MC simulation for β energy deposition on 3-D nanostructure

To design an optimal structure of BCs, MC simulation [30] was conducted to investigate the interaction of β radiation with 3-D ⁶³Ni-incorporated ZNRA structures. As the β-emitting source, this study selected ⁶³Ni with a half-life of approximately 100 years, average energy of 17.4 keV, and maximum energy of 66 keV. Two types of BCs were designed to compare the energy deposition simulations. As shown in Fig. 1a, a sandwich-type BC based on a 2-D ZnO bulk structure consisted of an Al-doped ZnO (AZO)/glass substrate, ZnO film, and ⁶³Ni/Ni layer deposited on the ZnO film. In contrast, the 3-D nanostructured BC consisted of an AZO/glass substrate, 3-D ZNRAs combined with ⁶³Ni, and a top Ni electrode (Fig. 1b).

Fig. 1Full-screen View

(Color online) Schematic diagram of BCs based on (a) 2-D ZnO bulk structure and (b) 3-D ZNRAs structure.

To qualitatively understand the interaction of incident β particles with ECSs, 2-D ZnO bulk and 3-D ZNRAs structures were modeled and simulated using the MC method. In this mode, the ZnO thickness was set to 3 μm, and the diameter and length of the ZnO nanorods were set to 300 nm and 5 μm, respectively. An electron beam with a diameter of 1.8 μm and incident angle of 45° along the horizontal direction was used as the exciting source, and consisted of 1000 β particles with an average energy of 17.4 keV for the β particles of ⁶³Ni. The scattering trajectories and energy deposition distribution of the β particles in both structures were simulated using the CASINO software, as shown in Fig. 2. As shown in Fig. 2a, a fraction of the incident β particles was reflected from the surface of the bulk ZnO, which led to certain energy loss in the 2-D ZnO bulk structure. The energy deposition distributions of the incident β particles in the 2-D ZnO bulk exhibited an incident depth of approximately 1.47 μm on the y–z plane (Fig. 2b) and an interaction radius of 2.9 μm on the x–y plane (Fig. 2c). Fig. 2d shows the dependence of the energy deposition density and deposition ratio on the depth of the 2-D ZnO bulk structure. The maximum depth of energy deposition is approximately 1.47 μm, and the energy deposition ratio reached 32%. In contrast, the MC simulation yielded different results for the ZNRAs. The scattering behaviors of the β particles in the ZNRAs imply a higher structure trapping for the β particles compared with the 2-D ZnO bulk structure (Fig. 2e). Moreover, the simulation results for the energy deposition distribution in the ZNRAs (Fig. 2f and 2g) reveal that incident energy can be deposited in a very deep range along the length of the nanorods owing to the multiple scattering and absorption of β particles. This can be further verified by the simulation results for the dependence of the energy deposition density and energy deposition ratio on the depth in 3-D ZNRAs structures, as shown in Fig. 2h. Compared with the 2-D ZnO structure, the energy deposition depth in the 3-D ZNRAs structure could reach approximately 4.5 μm with an energy deposition rate of approximately 85%. This result indicates that the 3-D ZNRAs can harvest more β energy in the ECSs compared with the 2-D ZnO structures, which results in the higher ECE and P_out of BCs.

Fig. 2Full-screen View

(Color online) MC modeling and simulation of 2-D ZnO bulk structure and 3-D ZNRAs structure under β irradiation. Scattering trajectories of β particles in (a) 2-D ZnO bulk structure and (e) 3-D ZNRA structure; energy deposition distributions of β particles in (b–c) 2-D ZnO bulk structure and (f–g) 3-D ZNRA structure on y–z plane and x–y plane; dependence of energy deposition density and deposition ratio on depth in (d) 2-D ZnO bulk structure and (h) 3-D ZNRAs structure.

