Download Vertical Multijunction Solar-Cell One

I E E E TRIIKSACTIONS O SDEVICES,
ELECTROS
ED-21,
VOL.
NO.
6,
JUNE
1974
351
[12] J. L. Fikartand P. A. Goud, “A t h y r y of oscillator noise and
its application toIMPATT diodes, J . Appl.Phys., vol. 44,
1967.
[22] K.Kurokawaand
F. M. Magelhaes, “-4n X-band10-Watt
nn ’?)3R4-370f3
Mnlt,inle-IMPATT Oscillator.” Proc. I E E E (Lett.).
vol. 59.
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,
p p ~ l 6 2 ~ 1 0 Jan.
3 , 1971.
[13] M. T. Vlaardingerbroek, “Output.spectrum of IMPATTdiodeoscillators,” Electron.Lett., vol. 5 , pp. 521-522, 1969; 1231 J. J. Goedbloed, “Determination of the intrinsic response time
of semiconductor avalanches from microwave measurements,”
J. J. Goedbloed and M. T. Ylaardingerbroek, “On the theory
Solid-StateElectron., vol. 15, pp. 635-647, 1972.
of noise and injectionphase locking of IMPATT-diode oscil[24] A. v.d. Ziel, LVoise. Englewood Cliffs, E.J.: Prentice-Hall,
lators,” Philips Res. Rep., vol. 25, pp. 452-471, 1970.
1954, pp. 321-322.
[14] J. J. Goedbloed, “FM noise of low-level operating IMPATT[25] R. J . McIntyre,L‘Multiplication noise inuniformavalanche
diodeoscillators,” Electron.Lett., vol. 7, pp. 445-446, 1971.
diodes,” IEEE. Trans. Electron Devices, vol. ED-13, pp. 164“On the upconverted noise of IMPATT-diode oscilla1151
. . ---,
168, Jan. 1966.
tors,” in Proc. MOGA Conf., Amsterdam, 1970, pp. 12.36-12.40.
[16] J. J. Goedbloed, “Intrinsic AM noise in singly tuned IMPATT[26] R . Zurmuhl, Matrizen. Berlin: Springer, 1964.
of IMPATT-diode oscillators,’’
1271
diodeoscillators.” I E E E Trans. Electron Devices. vol. ED-20,,
- G.Weidmann,“Modulation
Paper presented at ESSDERG, Munchen, 1971.
pp. 752-754, Auk. 1973.
[17] G.Convert, “Sur latheoriedubruit
des diodes avalanche,” [28] K. Mouthaan and H.
P. M. Rijpert,“Nonllnearityand noise
T
in the avalanche transit-time oscillator,” Philips
Res.
Rep.,
-Rm
.
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... ..... ... , vol.
- , nn. 419-471. 1971.
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. pp. 391-413, 1971.
[18] M . E. Hines, “Large-signal noise, frequency conversion, and
parametric instabilities inIMPATT-diode networks,” Proc. [29] J. J. Goedbloed, “Noise inIMPATT-diode oscillators,”Thesis,
PhilipsRes.Rep. Supplements.no. 7,1973.
I E E E , vol. 60, pp. 1534-1548, Dee. 1972.
[30] D. de Nobel, “IMPATT-diode technology,”presented at the
[19] B. B. v. Iperen and 15. Tjassens, “Novel and accurate methods
Avalanche Diode Workshop, New York, Dec. 1969.
for
measuring
small-signal and large-signal impedances of
[31] H. Tjassens, “Stable and low-nolse IRlPATT dlode oscillator
IMPATT-diodes,” Philips Res. Rep., vol. 27, pp, 38-75, ,l972.
for X band,’’ in Proc. 1973 European Microwave Conf., Brussels,
[20] J. C. Inkson, “Noisegenerationunder large-signal condltlons
vol. 1, 1973, paper A.1.2.
inthe Read microwave avalanche diode,’’ Int. J . Electron.,
V O ~ .25, pp. 1-16, 1968.
[32]
B. B. van Iperen, “Efficiency limitation by transverse instability
in IMPATT diodes,” Proc. I E E E . (Lett,.),vol. 62, pp. 284-285,
[21] D. E. Iglesias, “Circuit for testing high-efficiency IMPATTdiodes,” Proc. I E E E (Lett.), vol. 55, pp. 2065-2066, Nov.
