Question

In a light-emitting diode (LED), which is used in displays on electronic equipment, watches, and clocks, a voltage is imposed across an n–p semiconductor junction. The electrons on the n side combine with the holes on the p side and emit light at the frequency of the band gap. This process can also be described as the emission of light as electrons fall from levels in the conduction band to empty levels in the valence band. It is the reverse of the production of electric current by illumination of a semiconductor. Many LEDs are made from semiconductors that have the general composition ${\mathbf{GaAs}}_{\mathbf{1}\mathbf{-}\mathbf{x}}{\mathbf{P}}_{\mathbf{x}}\mathbf{.}$When x is varied between 0 and 1, the band gap changes and, with it, the color of light emitted by the diode. When $\mathit{X}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{.}\mathbf{4}$, the band gap is $\mathbf{2}\mathbf{.}\mathbf{9}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{19}}\mathbf{J}$. Determine the wavelength and color of the light emitted by this LED.

Short answer

The wavelength of 685 nm corresponds to the visible spectrum, and the color is red.

Step-by-step solution

The given data

The band gap value is $E_g=2.9\times {10}^{-19}J$

The given semiconductor is${\mathbf{GaAs}}_{0.6}{\mathbf{P}}_{0.4}\mathbf{.}$

Explanation

The energy carried by a photon is $\mathit{E}\mathbf{=}\mathit{h}\mathit{v}$, where h is Planck’s constant, $\nu$the photon frequency, c the speed of light, and role="math" localid="1663435872997" $\mathit{\lambda }$ the photon wavelength. For a photon to just excite an electron across the band gap, this energy must be equal to that band gap energy role="math" localid="1663435841623" ${\mathit{E}}_{\mathbf{g}}$ .

$\lambda =\frac{hc}{E_g}---\left(1\right)$

Calculation

Substitute the given values in formula (1), to get the wavelength:

$$\begin{gathered} \lambda =\frac{hc}{E_g} \\ =\frac{6.626\times {10}^{-34}J.\sec \times 3\times {10}^8 m/\sec }{2.9\times {10}^{-19} J} \\ =685\times {10}^{-9} nm \\ =685 nm \end{gathered}$$

This wavelength corresponds to the visible spectrum, and the color is red.