Carrier confinement

Carrier confinement is the trapping of injected electrons and holes in the active region of a semiconductor laser or LED by steps in the band edges at heterojunctions. The bandgap difference between the active layer and the wider-gap layers around it divides into a conduction-band step $\mathrm{\Delta }{E}_c$, which holds the electrons, and a valence-band step $\mathrm{\Delta }{E}_v$, which holds the holes, with $\mathrm{\Delta }{E}_g=\mathrm{\Delta }{E}_c+\mathrm{\Delta }{E}_v$. The steps must be several times the thermal energy, $k_{B}T$ = 25.9 meV at 300 K, to hold the carriers: a GaAs active layer between Al₀.₃Ga₀.₇As layers has steps of about 0.21 eV for electrons and 0.16 eV for holes. Confinement concentrates the carriers where the optical mode is, which raises the carrier density obtained per unit current and with it the optical gain and the rate of radiative recombination. In modern lasers the carriers are held in quantum wells a few nanometers thick.

Band offsets at heterojunctions

How the gap difference divides between the two bands is described by the conduction-band share $Q=\mathrm{\Delta }{E}_c\mathrm{/}\mathrm{\Delta }{E}_g$, which depends on the pair of materials and is harder to measure than the gaps themselves. The value for GaAs/AlGaAs was revised from 0.85 (Dingle et al. 1974) to 0.57 (Miller et al. 1984).

The gap differences follow from $E_g$ = 1.424 + 1.247$x$ eV for AlₓGa₁₋ₓAs at 300 K (Casey and Panish), 1.344 eV for InP (Sze and Ng) against 0.800 eV for a gap at 1.55 µm, and the low-temperature gaps of 6.25 and 3.51 eV for AlN and GaN (Vurgaftman and Meyer). The shares are from Miller et al. for GaAs/AlGaAs, from Zah et al. for the two InP systems, and, for AlN/GaN, from the valence-band step of 0.70 eV measured by x-ray photoemission (Martin et al. 1996). Alloys of the same families divide their steps in roughly the same proportions. In a 1.55 µm multi-quantum-well laser the wells sit between InGaAsP barriers of intermediate gap rather than InP, so the steps around the wells are smaller than those in the table.

Electrons are lighter than holes, so their confined levels sit higher and they escape more easily, and the conduction-band step usually decides how well a structure confines. In InP-based lasers it is the weak point: for the same gap difference AlGaInAs places 1.8 times as much of the step in the conduction band as InGaAsP, which holds the electrons more firmly at high temperature and suits lasers that run without temperature control (Zah et al. 1994).

Barrier height and temperature

The fraction of carriers with enough thermal energy to cross a barrier of height $\mathrm{\Delta }E$ scales as $\exp (-\mathrm{\Delta }E\mathrm{/}{k}_{B}T)$. A barrier of $5k_{B}T$ leaves a Boltzmann factor of 0.7% and one of $10k_{B}T$ a factor of $4.5\times {10}^{-5}$; at 300 K that range is 130 to 260 meV. The prefactor depends on the densities of states and on how quickly carriers that cross are carried away, so the Boltzmann factor gives the temperature scaling of the leakage rather than its size. Because the factor depends on $\mathrm{\Delta }E\mathrm{/}{k}_{B}T$, a laser specified to 85 °C, where $k_{B}T$ is 30.9 meV against 25.7 meV at 25 °C, needs barriers 20% higher for the same confinement, and for a 0.2 eV barrier the factor grows 3.7 times between the two temperatures. Leakage over the barriers is one of the reasons the threshold current rises with temperature, as described under characteristic temperature.

Double heterostructures

A double heterostructure places a narrow-gap active layer between two wider-gap layers, so that electrons injected from the n side and holes from the p side meet in the active layer and are held there by the band steps on either side. Kroemer proposed the structure in 1963, and double-heterostructure lasers reached continuous operation at room temperature in 1970 (Hayashi et al. 1970; Alferov's group in the same year), work recognized by the 2000 Nobel Prize in Physics to Alferov and Kroemer. The active layer of a bulk double heterostructure is a fraction of a micrometer thick, too thick for quantization to matter: in a GaAs layer 0.1 µm thick the infinite-well levels for electrons are 1.7 meV apart, a fifteenth of $k_{B}T$ at 300 K. Because the narrower-gap material usually also has the higher refractive index, the same structure guides the light along the active layer, so the carriers and the photons are confined together.

Carrier confinement in quantum wells

A quantum well is an active layer a few nanometers thick, thin enough that the motion of the carriers across it is quantized. The carriers then occupy subbands whose lowest level lies above the band edge of the well, so the barrier a carrier must overcome to escape is the band step minus its confinement energy. For a 10 nm GaAs well between Al₀.₃Ga₀.₇As barriers with a 0.21 eV conduction-band step, a finite-well calculation with electron effective masses of 0.067 and 0.092 $m_0$ places the first electron level 29 meV above the band edge of the well, leaving an effective barrier of 184 meV, 7.1 $k_{B}T$ at 300 K. Narrowing the well to 6 nm raises the level to 57 meV and lowers the effective barrier to 156 meV, 6.0 $k_{B}T$, so thin wells confine less well than their band offsets alone suggest.

