Quantum well

A quantum well is a layer of semiconductor a few nanometers thick grown between two layers of a semiconductor with a wider bandgap, the barriers. The band offsets at the two interfaces trap electrons and holes in the layer, as described under carrier confinement, and because the layer is comparable in thickness to the carriers' de Broglie wavelength, their motion across it is quantized: each carrier can have only one of a set of discrete energies for that motion, while it moves freely in the plane of the layer. Its energy levels, and with them its emission wavelength, therefore depend on the well's thickness as well as its composition. A 10 nm GaAs well between Al₀.₃Ga₀.₇As barriers has its first optical transition near 848 nm, where bulk GaAs absorbs and emits at 871 nm. Most semiconductor lasers and LEDs use one or more quantum wells as their active region, the gain medium of a laser, and electro-absorption modulators use them as the absorbing layer.

Energy levels

For a well of width $L_w$ with infinitely high barriers, the confined energies, measured from the band edge, are

$$E_n=\frac{n^2{\pi }^2{\mathrm{\hslash }}^2}{2m^{\ast }{L}_w^2},n=1,2,3,\dots$$

where $m^{\ast }$ is the effective mass of the carrier along the growth direction. The levels rise as the square of the index $n$, scale as the inverse square of the width, and are larger for lighter particles, so electrons are shifted much more than heavy holes. An electron dropping from the first electron level to the first heavy-hole level emits a photon of energy

$$h\nu =E_g+E_{e1}+E_{hh1},$$

so a narrower well emits at a shorter wavelength. Real barriers are finite and the wave functions extend into them, which lowers every level below the infinite-well value: for electrons in a 10 nm GaAs well the formula gives 56 meV, while the 240 meV conduction-band barrier of Al₀.₃Ga₀.₇As gives 30 meV. A finite well also holds only a limited number of levels; a carrier with more energy than the barrier is no longer confined and moves in three dimensions.

For GaAs wells between Al₀.₃Ga₀.₇As barriers at room temperature, a finite-well calculation gives:

The calculation takes band offsets of 240 meV for electrons and 134 meV for holes, a 64:36 division of the 374 meV step in gap, effective masses of 0.067 and 0.092 for electrons in the well and barrier, and 0.35 and 0.38 for heavy holes along the growth direction. Dividing the step 57:43 instead, as R. C. Miller et al. measured and the carrier confinement entry uses, lowers the first transition by 4 meV in a 4 nm well and by less than 1 meV in a 10 nm well. It leaves out the exciton binding energy, which lowers absorption peaks by several meV, and band filling, which raises the photon energy of a laser above the first transition. Thin wells are the most sensitive to their width: one monolayer of GaAs (0.28 nm) more or less moves the transition of a 4 nm well by about 5 nm of wavelength, and that of a 10 nm well by about 1 nm.

Density of states

Because the carriers move freely only in the plane, each subband has a constant density of states rather than one rising as the square root of energy as in bulk material,

$$g_{\text{2D}}=\frac{m^{\ast }}{\pi {\mathrm{\hslash }}^2}\text{per unit area, for each subband,}$$

which for electrons in GaAs is $2.8\times {10}^{13}$ cm⁻² eV⁻¹. The total is a staircase with a step at each level. At the bottom of the first subband the density of states jumps straight to its full value, so the injected carriers crowd into a narrow range of energies at the band edge and contribute to the gain near a single photon energy, instead of spreading over a band whose density of states starts from zero. The gain therefore rises steeply with carrier density near transparency and then flattens as the first subband fills, the logarithmic behavior described under modal gain.

Because the heavy-hole and light-hole levels separate in a well, the optical transitions also depend on polarization. The transition from the electron level to the heavy-hole level couples only to light polarized in the plane of the well, which is TE for a waveguide, while the electron to light-hole transition couples more strongly to TM light (TE and TM polarization). Unstrained and compressively strained wells, whose top valence level is a heavy-hole level, therefore give TE gain.

Quantum-well lasers

The current needed to reach transparency scales with the volume of the active region, and a well a few nanometers thick holds a small fraction of the carriers of a double-heterostructure active layer, which is typically 0.1 µm or more thick. A layer that thin cannot guide light on its own, so the well sits inside a separate-confinement heterostructure: a waveguide layer of intermediate bandgap, often graded in composition, that holds the optical mode while the well holds the carriers. The cost is a small confinement factor, 0.01 to 0.03 for a single well, so a single-well laser needs a high material gain and works best in a long, low-loss cavity. Lasers with short cavities or high losses, including most InP telecom lasers, use several wells in a multi-quantum-well active region, which multiplies the confinement factor. Strained InGaAs quantum-well lasers on GaAs reached threshold current densities below 50 A/cm² by 1991.

The high differential gain near transparency also raises the relaxation oscillation frequency and lowers the linewidth enhancement factor, so quantum-well lasers modulate faster and chirp less than lasers with bulk active layers.

