The open-circuit voltage developed across a Schottky-barrier solar cell is calculated in the presence of monochromatic light. The limitations of previous ...
Open-circuit voltage of a Schottky-barrier solar cell P. K. Dubey, and V. V. Paranjape
Citation: Journal of Applied Physics 48, 324 (1977); View online: https://doi.org/10.1063/1.323381 View Table of Contents: http://aip.scitation.org/toc/jap/48/1 Published by the American Institute of Physics
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Open-circuit voltage of a Schottky-barrier solar cell* P. K. Dubeyt Department of Electrical Engineering. University of Toronto. Toronto. Ontario. Canada
V. V. Paranjape Department of Physics. Lakehead University, Thunder Bay, Ontario, Canada (Received 12 July 1976)
The open-circuit voltage developed across a Schottky-barrier solar cell is calculated in the presence of monochromatic light. The limitations of previous treatments are brought out. Our analysis differs from the earlier ones in using appropriate boundary conditions. A tractable expression for the open-circuit voltage is obtained assuming low-level injection conditions, which is met by average solar radiation in AMO and AMI conditions. PACS numbers: 84.60.Jt, 73.40.Qv, 72.40. + w
I. INTRODUCTION interest 1- 3
In recent years, there is renewed in the Schottky-barrier (SB) solar cells as an alternative to the usual p-n junction solar cells. Solar cells of the SB or AMOS (antireflection-coated metal-thin oxidesemiconductor) types have recently been fabricated1 with efficiencies comparable to that of the p-n junction type, and due to their simple fabrication technology and low cost, these cells appear to be a viable alternative. In this paper, we present the calculation of the open-circuit voltage that is developed across an SB solar cell when monochromatic light is incident on it. Gartner 4 was the first to calculate the short-circuit current of an SB solar cell, which was later modified by Ahlstrom and Gartner 5 and by Tuzzolino et al. 6 This analysis is, however, unconvincing since the results are obtained using questionable boundary conditions Consequently, the analysis has a limited validity. This is discussed in detail in Sec. II of this paper. Li et al. 7 have further extended the treatment of Gartner to include the effect of the inversion layer adjacent to the metal- semiconductor interface; however, their approach follows the same basic treatment of Gartner. o
We have calculated the open-circuit voltage of an SB solar cell by using appropriate boundary conditions. SCHOTTKY CONTACT THIN TRANSPARENT
15~5!
~
+
II. LIMITATIONS OF THE PREVIOUS TREATMENTS The energy-band diagram of an SB cell is shown in Fig. 1. The plane x = 0 is defined as the interface where the metal makes Schottky contact to the n-type semiconductor, the plane x = W represents the edge of the depletion region, and the plane x = L is defined as the interface where the metal makes Ohmic contract to the semiconductor (providing an infinite sink for excess electrons and holes). The total current J T flowing through the cell under illumination can be expressed as the sum of electron and hole currents I n and J p at the edge of the depletion region x= W; (1)
Gartner associated In(W) to drift current in the depletion region and Jp(W) to diffusion current in the neutral region. The difference between our definitions of Eq. (1) with those of Gartner is purely due to nomenclature even though our terminology is more natural. OHMIC CONTACT
+
I
I
~I
N- SEMICONDUCTOR
'~!=I=I~'~'~-~l====~E~f======~~~
L1GHT~ •
~~~~
METAL
Ec
E
j
~~J------E~v------~I
~~!,------~--------~I
X= 0
I I I I
X=W
X=L
I I
I I I
I I I
324
This approach is outlined in Secs. III and IV, where an expression for the open-circuit voltage is obtained for the low-level injection condition.
Journal of Applied Physics, Vol. 48, No.1, January 1977
EQUILIBRIUM FERMI LEVEL
FIG. 1. Energy-band diagram of a Schottky-barrier solar
cell.
