Materials and device designs for thermophotovoltaic power conversion

Two IV–VI semiconductor alloys, Pb0.81Sn0.19Se and Pb0.8Sr0.2Se, are proposed for use in designing multiple quantum well (MQW) materials and devices for thermophotovoltaic (TPV) power conversion. These materials can be epitaxially grown on silicon substrates, so they offer the potential for a low cost TPV device manufacturing technology. MQW materials examples are provided for fabricating triple junction TPV devices for power conversion with a 1400°C radiator. Optimal n‐type and p‐type layer thicknesses for each junction were determined using internal quantum efficiency expressions derived assuming all photogenerated charge collection is by diffusion. Depending on MQW material quality and assumed optical absorption levels, predicted power generation for current matched triple junction devices ranged from 1.7 to 4.6 W/cm2 and power conversion efficiencies ranged from 15.9% to 35.7%. These device performance parameters were used in a levelized cost of energy (LCOE) calculation to predict the cost of energy provided by TPV devices with a molten silicon thermal energy storage system. Results show that even for TPV devices fabricated from low quality materials, such energy storage systems will have a low enough LCOE to be competitive with other forms of energy storage and power generation.


| INTRODUCTION
Thermal energy storage offers a low cost option for long duration energy storage at utility scale. An attractive thermal energy storage medium is molten silicon. 1 Silicon has an unusually large latent heat of fusion, 0.497 kWh/kg, at its 1414°C melting point and is among the most Earth abundant elements. Silicon offers both high energy storage density and low cost, and technologies for handling its molten form are well developed. The key challenge in developing useful thermal energy storage systems is having an effective way to convert stored thermal energy to electricity. In this paper we describe a materials technology that can be used to fabricate multijunction thermophotovoltaic (TPV) devices. 2 We show that multiple quantum well (MQW) materials composed of pseudomorphically strained Pb 0.81 Sn 0. 19 Se and Pb 0.8 Sr 0.2 Se layers can be used to fabricate multijunction TPV devices for conversion of thermal radiation to electrical power. We provide materials and device design models for fabricating multijunction TPV devices where the bandgap energy and layer thickness can be optimized for TPV power conversion applications. These IV-VI semiconductor materials can be grown as epitaxial layers on silicon substrates, which allows for low cost manufacturing. Based on known materials quality parameters for IV-VI semiconductor materials grown on silicon substrates, we show that it should be possible to fabricate triple junction TPV devices with optical-to-electrical power conversion efficiencies as high as 35.7% when converting radiation from a 1400°C thermal radiator, as would likely be the case for a molten silicon thermal energy storage system. A cost model for energy dispatched by a TPV-based thermal energy storage system is also provided, and it shows that the levelized cost of energy (LCOE) from a molten silicon thermal energy storage system can be below $0.10/kWh. The TPV materials and device technologies described in this paper offer a pathway for development of low cost energy storage technology. With no moving parts, other than pumps for circulating heat exchange fluids, TPV-based thermal energy storage systems will operate much like lithium ion batteries, but their cost will be much lower.
This paper begins with a materials design model involving use of alternating layers of Pb 0.81 Sn 0. 19 Se well and Pb 0.8 Sr 0.2 Se barrier alloys that comprise a MQW material. This model provides parameters for growth of MQW materials with bandgap energies between 400 and 700 meV. Analytical expressions are then derived for calculating the internal quantum efficiency (IQE) for electron and hole collection for a given material quality and layer thickness. An example material design is included for a triple junction TPV device designed for power conversion with a 1400°C thermal radiator. Optimal layer thicknesses are provided, and power generation densities and power conversion efficiencies are predicted. These predictions are then used in a cost model to estimate the LCOE for a molten silicon thermal storage system with IV-VI semiconductor TPV devices for power conversion.

