Nonreciprocal Photovoltaics: The Path to Conversion of Entire Power-Beam Exergy

Abstract

Nonreciprocal photonic management can shift the absorption-emission balance in favor of absorption and enhance the conversion efficiency beyond the detailed balance Shockley - Queisser limit. Nonreciprocal photovoltaic (PV) cells can provide the conversion of the entire exergy (Helmholtz free energy) of quasi-monochromatic radiation into electric power. Recent discoveries in electromagnetics have demonstrated the ability to break Kirchhoff’s reciprocity in a variety of ways. The absorption-emission nonreciprocity may be realized via dissipationless one-way optical components as well as via the greenhouse-type electron-photon kinetics that traps the low-energy near-bandgap photons in the cell. We calculate the limiting performance of the nonreciprocal dissipationless monochromatic converter and discuss the limiting efficiency of the nonreciprocal converter based on the greenhouse effect. We also perform detailed modeling of the greenhouse effect in the GaAs PV converter and determined its PV performance for conversion of 809 nm laser radiation. In perovskite PV cells the greenhouse filter establishes a sharp absorption edge and reduces conversion losses related to the distributed PV bandgap and laser-cell matching losses.

1. Introduction

Power delivery by a laser beam is an emerging technology with numerous potential applications. The capabilities of unmanned aerial vehicles, various robotic platforms, and sensor networks will be strongly enhanced due to remote charging. Currently all technological components for power beam delivery and conversion are commercially available. Integration of lasers with photovoltaic converters requires matching laser quanta to semiconductor material characteristics. In traditional design, the optimal bandgap value is determined by a tradeoff between the low near-bandgap absorption and thermalization losses [1, 2]. The optimal bandgap depends on laser power, optoelectronic properties of semiconductor material, and cell design [1]. Usually, the optimal bandgap wavelength exceeds the semiconductor bandgap by 20–80 nm. Even in optimized converters, the photoelectron thermalization and weak near-bandgap absorption produce notable losses [2]. Greenhouse-type filter that traps photons with wavelengths above the laser wavelength is a tool of choice to eliminate the laser-cell matching losses [3, 4]. Moreover, as it is shown in this work, the greenhouse filter can generate the greenhouse effect in the cell, which strongly reduces the emission from the cell and, in this way, increases the conversion efficiency beyond the detailed-balance Shockley - Queisser limit. The greenhouse PV effect is a special case of nonreciprocal photonic management, which violates Kirchhoff’s reciprocity between absorption and emission [5, 6].

The atmospheric greenhouse effect was discovered by Joseph Fourier, who calculated the balance between the incoming solar power and the outgoing power of emitted radiation and found the Earth’s average temperature near of $0°\mathrm{F}$ To shift the absorption-emission balance in a favor of absorption and get the average temperature of $\sim 60°\mathrm{F},$ Fourier proposed that the emitted radiation is trapped by the atmosphere similar to the way a greenhouse glass traps the heat [7]. The greenhouse glass transmits high frequency (high energy) radiation, which is absorbed by the greenhouse media and plants. The media convert the radiation into heat and emit thermal low frequency (low energy) radiation, which is reflected back to the greenhouse by the greenhouse glass. Plants use all high and low energy radiation and convert the radiation into biochemical energy. Photovoltaic conversion directly transfers radiation into electric power via the generation of electrons and holes, which lose part of their energy to crystalline lattice and are accumulated near the bandgap edge. Photocarriers can recombine via nonradiative processes, which convert solar energy into heat. Photocarriers can also recombine via the radiative process and emit bandgap photons. In the absence of nonradiative processes, the photocarrier density and PV performance are determined by the detailed balance between the absorbed and emitted radiation fluxes. As was shown by Shockley and Queisser (SQ), this absorption-emission balance defines a fundamental limit of the photovoltaic conversion efficiency, which in traditional cell design depends on a single material parameter, − the semiconductor bandgap [8].

Recently proposed greenhouse design [3] mimics the greenhouse effect and nonequilibrium greenhouse processes, such as trapping of near bandgap radiation and reusing it for conversion into electricity. Greenhouse filter is placed at the front surface of a cell and the back surface mirror. It establishes a photonic bandgap above the semiconductor bandgap and traps the photons with energies below the photonic bandgap (see Figure 1). The mirror may be a wideband Bragg reflector, or photonic crystal reflector, or metallic reflector with small absorption. In the greenhouse design, the photon emission from the converter is limited by recombination processes of hot photo carriers that emit photons with energy above the photonic bandgap. Bandgap photons emitted by the near-bandgap photocarriers are recycled by the filter and reused in the cell for PV conversion. In the PV greenhouse effect, the nonradiative recombination plays a role of a greenhouse media, which heats the greenhouse. The radiative processes play a role in photosynthesis, which convert solar energy into biochemical energy used by plants. Therefore, the greenhouse PV design requires high-quality PV materials with weak nonradiative recombination, i.e. materials with high internal quantum (radiative) efficiency. Fast progress in traditional semiconductor materials and the development of novel optoelectronic materials raises principle questions about photonic management for PV conversion.

