Temperature-Dependent Analysis of Solid-State Photon-Enhanced Thermionic Emission Solar Energy Converter

Abstract

Solid-state photon-enhanced thermionic emission (PETE) solar energy converters are newly proposed devices that can directly convert solar energy into electrical power at high temperatures. An analytical model based on a one-dimensional steady-state equation is developed to analyze the temperature-dependent performance of the solid-state PETE converter. The treatment used to derive the reverse saturation current density ($J_0$) and open-circuit voltage ($V_{oc}$) of the solid-state PETE converter is similar to that used in photovoltaic cells. Thus, their performances at elevated temperatures can be compared. Analysis results show that $J_0$ of the solid-state PETE converter with a GaAs absorption layer is approximately three orders of magnitude lower, and the decrease rate of open-circuit voltage ($-d{V}_{oc}/dT$) is smaller than that of a practical GaAs photovoltaic cell. The improved performance of the solid-state PETE converter at high temperatures is attributed to the simultaneous use of diffusion and ballistic transport to harvest photo-generated electrons. The results presented in this paper demonstrate that, besides using wide bandgap materials and increasing doping density, harvesting solar energy via PETE effect can effectively improve the performance of solar cells at elevated temperatures.

1. Introduction

For concentrator photovoltaic systems and solar arrays of near-solar space probes, solar cells are generally exposed to high temperatures [1,2]. Unfortunately, the performance of solar cells degrades with the increase in temperature [3]. Efficiency degradation is primarily due to the decrease in open-circuit voltage ($V_{oc}$) with the increase in temperature [1,2,3,4,5]. The reverse saturation current density ($J_0$) is a critical parameter because it exponentially increases with increasing temperature and decreases $V_{oc}$ rapidly [1,4]. $J_0$ is a material-dependent parameter that relies on the bandgap and doping level of the material [4,5]. Wide bandgap solar cells, including photovoltaic cells made from GaInP [6], GaP [7], GaN [8], and SiC [9], were developed for high-temperature near-solar operations to improve the efficiency at high temperatures.

Recently, a novel high-temperature solar conversion concept called photon-enhanced thermionic emission (PETE) has been proposed, in which carrier separation is achieved by thermionic emission of photo-generated electrons from a hot cathode to a cold anode [10]. PETE can simultaneously utilize photon and thermal energy to achieve a conversion efficiency higher than 40% at a temperature of 200 °C. However, the vacuum gap that separates the cathode and anode results in numerous challenges in fabricating practical devices [11,12,13]. Diamond-based PETE cathodes with nanotexturing light receiving surfaces and hydrogenation emitting surfaces have been designed for high temperature operation [14,15]. A solid-state PETE solar energy converter has been proposed, to avoid the complicated fabrication and encapsulation of vacuum PETE devices [16]. The vacuum gap is replaced by a wide bandgap semiconductor layer. Carrier separation in the solid-state PETE converter is due to the internal PETE effect at the interface of the absorber and barrier layers, rather than the built-in electric field of a p–n junction in photovoltaic cells [17]. The conversion efficiency of the solid-state PETE converter exceeds 30% at a temperature of 600 K and a flux concentration of 1000. However, the temperature dependence of parameters, such as $J_0$ and $V_{oc}$, and the performance of the solid-state PETE converter at elevated temperatures, need to be further studied in detail.

In this paper, an analytical model is proposed to investigate the temperature-dependent performance of the solid-state PETE converter. The critical parameters $J_0$ and $V_{oc}$ are derived on the basis of a one-dimensional steady-state continuity equation, which is a similar method for photovoltaic cells. We compare the temperature-dependent performance of the GaAs-based solid-state PETE converter and GaAs photovoltaic cells. Previous studies suggested that solar cells using wide bandgap and heavy doping materials would have better performance at high temperatures. The results presented in this study demonstrate that utilizing the PETE effect as a carrier separation mechanism is another effective method for reducing voltage degradation and improving the performance of solar cells at elevated temperatures.

