Radiation-induced carrier generation in 3D diode structures

2.2.1.1

Model for radiation-induced carrier generation rate in 3D diode structures

The distribution of the EHP generation rate in a betavoltaic nuclear battery is governed by the energy deposition of beta particles in the converter, which significantly affects the output performance. In the conventional planar diode structure depicted in Fig. 2(b), the energy deposition [E_dep(x)] along the radiation transport depth [x] in bulk SiC was calculated via a Monte Carlo simulation with the Geant4 radiation transport toolkit, using a rectangular ⁶³Ni source with a full energy spectrum. The ⁶³Ni source is characterized by the specific activity of 5.68 Ci/g, 100% abundance, and density of 8.9 g/cm³, with isotropic emission of beta particles. SiC has the bandgap width of 3.26 eV, relative dielectric constant of 9.7, density of 3.21 g/cm³, and intrinsic carrier concentration of 7.4×10^-9 cm^-3, derived using the widely employed formula [40]. G(x) is obtained and expressed as: $G\left(x\right)=\frac{A\cdot E_{\text{dep}}\left(x\right)}{\epsilon }=G_0\exp \left(-\alpha x\right),$ (2) where A is the activity of the radioisotope, ε is the average energy needed to generate an EHP, commonly considered as 6.88 eV for SiC [41], and G₀ and α are the surface EHP generation rate and absorption coefficient, respectively. G(x) of ⁶³Ni sources with varying thickness [t] is presented in Fig. 2(c), and the corresponding G₀ and α are derived by fitting G(x). The dependences of G₀ and α on the source thickness [t] are then fitted, as demonstrated in Fig. 2(d). Consequently, G(x) considering different source thicknesses is expressed as: $G\left(x,t\right)=A_1\left(1-{\text{e}}^{-{\mu }_{1}t}\right){\text{e}}^{-\left(A_2{\text{e}}^{-{\mu }_{2}t}+B_2\right)x}$, the fitting parameters for which are provided in Table 1. Fig. 2(c) shows that the saturation thickness of the ⁶³Ni source is 3 μm, consistent with previous work [24, 42]. The penetration depth of β particles emitted from ⁶³Ni is 10.2 μm in SiC, defined as the location where 99% of the total energy deposition occurs according to Alam [8].

Table 1Full-screen View

Exponential fitting parameters and R-squared values for G₀(t), α(t), and γ(t).

Equations	Fitting values	R2
$G_0\left(t\right)=A_1{\text{e}}^{-{\mu }_{1}t}+B_1$	A1=-8.14×1015 cm-3·s-1, μ1=1.35 μm-1, B1=8.69×1015 cm-3·s-1	0.999
$\alpha \left(t\right)=A_2{\text{e}}^{-{\mu }_{2}t}+B_2$	A2= 3638.76 cm-1, μ2=2.19 μm-1, B2=5216.14 cm-1	0.992
$\gamma \left(t\right)=A_3{\text{e}}^{-{\mu }_{3}t}+B_3$	A3= 8472.34 cm-1, μ3=2.06 μm-1, B3= 15344.09 cm-1	0.997

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Fig. 2Full-screen View

(Color online) Schematics of (a) 3D and (b) 2D converters; (c) EHP generation rates along penetration depth and (d) surface EHP generation rate and absorption coefficient for different source thicknesses in 2D converters; EHP generation rates in 3D converter with (e) 1.5 μm ridge spacing and 3 μm ridge width, (f) 1 μm ridge spacing and 3 μm ridge width, (g) 1 μm ridge spacing and 4 μm ridge width.

