Comparison of time-related electrical properties of PN junctions and Schottky diodes for ZnO-based betavoltaic batteries

Comparison of time-related electrical properties of PN junctions and Schottky diodes for ZnO-based betavoltaic batteries 1 first-author https://orcid.org/0000-0002-2193-6049 1 corresp Corresponding author, ljb@jlu.edu.cn Corresponding author, ljb@jlu.edu.cn ljb@jlu.edu.cn 1 1 1 https://orcid.org/0000-0002-9079-5366 2 1 College of Physics, Jilin University, Changchun 130012, China College of Physics, Jilin University, Changchun 130012, China East China University of Technology, Nanchang 330013, China East China University of Technology, Nanchang 330013, China Schottky diodes and PN junctions were utilized as energy converting structures in ZnO-based betavoltaic batteries, in which 0.101121 Ci 63 Ni was selected as the beta source. The time-related electrical properties were obtained using Monte Carlo simulations. For the n-type ZnO, the Pt/ZnO Schottky diode had the highest energy conversion efficiency, and the Ni/ZnO Schottky diode had the largest I sc . The overall electrical performance of PN junctions is better than that of Schottky diodes. The lifetimes of Pt/ZnO and Ni/ZnO are longer than for other Schottky devices, coming close to those of PN junctions. Considering that Schottky diodes are easier to fabricate and independent of p-type semiconductors, Pt/ZnO and Ni/ZnO diodes offer alternatives to PN-junction-based betavoltaic batteries. Beta voltaic effect Zinc oxide Time -related properties PN junction Schottky diode Monte Carlo simulation 1 Introduction 1 MEMS (microelectromechanical systems) is a fabrication technology by which a microfabricated miniature electromechanical structure can be added to an integrated circuit (IC) [ 1 1 ]. These devices have both mechanical and electronic parts, with size ranging from a few micrometers to millimeters. Their functions include sensing, control, and execution at micro scale, and have the ability to impact macroscopic domains [ 2 2 ]. However, the conventional power supplies used with MEMS have disadvantages. For example, chemical batteries have low energy density and short lifetime, solar cells cannot work properly in conditions without steady illumination [ 3 3 ], and fuel cells or other devices using fossil fuels, must replenish the liquid fuel supply while eliminating byproducts formed inside the electronics [ 4 4 ]. Under these conditions, nuclear micro batteries, especially betavoltaic batteries, have become promising candidates to supply power to MEMS. A nuclear battery has very high energy density (the total ene rgy content per unit mass): approximately 10 2 to 10 4 times higher than that of chemical or fossil fuels [ 5 5 ], while the power density is lower than that of conventional batteries. Based on semiconductor technology, some kinds of nuclear batteries can be miniaturized and integrated easily. Considering the long lifetimes of isotopes, and the spontaneity and stability of the decay process, such batteries can serve well under extreme conditions for many years. Generally, the domains of operation of a nuclear battery include long duration, no maintenance, low power, and high-energy power supply. The first nuclear battery was made by Mosely in 1913, which directly used the charges of beta particles (electrons) to derive a current [ 6 6 ]. Subsequently, the first study of a betavoltaic battery was conducted in 1951, by Ehrenberg [ 7 7 ]. This was followed by the first betavoltaic battery (fabricated by Rappaport in 1953) using 90 Sr/ 90 Y with a semiconductor junction. The efficiency obtained was 0.4% [ 8 8 ]. Since the 1960s, several experiments involving betavoltaic batteries have been carried out, some of which were used in pacemakers [ 9 9 ][ 10 10 ]. However, considering the high cost and the toxicity of the contaminating isotope 146 Pm, the nuclear batteries used in pacemakers were replaced with safer lithium-ion batteries [ 11 11 ]. In the 1990s, due to the continuous development of MEMS, betavoltaic batteries eventually became a major alternative among micro power sources. Various radioisotopes, semiconductors, and energy converting structures have been simulated and tested to gain additional output power. The candidate isotopes include 147 Pm [ 9 9 ], 35 S [ 12 12 ], 3 H [ 13 13 ], 63 Ni [ 3 3 ][ 14 14 ], 90 Sr [ 11 11 ], and 14 C [ 15 15 ]. The two main energy converting structures are PN junctions and Schottky devices. With a planar conversion structure, a large portion of the energy is wasted because of the angular distribution of emitted particles. Therefore, a nonplanar structure is utilized to increase the contact area between the source and the energy converting portion. This has enabled the used of additional energy [ 16 16 ]. More recently, betavoltaic batteries with nanostructures have also been proposed. One of thse batteries uses one-dimensional ZnO nanowires, and by applying a sandwich structure, the betavoltaic device achieved excellent electrical performance. With a source radiating 10 mCi 63 Ni/Ni, the open-circuit voltage, short-circuit current, and effective energy conversion efficiency was 2.74 V, 18.4 nA, and 27.92%, respectively [ 17 17 ]. The development of the semiconductor industry has also played an important role in promoting betavoltaic battery research. In the 1950s, although Si was the first candidate energy converting material because of its technical maturity, its conversion efficiency was relatively low and it was vulnerable to radiation damage (the radiation threshold is ~200–250 