Optical characterization of Schottky barrier solar cells

Solar Cells, 10 (1983) 7 - 15

7

OPTICAL CHARACTERIZATION OF SCHOTTKY BARRIER SOLAR CELLS F. DEMICHELIS and M. AGNELLO

Dipartimento di Fisica del Politecnico di Torino, Turin (Italy) (Received December 8, 1982; accepted March 1, 1983)

Summary A general study to optimize the photogenerated current density as a function of the optical properties of Schottky barrier cells is reported. The model takes into account the transmittance of the various film~ constituting the cell in order to determine the o p t i m u m thickness, consistent with their other physical properties, for maximizing the current density. Calculations for an Au/n-Si cell indicate that there are well-defined thicknesses giving o p t i m u m values of the current density.

1. Introduction There has recently been considerable interest in the development of metal-insulator-semiconductor (MIS) and semiconductor-insulatorsemiconductor (SIS) solar cells because of their simplicity, high radiation resistance and, most important, their adaptability to polycrystalline films and because, at the same time, they do not necessitate high temperature processing [1 - 4]. For all these reasons, MIS- and SIS-structured solar cells have also been considered a viable alternative to thermally diffused p - n junctions for use on polycrystalline thin film silicon and. on hydrogenated amorphous silicon (a-Si:H) substrates [ 5 - 7 ]. Theoretically the cell efficiency of the MIS and SIS structures should be better than those obtained with diffused junction technology, b u t so far experimental MIS and SIS cell efficiencies have been below the values obtained by theoretical analysis and below those achieved with diffused junction technology. There are indeed some problems that raise doubts concerning the feasibility of high efficiency MIS- and SIS-type devices for large-scale applications. Among the loss mechanisms [8 - 10] in MIS- and SIS-type devices that could account for the lower efficiencies, the transmission-reflectionabsorption of light associated with the thin metal or semiconductor top layer can be considered as one of the more important. With thin metal MIS cells 0379-6787]83/$3.00

© Elsevier Sequoia/Printed in The Netherlands

there is a strong trade-off between the sheet resistance and the optical transmission. Since very thin films can become discontinuous and would offer a very high sheet resistance, above a certain thickness the efficiency of the cell no longer increases as a result of the reduced series resistance but starts to decrease as a result of the reduced p h o t o n transmission into the semiconductor. This problem is less important in the transparent semiconductor top layer of SIS cells. Since the optical properties of the oxide films depend very little on the film thickness, the electrical sheet resistance can be minimized by depositing a thick film. In any case either the metal film or the thicker oxide film must be coated with an additional antireflection (AR) film to achieve lower reflection losses [ 11, 12 ]. As already mentioned, another very promising technique for photovoltaic applications is the use of films of a-Si:H and fluorinated a-Si:H in MIS solar cells. One of the possible ways o f improving the red light response of amorphous silicon (a-Si) solar cells is to increase the film thickness so that a larger part bf the long wavelength radiation is absorbed. However, for film thicknesses of over I pm the series resistance will become t o o great [ 13, 14]. This trade-off suggests that it is advisable to use highly reflecting back contacts and to optimize the film thicknesses. Relatively little attention has been paid to the optical optimization of MIS and SIS cells. The only optimization achieved t o d a y is by using AR coatings. From the optical viewpoint a Schottky barrier cell has essentially the form of a multilayer with absorbing and dielectric film stacks. The theoretical discusssions of multilayer interference filters only outline the essential points of the transmission concept and its application to the design of a particular type of filter such as a Schottky barrier cell. The refining method [15, 16] in the transmission optimization can be applied directly to the problem of improving the photogenerated current density in Schottky barrier cells. In this paper a design study to optimize the photogenerated current density as a function of the transmittance of the different layers, thereby giving the optimal thicknesses of the various layers that make up the cell, is reported. Many combinations of metals and semiconductors can be selected to form a cell for which the following analysis can be used. Although this work is primarily devoted to MIS and SIS cells, the equations and conclusions derived here can be applied to any other cells satisfying the conditions given below. To test the method, calculations for Au/n-Si systems were carried o u t and are presented in this paper.

