Introduction

BV batteries are regarded as an alternative power source due to their high energy density, long lifetime and strong environmental adaptability1,2. Conventional planar BV batteries consist of a beta radioactive source and a semiconductor energy converter. The operational principle of BV batteries is that the beta particles emitted from radioactive sources interact with semiconductors and generate large quantities of electron–hole pairs (EHPs) via impact ionization, and then these EHPs are separated by the built-in electric field of the energy converter and collected by the electrodes to form the radiation-induced current.

For conventional BV batteries, the energy conversion efficiency and maximum output power density are low due to the self-absorption of radioactive sources3,4,5. The shortcomings limit the application of BV batteries. In the past few years, most of the research focused on optimizing the structure parameters of the energy converter to reduce energy loss, augment energy deposition and improve the collection efficiency to raise energy conversion efficiency and maximum output power density6,7,8,9,10,11. The structure parameters include the thickness of the beta radioactive source, the thicknesses of p-type semiconductor and n-type semiconductor, the donor concentration and acceptor concentration. Wide bandgap semiconductor nuclear batteries have obtained increasing attention due to their higher energy conversion efficiency and corresponding higher maximum output power density11,12,13. However, the energy conversion efficiency remains relatively low. This is because the beta-radioactive source and semiconductor converter of conventional nuclear batteries are separated, resulting in most of the energy of the beta radiation source being wasted inside the radiation source and failing to fundamentally solve the problem of low energy conversion efficiency. Therefore, only by solving the problem of self-absorption of the beta-radioactive source can we fundamentally resolve the low energy conversion efficiency of conventional nuclear batteries. In addition, due to the self-absorption effect of the beta-radioactive source, the maximum output power density of the nuclear battery will reach a saturation value and can no longer increase when the beta-radioactive source reaches a certain thickness7. Recently, some researchers have changed the distribution of radioactive source and energy transducer, or have combined radioactive source and electrode or have combined radioactive source and energy transducer. These modifications have significantly improved energy conversion efficiency and maximum output power density. McNamee et al. and Wagner et al. utilized nanowire structures to reduce the self-absorption energy loss14,15. Yakimov et al. utilized 63Ni as the electrode to reduce the energy loss in conventional metal electrode16. Wang et al. and Yuan et al. combined 63Ni with NiO to form 63NiO17,18. Among these studies, the last two achieved the best energy conversion efficiency. But which semiconductor material combined with 63NiO forming corresponding heterojunction can achieve the optimum conversion efficiency is still a question, and the minimum thickness of 63NiO corresponding to saturation maximum output power density of the heterojunction is also an unknown question.

In this study, we selected six kinds of common semiconductor materials which are able to form heterojunctions with 63NiO19,20,21,22,23,24. These materials are Si, InP, GaAs, \(\hbox {Al}_{0.3}\hbox {Ga}_{0.7}\)As, GaP and diamond and they are arranged in sequence of increasing bandgap. Compared to other semiconductor materials, these semiconductor materials have the following advantages respectively. Si is dominant in electronic devices due to its mature manufacturing technology and low cost. InP is known for its high radiation resistance. GaAs is characterized by a low noise figure and good temperature stability. \(\hbox {Al}_{0.3}\hbox {Ga}_{0.7}\)As has a wider bandgap and higher carrier mobility than GaAs, which is advantageous for enhancing the open-circuit voltage of nuclear batteries. GaP has a high melting point and thus has the potential for high-temperature applications. Diamond stands out with its ultra-wide bandgap, high thermal conductivity and high carrier mobility. The energy deposition of these heterojunctions is simulated utilizing Geant4 and the J–V and P–V characteristics are simulated by COMSOL Multiphysics. Those heterojunctions are simulated in different thicknesses of 63NiO and different doping concentrations of the six materials to obtain an optimum maximum output power density and then to find an optimal heterojunction. To further improve the maximum output power density, we use graphene as an electrode to combine the two 63NiO/GaP heterojunctions in parallel connection. This doubled the maximum output power density compared with the single 63NiO/GaP heterojunction. Finally, we simulated the time-related performance of the parallel connection heterojunction within 200 years and gave a fitting formula of the maximum output power density.

