We have developed a Kramers–Kronig consistent analytical expression to fit the measured optical functions of hydrogenated amorphous silicon $\mathrm{(a-}\mathrm{Si:H})$ based alloys, i.e., the real and imaginary parts of the dielectric function ${\mathrm{(\epsilon }}_1{\mathrm{,\epsilon }}_2)$ (or the index of refraction n and absorption coefficient α) versus photon energy E for the alloys. The alloys of interest include amorphous silicon–germanium $\mathrm{(a-}{\mathrm{Si}}_{\text{1−x}}{\mathrm{Ge}}_x:\mathrm{H})$ and silicon–carbon $\mathrm{(a-}{\mathrm{Si}}_{\text{1−x}}{\mathrm{C}}_x:\mathrm{H}\mathrm{),}$ with band gaps ranging continuously from ∼1.30 to 1.95 eV. The analytical expression incorporates the minimum number of physically meaningful, E independent parameters required to fit ${\mathrm{(\epsilon }}_1{\mathrm{,\epsilon }}_2)$ versus E. The fit is performed simultaneously throughout the following three regions: (i) the below-band gap (or Urbach tail) region where α increases exponentially with E, (ii) the near-band gap region where transitions are assumed to occur between parabolic bands with constant dipole matrix element, and (iii) the above-band gap region where ${\mathrm{(\epsilon }}_1{\mathrm{,\epsilon }}_2)$ can be simulated assuming a single Lorentz oscillator. The expression developed here provides an improved description of ${\epsilon }_2$ (or α) in the below-band gap and near-band gap regions compared with previous approaches. Although the expression is more complicated analytically, it has numerous applications in the analysis and simulation of thin film $\mathrm{a-}\mathrm{Si:H}$ based $\mathrm{p-i-n}$ and $\mathrm{n-i-p}$ multilayer photovoltaic devices. First, we describe an approach whereby, from a single accessible measure of the optical band gap, the optical functions can be generated over the full solar spectrum for a sample set consisting of the highest quality intrinsic $\mathrm{a-}\mathrm{Si:H}$ based alloys prepared by plasma-enhanced chemical vapor deposition using the principle of maximal ${\mathrm{H}}_2$ dilution. Second, we describe quantitatively how such an approach can be modified for sample sets consisting of lower quality alloy materials. Finally, we demonstrate how the generated optical functions can be used in simulations of the absorption, reflection, and quantum efficiency spectra of $\mathrm{a-}\mathrm{Si:H}$ based single-junction and multijunction solar cells.

Most experimental research projects are executed with a combination of purchased hardware equipment, which may be modified in the laboratory and custom single-built equipment fabricated inhouse. However, the computer software that helps design and execute experiments and analyze data has an additional source: It can also be free and open-source software (FOSS) ([ 1 ][1]). FOSS has the advantage that the code is openly available for modification and is also often free of charge. In the past, customizing software has been much easier than custom-building equipment, which often can be quite costly because fabrication requires the skills of machinists, glassblowers, technicians, or outside suppliers. However, the open-source paradigm is now enabling creation of open-source scientific hardware by combining three-dimensional (3D) printing with open-source microcontrollers running on FOSS. These developments are illustrated below by several examples of equipment fabrication that can better meet particular specifications at substantially lower overall costs. ![Figure][2] Factory for one. A parametric (easily customized) filter-wheel holder is shown in the OpenSCAD program on the monitor ( left ), with the completed inside of the Arduino-controlled automated filter wheel ( center ). An Arduino-controlled RepRap 3D printer ( right ) is printing out a component of a case design. All of the hardware and software for both the filter wheel and the RepRap are open source. The FOSS movement emerged as a decentralized, participatory, and transparent system to develop software, in contrast to commercial software, which tends to be written anticipating user needs and does not allow modifications to the code, which is often proprietary ([ 2 ][3]). Although FOSS is a collaborative effort driven by user demands, this decentralized innovation process is still efficient and has been implemented in areas such as nanotechnology ([ 3 ][4]) and medicine ([ 4 ][5]), and the open and collaborative principles of FOSS have been readily transferred to hardware ([ 5 ][6]). A key enabling open-source hardware project is the Arduino electronic prototyping platform ([ 6 ][7]–[ 8 ][8]). The $20 to $30 Arduino is a versatile yet easy-to-learn microcontroller that can run a number of associated scientific instruments, including Arduino Geiger (radiation detector), pHduino (pH meter), Xoscillo (oscilloscope), and OpenPCR (DNA analysis). However, Arduino's most impressive enabling application is 3D printing. Open-source 3D printers can perform additive-layer manufacturing with polymers, ceramics, and metals. Such approaches have been popular in microfluidic lab-on-a-chip architectures, where flow paths are created layer by layer, but are adaptable to a much wider array of devices. The most popular fabrication tool is the RepRap, named because it is a partially self-replicating rapid prototyping machine. Currently, the <$1000 RepRap can fabricate about 50% of its own parts from acrylonitrile butyl styrene or polylactic acid polymers with no postprocessing and a 0.1-mm spatial precision. This ability for self-replication has resulted in an explosion of both RepRap users and evolutionary design improvements ([ 9 ][9]). Scientists with access to RepRaps have found many examples where it is less expensive to design and print research tools rather than buy them. A number of simple designs are flourishing in Thingiverse, a free and open repository for digital designs of physical objects ([ 10 ][10]). These include single-component prints such as vial racks (thing:25080), Buchner funnels (thing:25188), and microtiter plates (thing:11621). 3D printers have also been used to print custom chemical reactionware ([ 11 ][11]). For example, 3D printers can be outfitted with syringes to print with materials like acetoxysilicone to quickly make reactionware capable of in situ spectroelectrochemistry or easily alter reactor architecture to gauge the effects on chemical synthesis ([ 11 ][11]). The 3D printers can also be coupled with existing hardware tools such as the portable cell lysis device for DNA extraction (thing:18289), a 3D printable adapter that converts a Craftsman automatic hammer into a bead grinder for use with DNA extraction, or the DremelFuge chuck (thing:1483), a printable rotor for centrifuging standard microcentrifuge tubes and miniprep columns. These combination devices can radically reduce research costs. For example, the DremelFuge can be used in the laboratory or the field as an inexpensive centrifuge demonstrated to be effective up to 33,000 rpm or 52,000g. The price is ∼$50—primarily for the Dremel drill—compared with commercial centrifuge systems, which cost a few hundred dollars ([ 12 ][12]). The most aggressive research savings can come from coupling Arduinos with 3D printers to make full open-source scientific hardware. Consider the Arduino-controlled open-source orbital shaker (thing:5045) used for mammalian cell and tissue culture and bench-top science. The <$200 open-source orbital shaker fits inside a standard 37°C/5% CO2 cell incubator and replaces commercial versions that start at more than $1000. As the scientific tools that are open sourced gain complexity, the cost differential becomes even more substantial. For example, it is now possible to make a <$50 customizable automated filter wheel (thing:26553) that replaces $2500 commercial versions. For any given project, there may still be drawbacks to open-source 3D digital fabrication versus buying. Open-source scientific hardware is still at an early stage of the evolutionary process. Today, most major equipment is too complex to build in this manner (e.g., an entire nuclear magnetic resonance spectrometer versus a sample holder for one) or requires specialized materials that need to be fabricated in dedicated systems (e.g., ultrahigh vacuum semiconductor deposition). Commercial equipment may have longer lifetimes and,...