Genetic-Algorithm Discovery of a Direct-Gap and Optically Allowed Superstructure from Indirect-Gap Si and Ge Semiconductors

Combining two indirect-gap materials—with different electronic and optical gaps—to create a direct gap material represents an ongoing theoretical challenge with potentially rewarding practical implications, such as optoelectronics integration on a single wafer. We provide an unexpected solution to this classic problem, by spatially melding two indirect-gap materials (Si and Ge) into one strongly dipole-allowed direct-gap material. We leverage a combination of genetic algorithms with a pseudopotential Hamiltonian to search through the astronomic number of variants of ${\mathrm{Si}}_n/{\mathrm{Ge}}_m/\dots /{\mathrm{Si}}_p/{\mathrm{Ge}}_q$ superstructures grown on (001) ${\mathrm{Si}}_{1-x}{\mathrm{Ge}}_x$. The search reveals a robust configurational motif—${\mathrm{SiGe}}_2{\mathrm{Si}}_2{\mathrm{Ge}}_2{\mathrm{SiGe}}_n$ on (001) ${\mathrm{Si}}_x{\mathrm{Ge}}_{1-x}$ substrate ($x\le 0.4$) presenting a direct and dipole-allowed gap resulting from an enhanced $\Gamma \mathrm{\text{−}}X$ coupling at the band edges.

Figure 1

Band structures of the ${\alpha }_{n=12}$ magic sequence grown on ${\mathrm{Si}}_{0.4}{\mathrm{Ge}}_{0.6}$. It is wave vector direct at $\overline{\overrightarrow{\Gamma }}$ and presents a nonzero dipole moment between the band edges. Thus it is an optically active direct-gap silicon material.

Figure 2

Comparison of between ${\mathrm{Si}}_6{\mathrm{Ge}}_4$ superlattice [18, 19] and the magic sequence discovered in this Letter (with a Ge buffer of 12 monolayers and grown on ${\mathrm{Si}}_{0.4}{\mathrm{Ge}}_{0.6}$). The top panel shows the direct absorption spectra (computed with a 20 meV gaussian broadening). The middle panel represents the reciprocal space orbital character of the CBM, e.g., the components of the CBM in the zinc blende Brillouin zone. The bottom most panel shows the location in real-space of the CBM and the VBM along the growth direction, with silicon layers in gray and Germanium regions in white. The CBM of both superlattices are mostly located on Si, and both contain some $\overrightarrow{\Gamma }$ character necessary for optical coupling with the VBM. However, the absorption spectra of the magic sequence is by far the largest.

Figure 3

(a) A dipole-allowed direct-gap material must present two distinct and necessary properties: (i) CBM and VBM are at the same location in crystal momentum space, and (ii) the transition between CBM and VBM is optically allowed. These two conditions are illustrated here for ${\mathrm{Si}}_6{\mathrm{Ge}}_4$, ${\alpha }_{12}$ (magic sequence), ${\alpha }_{14}$, $\beta$, and $\gamma$ superlattices, with respect to the substrate ${\mathrm{Si}}_{1-x}{\mathrm{Ge}}_x$. The solid lines measure the energy difference between the conduction band ${\overrightarrow{\Gamma }}_c$ and at the in-plane ${\overrightarrow{\Delta }}_c$. It is positive only when (i) is satisfied. The dashed line represent the dipole elements between the VBM and conduction band at $\overrightarrow{\Gamma }$. The superlattices discovered in this Letter show dipole elements much larger than the previous state-of-the-art ${\mathrm{Si}}_6{\mathrm{Ge}}_4$. (b) Effect of interface mixing on dipole transitions: interface mixing is modeled by replacing pure Si with ${\mathrm{Si}}_{1-x}{\mathrm{Ge}}_x$ and Ge with ${\mathrm{Si}}_x{\mathrm{Ge}}_{1-x}$ within the magic pattern and its edge (defined as two monolayers). For $x=0$, there is no mixing, and at $x=0.5$, the pattern has disappeared completely, since there is no contrast between Si rich and Ge rich layers.