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The design of anisotropically etched III–V solar cells
Solar Cells, 11 (1984) 221 - 239
221
THE DESIGN OF ANISOTROPICALLY ETCHED III-V SOLAR CELLS
R. J. ROEDEL and P. M. HOLM
Department of Electrical and Computer Engineering, Arizona State University, Tempe, AZ 85287 (U.S.A.) (Received June 7, 1983; accepted September 26, 1983)
Summary A solar cell with negligible reflectivity requires a departure from conventional planar design. An anisotropically etched (grooved) surface modifies the geometry o f the cell to permit multiple reflections and enhanced absorption. Two distinct groove geometries can be generated in III-V semiconductors because of the lack of inversion s y m m e t r y in zinc blende lattices. The grooves are said to be "V shaped" or "inverted". In this paper, ray-tracing models are developed to analyze novel solar cells based on these two groove types. The models treat groove angle, spacing and aspect ratio as variables, and appropriate conclusions are drawn for the case where the grooves are formed by intersecting { l l l } A planes. It is shown (1) that, for a structure with V-grooves, the m a x i m u m conversion efficiency can be increased by about 38% with respect to a planar structure and (2) that, for a structure with inverted grooves, the efficiency increase can be as high as 26%.
1. Introduction To fabricate solar cells with m a x i m u m efficiency, it is necessary to reduce the reflectivity of the surface to zero, if possible. Most traditional solar cell designs have used polished planar surfaces, and the reflectivity for normally incident light (which is essentially independent o f wavelength for 0.5 ~zm < k0 < 1.0 pm) is approximately 35% for both silicon and GaAs. The use of antireflection coatings can reduce the reflectivity to approximately 10% (single-layer cells) or 7% (multiple-layer cells) in silicon cells [ 1 ]. Further reductions are possible only with modifications in the geometry of the cell. Recently, several reports have shown that reduced reflectivity can be achieved in silicon solar cells by preferentially etching the silicon surface [1 4]. The resulting surface is no longer planar and can be said to be "serrated" if the etching produces regularly spaced grooves or to be " t e x t u r e d " if the etching produces randomly spaced and sized etch features. Without antireflection coatings, the serrated silicon surface has an average -
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© Elsevier Sequoia/Printed in The Netherlands
222
reduced reflectivity of roughly 22% (for 35% coverage of the surface with grooves), and the textured silicon surface approximately 12% [1, 2]. The application of antireflection coatings reduces these values even further. Solar cells fabricated from these etched substrates demonstrate an enhanced shortcircuit current and spectral response. Clearly, the addition of one relatively simple etching step to the fabrication process can yield solar cells with dramatically improved performance. Solar cells constructed from I I I - V semiconductors (such as the Gal _xAlxAs/GaAs system) are promising candidates for cost~ffective photovoltaic energy conversion in high sunlight concentration applications [5]. The design of preferentially etched I I I - V solar cells is the subject of this paper. Unlike silicon, all III-V semiconductors possess a non-centrosymmetric lattice, and this has a major bearing on the geometry of etch figures produced by an anisotropic etchant. The preferentially etched silicon substrates (with an original (001) surface orientation) have (ll0)-oriented grooves with (111) sides in the serrated structures, or tetrahedra with {110)oriented bases and (111) sides in the textured structures. In contrast, because the (111) planes terminate on group III atoms and the (111) planes terminate on group V atoms in III-V semiconductors [6], and because of the enormous difference in the chemical reactivity of the (111) and the ( i i l ) planes [7], two distinct groove geometries can be obtained on (001) I I I - V surfaces. As shown in Fig. 1, when the mask stripes are oriented in the [ l i 0 ] direction, a " n o r m a l " V-groove with (111) and (:~11) side walls can be produced with an anisotropic etchant (such as 1% bromine in methanol) but, when the mask stripes are oriented in the [110] direction, an "inverted" Vgroove with (111) and ( 1 1 i ) side walls is formed. All four of these side wall habit planes are of the { l l l ) A type; the ( l l l } B planes are either not reached (V-groove) or not effective as stop etch planes (inverted groove). In general, the following rule is o b e y e d for anisotropic etchants in I I I - V semiconductors [ 7 ] :
Fig. 1. Groove geometries applicable for GaAs (001) subjected to an anisotropic etchant through (110)-oriented windows.
223 r {110}~ r{iii}B ~ r {100}~ r{lll}A
where r is the etchant rate. Either groove configuration can form the basis for a novel geometry I I I - V solar cell, as shown in Fig. 2. Both cells (constructed from GaAs for illustrative purposes) would require junction formation through p-type diffusion, a p-type wide band gap window layer and stripe metallization for efficient operation. An analysis of the total reflectivity for V-grooved and inverted-grooved cells is carried o u t in Sections 2 and 3; this is followed b y a discussion of the expected efficiency improvements for b o t h structures and a summary of the results. fp
- COl. x AIxAS
% ' p - GaAs DIFFUSED LAYER n -GaAs EPITAXIAL LAYER
n - GoAs SUBSTRATE
(a)
n - GoAs EPITAXIAL
LAYER
r,/p-
Gal. x AIxAs
~"p-
GoAs
DIFFUSED LAYER
n - GaAs SUBSTRATE
(b) Fig. 2. Cross-sectional views of anisotropically etched GaAs solar cells: (a) the normal Vgroove cell; (b) the inverted-groove cell. Both cells are shown with a diffused p-type layer and a p-type Ga]-xAlxAs window layer.
2. V-groove structure The major difference between a planar surface and a grooved surface is that there exists a possibility for multiple reflections o f the incident light rays with the serrated surface. Multiple reflections decrease the total reflectivity of the cell and thus permit a greater a m o u n t of photoabsorption. Although the V-grooves described in Section 1 are formed b y { l l l } A planes and have a fixed groove angle given by
av = 1SO° -- cos-I {(111)" ( i l l ) t
\i Fl -fi]
= 70.5 °
(1)
224 in this s e c t i o n we shall calculate t h e r e f l e c t i o n c o e f f i c i e n t for a serrated surface as a f u n c t i o n o f t h e g r o o v e angle a. Using these results, we shall find the t o t a l r e f l e c t i v i t y at ~ = a v When ~ = 180 °, t h e surface is planar, and i n c i d e n t rays can strike t h e surface o n l y once. In fact, f o r g r o o v e angles b e t w e e n 180 ° and 120 ° , t h e r e can b e o n l y o n e r e f l e c t i o n , for with specular r e f l e c t i o n and vertically i n c i d e n t light t h e o u t g o i n g rays c a n n o t i n t e r c e p t t h e o p p o s i t e side o f t h e Vgroove. T h e i n c i d e n t angle is given b y 01 = 90° -- -2
(2)
When ~ = 120 °, 01 -- 30°, and t h e o u t g o i n g ray is parallel t o t h e o p p o s i t e g r o o v e side. This is d e p i c t e d in Fig. 3(a). When ~ is r e d u c e d f r o m 120 °, say to 95 ° as s h o w n in Fig. 3(b), a f r a c t i o n o f t h e rays striking t h e left side o f t h e g r o o v e will be involved in a s e c o n d r e f l e c t i o n f r o m t h e right side o f the groove. Rays i n c i d e n t near t h e t o p w i l l n o t i n t e r c e p t t h e o p p o s i t e side. Using
/ /
(a)
/
/
//// / / / \/e,/
(h)~
\
/ ~ / /
=
"'///
Fig. 3. R a y - t r a c i n g m o d e l s f o r t h e V - g r o o v e s t r u c t u r e : ( a ) g r o o v e a n g l e 0t = 1 2 0 °, o n l y o n e r e f l e c t i o n w i t h 01 = 3 0 ° ; ( b ) g r o o v e a n g l e 0t = 9 5 °, a f r a c t i o n o f t h e r a y s are r e f l e c t e d t w i c e w i t h 01 = 4 2 . 5 ° a n d 02 = 4 7 . 5 ° •
225
t h e law o f sines it can b e s h o w n t h a t t h e f r a c t i o n o f rays t h a t can be r e f l e c t e d t w i c e is cos 02 fl
- - -
(3)
cos 0 ,
w h e r e 02 is t h e s e c o n d i n c i d e n t angle and is given b y 02=ct--O, 3o~ -
90 °
(4)
2 At a groove angle (~ of 90 °, all rays will be reflected twice, and the outgoing ray is vertical. Between the angles of 90 ° and 72 °, in fact, all rays are reflected twice. When ~ = 72 °, the outgoing ray is parallel to the left side of the groove. This general scheme is repeated and can be summarized in the following form. When 180 °
-
n = 1, 2 , . . .