Most lasers use several wells in a multi-quantum-well active region, separated by barriers thick enough to keep the states of neighboring wells apart and low enough for carriers to cross between them. The barrier height is then a compromise between holding the carriers in each well and letting them spread across the stack; holes, heavier and slower, spread least evenly.

Quantum dots

A quantum dot confines carriers in all three dimensions, in nanometer-scale islands of narrow-gap material, so the carriers occupy discrete levels like those of an atom. When the levels are separated by more than $k_{B}T$, few carriers are thermally excited out of the lowest ones, which is one reason quantum dot lasers have low thresholds that depend weakly on temperature.

Capture, escape and carrier leakage

Injected carriers reach the wells by drift and diffusion across the surrounding layers and are captured into them, and they can escape again by thermionic emission over the barrier. For a well of width $L_w$ and effective barrier $E_B$ the thermionic escape time is

$${\tau }_{\text{esc}}=\sqrt{\frac{2\pi m^{\ast }{L}_w^2}{k_{B}T}}\exp \text{\hspace{0.17em}⁣}\left(\frac{E_B}{k_{B}T}\right)$$

(Nagarajan et al. 1992), which for electrons in the 10 nm GaAs well above, with $E_B$ = 184 meV, is about 120 ps at 300 K and 34 ps at 85 °C. Escape does little harm when capture back into the well is much faster. The ratio of the capture time to the escape time sets the share of carriers waiting outside the wells, and Nagarajan and colleagues showed that this share lowers the effective differential gain and with it the modulation bandwidth. Carriers that escape and are not recaptured recombine in the barriers and confinement layers or cross into the cladding. This leakage current grows roughly as the Boltzmann factor, and together with Auger recombination it lowers the characteristic temperature of InP-based lasers.

At high injection, as the wells fill, carriers occupy higher subbands and spill over into the barriers. The gain then rises more slowly with current, and this overflow, together with heating, limits the continuous output power. More wells to share the carriers, deeper barriers, and strained wells, whose lighter holes reduce the carrier density needed for gain, all reduce it.

Carrier confinement vs optical confinement

Carrier confinement and optical confinement are separate effects. Band-edge steps localize the electrons and holes, while a refractive-index step localizes the optical mode, and the fraction of the mode that overlaps the active layer is the confinement factor. A double heterostructure provides both at once, since lower-gap materials usually have the higher index, but the two profiles need not coincide. A quantum well is far too thin to guide light on its own, so a separate-confinement heterostructure (SCH) surrounds the wells with a waveguide layer of intermediate gap, often graded in composition, that holds the mode while the wells hold the carriers. The SCH is the standard design of 1310 and 1550 nm DFB lasers.

The cladding layers outside the waveguide carry the current and keep the mode away from the absorbing contact layers. They must be thick enough for the evanescent tail of the mode to decay before it reaches the contact and conductive enough to keep the series resistance low. Heavy doping lowers the resistance but adds free-carrier absorption, which grows in proportion to the carrier density, so the usual compromise is a doping profile that is heaviest near the contacts and graded down toward the active region.

Lateral carrier confinement and diffusion

Within the active layer the carriers also diffuse sideways, over a diffusion length

$$L_D=\sqrt{D\tau },$$

where $D$ is the ambipolar diffusion coefficient and $\tau$ the carrier lifetime. In GaAs the hole mobility of pure material, 400 cm²/V·s (Sze and Ng), gives through the Einstein relation a hole diffusion coefficient of 10 cm²/s. Electrons diffuse much faster, so at high injection the ambipolar coefficient is close to twice the hole value, about 20 cm²/s, and $L_D$ is 1.4 µm for a carrier lifetime of 1 ns and 2.0 µm for 2 ns; mobilities fall at the carrier densities of a laser, so these are upper estimates. A ridge-waveguide laser channels the current and guides the light under the ridge, but carriers in its unetched active layer diffuse out beyond the ridge edges. A buried heterostructure etches the active stripe and regrows wider-gap material around it, adding heterojunctions at the sides that confine the carriers laterally as well. Where the active layer is etched through and left open, as in micro-LEDs and etched pillars, carriers within about a diffusion length of the sidewall recombine at the surface, and the efficiency falls as the device shrinks toward that size (non-radiative recombination).

Electron blocking layers in GaN LEDs

In InGaN/GaN emitters most of the band step lies in the conduction band, as for the AlN/GaN pair in the table, but the electrons are still the harder carriers to keep in the active region. They are light and mobile, while holes, supplied by magnesium-doped GaN, are scarce and heavy, so injected electrons can cross the active region and recombine on the p side before meeting holes, and the holes that are injected populate mainly the well nearest the p side (David et al. 2008). An AlGaN electron blocking layer (EBL) between the last well and the p-type GaN raises the conduction-band barrier against this overflow; its valence-band step also impedes the injection of holes. Electron leakage is one of the mechanisms proposed for efficiency droop, the fall in quantum efficiency of InGaN LEDs at high current density that also lowers their wall-plug efficiency (Piprek 2010). Auger recombination is another leading candidate, and Iveland et al. (2013), who detected Auger-generated hot electrons emitted from an operating LED, identified it as the dominant one.

