Investigation of carrier transport and collection characteristics for GaAs-based betavoltaic batteries

Betavoltaic batteries using radioactive isotopes emitting beta particles have been studied for powering the micro-devices due to their long service life, high power density, and strong environmental adaptability.^1,2 These batteries are composed of a semiconductor energy converter and a beta source, and their operational principle is similar to that of a photovoltaic battery. The beta particles interact with the semiconductors and create thousands of electron–hole pairs (EHPs) through impact ionization. Then, these EHPs can be collected in the energy converter through certain transport mechanisms, forming the radiation-induced current and converting the decay energy into electrical energy.

GaAs is a direct bandgap III–V compound semiconductor, which has the advantages of wide bandgap (1.424 eV), high electron mobility [∼8000 cm²/(V s)], and high threshold energy for radiation damage (270 keV).³ Due to its low intrinsic carrier concentration, the GaAs-based betavoltaic batteries can have a low leakage current and a high open-circuit voltage. Additionally, the growth and processing techniques of GaAs are relatively mature, and it is easy to fabricate a high quality betavoltaic device.⁴ These make GaAs an ideal energy conversion material for the betavoltaic batteries. In the past decade, the betavoltaic batteries with a p–n junction and Schottky barrier structures have been extensively studied.^5–10 The performances of these batteries were mainly investigated with the effects of structure parameters, such as doping concentrations and junction depth.^11–13 However, the experimental results of the fabricated batteries show very low energy conversion efficiency, and the performance prediction and optimization design are always unsatisfactory.^14–17 On the one hand, due to the self-absorption of the source, the apparent power density is much less than the total power density, resulting in the low utilization efficiency of the source and the small output power density of batteries. Fortunately, recent studies have shown that the transport process of beta particles in the energy conversion material can be well simulated by using the Monte Carlo codes, in which the self-absorption is considered.¹⁸ Therefore, through certain simulation calculations, the geometry of the source can be optimized to improve the utilization efficiency of the source in a betavoltaic battery. The battery performance predictions considering the self-absorption of the source are more accurate. On the other hand, theoretical calculations on the output parameters are almost based on the analytical expressions, which are obtained by solving the minority carrier diffusion equation with a lot of hypotheses,^11,12,19 eventually resulting in an unreliable performance prediction and even structure design. Although using the energy deposition distribution of the source simulated by Monte Carlo codes and then using the Klein formula to calculate the EHP generation rate distribution are proved to be reasonable,²⁰ the transport and collection processes of carriers inside the battery cannot be described in detail, especially through the analytical expressions for calculating the radiation-induced current. Therefore, further investigations of carrier transport and collection characteristics are needed for improving the accuracy of performance prediction and structure design of the betavoltaic batteries.

Recent research studies have shown that it is feasible to investigate the performance of a betavoltaic battery using device simulators, such as Synopsys Medici, technology computer-aided design (TCAD), and COMSOL Multiphysics.^21–27 In 2016, the performance of a 4H–SiC betavoltaic battery was predicted using the Monte Carlo code and Synopsys Medici device simulator.²² They investigated the effects of source thickness and the thickness and doping concentration of the p-region on the battery performance and proposed the optimal design. In 2020, a GaN-based betavoltaic battery with an AlGaN back-barrier layer was reported.²⁵ In this research, a three-dimensional (3D) TCAD simulator was used to optimize the finger structure, which can improve the output power density of the battery. In the same year, the betavoltaic generators based on the nanowire were presented, which are made of Si, GaAs, and GaP for ⁶³Ni and tritium sources.²⁶ The energy deposition distribution of beta particles was obtained by using the Monte Carlo simulations, and furthermore, the nanowire geometry was optimized. In addition, the current–voltage (I–V) characteristics and maximum power of the batteries were determined using COMSOL Multiphysics, and the optimal diode designs are suggested. According to these studies, the performance predictions of betavoltaic batteries by the device simulators are reliable and accurate. However, there is little discussion about the energy band structure and current density distribution inside the betavoltaic battery, which essentially determine the battery performance. It is necessary to take a detailed observation during the physical processes that take place inside the battery, especially carrier transport and collection characteristics. This can contribute to illuminating the effects of structure parameters on the battery performance, further guiding the optimization and fabrication.

