Efficiency analysis of betavoltaic elements
A.V. Sachenko, A.I. Shkrebtii, R.M. Korkishko, V.P. Kostylyov, M.R. Kulish, I.O. Sokolovskyi
1 Efficiency analysis of betavoltaic elements A.V. Sachenko , A.I. Shkrebtii , R.M. Korkishko , V.P. Kostylyov , M.R. Kulish , I.O. Sokolovskyi
1 1
V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine, 41 prospect Nauky, 03028 Kyiv, Ukraine University of Ontario Institute of Technology, Oshawa, ON, Canada * Corresponding author. E-mail: [email protected]: +38(044)525-57-34 Abstract The conversion of energy of electrons produced by a radioactive β-source into electricity in a Si and SiC np junctions is modeled. The features of the generation function that describes the electron-hole pair production by an electron flux and the emergence of a “dead layer” are dis-cussed. The collection efficiency Q that describes the rate of electron-hole pair production by incident beta particles, is calculated taking into account the presence of the dead layer. It is shown that in the case of high-grade Si np junctions, the collection efficiency of electron-hole pairs created by a high-energy electrons flux (such as, e.g., Pm-147 beta flux) is close or equal to unity in a wide range of electron energies. For SiC p – n junctions, Q is near unity only for elec-trons with relatively low energies of about 5 keV (produced, e.g., by a tritium source) and de-creases rapidly with further increase of electron energy. The conditions, under which the influ-ence of the dead layer on the collection efficiency is negligible, are determined. The open-circuit voltage is calculated for realistic values of the minority carriers’ diffusion coefficients and life-times in Si and SiC np junctions, irradiated by a high-energy electrons flux. Our calculations allow to estimate the attainable efficiency of betavoltaic elements. Keywords: betavoltaics; beta source; collection efficiency; open-circuit voltage
1. Introduction
Betavoltaic effect refers to conversion of energy from electrons generated in nuclear reac-tions into electricity in semiconductor np junctions. Long action periods (decades) and the ability to operate in a wide temperature range (from –50 to 150º C) make betavoltaic effect at-tractive for a number of technological applications, such as communication devices, sensors in hard-to-reach areas, and implantable medical devices [1]. In particular, a battery for heart pace-makers based on the Si np junction and Pm-147 source of beta particles was produced [2]. Other popular beta-source and recipient are tritium and A B semiconductors [3-5]. 2 After the pioneering work of Ehrenberg, et al. [6], betavoltaic conversion has been inten-sively investigated both experimentally [7-10] and theoretically [8], [9]. Of particular importance in this respect is the efficiency of a betavoltaic cell, defined as the ratio of the power generated by the cell to the power delivered by the incident electrons. The first estimates of this parameter were performed in Refs. [2, 10-12]. Betavoltaic and a more familiar photovoltaic effect are related and share several traits. In both cases, calculations of the limiting efficiency are made under the assumption that all elec-tron-hole pairs produced by the incident beta particles or photons contribute to the betavoltaic or photovoltaic current. When viewed as a function of the band gap, E g , the current generated by beta-particles decreases, whereas the open-circuit voltage grows with E g . Because the latter ef-fect dominates over the former, the overall betaconversion efficiency increases with E g [2]. In the case of photovoltaics, in contrast, the photocurrent decrease with band gap is stronger, result-ing in a maximum of photoconversion efficiency as a function of E g [13]. In this research the attainable betavoltaic conversion efficiency is modeled using realistic values of minority carriers’ lifetimes and diffusion coefficients to calculate the collection coeffi-cient Q and open circuit voltage V OC . The dominating recombination mechanisms were included in the formalism developed. The results were compared to the limiting photoconversion effi-ciency, which corresponds to the fundamental maximum of η. For the betavoltaic cell we assume that the β source is in the form of a foil [2]. This al-lows considering the one-dimensional (slab) geometry for our theoretical analysis. The cross-section of the betavoltaics cell and β-source is shown schematically in Fig. 1. The beta electrons flux is directed toward p – n junction, which separates generated electron-hole pairs, are shown. The dead layer, where the scattering the beta-electrons can be neglected, is x m thick, and is lo-cated close to the frontal surface of the sample and the β source. 3 Fig. 1. Cross-section of the betavoltaic system with β-source adjacent to p – n junction. The emitter and the base are d p and d b thick, and p – n junction width d = d p + d b . The dead layer is x m thick. In the beta battery, each beta-electron produces a number of electron-hole pairs by dissipating its energy E in the semiconductor. Electron energy is in turn averaged with respect to the decay spectrum for each beta emitter [14]. Experimental spectra for tritium from [15] and promethium from [16] were used in beta-conversion efficiency calculation. Electron-hole pair generation function by a beta electron can be written as [17].