To accurately model the energy deposition and EHP generation rate in a multi-groove betavoltaic nuclear battery, it is essential to consider the superposition contributions of all the isotope sources and ridges. This involves extending G(x,t) of the 2D structure to that of the 3D structure. Figure 2 (a) characterizes the penetration distance distribution of β particles released by ⁶³Ni isotope sources, where D_j denotes the jth layer converter (the jth ridge of the proposed battery) and S_j indicates the jth ⁶³Ni layer. At a random location of the EHPs generated in converter D_j (specified by reference site P), the distance between P and source S_j-1 is represented by [x]. Converter D_j is exposed to [j-1] layers of isotope sources on its left, and the β particles emitted from the ith (i<j) source must traverse a total source thickness of [(j-i-1)t] and a total converter thickness of [(j–i–1)d+x] to reach reference site P. Similarly, D_j is exposed to [n+1–j] layers of sources on its right, and the β particles emitted from the kth (j≤k ≤n) source must penetrate a total source thickness of [(k-j)t] and a total converter thickness of [(k+1-j)d-x] to reach P. During this process, the energy of the beta particles decays exponentially with penetration distance, and the EHP generation rate of D_j is given by: $\begin{array}{c}{G}_j\left(x,t\right)={\displaystyle \sum _{i=1}^{i=j-1}\left\{G_0\left(t\right){\text{e}}^{-\text{α}\left(t\right)\left(\left(j-i-1\right)d+x\right)}{\text{e}}^{-\text{γ}\left(t\right)\left(j-i-1\right)t}\right\}}\\ +{\displaystyle \sum _{k=j}^{k=n}\left\{G_0\left(t\right){\text{e}}^{-\text{α}\left(t\right)\left(\left(k+1-j\right)d-x\right)}{\text{e}}^{-\text{γ}\left(t\right)\left(k-j\right)t}\right\}},\end{array}$ (3) where n is the layers of sources, G₀(t) is the surface EHP generation rate, α(t) is the absorption coefficient of β-electron flux in SiC, and γ(t) is the absorption coefficient of β-electron flux in ⁶³Ni, which are acquired via the equation in Table 1 based on Monte Carlo code simulation. The EHP generation rate of the (n+1)th converter can be expressed as $G_{n+1}\left(x,t\right)=G_1\left(d-x,t\right)$, owing to the symmetry of the multi-groove structure. Table 1 shows that the R² (R-squared or coefficient of determination) values are greater than 0.99, indicating the excellent fitting performance of the model.

Model for radiation-induced current in 3D diode structures

Fig. 1(b) illustrates that radiation-induced EHPs generated within the depletion region can be collected with 100% efficiency, whereas those generated outside the depletion region can only be collected after diffusion to the PN junction boundary, the P⁺/P interface, or the N/N⁺ interface. The CE(y) was calculated using the equation [43]: $CE\left(y\right)=1-\tanh \frac{d\left(y\right)}{L},$ (4) where d(y) represents the distance from the depletion region boundary or the interfaces and is set to zero inside the depletion region. The electron and hole minority carrier diffusion lengths L are expressed as L_n1, L_n, L_p, and L_p1 in regions P⁺, P, N, and N⁺, respectively. The radiation-induced current density J_R can be expressed as: $J_{\text{R}}=\left({\displaystyle \sum _{i=1}^{\text{i}=n+1}q}\underset{0}{\overset{H_1}{{\displaystyle \int }}}{G}_j\left(x,t\right)\text{d}x\underset{0}{\overset{H\_\text{source}}{{\displaystyle \int }}}CE\left(y\right)\text{d}y\underset{0}{\overset{H_2}{{\displaystyle \int }}}\text{dz}\right)/S,$ (5) where q is the electron charge; H₁ is the ridge width [d]; and H₂ and S are the rectangular side length (1 cm) and area of the device (1 cm²), respectively.

According to the drift-diffusion theory, nonequilibrium carriers generated within the depletion region and the neutral region outside the depletion region boundary within a minority diffusion length can be collected, thus contributing to the current density (J_R). The effective charge collection region (ECR) length, represented by H_ECR= (W_d+ L_n1+2L_n+2L_p+ L_p1), determines J_R and can be maximized by increasing W_d, L_n1, L_n, L_p, and L_p1. Lower doping concentrations increase W_d and L, leading to higher EHP collection, as demonstrated in Figs. 3(a) and (b), where W_d decreases from 7.58 μm to 30 nm and L_n decreases rapidly from 77.34 to 11.06 μm with increasing doping concentration from 10¹⁴ to 10¹⁹ cm^-3. L_p decreases more gradually over the same doping range. The minimum doping concentration in the P- and N-regions was 10¹⁴ cm^-3 due to our facility’s capacity to process low doping in SiC materials; whereas, the minimum doping concentration of the heavily doped P⁺- and N⁺-regions required N_A =10¹⁹ cm^-3 and N_D =10¹⁸ cm^-3, respectively, to reduce the ohmic contact. To maximize H_ECR and J_R, the maximum values of W_d, L_n1, L_n, L_p, and L_p1 should be adopted, as listed in Table 2 and detailed in Supplementary Materials S1 and S2.

