3.4 Quantum wells

Fundamentals of quantum wells

A quantum well is a nanoscale structure that confines electrons (and holes) in one dimension, producing discrete energy levels and dramatically altered electronic and optical properties. These structures underpin much of modern optoelectronics, from semiconductor lasers to infrared detectors, and they serve as the cleanest example of quantum confinement in a solid-state system.

Definition and basic concepts

A quantum well consists of a thin semiconductor layer sandwiched between two layers of a wider-bandgap material. The classic example is a thin GaAs layer between two AlGaAs barriers. Because the bandgap of the well material is smaller, electrons and holes become trapped in the well layer, confined along the growth direction but free to move in the other two dimensions.

The well layer is typically a few nanometers to a few tens of nanometers thick. At these length scales, the confinement direction supports only a discrete set of energy levels rather than a continuous band, which is the defining feature of quantum well physics.

Quantum confinement effects

Confinement effects become significant when the well width approaches the de Broglie wavelength of the charge carriers (roughly 10–50 nm in typical semiconductors). At that point, the carrier's wave-like nature dominates, and you can no longer treat its motion classically along the confined direction.

The main consequences of confinement:

• Energy levels become quantized into discrete subbands
• The density of states changes from a smooth parabolic function (3D) to a step-like function (2D)
• Excitonic effects are enhanced because the electron and hole are forced closer together, increasing their Coulomb interaction
• Oscillator strengths for optical transitions increase, making quantum wells more optically active than bulk material of the same composition

Density of states

The density of states (DOS) counts the number of available electronic states per unit energy per unit area (in 2D). In a bulk 3D semiconductor, the DOS rises as $\sqrt{E}$. In a quantum well, each subband contributes a constant DOS of:

$g_{2D} = \frac{m^*}{\pi\hbar^2}$

per subband (per spin), where $m^*$ is the effective mass. Each time the energy crosses a new subband edge, the total DOS jumps up by another step. This step-like structure has direct consequences for optical absorption spectra and device threshold behavior.

Energy levels in quantum wells

Infinite potential well model

The simplest starting point assumes the barriers are infinitely high, so the particle is perfectly trapped. The confined energy levels are:

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

where $n = 1, 2, 3, \ldots$ is the quantum number, $m^*$ is the effective mass, and $L$ is the well width. The corresponding wave functions are standing sine waves:

$\psi_n(z) = \sqrt{\frac{2}{L}} \sin\!\left(\frac{n\pi z}{L}\right)$

inside the well, and identically zero outside. This model captures the key scaling: energy levels grow as $n^2$ and as $1/L^2$, so narrower wells have more widely spaced levels. It works best as an approximation for deep wells with large band offsets.

Finite potential well model

Real barriers have a finite height $V_0$, set by the conduction (or valence) band offset between the well and barrier materials. This changes things in two important ways:

• Wave functions penetrate into the barriers as exponentially decaying tails, so the particle "leaks" slightly outside the well.
• Energy levels are lower than the infinite-well predictions because the effective confinement length is larger than $L$.

Finding the energy levels requires solving transcendental equations that match the sinusoidal solution inside the well to the decaying exponentials outside. The number of bound states is finite and depends on both $L$ and $V_0$. A very shallow or very narrow well may support only a single bound state.

Bound states vs. continuum states

• Bound states have energies $E_n < V_0$. Their wave functions are localized in and near the well.
• Continuum states have energies $E > V_0$. They extend across the entire structure and behave like modified plane waves.
• Near the barrier edge, quasi-bound (resonant) states can appear. These are not truly confined but have enhanced probability density inside the well due to constructive interference, and they decay over time by tunneling out.

Electronic properties

Wave functions in quantum wells

Within the envelope function approximation, the total electron wave function is written as a slowly varying envelope function multiplied by the rapidly oscillating Bloch function of the host crystal. The envelope function captures the confinement physics.

For a symmetric well, the ground state ($n=1$) envelope is symmetric about the well center, the first excited state ($n=2$) is antisymmetric with one node, and so on. The probability density $|\psi_n(z)|^2$ tells you where the electron is most likely to be found. For the ground state, that's the center of the well; for higher states, the probability spreads out with nodes appearing inside the well.

Quantum well subbands

Each quantized level $E_n$ in the confinement direction becomes the bottom of a subband. Within each subband, electrons are free to move in the two unconfined directions (the plane of the well), giving a 2D dispersion:

$E(n, \mathbf{k}_\parallel) = E_n + \frac{\hbar^2 k_\parallel^2}{2m^*}$

where $\mathbf{k}_\parallel$ is the in-plane wave vector. Higher subbands sit at higher energies and have wave functions with more nodes. In practice, at low carrier densities and moderate temperatures, only the lowest one or two subbands are occupied.