2. Materials and Methods

In this study, for the single-junction photovoltaic cell and solid-state PETE converter, the effect of series resistance and shunt resistance is ignored, to simplify the problem. For a single-junction photovoltaic cell under steady state illumination, the current density–voltage ($J–V$) characteristics can be described by solving the one-dimensional steady-state continuity equation [18]:

$$J\left(V\right)=J_{ph}-J_{dark}\left(V\right)$$

(1)

where $J_{ph}$ represents the photogenerated current density, and $J_{dark}$ is the junction current density without illumination, which can be described as follows [18]:

$$J_{dark}\left(V\right)=J_0\left[exp\left(qV/n{k}_{B}T\right)-1\right]$$

(2)

where $J_0$ is the reverse saturation current density, $k_B$ is Boltzmann constant, $T$ is absolute temperature, and $n$ is the ideality factor. $J_{dark}$ is determined by several mechanisms. For high-quality materials, carrier recombination in the depletion region can be neglected. Thus, the ideality factor $n$ is close to unity [4,5]. $V_{oc}$ can be obtained from Equation (1) when $J=0$ [1]:

$$V_{oc}=\frac{k_{B}T}{q}ln\left(\frac{J_{sc}}{J_0}+1\right)$$

(3)

where $J_{sc}$ is the short-circuit current density, and $J_{sc}\approx J_{ph}$. For simplicity, $J_{sc}$ can be described as follows [1]:

$$J_{sc}=q{{\displaystyle \int }}_{E_g}^{\infty }\frac{I_{ph}}{dh\nu }d\left(h\nu \right)$$

(4)

where $I_{ph}$ is the initial photon flux, and $E_g$ is the bandgap of the absorber material. $E_g$ decreases with the increase in temperature, which leads to a slight increase in $J_{sc}$ with increasing $T$.

For a single-junction photovoltaic cell, $J_0$ can be represented as follows [19]:

$$J_0=q\left(\frac{D_n}{L_n{N}_A}+\frac{D_p}{L_p{N}_D}\right)n_i{}^2$$

(5)

where $n_i$ is the intrinsic carrier density, and $D_n$ and $D_p$ are diffusion constants of minority carriers in p and n regions, respectively. $N_A$ and $N_D$ are the densities of the acceptor and donor, respectively. $L_n$ and $L_p$ respectively denote the diffusion lengths of minority carriers in p and n regions.

The solid-state PETE converter consists of a heavy doped p-type narrow bandgap semiconductor as the absorber, a wide bandgap semiconductor as the barrier, and metal electrodes. Figure 1 shows the band structure of the converter. From the perspective of thermionic emission, the absorber is treated as a cathode with a barrier of $V_C$, whereas the electrode is treated as an anode with a barrier of $V_A$. $E_{g1}$ and $E_{g2}$ are the bandgaps of the absorber and barrier layer, respectively. $S_1$ and $S_2$ are the interface recombination velocity at the front and emission interfaces, respectively. $d$ is the absorber thickness. The carriers are generated in the absorber and diffused to the hetero-junction interface. The conduction band offset ($\Delta E_c$) at the interface is considerably smaller than the valence band offset ($\Delta E_v$). In this condition, only photo-generated electrons can overcome the barrier by thermionic emission, whereas the transportation of photo-generated holes is blocked [20]. The barrier width is in the range of 10–100 nm to prevent electron diffusion and tunneling [21].

The $J_0$ and $V_{oc}$ of the solid-state PETE converter are derived based on the one-dimensional steady-state continuity equation. We assume that the voltage difference is mainly across the barrier layer. Thus, the applied electron field inside the absorber is zero. The photo-generated carriers are assumed to support the local charge neutrality approximation. In the steady state, the continuity equation for photo-generated electrons is as follows [18]:

$$\frac{d^2\Delta n\left(x\right)}{d{x}^2}-\frac{\Delta n\left(x\right)}{L_n{}^2}+\frac{G}{D_n}=0$$

