12.2 Quantum wells, wires, and dots

Quantum confinement

Quantum confinement occurs when one or more dimensions of a material shrink to the nanoscale (typically comparable to the de Broglie wavelength of the electron). At this scale, charge carriers (electrons and holes) can no longer move freely in the confined direction(s), and their energy becomes quantized.

The degree of confinement depends on how many dimensions are restricted:

• Quantum well (2D system): confinement in one dimension, free motion in two
• Quantum wire (1D system): confinement in two dimensions, free motion in one
• Quantum dot (0D system): confinement in all three dimensions

Stronger confinement (smaller structures) produces larger energy level spacing and more pronounced quantum effects. This is the central principle behind all the nanostructures discussed here.

Density of states

Bulk vs. low-dimensional structures

The density of states (DOS) counts the number of available electronic states per unit energy per unit volume. It determines how carriers distribute across energies and directly shapes optical and transport behavior.

In a bulk (3D) semiconductor, the DOS varies continuously as $g(E) \propto E^{1/2}$. As you reduce dimensionality through confinement, the functional form of the DOS changes dramatically, and that change is what gives each nanostructure its distinctive physics.

Quantum wells

A quantum well is a thin layer of narrow-bandgap semiconductor sandwiched between two wider-bandgap barrier layers (e.g., GaAs between AlGaAs). Confinement along one axis quantizes the perpendicular motion into discrete subbands, while carriers remain free in the plane of the well.

The DOS in a quantum well is a staircase function: each step corresponds to the onset of a new subband, and within each step the DOS is constant (independent of energy). This flat, step-like DOS is a hallmark of 2D systems and is very different from the smooth $E^{1/2}$ curve of bulk material.

Quantum wires

Quantum wires confine carriers in two dimensions, leaving free motion along only one axis. The DOS for each 1D subband diverges as $g(E) \propto (E - E_n)^{-1/2}$ near the subband edge $E_n$, producing sharp van Hove singularities at each quantized level.

These peaked features in the DOS concentrate carriers at specific energies, which can enhance optical transitions and improve carrier mobility along the wire axis.

Quantum dots

Quantum dots confine carriers in all three dimensions. With no free directions remaining, the energy spectrum is fully discrete, much like an atom. The DOS reduces to a series of delta-function-like peaks, each at a specific quantized energy.

Because of this atom-like spectrum, quantum dots are sometimes called "artificial atoms." The spacing between levels is tunable by changing the dot size, giving experimentalists direct control over electronic and optical properties.

Electronic structure

Energy levels and wave functions

The quantized energies in these structures come from solving the Schrödinger equation with the appropriate confinement potential. For the simplest case of an infinite square well of width $L$, the confined energy levels are:

$E_n = \frac{n^2 \pi^2 \hbar^2}{2 m^* L^2}, \quad n = 1, 2, 3, \ldots$

where $m^*$ is the effective mass of the carrier. Two things to notice: energies scale as $n^2$ (so the spacing grows with quantum number), and they scale as $1/L^2$ (so shrinking the structure raises the energies and widens the gaps).

The corresponding wave functions are standing-wave-like and localized within the nanostructure. In real systems with finite barrier heights, the wave functions penetrate slightly into the barriers (evanescent tails), and the energy levels are somewhat lower than the infinite-well estimate.

Electron-hole recombination

Confining electrons and holes in the same small region increases their spatial overlap, which directly boosts the probability of recombination. Recombination can proceed through several channels:

• Radiative recombination: the electron-hole pair annihilates and emits a photon. This is the useful channel for LEDs and lasers.
• Non-radiative recombination: energy is released as phonons (heat) or transferred to a third carrier (Auger process). These pathways waste energy and reduce device efficiency.

The modified DOS in low-dimensional systems concentrates carriers at specific energies, which tends to enhance radiative rates relative to bulk. This is one reason quantum well and quantum dot emitters can be so efficient.

