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N+ dopant exponential profiles for silicon solar cells
Solar Energy Materials and Solar Cells 30 (1993) 291-300 North-Holland
Solar Energy Materials and Solar Cells
N + dopant exponential profiles for silicon solar cells L. Hernandez, R. Soler a n d A. M a r t e l Institute of Materials and Reagents for Electronics (IMRE), Uniuersity of Hat,ana, Hat,ana, Cuba Received 5 June 1992; in revised form 25 January 1993 A numerical model of solar cells is described, which provides optimum design rules for exponential N + dopant profiles for heavily doped emitter regions. The model gives a useful insight into the relative impact of surface and bulk recombination on device performance. Results agree well with theorical models.
I. Introduction The theoretical maximum efficiency of a silicon solar cell is limited as a consequence of heavy doping effects in the emitter region. Both the blue response and the open circuit voltage in silicon ceils of conventional design depend upon the properties of the heavily doped emitter region, related with bandgap narrowing and a high recombination at and near the surfaces. At the present time, the efforts to increase the efficiency have been focused appropriately on the problem of raising the open circuit voltage, since the current collection efficiency is approaching its theoretical limits [1]. To increase the conversion energy efficiency, it is necessary to obtain large values of the effective diffusion lengths, together with the highest possible donor concentrations, in order to reduce the sheet resistivity of the emitter layer. Moreover, the blue response of conventional cells is improved by reducing the junction depth, but the power loss owing to the sheet resistence increases. These effects work in opposite directions and it is thus evident that optimization is essential. The problem is complicated by imperfectly understood mechanisms or imprecise knowledge of the controlling technological parameters and by experimental difficulties in separating the effects of these mechanisms. The main difficulty when modelling an emitter region is the uncertainties in the values of the bandgap narrowing, mobility and lifetime parameters commonly used to characterize highly doped silicon. All of these parameters are affected by the dopant concentration profile and by the surface conditions. Several analytical models have been developed by different authors to provide insight into the transport mechanisms of minority carries injected into the heavily doped emitter, as well as recombination phenomena. Dumke [2] derived a Hermite 0927-0248/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
292
L. Hernandez et al. / N " dopant exponential pro]~les
polynomial expression for a Gaussian profile. He assumed that diffusion dominates the minority carrier current and that the diffusion coefficient is independent of doping. Shibib et al. [3] included majority carrier degeneracy and a finite surfacerecombination velocity S at the emitter surface in a transparent emitter model for thin emitters. Transparent emitter means that the average minority carrier diffusion length is longer than the junction depth and this implies that surface recombination can be the dominant recombination mechanism. Under the assumption of a transparent emmitter, many, efforts have been made to find a simple analytical aproximation. Fossum and Shibib [4] assumed diffusion dominant current near the surface where the effective doping concentration is nearly constant. After that, Amantea [5] and Del Alamo and Swahson [6] derived an expression for the emitter recombination current that keeps the position dependence of the effective doping concentration, diffusion coefficient and lifetime. The model of Rohatgi and Rai-Choudhury [7] is based on effective recombination velocity as a measure of the minority carrier losses in various regions of the device. The model includes the effect of bandgap narrowing, Auger recombination and recombination at the device surface. Morales-Acevedo [8] completed these models considering the electric field and series resistance effects upon the efficiency of the cell. Recently, a new analytical method for the study of solar cell emitters with arbitrary doping profiles have been developed [9]. Morales-Acevedo [10] has also reported that the quantum collection efficiency of heavily doped emitters in silicon solar cells is high and almost independent of the surface dopant concentration for thin emitters. Up to now the analytical models have been based on some specific assumptions about the transport mechanism or the geometry of the device [11]. Seeking a new aproach, in this paper, we report a numerical solution of the minority carrier transport equations in a nonuniform (exponential profile) heavily doped emitter. The solution is in the form of a power series of minority carriers as a coordinate function. The model includes the effects of bandgap shrinkage, dopant-depended mobility and lifetime, of minority carriers and surface recombination on the dark saturation current. We also consider the effect of surface impurity concentration upon the series resistance and efficiency conversion, but the electric field, resulting from the gradient of doping concentration is taken to be constant.
2. Theory A sketch of the emitter in a typical silicon cell is shown in fig. 1. We consider a n - p junction formed by diffusing n-type impurities into a uniform p-type substrate. In the emitter region, the minority carrier current density Jp is given by dp Jp = qtxppE - qDp d x
(1)
L. Hernandez et al. / N + dopant exponential profiles DOPING PROFILE ND°
CONTACTS
nnnnn
o
DEPLETION REGION W
z>
BASE
293
H
I
]
I
~=3
Idrcm -3
Fig. 1. Cross section of n + p solar cell and a graded doping profile used as a schematic diagram.