Energy deposition and electron–hole pair generation

The Monte Carlo software Geant4 (version: 11.0.3, September 2022) is utilized to simulate the energy deposition distribution of beta particles emitted from 63NiO of the six heterojunctions. Geant4 is a toolkit to create simulations of the particles or radiation through matter. Applications build on Geant4 can simulate any setup or detector and radiation source, and record chosen output of physical quantities due to source particles and secondaries interacting with the material of the setup. It is used by a large number of studies in a variety of application areas, including high energy physics, nuclear physics, medical physics and astrophysics25. In order to ensure complete energy deposition, the thicknesses of the six n-type semiconductors are all set to 0.1 cm (a cross-sectional area of \(1\times 1\) \(\hbox {cm}^2\)). The densities of 63NiO, Si, InP, GaAs, \(\hbox {Al}_{0.3}\hbox {Ga}_{0.7}\hbox {As}\), GaP and diamond are 7.06, 2.33, 4.81, 5.32, 4.85, 4.14 and \(3.52\, \hbox {g}\cdot \hbox {cm}^{-3}\) respectively. Three interaction models, including G4DecayPhysics, G4EmStandardPhysicsWVI and G4RadioactiveDecayPhysics, are set in the PhysicsList.cc files of the six heterojunctions.

The schematic diagram taking 63NiO/GaP heterojunction as an example is shown in Fig. 1a. Other heterojunctions have identical structures. The relationship between energy deposition in semiconductor materials and radiation transport depth taking 63NiO/GaP heterojunction as an example is shown in Fig. 1b. The thickness of 63NiO is \(3.2\, \upmu \hbox {m}\). Other heterojunctions’ are similar. Furthermore, the relationship between electron-hole pair (EHP) generation rate [G(x)] and radiation transport depth is obtained due to the following formula6:

$$\begin{aligned} G(x)=\frac{E(x)}{E_{\text {ehp}}}, \end{aligned}$$
(1)

where E(x) is the energy deposition rate, \(E_\text {ehp}\) is the mean ionizing energy. Herein, Bertuccio-Maiocchi-Barnett (BMB) relationship is utilized to calculate \(E_\text {ehp}\) which is given by26

$$\begin{aligned} E_\text {ehp}=1.54 \ E_\text {g}+1.89 \, \text {eV}, \end{aligned}$$
(2)

where \(E_\text {g}\) is the bandgap of the semiconductor material.

Fig. 1
figure 1

(a) Schematic diagram of 63NiO/GaP heterojunction. (b) Energy deposition rate vs radiation transport depth of 63NiO/GaP heterojunction. The thickness of 63NiO is 3.2 \(\upmu \hbox {m}\).

Full size image

Comparison and optimization

The finite element analysis software COSMSOL Multiphysics (version: 6.0, April 2022) is utilized to simulate the electrical characteristics of the heterojunction nuclear batteries. COMSOL Multiphysics is a powerful simulation software that integrates various physics and engineering disciplines, enabling users to model and analyze complex Multiphysics problems through finite element analysis. It is widely utilized in fields such as acoustics, electromagnetics, structural mechanics and semiconductor physics, allowing for detailed simulations that can predict system behavior under various conditions27. To make the simulation process efficient, the cross-sectional area of each heterojunction is set to \(1\times 1 \upmu \hbox {m}^2\), and the thicknesses of the six n-type semiconductors are all set to \(20\, \upmu \hbox {m}\). To explore the relation between the thickness of 63NiO and the kind of semiconductor material, the thicknesses of 63NiO are set as 0.05, 0.2, 0.8, 1.6 and \(3.2\, \upmu \hbox {m}\) respectively among the above six heterojunctions. The minimum \(0.05\, \upmu \hbox {m}\) is based on the preparation technique sol-gel spinning28. The x-coordinates of the left boundaries of 63NiO under the five thicknesses are all set to 0. The acceptor concentration of the p-type 63NiO is 10\(^{16}\) \(\hbox {cm}^{-3}\) and the donor concentration (\(N_\text {d}\)) of the five n-type semiconductors varies from \(10^{13}\) to \(10^{16}\) \(\hbox {cm}^{-3}\) except that the donor concentration of n-type diamond varies from \(10^{13}\) to \(10^{15}\) \(\hbox {cm}^{-3}\). These structure parameters will be optimized to maximize the maximum output power density.

