Spatially selective laser cooling of carriers in semiconductor quantum wells

A successive four-step model is proposed for spatially selective laser cooling of carriers in undoped semiconductor quantum wells. The four physical steps include the following processes: (1) cold electrons with nearly zero kinetic energy are initially excited across a bandgap in a coherent and resonant way by using a weak laser field; (2) the induced cold carriers in two different bands are heated via inelastic phonon scattering to higher-energy states above their chemical potentials; (3) the resulting hot electrons and holes radiatively recombine to release photons, thus extracting more power from the quantum well than that acquired during the weak pump process; and (4) hot phonons in two surrounding hot barrier regions thermally diffuse into the central cool quantum well, thereby cooling the entire lattice with time. Based on this model, a thermal-diffusion equation for phonons including source terms from the carrier-phonon inelastic scattering and the thermal radiation received by the lattice from the surrounding environment is derived to study the evolution of the lattice temperature. At the same time, an energy-balance equation is applied to adiabatically find the spatial dependence of the carrier temperature for a given lattice temperature at each moment. There are two interesting findings in this paper. First, a V-shape feature in the carrier temperature is predicted by numerical calculations, which becomes apparent only for initial lattice temperature above 150 K. Second, a thermal-drag of the carrier temperature is found as a result of the strong carrier-phonon scattering. The difference between the lattice and carrier temperatures resulting from the thermal-drag effect is larger in the barrier regions than in the well region.

Figure 1

An illustration of successive four steps (indicated by bracketed numbers) of photoluminescence (PL)-induced spatially selective laser cooling model for carriers in an undoped semiconductor quantum well, where ${\mu }_e\left({\mu }_h\right)$ is the chemical potential of electrons (holes) in the quantum well. The upper inset of the figure also displays the thermal diffusion of phonons and the quantum well potential profile in the $z$ direction, as well as the ground-state wave function of electrons (curve with a peak). The curve with a V-shape-like feature in the inset represents the spatial profile of the carrier temperature.

Figure 2

An illustration of the spatial profiles of the lattice temperatures $T_L\left(z\right)$ as a function of the position $z$ with (solid curve) and without (dashed curve) the phonon thermal diffusion after the photoluminescence in the well (surrounded by two barriers) starts. A V-shape feature (dashed curve) at the well center in the spatial profile of the lattice temperature can be seen. The phonon diffusion reduces the lattice temperature in the barrier and increases the lattice temperature inside the well at the same time. Finally, a uniform $T_L\left(z\right)$ is established by including the fast phonon-diffusion effect (solid curve). Although the lattice temperature becomes independent of $z$, the carrier temperature can still exhibit a V-shape feature at the well center due to the lack of the thermal diffusion of carriers in the $z$ direction.

Figure 3

3D plot of the calculated carrier temperature $T$ as functions of time $t$ and position $\mid z\mid$ at $T_0=40\mathrm{K}$, where $T$ is a symmetric function of $z$ with respect to $z=0$, and $\mid z\mid =22.5\mathrm{\AA }$ represents two interfaces of the GaAs quantum well sandwiched by two ${\mathrm{Al}}_{0.3}{\mathrm{Ga}}_{0.7}\mathrm{As}$ barrier layers. Here, the half$-$V-shape feature at $z=0$ in the spatial profile of the carrier temperature is clearly visible within a few radiative lifetimes (nanoseconds) after the pump is turned on, and it disappears after $6\mu \mathrm{s}$ from the start of the photoluminescence.

Figure 4

3D plot of the calculated carrier temperature $T$ as functions of time $t$ and position $\mid z\mid$ at $T_0=300\mathrm{K}$. Here, the visible half-$V$-shape feature at $z=0$ in the spatial profile of the carrier temperature is retained, even after $6\mu \mathrm{s}$ from the start of the photoluminescence.

Figure 5

3D plot of the calculated power-loss density $W_{sp}-W_{ab}$ as functions of time $t$ and position $\mid z\mid$ at $T_0=40\mathrm{K}$, where $W_{sp}-W_{ab}$ is also a symmetric function of $z$ with respect to $z=0$.

Figure 6

3D plot of the calculated difference between the lattice and carrier temperatures $T_L-T$ as functions of time $t$ and position $\mid z\mid$ at $T_0=40\mathrm{K}$, where $T_L-T$ is a symmetric function of $z$ with respect to $z=0$.

Figure 7

3D plot of the calculated difference between the lattice and carrier temperatures $T_{\mathrm{L}}-T$ as functions of time $t$ and position $\mid z\mid$ at $T_0=77\mathrm{K}$.