Since the minority carrier concentration p is non-degenerate, Einstein's relation can be used between the mobility /zp and the diffusion constant Dp. We assume full ionization of donor impurity atoms and low injection. The rest of the parameters in eq. (1) have their usual meaning. The differential equation describing the minority carrier concentration p in the quasi-neutral n emitter of a solar cell under illumination can be written as 1 d
q dxJv- G(x)
/~ + - - = 0,
(2)
%
where G(x) is the generation rate of electron-hole pairs due to light and ~-p the hole lifetime. We assume an impurity exponential profile in the emitter regions, U D = ND0 e x p ( - F x ) ,
(3)
where ND0 is the surface concentration. In LAMEL's Institute [12], the possibilities of the open tube diffusion technology have been explored. The dopant profiles that result from these advanced procedures present neither precipitates nor a platau near the surface and can be approximately represented by an exponential function with surface concentration No0. For the space variations of hole mobility and lifetime in the emitter layer we fit the experimental data for p~p and ~-p versus donor concentration reported in ref. [13]. The fit equations are /d,p = / ~ A N D ~B,
(4)
7"p = "rAND ~'u ,
(5)
Where /~a, /ZB, rA and ~'B are the fitting parameters. Hence, writing ~p and ~-p as a function of coordinates, eqs. (4) and (5) become /Xp =/x~ exp(~bX),
(6)
7p = % e x p ( % x ) ,
(7)
where /zb /Xa = / ~ A N D ~ B '
/ZB --
'rc~ = T A N D ~ ' " '
Tb "rB = T "
F
'
(8) (9)
L. tternandez et al. / N ~ dopant exponential profiles
294
Thus, we have expressed the exponential space dependence for/Zp and 7p. Putting eq. (1) into eq. (2) we obtain c l [ Dp(X) dxP(X) d 1 -ol(A)F(A) exp[-a(A)x] d-~ [Elzp(x)p(x)] - ~x + - -
-0,
%(x)
(10)
with the common boundary conditions
Jp I x=0 +qSp~lx=0 = O,
(x)lx=x =0, where a(A) is the absorption coefficient (data reported by Aspnes) [14], F(A) is the solar spectrum density at AM1, Sp is the surface recombination velocity. Introduction the non-dimensional variables, u =x/xj, 0 < u < 1, and ~-(u)= p(u)/n i, where x i is the junction depth and n i is the intrinsic carrier concentration, and combining eqs. (6) and (7), eq. (10) becomes Ou2"rr(u) +a~urr(u ) -b(U)~uTr(u ) = f ( u ) ,
(11)
where (12)
a = x j IZb --
2[ qE b ( u ) - xj [--~-/x b + - qE _
k T/x~ "
[ qe
t
f(u) = - - / ,-~--exp~ n i [ Klld'b
exp{-xj(/z b + 7b)U}] ,
--Xj(a+~b)U
} + -
qPo -
k T/z a%
(13)
exp{-xj(/~ b + rb)U}], (14)
with the solution ov
7r(u) = E ~rnun + 7r1(u) e x p { - x j ( a +/Zb)U } + n2(u) exp{--xj(tt b + %)u}. n--0
(15) More details about the mathematical handling can be found in the appendix. Thus, we determine the spatial dependence of minority carrier concentration through the emitter region solving eq. (10) by means of computer programs. Therefore, we get the photocurrent density at junction, taking into account eq. (1). The open circuit voltage is conditioned by the dark saturation current of the cell. The contribution made by the emitter region can be calculated by analogous
L. Hernandez et al. / N + dopant exponential profiles
295
Voc(v) 0.68" Xj =0.2p.m Sn=lOScm/s E = 103V/cm H = 2502~.m Ln = 230/~rn W=O.2#m Dn=13cma/s
0.660.64 0.62 0.60b)Sp=5-105cm/$
0.580.56
Fig. 2. Open circuit voltage versus donor concentration.
considerations. In this case, we hold G ( x ) = 0 and the boundary condition at junction is
P lx~x~= p o [ e x p ( q V / k t )
- 1].