Feb.
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Vertical Multijunction Solar-CellOne-DimensionalAnalysis
AVRAHAM GOVER,
STUDENT MEMBER, IEEE, AND
Abstract-The verticalmultijunction solar cellisa photovoltaic
device which may allow conversion efficiencies higher than conventional planar devices. A one-dimensionalmodel of thedevice is
presented here which allows a simple
and straightforward analysis
of device performance to be conducted.
The analysis covers the derivation of device short-circuit current,
saturation current, open-circuit voltage, andmaximumpower as a
function of illuminationspectra,devicegeometry,
and devicematerial properties.
LIST O F SYMBOLS
7a
n,p
Excess electrons and holes concentration ( ~ m - ~ ) .
j,,j,
Excess electrons and holes currentdensity ( ~ m - ~ ) 7,011
.
jL
Pairs generation rateatdepth x ( ~ m s-l)
- ~.
Y
rn,rp
Excess electron and hole lifetime (s) .
D,,D,
Electron and holediffusion constants (cmz/s).
L,,L,
Electron and holediffusion lengths (cm) .
N A , N D Acceptor and donor impurity concentrations
.
Manuscript received January 22, 1973; revised December 12, 1973.
This work was conducted under Air Force Contract F33615-72-C1310.
The authors are withHeliotek, Sylmar, Calif. 91342.
PAUL STELLA,
MEMBER, IEEE
Spectral response; defined as the output current
for one unit of input power a t wavelength X
(Am.
Short-circuit current (A).
Open-circuit current (V) .
Photon flux density (cmY2 s-l).
Absorption coefficient of the cell material [5]
(cm-l) .
Absorption efficiency; defined as the fract’ion of
impinging light which is absorbed by the cell.
Collectionefficiency;defined
as the fract’ion of
generated pairswhich is collected
by thejunctions.
Conversion efficiency; defined as the maximumoutput electrical power fraction of theinput
power.
Input radiation power density (W/cma).
p- and n-region widths (cm) .
Cell thickness (cm) .
Air-mass zero solar photon flux density (3.45 X
1017 photons/cm2. s) , The cell y-dimension width
is assumed to be 1 cm.
352
1974
JUNE
I E E E TRANSACTIOKS
DEVICES, ON ELECTRON
I. INTRODUCTION
T
H E vertical junction solar cell,shown schematicdly
in Fig.1, is potentially amore efficient device compared
tothe conventionalplanar cell. Theadvantage results
mainly from the possibility of using highly doped p and
n regions to provide high voltage factors, without incurring
severe reductionin the generatedcarrier collection t:fficiency, and
without
requiring
advances
in
material
properties.
Although junction voltage improvement can
be obtained
with the conventional planar cell by using high impurmity
concentration mat'erial, there is very little which can be
done to prevent a pronounced reduction of the collection
efficiency due t o the reduction of the excess carrier diffusion lengt'h. Thisloss can become severe enough t o camel
the improvement gained in the junction voltage. I n cclmparison, this loss can be avoided in the vertical junction
cell by reducing the distancebetween the junctions SO
that it is smaller t'han the reduced diffusion length. The
intent of this paper is t o investigate the vertical junction
cell parameters and to present' them in a useful model
whichwillallow
for simple calculation of the cell performance.
I n our model, the following assumptions havebeen made
for simplicity: aero front- and rear-surface recombination
velocities, homogeneous n and p regions, no front-surfme
reflection losses, andno ohmic losses. Furthermore, .;he
carrier recombination process in the junction space-charge
region is neglected.
11. COLLECTIONEFFICIENCYAND
SHORT-CIRCUITCURRENT
Although the carrier generation rate is x dependent (3ee
Fig. 1(a) ), the assumption of zero surface recombinat:lon
allows us to consider only the one-dimensional problem of
excess carrier currentflow in thex direction. Thiswill allow
for simple analytical solutions and expressions for the cell
parameters.
