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Two-dimensional collection and injection in grating solar cells
Solar Cells, 17 (1986) 363-371
363
TWO-DIMENSIONAL COLLECTION AND INJECTION IN G R A T I N G S O L A R CELLS
P. DE VISSCHERE RUG, Laboratorium voor Elektroniea, Sint-Pietersnieuwstraat 41, B-9000 Gent (Belgium)
(Received December 29, 1984; accepted in revised form September 5, 1985)
Summary A review is given of the parameters which control the electron flow (one-dimensional or two
1. Introduction The junction in a p-type silicon inversion layer (IL) solar cell is induced b y a positive charge density present in a dielectric layer covering the silicon substrate. Usually contact is made with this induced inversion layer by means of a metal-insulator-semiconductor (MIS) grid, resulting in an MIS:IL solar cell. If the inversion layer is "sufficiently strong" then the electron flow is the same as that in a classical n - p junction solar cell: electrons flow perpendicularly towards the junction and then along the inversion layer to the grid fingers. If, on the contrary, no inversion layer is present, then the electrons can only flow directly towards the grid fingers and the current flow will be two
© Elsevier Sequoia/Printed in The Netherlands
364 Recently an alternative two-dimensional model has been published by Miller and Olsen [4]. In their article our previous results are discarded as being incorrect, since we obtained an active area efficiency of 24.7% for a base doping level NA = 1016 cm -a, whereas the best possible efficiency is only about 19%. However, the latter figure is almost certainly an AM 1 efficiency, whereas in ref. 2, as a preliminary step, we only considered monochromatic illumination of the cell (section 2c of ref. 2), which, for = 0.7 pm, gives a best efficiency of 26%, in agreement with our calculations. Using our model (which now allows for an arbitrary spectrum, surface state effects and recombination in the space charge layer) we have been able to gain more physical insight into the operation of grating solar cells. The results of this analysis, together with a description of the physical principles underlying the model, can be found in ref. 5, while the mathematics are dealt with in detail in ref. 6. Probably the most interesting result concerns the two
2. Two~iimensional effects in grating solar cells Strictly speaking, the current flow in a grating solar cell is always twodimensional, due to shading by the grid lines. However, there are situations in which the current flow is quasi-one-dimensional, by which it is meant that the o u t p u t current of the cell can be calculated with reasonable accuracy (an error of a few percent being tolerable) using a one-dimensional model. In this case it will be assumed that the current flow is one-dimensional. For instance, if inhomogeneous illumination is the only two-dimensional effect, it can be shown that a one-dimensional calculation gives an exact result for the o u t p u t current [ 8 ]. The question then arises as to when the current flow can safely be considered one-dimensional and when it must be considered two-dimensional. Statements which have appeared in the literature on this subject have at best been vague [4], or the problem has been simply ignored [9]. Ideally it should be possible to decide a p r i o r i whether or not a two-dimensional treatment is needed. The following analysis points in that direction. Obviously, the number of oxide charges Nox has great influence on the type of current flow. If Nox is very large, such that the induced inversion layer has very little resistance, then a one-dimensional treatment is certainly sufficient. If, however, no charges are present, then usually (but not always) a two
(1)
where Pr~ is an effective mobility and Nn the number of electrons in the space charge layer. Besides depending on Nox, Ro also depends on the doping density NA and on the local position of the electron Fermi level with respect
365 to the bulk majority hole Fermi level. This latter difference is denoted q~n and ~, is called the local bias voltage. The adjective "local" is introduced to emphasize that ~n varies with position between two grid fingers. However, it turns out [5] that another, but related, quantity is more important than Ro, namely g
/~ne Nn -
RQ-1 -
~n
-
nd
-
(2)
q P n nd
