Plasmonic light trapping for enhanced light absorption in film-coupled ultrathin metamaterial thermophotovoltaic cells

Abstract

Ultrathin cells have gained increasing attention due to their potential for reduced weight, reduced cost and increased flexibility. However, the light absorption in ultrathin cells is usually very weak compared to the corresponding bulk cells. To achieve enhanced photon absorption in ultrathin thermophotovoltaic (TPV) cells, this work proposed a film-coupled metamaterial structure made of nanometer-thick gallium antimonide (GaSb) layer sandwiched by a top one-dimensional (1D) metallic grating and a bottom metal film. The spectral normal absorptance of the proposed structure was calculated using the rigorous coupled-wave algorithm (RCWA) and the absorption enhancement was elucidated to be attributed to the excitations of magnetic polariton (MP), surface plasmon polariton (SPP), and Fabry-Perot (FP) resonance. The mechanisms of MP, SPP, and FP were further confirmed by an inductor-capacitor circuit model, dispersion relation, and phase shift, respectively. Effects of grating period, width, spacer thickness, as well as incidence angle were discussed. Moreover, short-circuit current density, open-circuit voltage, output electric power, and conversion efficiency were evaluated for the ultrathin GaSb TPV cell with a film-coupled metamaterial structure. This work will facilitate the development of next-generation low-cost ultrathin infrared TPV cells.

Keywords

metamaterial / thermophotovoltaic / plasmonics / light trapping / selective absorption

Introduction

Much effort has been devoted to study solar cells with thicknesses of a few micrometers or even less for reduced cost, reduced weight, and increased flexibility [¹–⁵]. One of the major challenges in thin-film solar cells is weak light absorption. Therefore, a light-trapping structure to enhance the light absorption is usually required. In recent years, film-coupled metamaterials, consisting of sub-wavelength periodic gratings on a dielectric layer and a metallic ground plane, have drawn much attention for enhancing energy absorption or thermal emission [⁶–¹²]. Excitation of magnetic polaritons (MPs) [⁶] has been demonstrated as one of the main mechanisms responsible for spectrally enhanced photon absorption above the bandgap in these metamaterials. However, most of the published works focused on enhancing photon absorption for solar absorbers [⁹,¹³–¹⁵], solar cells [¹¹], and thermophotovoltaic (TPV) emitters [⁸,¹⁰,¹⁴]. Few studies have looked into how to enhance the photon absorption in ultrathin narrow-bandgap semiconductors, which could be used for infrared detectors or TPV cells [¹⁶].

This work aims to investigate the light trapping in an ultrathin narrow-bandgap TPV cell with a film-coupled metamaterial structure, as shown in Fig. 1. The top metallic grating and bottom metallic film cannot only excite MP modes, but also act as electrodes for collecting photon-generated free charges. Gallium antimonide (GaSb) with an arrow direct bandgap of 0.72 eV at room temperature, is considered here as a promising candidate for high-efficiency TPV cells. The spectral normal absorptance of the metamaterial structure will be numerically modeled by the rigorous coupled-wave algorithm (RCWA). Electromagnetic fields are plotted to elucidate the underlying mechanisms responsible for enhanced infrared light absorption and an inductor-capacitor (LC) circuit model is utilized to further confirm the excitation of MP. The dispersion relation and phase shift are used to help understand surface Plasmon polariton (SPP) behaviors and Fabry-Perot (FP) resonance, respectively. Effects of grating period, width, spacer thickness, as well as incidence angle will be discussed. Lastly, the performance of the TPV cell will be evaluated.