(5)
n
t h e o u t g o i n g ray is vertical, having u n d e r g o n e n reflections. When 180 °
-
n = 1, 2 , . . .
n + 1/2
(6)
t h e o u t g o i n g ray, having u n d e r g o n e n reflections, is parallel t o o n e o f t h e e t c h e d sides. This r a y is parallel t o t h e first r e f l e c t i n g side if n is even, a n d parallel t o t h e o p p o s i n g side if n is o d d . When t h e g r o o v e angle is given b y 180 °
n + 1/2
< ~ <
180 ° - -
n + 1
(7)
a f r a c t i o n o f the i n c i d e n t rays have n + 1 r e f l e c t i o n s , and t h e r e m a i n d e r n reflections. This f r a c t i o n is COS On+ , f.-
(8)
COS 0 1
w h e r e On+, is t h e (n + 1 ) t h i n c i d e n t angle and is f o u n d t o be
On÷,=
180
- - (2n + 1)~
(9)
It should b e n o t e d t h a t fn = 0 f o r ~ = 180°/(n + 1/2) and t h a t fn = 1 f o r ~ = 180°/(n + 1). Table 1 lists t h e critical angles f o r vertical and parallel final r e f l e c t i o n s a n d t h e applicable n u m b e r o f reflections. Once a g r o o v e angle is specified, t h e n u m b e r o f r e f l e c t i o n s can be specified using eqns. (5) - (9). T o calculate t h e t o t a l r e f l e c t i v i t y , we e m p l o y Fresnel's e q u a t i o n s f o r t h e o b l i q u e i n c i d e n c e o f light [8]. T h e electric field v e c t o r f o r t h e i n c i d e n t r a d i a t i o n can b e w r i t t e n as E o -- (Eo, pimp + Eo, Nt~N) e x p { - - i ( c o o t - - ko-r)}
(10)
226 TABLE 1
Critical angles for V-groove reflections n 1 2
n +1
a (deg)
2
1 1~
180
V
120
P
90 72
V P
60 51.43
V P
45
V
40
P
36
V
32.72
P
30
V
1 2~
3 1 3g 4 5
O u t g o i n g ray a
1 4~ 1 5~
6
N u m b e r m o f reflections 1
1 < m < 2
2 2
3
a V, vertical; P, parallel.
where E0. p and E0, N are the parallel and normal components o f the electric field, tip and tiN are the unit vectors along the two polarization directions, co 0 is the angular frequency and k0 is the wavenumber of the radiation. We shall assume that the incident radiation is initially unpolarized, so that ( E o, P)time average -- ( E 0 , N )time average
( 11 )
After one reflection, the electric field is then given by E1 = (El ,p~p + E l . N~N) e x p { - - i ( w 0 t --
where Ea, p and
El, N are
El, P = (Rp(1))l/2Eo, p
ko'r)}
(12)
found from Fresnel's equations: El, N = (RN(I))I/2Eo, N
(13)
and RpO) = tan2( 01 -- 0 i') tan2(01 + 01') (14)
RN(1)= s i n 2 ( 0 1 - - 0 1 ') sin2( 01 + 01') Rp (1) and RN (1) are the first reflection components, and 01 and 01' are the first incident angle and the first refracted angle respectively. They are related by the law of refraction sin 0 1 - - no (15) sin 01'
227 where no is the index of refraction for GaAs and is approximately equal to 3.80. The total reflection coefficient, designated by R ('), is t h e n c o m p u t e d to be R O) - IE112 IEol 2 = Rp(')IE0, pl 2 + RNO)[E0,N]2 IEo, pI 2 + IEo,NI 2
(16)
For unpolarized light the reflection coefficient becomes R(1) = RP ( D + R N (D 2
(17)
If the light rays undergo a second reflection, the electric field vector is written as E2 = (E2,pt~p + E2oNtIN) exp(--i(coot - - k o ' r ) } = (El, p(Rp(2))1/21~p "t- El, N(RN(2))I/2/~N) exp{--i(COot -- ko'r)}
(18)
where the second reflection c o m p o n e n t s are given by Rp(2)
=
t a n 2 ( 0 2 - - 0 2 ') tan2(02 + 02' ) (19)
RN(2) = sin2(02 -- 02') sin2(02 + 02') and, again, 02 and 02' are the second incident and second refracted angles. The total reflection coefficient, after two reflections, becomes
R(2)-
[E2I 2 IE,] 2 Rp(2)Rp(D + RN(2)RN(D
=
2
(20)
for initially unpolarized light [ 9]. In general then, after m reflections, the total reflection coefficient is
R(")
=1_ 2 (ifi = 1RP(i) + py,) m .
(21)
where Rp(i) = tan2(0i -- 0 / ) tan2(0~ + 0t') (22) RN(i) = sin2(0i --0t') sin2(0i + 0j')
228
and 0i and 0~' are the ith incident and refracted angles. For any specified groove angle, the range into which ~ falls must be determined: for range A, 180 ° - -
180 ° ~< ~ ~< -
n
-
(23)
n + 1/2
and, for range B, 180 ° -
-
180 ° <
~ <
- -
n+l/2
(7)
n+l
If ~ lies in range A, the reflection coefficient,is given by R(oO = R(')(cO
If ~ fails into range B, the reflection coefficient becomes R(ol) = R(')(o~)
+
fn{R ("+ 1)(~) _ R(n)(o())
where f , is the fraction given by eqn. (8). The total reflection coefficient is plotted v e r s u s groove angle for a serrated GaAs structure in Fig. 4. The step-like nature of the curve is a consequence of the abruptness of the grooves themselves; the number of reflections varies in a nearly discontinuous fashion with the groove angle. It is apparent from Fig. 4 that, for groove angles less than 20 °, the reflection 0.35
0.30
I.-- 0 . 2 5 Z u.I o u.
0.20 0 L) Z
o_
o.15
u LU J u~
OlO
0.05
0 180
I
I
I
L
160
140
120
I00
GROOVE
ANGLE
I 80
60
40
20
0
(DEG)
Fig. 4. R e f l e c t i o n c o e f f i c i e n t f o r a GaAs V-groove s t r u c t u r e as a f u n c t i o n o f groove angle.
229
coefficient is reduced virtually to zero. For the angle ( a v = 70.5 °) of the ( l l l ) - d e f i n e d groove, the reflection coefficient R(~v) is approximately 0.10, or 72% smaller than the reflection coefficient for a planar surface. On the average, approximately 11% o f the incident rays undergo three reflections, and the remainder two, in a V-groove structure with ~ = ~ v .
3. Inverted-groove structure A different approach will be utilized for the calculation of the reflection coefficient for the inverted-groove structure. We shall assume t h a t the inverted groove is formed with the ( l l l } A planes so t h a t the angular aspect o f the problem is fixed. The groove will be described by the width 2a of the opening and its depth d. Consideration of the etching process used to fashion the groove leads to the conclusions (1) t h a t the b o t t o m of the groove can be approximated by a circular arc with radius of curvature p, (2) that the center of the arc is a fixed point located a distance t from the original surface, (3) t h a t the fixed center may lie on, above or below the surface and (4) that, as the depth of groove is varied, the radius of curvature varies as p =d +t
(24)
It is convenient to define new quantities for this analysis: let 7 = d / 2 a be the aspect ratio, e = p / 2 a t h e radius ratio and ~ = t / 2 a the normalized center offset. Equation (24) then is written as e =7+~
(25)
In the calculation of the total reflection coefficient, the aspect ratio and the center offset will be variable. Figures 5(a) and 5(b) depict the ray-tracing models for the cases t > 0 and t < 0 respectively. A critical horizontal distance from the center O, labeled x*, is shown in these figures. If rays enter the groove at a position less than the distance x* from the center, t h e y will reflect once before exiting. Rays entering between x = x* and x = a will undergo multiple reflections within the groove and will effectively be trapped inside the groove. The distance x* is f o u n d from examining triangles OAB and BCD. Let ~ be the angle between the vertical and the radius of the arc for the critical case x = x*; it is also the angle of incidence. It is clear t h a t tan~ -
X*
(26)
h
and, using the law of sines, sin(2~)
sin(90 ° -- 2~) -
a+x*
(27)
h--t
Combining eqns. (26) and (27) to eliminate h yields x* = t sin(2~) + a cos(2~)
(28)
230
/
\'
(a)
c
t0\ 0 ....