(6)

where $\Delta n$ is the concentration of photo-generated electrons, and $G$ is the photon generation function. The boundary conditions for excess electrons are as follows [17]:

$${D_n\frac{d\Delta n\left(x\right)}{dx}|}_{x=0}=\Delta n\left(0\right)S_1$$

(7)

$${-q{D}_n\frac{d\Delta n\left(x\right)}{dx}|}_{x=d}=q\Delta n\left(d\right)S_2+J_{em}-J_{rev}$$

(8)

At the hetero-junction interface, the emission current from the absorber at a positive voltage $V$ is described as follows [22]:

$$J_{em}=qn\sqrt{\frac{q{k}_{B}T}{2\pi m_n}}\exp \left(-\frac{\Delta E_c}{k_{B}T}\right),\mathrm{when}V\le \left(V_C-V_A\right)/q.$$

(9)

$$J_{em}=qn\sqrt{\frac{q{k}_{B}T}{2\pi m_n}}exp\left(-\frac{\Delta E_c+\left(V_A+qV-V_C\right)}{k_{B}T}\right),\mathrm{when}V>\left(V_C-V_A\right)/q.$$

(10)

Here, $n$ is the total conduction band concentration, and $m_n$ is the electron effective mass. The reverse current from the electrode to the absorber can be calculated as follows [22]:

$$J_{rev}=A{T}^{2}exp\left(-\frac{V_C-qV}{k_{B}T}\right),\mathrm{when}V\le \left(V_C-V_A\right)/q.$$

(11)

$$J_{rev}=A{T}^{2}exp\left(-\frac{V_A}{k_{B}T}\right),\mathrm{when}V>\left(V_C-V_A\right)/q.$$

(12)

Here, $A$ is Richardson’s constant, and $120\mathrm{A}/{\mathrm{cm}}^2{\mathrm{K}}^2$ is used in the calculation. The output current of the solid-state PETE converter is given as follows:

$$J_{SPETE}=J_{em}-J_{rev}.$$

(13)

By solving the continuity equation with the boundary conditions derived above, the output current can be obtained. The output current of the solid-state PETE converter can be rewritten as Equation (1). When $G$ is set to zero, the dark current of the solid-state PETE converter is expressed as follows [23]:

$$J_{dark}\left(V\right)=\frac{q{D}_n{n}_{eq}}{\beta L_n}\left[exp\left(\frac{qV}{k_{B}T}\right)-1\right]$$

(14)

where

$$\beta =\frac{q{D}_n{N}_c}{L_{n}A{T}^2}exp\left(\frac{\Delta E_c}{k_{B}T}\right)+\frac{1}{a}\left[cosh\left(\frac{d}{L_n}\right)+\frac{L_n{S}_1}{D_n}sinh\left(\frac{d}{L_n}\right)\right]$$

$$a=sinh\left(\frac{d}{L_n}\right)+\frac{L_n{S}_1}{D_n}cosh\left(\frac{d}{L_n}\right)+\frac{b{L}_n{S}_2}{D_n}$$

$$b=cosh\left(\frac{d}{L_n}\right)+\frac{L_n{S}_1}{D_n}sinh\left(\frac{d}{L_n}\right).$$

A comparison of Equations (14) and (2) demonstrates that $J_0$ of the solid-state PETE converter differs from that of photovoltaic cells, as expressed by the following:

$$J_0=\frac{q{D}_n{n}_{eq}}{\beta L_n}.$$

(15)

The mechanism that determines $J_{sc}$ of the solid-state PETE converter is the same as that of photovoltaic cells. Hence, $J_{sc}$ of the solid-state PETE converter can be calculated using Equation (4), and $V_{oc}$ of the solid-state PETE converter can be calculated using Equations (3) and (15).

The conversion efficiency is computer as follows:

$$\mathsf{\eta }=\frac{J_m{V}_m}{P_{solar}}$$

(16)

where $P_{solar}$ is the power received from the sun, and $J_m$ and $V_m$ are the optimum operating current and voltage, respectively.