Numerical method

Figure 1 depicts the periodic film-coupled metamaterial structure under investigation, which is made of a one-dimensional (1D) Ag grating on a GaSb spacer layer and Ag substrate. The grating has a thickness of h = 30 nm, period of L = 200 nm, and width of w = 100 nm, while the thickness of the GaSb spacer is d = 30 nm. Note that the bottom Ag ground plane could be as thin as 100 nm to be opaque in the visible and near-infrared region. The RCWA method was employed to perform the numerical calculations for the radiative properties of the film-coupled metamaterial structure shown in Fig. 1. RCWA is a widely used method to obtain radiative properties of periodic grating nanostructures [⁸,¹⁰,¹⁷]. Based on Floquet’s theorem, this method solves for the solutions of periodic differential equations expanded with Bloch waves. After multiple scatterings in the periodic grating structure, the reflected electromagnetic wave is expanded to a series of diffracted ones with different orders. The algorithm details can be found in Refs. [¹⁸–²⁰]. Note that the convergence of RCWA strongly depends on the total diffraction orders, while a total of 101 diffraction orders were carefully checked to be sufficient and thus used here.

The dielectric function of Ag, e_Ag, is given by a Drude model [²¹]:

(1){\epsilon }_{\mathrm{Ag}}={\epsilon }_{\infty }-{\displaystyle \frac{{\omega }_{\mathrm{p}}^{\mathrm{2}}}{{\omega }^{\mathrm{2}}\mathrm{+i}\omega \mathrm{\Gamma }}},

where e_∞ = 3.7 is the infrared dielectric constant, w_p = 9.2 eV is the unscreened plasma frequency, and Г = 0.02 eV is the damping coefficient at room temperature. The optical properties of GaSb were taken from Palik’s tabulated data [²²].

Results and discussion

Radiative properties at normal incidence

Spectral normal absorptance of the film-coupled metamaterial TPV cell was calculated under transverse magnetic (TM) and transverse electric (TE) incident waves, as shown in Fig. 2(a). The shaded region represents the useful photon region above the GaSb bandgap. It is observed that there are three major absorption peaks (a = 0.77 at l = 1.69 mm, a = 0.96 at l = 0.87 mm, and a = 0.97 at l = 0.36 mm) under TM incident waves, while there is only one major absorption peak (a = 0.83 at l = 0.73 mm) under TE incident waves. All the absorption peaks are above the bandgap of GaSb and thus could effectively enhance the light absorption for photon-generated carriers. It is important to understand the physical mechanisms that are responsible for the enhanced light absorption in the film-coupled metamaterial structure.

Figure 2(b) compares the normal absorptance of three structures under TM incident waves: the film-coupled metamaterial structure, a GaSb-on-Ag structure, and a free-standing GaSb layer. All three structures have a 30-nm GaSb layer. It is found that without the upper Ag grating, the absorption peak at l = 1.69 mm as well as the peak at l = 0.36 mm disappear, while the peak at l = 0.87 mm still exists but slightly blueshifts. For a free-standing 30-nm GaSb layer, the normal absorptance is as low as 0.15 particularly in the wavelength range from l = 0.7 mm to the bandgap due to the small intrinsic absorption coefficient of GaSb.

Excitation of MP

The resonance peak at l = 1.69 mm is attributed to the excitation of MP. To help explain the underlying mechanism for the absorption peak, the distribution of the electromagnetic fields at the x-z cross section inside the structure is illustrated in Fig.3(a). The contour represents the logarithm of the squared magnetic field (normalized to the incident magnetic field), and the arrows indicate the electric field vectors. The Ag grating, the GaSb spacer, and the Ag substrate in one period are delineated. It can be clearly seen that the electric field inside the GaSb layer forms a current loop between the upper Ag grating and the lower Ag substrate. In addition, the magnetic field is confined in the GaSb layer with one order of magnitude higher than the incident one, indicating strong light confinement at the resonance wavelength of l = 1.69 mm. These features are the main characteristics of MP resonance, which have been discussed previously [⁶,⁸,¹⁰].

Moreover, a LC circuit model, as shown in Fig. 3(b) was used to theoretically further validate the existence of MP. The GaSb layer between the top grating ridge and the Ag substrate can be considered as a capacitor with capacitance C_m, while the top Ag grating and the bottom Ag substrate can be treated as inductors with inductances L_m and L_k to respectively represent the parallel-plate inductance and the kinetic one. The expressions for C_m, L_m, and L_k are listed below [¹²].

