9.1 Quantum dots

Quantum dots are nanoscale semiconductor structures, typically 2–10 nm across, that display discrete energy levels and size-tunable optical properties due to quantum confinement in all three dimensions. Often called "artificial atoms," they sit at the boundary between single-atom behavior and bulk semiconductor physics. Their tunable electronic and optical characteristics make them central to research in optoelectronics, biological imaging, and quantum computing.

Fundamentals of quantum dots

Quantum dots confine charge carriers in all three spatial dimensions, producing behavior that's qualitatively different from bulk semiconductors, quantum wells (2D confinement), or quantum wires (1D confinement). The physics that governs them draws on quantum mechanics, solid-state band theory, and electromagnetism all at once.

Definition and basic properties

Quantum dots are semiconductor nanocrystals with dimensions on the order of 2–10 nm. At this scale, the structure is small enough that quantum confinement dominates the electronic behavior, giving rise to discrete energy levels rather than the continuous bands you see in bulk materials.

• Composed of elements from groups II-VI (e.g., CdSe, CdTe), III-V (e.g., InAs, InP), or IV-VI (e.g., PbS, PbSe)
• Possess atom-like discrete energy spectra, which is why they're nicknamed "artificial atoms"
• Electronic and optical properties depend strongly on size, shape, and composition

Quantum confinement effect

Quantum confinement kicks in when the physical size of the dot approaches or drops below the exciton Bohr radius of the bulk material. The exciton Bohr radius is the natural length scale of an electron-hole pair in the semiconductor; once the dot is smaller than this, the carriers "feel" the boundaries and their allowed energies become quantized.

The simplest way to think about this is the particle-in-a-box model. A smaller box means higher ground-state energy and wider spacing between levels. The practical consequences:

• Smaller dots → larger effective bandgap → higher-energy (blue-shifted) emission
• Larger dots → smaller effective bandgap → lower-energy (red-shifted) emission

This size-tunability is one of the most powerful features of quantum dots. You can sweep emission across the visible spectrum just by changing the nanocrystal diameter during synthesis.

Density of states

The density of states (DOS) describes how many electronic states are available per unit energy. Dimensionality has a dramatic effect on the shape of the DOS:

• Bulk (3D): DOS goes as $\sqrt{E}$
• Quantum well (2D): step-function DOS
• Quantum wire (1D): DOS goes as $1/\sqrt{E}$
• Quantum dot (0D): delta-function-like DOS

For a quantum dot, the DOS is mathematically expressed as:

$D(E) = \sum_n \delta(E - E_n)$

where $E_n$ are the discrete energy levels. This means carriers can only occupy sharply defined energies, which is why quantum dot emission linewidths can be very narrow.

Fabrication techniques

Different fabrication routes offer trade-offs between size uniformity, crystalline quality, scalability, and compatibility with device architectures.

Colloidal synthesis

This is a solution-based, "wet chemistry" approach and the most common route for producing free-standing quantum dots.

1. Precursor compounds are injected into a hot solvent containing organic surfactant molecules.
2. Rapid thermal decomposition triggers nucleation of nanocrystal seeds.
3. Seeds grow as monomers deposit on their surfaces. Surfactant molecules cap the surface and control growth rate.
4. Size is tuned by adjusting reaction temperature and time: longer reaction times or higher temperatures yield larger dots.

Colloidal synthesis produces highly uniform size distributions (often < 5% standard deviation) and is readily scalable. CdSe quantum dots made this way are among the most thoroughly studied nanomaterials.

Epitaxial growth methods

Epitaxial techniques deposit semiconductor material layer by layer on a crystalline substrate, producing quantum dots embedded in a solid-state matrix. Two major methods:

• Molecular Beam Epitaxy (MBE): Uses ultra-high vacuum and precisely controlled atomic/molecular beams. Offers exceptional control over layer thickness (down to single monolayers) but is slow and expensive.
• Metal-Organic Chemical Vapor Deposition (MOCVD): Uses gaseous organometallic precursors that decompose on a heated substrate. Higher throughput than MBE and widely used in industrial III-V device fabrication.

Both methods produce quantum dots with well-defined crystalline interfaces, making them suitable for integration into optoelectronic devices.

Self-assembled quantum dots

Self-assembly exploits lattice mismatch between the deposited material and the substrate to spontaneously form nanoscale islands.

1. Growth begins in a layer-by-layer (Frank–van der Merwe) mode.
2. As the deposited layer thickens, strain energy from the lattice mismatch accumulates.
3. Beyond a critical thickness, the system lowers its energy by forming 3D islands on top of a thin wetting layer. This is the Stranski-Krastanov growth mode.
4. The resulting dots are typically pyramidal or lens-shaped, with dimensions of ~10–50 nm laterally and a few nm in height.