| IV-VI SEMICONDUCTORS
Unlike III-V semiconductors, IV-VI semiconductors have a strain relaxation mechanism that allows the growth of high crystalline quality epitaxial layers on silicon substrates. Misfit dislocations in IV-VI semiconductor crystals can move easily in the <110> directions along the {100} planes. 3 The {100} planes in (111)-oriented thin films are at inclined angles, so dislocations created due to temperature cycling with a thermal expansion mismatched substrate will be able to move completely out of the layer, which is observed as triangular crosshatching on epilayer surfaces following cool down from 300°C to 360°C molecular beam epitaxial (MBE) growth temperatures. 4 Measurement of narrow X-ray diffraction peak widths following more than 1000 temperature cycles from room temperature to 77 K 5 further show that these thin films are highly tolerant to thermal stresses even at low temperatures. III-V semiconductors do not have this ability to deform plastically without leaving crystalline damage. Thermal expansion matched III-V semiconductor substrates, which are significantly more costly than silicon, are therefore required for III-V semiconductor device fabrication. Procedures for obtaining high crystalline quality IV-VI semiconductor materials on silicon substrates using MBE growth methods have been reported. [6][7][8][9] Materials characterization shows that MBEgrown IV-VI semiconductor MQW materials on silicon can have high charge carrier mobilities and high photoluminescence (PL) intensities. Successful fabrication of mid-IR focal plane arrays 10 and mid-IR vertical cavity surface emitting lasers 11 from IV-VI semiconductor layers grown on silicon demonstrate the viability of this materials technology for optoelectronic device fabrication. Figure 1 is a bandgap versus lattice parameter plot for PbSe-based narrow bandgap IV-VI semiconductor materials. The wide solid solubility range of tin and strontium in PbSe allows growth of Pb 1 − x Sn x Se and Pb 1 − x Sr x Se ternary alloys, which can comprise the well and barrier materials, respectively, of a MQW material.
Two ternary alloys are indicated in Figure 1, Pb 0.81 Sn 0.19 Se and Pb 0.8 Sr 0.2 Se. The critical layer thickness at which misfit dislocations begin to form due to the 0.79% lattice mismatch between these two alloys, is in the range of 17.5 nm. 12 Strain-balanced MQW materials, with the same 6.127 Å lattice parameter, can thus be obtained if layer thicknesses are below this critical layer thickness. The bandgap energies of the well and barrier materials will be affected by the alternating tensile and compressive strains. The change in bandgap energy is given by Equation (1), where D d and D u are the dilatation and uniaxial acoustic deformation potentials, respectively, for PbSe and  ε and  ε are the in-plane and out-of-plane strains. 13 Table 1 summarizes the effect of this strain, where the room temperature bandgap energies of the strained well and barrier layers are provided. The deformation potentials for PbSe are from Zasavitskii et al., 13 16 MQW materials will thus have a band structure composed of a lower energy normal valley subband and three degenerate oblique valley subbands at a higher energy as indicated in Figure 2. With this subband structure, the effective density of states will be smaller than that for a bulk material with a similar bandgap energy, an effect that reduces reverse bias saturation current and increases open circuit voltage. 17 Figure 3 shows interband transition energies in Pb 0. 8  In addition to enabling a more than 30% improvement in power generation density, 17 which makes IV-VI semiconductor TPV devices comparable in performance to III-V semiconductor TPV devices, a MQW materials design is also attractive for TPV device manufacturing. Rather than being controlled by alloy composition, the bandgap energy is instead determined by quantum well width, which in turn is determined by the timing of shutter openings and closings during growth of the Pb 0.81 Sn 0.19 Se well layer. Only four effusion cells are needed for growing these MQW materials, PbSe, SnSe, Sr, and Se, where Se flux is used to control epilayer stoichiometry, and all effusion cells remain at stable T A B L E 1 Parameters for determining strain modified bandgap energies for Pb 0.81 Sn 0.19 Se and Pb 0.8 Sr 0.2 Se layers in a pseudomorphically strained MQW material at 300 K.

Bulk material properties
Strained MQW material temperatures with constant fluxes during growth of different bandgap materials. A complete multijunction TPV device structure can therefore be grown with stable MBE growth chamber conditions, a feature that helps to achieve consistently high quality epitaxial layer growth. Ability to obtain a wide range of bandgap energies with the same MBE growth conditions will provide important flexibility in optimizing TPV device designs for different applications. For example, TPV devices for power conversion from relatively low radiator temperatures can be made with relatively wide quantum wells and long shutter open times, while devices for matching the higher energy photon flux from a relatively hot radiator can be made using shorter shutter open times. Such manufacturing flexibility provides many options for the commercialization of TPV devices for a range of power conversion applications, a benefit that can expedite manufacturing scale-up and TPV device cost reduction. Layers with n-type and p-type conductivity can be obtained with the inclusion of two dopant cells in the MBE growth chamber. Bi 2 Se 3 for n-type doping and Tl 2 Se for p-type doping have been demonstrated. 18 Silver 19 and sodium 20 have also been used for p-type doping. Hall effect measurements show that impurity doping in IV-VI semiconductors can give electron and hole majority carrier densities between 2 × 10 17 and 8 × 10 18 cm −3 . Controlling electron and hole concentrations below about 1 × 10 17 cm −3 is difficult with these materials due to the wide solid solubility range of anion and cation vacancies, which act as donors and acceptors, respectively, in IV-VI semiconductor materials. 21 Degenerate doping, which is necessary for forming tunnel junctions and obtaining series connected multijunction device structures, typically occurs at carrier densities above 3 × 10 18 cm −3 in these materials, 22 which is within the controllable doping range. MBE technology is therefore available for the growth of broad spectral coverage multijunction device structures with bandgap energies that are well matched to thermal radiation spectra.

