A <double heterostructure laser> electrically pumps its central layer for <stimulated emission>. <Band offsets> confine <charge carriers>; <refractive index> contrast confines the optical <normal mode>.<Nobel Lecture: The double heterostructure concept and its applications in physics, electronics, and technology by Zhores Ivanovich Alferov>[§II, Fig. 1, p. 768]{ref}

\Image[/semiconductor-heterostructure-band-injection.png]
{title=Chosen <type-I band alignment>: <carrier injection> and optical readout}
{description=Original schematic using stipulated <energy> units. The upper panel plots electron <energy band> edges across the layers; the lower panel separates electrical pumping from the emerging <light>. Flat segments omit <band bending>. The curves do not reproduce a historical device.}
{source=semiconductor-heterostructure-band-injection.py}

For an original bookkeeping model, let $R$ be the injected pair rate per unit area, $d$ the active-layer thickness, $N$ its excess pair <number density>, $\tau$ a recombination lifetime, and $\tau_e$ an escape time. Before appreciable <stimulated emission>, suppose
$$
\frac{dN}{dt}=\frac{R}{d}-\frac{N}{\tau}-\frac{N}{\tau_e},\qquad
N_{\rm ss}=\frac{R\tau}{d}\frac{1}{1+\tau/\tau_e}.
$$
Choose two otherwise identical devices with $\tau_e=\tau$ and $\tau_e=9\tau$. The resulting <steady state> populations are $R\tau/(2d)$ and $9R\tau/(10d)$: improved confinement raises the population by a factor $1.8$ at fixed injection. This linear model assumes equal excess <Electron> and <electron hole> densities, fixed lifetimes, and no <laser gain clamping>.

Optical confinement enters separately. For a chosen optical <normal mode>, define $0<\Gamma\leq1$ as its gain-overlap factor with the active layer. With material <optical gain> $g$, internal loss $\alpha_i$, and mirror loss $\alpha_m$, a simple <laser threshold> condition is
$$
\Gamma g=\alpha_i+\alpha_m.
$$
For stipulated losses $\alpha_i+\alpha_m=3$ in inverse-length units, $\Gamma=1/2$ requires $g=6$, whereas $\Gamma=3/4$ requires $g=4$. This calculation varies optical overlap only; the preceding population calculation varies carrier escape only. Their combination explains the usefulness of controlling both the <energy band> profile and the optical <normal mode>.

<Herbert Kroemer> proposed the <double heterostructure laser> in 1963; he did not participate in its 1970 experimental realization.<Nobel Lecture: Quasielectric fields and band offsets: teaching electrons new tricks by Herbert Kroemer>[§IV, p. 787]{ref} <Zhores I. Alferov> describes his group's independent 1963 proposal and later experimental program.<Nobel Lecture: The double heterostructure concept and its applications in physics, electronics, and technology by Zhores Ivanovich Alferov>[§II, pp. 768–770]{ref}


 Double heterostructure laser (source code)