Figure 1.

PV converter with the greenhouse filter, which establishes the photonic bandgap above the semiconductor bandgap and traps photons with energy below the photonic bandgap inside the converter.

Can the greenhouse filter increase the solar light conversion efficiency beyond the SQ limit? The answer substantially depends on a form of the nonequilibrium distribution function of photo-generated carriers. In the greenhouse converter, we have two characteristic bandgaps. The accumulation and collection of photocarriers occur near the semiconductor bandgap. The photonic bandgap controls the absorption-emission balance in the cell. In this design, only photons emitted by hot electrons can leave the converter. The population of hot electrons and emission from the cell are controlled by the cooling of the hot electrons. If photocarriers near the semiconductor bandgap and hot energy photocarriers above the photonic bandgap have the same photo-induced chemical potential, the greenhouse filter just shifts the PV bandgap from the semiconductor bandgap to the photonic bandgap. In other words, in this quasi-equilibrium case (actually, in a chemical equilibrium between low energy and high energy photocarriers), the limiting conversion efficiency is the SQ efficiency with the PV bandgap equaled to the photonic bandgap. However, hot photocarriers usually rapidly lose their energy and, as a result, the density of hot photocarriers decreases, and their chemical potential is significantly reduced in comparison with the chemical potential of carriers near the semiconductor bandgap. In this nonequilibrium regime with fast photocarrier cooling, the conversion efficiency may exceed the SQ limit. To realize the nonequilibrium regime, the difference between photonic and semiconductor bandgaps should substantially exceed the thermal energy. Therefore, the photonic bandgap should at least be $2-3k_{B}T=50‐75$ meV higher than the semiconductor bandgap.

Let us highlight that in the traditional cell design all conversion processes occur in the narrow energy interval above the semiconductor bandgap. Therefore, the conversion efficiency turns out to be insensitive to details of electron, photon, and phonon processes. In particular, the detailed modeling of PV conversion as a function of characteristic photoelectron relaxation and extraction times has shown that optimization of traditional design and operating regimes does not allow for surpassing the SQ limit [9]. The greenhouse design provides a splitting of key conversion processes. The photocarrier accumulation and collection take place near the semiconductor bandgap, while photocarrier recombination with photon escape occurs above the photonic bandgap. The filter provides an effective tool to reduce cell emission and shift the absorption-emission balance in favor of absorption.

To avoid cell heating, the greenhouse design requires photovoltaic materials with low nonradiative losses, i.e. high quantum efficiencies. Among traditional photovoltaic materials, GaAs has the highest radiative efficiency. In particular, high-quality GaAs with internal quantum efficiency (also termed internal radiative efficiency) of 99.7% provides solar cells with external radiative efficiency (ERE) above 30% [10, 11, 12]. In silicon, the Auger recombination is stronger than radiative recombination and the corresponding ERE of 1% substantially limits the conversion efficiency. Recent progress in emerging photovoltaic materials, − organic materials, dye-sensitized, CuInGaSe (CIGS), and lead-halide perovskites, − has demonstrated strong improvements in ERE. In particular, the ERE of CIGS currently exceeds 24%. ERE is the integral characteristic, which may be calculated via the special averaging of external quantum efficiency (EQE) in the relatively narrow spectral range around the absorption threshold, which in traditional semiconductors coincides with the bandgap [13]. The perovskite materials demonstrate EQE values very close to unity in wide spectral ranges and gradual reduction of EQE near threshold energy [14, 15], which substantially limits the conversion efficiency of traditional PV cells.

In this work, we investigate and optimize greenhouse photonic management for photovoltaic conversion of monochromatic radiation. Recent progress in electromagnetics has demonstrated the ability to break the absorption-emission reciprocity in a variety of ways. To understand the advantages and limitations of greenhouse photonic management, in Section 2 we derive the monochromatic detailed-balance efficiency and in Section 3 we consider the limiting nonreciprocal monochromatic efficiency realized via dissipationless nonreciprocal optical components. In Section 4, we investigate the greenhouse photonic management and present results of simulations of conversion of $10\mathrm{W}/{\mathrm{cm}}^2$ laser radiation with a wavelength around 809 nm by GaAs PV cell with the greenhouse filter. In p-doped ${\mathrm{A}}_3{\mathrm{B}}_5$ semiconductors the electron cooling is realized due to energy transfer from hot electrons to holes [16]. We determine the corresponding non-equilibrium distribution with reduced chemical potential of hot photocarriers and calculate the cell performance. Finally, we briefly discuss enhanced PV conversion in perovskite cells due to greenhouse filtering, which establishes a sharp absorption edge.