Fig. 3Full-screen View

(a) Depletion region width and (b) minority diffusion length with doping concentration.

Battery output characteristic model and simulation

The electron (hole) concentration [n(p)] inside the semiconductor converter device is governed by the carrier continuity equation as follows: $\frac{\partial n}{\partial t}=\frac{1}{q}\nabla \cdot j_{\text{n}}+G-R_{\text{n}},$ (6) $\frac{\partial p}{\partial t}=-\frac{1}{q}\nabla \cdot j_{\text{p}}+G-R_{\text{p}},$ (7) where j_n (j_p) is the electron (hole) current density, G is the EHP generation rate derived from Eq. (3), and R_n (R_p) is the electron (hole) recombination rate. j_n (j_p) can be described by the drift and diffusion processes [44] as: $j_{\text{n}}=nq{\mu }_{\text{n}}E+q{D}_{\text{n}}\nabla n,$ (8) $j_{\text{p}}=pq{\mu }_{\text{p}}E-q{D}_{\text{p}}\nabla p,$ (9) where E is the sum of the external and internal electric fields generated by the diffusion of EHPs. These relationships are governed by the Poisson equation: $\frac{\partial E}{\partial x}=\frac{\text{q}\left(N_{\text{D}}-N_{\text{A}}+p-n\right)}{{ϵ}_0{ϵ}_{\text{r}}},$ (10) where ρ is the charge density, ϵ₀ is the vacuum permittivity, ϵ_r is the relative permittivity of the semiconductor, and N_D (N_A) is the concentration of donor (acceptor).

Optimizing ridge spacing and width

The output power of a betavoltaic nuclear battery depends on the amount of beta particle energy deposited in the converters and the efficient collection of radiation-induced EHPs. The coupling between the radiation source and the device is crucial for enhancing the output power. To illustrate the impacts of t and d on the output performance of the betavoltaic battery, the 3D surface contours of the short-circuit current density (J_SC), open-circuit voltage (V_OC), and maximum output power density (P_m) were plotted, as shown in Fig. 4.

Fig. 4Full-screen View

(Color online) Effects of the ridge spacing [t] and ridge width [d] on the (a) J_SC, (b) V_OC, (c) P_m; the dependences of (d) P_m, P_in, (e) d, and η_tot on t. N_a and N_d are both set to 1 ×10¹⁴ cm⁻³; H_P and H_N are set to 158 μm and 26 μm, respectively, approximately equal to (2L_{n_max} + W_{d_max} /2) and (2L_{p_max} + W_{d_max} /2).

As depicted in the 3D surface wireframe perpendicular to the d direction in Fig. 4(a), J_SC initially increases rapidly, but subsequently decreases as t increases, reaching a peak value with t of 0.1–2.2 μm. Additionally, the dependence of J_SC on d exhibits a similar trend, reaching its peak with d of 0.2–4.0 μm. The bottom-projected contour shows that the optimal values of J_SC are achieved when d is in the range of 0.2–3 μm and t ranges from 0.1 to 2 μm. The maximum J_SC of 8.57 μA/cm² is achieved by combining t=0.8 μm and d=1.2 μm.

Figure 4(b) demonstrates that V_OC initially increases as t increases and reaches saturation at t=2 μm, while it decreases with an increase in d. Nevertheless, overall, V_OC is not highly sensitive to variations in t and d, with a range of 2.36–2.46 V. Due to the minor variation in V_OC, P_m is primarily determined by J_SC. The relationship between P_m and variations in t and d exhibit similar trend to that of J_SC, with a rapid initial increase and a subsequent slow decrease with increasing t or d, as illustrated in Fig. 4(c). The maximum P_m value of 19.73 μW/cm² is attained at t=0.8 μm and d=1.2 μm.

