Temperature dependence of the open-circuit voltage of an N+-P-P+ silicon solar cell under high illumination levels

Solar Cells, 10 (1983) 155 - 175

155

TEMPERATURE DEPENDENCE OF THE OPEN-CIRCUIT VOLTAGE OF AN N÷-P-P + SILICON SOLAR CELL UNDER HIGH ILLUMINATION LEVELS

R. V. SINGH and C. M. SINGAL

Department of Physics, University of Roorkee, Roorkee 247672 (India) (Received July 8, 1982; accepted April 25, 1983)

Summary The effect of the temperature
1. Introduction When silicon solar cells are used under concentrated sunlight they tend to heat up, as a result of which the current and voltage generated by the solar cell are modified. Apart from this the excessive rise in temperature of the solar cell has degrading effects on the solar cell characteristics. Although a large number of investigations have been carried out on the performance of back-surface field (BSF) silicon solar cells for photovoltaic applications, the information available in the published literature on the performance of BSF cells as a function of temperature is scarce. However, some work in this direction has been carried out by a few researchers [1 - 4] in the last 0379-6787/83/$3.00

© Elsevier Sequoia/Printed in The Netherlands

156 two decades. Wysocki and Rappaport [1] have studied theoretically as well as experimentally the performance of conventional p - a junction solar cells as a function of temperature. Their results are limited owing to various approximations, i.e. the temperature dependence of the minority carrier lifetime or the carrier generation-recombination rate and hence the shortcircuit current were neglected. Mandelkorn and Lamneck [2] have reported some results of experimental investigations on the behaviour of conventional and BSF solar cells at various temperatures. Fossum and Burgess [3] have also reported the performance of conventional and BSF solar cells in concentrated sunlight as well as in high temperature environments and their results are based on their theory and experiment. Recently Sinha and Chattop a d h y a y a [4] calculated the open-circuit voltage of a BSF cell without incorporating a temperature dependence of the short-circuit current, minority carrier lifetime and energy band gap narrowing. They have considered the variation in the carrier mobility with temperature but in an incomplete manner. In the present paper we report an a t t e m p t to study the effect of temperature on all the voltage-controlling parameters, i.e. mobility, diffusivity, lifetime, energy band gap and its narrowing, carrier generation rate (through the absorption coefficient) and light-generated current, under the opencircuit conditions for an n + - p - p + cell at high illumination levels. All these parameters as functions of temperature at high intensities of illumination on the individual junction voltages and on the Dember potential of the n + - p - p + solar cell are investigated in the present analysis. This analysis is very useful in identifying design modifications to optimize cell performance at specific illumination levels and operating temperatures. In Section 2 theoretical formulations are given for the junction voltages, and in Section 3 the important results obtained from numerical evaluations and their practical significance are dealt with.

2. Theoretical analysis To design and optimize the silicon solar cell for concentrated sunlight applications it is necessary to know how the cell parameters vary with temperature so that the relative degradation in the cell efficiency can be minimized. As the illumination level is raised from 1 sun to several suns, the increase in the short-circuit current density causes an increase in the opencircuit voltage. Apart from this, the increase in illumination level causes an increase in the cell temperature, and the semiconductor material tends to become increasingly intrinsic as a result of which a degradation in the opencircuit voltage of the silicon solar cell occurs. In the present paper the analysis of our earlier paper [5] for the calculation o f open-circuit voltages in an n ÷ - p - p + silicon solar cell, as shown in Fig. 1, is extended to include the effects of temperature on the excess carrier lifetime, the photogeneration rate of carriers, the carrier mobility, the diffusivity, the energy band gap and

157

,

I -Wn.X=O

Wp Wp÷Wp.

x

wp

bw~.q,

I

p+

P

I

~/_'~__L

II

of

- ',=wp,,~

I-,-w

]

,

o

p.

P

-q~7*

~

,'

wp

i

EF

w~

(b)

(a)

Fig. 1. (a) T y p i c a l s t r u c t u r e o f a n n + - p - p + s o l a r cell; ( b ) e n e r g y b a n d d i a g r a m o f a n n + - p - p + s o l a r cell.

the band gap narrowing at high carrier concentrations. The ambipolar continuity equation is given by dn' dt

-g----

n' T*

+p'E--

dn' dx

+ D*~

d2n '

(1)

dx 2

where p* -

p--n

P/Pn +

D* -

(2)

n/pp

p+n p/Dn +n/Dp

(3)

and r* is as defined later. For steady state conditions the solution of eqn. (1) is obtained for all three regions (n ÷, p and p÷) subject to the following b o u n d a r y conditions, as in ref. 5. For the n÷-p junction at x = 0,

(P,÷')x=0-

Pn÷°+np°exp{--q(~°--Vj)/kT}lexp(qk~) - - 1 1 --exp(--2q(~b 0 - Vj)/kT}

f

(4)

and

(rtp')x = o =

np° +P"÷°exp(--q(¢°--Vj)/kT) t (qk~T) l 1 -- exp{--2q(~ o -- Vj)/hT) exp -- 1

(5)