(16)
The expressions reported by Hovel [15] to calculate the photocurrent density and the dark reverse saturation current density in the depletion and base regions were used. Similarly, we calculate open-circuit voltage and efficiency by the usual formulas, in the Schottky approximation. Reduction of the V0c value as a consequence of the heavily doped emitter is taken into account through the bandgap shrinkage corrected by the Fermi degeneracy as obtained empirically by Rohatgi et al. Finally, we evaluate the fill factor considering only sheet resistence by the equation found in ref. [16].
3. Results and discussions
The open circuit voltage was calculated with the method described in section 2. In fig. 2 we compare the V0c variations versus surface dopant concentration for passivated and non-passivated front (or top) surface. Oxide passivation induces improvement in V0c resulted from the decrease in surface recombination velocity, which helps in the reduction of the emitter dark current and hence the V0c value is dominated by the parameters of the base. Nevertheless, without any front surface passivation, the dark current is managed by emitter components. In the case of a passivated surface, upon increasing the concentration in the emitter, the V0c is almost constant up to 8 × 1018 cm -3 but it is quickly reduced to greater values of ND0. On the other hand, high dopant concentration in emitter favours V0~ in non-passivated solar cells. In heavy doped emitter Rohatgi and
296
L. Hernandez et al. / N ~ dopant exponential profile,s
20
(a)
18-
16-
b)Sp= 5" IC~cm/s
t4"
P2
Fig. 3. Conversion efficiency versus surface donor concentration (cell 2 × 2 c m
2
and 5 figures).
C h o u d b u r y [7] have reported an increase in V0c of about 15 m V due to oxide passivation, while M o r a l e s - A c e v e d o [9] found a difference of about 25 mV. O u r results coincide with the former. T h e V0c values calculated by us and by Cuevas et al. [17], for an exponential profile, show the same d e p e n d e n c e also. Conversion efficiency of the solar cell as a function of surface d o p a n t concentration is shown in fig. 3. As show, the efficiency is always greater in passivated cells than in non-passivated ones and the differences are almost constant. T h e improvem e n t in the efficiency due to oxide passivation is explained by an increase in Voc
~(%) 20-
16-
14-
(a)/
a)Xj = 0.2 xtrn
/
b) Xj = 0.5/xm c) Xj = 1.0 ~ m
12
Fig. 4. Conversion efficiency of a passivated solar cell versus surface donor concentration.
L. Hernandez et al. / N + dopant exponential profiles
297
~%) 20'
(a)
18"
16-
14C)
12
.....
1~"
~b = I0 -~.. cm
.......
(d a°
No.(c~ 3)
Fig. 5. Conversion efficiencyof a passivated solar cell as a function of surface donor recombination. and Jac" The efficiency reaches a saturation value as a result of the V0c reduction in the heavily doped emitter. Fig. 4 shows the influence of the junction depth in the maximum of conversion efficiency. When the X j decrease, the efficiency reaches a maximum at greater donor concentrations, and its absolute value is also greater. We obtain the optimum value of donor concentration for Xj = 0.2 ~ m of about 8 X 10 ~s cm -3 slightly greater than the reported by Morales-Acevedo, [8] who optimized the grid dimensions for surface dopant concentration. In fig. 5 is shown some other important results, when the substrate resistivity is taken as a parameter. The theory mentioned in Section 2, considers the influence of resistivity of the base in V0c and Jac" This calculation was performed in order to clear how Pb affects the optimum value of donor concentration for passivated solar cells. When the dark current density of the base increases, the efficiency maximum is reached at a greater value of the surface donor concentration, otherwise, when p~ decreases, the conversion efficiency obviously increases.
4. Conclusions
The understanding of the relevant physical mechanisms responsible for the influence of the emitter layer in solar cell characteristics have been studied by a numerical computer model, by including the effects of bandgap narrowing, experimental recombination lifetime and mobility, exponential profile of impurity concentration, and surface recombination in the emitter component of the dark saturation current density. In this paper, we have found that to improve the open circuit voltage, it is necessary to reduce the surface recombination by oxide passivation in the front surface, together with keeping a lightly doped emitter.
L. H e r n a n d e z et al. / N ~ d o p a n t e x p o n e n t i a l pr~[i'les
298
However, to reach maximum conversion efficiency, it is indispensable to take into account the sheet resistance and we found that with the above mentioned considerations, the optimum donor surface concentration is always greater than 1 × 10 ~9 cm
3.
Finally, we have also analyzed the influence of junction depth and base resistivity on the conversion efficiency of passivated and non-passivated solar cells and concluded that an optimization is essential.
Acknowledgement The authors would like to acknowledge the invaluable contribution of Professor Perez-Alvarez by means of a helpful review and advice.
