A Simple theoretical model for 63Ni betavoltaic battery

Applied Radiation and Isotopes 82 (2013) 119–125

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Review

A Simple theoretical model for

63

Ni betavoltaic battery

Guoping ZUO a,n, Jianliang ZHOU b, Guotu KE c a b c

School of Nuclear Science and Technology, University of South China, Hengyang, China School of Nuclear Science and Technology, University of South China, Hengyang, China China Institute of Atomic Energy, Beijing, China

H I G H L I G H T S


The energy deposition distribution is found following an approximate exponential decay law when beta particles emitted from 63Ni pass through a semiconductor.
A simple theoretical model for 63Ni betavoltaic battery is constructed based on the exponential decay law.
Theoretical model can be applied to the betavoltaic batteries which radioactive source has a similar energy spectrum with 63Ni, such as 147Pm.

art ic l e i nf o

a b s t r a c t

Article history: Received 17 February 2013 Received in revised form 24 July 2013 Accepted 29 July 2013 Available online 8 August 2013

A numerical simulation of the energy deposition distribution in semiconductors is performed for 63Ni beta particles. Results show that the energy deposition distribution exhibits an approximate exponential decay law. A simple theoretical model is developed for 63Ni betavoltaic battery based on the distribution characteristics. The correctness of the model is validated by two literature experiments. Results show that the theoretical short-circuit current agrees well with the experimental results, and the open-circuit voltage deviates from the experimental results in terms of the influence of the PN junction defects and the simplification of the source. The theoretical model can be applied to 63Ni and 147Pm betavoltaic batteries. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Theoretical model Nickel-63 Betavoltaic battery

Contents 1. 2. 3. 4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Energy deposition characteristics of 63Ni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical model for 63Ni betavoltaic battery. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Validation of the theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Case 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Case 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction A betavoltaic battery is a device that directly converts nuclear decay energy to electric power. The energy conversion operation of betavoltaic batteries is similar to that of photovoltaic ones.

n

Corresponding author. Tel./fax: +86 7348282251. E-mail addresses: [email protected], [email protected] (G. ZUO).

0969-8043/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.apradiso.2013.07.026

119 120 121 122 122 123 124 124 124

Electron–hole pairs (EHPs) are created by beta particles emitted from radioactive isotopes. When EHPs diffuse into the depletion region of the semiconductor p–n junction, an electric field sweeps them across the depletion region, thus generating a radiationinduced current (Manasse et al., 1976). A betavoltaic battery has the following characteristics: high energy density, long service life, easy small-scale fabrication, and negligent environmental effect. Betavoltaic batteries have become promising micro-power sources for MEMS power, medical applications, and execrable ambient

G. ZUO et al. / Applied Radiation and Isotopes 82 (2013) 119–125

2. Energy deposition characteristics of

63

Ni

Beta particles can interact with electrons and nuclei in the medium wherein they are traveling. Beta particles passing near nuclei will be deflected by coulomb forces and may lose kinetic energy (Rutherford scattering). The interactions of beta particles with orbital electrons are crucial Coulomb repulsion between beta particles and electrons frequently results in ionization. In the ionization process, beta particles lose an amount of energy equal to the kinetic energy of the electron and the energy used to free the electron from the atom. The total energy loss of beta particles equal to the sum of ionization and radiation losses can be expressed as the following (Friedlander et al., 1981; L’Annunziata. 2003; Zhihua et al., 1996):


 dE dE dE  ¼  þ  ð1Þ dx total dx ion dx rad The ionization energy loss of low energy beta particles can be calculated approximately by using the following formula (Friedlander et al., 1981; L’Annunziata. 2003; Zhihua et al., 1996):  
 dE 4πe4 2m0 v2  þ 1:2329 ð2Þ ZN ln ¼ dx ion I m0 v 2 The energy loss because of bremsstrahlung radiation can be expressed as follows (Friedlander et al., 1981; L’Annunziata. 2003; Zhihua et al., 1996):
 