I
I
Copyright © 1977 American Institute of Physics
324
Neglecting recombination in the depletion region, it follows from the electron continuity equation that (2)
A. Depletion region (0';;;;; x';;;;; WI.
In the depletion region, we have the built-in field
Eb(x) due to the do~or change density. Thus, under
illumination we have where G(x) is the generation rate of electron-hole pairs by the incident light and q is the magnitude of the electronic charge. Gartner's analysis requires the assumption that (3)
and hence In(W) is obtained from Eq. (2). Jp(W) is then evaluated by solving the continuity equation for holes in the neutral region of the semiconductor (W~x~L) subject to the boundary conditions that Ap(W) = 0
(4)
Ap(L) = O.
(5)
and
We thus infer that the previous treatment is limited in scope and is of questionable validity. In a proper treatment, one should use the natural boundary conditions at planes x = 0 and x = L, requiring continuity of p. The external boundary condition, namely, J T = 0 for open-circuit conditions or V(O) - V(L) = 0 for the short-circuit condition should also be required. This approach is followed in Secs. III and IV. III. CALCULATION OF THE OPEN-CIRCUIT VOLTAGE
In this section we calculate the open-circuit voltage V(L) - V(O) when monochromatic light is shined on the interface x = 0 of the cell. The boundary condition in this case is that the total current flowing through the cell must be zero. Taking drift and diffusion of the electrons and holes, this is represented by (6)
where nand p are the electron and hole concentration, Iln' Dn and II p, D p, are respectively the electron and hole mobility and diffusion coeffiCients, and E is the electric field. We now express the open-circuit voltage, Voe, as Voe = V(L) - V(O) = [V(L) - V(W)] + [V(W) - V(O)],
and obtain the values of the two brackets in Eq. (7) separately. 325
J. Appl. Phys., Vol. 48, No.1, January 1977
(7)
(8)
=Po(x) + Ap(x),
(10)
p
(9)
where E~(x) is the electric field associated with the generation of electron-hole pairs by light, no(x) and Po(x) are the equilibrium electron and hole densities, and An(x) and Ap(x) are the excess electron and hole densities. We are able to eliminate some terms from Eq. (6) using the requirement that in equilibrium (no light) no current flows. Thus
Here Ap is the excess hole concentration. We see that the procedure followed by Gartner entails the three assumptions (3)-(5) and one approximation (neglect of recombination in the depletion region). The only natural assumption provided for by the cell configuration is Eq. (5), while the assumptions (3) and (4) require justification. As will be shown later in Sec. IV D, both In(O) and Ap(W) will not be zero in general and thus make the earlier analysis limited in scope. It will follow that in this treatment if one wishes to include recombination, the procedure will fail to give a solution for In(W). Also the treatment does not lead to zero potential difference between the two ends, a condition which is implied by short-circuiting the two contacts.
+ Ec (x) , n = no(x) + An(x),
E = Eb(x)
JT=q(no + An) IlnE~ + q(po + Ap) IlpEI +qAnllnEb + qAPllp Eb
+ qDn
dAn dAP _ dx - qDp dx - 0,
(11
)
giving the expression for E1(x) as EIf(x)
=Dpe:: _
Dn d!:n JUno + An) Il n + (Po + Ap) II p]-1
Anlln + APll p Eb(x) (no + An) Il n + (Po + Ap) II p •
(12)
In Eq. (12) it is possible to eliminate An using the following considerations. The continuity equations (in steady state) for electrons and holes are 1 dJ ~ q dx
- -
+ G(x) -
Ap
-
Tp
= 0
(13)
and
! ~ +G(x)q dx
An =0 Tn'
(14)
where Tn and Tp are the electron and hole lifetimes. It is assumed here that the generation rate for electrons and holes is same. In steady state, the total current flowing through the bar JT=Jn +Jp must be constant everywhere, and we have
dJ T -0 dx - •
(15)
From Eqs. (13)-(15) it follows that in steady state (16) Equation (16) is in contrastS with the commonly used space-charge neutrality assumption An = Ap. Substitution of Eq. (16) in Eq. (12) and integrating from 0 to W then yields P.K. Dubey and V.V. Paranjape
325
f fW
V(W) - v(O) = -
W
C. Total open-circuit voltage
Eg(x) dx
Combining Eqs. (19) and (25) yields the total opencircuit voltage developed across the SB solar cell
o
= -
(Dp - D;:nJ
o
~~P [~lO + ~: AP)
+(PO+AP)flj -1 dx+
x
[(no +
~
W
0 ~:
;: AP) fln + (Po + Ap)
n
flJ
fl"
-1
Eb(x) dx. (17)
Equation (17) gives the open-circuit voltage developed across the depletion region due to the external generation mechanism. It can be simplified under the approximation and in the depletion region (18) This approximation would be reasonable under strong illumination. It may be pointed out that no(x),po(x) = 0 is a convenient approximation normally used elsewhere. However, the use of approximation (18) would not permit us to check the limiting cases when An, Ap - 0 (i. e. , under conditions when either light is switched off and/or the absorption coefficient of the semiconductor is zero) and may not be a good approximation under weak illumination. Use of approximation (18) in Eq. (17) yields
tP _ Dp - Dn Tn/Tp In Ap(W) flp + fln TJ! Tp Ap(O) ,
V(W) - V(O) = _
(19)
S
where tPs = iqNa W 2/Es is the surface potential, Na is the donor density, and Es is the dielectric permittivity of the semiconductor 0
B. Neutral region (W ~ X ~ L). In the neutral region, there is no built-in field and we have E=Eg(x),
(20)
n=Na +An,
(21)
P=
n~ -1.