2. The collection efficiency analysis
Efficiency of a betavoltaic converter can be written as a product of three terms [2] sc . (1) Here / NN (2) is the ratio of the beta-flux N , reaching the semiconductor surface, to the total flux N emit-ted; Qr c (3) is the coupling efficiency, given by the product of absorption probability of a beta-particle ( r is the coefficient of electron reflection from the semiconductor surface) and collection efficiency Q of electron-hole pairs; finally, the semiconductor efficiency is /FFqV OCs . (4) Here q is the elementary charge, OC V is the open circuit voltage, FF is the fill factor of the cur-rent-voltage characteristics, )5.08.2( g E eV is the energy needed to create one electron-hole pair, and g E is semiconductor band gap [18]. Previously, the maximal betavoltaic efficiency was obtained under an implicit assump-tion that each incident beta-particle produces electron-hole pairs, i.e., the collection efficiency Q = 1. To derive the collection efficiency Q, we initially also used the expression )exp()( xxg for the generation function for electron-hole pairs created by beta-electron flux (see [8, 11]), where is the absorption coefficient, g is the generation rate of electron-hole pairs per unit vol-ume, and x is the distance from the semiconductor front surface. This expression, however, is 4 valid only outside of the dead layer when m xx , while g(x) is assumed be zero at m xx [19]. Sufficiently general expressions for Q in this approximation are given in [8, 11], while the most general one is derived in [20]. It has the following form: np QQQ , (5) pppppp dpppdppdpppp pp LdLdLS eLLdeLdeLSLL LQ ppp coshsinh sinhcosh11)( , (6)
LdLDLdS erSrDerLdLDLdSLLeQ d ddddddddn p coshsinh 111sinhcosh)(1 LdLDLdS erLdLDLdLS d ddd coshsinh 1coshsinh . (7) Here, p Q and n Q are the collection coefficients of electron-hole pairs in the emitter and the base respectively, p L is the diffusion length in the emitter, p d is the thickness of the emitter, p is the bulk lifetime in the emitter, d r is the electron reflection coefficient from the back surface of the np junction, DL is the diffusion length in the base, D and b are the base diffu-sion coefficient and the bulk lifetime, respectively, d is the thickness of the base, S is the ef-fective surface recombination rate at the emitter surface, and d S is the effective surface recom-bination rate at the back surface of the base. In the case of a beta battery, each beta-electron dissipating its energy E in the semicon-ductor produces a number of electron-hole pairs. Electron energy is in turn averaged with respect to the decay spectrum in beta-conversion efficiency calculation. Electron-hole pair generation function by a beta electron can be written as [17] )()(21)(1)()( xEBxE NZqxJdxdExJxg . (8) Here J(x) is the density of electrons flux, )( dxdE is the energy dissipation per unit path length, N is the number of absorber atoms per cm , Z is the atomic number of the absorber mate-rial, and B(E) is the stopping number. The electron energy dissipation per unit path length can be 5 calculated utilizing (8). The transcendent solution of (8) multiplied by the electron energy distri-bution function and integrated over the energy is fit well by the exponential generation function in the form )exp( xE [21]. The factor in (5) - (7) describes the decay of excess concentration of electron-hole pairs generated by beta electrons. Comparing the averaged by decay energy left part of (8) to the exponential generation function, one can determine the α-value. This procedure yields
300 cm -1 for Pm-147/Si, 270 cm -1 for Pm-147/SiC, 3 cm -1 for T/Si, and 1.5 cm -1 for T/SiC. The collection efficiency of a np junction based on high-quality silicon with Shock-ley-Read-Hall recombination lifetime SR