  dE NEZðZ þ 1Þe4 2E 4 ð3Þ  ¼ 4 ln  2 2 dx rad 3 m0 c 137m0 c4 where N is the atomic density, Z is the mean atomic number of the absorber, E is the kinetic energy of the particle, β¼ v/c, v is the

velocity of the electron in the medium, and I is the mean ionization and excitation potential of absorbing atoms. At relativistic energies, the ratio of beta-particle energy loss via bremsstrahlung emission to energy loss via ionization and excitation is described by the following equation Wu et al., 1986:
 
 dE dE ZE  ð4Þ  ¼ dx rod dx ion 800 where E is the beta-particle energy in MeV, and Z is the atomic number of the absorber material. According to this equation, the energy loss caused by radiation is very small when the particle energy is approximately below 1 MeV. Thus, the energy loss caused by radiation can be neglected for soft beta sources such as 63Ni, which has a maximum energy that is equal to 0.067 MeV and a deposition energy approximately equal to the energy loss by excitation and ionization. The transport processes of beta particles in semiconductors have been simulated by the Monte Carlo N-particle Radiation transport code (Briesmeister, 1993). By calculating the energy loss per unit length along the GaN depth direction, the stopping ranges of 67 and 17 keV electrons are approximately 14 and 1.2 μm, respectively (Fig. 1). The peak value of energy loss rating is also exhibited along the GaN depth direction. The value and position of the peak becomes smaller and deeper with increasing particle energy, respectively. These results are consistent with those of other studies (San et al., 2013; Yao et al., 2012; XiaoBin et al., 2012), and can be explained by Eq. (2). According to a finite thickness 63Ni source, a beta energy spectrum that reaches a detector is a continuous spectrum

20

dE/dx (keV.μm-1)

processes (Lu et al., 2011a, 2011b; Cress et al., 2008; Wu et al., 2011; Ulmen and Despairs, 2009; Hang Guo, 2007). Suitable and practical radioisotope selection plays a critical part in the design of betavoltaic batteries. The factors affecting the performance of betavoltaic devices are half-life, specific activity, toxicity, and specific power, among others (Mohamadian et al., 2007). 63Ni is one of the most suitable radioisotope source because of its pure particle emission, long half-life (approximately 100 yr), and low beta radiation (maximum value of 67 keV) (Ulmen and Despairs, 2009; Zaijun Cheng, 2010; Zai-jun et al., 2011). The beta radiation of 63Ni is below the radiation damage threshold (approximately 200 keV for Si) (Ulmen and Despairs, 2009; Hang Guo, 2007) of common semiconductors such as Si, GaAs, SiC, and GaN. Several theoretical studies on betavoltaic batteries have been reported over the last decade (Haiyang et al., 2011; Chen et al., 2011; Xianggao Piao and Chu, 2007; WEI, 1974; Guan-quan et al., 2010; Hui et al., 2012; Qiao Da-Yong and Xue-Jiao, 2011). These theories provide references for the design and fabrication of betavoltaic batteries. However, most of these studies, the average energy of beta particles was used to determine the deposition energy and penetration depth (Haiyang et al., 2011; Chen et al., 2011; Xianggao Piao and Chu, 2007; WEI. 1974). Some studies showed that the spectrum of sources affects the deposition distribution of beta particles in absorbers, thus influencing the design and performance of the device (Li et al., 2012; San et al., 2013). The general beta energy spectrum of a radioisotope usually peaks at one-third the maximum energy of the emitted beta particle (Cross et al., 1983). However, 63Ni has a simple beta energy spectrum that exhibits an approximate linear decline with increasing beta energy. Thus, a simple model can be used to describe a 63 Ni betavoltaic battery. This study aims to develop a simple theoretical model for 63Ni betavoltaic batteries based on its energy spectrum characteristics and validate the model by experiments.

17keV 67keV

15

10

5

0 0

2

4

6

8

10

12

14

16

18

Depth (μm) Fig. 1. Energy deposition distribution of 17 keV and 67 keV beta particles passing through GaN.

0.14 0.12 0.10

#/nt

120

0.08 0.06 0.04 0.02 0.00 0

10

20

30

40

50

Energy (keV) (Average) Fig. 2. Beta energy spectrum of

147

Ni.