In this paper, we presented a simulation model to predict the performance of GaAs-based betavoltaic batteries with a p–n junction structure, in which the carrier transport and collection characteristics were studied. The electron–hole pair generation rate in the GaAs material under the irradiation of a ⁶³Ni source was calculated by using the Monte Carlo codes. To accurately predict the performance of GaAs-based batteries, the finite element analysis software COMSOL Multiphysics was used to investigate the carrier transport and collection characteristics. The proposed GaAs-based batteries have four layers: a top heavily doped p⁺-GaAs layer, a p-GaAs layer, an n-GaAs layer, and a bottom heavily doped n⁺-GaAs layer. In order to enhance the carrier collection and optimize the structure parameters of the batteries, the effects of thicknesses and doping concentrations of the p-region and n-region on the output parameters were analyzed. Specifically, the energy band structure, electric field distribution, current density distribution, and current density–voltage characteristics of the batteries were obtained, and finally, the optimized output performances, including short-circuit current density, open-circuit voltage, and maximum output power density, were achieved. These results have guiding significance for the performance improvement, optimization design, and experimental preparation of the GaAs-based betavoltaic batteries. Our simulation model can be extended to the betavoltaic batteries with other semiconductors and radioactive isotopes.

The simulation process consists of two parts: the simulation of energy deposition distribution of beta particles in the semiconductor energy converter and the simulation of electrical characteristics of devices. The energy deposition distribution of ⁶³Ni beta particles in the GaAs material is simulated by using the Monte Carlo codes, and furthermore, the electron–hole pair generation rate is obtained. The output performances of batteries are determined by using COMSOL Multiphysics in which the electron–hole pair generation rate from the Monte Carlo simulation is utilized as input.

In this study, a 2 μm-thick ⁶³Ni source (with a total activity density of 100 mCi/cm²) is selected for the GaAs-based batteries with a p–n junction structure. The energy deposition distribution of beta particles in the GaAs material determines the spatial distribution of the electron-hole pair generation rate. In this part, a rectangular ⁶³Ni source (1 cm × 1 cm × 2 µm) with a full energy spectrum is used to calculate the energy deposition along the radiation transport depth in GaAs bulk (1 × 1 × 0.5 cm³), and the prototype structure of the simulation model is shown in Fig. 1(a). Klein reported the average energy dissipated per electron-hole pair generated as E_ehp = 2.8 E_g + 0.5 eV, where E_g is the bandgap of the semiconductor.²⁰ Furthermore, the electron–hole pair generation rate [G(Y)] is obtained, and as shown in Fig. 1(b), it decreases exponentially with the increase in radiation transport depth (Y) and can be expressed using the following formula:

where E(Y) is the energy deposition rate in the GaAs material and G₀ and α are the exponential fitting parameters (G₀ = 1.5907 × 10²² m⁻³ s⁻¹ and α = 0.987 67 µm⁻¹).

COMSOL Multiphysics is used to model the GaAs-based betavoltaic batteries with a p–n junction structure, which have four layers: the top p⁺-GaAs and bottom n⁺-GaAs layers are heavily doped for better metal contact, and the p-GaAs and n-GaAs layers in the middle are used to form the depletion region. As shown in Fig. 2(a), the overall dimension of the battery is 1 × 1 × 10 μm³. Using 10 μm as the total thickness of the battery is based on the maximum radiation transport depth of ⁶³Ni beta particles in GaAs [Fig. 1(b)]. The thicknesses of the top p⁺-GaAs and bottom n⁺-GaAs layers are set to 0.1 μm. The thicknesses of the p-region and n-region (H_p-GaAs and H_n-GaAs) are variables, and they will be optimized in the following simulations. The most material properties of GaAs are imported from the COMSOL library, but some properties, such as minority carrier mobility and minority carrier lifetime, are not available in the library, so we manually added them from the literature.^28,29 To improve the accuracy of results and to obtain a faster computation time, a user-controlled mesh is defined for the 2D geometry with a thickness of 1 µm, as shown in Fig. 2(b). The maximum and minimum element sizes are 10 and 1 nm, respectively. The maximum element growth rate is set to 1.08 with a curvature factor of 0.25, and the resolution of narrow regions is specified to be 1.