A
t
h
(b) Fig. 5. Ray-tracing m o d e l s f o r t h e i n v e r t e d - g r o o v e s t r u c t u r e : (a) c e n t e r o f t h e arc a b o v e t h e surface, t > 0; (b) c e n t e r o f t h e arc b e l o w t h e surface, t < 0.
b u t since x* = p sin fl
(29)
we can eliminate x* f r o m eqn. (28) t o o b t a i n t h e f o l l o w i n g e q u a t i o n f o r ~: p sin ~ = t sin(2~) + a cos(2fl)
(30)
This m a y be r e w r i t t e n as 1 (7 + X) sin fl = X sin(2fl) + -- cos(2fl) 2
(31)
E q u a t i o n (31) is a t r a n s c e n d e n t a l e q u a t i o n f o r t h e critical angle (which t h e n yields t h e critical distance x*) in t e r m s o f t h e aspect ratio and c e n t e r offset. In t h e special case w h e r e t = 0 (~ = 0), eqn. (31) has t h e analytic s o l u t i o n
231
sin ~ = - - 7-- +
+
(32)
2
w h i c h t h e n implies t h a t
2a
-
--
- -
2
+
+
- -
(33)
When X is n o n - z e r o , eqn. (31) can be solved numerically. This has b e e n carried o u t f o r - - 0 . 3 0 ~< X ~< 0 . 3 0 and 0.50 ~< 3' ~< 4.0. Figure 6 shows a p l o t o f x * / 2 a versus 3' with X used as a p a r a m e t e r to g e n e r a t e a f a m i l y o f curves, T o calculate t h e t o t a l r e f l e c t i o n c o e f f i c i e n t , we shall use t h e a s s u m p t i o n t h a t rays entering t h e g r o o v e f o r - - x * ~< x ~< x* u n d e r g o just o n e r e f l e c t i o n and rays entering b e t w e e n --a < x ~< - - x * and x* ~< x ~ a are t o t a l l y a b s o r b e d . T h e r e f o r e t h e r e f l e c t i o n c o e f f i c i e n t f o r t h e region - - a ~< x < a is given b y R(x) = R o
- - x * <. x ~ x *
R(x) = 0
- - a <<.x <~ - - x *
x * ~ x <<.a
yielding a r e f l e c t i o n c o e f f i c i e n t f o r t h e g r o o v e d region given b y
// o o
c~
-I~
~ o d
~,--0 3
o o
%00
i
0.40
i
0.80
i
,2o
:.so
ASPECT
2'.00 2'.40 RATIO
I
2,o
312o 3:eo
4:00
d 2a
Fig. 6. One-half of the normalized reflection coefficient (or x*/2a) vs. aspect ratio d/2~ for one inverted groove, with the center offset t/2a as a parameter.
232 X X<
(34)
R =R o -a
where R0 is the reflection coefficient for the original planar surface. Hence, Fig. 6 is also a plot of R / 2 R o versus aspect ratio. It should be noted that, when x * / 2 a = 0.5, R = R o, i.e. the reflectance of the grooved area is identical with that of a planar region when x* = a. Figure 6 reveals two important trends concerning the aspect ratio, the center offset and the reflectance. For a given center offset ),, the quantity x * / 2 a approaches 0.5 (R --> R0) as the aspect ratio ~? increases, i.e. as the groove becomes sufficiently deep the curved b o t t o m surface acts no differently from a planar surface, and there is no ray trapping. Secondly, for a given aspect ratio 77 the reflection coefficient decreases with decreasing center offset X; in general, as X decreases, the b o t t o m surface curvature increases, thus increasing the a m o u n t of ray trapping. Hence, in order for the inverted-groove structure to be effective, a premium is placed on producing closely spaced, steeply curved and somewhat shallow grooves. The groove packing density is ultimately limited by the lateral undercut of the grooves, which is determined by the groove depth d and the angle o f the {111)A habit planes. If we call the center-to-center spacing of the grooves 2W and define the normalized half-spacing co as W / 2 a , simple geometric arguments lead to an expression for the maximum aspect ratio (i.e. depth) for a given co: ~/max = 3¢J 2 -- 2cJ +
(35)
Alternatively, for a given aspect ratio, eqn. (35) can be arranged to give the minimum half-spacing: COmin -
1 3
{1 + (3e + 1) 1/2}
(36)
where 1 e = 7 : -- -2
(37)
The total reflection coefficient can then be calculated for a surface serrated with inverted grooves by RT = (1 -- F ) R o + F R
(38)
where F is the fraction of the surface that is grooved: 2a F2W a
W
1 200
(39)
233
Combining eqns. (34), (38) and (39) results in
R T - R0 1
(40)
2co
As a representative case, we pick )~ = 0 and 7 = 1 (d = 2a); from eqn. (33), we have x * / a = 0.73; from eqn. (36), wmm = 0.86, or F~ax = 0.58. As a result, according to eqn. (40), the total reflectance varies in the range 0.84R0~< RT ~< R0 as F varies from F m a x ( 5 8 % ) to F m i n ( 0 % ) . Figure 7 shows total reflectance v e r s u s groove coverage for several different values of X and 7?.
~
~o.,
~-,
o tL
Z O
.J b. w
O
00
0110
0'
• 2o
o:3o
o:40
0:50
PACKING FRACTION
o'.eO
o'.',o
o.so ' "
0":'.90
,:00
F
Fig. 7. Reflection coefficient for a GaAs inverted-groove structure as a function of packing fraction of the grooves, with the aspect ratio and center offset as parameters.
4. Discussion
With both groove geometries, there is a reduced total reflectance and both cells should show power conversion efficiencies substantially larger than planar devices. The change in short-circuit current density will follow the relation AJsc
(1 - - R T ) - - ( 1 - - R o ) -
Jsc
1 --R T -
1 - - Ro
1
1 - - Ro
(41)
234
where R0 is the total reflection of the planar surface and R T the total reflectance of the grooved surface. However, the increase in the surface area of the device will also increase the dark current density J0 by a factor AJo
AT - - A 0
J0
A0 AT
-
1
(42)
A0 where AT is the total surface area o f the grooved device and A 0 the original planar area. This leads generally to a decrease in the open-circuit voltage since AVoc~-q
In
kTq
_
+ AJ0 /
l n \ i + +- ~ /
_
--ln
\ J 0 ]t
1
_ __kTql n I ( 1 - - R T ) / ( 1 - - R 0 )
~ 0
(43)
The area ratio AT/A o will typically be considerably larger than unity, while the numerator cannot be larger than approximately 1.5. Hence AVoc will be a negative quantity but not large, in general, because of the logarithmic dependence. Nevertheless, there is still a net gain in power conversion efficiency because A~
_
~7
AP P
-
~Jsc Js~
+
~Yoo Yo~
AJso Js~
(44)
This simplified calculation assumes that the photocarrier collection efficiency and the fill factor are both unity. A more realistic calculation o f the increase in conversion efficiency should include additional collection losses and fill factor losses associated with the new geometry.