Appendix To solve the differential equation (11)we assume a solution of its homogeneous part in the form of a power series, hence ac
•r(u)
oc
and
= Y'~ ~ ' . u "
b(u)
= ~
n=0
(A1)
%u'.
n=0
Therefore, the solution is given by
,~+a)(.+l)~-.+~+~(n+a)~-.+l n
E
bm~_~®(n--m . ' = 0 ,
m=O
where ® ( n - m ) =
0, 1,
if n < m , ifn>m.
Therefore %+~= (n+2)(n+l)
-a(n+l)Tr,+,+
Y' bm%_m®(n-m
,
m =0
which can be written as
(A2)
~- = C~1)+ C~:)Tr0, where - 1)C (1) +
~
bmCn_2_m(1)
C~1)
1 n ( n - 1)
-a(n
C~2)
n ( n 1-- 1)
- a ( n - l ~ C J .(2~ (2) 2 - m , - 1 + b . - 2 ~ b m C n-m=O
for n_>2.
,
m=0
L. Hernandez et al. / N + dopant exponential profiles
299
Assuming a particular solution, like T/'(/g) = T / ' I ( u ) e x p { - x j ( a +/Zb)U } + rr2(u) exp{--xj(lz b + Zb)U}, we obtain particular solution for eq. (11), oc
7r(u) = Y'~ ~-.u" + 7r'(u) e x p { - x j ( a +~b)U} n=0
+ rr2(u) exp{ +xj(/x b + Zb)U},
(A3)
where qaF(h) "wl(u) = --Xj kTl~. ~'l(u)
1
L
+ IZb)]2-axj(a + tzb) --b(u) '
xyqp o
1
nikTlz,~z [xi(z b +/Zb)] 2 -- axj(z b +/Xb) -- b(u) The solution is undetermined by only two constant, 7r 0 and 7r I because the others are automatically determined by the recurrent equation (A2). In order to find both constants we value the boundary conditions in eq. (A3), and get from the surface boundary condition 7r, = C~ 1) + Cl2)rro,
CI')= qxjSpp° k Ttx ,~ni
dTr I
zr' (0) [ - x j ( a + lZb) ] -- --d-uu
du .=0 + kT
+
kTtz.
.=0- ~2(0)[-x~(~b + ~.)]
(~r'(0) + ~2(0)),
c12' = - U e + ~ Finally, from the junction boundary condition: 00
Po _ 7rl(u ) e x p { - x j ( a +/Zb)U } -- 7r2(u) exp(--Xj(tXb + ~'b)U} -- • F/i
C (1)
n+l
1 + ~ C.tl) n=l
References [1] [2] [3] [4]
A. Rohatgi and P.R. Choudbury, IEEE Trans. Electron Dev. ED-31 (1984) 596. W.P. Dumke, Solid State Electron. 24 (1981) 155. M.A., Shibib, F.A. Lindholm and F. Therez, I E E E Trans. Electron Dev. ED-26 (1979) 959. J.G. Fessum and M.A. Shibib, IEEE Trans. Electron Dev. ED-28 (1981) 1018.
300 [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
L. Hernandez et al. / N ~ dopant exponential pr~files
R. Amantea, IEEE Trans Electron Dev. ED-27 (1980) 1231. J. del Alamo and R.M.S. Swanson, IEEE Trans. Electron Dev. ED-31 (1984) 1878. A. Rohatgi and P.R. Choudbury, Sol. Cells 17 (1986) 119. A. Morales-Acevedo, J. Appl. Phys. 60 (1986) 815. F.J. Bisschop, L. Averholt and W.C. Sinle, IEEE Trans. Electron Dev. ED-37 (1990) 358. A. Morales-Acevedo, Sol. Cells 28 (1990) 293. J.S., Park, A. Neughoschel and F.A. Lindholm, IEEE Trans. Electron Dev. ED-33 (1986) 240. G. Masetti, M. Severi and S. Solmi, IEEE Trans. Electron Dev. ED-30 (1983) 764. J. del Alamo and R.M. Swirhun, Solid State Electron. 28 (1985) 47. D.E. Aspnes and J.B. Theeten, J. Electrochem. Soc. 127 (1980) 1359. H.J. Hovel, Semiconductor and Semimetals, Vol. 11. Solar Cells (Academic Press, New York, 1975). [16] P. Diaz, L. Hernandez and R. Romero, Crystal Res. Technol. 17 (1982) 67. [17] A. Cuevas, M.A. Balbuena and R. Galloni, IEEE Trans. Electron Dev. ED-34 (1987) 918.