2. Detailed-balance limiting efficiency for monochromatic radiation

Let us start with the SQ detailed balance approach, which is based on two assumptions. The first assumption is the reciprocity of photonic (radiative) processes. In the equilibrium, the emitted radiation is exactly given by the absorbed radiation reversed in time. In other words, following Kirchhoff’s law, emissivity, $e\left(\lambda ,\mathbf{n}\right),$ and absorptivity, $\alpha \left(\lambda ,-\mathbf{n}\right),$ are equal for any photon wavelength and any propagation direction, n. Second, in PV conversion, photocarriers and emitted photons reach chemical equilibrium. Thus, photocarriers and photons have the same light-induced chemical potential. The photocarriers are collected in the narrow energy range above the semiconductor bandgap and photons are also emitted in the same range. Therefore, assumption about the chemical equilibrium between photocarriers and photons is only essential for this narrow range, which is of the order of thermal energy $k_B{T}_0.$.

The generalized SQ model is described by three parameters. The detailed balance is taken into account by the ratio of the absorbed flux to the equilibrium emitted flux, $N_{\mathit{ab}}/N_{\mathit{em}},$ where $N_{\mathit{ab}}$ is the absorbed flux, $N_{\mathit{em}}$ is emitted photonic flux in the equilibrium at the device operating temperature $T_0.$ The incoming flux and its absorption by the cell are characterized by the average photon energy in the absorbed flux, ${ϵ}^{\ast }.$ Nonradiative recombination losses and losses in the photocarrier collection are described by the external radiative efficiency of the cell, $k_{\mathit{ERE}}.$ The generalized SQ model allows for a rather simple analytical solution [17]. The SQ open-circuit voltage and conversion efficiency are given by

$$V_{\mathrm{OC}}=\frac{k_B{T}_0}{q}\cdot ln{A}_k,A_k=k_{\mathit{ERE}}\cdot \frac{N_{\mathit{ab}}}{N_{\mathit{em}}}$$E1

$${\eta }_{\mathit{SQ}}=\frac{k_B{T}_0}{{ϵ}^{\ast }}\cdot \left[\mathrm{LW}\left(A_k\cdot e\right)-2+\frac{1}{\mathrm{LW}\left(A_k\cdot e\right)}\right]\approx \frac{k_B{T}_0}{{ϵ}^{\ast }}\cdot \left[\mathrm{LW}\left(A_k\right)-1\right],$$E2

where e = 2.71828, and LW(x) is the Lambert W function, which asymptotic form is well described by three terms,

$$\mathrm{LW}\left(z\right)=ln\left(z\right)-lnln\left(z\right)+\frac{lnln\left(z\right)}{ln\left(z\right)}+.\dots ,z\gg 1.$$E3

The SQ efficiency (Eq. 2) is the efficiency for conversion of the radiation power. The thermodynamic efficiency of energy conversion at zero output power is reached in the open circuit regime (negligible current). In the quasi-monochromatic limit, the cell is eliminated by photons within a narrow bandwidth, $\mathrm{\Delta }\nu <k_B{T}_0/h,$ around the central frequency ν. For thermodynamic analysis, it is convenient to describe the power of the monochromatic radiation by the temperature $T_m.$ Assuming that the semiconductor bandgap matches the photon energy, we obtained the thermodynamic conversion efficiency at zero output power,

$${\eta }_{\mathit{th}}\equiv \frac{{\mathit{qV}}_{\mathit{OC}}}{{ϵ}^{\ast }}=\left(1-\frac{T_0}{T_m}\right)-ln\frac{1}{k_{\mathit{ERE}}}.$$E4

As expected, for the ideal cell $\left(k_{\mathit{ERE}}=1\right)$ the thermodynamic efficiency is the Carnot efficiency, $1-T_0/T_m$ [18].

Analytical solution for the monochromatic energy conversion may be found, when the photons in the incoming flux may be described by the Boltzmann statistics, i.e. $\mathit{h\nu }>k_B{T}_m$ (for example, for photons with energy of 1.4 eV it means that the photon temperature is limited by 18,000 K). Using Eqs. 1–3 we find the monochromatic SQ conversion efficiency for the ideal cell,

$${\eta }_{\mathit{SQ}}=\left(1-\frac{T_0}{T_m}\right)\cdot \left(1-\frac{ln\left(B\right)}{B}+\frac{ln\left(B\right)}{B^2}\right)$$

$$B=\frac{\mathit{hv}}{k_B{T}_0}\cdot \left(1-\frac{T_0}{T_m}\right).$$E5

This analytical solution is illustrated in Figure 2. According to Eq. 5, the detailed-balance SQ conversion efficiency of the monochromatic radiation increases with an increase of the photon energy and approaches the Carnot efficiency at high frequencies.

Figure 2.

The limiting detailed-balance SQ monochromatic efficiency normalized by the Carnot efficiency as a function of the photon energy for the monochromatic radiation with the temperature of 1000 K (red) and 10,000 K (blue).

