Near-Infrared Quantum Cascade Lasers Designed with van der Waals Materials

Quantum cascade lasers (QCLs) constitute a leading source of coherent radiation in the mid-IR region. However, their performance outside this region remains unsatisfactory. Indeed, there are currently no QCLs in the near-IR region. Here, we propose that a superlattice of atomically thin layers held together by van der Waals forces may operate near room temperature as a compact and powerful near-IR QCL emitting at a wavelength of $1.66\mu \mathrm{m}$. It can compress over 100 stages within $0.5\mu \mathrm{m}$. The electric field required for operation is about $3\times {10}^6{\mathrm{V~cm}}^{-1}$, while the lasing threshold current density is about $22.4{\mathrm{kA~cm}}^{-2}$ depending on parameters. Rate-equation analysis reveals that the peak power per unit volume can reach over $0.1\mathrm{mW}\mu {\mathrm{m}}^{-3}$ in cw operation. Unlike existing QCLs, our device is $p$-type working with holes.

Figure 1

Schematic structure of the van der Waals quantum cascade laser. (a) Dimensions of the device. Emitted photons (green wiggly lines) are polarized along the thickness direction ($z$, double arrowed yellow lines) and propagate normal to it. A constant and uniform electric field $E$ is applied along positive $z$. (b) Side view of the vdW layers stacked into $N_s$ stages between two electrodes (yellow pads), which also use vdW layers and allow holes to be injected into the device. (c) Materials composition of a stage, containing an active region made of a two-layer ${\mathrm{Mo}\mathrm{S}}_2$ and an injector made of a three-layer ${\mathrm{Mo}\mathrm{Se}}_2$, which are separated by $\mathrm{In}\mathrm{Se}$ layers that serve as tunneling barriers. An active region is separated from the adjacent downstream (along the direction of $E$) injector by a monolayer $\mathrm{In}\mathrm{Se}$ but from the upstream one by a two-layer $\mathrm{In}\mathrm{Se}$ to achieve level alignment by the field $E$. The thickness of a stage is $d_s\approx 4.9\mathrm{nm}$. The whole structure is placed in an optical cavity.

Figure 2

Relative positions of the energy levels in a stage. (a) Alignment of the levels at the $\mathrm{\Gamma }$ point in the first Brillouin zone (1BZ) for $E=0$. The active region (two-layer ${\mathrm{Mo}\mathrm{S}}_2$) has two levels labeled $1$ and $2$, the injector (three-layer ${\mathrm{Mo}\mathrm{Se}}_2$) with three levels $1^{\prime }$, $2^{\prime }$, and $3^{\prime }$ while the barrier (monolayer $\mathrm{In}\mathrm{Se}$) with one lying below level $1$ by $\mathrm{\Delta }\approx 1\mathrm{eV}$. Level $3^{\prime }$ is irrelevant for the QCL. (b) Alignment of energy levels and (c) energy sub-band structure (computed using a $\mathbf{k}\cdot \mathbf{p}$ tight-binding model, well fitted by parabolic curves) for $E=300\mathrm{mV/nm}$ ($=3\times {10}^6{\mathrm{V~cm}}^{-1}$). The frequency of the emitted photons is primarily determined by the gap between $1$ and $2$, $\hslash {\omega }_0\approx 711\mathrm{meV}$. Only a tiny fraction approximately ${10}^{-6}$ of the 1BZ about the $\mathrm{\Gamma }$ point participates in the lasing action. The electric field $E$ shifts the levels so that they match in the manner as indicated. As the gap between $1$ and $2^{\prime }$ is about a half that between $2$ and $1^{\prime }$, the barrier separating the active region from the next downstream injector is thinner than that spacing it from the upstream one. Horizontal arrowed lines: quantum tunneling through barriers (${\gamma }_{1^{\prime }2}$ and ${\gamma }_{{12}^{\prime }}$). Vertical dashed lines: phonon-induced transitions (${\tau }_{21}^{-1}$ and ${\tau }_{2^{\prime }{1}^{\prime }}^{-1}$). Vertical solid lines: radiative transitions (${\tau }_R^{-1}$). Inset in (c): 1BZ. $k_M$: wave number at the $M$ point.

Figure 3

Lasing action in the device. Left axis: population inversion, $\delta f=f_2-f_1$, where $f_{1/2}$ is the population of holes in sub-band $1/2$. Right axis: number of photons in the cavity, $n$. $J$ is the injection current while $J_{\mathrm{th}}$ is the threshold current, and $\delta f_{\mathrm{th}}$ is the limit of $\delta f$ at large $J$. Lasing sets in for $J>J_{\mathrm{th}}$, where $n$ drastically increases as $\delta f$ approaches $\delta f_{\mathrm{th}}$. Parameters: photon frequency $\omega ={\omega }_0$, photon life time $\tau =100\mathrm{ps}$, and device area $W{L}^{\prime }=305\mu {\mathrm{m}}^2$, for which $J_{\mathrm{th}}\approx 67\mathrm{mA}$. Inset: sketch of the device.

Figure 4

Computed emission spectrum for a Fabry-Perot cavity. Intensity $P(\omega )$ versus photon frequency $\omega$ in (a) logarithmic and (b) linear scale for these parameters: out-coupling efficiency $\alpha =1$, mode confinement $\mathrm{\Lambda }=1$, photon lifetime $\tau =100\mathrm{ps}$, number of stages $N_s=100$, and dimensions $W=L^{\prime }=17.5\mu \mathrm{m}$, for which $J_{\mathrm{th}}\approx 67\mathrm{mA}$. Extreme line narrowing occurs to the four lasing modes for dramatic increase in the number of photons $n$. The dimension $L^{\prime }$ is chosen so that ${\omega }_0$ coincides with one of the eigenfrequencies of the cavity.