Solar Energy Materials and Solar Cells
N + dopant exponential profiles for silicon solar cells L. Hernandez, R. Soler a n d A. M a r t e l Institute of Materials and Reagents for Electronics (IMRE), Uniuersity of Hat,ana, Hat,ana, Cuba Received 5 June 1992; in revised form 25 January 1993 A numerical model of solar cells is described, which provides optimum design rules for exponential N + dopant profiles for heavily doped emitter regions. The model gives a useful insight into the relative impact of surface and bulk recombination on device performance. Results agree well with theorical models.
I. Introduction The theoretical maximum efficiency of a silicon solar cell is limited as a consequence of heavy doping effects in the emitter region. Both the blue response and the open circuit voltage in silicon ceils of conventional design depend upon the properties of the heavily doped emitter region, related with bandgap narrowing and a high recombination at and near the surfaces. At the present time, the efforts to increase the efficiency have been focused appropriately on the problem of raising the open circuit voltage, since the current collection efficiency is approaching its theoretical limits [1]. To increase the conversion energy efficiency, it is necessary to obtain large values of the effective diffusion lengths, together with the highest possible donor concentrations, in order to reduce the sheet resistivity of the emitter layer. Moreover, the blue response of conventional cells is improved by reducing the junction depth, but the power loss owing to the sheet resistence increases. These effects work in opposite directions and it is thus evident that optimization is essential. The problem is complicated by imperfectly understood mechanisms or imprecise knowledge of the controlling technological parameters and by experimental difficulties in separating the effects of these mechanisms. The main difficulty when modelling an emitter region is the uncertainties in the values of the bandgap narrowing, mobility and lifetime parameters commonly used to characterize highly doped silicon. All of these parameters are affected by the dopant concentration profile and by the surface conditions. Several analytical models have been developed by different authors to provide insight into the transport mechanisms of minority carries injected into the heavily doped emitter, as well as recombination phenomena. Dumke [2] derived a Hermite 0927-0248/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
292
L. Hernandez et al. / N " dopant exponential pro]~les
polynomial expression for a Gaussian profile. He assumed that diffusion dominates the minority carrier current and that the diffusion coefficient is independent of doping. Shibib et al. [3] included majority carrier degeneracy and a finite surfacerecombination velocity S at the emitter surface in a transparent emitter model for thin emitters. Transparent emitter means that the average minority carrier diffusion length is longer than the junction depth and this implies that surface recombination can be the dominant recombination mechanism. Under the assumption of a transparent emmitter, many, efforts have been made to find a simple analytical aproximation. Fossum and Shibib [4] assumed diffusion dominant current near the surface where the effective doping concentration is nearly constant. After that, Amantea [5] and Del Alamo and Swahson [6] derived an expression for the emitter recombination current that keeps the position dependence of the effective doping concentration, diffusion coefficient and lifetime. The model of Rohatgi and Rai-Choudhury [7] is based on effective recombination velocity as a measure of the minority carrier losses in various regions of the device. The model includes the effect of bandgap narrowing, Auger recombination and recombination at the device surface. Morales-Acevedo [8] completed these models considering the electric field and series resistance effects upon the efficiency of the cell. Recently, a new analytical method for the study of solar cell emitters with arbitrary doping profiles have been developed [9]. Morales-Acevedo [10] has also reported that the quantum collection efficiency of heavily doped emitters in silicon solar cells is high and almost independent of the surface dopant concentration for thin emitters. Up to now the analytical models have been based on some specific assumptions about the transport mechanism or the geometry of the device [11]. Seeking a new aproach, in this paper, we report a numerical solution of the minority carrier transport equations in a nonuniform (exponential profile) heavily doped emitter. The solution is in the form of a power series of minority carriers as a coordinate function. The model includes the effects of bandgap shrinkage, dopant-depended mobility and lifetime, of minority carriers and surface recombination on the dark saturation current. We also consider the effect of surface impurity concentration upon the series resistance and efficiency conversion, but the electric field, resulting from the gradient of doping concentration is taken to be constant.
2. Theory A sketch of the emitter in a typical silicon cell is shown in fig. 1. We consider a n - p junction formed by diffusing n-type impurities into a uniform p-type substrate. In the emitter region, the minority carrier current density Jp is given by dp Jp = qtxppE - qDp d x
(1)
L. Hernandez et al. / N + dopant exponential profiles DOPING PROFILE ND°
CONTACTS
nnnnn
o
DEPLETION REGION W
z>
BASE
293
H
I
]
I
~=3
Idrcm -3
Fig. 1. Cross section of n + p solar cell and a graded doping profile used as a schematic diagram.