3. Nonreciprocal monochromatic conversion limit

In this section, we consider the thermodynamic limit and material-determined limit of PV converters with negligible emission realized via nonreciprocal dissipationless photonic management. Let us start with the endoreversible thermodynamics of an engine that receives power from an emitter with temperature $T_{\mathit{em}}$ and operates between temperatures $T_{\mathit{hot}}$ and $T_{\mathit{\text{cold}}}$ with the Carnot efficiency, ${\eta }_{\mathit{\text{Carnot}}}=1-T_{\mathit{\text{cold}}}/T_{\mathit{hot}}$ (Figure 3a). It is well understood that for this converter the conversion efficiency of the heat power from the emitter into useful mechanical or electrical power is below the Carnot efficiency due to the emission from the hot sink to the emitter. These emission losses are not directly related to the engine operation. The emission is associated with Kirchhoff’s absorption-emission symmetry, which leads to losses in the delivery of heat power from the emitter to the engine. Let us highlight that Kirchhoff’s law is not a thermodynamic law. According to the Onsager - Casimir reciprocity relations, the time-reversal asymmetry in the system may be generated by a magnetic field, magnetization, electric currents, and time-modulation of optical properties [5, 6]. Nonreciprocal optical components can provide absorption channels with near-zero emissivity and emission channels with near-zero absorptivity. As it is shown in Figure 3b, we can split absorption channels into two parts. The first part is used for the absorption of incoming light. The second part of the absorption channels is connected with the emission channels and provides 100% reuse of the emitted photons [19]. Also, the emission can be completely suppressed due to quantum effects [20]. Both effects, − the suppressed emission [20] or recycling the emitted radiation to the engine [19], − eliminate the emission losses and increase the conversion efficiency up to the thermodynamic limit given by the Carnot formula.

Figure 3.

Conversion of the heat power by the engine that receives heat power from an emitter with temperature $T_{\mathit{em}}$ and operates between heat sinks with temperatures $T_{\mathit{hot}}$ and $T_{\mathit{\text{cold}}}:$ (a) a traditional converter with the losses due to emission from the hot sink to the emitter and (b) the nonreciprocal converter with suppressed emission [19, 20].

The above consideration in the frame of endoreversible thermodynamics assumes that the hot end of an engine may be described by the light-increased temperature. Therefore, it does not apply to the nonequilibrium states of semiconductors, which are described by the light-induced chemical potential. Let us also note, that the detailed-balance approach also cannot be employed for the nonreciprocal conversion with zero emission from the cell. Formally, Eq. 2 in this limit gives a divergent result. To determine the nonreciprocal conversion limit we will directly employ the second thermodynamic law. In the general form applicable to non-temperature distributions, the distribution function of photons emitted by electrons in the cell cannot exceed the distribution function of incoming photons. In the quasi-monochromatic limit, we are interested in values of these functions in the narrow bandwidth near the energy hv and, therefore, the limiting value of the light-induced chemical potential may be found from the following equation,

$${\left(exp\left(\frac{\mathit{hv}}{k_B{T}_m}\right)-1\right)}^{-1}={\left(exp\left(\frac{\mathit{hv}-\mu }{k_B{T}_0}\right)-1\right)}^{-1},$$E6

where $T_m$ is the temperature of the quasi-monochromatic radiation and $T_0$ is the cell operating temperature. Thus, the thermodynamic limit of the light-induced chemical potential in the nonreciprocal quasi-monochromatic converter is

$$\mu =\mathit{hv}\cdot \left(1-\frac{T_0}{T_m}\right).$$E7

Taking into account that emission from the nonreciprocal converter is absent and every absorbed photon generates an electron in the output circuit, we see that an ideal nonreciprocal converter provides entire conversion of the photon exergy, $\mathit{h\nu }\cdot \left(1-T_0/T_m\right).$ Thus, the conversion efficiency of the nonreciprocal monochromatic converter is the Carnot efficiency,

$$v_{\mathit{nr}}=\left(1-\frac{T_0}{T_m}\right).$$E8

As the nonreciprocal photonic management provides 100% reuse of the emitted photons, it is interesting to compare the nonreciprocal limiting efficiency (Eq. 8) with the efficiency of the thermophotovoltaic conversion, where the emitted photons are reabsorbed by the emitter and the corresponding energy is reused in conversion. For the monochromatic emitter, the output electric power may be presented as [21],

$$P=\frac{2{\mathit{Hv}}^3}{c^2}\cdot \left[N\left(T_m,\mu =0\right)-N\left(T_0,\mu \right)\right]\cdot h\mathrm{\Delta }v\cdot \left(1-\frac{T_0}{T_m}\right),$$E9