4.1. V-groove structure As shown in Fig. 4, R T ~ 0 for a groove angle ~ of a b o u t 20 °. The total surface area o f the groove is given by
235 AT
2 sin(90
A0
° --
a/2)
sin a
(45)
In this case AJsc
1
J~c
1 -- Ro = 0.54
and, since AT/Ao = 5.79,
kT ln(1.54 t q \5-~/
AVo~ ~ - -
--0.033 V at r o o m temperature. Thus, the maximum conversion efficiency change will be At? --
51%
~
where it has been assumed that Vo¢ ~ 1 V. This improvement is substantial, b u t it must be pointed o u t that significant processing difficulties must be overcome before such a structure could be fabricated. In the first place, etching uniform grooves with a 20 ° groove angle, or any angle other than one formed by the intersection of low index planes, is an u n k n o w n processing step at the present time. Moreover, overlay metaUization will most probably be unable to cover conformally the sharp peaks and valleys of this structure. However, there are no substantial fabrication problems associated with V-grooves whose groove angle C~v is 70.5 °. Etchants that reveal ( l l l } A habit planes are well k n o w n [7, 10], and either overlay metaUization or angleshadow metallization [3] can be employed. When a = ~v, the performance enhancement becomes AJsc
1 --
Jsc
0.098
1 -- 0.35 = 0.39
A J0
Jo
- 0.73
AVo¢ ~
Vo¢ yielding
--0.005
236 A~ --
~
38%
This factor, o f course, represents the best possible efficiency increase for this structure.
4.2. Inverted-groove structure One i m p o r t a n t f a c t o r involved in the analysis o f t h e inverted-groove cell is the center o f f s e t ?,. In the absence o f experimental results the analysis included a positive, zero or negative offset. In all likelihood, t h e preferential e t c h a n t p r o b a b l y behaves as an isotropic e t c h a n t for small variations in the etching direction f r o m the [001] direction. In this case the etching " c e n t e r " is m o s t p r o b a b l y on t h e original surface. The ratio o f t h e total surface area o f t h e groove to t h e original planar area is AT / cos 0 ) Aoo - 2 7 [ s i n 5-4-7 ° + 0
(46)
where 0 is one-half o f the angle s u b t e n d e d by the b o t t o m surface and is found from sin 0 -- cos 0 -
1
(47)
27 Picking 7 = 1 (i.e. t h e groove d e p t h equal to t h e groove width), we have RT(min ) = 0 . 8 4 R 0 = 0.28 AJ~c
1
J~
--
0.28
1 -- 0.35
--1
= 0.10 A J0
AT
J0
A0 = 2.3
(0 ~ 65.9 ° )
AVoc ~ - - 0 . 0 2 6 V yielding --
~
7.6%
77 Additional i m p r o v e m e n t s are f o u n d by decreasing the aspect ratio 7 further. In fact, a m o s t curious case is o b t a i n e d w h e n the inverted groove has the parameters k = 0 and 7 = 1/2. In this instance, w h e n the groove d e p t h equals
237
half of the spacing, the groove actually is semicircular in cross section. For zero center offset this turns o u t to be the o p t i m u m case for inverted-groove geometry! As shown in Fig. 6, the normalized reflection coefficient for ~ = 1/2 is 0.5. According to eqns. (36) and (39), the groove fraction F can, in principle, be unity. This would necessarily be reduced below unity in practice because o f the need for metallization stripes o f finite width. Nevertheless, using eqns. (40) - (44), we obtain RT(min ) = 0.5R 0 = 0.175 AJs¢
1 --
Js¢
0.175
1 -- 0.35 = 0.27
AJ 0 -
J0
1
2 = 0.57
Voc ~ --0.005 V yielding A~
~ 26% 77 This is a dramatic increase from the first inverted-groove example (7 = 1; A~/~? = 7.4%), and it indicates again the importance o f employing shallow grooves. Additional increases would be possible if, during processing, we could force the center offset to be negative. Substantial ray trapping occurs for X < 0, as depicted in Fig. 6, leading to an extremely low total reflectance.
5. Summarizing remarks The use of an anisotropic etchant during the processing of semic o n d u c t o r solar cells can produce cells with non-planar grooved surfaces. The grooved surface permits multiple reflections to take place; this, in turn, enhances the total solar absorption and power conversion efficiency. In I I I - V semiconductors, whose lattices lack inversion s y m m e t r y , the grooves can be either V shaped or inverted depending on the groove orientation. Ray-tracing models were developed to help to determine the total reflectivity of the etched surfaces. For the V-groove structure, it was shown that the total reflectivity (a) remains the same as a planar surface if the groove angle is larger than 120 °, (b) is essentially zero if the groove angle is less than
238 20 ° and (c) is reduced by about 72% from the planar structure if the groove is formed by the (111}A planes (c~v = 70.5°). The change in the conversion efficiency, under ideal circumstances, can be approximately +38% for the 70.5 ° groove and as high as +51% for the 20 ° groove. Although 70.5 ° Vgrooved silicon solar cells have been reported to show an efficiency gain of only 13% [1], optimization of the junction depth and other cell parameters should push this improvement value closer to the theoretical limit. As pointed out, the construction of a 20 ° V-groove sample awaits the development of new etching and processing techniques. In general, the inverted-groove structure is predicted to show an enhanced efficiency when compared with a similar planar structure; the enhancement, however, would be smaller than for a V-groove structure with similar design rules. An inverted-groove cell with a depth-to-width ratio of unity may show a conversion efficiency enhancement of only 7.5% with respect to a planar sample. Yet the inverted-groove analysis shows that, if a perfectly isotropic etchant is employed to produce a groove with a semicircular cross section, the conversion efficiency can increase by as much as 26%. This suggests that a simple way to improve any solar cell is to allow the metallization grid to act as a mask and to subject the cell to an isotropic etchant. The resulting round grooves in the unmetallized regions will decrease the reflectivity and increase the cell's efficiency. In short, there is no compelling reason to "remain planar", when one final processing step can produce a clear-cut performance advantage. A note should be added concerning the dimensions of the grooves. We have used a ray-tracing model in this analysis, and in order for it to have general validity it is necessary that the groove dimensions be larger than the wavelength of light. We expect that devices with groove spacings of the order of 5 pm or larger (depths of 7 pm or larger) will be adequately covered by the analysis presented in this paper. Furthermore, for groove dimensions approaching the minority carrier diffusion lengths (which can be 1 pm or more in GaAs), a two-dimensional solution of the diffusion equation, rather than a one-dimensional analysis, will be required. The fabrication of both 7 0 . 5 ° V-groove and inverted-groove GaAs/ Gal _xAlxAs solar cells is under way in our laboratory. Both groove structures will use 32 pm spacing design rules. The processing procedure, performance of the finished cells and comparison with this analysis will be presented in a future paper.
References
1 R. A. Arndt, J. F. Allison, J. G. Haynos and A. Mealenberg, Jr., Optical properties of the COMSAT non-reflective cell, Proc. 1 l th Photovoltaic Specialists' Con f , Phoenix, A Z , May 6 - 8, 1975, IEEE, New York, 1975, p. 40. 2 C. R. Baraona and H. W. Brandhorst, V-grooved silicon solar cells, Proc. 11th Photovoltaic Specialists' C o n f , Phoenix, A Z , May 6 - 8, 1975, IEEE, New York, 1975, p. 44.
239
3 P. G. Borden and R. V. Walsh, Silicon solar cell with a novel low-resistance emitter structure, Appl. Phys. Left., 41 ( 1 9 8 2 ) 6 4 9 . 4 B. Dale and H. G. Rundeberg, Photovoltaic conversion, 1, High efficiency silicon solar cells, Proc. 14th Annu. Power Sources Conf., U.S. A r m y Signal Research and Development Laboratory, Fort Monmouth, NJ, May 17 - 19, 1960. 5 L. W. James and R. L. Moon, GaAs concentrator solar cell, Appl. Phys. Lett., 26 (1975) 467. 6 H. C. Gatos and M. C. Lavine, Etching and inhibition of the {111} surfaces o f the IIIV intermetallic compounds: InSb, J. Phys. Chem. Solids, 14 (1960) 169. 7 Y. Tarui, Y. Komiya and Y. Harada, Preferential etching and etched profile of GaAs, J. Electrochem. Soc., 118 (1971) 118. 8 D. Corson and P. Lorrain, Introduction to Electromagnetic Fields and Waves, Freeman, San Francisco, CA, 1962, Chap. 11. 9 B. L. Sopori and R. A. Pryor, Design of antireflection coatings for textured silicon solar cells, Sol. Cells, 8 (1983) 249. 10 W. T. Tsang and S. Wang, Profile and groove-depth control in GaAs diffraction gratings fabricated by preferential chemical etching in H2SO4-H202-H20 systems, Appl. Phys. Lett., 28 ( 1 9 7 6 ) 4 4 .