For the p-p+ junction at x = Wp,

(nP')x=wP=PP°+np÷°exp{--q(~J°--Vm)/kT}t ( ~ ) 1 -- e x p ( - - 2 q ( ~ o - - Vlh)/kT} exp

I

(6)

Vm)/kT}~exp [qVm\ t l ~kT ] -- 1

(7)

-- 1

and ' . (np÷)x=wp

np*0 + Pp0 e x p { - - q ( ~ 0 - . . . . . 1--exp(--2q(~o--Vm)/kT)

158 where ni 2 t81}

Pn+o -

ND ni 2

npo-

(9)

NA ni 2

np+o -

(10) NAA

NA

PpO =

kT

¢o---q

(11)

[NAN D\

(12)

J

kTInINAAI Co= q ~NA ]

(13)

T h e surface c o n d i t i o n s at x = --Wn+ and x = Wp + Wp+ are as follows:

=Sn+(Pn+')x:_Wn +

D'n+ (dPn÷' t

(14)

\ dx ix=--Wn+

and (15)

(np+t)x = Wp + Wp+ = 0

T h e last c o n d i t i o n implies t h a t t h e r e is an infinite surface r e c o m b i n a t i o n v e l o c i t y o n t h e back surface o f t h e p+ layer. T h e expressions f o r the n ÷ - p j u n c t i o n c u r r e n t d e n s i t y J and t h e l o w - h i g h ( p - p ÷ ) j u n c t i o n c u r r e n t d e n s i t y J ' were f o u n d t o be [5]

J° + J°' exp{--q(¢°-- Vj)/kT} t (qk~) 1 -- e x p ( - - 2 q ( ¢ o - - Vj)/kT} e x p +

Jo" +_Jo"exp(--q(+o = V,h)ikT) 1 1 -- exp(--2q(~ o --

-- 1

tqVmt

+

V~h)/kT} f exp\-~-~-] - 1

\PpP~p.nn]n+ \#Xpp+.nn)pf-i]x: o and

l

f + JL i × (16)

159

1 -- exp {--2q(gbo -- Vj )/kT}

[

--

Jwp " +Jwp "' exp(--q(@o-- Vm)/kT}~ /qVm -- 1 , exp ~ 1 - - e x p { - - 2 q ( @ o - Vlh)/kT} 1 1 ] ~+ JL']1 x

( pnn t - t _itnn_ X[l\pnn+ppp] p \#nn+PpP

)p+}-llx

(17)

=Wp

where

J°=

qD*n+ qD*p (Wp) L'n+ 7Pn+° + L*p np° coth

Jo' =

qD*

qD*p

n +

[ Wp \

7npo + ---:--Pn+0 c o t h | 7 7 - | L*p \L p] L'n+

(18)

(19)

jo. - qD*p Ppo cosech (Wp)

(20)

Jo"'- qD*p np÷ocosech(-W-~/ L*p \L p]

(21)

L*p

J L = g . ÷qD*.÷ ~r ), =

=

.

Wp Wp . qD*pv.plcotht__l_cosechIT_~_tt .÷('),--0) +gp L*---~ ~ \L'p/ \Lp]~

Sn+cosh(Wn+/L*n÷)+ (D*n÷/L*n÷)sinh(Wn÷/L-n÷)

Sn÷sinh(Wn÷/L*n÷)+ (D*n÷/L*n÷)c°sh(Wn÷/L*n÷) S~ Sn÷sinh(Wn÷/L*n*)+ (D*n÷/L*n÷)c°sh(Wn÷/L*n÷)

(22)

(23)

(24)

Jwp _qD*pnpocosech(Wp)

(25)

j ,Wp _ qD*p L*p Pn÷o cosech \l L- ~ lp ]

(26)

L*---~

L~p

160

Jwp

,,_qD*p 'I --~- ] qD*p÷ / Wp+'~ L* Ppo coth + np+0 coth p

\L p ]

= L---7 - np+0 coth

Jwp

p

\L p ]

L'p,

+ qD - - - 7p-+p_p 0 c o l.;.n /[~ W : - -p.+

L'p+

- ~ p+] t -- c o s e c h ( ~ ) JL' = gp+ qD*"-----:+ L'p+ T'p+ t coth (\L

\L'p+

(29)

L*p

~

k T ln t NDN A f

- -

q

(28)

}+

+ gp

(~0

(27)

((ni)n+(ni)p

(30)

and ~0 = -~- In NAA(ni)p (31 )

NA (n i )p+

Equations (16) and (17) are solved numerically simultaneously under opencircuit conditions, i.e. J = J ' = 0, using the Newton-Raphson iterative method to determine the n+-p junction voltage Vj and the p-p+ junction voltage Via. Using these two voltages, the Dember potential VD appearing across the quasi-neutral p-type base region is calculated using the expression given by Sabnis [ 6 ] : VD -

kT b -- 1 In I N A + (b + 1)np'(X -- 0) t q

b+ 1

NA + ~ - + ~ - - ~ - p ) ~

(32)

where b is the ratio of the electron mobility to the hole mobility, N A is the doping density in the p-type base region and np' is the excess carrier concentration at the n+-p and p-p÷ junctions in the p-type base region. The total open~ircuit voltage Voc is the algebraic sum of Vj, Vlh and VD. The temperature dependence of these voltages arises through various parameters such as the excess carrier lifetime T, the mobility/~, the intrinsic carrier concentration n i and the carrier generation rate g. We discuss the temperature dependence of these quantities separately in Sections 2.1 - 2.4.