In this study, our focus is on the carrier transport and collection simulations; thus, the finite element method is used to solve the coupled Poisson and carrier continuity equations^30,31 as follows:

where V is the electrostatic potential, ρ is the charge density, ϵ₀ is the vacuum permittivity, ϵ_r is the relative permittivity of the material, q is the electron charge, G is the electron–hole pair generation rate, and R_n (R_p) is the electron (hole) recombination rate. j_n (j_p) is the electron (hole) current density, which can be expressed as³⁰ 

where μ_n (μ_p) is the electron (hole) mobility, k is Boltzmann’s constant, T is the absolute temperature, and n (p) is the electron (hole) concentration.

The battery is assumed to be operating at room temperature (300 K), and various physical models are employed in the simulations. First of all, the COMSOL calculations utilize the electron-hole pair generation rate from the Monte Carlo simulation as input. Second, the analytic doping model features are used to define the background doping and the main p and n sections of the device. The geometric doping models are used to create highly doped layers at the top and bottom of the device. The low-field mobility model is used to calculate the minority carrier mobility, which is a function of doping concentration,²⁸ 

where N is the net doping concentration and μ_a, μ_b, N_ref, and d are the fitting parameters that can be obtained from the literature.²⁸ In addition, the Shockley–Read–Hall (SRH) model is used to define the trap-assisted recombination, with associated parameters, such as low-injection minority carrier lifetime, which can be expressed as²⁹ 

where τ₀ is the intrinsic lifetime. The semiconductor properties of GaAs used in COMSOL Multiphysics are listed in Table I.^28,29

To maximize the output power density and optimize the structure parameters of the GaAs-based betavoltaic batteries, the parametric sweep is used in the COMSOL model to tune the variables including the thicknesses of the p-region and n-region (H_p-GaAs and H_n-GaAs), the acceptor concentration of the p-region (N_a), and the donor concentration of the n-region (N_d). The feasible H_p-GaAs value ranged from 0.1 to 5 μm, H_n-GaAs = 10 μm–H_p-GaAs, the feasible N_a value ranged from 1 × 10¹⁵ to 1 × 10¹⁸ cm⁻³, and the feasible N_d value ranged from 1 × 10¹⁴ to 1 × 10¹⁵ cm⁻³. In addition, the H_p-GaAs, H_n-GaAs, N_a, and N_d values of the structure as initial values are 0.5 μm, 9.5 μm, 1 × 10¹⁶ cm⁻³, and 1 × 10¹⁵ cm⁻³, respectively. The acceptor concentration and donor concentration of the heavily doped top p⁺-GaAs and bottom n⁺-GaAs layers (N_A and N_D) are set to 1 × 10¹⁸ and 1 × 10¹⁷ cm⁻³, respectively.

Figure 3(a) shows the energy diagram of the battery at thermodynamic equilibrium. For the battery without top and bottom heavily doped layers, the relative positions of the conduction and valence bands (E_c and E_v) between the p-region and n-region change with the position of the Fermi energy level (E_F), and as a result, the conduction band and valence band in the space charge region will be bent.³² Meanwhile, an energy potential barrier (built-in potential barrier) of 1.04 eV is formed, and it blocks the electrons in the n-region from moving into the p-region and maintains the equilibrium of carrier diffusion and drift. When the top and bottom layers are heavily doped, the Fermi energy level in the p⁺-GaAs layer is close to the valence band, and the Fermi energy level in the n⁺-GaAs layer is close to the conductor band. The higher energy potential barrier contributes to the larger open-circuit voltage of the battery.

