Optimal algorithm to improve the calculation accuracy ...

Optimal algorithm to improve the calculation accuracy of energy deposition for betavoltaic MEMS batteries design Li Sui-xian* a,b a

Chen Haiyang a

Sun Min a

Cheng Zaijun c

School of Mechanical and Vehicle, Beijing Institute of Technology, Beijing, 100081,

China; b China Space Technology Academic , Beijing ,100086， China; c MEMS centrer of Xia Men University,Xiamen ,361005， China

ABSTRACT Aimed at improving the calculation accuracy when calculating the energy deposition of electrons traveling in solids, a method we call optimal subdivision number searching algorithm is proposed. When treating the energy deposition of electrons traveling in solids, large calculation errors are found, we are conscious of that it is the result of dividing and summing when calculating the integral. Based on the results of former research, we prop ose a further subdividing and summing method. For β particles with the energy in the entire spectrum span, multiple of keV,

the energy data is set only to be the integral

and the subdivision number is set to be from 1 to 30, then the energy deposition

calculation error collections are obtained. Searching for the minimum error in the collections, we can obtain the corresponding energy and subdivision number pairs, as well as the optimal subdivision number. The method is carried out in four kinds of solid materials, Al, Si, Ni and Au to calculate energy deposition. The result shows that the calculation error is reduced by one order with the improved algorithm. Key words :β energy spectrum, low energy electrons， solid material, calculation error， energy deposition, optimal subdivision number searching algorithm， betavoltaic MEMS battery 1

Introduction

MEMS betavoltaic microbatteries are the devices that extract power from the current which is the result of the division of β-induced EHPs due to the potential in the depletion region of the semiconductor pn-junction. The application of betavoltaic effect dates from early 1950s, the study of Ehrenberg[1] and Rappaport and Coworkers[2]. In the subsequent decades, the development was not fast until the appearance of MEMS with the development of semiconductor technology, IC and micro-processing and assembly technology. The miniaturization of MEMS makes the requirement of

2009 International Conference on Optical Instruments and Technology: MEMS/NEMS Technology and Applications, edited by Zhaoying Zhou, Toshio Fukuda, Helmut Seidel, Xinxin Li, Haixia Zhang, Tianhong Cui, Proc. of SPIE Vol. 7510, 751008 · © 2009 SPIE · CCC code: 0277-786X/09/$18 · doi: 10.1117/12.838219 Proc. of SPIE Vol. 7510 751008-1

power output in some occasions no longer harsh, and microbatteries based on betavoltaic effects have some characteristics that are prominent compatible with MEMS power. So the research about *[email protected] ; phone:086-13261852506;adress:Room 1635,NO.34 South Road of HaiDian,HaiDian district of Beijing, Beijing,Chi na,100080

betavoltaic MEMS batteries has made considerable progress in recent years [ 3 - 1 3 ] . Modern betavoltaic microbatteries have the features as follows, long life-time, high energy density, small volume, good environment adaptability and reliability, and safety for human beings. The half life of radioisotopes used in MEMS microbatteries are generally

several decades, thus the batteries

can keep working for a long time without extra energ y addition. The energy of radioisotopes is released from atomic radiation and the energy density is hundreds of times more than that of general fuel cells or chemical cells. Because of the high energy density, the volume of batteries can greatly reduce without reducing the output. So, the batteries and MEMS integration can be realized by the micro-machining and semiconductor manufacturing process, allowing the whole system volume down to the order of mm, even micron scales. Because the physics and chemistr y state like pressure, temperature has no influence on the radioactivit y of radioactive elements, the batteries show good environment tolerance. Besides, the batteries are safe to human bod y because the energy of β particles emitted from

63

Ni and 3 H is very low.