1 ms can be calculated using (5) - (7). Due to the low currents in betavoltaic elements, the excess concentration of electron-hole pairs, p , is much smaller than the equilibrium majority charge carriers (holes) concentration in the base even for very long lifetimes. Considering the Shockley-Read-Hall lifetime SR in the base, radiative recombination coefficient r B , and Auger interband recombination coefficient Auger C , the resultant bulk carrier lifetime b can be written as pCpB AugerrSRb , (9) where p is the equilibrium majority carriers’ concentration in the base. Figure 2 (a) shows the collection coefficient Q decrease with the increase in the effective surface recombination rate d S for silicon np junctions. In this case, the mean energy of Pm-147 electrons is 61.9 keV. The parameters used to calculate Si and SiC collection coefficients are given in Table 1. Absorption coefficient is taken from (8), using the approach of [17]. For the above considered parameters, the collection coefficient depends very weakly on the recombina-tion rate S d and always exceeds the rather high value of 0.85. The parameters of beta-sources are given in Table 2. For tritium as the source of beta-electrons with the mean energy of 5.7 keV, is assumed to be 1,5·10 cm -1 for SiC [17], and Q for this case very close to unity (see Fig. 2(b)). Since the holes lifetime is higher than the one for electrons in SiC [22], n-type base was considered in this case to ensure large diffusion length. Standard p-type base was used in Si. 6 SR =5 10 -5 s SR =10 -4 s C o ll e c t i on c oe ff i c i en t Surface recombination, cm/s SR =10 -3 s(a) Promethium-147 SR =5 10 -5 s SR =10 -4 s C o ll e c t i on c oe ff i c i en t Surface recombination, cm/s SR =10 -3 s(b) Tritium Fig. 2. The collection coefficient Q vs. the effective surface recombination velocity for silicon np junctions with Pm-147 (a) and tritium (b) as the electrons source. L p , cm τ p , s d p , s p , cm d, cm D, cm /s r d B r , cm /s C Auger , cm /s Si 10 -3 -7 -5 -2
30 0.6 6 10 -15 -31 SiC 6 -6 -10 -6 -2
1 0.6 1.5 -12 -31 Table 1: Set of the semiconductor parameter used in (5)-(7). Source α (Si), cm -1 α (SiC), cm -1 Mean energy, keV Maximal Energy, keV Pm-147 300 270 61.9 224,6 T 30 000 15 000 5.7 18,6 Table 2: The absorption coefficients α for different combinations of beta-source and semicon-ductors used in the calculations, and the mean and the maximum energies of beta-particles from the Pm-147 and T sources. Figure 3 (a) shows the collection coefficient Q as a function of the effective surface re-combination velocity d S for SiC np junctions. The dependences were calculated using the parameters from Tables 1 and 2. The figures demonstrate that the collection coefficients Q for 7 this case are small compared to unity and do not depend on d S , because the dL condition is satisfied. This criterion means high losses of minority carriers in the bulk. Therefore, the attain-able efficiency of the SiC betavoltaic battery with Pm is smaller than the efficiency of the high-quality silicon battery. SR =10 -9 s SR =10 -8 s C o ll e c t i on c oe ff i c i en t Surface recombination, cm/s SR =10 -7 s(a) Pm-147 SR =10 -9 s SR =10 -8 s C o ll e c t i on c oe ff i c i en t Surface recombination, cm/s SR =10 -7 s(b) Tritium Fig. 3. The collection coefficient Q vs. the effective surface recombination velocity for SiC np junctions with Pm-147 (a) and tritium (b) as the electrons source. The graphs in Fig. 3 (b) were calculated for tritium as the source of beta-electrons and us-ing the effective surface recombination velocity d S . Other parameters are the same as for Fig. 3 (a). Collection coefficient Q in this case is practically independent from d S , being rather high even for short Shockley-Read-Hall recombination time SR -9 s, resulting in Q = 0.67. This value of collection coefficient Q is acceptable for practical applications. Therefore, SiC battery efficiency increase