  Due to their narrow band gap and compatibility with silicon processing,
germanium-tin (Ge$_{1-x}$Sn$_x$) alloys are a versatile platform for scalable
integrated mid-infrared photonics. These semiconductors are typically grown on
silicon wafers using Ge as an interlayer. However, the large lattice mismatch
in this heteroepitaxy protocol leads to the build-up of compressive strain in
the grown layers. This compressive strain limits the material quality and its
thermal stability besides expanding the band gap, thereby increasing the Sn
content needed to cover a broader range in the mid-infrared. Released
Ge$_{1-x}$Sn$_x$ membranes provide an effective way to mitigate these harmful
effects of the epitaxial strain and control the band gap energy while enabling
the hybrid integration onto different substrates. With this perspective, herein
strain-relaxed Ge$_{0.89}$Sn$_{0.11}$ membranes are fabricated and subsequently
transfer-printed with metal contacts to create multiple photodetectors in a
single transfer step. The resulting photodetectors exhibit an extended
photodetection cutoff reaching a wavelength of $3.1 \,\mu$m, coupled with a
significant reduction in the dark current of two orders of magnitude as
compared to as-grown photoconductive devices. The latter yields a reduced
cutoff of $2.8 \,\mu$m due to the inherent compressive strain. Furthermore, the
impact of chemical treatment and annealing on the device performance was also
investigated showing a further reduction in the dark current. The demonstrated
transfer printing, along with the use of an adhesive layer, would allow the
transfer of multiple GeSn membranes onto virtually any substrate. This approach
paves the way for scalable fabrication of hybrid optoelectronic devices
leveraging the tunable band gap of Ge$_{1-x}$Sn$_x$ in the mid-wave infrared
range.
