Quantum Dot

Energy levels, optical absorption and quantum confinement in nanostructures

Parameters
Energy Levels
1. What is a Quantum Dot?

A Quantum Dot (QD) is a nanoscale semiconductor crystal — typically 1–10 nm in diameter — where electrons are confined in all three spatial dimensions. It is essentially an artificial atom: energy is quantized, but unlike a real atom, the energy levels can be tuned by changing the size. CdSe is the most studied QD material: bulk bandgap Eg = 1.74 eV, effective masses me* = 0.13 m₀, mh* = 0.45 m₀, dielectric constant εr = 9.4.

2. Quantum Confinement

In a bulk semiconductor, electrons move freely. When the crystal size approaches the exciton Bohr radius (a_B), electrons "feel" the boundaries and energy becomes quantized. For CdSe: a_B ≈ 4.9 nm.

R ≫ a_B Bulk behavior — quantum effect negligible
R ~ a_B Weak confinement regime
R ≪ a_B Strong confinement — energy strongly increases
3. Spherical QD — Particle in a Sphere

A spherical QD is modeled as a spherical box: potential is zero inside (r < R) and infinite outside. The Schrödinger equation solution in spherical coordinates:

$$ \psi_{nlm}(r,\theta,\phi) = A_{nl}\, j_l\!\left(\frac{x_{nl}}{R}\,r\right) Y_l^m(\theta,\phi) $$

where j_l is the spherical Bessel function, x_nl its n-th zero, Y_l^m spherical harmonics. Energy levels:

$$ E_{nl} = \frac{\hbar^2\, x_{nl}^2}{2m^* R^2} $$

Key result: Energy is inversely proportional to R² — smaller QD means higher energy, shorter wavelength photon (blue shift).

Level l x_nl Orbital
103.14161s
214.49341p
325.76351d
406.28322s
536.98791f
4. Brus Equation

The most widely used formula in practice — accounts for electron, hole, and Coulomb interaction:

$$ E_{QD} = E_g^{bulk} + \frac{\hbar^2\pi^2}{2\mu R^2} - \frac{1.786\,e^2}{4\pi\varepsilon_0\varepsilon_r R} $$

First term — bulk bandgap. Second — quantum confinement (positive, ~1/R²). Third — Coulomb correction (excitonic binding, negative, ~1/R). μ = me*·mh*/(me*+mh*) — reduced mass. CdSe R=3 nm: confinement +0.64 eV, Coulomb −0.16 eV, E_QD ≈ 2.22 eV (λ ≈ 558 nm, green).

5. Optical Properties and Exciton

When a QD absorbs a photon, an exciton forms — an electron-hole pair bound by Coulomb force. In a QD, the exciton is fully confined. Each transition gives a Lorentzian peak:

$$ A(E) = \sum_i \frac{f_i\,\Gamma/2}{(E-E_i)^2 + (\Gamma/2)^2}, \qquad f_i \propto \frac{1}{n^2} $$

where f_i is oscillator strength, Γ is linewidth. Size → Color:

R (ნმ) E (eV) λ (ნმ) Color
1.5~3.1~400Violet
2.0~2.5~496Cyan-green
3.0~2.1~600Orange
4.0~1.9~650Red
6.0~1.8~690Deep red
6. Sphere vs Cube — Comparison

For a QD of equivalent "size" (L = 2R):

$$ E_\text{sphere} = \frac{\hbar^2\pi^2}{2m^* R^2}, \qquad E_\text{box} = \frac{3\hbar^2\pi^2}{2m^* L^2} = \frac{3\hbar^2\pi^2}{8m^* R^2} $$

Ratio: E_sphere/E_box = 4/3 — sphere gives stronger confinement because the first Bessel zero (π) differs.

7. Quantities Reference
E_gBulk bandgap [eV]
x_nln-th zero of spherical Bessel function j_l
m* = m_e*, m_h*Electron and hole effective masses [m₀]
μReduced mass: μ = me*·mh*/(me*+mh*)
a_BExciton Bohr radius [nm]
ε_rRelative dielectric constant
ΓSpectral linewidth — peak width [eV]
PVRLorentzian: A(E) = Σ f_i·(Γ/2)/[(E−Ei)²+(Γ/2)²]
8. Materials Comparison
Material Eg (eV) Spectrum Application
CdSe1.74VisibleQLED, bio-imaging
InAs0.354IRtelecom, IR detector
InP1.344VisibleCd-free QLED
GaAs1.424Visible/NIRlaser, solar
ZnS3.54UVCdSe/ZnS shell
PbS0.41NIRsolar, IR

CdSe/ZnS core-shell is the most common structure — ZnS shell reduces surface defects and increases quantum yield to 80–90%.

9. Real-world Applications
QLED TVRed/green QD filters on LCD — brighter, purer colors
Solar CellMEG (multi-exciton generation) — one photon → multiple excitons
Bio-imagingFluorescent QD on antibodies — tumor visualization
Quantum ComputingSpin qubits — electron spin in QD as qubit
QD LaserLow threshold, temperature-stable — optical communications
Beginner Guide

Step 1 — Material
Choose CdSe (most common). Its bulk bandgap 1.74 eV corresponds to 713 nm (red). This is the QD "base color" at large size.

Step 2 — Radius
Start with R = 3.0 nm. After calculation, note the 1s level energy. Then change to R = 2.0 nm — you will see energy increase and blue shift.

Step 3 — Optical Absorption
Switch to "Optical Absorption" mode. Check the first peak wavelength in nm — this is the QD emission color.

Step 4 — Comparison
Switch to "Confinement" mode. See sphere and box curves — sphere is always slightly higher. Also see the wavelength curve: as R decreases, color shifts blue.

Advanced Guide

InAs — IR QD
InAs bulk Eg = 0.354 eV (IR). At R = 3 nm, confinement raises energy into visible range. Effective mass me* = 0.026 m₀ — very small, so confinement is strong.

Coulomb Correction
Results show the E_coulomb value. At small R (~1-2 nm) Coulomb subtracts 0.3–0.5 eV — significant correction, especially for InAs and PbS (large ε_r).

R/a_B Ratio
Results show confinement_ratio = R/a_B. If < 0.5 — strong confinement, Brus is accurate. 0.5–2 — intermediate. > 2 — weak confinement, bulk model is more appropriate.

Quantum Dot — Quiz

1. CdSe QD radius decreases from 3 nm to 1.5 nm. Emission wavelength:

2. In the Brus equation, the Coulomb term:

3. In CdSe/ZnS core-shell, the role of ZnS shell is:

4. In the "particle in a sphere" model, energy levels are determined by: