Polarization-dependent absorption in Ge/SiGe multiple quantum wells: Theory and experiment

Polarization resolved absorption spectra of a strain-compensated Ge multiple quantum well (MQW) structure with Ge-rich SiGe barriers have been calculated with an $s{p}^3{d}^5{s}^{\ast }$ tight-binding model and measured for light propagating perpendicular to the growth direction. The MQW was grown by low-energy plasma-enhanced chemical vapor deposition and consists of 50 Ge quantum wells deposited onto a thick graded ${\text{Si}}_{1-x}{\text{Ge}}_x$ buffer layer. The MQW was structurally characterized by high-resolution x-ray diffraction. The measured absorption spectra show clear quantum confined excitonic transitions related to the $\text{Ge}\Gamma$ point band gap, and strong dependence on the incident light polarization, as expected from selection rules for type I direct gap quantum confined systems. A good agreement between theoretically predicted spectra and experimental data is found, demonstrating light and heavy hole polarization-dependent selection rules in Ge MQWs.

Figure 1

(a) Layer stack of the MQW structure; (b) shape of the processed MQW sample (not to scale): the arrows show the optical path of the transmitted light. The angles $\alpha$ and $\gamma$ are $22°$ and $37°$, respectively (see text).

Figure 2

(Color online) Calculated band structure for the system discussed in the text. QW conduction subbands close to the $L$ and $\Gamma$ points are reported in green and violet, respectively. Subbands in the valence band are shown in red (heavy hole) and blue (light hole). The bulk band edge profiles, evaluated for the strained Ge and SiGe materials, are sketched together with the energies of the MQW states at the $L$ and $\Gamma$ points. In the inset the square modulus of the wave functions of the near-gap electron and hole states are shown. Folded states (black, see text) are present in the conduction band near the $\Gamma$ point.

Figure 3

Transmission spectra of the MQW sample measured at 300 K (dashed line) and at 5 K (full line) with unpolarized light (left-hand $y$ axis). The prism configuration of Fig. 1b was adopted. The low temperature tight-binding theoretical absorption coefficient (dotted line) is also reported (right-hand $y$ axis). The arrows indicate the $\text{HH}n\text{−}c\Gamma n$ and $\text{LH}n\text{−}c\Gamma n$ transitions; the low temperature bulk Ge direct gap energy is also reported for reference.

Figure 4

Low temperature measured OD (solid line) and tight-binding theoretical absorption spectra (dashed line) in TE (left-hand panel) and TM (right-hand panel) polarization. Vertical lines represent, in arbitrary units, the square modulus of the dipole matrix elements calculated between near gap valence and conduction states at the $\Gamma$ point. The weak transition at about 1.04 eV (not labeled) involves the SO1 and $c\Gamma 1$ states.