should be expected if tritium source is used. The L criterion ensures the effective electron-hole pairs collection in this case. In the case of silicon, this criterion is well satisfied for both Pm-147 and for tritium sources. However for SiC, L is close to 1 only for tritium source. In the beginning of this section we initially wrote the electron-holes generation function as )exp()( xxg . Since according to [8], the generation function has a maximum at m x , the 8 )exp( x form is correct for distances exceeding m x . This maximum is due to negligible elec-tron scattering until m xx . For gallium arsenide under beta irradiation, m x is in the range of 10 -5 ÷ 10 -4 cm [19]. The greater the electron energy, the greater m x is. The area of m xx , is termed the dead layer [19]. The collection efficiency can be significantly reduced for the case m xL . To illustrate this, an expression for the collection efficiency Q was derived under the assumption that the electron-hole pair generation is absent for m xx and is described as )exp()( xIxg for m xx . We also assume that mp xd , d and . Then, solv-ing the continuity equation for the excess electron-hole pairs concentration p in the region 1 (where m xx ) and in the region 2 (where m xx ) with the boundary conditions p dxp , dxp and matching the solutions, we find the integration constants. Collection effi-ciency is then determined from the expression /)/( IdxpdDQ at the point p dx and it has the following form: L dxLLQ pm exp1 . (10) Figure 4 shows the dependence of Q on the diffusion length, calculated by (10). The depth of the np junction was fixed ( p d =10 -5 cm), while the beta-electron energy (and so the absorption coefficient ) and m x were varied. The dashed curves are plotted using )1/( LLQ expression, which becomes correct in the absence of the dead layer. Figure 4(a) shows that biggest collection efficiency reduction occurs for the upper curves with tritium as an electron source. The discrepancy between Q values calculated with and with-out taking the dead layer into account is minimal for lowest curves, for which Pm is a source of electrons. In this case, to obtain sufficiently large collection efficiency Q L must exceed 35 µm. High Q values ( Q ~ 1) can only be achieved in sili-con np junctions with large minority charge carriers lifetime. The graphs in Fig 4 (b) are plotted for m x = 10 -5 cm and =4·10 cm, which corresponds to the tritium decay. The figure demonstrates that the Q increases with L and when the junction depth is approaching m x . This effect is particularly significant for small diffusion lengths. Thus, the analysis shows that diffusion length L higher than m x is needed for sufficiently high elec-tron-hole pairs collection efficiency. An alternative way to increase Q is to use a deep np junction with mp xd . 9 m =1,5 10 -5 cmx m =1,5 10 -5 cmx m =10 -4 cm =3 10 cm -1 =2 10 cm -1 C o ll e c t i on c oe ff i c i en t Diffusion length, m =4 10 cm -1 (a) p =10 -5 cmd p =7 10 -6 cmd p =4 10 -6 cmd p =10 -6 cm C o ll e c t i on c oe ff i c i en t Diffusion length, m (b) Fig. 4. Collection efficiency Q dependence on the minority carriers’ diffusion length in the base L for the SiC p – n junction, calculated taking into account existence of the dead layer (solid lines). Pm-147 is the beta-electron source in (a), while the tritium source is considered in (b). The results of Q calculations without considering the dead layer shown in Figs. 2 and 3 are accurate enough for mp xd . Indeed, Fig. 4(a) shows that Q does not change substantially for mp xd in SiC np junctions. With SiC and tritium, Q from Fig. 3(b) is reduced by a fac-tor of Ldx pm /exp for mp xd . Therefore, the Q value is 0.83 for the base lifetime of 10 -8 s, m x =2·10 -5 сm, p d =10 -6 сm and L =10 -4 сm. This reduction does not lead to drastic changes of the Q values shown in Fig. 3(b).