The BV batteries are set as operating at room temperature (300 K), and several physical models are employed in the simulation. Firstly, the EHP generation rate is defined based on the Geant4 simulation results. Secondly, the analytic doping model is utilized to define the doping of p-type and n-type regions of the heterojunctions. Thirdly, the low-field mobility model is utilized to calculate the minority hole mobility (\(\mu _\text {p}\)),which is a function of \(N_\text {d}\) and is given by6

$$\begin{aligned} \mu _\text {p}=\mu _\text {a}+\frac{\mu _\text {b}-\mu _\text {a}}{1+\left( \frac{N_\text {d}}{N_\text {ref}}\right) ^d}, \end{aligned}$$
(3)

where \(\mu _\text {a}\), \(\mu _\text {b}\), \(N_\text {ref}\) and d are the fitting parameters29,30,31. In addition, the Shockley–Read–Hall (SRH) model is utilized to define the trap-assisted recombination. The minority hole lifetime (\(\tau\)) is also a function of \(N_\text {d}\) and can be expressed as6

$$\begin{aligned} \tau _\text {p}=\frac{\tau _0}{1+\frac{N_\text {d}}{N_\text {R}}}, \end{aligned}$$
(4)

where \(\tau _0\) is the intrinsic carrier lifetime and \(N_\text {R}\) is the fitting parameter32,33,34,35,36,37,38. The semiconductor materials’ properties utilized in COMSOL Multiphysics are listed in Tables 1 and 239,40,41.

Table 1 Properties of 63NiO utilized in COMSOL Multiphysics.
Full size table
Table 2 Properties of Si, InP, GaAs, \(\hbox {Al}_{0.3}\hbox {Ga}_{0.7}\)As, GaP and diamond utilized in COMSOL Multiphysics.
Full size table

The simulation results show that for different heterojunctions, the optimized donor concentrations corresponding to maximum value of maximum output power density (\(P_\text {m}\)) are not totally identical. To be specific, the optimized concentrations are \(10^{16}\), \(10^{16}\), \(10^{16}\), \(10^{16}\), \(10^{15}\) and \(10^{13}\) \(\hbox {cm}^{-3}\) for 63NiO/Si, 63NiO/InP, 63NiO/GaAs, 63NiO/\(\hbox {Al}_{0.3}\hbox {Ga}_{0.7}\)As, 63NiO/GaP and 63NiO/diamond heterojunctions respectively. The \(P_\text {m}\) depending on the thicknesses of 63NiO for the heterojunctions is shown in Fig. 2. It shows that for different thicknesses of 63NiO, the optimal heterojunction is not always the same one based on \(P_\text {m}\). To be specific, when the thickness of 63NiO is less than \(1.6 \,\upmu \hbox {m}\), the optimal heterojunction is 63NiO/diamond. However, when the thickness of 63NiO is more than \(1.6\, \upmu \hbox {m}\), the optimal heterojunction is 63NiO/GaP. It also indicates that the \(P_\text {m}\) increases with increasing thickness of 63NiO.

Fig. 2
figure 2

\(P_\text {m}\) vs thickness of 63NiO for the six heterojunctions.

Full size image

Figure 3a shows the energy diagram of vacuum level of 63NiO/GaP heterojunction for the different thicknesses of 63NiO at thermodynamic equilibrium. In the depletion region, the vacuum level is bent and the depletion region width is \(2.54\, \upmu \hbox {m}\). The built-in energy barrier of 1.03 eV is formed. The position of metallurgical junction moves right with increasing thickness of 63NiO. Figure 3b shows the electric field distribution of 63NiO/GaP heterojunction for the different thicknesses of 63NiO. The electric field is mainly distributed in the depletion region where the radiation-induced electron–hole pairs can be separated and it means that the electron–hole pairs generated in the depletion region can be collected more effectively.

Fig. 3
figure 3

(a) Energy diagram of vacuum level of 63NiO/GaP heterojunction for the different thicknesses of 63NiO. (b) Electric field distribution of x component of 63NiO/GaP heterojunction for the different thicknesses of 63NiO.

Full size image

Figure 4a shows the EHP generation rate distribution of 63NiO/GaP heterojunction for the different thicknesses of 63NiO. The EHP generation rate increases with increasing thickness of 63NiO due to the radioactivity increasing with increasing thickness of 63NiO. The radioactivity increasing leads to energy deposition increasing. In addition, the EHP generation rate increases first and then decreases in 63NiO and decreases exponentially in GaP for the same thickness of 63NiO. Figure 4b shows the SRH recombination rate distribution of 63NiO/GaP heterojunction for the different thicknesses of 63NiO. The SRH recombination rate of 63NiO is higher than GaP as a result of the higher EHP generation rate of 63NiO11. The SRH recombination is lower in the depletion region because the EHPs generated in it can be directly separated by the built-in electric field while the EHPs generated out of the depletion region diffuse into it and then be separated17. Figure 4c shows the total rate distribution of 63NiO/GaP heterojunction for the different thicknesses of 63NiO. The total rate is defined as EHP generation rate minus SRH recombination rate. The total rate increases with increasing thickness of 63NiO so that the short circuit current density (\(J_\text {sc}\)) will increase with increasing thickness of 63NiO.