Figure 3 shows the working principle of the BCs based on ⁶³Ni@ZNRAs. A schematic energy band diagram is shown in Fig. 3a. An extended 3-D heterojunction of ⁶³Ni/ZnO is formed with a Schottky contact (Φ_SBH = W_m – χ) in the interface of ⁶³Ni/ZnO and an ohmic contact in the interface of AZO/ZnO. The β particles emitted from the ⁶³Ni source are incident on the ZnO lattice, where impact ionization results in the energy deposition and generation of electron-hole pairs (EHPs). The EHPs are separated by a built-in electric field in the depletion region of the ⁶³Ni@ZnO Schottky junction. In this process, electrons migrate from the conduction band (with energy level E_C) of ZnO to the AZO substrate, while holes are transferred from the valence band (with energy level E_V) of ZnO to the ⁶³Ni electrode [31]. Figure 3b shows the horizontal EHP separation and longitudinal carrier transport.

Fig. 3Full-screen View

(Color online) (a) Schematic energy band diagram of BC based on ⁶³Ni@ZNRAs and (b) transferring process of β excited carriers in ⁶³Ni@ZNRAs; (c) energy deposition and deposition ratio of incident β particles of devices based on ⁶³Ni@ZNRAs with different nanorod spacings; (d) energy deposition distribution and deposition ratio of incident β particles of devices based on ⁶³Ni@ZNRAs with different nanorod heights.

To simulate the β energy deposition in the ZNRAs with satisfactory accuracy, the MCNP software was used to establish the MC model of the ⁶³Ni@ZNRAs. This study used ⁶³Ni as an isotropic volumetric source integrated into the gap of the nanorods, and the number of incident β particles was set to 1×10⁵. The β energy deposition in the ZNRAs by radiation from the ⁶³Ni source was simulated using the actual spectrum of the emitted electrons. The energy deposition in the ZNRAs was calculated using the F8 pulse counting in the MCNP program, and the deposited energy could then be employed to calculate the actual current density using Eq. (2), as described in Sect. 2.2. The dependence of the energy deposition and deposition ratio of the incident β particles on the spacing and height of the nanorods is shown in Fig. 3c and 3d, respectively. As the spacing of the nanorods increased, the loading amount of ⁶³Ni increased, which led to an increase in the total β-emitting energy. However, the number of nanorods per unit area decreased as the spacing increased. Consequently, the energy deposition rate decreased. The trade-off reached a balance at the spacing of approximately 200 nm, where the maximum energy deposition was achieved (Fig. 3c). As the nanorod height increased, as shown in Fig. 3d, both the deposited energy and energy deposition ratio increased, and the energy deposition rate exhibited a trend of slow increase when the height exceeded 4 μm.

Quantitative modeling for BCs based on 3-D ZNRAs

A quantitative model should be established to accurately evaluate the performance of the BCs. Because the working principle of BCs is similar to that of solar cells, a conventional theoretical model of solar cells can be improved to establish a model of 3-D nanostructured BCs. As high-energy incident β particles on the ECSs, multiple electron-hole pairs (EHPs) are generated by the impact ionization of the β particles with the semiconductors. The EHPs are effectively separated and drawn as the theoretical maximum current density (I_max), which can be calculated as follows [32]: $I_{\max }=e{N}_{\beta }(\overline{E_{\beta }}/\psi )$ (1) $\psi =2.8E_{\text{g}}{+}_{}{0.5}_{}\text{eV}$ (2)

where ψ is the ionization energy that generates an EHP; E_g is the bandgap of the semiconductor; N_β is the β-emitting flux (N_β = A × 3.7 × 10⁷); A is the activity (mCi) of the radioactive source; Ē_β is the average energy of the β particles. The radiation-generated current (I_β) in the BCs is given as follows: $I_{\beta }=\phi (1-\gamma )(1-\mu )(1-\sigma )Q{I}_{\max }=Qe{E}_{\text{s}}/\psi ,$ (3)

where γ is the reflection coefficient of the β particles from the semiconductor surface; μ is the interaction coefficient between the external medium (air or electrode) and incident β particles; σ is the self-absorption coefficient of the radioactive source, which is related to the thickness of the radioactive source; ϕ is the scattering angle coefficient of the radioactive source, which is the ratio of the number of incident particles on the semiconductor to the total number of emitted β particles. For 3-D ZNRAs, γ = 0, μ = 0, ϕ = 1, e is the electron charge (e = 1.6 10^-19 C), and E_s is the β energy deposited in the semiconductors; Q is the collection efficiency, which is the ratio between the number of effective carriers for forming a current in the devices and the total number of carriers produced by the impact ionization of the incident β particles with the semiconductors. Here, Q is the collection efficiency of carriers in the space-charge region of the Schottky junction, which is approximately 100%; the collection efficiency of the carriers outside of the space charge region can be calculated as follows [33]: $Q=1-\tanh \text{(}d/L\text{),}$ (4)

where d is the distance of the carrier moving to the boundary of the depletion region and L is the minority carrier diffusion length.