3. Open circuit voltage analysis
It should be noted that the calculated in [2,11] open circuit voltage, collection efficiency, and therefore the limiting efficiency as a function of band gap were overestimated. However, it will be shown below that the efficiency overestimation is particularly high for wide-gap semi-conductors having much lower bulk lifetimes than for high-quality silicon. For two-component 10 wide-gap semiconductors a significant degree of compensation is also common, which decreases OC V . To find the open-circuit voltage V OC in the realistic case (with the system parameters close to the experimentally achievable) and taking into account such recombination mechanisms as recombination in the quasi-neutral region and recombination in the space charge region (SCR) with the width w, the generation-recombination balance equation should be written as: **0 tanh pqVpSLdLDqJ SC . (11) Here g EQJJ is the density of the beta-electron-excited current, J is the density of current generated by beta-particles in silicon, S is the effective surface recombination velocity at the front surface of the cell, Δp * = Δp| w , b DL , and )2)( pn rr y iyad iiyad adSRDSC dyFenepNN nbeneNN NNLpV is recombination in the SCR. Here F is the dimensionless electric field in the space-charge re-gion, i adai adada n NNn NNyNNyF lnexp1lnlnexp1ln yd ad eN NN , np CCb / . kTE aa / and kTE rr / are dimensionless energies of the acceptor and recombination lev-els with respect to the middle of the band gap, d N and a N are donor and acceptor concentra-tions, y is the dimensionless current potential, pn y is the dimensionless potential at the np interface. L and D are the diffusion length and diffusion coefficient of the minority carriers, rpSR NC is the Shockley-Read-Hall lifetime, r N is the concentration of deep recombina-tion centers, ad NN is the equilibrium concentration of majority carriers in quasi-neutral re-gion of the base, p is the excess concentration of minority carriers in the base on the boundary between the space-charge region and the quasi-neutral region. The value of J usually varies in the range of 1 µA / cm [2]. Open circuit voltage is described by the relation 11 ln iOC n pnqkTV . (12) Here i n is the concentration of charge carriers in an intrinsic semiconductor. To calculate OC V the solution of (11) should be substituted in (12). Fig. 5 shows the OC V dependence on the base doping level p for silicon np junction. Shockley-Read-Hall recombination, radiative recombination, and interband Auger recombination were taken into account in our calculations. No compensation was assumed, while the Shockley-Read-Hall lifetime values of 10 -3 , 10 -4 and 5·10 -5 s were used. Calculations were carried out for the temperature of 310 K, which corresponds to the human body temperature, and was taken as constant. One can see from the figure that )( pV OC has the maximum at high lifetimes, which disappears with lifetime decrease. The highest OC V = 0.58 V is reached at SR = 10 -3 s and p = 10 cm -3 . The highest OC V is reached at the current density of 100 µA / cm for silicon. Such current density is more than two orders of magnitude smaller than the typical photocurrent in the silicon solar cells. SR =5 10 -5 s SR =10 -4 s O pen c i r c u i t v o l t age , V Doping, cm -3 SR =10 -3 s Fig. 5. Open circuit voltage vs. doping level for silicon np junction. The following parameters were used: J = 100 µA/ cm , D = 30 cm /s, T = 310 K, d = 300 µm. SR =10 -9 s SR =10 -8 s O pen c i r c u i t v o l t age , V Doping, cm -3 SR =10 -7 s Fig. 6. The dependence of the open circuit voltage on doping level for SiC np junction. The following parameters were used: J = 100 µA /cm , D = 1 cm / s, T = 310 K, d = 300 µm. 12 Fig. 6 shows the open-circuit voltage OC V vs. base doping level d N for SiC np junc-tion. Compensation was not taken into account in the calculation, i.e., a N value was assumed to be zero. Bulk lifetimes of 10 -7 , 10 -8 and 10 -9 s were used, g E value was assumed to be 3 eV. In contrast to Si, the )( dOC NV curve increases logarithmically in this case. For the acceptor levels energy close to the valence band, the compensation was taken into account by using of the OC V value calculated for ad NNn . Therefore, at high compensation level taking, e.g., d N = 10 cm -3 and a N = 9.99·10 cm -3 , OC V should be calculated for n = 10 cm -3 . For example, open-circuit voltage OC V = 2.35 V for the top curve is at n = 10 cm -3 ; for n = 10 cm -3 open cir-cuit voltage V OC = 2.16 V is significantly lower. The open circuit OC V value of 2.16 V is reached for q J = 1 μA / cm and n =10 cm -3 , while OC V = 1.96 V for n = 10 cm -3 .
