A simple proton damage model for GaAs solar cells is derived and compared to experimental values of change in short circuit currents. The recombination cross section associated with the defects was determined from the experimental comparison to be approximately 1.2 x 10 to the -13th power sq cm in fair agreement with values determined from the deep level transient spectroscopy technique

Outlaw, R. A.

Walker, G. H.

Wilson, J. W.

English

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	 NASA Technical Memorandum SM
(NASA-TN-84495) A SIUIZ SOUL OF VACTCB
	 182-30707
CANAGE IN GaAs SOLAR CELLS (MASA) 18 p
8C A02/NF A01
	 CSCL 101
Oncla s
G3/48 28693
A S IMPLE MODEL OF PROTON DAMAGE IN
GaAs SOLAR CELLS
John W. Wilson, Gilbert H. Walker, and R. A. Outlaw
June 1982
0
Table of Contents
NOMENCLATURE .............................................................. ii
SUMMARY ................................................................... 	 1
INTRODUCTION ...................................................... I....... 	 2
THEORY OF DEFECT FORMATION ................................................ 	 3
RESULTS ................................................................... 	 8
REFERENCES ................................................................ 	 9
FIGURES ................................................................... 10
I
i
NOMENCLATURE
ao Bohr's radius, 5.29 x 10- 5
 
on
0(E) total number of displacements per proton of energy E
Dv(E) number of displacements per unit volume, Nm-3
E proton energy, KeV
ED displacement damage threshold, eV
ER Rydberg's constant, 13.6 eV
f(u) efficiency for diffusion to junction along direction defined
by u
F(x) efficiency function for diffusion from depth x to the
junction
-tp(E) displacement mean free path for proton of energy E, w
L minority carrier diffusion length, um
M2 average mass of gallium and arsenic nucleus, amu.
n nuclear density in GaAs crystal, wn-3
P distance traveled along proton path, vm
P(E) pathlength of proton of energy E, um
r minority carrier rms distance traveled before recombination,
um
R(E)	 average projected range of a proton of energy E, um
T	 energy transfered to struck nucleus, keV
Tm	 maximum energy transferred to struck nucleus, keV
x	 penetration depth into solar cell, um
Z2	average charge number of GaAs nucleus
sn	 change in collection efficiency
a	 photon absorption pathlength, um
u	 cosine of direction of diffusion in reaching junction
v(E)	 average direction cosine of scattered proton of energy E
ii
1
^.	 t
v(E)	 average number of displacements formed by one proton
scattering event
CD(E)
	 number of displacele nts per unit pathlength for a proton
of energy E, (wm)'
P(X)	 cel collection efficiency for a normal unirradiated cell,
op(E)
	 displacement cross section for protons, w
or	 recombination cross section,
	 2wn
A
_
E
y^
i
b
iii
i
i
A Simple Model of Proton Damage in GaAs Solar Cells
John W. Wilson, Gilbert H. Walker, and R. A. Outlaw
Langley Research Center
SUMMARY
A simple proton damage model for GaAs solar cells is derived and
compared to experimental values of change in short circuit currents. The
recombination cross section associated with tIs defects was determined from
the experimental comparison to be M 1.2 x 10- cm in fair agreement with
values determined from the deep level transient spectroscopy technique.
i
Gallium arsenide solar cells have received considerable attention due
to their potential usefulness in high-power space-energy systems as well as
special space probe applications where high operating temperature is a
factor (ref. 1). However, space radiation damage to the cell may be a
limiting factor. Consequently, radiation damage experimental studies have
been cnnt-- •ted and the present report is an attempt to develop a simple
model in wmich Lhese experimental results can be understood.
Earlier models for electron radiation damage assumed the defects to be
produced uniformly throughout the cell volume, and modeled the cell perfor-
mance in terms of cell-averaged diffusion lengths of the minority carriers
(refs. 2 and 3). In view of the short pathlengths of the protons used in
radiation damage studies and the specific dependence of the cell efficiency
on the proton energy, the assumption of uniform damage within the cell
seems unwarranted for low-energy proton exposure. Consequently; the pres-
ent report treats geometric distribution of the displacement damage in
detail and cell performance is not treated by a cell-averaged parameter
such as diffusion length. Cell performance is evaluated in terms of the
cell-averaged minority carrier recombination probability in diffusion to
the junction of the cell. The average of the minority recombination proba-
bility over the cell active region weighted according to the solar-averaged
photoabsorption rate is used to estimate the decrement in the short circuit
current.
2
ORIGI14AL F' ^E l
OF POOR QUALITY
Theory of Defect Formation
Atomic displacements caused by proton impact with atomic nuclei result
in crystal defects. The formation rate of these defects is related to
Rutherford's cross section (ref. 4)
2 2 2
^D(E) 
4^ 
ate° R--2- (^ - T )
2	 0	 m
where ao is Bohr's radius, ER is Rydberg's constant, Z 2 is the
atomic number of the struck nucleus, M2
 the corresponding nuclear mass
number, E is the proton kinetic energy, ED is the energy required to
displace the nucleus from it's lattice site, and Tm is the maximum
energy transfer in the collision
4M2
WTM (1+M^ E
The threshold for displacement requires Tm > ED. The value of
Ep - 25 eV insures that only close collisions result in displacem ent so
that screening corrections to the Rutherford formula are unimportasit
(ref. 4). If the atonic recoil energy is sufficiently large ( T»ED) then
additional displacements can be produced by the recoiling nucleus before
coming to rest at an interstitial site. The average number of recoils
produced by one initiating proton collision event is given as a function of
the maximum energy transfer by
T
1 +
	
