Figure 1

(Color online) Cartoon of population injection in midinfrared (left) vs terahertz (right) structures. Population inversion can be easily achieved on the left but not on the right. In this case, the electrons can tunnel to either the upper or lower levels: it is difficult to achieve population inversion in terahertz intersubband–based devices.Reuse & Permissions

Figure 2

(Color online) The main part of the plot shows the hole occupation functions (probability of occupation of the subband at a given point in $k$ space) of the top two valence subbands of a 10 nm ${\text{GaAs-Al}}_{0.3}{\text{Ga}}_{0.7}\text{As}$ quantum well. The electronic temperature in both subbands is $T=300\text{K}$ and both have the same occupation $n_1=n_2=2\times {10}^{12}\text{carriers}/{\text{cm}}^2$. The dashed and solid curves are respectively for $n_1\left(k\right)$ and $n_2\left(k\right)$ with the corresponding valence bands shown in the inset with the same convention for curves. The dot-dashed line is the transition dipole in nm $\left[d_{12}\left(k\right)={\wp }_{12}\left(k\right)/e\right]$ scaled down by a factor of 10 to fit in the figure. The dipole is highly concentrated around $k=1.5$ leading to the $k$-space filter effect discussed in the main body of the paper.Reuse & Permissions

Figure 3

(Color online) TE absorption spectrum. The electronic temperature in both subbands is $T=300\text{K}$ and both have the same occupation $n_1=n_2=2\times {10}^{12}\text{carriers}/{\text{cm}}^2$, which correspond, respectively, to the solid and dashed curves in Fig. 2. Here, the solid absorption curve has been calculated with the full momentum dependence for all transitions; the (2,1) dipole is illustrated by a dot-dashed curve in Fig. 2. In the dashed curve the actual dipoles have been replaced by constants, i.e., the maximum value of each transition dipole. The inset shows in detail the low-frequency region where gain $\left(\text{absorption}\times -1\right)$ develops. In other words, $k$-space filtering is included in the solid curve by means of the actual dipole moment leading to terahertz gain. It is not taken into account in the dashed curve and the gain disappears. The horizontal dotted line is a guide to eyes. It separates the absorption from the gain region between 0 and 3.67 THz.Reuse & Permissions

Figure 4

(Color online) TE gain spectrum. (a) Both subbands have the same carrier density $n_1=n_2=2\times {10}^{12}\text{carriers}/{\text{cm}}^2$. The top hole subband temperature is $T_2=300\text{K}$. The solid, dashed, and dot-dashed curves correspond, respectively, to lower subband temperatures $T_1=300$, 310, and 320 K. (b) Here the holes at both subbands are at the same temperature $T_1=T_2=300\text{K}$. The top subband has a density $n_2=2\times {10}^{12}\text{carriers}/{\text{cm}}^2$, while from top to bottom, the solid, dashed, dot-dashed, and dotted curves correspond, respectively, to lower subband densities $n_1=2$, 2.1, 2.2, and $2.3\times {10}^{12}\text{carriers}/{\text{cm}}^2$. (c) All curves have the first and second subband occupations given, respectively, by $n_1=2.3\times {10}^{12}\text{carriers}/{\text{cm}}^2$ and $n_2=2\times {10}^{12}\text{carriers}/{\text{cm}}^2$ and $T_2=300\text{K}$. The dotted line represents the same curve as in (b), i.e., $T_1=300$. The lower subband temperature is progressively raised from bottom to top. The dot-dashed, dashed, and solid curves correspond, respectively, to $T_1=310$, 320, and 330 K.Reuse & Permissions

Figure 5

TE gain spectrum. (a) The carriers in both subband have the same temperature, $T_1=T_2=300\text{K}$, and the same total carrier density. From top to bottom the total carrier density carriers per subband is decreased by $n_1=n_2=2$, 1.6, 1.2, 0.8, 0.4, and $0.2\times {10}^{12}\text{carriers}/{\text{cm}}^2$.Reuse & Permissions

Figure 6

(Color online) TE absorption spectrum. (a) The carriers in both subband have the same temperature, $T_1=T_2=300\text{K}$, and the same total carrier density $n_1=n_2=2\times {10}^{11}\text{carriers}/{\text{cm}}^2$. The solid curve includes all transitions and from left to right, the dashed, dash-double-dotted, dotted, dot–dashed and dot-double-dashed curves correspond, respectively, to transitions between the final $j$ and initial $i$ subbands $\left(j,i\right)=\left(2,1\right)$, (3,2), (4,2), (3,1), and (4,1).Reuse & Permissions