“Know how to solve every problem that has been solved.” “What I cannot create, I do not understand.” — Richard Feynman

The Particle in a Box by Hand — An infrared detector from a quantum well

Exercises

Introductory Quantum Mechanics › Unit 2 · Bound states in one dimension › The Particle in a Box by Hand › all problems

Practice Problem 2 of 10

An infrared detector from a quantum well

Problem

A semiconductor quantum well confines an electron (use the free mass) in

L = 5

nm. Find

E_{1}

E_{2}

, and the

1 \to 2

transition energy in meV and as a frequency. Which band of the spectrum does a detector built on this transition see?

Solution

Predict before reading on. Same move as the worked example’s first two steps — compute $E_{1}$ for the new $L$ , then difference two levels. Only the fill-the-levels step disappears (one electron). Which factor changes by how much when $L$ goes from 0.70 nm to 5 nm?

E_{1} = \frac{h ^{2}}{8 m _{e} L ^{2}} = 15.0 meV, E_{2} = 4 E_{1} = 60.2 meV

Δ E_{1 \to 2} = 3 E_{1} = 45.1 meV \Rightarrow f = Δ E / h = 10.9 THz

Far infrared / terahertz. This is a real device family (QWIPs — quantum-well infrared photodetectors): the box length is a lithography knob that dials the detection band.

Result

E_{1} = 15.0 meV, Δ E_{1 \to 2} = 45.1 meV \approx 10.9 THz