60

70

G. ZUO et al. / Applied Radiation and Isotopes 82 (2013) 119–125

(Fig. 2). Spectrum characteristics have a significant effect on the energy deposition distribution of 63Ni in the absorber (Fig. 3). Fig. 3 shows the energy deposition distribution of 63Ni in GaN and Si semiconductors. A comparison of Figs. 1 and 3 reveals that a significant difference exists between the single energy beta particles and beta energy spectra. The relationship between the energy deposition and penetration depth approximately follows an exponential decay law (Fig. 3). Although a deviation exists at the end of the range, the values are very small, i.e., 0.5 eV mm  1 for Si; thus, the influence of radiationinduced current can be ignored. The relationship between energy deposition and penetration depth can be expressed as the following

 Z dE dE i ¼ f ðEÞ  dE ¼ a1 expða2 xÞ ð5Þ  dx 63 N dx ion E 102

dE/dx (MeV.cm-1)

101 10

where f (E) is the emission probability of the beta particle with energy E, and α1 and α2 are the fitting coefficients that depend on the physical properties and geometric configurations of the device. This phenomenon is strongly associated with the energy spectrum characteristics of 63Ni radioactive sources, which approximately follows an exponential decay distribution. Different phenomena will occur when β rays emitted from different radioactive sources passes through matter. For example, the energy spectrum of 32P (Fig. 4) is significantly different with that of 63Ni, and the energy deposition distribution of 32P in GaN (Fig. 5) is different with that of 63Ni. Therefore, the exponential decay law only applies to specific radioactive sources.

3. Theoretical model for

10-1 10-2 10-3

Dp

10-4 0

5

10

15

pn pn0 ∂ 2 pn þ GðxÞ ¼0 τp ∂x2

102

GðxÞ ¼ A

dE/dx Exponential Fit

ð6Þ

ðdE=dxÞ63 N

ð7Þ

Eion

100

110 10-1 10-2 10-3 10-4

0

5

10

15

20

25

Depth (μm) Fig. 3. 63Ni source energy deposition along the thickness of GaN and Si. (a) Energy deposition in GaN and (b) energy deposition in Si.

dE/dx (MeV.cm-1)

dE/dx (MeV.cm-1)

10

Ni betavoltaic battery.

where pn and pn0 are the hole concentrations with and without nuclear radiation, respectively; Dp is the hole diffusion coefficient; G refers to the electron–hole generation rate, which can be expressed as follows:

20

Depth (μm)

1

63

The diagram of the p–n junction betavoltaic battery is shown in Fig. 6, where β is the beta particle emitted from radioactive isotopes, Xn is the junction depth, W is the width of the depletion region, and H is the thickness of the whole battery. Similar to the physics of solar cells, the analytical expression of the short-circuit current density of a betavoltaic battery can be obtained by solving the minority carrier diffusion equation Ulmen, 2009. In the N doped region, the solution for the minority carrier diffusion equation is expressed as the following:

dE/dx Exponential Fit

0

121

105 100 95 90 85 80 0.000

0.08

0.002

0.003

Fig. 5.

32

P source energy deposition along the thickness of GaN.

0.04 0.02 0.00 0.0

0.004

Depth (cm)

0.06

#/nt

0.001

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Energy (Mev) (Average) Fig. 4. Beta energy spectrum of

32

P.

Fig. 6. Schematic diagram of betavoltaic battery.

122

G. ZUO et al. / Applied Radiation and Isotopes 82 (2013) 119–125

where A is the activity of 63Ni, and Eion is the average energy expended to create EHPs in the absorber. From Eq. (5), Eq. (7) can be rewritten as the following: a1 expða2 xÞ GðxÞ ¼ A Eion

ð8Þ

Referring to Fig. 6, the two boundary conditions of Eq. (6) are as follows: 1) At the junction edge of the base region, excess electron density is under the short-circuit condition: pn pn0 ¼ 0;

at x ¼ xn

H’ is defined as the following: H′ ¼ xh xp ¼ Hxn W

The total radiation-induced current is the sum of the current generated in the n-type, p-type, and depletion regions. J ion ¼ J n þ J c þ J p