Looking first a t p-type material, the minority carrier
diffusion current in the x direction a t some fixed depth x
is (see Fig. l ( a ) ) [1]
The continuity equation is then
Formonochromaticillumination
equation is
j~
= cr(X)N(A)
of wavelength
(C)
Fig. 1. Vertical junction cell configurations.
where
At the junctions, the short-circuitconditiondetermines
the excess minoritycarrierboundaryvalues(usingthe
cell center line as the origin) as
The solution of (4)with (6) is, for any fixed x,
This distribution is illustrated in Fig. 2.
The generated current, which is collected per unit length
a t depth x from the surface in each junction, is
For an n-type region, the sameresultsapplywith
the
subscript p substituted for n and W z substituted for W 1 .
The solution of the differential equation is valid for any
of the homogeneous n- or p-type regions in each of the
configurations presented in Fig. 1, since (6) applies both
a t a junction or at anohmic contact. Thedifference is that
the electron-hole pairs arriving at the junction separate
and contribute useful current, while pairs arriving at the
ohmic contact recombine and do notcontributeany
externally useful current.
Considering 1 (a), the total current collected per unit
length a t depth x is
x, the
~
exp
(-ax).
(3)
"1 .
+ L, tanh 2LP
(9)
We define the function vc (plotted in Fig. 3) as
Equations (1) and ( 2 ) are then combined to yield
1
~ c ( U=
)
(4)
and call
vCl =
-
U
tanh u
vo( W1/2L,) and vcz = vc( W2/2L,) the col-
GOYLR AXD STELLA:VERTICALMULTIJUNCTIONSOLARCELL
$
z
-
t
10
.
A O 0 0 U M \
PLANAR
c
WAVELENGTH (JM)
Fig. 4.
Fig. 2. Minority carrier distribution for the configuration
of Fig. l(a).
'';h
Vertical junction cell spectral response.
dependentfunction)
does not change the junction-tocarrier distance. In the conventional planar cell, the junction-to-carrier distance varies with the absorption depth.
The spectral response of the cell is by definition,
.7
Substituting in (15), we find
0 I.
0
I
.
I
1
1
2
3
4
5
,
6
I
7
I
8
9
1
0
I
and using in (16) , we get
U
Fig. 3.
Collection efficiency (-qc).
The spectralresponse of the vertical cell is plotted inFig. 4
for various values of thickness ( h ). In comparison to the
planar cell, the vertical solar-cell spectral response is better
j ( z ) = q j L ( z ) [wml
WZ11cZl.
(11) in both thelong and shortwavelengths, and it tends to the
ideal response shape for largevalues of h.
Thetotalcurrent
collected in the cellis obtainedby
All the expressions derived in the preceeding have been
integrating over z. Integrating (11), using (3) for j L ( x ) ,
calculated for the cell configuration of Fig. 1(a). However,
we obtainthe expression for the short-circuit current
it is quite easy t o modify them to the other
configurations.
collected by the cell when illuminated by light of waveFor the cell 1(b) , the collection efficiency is written as
length X, as seen by
WlVCl WZVCZ
I s c ( X ) = qN(X)[l - exp [ - a ( X ) h ] ] ( W ~ m W Z ~ ) .
Vcoll =
(19)
w1
(12)
and the short-circuit current as
Using the definitions of absorption efficiency, va, and
collection efficiency, ~ ~ ~ we
1 1 find
,
lection efficiency factors for the p and n regions, respectively. Then,
+
+
+ wz
+
va = 1 - exp (-ah)
(13)
m
= - q N ( h ) (1 - exp [-ah]) ( W m ,
2
The short-circuit current can be rewrittenas
Iso(X)
=
q11aTooll(W1
+ 2TiVz)N(X)*
+ Wzm)
(20)
(15)
Interestingly, the collection efficiency of the vertical junction cell is found t o be independent of wavelength, in contrast to the planar cell. This is due to the fact that the
depth a t which radiation is absorbed (a wavelength-
where m is the numberof junctions in thecell. The spectral
response is still described in (18) and Fig. 4, but vcou in
that equation is given by (19). S i c e all the excess carrier
pairs which arrive at the ohmic contacts recombine and
do not contribute external current, the generated current
354
IEEE TRANSACTIOKS ON ELECTROK DEVICES,
collected in the cell configuration 1(c) is essent'ially half
the current collected by the cell 1(b) . Therefore, its collection efficiency, the short-circuit current, and the spectral
response will be exactly half of (19), (20),and(18),
respectively.
