High efficiency 4H-SiC Betavoltaic Power Sources using Tritium
Radioisotopes
Supplemental Information
Christopher Thomas1, Samuel Portnoff 1, and M. G. Spencer2
1
2
Widetronix Corp, Ithaca, New York, 14850 USA
Dept. of Electrical and Computer Engineering, Cornell University, Ithaca, New York, 14850 USA
A. Calculation of Beta Spectrum
The Fermi approximation is used to calculate the beta spectrum1
4𝜋
(S1) 𝜂𝛽 (𝜖)𝑑𝜖 = 137
𝜖
4𝜋
1
(𝜖 2 −1)2
{1 − 𝑒𝑥𝑝 (− 137
𝜖
1
(𝜖 2 −1)2
)} ((𝜖 2 − 1))𝜖(𝜖𝑚𝑎𝑥 − 𝜖)2 𝑑𝜖
where 𝜖 =kinetic + rest energy of emitted beta particle and 𝜖𝑚𝑎𝑥 =max kinetic energy +rest
energy of the emitted beta particle (in units of rest energy). Substituting 𝐸 ′ for 𝜖 ′ where 𝐸 ′ =1+ 𝜖
/moec2 and moe is the mass of an electron 𝐸 ′ 𝑚𝑎𝑥 =1+ 𝜖 max/ moec2
2
(S2) 𝜂𝛽 (𝜖𝛽 )𝑑𝜖𝛽 = 𝐶𝑁 𝑓(𝑍, 𝐸 ′ )(𝐸 ′ − 1)𝐸 ′ (𝐸 ′ 𝑚𝑎𝑥 − 𝐸 ′ )2 𝑑𝜖𝛽
where f is the Fermi function. The coefficient CN is calculated so that the distribution function is
normalized
𝜖
(S3) 1 = ∫0 𝛽𝑚𝑎𝑥 𝜂𝛽 (𝜖𝛽 )𝑑𝜖𝛽
The average value of energy is calculated by
𝐸 ′ 𝑚𝑎𝑥
(S4) 〈𝐸 ′ − 1〉𝐴𝑣𝑒 = ∫1
(𝐸 ′ − 1)𝜂𝛽 (𝜖𝛽 )𝑑𝜖𝛽
Fig S1 (left) shows the calculated beta spectrum for 1ML of TiH32 while (right) shows the beta
spectrum for .4 microns of TiH32 note the shift in the position of the maximum energy when
multiple monolayers are used. This shift is due to self-absorption and will be discussed in B.
1
1.5 10
-3
Probability (arb. units)
Beta Distribution for
one ML Hydride Thickness
1 10
-3
5 10
-4
Beta Distribution
Hydride Thickness .4u
0
0
5
10
15
20
Beta Energy (keV)
Figure S1 Calculated beta distribution function for one monolayer of TiH 32 and for TiH32 of thickness .4 microns
B. Calculation of Self Absorption in TiH32
The maximum absorption range of electrons in TiH32 and SiC was calculated using the
relationship given by Everhart 2
4
(S5) 𝑅(𝐸) = 𝐸01.75 100𝜌
where R(E) is the range in microns, E0 is the incident energy of the electron in keV and 𝜌 is the
density of the TiH32 in g/cm3 . The empirical equations of Kanaya and Okayama3 were used to
calculate the beta energy spectrum from a TiH32 of thickness d. The TiH32 was divided in to n
monolayers
(S6) 𝑛 = 𝑀𝐿
𝑑
𝑇ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠
Where MLThickness is the thickness of one monolayer of TiH32. Individual electrons will transverse
a distance equal to ML*n. For incident energies in the beta spectrum a Matlab program was used
to calculate the reduced energy of the electrons originating from the nth monolayer at the edge of
the foil (S6). These electrons can be either reflected, absorbed or transmitted using (S7) to
calculate the transmission percentage at the edge of the TiH32.
2
(S7) 𝐸𝑛 = (1 − 𝑦)2/3 𝐸0
where y is the reduced depth x/R(E0) and E0 is the starting energy of the electron from the nth
monolayer
𝛾𝑦
(S8) 𝜂𝑇 = exp(− 1−𝑦)
2
Where 𝛾 = .187𝑍 3
and Z is the atomic number of the absorbing material
Fig S2 shows the calculated beta spectrum for several thickness of TiH32 100% conversion of the
titanium is assumed. The average beta energies and total particle flux are computed for each
thickness, notice the shift of average energy to higher values as the thickness increases due to
absorption of the lower energy betas. Also note the saturation in particle flux for thickness
greater than .4um. The saturation in particle flux is because .4um is approximately the maximum
range for tritium betas
8.5
7 108
7.5
6 10
8
5 10
8
4 10
8
-2
Beta Flux (cm )
Avg Beta Energy (keV)
8
7
6.5
3 108
2 108
6
1 10
5.5
8
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Titanium Hydride Thickness (microns)
0
0.1
0.2
0.3
0.4
0.5
0.6
Titanium Hydride Thickness (Microns)
Figure S2a,b (left) Average energy of electrons leaving foil of TiH32as a function of foil thickness. (right)
Flux of electrons leaving foil of TiH32 as a function of foil thickness
C. Measurement of Input Power
The input power is calculated from (S9)
(S9) 𝑃𝑖𝑛𝑝𝑢𝑡 = 𝐽𝛽 〈𝜖𝛽 〉 where 𝐽𝛽 is the beta flux.