When designing the batteries, it’s necessar y to calculate the spatial distribution of energy in the target material. It is of great significance to definite batteries’ structures, such as the depth and the width of pn-junction. The thickness of the source metal

63

Ni should be rationally designed, because the energ y

emitted from isotopes tends to be saturable when the thickness comes to critical point. Besides, to improve the output and conversion efficiency, we have to prop ose reasonable design of depletion region to collect the energy of particles as much as possible. Althou gh the path of electrons in solids is ver y tortuous, the path length is definite for the electrons with the specific energy in a particular material. This paper intends to solve the issues of the energ y deposition of β particles along the track direction in solids. Without taking into account the specific spatial location and the direction change during the moving of particles, we only seek the energy loss (deposition) of β particles along the tracks. Hence, we can simply assume that the particles move straight and seek the energy deposition during each small trip. 2 2.1

Principle Energy loss of low-energy electrons traveling in solids

β particles are essentially high-speed electrons, therefore the interaction between β particles and solid atoms can be explained according to the interaction between electrons and solid atoms. The electrons interact with solid atoms when they move in solids, accompanied by the energy loss. The rate of

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energy loss on unit path describes the physical characterization of solid materials to stop a certain energy electrons. The more the energy loses on unit path, the more the solid materials can stop or slow down the electrons, thus we call the energy loss on unit path stopping power. When fast electrons move in solids, inelastic collision ma y occur between fast electrons and extranuclear electrons of solid atoms, and a portion of energy ma y be transfered to the extranuclear electrons, so that the atoms can be ionized or excited. Ionization energy loss is an important way of losing energy for β particles. Inelastic collision can cause a large amount of energy loss, possibly up to half of the energy of particles for one time. Generally, the energy transfer of a single collision is a few keV and the direction of incident β particles may have a great change. The energy loss caused by collisions is called collision stopping power. For low-energy incident β particles, the energy loss cause by collisions accou nts for the main portion of the energy loss. Another wa y of energy loss is electromagnetic radiation, caused by bremsstrahlung. Bremsstrahlung is classical electromagnetic radiation which occurs due to the Coulomb drag on the incident β particles when getting close to nuclei. Bremsstrahlung stopping power is more evident for high-energy incident β particles. Quantitative analysis results can be obtained from Fig. 1 and Fig. 2, getting from ESTAR database

[14]

with minor amendments. We can see that the total stopping power is approximately equal with collision stopping power for incident β particles below 100keV while it’s approximatel y equal with bremsstrahlung stopping power for β particles above hundreds of MeV / GeV.

Fig.1. Stopping power of silicon

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The energies of β particles emitted from radioisotopes commonly used for betavoltaic effect batteries are below 100keV. For example, the maximum energy of β particles emitted from

63

Ni is 66.7keV, and

3

for H, 18.6keV. Therefore, we assume that the total stopping power is equal with collision stopping power, which means that we ignore the bremsstrahlung stopping power and only consider energy loss caused by collisions. The path length of β particles can calculated from the following expression n

E 0 = ∑ ΔEl i

(1)

i =1

Where, E 0 is the initial energy of incident β particles, and i is the number of sub-path,

ΔE l i is the

energy deposition on the ith sub-path length. How to divide the path, as well as the number of subdivision, there are different methods in literature. Bishop raised that subdividing for 25 times can reach sufficient accuracy for gold (heavy atoms) [ 9 ] to calculate energy deposition; G. Love divided total path length into 100 segments to calculate energy deposition

[9]

; Mao et al

[15]

divided total path length into 50 segments to calculate the backscattering

coefficient of electrons of tens of keV and obt ained results consistent with experiments. After determined the number of subdivision, we can calculate the energy deposition on each sub-path as follows

ΔE li = E l i − E l i-1 =

Where -

li

dE dl dl l i −1

∫

（2 )

dE is the stopping power, and i is the number of sub-path. In fact, because of the difficulty dl

of direct integration of expression (2), usuall y numerical method is used to solve the problem, thus the energy deposition on each sub-path is expressed as

ΔEli = (

Where

dE ) dx

E = Ei

Δl i

（ 3）

Δl i is the length of ith sub-path.

