The Finite Element Semiconductor Three-dimensional Analyzer by Ralph O. Clark (FIESTA ROC) is a computational tool for investigating in detail the performance of arbitrary solar cell structures. As its name indicates, it uses the finite element technique to solve the fundamental semiconductor equations in the cell. It may be used for predicting the performance (thereby dictating the design parameters) of a proposed cell or for investigating the limiting factors in an established design

Clark, Ralph O.

English

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N91-30234
FIESTA ROC: a New Finite Element Analysis Program for Solar Cell Simulation
Ralph O. Clark
Space Photovoltaic Research Center,* Electrical Engineering Department
Cleveland State University
Cleveland, Ohio
INTRODUCTION
FIESTA ROC (Flnite Element Semiconductor Three-dimensional Analyzer by Ralph O. Clark) is a computational tool
for investigating in detail the performance of arbitrary solar cell structures. As its name indicates, it uses the finite element
technique to solve the fundamental semiconductor equations in the cell. It may be used for predicting the performance
(thereby dictating the design parameters) of a proposed cell or for investigating the limiting factors in an established design.
THE FINITE ELEMENT METHOD
The fundamental semiconductor equations solved by FIESTA ROC are [i]
_7. (E_') = p (Poisson's Equation)
_7-._'n = G-/_ (Electron Continuity Equation)
V • _p = G - R (Hole Continuity Equation)
(la)
(lb)
(lc).
Here _ is the electric field, _ the permittivity, p the volume charge density, G- R the net carrier generation rate (generation
minus recombination), and ._'_ and _'p the electron and hole flux densities.
electrostatic potential tb and the quasi-Fermi potentials ¢,_, ep by
E = -_7¢
_n = nipn e(¢-c")/vr Vfbn
._p = -niltpe (¢_p-*b )lvr V q_p
In turn, C, Fn, and .Tp are related to the
(2a)
(2c).
As boundary conditions, we assume that the domain of simulation, £Z, has a boundary 0_ that can be partitioned into
two segments: the Dirichlet boundary c3_D, on which _b, ¢,_, and cp are prescribed, and the Neumann boundary oq£/N, on
which the components of _7_/,, _'q_n, and _'_v normal to the surface all vanish. In particular, on the Dirichlet boundary,
which corresponds to the contacts of a device, an infinite surface recombination velocity is assumed, pinning n and p to their
levels at thermal equilibrium and q_n and _p both to _b - _b0. On the Neumann boundary, arbitrary surface recombination
velocities are possible. Note that, for a one-dimensional model of a two-terminal device under these assumptions, we cannot
model finite surface recombination velocities at the ends of the device because both ends represent contacts, i.e. Dirichlet
boundaries.
Equations (1) can all be written in the generic form
vf - = 0 (3).
To solve (3) by the finite element method, we first write it in its weak form, as follows. Let w be an arbitrary piecewise-
continuous function on f_, vanishing identically on Cgf2O. Multiplying (3) by w, integrating over the entire volume, and
integrating the first term by parts, we obtain
O = /_w(XY " f -- s) = fafl wf . n -- /f (_Yw . f + ws),
*funded by NASA Lewis Research Center, Cleveland, Ohio
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https://ntrs.nasa.gov/search.jsp?R=19910020920 2020-03-19T17:02:48+00:00Z
where n is the unit outward normal to the surface. Note that since w vanishes on the Dirichlet boundary and f. n on the
Neumann boundary, the surface integral vanishes, leaving us with the weak form of equation (3):
(Vw. f + ws) - 0 (4).
We now dlscretize the problem by fixing N points, called nodes, between which w, _, _n, and _p all vary in a
continuous way. In particular, we express w(t_, _n, _p} as a linear combination of continuous, plecewise smooth functions
Wi{_i,_?,_}, i - 1 ..... N, each of which assumes the value of 1 at node i and zero at all other nodes. Thus,
W --_ EN=I wiWi, _J ._ EN__I 1_i@i, _n -" EN=I _ (I)n, ¢_p -_ EN=I _v/(_, where the wi are arbitrary coefficients (to keep w
arbitrary) and _'i, _, _ are just the values assumed by _, @,,, and _p at node i. Now equation (4) becomes
N
f (vw,. f + = o
i=1
Since the N coefficients wi are arbitrary, equation (5) really represents the system of N equations
n(VWi, f+ Wis) = O, i = 1,...,N (6).
In fact, with our three coupled equations (1) and three dependent variables t_, _bn, _v, we now have 3N coupled non-
linear equations in the 3N unknowns _'i, _, _', i = 1,..., N. These are solved by the finite element code by means of a
generalized Newton's method.
ADVANTAGES OF FINITE ELEMENT
Because the finite element method is an integral method, with only first derivatives appearing, many complications
associated with boundary value problems disappear. First of all, surface effects, whether at a boundary between regions
of the device (e.g. the junction) or at the physical surface of the device (e.g. surface recombination) are effectively a
special case of bulk effects where the integrand includes a Dirac delta function to localize the effect to the boundary. The
delta function converts volume integrals into surface integrals. Thus, surface recombination and interface charge density
may be handled easily by including surface integrals in equation (6). In addition, the perennial problem of matching the
electric field at a jump discontinuity of c does not even come up, having been integrated out of existence. Thirdly, multiple