	 m	 log (Tm/ED)	 Tm > 2ED
2 ( m- ED)
v(E) 3	 (3)
1.0	 2 E D > Tm > ED
assuming half of the recoil energy produces further displacements and the
other half is dissipated in other processes (ref. 4). These quantities
allow the calculation of the number of displacements produced per unit
•	 distance traveled by a proton of fixed energy.
In passing through a crystal, most of the energy of a proton is
transferred to orbital electronsref. 5 .	 The(	 )	 pathlength traveled in
coming to rest is found by fitting the data of reference 5 as
P(E) - 0.077 E0 ' 5
 + 1.125 x 10-4 E1.64	 (4)
where E is in keV and P is in um. As derived from slowing down theory,
a unique value of kinetic energy can be associated with each position along
a proton's trajectory. The proton energy as a function of the distance P
yet to be traveled before coming to rest is given by
3
(1)
(2)
E - 209.6 
P2.08/(1
 + 1.055 P1.43)	 (5)
as determined from the stopping power data of Andersen and Ziel1lr
(ref. 5). In coming to rest, the proton undergoes multiple scatterings
from atomic nuclei (of which a few result in displacements) which alters
ever so slightly its direction of motion. The depth of penetration R and
pathlength P are approximately related by (ref. 5)
P-1 R = [1 - exp (- 0.084 E0.55 	 (6)
This ratio is related to the average deviation in the direction of motion
and is most important at low energies. The average depth of penetration
and initial energy as related through equations (4) and (6) can be
approximated by
R(E) - 0.0062 E + 2.92 x 10-5 E 1.77	 (7)
There is no unique energy associated with a given depth of penetration due
to multiple scattering. However, the average energy of protons whici,
penetrate and stop at a depth R is
E - 593 R1.5/(1 + 3.71 R0.5)	 (8)
The above quantities were used to determine the displacement density within
a GaAs crystal.
A proton of energy Eo incident on the face of the crystal will
travel a distance
Po - P ( E0 )	 (9)
before coming to rest. After traveling a distance p the energy will be
reduced to
E - 209.6 (Po - p)2.08/[1 + 1.055(Po - p)1.431
	
(10)
At this position p the displacement mean free path is
tD(E) - 1/n OD(E)
	 (11)
w
whiroe 3n	 is the density of scattering centers in the crystal (4.42
	
x
10 /um) and oD is the displacement cross section averaged for GaAs
(M2 - 72.5 and Z2 - 32).	 The average number of displacements . per unit
pathlength is
ED(E) - v (E)/#.O(E)
	