where n d = no e x p ( q 4 p , / k T ) is the electron density at the edge of the space charge layer, with no the equilibrium bulk electron density. The meaning of g will be made clear below. First we analyse the dependence of R~ (and g) on @,. This dependence can be approximated as follows. In the neutrality equation Nox =
WN A + g n
(3)
where W is the width of the space charge region, we neglect either the fixed impurity charges WNA or the mobile charges N.. In the former case ( W N a Nn), which corresponds with inversion and we find immediately RD-I ----ql~,~Nox
(4)
independent of ~.. In the latter case (no inversion, W N A >>Nn) we use the classical abrupt space charge layer approximation. If we assume that q ~ s / k T = ½ ( W / L D ) 2 >~>1, where L D -- ( k T e / q 2 N A ) 1/2 is the Debye length and ~s the surface potential, then N ~ n s L ~ / W where n~ is the electron concentration at the surface. Since, from eqn. (3), W = N o x / N A in this case, we find RQ-1 = qPne ~xoxLD2 e x P L 2 \ L D ']NA q- k T J
(5)
The relations (4), (5) and (2) are shown in Figs. 1 and 2 for NA = 10 21 m -3. N o w we return to the meaning of g, defined in eqn. (2). The current It flowing through the space charge layer towards a grid finger (see Fig. 2 of ref. 5), can be written as dnd It = q D n g -
dy
(6)
where y is directed along the surface. Thus g determines the gradient dnd/dy that is needed to force a given current through the space charge layer. Due to this gradient a lateral current also flows in the base. This current can be estimated as dnd Iht = qDngo d y
(7)
where go equals the diffusion length Ln or the base thickness a, depending on which is the smaller. If g ,~ go then I t ,~ Iht, which means that the current flow is two~iimensional. On the other hand if g >>go then the lateral current
366
t\l
105
l l
I i I I 6.1012
104 6.1013 10361014 10261015 10Nox = 610 IGm-2
10.1
'
t
I
025
J '
I
t
050
I
,
,
:
',
075
Cn
(~o
It)
Fig. 1. Sheet resistance Ra as a function of the local bias voltage (bn for grating solar cells with N A = 1021 m -3 and/2ne = 0.064 m 2 V -1 s-1 and for different values of the number of oxide charges Nox,
g(m)
.I
~
III
III
II
II
II
~+..¢
1o3_
I
610 I/*
ldS.
1~6.
Ln-- 1oo~m
Ln =10,,um 6.1013 6.1012 I I I I I I
o zs
050
075
Cn(vo#)
Fig. 2. D i f f u s i o n thickness o f the space charge layer g as a f u n c t i o n o f the local bias voltage ~n for the same cells as in Fig. 1, w i t h / 2 n = 2~ne.
in t h e base can be n e g l e c t e d a n d it c a n be p r o v e d t h a t in t h i s case t h e c u r r e n t p a t t e r n is o n e - d i m e n s i o n a l . As seen in Fig. 2, g s a t u r a t e s f o r l o w values o f t h e l o c a l bias v o l t a g e . If t h i s s a t u r a t i o n value is less t h a n Ln (we a s s u m e go = L n ) , t h e n t h e c u r r e n t f l o w in t h e cell will a l w a y s be t w o - d i m e n s i o n a l . In t h i s case w e have a real
367 2D-cell. The n u m b e r of oxide charges needed to overcome this situation is easily found, namely
LDNA ]
\LDNA]
or approximately
If this condition is met then, considering normal operation of the cell, collection of carriers is one
(10)
If this happens then further away from the grid fingers only two-dimensional collection is possible. Notice that, again considering normal operation, the local bias voltage ~b= cannot surpass the one-dimensional open-circuit voltage ~o¢. Thus, if ~boc< ¢,(L=), the grating cell can be modelled using a quasi-onedimensional model [71, whatever the finger spacing. But if ~o¢ > ~bg(L=), then a two-dimensional treatment will be necessary b e y o n d some value of the finger spacing. Moreover, if ¢o¢ > ~bg(L=), then the cell will behave as a 2Dcell if the applied voltage V exceeds ¢,(L=), whatever the finger spacing. With the above ideas the performance of grating solar cells can be adequately explained [5]. 3. Dark current-voltage characteristic It is interesting to investigate how two-dimensional effects alter the dark current-voltage characteristic of a grating solar cell. Again, if condition (8) is not fulfilled, the current pattern is always two-dimensional, which means that electrons, injected from the MIS grid fingers, diffuse into the base in all directions. But if condition (8) is fulfilled and the applied voltage V is less than ~bg(Ln), then ~b= < ~s(Ln) everywhere, since ~n is now at a maxim u m at the grid fingers. In this case the current pattern is one-dimensional throughout the cell and irrespective of the finger spacing. In other words electrons flow along the induced space charge layer and are then injected vertically into the base. Thus a quasi-one-dimensional model is again sufficient for calculating the I - V characteristic. For small bias voltage values this /-V-characteristic is ideal with an ideality factor of unity and a saturation current density a