2.1. Effect of temperature on intrinsic carrier concentration The temperature dependence of n i is very important in determining the variation in the open,circuit voltage. Its dependence on temperature at high levels of illumination is also related to the energy band gap shrinkage which comes into play when the carrier concentration becomes large, i.e. greater

161

than 10 is cm-3, as has been pointed out by us [5]. The intrinsic carrier concentration n i at two different temperatures To and T is given by hi(To) = NoTo 3n expt

ni(T) = No T3n exp I

Eg(T°) I

(33)

Eg(T) 2-~ t

(34)

where Eg(T) is the energy band gap of the semiconductor at a temperature T. When these two equations are combined and TO= 300 K, ni(To) = 1.45 )< 10 'o cm-3 and Eg(To) = 1.1 eV are substituted, n i is given as

/

T

\3/2

ni(T ) = 3.082 )< 1 0 ' 9 ( 3 ~ )

exp

{ Eg(T)I~t

(35)

This form of ni(T) takes into account both the explicit temperature dependence of nj and the implicit dependence on temperature through the variation in Eg with T. The value of Eg(T) at different temperatures is taken to be 7.021 )< 10-4T: Eg(T) = 1 . 1 5 5 7 (36) T + 1108 as has been calculated by Rajkanan et al. [7]. The value of ni is further modified by the effect of the carrier-concentration-dependent band gap shrinkage AEg, as derived by Lanyon and Tuft [8], with AEg = 0.389 T)< 10 's

eV

(37)

for non-degenerate semiconductors and / N ~1/6 AEg = 0.16211--~ ) eV

(38)

for degenerate semiconductors, where N is the carrier concentration,/.e, the sum of the doping concentration and the concentration of excess carriers generated by light. The modified equation for r/i now becomes T )3/: ni(T)=3.082)< 1019(3-~ exp{ E~(~--~TAEg }

(39)

2.2. Effect of temperature on mobility The carrier mobility p also depends on temperature, p is known to be the composite of the impurity scattering mobility p~, the lattice scattering mobility #L and the charge carrier mobility Pc and is given by Wolf [9] as

162 1

1

/J

PI

1

+ --

PL

1

+ p

(40)

Pc

where

27/2(esieo)2(kT) 3/2 12I = ~ 3 / 2 C B q 3 m e f f l / 2

ln{(3esieokT/q2CB

l/a) +

1}

(41}

tZL~ = 2.1 × 109 T -2's

(42)

PLp = 2.3 × 109T -2"7

(43)

and

(-~n

1 1 ) 1/2 3(esie°)2(kT)3/2 + ~m p 23/27rl/2q2N ln[{1 + (4esieokT/q2N1/a)) 2]

#e =

(44)

N is the total density o f charged particles, mn, mp and meff are the effective masses of electrons and holes or either respectively, CB is the impurity concentration and esi is the dielectric constant of silicon. On the application of the Einstein relation D = (kT/q)p, the temperature dependence of the diffusion coefficient D can also be calculated.

2.3. Effect of temperature on carrier generation rate The generation of carriers is also affected by temperature. However, it is seen through numerical evaluations t h a t there is only a slight increase in the carrier generation rate g with increasing temperature. The generation of carriers is assumed to be uniform t h r o u g h o u t the cell after an average has been taken over the cell thickness. The dependence of g on temperature can be calculated as follows. The number of photons per unit area per second for a small wavelength range from X to X + dX is N~ dX. Now the intensity of photons, at a depth equal to the total thickness D of the cell, is Nx e x p ( - anD) dX and the n u m b e r of photons absorbed in this thickness is dN = Nx dX --N~ exp(--~xD) d~

(45)

The total number of absorbed photons with energy hc/X ~> Eg can be obtained by integrating this equation over the visible spectrum with 0.3 pm ~< ~< 1.1 pm: 1.1/~m

N = f

Nx(1 -- exp(--a~D)} d~

(46)

0.3Din

On the assumption of a one-to-one correspondence between absorbed photons and generated carriers, the average carrier generation rate required for our theoretical analysis is taken to be 1

1.1 pm

= -- f Nx{1 --exp(--c~D)) dX g D 0.3"J#m

(47)