In our simulations, the Y-component (normal to the junction plane) of each physical quantity is significant. Thus, Fig. 3(b) shows the Y-component distribution of the electric field across the battery (the n-region pointing to the p-region is the positive direction of the electric field). It can be observed that for the battery without top and bottom heavily doped layers, the electric field is more intensive in the depletion region and reaches its maximum at the metallurgical junction as expected. The radiation-induced electron–hole pairs generated in this region can be collected by the drift mechanism and form the drift current. When the top and bottom layers are heavily doped, the drift fields will be formed between the p⁺-GaAs (n⁺-GaAs) and p-GaAs (n-GaAs) layers and increase the radiation-induced current by boosting the minority carrier transport in the battery. The Y-component distribution of the electric field of the battery is also shown in 2D [Fig. 3(c)], where the top and bottom layers are heavily doped.

Figure 4 shows the electron (n) and hole (p) concentration distributions across the battery without and with ⁶³Ni source irradiation. Due to the donor doping in the n-region and acceptor doping in the p-region, the electron and hole concentrations have a very large concentration gradient at the metallurgical junction. Meanwhile, the diffusion current driven by the concentration gradient and the drift current driven by the built-in electric field will remain in balance, and as a result, the total current density across the battery is 0. When the GaAs p–n junction is irradiated by the ⁶³Ni source, thousands of electron-hole pairs (excess electrons and holes) are generated, as shown by the red line in Fig. 4, and the minority electron concentration in the p-region and the minority hole concentration in the n-region will increase significantly. In the condition of short circuit, the balance between carrier diffusion and drift is broken. These radiation-induced excess electrons and holes will be separated by the built-in electric field, forming the electron drift current and hole drift current. The lower minority carrier concentration in the depletion region and near the cell surface can be explained as the carrier loss due to carrier drift, diffusion, and surface recombination.

For a betavoltaic battery, in the condition of short circuit, the radiation-induced current consists of the electron current and the hole current, which also include the drift current and diffusion current components.³³ The Y-components of current densities of the battery (without heavily doped layers), which change following the relative location, are shown in Fig. 5 (the n-region pointing to the p-region is the positive direction of current density). It can be seen that in the range of ∼0.1 to ∼3.4 μm, both electrons and holes contribute to the total current and the short-circuit current density (J_sc) reaches 0.232 μA/cm². However, in the range of the top electrode to ∼0.1 μm, only holes contribute to the total current. In the range of ∼3.4 μm to the bottom electrode, only electrons contribute to the total current.

Figure 5(b) gives further details of electron drift and diffusion current density distributions. Due to the diffusion of electrons to the top electrode, the movement of electrons in the range of the top electrode to ∼0.1 μm has no contribution, and as a result, the electron diffusion current density is negative. In the depletion region, the drift of electrons is dominant compared with the diffusion; thus, the positive total electron current density is achieved, which contributes to J_sc. In the n-region, the electrons are the majority carriers, and they move mainly in the drift mechanism. Both the electron drift current density and the electron diffusion current density resulting from the diffusion to the bottom electrode contribute to J_sc. It is worth noting that the flat band region (quasi-neutral region) actually has an electric field, but its intensity is very weak. The drift of the minority carriers is of low importance, while the drift of the majority carriers cannot be ignored.

The hole drift and diffusion current density distributions are shown in Fig. 5(c), which are similar to those of the electron. In the range of ∼3.4 μm to the bottom electrode, the holes diffuse toward the bottom electrode and the hole diffusion current density is negative. However, as the holes approach the depletion region, the diffusion to the depletion region becomes dominant and the hole diffusion current density becomes positive. In the depletion region, the drift of holes is dominant compared with the diffusion, and the holes are constantly being swept into the p-region. In the p-region, the holes are the majority carriers, and they drift toward the top electrode. The same as other current components mentioned above, they also contribute to J_sc.