Na
x
+flp) Ap
rlfl
L~ p
+
fln Tn\-1 In Ap(W) _ AP(W)] Tp Ap(O) Nafln
1
where we have used the boundary condition (5) that at the plane x = L, the Ohmic contact provides an infinite sink for excess holes. Complete determination of Vac requires that Ap(O) and Ap(W) be known; this is achieved in Sec. IV where the continuity equation for holes is solved in the two regions and matched at the junction x=W.
IV. SOLUTION OF THE CONTINUITY EQUATION In steady-state conditions, the continuity equation for holes reads
_.!
~ + ipla exp(- ax) _
q dx
Ap = 0, TI>
where ni is the intrinsic carrier concentration. Substituting Eqs. (20)- (22) in Eq. (11) and limiting ourselves to the low-level injection condition, i. e. , (23)
An, Ap «Na »nUNa,
we obtain E ( ) = Dp - Dn Tn/T~ dAp gX Nil dx' a"'n
(24)
where we have also used Eq. (16). Thus integrating Eq. (24) between the limits Wand L, we obtain the open-circuit voltage developed across the neutral region
(27)
where we have used the fact that the generation rate G(x) is given by G(x) = ip{l - R(A) - alA)} f(A)
0'
exp(- ax)
= ipla exp(- ax).
(28)
Here ip is the number of photons striking unit area of the cell in unit time, R(A) is the reflection coefficient as a function of wavelength from the metal surface, alA) is the absorption in the thin metal film (normalized to the incident radiation) as a function of wavelength, f(A) is the probability that a photon will produce an electron-hole pair and 0' is absorption coefficient of the semiconductor. The transmission coefficient through the cell is assumed to be zero. Equation (27) will now be solved separately in the two regions. A. Depletion region (0
~
X
~ W)
In this region, we can approximate J p as (Po« Ap):
(22)
+Ap,
(26)
(29) The field E = Eb(x) + Eg(x) in this region under the given boundary condition, J T = 0, is given by Eq. (12), and using the approximation (18), we have E=A dAp dx '
(30)
where A = (D" TI> - Dn Tn)/(fl" T" + fln Tn).
(31)
Substitution of Eqs. (29) and (30) in Eq. (27) yields VeL) - V(W) = -
lL
2
d AP D; -::>::r dx
Eg(x)dx
W
Ap Tp
+ ipla exp(- ax) = 0,
(32)
where (25)
326
J. Appl. Phys., Vol. 48, No.1, January 1977
(33) P.K. Dubey and V.V. Paranjape
326
The solution of Eq. (33) is given by
Utilizing Eqs. (44)-(47) in Eqs. (34) and (39), we obtain the constants C1 and C2 according to C _ b 1b 2 b 3 exp(- 0 W) - Q 1b4 b 5 exp(- WiLt) t b 3b s exp(W/L;) + b 5b 7 exp(- W/L;)
(34)
where
(48)
and
L;2 =D;Tp,
(35)
Qt =