260 280 300 3200,91,01,11,2 SR =5 10 -5 s SR =10 -4 s O pen c i r c u i t v o l t age Temperature, K SR =10 -3 s (a)
260 280 300 3200,951,001,051,10 SR =5 10 -9 s SR =10 -8 s O pen c i r c u i t v o l t age Temperature, K SR =10 -7 s (b) Fig. 7. The open circuit voltage V OC (normalized to its value at 300 K) vs. the external tempera-ture. (a) Silicon np junction. (b) SiC np junction. Parameter used: a p = d N = 10 cm -3 , q J = 100 μA /cm , d = 300 μm. The values of D and g E in 7(a) are 30 cm /s and 1.12 eV; and in 7 (b) they are 1 cm /s and 3 eV. 13 When the external temperature is changing, OC V value decreases with temperature. Fig-ure 7 shows the temperature dependence of OC V normalized to the value at 300 K ( )300(/)()( KVTVTv
OCOC ). The temperature is varied in the range from -40 to 40º C. The temperature dependence of V OC is predicted to be stronger in the case of Si p – n junctions than for SiC ones, because of the smaller band gap in Si. It should be noted that the slope of )(TV OC is defined by semiconductor band gap and minority carriers’ lifetime. Fig. 7 shows that the high-er minority carriers’ lifetime the smaller )(TV OC slope. Therefore, both b value and the compensation factor affect OC V for the realistic parame-ters of the battery: the lower lifetime and the higher compensation, the lower OC V . The attainable conversion efficiency for the realistic systems can be written as limlim OCOCreal
VVQ , (13) where lim is the limiting efficiency and limOC V is the limiting open circuit voltage. The limiting efficiency lim was calculated as 5.08.2 limlimlim gOC E FFqV . (14) Here lim FF is the limiting fill factor defined as limlimlimlim kTVkTVkTVFF OCOCOC . (15) lim , limOC V and lim FF were calculated using the modified Shockley-Queisser approach [23] us-ing r , Q and . Figure 8 shows the limiting efficiency η lim vs. bandgap E g . The ef-ficiency was calculated by (14) using silicon radiative recombination parameter and effective densities of states for -arbitrary bandgap. The similar dependence from [2] (Fig. 4) is also in-cluded for comparison. L i m i t e ff i c i en cy , % Bandgap, eV
Olsen et al. [2]Calculation by (14)
14 Fig. 8. The limiting betaconversion efficiency η lim calculated by (14) vs. semiconductor bandgap E g , solid line. The dashed line shows the same dependence as calculated in [2]. Semiconductor /emitter V OClim , V V OCreal , V lim , % real ,% ( SR , s) real , % ( SR , s) real ,% ( SR , s) Si/T 0.53 0.48 11.8 10.7 (10 -3 ) 9.8 (10 -4 ) 9.6 (5 10 -5 ) Si/Pm-147 0.65 0.60 14.9 13.6 (10 -3 ) 12.4 (10 -4 ) 12 (5 10 -5 ) SiC/T 2.21 2.1 23.6 21.5 (10 -7 ) 18.9 (10 -8 ) 14.6 (10 -9 ) SiC/Pm-147 2.33 2.30 24.9 5.3 (10 -7 ) 1.9 (10 -8 ) 0.65 (10 -8 ) Table 3. The limiting and achievable parameters of a betavoltaic converter The, attainable and limiting betaconversion efficiencies calculated by (13) and (14) using Si and SiC parameters are shown in Table 3. The obtainable efficiencies real are calculated using collection coefficients Q from Figs. 1 and 2. The lifetimes of 10 -3 , 10 -4 and 5·10 -5 s were used for Si, while for SiC the lifetimes of 10 -7 , 10 -8 and 10 -9 s were used. The table shows that real decreases slightly with the decrease of SR . real is significantly lower for SiC with Pm-147 as the source of electrons than for the three remaining source/material combinations. The similar situation is expected for direct gap semiconductors with lifetimes of 10 -7 – 10 -9 s. The reason is the change of the L inequality needed for the full electron-hole collection to the L inequality. Note that the experimental efficiency is 4% for Si/Pm-147 system with two-sided irradia-tion [11]. The attainable value for Si/Pm-147 system is 13.6 % (see Table 1).