(2)C_{\mathrm{m}}={\displaystyle \frac{c_{\mathrm{1}}{\epsilon }_{\mathrm{0}}{\epsilon }_{\mathrm{d}}w}{d}},

(3)L_{\mathrm{m}}=\mathrm{0.5}{\mu }_{\mathrm{0}}wd,

(4)L_{\mathrm{k}}={\displaystyle \frac{-w}{{\omega }^{\mathrm{2}}{\epsilon }_{\mathrm{0}}{{\epsilon }^{\prime }}_{\mathrm{m}}\delta }},

where c₁ is a coefficient accounting for non-uniform charge distribution at the metal surface and it is in the range 0.2≤c₁≤0.3 [²³], e₀ is the permittivity of vacuum, e_d is the permittivity of the GaSb spacer, m₀ is the permeability of vacuum, and {{\epsilon }^{\prime }}_{\mathrm{m}} is the real part of the permittivity of Ag. \delta =\lambda /\mathrm{4}\pi kis the penetration depth [¹²], where l is the wavelength of the incident wave and k is the extinction coefficient of Ag. Note that in comparison to previous studies of similar film-coupled metamaterial structures [⁶,⁸,¹⁰], the gap capacitor, which considers the capacitance inside the gap between two nearby grating ridges, is neglected here because it is much smaller than the spacer capacitor C_m with the geometric parameters and wavelength ranges considered here.

The total impedance of the LC circuit model is

(5)Z_{\mathrm{total}}=\mathrm{2}\omega (L_{\mathrm{m}}+L_{\mathrm{k}})-{\displaystyle \frac{\mathrm{2}}{\omega C_{\mathrm{m}}}}.

Magnetic resonance occurs when the total impedance of the circuit is zero, i.e., Z_total = 0, which yields the analytical MP resonance wavelength:

(6){\lambda }_{\mathrm{MP}}=\mathrm{2}\pi c_{\mathrm{0}}\sqrt{C_{\mathrm{m}}(L_{\mathrm{m}}+L_{\mathrm{k}})}.

With the base geometric values for the film-coupled metamaterial structure, the LC circuit model predicts that MP will be excited at the wavelength l_MP between 1.58 mm (c₁ = 0.2) and 1.89 mm (c₁ = 0.3). The absorption peak positon of l = 1.69 mm from the RCWA simulation is in the LC circuit model prediction range, which confirms the excitation of MP inside the film-coupled metamaterial structure.

Excitation of SPP

The absorption peak at l = 0.36 mm for the proposed film-coupled metamaterial structure under TM waves is due to the excitation of SPP at the air-Ag interface. SPP represents the interaction between the electromagnetic waves and the oscillatory movement of free charges near the surface of the metal materials [²⁴]. The excitation of SPP between two nonmagnetic materials is determined by the dispersion relation at TM polarization [¹²]:

(7)k_{\mathrm{spp}}=(\omega /c_{\mathrm{0}})\sqrt{{\epsilon }_{\mathrm{1}}{\epsilon }_{\mathrm{2}}/({\epsilon }_{\mathrm{1}}+{\epsilon }_{\mathrm{2}})},

where k_spp represents the wavevector of the SPP wave, e₁ and e₂ are the dielectric functions of the materials at each side of the interface respectively, whose real parts should have opposite signs to excite SPP, w is the angle frequency, and c₀ is the speed of light in air. The SPP dispersion curve for a planar interface usually lies at the right side of the light line due to the nature of evanescent waves. However, SPP can be excited with propagation waves with periodic gratings because large in-plane wavevectors can be matched by high-order diffracted waves according to the Bloch-Floquet condition [²⁴]:

(8)k_{x,j}=k_x+{\displaystyle \frac{\mathrm{2}\pi j}{\mathrm{\Lambda }}},

where j is the diffraction order in the x direction and k_x=\omega /c is the parallel component of the wavevector in vacuum. The resonance frequency for SPP modes can be theoretically obtained by solving \left|k_{\mathrm{SPP}}\right|=\left|k_{x,j}\right|. The wavelength at which SPP is excited at the air-Ag interface is predicted to be l_SPP = 0.31 mm, which matches well with l = 0.36 mm from the RCWA simulation.