For low-energy electrons, the stopping power is equal with collision stopping power of which the expression is the well-known Bethe formula, that is proposed by Hans Bethe in 1930’s. The expression is as follows

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S=−

(e / 2) E dE Zρ ln( ) = 2π e04 N 0 dx AE I (in eV/A0)

（ 4）

Zρ 1.166 E = 785 ln( ) AE I Where e(=2.718) is the base of natural logarithm, N 0 is Avogadro constant, Z is the atomic number, E is the current energy of incident β particles(eV), A is the mass of solid atoms, I is the average excitation energy(eV),

(e / 2)=1.166 is the electronic interchange effect coefficient.

The stopping power of electrons obtained according to Bethe formula can be found in authoritative database [ 1 4 ] . The uncertainty of the data is 1-2% for electrons above 100keV, 2-3% for low-Z materials, 5-10% for high-Z materials [ 1 4 ] ..

Fig.3. The energy deposition when the total path is divided into 50 copies (The numbers of abscissa denotes the type of solids.)

Fig.4. The relative errors of energy deposition when the total path is divided into 50 segments (The numbers ofabscissa denotes the type of solids.)

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Because of this limitation, the data can only be used for electrons above 10keV. In addition, we can see from Equation (4) that the result of logarithm is negative when

E < J /1.166 , which is

inconsistent with the physical fact. Bethe formula can be used to calculate stopping power for electrons above 10keV, while stopping power of electrons below 10keV can be found from the data having been published. Moreover, the stopping power of electrons above 10keV can also be found from the database for some solids. 2.2 Calculation accuracy of energy deposition of single-energy β particle on the path First, we consider the energy deposition of single-energy β particle. We intend to divide total path length into 50 segments and calculate the energy deposition on each sub-path length. As an example, we calculated the energy deposition of β-particles wih the initial energy of 66.7keV emitted from 6 3 Ni, and the result is shown in Fig. 3. We can see that the energy depositions are different for different solids even though the incident energies of incident particles are the same (66.7keV), and relative errors that the energy depositions deviate from the initial value, are also different. Fig. 4 shows the relative errors, which are all more than 4%. The calculation error comes from the numerical integration itself. Calculation error can be reduced a lot when the number of evaluation of the integrand is increased. 2.3 The way of reducing calculation error and improve calculation accuracy of energy deposition Based on the calculation of dividing the total path length into 50 segments, we subdivide each sub-path, calculate the energy deposition on the sub-path respectively using the Equation (3), sum the energy of 50 segments and finally Fig. out the “ arithmetic relative error ” (the absolute value of relative error).

Rel at i ve Error/ %

10 9 8 7 6 5 4 3 2 1 0 0

5

10

15

20

25

30

Subdi vi de Number

Fig.5. The energy deposition for Si material with further subdivision when the total path length is divided into 50 segments

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Here, the number of subdivision is respectively 1, 2, 3 ... 10, 15, 18, 20, 25, 30; the number 1 means no subdivision is carried out. As an example, the result of Si is shown in Fig. 5. We can see that the accurac y increases but does not increase monotonically with number of subdivision. Hence, to achieve high accuracy an optical value of the subdivision number is required. Fig. 5 gives out the information about Si material, and it can be inferred that the same conclusion exists to other materials. 3 The optimized subdivision searching algorithm and the result of energy deposition 3.1 The optimized subdivision searching algorithm The total path length of βparticles varies with particle energy and material, therefore the subdivision number is optimally selected based on the calculation accurac y using different subdivision numbers. Here, we still take the "arithmetic relative error" of total energy deposition relative to the particles’ initial energy as the evaluation index. Take Fig. 5 as an example, without considering the calculation speed, the optimal subdivision number is 20 with relative error 0.1%. If considering the calculation speed, the optimal number is 8 with relative error 0.5%. We call this method subdivision number searching algorithm.