Since the minority carrier concentration p is non-degenerate, Einstein's relation can be used between the mobility /zp and the diffusion constant Dp. We assume full ionization of donor impurity atoms and low injection. The rest of the parameters in eq. (1) have their usual meaning. The differential equation describing the minority carrier concentration p in the quasi-neutral n emitter of a solar cell under illumination can be written as 1 d
q dxJv- G(x)
/~ + - - = 0,
(2)
%
where G(x) is the generation rate of electron-hole pairs due to light and ~-p the hole lifetime. We assume an impurity exponential profile in the emitter regions, U D = ND0 e x p ( - F x ) ,
(3)
where ND0 is the surface concentration. In LAMEL's Institute [12], the possibilities of the open tube diffusion technology have been explored. The dopant profiles that result from these advanced procedures present neither precipitates nor a platau near the surface and can be approximately represented by an exponential function with surface concentration No0. For the space variations of hole mobility and lifetime in the emitter layer we fit the experimental data for p~p and ~-p versus donor concentration reported in ref. [13]. The fit equations are /d,p = / ~ A N D ~B,
(4)
7"p = "rAND ~'u ,
(5)
Where /~a, /ZB, rA and ~'B are the fitting parameters. Hence, writing ~p and ~-p as a function of coordinates, eqs. (4) and (5) become /Xp =/x~ exp(~bX),
(6)
7p = % e x p ( % x ) ,
(7)
where /zb /Xa = / ~ A N D ~ B '
/ZB --
'rc~ = T A N D ~ ' " '
Tb "rB = T "
F
'
(8) (9)
L. tternandez et al. / N ~ dopant exponential profiles
294
Thus, we have expressed the exponential space dependence for/Zp and 7p. Putting eq. (1) into eq. (2) we obtain c l [ Dp(X) dxP(X) d 1 -ol(A)F(A) exp[-a(A)x] d-~ [Elzp(x)p(x)] - ~x + - -
-0,
%(x)
(10)
with the common boundary conditions
Jp I x=0 +qSp~lx=0 = O,
(x)lx=x =0, where a(A) is the absorption coefficient (data reported by Aspnes) [14], F(A) is the solar spectrum density at AM1, Sp is the surface recombination velocity. Introduction the non-dimensional variables, u =x/xj, 0 < u < 1, and ~-(u)= p(u)/n i, where x i is the junction depth and n i is the intrinsic carrier concentration, and combining eqs. (6) and (7), eq. (10) becomes Ou2"rr(u) +a~urr(u ) -b(U)~uTr(u ) = f ( u ) ,
(11)
where (12)
a = x j IZb --
2[ qE b ( u ) - xj [--~-/x b + - qE _
k T/x~ "
[ qe
t
f(u) = - - / ,-~--exp~ n i [ Klld'b
exp{-xj(/z b + 7b)U}] ,
--Xj(a+~b)U
} + -
qPo -
k T/z a%
(13)
exp{-xj(/~ b + rb)U}], (14)
with the solution ov
7r(u) = E ~rnun + 7r1(u) e x p { - x j ( a +/Zb)U } + n2(u) exp{--xj(tt b + %)u}. n--0
(15) More details about the mathematical handling can be found in the appendix. Thus, we determine the spatial dependence of minority carrier concentration through the emitter region solving eq. (10) by means of computer programs. Therefore, we get the photocurrent density at junction, taking into account eq. (1). The open circuit voltage is conditioned by the dark saturation current of the cell. The contribution made by the emitter region can be calculated by analogous
L. Hernandez et al. / N + dopant exponential profiles
295
Voc(v) 0.68" Xj =0.2p.m Sn=lOScm/s E = 103V/cm H = 2502~.m Ln = 230/~rn W=O.2#m Dn=13cma/s
0.660.64 0.62 0.60b)Sp=5-105cm/$
0.580.56
Fig. 2. Open circuit voltage versus donor concentration.
considerations. In this case, we hold G ( x ) = 0 and the boundary condition at junction is
P lx~x~= p o [ e x p ( q V / k t )
- 1].
(16)
The expressions reported by Hovel [15] to calculate the photocurrent density and the dark reverse saturation current density in the depletion and base regions were used. Similarly, we calculate open-circuit voltage and efficiency by the usual formulas, in the Schottky approximation. Reduction of the V0c value as a consequence of the heavily doped emitter is taken into account through the bandgap shrinkage corrected by the Fermi degeneracy as obtained empirically by Rohatgi et al. Finally, we evaluate the fill factor considering only sheet resistence by the equation found in ref. [16].