keV; see [ 18 18 ]). Since then, a variety of semiconductors have been extensively researched. One of the main research achievements is the determination that the limit of conversion efficiency is a function of the bandgap [ 19 19 ]; specifically, that a betavoltaic battery using wide-bandgap semiconductors has higher conversion efficiency. Meanwhile, the open-circuit voltage and short-circuit current have also become higher, and new conductors are more radiation tolerant [ 20 20 ]. Consequently, third-generation semiconductors have become the focus of more recent research. SiC [ 3 3 ] [ 20 20 ][ 21 21 ][ 22 22 ] and GaN [ 23 23 ][ 24 24 ], in particular, are the most popular objects of research utilizing both Monte Carlo simulation and experimental methods. However, the growth of both materials is expensive and occurs at high temperature; therefore, a lower-cost semiconductor is preferred. Until now, there have been many studies about PN junction or Schottky diode-based betavoltaic batteries individually, but studies comparing electrical properties using these two structures on the same semiconductor are very few. Tarelkin et al. compared different metals in a Schottky barrier betavoltaic battery [ 25 25 ] and Rahmani compared Si-based PN junctions and Ni/Si Schottky barriers [ 26 26 ]. The performance of Schottky diodes is determined by the properties of the metal and the semiconductor; therefore, to obtain an extensive view of the comparison, the Schottky contact should be formed using the same semiconductor with different metals. Thereafter, different Schottky diodes and PN junctions can be comparable. Unfortunately, such an extended comparison is still lacking. For an advanced material, the doping method may not be mature enough, but this comparison is necessary to determine the potential performance of different devices and to find the best possible technology for fabricating betavoltaic batteries. Moreover, although it is well known that radioisotopes decay incessantly, the influence of time on the performance of betavoltaic batteries has not been explored. In fact, the decay process impacts the electrical performance profoundly, especially for isotopes with gaseous products (i.e., 3 H), cascade decays (i.e., 90 Sr- 90 Y), obvious density changes (i.e., 147 Pm), or short lifetimes (i.e., 35 S). In this paper, a cost-effective third-generation (i.e., wide bandgap) semiconductor (ZnO) was utilized as the energy converting material of a betavoltaic battery. PN junctions and Schottky diodes with different metals were also used as energy converting structures, and 63 Ni was selected as the beta source. To utilize fully the energy produced, the doping concentrations were modulated to match the depletion width and energy deposition range. According to the decay formula of 63 Ni, we analyzed the factors that influence the self-absorption effect and energy deposition. Using the Monte Carlo method, the time-related self-absorption process of the source and energy depositions were explored. Finally, the time-related electrical properties for all the devices designed in this work were obtained and compared. 2 Research Status on the Properties and Fabrication of ZnO Devices 1 ZnO is a material with great potential for use in the semiconductor industry. It is a direct bandgap semiconductor with a wide gap (3.37 eV) and is among the third-generation semiconductors (with SiC and GaN). Because of its excellent properties, it has been applied as a piezoelectric material, in varistors, transmittance conductive oxide film, fluorescent material, and in many other applications [ 27 27 ]. The direct and wide band gap enables optoelectronic applications in the blue/UV region, and for excitonic-effects-based optical devices, its high exciton binding energy makes it a favorable candidate. In comparison with GaN, the growth of ZnO is much easier. The large and high-quality ZnO bulk substrate is commercially available [ 28 28 ]. Moreover, by a pplying the homoepitaxial growth method, ZnO can be grown on native substrates, which can reduce the defect concentration and enhance film quality. In contrast, for GaN there are no native substrates. This means that a ZnO-based device of high quality could surpass the efficiencies obtained using GaN-based devices. Moreover, ZnO can be grown at much lower temperature, and the wet chemical etching method is available for it, which mean that the processing, design, and integration of electronic and optoelectronic devices would be more flexible [ 29 29 ]. For a betavoltaic battery, the conversion structure must work for a long time in radiation, therefore the radiation resistance of a material is of great importance. In fact, radiation damage is one of the major factors leading to degradation of betavoltaic batteries [ 30 30 ][ 31 31 ], but the extraordinary radiation hardness of ZnO can meet this challenge easily. Under bombardment from high-energy electron flux, no permanent defects are formed in the ZnO lattice [ 27 27 ], which result surpasses those with other common semiconductor materials, including Si, GaAs, CdS, and GaN. This indicates that ZnO devices would be very suitable for space applications [ 27 27 ] [ 29 29 ]. In addition, ZnO is inexpensive, nontoxic, has high breakdown strength, and good chemical stability, which make it a favorable semiconductor for use in radiation tolerant electronics [ 28 28 ]. Two electronic devices important for betavoltaic batteries are PN junctions and Schottky diodes, of which the former has been thoroughly researched. The devices used as the energy