Evaluation of a Silicon 90Sr Betavoltaic Power Source

Betavoltaic energy converters (i.e., β-batteries) are attractive power sources because of their potential for high energy densities (>200 MWh/kg) and long duration continuous discharge (>1 year). However, conversion efficiencies have been historically low (<3%). High efficiency devices can be achieved by matching β-radiation transport length scales with the device physics length scales. In this work, the efficiency of c-Si devices using high-energy (>1 MeV) electrons emitted from 90Sr as a power source is investigated. We propose a design for a >10% efficient betavoltaic device, which generates 1 W of power. A Varian Clinac iX is used to simulate the high-energy electrons emitted from 90Sr, and a high efficiency c-Si photovoltaic cell is used as the converter. The measured conversion efficiency is 16%. This relatively high value is attributed to proper length scale matching and the generation of secondary electrons in c-Si by the primary β-particles.

where m e c 2 is the rest mass energy of the electron with regard to the relativistic nature of the high energy β-emission, c is the speed of light in vacuum and E β,max is the difference between the final and initial mass energies (end-point energy). The Fermi function F(Z',W), which accounts for the relativistic coulomb effects as the β-particle exits the envelope of the nucleus, is given by the following simplified expression as put forward by Venkataramaiah et al. S2 1/2 In Eqn. S2, W is the total energy of the β-particle at a particular momentum (kinetic energy + rest mass energy), and A and B are constants corresponding to the atomic number of the daughter nucleus Z'. For 90 Y, A and B are 8.6994 and 4.7116 respectively. The same parameters are 9.3330 and 5.2025 respectively for 90 Zr. The nuclear matrix element |M fi | 2 accounts for the transition between initial and final nuclear states. The nuclear matrix element is a result of differences in the nuclear wave functions and thus, differences in the comparative half-life values. The nuclear matrix element is given as, S1 where g is the β-decay strength constant (g = 1.3192), m e is the mass of the electron, c is the speed of light in vacuum, ! is the reduced Planck constant (i.e., h-bar), and ft 1/2 is the comparative half-life. The comparative half-life is a measure of the β-decay probabilities in different nuclei. S1 Finally, the shape factor S(p,q) is used to account for the first forbidden transition of both 90 Sr and 90 Y and is given as, Using the above terms, the spectrum for both 90 Sr and 90 Y can be calculated. Since the two parent-daughter radioisotopes are in secular equilibrium, the cumulative spectrum can be evaluated by simply adding the normalized individual spectra. This culminates in Fig. 1(a).

Section 2: Efficiency of energy collection of a semi-infinite semiconductor material
As the β-particles penetrate into the semi-infinite semiconductor material, the β-particles dissipate their energy by generating electron-hole pairs and Bremsstrahlung x-rays. Being a material with low-Z value, the amount of Bremsstrahlung produced in c-Si is small (~0.43%); this is further addressed in Section 3. Furthermore, on account of the semi-infinite material, the energy dissipation mechanisms are limited to electron-hole generation and energy transfer to phonons. Ignoring recombination losses, the efficiency of energy collection η is then given by, S3 where η dp is the ratio of the open circuit voltage to the bandgap, η d represents the fraction of incident energy deposited in the depletion region, η pp is the efficiency of electron-hole pair production, and FF is the fill factor of the semiconductor device. In the limit of a perfect semiinfinite device (i.e., FF=1, η dp =1, η d =1) a theoretical maximum efficiency of energy collection can be defined as, Considering the cumulative number distribution of β-particles, N cum (E β ), emitted by 90 Sr + 90 Y, the total energy of the particles is given by, Among the β-particles, those with energies greater than the bandgap could contribute to electronhole pair production. On average, it costs the particles the mean ionization energy to generate the pair. 4 For a material with bandgap E g and mean ionization energy E ion , the energy collected as electron-hole pairs from the incident β-particles is given as, The approximation is valid because the minimum energy of the cumulative β-spectrum of 90 Sr and 90 Y is orders of magnitude larger than the bandgap. The mean ionization energy, as stated by Klein, is the sum of the energy contributed to generating the electron-hole pair (1E g eV), energy lost to acoustic phonons and optical phonons in the pair production process (1.8E g eV and 0.5 eV, respectively). S4