Jo = q Ln NA coth L~
368 (We consider here an ideal grating cell where all other recombination processes are negligible and where the MIS-contact forms no limitation for the electron current.) Possibly for larger values of the bias voltage the ideality factor becomes 2, due to the distributed resistance of the space charge layer [10]. This transition occurs approximately [11] for
kT q
V= V12=--ln--
2kT qRQJo12
(11)
where 2l is the gap between two consecutive grid fingers and with R= given by eqn. (4). If the applied voltage V is increased beyond Cg(Ln) then, at least near the grid fingers, the electrons will again diffuse by preference directly into the base instead of flowing into the space charge layer, since we have g ,~ Ln. Further away from the grid fingers, where Cn is still less than Cg(Ln) and g is thus larger than Ln, the current pattern remains one-dimensional. However, since at the grid fingers we have g ~ L~, the current injected directly into the base dominates over the current flowing along the space charge layer. Thus for V > ~bg(L,) the current-voltage characteristic again has an ideality factor of 1 but now the saturation current density will be less than the one-dimensional value; thus D n r/i 2
a
J0 ( q - coth - Ln NA Ln As an example Fig. 3 shows the current-voltage characteristics of some grating solar cells in the dark, calculated with our exact, two-dimensional model [5]. The approximated sheet resistance R~ and g of these cells are shown in Figs. 1 and 2 respectively. With a doping density NA = 1021 m -3, a mobility Pn = 0.128 m 2 V-1 s-l, a diffusion length L n = 10 pm and a thickness a = 25 #m, the ideal saturation current density of these cells amounts to 11.7 × 10 -7 mA cm -2. The minimum number of oxide charges needed for influencing the current pattern in this cell is, according to eqn. (8), with p , / ~ = 2, 4.5 × 1014 m -2. Therefore the lowest curve in Fig. 3 (No× = 6 × 1013 m -2) corresponds to a 2D-cell. The ideality factor A - 1, but due to two-dimensional injection and since l = 100 pm >~ Ln = 10 pm and 21 >~ finger width = 10 pm the saturation current density is roughly 10 times less than the ideal one
369
//
Nox=6.1015rr£2 t~ /
Nox:6.1014rr~2 ~ ,
v ( vo#)
0.25 qV --2.--
--3.--
050
0.75
qF t.3 108ekT mA cm2
" 13m-2 Nox~<6.10
-5 Fig. 3. Calculated dark J-V-characteristics of the grating solar cells in Fig. I. Only minority carrier injection is taken into account. L n = 10 pro, gridline spacing 21 = 200 p m and substrate thickness a = 25/~m.
by eqn. (9) and is indicated in Fig. 3 by full circles. Thus, the simplified analysis given above appears to agree well with the exact calculations. Additional insight may be gained if we look at the decay of the local bias voltage ~bn along the space charge layer (Fig. 4). For the 2D-cell (Nox = 6 × 10 la m -2) Cn decays approximately as a straight line (independent of the applied voltage) with a slope kT/qL n (2.5 mV # m -1 in the present case). This is a direct consequence of the lateral two-dimensional injection of electrons in the base. For the other t w o cells (Nox = 6 X 1014 m -2 and 6 × 10 is m -2) the slope of the decay curve increases with applied voltage and towards the grid fingers, which reveals quasi-one-dimensional injection into the base. However, if C n > 4 7 5 m V (for N o ~ = 6 × 1 0 1 4 m -2) or C n > 5 2 5 m V (for No~ = 6 X 10 is m -2) then the decay curve merges with the decay curve of the 2D-cell. This is in accordance with Fig. 2 and reveals that electrons are again
370
Avo#) 0.75
075
i ; 6.,o'5 / 5 , 0 ' 4 . . . /
V = ~
~50-
050 •
/
610/''''/
o.25-
"6.10'°
/
61015
v:o.25Vo#
025
Nox=6.101%-2
~......~//
lO~m
Fig. 4. Decay of the local bias voltage Cn for the cells of Fig. 3, for various values of the applied voltage.
injected directly into the base. Summarizing we can say that the space charge layer will support the injected electron current as long as the resulting slope of the electron Fermi level does not exceed the m a x i m u m value k T / q L n .