Fig. 4
figure 4

(a) EHP generation rate distribution of 63NiO/GaP heterojunction for the different thicknesses of 63NiO. (b) SRH rate distribution of 63NiO/GaP heterojunction for the different thicknesses of 63NiO. (c) Total rate distribution of 63NiO/GaP heterojunction for the different thicknesses of 63NiO.

Full size image

Figure 5a shows the current density–voltage (J–V) characteristics of 63NiO/GaP heterojunction for the different thicknesses of 63NiO. It can be seen that the short current density increases with increasing thickness of 63NiO which can be attributed to the total rate increasing with increasing thickness of 63NiO. In addition, the open circuit voltage (\(V_\text {oc}\)) increases with increasing thickness of 63NiO which can be owing to the following formula42:

$$\begin{aligned} V_\text {oc}=\frac{kT}{q}\ln \left( {\frac{J_\text {sc}}{J_0}+1}\right) , \end{aligned}$$
(5)

where k is the Boltzmann constant, T is the temperature, q is the elementary charge and \(J_0\) is the reverse saturation current density. It can be seen that the \(V_\text {oc}\) increases with increasing \(J_\text {sc}\). Figure 5b shows the electron current density distribution of 63NiO/GaP heterojunction for the different thicknesses of 63NiO. In the region of p-region left boundary, the diffusion current density is dominant compared with the drift current density for the minority carrier electron due to the higher EHP generation rate of the 63NiO part compared with the GaP part so that the electron density is negative. When it is close to the left boundary of the depletion region, the drift current density becomes dominant in comparison with the diffusion current density owing to the built-in electric field. Figure 5c shows the hole current density distribution of 63NiO/GaP heterojunction for the different thicknesses of 63NiO. In the p-region, the drift current density is dominant compared with the diffusion current density for the majority hole and the EHP generation is higher than n-region so that the hole current density is larger. Figure 5d shows the total current density distribution of 63NiO/GaP heterojunction for the different thicknesses of 63NiO. The total current density is electron current density plus hole current density. We can see that the total current density is consistent with the \(J_\text {sc}\) in Fig. 5a.

Fig. 5
figure 5

(a) J–V characteristics of 63NiO/GaP for the different thicknesses of 63NiO. (b) Electron current density distribution of 63NiO/GaP for the different thicknesses of 63NiO. (c) Hole current density distribution of 63NiO/GaP for the different thicknesses of 63NiO. (d) Total current density distribution of 63NiO/GaP for the different thicknesses of 63NiO.

Full size image

We further studied the saturation \(P_\text {m}\) based on 63NiO/GaP heterojunction. Figure 6 shows that \(P_\text {m}\) arrives at the saturation value when the thickness of 63NiO is \(30\, \upmu \hbox {m}\). With increasing thickness of 63NiO, the EHP pairs generated near the left boundary of 63NiO become more difficult to collect owing to recombination so that there will be a saturation value of \(P_\text {m}\).

Fig. 6
figure 6

\(P_\text {m}\) vs Thickness of 63NiO for 63NiO/GaP heterojunction.

Full size image

Parallel connection structure

In the above study, the converted energy is restricted by the single side of the beta particles emitted from 63NiO. To fully utilize the decay energy and further improve \(P_\text {m}\), the two 63NiO/GaP heterojunctions were combined in parallel connection with graphene. Graphene has the advantages of large specific surface area, impressive electrical conductivity, high thermal conductivity, exceptional mechanical strength and excellent corrosion resistance. It has been used as an electrode material in electrochemical sensors, \(\hbox {NO}_{2}\) gas sensors, supercapacitors, triboelectric nanogenerators43,44,45,46,47. Meanwhile, the atomic number of graphene is small so that it is able to minimally deposit the energy of beta particles when they pass through the middle layer compared with metal materials. As shown in Fig. 7, the thickness of 63NiO is \(30\, \upmu \hbox {m}\), the thickness of GaP is \(20\, \upmu \hbox {m}\) and the thickness of single layer graphene is 0.345 nm48. Taking the practical processing technique into consideration, we simulated the performance of the nuclear batteries with different layer numbers of graphene. The results are shown in Table 3. It indicates that the layer number has slight influence on the \(P_\text {m}\) when the layer number is less than 10000. The comparison of the performance among the parallel connection 63NiO/GaP, the optimal single 63NiO/GaP, the conventional 63Ni-NiO/GaP and the conventional 63Ni-NiO/diamond heterojunction nuclear batteries is presented in Table 4. The thicknesses of 63Ni of the two conventional heterojunction nuclear batteries are both \(5\, \upmu \hbox {m}\)7 and the corresponding radioactivity is \(9.918\times 10^{9}\) Bq.