| TPV MATERIALS DESIGN
A cross-sectional drawing of a triple junction TPV device structure with thermal radiation incident on the top junction is shown in Figure 4. The bandgap of each junction is selected to match a portion of the emission spectrum of a thermal radiator with the top junction having the largest bandgap energy and the bottom junction having the smallest. Figure 5   F I G U R E 5 Portions of the thermal emission spectrum that can be collected with a triple junction thermophotovoltaic device with bandgap energies of 707, 559, and 443 meV. The number of absorbable photons in each junction is indicated for a 1400°C radiator with an emissivity of 0.9. specific example for a triple junction TPV device with materials designed for absorption of radiation from a 1400°C radiator. Three spectral bands with approximately equal photon fluxes are obtained with top, middle, and bottom junction bandgap energies of 707, 559, and 443 meV, respectively. These bandgap energies can be obtained with Pb 0.8 Sr 0.2 Se/Pb 0.81 Sn 0.19 Se/Pb 0.8 Sr 0.2 Se MQW materials with well widths of 1.4, 2.45, and 3.5 nm, respectively. Assuming an upper limit of 1.1 eV for photon absorption by the top junction, these three junctions are illuminated with a total of 21.7 W/cm 2 of absorbable optical power, which is 54% of the total radiant optical power from a 1400°C radiator.
As shown in Figure 4, each junction has a lightly doped n-type side on a heavily doped p + side, which also serves as the p + side of two tunnel junctions. Two thin, heavily doped n + layers serve as the n + sides of the tunnel junctions, which provide a series connection of the three junctions through the middle junction. Table 2 provides materials parameters for the top, middle, and bottom junctions, each with a lightly doped n-type side and the more heavily doped p + side. Table 2 23  and depletion width, , for each junction are also provided.

| ABSORPTION AND CHARGE CARRIER GENERATION
The photogeneration rate for holes and electrons can be obtained from the Beer-Lambert law for photon absorp- is the photon flux as a function of depth, x, into the material, α is the absorption coefficient, and N o is the flux of absorbable photons (N o = incident photon flux − reflected photon flux). Assuming photon flux decrease is due to electron-hole pair generation across the semiconductor bandgap, the electron-hole pair generation rate can be expressed as the change in photon flux over distance, Optical absorption in the MQW material will be determined by the joint density of states associated with direct interband transitions between normal and oblique valley states. All four L-valleys will participate in optical absorption when photon energies exceed the oblique valley bandgap energy, so the joint density of states at the bottom of the oblique valley subband will be similar to that for a chemically similar bulk material. An estimate for MQW absorption coefficient can therefore be made from measured absorption coefficients for PbSe. Figure 6 shows plots of αe αx − for different absorption coefficients corresponding to photon energies that are up to 110 meV above the bandgap energy of PbSe. IV-VI semiconductors have relatively large absorption coefficients due to their multivalley band structure, 24 and this is reflected in the large generation rates within the first 2-3 μm of a IV-VI semiconductor material even for near-band edge generation where absorption is weakest.
T A B L E 2 MQW materials parameters at room temperature for three different bandgap materials. Note: Electron densities of 3 × 10 17 cm −3 and hole densities of 3 × 10 18 cm −3 are assumed for the n-type and p + -type sides of each junction, respectively. A relative permittivity of 22.9 is assumed for depletion width calculation. Table 3 lists the absorption band width for the top, middle, and bottom junctions described in Table 2. The middle and bottom junctions have spectral bands defined by their bandgaps and the bandgap(s) of the layer(s) above. In the case of the top junction, our model assumes zero absorption of photons at energies above 1100 meV, an energy range where the photon flux from a 1400°C radiator is relatively low and decreasing. Table 3 also lists the bandgap energy difference between normal and oblique valley direct interband transitions, the median photon energy above the bandgap in each spectral band for a 1400°C radiator, and the absorption coefficient of PbSe at the same above bandgap photon energy. The top junction has the broadest absorption band with a median photon energy of 903 meV, 196 meV above the MQW bandgap energy, where the representative absorption coefficient is 1.34 × 10 4 cm −1 according to Figure 6. With more narrow absorption bands and lower median photon energies above the normal valley bandgap, the middle and bottom junctions have lower representative absorption coefficients of 0.82 × 10 4 and 0.72 × 10 4 cm −1 , respectively.
Estimates for absorbable photon flux by each MQW material, N o , are provided in Table 4. Nonoptimized and optimized optical coatings are considered. The amount of optical absorption by a IV-VI semiconductor material can be estimated using the Fresnel equations, photon incidence angle, and index of refraction. Assuming an index of refraction of 4.5 for the top junction material, 25 the amount of light transmitted into the MQW material is about 60% up to an incidence angle of 75°from the normal. Absorption decreases dramatically for glancing incident photons, falling to below 45% for an incidence angle of 85°. A relatively large fraction of the photon flux from a planar radiator will be at these larger incident angles, so to account for this effect, which has been observed in TPV generator design, 26 we will assume 50% optical absorption for TPV devices fabricated without optimized optical coatings.
Optimization of optical coatings on both the TPV device and the radiator can enhance optical absorption. For example, PbSe mid-IR detectors fabricated with CaF 2 antireflective coatings exhibited more than 75% absorption across spectral bandwidths similar to those for the junctions modeled here. 27 Implementing photonic bandgap 28 and meta-optics 29 technologies, for example, can further enhance absorption by shaping, both spectrally and spatially, optical emission from a thermal radiator to match the absorption spectrum and geometry of the TPV device. For optimized optical coatings, absorption is assumed to increase to 60% for the top junction and to 80% for the middle and bottom junctions, where the more narrow and well defined absorption bands should allow more effective optical engineering. The incident photon flux for each junction is calculated using Plank's radiation law and assumes a 1400°C thermal radiator with an emissivity of 0.9, as would likely be the case for a molten silicon storage system and a thermal radiator fabricated from high emissivity silicon carbide materials.
absorption coefficients. The inset shows measured PbSe absorption coefficient as a function of photon energy. 24 Representative absorption coefficients for the above bandgap photon energies are indicated.
T A B L E 3 Absorption band width, bandgap difference between normal and oblique valleys, median photon energy above MQW bandgap for radiation from a 1400°C thermal radiator, and representative absorption coefficients for each junction.