4. Conclusions
Whether electrons in a p-type grating solar cell are injected (or collected) one- or twodiimensionally depends on the relative value of a property g of the induced space charge layer, with respect to a similar property go of the base. Whereas go is constant for a given cell, g depends strongly on the local bias voltage, which is the local separation between the quasi Fermi levels. This latter separation, in turn, depends on the sheet-resistance of the induced space charge layer. If a sufficiently high positive bias voltage is applied to a grating solar cell, then the current pattern will always be two-dimensional. As a consequence grating solar cells do not show the distributed series resistance induced ideality factor A = 2, except perhaps in a small transition region.
References 1 C. E. Norman and R. E. Thomas, Detailed modeling of inversion layer solar cells, IEEE Trans. Electron Devices, ED-27 (1980) 731-737.
371 2 P. De Visschere, A two-dimensional model of an MIS inversion layer solar cell, Proc.
14th IEEE Photovoltaic Specialists' Conf. ( 1 9 8 0 ) 8 6 - 9 2 . 3 P. De Vissehere, A two-dimensional model for the p-Si metal insulator semiconductor inversion layer (MIS-IL) solar cell, Proc. 2nd NASECODE Conf., Dublin, June 1 7-19, 1981, Boole Press, Dublin, 1981, pp. 177-181. 4 W. A. Miller and L. C. Olsen, Model calculations for silicon inversion layer solar cells, Sol. Cells, 8 (1983) 371. 5 P. De Visschere, Two-dimensional modelling of the MIS grating solar cell, IEEE ~Pfans. Electron Devices, ED-30 (1983) 840-849. 6 P. De Visschere, Simulation of solar cells using the boundary element method, Compel, 2 (1983) 161-177. 7 R . B . Godfrey and M. A. Green, High-efficiency silicon min MIS solar cells - - design and experimental results, IEEE Trans. Electron Devices, ED-27 (1980) 737-745. 8 P. De Visschere, Calculation o f the base collection efficiency in mono- and polycrystalline solar cells by separation o f variables, Sol. Cells, to be published. 9 E . L . Heasell, An analysis of the collection mechanisms in inversion layer solar cells, Solid-State Electron., 27 (1984) 475-483. 10 L. D. Nielsen, Distributed series resistance effects in solar cells, IEEE Trans. Electron Devices, ED-29 (1982) 821-827. 11 A. De Vos, The distributed series resistance problem in solar cells, Sol. Cells, 12 (1984) 311-327. 12 A. K. Kong and M. A. Green, The efficiency of grating solar cells, J. Appl. Phys., 49 (1978) 437-442.
