Quantum-Well Emission: Infinite-Well Estimate Versus Finite Barrier
A semiconductor quantum well is modeled as an electron in a one-dimensional infinite square well of width L = 8.0 nm. The electron effective mass is m* = 0.067 m_e. An engineer wants the n = 2 to n = 1 transition to emit at λ = 1.55 μm for fiber-optic communications. The actual well has finite barriers of height 0.30 eV, and the electron states are confined by those barriers. The engineer uses the infinite-well formula E_n = n²π²ħ²/(2m*L²) and concludes that an 8.0 nm well is close to the target, so only a small width adjustment is needed. A colleague warns that the infinite-well estimate may not be valid because the upper state could be near or above the barrier. The team needs the infinite-well transition wavelength for L = 8.0 nm, the well width that the infinite-well model would require for 1.55 μm, and a judgement about whether the finite barrier invalidates the model at that required width. Use hc = 1240 eV·nm and neglect exciton effects.
1. Using the infinite-well model, calculate the n=2 to n=1 transition wavelength for L=8.0 nm. State the energy expression and numerical result.
Read the worked answer
For an infinite well, E_n = n²π²ħ²/(2mL²). The transition energy is E2-E1 = 3π²ħ²/(2mL²). Using ħc=197 eV·nm and m*c²=0.067×511,000 eV, E1 = π²(197)²/[2(0.067×511,000)(8.0)²] = 0.088 eV. Therefore ΔE = 3×0.088 = 0.26 eV, and λ = hc/ΔE = 1240 eV·nm/0.26 eV = 4.7×10^3 nm = 4.7 μm.
This is far longer than 1.55 μm; the 8.0 nm well is not close to the target in the infinite-well model.
2. Find the well width that the infinite-well model would require for a 1.55 μm transition, and test whether the n=2 state remains below the 0.30 eV barrier at that width.
Read the worked answer
For λ=1.55 μm, ΔE = 1240/1550 = 0.800 eV. Since ΔE=3E1, E1=0.267 eV. Because E1 scales as L^-2, L_new = 8.0 nm × sqrt(0.088/0.267) = 4.6 nm.
At this width the infinite-well n=2 energy is E2=4E1=1.07 eV, well above the 0.30 eV barrier. A state above the barrier is not confined, so the infinite-well formula is not self-consistent here. The finite barrier also means the true bound-state energies are lower and the wavefunctions leak into the barriers; a numerical finite-well solution is required.
3. Given the finite barrier, what is the maximum photon energy for a transition between two bound states in this well? What does that imply for the 1.55 μm goal, and will the actual transition wavelength at 8.0 nm be longer or shorter than the infinite-well estimate?
Read the worked answer
Both bound-state energies must lie between 0 and V0=0.30 eV, so any bound-to-bound transition energy is less than 0.30 eV. The minimum possible wavelength is therefore λ_min = hc/(0.30 eV) = 1240/0.30 = 4.1×10^3 nm = 4.1 μm, in the mid-infrared. A 1.55 μm photon has 0.80 eV, which exceeds the entire well depth; no bound-to-bound transition in this 0.30 eV well can produce it.
For the 8.0 nm well, the actual finite-well transition wavelength will be longer than the infinite-well 4.7 μm estimate because barrier penetration effectively widens the well and lowers the level spacing. Thus the infinite-well conclusion is qualitatively misleading for this design.