Effective mass approximation

The effective mass approximation replaces the full band structure with a parabolic dispersion characterized by $m^*$. This works well near subband edges where the dispersion is approximately parabolic.

Keep in mind that $m^*$ can differ between the well and barrier materials, between conduction and valence bands, and even between heavy-hole and light-hole subbands. In strained quantum wells, the effective mass can be further modified. Despite these complications, the effective mass picture is the workhorse of quantum well device modeling.

Optical properties

Interband transitions

Interband transitions promote an electron from a valence-band subband to a conduction-band subband (absorption) or the reverse (emission). The photon energy equals the transition energy, which is the bandgap plus the confinement energies of both the electron and hole subbands involved.

Selection rules in a symmetric well require that the quantum numbers of the initial and final states have the same parity ($\Delta n = 0$ for the dominant transitions: $n=1 \to n=1$, $n=2 \to n=2$, etc.). Because of the step-like 2D DOS, the absorption spectrum rises in discrete steps rather than the smooth $\sqrt{E}$ onset seen in bulk semiconductors.

Intersubband transitions

These transitions occur between subbands within the same band (e.g., $n=1 \to n=2$ in the conduction band). They typically fall in the mid-infrared to terahertz range because the energy spacing between subbands is much smaller than the bandgap.

A critical selection rule applies: only light polarized along the confinement direction (perpendicular to the well plane) can drive these transitions. This means normal-incidence light, which is polarized in-plane, cannot excite intersubband absorption directly. Device designs use gratings, angled facets, or waveguide geometries to couple light into the correct polarization. Intersubband transitions are the operating principle behind quantum cascade lasers and quantum well infrared photodetectors.

Excitons in quantum wells

An exciton is a bound electron-hole pair held together by Coulomb attraction. In a quantum well, confinement squeezes the electron and hole closer together, which increases the binding energy and oscillator strength compared to bulk excitons.

• In bulk GaAs, the exciton binding energy is about 4.2 meV. In a narrow GaAs/AlGaAs quantum well, it can exceed 10 meV.
• The enhanced binding means excitons can survive to higher temperatures. In wide-gap systems like GaN/AlGaN, quantum well excitons are stable well above room temperature.
• Excitonic absorption peaks appear as sharp features just below the subband-edge steps, and they dominate the near-band-edge optical response at low temperatures.

Fabrication techniques

Molecular beam epitaxy

Molecular beam epitaxy (MBE) grows quantum wells one atomic layer at a time in ultra-high vacuum ($\sim 10^{-10}$ torr). Elemental sources (Ga, Al, As, etc.) are heated in effusion cells, and their molecular beams impinge on a heated substrate.

• Layer thickness is controlled to single-monolayer precision by opening and closing mechanical shutters over each cell.
• RHEED (reflection high-energy electron diffraction) provides real-time feedback on surface reconstruction and growth rate during deposition.
• MBE produces the sharpest interfaces and highest-purity quantum wells, making it the method of choice for research-grade structures.
• The main drawback is low throughput: growth rates are typically around 1 µm/hour.

Metal-organic chemical vapor deposition

MOCVD (also called MOVPE) uses metal-organic precursors (e.g., trimethylgallium) and hydrides (e.g., arsine) that react on a heated substrate to deposit semiconductor layers. It operates at much higher pressures than MBE (typically 10–760 torr).

• Growth rates are higher than MBE, and the technique scales well to large wafers, making it the standard for commercial quantum well device production.
• Layer composition and thickness are controlled by adjusting precursor gas flow rates and substrate temperature.
• MOCVD handles a wide range of III-V and II-VI materials, including the nitride family (GaN, InGaN, AlGaN) used in blue LEDs and laser diodes.

Atomic layer deposition

ALD deposits material one atomic layer per cycle using alternating, self-limiting surface reactions. Each cycle consists of a precursor pulse, a purge, a reactant pulse, and another purge.

• The self-limiting nature gives excellent thickness uniformity and conformality, even over non-planar substrates.
• ALD is most commonly used for dielectric and barrier layers rather than the active quantum well layer itself.
• Growth rates are slow (fractions of an angstrom per cycle), so ALD is practical mainly for very thin layers where uniformity matters more than speed.