163 where an is the temperature-dependent absorption coefficient as calculated by Rajkanan et al. [7] :

oLk( T) = ~ CiAi [{~¢o --Egi(T ) + E p i } 2 i =1,2

[

+

exp(Epi/kT ) -- 1

{hco

-- EgI(T)

--

Epi) 2 ] +

1 -- exp(--Epl/kT)

1=1,2

+ Ad {hCO -- Egd(T)} 1/2

(48)

In the above expression, co = 2~c/k, the suffix i refers to various possible phonons of energy Ep, the suffix j refers to different indirect band gaps Eg i which may be active in the p h o t o n absorption process, Egd is the direct band gap energy and Ci, Aj. and A d are the strength coefficients for the various optical transitions, as given by Rajkanan et al. [7]

2.4. Effect of temperature on carrier lifetime The lifetime of excess carriers is very much affected by the temperature and illumination level. In fact, it has been found by Singal et al. [10] to increase with increasing temperature and illumination level. The excess carrier lifetime r* has been obtained by Singal et al. as nf

T* = -R

(49)

where n' and R are the excess carrier concentration and the carrier recombination rate given by {EFn - - E F p l l ~/2 ~-~ ] ~ --

n'= ~ I I (NA-- ND)2 + 4hi 2 exp ~.

]

-- ND) 2 + 4ni2}1/2[

(50) and R-

NROV 25 × In

k Tni2[ exp { (Efn -- Efp)/k T) -- 1 ] × ((NA --ND) 2 + 4ni2[exp{(Efn --EFp)/kT) -- 1]) 1/2 X+Ytanh{(E a+5)/2kT) X - - Y t a n h ( ( E a + 5)/2kT)

X ~ _ Y tanh((Ea _--.5)/2kT) ] X + Y t a n h ( ( E a - - 5)/2kT) ]

(51)

where X =

t (N h --ND)

2 + 4ni 2

exp~[EFn--EFp)fl/2 "k-'T + 2ni

(51a)

(51b)

164

o is the capture cross section of electrons or holes; v is the thermal velocity of carriers, given by v = ( 3 k T / m ) l / 2 ; n ' is the excess carrier concentration; N A and ND are the acceptor and donor concentrations in the semiconductor~ the defect states giving the recombination centres are uniformly distributed in an energy range from El + Ea -- 6 to Ei + Ea + 6 and their total number per unit volume is NR; EFn and EFp a r e the quasi-Fermi levels for electrons and holes. For the present analysis we determine the value of ~ in a particular semiconductor region of the solar cell by first equating the generation rate g from eqn. (47) to the recombination rate R and finding EFn - - E F p from eqn. (51). Using this value of E F n - - E F p , the excess carrier concentration is obtained from eqn. (50) and then T is determined from eqn. (49).

3. Results and discussion For numerical calculations we take fixed values of some of the device parameters, as given in Table 1. With these fixed parameters the temperature of the cell was varied from 273 to 373 K and the illumination level was increased from 0.01 sun at air mass (AM) 1 to 107 suns at AM 1 to examine the effect of temperature at different levels of illumination on the n÷-p junction voltage Vj, the p-p+ low-high junction voltage Vlh, the Dember potential VD and the total open-circuit voltage Voc o f the cell. In Fig. 2 the variation in Vj with temperature is shown for various illumination levels. Vj was found to decrease linearly with increasing cell temperature and it may be written empirically as

Vi = Vj(0) + %t

(52)

where % is the temperature coefficient of Vi and t is the temperature in degrees Celsius. This temperature coefficient is negative and decreases in TABLE

1

Structure and material parameters used for the solar cell analysis Front n + and back p+ doping Base p-type region doping Front-surface recombination velocity Back-surface recombination velocity Thickness o f n + region Thickness of p region Thickness of p÷ region Capture cross section a Density N R of recombination centres p region n + region and p+ region Energy distribution parameters for recombination centres Ea

6

1019 cm-3 1016 cm-3 10 3 cm s- I oo

0.25 pm 150 pm 0.50 pm 10-1s cm 2 1013 cm-3 101s cm-3 0.2 eV 0.2 eV

165 1.0

I

I

1

I

I

I

I

I

I

I 20

I 30

I 40

I 50

I 60

I 70

I IBO

I gO

O.c~

0.~

0.7

0.6~ 0.5~

}

0.4

O2

0.2

0.1

0 0

I IO

T

I00

~°C)

Fig. 2. The n +-p junction " " voltage Vj vs. cell temperature T for various illumination levels. magnitude with increasing sunlight concentration; aj is given in Table 2 for various levels of illumination. At low and medium levels of sunlight concentration the decrease in the magnitude of o~ is nearly 0.20 mV °C-1 for every tenfold increase in the sunlight concentration. The degradation in V~ with temperature is very much expected because of the tendency of the extrinsic semiconductor to become intrinsic in character as a result of the shifting of Fermi levels lying either above or below the intrinsic level towards the middle of the band gap; this shifting is due to the increase in the intrinsic carrier concentration with increasing cell temperature. A plausible justification of the variation in o~ with illumination level for low and medium sunlight concentrations can be given by the following simple derivation. Neglecting the low-high junction effects, we may write the junction voltage Vi under open-circuit conditions (J = 0) from eqn. (16) very approximately as Vj=

kT q __In(j~)