2. Absorbing and dielectric multilayer optics Figure 1 shows a multilayer representation for MIS An electric field E is normal to the plane of incidence and is parallel to the plane of incidence. The Poynting vector wavefront. The components E; and H; on the jth layer

or SIS solar cells. a magnetic field H S is normal to the are related to the

C

J

H,

antireftecting coating Nj = ni - i kj

top

conductor

insulator base

semiconductor back

Fig. 1.

contact

Multilayer representation for MIS or SIS solar cells.

c o m p o n e n t s E# _ l and Hi_ 1 on the (j -- 1)th layer by the relationship -- m :OS

~j

• c## sin ~i 1

E~ (1)

I-/] - 1

i r~j sin ¢i c/~j

COS

¢~]

Hi

where 27r

~i = ~ - 7i6 cos 0i ~?# = n i -- ik 1 is the c o m p l e x index o f refraction (n is the real part and k is the extinction coefficient which depends o n the material absorption) = I ~j/cos Oj ~J ! 7 7 ; c o s 0 i

for a p wave for an s wave

#j is the magnetic permeability o f the jth layer, c is the velocity o f light and Ij is the thickness o f the jth layer while

", I

1/2

cos 0 r

P1 = 1 +

2 /no sin 00\ 2 (ki 2 - ni )~-~ +-k7 )

(2)

I0

q; =

-

-

/n o sin 00\ 2 2n;ki | 5--2 --:-~J

(2)

\ ni + kj /

with Oo the incidence angle and n o the real part of the index of refraction of the medium on which the light is incident. The matrix product of M matrices corresponding to M multilayers is given by

I All*

A121 .-- IAll cOs ~l cOs ~2 "'" cOS~M A12cos~I cos~2"" COS~M I

IA21*

A22:]

D21 cos

cos

cos

A22 cos

cos

cos SMI

(3) where M-I

M

A,,=I + ~An n=l

M-3

~ m=n+l

m=n+l

~

4 ~_~ Ai...

m=n+l

M-Z 1

A,

~M

A;'"

j=m+l

Z

BI+...+{Q}

i='+

Bq

for o d d M

q=p+l

M-,

Z

M

~, Aj ~ j=rn+ I

p=l+l

:

B,n

M-I

Bm

m=n+l

j=m+l

:

~_An

~

n=l

3 ~.~ Sm

tn!lAn

(Q}

M--2

Bm+ ~ A , ,

M

Ap

p=l+t

(4)

~.j Bq

for e v e n M

q=P+l

The expression for A 22 is obtained by exchanging A with B. The off-diagonal matrix element A21 is given by

M M 2 M-I M A : I : N Sn + ~, B,, ~, Am ~, Bj+ n=l

n=l

M

4

M--3

+ ~_~ B n n=l

= {w)

m=n+l

j=m+l

M--2

M--I

M

~ A m ~ Bj ~ A t ~_, B,+...+(W} mfn+l j= rn+ 1 ifj+l 1=i+1

~., Am ... ~, A, m= n+l p=/ +1

~

Bq

(5)

for e v e n M

q~p+l

(6)

=

2

M--I

M

~, Am... ~ A v ~ Bq m=n+l p=/ +1 q=P+l

for o d d M

The expression for A 12 is obtained by exchanging A with B. In addition ic/~ sin ~l Ai ~, cos ~i

(7)

11

i~?, sin ~i

B~-

(7)

C~i COS ~i

The transmittance T and the reflectance R are given b y T=

4Re(~s) Re(~o)lT 1 + T2[ 2

R =

(8)

IT1 -- T212 IT 1 + T2[ 2

where Re( ) indicates " t h e real part o f " and Tl = Al1" + A12*~s T2 =

A21" + A:2*~s

(9)

The above expressions refer to the most general case when all the layers have complex indices of refraction. Obviously they also hold when some layers are dielectric with a real index o f refraction.