| DIFFUSION COLLECTION MODEL
The charge generation curves shown in Figure 6 suggest that optimal junction thicknesses for PbSe-based TPV devices will be in the range of 2-3 μm. Photogenerated electrons and holes contribute to TPV device photocurrents when they drift through the depletion region. Photogenerated electrons and holes generated outside of the depletion region will diffuse until they either reach the depletion region edge and drift through it or they recombine before reaching the depletion region. Electrons and holes generated inside the depletion region will immediately drift and thus will all be collected, giving an IQE of 100% for this region of the junction material. However, depletion widths are less than 80 nm in these materials, so most photogeneration will be outside of the depletion region. A model for photocurrent calculation can be developed by assuming all charge collection is by diffusion of minority charge carriers to the metallurgical junction. In this model, the depletion width is assumed to be zero, and all charge collection occurs at x = d, as shown in Figure 4 for the top junction. This assumption, which introduces only a small underestimate of photocurrent since it neglects the advantage of 100% IQE for charge generated inside a depletion region, allows the derivation of analytical expressions for short circuit photocurrent as a function of materials and device design parameters. Note that this diffusion-only charge collection model can be used in general to analyze IV-VI semiconductor TPV devices made from either MQW or bulk materials.

| Hole collection from n-type top layer
A collection of holes generated in the n-type top layer, which has a thickness d, as shown in Figure 4, occurs when they diffuse though the layer and reach the junction where they drift into the p-type bottom layer and contribute to photocurrent. Hole collection probability, c(x), as a function of depth, x, into the layer is obtained from the minority carrier continuity equation 30 c x e e e e ( ) = + + , where L p , D p , and τ p are the hole diffusion length, hole diffusion coefficient, and hole lifetime, respectively. Equation (3) assumes that surface recombination velocity is negligible, less than about 5000 cm/s. The fraction of holes that are able to diffuse and become collected by the p-type side increases as x increases, reaching 100% for holes generated at the junction where x = d, the n-type layer thickness. Hole collection rate can be expressed as Substituting the hole generation rate, Equation (2), and the hole collection probability, Equation (3), into Equation (5) gives and after integrating  This rate of hole collection divided by the rate of absorbed photons, N o , is the internal quantum efficiency T A B L E 4 Estimates for absorbable photon fluxes from a 1400°C radiator with a 0.9 emissivity in the top, middle, and bottom junctions for a triple junction TPV device.  Figure 7 shows IQE h values obtained using Equation (8) and an absorption coefficient of 1.34 × 10 4 cm −1 as a function of n-type top layer thickness for three different hole diffusion lengths. Optimal IQE h values range from 37% for a low quality material with a hole diffusion length of 0.5 μm to 86% for a high quality material with a hole diffusion length of 4.5 μm. Figure 8 shows the calculated IQE h values for a material with an absorption coefficient of 0.72 × 10 4 cm −1 . Optimal IQE h values are slightly lower, ranging from 24% for a low quality material to 75% for a high quality material. Optimal layer thicknesses for low quality material range between 0.60 and 0.73 μm depending on absorption coefficient. High quality material, which can have two to three times larger IQE h values, will have optimal layer thicknesses between 2.1 and 2.8 μm. Table 5 summarizes IQE h and optimal layer thickness values for the top, middle, and bottom junctions for low, medium, and high quality materials with hole diffusion lengths of 0.5, 2.0, and 4.5 μm, respectively.