Applications of quantum wells

Quantum well lasers

Quantum well lasers use interband transitions in one or more quantum wells as the gain medium. Compared to bulk double-heterostructure lasers, they offer:

• Lower threshold currents, because the step-like DOS concentrates carriers at the band edge more efficiently
• Better temperature stability of the threshold current
• Wavelength tunability by adjusting well width and alloy composition

The most common design is the separate confinement heterostructure (SCH), where the quantum well(s) sit inside a broader waveguide region that provides optical confinement independently of carrier confinement. Multiple quantum well (MQW) active regions increase the optical gain. These lasers are ubiquitous in fiber-optic communications (1.3 and 1.55 µm InGaAsP/InP systems), optical disc drives, and laser pointers.

Photodetectors and infrared sensors

Quantum well infrared photodetectors (QWIPs) exploit intersubband absorption to detect mid- and long-wave infrared radiation (roughly 3–20 µm). The detection wavelength is set by the subband spacing, which is engineered through well width and barrier height.

• QWIPs offer fast response times and uniform large-format focal plane arrays for thermal imaging.
• Multi-color detection is achieved by stacking quantum wells with different transition energies.
• Applications include night vision, surveillance, environmental monitoring, and industrial process control.
• A limitation is that QWIPs require cooling (typically to 77 K) for good signal-to-noise performance, unlike competing technologies such as HgCdTe detectors.

High-electron-mobility transistors

HEMTs exploit the two-dimensional electron gas (2DEG) that forms at a quantum well interface (e.g., AlGaAs/GaAs or AlGaN/GaN). Donors are placed in the wide-bandgap barrier layer, and the electrons they release fall into the quantum well, spatially separating carriers from ionized impurities.

This modulation doping scheme dramatically reduces impurity scattering, yielding very high electron mobilities (exceeding $10^5$ cm$^2$/V·s in GaAs-based structures at low temperature). HEMTs are the transistor of choice for low-noise amplifiers, millimeter-wave radar, satellite receivers, and 5G wireless infrastructure.

Multi-quantum well structures

Superlattices vs. multiple quantum wells

The distinction comes down to barrier thickness:

• In a multiple quantum well (MQW) structure, the barriers are thick enough that adjacent wells are essentially independent. Each well has its own set of discrete energy levels.
• In a superlattice, the barriers are thin enough that wave functions in neighboring wells overlap significantly, and the discrete levels broaden into minibands.

The crossover between these regimes depends on both the barrier thickness and height. As a rough guide, barriers thinner than about 3–5 nm in GaAs/AlGaAs begin to show significant inter-well coupling.

Miniband formation

When wells are close enough to couple, the $N$ degenerate levels of $N$ identical wells split into a band of $N$ closely spaced states. In the limit of an infinite superlattice, this becomes a continuous miniband with a well-defined width $\Delta$ and minigaps separating successive minibands.

• Miniband width increases with stronger inter-well coupling (thinner or lower barriers).
• Carriers can transport vertically through the superlattice via miniband conduction, unlike in MQWs where vertical transport requires tunneling through isolated barriers.
• Superlattice band structure is tunable by design, enabling engineered electronic and optical properties not available in any single bulk material.

Quantum cascade devices

A quantum cascade laser (QCL) chains together tens of coupled quantum well stages. In each stage, an electron undergoes an intersubband transition, emits a photon, and then tunnels into the next stage where it repeats the process. A single electron can therefore emit dozens of photons as it cascades through the structure.

• QCLs operate in the mid-infrared ($\sim$3–25 µm) and terahertz ($\sim$60–300 µm) ranges, wavelengths difficult to reach with interband lasers.
• Room-temperature continuous-wave operation is now routine in the mid-infrared.
• Quantum cascade detectors (QCDs) use the same multi-well architecture in reverse, converting absorbed infrared photons into a photocurrent without requiring an external bias.

Quantum wells in different materials

III-V semiconductor quantum wells

The III-V family dominates quantum well technology:

• GaAs/AlGaAs: The most studied system. Nearly lattice-matched, so wells are essentially strain-free. Emission in the near-infrared (~800–870 nm).
• InGaAs/InP: Used for telecom-wavelength (1.3 and 1.55 µm) lasers and detectors. Lattice matching to InP substrates requires specific In/Ga ratios.
• GaN/AlGaN: Wide bandgap system for blue/UV emitters and high-power HEMTs. Strong piezoelectric fields in these polar wells create internal electric fields that tilt the band profile (the quantum-confined Stark effect).

All of these systems have direct bandgaps (except certain AlGaAs compositions above ~45% Al), enabling efficient radiative transitions.

II-VI semiconductor quantum wells

Materials like CdTe/CdZnTe and ZnSe/ZnCdSe access shorter wavelengths in the blue, green, and UV. Their larger exciton binding energies (tens of meV) make excitonic features prominent even at room temperature.