3
The beta flux is measured using a surface activity monitor adapted from Liu4 . The measured
data from an example foil is plotted in Fig 3. The voltage on the plates is increased until current
saturation occurs. The current at saturation is given by (S10)
(S10) 𝐽𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 = 𝐽𝛽𝑒𝑡𝑎 𝜖
〈𝜖𝛽 〉
𝑖𝑜𝑛𝑖𝑠𝑎𝑡𝑖𝑜𝑛 𝑎𝑖𝑟
where the ionization of air occurs at
ionization
air=33.7eV
Based on these measurements, the saturation current (at 100V) is 7.83na. The area of the foil is
.36cm-2. Pinput is calculated as 732.9nW cm-2. Using a average energy of 8.25keV the beta flux
produced is 5.54x108 cm-2 which correspond to a specific activity of 14.97 mCi. This foil
produced 85% of the max beta flux calculated in B.
Ionization Current (Amps)
1 10
-8
8 10-9
6 10-9
4 10-9
2 10-9
0
0
50
100
150
200
250
Voltage (V)
Figure S3 Air ionization current as a function of applied voltage between electrodes forming an air
gap. Test sample is placed on bottom electrode
D. Correction of Power due to Absorption in Palladium layer
The schematic of our TiH32 layer is shown in Fig 2. I n the main text. Equal numbers of beta
electrons are emitted up as down. However the electrons travelling in an upward direction have
an extra absorption due to the presence of a 20nm thick Pd layer. In our calculation of efficiency
this will cause an underestimation of the electrons traveling in the downward direction. Using
4
the simulation developed in B we calculation the ratio of the output power from electrons
traveling up/ electrons traveling down. The calculated value was .62 this value is used to correct
the efficiency calculation.
E. Calculation of the beta backscattering percentage
The backscatter correction to the calculation of betavoltaic conversion efficiency given by (S11)
(S11)
〈𝐸𝐵 〉
𝜂 = 𝜂𝐵 〈𝐸
𝑀𝑒𝑎𝑛 〉
Where 𝜂𝐵 is the percentage of the total electrons backscattered. Experimentally it has been
determined that the backscattering electron energy is significantly less than the incoming energy.
〈𝐸𝐵 〉
Therefore the backscattering correction term is given by 𝜂𝐵 〈𝐸
𝑀𝑒𝑎𝑛 〉
〈𝐸𝐵 〉
backscattering 〈𝐸
The mean energy loss from
has experimentally been found to be ~.6 for a Z number of 20. From K&
𝑀𝑒𝑎𝑛 〉
S3 the total backscatter percentage is given by (S12)
6
𝑦
(S12) 𝜂𝐵 = 5 ∫0
𝛾𝐵
𝛾𝐵 𝑦
exp (− 1−𝑦
) 𝑑𝑦
(1−𝑦)7/6
6
𝛾 𝑦
𝐵
− 5𝑥25/6 {1 − exp (− 1−𝑦
)}
When (S12) is evaluated at y=1 (the maximum range of the incident electron)
=
6
6
𝛾𝐵 exp(−𝛾𝐵 ) . 6435 −
5
5𝑥25/6
Where 𝛾𝐵 = 1.9𝛾 Table I shows parameters used for the calculation of 𝜂
Material
Z#
𝜸𝑩
𝜼𝑩
𝜼
SiC
20
2.61
.26
.16
Table SI Parameters used for calculation of electron backscatter correction
F. Calculation of Generation Function
Using the beta spectrum found form the TiH32 as input we calculate the electron hole pair
generation as a function of distance using (S13)
5
𝜖
1
(S13) 𝐺(𝑥 ′ ) = ∑0𝛽𝑚𝑎𝑥 𝑆(𝜖𝛽 ) 7.28𝑒𝑉
where
𝑑𝐸𝐴 (𝑥)
𝑑𝑥
𝑑𝐸𝐴 (𝑥)
𝑑𝑥
|
𝑥=𝑥 ′
𝑑𝑥
is the change in energy of the electron due to absorption evaluated at x’ and 7.28eV
is the mean ionization energy in SiC 5 and 𝑆(𝜖𝛽 ) is the spectral function calculated from B.