| Electron collection from p-type bottom layer
A collection of electrons generated in the p-type bottom layer, which has a thickness d, occurs when they diffuse though the layer and reach the junction where they then drift into the n-type top layer and contribute to photocurrent. Electron collection probability, c(x), as a F I G U R E 7 IQE h versus n-type layer thickness for different hole diffusion lengths in a high absorption (α = 1.34 × 10 4 cm −1 ) material.
where L n , D n , τ n , and d t are the electron diffusion length, electron diffusion coefficient, electron lifetime, and top layer thickness, respectively. The fraction of electrons that are able to diffuse in the negative x direction and become collected by the n-type side at x = d t increases as x decreases, reaching 100% at x = d t . The rate of electron collection from the p-type bottom layer as shown in Figure 9 can be expressed as Substituting the electron generation rate, Equation (2), and the electron collection probability, Equation (9) Figure 10 shows IQE e values obtained using Equation (14) for a high absorption coefficient material, α = 1.34 × 10 4 cm −1 , as would be the case for the top junction material according to data in Table 3. Plots are shown for three different electron diffusion lengths representing different quality materials. Values used for the top layer thicknesses, d t , are the optimal layer thickness for the n-type top layer as shown in Table 5. The maximum IQE e for a highquality material with a 4.5 μm electron diffusion length is 5.4%, whereas a low-quality material with a 0.5 μm electron diffusion length has a maximum IQE e of 19%. The influence of the top layer thickness is evident in these data. A lower quality material will have a thinner top layer, allowing the p-type bottom layer to be exposed to a larger portion of absorbable photon flux, and this allows the p-type bottom layer to participate more in overall junction performance. Figure 11 shows IQE e versus layer thickness for a F I G U R E 9 Junction parameters for modeling electron collection from the p-type bottom layer with thickness d and n-type top layer thickness d t .
F I G U R E 10 IQE e as a function of p-type layer thickness for different electron diffusion lengths in a high absorption (α = 1. 34 × 10 4 cm −1 ) material.

MCCANN and KHODR
| 2477 low absorption coefficient junction material, α = 0.72 × 10 4 cm −1 , as would be the case for a bottom junction material. Plots are shown for different electron diffusion lengths where the optimal top layer thicknesses, obtained from Table 5, are used for the d t values. The maximum IQE e for a high-quality material with a 4.5 μm electron diffusion length is 9.3%, whereas a low-quality material with a 0.5 μm electron diffusion length has a maximum IQE e of 16%. As with the high absorption coefficient material, the top layer thickness has a strong influence on how much charge collection is provided by the bottom p-type layer. Table 6 summarizes optimal layer thickness and IQE e data for high, medium, and low quality materials.

| SHORT CIRCUIT CURRENT
Short circuit current for each junction is the elementary charge times the sum of the hole and electron collection rates, which can be expressed in terms of absorbed photon flux and the internal quantum efficiencies for hole and electron collection, Absorbed photon flux data, as shown in Table 3, and Equations (8) and (14) can be used to calculate J sc values for triple junction IV-VI semiconductor MQW TPV devices. Optimizing triple junction device performance requires the matching of photogeneration currents for each junction. Layer thickness can be used as an adjustable parameter for achieving current matching conditions. Table 7 provides device design parameters for fabricating current-matched triple junction TPV devices with nonoptimized optical coatings. Table 8 provides the same parameters for devices with optimized optical coatings.

| ELECTRIC POWER GENERATION
Electric power delivered to a load by photovoltaic power generation is the product of the pn junction's short circuit current, open circuit voltage, and fill factor (FF) The open circuit voltage for each junction can be determined from the short circuit current, J sc , where J o is the reverse bias saturation current, . This expression assumes parasitic series and shunt resistances are zero and infinite, respectively, and that the pn junction has an ideality factor of 1. Table 9 provides a summary of predicted TPV device performance parameters for low and high quality materials with nonoptimized optical coatings, and Table 10 provides the same parameters for low and high quality materials with optimized optical coatings. Charge carrier mobility, μ, and minority carrier lifetime, τ, determine minority charge carrier diffusion length, T A B L E 7 Layer thicknesses and short circuit currents for current-matched junctions in a triple junction TPV device with nonoptimized optical coatings.
Low quality, L = 0.5 μm High quality, L = 4.5 μm T A B L E 9 Performance parameters for electrical power generation by a triple junction TPV device with a 1400°C thermal radiator.