converting structure of a nuclear battery should have high performance and long-term stability because they would operate for a long time in radiation. Often, wide bandgap semiconductors suffer from doping asymmetry problems, which means that a semiconductor can be doped to achieve one type easily, but to obtain a contrary type is rather difficult. ZnO can easily be doped to achieve n-type, but creation of p-type ZnO has remained a challenge for years. In spite of many difficulties, researchers have succeeded in using various dopants and different growth techniques to grow p-type ZnO and even a ZnO homojunction. All these achievements now make it possible to fabricate ZnO-based betavoltaic batteries [ 32 32 ]. Moreover, the fabrication of ZnO homojunction diodes has also evolved in positive ways, adding to the potential for fabricating ZnO-based nuclear batteries. At present, well-maintained p-type ZnO films show no obvious degradation after a year [ 33 33 ]. By adding ZnMgO asymmetric double barriers, the homojunction performance can be enhanced substantially [ 34 34 ][ 35 35 ]. A series of novel and simpler fabrication methods have also been reported recently. By employing a surface pulsed laser irradiation method, p- and n-type ZnO can be fabricated with only one dopant in a single layer of NZO film [ 36 36 ]. A simple and controllable method at low temperature and in aqueous solution was also put forward, to create an Sb-doped ZnO (SZO)/ZnO film-based homojunction [ 37 37 ]. The wet chemistry method has also been utilized to fabricate ZnO homojunctions that show high rectification ratios [ 38 38 ]. In the mid-1960s, the study of ZnO-Schottky contacts was initiated. In comparison with ZnO homojunctions, the research on understanding and control of Schottky contacts on ZnO are relatively mature, especially for n-type ZnO. Experimental results show that Au, Ir, Pt, Pd, and Ag are the main metals used to provide Schottky contact on n-type ZnO, and the Schottky barrier height and ideality factor varies with different chemical treatments on a surface. The Schottky barrier height and ideality factor are usually within the range 0.4–1.2 eV and 1.0–3.57, respectively. Ideally, to get clean ZnO-metal contacts, the surfaces should be prepared under clean conditions. However, for a long time, several studies had not considered this; hence, the reported barrier heights were inconsistent. Even now, there is still no predictable method [ 39 39 ]. For p-type ZnO, Pd is suitable to form a Schottky contact [ 40 40 ], but generally, this research is still in its infancy. In short, ZnO has definite potential as a commendable energy converting material for use in betavoltaic batteries. 3 Principles for the Use of Schottky Devices and PN Junctions in Betavoltaic Batteries 1 Schematic diagrams of PN junction and Schottky device-based betavoltaic batteries are shown in Fig. 1 Fig. 1 . PN junctions play an extremely important role in electronic and optical applications. By joining together n- and p-type semiconductors, a PN junction is formed at the interface. Because of the existence of a concentration gradient, electrons diffuse into the p-region and holes diffuse into the n-region. An electric field that counteracts the diffusion can be created and this electric field can also sweep away mobile charges and obtain a space-charge region [ 41 41 ]. The movable charge carriers can hardly exist in this region; hence, this region is referred to as the depletion region. In fabricating semiconductor devices, there are metallization techniques by which to form Schottky and Ohmic contacts. A Schottky contact is formed using a metal and a semiconductor, and has a rectifying function similar to that of a PN junction [ 42 42 ]. By selecting the proper metals and doping concentration, a potential energy barrier between metal and semiconductor can be formed, analogous to that of a PN junction. When beta particles are emitted into a semiconductor, their energy will produce electron-hole pairs by ionization or transform the energy to heat. Electron-hole pairs can be produced either inside or outside the depletion region. In the depletion region, there is an electrical field that can separate electron-hole pairs, after which the charge carriers will be driven to produce current. However, outside the depletion region, there are no electrical fields to separate the electron-hole pairs, and instead, most of the charge carriers will recombine or become trapped by defects. Some charge carriers can drift into the depletion region, where they become part of the cell current [ 43 43 ]. 4 Choice of Isotope and Corresponding Time-Related Behavior 1 Usually pure beta emitters are utilized for betavoltaic batteries, because beta particles are easier to shield than is gamma radiation; thus, the most common isotopes considered are 3 H, 35 S, 63 Ni, 147 Pm, and 90 Sr. Among these isotopes, gaseous 3 H is inconvenient to store and load, and its energy is relatively low (average β energy 5.682 keV). The 147 Pm source has relatively short half-life (approximately 2.62 a), which is unsuited to long-term, no-refill situations. Moreover, the 147 Pm used in betavoltaic batteries usually contains the impurity 146 Pm, which is toxic and emits energetic gamma rays (0.75 MeV). The half-life of 35 S is only 87.37 d, even shorter than that of 147 Pm. The strontium isotope ( 90 Sr) and its product 90 Y emit high-energy beta particles (average β energy: 195.8 keV for 90 Sr and 933.7 keV for 90 Y). However, to utilize fully the energy released, the device would have to