where H is the etendue of incoming and emitted photon bundle (we assume that absorption and emission angles are the same), $N\left(T_m,\mu =0\right)$ is the Bose distribution function of incoming photons with the emitter temperature and zero chemical potential, and $N\left(T_0,\mu \right)$ is the Bose distribution function of emitted photons with the cell temperature and light-induced chemical potential. Eq. 9 is the direct consequence of the endoreversible thermodynamics, according to which the limiting output power is given by a product of the Carnot efficiency of the conversion engine and the photon power flux delivered to the engine. The delivered flux is the difference between the absorption and emission fluxes. Assuming that the emitted flux returns to the emitter for later use, the PV conversion efficiency is given by the thermodynamic Carnot efficiency (see Eq. 4.27 in [21]). Thus, the photon reuse via the nonreciprocal photonic management (Figure 3) provides the same maximal Carnot efficiency as the photon reuse in the thermophotovoltaic system [21]. Let us highlight, that the Carnot efficiency may be also reached in the PV cell, where the radiative emission processes in the cell are suppressed by quantum effects [20].

Finally, we discuss the material limit of the nonreciprocal (time asymmetric) PV converter. If all emitted photons are reabsorbed by the cell, the radiative recombination lifetime of photocarriers approaches infinity, and photocarrier recombination is realized solely via nonradiative processes. In this case, we can employ the detailed balance SQ approach and corresponding analytical solution given by Eqs. 1 and 2, where the parameter $A_k$ should be changed by

$$A_{\mathit{nr}}=\frac{N_{\mathit{ab}}}{R_{\mathit{nr}}}=A_k\frac{1}{1-k_{\mathit{ERE}}}$$E10

where $R_{\mathit{nr}}$ is the equilibrium nonradiative recombination rate, $R_{\mathit{nr}}=n_{0}d/{\tau }_{\mathit{nr}},$ where $n_0$ is the equilibrium concentration of carriers, d is the cell thickness, and ${\tau }_{\mathit{nr}}$ is the nonradiative recombination time. In the cell design with the nonreciprocal photonic management, effective photon trapping and high photon absorption may be realized via the same external nonreciprocal recycling, which returns radiation to the cell (Figure 3). Therefore, the cell thickness may be significantly reduced, which leads to reduction of nonradiative recombination. Also, in high-quality optoelectronic materials the nonradiative recombination time significantly exceeds the radiative time, e.g. in GaAs the ratio ${\tau }_{\mathit{nr}}/{\tau }_r\sim 300$ [11]. Thus, the nonreciprocal management with strong suppression of the emission from the GaAs converter can increase the detailed balance coefficient $A_k$ by three orders in magnitude or more, which increases the open circuit voltage at least by 180 mV.

Let us note that emission suppression due to nonreciprocal photonic management strongly enhances photon recycling in the system. The limiting Carnot efficiency of the nonreciprocal converter (Eq. 8) corresponds to the zero-emission and infinite intrinsic photon recycling. Any negligible losses (photon leakage or nonradiative recombination) regularize the solution of the SQ model (Eq. 2). This is a general resolution of thermodynamic paradox related to nonreciprocal power converters and nonreciprocal transfer of electromagnetic energy. In this way, the optical diode paradox was resolved by Ishimaru for the nonreciprocal ferrite-loaded waveguide in 1962 [22]. It was shown that any negligible material loss in ferrites leads to the convergent solution, which does not contradict the second law of thermodynamics [23] (see also a review [5]). The Ishimarus results and our analysis show that such paradoxes appear, when the Maxwell equations or photon balance equations are applied to a completely lossless medium and systems. Any negligible dissipation eliminates divergent solutions.

4. Photovoltaic greenhouse effect

The Kirchhoff’s law is valid for opaque bodies in thermodynamic equilibrium with the environment. As it is highlighted in Ref. [24], these two assumptions are often not satisfied. In particular, any photovoltaic converter operates in strongly nonlinear regime far from equilibrium. In this section, we investigate the nonreciprocal photonic management realized due to greenhouse type filter, which mimics the greenhouse operation, where the greenhouse glass/plastic reduces thermal emission and preserves more thermal energy in the greenhouse. The greenhouse filter is placed at the front surface of the cell. It reflects low energy photons in some narrower energy interval (∼ 50–150 meV) above the semiconductor bandgap (see Figure 1). In other words, the filter and back surface mirror establishes the photonic bandgap, ${\sigma }_{\mathit{ph}},$ above the semiconductor bandgap, ${\sigma }_g.$ The greenhouse filter prevents escape of photons with energies in the range between semiconductor and photonics bandgaps. The filter separates main electronic and photonic processes in the PV converter. Absorption-emission balance and conversion efficiency of this converter drastically depend on the light-induced concentration of hot photocarriers above the photonic bandgap established by the greenhouse filter. For the monochromatic conversion, the photonic bandgap, should correspond to the energy of radiation quanta, ${\sigma }_{\mathit{ph}}=\mathit{hv}.$ As the photonic bandgap exceeds the semiconductor bandgap by ∼100–150 meV, the absorption at the photonic bandgap strongly exceeds the absorption near the semiconductor bandgap. To achieve negligible nonreciprocal emission above the photonic bandgap, the population of hot electrons with energy above the photonic bandgap should be strongly reduced by fast cooling processes that dominate over thermo-excitation processes of photocarriers accumulated above the semiconductor bandgap, ${\sigma }_g.$ In the limit of strong greenhouse effect, i.e. negligible emission above the photonic bandgap, we can repeat consideration of the previous section and obtain the limiting conversion efficiency of the greenhouse PV converter,

