Theory and design of quantum cascade lasers in (111) $n$-type Si/SiGe

Although most work toward the realization of group IV quantum cascade lasers (QCLs) has focused on valence-band transitions, there are many desirable properties associated with the conduction band. We show that the commonly cited shortcomings of $n$-type Si/SiGe heterostructures can be overcome by moving to the (111) growth direction. Specifically, a large band offset and low effective mass are achievable and subband degeneracy is preserved. We predict net gain up to lattice temperatures of 90 K in a bound-to-continuum QCL with a double-metal waveguide, and show that a Ge interdiffusion length of at least $8\text{Å}$ across interfaces is tolerable.

Figure 1

Energy band schematic (not to scale) for a strained alloy layer on a rigid substrate. To find the potentials of the $\Delta$ valley minima in the strained layer, the difference in valence-band energy and band gap are calculated before adding the hydrostatic and uniaxial strain effects.

Figure 2

Usable range of $\Delta$-valley offsets for (111) and (001) oriented Si wells with ${\text{Si}}_{0.5}{\text{Ge}}_{0.5}$ barriers as a function of Ge fraction in the substrate. For the (111) orientation the entire quantum well may be used, whereas in (001) the ${\Delta }_{xy}$ well potential defines the upper limit.

Figure 3

A seven-well bound-to-continuum QCL at a biasof $7\text{kV}{\text{cm}}^{-1}$ with layer widths of $\mathbf{3.2}$/3.7/$\mathbf{0.8}$/6.4/$\mathbf{1.6}$/5.7/$\mathbf{1.8}$/5.1/$\mathbf{2.0}$/4.7/$\mathbf{2.0}$/4.3/$\mathbf{2.2}$/4.1, where boldface denotes 40% Ge barriers and lightface denotes pure Si wells. Dopants are spread evenly through the structure with a concentration of $5\times {10}^{16}{\text{cm}}^{-3}$. The conduction-band potential (solid line) is shown, with spatially dependent probability densities superimposed at each subband minimum. The upper laser subband is shown in bold, while other subbands are shown as dashed lines.

Figure 4

Active region gain at $7\text{kV}{\text{cm}}^{-1}$, and 4 K lattice temperature as a function of terahertz frequency for the design in Fig. 3. Arrows indicate linearly increasing diffusion lengths from 0 to $8\text{Å}$. The resulting frequency shift of the largest gain peak is shown inset.

Figure 5

Current density as a function of applied electric field for the QCL design in Fig. 3. Results are shown for linearly increasing lattice temperatures, in the direction of the arrow, between 4 and 100 K.

Figure 6

Peak gain near 5 THz as a function of lattice temperature for the QCL design in Fig. 3. Results are given for linewidths of 1.5, 2.0, and 2.5 meV. The threshold gains for 10 and $15\mu \text{m}$ active region thicknesses are shown as dotted and dashed lines, respectively.