The width of the Schottky junction space-charge region can be calculated as follows [34]: $W=\sqrt{2{\epsilon }_0{\epsilon }_{\text{s}}({\Phi }_{\text{SBH}}-V)/(e{N}_{\text{D}})}$ (5)

where ε₀ is the dielectric constant of the vacuum; ε_s is the relative dielectric constant of the semiconductor; N_D is the ZnO carrier concentration (approximately 2.1×10¹⁵ cm^-3); Φ_SBH is the height of the Schottky barrier of Ni@ZnO (Φ_SBH = W_m − χ = 0.8 eV), which is, the theoretical difference between the work function of Ni (W_m = 5.15 eV) and the electron affinity energy of ZnO (χ = 4.35 eV).

From Eq. (5), the width of the depletion region of the Ni@ZnO Schottky barrier can be calculated as approximately 6.04×10^-1 μm. Because the average diameter of the ZnO nanorods (approximately 0.3 μm) grown using the hydrothermal method is smaller than the depletion region width, Q is approximately equal to 1 for BCs based on the ⁶³Ni@ZNRAs structure.

Figure S1 shows the equivalent circuit model of the BCs (Supporting Information). The current density (I) delivered to the load resistor (R_L) at voltage V can be calculated as follows [35]: $I=I_{\beta }-\delta I_{\text{o}}[\exp (e(V+I\alpha R_{\text{s}})/(nKT))-1]-(V+I\alpha R_{\text{s}})/R_{\text{sh}}$ (6) $\delta =S/S_{\text{o}}\text{, }\alpha \text{=(}{D}_{\text{o}}H\text{)/(}D{H}_{\text{o}}\text{),}$ (7)

where R_s and R_sh are the series and shunt resistances of the equivalent circuit of the BC, respectively. For the specific BC based on 3-D ZNRAs that was prepared for the subsequent experiment, I_o, n, and R_s were extracted from its I-V curve measured under dark conditions, and the ideal R_sh was set to infinity. The greater height and smaller density of the nanorods will result in greater R_s. Considering that both I_o and R_s are variables related to the structural parameters of 3-D ZNRAs, such as the area of the Schottky junction (S), nanorod array density (D), and nanorod height (H), two factors (δ and α) are combined with I_o and R_s in Eqs. (6–7). Thus, H_o and D_o are the structural parameters of the specific 3-D ZNRAs structure.

The maximum short-circuit current (I_sc) of the BCs was calculated using Eq. (6). The open circuit voltage (V_oc) of the BCs can be calculated as follows: ADDIN NE.Ref.{8D4F474E-6FFD-4538-9213-F61B71B18B0F} $V_{\text{oc}}=nkT\ln (I_{\text{sc}}/I_{\text{o}}+1)/e,$ (8)

where k is the Boltzmann constant (k =1.38 × 10^-23 J/K), T is the thermodynamic temperature at room temperature (T = 300 K), I_o is the reverse saturation current, and n is an ideal factor.

The maximum output power (P_max), filling factor (FF), and energy conversion efficiency (η) can be expressed as follows [36]: $P_{\max }=I_{\text{sc}}{V}_{\text{oc}}FF$ (9) $FF=(I_{\text{mp}}{V}_{\text{mp}})/(I_{\text{sc}}{V}_{\text{oc}})$ (10) $\eta =P_{\max }/P_{\beta }\times 100\%=P_{\max }/(e{N}_{\text{o}}\overline{E_{\beta }})\times 100\%$ (11)

where I_mp and V_mp are the current and voltage, respectively, corresponding to P_max; P_β is the total power of the radioactive source; N_o is the total amount of β particles (N_o = 3.7×10⁷×Φ, where Φ is the activity of the radioactive source) emitted from the radioactive source.