be large and would undergo serious radiation damage. For these reasons, 63 Ni is preferred for its solid state, moderate energy (average β energy 17.425 keV), and relatively long half-life (101.2 a). However, for an actual solid-state source, the beta particles can be absorbed by the source itself (the self-absorption effect), which is inevitable. This effect may cause a rise in temperature, energy waste, and performance degradation. Moreover, the source decays continuously, which results in decrease of the radioactivity and changes the components in the source. Consequently, the regular pattern of the self-absorption effect is also changed. By choosing the proper level of radioactivity, the self-absorption effect can be reduced. In this work, the impact of time on the self-absorption effect was also taken into consideration. The MCNP5 code was utilized in this work, and the number of particles in all the simulating processes was 10 7 . In this simulation, a 1×1 cm rectangular source with changeable height was used as the energy depositing model, and the self-absorption rate <math id="M1"><mi>η</mi></math> was defined as: <math display="block" id="M2"><mrow><mi>η</mi><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mrow><mtext>in</mtext></mrow></msub></mrow><mrow><msub><mi>P</mi><mrow><mtext>total</mtext></mrow></msub></mrow></mfrac><mo>,</mo></mrow></math> where <math id="M3"><mrow><msub><mi>P</mi><mrow><mtext>in</mtext></mrow></msub></mrow></math> is the deposition power in the source, and <math id="M4"><mrow><msub><mi>P</mi><mrow><mtext>total</mtext></mrow></msub></mrow></math> is the total power. Fig. 2a Fig. 2a depicts the surface radioactivity and self-absorption rate at initiation, in relation to thickness. With increasing thickness of the source, there is less increase of the surface power, while the self-absorption rate can be very high (approaching 100%). This means that the extra thickness of the source makes no contribution to increase in the surface power. At 2.00337 μm, the surface power is almost unchanged, and the self-absorption rate is comparatively low; thus it was chosen as the proper thickness. The relevant absolute radioactivity, surface power, and self-absorption rate was 0.101121 Ci, 3.38845 μW, and 67.470%, respectively. The decay formula of 63 Ni is <math id="M5"><mrow><mmultiscripts><mtext>N</mtext><mprescripts/><mrow><mtext>28</mtext></mrow><mrow><mtext>63</mtext></mrow></mmultiscripts><mtext>i</mtext><mo>→</mo><mmultiscripts><mtext>C</mtext><mprescripts/><mrow><mtext>29</mtext></mrow><mrow><mtext>63</mtext></mrow></mmultiscripts><mtext>u</mtext><mo>+</mo><msup><mi>e</mi><mo>−</mo></msup><mo>+</mo><msub><mover accent="true"><mi>ν</mi><mo>¯</mo></mover><mtext>e</mtext></msub></mrow></math> . Eventually, the 63 Ni source changes into a mixture of 63 Ni and 63 Cu. In this process, the product 63 Cu is also solid, and there are no other reactions that cause decrease in the radioactivity. For the 2.00337 μm 63 Ni source mentioned above, the changes of time-related surface power and self-absorption rate are shown in Fig. 2b Fig. 2b . As time goes by, the surface power-decline approximately follows the exponential decay law, while the self-absorption rate rises slightly. This is because 63 Ni and its decay product 63 Cu have almost the same density, and the difference in their atomic numbers is very small, which means it has little effect on the regular pattern of surface power decrease. The rise of the self-absorption rate is caused by a rise in the average electron number. 5 Choice of Metals for Schottky and Ohmic Contacts 1 For this work, n-type ZnO was selected to obtain the theoretical electrical behavior of Schottky devices because the fabrication of n-type ZnO is easier, and there are more studies about Schottky barriers on n-type ZnO. The influence of the surface state is neglected in the simulation. Theoretically, when a metal is brought into intimate contact with a semiconductor, whether it is an Ohmic or Schottky contact is defined by the work function of the metal and electron affinity of the semiconductor [ 39 39 ]. This can be expressed by the Schottky-Mott rule [ 44 44 ]: <math display="block" id="M6"><mrow><msub><mi>ϕ</mi><mtext>B</mtext></msub><mo>=</mo><msub><mi>ϕ</mi><mtext>m</mtext></msub><mo>−</mo><mi>χ</mi><mo>.</mo></mrow></math> This form is only available for Schottky barriers formed on n-type semiconductors, where <math id="M7"><mrow><msub><mi>ϕ</mi><mtext>B</mtext></msub></mrow></math> is the Schottky barrier height, which is defined as the difference in electrical potential between the Fermi energy of the metal and the band edge where the majority carrier resides. The term <math id="M8"><mrow><msub><mi>ϕ</mi><mtext>m</mtext></msub></mrow></math> is the metal work function, which is the energy needed to move an electron from the Fermi level to vacuum. Here, χ is the electron affinity of n-type ZnO (4.1 eV), which is the difference in potential between the conduction-band edge and the vacuum level. The densities, work functions of metals, and ideal Schottky barrier heights are presented in Table 1 Table 1 . To form a Schottky contact, the Schottky barrier height should be larger, and the doping concentration should be lower, than in the conduction band or valence band density of states. For n-type ZnO, metals with higher <math id="M9"><mrow><msub><mi>ϕ</mi><mtext>m</mtext></msub></mrow></math> are appropriate (i.e., Au, Pd, Ni, and Pt) and the conduction band density of states limits the doping concentration. 