(53)

TABLE

2

aj ( m V °C - 1 ) ~lh ( m V ° C - I ) a D ( m V °C - l ) a ( m Y °C-I)

--2.351 -. . --2.351

0.01 s u n

--2.155 -. --2.153

0.1 s u n

. --1.951

--1.951 --

1 sun

The coefficients ~ at various illumination levels

--1.748 +0.021 0.005 --1.732

10 s u n s

--1.555 +0.094 --0.018 --1.479

102suns

--1.358 +0.189 --0.062 --1.231

103 s u n s

--1.131 +0.275 --0.109 --0.965

104 s u n s --0.609 +0.179 --0.174 --0.605

10 s s u n s

--0.803 +0.370 --0.181 --0.614

10 6 suns

--0.885 +0.330 --0.300 --0.855

107suns

Ob

167 where J0 may again be written from eqn. (18) as approximately J0 = A exp

Eg

--/kEg) kT

(54)

with A a constant. Combining these two equations and simplifying, we obtain

Vj= kT__ q In

+

(55)

q

which on differentiation with respect to temperature T gives aEg _ dVj _ k In ( ~ ) 1 + o~ dT q q aT

1 a(AEg) q

(56)

aT

Substituting for A from eqn. (54) and for J0 from eqn. (53), we get ~j -

Vj T

Eg -- AEg qT

+

1 a(Eg -- AEg) q aT

(57)

For moderate illumination levels and doping concentration when the band gap shrinkage AEg can be neglected, substituting Eg from eqn. (36) (at T = 300 K), we find that aj = --2.137 mV °C-1 if Vj = 550 mV and oq = - - 1 . 9 7 0 mV °C -1 if Vj = 600 mV for a silicon solar cell. This approximate analysis indicates that the temperature coefficient aj depends directly on the opencircuit voltage of the solar cell. If ajl and aj2 are the temperature coefficients at t w o different levels of illumination, corresponding to the light-generated current densities JL1 and JL2 respectively, then we have k In ajl = q

(__~)

+

1 aEg

1 a(AEgl)

q

q

aT

(58)

aT

and k In ~2 = q

(J.~)

+

1 aEg

1 a(hE,2 )

q

q

aT

(59)

aT

which give oq2 = o~, + - - In -- -q \JL, ] q

(60) aT

or

1 a(AEg 2 -- AEg,) (~j2 = (°ql + 0 . 2 ) m V °C -~ - - - q

(61) aT

168 0.30

I

I

l

r

i

f

iO 7

I

i

SUN

O25

/

, 0 6 SUN

0.20

10 5 S U N

/

0.15

>

IO4 S U N > 0.10

103SUN

_...____---.

0.0 5

I 02 SUN

I 0 j SUN I

0

IO

I 20

I 30

I 40

I 50

I 60

f 70

I SO

J gO

100

T(oc )

Fig. 3. The p-p+ junction voltage Vlh v s . cell temperature T for various illumination levels.

if JL2 = 10 J m , i.e. for a tenfold increase in illumination level, where AEg 2 is the band gap shrinkage at a tenfold increase in illumination level. We observed from our detailed numerical evaluations at low and medium sunlight concentrations that ~i2 does increase from 0~1 by nearly 0.2 mV °C- l when the effects of band gap shrinkage are negligible. At very high sunlight concentrations, e.g. 104 suns and higher, the excess carrier concentration becomes very large, i.e. 1019 cm -a or higher, and the effects of the carrierconcentration-dependent band gap shrinkage begin to become evident, e.g. the reduction in the value of Vj with increasing illumination level as has been discussed by us in ref. 5. As a result, when the carrier concentration is such that the non
169 -0.20

t"

"I

I

I

I

I

I

J

I

-0.~5

-0.10 o >

107Sun •.~.0 5

......

i: -Sun 10 2 Sun

' I0

/ 50

410

20

3o

l 60

I 70

| 80

' gO

I00

T("C )

Fig. 4. The Dember potential VD vs. cell temperature T for various illumination levels. in this figure. Unlike Vj the voltage Vlh increases linearly with increasing temperature and may be written as V l h = V l h ( 0 ) + O~lht