Normally we desire to know the cell's performance relativeto some given lightspectrum N ( X ) . This can b?
accomplished by integrating either (12) or ( 2 0 ) over all
wavelengths X shorter than the bandgap wavelength X,,.
Inthe numerical calculation which
follows, we use the
air-mass zero solar spectrum reported by Thekakara [21,
and assume for silicon X 0 = 1.1 b. Hence, we get
I
, , ~
.977A,~-
.6
>
10
l
20
50
100
THICKNESS, h
(21)
for the p-type region, and
Then,
for the n region where
n,
=
ni2
-,
nA
the
rn
=-
2
(wl f
VcollllAqNo.
w2)
(25)
200
500
(UM)
Fig. 5 . Absorption efficiency (sa) versus cell thickness.
Let
Isc
1974
JTJKE
ni2
pa=-.
nD
In a similar fashion, the n-region contribution
the to
tioncurrent density is
qDp tanh ( W 2 / 2 L p )
The collection efficiency qcolll being independent of wa7.e= [TPjL(X,Z) - P l l
*
islength, the in
same as
monochromatic illumination
L P
case, and is given by (14) or (19). The absorption e:%Hence, the total current density per junction is
ciency, ?A, for air-mass zero spectrum is given in Fig, 5
for various values of cell thickness ( h ), where the param- j ( z ) = j n ( x )
j , (z)
eter No calculated from (22) is
junc-
(31)
+
N o = 3.45 X
lo1' photons/cm2.s.
111. CURRENT-VOLTAGE CHARACTERISTICS
AND CONVERSION EFFICIENCY
If we now allow a voltage across the junctions, (4)i s
still valid, but (6) must
be
changed to
(3),
(32)
where we made use of ( 5 ) and (10). The total current per
junctionisobtained
byintegrating (32) Over using
(26), ( 2 7 ) , and (28), as seen by
GOVER ANDSTELLA:
VERTICAL MULTIJUNCTION SOLAR CELL
355
Proper summationof the contributions'from each junction
yields, for all configurations, the dependence
I
=
( 1; )
I,, - Io exp - - 1
(34)
where, for configuration 1( a ) ,
(35)
with the I,, given by (12) or (15), and for configuration
1 (b) ,
Io = 5
m qni2h
[-
W
l
l
l
O
l
NATn
-1
+ wzllcz
NDrp
15
20
25
1
30
35
40
I
X0
(36)
with I,, given by (20).
For polychrom& illumination, integration over X must
also be carried out.Equations (34), (35), and (36) are
still valid in thiscase, but theexpressions for I,, t o be used
in (34) are (24) and (25) for configurations 1(a) and 1 (b),
respectively.
Fig. 6. Solution to (44) (maximum power condition).
quite dependent on the cell thickness with an increasing
h providing a decreasing Voc.
B. Power Output
The maximum power available from the solar cell is
obtained when
A . Open-circuit Voltage
d
From (34), the open-circuit voltage is
The solution of this equation is
(37)
Substituting (14), (24), and (35), or alternatively (19),
(25), and (36), we have for configurations (a) and (b)
kT X ,
(maximum power voltage)
Q
Im = IBa- Io exp [X,]
(maximum power current)
V,
=
(43)
where X , is the solution of the transcendental equation
X
For 1 (a) with W z<< WI and N D >> N A , the n-region
contribution t o both the generated current and the diode
saturation current is negligible, so to a very good approximation,
(39)
For the samecell configuration, but with opposite conductivity, i.e., the base is an n type,
+ In (1 + X) = X .
and
x
- P
-vo.
kT
(45)
Equation (44) was numerically solved for various values
of X,, and the parameter X , is plotted as function of X ,
in Fig. 6. For a given cell structure, Vo and X . may be
calculated using (38) and (45) and, with the aid of Fig. 6,
V m and Im are calculated from (42) and (43). The maximum power available from the cell is then
Pm = ImVm.