Size and density are controlled through growth parameters like substrate temperature and deposition rate. InAs dots on GaAs substrates are a classic example.

Electronic structure

Energy levels and quantization

Because carriers are confined in all three dimensions, the allowed energies are fully quantized. The simplest model treats the dot as a spherical potential well with infinite barriers. For a particle of effective mass $m^*$ in a sphere of radius $R$:

$E_n = \frac{\hbar^2 \pi^2 n^2}{2m^* R^2}$

This is the same functional form as the 1D particle-in-a-box, scaled to three dimensions. Two things to note:

• Energy scales as $1/R^2$, so shrinking the dot rapidly increases level spacing.
• The spacing between successive levels grows with $n$, unlike the hydrogen atom where levels converge at high $n$.

Real quantum dots have finite barriers and non-spherical shapes, so the actual level structure is more complex, but this model captures the essential physics.

Excitons in quantum dots

An exciton is a bound electron-hole pair held together by Coulomb attraction. In a quantum dot, spatial confinement forces the electron and hole closer together than they would be in the bulk, which enhances the exciton binding energy.

When the exciton recombines (the electron drops back into the hole), a photon is emitted. This radiative recombination is the basis for quantum dot luminescence and most optical applications.

The relevant length scale is the exciton Bohr radius $a_B$ of the bulk material. When $R < a_B$, you're in the strong confinement regime and the confinement energy dominates over the Coulomb interaction.

Coulomb blockade effect

In transport experiments, you can connect a quantum dot to source and drain electrodes and measure current flow. Because the dot is so small, adding even a single extra electron costs a significant charging energy $E_C \approx e^2 / 2C$, where $C$ is the dot's capacitance.

At low temperatures, this charging energy blocks current flow unless the gate voltage is tuned to a value where adding an electron costs zero net energy. The result is a staircase pattern in the current-voltage characteristics, with electrons tunneling through the dot one at a time.

This effect enables single-electron transistors and is directly relevant to charge-based quantum dot qubits.

Optical properties

Photoluminescence and absorption

Quantum dots absorb photons with energies above their bandgap and emit photons at a slightly lower energy. The difference between the absorption edge and the emission peak is the Stokes shift, caused by rapid relaxation of carriers to the lowest excited state before emission.

Two features distinguish quantum dot spectra from bulk:

• Broad absorption: Quantum dots absorb efficiently across a wide range of energies above the bandgap, since many higher-lying states are available. This is valuable for solar energy harvesting.
• Narrow emission: Emission comes primarily from the lowest exciton transition, producing a narrow, well-defined peak. This is what makes quantum dots attractive for displays, where saturated colors require spectrally pure emission.

Size-dependent emission

The relationship between dot size and bandgap energy is quantitatively described by the Brus equation:

$E_g(R) = E_g(\text{bulk}) + \frac{\hbar^2 \pi^2}{2R^2}\left(\frac{1}{m_e^*} + \frac{1}{m_h^*}\right) - \frac{1.8\,e^2}{4\pi\epsilon_0 \epsilon_r R}$

The three terms represent:

1. The bulk bandgap energy
2. The quantum confinement energy (kinetic energy increase from confining both electron and hole), which scales as $1/R^2$
3. The Coulomb attraction between electron and hole, which scales as $1/R$

Because the confinement term grows faster than the Coulomb term as $R$ shrinks, smaller dots always have a larger effective bandgap. For CdSe quantum dots, this allows continuous tuning of emission from red (~650 nm, diameter ~6 nm) through blue (~450 nm, diameter ~2 nm).

Quantum yield and blinking

Quantum yield (QY) is the ratio of photons emitted to photons absorbed. A QY of 1.0 means every absorbed photon produces an emitted photon. Surface defects act as non-radiative recombination centers that lower QY.

Core-shell structures (discussed below) dramatically improve QY by passivating dangling bonds at the core surface. Modern CdSe/ZnS core-shell dots routinely achieve QY > 80%.

Blinking (fluorescence intermittency) is a phenomenon where a single quantum dot randomly switches between bright ("on") and dark ("off") states. The leading explanation involves random charging events: when the dot acquires an extra charge, non-radiative Auger recombination quenches emission. Blinking can be suppressed through thick or graded shells and careful surface engineering.

Applications of quantum dots

Optoelectronic devices

• Displays: Quantum dot LEDs (QLEDs) and quantum dot enhancement films produce highly saturated colors with wide color gamut. Samsung's QLED TVs use this technology.
• Solar cells: Broad absorption and the possibility of multiple exciton generation make quantum dots promising for next-generation photovoltaics.
• Photodetectors: Spectral sensitivity can be tuned by choosing the dot size, enabling detectors optimized for specific wavelength ranges including the infrared.
• Lasers: Quantum dot gain media offer low threshold currents and reduced temperature sensitivity compared to quantum well lasers.