Optical properties

Absorption and emission

Optical transitions in confined structures occur between quantized levels, subject to selection rules. Because the energy levels are discrete (or quasi-discrete), the absorption and emission spectra show sharp, well-defined peaks rather than the broad edges seen in bulk semiconductors.

The peak positions are tunable: shrinking the structure increases confinement energy and blue-shifts the transitions, while enlarging it red-shifts them. This size-tunability is especially striking in quantum dots, where changing the diameter by just a few nanometers can shift emission across the visible spectrum.

Excitons in quantum structures

An exciton is a bound electron-hole pair held together by Coulomb attraction. In bulk semiconductors, exciton binding energies are typically small (a few meV), so excitons are fragile and only observable at low temperatures.

Confinement changes this picture. Squeezing the electron and hole into a smaller volume forces them closer together, which increases the Coulomb interaction and raises the binding energy. In a GaAs quantum well, for example, the exciton binding energy roughly doubles compared to bulk GaAs. In quantum dots, binding energies can be an order of magnitude larger.

The enhanced binding energy and oscillator strength mean that excitonic absorption peaks are sharper and persist to higher temperatures in confined systems. This makes excitons practically important for device design, not just a low-temperature curiosity.

Transport properties

Carrier scattering and mobility

Carriers in low-dimensional systems experience several scattering mechanisms:

• Phonon scattering (acoustic and optical)
• Impurity scattering (ionized dopants, charged defects)
• Interface roughness scattering (unique to heterostructures)

Reduced dimensionality can suppress certain scattering channels. In a quantum well, for instance, modulation doping spatially separates carriers from their parent dopant ions, dramatically reducing impurity scattering. This is the basis of the high-electron-mobility transistor (HEMT).

Confinement also modifies the phonon spectrum itself and the strength of electron-phonon coupling, which can further alter scattering rates.

Conductivity in low-dimensional systems

• Quantum wells can host a two-dimensional electron gas (2DEG) with very high mobility, enabling phenomena like the quantum Hall effect.
• Quantum wires, when shorter than the mean free path, can exhibit ballistic transport where carriers travel without scattering. Conductance becomes quantized in units of $\frac{2e^2}{h}$ per conducting channel.
• Quantum dots, being fully confined, show Coulomb blockade: adding a single extra electron costs a measurable charging energy $E_C = \frac{e^2}{2C}$, where $C$ is the dot's capacitance. Current flows only when gate voltage aligns with a discrete level, producing sharp conductance peaks.

Fabrication techniques

Molecular beam epitaxy (MBE)

MBE grows crystalline films one atomic layer at a time in ultra-high vacuum ($\sim 10^{-10}$ torr). Molecular beams of constituent elements (e.g., Ga, As, Al) are directed at a heated substrate, and shutters control which beams reach the surface at any moment.

Key advantages for nanostructure fabrication:

• Atomic-level control over layer thickness (sub-monolayer precision)
• Extremely abrupt interfaces between different materials
• In-situ monitoring via reflection high-energy electron diffraction (RHEED)

MBE is the workhorse for growing high-quality quantum wells (e.g., GaAs/AlGaAs) and can also produce quantum dots through the Stranski-Krastanov self-assembly mode, where strain drives the spontaneous formation of nanoscale islands.

Chemical vapor deposition (CVD)

In CVD, gaseous precursors flow into a heated reaction chamber, decompose on the substrate surface, and deposit the desired material. Compared to MBE, CVD offers higher throughput and is more scalable for industrial production.

Important variants include:

• Metal-organic CVD (MOCVD): uses metal-organic precursors (e.g., trimethylgallium). This is the dominant technique for growing III-V quantum well structures used in commercial LEDs and laser diodes.
• Plasma-enhanced CVD (PECVD): uses plasma to lower deposition temperatures, useful for temperature-sensitive substrates.

CVD techniques can produce quantum wells, wires, and dots with good control over size and composition, though interface abruptness is generally slightly less than MBE.