3. Formulation of the photocurrent density in Schottky barrier cells introducing optical parameters The current density J v h p r o d u c e d by the incident photons is the superposition of hole and electron c o m p o n e n t s according to [ 17, 1 8]

Jph =

J t + Ji +

Jde +

Jb

(10)

The current density Jt is due to the absorption in the metal film (MIS) or in the t o p semiconductor film (SIS) of radiation energy between the limits of the m e t a l - s e m i c o n d u c t o r or s e m i c o n d u c t o r - s e m i c o n d u c t o r barrier height and the solar spectrum edge (4.1 eV). The current density in metal films can be calculated by Fowler's m e t h o d [19]. Its value is of the order of 10 -3 times those of the other components. As regards the SIS cells, the t o p semic o n d u c t o r is generally a degenerate oxide with a band gap greater than 3 eV. In this case the whole solar spectrum of radiation can be considered to be absorbed by the base semiconductor. For band gaps less than 3 eV, radiation from part of the solar spectrum is lost in the t o p semiconductor w i t h o u t contributing anything to the p h o t o c u r r e n t [20]. The current density Ji arises from p h o t o n absorption within the inversion layer. On the assumption that recombination in the inversion layer affects the total photogenerated current only slightly, Ji can be written as J i ( k ) = q N ( ; k ) T t ( ~ . X 1 - - Ti(k)}

(1 1 )

12

where N(X) is the incident p h o t o n flux over the surface of the cell, Tt(~ ) is the transmittance of the AR and metal films or the AR and t o p semiconductor films and Ti{k) is the transmittance of the inversion layer. The current density Jde arises from p h o t o n absorption within the depletion region. On the assumption that in this region also recombination has only a slight effect on the total photogenerated current, Jde can be written as Jde = qN(k)Txi(k){1 -- Tde()k)}

(12)

where Txi(k) is the transmittance of the AR layer/metal/inversion layer system and Tde(k) is the transmittance of the depletion region. The bulk c o m p o n e n t of the photogenerated current is given by -- 1 Jb = q N(~')Txi ~ AJ~'Lp" ' de(~')°~(~')LP 7

p ( - - L / L/LP) Lp) c~(k)Lp -- 1 + 2 exp(-- o~(k)L}-- --e xexp(-(13)

where Txi,de (k) is the transmittance of the AR layer/metal/inversion layer/ depletion region multilayer system, L is the thickness of the bulk, Lp is the hole diffusion length and ~(k) is the bulk absorption coefficient. Since the c o m p o n e n t s Ji and '/de can be written as Ji = q N ( ~ ) { T t ( k ) -

Txi(~)}

(14)

Jd~ = qN(~,){Txi(~,)- T x i , d e ( k ) }

(15)

the total photogenerated current density is given by Jph = qg(X) Tt(X) -- Txi,de(X) 1 -- {Ol(X)}2Lp2 _ 1 × x I.(X)Lp-- 1 + 2 e x p { - - ° l ( k ) L } - - e x p ( - - L / L p ) ] t l

(16)

The total current density is J~t =

?

Jph(k) dk (17) 0 when the band gap wavelength kc, and hence the band gap energy Vg, is the same for the inversion layer, depletion region and bulk. The general expression for the total current density is kc,de Jtot = q 0 +

a¢,b

N(~,)Txi,de(k)a(k)Lp {a(k))2Lp 2 -- 1

×

0

X o~(X)Lp -- 1 + 2

e x p ( L / L p ) -- exp(-- L / L p )

d

(18)

13

Accurate methods to determine the optimum thicknesses of the multilayer films to maximize the transmittance have been proposed by several researchers [12, 15, 16]. Similarly, a computer programming method was employed here to optimize, through the transmittances, the photogenerated current density by the adjustment of different layer thicknesses, taking into account that they must satisfy other physical conditions of the problem such as the sheet resistance and the donor density. The inversion layer thickness xi and depletion layer thickness W that maximize the current density can provide, in turn, useful information about some parameters of the cells stlch as the donor density [21].