(62)

where ale is the temperature coefficient of Via. The temperature coefficient c~m at low levels of illumination is small compared with that at high illuminations. As has been explained by us in ref. 5, Vlh is enhanced as the intrinsic carrier concentration ni increases. At medium levels of illumination the increase in ni with increasing temperature occurs according to e x p ( - - E g / 2 k T ) and therefore a slow increase in V m is observed with increasing temperature. ~lh is dependent on the illumination level and increases by nearly 0.1 mV °C-~ for every tenfold increase in the sunlight concentration. At very high sunlight concentrations, e.g. 104 suns and higher, the effect of the carrierconcentration~lependent band gap shrinkage AEg begins to be important and, when AE~ is temperature dependent according to the non-degenerate situation given by eqn. (37), alh decreases sharply with increasing illumination level. At even higher illumination levels such as 106 and 107 suns, AEg is given by eqn. (38) and is temperature independent and thus C~lhbegins to vary only slowly with illumination level. In Fig. 4 we plotted the variation with temperature in the Dember potential VD arising in the p-type base region of the cell for various levels of illumination. It can be seen that the values of VD are quite small, but they increase in magnitude when the temperature of the cell rises. Since VD is negative for p-type base regions in n*--p-p + solar cell structures, this should cause a reduction in the open,circuit voltage of the cell with increasing temperature. The temperature dependence of VD is again linear and as earlier we may write

170 1.0 ¸

V ......

T ........

~

.........

~ - - - - - ~

......

• ...........

I

T ..............

0.9 IO6 sun

0.8

0.7

I0 2 sun

IO Sun

0.6

I

su~

0.5

q~un O u >o

0,4

"~'-'----...,..~. 01 sun

0.~

0.2

i 0.1

I ro

J 20

I :~0

I 40

I 50 T

Fig. 5. The open-circuit voltage Voc

VD = VD(0) + aDt

vs.

I 60

I 70

J 80

1 90

100

~,°C )

cell temperature T for various illumination levels.

(63)

where aD is the temperature coefficient of VD. aD is quite small but increases in magnitude by nearly 0.04 mV °C-1 for every tenfold increase in the sunlight concentration. The combined effect of the temperature dependences of Vj, Vlh and V D is given by Voc = Vj + Vm + VD, which is shown in Fig. 5 for various levels of illumination. The temperature dependence of Voc is again linear and may be written as Voc = Voc(0) + a t

(64)

where a is the temperature coefficient of Voc. This temperature coefficient again depends on the illumination level and increases by nearly 0.20 mV °C-1 for every tenfold increase in illumination for the range of illumination

171

intensities which do n o t generate any Vlh and VD. At higher intensities of illumination when both Vlh and VD begin to contribute, the temperature coefficient ~ increases by nearly 0.15 mV °C-I for every tenfold increase in the sunlight concentration. Our analysis shows that the temperature variation in the open-circuit voltage of an n + - p - p ÷ silicon solar cell arises mainly from the temperature variations in the intrinsic carrier concentration n~ directly through the temperature dependence of ni cc e x p ( - - E g / 2 k T ) as well as indirectly through the temperature dependence of the energy band gap and its shrinkage. The temperature variation in Vo¢ also depends on the illumination level and the cell structure; in particular the temperature coefficient will be lower in magnitude for a cell with a back-surface low-high p-p÷ junction and will also be lower at high illumination levels. Experimentally it has been observed by Mandelkorn and Lamneck [2] that the temperature coefficient of Voc for an n+-p cell at 1 sun is --2.3 mV °C -1 and for an n + - p - p + cell at 1 sun it is --2.13 mV °C-1 for cells made with a 10 ~2 cm p-type base region. This observation agrees with our deduction that when the p-p÷ junction begins to contribute to the open-circuit voltage of the cell the magnitude of ~ decreases. Further, for a cell with this junction the temperature coefficient of Voc will change with illumination level at a different rate for low and high levels of illumination as can be seen from Table 2. In another experimental study it has been seen by Yasui and Schmidt [11] and Luft [12] t h a t for an n÷-p cell with a 2 ~2 cm base ~ is --2.2 mV °C -1 and for an n+-p cell with a 10 ~ cm base a is --2.5 mV °C-1. It should be noted t h a t solar cells with a lower resistivity base region normally give higher open-circuit voltages which, as is indicated by our eqn. (57), should lead to smaller magnitudes of the temperature coefficient a. In particular, it is seen from eqn. (57) that for a junction voltage Vj of 500 mV a coefficient ~j of --2.303 mV °C-1 is possible and for a Vj value of 600 mV a coefficient aj of --1.970 mV °C-1 is likely. Our detailed analysis yields a = --1.951 mV °C-1 (Table 2) for a cell with a 2 ~2 cm base resistivity whose open-circuit voltage at 1 sun and at T = 300 K is 588 mV and gives an approximate value o f a = --2.010 mV °C-I from eqn. (57), which are in near agreement with each other. Experimental observations have also been made by Agarwala e t al. [13] for n+-p cells and t h e y f o u n d a = --2.46 mV °C-1 at 1 sun illumination, their open-circuit voltage at 1 sun being 542 mV. For this open-circuit voltage our a estimated from eqn. (57) is --2.17 mV °C-1. The experimental values of Agarwala et al. [13] are somewhat higher than expected; this could be due to the experimental difficulties of accurately determining the solar cell junction temperature. Fossum and Burgess [3] have calculated the open-circuit voltage of n÷-p and n + - p - p ÷ cells with a 10 ~ cm base resistivity at T = 27 °C and T = 100 °C for 1 sun and 40 suns illuminations. Using their results we f o u n d t h a t the temperature coefficients of n+-p cells are --2.62 mV °C-~ and --2.37 mV °C -~ at 1 sun and at 40 suns with the room temperature open-circuit voltages of Fossum and Burgess of 541 mV and 605 mV respectively. It can be seen t h a t an increase in the illumination level reduces the temperature coefficient, in agreement with our