From the K&S equations
(S14)
𝑑𝐸𝐴 (𝑥)
𝑑𝑥
=
𝜌𝐸0
𝑅
𝑑𝐸𝐴 (𝑥)
𝑑𝑥
is given by (S14)
1
𝛾𝑦
𝛾
3
{(1−𝑦)2/5 𝑒𝑥𝑝 (− 1−𝑦) (1−𝑦 + 5) + 𝐸𝐵
6𝑥1.9
5
𝛾
1.9𝛾𝑦
𝑒𝑥𝑝 (− 1−𝑦 ) −
(1−𝑦)2
1
(25/6 − (1 − 𝑦)5/6 )}
The generation function is shown as Fig. S4 in the text
G Calculation of the steady state carrier concentration and short circuit current density
The steady minority carrier concentration and short circuit current density were calculated by
solving the continuity equation for the region of the device between the edge of the depleted
surface (xsurface
depl)
and the edge of the P+N junction depletion region (xd) for calculation
convenience we set x surface depletion =0. For a PN junction in steady state neglecting drift
transport the continuity equation for the top surface is given by (S15)
(S15) 𝐷𝑛
𝜕(𝑛𝑝 −𝑛𝑝0 )
𝑑𝑥 2
+ 𝐺(𝑥) −
(𝑛𝑝 −𝑛𝑝0 )
𝜏𝑛
=0
Where Dn is the diffusion constant for electrons and 𝜏𝑛 is the minority carrier lifetime and G(x)
is the generation function calculated in F. We approximate G(x) with a Gaussian distribution
and (S15) in terms of constants A, C, D, a, and b becomes (S16)
(S16) 𝐴
𝜕(𝑛𝑝 −𝑛𝑝0 )
𝑑𝑥 2
+ 𝐶𝑒𝑥𝑝 (−
(𝑎−𝑥)2
𝑏
) − 𝐷(𝑛𝑝 − 𝑛𝑝0 )
The close form solution is (S17) (𝑆17) (𝑛𝑝 − 𝑛𝑝0 ) = −√𝜋√𝑏𝐶𝑒𝑥𝑝 (
√𝐷𝑥
√𝐴
) erf(
𝑏√𝐷−2√𝐴(𝑎−𝑥)
2√𝐴√𝑏
1
) 4√𝐴√𝐷 − √𝜋√𝑏𝐶𝑒𝑥𝑝 (
𝑎√𝐷
√𝐴
𝑏𝐷
+ 4𝐴 −
√𝐷𝑥
√𝐴
) erf(
𝑎√𝐷
√𝐴
𝑏𝐷
+ 4𝐴 +
2√𝐴(𝑎−𝑥)+𝑏√𝐷
2√𝐴√𝑏
1
) 4√𝐴√𝐷
6
The equation is solved subject to the boundary conditions (S18) and (S19)
(S18) 𝑛𝑝 = 0 at x=xd
(S19) 𝐷𝑛
𝜕(𝑛𝑝 −𝑛𝑝0 )
= 𝑆𝑛
𝑑𝑥
Where Sn is the effective surface recombination velocity for electrons The simulation calculates
Jn the junction current, JSrec the recombination current flowing to the surface and JBulk the current
due to recombination in the bulk
(S20) 𝐽𝑛 = 𝑞𝐷𝑛
𝜕(𝑛𝑝 −𝑛𝑝0 )
𝑑𝑥
|
𝑥=𝑥𝑑
𝐽𝑆𝑅𝑒𝑐 = 𝑞𝑆𝑛 (𝑛𝑝 − 𝑛𝑝0 )
Results of the calculations for Sn =107 cm/sec for a junction thickness of 2um (left) and .2um
(right) are shown in Fig S4. For Sn=.1 cm/sec
and junction thickness of 2um and using a
minority carrier lifetime of .5-1usec JBulk is not significant. Therefore the maximum junction
current JnMax was set equal to Jn calculated for a junction thickness of 2um (insuring total beta
absorption) and Sn=.1 cm/sec. For junction thicknesses less than 2um and Sn=.1 cm/sec the
difference JnMax - Jn was assumed to be due to absorption of beta’s in the junction depletion
region or the N region. Fig S5 in the text was calculated by (S21)
(S21)
𝐶𝑒𝑓𝑓 =
𝐽𝑛𝑀𝑎𝑥
𝐽𝑛
7
1.5 10
6
2.5 105
-3
Minority Carrier Density (cm )
-3
6
Minority Carrier Concentration (cm )
2 10
1 10
5 10
6
5
2 105
1.5 10
5
1 105
5 10
4
0
0
0
0.5
1
Thickness (microns)
1.5
0
0.05
0.1
0.15
Thickness (microns)
Figure S4 Minority carrier density vs top layer junction thickness (junction depletion
edge-surface depletion edge) for a minority carrier lifetime of .5usec and surface
recombination velocity of 107 cm/sec. (left) junction thickness of 2um (right) junction
thickness of .2um
References
S1. Polymers, Phosphors and Voltaics for Radioisotope Microbatteries edited by Kenneth E.
Bower, Yuri A. Barbanel , Yuri G. Shreter, and George W. Bohnert Chap 2 p55 and
references therein
S2. Everhart, TE, Hoff, PH Journal of Applied Physics 42 pp. 5837-5846 (1971)
S3. Kanaya, K and Okayama, S Journal of Physics D 5 pp. 43-58 (1972)
S4. Liu, Baojun, Chen, KP, Kherani, NP, Zukotynski, S, Antoniazzi, AB Applied Physics
Letters 92 083511 (2008)
S5. Sandeep K Chaudhuri, Kelvin, J. Zavalla and Krishna C. Mandal, Applied Physics
Letters 102, 03119 (2013)
8