Excitation of FP resonance

When the ultrathin GaSb layer was placed on a Ag substrate to form a GaSb-on-Ag structure, it is observed that there exists an absorption peak at l = 0.84 mm with an absorptance a = 0.90, as shown in Fig. 2(b). This peak is attributed to interference inside the GaSb layer. The spectral normal reflectance (R_l) of the GaSb-on-Ag structure can also be analytically calculated by the thin-film optics method [²⁴]:

(9)R_{\lambda }={\left|{\displaystyle \frac{r_{\mathrm{12}}+r_{\mathrm{23}}{\mathrm{e}}^{\mathrm{2}\mathrm{i}\beta }}{\mathrm{1}+r_{\mathrm{12}}{r}_{\mathrm{23}}{\mathrm{e}}^{\mathrm{2}\mathrm{i}\beta }}}\right|}^{\mathrm{2}},

where r₁₂ is the reflection coefficient at the interface from air to GaSb and r₂₃ is the reflection coefficient at the interface from GaSb to Ag. β=2πn₂dcosθ₂/λ is the phase shift inside the ultrathin GaSb layer, where n₂ is the refractive index of GaSb and q₂ is the refraction angle inside GaSb. The absorptance of the GaSb-on-Ag structure can be calculated by A_λ=1–R_λ due to its opaqueness. The analytical thin-film optics method yields an absorption peak due to interference at l = 0.84 mm with a = 0.90, which is in good agreement with the RCWA simulation. To verify that the absorption peak at l = 0.84 mm is caused by the excitation of FP resonance, the total phase shift \psi in the ultrathin GaSb layer is calculated by [²⁴]

(10)\psi ={\varphi }_{\mathrm{1}}+{\varphi }_{\mathrm{2}}\mathrm{+2}\beta ,

where ϕ₁= arg (r₂₁) and ϕ₂= arg (r₂₃) are the phase shifts caused by the reflection at the interface from GaSb to air and the interface from GaSb to Ag, respectively. FP resonance occurs when the total phase shift in the GaSb layer is a multiple of 2p, leading to standing waves existing in the cavity. The calculated phase shift \psi at l = 0.84 mm is zero, which means that there exists a constructive interference effect inside the ultrathin GaSb layer thus enhancing the light absorption.

When a 1D subwavelength Ag grating is added onto the GaSb-on-Ag structure to form a film-coupled metamaterial structure, the upper Ag grating layer with a thickness of 30 nm modifies the reflection coefficient r₁₂ at the top interface of GaSb, which becomes air-Ag-GaSb interface, resulting in a slight peak shift from l= 0.84 mm to 0.87 mm under TM waves as shown in Fig. 2(b). Therefore, it is reasonable to deduce that this absorption peak at l= 0.87 mm in the film-coupled metamaterial structure is also caused by FP resonance [²⁵]. Note that, FP resonance also results in a similar absorption peak for TE waves but at a slightly differently wavelength of l = 0.74 mm, as shown in Fig. 2(a). To confirm this, effective medium theory (EMT) was employed to homogenize the top 1D Ag grating layer, whose effective dielectric functions for TE and TM waves can be obtained from 0th order approximation by [²⁴]

(11){{\displaystyle \epsilon }}_{\mathrm{T}\mathrm{E}}=f{{\displaystyle \epsilon }}_{\mathrm{A}\mathrm{g}}+(\mathrm{1}-f){{\displaystyle \epsilon }}_{\mathrm{a}\mathrm{i}\mathrm{r}}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{T}\mathrm{E}\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{s},

