change in band gap

change in band gap

1. What Changes — Quantum Confinement Widens Gap

In a quantum dot (size $L$ comparable to exciton Bohr radius $a_B∼1$–$10$ nm), carriers are confined as particle in a box:

adds to band gap.

Total effective gap:

(Brus equation: confinement term $∝1/L²$ dominates at small $L$, Coulomb term $∝1/L$ reduces).

• Result: as $L$ shrinks, $E_g$ increases ⇒ blue shift of absorption/emission.

2. Experimental Signature

• CdSe quantum dots: $E_g^{bulk}=1.74$ eV (red, 713 nm) → $L=5$ nm ⇒ $E_g≈2.1$ eV (590 nm orange), $L=2$ nm ⇒ $2.5$ eV (496 nm blue-green). Same material, colour tunable by size — used for QLED displays.

• Silicon: bulk $1.12$ eV (IR), nano Si $∼2$–$3$ eV (visible photoluminescence due to confinement).

3. Why Bulk Gap Fixed But Nano Tunable

Bulk gap set by periodic lattice; nano adds extra kinetic energy due to confinement (Heisenberg $Δp∼ħ/L$ ⇒ $E∼(Δp)²/2m∼ħ²/2mL²$). So gap is size-dependent, not fixed.

4. Worked Estimate

Particle in box for electron m*=0.1 m0, L=5 nm: ΔE≈ h²/(8·0.1·9.1e-31·25e-18)= (6.6e-34)²/(...)? Compute: h²≈4.36e-67, denominator 8·0.91e-31·25e-18≈1.82e-47 ⇒ ΔE≈2.4e-20 J≈0.15 eV per carrier ⇒ total ~0.3 eV increase ⇒ matches Brus.

Coulomb term ~1.8e²/4π ε L ≈1.8·1.44e-9/ (ε_r L)?? For ε_r≈10, L=5 nm ⇒ ~0.05 eV smaller.

5. Exam Points

• State $E_g^{nano}=E_g^{bulk}+h²/8L²(1/m_e+1/m_h)$ (or $∝1/L²$) — 5 marks, very frequent (the most asked nano property).
• Explain blue shift with decreasing size and give CdSe example — 5 marks, very frequent.
• Mention Brus equation terms — 3-5 marks, frequent.
• Link to particle in a box — 3 marks.