1 1 Metal 1 1 Al 1 1 Ag 1 1 Ti 1 1 Au 1 1 Pd 1 1 Ni 1 1 Pt 1 1 Density/g·cm -3 1 1 2.7 1 1 10.5 1 1 4.51 1 1 19.32 1 1 12.02 1 1 8.9 1 1 21.45 1 1 φ m (eV) 1 1 4.28 1 1 4.26 1 1 4.33 1 1 5.1 1 1 5.12 1 1 5.15 1 1 5.64 1 1 φ B (eV) 1 1 0.18 1 1 0.16 1 1 0.23 1 1 1.0 1 1 1.02 1 1 1.05 1 1 1.54 Table 1 Table 1 can also be used to select metals to produce electrodes. Another metal-semiconductor contact, namely the Ohmic contact (which has relatively low contact resistance to the semiconductor bulk or series resistance), is appropriative for fabricating electrodes. In an actual Ohmic contact with gratifying performance, there should not be obvious performance degradation of devices, and the voltage drop should be negligible in relation to the drop across the device’s active region. An ideal Ohmic contact is formed when the barrier is zero, and in this condition the carriers are free to flow in or out of the semiconductor to ensure that the contact has minimum resistance [ 39 39 ]. For an n-type ZnO, the work function of the selected metal should be close to or smaller than the electron affinity of ZnO. It can be seen that Al, Ag, and Ti have relatively low work functions, indicating that they are adequate for this kind of contact. To form an Ohmic contact on ZnO with low contact resistivity, increase of the ZnO surface doping concentration is beneficial, along with reducing the Schottky barrier height. By applying these two methods, the barrier width becomes very thin, and the carriers are able to tunnel through. 6 Energy Deposition 1 The simulation model is presented in Fig. 3 Fig. 3 . ZnO bulk (1×1×0.5 cm 3 ) was utilized for energy deposition. In this model, a layer of 200 nm is presented that refers to the dead layer (in which the deposited energy cannot be utilized). This includes the p-type region for the PN junction and metal layer for the Schottky diodes. The source was set on the bottom surface of the dead layer, and all particles were vertically incident. The energy deposition at initiation (time zero) in the dead layer and effective area (from the upper surface of the metal layer/p-type ZnO layer to the area of stratification) are presented in Fig. 4a Fig. 4a . For all kinds of devices, more than 99% of the energy was deposited 6 μm from the upper surface of the source to the stratification area; therefore, the effective area was set. To utilize fully the energy produced, the width of the depletion layer was set at the same thickness as the effective area, which was 5.8 μm. Within the dead layer, the energy deposition remained almost unchanged throughout the simulations. Fig. 4b Fig. 4b shows the change of time-related energy deposition, which has a slight downward trend in both the effective area and the dead layer. The energy deposition comparisons of Schottky diodes and PN junctions are also shown in Fig. 4 Fig. 4 . In the dead layer, energy deposition takes the order E Pt > E Au > E Pd > E Ni > E ZnO , while in the effective area, the situation is completely opposite. Clearly, from the energy deposition point of view, Ni is the most suitable metal for a Schottky diode among these four metals because it has relatively lower density (shown in Table 1 Table 1 ). This leads to less energy deposited in the metal itself and more in the effective area. Moreover, a ZnO homojunction has the most energy deposition in the effective area among all these devices because ZnO has the lowest density (5.6 g/cm 3 ). 7 Analyses of Built-in Potential and Depletion Width in Schottky Diodes and PN Junctions 1 In a betavoltaic battery, electron-hole pairs are separated by the potential built into the depletion region. To exploit the beta source energy, the depletion width and the energy deposition range should be well matched. The depletion width can be changed by adjusting the doping concentrations. For Schottky diodes, the built-in potential is given by the following relation [ 20 20 ]: <math display="block" id="M10"><mrow><msub><mi>V</mi><mrow><mtext>bi</mtext></mrow></msub><mo>=</mo><msub><mi>ϕ</mi><mtext>B</mtext></msub><mo>−</mo><msub><mi>V</mi><mtext>t</mtext></msub><mi>ln</mi><mfrac><mrow><msub><mi>N</mi><mtext>c</mtext></msub></mrow><mrow><msub><mi>N</mi><mtext>D</mtext></msub></mrow></mfrac><mo>,</mo></mrow></math> where V t is the thermal voltage, and V t = 0.0259 V at ambient temperature. Here, <math id="M11"><mrow><msub><mi>N</mi><mtext>c</mtext></msub></mrow></math> is the conduction band effective density of states, and N D is the doping concentration of n-type ZnO (cm -3 ). At zero bias, the depletion width is given by [ 20 20 ]: <math display="block" id="M12"><mrow><mi>W</mi><mo>=</mo><msqrt><mrow><mfrac><mrow><mn>2</mn><msub><mi>ε</mi><mtext>s</mtext></msub><msub><mi>ε</mi><mn>0</mn></msub><msub><mi>ϕ</mi><mtext>i</mtext></msub></mrow><mrow><mi>q</mi><msub><mi>N</mi><mtext>D</mtext></msub></mrow></mfrac></mrow></msqrt><mo>,</mo></mrow></math> where <math id="M13"><mrow><msub><mi>ε</mi><mtext>s</mtext></msub></mrow></math> is 8.75, the relative permittivity of ZnO, <math id="M14"><mrow><msub><mi>ε</mi><mn>0</mn></msub></mrow></math> is the vacuum permittivity, and <math id="M15"><mi>q</mi></math> is the electron charge. In equilibrium condition, the built-in potential for PN junctions is given by the following relation [ 45 45 ]: <math display="block" id="M16"><mrow><msub><mi>V</mi><mrow><mtext>bi</mtext></mrow></msub><mo>=</mo><msub><mi>V</mi><mtext>t</mtext></msub><mo>⋅</mo><mi>ln</mi><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>N</mi><mtext>A</mtext></msub><msub><mi>N</mi><mtext>D</mtext></msub></mrow><mrow><msubsup><mi>n</mi><mtext>i</mtext><mtext>2</mtext></msubsup></mrow></mfrac></mrow><mo>)</mo></mrow><mo>.