3. Results and Discussion

In this study, GaAs with a doping density of $1\times {10}^{19}{\mathrm{cm}}^{-3}$ was used as the absorber, and AlGaAs was used as the barrier layer in the solid-state PETE converter [17]. The model described above was implemented and solved numerically in MATLAB. For simplicity, diffusion lengths and diffusion coefficients were assumed to be temperature-independent in the simulation, following the same assumption as [11] and [24]. When simulating the performance of the solid-state PETE converter, the absorber thickness and conduction band offset took the optimal values. The surface recombination velocities of the front and emission interfaces were set to 10³ cm/s, which can be achieved in a high-quality heterostructure or a passivated surface [25,26]. For comparison, the temperature-dependent performance of a GaAs photovoltaic cell was also calculated. The temperature dependence curves of $J_0$ for the solid-state PETE converter and photovoltaic cell were obtained using Equations (14)–(15) and Equation (5), respectively. Combining Equations (3)–(5) and (15), the temperature dependence of $V_{oc}$ was calculated. The conversion efficiencies were obtained using Equation (16) under different temperatures and solar concentrations. For both the solid-state PETE converter and photovoltaic cells, the temperature dependence of the bandgap of GaAs was considered. The calculations used the AM1.5 global solar spectrum.

Figure 2 shows the temperature dependence curve of $J_0$ for the solid-state PETE converter with the GaAs absorption layer and GaAs photovoltaic cell with different doping concentrations. $J_0$ of all devices increased exponentially with the increase in temperature. The solid-state PETE converter achieved the minimum $J_0$ in the temperature range of 300–600 K. At the same temperature, $J_0$ of the photovoltaic cell with high doping concentration was small, which is consistent with the previous study [4]. This result is mainly due to the increase in doping concentration that causes a reduction in the equilibrium carrier concentration in the material, which is beneficial for reducing $J_0$ according to Equation (5). Figure 2 demonstrates that $J_0$ of the solid-state PETE converter was approximately one order of magnitude smaller than that of the GaAs photovoltaic cell with a doping density of $1\times {10}^{19} {\mathrm{cm}}^{-3}$. Given that the absorption material of the solid-state PETE converter had the same band gap (GaAs) and doping density ($1\times {10}^{19} {\mathrm{cm}}^{-3}$), only the difference in their working mechanisms can explain the difference in $J_0$. In photovoltaic cells, the photo-generated carrier transport mainly involves diffusion. Thus, the expression of $J_0$ included only the diffusion parameters [1]. For the solid-state PETE converter, the carriers generated in the absorber diffused to the hetero-junction interface. Given that the barrier thickness was less than the mean free path of the carrier in the barrier material, photo-generated carriers traversed over the barrier by thermionic emission, which is a ballistic transport process [17]. The carrier separation and extraction in the solid-state PETE converter occurred via both diffusion and ballistic transport. Hence, its $J_0$ was determined by diffusion and thermal emission parameters, as shown in Equations (14) and (15).

Heavily doped semiconductor bulk materials are unsuitable to make p–n homo-junction for practical photovoltaic cells. Carrier transport in photovoltaic cells mainly occurs through diffusion. With an increase in doping density, additional impurities and defects are introduced into semiconductor bulk materials, which leads to a decrease in carrier mobility, diffusion coefficient, minority carrier life, and diffusion length. These conditions affect the transport and collection of photo-generated carriers and reduce their efficiency [27,28,29]. Therefore, the doping density of practical photovoltaic cells is usually less than $1\times {10}^{18}{\mathrm{cm}}^{-3}$. Increasing the doping density is ineffective in further improving the performance of photovoltaic cells at high temperatures. Compared with the practical GaAs photovoltaic cell with a doping density of $1\times {10}^{17}{\mathrm{cm}}^{-3}$ [1,30], $J_0$ of the solid-state PETE converter was approximately three orders of magnitude lower, as shown in Figure 2.