Preparation of BCs based on 3-D ZNRAs nanostructure

The preparation process of the Ni-incorporated ZNRAs structure is shown in Fig. 4a. The ZNRAs were grown on Al-doped ZnO (AZO) conductive glass (R ≤ 6 Ω, Nanotech, China) using a hydrothermal method [16]. First, the AZO conductive glass was cleaned using CH₃COCH₃, ethanol (C₂H₅OH), and deionized water in an ultrasonic bath for 15 min. After drying, the specified electrode region in the AZO glass substrates was covered with high temperature tape, and the substrates were then transferred to a reactor with 10 ml of growth solution consisting of 0.05 M Zinc nitrate hexahydrate [Zn(NO₃)₂·6H₂O], 0.05 M hexamethylenetetramine (CH₂)₆N_4, HMTA), 0.02 M polyetherimide (PEI), and deionized water. The temperature of the hydrothermal treatment was maintained at 95 ℃ for 4 h. Next, the samples were cleaned with deionized water and dried at 150 ℃ for 10 min. Finally, the samples were soaked in 10% concentration of hydrogen peroxide (H₂O₂) for 1 min for surface passivation, and then annealed in air at 400 ℃ for 2 h to obtain the ZNRAs [37]. Passivation treatment with H₂O₂ can reduce the surface defects of the ZnO nanorods, which favors the improvement of the Schottky diode characteristics. The incorporation of Ni into the ZNRAs was performed using a direct current (DC) magnetron sputtering technique. The vacuum chamber was evacuated to a pressure of 5×10^-6 Pa, and Ar gas was then introduced into the chamber to reach the constant pressure of 0.6 Pa. Subsequently, Ni nanoparticles were deposited onto the ZNRAs using a DC power of 400 W at room temperature for 9 min. Fig. 4b shows a 3-D explosive diagram of the device based on the Ni-incorporated ZNRAs structure. To perform the I-V measurements, a planar Ni sheet was mechanically compressed onto the surface of the as-prepared samples to form a sandwich-type device with a Ni/Ni@ZNRAs/AZO/glass structure.

Fig. 4Full-screen View

(Color online) (a) Schematic illustration of preparation process flow of Ni-incorporated ZNRAs structure; (b) 3-D explosive diagram of device based on Ni-incorporated ZNRAs structure; (c) photographs of (i) AZO glass, (ii) ZNRAs/AZO glass, and (iii) Ni-incorporated ZNRAs/AZO/glass samples.

Material characterization and device measurements

Scanning electron microscopy (SEM, ZEISS microscope, Germany) and energy-dispersive X-ray spectroscopy (EDS) were used to analyze the morphology and chemical composition of the samples. The crystal structures were characterized by X-ray diffraction (XRD, D8 Advance, Bruker, Germany), and I-V measurements were performed in a Faraday cage using a Keithley Model 2450 source meter.

Experimental results and discussion

The XRD pattern of the Ni@ZNRA structure is shown in Fig. 5a. All diffraction peaks can be indexed to the hexagonal structure of wurtzite ZnO (JCPDS card No. 36-1451) and cubic structure of Ni (JCPDS card No. 70-1849). The typical 2θ peaks at 34.4° and 36.3° are assigned to the (002) and (101) planes of ZnO, respectively. A new diffraction peak was not observed in the Ni@ZNRA structure, which indicates that the introduction of Ni did not change the crystal structure of the ZnO nanorods. The growth direction of the (002) crystal plane of ZnO is preferred to the c-axis orientation perpendicular to the substrate, as can be verified from the cross-sectional and top-view SEM images of the ZNRAs (Fig. 5b and Fig. 5c). The average diameter and height of the nanorods is 300 nm and 4 µm, respectively. The cross-section-view and top-view SEM images of the Ni@ZNRAs are shown in Fig. 5d and 5e, respectively. As can be clearly observed, Ni particles were uniformly filled into the gap between the ZnO nanorods and stacked to form a continuous Ni layer covering the surface of the ZNRAs. The SEM-EDS mappings show that Ni was adequately distributed on the surface of the ZnO nanorods, forming a core-shell structure (Fig. 5f–h).

Fig. 5Full-screen View

(Color online) (a) XRD spectra of Ni@ZNRAs structure; (b) cross-section-view and (c) top-view SEM images of ZNRAs; (d) cross-section-view and (e) top-view SEM images of Ni@ZNRAs structure; (f–h) EDS mappings of Zn, Ni, and O.