</mo></mrow></math> Here, <math id="M17"><mrow><msub><mi>N</mi><mtext>A</mtext></msub></mrow></math> and <math id="M18"><mrow><msub><mi>N</mi><mtext>D</mtext></msub></mrow></math> are doping concentrations of the p and n-types (cm -3 ). At normal atmospheric temperature, they approximate the carrier intensities, and <math id="M19"><mrow><msub><mi>n</mi><mtext>i</mtext></msub></mrow></math> is the intrinsic carrier concentration. The depletion width of a PN junction can be calculated using [ 45 45 ]: <math display="block" id="M20"><mrow><mi>W</mi><mo>=</mo><msqrt><mrow><mfrac><mrow><mn>2</mn><msub><mi>ε</mi><mtext>s</mtext></msub><msub><mi>ε</mi><mn>0</mn></msub></mrow><mi>q</mi></mfrac><mo>⋅</mo><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>N</mi><mtext>A</mtext></msub><mo>+</mo><msub><mi>N</mi><mtext>D</mtext></msub></mrow><mrow><msub><mi>N</mi><mtext>A</mtext></msub><msub><mi>N</mi><mtext>D</mtext></msub></mrow></mfrac></mrow><mo>)</mo></mrow><msub><mi>V</mi><mrow><mtext>bi</mtext></mrow></msub></mrow></msqrt><mo>.</mo></mrow></math> Theoretically, the intrinsic carrier concentration can be expressed by: <math display="block" id="M21"><mrow><msub><mi>n</mi><mtext>i</mtext></msub><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><msub><mi>N</mi><mtext>c</mtext></msub><msub><mi>N</mi><mtext>v</mtext></msub><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mi>exp</mi><mo stretchy="false">(</mo><mo>−</mo><mfrac><mrow><msub><mi>E</mi><mtext>g</mtext></msub></mrow><mrow><mn>2</mn><msub><mi>k</mi><mn>0</mn></msub><mi>T</mi></mrow></mfrac><mo stretchy="false">)</mo><mo>.</mo></mrow></math> The term E g is the band gap of ZnO, and <math id="M22"><mrow><msub><mi>N</mi><mtext>c</mtext></msub></mrow></math> and <math id="M23"><mrow><msub><mi>N</mi><mtext>v</mtext></msub></mrow></math> are effective state densities of the conduction and valence bands, which can be calculated using <math id="M24"><mrow><msub><mi>N</mi><mtext>c</mtext></msub><mo>=</mo><mn>2</mn><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>π</mi><msubsup><mi>m</mi><mi>n</mi><mo>*</mo></msubsup><msub><mi>k</mi><mn>0</mn></msub><mi>T</mi></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup></mrow><mrow><msup><mi>h</mi><mn>3</mn></msup></mrow></mfrac></mrow></math> and <math id="M25"><mrow><msub><mi>N</mi><mtext>v</mtext></msub><mo>=</mo><mn>2</mn><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>π</mi><msubsup><mi>m</mi><mtext>p</mtext><mo>*</mo></msubsup><msub><mi>k</mi><mn>0</mn></msub><mi>T</mi></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup></mrow><mrow><msup><mi>h</mi><mn>3</mn></msup></mrow></mfrac></mrow></math> , where <math id="M26"><mrow><msubsup><mi>m</mi><mtext>n</mtext><mtext>*</mtext></msubsup></mrow></math> and <math id="M27"><mrow><msubsup><mi>m</mi><mtext>p</mtext><mo>*</mo></msubsup></mrow></math> indicate the effective masses of electrons and holes, k 0 indicates the Boltzmann constant, and T is the absolute temperature. In ZnO material, <math id="M28"><mrow><msubsup><mi>m</mi><mtext>n</mtext><mtext>*</mtext></msubsup></mrow></math> and <math id="M29"><mrow><msubsup><mi>m</mi><mtext>p</mtext><mo>*</mo></msubsup></mrow></math> are 0.27 m 0 and 0.64 m 0 , respectively [ 46 46 ]. It was determined that <math id="M30"><mrow><msub><mi>N</mi><mtext>c</mtext></msub></mrow></math> = 3.52×10 18 cm -3 , <math id="M31"><mrow><msub><mi>N</mi><mtext>v</mtext></msub></mrow></math> = 1.28×10 19 cm -3 , and n i = 3.74×10 -10 cm -3 . These equations are only adequate for non-degenerate semiconductors ( N D < < N c and N A < < N v ). Thus, for ZnO material, N D and N A should be no more than 1×10 18 and 5×10 18 cm -3 , respectively. The <math id="M32"><mrow><msubsup><mi>m</mi><mtext>n</mtext><mo>*</mo></msubsup></mrow></math> and <math id="M33"><mrow><msubsup><mi>m</mi><mtext>p</mtext><mtext>*</mtext></msubsup></mrow></math> selected here are a little different from those in our previous work, because we had to match the universally acknowledged value of the effective Richardson’s constant of ZnO [ 46 46 ] (that is, 32 A cm -2 K -2 while <math id="M34"><mrow><msubsup><mi>m</mi><mtext>n</mtext><mtext>*</mtext></msubsup></mrow></math> = 0.27 m 0 ). These differences led to small changes in the values of <math id="M35"><mrow><msub><mi>N</mi><mtext>c</mtext></msub></mrow></math> and <math id="M36"><mrow><msub><mi>N</mi><mtext>v</mtext></msub></mrow></math> ; nonetheless, the doping concentration ranges for non-degenerate semiconductors remained the same. In this work, the depletion width was 5.8 μm. For Schottky diodes, the relationships of V bi and W versus N D are shown in Fig. 5a Fig. 5a . With increase in the doping concentration, the built-in potential increases slightly while the depletion width decreases substantially. For the PN junctions, N D was set at 10 16 cm -3 , and the relationships of V bi and W versus N A are shown in Fig. 5b Fig. 5b . These resemble the regular pattern of a Schottky diode. The selected doping concentrations and built-in potentials are listed in Table 2 Table 2 . 