Strain

A well made of a material whose lattice constant differs from the substrate's can still be grown without dislocations if it is thinner than a critical thickness: the layer is compressed or stretched in the plane to fit. Adams, and Yablonovitch and Kane, proposed strain as a design tool in 1986. Compressive strain lifts the heavy-hole level further above the light-hole level and makes the in-plane hole mass lighter, bringing the valence-band density of states closer to that of the conduction band. Transparency is then reached at a lower carrier density, the differential gain rises, and loss processes that depend on the hole population, such as intervalence-band absorption, are reduced. Tensile strain puts the light holes on top instead and favors TM gain; polarization-insensitive semiconductor optical amplifiers use it to balance the TE and TM gain. Strain also extends the wavelengths a substrate can reach: compressively strained InGaAs wells on GaAs cover 900 to 1100 nm, including the 980 nm pump lasers of erbium-doped fiber amplifiers, and the critical thickness is what stops them near the long end of that range.

Materials and wavelengths

The two InP systems cover the same wavelengths but divide the band offset differently. In AlGaInAs about 70% of it falls in the conduction band, against about 40% in InGaAsP, so electrons are held more firmly at high temperature and the threshold rises less with heat: characteristic temperatures of 70 to 90 K against 50 to 70 K. This is why AlGaInAs wells are common in uncooled lasers. InGaN wells in blue and violet emitters are kept thin, because the piezoelectric field produced by their strain pulls the electrons and holes to opposite sides of a thicker well and reduces their overlap; efficiency falls as the indium content rises toward green.

Modulators and intersubband devices

Excitons, electron-hole pairs bound by their Coulomb attraction, are squeezed by the well and bound more tightly than in bulk material, so a sharp excitonic absorption peak at the absorption edge survives at room temperature. An electric field across the well pulls the electron and the hole toward opposite sides and shifts that peak to lower energy, while the barriers keep the pair from separating. This quantum-confined Stark effect, reported by Miller and colleagues in 1984, is the mechanism of electro-absorption modulators.

Transitions between two subbands of the same band, rather than between the conduction and valence bands, have energies set by the well design alone and lie mostly in the mid- and far-infrared. Quantum cascade lasers use them for emission and quantum-well infrared photodetectors for detection. Intersubband transitions in the conduction band couple only to light polarized along the growth direction, so these detectors need a grating or an angled facet to absorb light arriving at normal incidence.

Common questions

What is a quantum well?

A semiconductor layer, usually a few nanometers to about 20 nm thick, grown between two layers of wider bandgap, thin enough that the electrons and holes in it have discrete energy levels for motion across the layer. The levels depend on the thickness, so the emission wavelength can be set by the well width as well as by the composition.

Why do quantum-well lasers have lower threshold currents?

The thin well needs far fewer carriers to reach transparency than a bulk active layer, and its step-like density of states concentrates the injected carriers at the band edge, where they contribute to the gain. Compressive strain lowers the threshold further by lightening the holes.

How does the well width change the wavelength?

A narrower well raises the confined energies, which scale roughly as $1\mathrm{/}{L}_w^2$, and so shortens the wavelength. A GaAs well between Al₀.₃Ga₀.₇As barriers has its first transition near 848 nm at a width of 10 nm and near 801 nm at 4 nm.

What is the difference between a quantum well, a quantum wire and a quantum dot?

The number of confined directions. A well confines carriers in one direction and has a staircase density of states, a wire confines them in two and has peaks in the density of states at each level, and a dot confines them in all three and has discrete levels like an atom. Quantum-dot lasers use the discrete levels to make the threshold less sensitive to temperature.

References: R. Dingle, W. Wiegmann and C. H. Henry, "Quantum states of confined carriers in very thin AlₓGa₁₋ₓAs-GaAs-AlₓGa₁₋ₓAs heterostructures," Physical Review Letters 33, 827 (1974). W. T. Tsang, "Extremely low threshold (AlGa)As graded-index waveguide separate-confinement heterostructure lasers grown by molecular beam epitaxy," Applied Physics Letters 40, 217 (1982). D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood and C. A. Burrus, "Band-edge electroabsorption in quantum well structures: the quantum-confined Stark effect," Physical Review Letters 53, 2173 (1984). R. C. Miller, D. A. Kleinman and A. C. Gossard, "Energy-gap discontinuities and effective masses for GaAs-AlₓGa₁₋ₓAs quantum wells," Physical Review B 29, 7085 (1984). A. R. Adams, "Band-structure engineering for low-threshold high-efficiency semiconductor lasers," Electronics Letters 22, 249 (1986). E. Yablonovitch and E. O. Kane, "Reduction of lasing threshold current density by the lowering of valence band effective mass," Journal of Lightwave Technology 4, 504 (1986). N. Chand et al., "Excellent uniformity and very low (<50 A/cm²) threshold current density strained InGaAs quantum well diode lasers on GaAs substrate," Applied Physics Letters 58, 1704 (1991). I. Vurgaftman, J. R. Meyer and L. R. Ram-Mohan, "Band parameters for III-V compound semiconductors and their alloys," Journal of Applied Physics 89, 5815 (2001). L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012).