3. Results and discussions
The open circuit voltage was calculated with the method described in section 2. In fig. 2 we compare the V0c variations versus surface dopant concentration for passivated and non-passivated front (or top) surface. Oxide passivation induces improvement in V0c resulted from the decrease in surface recombination velocity, which helps in the reduction of the emitter dark current and hence the V0c value is dominated by the parameters of the base. Nevertheless, without any front surface passivation, the dark current is managed by emitter components. In the case of a passivated surface, upon increasing the concentration in the emitter, the V0c is almost constant up to 8 × 1018 cm -3 but it is quickly reduced to greater values of ND0. On the other hand, high dopant concentration in emitter favours V0~ in non-passivated solar cells. In heavy doped emitter Rohatgi and
296
L. Hernandez et al. / N ~ dopant exponential profile,s
20
(a)
18-
16-
b)Sp= 5" IC~cm/s
t4"
P2
Fig. 3. Conversion efficiency versus surface donor concentration (cell 2 × 2 c m
2
and 5 figures).
C h o u d b u r y [7] have reported an increase in V0c of about 15 m V due to oxide passivation, while M o r a l e s - A c e v e d o [9] found a difference of about 25 mV. O u r results coincide with the former. T h e V0c values calculated by us and by Cuevas et al. [17], for an exponential profile, show the same d e p e n d e n c e also. Conversion efficiency of the solar cell as a function of surface d o p a n t concentration is shown in fig. 3. As show, the efficiency is always greater in passivated cells than in non-passivated ones and the differences are almost constant. T h e improvem e n t in the efficiency due to oxide passivation is explained by an increase in Voc
~(%) 20-
16-
14-
(a)/
a)Xj = 0.2 xtrn
/
b) Xj = 0.5/xm c) Xj = 1.0 ~ m
12
Fig. 4. Conversion efficiency of a passivated solar cell versus surface donor concentration.
L. Hernandez et al. / N + dopant exponential profiles
297
~%) 20'
(a)
18"
16-
14C)
12
.....
1~"
~b = I0 -~.. cm
.......
(d a°
No.(c~ 3)
Fig. 5. Conversion efficiencyof a passivated solar cell as a function of surface donor recombination. and Jac" The efficiency reaches a saturation value as a result of the V0c reduction in the heavily doped emitter. Fig. 4 shows the influence of the junction depth in the maximum of conversion efficiency. When the X j decrease, the efficiency reaches a maximum at greater donor concentrations, and its absolute value is also greater. We obtain the optimum value of donor concentration for Xj = 0.2 ~ m of about 8 X 10 ~s cm -3 slightly greater than the reported by Morales-Acevedo, [8] who optimized the grid dimensions for surface dopant concentration. In fig. 5 is shown some other important results, when the substrate resistivity is taken as a parameter. The theory mentioned in Section 2, considers the influence of resistivity of the base in V0c and Jac" This calculation was performed in order to clear how Pb affects the optimum value of donor concentration for passivated solar cells. When the dark current density of the base increases, the efficiency maximum is reached at a greater value of the surface donor concentration, otherwise, when p~ decreases, the conversion efficiency obviously increases.
4. Conclusions
The understanding of the relevant physical mechanisms responsible for the influence of the emitter layer in solar cell characteristics have been studied by a numerical computer model, by including the effects of bandgap narrowing, experimental recombination lifetime and mobility, exponential profile of impurity concentration, and surface recombination in the emitter component of the dark saturation current density. In this paper, we have found that to improve the open circuit voltage, it is necessary to reduce the surface recombination by oxide passivation in the front surface, together with keeping a lightly doped emitter.
L. H e r n a n d e z et al. / N ~ d o p a n t e x p o n e n t i a l pr~[i'les
298
However, to reach maximum conversion efficiency, it is indispensable to take into account the sheet resistance and we found that with the above mentioned considerations, the optimum donor surface concentration is always greater than 1 × 10 ~9 cm
3.
Finally, we have also analyzed the influence of junction depth and base resistivity on the conversion efficiency of passivated and non-passivated solar cells and concluded that an optimization is essential.
Acknowledgement The authors would like to acknowledge the invaluable contribution of Professor Perez-Alvarez by means of a helpful review and advice.