Figure 6 shows the dark characteristics of the devices based on the Ni@ZNRAs/AZO structure. A typical I-V curve of the Schottky diode is shown in Fig. 6a, confirming the existence of the Ni/ZnO Schottky junction. The reverse saturation current of the device is 4.2×10^-11 A/cm^-2, as determined by extrapolating the linear region of the logarithm-scale I-V curve in the forward bias to zero bias voltage (Fig. 6b). The ideal factor n and series resistance R_s of the Schottky diode can be extracted from the dependence of dV/d(lnI) on the current (Fig. 6c) [38]. The extracted n and R_s are 6.96 and 248.45 Ω, respectively. This ideal factor n is larger than 2, which indicates that the junction quality of Ni@ZNRAs is not perfect owing to the crystal defects or insufficient filling of Ni in the ZNRAs [39–40]. Fig. S2 (Supporting Information) shows the I-V curve of the devices based on the Ni@ZNRAs/AZO structure before H₂O₂ passivation and the calculated n value (n = 11.2). As can be seen, the n value before passivation is significantly larger compared with that after passivation, indicating that H₂O₂ passivation can significantly improve the diode quality. The presence of a large amount of interface defects in the ZnO nanorods may deteriorate the rectification characteristics of Schottky devices, leading to an increase in the reverse saturation current and ideal factor. In turn, the increase in reverse saturation current will decrease the open-circuit voltage, resulting in the decrease of output power, which can be calculated using Eq. 8. This indicates that H₂O₂ passivation improves the output power of the BCs. To evaluate the electrical performance of BCs based on the ⁶³Ni@ZNRAs structure, the above-mentioned parameter values of the Schottky diode are used in the quantitative model of BCs, as described in Sect. 2.2.

Fig. 6Full-screen View

Dark characteristics of devices based on Ni@ZNRAs/AZO structure: (a) linear-scale I-V curve in bias voltage range of -2.5–2.5 V; (b) logarithm-scale I-V curve in forward bias; (c) dV/d(lnI)-I curve.

Table 1 lists the parameter values applied to the quantitative model of BCs. Notably, these parameter values can be controlled and adjusted when growing ZNRAs using the hydrothermal method. Tao et al. [41] developed a combined lithography-assisted and hydrothermal growth approach to selectively control the spacing and size of ZNRAs on the ZnO seed layer. Sinem et al. [42] prepared ZNRAs with nanorod lengths greater than 60 μm by adjusting the contents of ammonium hydroxide and polyethyleneimine using a hydrothermal synthesis method. This means that the optimal values of nanorod parameters, such as the diameter, height, and density, can be prepared by adjusting process parameters such as the density of the ZnO seeds, concentration of the reactants, time and temperature of reactions, and pH values [43].

Simulated results and discussion

Using the fixed parameter values listed in Table 1 and an average nanorod height of 4 μm, the calculated I-V and P-V characteristics of BCs based on the ⁶³Ni@ ZNRAs structure with different nanorod spacings are as shown in Fig. 7a and Fig. 7b, respectively. By extracting P_max from Fig. 7b and calculating the ECE according to the total activity of ⁶³Ni incorporated into the gap of the ZNRAs, the dependences of P_max and ECE on the variable spacing of the nanorods can be obtained as shown in Fig. 7c. As can be seen, as the nanorod spacing increases, the activity of the ⁶³Ni incorporated into the gap of the nanorods also increases, resulting in the rapid increase of output power. However, as the nanorod spacing increases further, the nanorod density is reduced, which reduces the energy deposited on the ZNRAs and thus reduces the output power. Moreover, a maximum P_out at 875.4 nW/cm² corresponding to I_sc=676.2 nA/cm² and V_oc=1.85 V was achieved with FF=0.699 and η=6.07%. In contrast, Figure 7c shows a monotonic decline in the ECE curve as the nanorod spacing increased. The ECE reached a maximum value of 9.3% when the spacing was set to 40 nm.