Biological imaging

Quantum dots are attractive fluorescent labels because they're much brighter and more photostable than traditional organic dyes.

• Different-sized dots emit at different wavelengths, enabling multiplexed imaging where multiple biological targets are labeled and detected simultaneously.
• Near-infrared-emitting dots (e.g., PbS, InAs) penetrate tissue more deeply than visible-wavelength probes.
• Surface functionalization with antibodies, peptides, or other biomolecules allows targeting of specific cells or proteins.

Quantum computing

• Quantum dots can serve as qubits, with quantum information encoded in the spin state of a confined electron.
• Spin qubits in gate-defined quantum dots (typically in GaAs or Si/SiGe heterostructures) offer relatively long coherence times and are compatible with existing semiconductor fabrication.
• Coupled quantum dots enable two-qubit gates through controlled exchange interaction.
• Scalability is a major advantage: semiconductor fabrication techniques could, in principle, produce large arrays of coupled dot qubits.

Characterization methods

Spectroscopy techniques

• UV-Vis absorption spectroscopy measures the absorption spectrum, revealing the bandgap and higher-energy transitions. The position of the first absorption peak gives a direct estimate of dot size.
• Photoluminescence (PL) spectroscopy measures emission wavelength, linewidth, and quantum yield.
• Time-resolved PL tracks carrier dynamics and recombination lifetimes on picosecond to nanosecond timescales.
• X-ray photoelectron spectroscopy (XPS) probes elemental composition and oxidation states at the surface.

Microscopy for quantum dots

• Transmission Electron Microscopy (TEM): Provides direct images of individual dots with sub-nanometer resolution. High-resolution TEM reveals crystal lattice fringes and can confirm crystal structure.
• Scanning Tunneling Microscopy (STM): Enables atomic-scale imaging and local spectroscopy (measuring the local density of states) on individual dots.
• Atomic Force Microscopy (AFM): Maps surface topography and can measure dot height distributions.
• Confocal fluorescence microscopy: Isolates emission from single quantum dots, enabling studies of blinking, spectral diffusion, and single-photon emission.

Electrical measurements

• Current-voltage (I-V) characteristics reveal Coulomb blockade steps and tunneling transport.
• Capacitance-voltage (C-V) measurements probe charge states and can extract energy level spacings.
• Hall effect measurements on quantum dot films determine carrier type, concentration, and mobility.
• Scanning gate microscopy (SGM) uses a biased AFM tip as a local gate to map the spatial distribution of electronic states.

Quantum dots vs bulk semiconductors

Band structure differences

In bulk semiconductors, the large number of atoms produces quasi-continuous energy bands. In quantum dots, confinement discretizes these bands into atom-like levels. Key differences:

• The effective bandgap in a quantum dot is always larger than the bulk value, and it increases as the dot shrinks.
• The density of states changes from a continuous $\sqrt{E}$ function (bulk) to a series of delta functions (dot).
• Excitonic effects are much stronger in dots because the electron and hole wavefunctions overlap more in a confined space.

Carrier confinement effects

• Carriers are confined in all three dimensions, quantizing both energy and momentum.
• Reduced dimensionality suppresses certain phonon scattering channels, which can increase carrier lifetimes.
• Coulomb interactions between carriers are enhanced because they can't spatially separate as they would in bulk.
• The "phonon bottleneck" (predicted slowing of carrier relaxation between widely spaced levels) is a topic of ongoing research, though in practice relaxation remains fast due to alternative mechanisms like Auger processes.

Surface-to-volume ratio impact

A 3 nm diameter quantum dot has roughly 50% of its atoms at the surface. This enormous surface-to-volume ratio means:

• Surface states can dominate electronic and optical behavior, often acting as trap states that quench luminescence.
• Surface passivation (through ligands or epitaxial shells) is not optional; it's essential for useful optical properties.
• Quantum dots are highly sensitive to their chemical environment (solvent polarity, pH, adsorbed species), which is both a challenge for stability and an opportunity for sensing applications.

Advanced quantum dot structures

Core-shell quantum dots

A core-shell quantum dot consists of a semiconductor core surrounded by a shell of a different semiconductor. The shell serves two purposes: it passivates surface dangling bonds and it modifies the confinement potential.

• Type-I (e.g., CdSe core / ZnS shell): Both electron and hole are confined to the core. The wider-bandgap shell acts as a potential barrier, keeping carriers away from surface traps. This maximizes quantum yield.
• Type-II (e.g., CdTe core / CdSe shell): The band alignment staggers so that the electron localizes in one material and the hole in the other. This spatial separation of charges is useful for photovoltaic applications where you want to extract carriers before they recombine.