Appendix To solve the differential equation (11)we assume a solution of its homogeneous part in the form of a power series, hence ac
•r(u)
oc
and
= Y'~ ~ ' . u "
b(u)
= ~
n=0
(A1)
%u'.
n=0
Therefore, the solution is given by
,~+a)(.+l)~-.+~+~(n+a)~-.+l n
E
bm~_~®(n--m . ' = 0 ,
m=O
where ® ( n - m ) =
0, 1,
if n < m , ifn>m.
Therefore %+~= (n+2)(n+l)
-a(n+l)Tr,+,+
Y' bm%_m®(n-m
,
m =0
which can be written as
(A2)
~- = C~1)+ C~:)Tr0, where - 1)C (1) +
~
bmCn_2_m(1)
C~1)
1 n ( n - 1)
-a(n
C~2)
n ( n 1-- 1)
- a ( n - l ~ C J .(2~ (2) 2 - m , - 1 + b . - 2 ~ b m C n-m=O
for n_>2.
,
m=0
L. Hernandez et al. / N + dopant exponential profiles
299
Assuming a particular solution, like T/'(/g) = T / ' I ( u ) e x p { - x j ( a +/Zb)U } + rr2(u) exp{--xj(lz b + Zb)U}, we obtain particular solution for eq. (11), oc
7r(u) = Y'~ ~-.u" + 7r'(u) e x p { - x j ( a +~b)U} n=0
+ rr2(u) exp{ +xj(/x b + Zb)U},
(A3)
where qaF(h) "wl(u) = --Xj kTl~. ~'l(u)
1
L
+ IZb)]2-axj(a + tzb) --b(u) '
xyqp o
1
nikTlz,~z [xi(z b +/Zb)] 2 -- axj(z b +/Xb) -- b(u) The solution is undetermined by only two constant, 7r 0 and 7r I because the others are automatically determined by the recurrent equation (A2). In order to find both constants we value the boundary conditions in eq. (A3), and get from the surface boundary condition 7r, = C~ 1) + Cl2)rro,
CI')= qxjSpp° k Ttx ,~ni
dTr I
zr' (0) [ - x j ( a + lZb) ] -- --d-uu
du .=0 + kT
+
kTtz.
.=0- ~2(0)[-x~(~b + ~.)]
(~r'(0) + ~2(0)),
c12' = - U e + ~ Finally, from the junction boundary condition: 00
Po _ 7rl(u ) e x p { - x j ( a +/Zb)U } -- 7r2(u) exp(--Xj(tXb + ~'b)U} -- • F/i
C (1)
n+l
1 + ~ C.tl) n=l
References [1] [2] [3] [4]
A. Rohatgi and P.R. Choudbury, IEEE Trans. Electron Dev. ED-31 (1984) 596. W.P. Dumke, Solid State Electron. 24 (1981) 155. M.A., Shibib, F.A. Lindholm and F. Therez, I E E E Trans. Electron Dev. ED-26 (1979) 959. J.G. Fessum and M.A. Shibib, IEEE Trans. Electron Dev. ED-28 (1981) 1018.
300 [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
L. Hernandez et al. / N ~ dopant exponential pr~files
R. Amantea, IEEE Trans Electron Dev. ED-27 (1980) 1231. J. del Alamo and R.M.S. Swanson, IEEE Trans. Electron Dev. ED-31 (1984) 1878. A. Rohatgi and P.R. Choudbury, Sol. Cells 17 (1986) 119. A. Morales-Acevedo, J. Appl. Phys. 60 (1986) 815. F.J. Bisschop, L. Averholt and W.C. Sinle, IEEE Trans. Electron Dev. ED-37 (1990) 358. A. Morales-Acevedo, Sol. Cells 28 (1990) 293. J.S., Park, A. Neughoschel and F.A. Lindholm, IEEE Trans. Electron Dev. ED-33 (1986) 240. G. Masetti, M. Severi and S. Solmi, IEEE Trans. Electron Dev. ED-30 (1983) 764. J. del Alamo and R.M. Swirhun, Solid State Electron. 28 (1985) 47. D.E. Aspnes and J.B. Theeten, J. Electrochem. Soc. 127 (1980) 1359. H.J. Hovel, Semiconductor and Semimetals, Vol. 11. Solar Cells (Academic Press, New York, 1975). [16] P. Diaz, L. Hernandez and R. Romero, Crystal Res. Technol. 17 (1982) 67. [17] A. Cuevas, M.A. Balbuena and R. Galloni, IEEE Trans. Electron Dev. ED-34 (1987) 918.