(12){\displaystyle \frac{\mathrm{1}}{{{\displaystyle \epsilon }}_{\mathrm{T}\mathrm{M}}}}={\displaystyle \frac{f}{{{\displaystyle \epsilon }}_{\mathrm{A}\mathrm{g}}}}+{\displaystyle \frac{\mathrm{1}-f}{{{\displaystyle \epsilon }}_{\mathrm{a}\mathrm{i}\mathrm{r}}}}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{T}\mathrm{M}\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{s},

where f = w/L is filling ratio of Ag, and e_Ag and e_air = 1 are the dielectric function of Ag and air, respectively. Apparently, the effective dielectric function of the top 1D Ag grating is different for TE and TM waves, which would result in different reflection coefficients r₂₁ and thereby different FP wavelength. Figure 4 compares the spectral normal absorptance of the proposed metamaterial structure from RCWA calculation for TE and TM waves, and the normal absorptance of a multilayer structure with the homogeneous 1D Ag grating layer descried by the effective dielectric functions. Clearly, the FP resonance wavelengths of a multilayer structure with the homogeneous 1D Ag grating layer described by the effective dielectric function agrees well with that of the proposed metamaterial structure from RCWA simulation for TE and TM waves. This confirms that the spectral absorption peaks of proposed film-coupled metamaterial TPV cell (at l = 0.87 mm for TM waves and l = 0.74 mm for TE waves) are due to the FP resonances from effective medium behavior. On the other hand, EMT cannot capture the MP resonance, which is clearly predicted by RCWA at TM waves.

Geometric dependence of MP, SPP, and FP resonances

The effects of geometric parameters on MP, SPP, as well as FP were considered. The spectral normal absorptance of the film-coupled metamaterial structure was calculated by independently varying grating width, period, and spacer thickness, while all other geometric parameters remain the same. Figure 5(a) shows the spectral absorptance of the structure with grating width changing from 80 nm to 120 nm. It can be observed that the absorption peak due to the excitation of MP shifts to longer wavelengths when the grating width increases, which is consistent with previous results [¹²]. Fig.5(b) shows the comparison of the MP resonance wavelength between the RCWA simulation and the LC circuit model prediction as a function of grating width. Clearly, the RCWA results agree well with the LC circuit model, which also validates the resonability of the RCWA simulation. As predicted by the LC circuit model, the MP resonance wavelength monotonically increases with the grating width, which is due to the increase of the capacitance and inductance values according to Eqs. (2–4).

On the other hand, the absorption peak associated with SPP and the one associated with FP are little affected by the grating width, as SPP peaks only depend on the grating period according to the dispersion relation and FP peaks only depend on the spacer thickness according to the phase shift. Fig. 5(c) shows the spectral absorptance with the grating period changing from 150 nm to 250 nm. Note that here the SPP peak barely changes with the grating period, which is different from the previous results [¹²,¹⁷] where the SPP resonance wavelength increases with the grating period. Figure 5(d) shows the dispersion curve (unfolded) to excite SPP at the air-Ag interface. k_x increases quickly as w increases and should reach an asymptote at \omega ={\omega }_{\mathrm{p}}/\sqrt{\mathrm{1}+{\epsilon }_{\infty }} when the real part of the dielectric function of Ag approaches -1 [²⁴]. Based on the Bloch-Floquet condition, the dispersion relations of SPP can be folded at kx = jp/L into the region kx≤p/L, and are shown as the dashed curves in Fig. 5(d). For normal incidence, the excitations of SPP are determined by the intersection location of the dispersion curves with kx = 0. It can be seen that as the grating period decreases, the SPP resonance frequency will reach the asymptote \omega ={\omega }_{\mathrm{p}}/\sqrt{\mathrm{1}+{\epsilon }_{\infty }}. Since the grating period values of 150, 200, 250 nm considered in this work are much smaller than 1 mm in Ref. [¹²] or 7 mm in Ref. [¹⁷], the folding occurs at large kx values or corresponding frequencies close to the SPP asymptote, which becomes independent on the frequency at large kx. Therefore, the influence of the grating period on the SPP resonance wavelength is negligible. On the other hand, the grating period does not affect the FP peak wavelength but slightly shifts the MP peak location as observed in Fig. 5(c). This is because the grating period only affects the gap capacitance between the neighboring gratings, which is much smaller than the spacer capacitance between the top grating ridge and the Ag substrate [¹²].