1 1   1 1 PN junction 1 1 Pt/ZnO 1 1 Au/ZnO 1 1 Ni/ZnO 1 1 Pd/ZnO 1 1 Doping concentrations (×10 13 cm -3 ) 1 1 N D =10 3 1 1 N D = 3.57447  1 1 N D = 1.9765  1 1 N D =2.1258 1 1 N D =1.8978 1 1 1 1 N A = 8.44 1 1 1 1 1 1 1 1 1 1 Built-in potential (V) 1 1 2.90913 1 1 1.24224 1 1 0.68690 1 1 0.73878 1 1 0.70585 8 Electrical Performance Simulations 1 In the following calculations, it was assumed that the electron-hole pairs in the depletion region are completely collected, and that outside this region, none of them are collected. With these assumptions, the short-circuit current can be expressed by the following equation [ 45 45 ]: <math display="block" id="M37"><mrow><msub><mi>I</mi><mrow><mtext>SC</mtext></mrow></msub><mo>=</mo><mfrac><mrow><mi>A</mi><mi>q</mi></mrow><mrow><mi>E</mi><mmultiscripts><mrow/><mprescripts/><mtext>e-h</mtext><mi/></mmultiscripts></mrow></mfrac><mo>⋅</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mi>C</mi><mi>E</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></mstyle><mo>×</mo><mi>E</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mn>,</mn></mrow></math> where CE ( n ) is the collection rate of electron-hole pairs in layer n of the sample, E ( n ) is the deposition energy in layer n , A is the absolute radioactivity, q is the electron charge, and E e-h is the average ionization threshold. In Section 6, we obtained the energy deposition in the depletion region, E ( n ), and according to the condition above, the relevant CE ( n ) was 1. In our previous work, we used an empirical equation to estimate E e-h because the universally acknowledged value was not presented [ 47 47 ]: <math display="block" id="M38"><mrow><msub><mi>E</mi><mrow><mtext>e-h</mtext></mrow></msub><mo>=</mo><mn>2.8</mn><msub><mi>E</mi><mtext>g</mtext></msub><mo>+</mo><mn>0.5</mn><mo>,</mo></mrow></math> where E g is the band gap and E e-h as achieved was 9.9 eV. To evaluate device quality, reverse saturation current is often utilized. If a device has low reverse saturation, it has fine quality. For Schottky diodes, the density of a reverse saturation current (A/cm 2 ) can be expressed by [ 20 20 ]: <math display="block" id="M39"><mrow><msub><mi>J</mi><mn>0</mn></msub><mo>=</mo><msup><mi>A</mi><mo>∗</mo></msup><msup><mi>T</mi><mn>2</mn></msup><mi>exp</mi><mo stretchy="false">(</mo><mo>−</mo><msub><mi>ϕ</mi><mtext>B</mtext></msub><mo>/</mo><msub><mi>V</mi><mtext>t</mtext></msub><mo stretchy="false">)</mo><mo>,</mo></mrow></math> where A * is the effective Richardson’s constant. For ZnO, the universally acknowledged value is <math id="M40"><mrow><msubsup><mi>m</mi><mtext>n</mtext><mtext>*</mtext></msubsup></mrow></math> = 0.27 m 0 , A * is 32 A cm -2 K -2 . For PN junctions, it can be determined by [ 47 47 ]: <math display="block" id="M41"><mrow><msub><mi>J</mi><mn>0</mn></msub><mo>=</mo><mn>1.5</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>5</mn></msup><mi>exp</mi><mo stretchy="false">(</mo><mo>−</mo><msub><mi>E</mi><mtext>g</mtext></msub><mo>/</mo><msub><mi>k</mi><mn>0</mn></msub><mi>T</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></math> where k 0 indicates the Boltzmann constant. The open-circuit voltage can be obtained by the following formula: <math display="block" id="M42"><mrow><msub><mi>V</mi><mrow><mtext>oc</mtext></mrow></msub><mo>=</mo><mfrac><mrow><mi>n</mi><msub><mi>k</mi><mn>0</mn></msub><mi>T</mi></mrow><mi>q</mi></mfrac><mi>ln</mi><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><msub><mi>J</mi><mrow><mtext>sc</mtext></mrow></msub></mrow><mrow><msub><mi>J</mi><mn>0</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math> where <math id="M43"><mi>n</mi></math> is the ideal factor. In this work, it is considered to be 1, which indicates an ideal PN junction or Schottky device. Here, <math id="M44"><mrow><msub><mi>J</mi><mrow><mtext>sc</mtext></mrow></msub></mrow></math> is the short-circuit current density. One of the key parameters that evaluates the performance of a betavoltaic battery is the fill factor ( FF ). A battery with higher fill factor has output power that comes closer to its theoretical maximum, thus its efficiency is higher. The definition of the fill factor is the ratio of the maximum output power ( P m ) to the product of the relative open-circuit voltage ( V oc ) and the short-circuit current ( I sc ): <math display="block" id="M45"><mrow><mi>F</mi><mi>F</mi><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mtext>m</mtext></msub></mrow><mrow><msub><mi>V</mi><mrow><mtext>oc</mtext></mrow></msub><mo>⋅</mo><msub><mi>I</mi><mrow><mtext>sc</mtext></mrow></msub></mrow></mfrac><mo>.</mo></mrow></math> To evaluate the fill factor, an empirical equation can be utilized [ 47 47 ]: <math display="block" id="M46"><mrow><mi>F</mi><mi>F</mi><mo>=</mo><mfrac><mrow><msub><mi>v</mi><mrow><mtext>oc</mtext></mrow></msub><mo>−</mo><mi>ln</mi><mrow><mo>(</mo><mrow><msub><mi>v</mi><mrow><mtext>oc</mtext></mrow></msub><mo>+</mo><mn>0.72</mn></mrow><mo>)</mo></mrow></mrow><mrow><msub><mi>v</mi><mrow><mtext>oc</mtext></mrow></msub><mo>+</mo><mn>1</mn></mrow></mfrac><mo>,</mo></mrow></math> where v oc = V oc / ( k 0 T / q ). The definition of maximum output power is as follows [ 47 47 ]: <math display="block" id="M47"><mrow><msub><mi>P</mi><mtext>m</mtext></msub><mo>=</mo><mi>F</mi><mi>F</mi><mo>×</mo><msub><mi>V</mi><mrow><mtext>oc</mtext></mrow></msub><mo>×</mo><msub><mi>I</mi><mrow><mtext>sc</mtext></mrow></msub><mo>.</mo></mrow></math> In this work, we used the device converting efficiency to evaluate the total performance of the betavoltaic battery, which can be defined as below [ 48 48 ]: <math display="block" id="M48"><mrow><mi>η</mi><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mtext>m</mtext></msub></mrow><mrow><msub><mi>P</mi><mrow><mtext>source</mtext></mrow></msub></mrow></mfrac><mo>,</mo></mrow></math> where P source is the surface power of the source. 