The factors affecting the output power mainly include A, η_s, (1-r), Q, V_OC, and FF. With varying t and d, A, η_s, (1-r), V_OC, and FF exhibit relative increases of 9844.28%, 578.19%, 1956.94%, 4.14% and 0.21%, respectively, while Q remains unchanged at 45.64%. Therefore, the key factor affecting the output power is the coupling of A, η_s, and r. For the detailed calculation process, please refer to Supplementary Material S4.

The input power [P_in] can be calculated by combining A, η_s, and r, as P_in= AE_avgη_s(1-r). Figures 4(d)–(e) present the maximum output power [P_m] and its corresponding optimized ridge width [d] of 0.2–4.0 μm, for different source thicknesses [t]. We found that P_m is determined by the coupling of A, η_s and r, i.e., P_in. The optimized value of d gradually saturates with increasing t, consistent with the phenomenon of energy deposition saturation in the converter with increasing d. Additionally, the overall conversion efficiency [η_tot] increases and then decreases with t, reaching a saturated value of approximately 1.5%, and η_tot corresponding to maximum P_in and P_m is 4.58%.

In the proposed battery, smaller ridge spacing and ridge width can improve the source activity and reduce the self-absorption effect. However, excessively thin converters may result in a lower (1-r) and do not match the particle penetration depth, leading to a reduced P_in. Therefore, a tradeoff between t and d is necessary to maximize the power density.

Optimizing widths of P-region and N-region

After optimizing t and d, we investigated the dependence of J_SC, V_OC, and P_m on the widths of the P- and N-regions for betavoltaic nuclear batteries, as shown in Figs. 4(a)–(c). Increasing H_P results in an initial increase and subsequent saturation of J_SC, V_OC, and P_m. This trend can be attributed to the increase in the radioisotope source activity with H_P, which generates more EHPs in the ECR. However, when H_P exceeds the ECR length, the performance metrics reach saturation. Specifically, J_SC, V_OC, and P_m saturate when H_P reaches 160 μm, and an additional increase of 10 μm results in a negligible increase of less than 1%. Similarly, J_SC, V_OC, and P_m present a similar trend with H_N, showing a gradual rise and saturation at H_N =24 μm. This phenomenon can be attributed to the longer minority carrier diffusion length (L_n=77.34 μm) of the P-region compared with that of the N-region (L_p=11.44 μm), and their saturation values of H_P and H_N are around twice L_n and L_p, respectively. Therefore, H_source should not exceed 200 μm, where H_P⁺ and H_N⁺ are 6 and 10 μm, respectively, and H_P and H_N should not exceed their saturation values, corresponding to 160 and 24 μm.

Figure 5(d) shows that A linearly increases with H_P from 0.51 to 6.59 Ci (by 1204.00%), while Q first increases and then decreases with the relative change rate of 164.92%. The optimization of H_P reveals that factor A is the dominant factor affecting the output power, and factor Q is secondary. Although A and Q show different trends with changes in H_P, their products, A·Q and P_m, exhibit consistent trends, further confirming that A and Q are the key factors affecting P_m, as shown in Fig. 5(e). However, V_OC and FF increase minimally by 1.81% and 0.09%, while η_s and (1-r) remain unchanged at 61.53% and 48.77%, respectively. Please refer to Supplementary Material S4 for detailed calculations.

Fig. 5Full-screen View

(Color online) Effects of H_P and H_N on (a) J_SC, (b) V_OC, and (c) P_m with varying H_source; dependences of (d) A, Q, (e) P_m, and A·Q on H_P with H_N fixed at 24 μm; (f) dependence of P_m on H_P with H_source = 200 μm. N_a and N_d are both set to 1 ×10¹⁴ cm⁻³, t=0.8 μm, and d=1.2 μm.

As depicted in Fig. 5(f), for the structure with H_source = 200 μm, J_SC and P_m initially increase with H_P and then decline instead of continuously increasing. This behavior is due to the contribution of H_N to J_SC, and P_m outweighs that of H_P when H_P approaches its saturation thickness. Therefore, when optimizing the widths of the P- and N-regions, the balance between H_P and H_N should be considered, as the combination of H_P =156 μm and H_N =28 μm achieves the maximum P_m of 19.74 μW/cm². A longer P-region is more suitable for a betavoltaic nuclear battery with a P⁺PNN⁺ junction structure because it is more conducive to enhancing the collection of EHPs owing to the larger minority carrier lifetime and mobility compared with the N-region, as shown in Fig. S2 in the Supplementary Material.