non-interacting recombination levels are dealt with simply by including additional terms in R (which appears in the s in
equation (6)). Finally, non-rectilinear elements are readily handled by performing the integrations in (6) numerically.
IMPLEMENTATION
We have implemented a one-dimensional prototype of FIESTA ROC in C, running on an IBM-PC compatible and on a
Cray XMP. The prototype features automatic mesh generation and automatic dark and light I-V simulations, as well as
spectral response. Arbitrary numbers of recombination levels, degenerate statistics, and heavy doping effects are supported.
Finite surface recombination velocities, as noted earlier, are imcompatible with our choice of boundary conditions in the
one-dimensional case, so they have been left out of the prototype. Once the solution to equation (6) has been found for a
given bias and illumination, many quantities of interest can be examined at any point in the cell. These include the electron
and hole current densities, electric field, electric potential and quasi-Fermi potentials, net recombination, n, and p.
EXAMPLE
As an example of the use of FIESTA ROC, we have modeled an InP shallow homojunction n+pp + cell from Spire
Corporation which has been extensively studied with analytical models [2]. The growth parameters and modeling parameters
found in [2] are reproduced in Tables I and II. Without any change in the parameters of either table, the results of light I-V
and spectral response simulations by the prototype are shown in Figures 1 and 2. In addition, the measured and calculated
solar cell parameters are shown in Table III.
31-2
Table h Growth Parameters for Spire 6 Cell
emitter width 400 A
n+ emitter doping (Si) 1 x 1018 cm-s
base width 3 pm
p base doping (Zn) 2 x 10 TM cm -s
BSF width 250 pm
p+ BSF doping (Zn) 5 x 10 TM cm -s
cell area 0.25 cm_
grid shadowing 4.8%
Table Ih Modeling Parameters for Spire 6 Cell [2]
indirect lifetime of holes in emitter 2.0 ns
hole mobility in emitter 75 cm2/V-s
indirect lifetime of electrons in base 150 ns
electron mobility in base 3988 cm_/V-s
indirect lifetime of electrons in BSF 0.60 ns
electron mobility in BSF 2456 cm2/V-s
The real utility of a numerical solver, however, lies in its ability to show what is happening in various regions of the
cell. In this respect, it functions as a sort of computational microscope, allowing the investigator to probe all regions of
the cell to determine, for example, which contribute most to the total recombination, light-generated current, and so on.
In Figure 3, we show the calculated net recombination for three different bias points---short circuit, maximum power, and
open circuit--plotted against the spatial coordinate z. Here z = 0 represents the metallurgical p.n junction and x = 3pm
the base-BSF low-high junction. The emitter, being only .04pm wide, is invisible on this scale. The junction space charge
region is visible as a low-recombination valley at short circuit and a high-recombination peak under forward bias. Note,
however, the additional peak in recombination in the BSF region. In Figure 4, we show the same curve, at maximum power
only, calculated for the same cell and for another differing only in the doping in the BSF, namely 5 x 1017 rather than
5 X 10 TM cm -3. The indirect recombination lifetime in the BSF is also correspondingly higher (6.0 ns instead of .60). The
peak in the BSF almost disappears when the BSF doping is decreased. Such an analysis can aid cell fabricators in designing
and producing more efficient cells.
CONCLUSIONS
We have demonstrated a one-dimensional prototype of a flexible finite element package that will enable cell designers to
simulate a variety of effects and to pinpoint problems in proposed or existing cells. The two- and three-dlmensional versions
are currently under development.
REFERENCES
[1] Markowich, Peter. The Stationary Semiconductor Device Equations, Springer-Verlag, Vienna, 1986.
[2] Thesling, William. "Theoretical Modeling, Optimal Design and Performance Predictions of the Shallow Homojunction
Indium Phosphide Solar Cell in Space Applications", Master's Thesis, Cleveland State University, Cleveland, Ohio, August
1990.
31-3
Table II1: Solar Cell Parameters for Spire 6 Cell
Measured Calculated
Isc,mA 8.47 8.37
Foe,mV 868 869
I,_=_,mA 8.19 8.03
Vm_, mV 751 754
FF, % 83.8 83.3
_,% 17.94 17.64
at 1AM0, 25°C
10
<
E
c-
o
k,,
k.,
-I(3
8
6
4
2
--t- + 4- + +
-- Measured
0
0 200 400 600 800 1000
Voltage (mV)
Figure 1: Illuminated I-l/characteristic for cell Spire 6, with calculated results by FIESTA ROC prototype.
31-4
E
<
E
0.7
0.6
0.5
0.2
0.1
I 1 I _ I _ I _ 1 ' I '
+
+
0,0 , I L I 1 I I I I I L I
0.3 0.4 0.5 0.6 0.7 0.8 o.g
t
1.0
Wavelength (p,m)
Figure 2: Spectral response for cell Spire 6, with calculated results by FIESTA ROC prototype.
10 =1
0
O_
10 =°
10 19
10 TM
n-
10 17
open
short
circuit
oower point
circuit
10 _a
-'10 0 10 20 30 40 50
x (microns)
Figure 3: Net recombination calculated by FIESTA ROC prototype at three bias conditions. The n+# junction is at z = 0
and the pp+ junction at x = 3#m.
31-5
Ao
n-
I 011
I 0 t°
10 _,
10 TM
10 17
• "_ • i ] v _ • • | • i i" i ] • i' i" i l _ z i i
at max. power point
\5E17
i|*llll**JLJJ|JlsJlll*|JlJa||
0 10 20 30 40 50
x (microns)
Figure 4: Net recombination calculated at max power voltage for cells with different BSF dopings.
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FIESTA ROC: A new finite element analysis program for solar cell simulation

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