The main challenges are achieving reliable p-type doping and controlling defect densities, which have historically limited device lifetimes. Despite this, II-VI quantum wells remain important for fundamental studies of excitonic physics and for niche applications in UV detection.

Silicon-based quantum wells

Si/SiGe heterostructures provide quantum confinement compatible with mainstream CMOS fabrication. Strain from the lattice mismatch between Si and SiGe is used to modify the band structure and enhance carrier mobility.

The indirect bandgap of silicon severely limits radiative efficiency, so Si-based quantum wells are not competitive for light emission. Their primary applications are:

• High-mobility strained-Si channels in advanced MOSFETs
• SiGe HBTs (heterojunction bipolar transistors) for RF circuits
• Emerging silicon photonics platforms where Ge-rich quantum wells provide electroabsorption modulation

Characterization methods

Photoluminescence spectroscopy

Photoluminescence (PL) is the go-to non-destructive probe of quantum well quality. A laser excites carriers above the bandgap, and you measure the spectrum of light emitted as they recombine.

• Peak positions reveal subband energies and well width uniformity.
• Linewidths indicate interface roughness and alloy disorder (broader peaks mean rougher interfaces).
• Temperature-dependent PL extracts exciton binding energies and identifies non-radiative recombination channels.
• Time-resolved PL measures carrier lifetimes, distinguishing radiative from non-radiative recombination.

Transmission electron microscopy

TEM provides direct atomic-scale images of quantum well cross-sections. High-resolution TEM can resolve individual atomic columns, letting you measure layer thicknesses and spot interface steps or dislocations.

• STEM with Z-contrast (HAADF) imaging produces contrast proportional to atomic number, making compositional variations across interfaces visible.
• EELS (electron energy loss spectroscopy) maps elemental composition and local electronic structure with sub-nanometer resolution.
• The main drawback is sample preparation: specimens must be thinned to electron transparency (~50–100 nm), which is destructive and time-consuming.

X-ray diffraction analysis

High-resolution XRD is the standard non-destructive structural characterization tool for quantum well heterostructures.

• Satellite peaks around the substrate Bragg reflection encode the superlattice period and individual layer thicknesses.
• Peak positions and spacings yield composition and average strain.
• Reciprocal space maps separate the effects of lattice mismatch from strain relaxation, telling you whether the structure is coherently strained or partially relaxed.
• X-ray reflectivity (XRR) measures interface roughness and layer density with sub-nanometer sensitivity.

Advanced concepts

Quantum well intermixing

Quantum well intermixing (QWI) is a post-growth technique that intentionally blurs the sharp interface between well and barrier by promoting atomic interdiffusion. This shifts the effective composition profile, increasing the ground-state energy and blue-shifting the emission wavelength.

Intermixing can be induced by:

1. Thermal annealing with or without a dielectric cap
2. Ion implantation followed by annealing (the implantation damage enhances diffusion)
3. Laser irradiation for localized heating

By selectively intermixing certain regions of a wafer while protecting others, you can create areas with different bandgaps on the same chip. This is valuable for photonic integrated circuits that need passive waveguides (larger bandgap, transparent) alongside active gain sections (original bandgap).

Strain effects in quantum wells

When the well material has a different lattice constant than the barrier, the well layer is strained to match the barrier lattice (assuming the well is thin enough to remain coherent). This strain modifies the band structure in important ways:

• Compressive strain (well lattice constant > barrier) splits the heavy-hole and light-hole bands, pushing the heavy-hole band up. This reduces the hole effective mass in-plane and lowers laser threshold currents.
• Tensile strain (well lattice constant < barrier) pushes the light-hole band up instead, and in extreme cases can convert a direct-gap material to indirect.
• Strain also modifies effective masses and piezoelectric fields (especially in wurtzite III-nitrides).

Strain engineering is a standard design tool for optimizing quantum well laser and HEMT performance.

Magnetic field effects

Applying a magnetic field perpendicular to the quantum well plane quantizes the in-plane motion into Landau levels, with energies:

$E_{n,\ell} = E_n + \left(\ell + \frac{1}{2}\right)\hbar\omega_c$

where $\ell = 0, 1, 2, \ldots$ indexes the Landau level and $\omega_c = eB/m^*$ is the cyclotron frequency. The 2D system is now fully quantized in all directions.

• At high fields and low temperatures in high-mobility 2DEGs, this leads to the integer and fractional quantum Hall effects, where the Hall resistance is quantized in units of $h/e^2$.
• Zeeman splitting of spin-degenerate levels provides a direct measurement of the electron g-factor.
• Magneto-excitons form when the magnetic length becomes comparable to the exciton Bohr radius, further enhancing binding energies.