Observe that the open-circuitvoltage is inthis case
independent of the cell width. This is because thecollected
and saturation currents both have the same
functional
dependence on width, thereby
cancelling any width dependence from the V,, expression. However, the Voois
(44)
(46)
C . Conversion Eficiency
The power input into the solar cell is PinA, where A is
the cell area, and Pin is the radiation power density shining
on the cell. For air-mass zero solar radiation [2],
Pin = 135.3 mW/cm2.
(47)
356
IEBE TRANSACTIOKS
ON
The conversion efficiency is given forthe configuration ( a )
by
ELECTRON DEVICES, JUKE
1974
and the conversion efficiency is
P,
- 19.7 percent.
PinW
y=--
(48)
V. CONCLUSION
and for configuration (b) by
The above model provides a description of vertical junc-
(4!)) tion solar-cell behavior. It is proposed that such a structure
IV. EXAMPLES
To demonstrate the use of the model analysis and to
display the numerical value of the vertical junction performance parameters, a sample calculation is shown. PJe
chose a p-base cell of configuration 1( a ) , where
h
=
100 pm,
W1
=
20 pm,
5 X lo1’ cm-3.
N A =
From [3] and [4] for this impurit’y concentration,we have
r, =
1.1 ps, D, = 10 cm2.s-1, L,
=
33 pm.
If we assume W2 << W1, then (19) reduces into
Ilcoll = I l c l =
?IC
(2)
=
0.97
where Fig. 3 is used to evaluate yc. The absorption eiEciency for air-mass zero solar illumination is found from
Fig. 5 and seen as
VA =
0.89.
The short-circuit current and the saturation current we
found from (24) and (35) and seen as
I,,
=
0.1 mA
Io = 8.5 X 10-13 mA.
Using (39) and (45), we have
x,= 25.5
Vo = 662 mV.
Then with the aid of Fig. 6, using (42) and (43), we have
X,
=
22.3
Vm = 580 mV
Im = 0.09 mA.
The maximum power output of the cell is therefore
P,
=
ImVm= 0.05 mW
can provide hitherto unobtained conversionefficiencies.
Efficiencies which are predict’ed for ideal planar cells (with
exceptional materialproperties
and lifetimes [6]) are
achievable inthe
vertical cell, utilizing present-day
state-of-the-art material properties.
The conversion efficiency of the cell calculated in the
example in the previous section exceeds the conversion
efficiency of recently reported high-efficiency planar cells
[7] by more than 40 percent. Yet the freedom in choosing
thestructural parameters may allow even higher efficiencies.
The impact of additional effects listed briefly in the
introduction, but not discussed in the present paper, will
reduce some of theadvantages predicted bythe onedimensional model. At present, the authors are preparing
an additional paper which will include an analysis of these
effects and their expected impact. Of prime importance is
the realization of fabrication techniques capable of providing the high junction density required for best operation. It is possible that techniques such as preferential
etching and epitaxy can provide optimized structures with
a few year’s effort.
ACKNOWLEDGMENT
The authors wish to thank W. P. Rahilly, M. Wolf, T.
Chadda, and J. Scott-Monck for their helpful comments
and assitance during this program.
REFERENCES
[I] William Shockley, Electrons andHoles in Semiconductors. New
York, N. Y.; Van Nostrand, 1950.
[a]M. P. Thekaekara, “Evaluating the light from the sun,” Opt.
Spectra, p. 32, Mar. 1972.
[3] S.91.Sze and J. C . Irvin, “Resistivity, mobility, and impurity
levels in GaAs, Ge, and Si at 300”K,” Solid-state Electron., vol.
11, p. 599, 1968.
[4] B. Ross and J. R. Madigan, “Thermal generation of recombination centersin silicon,” Phys. Rev., vol. 108, no. 6, p. 1428-33,
Dec. 1957.
[5] H. Y. Fan et al., Infra-Red Absorption and Energy-Bond Structure
of Germanium and Silicon, New York, N. Y.; Wiley, 1956, p. 184.
[6] M. Wolf, “A new look at silicon solar cell performance,” in 8th
I E E E Photovoltaic Specialists Conf.,. 1970, p. 360.
[7] J. Lindmayer and J. Allison, “An improved slhcon solar cell,
the violet cell,” in 9th I E E E Photovoltaic Specialists Conf., 1972,
p. 83.