221
THE DESIGN OF ANISOTROPICALLY ETCHED III-V SOLAR CELLS
R. J. ROEDEL and P. M. HOLM
Department of Electrical and Computer Engineering, Arizona State University, Tempe, AZ 85287 (U.S.A.) (Received June 7, 1983; accepted September 26, 1983)
Summary A solar cell with negligible reflectivity requires a departure from conventional planar design. An anisotropically etched (grooved) surface modifies the geometry o f the cell to permit multiple reflections and enhanced absorption. Two distinct groove geometries can be generated in III-V semiconductors because of the lack of inversion s y m m e t r y in zinc blende lattices. The grooves are said to be "V shaped" or "inverted". In this paper, ray-tracing models are developed to analyze novel solar cells based on these two groove types. The models treat groove angle, spacing and aspect ratio as variables, and appropriate conclusions are drawn for the case where the grooves are formed by intersecting { l l l } A planes. It is shown (1) that, for a structure with V-grooves, the m a x i m u m conversion efficiency can be increased by about 38% with respect to a planar structure and (2) that, for a structure with inverted grooves, the efficiency increase can be as high as 26%.
1. Introduction To fabricate solar cells with m a x i m u m efficiency, it is necessary to reduce the reflectivity of the surface to zero, if possible. Most traditional solar cell designs have used polished planar surfaces, and the reflectivity for normally incident light (which is essentially independent o f wavelength for 0.5 ~zm < k0 < 1.0 pm) is approximately 35% for both silicon and GaAs. The use of antireflection coatings can reduce the reflectivity to approximately 10% (single-layer cells) or 7% (multiple-layer cells) in silicon cells [ 1 ]. Further reductions are possible only with modifications in the geometry of the cell. Recently, several reports have shown that reduced reflectivity can be achieved in silicon solar cells by preferentially etching the silicon surface [1 4]. The resulting surface is no longer planar and can be said to be "serrated" if the etching produces regularly spaced grooves or to be " t e x t u r e d " if the etching produces randomly spaced and sized etch features. Without antireflection coatings, the serrated silicon surface has an average -
0379-6787/84/$3.00
© Elsevier Sequoia/Printed in The Netherlands
222
reduced reflectivity of roughly 22% (for 35% coverage of the surface with grooves), and the textured silicon surface approximately 12% [1, 2]. The application of antireflection coatings reduces these values even further. Solar cells fabricated from these etched substrates demonstrate an enhanced shortcircuit current and spectral response. Clearly, the addition of one relatively simple etching step to the fabrication process can yield solar cells with dramatically improved performance. Solar cells constructed from I I I - V semiconductors (such as the Gal _xAlxAs/GaAs system) are promising candidates for cost~ffective photovoltaic energy conversion in high sunlight concentration applications [5]. The design of preferentially etched I I I - V solar cells is the subject of this paper. Unlike silicon, all III-V semiconductors possess a non-centrosymmetric lattice, and this has a major bearing on the geometry of etch figures produced by an anisotropic etchant. The preferentially etched silicon substrates (with an original (001) surface orientation) have (ll0)-oriented grooves with (111) sides in the serrated structures, or tetrahedra with {110)oriented bases and (111) sides in the textured structures. In contrast, because the (111) planes terminate on group III atoms and the (111) planes terminate on group V atoms in III-V semiconductors [6], and because of the enormous difference in the chemical reactivity of the (111) and the ( i i l ) planes [7], two distinct groove geometries can be obtained on (001) I I I - V surfaces. As shown in Fig. 1, when the mask stripes are oriented in the [ l i 0 ] direction, a " n o r m a l " V-groove with (111) and (:~11) side walls can be produced with an anisotropic etchant (such as 1% bromine in methanol) but, when the mask stripes are oriented in the [110] direction, an "inverted" Vgroove with (111) and ( 1 1 i ) side walls is formed. All four of these side wall habit planes are of the { l l l ) A type; the ( l l l } B planes are either not reached (V-groove) or not effective as stop etch planes (inverted groove). In general, the following rule is o b e y e d for anisotropic etchants in I I I - V semiconductors [ 7 ] :
Fig. 1. Groove geometries applicable for GaAs (001) subjected to an anisotropic etchant through (110)-oriented windows.
223 r {110}~ r{iii}B ~ r {100}~ r{lll}A
where r is the etchant rate. Either groove configuration can form the basis for a novel geometry I I I - V solar cell, as shown in Fig. 2. Both cells (constructed from GaAs for illustrative purposes) would require junction formation through p-type diffusion, a p-type wide band gap window layer and stripe metallization for efficient operation. An analysis of the total reflectivity for V-grooved and inverted-grooved cells is carried o u t in Sections 2 and 3; this is followed b y a discussion of the expected efficiency improvements for b o t h structures and a summary of the results. fp
- COl. x AIxAS
% ' p - GaAs DIFFUSED LAYER n -GaAs EPITAXIAL LAYER
n - GoAs SUBSTRATE
(a)
n - GoAs EPITAXIAL
LAYER
r,/p-
Gal. x AIxAs
~"p-
GoAs
DIFFUSED LAYER
n - GaAs SUBSTRATE
(b) Fig. 2. Cross-sectional views of anisotropically etched GaAs solar cells: (a) the normal Vgroove cell; (b) the inverted-groove cell. Both cells are shown with a diffused p-type layer and a p-type Ga]-xAlxAs window layer.
2. V-groove structure The major difference between a planar surface and a grooved surface is that there exists a possibility for multiple reflections o f the incident light rays with the serrated surface. Multiple reflections decrease the total reflectivity of the cell and thus permit a greater a m o u n t of photoabsorption. Although the V-grooves described in Section 1 are formed b y { l l l } A planes and have a fixed groove angle given by
av = 1SO° -- cos-I {(111)" ( i l l ) t
\i Fl -fi]
= 70.5 °
(1)
224 in this s e c t i o n we shall calculate t h e r e f l e c t i o n c o e f f i c i e n t for a serrated surface as a f u n c t i o n o f t h e g r o o v e angle a. Using these results, we shall find the t o t a l r e f l e c t i v i t y at ~ = a v When ~ = 180 °, t h e surface is planar, and i n c i d e n t rays can strike t h e surface o n l y once. In fact, f o r g r o o v e angles b e t w e e n 180 ° and 120 ° , t h e r e can b e o n l y o n e r e f l e c t i o n , for with specular r e f l e c t i o n and vertically i n c i d e n t light t h e o u t g o i n g rays c a n n o t i n t e r c e p t t h e o p p o s i t e side o f t h e Vgroove. T h e i n c i d e n t angle is given b y 01 = 90° -- -2
(2)
When ~ = 120 °, 01 -- 30°, and t h e o u t g o i n g ray is parallel t o t h e o p p o s i t e g r o o v e side. This is d e p i c t e d in Fig. 3(a). When ~ is r e d u c e d f r o m 120 °, say to 95 ° as s h o w n in Fig. 3(b), a f r a c t i o n o f t h e rays striking t h e left side o f t h e g r o o v e will be involved in a s e c o n d r e f l e c t i o n f r o m t h e right side o f the groove. Rays i n c i d e n t near t h e t o p w i l l n o t i n t e r c e p t t h e o p p o s i t e side. Using
/ /
(a)
/
/
//// / / / \/e,/
(h)~
\
/ ~ / /
=
"'///
Fig. 3. R a y - t r a c i n g m o d e l s f o r t h e V - g r o o v e s t r u c t u r e : ( a ) g r o o v e a n g l e 0t = 1 2 0 °, o n l y o n e r e f l e c t i o n w i t h 01 = 3 0 ° ; ( b ) g r o o v e a n g l e 0t = 9 5 °, a f r a c t i o n o f t h e r a y s are r e f l e c t e d t w i c e w i t h 01 = 4 2 . 5 ° a n d 02 = 4 7 . 5 ° •
225
t h e law o f sines it can b e s h o w n t h a t t h e f r a c t i o n o f rays t h a t can be r e f l e c t e d t w i c e is cos 02 fl
- - -
(3)
cos 0 ,
w h e r e 02 is t h e s e c o n d i n c i d e n t angle and is given b y 02=ct--O, 3o~ -
90 °
(4)
2 At a groove angle (~ of 90 °, all rays will be reflected twice, and the outgoing ray is vertical. Between the angles of 90 ° and 72 °, in fact, all rays are reflected twice. When ~ = 72 °, the outgoing ray is parallel to the left side of the groove. This general scheme is repeated and can be summarized in the following form. When 180 °
-
n = 1, 2 , . . .