Figure 5(e) shows the spectral absorptance with the spacer thickness changing from 20 nm to 50 nm. It can be seen that the FP peak shifts to longer wavelengths when the spacer thickness increases, agreeing well with the result predicted by the phase shift in Eq. (3), as shown in Fig. 5(f). This is due to the increase of the phase shift in the ultrathin GaSb layer. On the other hand, the SPP peak is not affected by the spacer thickness and the MP peak barely changes when d is larger than 30 nm.

Behaviors of MP, SPP and FP at oblique incidences

To understand the behaviors of MP, SPP, and FP at oblique incidences, contours of the spectral-directional absorptance of the film-coupled metamaterial TPV cell as a function of wavenumber  and x-direction wavevector kx were plotted for TM and TE incident waves in Fig. 6. Note that the results for kx= 0 is for normal incidence. The inclined blue dashed lines represent the results for an incidence angle of 10°, 30°,and 60°. As shown in Fig. 6(a), there are three bright bands under TM waves, which indicate three major absorption peaks. The first one around v= 6000 cm-1 is attributed to the excitation of MP. The flatness of the band indicates that the excitation wavelength of MP does not change at oblique incidence angles, which is consistent with the observations from previous studies [¹²]. The second one around v = 11200 cm⁻¹ is associated with FP resonance. It can be seen that the FP resonance shifts slightly toward higher wavenumbers (i.e., shorter wavelengths) when increasing the incidence angle. This is mainly due to the cosθ2 term in the phase shift β. The third one around v = 28000 cm⁻¹ is because of the excitation of SPP. Note that SPP resonance condition is insensitive to the incidence angle which is due to the small grating period in this work. One the other hand, as shown in Fig. 6(b), under TE waves there is only the bright band due to FP resonance, as MP and SPP can only be excited under TM waves for 1D grating structures. Note that the FP resonance is insensitive to the incidence angle. Moreover, the absorption due to FP resonance under TE waves is much weaker than that under TM waves.

Energy absorbed by the ultrathin GaSblayer

Although light absorption can be significantly enhanced in the film-coupled metamaterial structure by exciting MP, SPP, and FP, only the energy absorbed by the photovoltaic layer can contribute to the photon-generated carriers, while the energy absorbed by metals in the structure is lost. Therefore, it is important to evaluate the amount of energy absorbed by the active layer rather than that by the entire structure. The entire structure can be divided into three layers: the first layer is the Ag grating, the second one is the GaSb layer, and the third one is the Ag substrate. The energy absorbed per unit volume inside each layer can be obtained by [²⁶]

(13)P_i=\mathrm{0.5}{\epsilon }_{\mathrm{0}}{{\epsilon }^{″}}_i\omega {\left|E_i\right|}^{\mathrm{2}},

and the absorbed power in each layer can be normalized to the incident power [²⁶]:

(14){\alpha }_i={\displaystyle \frac{{\displaystyle \int P_{i}d{V}_i}}{\mathrm{0.5}{c}_{\mathrm{0}}{\epsilon }_{\mathrm{0}}{\left|E_{\mathrm{inc}}\right|}^{\mathrm{2}}A}},

where {{\epsilon }^{″}}_i is the imaginary part of the relative permittivity of the layer i, V_i is the volume of layer i, A is the area that the light source is incident upon, E_i is the electric field inside layer i, and E_inc is the incident electric field. The energy absorbed by the entire structure is simply the sum of the energy absorbed by each layer. Figure 7 shows the normalized energy absorption in the GaSb layer, the metals, and the entire structure. It can be seen that most of the energy is absorbed by the GaSb layer, while some non-negligible energy between 1.5 μm to 2μm is absorbed by Ag.