V oc ¼ η¼

where Sp is the surface recombination velocity of the hole. In the p-type doped region, the minority carrier diffusion equation and boundary conditions are expressed as follows: Dn

np np0 ∂ 2 np þ GðxÞ ¼0 τn ∂x2 dðnp np0 Þ ¼ Sn ðnp np0 Þ  x¼H dx

np np0 ¼ 0;

x ¼ xn þ W

P out J sc V oc ¼ FF P in AEave

ð21Þ

where k is the Boltzmann constant, T is the absolute temperature, Jsc is the circuit current, which is equal to Jion in an ideal situation, FF is the fill factor, Pin is the incident power of the radioactive isotope, Eave is the average energy of electrons and A is the radioactivity.

4. Validation of the theoretical model

ð12Þ

4.1. Case 1

ð13Þ

The experimental results of the planar 63Ni–Si p–n junction betavoltaic batteries in (Guo et al., 2003) are chosen as the validation object of the theoretical model. The structural parameters of the device are listed in Table 1. The theoretically calculated results and literature experimental data are listed in Table 2. The results show that the theoretical short-circuit current agrees well with the experimental values, with a maximum relative error of less than 5%. However, a significant difference exists in the open-circuit voltage, particularly for Device No. 1, which has more than twice the theoretical value of the experimental results. The difference in the opencircuit voltage is most likely caused by the defects created by the heavily doped p-type glass-source, thus resulting in a big reverse saturation current (Briesmeister, 1993). The experimental value of the heavily doped p-type glass-source at 24 pA is higher than the

Lp eAα1  Eion α22 L2p 1   ððSp Lp =Dp Þ þ α2 Lp Þea2 xn ððSp Lp =Dp Þchðxn =Lp Þ þ shðxn =Lp ÞÞ  α2 Lp ea2 xn ðSp Lp =Dp Þshðxn =Lp Þ þ chðxn =Lp Þ

ð14Þ Jn ¼

ð20Þ

ð11Þ

where Dn is the electron diffusion coefficient, Sn is the surface recombination velocity of the electron, τ is the charge carrier lifetime, and np and np0 are the electron concentration with and without nuclear radiation, respectively. The expressions of generated current density in the emitter and base can be derived from the solutions of Eqs. (6) and (13), respectively: Jp ¼

nkT J sc ln q J0

In this section, two betavoltaic devices from previous literature are selected as objects for our theoretical study to check the correctness of the model described in Section 3. The selected betavoltaic devices are 63Ni–Si and 147Pm–Si betavoltaic batteries, which have been fabricated and tested in previous literature.

and Dn

ð19Þ

The open-circuit voltage and conversion efficiency are expressed by the following equation (Antonio and Steven, 2003):

ð9Þ

2) At the back contact, the surface is characterized by a recombination velocity Sp; thus, the excess electron density satisfies the following equation: dðpn pn0 Þ ¼ Sp ðpn pn0 Þ at x ¼ 0; ð10Þ Dp  x¼0 dx

ð18Þ

eAa1 Ln eα2 ðxn þWÞ  Eion α22 L2n 1   ðSn Ln =Dn ÞðchðH′=Ln Þeα2 H′ Þ þ shðH′=Ln Þ þ α2 Ln eα2 H′  α2 Ln  ðSn Ln =Dn ÞshðH′=Ln Þ þ chðH′=Ln Þ ð15Þ

The current density generated in the depletion region is defined by the following: Z Wþxn qa1  a2 xn a2 ðxn þWÞ Jc ¼ qGðxÞdx ¼ e e ð16Þ a2 xn The leakage current is expressed as follows (Kurtz et al., 1990):

 qn2 Dp ðDp =Lp Þshðxn =Lp Þ þ Sp chðxn =Lp Þ  J0 ¼ i

N D Lp ðDp =Lp Þchðxn =Lp Þ þ Sp shðxn =Lp Þ

 qn2i Dn ðDn =Ln ÞshðH′=Ln Þ þ Sn chðH′=Ln Þ

 ð17Þ þ N A Ln ðDn =Ln ÞchðH′Ln Þ þ Sn shðH′=Ln Þ where Lp and Ln are the minority carrier diffusion lengths, NA and ND are the acceptor and donor doping concentrations, respectively.