Quantum dot molecules

When two or more quantum dots are placed close enough to interact through quantum mechanical tunneling, they form a quantum dot molecule. The individual dot energy levels hybridize into bonding and antibonding molecular-like orbitals, analogous to how atomic orbitals combine in real molecules.

• Can be fabricated by vertically stacking self-assembled dots (separated by a thin barrier) or by lithographic patterning.
• Enable controlled entanglement of electron spins across dots.
• Relevant to two-qubit gate operations in quantum computing and to fundamental studies of artificial molecular physics.

Quantum dot superlattices

Ordered arrays of quantum dots, arranged with regular spacing, form a superlattice. When the dots are close enough, their wavefunctions overlap and the discrete dot levels broaden into minibands.

• Miniband formation enables coherent carrier transport across the array, unlike the hopping transport seen in disordered dot films.
• Can be created through self-assembly of colloidal dots (using controlled evaporation) or through repeated epitaxial growth cycles.
• Applications include thermoelectric materials (where miniband engineering can optimize the Seebeck coefficient) and intermediate-band solar cells.

Theoretical models

Effective mass approximation

The simplest and most widely used model. It replaces the full periodic crystal potential with a single parameter: the effective mass $m^*$ of the carrier.

The Hamiltonian for a carrier in a spherical quantum dot is:

$H = -\frac{\hbar^2}{2m^*}\nabla^2 + V(r)$

where $V(r)$ is the confinement potential (zero inside the dot, infinite or finite at the boundary).

This model works well when the dot is significantly larger than the lattice constant of the material, so that the envelope function varies slowly over many unit cells. It breaks down for very small dots (below ~2 nm) where the atomic-scale structure matters.

Tight-binding model

Instead of treating carriers as nearly free particles, the tight-binding approach builds electronic states from linear combinations of atomic orbitals (LCAO) centered on each atom in the dot.

• Naturally incorporates the discrete atomic structure and chemical bonding.
• The Hamiltonian is constructed from on-site energies and hopping integrals between neighboring atoms.
• More accurate than the effective mass approximation for small dots and for capturing details of the band structure away from the zone center.
• Can handle complex geometries, alloy compositions, and heterostructure interfaces.

The trade-off is computational cost: the matrix dimension scales with the number of atoms, which can reach thousands even for a modest quantum dot.

Configuration interaction method

For problems where electron-electron interactions matter (multi-exciton states, charged dots, optical spectra of few-electron systems), you need a many-body approach.

1. Start with single-particle states obtained from one of the models above.
2. Construct many-electron basis states as Slater determinants (antisymmetrized products of single-particle states).
3. Build and diagonalize the many-body Hamiltonian, which includes Coulomb interaction matrix elements, in this basis.

Configuration interaction (CI) captures correlation effects that mean-field approaches miss, such as biexciton binding energies and the fine structure of emission spectra. The computational cost grows rapidly with particle number, so CI is typically applied to systems with fewer than ~10 interacting carriers.

Environmental and health considerations

Toxicity of quantum dot materials

Many of the best-performing quantum dots contain heavy metals. CdSe and PbS dots, for example, incorporate cadmium and lead, both of which are toxic to biological systems.

• Toxicity depends on whether the toxic core is exposed or protected by a shell and surface ligands.
• Degradation (through oxidation, UV exposure, or low pH) can release free $\text{Cd}^{2+}$ or $\text{Pb}^{2+}$ ions, which are the primary toxic agents.
• In vitro studies show dose-dependent cytotoxicity; in vivo studies in animal models show accumulation in liver, spleen, and kidneys.

Biocompatibility issues

For biological applications, the quantum dot surface must be engineered for compatibility with aqueous, physiological environments.

• Hydrophilic ligands (e.g., PEG coatings, mercaptoacetic acid) are used to make dots water-soluble and reduce nonspecific binding.
• In biological fluids, proteins rapidly adsorb onto the dot surface, forming a protein corona that changes the dot's effective size, charge, and biological identity.
• Long-term fate and clearance pathways for quantum dots in living organisms remain active areas of investigation.

Disposal and recycling challenges

• Quantum dot-containing products (displays, solar cells) require proper end-of-life handling to prevent heavy metal contamination.
• Recycling is complicated by the small quantities of material and the difficulty of separating dots from device matrices.
• Research into heavy-metal-free alternatives is a major thrust: InP-based dots (already used commercially in displays), carbon dots, and silicon dots offer reduced toxicity while approaching the optical performance of cadmium-based systems.