Section 3: Penetration depth, secondary electrons and Bremsstrahlung
Utilizing the energy spectrum, the penetration depth of the beta electrons can be calculated to determine the amount of energy deposited within the device as a function of device thickness. The first step in developing this relationship is to examine the interactions of an electron at a discrete energy as it travels through a material. The two primary modes of interaction are direct collisions and radiative emissions or Bremsstrahlung. The penetration depth is obtained in terms of the stopping power, -dE/dx, as given below S5 which is a function of the collisional stopping power, (-dE/dx) col , and the stopping power associated with radiative emissions, (-dE/dx) rad . The Constant Slowing Down Approximation assumes that charged particles slow continuously with all its energy deposited locally along the particle paths. S1 In this approximation and for low radiative yield (which is true for β-particles through c-Si), (-dE/dx) rad and (-dE/dx) col are respectively given as, S5 where Z is the atomic number of the material through which the β-particle is traversing, W β is the total energy of the β-particle, I is the mean excitation energy for silicon, k 0 is the Coulomb constant, e is the charge of an electron, n e is the number of electrons in the substrate per unit volume, m e is the mass of an electron, c is the speed of light and v is the relativistic speed of the β-particle. F -(v) is the function, S5 ( ) The CSDA approximation treats any energy lost by the β-particle via collisions to be infinitesimal compared to the energy of the β-particle. S6 However, the CSDA approximation does not account for the generation of secondary electrons. In our case, the incident β-particles are several orders of magnitude more energetic than the bandgap and thus, secondary electrons are crucial to determining the penetration depth. The concept of secondary electrons relies on the high kinetic energies of the incident β-particles. Upon collisions with electrons, the β-particles are capable of imparting as much as half of their energy to the electrons (several orders of magnitude greater than the bandgap). These energetic electrons are now capable of generating their own electron-hole pairs or more energetic electrons. This type of collisions leads to a "particle shower". The probability of collisions P m (E, E loss ) by an electron of energy E leading to an energy loss E loss , is given by the Møller's relativistic equation, which also accounts for the generation of secondary electrons. S6 As per the definition of a secondary electron, E loss ≤ E/2. Both the previous terms are expressed in mc 2 units. This expression is one of the main deviations from the CSDA. We adopted Spencer and Fano's treatise S6 on the energy spectrum of secondary electrons for an arbitrary β-particle (Fig. S1). We convolved this treatment with the β-spectrum of 90 Sr+ 90 Y which resulted in Fig. 1(d). The secondary electron spectrum also represents the average track length of the electron i.e., the distance travelled by an electron as its energy varies from E+dE to E. A probabilistic weighting of all the track lengths associated with the 90 Sr+ 90 Y β-spectrum allows us to determine the energy deposition in c-Si and 90 Sr, as illustrated in Fig. 1(b).
To calculate the amount of Bremsstrahlung generated, we used the penetration depth associated with radiation emission as given in Eqn. 11. The percentage of radiation yield Y, is given as, S5 4 4 (6 10 ) 1 (6 10 ) S13) Figure S1. Normalized energy spectrum of electrons generated in c-Si for different incident β-particle energies. The area between a colored line and the black CSDA plot denotes the surplus secondary electrons generated in c-Si for the specified energy of incident β-particles. The y-axis, in addition to representing the average track length an electron travels as its energy decreases from E+dE to E, also describes the spectrum of the electrons generated by a mono-energetic source of β-particles.
where Z is the atomic number of the material through which the β-particle with kinetic energy E β is traversing. Figure S2 shows that β-particles generated from the source material and penetrating into 90 Sr would generate Bremsstrahlung radiation worth 1.15% of the energy of the β-particles. However, through c-Si, the same value is only 0.43%. The remaining interactions are dominantly collisional, which result in the generation of lower energy secondary electrons and then ultimately, in electron-hole pair generation within the semiconductor. Sr are only 0.43% and 1.15% of the total energy of the β-particles emitted, respectively. Si is a low-Z material and therefore produces less Bremsstrahlung compared to 90 Sr, which is a high-Z material.

Section 4: UCEP cell parameters
Table S1 below shows the specifications of the UCEP cells used in the experiment as determined from photovoltaic characterization.