4. Results and discussion An Au/n-Si cell was used to verify the validity of the model. All the computations were performed with an IBM 370 computer. The optical parameters n, k and a of silicon were taken from ref. 22 and those of the gold film were from ref. 23. The integration covers only the wavelength region 0.4 - 1.1 #m since the contribution of the longer wavelength photons to the photocurrent is negligible. We should point out that it is important to choose a coating thickness which maximizes the transmittance rather than a thickness which minimizes the reflectance of the system. Designs for optimum AR coatings on silicon or GaAs or on Schottky barrier cells have been reported by many researchers 1.o-

~a 45 b

OG 05

;~(pm)

I 1.0

Fig. 2. Transmittance T through the following structures v s . wavelength )~: curve a, AR layer (0.040 ~m)/Au (0.007/~rn); curve h, AR layer (0.040 #m)/Au (0.007/~m)/inversion layer (0.040 /Jzn)/depletion region (3.5 p-m); curve c, AR layer (0.040/~m)/Au (0.007 pm)/inversion layer (0.078 prn)/depletion region (6.5/~m).

14

[12, 16]. For this reason we did not pay particular attention to the AR layer optimization. Figure 2 shows the numerically calculated transmittance through the AR layer/metal film and through the AR layer/metal/inversion layer/depletion region system. Table 1 summarizes the magnitude of the photogenerated current density, under simulated air mass 0 conditions, for various thicknesses of the layers and an angle of incidence ~b = 0. It was found that reasonable current densities could only be obtained for b o u n d e d values of the thicknesses of the different layers. Table 2 shows the magnitude of the photogenerated current density for various angles of incidence. Only small changes in the current density are observed. TABLE 1 The current density for various structures

Current density

Thickness o f the following regions (/~m) AR layer a

Metal film

Inversion layer

Depletion region

Bulk

0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04

0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.012 0.014 0.019 0.019

0.001 0.002 0.010 0.010 0.010 0.010 0.017 0.056 0.080 0.080 0.080 0.080 0.080 0.001 0.002 0.001 0.002

0.50 3.00 0.50 3.00 3.05 8.15 0.50 0.50 0.50 2.30 3.50 4.10 5.90 0.50 3.00 0.50 3.00

200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200

( m A crn --~)

25.5 28.1 26.3 28.7 28.7 31.0 26.9 28.8 29.0 30.6 31.4 31.7 32.4 18.8 19.5 12.7 14.5

ansio2 = 1.55. TABLE 2 Current density for various angles o f incidence ¢

Thickness o f the following regions (/~m) AR layer a 0.04 0.04 0.04

Metal film 0.007 0.007 0.007

a n s i 0 2 -- 1 . 5 5 .

Inversion layer 0.002 0.010 0.010

Depletion region 3.00 3.00 8.15

BUlk

200 200 200

Current density ( m A c m -2) for the following angles o f incidence ¢=0 °

dp=15 °

¢=30 °

28.1 28.7 31.0

28.0 28.6 30.9

27.7 28.3 30.6

15 5. C o n c l u s i o n O n e r e a s o n f o r t h e p r e s e n ~ d a y l o w e f f i c i e n c y o f S c h o t t k y cells is excessive o p t i c a l losses. A n o p t i m i z e d design w h i c h utilizes b a c k - s u r f a c e r e f l e c t i o n a n d a p p r o p r i a t e o p t i m u m t h i c k n e s s e s o f t h e S c h o t t k y cell c o m p o n e n t s is c a p a b l e o f achieving t h e t r u e p o t e n t i a l o f such cells. T h e m o d e l p r e s e n t e d in this p a p e r o p t i m i z e s t h e p h o t o g e n e r a t e d c u r r e n t d e n s i t y as a f u n c t i o n o f t h e o p t i c a l p r o p e r t i e s o f t h e device. T h e c o m p u t e r p r o g r a m t h a t we p r o p o s e c a n b e easily a d a p t e d f o r c a l c u l a t i o n s o n m o r e c o m p l i c a t e d conf i g u r a t i o n s , such as m u l t i j u n c t i o n cells a n d t a n d e m cells as well as p o l y c r y s talline a n d a-Si cells, allowing p r e d i c t i o n s o f t h e p e r f o r m a n c e o f such cells t o be m a d e . Illustrative c a l c u l a t i o n s f o r t h e A u / n - S i s y s t e m give values o f t h e p h o t o g e n e r a t e d c u r r e n t d e n s i t y in g o o d a g r e e m e n t w i t h t h e results o f refs. 17 a n d 18.

References

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