363
TWO-DIMENSIONAL COLLECTION AND INJECTION IN G R A T I N G S O L A R CELLS
P. DE VISSCHERE RUG, Laboratorium voor Elektroniea, Sint-Pietersnieuwstraat 41, B-9000 Gent (Belgium)
(Received December 29, 1984; accepted in revised form September 5, 1985)
Summary A review is given of the parameters which control the electron flow (one-dimensional or two
1. Introduction The junction in a p-type silicon inversion layer (IL) solar cell is induced b y a positive charge density present in a dielectric layer covering the silicon substrate. Usually contact is made with this induced inversion layer by means of a metal-insulator-semiconductor (MIS) grid, resulting in an MIS:IL solar cell. If the inversion layer is "sufficiently strong" then the electron flow is the same as that in a classical n - p junction solar cell: electrons flow perpendicularly towards the junction and then along the inversion layer to the grid fingers. If, on the contrary, no inversion layer is present, then the electrons can only flow directly towards the grid fingers and the current flow will be two
© Elsevier Sequoia/Printed in The Netherlands
364 Recently an alternative two-dimensional model has been published by Miller and Olsen [4]. In their article our previous results are discarded as being incorrect, since we obtained an active area efficiency of 24.7% for a base doping level NA = 1016 cm -a, whereas the best possible efficiency is only about 19%. However, the latter figure is almost certainly an AM 1 efficiency, whereas in ref. 2, as a preliminary step, we only considered monochromatic illumination of the cell (section 2c of ref. 2), which, for = 0.7 pm, gives a best efficiency of 26%, in agreement with our calculations. Using our model (which now allows for an arbitrary spectrum, surface state effects and recombination in the space charge layer) we have been able to gain more physical insight into the operation of grating solar cells. The results of this analysis, together with a description of the physical principles underlying the model, can be found in ref. 5, while the mathematics are dealt with in detail in ref. 6. Probably the most interesting result concerns the two
2. Two~iimensional effects in grating solar cells Strictly speaking, the current flow in a grating solar cell is always twodimensional, due to shading by the grid lines. However, there are situations in which the current flow is quasi-one-dimensional, by which it is meant that the o u t p u t current of the cell can be calculated with reasonable accuracy (an error of a few percent being tolerable) using a one-dimensional model. In this case it will be assumed that the current flow is one-dimensional. For instance, if inhomogeneous illumination is the only two-dimensional effect, it can be shown that a one-dimensional calculation gives an exact result for the o u t p u t current [ 8 ]. The question then arises as to when the current flow can safely be considered one-dimensional and when it must be considered two-dimensional. Statements which have appeared in the literature on this subject have at best been vague [4], or the problem has been simply ignored [9]. Ideally it should be possible to decide a p r i o r i whether or not a two-dimensional treatment is needed. The following analysis points in that direction. Obviously, the number of oxide charges Nox has great influence on the type of current flow. If Nox is very large, such that the induced inversion layer has very little resistance, then a one-dimensional treatment is certainly sufficient. If, however, no charges are present, then usually (but not always) a two
(1)
where Pr~ is an effective mobility and Nn the number of electrons in the space charge layer. Besides depending on Nox, Ro also depends on the doping density NA and on the local position of the electron Fermi level with respect
365 to the bulk majority hole Fermi level. This latter difference is denoted q~n and ~, is called the local bias voltage. The adjective "local" is introduced to emphasize that ~n varies with position between two grid fingers. However, it turns out [5] that another, but related, quantity is more important than Ro, namely g
/~ne Nn -
RQ-1 -
~n
-
nd
-
(2)
q P n nd
where n d = no e x p ( q 4 p , / k T ) is the electron density at the edge of the space charge layer, with no the equilibrium bulk electron density. The meaning of g will be made clear below. First we analyse the dependence of R~ (and g) on @,. This dependence can be approximated as follows. In the neutrality equation Nox =
WN A + g n
(3)
where W is the width of the space charge region, we neglect either the fixed impurity charges WNA or the mobile charges N.. In the former case ( W N a Nn), which corresponds with inversion and we find immediately RD-I ----ql~,~Nox
(4)
independent of ~.. In the latter case (no inversion, W N A >>Nn) we use the classical abrupt space charge layer approximation. If we assume that q ~ s / k T = ½ ( W / L D ) 2 >~>1, where L D -- ( k T e / q 2 N A ) 1/2 is the Debye length and ~s the surface potential, then N ~ n s L ~ / W where n~ is the electron concentration at the surface. Since, from eqn. (3), W = N o x / N A in this case, we find RQ-1 = qPne ~xoxLD2 e x P L 2 \ L D ']NA q- k T J
(5)
The relations (4), (5) and (2) are shown in Figs. 1 and 2 for NA = 10 21 m -3. N o w we return to the meaning of g, defined in eqn. (2). The current It flowing through the space charge layer towards a grid finger (see Fig. 2 of ref. 5), can be written as dnd It = q D n g -
dy
(6)
where y is directed along the surface. Thus g determines the gradient dnd/dy that is needed to force a given current through the space charge layer. Due to this gradient a lateral current also flows in the base. This current can be estimated as dnd Iht = qDngo d y
(7)
where go equals the diffusion length Ln or the base thickness a, depending on which is the smaller. If g ,~ go then I t ,~ Iht, which means that the current flow is two~iimensional. On the other hand if g >>go then the lateral current
366
t\l
105
l l
I i I I 6.1012
104 6.1013 10361014 10261015 10Nox = 610 IGm-2
10.1
'
t
I
025
J '
I
t
050
I
,
,
:
',
075
Cn
(~o
It)
Fig. 1. Sheet resistance Ra as a function of the local bias voltage (bn for grating solar cells with N A = 1021 m -3 and/2ne = 0.064 m 2 V -1 s-1 and for different values of the number of oxide charges Nox,
g(m)
.I
~
III
III
II
II
II
~+..¢
1o3_
I
610 I/*
ldS.