172 analysis. Similarly we found that for n + - p - p + cells the temperature coefficients are --2.30 mV °C -1 and --1.93 mV °C --1 at 1 sun and at 40 suns with their room temperature open-circuit voltages of 606 mV and 721 mV respectively, which again shows that when the p-p+ junction contributes to the open-circuit voltage the temperature coefficient is reduced. The effect of temperature variations in the mobility and the absorption coefficient on the temperature dependence of the open-circuit voltage has been found to be minimal. However, the effect of the temperature variation in the excess carrier lifetime T on the temperature dependence of the opencircuit voltage was found to be substantial, as can be seen by comparing the results in Table 2 with the temperature coefficients of the voltages given in Table 3 which were evaluated by taking T as independent of temperature. The lifetime r is expected to increase with temperature and illumination level [10] as shown in Table 4. This increase in T with temperature which has also been observed experimentally b y Agarwala e t al. [13] tends to reduce the diode saturation current and thereby to increase the open-circuit voltage of the cell and thus to reduce the magnitude of the temperature coefficient of the open-circuit voltage. This is clearly shown in Tables 2 and 3, since the values of a in Table 2 are lower than those in Table 3 for most of the illumination levels. From a practical standpoint, the facts that the magnitude of the temperature coefficient of Voc is much less at high levels of illumination than at 1 sun and that it is further reduced for a cell with a p-p+ backsurface junction, imply that cells with an n ÷ - p - p ÷ structure can be operated at elevated working temperatures with much less degradation in the opencircuit voltage at high illumination levels.

4. Conclusions (1) The n÷-p junction voltage decreases with increasing cell temperature. The temperature coefficient ~j is dependent on Vj, i.e. o~ is higher for lower Vj and c~j is lower for higher Vj at the same temperature. (2) At higher levels o f illumination the degradation in the open-circuit voltage due to increase in the cell temperature is small compared with that at low and medium levels o f illumination. (3) The increase in the p - p + junction voltage, under open-circuit conditions, with increasing temperature is very small at low and medium levels of illumination while at high illumination levels it increases rapidly. (4) The reduction in the D e m b e r potential with increasing temperature is significant at high levels of illumination and is negligible at low levels of illumination. (5) A solar cell with the n + - p - p + structure exhibits a lower temperature coefficient than that for the n+-p structure at illumination levels where the p - p + junction contributes significantly to the voltage.

--2.386 -. --2.386

.

--2.187 -. --2.187

0.1 sun

. --1.987

--1.987 --

1 sun

0.01 0.10 1.00 10.0 100.0 1000.0

(suns)

Illumination

9.79 9.81 9.96 11.39 16.14 17.74

0 °C

9.83 9.84 9.99 11.41 15.97 17.44

I 0 °C

9.89 9.90 10.05 11.46 15.84 17.17

20 °C 9.97 9.98 10.13 11.54 15.76 16.91

10.07 10.08 10.24 11.65 15.72 16.69

40 °C

--1.777 +0.017 0.004 --1.764

lO suns

30 °C

r (ps) for the following temperatures

Variation in r with temperature and illumination level

TABLE 4

o~j (mV °C-1) 0lib (mV °C-1) olD (mV °C-1) ol (mV "C- z )

0.01 sun

10.20 10.21 10.37 11.79 15.71 16.48

50 °C

--1.556 +0.094 --0.025 --1.487

102suns

10.34 10.36 10.51 11.95 15.74 16.30

60 °C

--1.352 +0.197 --0.065 --1.220

lOa suns

The coefficients ~ at various illumination levels taking r to be independent o f temperature

TABLE 3

10.51 10.53 10.69 12.14 15.80 16.14

70 °C

--1.144 +0.288 --0.111 --0.967

104suns

10.70 10.71 10.88 12.35 15.88 16.01

80 °C

--0.929 +0.359 --0.161 --0.731

lOS suns

10.91 10.93 11.10 12.59 15.99 15.90

90 °C

--0.991 +0.504 --0.177 --0.664

106suns

11.15 11.16 11.34 12.86 16.13 15.81

100 °C

--0.973 +0.622 --0.211 - - 0. 562

107suns

174 Acknowledgments Financial s u p p o r t for this w o r k f r o m the University Grants Commission, India, is gratefully a c k n o w l e d g e d . T h e a u t h o r s also wish to a c k n o w l e d g e with t h a n k s the facilities provided b y t h e D e p a r t m e n t o f Physics, University o f R o o r k e e , R o o r k e e , for c a r r y i n g o u t this w o r k .