Table 1 The device structure parameters (Briesmeister, 1993). Parameter

Device No. 1

3

ND (cm ) NA (cm  3) p-Type thickness (mm) Area (cm2) Activity (mCi)

Device No. 2

20

1020 1017 1 44 1

10 1017 1 44 0.25

Table 2 Experimental and theoretical short-circuit current and open-circuit voltage.

Isc (nA) Voc (mV)

Device No. 1 (0.25 mCi)

Device No. 2 (1 mCi)

Exp.

Comp.

Exp.

Comp.

0.71 64

0.73 130

2.41 115

2.44 161

G. ZUO et al. / Applied Radiation and Isotopes 82 (2013) 119–125

theoretical value of 4.8 pA. Thus, a smaller open-circuit voltage is induced according to Eq. (20). If the experimental result is substituted into Eq. (20), the ideal factors of 0.72 and 0.96 are obtained for Devices No. 1 and No. 2, respectively. However, the ideal factor equal to unit for an ideal p–n junction diode, which is used in our theoretical calculation. If the effect of defects is considered and the theoretical reverse saturation current is replaced with experimental values, the computed open-circuit voltage will be 65 and 120 mV for Devices No. 1 and No. 2, respectively, which are very close to the experimental results. Another possible cause of the deviation is the simplification of the source in our theoretical simulation. The radioactivity values of 0.25 and 1 mCi are not the surface activities in reference (Guo et al., 2003) and are greater than the experimental values for the source’s self-absorption. Although we simulated the self-absorption effect of the source according to reference (Guo et al., 2003), the deviation still exists for the simplification.

4.2. Case 2 Although the above theoretical model is established for the Ni betavoltaic battery, the model can also be applied to any kind of radioisotope with a similar exponential decay distribution. The beta energy spectrum and energy deposition distribution of 147Pm is similar to those of 63Ni (Figs. 7 and 8). Thus, the 147Pm betavoltaic battery can be adopted as the second case. The 63

0.00 0.15

0.02

0.04

0.06

147Pm 63Ni

#/nt

0.10

0.05

0.00 0.00

0.05

0.10

0.15

0.20

0.25

Energy (MeV) Fig. 7. Beta energy spectrum of

102

0

50

100

147

150

63

Ni.

200 63Ni 63Ni Fitted 147Pm 147Pm Fitted

101

dE/dx (MeV/cm)

Pm and

100

250 1 10 100 10-1

10-1

10-2

10-2

10-3

10-3

10-4

10-4 0

5

10

15

20

Depth (μm) Fig. 8. Energy deposition of

147

Pm and

63

Ni in Si.

10-5 25

123

Table 3 Device parameters (WEI 1974).

ND (cm  3) NA (cm  3) n-Type thickness (mm) Total thickness(mm) Area (cm2) Curie content (Ci cm  2)

Device No. 1

Device No. 2

Device No. 3

1  1020 3  1016 0.8 180 2.85 0.8–1.9

1  1020 6  1015 0.8 180 2.85 0.8–1.9

1  1020 2  1015 0.8 180 2.85 0.8–1.9

Table 4 Experimental and theoretical short-circuit current, mA cm  2. Activity

Device No. 1

Device No. 2

Device No. 3

Ci cm  2

Exp

Comp

Err (%)

Exp

Comp

Err (%)

Exp

Comp

Err (%)

