Quantum Dots and Size-Tunable Band Gaps

Introduction

Quantum dots are semiconductor nanocrystals whose optical properties can change markedly with size. A vial of small dots and a vial of larger dots made from the same semiconductor may absorb and emit different colours. This is useful only if we keep three ideas separate: the bulk band gap, the confinement of the electron and hole, and additional effects of their attraction and the surface. The Brus model combines the first two with an approximate attraction term.

Core explanation

For a simplified spherical dot of radius R, one common effective-mass form is E(R) ≈ E bulk + h²/(8R²)(1/m e + 1/m h ) − 1.786e²/(4πε₀ε rR) . Here E bulk is the bulk semiconductor's gap, m e and m h are electron and hole effective masses, and ε r is a relative dielectric constant. The positive term represents kinetic energy required to confine carriers. Its leading dependence is 1/R². The negative term represents electron–hole Coulomb attraction and has approximate 1/R dependence. The equation is an approximation for a spherical particle with idealised confinement and bulk-like parameters; real boundaries, dielectric screening, band structure and surface states can change the result.

As R decreases, the positive confinement term usually grows faster than the attractive term. The lowest allowed optical excitation therefore generally shifts upward in energy for smaller dots of the same material and comparable surface treatment. Since E photon = hc/λ , higher energy means shorter wavelength, often described as a blue shift. This is a comparison within a material family; a small dot of one material need not emit bluer light than a large dot of another.

An absorption spectrum and an emission spectrum measure different processes. Absorption requires a photon to create an excited state. Emission follows relaxation and radiative recombination. Energy can be lost before emission, producing a Stokes shift to longer wavelength. Surface traps may produce even broader or lower-energy emission. Thus a photograph of a glowing sample does not by itself reveal its band-edge absorption energy or the exact radius.

Quantum-dot samples also have size distributions. Different dots absorb at slightly different energies, broadening spectral peaks. A narrow distribution and passivated surfaces often yield sharper band-edge features. The Brus expression is best used to reason about direction and scale, or to fit data after its assumptions have been justified; one should not claim an exact diameter from a single colour without calibration.

Step-by-step reasoning

First identify the semiconductor and whether the dots have similar chemistry and surface coverage. Next compare radii: smaller R increases both confinement and attraction in magnitude, but the ideal positive term grows as 1/R². Determine the expected shift of the absorption edge. Convert between wavelength and energy with E(eV) ≈ 1240/λ(nm). Finally ask whether the stated colour describes absorption or emission, because relaxation can separate the two.

Visual explanation

Sketch bulk valence and conduction bands with a fixed gap. Beside them draw a large dot with modestly separated electron and hole levels and a small dot with a larger lowest transition. Put a short-wavelength arrow above the small dot and a long-wavelength arrow above the large one. Add a second downward arrow for emission after relaxation to show why emitted photons may be less energetic than initially absorbed photons.

Real-world analogy

Imagine confining two moving objects in a room whose walls progressively move inward. Restricting their motion raises the kinetic cost, while an attractive cord between them partly compensates. The analogy explains the competing signs in the model, though electrons and holes are quantum waves and their effective masses are properties of the solid, not ordinary mechanical masses.

Real-world example

In a synthesis that makes different sizes of one semiconductor quantum dot, researchers can record the wavelength of the first absorption feature after each growth interval. If the particles grow while their composition remains similar, that feature often moves to longer wavelength. Comparing spectra with electron microscopy gives a size–energy calibration. Such calibration is especially important if ligand changes also alter surface traps or the dielectric environment.

Why?

Why is size a colour control knob? Spatial confinement changes the allowed electron and hole energies without requiring a different elemental composition. Why does the model include an attractive correction? The optically created electron and hole carry opposite charges and interact. Ignoring that attraction would overestimate the simplest optical transition energy in this approximation.

Common misconception

The gap does not simply double when dot diameter halves. The measured transition includes a fixed bulk contribution, a radius-squared confinement term and other corrections. It is also incorrect to equate an emission maximum with the absorption edge in every sample; relaxation, traps and Stokes shifts can make them differ.

Worked example

Question: Two samples of the same dot family have first absorption features at 620 nm and 500 nm. Which has the larger optical transition energy, and by how much?

Reasoning: Using E(eV) ≈ 1240/λ(nm), the 620 nm feature corresponds to 2.00 eV; the 500 nm feature corresponds to 2.48 eV. The difference is 0.48 eV. Under comparable composition and surface conditions, the 500 nm sample is consistent with smaller dots, although spectroscopy alone is not an exact size measurement.

Answer: The 500 nm sample is higher by about 0.48 eV and is expected to have smaller dots under the stated comparison.

Quick check

1. Which Brus-model contribution grows approximately as the inverse square of dot radius? Answer: The positive electron and hole confinement-energy contribution.

Exam focus

State the assumptions when using a simplified Brus expression. Show both powers of radius and both signs. For a wavelength calculation, report units and distinguish an absorption transition from an emission peak. In qualitative questions, hold composition constant before attributing a colour trend to size.

Advanced insight

At very small radii the effective-mass approximation can fail because its parameters were derived for a bulk crystal. Finite barrier heights let wavefunctions extend into the surroundings, and the dielectric constant may vary across the dot boundary. Even a good fit to an absorption series does not prove each physical term individually. Time-resolved emission, size measurements and surface analysis provide independent constraints. The original Brus analysis focused on electron–electron and electron–hole interactions in small semiconductor crystallites, establishing the importance of confinement and Coulomb effects together.

Summary

The quantum-dot optical transition starts from the bulk gap, gains a positive confinement contribution and receives a negative electron–hole attraction correction. For similar dots of one material, smaller radii usually yield higher-energy, shorter-wavelength absorption. Emission may shift after relaxation. The Brus model makes this trend intelligible, but precise spectra require material and surface details.

Practice questions

1. A dot's radius changes from 6 nm to 3 nm. By what factor does the ideal 1/R² confinement term change if effective masses stay fixed? Answer: It becomes four times as large because (6/3)² = 4; the total optical gap does not become four times as large.

2. Why is the electron–hole term in the simple Brus equation negative? Answer: Opposite charges attract, lowering the energy of the electron–hole excitation relative to noninteracting confined carriers.

3. Calculate the photon energy for a 550 nm absorption feature. Answer: E ≈ 1240/550 = 2.25 eV, to three significant figures.

4. A sample emits at longer wavelength than it absorbs most strongly near its edge. Does this disprove confinement? Answer: No. Relaxation before emission and possible surface states can produce a Stokes shift while the absorption energy still depends on confinement.

Sources: Brus, Journal of Chemical Physics (1984); National Nanotechnology Initiative, nanotechnology overview.