The temperature dependence of $V_{oc}$ is determined by $J_0$. For the device with low $J_0$, the decrease rate of $V_{oc}$ with temperature, that is, $-d{V}_{oc}/dT$, was small. Figure 3 shows $V_{oc}$ of the solid-state PETE converter with GaAs absorption layer and practical GaAs photovoltaic cell as a function of operating temperature under different flux concentrations (i.e., the concentration ratios of the incident solar radiation flux). At a flux concentration of 1 sun, $V_{oc}$ of the solid-state PETE converter decreased less with the increase in temperature, compared with that of practical GaAs photovoltaic cell, and $-d{V}_{oc}/dT$ of the solid-state PETE converter was approximately 70% that of the photovoltaic cell. Figure 3 shows that $V_{oc}$ of the solid-state PETE converter was higher than that of photovoltaic cells, which is due to the use of heavily doped material as absorber layer, which suppresses the equilibrium carrier concentration.

Increasing the irradiance concentration increased $V_{oc}$ and decreased $-d{V}_{oc}/dT$ for photovoltaic cells [31,32]. This trend was also valid for the solid-state PETE converter. Figure 3 shows that as the flux concentration increased from 1 sun to 1000 sun,$V_{oc}$ at a certain temperature increased, and $-d{V}_{oc}/dT$ decreased from 1.60 mV/K to 1.01 mV/K. Notably, $V_{oc}$ and $-d{V}_{oc}/dT$ of the photovoltaic cell at 1000 sun were similar to those of the solid-state PETE converter at 1 sun. This result shows that by utilizing the novel conversion mechanism of PETE effect, the performance of solar energy converters can be effectively improved at high temperatures.

Since $J_{sc}$ increased slightly with increasing temperature, the degradation of conversion efficiency is mainly due to a decrease in $V_{oc}$. The conversion efficiencies of the solid-state PETE converter with GaAs absorption layer and practical GaAs photovoltaic cell as functions of temperature and incident solar radiation concentration are shown in Figure 4. The efficiency decreased as temperature increased, and increased as incident light intensity increased, which is consistent with the change trend of $V_{oc}$ with temperature and flux concentration. Under the same temperature and concentration conditions, the efficiency of the solid-state PETE converter with GaAs absorption layer was higher than that of GaAs photovoltaic cell. The conversion efficiency of the solid-state PETE converter was 28% at 600 K and a flux concentration of 1000.

Solid-state PETE converters are suitable for working conditions under high temperatures and high light intensity. For concentrator solar systems, replacing photovoltaic cells with solid-state PETE converters can achieve increased efficiency. For concentrator hybrid systems combining solar cells and secondary thermal converters, solid-state PETE converters can function at high temperatures. Thus, additional heat energy can be delivered to the thermal cycle, boosting the total efficiency. For near-solar space missions, solar cells with improved performance at high temperatures are desirable over techniques, such as off-pointing array, increasing reflection, and active cooling, to reduce the temperature, but at the expense of losing total performance [2]. Harvesting solar energy via the PETE effect, combined with the use of wide bandgap and heavily doped materials, is a feasible strategy to achieve such high-temperature solar cells. Hetero-junction materials such as GaAs/AlGaAs and Si/diamond can be potential candidate materials.

4. Conclusions

An analytical model based on one-dimensional steady-state continuity equation is presented to investigate the temperature-dependent performance of the solid-state PETE solar energy converter. The treatment to derive $J_0$ and $V_{oc}$ of the solid-state PETE converter in this study is similar to that used in photovoltaic cells. The analysis results show that compared with the photovoltaic cells, the performance of the solid-state PETE converter at elevated temperatures is remarkably improved, due to its simultaneous use of diffusion and ballistic transport to harvest photo-generated electrons. $J_0$ of the solid-state PETE converter with GaAs absorption layer is approximately three orders of magnitude lower than that of the practical GaAs photovoltaic cell. At 1 sun, $-d{V}_{oc}/dT$ of the solid-state PETE converter is approximately 70% that of photovoltaic cell. As the flux concentration increases from 1 sun to 1000 sun, $V_{oc}$ of the solid-state PETE converter at a certain temperatures increases, and $-d{V}_{oc}/dT$ decreases from 1.60 mV/K to 1.01 mV/K. The results presented in this paper show the feasibility of the solid-state PETE converter as a candidate device for concentrator solar systems and solar array of near-solar space probes to further improve their performances at high temperatures.