Red light-emitting diodes are made from GaAs and GaP solid...

Red light-emitting diodes are made from GaAs and GaP solid solutions, GaP $_{x} A s_{1-x}($ see Exercise 12.79$) .$ The original red LEDs emitted light with a wavelength of 660 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.

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Key Concepts

Energy-Wavelength Relationship

In the context of light emission from semiconductors, there is a direct relationship between the emitted photon energy and its wavelength, given by the equation E = hc/?. This relation allows one to connect the band gap energy of the material directly to the wavelength of light it emits, which is crucial for designing LEDs with specific color outputs.

Semiconductor Band Gap Engineering

This concept involves modifying the energy difference between the valence band and the conduction band in semiconductor materials by altering their composition. By mixing two semiconductors with different band gaps, such as GaAs and GaP, the resulting alloy shows intermediate properties that can be tuned to achieve desired optical or electrical characteristics.

Linear Interpolation of Material Properties

When the physical properties of a semiconductor alloy, such as its band gap, are assumed to vary linearly with composition, linear interpolation can be used to estimate the property at any given composition. This simplification, often referred to as Vegard's Law in some contexts, helps predict intermediate values between the end-member compounds.

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