(12)
I
4
...
ORIGINAL
	 rJ
OF POOR QUALITY
The use of equations (10) and (12) allow appropriate partitioning of the
proton's energy into electronic excitation and displacements everywhere
along its path.
The number of displacements along the protons -path is now to be
related to the displacement damage in the crystal. For normal incident
protons of energy Eo on the face of a crystal, the number of
displacements along its path are given by equations (10) and (12).
However, by the time its energy is reduced to E, it has penetrated to an
average depth x given by
x = R(E0) - R(E)	 (13)
The pathlength and penetration depth are related to the average direction
cosine (ref. 6) and is approximated here by finding the ratio 'u(E)
dP(E)/dR(E) using equations (4) and (6). In terms of P(E), the average
number of displacements per unit depth is
Zx = v(E) ED (E)
	
(14)
where x is found from relation (13). The effects of multiple scattering
are demonstrated in fig. 1. The results of equation (14) for the average
proton path is shown as the full line in comparison to calculations
neglecting multiple scattering according to equation (12) shown as the
dashed line.
The total number of displacements formed along the path of a proton
with initial energy Eo is
D(Eo ) = jE0 %(E) dPd E	 (15)0
The numerical evaluation of equation (15) is found to be
0	 Eo < .728
D(E0 ) _	 ( 16)
1.6 + 14.5(1 + 0.005 Eo ' 68) loglo Eo	Eo > .728
where the threshold displacement energy for the proton is 0.728 MeV.
Equation (15) was also evaluated using the displacement theory of Lindhard,
Scharff and Schiott (LSS) as discussed in reference 7. The numerical
values are shown in comparison to equation (16) in figure 2.
It is assumed that these displacements form recombination centers for
electron-hole pairs produced by photon absorption. A pair once formed
undergoes thermal diffusion until it recombines or is separated at the
junction. The relation between the rms distance traveled in moving to a
position a distance L away from the source point is (ref. 8)
5
rORIGINAL PAGE [S
OF POOR  QUALI'T'Y
F - Ib L	 (17)
If or is the recombination cross section and t the distance along an
arbitrary straight line path to the junction, then the fractional loss of
lairs due to recombination it. reaching the junction along a fixed direction
[1 - exp(-)x ar Dv(x)*r6 dx/u)]	 x > xi
xi
f(u)	 {18)
[1 - exp (- jx J 0
.
 Ov (x) 
,%
Ir
 