$${\eta }_{\mathit{gh}}=\frac{{\sigma }_g}{\mathit{hv}}\left(1-\frac{T_0}{T_m}\right)=\frac{{\sigma }_g}{{\sigma }_{\mathit{ph}}}\left(1-\frac{T_0}{T_m}\right).$$E11

The limiting efficiency of the greenhouse PV converter is smaller than the Carnot efficiency by the factor of ${\sigma }_g/{\sigma }_{\mathit{ph}},$ because the nonreciprocity is realized via nonreversible dissipative processes. As all kinetic processes are essentially reversible in equilibrium, the nonreversible relaxation of photocarriers between photonic bandgap and semiconductor bandgap levels requires that the level separation substantially exceeds the thermal energy, i.e. ${\sigma }_{\mathit{ph}}-{\sigma }_g\ge 2-3k_B{T}_0.$ In other words, the dissipated energy should exceed $\sim 2-3k_B{T}_0$ per photocarrier to reach electromagnetic nonreciprocity via relaxation processes. Without such separation, we will return to the results of Ref. [9], which show that the PV conversion is insensitive to photocarrier kinetics within the thermal energy scale.

As we discussed above, the efficiency of the greenhouse PV converter strongly depends on cooling mechanisms of hot photocarriers. Let us consider kinetics of the GaAs cell. The photon with energy above the semiconductor band gap creates electron and hole with energies

$$E_e=\left(\mathit{hv}-{\sigma }_g\right)\cdot \frac{m_h}{m_h+m_e}$$E12

$$E_h=\left(\mathit{hv}-{\sigma }_g\right)\cdot \frac{m_e}{m_h+m_e}.$$E13

Usually in semiconductor materials, the whole mass strongly exceeds the electron mass and practically whole photon energy is transferred to photoelectron. In GaAs these effective masses are $m_e=0.067m_0$ and $m_h=0.45m_0,$ where $m_0$ is the free electron mass. Therefore, to manage kinetics of hot photocarriers we should choose the p-doped GaAs.

Photoelectrons accumulated above semiconductor bandgap may be described by the Boltzmann distribution function with the light-induced chemical potential ${\mu }_{\mathit{sc}},$

$$f\left(ϵ\approx {\sigma }_g\right)=exp\left(\frac{{\mu }_g-ϵ}{k_B{T}_0}\right)$$E14

These photoelectrons are collected and produce the output voltage $V={\mu }_g/q.$ In the narrow range of the order of thermal energy above the photonic bandgap, ${\sigma }_{\mathit{ph}},$ the distribution function may be approximated by the chemical potential ${\mu }_{\mathit{ph}},$

$$f\left(ϵ\approx {\sigma }_{\mathit{ph}}\right)=exp\left(\frac{{\mu }_{\mathit{ph}}-ϵ}{k_B{T}_0}\right)$$E15

If inter-electron interaction dominates over other processes in photoelectron kinetics, the whole system is described by the same chemical potential. The conversion efficiency is given by Eq. 2 with the absorption-emission balance established above the photonic bandgap. In this case, the greenhouse filter only suppresses the matching losses. If cooling of photoelectrons above the photonic bandgap dominates over phonon-induced thermo-excitation of photoelectrons (see Figure 4), the chemical potential of hot photoelectrons, ${\mu }_{\mathit{ph}},$ is much smaller than ${\mu }_g.$ A reduced population of hot photoelectrons directly reduces the generation of photons with energy above the photonic bandgap. Because only these photons can leave the converter, emission from the converter is suppressed by the factor of $exp\left({\mu }_g/{\mu }_{\mathit{ph}}\right).$.

Figure 4.

Fast energy relaxation of the beam-generated photoelectrons depopulates energy levels above the photonic bandgap. As a result, the chemical potential of hot electrons and emitted photons that leave converter is much less than the chemical potential of photoelectrons accumulated near the semiconductor bandgap.