Junction Layers
Low material quality High material quality μ = 200 cm 2 /Vs, τ = 0.5 ns, L = 0.5 μm μ = 800 cm 2 /Vs, τ = 10 ns, L = 4.5 μm Specific values for μ and τ are provided in Tables 9  and 10 as examples for low and high quality materials. A low quality material with a charge carrier mobility of 200 cm 2 /Vs and a minority carrier lifetime of 0.5 ns, for example, will have a minority carrier diffusion length of 0.5 μm. A high quality material with a charge carrier mobility of 800 cm 2 /Vs and a minority carrier lifetime of 10 ns, will have a minority carrier diffusion length of 4.5 μm.
TPV device power conversion efficiency is defined as electrical power generated divided by optical power absorbed by the TPV device. Optical power not absorbed by the TPV device, which includes reflection from the surface and below bandgap reflection by a back reflector, is assumed to return to the radiation source and is thus not considered a loss for efficiency calculations. Absorbed optical power includes absorption of photons by direct interband electronic transitions in the MQW material and by indirect phonon-assisted intervalley electronic transitions in the conduction and valence bands of n-type and p-type materials, respectively. An estimate for intervalley absorption can be made using the measured absorption coefficients for low energy photons in n-type GaSb where the primary photon absorption mechanism is by indirect phonon-assisted Γ-valley to L-valley electronic transitions, 32 which are similar to the normal-to-oblique intervalley electronic transitions in IV-VI semiconductor MQW materials. With the same absorption mechanism in both the conduction and valence bands in IV-VI semiconductors, the same absorption coefficients for n-type and p-type materials can be used. (This is not the case for p-type III-V semiconductors where the dominant photon absorption mechanism in the valence band is direct intervalence band transitions between the light, heavy, and split-off bands near the Γ point, an effect that makes p-type III-V semiconductors much more absorptive of low energy photons. 33 GaSb p-type samples, when compared to n-type samples with comparable charge carrier concentrations, have absorption coefficients that are three times larger at 500 meV and 10 times larger at 150 meV. 32 ) Measurements of n-type GaSb with a 2.2 × 10 17 cm −3 electron concentration showed absorption coefficients increased from 10 cm −1 at 443 meV to 25 cm −1 at 150 meV, a spectral region that covers 90% of the 16.3 W/cm 2 of radiation below 443 meV from a 1400°C radiator with an emissivity of 0.9. Absorption coefficients for n-type GaSb with a 1.3 × 10 18 cm −3 electron concentration increased from 50 to 150 cm −1 over the same spectral range. Assuming an absorption coefficient of 20 cm −1 for 8 μm of low charge carrier density IV-VI semiconductor MQW material and 150 cm −1 for 0.4 μm of high charge carrier density tunnel junction material gives a total absorption of 2.2%. A TPV device with a 443 meV bottom junction bandgap and nonoptimized optical coatings will have 8.2 W/cm 2 of below bandgap radiation transmitted into the device material, of which 179 mW/cm 2 will be dissipated as heat. When added to the 10.8 W/cm 2 of absorbed optical power within the MQW subbands, the total absorbed power increases by 1.7%. This estimate should be considered an upper limit for this below bandgap absorption effect since the assumed absorption coefficients are in the high range for the GaSb measurements and a generous thickness for the two tunnel junction is assumed. This optical absorption loss, being small, especially for devices with optimized doping and layer thicknesses, is not considered in the power conversion efficiencies discussed below.
Data for efficiency calculations can be obtained from Tables 4, 9 and 10. For example, a device with low quality material and nonoptimized optical coatings will generate 1.72 W/cm 2 of electrical power for a power conversion efficiency of 15.9%, while a high quality material will generate 3.86 W/cm 2 of electrical power for a power conversion efficiency of 35.7%. There will be more electrical power generated by devices with optimized optical coatings, 2.30 W/cm 2 and 4.61 W/cm 2 , for low and high quality materials, respectively, but with more optical power absorbed, 15.7 W/cm 2 , power conversion T A B L E 10 Performance parameters for electrical power generation by a triple junction TPV device with a 1400°C thermal radiator.

Low material quality
High material quality μ n = 200 cm 2 /Vs, τ = 0.5 ns, L = 0.5 μm μ = 800 cm 2 /Vs, τ = 10 ns, L = 4.5 μm efficiencies are lower, 14.6% and 29.4%, respectively. Absorption of more optical power reduces power conversion efficiency since there is more heat dissipation in the TPV device, which does not occur if incident optical power is reflected back to the radiator.
The photogenerated charge collection model described here shows that short circuit current densities with a 1400°C radiator are expected to be in the range of 2.3 to 5.3 A/cm 2 depending on material quality and optical coatings. As shown previously, 17 open circuit voltage for small bandgap materials can be a large fraction of E g /q when short circuit densities are large. For example, a 450 meV bandgap material will have an open circuit voltage of 182.3 mV at a short circuit current density of 0.5 A/cm 2 , while at a short circuit current density of 5 A/cm 2 open circuit voltage increases more than 30% to 241.8 mV. As described in Khodr et al.,17 this effect can make it worthwhile to incorporate small bandgap materials into TPV device designs when large short circuit densities are possible.
Minority charge carrier lifetime is a key materials parameter that determines TPV device performance. As with short circuit current density, there is a strong relationship between open circuit voltage and minority charge carrier lifetime. 17  For example, a 730 meV bandgap GaSb TPV device exposed to a SiC radiator at 1380°C exhibited a J sc of 3.92 A/cm 2 and a V oc of 0.44 V. 34 By comparison, our model predicts, for a similar bandgap 707 meV top junction material, J sc values between 2.3 and 5.3 A/cm 2 and V oc values between 0.39 V and 0.43 V. Open circuit voltage has a strong dependence on intrinsic charge carrier density, which is determined by the effective density of states in the conduction and valence bands of the MQW material. The (111)-oriented quantum confinement in MQW materials removes L-valley degeneracy and gives IV-VI semiconductors an intrinsic charge carrier density similar to that for a III-V semiconductor like GaSb. A good agreement between these modeling predictions and the results from experimental measurement of a similar bandgap material with a similar conduction band structure under similar operating conditions provides validation for the IV-VI semiconductor TPV device model presented here.