Fig.6. The optimal subdivision number for β particles emitted from

63

Ni in solids

We select an arbitrar y natural number from 1 to 30 as the subdivision number and take the integer from 1keV to 67keV as the energy of β particles, to be similar to the spectrum of

63

Ni. For β particles

of different energies, using the method above, we can calculate the arithmetic relative error corresponding to each subdivision number. Searching through the results of all the energies, the number which the arithmetic relative error is corresponding to is the optimal subdivision number. The optical subdivision numbers of Al, Si, Ni and Au are shown in Fig. 6.

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Fig.6 shows that the optimal subdivision numbers for β particles emitted from

63

Ni in the four kinds

of solids are all below 20. Generally, the larger the energy is, the larger the number is; for low-energy particles, especially in gold, the number is 1, which means that subdivision can’t increase the calculation accuracy.

Fig.7. The relative error of energy deposition before and after subdivision for Al material

Fig.8. The relative error of energy deposition before and after subdivision for Si solid material

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Fig.10. The relative error of energy deposition before and after subdivision for Au solid material 3.2 The calculation accuracy of the improved algorithm When carrying out the optimal subdivision searching algorithm, we recorded the relative error of energy deposition before and after subdivision, and the results are shown in the following Figures(Fig.7 to Fig.10).

Fig.11. The spatial distribution of energy deposition in the depth direction of β particles emitted from 63Ni in Si solid material

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10 5 β particles is generated according to the probabilit y distribution of spectrum of 63Ni and then we calculate the path location and energy deposition in solid of ever y particle. The simulation program recorded 5000 thousand locations on the 10 5 paths each, as well as the corresponding depositional energy. The results are as follows. Fig.11 gives out the spatial distribution of energy deposition in the depth direction of β particles emitted from 63Ni in Si material. Fig.11 shows that the energy deposition density is high in the incident point but the energy value is low, while high energy depositions locate in deep point with low density. However, it’s difficult to completely determine the spatial distribution in the depth direction solely on the basis of Fig.11. The cumulative percentage of energy deposition in the depth direction can reflect the energy distribution more clearly, so we calculated the percentage of 100,000 particles and the statistical result is shown in Fig.12. From Fig.12, we can see that the rate of change of the energy deposition on depth decreases with the increasing of depth. Quantitively, the energy deposition accounts for about 40% of total quantity within 1/10 depth from the surface, 80% within 1/3 depth and 90% within 1/2 depth. 4.2 The experimental verification Fig.13 is the percentage distribution of the energy deposition in the depth direction for β particles emitted from

63

Ni in Si material. The data comes from the published experimental data which is

measured by Semiconductor Analyzer when

63

Ni irradiating Si-based pn-junction. In this paper, we

normalize the experimental data and the result is shown in Fig.13.

Fig.13. The experimental results of percentage distribution of energy deposition in the depth direction for β particles

emitted from

63

Ni in Si material

We can see from Fi g.13 that the simulation result in section 4.1 is consistent with the result published

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by Hang Guo of Cornell University. The experimental value is lower. For instance, the energy deposition in the pre-5μm depth accounts for 54% according to simulation results while 46% according to experiment, and in the pre-15μm depth the value is 91% and 79% respectively. That is reasonable because the pre-3μm of pn-junction is not active and the EHPs can’t be collected completely. 5

Analysis and conclusions

We can see from the Fig.s from 7 to 10 that, using the optimal subdivision searching algorithm, the calculation error for most of the β particles emitted from

63

Ni is significantly decreased b y one order

of magnitude, less than 1%. It means that the algorithm is effective. Moreover, the consistency between the result of Monte Carlo simulation, in which the algorithm is introduced, and the experimental data indicates the rationality of the algorithm. There are other conclusions worth to pa y attention to. For a few low-energy β particles and relatively low–energy particles in heavy metals like gold, the algorithm is noneffective, thus it can’t improve the calculation accuracy. The failure is not due to errors caused by quadrature and possibly due to the accurac y of expression of stopping power for low energies. Obviously, the classical Bethe formula is an important reason. However, the optimal subdivision searching algorithm can still improve the calculation accuracy of energy deposition.

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