Fig. 7Full-screen View

(Color online) (a) I-V and (b) P-V characteristics of devices based on ⁶³Ni@ZNRAs with different nanorod spacings; (c) dependences of P_max and ECE of devices based on ⁶³Ni@ZNRAs on nanorod spacing; (d) I-V and (e) P-V characteristics of devices based on ⁶³Ni@ZNRAs with different nanorod height; (f) dependences of P_max and ECE of devices based on ⁶³Ni@ZNRAs on nanorod height.

The effects of the nanorod height on the ECE and P_max of the BCs were investigated under the optimal nanorod spacing. Fig. 7d and 7e shows the calculated I-V and P-V characteristics of the BCs based on the ⁶³Ni@ ZNRAs structure with different nanorod heights. As can be seen, I_sc increased with the nanorod height, while V_oc was kept at 1.62 V without any change. Fig. 7f shows the dependence of the P_max and ECE of the BCs on the nanorod height. The energy deposition ratio increased slowly when the nanorod height exceeded 4 µm (Fig. 3d), resulting in a slow increase in the saturation level of the energy conversion efficiency. As the nanorod height increased, the ECE tended toward the saturation value of approximately 10%, and the maximum P_out of 1185.9 nW/cm² was achieved when the height was 10 µm, corresponding to a loading amount of 114.37 mCi, I_sc of 1081.2 nA/cm², and FF of 0.673. Although long nanorods are beneficial for increasing the output power, it is difficult to fully deposit ⁶³Ni into a high-aspect-ratio ZNRAs structure in practical operation. Hence, experimental investigation is required to determine the practical deposition depth of ⁶³Ni on the ZNRAs.

To verify the advantages of 3-D nanostructures in enhancing the betavoltaic effect, a BC based on a 2-D ZnO bulk structure was investigated for comparison to the BCs based on a 3-D ZNRAs structure, as shown in Fig. 1a. According to the MC simulation, the energy deposition depth of incident β particles in 2-D ZnO bulk is approximately 1.5 μm. Moreover, the dependence of the energy deposition and the deposition ratio of the incident β particles on the thickness of ⁶³Ni are shown in Fig. 8a. As the thickness of ⁶³Ni increases, the self-absorption effect of the radiation source shields the β particles emitted from the deep location of the radiation source, resulting in deposition ratio reduction. As can be seen, the energy deposition tended to saturate when the thickness exceeded 1.5 μm. The dependences of P_max and ECE of the BCs based on 2-D ZnO bulk on the thickness of ⁶³Ni were investigated, as shown in Fig. 8b. As the thickness of ⁶³Ni increased from 50 nm to 150 nm, the energy conversion efficiency increased from 4.05% to 4.69% owing to the deposited energy increase. As the thickness of ⁶³Ni increased further and exceeded 150 nm, the self-shielding effect of the radiation source became significant, which reduced the energy deposition ratio and thus reduced the energy conversion efficiency. It can be observed that P_max tended toward saturation (approximately 135.2 nW/cm²) when the ⁶³Ni thickness reached 2 μm. Beyond this thickness, the self-shielding effect of the radioactive source limits the continued increase of output power. Moreover, the maximum ECE reached 4.69% when the ⁶³Ni thickness was 150 nm, as expressed by Eq. S8 (Supporting Information).

Fig. 8Full-screen View

(a) Energy deposition and deposition ratio of incident β particles of devices based on 2-D ZnO bulk with different ⁶³Ni thickness; (b) dependences of P_max and ECE of BCs based on 2-D ZnO bulk on ⁶³Ni thickness.

The performance comparison between the BCs based on 3-D ZNRAs and those based on 2-D ZnO bulk is presented in Table 2. The BCs based on the 2-D ZnO film generated higher V_oc (1.87 V), which is attributed to the generation of lower reverse saturation current compared with the 3-D ZNRAs. Owing to the low utilization efficiency of the radioactive source, the β energy deposited in the semiconductor decreased dramatically, resulting in lower I_sc (28.1 nA/cm²) and P_max (36.8 nW/cm²), which led to a low ECE of 4.69% compared with the BCs based on the 3-D ZNRAs. According to the Shockley–Quaiser limit of photovoltaics [43], the efficiency cap of BCs is predicted to be as high as 13%, as can be calculated by η = V_ocFF/ψ. This means that 3-D ZNRAs with optimal structural design are significantly superior and can be used to prepare BCs close to their upper limits.