0.8 1.0 1.2 1.6 1.9

27 29 36 39 47

27.21 30.15 34.54 39.37 45.06

0.78 3.97  4.05 0.95  4.12

26 28 34 37 44

27.34 30.3 35.02 39.56 45.28

5.15 8.21 3.0 6.92 2.90

26 31 35 41 45

27.41 30.40 35.61 40.51 45.36

5.42  1.94 1.74  1.20 0.81

experimental results of the 147Pm betavoltaic battery can also be used to check the correctness of the proposed model. Several Pm2O3–Si betavoltaic batteries have been demonstrated in (WEI, 1974). The structural parameters of the Pm2O3– Si betavoltaic batteries are listed in Table 3. To investigate the effects of activity on device performance, five different activities of Pm2O3 (Table 4) are used as radiation sources in the experiment. The device parameters listed in Table 3 are used as theoretical input data to calculate the performance parameters of the Pm2O3– Si betavoltaic batteries. The performance parameters include short-circuit current, open-circuit voltage, fill factor, and conversion efficiency. The experimental and theoretical values of shortcircuit current are listed in Table 4. The result shows that the theoretical values agree well with the experimental results and that the maximum deviation is less than 10%. Fig. 9 shows the short-circuit current values of the experimental and theoretical data of WEI (1974), as well as our theoretical results. The theoretical short-circuit current determined from the proposed model shows the least deviation from the experimental results. The small deviation of the theoretical short-circuit current is due to the difference of dealing with radiation-induced current. In WEI (1974), total radiation-induced current is the sum of the diffusion current of the p-type and ntype regions and ignores the role of the depletion region. The total radiation-induced current is significant for the p–n junction betavoltaic batteries, where radiation-induced EHPs are separated and cast aside by the built-in electric field. The collection efficiency of EHPs in the depletion region is higher than that in the p-type and n-type regions. Therefore, for a high-performance p–n junction betavoltaic battery, the depletion region should be designed with proper thickness and depth to improve the characteristics of the output current significantly. However, this approach is not applicable for the devices reported in WEI (1974). The thickness and junction of the n-type region is very large and very deep, respectively; thus, most of the EHPs created are in the n-type region, and few EHPs are created in the depletion and p-type regions. For example, the radiation-induced currents of the n-type, p-type, and depletion regions are 3.41  10  5 A cm  2, 7.66  10  7 A cm  2, and 4.38  10  7 A cm  2, respectively, when Device No. 1 is irradiated by the 1.2 Ci cm  2 source. Therefore, no significant deviation occurs for the thicker top layer of Device No.

124

G. ZUO et al. / Applied Radiation and Isotopes 82 (2013) 119–125

50 45

Jsc (uA/cm2)

for short-circuit currents. For open-circuit voltage, the results deviate from the experimental values in terms of the influence of the PN junction defects and the simplification of the source. The open-circuit voltage will agree with the experimental results if the defects are considered. The results also show that the theoretical model can be applied to 63Ni and 147Pm betavoltaic batteries.

Exp values of No.1 Comp values of this paper Comp values of ref 14

40 35 30

Acknowledgment

25

This study is supported by the National High Technology Research and Development Program of China (Grant no. 2009AA050701), Natural Science Foundation of Hunan Province (Grant no. 10JJ9014), and Science and Technology Project of Hunan Province (Grant no. 2011TT2039).

20 0.8

1.0

1.2

1.4

1.6

1.8

2.0

Activity (Ci/cm2) 50

Exp values of No.2 Comp values of this paper Comp values of ref 14

Jsc (uA/cm2)

45

References

40 35 30 25 20 0.8

1.0

1.2

Activity 50

1.6

1.8

2.0

(Ci/cm2)

Exp values of No.3 Comp values of this paper Comp values of ref 14

45

Jsc (uA/cm2)

1.4

40 35 30 25 20 0.8

1.0

1.2

1.4

1.6

1.8

2.0

Activity (Ci/cm2) Fig. 9. Relationships of the short-circuit current and the Curie content of sources. (a) Device No.1, (b) Device No.1 and (c) Device No.1.

1 when the role of the depletion region is ignored. However, Device No. 1 exhibits significant deviation for its thinner top layer and larger particle range.

5. Conclusion A numerical simulation of the energy deposition distribution in semiconductors is performed for 63Ni beta particles. The results show that the energy deposition distribution approximately follows an exponential decay law. A simple theoretical model for the 63 Ni betavoltaic battery is constructed based on the distribution characteristics. The proposed model is then applied to two case studies, and the results agree well with the experimental results

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