1~6.
Ln-- 1oo~m
Ln =10,,um 6.1013 6.1012 I I I I I I
o zs
050
075
Cn(vo#)
Fig. 2. D i f f u s i o n thickness o f the space charge layer g as a f u n c t i o n o f the local bias voltage ~n for the same cells as in Fig. 1, w i t h / 2 n = 2~ne.
in t h e base can be n e g l e c t e d a n d it c a n be p r o v e d t h a t in t h i s case t h e c u r r e n t p a t t e r n is o n e - d i m e n s i o n a l . As seen in Fig. 2, g s a t u r a t e s f o r l o w values o f t h e l o c a l bias v o l t a g e . If t h i s s a t u r a t i o n value is less t h a n Ln (we a s s u m e go = L n ) , t h e n t h e c u r r e n t f l o w in t h e cell will a l w a y s be t w o - d i m e n s i o n a l . In t h i s case w e have a real
367 2D-cell. The n u m b e r of oxide charges needed to overcome this situation is easily found, namely
LDNA ]
\LDNA]
or approximately
If this condition is met then, considering normal operation of the cell, collection of carriers is one
(10)
If this happens then further away from the grid fingers only two-dimensional collection is possible. Notice that, again considering normal operation, the local bias voltage ~b= cannot surpass the one-dimensional open-circuit voltage ~o¢. Thus, if ~boc< ¢,(L=), the grating cell can be modelled using a quasi-onedimensional model [71, whatever the finger spacing. But if ~o¢ > ~bg(L=), then a two-dimensional treatment will be necessary b e y o n d some value of the finger spacing. Moreover, if ¢o¢ > ~bg(L=), then the cell will behave as a 2Dcell if the applied voltage V exceeds ¢,(L=), whatever the finger spacing. With the above ideas the performance of grating solar cells can be adequately explained [5]. 3. Dark current-voltage characteristic It is interesting to investigate how two-dimensional effects alter the dark current-voltage characteristic of a grating solar cell. Again, if condition (8) is not fulfilled, the current pattern is always two-dimensional, which means that electrons, injected from the MIS grid fingers, diffuse into the base in all directions. But if condition (8) is fulfilled and the applied voltage V is less than ~bg(Ln), then ~b= < ~s(Ln) everywhere, since ~n is now at a maxim u m at the grid fingers. In this case the current pattern is one-dimensional throughout the cell and irrespective of the finger spacing. In other words electrons flow along the induced space charge layer and are then injected vertically into the base. Thus a quasi-one-dimensional model is again sufficient for calculating the I - V characteristic. For small bias voltage values this /-V-characteristic is ideal with an ideality factor of unity and a saturation current density a
Jo = q Ln NA coth L~
368 (We consider here an ideal grating cell where all other recombination processes are negligible and where the MIS-contact forms no limitation for the electron current.) Possibly for larger values of the bias voltage the ideality factor becomes 2, due to the distributed resistance of the space charge layer [10]. This transition occurs approximately [11] for
kT q
V= V12=--ln--
2kT qRQJo12
(11)
where 2l is the gap between two consecutive grid fingers and with R= given by eqn. (4). If the applied voltage V is increased beyond Cg(Ln) then, at least near the grid fingers, the electrons will again diffuse by preference directly into the base instead of flowing into the space charge layer, since we have g ,~ Ln. Further away from the grid fingers, where Cn is still less than Cg(Ln) and g is thus larger than Ln, the current pattern remains one-dimensional. However, since at the grid fingers we have g ~ L~, the current injected directly into the base dominates over the current flowing along the space charge layer. Thus for V > ~bg(L,) the current-voltage characteristic again has an ideality factor of 1 but now the saturation current density will be less than the one-dimensional value; thus D n r/i 2