So far, the Y-component distributions of current densities inside the battery in the condition of short circuit are obtained. Furthermore, by sweeping the forward voltage across the device and recording the terminal current, we can obtain the current density–voltage (J–V) and power density–voltage (P–V) characteristics of the betavoltaic battery. Figure 6 shows the J–V and P–V characteristics of the battery for two structures without and with heavily doped layers. For the battery without heavily doped layers, when the structure parameters are H_p-GaAs = 0.5 μm, H_n-GaAs = 9.5 μm, N_a = 1.58 × 10¹⁵ cm⁻³, and N_d = 1 × 10¹⁵ cm⁻³ (the doping concentrations are also the optimized values, which will be presented later), the short-circuit current density, open-circuit voltage, and maximum output power density can reach 0.232 μA/cm², 0.44 V, and 0.072 μW/cm², respectively. When the top and bottom layers are heavily doped (N_A = 1 × 10¹⁸ cm⁻³ and N_D = 1 × 10¹⁷ cm⁻³), the wider electric field region and higher energy potential barrier contribute to the larger short-circuit current density and higher open-circuit voltage. The increased drift current densities finally result in a larger short-circuit current density of 0.246 μA/cm², an open-circuit voltage of 0.50 V, and a maximum output power density of 0.086 μW/cm².

Obviously, the thicknesses and doping concentrations of each region determine the energy band structure and electric field distribution inside a p–n junction and further affect the output performance of the battery. In order to optimize the structure parameters of the GaAs-based battery, the J–V characteristics, short-circuit current density (J_sc), open-circuit voltage (V_oc), and maximum output power density (P_m) are investigated with a variation of H_p-GaAs, N_a, and N_d. Figure 7(a) shows the J–V characteristics of the battery without heavily doped layers whose H_p-GaAs, H_n-GaAs, and N_d values are 0.5 μm, 9.5 μm, and 1 × 10¹⁵ cm⁻³, respectively. It can be seen that as N_a increases, the current density changes significantly, to be specific, J_sc decreases, while V_oc increases. Figures 7(b) and 7(c) show the energy diagram and Y-component distribution of the electric field of the battery, respectively. The increase in the doping concentration N_a results in the decrease in the depletion region width; meanwhile, the depletion region is away from the surface of the p-region where the generation rate of electron-hole pairs is higher. Thus, the drift current densities decrease and the short-circuit current density J_sc decreases with increasing N_a. Conversely, the larger N_a value is beneficial to form the higher built-in potential barrier, which can enhance V_oc.

Furthermore, Figs. 8(a)–8(c) show the effects of H_p-GaAs and N_a on J_sc, V_oc, and P_m. When H_p-GaAs is changed, H_n-GaAs = 10 μm–H_p-GaAs, and when N_a is changed, N_d is fixed as 1 × 10¹⁵ cm⁻³. J_sc decreases with increasing H_p-GaAs, and this can be explained as larger H_p-GaAs, leading to the deeper junction, which has a lower collection efficiency of electron–hole pairs. As shown in Fig. 8(b), the open-circuit voltage (V_oc) increases with increasing N_a and shows the same trend for different H_p-GaAs values. Ultimately, the maximum output power density (P_m) first increases with increasing N_a and then decreases. For the battery with an H_p-GaAs value of 0.1 μm, the largest P_m value of 0.080 μW/cm² can be achieved when the doping concentrations are N_a = 3.98 × 10¹⁶ cm⁻³ and N_d = 1 × 10¹⁵ cm⁻³. When the H_p-GaAs value increases to 0.5 μm, the P_m value decreases to 0.072 μW/cm² with the optimized doping concentrations N_a = 1.58 × 10¹⁵ cm⁻³ and N_d = 1 × 10¹⁵ cm⁻³.