Optimizing doping concentration in P- and N-regions

Based on the analysis in Sect. 2.2.2, the doping concentration plays a crucial role in determining the depletion width and minority carrier diffusion length of the betavoltaic nuclear battery, which significantly affects the collection efficiency of the EHPs and the output performance of the battery.

Figures 5(a)–6(c) illustrate the effects of N_a and N_d on J_SC, V_OC, and P_m for the proposed battery. As shown in Fig. 6(a), J_SC increases with a decrease in N_a owing to its beneficial effect of expanding the minority carrier diffusion length and depletion region width to promote EHP collection, as depicted in Fig. 3. The variation in J_SC with N_d was small compared with N_a because L_p is much smaller than L_n, resulting in less changes in the collection efficiency of EHPs.

Fig. 6Full-screen View

(Color online) Influence of N_a and N_d on (a) J_SC, (b) V_OC, and (c) P_m; effect of N_a on (d) H_ECR, Q, and (e) P_m. H_P = 156 μm, H_N= 28 μm, t=0.8 μm, and d=1.2 μm.

Figure 6(b) shows that V_OC initially increased rapidly but subsequently decreased slowly as N_a increased, as depicted in the 3D surface wireframe perpendicular to the N_d direction. The dependence of V_OC on N_d exhibits a similar trend. The maximum V_OC of 2.66 V is obtained by combining N_a =3.16×10¹⁸ cm^-3 and N_d =7.94×10¹⁷ cm^-3. The bottom-projected contour indicates that the optimal values of V_OC are achieved at higher doping concentrations, which is attributed to the reduction in the leakage current [J₀] owing to the higher doping concentration, as demonstrated in Fig. S3 in Supplementary Material.

Although J_SC and V_OC exhibit opposite dependencies on doping concentration, P_m varies in the same way as J_SC because the variation range of V_OC with doping concentration is small, of 2.44–2.66 V with a relative increase of 8.98%. The optimal doping concentration combination is N_a =1×10¹⁴ cm^-3 and N_d = 1×10¹⁴ cm^-3, yielding the maximum output power density of 19.74 μW/cm².

As shown in Fig. 6(d), low doping increased Q from 13.48% to 45.65% owing to the widened depletion region and diffusion length, leading to longer ECR length [H_ECR] and improved EHP collection, and resulting in a higher power density. H_ECR and P_m follow a similar trend as Q, confirming that low doping enhances the power density by increasing H_ECR for EHP collection. However, at the highest doping concentration, H_ECR and P_m are minimum, of 53 μm and 6.37 μW/cm², respectively, which are only slightly better than the power of 5.80 μW/cm² with the H_source of 53 μm. P_m only marginally improves even with a nearly four-fold increase in the source activity [A], indicating the critical role of H_ECR in the design of the 3D battery and that H_source should not exceed H_ECR.

Additionally, FF ranges from 94.18% to 94.57%, with a tiny relative increase of 0.42%, while A, η_s, and (1-r) remain constant at 4.04 Ci, 61.53%, and 48.77%, respectively, resulting in constant P_in of 129.16 μW/cm². Therefore, Q is a significant factor affecting the output power density, and a low doping concentration in the P- and N-regions is recommended to enhance it, hence maximizing the short current and output power density. Detailed calculations are provided in Supplementary Material S4.

Critical parameters of device structure

To achieve the best device performance, we comprehensively optimized the geometric dimensions (optimization procedure #1), doping concentration (optimization procedure #2), and width of each doping layer (optimization procedure #3) of the semiconductor materials. The final optimized parameters for the battery are N_a = N_d =1 ×10¹⁴ cm⁻³, H_P =156 μm, H_N=28 μm, t=0.8 μm, and d=1.2 μm.