a
J0 ( q - coth - Ln NA Ln As an example Fig. 3 shows the current-voltage characteristics of some grating solar cells in the dark, calculated with our exact, two-dimensional model [5]. The approximated sheet resistance R~ and g of these cells are shown in Figs. 1 and 2 respectively. With a doping density NA = 1021 m -3, a mobility Pn = 0.128 m 2 V-1 s-l, a diffusion length L n = 10 pm and a thickness a = 25 #m, the ideal saturation current density of these cells amounts to 11.7 × 10 -7 mA cm -2. The minimum number of oxide charges needed for influencing the current pattern in this cell is, according to eqn. (8), with p , / ~ = 2, 4.5 × 1014 m -2. Therefore the lowest curve in Fig. 3 (No× = 6 × 1013 m -2) corresponds to a 2D-cell. The ideality factor A - 1, but due to two-dimensional injection and since l = 100 pm >~ Ln = 10 pm and 21 >~ finger width = 10 pm the saturation current density is roughly 10 times less than the ideal one
369
//
Nox=6.1015rr£2 t~ /
Nox:6.1014rr~2 ~ ,
v ( vo#)
0.25 qV --2.--
--3.--
050
0.75
qF t.3 108ekT mA cm2
" 13m-2 Nox~<6.10
-5 Fig. 3. Calculated dark J-V-characteristics of the grating solar cells in Fig. I. Only minority carrier injection is taken into account. L n = 10 pro, gridline spacing 21 = 200 p m and substrate thickness a = 25/~m.
by eqn. (9) and is indicated in Fig. 3 by full circles. Thus, the simplified analysis given above appears to agree well with the exact calculations. Additional insight may be gained if we look at the decay of the local bias voltage ~bn along the space charge layer (Fig. 4). For the 2D-cell (Nox = 6 × 10 la m -2) Cn decays approximately as a straight line (independent of the applied voltage) with a slope kT/qL n (2.5 mV # m -1 in the present case). This is a direct consequence of the lateral two-dimensional injection of electrons in the base. For the other t w o cells (Nox = 6 X 1014 m -2 and 6 × 10 is m -2) the slope of the decay curve increases with applied voltage and towards the grid fingers, which reveals quasi-one-dimensional injection into the base. However, if C n > 4 7 5 m V (for N o ~ = 6 × 1 0 1 4 m -2) or C n > 5 2 5 m V (for No~ = 6 X 10 is m -2) then the decay curve merges with the decay curve of the 2D-cell. This is in accordance with Fig. 2 and reveals that electrons are again
370
Avo#) 0.75
075
i ; 6.,o'5 / 5 , 0 ' 4 . . . /
V = ~
~50-
050 •
/
610/''''/
o.25-
"6.10'°
/
61015
v:o.25Vo#
025
Nox=6.101%-2
~......~//
lO~m
Fig. 4. Decay of the local bias voltage Cn for the cells of Fig. 3, for various values of the applied voltage.
injected directly into the base. Summarizing we can say that the space charge layer will support the injected electron current as long as the resulting slope of the electron Fermi level does not exceed the m a x i m u m value k T / q L n .
4. Conclusions
Whether electrons in a p-type grating solar cell are injected (or collected) one- or twodiimensionally depends on the relative value of a property g of the induced space charge layer, with respect to a similar property go of the base. Whereas go is constant for a given cell, g depends strongly on the local bias voltage, which is the local separation between the quasi Fermi levels. This latter separation, in turn, depends on the sheet-resistance of the induced space charge layer. If a sufficiently high positive bias voltage is applied to a grating solar cell, then the current pattern will always be two-dimensional. As a consequence grating solar cells do not show the distributed series resistance induced ideality factor A = 2, except perhaps in a small transition region.
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