4. Conclusions
We develop a formalism to determine the maximum attainable conversion efficiency η lim in the Si and SiC based betavoltaics batteries, using
Pm and tritium as the source of beta-electrons. The formalism of carrier transport and collection is based on the similarity of betavol-taic and photovoltaic processes, however, several important differences have been included in the description of the physical processes in the beta-voltaics systems. The realistic experimen-tally achieved parameters of the betavoltaic systems were included in the analytical formalism developed. This allows us to also calculate the realistically achievable betavoltaic efficiencies. We calculate the collection coefficient achievable Q and open-circuit voltage OC V . The betavoltaic conversion efficiency obtained for high-quality Si np junction is higher than for 15 SiC np junction with Pm source. This is due to the high electron-hole pairs’ generation depth and low diffusion length in SiC. Therefore, the majority of the generated electron-hole pairs recombine and do not reach the np junction. The situation for direct-gap semiconductors (particularly GaAs) with low minority carriers lifetime and low diffusion length is similar. Tritium with low beta-electron energy and low pairs generation depth is a better source for SiC with low diffusion length. The collection coefficient for this case is high enough. There-fore, SiC/T conversion efficiency can be greater than Si/T efficiency because of the higher SiC band gap, and consequently higher open circuit voltage. The influence of the dead layer on the collection efficiency value is also included in the formalism and analyzed. The dead layer is associated with a nonmonotonic electron-hole pair generation function in the semiconductor under electrons flow irradiation. The dependence of the collection coefficient Q on the dead layer thickness m x is established. This dependence is weak for the case of diffusion length L significantly greater than m x value. The calculated efficiencies indicate a limit to the maximum possible performance of the betavoltaic systems, e.g., η = 8% for the tritium based H/Si system. While being comparable to experimentally achieved efficiencies, our results demonstrate that there is still sufficient room for efficiency increase using optimized materials parameters and the system design. 16 References 1. L.C. Olsen, P. Cabauy, and B.J. Elkind, Betavoltaic power sources, Physics Today, December, 2012, p. 35. 2. L.C. Olsen, “Review of betavoltaic energy conversion”, NASA TechDoc 19940006935, http://hdl.handle.net/2060/19940006935 3. V.M. Andreev, A.G. Kevetsky, V.S. Kaiinovsky, V.P. Khvostikov, V.R. Larionov, V.D. Rumyantsev, M.Z. Shvarts, E.V. Yakimova, V.A. Ustinov, Tritium-powered betacells based on Al x Ga As, Conference Record of the Twenty-Eighth IEEE Photovoltaic Specialists Conference -2000, IEEE, Piscataway, NJ (2000), pp.1253-1256, DOI:
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