dx/u) ]	 x ( x,
x
mere u is the cosine of the direction to the junction. Averaging the
fractional loss over all directions towards the junction
NO - f ' f(u) du	 (19)
0
and noting
Ix^ Dv (x)dx - /[D(Ex ) - D( Ej)]	 (20)
x
where Dv(x)	 is displacement density and 1 is the proton fIuence
results in
1 - E 2
 [-%b or
 t ( D( Ex) - D( E^) )] :
	 Ex > E 
F(x) -	 (21)
1 - E2 [-Nr6 or 4 (D(E^) - D( Ex) ) ]	 E  > Ex
Assuming the trajectory of an average proton as defined by equation (8),
Ex and Ej become
E	 593 N - x)3/2	
(22)-Ex 	 1♦3. IN- x)
and
593 (R - x )3/2
-
o	 (Z3)
1 ♦ 3.71 (% - x^) 12
6
sORIGINAL PAGE IS
OF POOR QUALITY
Complete neglect of multiple scattering effects is accomplished by
replacing equations (22) and (23) by
E n 209.6(P - x)2.08/[1 + 1.055(P - x)1.431
	 (24)
x	 o	 0
and
Ej n 209.6(Po - xj) 2.064 1 + 1.05500 - xi) 1.431	 (25)
If o(x) is the normal efficiency function for producing and collecting
electron-hole pairs at the junction, then the efficiency change for the
proton damaged crystal is
t	 t
an • j o(x) F(x) dx/ j P(x) d(x)	 (26)
0	 0
where t is the maximum depth of the active region of the cell. To
simplify the present calculation without serious error, it is assumed that
P(x) is proportional to the photoabsorption density normalized to unity as
P(x)	 a-x/^	 (27)
where the solar absorption length a is 0.714 w (ref. 9).
i
RESULTS
The geometry of the solar cell is shown in figure 3. The change in
the cell current collection efficiency as given by equations (24) and (25)
were evaluated numerically for solar cell parameters shown. In traversing
the cell, the protons follow neither the trajectory of the average proton
nor the trajectory in which multiple scattering is neglected as shown in
figure 1. The effects of multiple scattering were estimated by averaging
the cell damage for the two functions shown in figure 1 in which some
effects of deviations about the average trajectory are included. It is
clear that an understanding of the low-energy experimental data requires
detailed modeling of multiple scattering effects. The window thickness
parameter which varied from cell to cell in experimental tests
(refs. 10 and 11) were assumed to be governed by a uniform distribution in
the present calculations. The model results averaged over the window
thickness are compared to short circuit current measurements in irradiated
cells in figure 4. The best value of recombination cross section is found
to be
°r - 1.2 x 10-13 W2
which is in f a^r3 ag;eement with the estimated average cross section
or - 1.06 x 10-	 cm determined from deep-level transient spectroscopy(ref.  12) .
8
References
I. Anonymous: Solar Power Satellite Concept Evaluation. NASA TN-74820,
July 1977.
2. Heinbockel, J. H.; Conway, E. J.; and Walker, 6. H.: Simultaneous
Radiation Damage and Annealing of GaAs Solar Calls. Conference
Record of the Fourteenth IEEE Photovoltaic Specialists
Conference - 1990 9 80Cii1508-1, 1980 9 pp. 1085-1089.
3. Conway, E. J.; Walker, 6. H.; and Heinbcckel, J. H.: A Thermochemicai
Nodel of Radiation Damage and Annealing Applied to GaAs Solar
Cells. Conference Record of the Fifteenth IEEE Photovoltaic
Specialists Conference - 1981, SICH1644-4 9 pp. 38-44.
4. Dienes, G. J.; and Vineyard, G. H.: Radiation Effects in Solids.
Interscience Publications, New York, 1957, pp. 12-32.
5. Andersen, H. H.; and Ziegler, J. F.: Hydrogen Stopping Powers in All
Elements. Pergamon Press, New York, 1977.
6. Janni, J. F.: Calculations of Energy Loss, Range, Pathlength, Stragg-
ling, Multiple Scattering, and the Probability of Inelastic Nuclear
Collisiors for 0.1 to 1000 MeV Protons. AFWL-TR65-150, September
1966.
7. Peterson, N. L.; and Harkness, S. D., editors: Radiation Damage in
Metals. American Society for Metals, Ohio, 1976.
8. Liverhant, S. E.: Nuclear Reactor Physics. Wiley and Sons, New York,
1960, P. 252.
9. Loferski, J. J.: Recent Research on Photovoltaic Solar Energy Con-
verters. Proceedings of the IEEE, vol. 51, 1963, pp. 667-674.
10. Loo, R.; Knechtli, R. C.; and Kamath, G. S.: Enhanced Annealing of
GaAs Solar Radiation Damage. Conference Record of the Fifteenth
IEEE Photovoltaic Specialists Conference - 1981, SICH1644-4 1981,
pp. 33-37.
11. Kamath, G. S.: GaAs Solar Cells for Space Applications. Proceedings
of the Sixteenth Intersociety Energy Conversion Engineering
Conference, 81-65380, 1981, pp. 416-421
12. Li, S. S.; Chiu, T. T.; Schoenfeld, D. W.; and Loo, R. Y.: Defect
Characterization anus Thermal Annealing Study of 200 KeV Proton
Irradiated n-GaAs L?E Layers. Institute of Physics Conference
Series. Number 59, chapter 6, 1981, pp. 335-340.
9
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A simple model of proton damage in GaAs solar cells

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