(5)
n
t h e o u t g o i n g ray is vertical, having u n d e r g o n e n reflections. When 180 °
-
n = 1, 2 , . . .
n + 1/2
(6)
t h e o u t g o i n g ray, having u n d e r g o n e n reflections, is parallel t o o n e o f t h e e t c h e d sides. This r a y is parallel t o t h e first r e f l e c t i n g side if n is even, a n d parallel t o t h e o p p o s i n g side if n is o d d . When t h e g r o o v e angle is given b y 180 °
n + 1/2
< ~ <
180 ° - -
n + 1
(7)
a f r a c t i o n o f the i n c i d e n t rays have n + 1 r e f l e c t i o n s , and t h e r e m a i n d e r n reflections. This f r a c t i o n is COS On+ , f.-
(8)
COS 0 1
w h e r e On+, is t h e (n + 1 ) t h i n c i d e n t angle and is f o u n d t o be
On÷,=
180
- - (2n + 1)~
(9)
It should b e n o t e d t h a t fn = 0 f o r ~ = 180°/(n + 1/2) and t h a t fn = 1 f o r ~ = 180°/(n + 1). Table 1 lists t h e critical angles f o r vertical and parallel final r e f l e c t i o n s a n d t h e applicable n u m b e r o f reflections. Once a g r o o v e angle is specified, t h e n u m b e r o f r e f l e c t i o n s can be specified using eqns. (5) - (9). T o calculate t h e t o t a l r e f l e c t i v i t y , we e m p l o y Fresnel's e q u a t i o n s f o r t h e o b l i q u e i n c i d e n c e o f light [8]. T h e electric field v e c t o r f o r t h e i n c i d e n t r a d i a t i o n can b e w r i t t e n as E o -- (Eo, pimp + Eo, Nt~N) e x p { - - i ( c o o t - - ko-r)}
(10)
226 TABLE 1
Critical angles for V-groove reflections n 1 2
n +1
a (deg)
2
1 1~
180
V
120
P
90 72
V P
60 51.43
V P
45
V
40
P
36
V
32.72
P
30
V
1 2~
3 1 3g 4 5
O u t g o i n g ray a
1 4~ 1 5~
6
N u m b e r m o f reflections 1
1 < m < 2
2 2
3
a V, vertical; P, parallel.
where E0. p and E0, N are the parallel and normal components o f the electric field, tip and tiN are the unit vectors along the two polarization directions, co 0 is the angular frequency and k0 is the wavenumber of the radiation. We shall assume that the incident radiation is initially unpolarized, so that ( E o, P)time average -- ( E 0 , N )time average
( 11 )
After one reflection, the electric field is then given by E1 = (El ,p~p + E l . N~N) e x p { - - i ( w 0 t --
where Ea, p and
El, N are
El, P = (Rp(1))l/2Eo, p
ko'r)}
(12)
found from Fresnel's equations: El, N = (RN(I))I/2Eo, N
(13)
and RpO) = tan2( 01 -- 0 i') tan2(01 + 01') (14)
RN(1)= s i n 2 ( 0 1 - - 0 1 ') sin2( 01 + 01') Rp (1) and RN (1) are the first reflection components, and 01 and 01' are the first incident angle and the first refracted angle respectively. They are related by the law of refraction sin 0 1 - - no (15) sin 01'
227 where no is the index of refraction for GaAs and is approximately equal to 3.80. The total reflection coefficient, designated by R ('), is t h e n c o m p u t e d to be R O) - IE112 IEol 2 = Rp(')IE0, pl 2 + RNO)[E0,N]2 IEo, pI 2 + IEo,NI 2
(16)
For unpolarized light the reflection coefficient becomes R(1) = RP ( D + R N (D 2
(17)
If the light rays undergo a second reflection, the electric field vector is written as E2 = (E2,pt~p + E2oNtIN) exp(--i(coot - - k o ' r ) } = (El, p(Rp(2))1/21~p "t- El, N(RN(2))I/2/~N) exp{--i(COot -- ko'r)}
(18)
where the second reflection c o m p o n e n t s are given by Rp(2)
=
t a n 2 ( 0 2 - - 0 2 ') tan2(02 + 02' ) (19)
RN(2) = sin2(02 -- 02') sin2(02 + 02') and, again, 02 and 02' are the second incident and second refracted angles. The total reflection coefficient, after two reflections, becomes
R(2)-
[E2I 2 IE,] 2 Rp(2)Rp(D + RN(2)RN(D
=
2
(20)
for initially unpolarized light [ 9]. In general then, after m reflections, the total reflection coefficient is
R(")
=1_ 2 (ifi = 1RP(i) + py,) m .
(21)
where Rp(i) = tan2(0i -- 0 / ) tan2(0~ + 0t') (22) RN(i) = sin2(0i --0t') sin2(0i + 0j')
228
and 0i and 0~' are the ith incident and refracted angles. For any specified groove angle, the range into which ~ falls must be determined: for range A, 180 ° - -
180 ° ~< ~ ~< -
n
-
(23)
n + 1/2
and, for range B, 180 ° -
-
180 ° <
~ <
- -
n+l/2
(7)
n+l
If ~ lies in range A, the reflection coefficient,is given by R(oO = R(')(cO
If ~ fails into range B, the reflection coefficient becomes R(ol) = R(')(o~)
+
fn{R ("+ 1)(~) _ R(n)(o())
where f , is the fraction given by eqn. (8). The total reflection coefficient is plotted v e r s u s groove angle for a serrated GaAs structure in Fig. 4. The step-like nature of the curve is a consequence of the abruptness of the grooves themselves; the number of reflections varies in a nearly discontinuous fashion with the groove angle. It is apparent from Fig. 4 that, for groove angles less than 20 °, the reflection 0.35
0.30
I.-- 0 . 2 5 Z u.I o u.
0.20 0 L) Z
o_
o.15
u LU J u~
OlO
0.05
0 180
I
I
I
L
160
140
120
I00
GROOVE
ANGLE
I 80
60
40
20
0
(DEG)
Fig. 4. R e f l e c t i o n c o e f f i c i e n t f o r a GaAs V-groove s t r u c t u r e as a f u n c t i o n o f groove angle.
229
coefficient is reduced virtually to zero. For the angle ( a v = 70.5 °) of the ( l l l ) - d e f i n e d groove, the reflection coefficient R(~v) is approximately 0.10, or 72% smaller than the reflection coefficient for a planar surface. On the average, approximately 11% o f the incident rays undergo three reflections, and the remainder two, in a V-groove structure with ~ = ~ v .
3. Inverted-groove structure A different approach will be utilized for the calculation of the reflection coefficient for the inverted-groove structure. We shall assume t h a t the inverted groove is formed with the ( l l l } A planes so t h a t the angular aspect o f the problem is fixed. The groove will be described by the width 2a of the opening and its depth d. Consideration of the etching process used to fashion the groove leads to the conclusions (1) t h a t the b o t t o m of the groove can be approximated by a circular arc with radius of curvature p, (2) that the center of the arc is a fixed point located a distance t from the original surface, (3) t h a t the fixed center may lie on, above or below the surface and (4) that, as the depth of groove is varied, the radius of curvature varies as p =d +t
(24)
It is convenient to define new quantities for this analysis: let 7 = d / 2 a be the aspect ratio, e = p / 2 a t h e radius ratio and ~ = t / 2 a the normalized center offset. Equation (24) then is written as e =7+~
(25)
In the calculation of the total reflection coefficient, the aspect ratio and the center offset will be variable. Figures 5(a) and 5(b) depict the ray-tracing models for the cases t > 0 and t < 0 respectively. A critical horizontal distance from the center O, labeled x*, is shown in these figures. If rays enter the groove at a position less than the distance x* from the center, t h e y will reflect once before exiting. Rays entering between x = x* and x = a will undergo multiple reflections within the groove and will effectively be trapped inside the groove. The distance x* is f o u n d from examining triangles OAB and BCD. Let ~ be the angle between the vertical and the radius of the arc for the critical case x = x*; it is also the angle of incidence. It is clear t h a t tan~ -
X*
(26)
h
and, using the law of sines, sin(2~)
sin(90 ° -- 2~) -
a+x*
(27)
h--t
Combining eqns. (26) and (27) to eliminate h yields x* = t sin(2~) + a cos(2~)
(28)
230
/
\'
(a)
c
t0\ 0 ....