| COST MODEL
A molten silicon thermal energy storage system will be relatively low cost because its high latent heat energy density allows relatively small storage vessels requiring less insulation, which can be the most costly portion of a thermal energy storage vessel. Using cost estimates provided by Amy et al., 35 the total materials cost for a molten silicon thermal energy storage system is estimated to be in the range of $6.75 per kWh of energy storage capacity. Using this cost and assuming a roundtrip efficiency of 35.7%, 10 h of power dispatched per day, and a 30 year lifetime, the energy storage hardware portion of the system will contribute $0.0017/kWh to the LCOE. If power conversion efficiency is less because of low quality TPV device material, then a larger amount of thermal energy storage material would be needed. For example, energy storage hardware will contribute $0.0039/kWh to the LCOE for a system with 15.9% power conversion efficiency.
Costs for TPV power conversion hardware include the TPV device, TPV device cooling hardware, and power inverter electronics. Cooling hardware cost is estimated to be $0.08/W, and power inverter electronics cost is estimated to be $0.07/W. 35 TPV device cost is primarily the cost of epitaxial layer growth, which scales with thickness. Based on a TPV device fabrication cost analysis that assumes reuse of costly III-V semiconductor growth substrates, the cost for epitaxial growth of III-V semiconductor materials has been estimated to be $1/cm 2 . 35 The optimal III-V semiconductor layer thickness for a 1400°C radiator is in the range of 8 μm, 36 which is the same thickness for an optimized triple junction IV-VI semiconductor TPV device fabricated from high quality material, see Tables 7 and 8. Assuming a $1/cm 2 manufacturing cost, a IV-VI semiconductor TPV device made from high quality MQW material will have a cost between $0.22/W and $0.26/W depending on optical coating effectiveness. In the case of TPV devices made from low quality material, which perform optimally with thinner layers that match shorter minority carrier diffusion lengths, TPV device manufacturing cost will be less. A TPV device without optimized optical coatings and fabricated from low quality material, for example, will have a thickness of 3.36 μm, less than half of the 7.9 μm thickness of a TPV device made from high quality material. Assuming a $0.50/cm 2 cost for TPV devices fabricated with thinner MBE layers gives an estimated cost between $0.22/W and $0.29/W, depending on optical coating effectiveness. Table 11 provides example calculations for the contribution of TPV power conversion to the LCOE for a thermal energy storage system. TPV power conversion cost for dispatched energy from a thermal energy storage system is the cost of TPV power conversion hardware plus purchased energy cost divided by power conversion efficiency. For example, TPV power conversion hardware costing $0.40/W (assuming $0.25/W for the TPV device plus $0.15/W for device cooling and invertor electronics) with a 10 year service life will contribute $0.011/kWh to the LCOE, while a service life of 3 years will contribute $0.036/kWh. Assuming a purchased energy cost of $0.026/ kWh, which is the current low-range cost of wind power, 37 the marginal cost of dispatched energy is $0.073/kWh for TPV devices with 35.7% power conversion efficiency and $0.163/kWh for TPV devices with 15.9% power conversion efficiency. Adding TPV and thermal energy storage hardware costs gives LCOE values ranging from $0.086/ kWh to $0.203/kWh. Operational and maintenance costs are expected to be negligible with a TPV-based latent heat energy storage system. It remains at a constant temperature during charge and discharge cycles, and the only moving parts in the entire energy storage system are for pumps that circulate heat exchange fluids. These LCOE calculations assume a purchased energy cost of $0.026/ kWh. Lower purchased energy costs can significantly reduce these LCOE values. For example, if energy can be purchased at $0.013/kWh, which is predicted for future utility-scale solar photovoltaic systems, 38 then a TPV device with 15.9% power conversion efficiency, and a 3year operational lifetime will give a thermal energy storage LCOE of $0.12/kWh. This cost analysis shows that there will be economic viability even for TPV devices that are not fully optimized in terms of power conversion efficiency and operational lifetime.