Applications

Quantum well lasers

Quantum well lasers replace the bulk active region of a conventional semiconductor laser with one or more quantum wells. The step-like DOS concentrates carriers at the band edge, which provides several benefits:

• Lower threshold current (fewer carriers needed to reach population inversion)
• Higher differential gain and faster modulation speeds
• Tunable emission wavelength by adjusting well width and composition

These lasers are the standard in fiber-optic telecommunications, optical disc drives, and laser pointers. Multiple quantum well (MQW) designs stack several wells to increase the total gain.

Quantum dot solar cells

Quantum dots offer two features attractive for photovoltaics: a size-tunable bandgap and the theoretical possibility of multiple exciton generation (MEG), where a single high-energy photon creates more than one electron-hole pair.

By incorporating dots of different sizes, a solar cell can absorb across a broader range of the solar spectrum than a single-junction device. In principle, MEG could push efficiency beyond the Shockley-Queisser limit of about 33% for a single junction. Achieving this in practice remains challenging, but colloidal quantum dot solar cells have shown steady efficiency improvements.

Single-electron transistors

A single-electron transistor (SET) consists of a quantum dot (the "island") coupled to source and drain electrodes through tunnel junctions, with a capacitively coupled gate.

Operation relies on Coulomb blockade: at low temperatures, the charging energy $E_C$ required to add one electron to the dot exceeds the thermal energy $k_B T$, so current is blocked unless the gate voltage brings a dot energy level into the bias window. The result is a series of sharp current peaks as the gate voltage is swept, each corresponding to the addition of exactly one electron.

SETs are extraordinarily sensitive electrometers and are being explored for low-power logic and as charge-sensing readout devices for spin qubits.

Challenges and limitations

Interface quality and defects

The performance of all these structures depends critically on interface quality. Common interface problems include:

• Roughness: atomic-scale thickness fluctuations broaden energy levels and scatter carriers
• Intermixing: diffusion of atoms across the interface blurs the composition profile
• Dangling bonds and traps: unsatisfied bonds at the interface create mid-gap states that act as non-radiative recombination centers

Minimizing these defects requires precise growth control, optimized substrate preparation, and often post-growth annealing. Advanced characterization tools like cross-sectional TEM and photoluminescence mapping help identify and quantify interface imperfections.

Strain and lattice mismatch

Growing one crystalline material on another with a different lattice constant introduces epitaxial strain. Small mismatch (below ~1-2%) can be accommodated elastically, and the resulting strain actually shifts band edges in useful ways (strained quantum wells are used to engineer laser wavelengths and effective masses).

Beyond a critical thickness, however, the strain relaxes through the formation of misfit dislocations. These threading defects propagate through the structure, act as non-radiative recombination centers, and severely degrade optical efficiency and device lifetime. Managing strain through buffer layers, graded compositions, and careful thickness control is a central challenge in heterostructure design.

Current research trends

Topological quantum structures

Topological insulators and topological superconductors represent a frontier where low-dimensional confinement meets topology. These materials host protected surface or edge states whose existence is guaranteed by topological invariants rather than material details. The states are robust against non-magnetic disorder and backscattering.

Confining topological materials into quantum wells or nanowires can induce topological phase transitions and potentially host Majorana zero modes, quasiparticles that are their own antiparticles. Majorana modes are of intense interest because they could serve as the basis for topological quantum computing, which would be inherently resistant to certain types of decoherence.

Quantum information processing

Quantum dots are among the leading platforms for solid-state qubits. Information can be encoded in:

• Spin states of a single electron confined in a dot (spin qubit)
• Charge states corresponding to the electron's position in a double-dot system (charge qubit)

Spin qubits in silicon quantum dots have demonstrated coherence times exceeding milliseconds and single- and two-qubit gate fidelities above 99%. Current research focuses on scaling up to arrays of many coupled dots, integrating qubit control with classical CMOS electronics, and developing error correction protocols compatible with the solid-state environment.