A
t
h
(b) Fig. 5. Ray-tracing m o d e l s f o r t h e i n v e r t e d - g r o o v e s t r u c t u r e : (a) c e n t e r o f t h e arc a b o v e t h e surface, t > 0; (b) c e n t e r o f t h e arc b e l o w t h e surface, t < 0.
b u t since x* = p sin fl
(29)
we can eliminate x* f r o m eqn. (28) t o o b t a i n t h e f o l l o w i n g e q u a t i o n f o r ~: p sin ~ = t sin(2~) + a cos(2fl)
(30)
This m a y be r e w r i t t e n as 1 (7 + X) sin fl = X sin(2fl) + -- cos(2fl) 2
(31)
E q u a t i o n (31) is a t r a n s c e n d e n t a l e q u a t i o n f o r t h e critical angle (which t h e n yields t h e critical distance x*) in t e r m s o f t h e aspect ratio and c e n t e r offset. In t h e special case w h e r e t = 0 (~ = 0), eqn. (31) has t h e analytic s o l u t i o n
231
sin ~ = - - 7-- +
+
(32)
2
w h i c h t h e n implies t h a t
2a
-
--
- -
2
+
+
- -
(33)
When X is n o n - z e r o , eqn. (31) can be solved numerically. This has b e e n carried o u t f o r - - 0 . 3 0 ~< X ~< 0 . 3 0 and 0.50 ~< 3' ~< 4.0. Figure 6 shows a p l o t o f x * / 2 a versus 3' with X used as a p a r a m e t e r to g e n e r a t e a f a m i l y o f curves, T o calculate t h e t o t a l r e f l e c t i o n c o e f f i c i e n t , we shall use t h e a s s u m p t i o n t h a t rays entering t h e g r o o v e f o r - - x * ~< x ~< x* u n d e r g o just o n e r e f l e c t i o n and rays entering b e t w e e n --a < x ~< - - x * and x* ~< x ~ a are t o t a l l y a b s o r b e d . T h e r e f o r e t h e r e f l e c t i o n c o e f f i c i e n t f o r t h e region - - a ~< x < a is given b y R(x) = R o
- - x * <. x ~ x *
R(x) = 0
- - a <<.x <~ - - x *
x * ~ x <<.a
yielding a r e f l e c t i o n c o e f f i c i e n t f o r t h e g r o o v e d region given b y
// o o
c~
-I~
~ o d
~,--0 3
o o
%00
i
0.40
i
0.80
i
,2o
:.so
ASPECT
2'.00 2'.40 RATIO
I
2,o
312o 3:eo
4:00
d 2a
Fig. 6. One-half of the normalized reflection coefficient (or x*/2a) vs. aspect ratio d/2~ for one inverted groove, with the center offset t/2a as a parameter.
232 X X<
(34)
R =R o -a
where R0 is the reflection coefficient for the original planar surface. Hence, Fig. 6 is also a plot of R / 2 R o versus aspect ratio. It should be noted that, when x * / 2 a = 0.5, R = R o, i.e. the reflectance of the grooved area is identical with that of a planar region when x* = a. Figure 6 reveals two important trends concerning the aspect ratio, the center offset and the reflectance. For a given center offset ),, the quantity x * / 2 a approaches 0.5 (R --> R0) as the aspect ratio ~? increases, i.e. as the groove becomes sufficiently deep the curved b o t t o m surface acts no differently from a planar surface, and there is no ray trapping. Secondly, for a given aspect ratio 77 the reflection coefficient decreases with decreasing center offset X; in general, as X decreases, the b o t t o m surface curvature increases, thus increasing the a m o u n t of ray trapping. Hence, in order for the inverted-groove structure to be effective, a premium is placed on producing closely spaced, steeply curved and somewhat shallow grooves. The groove packing density is ultimately limited by the lateral undercut of the grooves, which is determined by the groove depth d and the angle o f the {111)A habit planes. If we call the center-to-center spacing of the grooves 2W and define the normalized half-spacing co as W / 2 a , simple geometric arguments lead to an expression for the maximum aspect ratio (i.e. depth) for a given co: ~/max = 3¢J 2 -- 2cJ +
(35)
Alternatively, for a given aspect ratio, eqn. (35) can be arranged to give the minimum half-spacing: COmin -
1 3
{1 + (3e + 1) 1/2}
(36)
where 1 e = 7 : -- -2
(37)
The total reflection coefficient can then be calculated for a surface serrated with inverted grooves by RT = (1 -- F ) R o + F R
(38)
where F is the fraction of the surface that is grooved: 2a F2W a
W
1 200
(39)
233
Combining eqns. (34), (38) and (39) results in
R T - R0 1
(40)
2co
As a representative case, we pick )~ = 0 and 7 = 1 (d = 2a); from eqn. (33), we have x * / a = 0.73; from eqn. (36), wmm = 0.86, or F~ax = 0.58. As a result, according to eqn. (40), the total reflectance varies in the range 0.84R0~< RT ~< R0 as F varies from F m a x ( 5 8 % ) to F m i n ( 0 % ) . Figure 7 shows total reflectance v e r s u s groove coverage for several different values of X and 7?.
~
~o.,
~-,
o tL
Z O
.J b. w
O
00
0110
0'
• 2o
o:3o
o:40
0:50
PACKING FRACTION
o'.eO
o'.',o
o.so ' "
0":'.90
,:00
F
Fig. 7. Reflection coefficient for a GaAs inverted-groove structure as a function of packing fraction of the grooves, with the aspect ratio and center offset as parameters.
4. Discussion
With both groove geometries, there is a reduced total reflectance and both cells should show power conversion efficiencies substantially larger than planar devices. The change in short-circuit current density will follow the relation AJsc
(1 - - R T ) - - ( 1 - - R o ) -
Jsc
1 --R T -
1 - - Ro
1
1 - - Ro
(41)
234
where R0 is the total reflection of the planar surface and R T the total reflectance of the grooved surface. However, the increase in the surface area of the device will also increase the dark current density J0 by a factor AJo
AT - - A 0
J0
A0 AT
-
1
(42)
A0 where AT is the total surface area o f the grooved device and A 0 the original planar area. This leads generally to a decrease in the open-circuit voltage since AVoc~-q
In
kTq
_
+ AJ0 /
l n \ i + +- ~ /
_
--ln
\ J 0 ]t
1
_ __kTql n I ( 1 - - R T ) / ( 1 - - R 0 )
~ 0
(43)
The area ratio AT/A o will typically be considerably larger than unity, while the numerator cannot be larger than approximately 1.5. Hence AVoc will be a negative quantity but not large, in general, because of the logarithmic dependence. Nevertheless, there is still a net gain in power conversion efficiency because A~
_
~7
AP P
-
~Jsc Js~
+
~Yoo Yo~
AJso Js~
(44)
This simplified calculation assumes that the photocarrier collection efficiency and the fill factor are both unity. A more realistic calculation o f the increase in conversion efficiency should include additional collection losses and fill factor losses associated with the new geometry.