| DISCUSSION
The ability to grow high crystalline quality narrow bandgap IV-VI semiconductor materials on low cost silicon substrates is the key technological capability that enables development of the low cost energy storage technology described above. Uniformly high quality IV-VI semiconductor material, as assessed by PL intensity, X-ray diffraction, and Hall effect measurements, across entire three-inch diameter (111)-oriented silicon wafers can be typically achieved with the appropriate MBE growth conditions, which include good oxide desorption and growth of a thin CaF 2 layer before IV-VI semiconductor layer growth, as described previously. 6 Unpublished Hall effect measurements have shown that room temperature electron and hole mobilities are as high as 1500 cm 2 /Vs for Pb 1 − x Sn x Se epilayers with a 10% tin content on the group IV sublattice, while they decrease to about 200 cm 2 /Vs for Pb 1 − x Sr x Se epilayers with a 20% strontium content on the group IV sublattice. The 800 cm 2 /s charge carrier mobility assumed for high quality MQW materials is near the average of these two mobilities, so there is experimental support for this assumption. There is also experimental support for the assumption of a 10 ns minority carrier lifetime for high quality material. For example, time-resolved PL measurements of PbSe epilayers grown on thermal expansion mismatched (111)-oriented InAs substrates showed minority carrier lifetimes of 20 ns, 39 and measurements of optical pumping lasing thresholds for PbSrS/PbS/PbSrS heterostructures grown on (111)-oriented silicon showed minority carrier lifetimes of 10 ns. 11 Therefore, the predictions for electrical power generation shown in Tables 9 and 10 should be considered realistic estimates for the performance of TPV devices made from these materials.
A TPV device fabricated with high quality material and without optimized optical coatings when exposed to a 1400°C radiator will absorb 10.8 W/cm 2 of optical power, of which 3.9 W/cm 2 will be converted to electrical power leaving 6.9 W/cm 2 of heat generation in the TPV device material. Most of this heat will be generated in the top junction material where the median absorbed photon T A B L E 11 Levelized cost of energy (LCOE) calculations for molten silicon thermal energy storage with IV-VI semiconductor TPV devices fabricated from low and high quality materials and with two different operational lifetimes.  9 Ability to tolerate thermal cycling strains when epilayer surfaces are repeatedly heated from room temperature to 83.5°C suggest that they will be reliable when used as TPV device materials in a thermal energy storage system. As discussed above, the same easy dislocation glide system that allows (111)-oriented IV-VI semiconductor thin films to deform plastically following MBE growth at 350°C also allows them to maintain high crystalline quality when used in a high temperature application. The TPV materials and device models presented here can be used generally to optimize layer thicknesses based on measured material parameters (charge carrier mobility from Hall effect measurements and minority charge carrier lifetime from reverse bias saturation current measurements) and predict device performance for different TPV power conversion applications. The device model here can be refined to incorporate the effects of a nonzero series resistance, which is likely for wider bandgap MQW materials where donor and acceptor doping can be less effective. Future device fabrication work should therefore focus on doping optimization of wider bandgap materials. This work can include investigation of PbEuSe alloys, 18 which have a similar bandgap energy dependence on alloy composition as PbSrSe, for use as the barrier layer in MQW materials. Device fabrication will require patterned etching and metallization of MBE-grown material. Of particular concern is the possibility of creating surface states on etched IV-VI semiconductor sidewalls since this could make surface recombination nonnegligible. The next step in IV-VI semiconductor TPV device development is to fabricate etched and metallized layer structures and demonstrate TPV power generation from fabricated devices. Device performance parameters can be used to determine materials quality parameters, which can then be used with the device model to optimize layer thicknesses. Although this paper highlighted a specific thermal energy storage application with a 1400°C radiator, there are many other applications such as in combined heat and power (CHP) where lower radiator temperatures will likely be involved. The MQW materials design model described in this paper offers flexibility in obtaining semiconductor materials, grown using manufacturingfriendly MBE growth methods, that will have optimal bandgaps for many different TPV device applications. Both the materials and device design models can also be used to understand the effects of lattice heating. The data in Tables 9 and 10 are for a TPV device with a lattice temperature of 25°C. The effects of a hotter lattice temperature, for example, are shown through Equations (17) and (18) and the intrinsic carrier density. At a lattice temperature of 52°C (325 K), which is expected to be the case for a TPV device exposed to a 1400°C radiator, the reverse bias saturation current increases by a factor of 4 in the bottom junction and by more than a factor of 8 in the top junction. These increases, which are due to the increased thermal generation of electron-hole pairs, reduce the open circuit voltage and power density by about 8%. Future modeling efforts can be built upon the basic models presented here. For example, finite element thermal modeling can be performed to assess TPV device performance for different thermal management approaches. Results from this future modeling work will help to guide the development of optimized TPV materials and devices for power conversion applications.
MBE growth of IV-VI semiconductor layers on silicon typically involves the use of a thin CaF 2 buffer layer. 6 An insulating CaF 2 layer under the TPV device material is attractive since it allows the complete electrical isolation of etched IV-VI semiconductor structures. Photolithographically patterned processing steps involving a mesa isolation etch down to the CaF 2 buffer layer, a contact via etch down to the bottom p-type layer, insulator deposition, contact via etches through the insulator to the n-type and p-type layers, and metal deposition allow fabrication of series connected TPV devices from these materials. Such monolithic series connection has been demonstrated for TPV devices fabricated from narrow bandgap III-V semiconductor materials. 43 In this case, however, additional processing steps involving removal of the GaSb growth substrate and transferring the epilayer to a semi-insulating GaAs handle wafer were needed so that fully electrically isolated devices could be obtained. On the other hand, CaF 2 -coated silicon growth substrates used for IV-VI semiconductor MQW materials growth allow chip-