Typically, cooling processes of hot photoelectrons are rather fast and, therefore, the chemical potential ${\mu }_{\mathit{ph}}$ is much smaller than ${\mu }_g.$ Cooling processes are exceptionally strong in ${\mathrm{A}}_3{\mathrm{B}}_5$ semiconductors [16, 25, 26, 27, 28], where the valence band consists of light and heavy holes, and photo-induced hot electrons transfer their energy to heavy holes created by p-doping. The electron cooling may be realized via inelastic scattering of electrons on heavy holes that is accompanied by the transition of a heavy hole to the light-hole band [25] as well as via inelastic electron scattering on oscillations of multicomponent hole plasma with heavy and light holes [28]. Evaluations show that the many body processes dominate in hot carrier cooling. Using the results of Ref. 28, in our modeling, we employ the following equation for the energy relaxation rate of hot electrons due to energy transfer to hole plasmons,

$$\frac{1}{{\tau }_{e-h}}=\frac{\pi {\mathrm{\hslash }}^2\cdot p_0}{m_{\mathit{lh}}^{3/2}\sqrt{{ϵ}_e}}$$E16

where $p_0$ is the hole concentration due to doping, $m_{\mathit{lh}}$ is the mass of the light hole, and ${ϵ}_e$ is the electron energy. Eq. 16 applies to electrons with ${ϵ}_e\ge 15$ meV above the semiconductor bandgap for the doping levels below ${310}^{18}{\mathrm{cm}}^{-3}.$ The energy transfer from hot electrons to holes is not limited by thermal energy. The characteristic transferred energy changes from $5{\mathrm{\hslash }}^2{p}_0^{3/2}/m_{\mathit{hh}}$ ($m_{\mathit{hh}}$ is the mass of the heavy hole) to $5{\mathrm{\hslash }}^2{p}_0^{3/2}/m_{\mathit{lh}}$ and substantially exceeds thermal energy. As a result, the electron energy relaxation due to interaction with hole plasma oscillations is very fast. Femtosecond electron cooling in GaAs and other ${\mathrm{A}}_3{\mathrm{B}}_5$ materials was observed in numerous experimental investigations [16, 29, 30, 31]. The femtosecond electron–hole relaxation strongly dominates over electron–phonon processes with a characteristic time of 1–2 ps.

To illustrate the operation of the greenhouse converter we perform simulations of conversion efficiency of $10\mathrm{W}/{\mathrm{cm}}^2$ laser radiation with a wavelength around 809 nm by GaAs cell with the greenhouse filter that establishes the photonic bandgap equaled to the laser energy quanta. The integration of GaAs cell with this laser is widely studied for power beaming [32]. To enhance absorption we add the Lambertian scattering layer placed between the filter and the cell (the same effect may be reached by the curved or textured mirror). Lambertian scattering significantly enhances light absorption by the cell. The corresponding absorption coefficient is given by.

$$A=\frac{1-exp\left(-4\mathit{\alpha d}\right)}{1-\left(1-b\right)\left(1-1/n^2\right)\cdot exp\left(-4\mathit{\alpha d}\right)},$$E17

where d is the cell thickness, $\alpha$ is the GaAs absorption at the laser wavelength, n is the refractive index, and b is the absorption of the back surface mirror.

The modeling of PV performance is based on analytical SQ solution (Eqs. 1 and 2), which was described in details for GaAs cell in Ref. cite4 and for various thermophotovoltaic cells in Ref. [33]. In Ref. [3] we investigated the same power beaming conversion in quasi-equilibrium approximation ${\mu }_{\mathit{ph}}={\mu }_g.$ In this modeling, we take into account the greenhouse PV effect due to fast electron cooling and, using the Boltzmann equation we calculate the reduced emission flux.

The main results of our modeling are presented in Figure 5, which shows the increase in the conversion efficiency due to the greenhouse filter as a function of internal quantum efficiency (IQE), $k_{\mathit{int}}$ and normalized cell thickness, $\mathit{d\alpha }.$ As seen, the increase in efficiency can reach 12–14%, but significant improvements require high-quality PV materials with high internal quantum efficiency.

Figure 5.

Increase in the conversion efficiency of the laser light with 809 nm wavelength and power of $\mathit{10}W/{\mathit{cm}}^{\mathit{2}}$ due to the PV greenhouse effect as a function of internal quantum efficiency (IQE) and normalized cell thickness, $\mathit{d\alpha }$.

To distinguish the nonequilibrium greenhouse effect in PV conversion, in Figure 6 we present the conversion efficiency calculated in the nonequilibrium model with fast photoelectron cooling, quasi-equilibrium approximation, and for traditional cell design without the greenhouse filter. The efficiency as a function of the dimensionless cell thickness, $\mathit{d\alpha },$ is shown for several values of the internal quantum efficiency, $k_{\mathit{int}}.$ For available GaAs materials, the IQE may reach 0.997 [11] and the greenhouse filter adds 7% to the conversion efficiency, where 1% is added due to nonequilibrium effects (red, blue, and green solid lines). At $k_{\mathit{int}}=0.998$ (yellow line) the efficiency exceeds the SQ power beaming efficiency (blue dashed line). The limiting efficiency (green dashed line) exceeds the SQ efficiency by 7%.

Figure 6.