4.1. V-groove structure As shown in Fig. 4, R T ~ 0 for a groove angle ~ of a b o u t 20 °. The total surface area o f the groove is given by
235 AT
2 sin(90
A0
° --
a/2)
sin a
(45)
In this case AJsc
1
J~c
1 -- Ro = 0.54
and, since AT/Ao = 5.79,
kT ln(1.54 t q \5-~/
AVo~ ~ - -
--0.033 V at r o o m temperature. Thus, the maximum conversion efficiency change will be At? --
51%
~
where it has been assumed that Vo¢ ~ 1 V. This improvement is substantial, b u t it must be pointed o u t that significant processing difficulties must be overcome before such a structure could be fabricated. In the first place, etching uniform grooves with a 20 ° groove angle, or any angle other than one formed by the intersection of low index planes, is an u n k n o w n processing step at the present time. Moreover, overlay metaUization will most probably be unable to cover conformally the sharp peaks and valleys of this structure. However, there are no substantial fabrication problems associated with V-grooves whose groove angle C~v is 70.5 °. Etchants that reveal ( l l l } A habit planes are well k n o w n [7, 10], and either overlay metaUization or angleshadow metallization [3] can be employed. When a = ~v, the performance enhancement becomes AJsc
1 --
Jsc
0.098
1 -- 0.35 = 0.39
A J0
Jo
- 0.73
AVo¢ ~
Vo¢ yielding
--0.005
236 A~ --
~
38%
This factor, o f course, represents the best possible efficiency increase for this structure.
4.2. Inverted-groove structure One i m p o r t a n t f a c t o r involved in the analysis o f t h e inverted-groove cell is the center o f f s e t ?,. In the absence o f experimental results the analysis included a positive, zero or negative offset. In all likelihood, t h e preferential e t c h a n t p r o b a b l y behaves as an isotropic e t c h a n t for small variations in the etching direction f r o m the [001] direction. In this case the etching " c e n t e r " is m o s t p r o b a b l y on t h e original surface. The ratio o f t h e total surface area o f t h e groove to t h e original planar area is AT / cos 0 ) Aoo - 2 7 [ s i n 5-4-7 ° + 0
(46)
where 0 is one-half o f the angle s u b t e n d e d by the b o t t o m surface and is found from sin 0 -- cos 0 -
1
(47)
27 Picking 7 = 1 (i.e. t h e groove d e p t h equal to t h e groove width), we have RT(min ) = 0 . 8 4 R 0 = 0.28 AJ~c
1
J~
--
0.28
1 -- 0.35
--1
= 0.10 A J0
AT
J0
A0 = 2.3
(0 ~ 65.9 ° )
AVoc ~ - - 0 . 0 2 6 V yielding --
~
7.6%
77 Additional i m p r o v e m e n t s are f o u n d by decreasing the aspect ratio 7 further. In fact, a m o s t curious case is o b t a i n e d w h e n the inverted groove has the parameters k = 0 and 7 = 1/2. In this instance, w h e n the groove d e p t h equals
237
half of the spacing, the groove actually is semicircular in cross section. For zero center offset this turns o u t to be the o p t i m u m case for inverted-groove geometry! As shown in Fig. 6, the normalized reflection coefficient for ~ = 1/2 is 0.5. According to eqns. (36) and (39), the groove fraction F can, in principle, be unity. This would necessarily be reduced below unity in practice because o f the need for metallization stripes o f finite width. Nevertheless, using eqns. (40) - (44), we obtain RT(min ) = 0.5R 0 = 0.175 AJs¢
1 --
Js¢
0.175
1 -- 0.35 = 0.27
AJ 0 -
J0
1
2 = 0.57
Voc ~ --0.005 V yielding A~
~ 26% 77 This is a dramatic increase from the first inverted-groove example (7 = 1; A~/~? = 7.4%), and it indicates again the importance o f employing shallow grooves. Additional increases would be possible if, during processing, we could force the center offset to be negative. Substantial ray trapping occurs for X < 0, as depicted in Fig. 6, leading to an extremely low total reflectance.
5. Summarizing remarks The use of an anisotropic etchant during the processing of semic o n d u c t o r solar cells can produce cells with non-planar grooved surfaces. The grooved surface permits multiple reflections to take place; this, in turn, enhances the total solar absorption and power conversion efficiency. In I I I - V semiconductors, whose lattices lack inversion s y m m e t r y , the grooves can be either V shaped or inverted depending on the groove orientation. Ray-tracing models were developed to help to determine the total reflectivity of the etched surfaces. For the V-groove structure, it was shown that the total reflectivity (a) remains the same as a planar surface if the groove angle is larger than 120 °, (b) is essentially zero if the groove angle is less than
238 20 ° and (c) is reduced by about 72% from the planar structure if the groove is formed by the (111}A planes (c~v = 70.5°). The change in the conversion efficiency, under ideal circumstances, can be approximately +38% for the 70.5 ° groove and as high as +51% for the 20 ° groove. Although 70.5 ° Vgrooved silicon solar cells have been reported to show an efficiency gain of only 13% [1], optimization of the junction depth and other cell parameters should push this improvement value closer to the theoretical limit. As pointed out, the construction of a 20 ° V-groove sample awaits the development of new etching and processing techniques. In general, the inverted-groove structure is predicted to show an enhanced efficiency when compared with a similar planar structure; the enhancement, however, would be smaller than for a V-groove structure with similar design rules. An inverted-groove cell with a depth-to-width ratio of unity may show a conversion efficiency enhancement of only 7.5% with respect to a planar sample. Yet the inverted-groove analysis shows that, if a perfectly isotropic etchant is employed to produce a groove with a semicircular cross section, the conversion efficiency can increase by as much as 26%. This suggests that a simple way to improve any solar cell is to allow the metallization grid to act as a mask and to subject the cell to an isotropic etchant. The resulting round grooves in the unmetallized regions will decrease the reflectivity and increase the cell's efficiency. In short, there is no compelling reason to "remain planar", when one final processing step can produce a clear-cut performance advantage. A note should be added concerning the dimensions of the grooves. We have used a ray-tracing model in this analysis, and in order for it to have general validity it is necessary that the groove dimensions be larger than the wavelength of light. We expect that devices with groove spacings of the order of 5 pm or larger (depths of 7 pm or larger) will be adequately covered by the analysis presented in this paper. Furthermore, for groove dimensions approaching the minority carrier diffusion lengths (which can be 1 pm or more in GaAs), a two-dimensional solution of the diffusion equation, rather than a one-dimensional analysis, will be required. The fabrication of both 7 0 . 5 ° V-groove and inverted-groove GaAs/ Gal _xAlxAs solar cells is under way in our laboratory. Both groove structures will use 32 pm spacing design rules. The processing procedure, performance of the finished cells and comparison with this analysis will be presented in a future paper.
References
1 R. A. Arndt, J. F. Allison, J. G. Haynos and A. Mealenberg, Jr., Optical properties of the COMSAT non-reflective cell, Proc. 1 l th Photovoltaic Specialists' Con f , Phoenix, A Z , May 6 - 8, 1975, IEEE, New York, 1975, p. 40. 2 C. R. Baraona and H. W. Brandhorst, V-grooved silicon solar cells, Proc. 11th Photovoltaic Specialists' C o n f , Phoenix, A Z , May 6 - 8, 1975, IEEE, New York, 1975, p. 44.
239
3 P. G. Borden and R. V. Walsh, Silicon solar cell with a novel low-resistance emitter structure, Appl. Phys. Left., 41 ( 1 9 8 2 ) 6 4 9 . 4 B. Dale and H. G. Rundeberg, Photovoltaic conversion, 1, High efficiency silicon solar cells, Proc. 14th Annu. Power Sources Conf., U.S. A r m y Signal Research and Development Laboratory, Fort Monmouth, NJ, May 17 - 19, 1960. 5 L. W. James and R. L. Moon, GaAs concentrator solar cell, Appl. Phys. Lett., 26 (1975) 467. 6 H. C. Gatos and M. C. Lavine, Etching and inhibition of the {111} surfaces o f the IIIV intermetallic compounds: InSb, J. Phys. Chem. Solids, 14 (1960) 169. 7 Y. Tarui, Y. Komiya and Y. Harada, Preferential etching and etched profile of GaAs, J. Electrochem. Soc., 118 (1971) 118. 8 D. Corson and P. Lorrain, Introduction to Electromagnetic Fields and Waves, Freeman, San Francisco, CA, 1962, Chap. 11. 9 B. L. Sopori and R. A. Pryor, Design of antireflection coatings for textured silicon solar cells, Sol. Cells, 8 (1983) 249. 10 W. T. Tsang and S. Wang, Profile and groove-depth control in GaAs diffraction gratings fabricated by preferential chemical etching in H2SO4-H202-H20 systems, Appl. Phys. Lett., 28 ( 1 9 7 6 ) 4 4 .