varies linearly with composition between \(x = 0\) and \(x = 1\). We can write this relation as: \(E(x) = E_{GaAs}(1 - x) + E_{GaP} x\) Now, we can find the energies for GaAs (\(E_{GaAs}\)) and GaP (\(E_{GaP}\)) through literature or external sources. \(E_{GaAs} = 1.43 eV\) \(E_{GaP} = 2.26 eV\) Convert these energies to Joules: \(E_{GaAs} = 2.29 \times 10^{-19} J\) \(E_{GaP} = 3.62 \times 10^{-19} J\)

Step 3: Plug the energy values into the linear relation and solve for x

We will use the energy of the emitted light E, the energies for GaAs and GaP, and the linear relation in the previous step to solve for x. First, we rewrite the linear relation using the calculated energy and the energies for GaAs and GaP: \(3.374 \times 10^{-19} J = (2.29 \times 10^{-19} J)(1 - x) + (3.62 \times 10^{-19} J) x\) Now, solve for x: \(x = \dfrac{3.374 \times 10^{-19} J - 2.29 \times 10^{-19} J}{3.62 \times 10^{-19} J - 2.29 \times 10^{-19} J}\) \(x \approx 0.62\)

Step 4: Conclusion

The composition (the value of x) used in orange LEDs made from GaAs and GaP solid solutions is approximately \(0.62\). This means that the GaP solid solution contains 62% GaP and 38% GaAs.

Key Concepts

Band Gap Energy

Band gap energy is a crucial concept in understanding how light-emitting diodes (LEDs) work. It refers to the energy difference between the valence band (where the electrons are present) and the conduction band (where they can move freely) in a semiconductor. The band gap determines the color of light the LED emits. A smaller band gap results in the emission of longer wavelength (red light), and a larger band gap results in shorter wavelength light (blue light). In the case of orange LEDs, the semiconductor material must have a band gap that allows for the emission of light with a wavelength around 590 nm. This specific energy is what allows the LED to emit visible orange light.

Photon Energy Calculation

Calculating the energy of a photon is essential to understanding the color output of LEDs. This energy is given by Planck's equation: \[ E = \dfrac{h c}{\lambda} \]where:

• \( E \) is the energy of the photon,
• \( h \) is Planck's constant \(6.626 \times 10^{-34} \text{J s}\),
• \( c \) is the speed of light \(3 \times 10^8 \text{m/s}\),
• \( \lambda \) is the wavelength of the light \( 590 \text{nm} = 590 \times 10^{-9} \text{m} \).

Converting the wavelength into photon energy helps us understand what band gap energy is needed for a specific light color. Therefore, for orange light with a 590 nm wavelength, the energy is calculated to be \(3.374 \times 10^{-19} \text{J}\). This value is used to relate the composition of the semiconductor materials in the LED.

LED Composition Analysis

The composition of an LED is vital in ensuring it emits the desired color. In orange LEDs, a mixture of Gallium Arsenide (GaAs) and Gallium Phosphide (GaP) is used. The formula \( \text{GaP}_x \text{As}_{1-x} \) represents a solid solution where \( x \) determines the proportion of GaP and GaAs. To find the right mix for orange light, it’s necessary to use a linear relation for the band gap energy based on the composition:\[ E(x) = E_{\text{GaAs}}(1 - x) + E_{\text{GaP}} x \]Given the energies:

• \( E_{\text{GaAs}} = 1.43 \text{eV} = 2.29 \times 10^{-19} \text{J} \)
• \( E_{\text{GaP}} = 2.26 \text{eV} = 3.62 \times 10^{-19} \text{J} \)

By substituting the known values into this equation, the composition for maximum efficiency in orange light emission is found to be \( x \approx 0.62 \). This means the LED composition should be roughly 62% GaP and 38% GaAs to achieve the desired orange emission.