References 1 J. J. Wysocki and P. Rappaport, Effect of temperature on photovoltaic solar energy conversion, J. Appl. Phys., 31 (1960) 571 - 578. 2 J. Mandelkorn and J. H. Lamneck, Proc. 11th Photovoltaic Specialists' Conf., Phoenix, AZ, May 1975, IEEE, New York, 1975, pp. 36 - 39. 3 J. G. Fossum and E. L. Burgess, Silicon solar cell development for concentrated sunlight, high temperature applications, Proc. 12th Photovoltaic Specialists' Conf., Baton Rouge, LA, November 15 - 18, 1976, IEEE, New York, 1976, Paper 11.1. 4 A. Sinha and S. K. Chattopadhyaya, Temperature dependence of open circuit photovoltage of a back-surface-field semiconductor junction, Solid-State Electron., 22 (1979) 849 - 852. 5 R. V, Singh and C. M. Singal, Open-circuit voltages across two junctions in n+-p-p + solar cells under high illumination levels, Sol. Cells, 8 (2) (1983) 97 - 123. 6 A. G. Sabnis, Junction potentials of strongly illuminated n+-p-p ÷ solar cells, SolidState Electron., 21 (1978) 581 - 587. 7 K. Rajkanan, R. Singh and J. Shewchun, Absorption coefficient of silicon solar cell calculation, Solid-State Electron., 22 (1979) 793 - 795. 8 H. P. D. Lanyon and R. A. Tuft, Band gap narrowing in moderately to heavily doped silicon, I EE E Trans. Electron Devices, 26 (1979) 1014 - 1018. 9 H. F. Wolf, Silicon Semiconductor Data, Pergamon, Oxford, 1969, pp. 76 - 95. 10 C. M. Singal, S. C. Joshi and R. V. Singh, Variation of excess carrier lifetime in silicon with temperature, illumination level and doping concentration, submitted to Int. J. Electron.

11 R. K. Yasui and L. W. Schmidt, Proc. 8th Photovoltaic Specialists' Conf., Seattle, WA, 1970, IEEE, New York, 1970, p. 110. 12 W. Luft, IEEE Trans. Aerosp. Electron. Syst., 7 (1971) 332. 13 A. Agarwala, V. K. Tewary, S. K. Agarwal and S. C. Jain, Temperature effects in silicon solar cells, Solid-State Electron., 23 (1980) 1021 - 1028.

Appendix A: Nomenclature b D D* Dn, Dp E Ea EFt, E~, E i

ratio o f e l e c t r o n m o b i l i t y t o hole m o b i l i t y t o t a l thickness o f t h e cell (cm) a m b i p o l a r carrier d i f f u s i o n c o e f f i c i e n t (cm 2 s -1) e l e c t r o n diffusivity a n d hole diffusivity respectively (cm 2 s-1) electric field (V c m -1) e n e r g y d i s t r i b u t i o n p a r a m e t e r f o r r e c o m b i n a t i o n centres (eV) F e r m i level f o r electrons, holes a n d intrinsic carriers respectively (eV)

175

Eg g k L*

me ff m n ,mp n no

ni NA, NAA, ND

energy band gap (eV) rate of carrier generation (cm -3 s-l) 8.62 × 10 -s eV K-1, Boltzmann's constant ambipolar carrier diffusion length (cm) effective mass of a carrier (g) effective mass of electrons and holes respectively (g) electron concentration (cm -3) electron concentration in dark conditions (cm -3) intrinsic carrier concentration (cm -3) doping concentration of p, p+ and n ÷ regions respectively (cm -3)

density of recombination centres (cm-3) hole concentration (cm -3) P hole concentration in dark conditions (cm -3) Po q 1.6 × 10 -19 C, electronic charge R rate of carrier recombination (cm -3 s-1) front-surface recombination velocity (cm s-1) sn÷ T absolute temperature (K) potential developed due to light intensity across the p-type YD, Yj, Ylh base region, n÷-p junction and p-p+ junction respectively (V) open~ircuit voltage (V) Yoc Wn*, Wp, Wp÷ thickness of the n ÷, p and p+ regions respectively along an arbitrary x axis (cm) O~ absorption coefficient of silicon 5 energy distribution parameter for recombination centres (eV) /kEg energy band gap shrinkage (eV) permittivity of free space ~o dielectric constant of silicon Csi wavelength of photons (cm) /.t* ambipolar carrier mobility (cm 2 V -1 s-1) electron mobility and hole mobility respectively (cm 2 V s-1) U carrier capture cross section (cm 2) 7"* ambipolar carrier lifetime (s) equilibrium barrier potential of n+-p and p-p÷ junctions re~0, ~0 spectively in the absence of light (V)

NR