9 Results and Discussion 1 According to Eqs. (10) and (11), the short-circuit current densities are constants without time-related changes, in theory. The short-circuit current density for a ZnO-based PN junction and for Pt, Au, Pd, and Ni/ZnO Schottky diodes is 4.65055×10 -52 , 4.32989×10 -20 , 4.91208×10 -11 , 2.26936×10 -11 , and 7.12622×10 -12 A/cm 2 , respectively. These current densities indicate that the PN junction has the best quality among all these devices, while the Pt/ZnO Schottky device has the best quality among the Schottky devices. Time-related electrical properties of the PN junction and of Schottky diodes are presented in Fig. 6(a)-(e) Fig. 6(a)-(e) . At the start (time zero), the ZnO-based PN junction has the highest open-circuit voltage, short-circuit current, fill factor, maximum output power, and conversion efficiency (up to 2.64452 V, 0.10259 μA, 94.5366%, 0.25647 μW, and 7.584%, respectively). These are much higher than with other Schottky devices. The reason is that the PN junction has very low reverse saturation current density, which leads to relatively higher V oc according to Eq. (12). Meanwhile, the density of ZnO is lower than all the Schottky metals mentioned above, leading to more energy deposition in the effective area. Among these Schottky diodes, the Pt/ZnO device has the highest V oc , P m , FF , and η due to its highest <math id="M49"><mrow><msub><mi>ϕ</mi><mtext>m</mtext></msub></mrow></math> , which leads to the highest <math id="M50"><mrow><msub><mi>ϕ</mi><mtext>B</mtext></msub></mrow></math> and lowest J 0 . In addition, the Ni/ZnO device has the largest I sc because Ni has the smallest atomic number and density among all the selected metals, which lead to the most energy deposition in the effective area. All the electrical properties (in a battery) decline as time goes by. The declines of I sc and P m follow the exponential law, while the declines of V oc , FF , and η show approximately linear patterns. Among all the electrical properties, the decline of P m is the most evident. Hence, P m was selected to evaluate the validity of the battery concept. Taking the ZnO-based PN junction as an example, in 200 years the decline rates of V oc , I sc , P m , FF , and η would be 1.365, 75.189, 0.065, 75.544, and 0.748%, respectively. While for Pt, Au, Pd, and Ni/ZnO Schottky diodes, the decline of P m becomes more evident (76.523, 81.589, 80.636, and 79.862%, respectively). Thus, from the viewpoint of long-term service, Pt/ZnO and Ni/ZnO Schottky devices are more suitable than PN junctions, whereas Au/ZnO and Pd/ZnO Schottky devices are less adaptable. A Pt/ZnO Schottky diode could be utilized in devices that need high V oc and P m , and a Ni/ZnO Schottky diode would be more suitable for devices needing higher electric current. In short, irrespective of whether the time factor is considered or not, PN junctions have the highest V oc , I sc , P m , FF , and η (that is, the best electrical performance) among all the competitors discussed here. Regardless of long-term radiation damage, PN junction-based devices also have the longest active period. Nevertheless, Schottky diodes have their own advantages. The fabrication of Schottky diodes requires only one type of extrinsic semiconductor, which is very convenient for semiconductors with asymmetry doping problems (i.e., for GaN and ZnO obtaining n-type semiconductors is easier, while for diamond it is easier to get p-type ones). Compared with PN junction-based devices, Schottky diodes offer additional choices of state-of-the-art semiconductors, and the manufacturing processes are simpler. When fabricating Schottky diode-based betavoltaic batteries, the radiation source is in contact with the metal layer of the Schottky diode, which makes it less susceptible to radiation damage. In contrast, with a PN junction, the semiconductor lattice is prone to displacement due to radiation damage, because the radiation source is in direct contact with the semiconductor. All of these advantages indicate that Schottky diodes are a good alternative to PN junction-based betavoltaic batteries. 10 Conclusion 1 In summary, Schottky contacts using different metals for n-type ZnO and ZnO-based PN junctions were utilized as energy converting structures for betavoltaic batteries, with 0.101121 Ci 63 Ni as the selected beta source. Based on Monte Carlo simulations, the time-related self-absorption rate and electrical properties of a number of devices were obtained and compared. For a 63 Ni source, the self-absorption rate showed a slightly upward trend as time went by. The optimal doping concentration for Schottky diodes was N D = 1.9765–3.57447×10 13 cm -3 , and for PN junctions it was N A = 8.44×10 13 cm -3 while N D = 10 16 cm -3 . Generally, the electrical performance of PN junctions is better than that of Schottky diodes. 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Bin Lu , X.Y. Li , et al ., Theoretical prediction of diamond betavoltaic batteries performance using 63Ni , Chinese Phys. Lett . 35 , 5 - 9 ( 2018 ). https://doi.org/10.1088/0256-307X/35/7/072301 https://doi.org/10.1088/0256-307X/35/7/072301 . Theoretical prediction of diamond betavoltaic batteries performance using 63Ni 10.1007/s41365-020-0723-y This work was supported by the National Major Scientific Instruments and Equipment Development Projects (No. 2012YQ240121) and National Natural Science Foundation of China (No.11075064). 2019-10-10 2019-12-04 2019-12-16 2020-02-03 2020-02-01 2021-04-10 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2020 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-020-0723-y http://dx.doi.org/10.1007/s41365-020-0723-y . 2020 Nuclear Science and Techniques 2 31