# Peak wavelength and Spectral Bandwidth

## Homework Statement

Calculate the expected peak wavelength and spectral bandwidth (in units of wavelength) of the
emission for both a GaAs and silicon LED at liquid nitrogen temperature (77 K) and room temperature (300 K). Which of these cases would you expect to result in the best emitter and why?

## Homework Equations

λg = hc/Eg
λg: maximum wavelength
λg [μm] = 1.24/Eg

Spectral bandwidth = (1.8kbT) / ħ

## The Attempt at a Solution

λg: band gap represents minimum energy, or maximum wavelength for which an electron-hole pair can be excited

GaAs Eg: 1.43 eV
Si Eg: 1.14 eV

λg [μm] = 1.24/1.43 = 0.867 μm (maximum wavelength for GaAs)

λg [μm] = 1.24/1.14 = 1.088 μm (maximum wavelength for Si)

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Spectral bandwidth = (1.8kb*300k) / ħ = 3.928*1013Hz
Expressing this in units of wavelength, I've used the relation between frequency and wavelength:

λ=c/f = c / 3.928*1013Hz
=7.6*10-6m

Spectral bandwidth = (1.8kb*77k) / ħ = 1.008*1013Hz = 2.9*10-5m/s
Expressing this in units of wavelength, I've used the relation between frequency and wavelength:

λ=c/f = c / 1.008*1013Hz
= 2.9*10-5m

So I think I've got the correct peak wavelength and spectral bandwidth, I'm not sure about the last question though or how I can quantify if one is a better emitter than the other. Any help would be much appreciated!

## Answers and Replies

RPinPA
Science Advisor
Homework Helper
I would interpret "better emitter" as "brighter", meaning more total energy is emitted. You do have relevant information here.

What does the peak wavelength measure? Peak of what?
What does the spectral bandwidth measure? Bandwidth of what?

I'm guessing greater energy = better emitter = lower wavelength

Question is asked in regards to LEDs

RPinPA
Science Advisor
Homework Helper
Are you saying that longer wavelengths always equal more energy? So the energy given off by a nuclear explosion in gamma rays is less than the energy given off by a glowing coal in the infrared? Are you sure?

Again, what does the peak wavelength measure? Peak of what? What does the spectral bandwidth measure? Bandwidth of what? How are those things defined?

I know it's about LEDs. I'm asking you to think through the meaning of those parameters. "It's the bandwidth" and "it's about LEDs" are not an explanation. What is it that has a width? What are the axes of the associated graph?