Similarly, Fig. 9(a) shows the effects of H_p-GaAs and N_d on P_m. When H_p-GaAs is changed, H_n-GaAs = 10 μm–H_p-GaAs, and when N_d is changed, N_a is fixed as the optimized doping concentration for different H_p-GaAs values. It can be seen that P_m increases with increasing N_d and slows down at last. This indicates that larger N_d brings the higher built-in potential barrier and eventually contributes to higher V_oc and larger P_m. The decrease in P_m with increasing H_p-GaAs can also be explained as larger H_p-GaAs, leading to the deeper junction, which is not conducive to the separation and collection of electron–hole pairs. When H_p-GaAs is in the range of 0.1–5 μm and the doping concentrations (N_a and N_d) are optimized values, the relationship between P_m and H_p-GaAs is shown in Fig. 9(b). To be specific, when H_p-GaAs increases from 0.1 to 0.5 μm, P_m decreases from 0.080 to 0.072 μW/cm². When H_p-GaAs increases to 5 μm, P_m is only 0.017 μW/cm².

For the battery with top and bottom heavily doped layers, the relationships among J_sc, V_oc, and P_m and H_p-GaAs, N_a, and N_d have also been investigated. The comparison of optimized output performances of the GaAs-based batteries is shown in Table II. For the sake of comparison, the optimized doping concentrations (N_a and N_d) of the battery for two structures without and with heavily doped layers are taken to be the same value. It can be seen that when the top and bottom heavily doped layers (N_A = 1 × 10¹⁸ cm⁻³ and N_D = 1 × 10¹⁷ cm⁻³) are introduced, J_sc increases from 0.234 to 0.247 μA/cm², and P_m increases from 0.080 to 0.086 μW/cm².

In summary, the performances of GaAs-based betavoltaic batteries with a p–n junction structure were predicted by our presented simulation model, and the carrier transport and collection characteristics were investigated. The Monte Carlo codes were used to calculate the electron-hole pair generation rate in the GaAs material under the irradiation of a ⁶³Ni source, and the finite element analysis software COMSOL Multiphysics was used to predict the battery performance. First, for the batteries without heavily doped layers, only in some areas, both the electron and hole drift current densities contribute to J_sc. This depends on the structure parameters. The J_sc, V_oc, P_m, and η values of the batteries are significantly affected by H_p-GaAs, H_n-GaAs, N_a, and N_d. The lower N_a value is beneficial to obtain a wider depletion region, and furthermore, a larger J_sc value can be achieved. However, the higher N_a value contributes to the higher built-in potential barrier, which can enhance V_oc. Thus, the largest P_m value of 0.080 μW/cm² can be achieved when the thicknesses and doping concentrations of each region are H_p-GaAs = 0.1 μm, H_n-GaAs = 9.9 μm, N_a = 3.98 × 10¹⁶ cm⁻³, and N_d = 1 × 10¹⁵ cm⁻³. The related J_sc, V_oc, and η values are 0.234 μA/cm², 0.49 V, and 1.55%, respectively. Second, when the top and bottom layers are heavily doped (N_A = 1 × 10¹⁸ cm⁻³ and N_D = 1 × 10¹⁷ cm⁻³), the drift fields are formed and the higher energy potential barrier enhances the battery performance. The optimized J_sc, V_oc, P_m, and η values are 0.247 μA/cm², 0.50 V, 0.086 μW/cm², and 1.67%, respectively. Although the GaAs-based betavoltaic batteries were investigated in this study, our simulation model can be extended to the betavoltaic batteries with other semiconductors and isotopes for the optimization and fabrication. Finally, we will fabricate the proposed GaAs-based betavoltaic batteries using the metalorganic chemical vapor deposition (MOCVD) and validate the simulation model in the future work.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11075064 and U1867210) and the National Major Scientific Instruments and Equipment Development Projects (Grant No. 2012YQ240121).

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.