PV efficiency of GaAs device, which converts laser light with 809 nm wavelength and power of $\mathit{10}W/{\mathit{cm}}^2$ vs. cell thickness. Cell without greenhouse filter: Red lines $\left(\mathit{\text{dashed}}{k}_{\mathit{int}}=\mathit{1},\mathit{\text{solid}}{k}_{\mathit{int}}=\mathit{0.997}\right);$ cell with the filter in quasi-equilibrium approximation: Blue lines $\left(\mathit{\text{dashed}}{k}_{\mathit{int}}=\mathit{1},\mathit{\text{solid}}{k}_{\mathit{int}}=\mathit{0.997}\right);$ cell with the filter and the greenhouse kinetics: Green lines $\left(\mathit{\text{dashed}}{k}_{\mathit{int}}=\mathit{1},\mathit{\text{solid}}{k}_{\mathit{int}}=\mathit{0.997}\right)$ and yellow line $\left(k_{\mathit{int}}=\mathit{0.998}\right).$.

Figure 7 demonstrates the conversion efficiency of 809 nm laser radiation as a function of the internal quantum efficiency (Figure 7a) and the laser power (Figure 7b) for the greenhouse GaAs PV converter (green lines), the quasi-equilibrium approximation for the same converter (blue lines), and traditional converter without the greenhouse filter. As it is shown in Figure 7b, all converters have the same, weak (logarithmic) dependence of the efficiency on the laser power. Dependencies of efficiency on the material IQE are substantially different. While the efficiency of the traditional converter has a rather weak, linear dependence on IQE, the performance of the greenhouse converter in the quasi-equilibrium approximation and especially in the model with photoelectron cooling strongly depends on the IQE. Our modeling shows that for suppression of the laser-cell matching losses (the quasi-equilibrium model) we need good PV materials with IQE better than 0.8. The efficiency improvement due to suppression of emission losses via cooling of photoelectrons requires high-quality PV materials with IQE better than 0.99. Otherwise, the nonradiative recombination dominates over the radiative component and determines the converter performance.

Figure 7.

Increase in the conversion efficiency of the laser light with 809 nm wavelength and power of $\mathit{10}W/{\mathit{cm}}^2$ due to the greenhouse PV effect as a function of internal quantum efficiency (IQE) and normalized cell thickness, $\mathit{d\alpha }$.

Perovskites are low-cost and rather nonhomogeneous materials. These materials show a very smooth absorption edge with the width ∼ 100 nm [14]. While above this range perovskites demonstrate excellent, very close to unity EQE, the smooth absorption drastically reduces the photovoltaic performance [14, 15]. Rau et al. [34] proposed that perovskites may be considered as a semiconductor with the distributed PV bandgap [35]. This model is widely applied to perovskite cells and successfully explains a significant reduction of conversion efficiency with respect to SQ limit [34, 35, 36]. In particular, according to the distributed bandgap model, the 100 meV standard deviation from the mean bandgap reduces the conversion efficiency by 6%. The greenhouse filter is a valuable tool to establish the sharp absorption edge above the smooth material absorption edge. Thus, for the power beaming with perovskite cells, the greenhouse design is expected to increase the conversion efficiency due to the suppression of both the laser-cell matching losses and the distributed bandgap losses.

5. Conclusions

Photonic management of radiative processes is an effective tool to enhance the performance of photovoltaic converters. An ideal nonreciprocal converter provides the entire conversion of the photon exergy, $\mathit{h\nu }\cdot \left(1-T_0/T_m\right).$ Nonreciprocal management is the most radical way to change the absorptionemission balance in favor of the absorption. Suppression of spontaneous emission via quantum interference effects was proposed in seminal works of Scully to increase the PV efficiency of quantum photocell [20]. Here we have proposed and investigated a more practical way to suppress emission due to a narrow bandwidth filter, which generates the greenhouse effect in a cell. As an ordinary greenhouse effect, the greenhouse photovoltaic effect requires fast cooling of photocarriers and strong trapping of low-energy photons, which are emitted by photoelectrons near the semiconductor bandgap. In ${\mathrm{A}}_3{\mathrm{B}}_5$ semiconductors, effective electron cooling is realized by energy transfer from hot photoelectrons to the plasma oscillations of holes. Let us note, that the greenhouse effect may be also realized in PV design with a 3D photonic crystal, which has a photonic bandgap that overlaps the electronic band edge. In this design, the emission inside the photonic bandgap is rigorously forbidden [37]. Greenhouse photonic management has a strong potential to substantially increase the conversion efficiency due to the reduction of the laser-cell matching losses, the radiative losses, and the distributed bandgap losses in low-cost perovskites and organic materials. Nonreciprocal photovoltaics requires materials with high internal quantum efficiency and high-quality optical components, which provide enhanced photon recycling.

Acknowledgments

The work is supported by the Army Research Laboratory. Research of AS was accomplished under Cooperative Agreement No. W911NF-18-2-0222. The work of KS is supported by Texas A&M University. The views and conclusions contained in this paper are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the US Government.