In Optimization procedure #1, A, η_s, and (1-r) exhibited significant relative increases, while the relative increases in Q, V_OC, and FF were negligible, as shown in Fig. 7(a). Therefore, the key factor affecting the output power is the coupling of A, η_s, and (1-r), and a tradeoff between them is required to maximize the power density, as depicted by their optimal parameters corresponding to the maximum output power.

Fig. 7Full-screen View

(Color online) Optimal parameters and their relative increase considering different optimization procedures: (a) Optimizing t and d; (b) Optimizing H_P and H_N; (c) Optimizing N_a and N_d. Relative increase is calculated as the difference between the maximum and minimum values divided by the minimum value. The bars show the minimum values only, indicating that the maximum value is equal to the minimum.

In Optimization procedure #2, A was identified as the dominant factor affecting the output power, with the significant relative increase of 1204.00%, and Q was a secondary factor with the relative increase of 164.92%, as depicted in Fig. 7(b). Increasing H_P and H_N can enhance the source activity [A] and ECR length. However, increasing H_P and H_N beyond the ECR leads to Q, as EHPs outside the ECR become difficult to collect. The tradeoff between A increasing and Q dropping results in saturation of the output power.

In Optimization procedure #3, we found that Q is the key factor affecting the output power; whereas, the other factors have negligible relative increases, as shown in Fig. 7(c). To optimize the output power density, low doping concentrations should be used in both P- and N-regions. This enables larger diffusion lengths and wider ECRs, resulting in a higher collection efficiency of EHPs, particularly through an extended H_ECR.

Among these parameters, the coupling of A, η_s, and (1-r) has the greatest impact on the output performance, which is related to the activity, self-absorption, and backscattering of the radioactive source. The second most influential parameter is Q, which depends on the depletion region width and diffusion length controlled by the doping concentration, as well as the ECR length governed by H_P, H_N, and the doping concentration. To maximize the output power density, use of thinner sources and converters, lower doping concentrations, and larger H_P, of approximately twice the diffusion length, are recommended.

To validate the numerical model established in this study, we performed calculations on planar batteries with the same geometric dimensions and semiconductor parameters as those in references [24] and [16]; good agreement was apparent regarding the output power density, as shown in Table 3. It is worth noting that the power density of battery #1 was converted based on the 100% abundance of ⁶³Ni source, in contrast to the 20% abundance mentioned in reference [24]. The difference in the results between reference [16] and this study (#5 battery) is relatively large because the previous work did not consider the energy loss caused by the collection efficiency Q, leading to an overestimation of P_m. Using our method, we can further provide suggestions for optimizing the batteries in references [24] and [16] by adjusting N_a, N_d, H_P and H_N. For reference [24], setting H_P, H_N, N_a, and N_d to 4.9 μm, 19.9 μm, 1×10¹⁴ cm⁻³, and 3.98×10¹⁴ cm⁻³, respectively, the maximum output power can be increased to 296.5 nW/cm², representing a 12.6% improvement as depicted by battery #3. For reference [16], by combining H_P =0.1 μm, N_a and N_d at 4.7×10¹³ cm⁻³ and 3×10¹⁶ cm⁻³, respectively, the output power can be increased to 378 nW/cm², with a 10.5% improvement as depicted by battery #6.

Finally, we conducted a comparative analysis of the P⁺PNN⁺ and PN structures, assessing their respective output power densities in the 2D (#7 and #8) and 3D (#9 and #10) batteries. To achieve this, we substituted the P⁺ and N⁺ layers in the P⁺PNN⁺ structure with P and N layers of the same size to obtain the PN structure, as shown in Table 3. Our results indicate that the P⁺PNN⁺ structure outperforms the PN structure, with a 10% increase in P_m in the 2D configuration and a 39% increase in the 3D configuration. These findings highlight the superior performance of the P⁺PNN⁺ structure, particularly in 3D configurations.

In this study, the diffusion lengths were calculated through commonly used equations [17, 42], and the minority carrier lifetimes (τ_n and τ_p) adopted in these calculations (shown in Supplementary Material S4) fall within the range of experimental values (τ_n of 0.9–10 μs and τ_p of 0.05–2.1 μs) [47-50]. Moreover, experimental diffusion lengths for SiC were reported to be 30–100 μm [49, 51], which provides further validation for the calculated electron and hole diffusion lengths presented in this article.