Common questions

What is carrier confinement in a laser diode?

The trapping of the injected electrons and holes in the thin active layer by the band-edge steps at the heterojunctions on either side. It keeps the carrier density high where the optical mode is, so that a small current gives enough gain to reach threshold.

What is the difference between carrier confinement and optical confinement?

Carrier confinement holds electrons and holes in the active layer through band-edge steps; optical confinement holds the light in a waveguide through refractive-index steps. A separate-confinement heterostructure uses quantum wells a few nanometers thick for the first and a much thicker waveguide layer of intermediate bandgap for the second.

How high must a confining barrier be?

Several times the thermal energy. Barriers of 5 to 10 $k_{B}T$, 130 to 260 meV at 300 K, reduce the Boltzmann factor for escape to between 0.7% and 0.005%. Lasers specified to 85 °C need about 20% more, and thin quantum wells need extra margin because their confined levels sit above the band edge.

Why are InP-based lasers more sensitive to temperature than GaAs lasers?

InGaAsP places only about 40% of the gap difference in the conduction band, so electrons leak over the barriers as the temperature rises, and Auger recombination, strong in narrow-gap materials, adds to the loss. AlGaInAs, with about 72% of the step in the conduction band, holds the electrons better and is used for uncooled lasers.

What does an electron blocking layer do?

It is a thin wider-gap layer, usually AlGaN in nitride emitters, placed between the active region and the p side to keep electrons from overflowing the wells into the p-type material. Its valence-band step also hinders the injection of holes, so its composition and doping are chosen as a compromise between the two.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 17; Coldren, Corzine & Mašanović, Diode Lasers and Photonic Integrated Circuits (2nd ed., Wiley, 2012), ch. 1 and appendix 2, on heterostructure confinement and carrier leakage; Piprek, Semiconductor Optoelectronic Devices (Academic Press, 2003) for the device-physics derivations; Yu & Cardona, Fundamentals of Semiconductors (4th ed., Springer, 2010) for the band-offset theory; H. C. Casey Jr. and M. B. Panish, Heterostructure Lasers, Part A: Fundamental Principles (Academic Press, 1978); S. M. Sze and K. K. Ng, Physics of Semiconductor Devices (3rd ed., Wiley, 2007); R. Dingle, W. Wiegmann and C. H. Henry, "Quantum states of confined carriers in very thin AlₓGa₁₋ₓAs-GaAs-AlₓGa₁₋ₓAs heterostructures," Phys. Rev. Lett. 33, 827 (1974); R. C. Miller, D. A. Kleinman and A. C. Gossard, "Energy-gap discontinuities and effective masses for GaAs-AlₓGa₁₋ₓAs quantum wells," Phys. Rev. B 29, 7085 (1984); I. Vurgaftman, J. R. Meyer and L. R. Ram-Mohan, "Band parameters for III-V compound semiconductors and their alloys," J. Appl. Phys. 89, 5815 (2001), for the effective masses; I. Vurgaftman and J. R. Meyer, "Band parameters for nitrogen-containing semiconductors," J. Appl. Phys. 94, 3675 (2003); G. Martin, A. Botchkarev, A. Rockett and H. Morkoç, "Valence-band discontinuities of wurtzite GaN, AlN, and InN heterojunctions measured by x-ray photoemission spectroscopy," Appl. Phys. Lett. 68, 2541 (1996); C. E. Zah et al., "High-performance uncooled 1.3-µm AlₓGaᵧIn₁₋ₓ₋ᵧAs/InP strained-layer quantum-well lasers for subscriber loop applications," IEEE J. Quantum Electron. 30, 511 (1994); H. Kroemer, "A proposed class of hetero-junction injection lasers," Proc. IEEE 51, 1782 (1963); I. Hayashi, M. B. Panish, P. W. Foy and S. Sumski, "Junction lasers which operate continuously at room temperature," Appl. Phys. Lett. 17, 109 (1970); Zh. I. Alferov, "Nobel Lecture: The double heterostructure concept and its applications in physics, electronics, and technology," Rev. Mod. Phys. 73, 767 (2001); R. Nagarajan, M. Ishikawa, T. Fukushima, R. S. Geels and J. E. Bowers, "High speed quantum-well lasers and carrier transport effects," IEEE J. Quantum Electron. 28, 1990 (1992); A. David, M. J. Grundmann, J. F. Kaeding, N. F. Gardner, T. G. Mihopoulos and M. R. Krames, "Carrier distribution in (0001)InGaN/GaN multiple quantum well light-emitting diodes," Appl. Phys. Lett. 92, 053502 (2008); J. Piprek, "Efficiency droop in nitride-based light-emitting diodes," Phys. Status Solidi A 207, 2217 (2010); J. Iveland, L. Martinelli, J. Peretti, J. S. Speck and C. Weisbuch, "Direct measurement of Auger electrons emitted from a semiconductor light-emitting diode under electrical injection: identification of the dominant mechanism for efficiency droop," Phys. Rev. Lett. 110, 177406 (2013).