Electrical properties and TPV performance of the ultrathin GaSb cell

To quantitatively evaluate the performance of the film-coupled metamaterial structure as an ultrathin TPV cell, short-circuit current density, open-circuit voltage, output electric power, and cell efficiency were calculated. The short-circuit current density J_sc (A/cm²) is calculated by [²⁷]

(15)J_{\mathrm{sc}}={\displaystyle {\int }_{\mathrm{0}}^{\frac{h{c}_{\mathrm{0}}}{E_{\mathrm{g}}}}{\displaystyle \frac{e\lambda }{h{c}_{\mathrm{0}}}}\alpha (\lambda ){\eta }_{\mathrm{i}}(\lambda )}q(\lambda )\mathrm{d}\lambda ,

where h is Planck’s constant, E_g is the bandgap of the GaSb cell (0.72 eV), a(l) is the normalized energy absorption in the GaSb layer only (average value over TM and TE waves), e is an elementary charge, η_i(λ) is the internal quantum efficiency (IQE) of the GaSb cell, and q(λ)is the radiative heat flux from a TPV emitter. The IQE of the ultrathin GaSb cell in this calculation is assumed to be 100%. Tong et al. [²⁸] assumes the IQE of the ultrathin cell to be 100% because bulk recombination losses for minority carriers are significantly reduced and the surface recombination losses can also be significantly reduced using a good passivation layer. The radiative heat flux from a blackbody emitter can be calculated by [²⁹]

(16)E_{\mathrm{b}\lambda }(T)={\displaystyle \frac{C_{\mathrm{1}}}{{\lambda }^{\mathrm{5}}[\mathrm{e}\mathrm{x}\mathrm{p}\hspace{0.17em}(C_{\mathrm{2}}/\lambda T)-\mathrm{1}]}},

where C₁ =\mathrm{3.742}\times {\mathrm{10}}^{\mathrm{8}}W\cdot {\mathrm{\mu m}}^{\mathrm{4}}/{\mathrm{m}}^{\mathrm{2}}, C₂ =\mathrm{1.439}\times {\mathrm{10}}^{\mathrm{4}}\mathrm{\mu m}\cdot \mathrm{K}, and T is the blackbody temperature. The total incident radiative flux onto the TPV cell P_in can be calculated by

(17)P_{\mathrm{in}}={\displaystyle {\int }_{\mathrm{0}}^{\infty }{E}_{\mathrm{b}\lambda }(T)}{\epsilon }_{\lambda }\mathrm{d}\lambda ,

where e_l is the spectral emissivity of a diffuse TPV emitter.

The open-circuit voltage V_oc is calculated by [²⁷]

(18)V_{\mathrm{oc}}\mathrm{=(}{k}_{\mathrm{B}}{T}_{\mathrm{c}}/e\mathrm{)ln}\hspace{0.17em}(J_{\mathrm{sc}}/J_{\mathrm{0}}\mathrm{+1),}

where k_B is the Boltzmann constant, T_c is the cell temperature which is assumed to be 300 K, and J₀ is the dark current which can be calculated by [²⁷]

(19)J_{\mathrm{0}}=e\left({\displaystyle \frac{n_{\mathrm{i}}^{\mathrm{2}}{D}_{\mathrm{h}}}{L_{\mathrm{h}}{N}_{\mathrm{D}}}}+{\displaystyle \frac{n_{\mathrm{i}}^{\mathrm{2}}{D}_{\mathrm{e}}}{L_{\mathrm{e}}{N}_{\mathrm{A}}}}\right),

where n_i is the intrinsic carrier concentration of GaSb, N_D and N_A are respectively the donor concentration and acceptor concentration, D_h and D_e are respectively the hole diffusion coefficient and electron diffusion coefficient, and L_h and L_e are respectively the hole and electron lifetime. The parameters for calculating J₀ are from Ref. [³⁰]. The output electric power P_cell can be calculated by [²⁷]