Fig. 7
figure 7

Schematic diagram of the parallel connection structure of double 63NiO/GaP heterojunctions with graphene.

Full size image
Table 3 Simulation results of \(J_\text {sc}\), \(V_\text {oc}\) and \(P_\text {m}\) with different layer numbers of graphene.
Full size table
Table 4 The comparison of the performance among the parallel connection 63NiO/GaP, the optimal single 63NiO/GaP, the conventional 63Ni-NiO/GaP and the conventional 63Ni-NiO/diamond heterojunction nuclear batteries.
Full size table

Time-related performance

With time going by, the radioactivity of 63NiO decreases, and the electrical performance of the BV changes subsequently. The time-related performance of the parallel connection structure with single layer graphene is simulated. As shown in Fig. 8a,b, \(J_\text {sc}\), \(V_\text {oc}\) and \(P_\text {m}\) decrease with increasing time because the deposited energy decreases with decreasing radioactivity EHP generation rate. As shown in Fig. 8c, FF increases with increasing time and \(\eta\) remains constant. The FF is given by Ref.49

$$\begin{aligned} FF=\frac{P_\text {m}}{V_\text {oc}J_\text {sc}}\times 100\%, \end{aligned}$$
(6)

and the \(\eta\) is given by31

$$\begin{aligned} \eta =\frac{P_\text {m}S}{1.602\times 10^{-19}AE_\text {ave}}\times 100\%, \end{aligned}$$
(7)

where S is the cross-sectional area (\(S = 1 \hbox {cm}^2\)), A is the radioactivity of 63NiO (the initial value of A is \(7.001\times 10^{10}\) Bq) and \(E_\text {ave}\) is the average energy of beta particles emitted from 63NiO (\(E_\text {ave} = 17.425\hbox { keV}\)).

In Fig. 8b, the formula of \(P_\text {m}(t)\) is obtained through fitting, which can be utilized to predict the performance of the parallel connection BV battery within 200 years. We can see that the \(P_\text {m}\) decreases from \(5236.2 \hbox { nW}\cdot \hbox {cm}^{-2}\) at 0 y to 3717.6, 2639.4 and \(1330.5 \hbox { nW}\cdot \hbox {cm}^{-2}\) at 50, 100 and 200 y respectively. Those value are 71%, 50% and 25% of the initial value. When the output power decreases significantly, the equipment may not be able to obtain enough power, resulting in unstable operation, or even malfunctions or shutdowns, affecting the normal function of the equipment and the completion of tasks. Since the decay of the radioactive source is not affected by the external environment, the decrease in activity caused by the decay of the radioactive source is inevitable, and the output power of the nuclear battery is inevitably reduced. Therefore, the potential mitigation strategy may only be to replace the nuclear battery with a new one.

Fig. 8
figure 8

(a) Time-related \(J_\text {sc}\) and \(V_\text {oc}\). (b) Time-related \(P_\text {m}\). (c) Time-related FF and \(\eta\).

Full size image

Conclusions

In this paper, to overcome the shortcomings of conventional planar BV batteries due to low efficiency and power density, a parallel connection structure of two 63NiO/GaP heterojunctions with a graphene layer was designed. The results exhibited a remarkable improvement of efficiency and power density. Herein, 63NiO/GaP heterojunction as one part of the parallel connection structure is based on the comparative research of six 63NiO-related heterojunctions. Through the Monte Carlo software Geant4 and the finite element analysis software COMSOL Multiphysics, 63NiO/Si, 63NiO/InP, 63NiO/GaAs, 63NiO/\(\hbox {Al}_{0.3}\hbox {Ga}_{0.7}\)As, 63NiO/GaP and 63NiO/diamond heterojunctions were investigated and compared on \(P_\text {m}\). When the thickness of 63NiO is more than 1.6 \(\upmu\)m, the optimal heterojunction is 63NiO/GaP. It also indicates that the \(P_\text {m}\) increases with increasing thickness of 63NiO, and the \(P_\text {m}\) arrives at the saturation value when the thickness of 63NiO is \(30\, \upmu \hbox {m}\). To further improve \(P_\text {m}\), the two 63NiO/GaP heterojunctions were combined in parallel connection with graphene. In addition, the time-related performance was researched within 200 years based on the parallel connection nuclear battery. The \(P_\text {m}\) decreases from \(5236.2 \hbox { nW}\cdot \hbox {cm}^{-2}\) at 0 y to \(1330.5\hbox { nW}\cdot \hbox {cm}^{-2}\) at 200 y. The final value is 25% of the initial value.