(20)P_{\mathrm{cell}}=J_{\mathrm{sc}}{V}_{\mathrm{oc}}(\mathrm{1}-\mathrm{1}/y)[\mathrm{1}-\ln \hspace{0.17em}(y)/y],

where y=\ln \hspace{0.17em}(J_{\mathrm{sc}}/J_{\mathrm{0}}). Finally, the heat-to-electricity conversion efficiency for this ultrathin GaSb TPV cell can be expressed as the ratio of the output electric power P_cell to the total incident radiative heat flux P_in

(21)\eta ={\displaystyle \frac{P_{\mathrm{cell}}}{P_{\mathrm{in}}}}.

Figure 8(a) shows the spectral distribution of radiative heat fluxes from a blackbody emitter at temperatures from 1000 K to 2000 K, along with the spectral emittance for an ideal selective TPV emitter, which is unity in the wavelength range from l₁ to l₂ and zero outside this spectral range. The purpose of selective emitters is to avoid wasted photons with energy below the bandgap, which cannot generate electron-hole pairs, as well as to reduce the excess energy above the bandgap, thus enhancing the TPV efficiency. Figure 8(b) shows the short-circuit current of the ultrathin GaSb cell under blackbody radiation at different temperatures. Clearly, the short-circuit current can be greatly enhanced with the film-coupled metamaterial structure and increases with the blackbody temperature. For example, when the blackbody temperature is 2000 K, the short-circuit current of the ultrathin GaSb cell with a film-coupled metamaterial structure is 10.21 A/cm², which is two times the short-circuit current of a free-standing GaSb layer.

Figure 8(c) and (d) show the effect of the emittance bandwidth on the performance of the GaSb cell with a selective emitter at 2000 K. Here, the emittance long cutoff wavelength l₂ is fixed at 1.73 mm which corresponds to the GaSb bandgap, while the varying value of emittance short cutoff wavelength l₁ impacts the performance of the GaSb cell. It is obvious that the selective emittance bandwidth has a significant influence on the cell performance. As the bandwidth decreases (i.e., l₁ increases), short-circuit current J_sc decreases, while open-circuit voltage V_oc barely changes. The incident radiative heat flux P_in and output electric power P_cell also decrease when decreasing the bandwidth with P_in decreasing at a faster rate, resulting in increasing cell efficiency h. The maximum cell efficiency can be obtained as h_max = 11.3% when l₁ is around 1.63 mm, which is related to the MP peak position at 1.69 mm because the SPP peak and FP peak will not contribute to the cell efficiency with a selective emitter.

Figure 9(a) and (b) show the output electric power and the cell efficiency as a function of the selective emitter temperature T and the emittance short cutoff wavelength l₁. It can be found that as the T increases, both the output electric power and cell efficiency increase. As the emittance bandwidth decreases (i.e., l₁ increases), the output electric power decreases while the cell efficiency increases. Therefore, a selective emitter with a narrow emittance band and high temperature are usually needed to reach high efficiency.

Conclusions

We have numerically demonstrated that the absorption of visible and near-infrared photons in an ultrathin GaSb layer can be greatly enhanced with a film-coupled metamaterial structure. The underlying mechanism for the absorption peak near the bandgap is shown to be the excitation of MP, which has been elucidated with the electromagnetic field distribution and confirmed with the LC circuit model. Moreover, SPP and FP resonance are excited at shorter wavelengths for selective absorption, and their mechanisms are verified respectively by the dispersion relation and phase shift. It is found that the MP resonance wavelength can be easily tuned by changing the grating ridge width, and the FP resonance wavelength can be tuned by changing the spacer thickness. With the film-coupled metamaterial structure, the short-circuit current in the ultrathin GaSb cell is greatly enhanced. The cell conversion efficiency can be further increased up to 11.3% by a selective emitter with a narrow unity emittance band. The results and understanding gained here will facilitate the development of next-generation low-cost high-efficiency ultrathin infrared photovoltaic cells and detectors.

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